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zai-org__GLM-OCR-api/arxiv_math/2502.15977_pg21_pg1_repeat1.md

@@ -0,0 +1,49 @@
+![](page=0,bbox=[376, 133, 716, 386])
+
+<div align="center">
+
+FIGURE 3. The decorated fan of an action of $ Q (1)^{2} $ on $ \mathbb{P}^{2|2} $
+
+</div>
+
+Example 4.28. Let us use Proposition 4.26 and Corollary 4.27 to classify toric supervarieties with supertorus $ Q(1)^{n} $ and underlying variety $ \mathbb{P}^{n}\cong X_{\Sigma} $ for the complete fan whose rays are $ \rho_{i}=\mathbb{R}_{+} x_{i} $ for $ i=1,...,n $ and $ \rho_{0}=\mathbb{R}_{+}(-x_{1}-...-x_{n}). $
+
+For $i = 1, \dots, n$, the ray $\rho_{i}$ must be decorated by a subspace $V_{\rho_{i}}$ such that $[V_{\rho_{i}}, V_{\rho_{i}}] \subseteq \mathbb{C}x_{i}$. Hence $V_{\rho_{i}} = \mathbb{C}\theta_{i}$ or 0. Likewise, unless $V_{\rho_{0}} = 0$, we have $V_{\rho_{0}} = \mathbb{C}(\theta_{1} \pm \theta_{2} \pm \dots \pm \theta_{n})$ for some $2^{n-1}$ choices of $ \pm$.
+
+It is straightforward to verify that condition (b) of the definition of a large-orbit decorated fan holds regardless of which subspaces are chosen. We therefore obtain a collection of $ 2^{n}(1+2^{n-1}) $ toric supervarieties which are not equivariantly isomorphic. Many, however, are isomorphic via toric morphisms (to be defined in the following section).
+
+Notice that if all but one of these decorations is nonzero, then the decorated fan describes a supervariety isomorphic to projective superspace $\mathbb{P}^{n|n}$. For instance, if $V_{\rho_i} = \mathbb{C}\theta_i$ and $V_{\rho_0} = 0$, then the coordinate superalgebras of the affine charts can be written as
+
+$$
+\mathbb {C} \left[ t _ {1}, \dots , t _ {n}, t _ {1} \xi_ {1}, \dots , t _ {n} \xi_ {n} \right]
+$$
+
+and
+
+$$
+\mathbb {C} \left[ t _ {i} ^ {- 1} t _ {1}, \dots , t _ {i} ^ {- 1}, \dots , t _ {i} ^ {- 1} t _ {n} \xi_ {1}, \dots , \xi_ {i}, \dots , t _ {i} ^ {- 1} t _ {n} \xi_ {n} \right].
+$$
+
+Figure 3 depicts the corresponding decorated fan for $ n=2 $
+
+If instead we change the decoration of $ \rho_{0} $ to $ \theta_{1}+\ldots+\theta_{n} $ , then the resulting affine charts have coordinate superalgebras
+
+$$
+\mathbb {C} \left[ t _ {1}, \dots , t _ {n}, t _ {1} \xi_ {1}, \dots , t _ {n} \xi_ {n} \right]
+$$
+
+and
+
+$$
+\mathbb {C} \left[ t _ {i} ^ {- 1} t _ {1} \left(1 + \xi_ {i} \xi_ {1}\right), \dots , t _ {i} ^ {- 1}, \dots , t _ {i} ^ {- 1} t _ {n} \left(1 + \xi_ {i} \xi_ {n}\right), t _ {i} ^ {- 1} t _ {1} \left(\xi_ {i} - \xi_ {1}\right), \dots , t _ {i} ^ {- 1} \xi_ {i}, \dots , t _ {i} ^ {- 1} t _ {n} \left(\xi_ {i} - \xi_ {n}\right) \right],
+$$
+
+so the supervariety is decidedly not isomorphic to projective superspace. Figure 4(B) depicts corresponding decorated fan for n=1.
+
+In general, when $ T = Q(1)^{n} $ , there are finitely many possible decorations for each ray. Namely, for $ \rho = \mathbb{R}_{+}\left(a_{1}x_{1}+\dots+a_{n}x_{n}\right) $ , there are $ 2^{d-1} $ many possible "square root subspaces" $ \mathbb{C}\left(\sqrt{a_{1}}\theta_{1}\pm \dots \pm \sqrt{a_{n}}\theta_{n}\right) $ , where $ d $ is the number of indices $ i=1,...,n $ for which $ a_{i}\neq 0 $ .
+
+The prior example admitted no issues of compatibility between different rays of the same cone. This is not ordinarily the case; if $ \rho_{1}=\mathbb{R}_{+}\left(a_{1}x_{1}+\dots+a_{n}x_{n}\right) $ and $ \rho_{2}=\mathbb{R}_{+}\left(b_{1}x_{1}+\dots+b_{n}x_{n}\right) $ , compatibility is less common if the linear matroid on the n vectors
+
+$$
+\left( \begin{array}{c} a _ {1} \\ b _ {1} \end{array} \right), \dots , \left( \begin{array}{c} a _ {n} \\ b _ {n} \end{array} \right)
+$$

zai-org__GLM-OCR-api/arxiv_math/2502.15977_pg21_pg1_repeat2.md

@@ -0,0 +1,49 @@
+![](page=0,bbox=[376, 133, 716, 386])
+
+<div align="center">
+
+FIGURE 3. The decorated fan of an action of $ Q (1)^{2} $ on $ \mathbb{P}^{2|2} $
+
+</div>
+
+Example 4.28. Let us use Proposition 4.26 and Corollary 4.27 to classify toric supervarieties with supertorus $ Q(1)^{n} $ and underlying variety $ \mathbb{P}^{n}\cong X_{\Sigma} $ for the complete fan whose rays are $ \rho_{i}=\mathbb{R}_{+} x_{i} $ for $ i=1,...,n $ and $ \rho_{0}=\mathbb{R}_{+}(-x_{1}-\dots-x_{n}). $
+
+For $i = 1, \dots, n$, the ray $\rho_{i}$ must be decorated by a subspace $V_{\rho_{i}}$ such that $[V_{\rho_{i}}, V_{\rho_{i}}] \subseteq \mathbb{C}x_{i}$. Hence $V_{\rho_{i}} = \mathbb{C}\theta_{i}$ or 0. Likewise, unless $V_{\rho_{0}} = 0$, we have $V_{\rho_{0}} = \mathbb{C}(\theta_{1} \pm \theta_{2} \pm \dots \pm \theta_{n})$ for some $2^{n-1}$ choices of $ \pm$.
+
+It is straightforward to verify that condition (b) of the definition of a large-orbit decorated fan holds regardless of which subspaces are chosen. We therefore obtain a collection of $ 2^{n}(1+2^{n-1}) $ toric supervarieties which are not equivariantly isomorphic. Many, however, are isomorphic via toric morphisms (to be defined in the following section).
+
+Notice that if all but one of these decorations is nonzero, then the decorated fan describes a supervariety isomorphic to projective superspace $\mathbb{P}^{n|n}$. For instance, if $V_{\rho_i} = \mathbb{C}\theta_i$ and $V_{\rho_0} = 0$, then the coordinate superalgebras of the affine charts can be written as
+
+$$
+\mathbb {C} \left[ t _ {1}, \dots , t _ {n}, t _ {1} \xi_ {1}, \dots , t _ {n} \xi_ {n} \right]
+$$
+
+and
+
+$$
+\mathbb {C} \left[ t _ {i} ^ {- 1} t _ {1}, \dots , t _ {i} ^ {- 1}, \dots , t _ {i} ^ {- 1} t _ {n} \xi_ {1}, \dots , \xi_ {i}, \dots , t _ {i} ^ {- 1} t _ {n} \xi_ {n} \right].
+$$
+
+Figure 3 depicts the corresponding decorated fan for $ n=2 $
+
+If instead we change the decoration of $ \rho_{0} $ to $ \theta_{1}+\ldots+\theta_{n} $ , then the resulting affine charts have coordinate superalgebras
+
+$$
+\mathbb {C} \left[ t _ {1}, \dots , t _ {n}, t _ {1} \xi_ {1}, \dots , t _ {n} \xi_ {n} \right]
+$$
+
+and
+
+$$
+\mathbb {C} \left[ t _ {i} ^ {- 1} t _ {1} \left(1 + \xi_ {i} \xi_ {1}\right), \dots , t _ {i} ^ {- 1}, \dots , t _ {i} ^ {- 1} t _ {n} \left(1 + \xi_ {i} \xi_ {n}\right), t _ {i} ^ {- 1} t _ {1} \left(\xi_ {i} - \xi_ {1}\right), \dots , t _ {i} ^ {- 1} \xi_ {i}, \dots , t _ {i} ^ {- 1} t _ {n} \left(\xi_ {i} - \xi_ {n}\right) \right],
+$$
+
+so the supervariety is decidedly not isomorphic to projective superspace. Figure 4(B) depicts corresponding decorated fan for $ n=1 $
+
+In general, when $ T = Q(1)^{n} $ , there are finitely many possible decorations for each ray. Namely, for $ \rho = \mathbb{R}_{+}\left(a_{1}x_{1}+\dots+a_{n}x_{n}\right) $ , there are $ 2^{d-1} $ many possible "square root subspaces" $ \mathbb{C}\left(\sqrt{a_{1}}\theta_{1}\pm \dots \pm \sqrt{a_{n}}\theta_{n}\right) $ , where $ d $ is the number of indices $ i=1,...,n $ for which $ a_{i}\neq 0. $
+
+The prior example admitted no issues of compatibility between different rays of the same cone. This is not ordinarily the case; if $ \rho_{1}=\mathbb{R}_{+}\left(a_{1}x_{1}+\dots+a_{n}x_{n}\right) $ and $ \rho_{2}=\mathbb{R}_{+}\left(b_{1}x_{1}+\dots+b_{n}x_{n}\right) $ , compatibility is less common if the linear matroid on the n vectors
+
+$$
+\left( \begin{array}{c} a _ {1} \\ b _ {1} \end{array} \right), \dots , \left( \begin{array}{c} a _ {n} \\ b _ {n} \end{array} \right)
+$$

zai-org__GLM-OCR-api/arxiv_math/2502.15977_pg21_pg1_repeat3.md

@@ -0,0 +1,49 @@
+![](page=0,bbox=[376, 133, 716, 386])
+
+<div align="center">
+
+FIGURE 3. The decorated fan of an action of $ Q (1)^{2} $ on $ \mathbb{P}^{2|2} $
+
+</div>
+
+Example 4.28. Let us use Proposition 4.26 and Corollary 4.27 to classify toric supervarieties with supertorus $ Q(1)^{n} $ and underlying variety $ \mathbb{P}^{n}\cong X_{\Sigma} $ for the complete fan whose rays are $ \rho_{i}=\mathbb{R}_{+} x_{i} $ for $ i=1,...,n $ and $ \rho_{0}=\mathbb{R}_{+}(-x_{1}-\dots-x_{n}). $
+
+For $i = 1, \dots, n$, the ray $\rho_{i}$ must be decorated by a subspace $V_{\rho_{i}}$ such that $[V_{\rho_{i}}, V_{\rho_{i}}] \subseteq \mathbb{C}x_{i}$. Hence $V_{\rho_{i}} = \mathbb{C}\theta_{i}$ or 0. Likewise, unless $V_{\rho_{0}} = 0$, we have $V_{\rho_{0}} = \mathbb{C}(\theta_{1} \pm \theta_{2} \pm \dots \pm \theta_{n})$ for some $2^{n-1}$ choices of $ \pm$.
+
+It is straightforward to verify that condition (b) of the definition of a large-orbit decorated fan holds regardless of which subspaces are chosen. We therefore obtain a collection of $ 2^{n}(1+2^{n-1}) $ toric supervarieties which are not equivariantly isomorphic. Many, however, are isomorphic via toric morphisms (to be defined in the following section).
+
+Notice that if all but one of these decorations is nonzero, then the decorated fan describes a supervariety isomorphic to projective superspace $\mathbb{P}^{n|n}$. For instance, if $V_{\rho_i} = \mathbb{C}\theta_i$ and $V_{\rho_0} = 0$, then the coordinate superalgebras of the affine charts can be written as
+
+$$
+\mathbb {C} \left[ t _ {1}, \dots , t _ {n}, t _ {1} \xi_ {1}, \dots , t _ {n} \xi_ {n} \right]
+$$
+
+and
+
+$$
+\mathbb {C} \left[ t _ {i} ^ {- 1} t _ {1}, \dots , t _ {i} ^ {- 1}, \dots , t _ {i} ^ {- 1} t _ {n} \xi_ {1}, \dots , \xi_ {i}, \dots , t _ {i} ^ {- 1} t _ {n} \xi_ {n} \right].
+$$
+
+Figure 3 depicts the corresponding decorated fan for $ n=2 $
+
+If instead we change the decoration of $ \rho_{0} $ to $ \theta_{1}+\ldots+\theta_{n} $ , then the resulting affine charts have coordinate superalgebras
+
+$$
+\mathbb {C} \left[ t _ {1}, \dots , t _ {n}, t _ {1} \xi_ {1}, \dots , t _ {n} \xi_ {n} \right]
+$$
+
+and
+
+$$
+\mathbb {C} \left[ t _ {i} ^ {- 1} t _ {1} \left(1 + \xi_ {i} \xi_ {1}\right), \dots , t _ {i} ^ {- 1}, \dots , t _ {i} ^ {- 1} t _ {n} \left(1 + \xi_ {i} \xi_ {n}\right), t _ {i} ^ {- 1} t _ {1} \left(\xi_ {i} - \xi_ {1}\right), \dots , t _ {i} ^ {- 1} \xi_ {i}, \dots , t _ {i} ^ {- 1} t _ {n} \left(\xi_ {i} - \xi_ {n}\right) \right],
+$$
+
+so the supervariety is decidedly not isomorphic to projective superspace. Figure 4(B) depicts corresponding decorated fan for $ n=1 $
+
+In general, when $ T = Q(1)^{n} $ , there are finitely many possible decorations for each ray. Namely, for $ \rho = \mathbb{R}_{+}\left(a_{1}x_{1}+\dots+a_{n}x_{n}\right) $ , there are $ 2^{d-1} $ many possible "square root subspaces" $ \mathbb{C}\left(\sqrt{a_{1}}\theta_{1}\pm \dots \pm \sqrt{a_{n}}\theta_{n}\right) $ , where $ d $ is the number of indices $ i=1,...,n $ for which $ a_{i}\neq 0. $
+
+The prior example admitted no issues of compatibility between different rays of the same cone. This is not ordinarily the case; if $ \rho_{1}=\mathbb{R}_{+}\left(a_{1}x_{1}+\dots+a_{n}x_{n}\right) $ and $ \rho_{2}=\mathbb{R}_{+}\left(b_{1}x_{1}+\dots+b_{n}x_{n}\right) $ , compatibility is less common if the linear matroid on the n vectors
+
+$$
+\left( \begin{array}{c} a _ {1} \\ b _ {1} \end{array} \right), \dots , \left( \begin{array}{c} a _ {n} \\ b _ {n} \end{array} \right)
+$$

zai-org__GLM-OCR-api/arxiv_math/2503.02004_pg9_pg1_repeat1.md

@@ -0,0 +1,49 @@
+The key advance of the MILP model is its global optimality guarantee, since it can explore all possible combinations via branch-and-bound. Furthermore, by employing upper and lower bound pruning strategies, MILP significantly reduces ineffective search efforts, thereby enhancing computational efficiency and guaranteeing the identification of the global optimality. However, despite its global optimality, its computational complexity still grows rapidly as the problem size increases with the time complexity $ \mathcal{O}(2^{M}\cdot K). $ Empirical results show that when $ M > 50 $ and $ K > 100 $ , the size of the branch-and-bound search tree leads to memory and time costs that exceed practical limits.
+
+**Algorithm 2 GRSIP:** Greedy Row Selection with Isolated Preselection.
+
+**Require:** $ \mathbf{G}, N_{r}, N_{\mathrm{init}}, \Delta n $
+
+**Ensure:** $ \mathcal{I}_{s} $
+
+1: Initialize: $ l=0, T_{(0)}=\{\varnothing\} $
+
+2: **while** $ l \leq N_{\mathrm{init}} $ **do**
+
+3: $ i=\operatorname* {arg} \max_{i \in T_{(l)}^{c}} \bar{\mathbf{g}}_{i} $ s.t. $ D_{1}(\{i\}, T_{(l)}) \leq \Delta n $
+
+4: $ l=l+1 $
+
+5: $ T_{(l)}=T_{(l)} \cup \{i\} $
+
+6: **end while**
+
+7: **while** $ l \leq N_{r} $ **do**
+
+8: $ i=\operatorname* {arg} \max_{i \in T_{(l)}^{c}} \min_{k} \| \mathbf{h}_{k} \|_{2} $ s.t. $ \mathbf{h}_{k} \in \operatorname{Col}\left( \mathbf{H}_{T_{(l)} \cup \{i\}} \right) $
+
+9: $ l=l+1 $
+
+10: $ T_{(l)}=T_{(l)} \cup \{i\} $
+
+11: **end while**
+
+12: **return** $ \mathcal{I}_{s}=T_{(l)} $
+
+Hence, in the following we also propose a greedy algorithm in Algorithm 2 with the time complexity $ \mathcal{O}(M K N_r) $ . The algorithm starts by choosing $ N_{\mathrm{init}} $ positions with the top average channel gains for each subcarrier, maintaining a minimum separation of $ \Delta n $ between them, where $ D_{1}(\cdot ,\cdot) $ represents the minimal $ \ell_{1} $ -norm distance between two point
+
+sets. Then it incrementally adds indices to the candidate set, following the principle of maximizing the minimum subcarrier gain, until $ N_{r} $ positions are chosen. Note that $ \mathbf{H}_{T} $ is formed by the row vectors of $ \mathbf{G} $ associated with row indices in $ T $ .
+
+Remark 2 (Applications to Other Models). The theoretical analysis and algorithms of the two-step framework for FAS proposed in this work could be extended to other problems besides FAS. First, the proposed two-step framework could be directly extended to the antenna selection problem with discrete positions, regardless of whether the exact antennas deployment [38] - [41]. The group-sparse recovery formulation and D-GRIP analysis can be directly adapted to delay-Doppler domain channel estimation [42], [43], where structures induce similar group-wise sparsity patterns in reconstruction. The DC-GOMP algorithm employs a correlation-aware selection mechanism to dynamically resolve coherence conflicts, offering a systematic and efficient approach to sparse event detection. Then, MILP-based spatial equalization offers new insights for the resource-constrained optimization in RIS configuration on discrete phase [44]. These potential extensions highlight that our methodology effectively tackles the unified challenge of sparsity-aware optimization under structured constraints, making it applicable to a wide range of domains, including computational sensing, adaptive control, and beyond.
+
+## V. SIMULATION RESULTS
+
+In this section, we present the performance of the proposed group-sparsity based frequency-space channel estimation algorithm, i.e., DC-GOMP, in comparison to two traditional algorithms (OMP, GOMP), under FAS-assisted wideband SIMO system. The proposed positions optimization methods, i.e., MILP and GRSIP, are also evaluated through the physical layer simulations and in terms of BER.
+
+![](page=0,bbox=[129, 945, 950, 1358])
+
+<div align="center">
+
+Figure 2. (1) The first row has four expressions in frequency-space domain. The first one represents the original SFG and the last three represent the recovered version by three different algorithm (our proposed DC-GOMP, OMP and GOMP). (2) Delay-wavenumber domain expressions corresponding to ones above. Black boxes denote the low power regions and red boxes denote the regions failing to correctly allocate the energy.
+
+</div>

zai-org__GLM-OCR-api/arxiv_math/2503.02004_pg9_pg1_repeat2.md

@@ -0,0 +1,49 @@
+The key advance of the MILP model is its global optimality guarantee, since it can explore all possible combinations via branch-and-bound. Furthermore, by employing upper and lower bound pruning strategies, MILP significantly reduces ineffective search efforts, thereby enhancing computational efficiency and guaranteeing the identification of the global optimality. However, despite its global optimality, its computational complexity still grows rapidly as the problem size increases with the time complexity $ \mathcal{O}(2^{M}\cdot K). $ Empirical results show that when $ M > 50 $ and $ K > 100 $ , the size of the branch-and-bound search tree leads to memory and time costs that exceed practical limits.
+
+**Algorithm 2 GRSIP:** Greedy Row Selection with Isolated Preselection.
+
+**Require:** $ \mathbf{G}, N_{r}, N_{\mathrm{init}}, \Delta n $
+
+**Ensure:** $ \mathcal{I}_{s} $
+
+1: Initialize: $ l=0, T_{(0)}=\{\varnothing\} $
+
+2: **while** $ l \leq N_{\mathrm{init}} $ **do**
+
+3: $ i=\operatorname* {arg} \max_{i \in T_{(l)}^{c}} \bar{\mathbf{g}}_{i} $ , s.t. $ D_{1}(\{i\}, T_{(l)}) \leq \Delta n $
+
+4: $ l=l+1 $
+
+5: $ T_{(l)}=T_{(l)} \cup \{i\} $
+
+6: **end while**
+
+7: **while** $ l \leq N_{r} $ **do**
+
+8: $ i=\operatorname* {arg} \max_{i \in T_{(l)}^{c}} \min_{k} \| \mathbf{h}_{k} \|_{2} $ , s.t. $ \mathbf{h}_{k} \in \operatorname{Col}\left( \mathbf{H}_{T_{(l)} \cup \{i\}} \right) $
+
+9: $ l=l+1 $
+
+10: $ T_{(l)}=T_{(l)} \cup \{i\} $
+
+11: **end while**
+
+12: **return** $ \mathcal{I}_{s}=T_{(l)} $
+
+Hence, in the following we also propose a greedy algorithm in Algorithm 2 with the time complexity $ \mathcal{O}(M K N_r) $ . The algorithm starts by choosing $ N_{\mathrm{init}} $ positions with the top average channel gains for each subcarrier, maintaining a minimum separation of $ \Delta n $ between them, where $ D_{1}(\cdot ,\cdot) $ represents the minimal $ \ell_{1} $ -norm distance between two point
+
+sets. Then it incrementally adds indices to the candidate set, following the principle of maximizing the minimum subcarrier gain, until $ N_{r} $ positions are chosen. Note that $ \mathbf{H}_{T} $ is formed by the row vectors of $ \mathbf{G} $ associated with row indices in $ T $ .
+
+Remark 2 (Applications to Other Models). The theoretical analysis and algorithms of the two-step framework for FAS proposed in this work could be extended to other problems besides FAS. First, the proposed two-step framework could be directly extended to the antenna selection problem with discrete positions, regardless of whether the exact antennas deployment [38] - [41]. The group-sparse recovery formulation and D-GRIP analysis can be directly adapted to delay-Doppler domain channel estimation [42], [43], where structures induce similar group-wise sparsity patterns in reconstruction. The DC-GOMP algorithm employs a correlation-aware selection mechanism to dynamically resolve coherence conflicts, offering a systematic and efficient approach to sparse event detection. Then, MILP-based spatial equalization offers new insights for the resource-constrained optimization in RIS configuration on discrete phase [44]. These potential extensions highlight that our methodology effectively tackles the unified challenge of sparsity-aware optimization under structured constraints, making it applicable to a wide range of domains, including computational sensing, adaptive control, and beyond.
+
+## V. SIMULATION RESULTS
+
+In this section, we present the performance of the proposed group-sparsity based frequency-space channel estimation algorithm, i.e., DC-GOMP, in comparison to two traditional algorithms (OMP, GOMP), under FAS-assisted wideband SIMO system. The proposed positions optimization methods, i.e., MILP and GRSIP, are also evaluated through the physical layer simulations and in terms of BER.
+
+![](page=0,bbox=[129, 945, 950, 1358])
+
+<div align="center">
+
+Figure 2. (1) The first row has four expressions in frequency-space domain. The first one represents the original SFG and the last three represent the recovered version by three different algorithm (our proposed DC-GOMP, OMP and GOMP). (2) Delay-wavenumber domain expressions corresponding to ones above. Black boxes denote the low power regions and red boxes denote the regions failing to correctly allocate the energy.
+
+</div>

zai-org__GLM-OCR-api/arxiv_math/2503.02004_pg9_pg1_repeat3.md

@@ -0,0 +1,49 @@
+The key advance of the MILP model is its global optimality guarantee, since it can explore all possible combinations via branch-and-bound. Furthermore, by employing upper and lower bound pruning strategies, MILP significantly reduces ineffective search efforts, thereby enhancing computational efficiency and guaranteeing the identification of the global optimality. However, despite its global optimality, its computational complexity still grows rapidly as the problem size increases with the time complexity $ \mathcal{O}(2^{M}\cdot K). $ Empirical results show that when $ M > 50 $ and $ K > 100 $ , the size of the branch-and-bound search tree leads to memory and time costs that exceed practical limits.
+
+**Algorithm 2 GRSIP:** Greedy Row Selection with Isolated Preselection.
+
+**Require:** $ \mathbf{G}, N_{r}, N_{\mathrm{init}}, \Delta n $
+
+**Ensure:** $ \mathcal{I}_{s} $
+
+1: Initialize: $ l=0, T_{(0)}=\{\varnothing\} $
+
+2: **while** $ l \leq N_{\mathrm{init}} $ **do**
+
+3: $ i=\operatorname* {arg} \max_{i \in T_{(l)}^{c}} \bar{\mathbf{g}}_{i} $ , s.t. $ D_{1}(\{i\}, T_{(l)}) \leq \Delta n $
+
+4: $ l=l+1 $
+
+5: $ T_{(l)}=T_{(l)} \cup \{i\} $
+
+6: **end while**
+
+7: **while** $ l \leq N_{r} $ **do**
+
+8: $ i=\operatorname* {arg} \max_{i \in T_{(l)}^{c}} \min_{k} \| \mathbf{h}_{k} \|_{2} $ , s.t. $ \mathbf{h}_{k} \in \operatorname{Col}\left( \mathbf{H}_{T_{(l)} \cup \{i\}} \right) $
+
+9: $ l=l+1 $
+
+10: $ T_{(l)}=T_{(l)} \cup \{i\} $
+
+11: **end while**
+
+12: **return** $ \mathcal{I}_{s}=T_{(l)} $
+
+Hence, in the following we also propose a greedy algorithm in Algorithm 2 with the time complexity $ \mathcal{O}(M K N_r) $ . The algorithm starts by choosing $ N_{\mathrm{init}} $ positions with the top average channel gains for each subcarrier, maintaining a minimum separation of $ \Delta n $ between them, where $ D_{1}(\cdot ,\cdot) $ represents the minimal $ \ell_{1} $ -norm distance between two point
+
+sets. Then it incrementally adds indices to the candidate set, following the principle of maximizing the minimum subcarrier gain, until $ N_{r} $ positions are chosen. Note that $ \mathbf{H}_{T} $ is formed by the row vectors of $ \mathbf{G} $ associated with row indices in $ T $ .
+
+Remark 2 (Applications to Other Models). The theoretical analysis and algorithms of the two-step framework for FAS proposed in this work could be extended to other problems besides FAS. First, the proposed two-step framework could be directly extended to the antenna selection problem with discrete positions, regardless of whether the exact antennas deployment [38] - [41]. The group-sparse recovery formulation and D-GRIP analysis can be directly adapted to delay-Doppler domain channel estimation [42], [43], where structures induce similar group-wise sparsity patterns in reconstruction. The DC-GOMP algorithm employs a correlation-aware selection mechanism to dynamically resolve coherence conflicts, offering a systematic and efficient approach to sparse event detection. Then, MILP-based spatial equalization offers new insights for the resource-constrained optimization in RIS configuration on discrete phase [44]. These potential extensions highlight that our methodology effectively tackles the unified challenge of sparsity-aware optimization under structured constraints, making it applicable to a wide range of domains, including computational sensing, adaptive control, and beyond.
+
+## V. SIMULATION RESULTS
+
+In this section, we present the performance of the proposed group-sparsity based frequency-space channel estimation algorithm, i.e., DC-GOMP, in comparison to two traditional algorithms (OMP, GOMP), under FAS-assisted wideband SIMO system. The proposed positions optimization methods, i.e., MILP and GRSIP, are also evaluated through the physical layer simulations and in terms of BER.
+
+![](page=0,bbox=[129, 945, 950, 1358])
+
+<div align="center">
+
+Figure 2. (1) The first row has four expressions in frequency-space domain. The first one represents the original SFG and the last three represent the recovered version by three different algorithm (our proposed DC-GOMP, OMP and GOMP). (2) Delay-wavenumber domain expressions corresponding to ones above. Black boxes denote the low power regions and red boxes denote the regions failing to correctly allocate the energy.
+
+</div>

zai-org__GLM-OCR-api/arxiv_math/2503.03754_pg10_pg1_repeat1.md

@@ -0,0 +1,49 @@
+[20] P. Skrzypczyk, N. Brunner, and S. Popescu, "Emergence of quantum correlations from nonlocality swapping," Physical Review Letters, vol. 102, no. 11, p. 110402, 2009.
+
+[21] A. J. Short, S. Popescu, and N. Gisin, "Entanglement swapping for generalized nonlocal correlations," Physical Review A—Atomic, Molecular, and Optical Physics, vol. 73, no. 1, p. 012101, 2006.
+
+[22] J. Barrett, "Information processing in generalized probabilistic theories," Physical Review A-Atomic, Molecular, and Optical Physics, vol. 75, no. 3, p. 032304, 2007.
+
+## A Proof of Theorem 3
+
+We prove the second part of the theorem first. Take some arbitrary $ \Phi $ in $ \mathcal{F}_{2} $ . We can represent a Z-channel source as follows
+
+$$
+P _ {X Y} = \left( \begin{array}{c c} 1 - s & 0 \\ s d & s (1 - d) \end{array} \right)
+$$
+
+where $s, d \in [0, 1]$. Let $u = f_X(0)$ and $v = f_X(1)$ for some $u, v \geq 0$. Assume that $\mathbb{E}[f] = m = (1 - s)u + sv$.
+
+Then $u = \frac{m - sv}{1 - s}$. One can verify directly that
+
+$$
+g (v, m) := \frac {H _ {\Phi} \left(\mathbb {E} [ f | Y ]\right)}{H _ {\Phi} (f)} = \frac {(1 - s (1 - d)) \Phi \left(\frac {1 - s}{1 - s (1 - d)} u + \frac {s d}{1 - s (1 - d)} v\right) + s (1 - d) \Phi (v) - \Phi ((1 - s) u + s v)}{(1 - s) \Phi (u) + s \Phi (v) - \Phi ((1 - s) u + s v)}
+$$
+
+$$
+= \frac {(1 - s (1 - d)) \Phi \left(\frac {m - s (1 - d) v}{1 - s (1 - d)}\right) + s (1 - d) \Phi (v) - \Phi (m)}{(1 - s) \Phi \left(\frac {m - s v}{1 - s}\right) + s \Phi (v) - \Phi (m)}.
+$$
+
+Then, we claim that if (10) holds, then $g(v,m)$ is decreasing in $v$ for every fixed $m$. Therefore, the maximum of $g(v,m)$ would occur when $v = 0$. This would complete the proof. Taking the partial derivative of $\log(g(v,m))$ with respect to $v$, we need to show that
+
+$$
+\frac {s \Phi^ {\prime} (v) - s \Phi^ {\prime} \left(\frac {m - s v}{1 - s}\right)}{(1 - s) \Phi \left(\frac {m - s v}{1 - s}\right) + s \Phi (v) - \Phi (m)} \geq \frac {s (1 - d) \Phi^ {\prime} (v) - s (1 - d) \Phi^ {\prime} \left(\frac {m - s (1 - d) v}{1 - s (1 - d)}\right)}{(1 - s (1 - d)) \Phi \left(\frac {m - s (1 - d) v}{1 - s (1 - d)}\right) + s (1 - d) \Phi (v) - \Phi (m)}.
+$$
+
+For any $ t\in[0,1] $ , define
+
+$$
+\begin{array}{l} k (t) = \frac {s t \Phi^ {\prime} (v) - s t \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right)}{(1 - s t) \Phi \left(\frac {m - s v t}{1 - s t}\right) + s t \Phi (v) - \Phi (m)} \\ = \frac {\Phi^ {\prime} (v) - \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right)}{\left(\frac {1}{s t} - 1\right) \Phi \left(\frac {m - s v t}{1 - s t}\right) + \Phi (v) - \frac {1}{s t} \Phi (m)}. \\ \end{array}
+$$
+
+Then, (15) can be written as $ k (1)\geq k (1-d). $ We would be done if we can show that $ k (t) $ is an increasing function. Showing $ k^{\prime}(t)\geq 0 $ is equivalent with
+
+$$
+\begin{array}{l} C (t) = - \frac {1}{s t ^ {2}} \left(\Phi^ {\prime} (v) - \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right)\right) \left(- \Phi \left(\frac {m - s v t}{1 - s t}\right) + \frac {(m - v) s t}{1 - s t} \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right) + \Phi (m)\right) \\ - \frac {s (m - v)}{(1 - s t) ^ {2}} \Phi^ {\prime \prime} \left(\frac {m - s v t}{1 - s t}\right) \left(\Phi (v) + \frac {1 - s t}{s t} \Phi \left(\frac {m - s v t}{1 - s t}\right) - \frac {1}{s t} \Phi (m)\right) \geq 0. \\ \end{array}
+$$
+
+Let $x_{1} = v, x_{2} = \frac{m - svt}{1 - st}$. Then we can compute $s$ from $x_{1}$ and $x_{2}$ as follows:
+
+$$
+s = \frac {m - x _ {2}}{t \left(x _ {1} - x _ {2}\right)}.
+$$

zai-org__GLM-OCR-api/arxiv_math/2503.03754_pg10_pg1_repeat2.md

@@ -0,0 +1,47 @@
+[20] P. Skrzypczyk, N. Brunner, and S. Popescu, "Emergence of quantum correlations from nonlocality swapping," Physical Review Letters, vol. 102, no. 11, p. 110402, 2009.
+
+[21] A. J. Short, S. Popescu, and N. Gisin, "Entanglement swapping for generalized nonlocal correlations," Physical Review A—Atomic, Molecular, and Optical Physics, vol. 73, no. 1, p. 012101, 2006.
+
+[22] J. Barrett, "Information processing in generalized probabilistic theories," Physical Review A—Atomic, Molecular, and Optical Physics, vol. 75, no. 3, p. 032304, 2007.
+
+## A Proof of Theorem 3
+
+We prove the second part of the theorem first. Take some arbitrary $ \Phi $ in $ \mathcal{F}_{2} $ . We can represent a Z-channel source as follows
+
+$$
+P _ {X Y} = \left( \begin{array}{c c} 1 - s & 0 \\ s d & s (1 - d) \end{array} \right)
+$$
+
+where $s, d \in [0, 1]$. Let $u = f_X(0)$ and $v = f_X(1)$ for some $u, v \geq 0$. Assume that $ \mathbb{E}[f] = m = (1 - s)u + sv. $ Then $ u = \frac{m - sv}{1 - s} $ . One can verify directly that
+
+$$
+g (v, m) := \frac {H _ {\Phi} \left(\mathbb {E} [ f | Y ]\right)}{H _ {\Phi} (f)} = \frac {(1 - s (1 - d)) \Phi \left(\frac {1 - s}{1 - s (1 - d)} u + \frac {s d}{1 - s (1 - d)} v\right) + s (1 - d) \Phi (v) - \Phi ((1 - s) u + s v)}{(1 - s) \Phi (u) + s \Phi (v) - \Phi ((1 - s) u + s v)}
+$$
+
+$$
+= \frac {(1 - s (1 - d)) \Phi \left(\frac {m - s (1 - d) v}{1 - s (1 - d)}\right) + s (1 - d) \Phi (v) - \Phi (m)}{(1 - s) \Phi \left(\frac {m - s v}{1 - s}\right) + s \Phi (v) - \Phi (m)}.
+$$
+
+Then, we claim that if (10) holds, then $g(v,m)$ is decreasing in $v$ for every fixed $m$. Therefore, the maximum of $g(v,m)$ would occur when $v = 0$. This would complete the proof. Taking the partial derivative of $\log(g(v,m))$ with respect to $v$, we need to show that
+
+$$
+\frac {s \Phi^ {\prime} (v) - s \Phi^ {\prime} \left(\frac {m - s v}{1 - s}\right)}{(1 - s) \Phi \left(\frac {m - s v}{1 - s}\right) + s \Phi (v) - \Phi (m)} \geq \frac {s (1 - d) \Phi^ {\prime} (v) - s (1 - d) \Phi^ {\prime} \left(\frac {m - s (1 - d) v}{1 - s (1 - d)}\right)}{(1 - s (1 - d)) \Phi \left(\frac {m - s (1 - d) v}{1 - s (1 - d)}\right) + s (1 - d) \Phi (v) - \Phi (m)}.
+$$
+
+For any $ t\in[0,1] $ , define
+
+$$
+\begin{array}{l} k (t) = \frac {s t \Phi^ {\prime} (v) - s t \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right)}{(1 - s t) \Phi \left(\frac {m - s v t}{1 - s t}\right) + s t \Phi (v) - \Phi (m)} \\ = \frac {\Phi^ {\prime} (v) - \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right)}{\left(\frac {1}{s t} - 1\right) \Phi \left(\frac {m - s v t}{1 - s t}\right) + \Phi (v) - \frac {1}{s t} \Phi (m)}. \\ \end{array}
+$$
+
+Then, (15) can be written as $ k (1)\geq k (1-d). $ We would be done if we can show that $ k (t) $ is an increasing function. Showing $ k^{\prime}(t)\geq 0 $ is equivalent with
+
+$$
+\begin{array}{l} C (t) = - \frac {1}{s t ^ {2}} \left(\Phi^ {\prime} (v) - \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right)\right) \left(- \Phi \left(\frac {m - s v t}{1 - s t}\right) + \frac {(m - v) s t}{1 - s t} \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right) + \Phi (m)\right) \\ - \frac {s (m - v)}{(1 - s t) ^ {2}} \Phi^ {\prime \prime} \left(\frac {m - s v t}{1 - s t}\right) \left(\Phi (v) + \frac {1 - s t}{s t} \Phi \left(\frac {m - s v t}{1 - s t}\right) - \frac {1}{s t} \Phi (m)\right) \geq 0. \\ \end{array}
+$$
+
+Let $x_{1} = v, x_{2} = \frac{m - svt}{1 - st}$. Then we can compute $s$ from $x_{1}$ and $x_{2}$ as follows:
+
+$$
+s = \frac {m - x _ {2}}{t \left(x _ {1} - x _ {2}\right)}.
+$$

zai-org__GLM-OCR-api/arxiv_math/2503.03754_pg10_pg1_repeat3.md

@@ -0,0 +1,47 @@
+[20] P. Skrzypczyk, N. Brunner, and S. Popescu, "Emergence of quantum correlations from nonlocality swapping," Physical Review Letters, vol. 102, no. 11, p. 110402, 2009.
+
+[21] A. J. Short, S. Popescu, and N. Gisin, "Entanglement swapping for generalized nonlocal correlations," Physical Review A—Atomic, Molecular, and Optical Physics, vol. 73, no. 1, p. 012101, 2006.
+
+[22] J. Barrett, "Information processing in generalized probabilistic theories," Physical Review A-Atomic, Molecular, and Optical Physics, vol. 75, no. 3, p. 032304, 2007.
+
+## A Proof of Theorem 3
+
+We prove the second part of the theorem first. Take some arbitrary $ \Phi $ in $ \mathcal{F}_{2} $ . We can represent a Z-channel source as follows
+
+$$
+P _ {X Y} = \left( \begin{array}{c c} 1 - s & 0 \\ s d & s (1 - d) \end{array} \right)
+$$
+
+where $s, d \in [0, 1]$. Let $u = f_X(0)$ and $v = f_X(1)$ for some $u, v \geq 0$. Assume that $ \mathbb{E}[f] = m = (1 - s)u + sv. $ Then $ u = \frac{m - sv}{1 - s} $ . One can verify directly that
+
+$$
+g (v, m) := \frac {H _ {\Phi} \left(\mathbb {E} [ f | Y ]\right)}{H _ {\Phi} (f)} = \frac {(1 - s (1 - d)) \Phi \left(\frac {1 - s}{1 - s (1 - d)} u + \frac {s d}{1 - s (1 - d)} v\right) + s (1 - d) \Phi (v) - \Phi ((1 - s) u + s v)}{(1 - s) \Phi (u) + s \Phi (v) - \Phi ((1 - s) u + s v)}
+$$
+
+$$
+= \frac {(1 - s (1 - d)) \Phi \left(\frac {m - s (1 - d) v}{1 - s (1 - d)}\right) + s (1 - d) \Phi (v) - \Phi (m)}{(1 - s) \Phi \left(\frac {m - s v}{1 - s}\right) + s \Phi (v) - \Phi (m)}.
+$$
+
+Then, we claim that if (10) holds, then $g(v,m)$ is decreasing in $v$ for every fixed $m$. Therefore, the maximum of $g(v,m)$ would occur when $v = 0$. This would complete the proof. Taking the partial derivative of $\log(g(v,m))$ with respect to $v$, we need to show that
+
+$$
+\frac {s \Phi^ {\prime} (v) - s \Phi^ {\prime} \left(\frac {m - s v}{1 - s}\right)}{(1 - s) \Phi \left(\frac {m - s v}{1 - s}\right) + s \Phi (v) - \Phi (m)} \geq \frac {s (1 - d) \Phi^ {\prime} (v) - s (1 - d) \Phi^ {\prime} \left(\frac {m - s (1 - d) v}{1 - s (1 - d)}\right)}{(1 - s (1 - d)) \Phi \left(\frac {m - s (1 - d) v}{1 - s (1 - d)}\right) + s (1 - d) \Phi (v) - \Phi (m)}.
+$$
+
+For any $ t\in[0,1] $ , define
+
+$$
+\begin{array}{l} k (t) = \frac {s t \Phi^ {\prime} (v) - s t \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right)}{(1 - s t) \Phi \left(\frac {m - s v t}{1 - s t}\right) + s t \Phi (v) - \Phi (m)} \\ = \frac {\Phi^ {\prime} (v) - \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right)}{\left(\frac {1}{s t} - 1\right) \Phi \left(\frac {m - s v t}{1 - s t}\right) + \Phi (v) - \frac {1}{s t} \Phi (m)}. \\ \end{array}
+$$
+
+Then, (15) can be written as $ k (1)\geq k (1-d). $ We would be done if we can show that $ k (t) $ is an increasing function. Showing $ k^{\prime}(t)\geq 0 $ is equivalent with
+
+$$
+\begin{array}{l} C (t) = - \frac {1}{s t ^ {2}} \left(\Phi^ {\prime} (v) - \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right)\right) \left(- \Phi \left(\frac {m - s v t}{1 - s t}\right) + \frac {(m - v) s t}{1 - s t} \Phi^ {\prime} \left(\frac {m - s v t}{1 - s t}\right) + \Phi (m)\right) \\ - \frac {s (m - v)}{(1 - s t) ^ {2}} \Phi^ {\prime \prime} \left(\frac {m - s v t}{1 - s t}\right) \left(\Phi (v) + \frac {1 - s t}{s t} \Phi \left(\frac {m - s v t}{1 - s t}\right) - \frac {1}{s t} \Phi (m)\right) \geq 0. \\ \end{array}
+$$
+
+Let $x_{1} = v, x_{2} = \frac{m - svt}{1 - st}$. Then we can compute $s$ from $x_{1}$ and $x_{2}$ as follows:
+
+$$
+s = \frac {m - x _ {2}}{t \left(x _ {1} - x _ {2}\right)}.
+$$

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@@ -0,0 +1,23 @@
+certain types of correlation that cannot be captured using real-valued probabilities alone.
+
+Consider a complex random variable $ Z $ whose probability distribution is defined over a discrete set of outcomes. The probability mass function $ P(z) $ is now complex-valued, meaning that $ P(z)\in \mathbb{C} $ . This introduces a fundamental shift in how entropy is conceptualized, as the standard Shannon entropy is defined solely for real, nonnegative probabilities. To extend this concept, a suitable framework must account for both the magnitude and phase of complex probabilities.
+
+A plausible extension of Shannon entropy to complex probability distributions can be expressed as
+
+$$
+H (Z) = \mathbb {E} [ I (Z) ] = \sum_ {z} P (z) I (z) = - \sum_ {z} P (z) \log P (z),
+$$
+
+where $ P ( z ) $ is complex. The challenge lies in the interpretation of the logarithm $ \log P ( z ) $ for complex values, as the logarithm of a complex number is inherently multivalued. To resolve this, $ \log P ( z ) $ is typically expressed in terms of its polar form
+
+$$
+\log P (z) = \log | P (z) | + i \theta (P (z)),
+$$
+
+where $ | P (z)| $ is the modulus (or absolute value) of $ P (z) $ , and $ \theta (P (z)) $ denotes the phase (or argument) of the complex probability. Here, $ \log | P (z)| $ captures the traditional magnitude-based contribution to entropy, while $ i \theta (P (z)) $ incorporates the phase information intrinsic to complex probabilities. Substituting the polar form of $ \log P (z) $ into the expression for $ H (Z) $ , we obtain the complex form of Shannon entropy:
+
+$$
+\begin{array}{l} H (Z) = - \sum_ {z} P (z) \log P (z) \\ = - \sum_ {z} P (z) \left(\log | P (z) | + i \theta (P (z))\right) \\ = - \sum_ {z} P (z) \log | P (z) | - i \sum_ {z} P (z) \theta (P (z)). \\ \end{array}
+$$
+
+This formulation reveals two distinct components of the complex entropy. The first component, $ - \sum_{z} P(z)\log |P(z)| $ , resembles the standard Shannon entropy but now incorporates the magnitudes of the complex probabilities $ |P(z)| $ . This term quantifies the uncertainty or information content in the distribution of probability magnitudes, maintaining a familiar structure while adapting to the complex domain. The second component, $ - i\sum_{z} P(z)\theta(P(z)) $ , introduces a phase-dependent contribution to the entropy, reflecting the coherence, interference, or relative phase relationships among the events. Unlike classical entropy, which focuses solely on magnitude, this phase term highlights the informational significance of relative phases, an essential feature in systems with quantum mechanical or wave-like properties.

zai-org__GLM-OCR-api/arxiv_math/2503.03759_pg9_pg1_repeat2.md

@@ -0,0 +1,23 @@
+certain types of correlation that cannot be captured using real-valued probabilities alone.
+
+Consider a complex random variable $ Z $ whose probability distribution is defined over a discrete set of outcomes. The probability mass function $ P(z) $ is now complex-valued, meaning that $ P(z)\in \mathbb{C} $ . This introduces a fundamental shift in how entropy is conceptualized, as the standard Shannon entropy is defined solely for real, nonnegative probabilities. To extend this concept, a suitable framework must account for both the magnitude and phase of complex probabilities.
+
+A plausible extension of Shannon entropy to complex probability distributions can be expressed as
+
+$$
+H (Z) = \mathbb {E} [ I (Z) ] = \sum_ {z} P (z) I (z) = - \sum_ {z} P (z) \log P (z),
+$$
+
+where $ P ( z ) $ is complex. The challenge lies in the interpretation of the logarithm $ \log P ( z ) $ for complex values, as the logarithm of a complex number is inherently multivalued. To resolve this, $ \log P ( z ) $ is typically expressed in terms of its polar form
+
+$$
+\log P (z) = \log | P (z) | + i \theta (P (z)),
+$$
+
+where $ | P (z)| $ is the modulus (or absolute value) of $ P (z) $ , and $ \theta (P (z)) $ denotes the phase (or argument) of the complex probability. Here, $ \log | P (z)| $ captures the traditional magnitude-based contribution to entropy, while $ i \theta (P (z)) $ incorporates the phase information intrinsic to complex probabilities. Substituting the polar form of $ \log P (z) $ into the expression for $ H (Z) $ , we obtain the complex form of Shannon entropy:
+
+$$
+\begin{array}{l} H (Z) = - \sum_ {z} P (z) \log P (z) \\ = - \sum_ {z} P (z) \left(\log | P (z) | + i \theta (P (z))\right) \\ = - \sum_ {z} P (z) \log | P (z) | - i \sum_ {z} P (z) \theta (P (z)). \\ \end{array}
+$$
+
+This formulation reveals two distinct components of the complex entropy. The first component, $ - \sum_{z} P(z)\log |P(z)| $ , resembles the standard Shannon entropy but now incorporates the magnitudes of the complex probabilities $ |P(z)| $ . This term quantifies the uncertainty or information content in the distribution of probability magnitudes, maintaining a familiar structure while adapting to the complex domain. The second component, $ - i\sum_{z} P(z)\theta(P(z)) $ , introduces a phase-dependent contribution to the entropy, reflecting the coherence, interference, or relative phase relationships among the events. Unlike classical entropy, which focuses solely on magnitude, this phase term highlights the informational significance of relative phases, an essential feature in systems with quantum mechanical or wave-like properties.

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@@ -0,0 +1,23 @@
+certain types of correlation that cannot be captured using real-valued probabilities alone.
+
+Consider a complex random variable $ Z $ whose probability distribution is defined over a discrete set of outcomes. The probability mass function $ P(z) $ is now complex-valued, meaning that $ P(z)\in \mathbb{C} $ . This introduces a fundamental shift in how entropy is conceptualized, as the standard Shannon entropy is defined solely for real, nonnegative probabilities. To extend this concept, a suitable framework must account for both the magnitude and phase of complex probabilities.
+
+A plausible extension of Shannon entropy to complex probability distributions can be expressed as
+
+$$
+H (Z) = \mathbb {E} [ I (Z) ] = \sum_ {z} P (z) I (z) = - \sum_ {z} P (z) \log P (z),
+$$
+
+where $ P ( z ) $ is complex. The challenge lies in the interpretation of the logarithm $ \log P ( z ) $ for complex values, as the logarithm of a complex number is inherently multivalued. To resolve this, $ \log P ( z ) $ is typically expressed in terms of its polar form
+
+$$
+\log P (z) = \log | P (z) | + i \theta (P (z)),
+$$
+
+where $ | P (z)| $ is the modulus (or absolute value) of $ P (z) $ , and $ \theta (P (z)) $ denotes the phase (or argument) of the complex probability. Here, $ \log | P (z)| $ captures the traditional magnitude-based contribution to entropy, while $ i \theta (P (z)) $ incorporates the phase information intrinsic to complex probabilities. Substituting the polar form of $ \log P (z) $ into the expression for $ H (Z) $ , we obtain the complex form of Shannon entropy:
+
+$$
+\begin{array}{l} H (Z) = - \sum_ {z} P (z) \log P (z) \\ = - \sum_ {z} P (z) \left(\log | P (z) | + i \theta (P (z))\right) \\ = - \sum_ {z} P (z) \log | P (z) | - i \sum_ {z} P (z) \theta (P (z)). \\ \end{array}
+$$
+
+This formulation reveals two distinct components of the complex entropy. The first component, $ - \sum_{z} P(z)\log |P(z)| $ , resembles the standard Shannon entropy but now incorporates the magnitudes of the complex probabilities $ |P(z)| $ . This term quantifies the uncertainty or information content in the distribution of probability magnitudes, maintaining a familiar structure while adapting to the complex domain. The second component, $ - i\sum_{z} P(z)\theta(P(z)) $ , introduces a phase-dependent contribution to the entropy, reflecting the coherence, interference, or relative phase relationships among the events. Unlike classical entropy, which focuses solely on magnitude, this phase term highlights the informational significance of relative phases, an essential feature in systems with quantum mechanical or wave-like properties.

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@@ -0,0 +1,29 @@
+<div align="center">
+
+# DISPROVING SOME THEOREMS IN SHARMA AND CHAUHAN et al. (2018, 2021) $ ^{*} $
+
+</div>
+
+Ramy Takieldin Faculty of Engineering, Ain Shams University, Cairo, Egypt Egypt University of Informatics, New Capital, Cairo, Egypt ramy.farouk@eng.asu.edu.eg
+
+Patrick Sole I2M (CNRS, University of Aix-Marseille), 13009 Marseilles, France patrick.sole@telecom-paris.fr
+
+## ABSTRACT
+
+The main objective of this work is to show, through counterexamples, that some of the theorems presented in the papers of Sharma et al. (2018) and Chauhan et al. (2021) are incorrect. Although they used these theorems to establish a sufficient condition for a multi-twisted (MT) code to be linear complementary dual (LCD), we show that this condition itself remains valid. We further improve this condition by removing the restrictions on the shift constants and relaxing the required coprimality condition. We show that compared to the previous condition, the modified condition is able to identify more LCD MT codes. Furthermore, without the need for a normalized set of generators, we develop a formula to determine the dimension of any $\rho$-generator MT code.
+
+Keywords Multi-twisted code · linear complementary dual · Determinantal divisors · Algebraic coding MSC: 94B05, 94B60, 11T71
+
+## 1 Introduction
+
+Multi-twisted (MT) codes over a finite field $\mathbb{F}_q$ constitute a significant and comprehensive class of linear codes. This class contains several well-known subclasses, including cyclic, constacyclic, quasi-cyclic, quasi-twisted, and generalized quasi-cyclic codes. For some integer $\ell \geq 1$, let $0 \neq \lambda_i \in \mathbb{F}_q$ and $m_i \geq 1$ for $1 \leq i \leq \ell$. If $\Lambda = (\lambda_1, \lambda_2, \dots, \lambda_\ell)$, then a $\Lambda$-MT code $C$ with block lengths $(m_1, m_2, \dots, m_\ell)$ is defined in [1, Definition 3.1] as a linear code of length $n = m_1 + m_2 + \dots + m_\ell$ that remains invariant under the $\Lambda$-MT linear transformation
+
+$$
+\begin{array}{l} T _ {\Lambda}: \left(c _ {1, 0}, c _ {1, 1}, \dots , c _ {1, m _ {1} - 1}; c _ {2, 0}, c _ {2, 1}, \dots , c _ {2, m _ {2} - 1}; \dots ; c _ {\ell , 0}, c _ {\ell , 1}, \dots , c _ {\ell , m _ {\ell} - 1}\right) \mapsto \\ \left(\lambda_ {1} c _ {1, m _ {1} - 1}, c _ {1, 0}, \dots , c _ {1, m _ {1} - 2}; \lambda_ {2} c _ {2, m _ {2} - 1}, c _ {2, 0}, \dots , c _ {2, m _ {2} - 2}; \dots ; \lambda_ {\ell} c _ {\ell , m _ {\ell} - 1}, c _ {\ell , 0}, \dots , c _ {\ell , m _ {\ell} - 2}\right). \\ \end{array}
+$$
+
+Throughout this paper, we adopt the same notations as in [1, 2]. Thus, $ \mathcal{C} $ denotes a $ \Lambda $ -MT code over $ \mathbb{F}_{q} $ with block lengths $ (m_{1}, m_{2}, \dots , m_{\ell}) $ . The Euclidean dual $ \mathcal{C}^{\perp} $ of $ \mathcal{C} $ is a $ (\lambda_{1}^{-1}, \lambda_{2}^{-1}, \dots , \lambda_{\ell}^{-1}) $ -MT code with the same block lengths. By using polynomial representation for blocks, $ \mathcal{C} $ can be regarded as an $ \mathbb{F}_{q}[x] $ -submodule of the $ \Lambda $ -MT module
+
+$$
+V = \bigoplus_ {i = 1} ^ {\ell} \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {i}} - \lambda_ {i} \rangle} = \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {1}} - \lambda_ {1} \rangle} \oplus \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {2}} - \lambda_ {2} \rangle} \oplus \dots \oplus \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {\ell}} - \lambda_ {\ell} \rangle}.
+$$

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+<div align="center">
+
+# DISPROVING SOME THEOREMS IN SHARMA AND CHAUHAN et al. (2018, 2021) $ ^{*} $
+
+</div>
+
+Ramy Takieldin Faculty of Engineering, Ain Shams University, Cairo, Egypt Egypt University of Informatics, New Capital, Cairo, Egypt ramy.farouk@eng.asu.edu.eg
+
+Patrick Sole I2M (CNRS, University of Aix-Marseille), 13009 Marseilles, France patrick.sole@telecom-paris.fr
+
+## ABSTRACT
+
+The main objective of this work is to show, through counterexamples, that some of the theorems presented in the papers of Sharma et al. (2018) and Chauhan et al. (2021) are incorrect. Although they used these theorems to establish a sufficient condition for a multi-twisted (MT) code to be linear complementary dual (LCD), we show that this condition itself remains valid. We further improve this condition by removing the restrictions on the shift constants and relaxing the required coprimality condition. We show that compared to the previous condition, the modified condition is able to identify more LCD MT codes. Furthermore, without the need for a normalized set of generators, we develop a formula to determine the dimension of any $\rho$-generator MT code.
+
+Keywords Multi-twisted code · linear complementary dual · Determinantal divisors · Algebraic coding MSC: 94B05, 94B60, 11T71
+
+## 1 Introduction
+
+Multi-twisted (MT) codes over a finite field $\mathbb{F}_q$ constitute a significant and comprehensive class of linear codes. This class contains several well-known subclasses, including cyclic, constacyclic, quasi-cyclic, quasi-twisted, and generalized quasi-cyclic codes. For some integer $\ell \geq 1$, let $0 \neq \lambda_i \in \mathbb{F}_q$ and $m_i \geq 1$ for $1 \leq i \leq \ell$. If $\Lambda = (\lambda_1, \lambda_2, \dots, \lambda_\ell)$, then a $\Lambda$-MT code $C$ with block lengths $(m_1, m_2, \dots, m_\ell)$ is defined in [1, Definition 3.1] as a linear code of length $n = m_1 + m_2 + \dots + m_\ell$ that remains invariant under the $\Lambda$-MT linear transformation
+
+$$
+\begin{array}{l} T _ {\Lambda}: \left(c _ {1, 0}, c _ {1, 1}, \dots , c _ {1, m _ {1} - 1}; c _ {2, 0}, c _ {2, 1}, \dots , c _ {2, m _ {2} - 1}; \dots ; c _ {\ell , 0}, c _ {\ell , 1}, \dots , c _ {\ell , m _ {\ell} - 1}\right) \mapsto \\ \left(\lambda_ {1} c _ {1, m _ {1} - 1}, c _ {1, 0}, \dots , c _ {1, m _ {1} - 2}; \lambda_ {2} c _ {2, m _ {2} - 1}, c _ {2, 0}, \dots , c _ {2, m _ {2} - 2}; \dots ; \lambda_ {\ell} c _ {\ell , m _ {\ell} - 1}, c _ {\ell , 0}, \dots , c _ {\ell , m _ {\ell} - 2}\right). \\ \end{array}
+$$
+
+Throughout this paper, we adopt the same notations as in [1, 2]. Thus, $ \mathcal{C} $ denotes a $ \Lambda $ -MT code over $ \mathbb{F}_{q} $ with block lengths $ (m_{1}, m_{2}, \dots , m_{\ell}) $ . The Euclidean dual $ \mathcal{C}^{\perp} $ of $ \mathcal{C} $ is a $ (\lambda_{1}^{-1}, \lambda_{2}^{-1}, \dots , \lambda_{\ell}^{-1}) $ -MT code with the same block lengths. By using polynomial representation for blocks, $ \mathcal{C} $ can be regarded as an $ \mathbb{F}_{q}[x] $ -submodule of the $ \Lambda $ -MT module
+
+$$
+V = \bigoplus_ {i = 1} ^ {\ell} \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {i}} - \lambda_ {i} \rangle} = \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {1}} - \lambda_ {1} \rangle} \oplus \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {2}} - \lambda_ {2} \rangle} \oplus \dots \oplus \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {\ell}} - \lambda_ {\ell} \rangle}.
+$$

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@@ -0,0 +1,29 @@
+<div align="center">
+
+# DISPROVING SOME THEOREMS IN SHARMA AND CHAUHAN et al. (2018, 2021) $ ^{*} $
+
+</div>
+
+Ramy Takieldin Faculty of Engineering, Ain Shams University, Cairo, Egypt Egypt University of Informatics, New Capital, Cairo, Egypt ramy.farouk@eng.asu.edu.eg
+
+Patrick Sole I2M (CNRS, University of Aix-Marseille), 13009 Marseilles, France patrick.sole@telecom-paris.fr
+
+## ABSTRACT
+
+The main objective of this work is to show, through counterexamples, that some of the theorems presented in the papers of Sharma et al. (2018) and Chauhan et al. (2021) are incorrect. Although they used these theorems to establish a sufficient condition for a multi-twisted (MT) code to be linear complementary dual (LCD), we show that this condition itself remains valid. We further improve this condition by removing the restrictions on the shift constants and relaxing the required coprimality condition. We show that compared to the previous condition, the modified condition is able to identify more LCD MT codes. Furthermore, without the need for a normalized set of generators, we develop a formula to determine the dimension of any $\rho$-generator MT code.
+
+Keywords Multi-twisted code · linear complementary dual · Determinantal divisors · Algebraic coding MSC: 94B05, 94B60, 11T71
+
+## 1 Introduction
+
+Multi-twisted (MT) codes over a finite field $\mathbb{F}_q$ constitute a significant and comprehensive class of linear codes. This class contains several well-known subclasses, including cyclic, constacyclic, quasi-cyclic, quasi-twisted, and generalized quasi-cyclic codes. For some integer $\ell \geq 1$, let $0 \neq \lambda_i \in \mathbb{F}_q$ and $m_i \geq 1$ for $1 \leq i \leq \ell$. If $\Lambda = (\lambda_1, \lambda_2, \dots, \lambda_\ell)$, then a $\Lambda$-MT code $C$ with block lengths $(m_1, m_2, \dots, m_\ell)$ is defined in [1, Definition 3.1] as a linear code of length $n = m_1 + m_2 + \dots + m_\ell$ that remains invariant under the $\Lambda$-MT linear transformation
+
+$$
+\begin{array}{l} T _ {\Lambda}: \left(c _ {1, 0}, c _ {1, 1}, \dots , c _ {1, m _ {1} - 1}; c _ {2, 0}, c _ {2, 1}, \dots , c _ {2, m _ {2} - 1}; \dots ; c _ {\ell , 0}, c _ {\ell , 1}, \dots , c _ {\ell , m _ {\ell} - 1}\right) \mapsto \\ \left(\lambda_ {1} c _ {1, m _ {1} - 1}, c _ {1, 0}, \dots , c _ {1, m _ {1} - 2}; \lambda_ {2} c _ {2, m _ {2} - 1}, c _ {2, 0}, \dots , c _ {2, m _ {2} - 2}; \dots ; \lambda_ {\ell} c _ {\ell , m _ {\ell} - 1}, c _ {\ell , 0}, \dots , c _ {\ell , m _ {\ell} - 2}\right). \\ \end{array}
+$$
+
+Throughout this paper, we adopt the same notations as in [1, 2]. Thus, $ \mathcal{C} $ denotes a $ \Lambda $ -MT code over $ \mathbb{F}_{q} $ with block lengths $ (m_{1}, m_{2}, \dots , m_{\ell}) $ . The Euclidean dual $ \mathcal{C}^{\perp} $ of $ \mathcal{C} $ is a $ (\lambda_{1}^{-1}, \lambda_{2}^{-1}, \dots , \lambda_{\ell}^{-1}) $ -MT code with the same block lengths. By using polynomial representation for blocks, $ \mathcal{C} $ can be regarded as an $ \mathbb{F}_{q}[x] $ -submodule of the $ \Lambda $ -MT module
+
+$$
+V = \bigoplus_ {i = 1} ^ {\ell} \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {i}} - \lambda_ {i} \rangle} = \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {1}} - \lambda_ {1} \rangle} \oplus \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {2}} - \lambda_ {2} \rangle} \oplus \dots \oplus \frac {\mathbb {F} _ {q} [ x ]}{\langle x ^ {m _ {\ell}} - \lambda_ {\ell} \rangle}.
+$$

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+<div align="center">
+
+# STABLE RECOVERY GUARANTEES FOR BLIND DECONVOLUTION UNDER RANDOM MASK ASSUMPTION
+
+</div>
+
+SONG LI AND YU XIA
+
+ABSTRACT. This study addresses the blind deconvolution problem with modulated inputs, focusing on a measurement model where an unknown blurring kernel $h$ is convolved with multiple random modulations $\{d_{l}\}_{l=1}^{L}$ (coded masks) of a signal $x$, subject to $\ell_{2}$-bounded noise. We introduce a more generalized framework for coded masks, enhancing the versatility of our approach. Our work begins within a constrained least squares framework, where we establish a robust recovery bound for both $h$ and $x$, demonstrating its near-optimality up to a logarithmic factor. Additionally, we present a new recovery scheme that leverages sparsity constraints on $x$. This approach significantly reduces the sampling complexity to the order of $L = O(\log n)$ when the non-zero elements of $x$ are sufficiently separated. Furthermore, we demonstrate that incorporating sparsity constraints yields a refined error bound compared to the traditional constrained least squares model. The proposed method results in more robust and precise signal recovery, as evidenced by both theoretical analysis and numerical simulations. These findings contribute to advancing the field of blind deconvolution and offer potential improvements in various applications requiring signal reconstruction from modulated inputs.
+
+## 1. INTRODUCTION
+
+1. 1. Problem Setup. Blind deconvolution is an inverse problem that aims to reconstruct two unknown signals, $ \boldsymbol{h}, \boldsymbol{x} \in \mathbb{C}^{n} $ , from their circular convolution $ \boldsymbol{y} \in \mathbb{C}^{n} $ , defined as $ \boldsymbol{y} := \boldsymbol{h} \circledast \boldsymbol{x} $ , where $ \circledast $ denotes the circular convolution operator. This operation can be equivalently expressed in matrix form as $ \boldsymbol{y} = \boldsymbol{h} \circledast \boldsymbol{x} = C_{\boldsymbol{h}}\boldsymbol{x} $ , where $ C_{\boldsymbol{h}} $ is the circulant matrix generated by $ \boldsymbol{h} = [h_{1},\dots,h_{n}]^{T} $ , defined as:
+
+$$
+C _ {h} = \left[ \begin{array}{c c c c} h _ {1} & h _ {n} & \dots & h _ {2} \\ h _ {2} & h _ {1} & \dots & h _ {3} \\ \vdots & \vdots & \ddots & \vdots \\ h _ {n} & h _ {n - 1} & \dots & h _ {1} \end{array} \right].
+$$
+
+This problem arises in numerous fields, including astronomy, optics, image processing, and communications engineering [17, 11, 21, 33]. The blind deconvolution problem is inherently ill-posed due to the presence of scaled-shift symmetry, which implies that there are infinitely many signal pairs that can yield the same convolution result. Consequently, incorporating prior information is crucial to overcoming this ill-posedness. For example, one might impose a subspace condition on $ \boldsymbol{x} $ [2], or enforce a short support condition on $ \boldsymbol{h} $ in combination with a sparsity constraint on $ \boldsymbol{x} $ [19, 31].
+
+In this work, we examine a related class of blind deconvolution problems, where the blur kernel $ \boldsymbol{h}\in\mathbb{C}^{n} $ is convolved with multiple modulated inputs. Specifically, the observations $ \boldsymbol{y}_{l}\in\mathbb{C}^{n} $

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@@ -0,0 +1,21 @@
+<div align="center">
+
+# STABLE RECOVERY GUARANTEES FOR BLIND DECONVOLUTION UNDER RANDOM MASK ASSUMPTION
+
+</div>
+
+SONG LI AND YU XIA
+
+ABSTRACT. This study addresses the blind deconvolution problem with modulated inputs, focusing on a measurement model where an unknown blurring kernel $h$ is convolved with multiple random modulations $\{d_{l}\}_{l=1}^{L}$ (coded masks) of a signal $x$, subject to $\ell_{2}$-bounded noise. We introduce a more generalized framework for coded masks, enhancing the versatility of our approach. Our work begins within a constrained least squares framework, where we establish a robust recovery bound for both $h$ and $x$, demonstrating its near-optimality up to a logarithmic factor. Additionally, we present a new recovery scheme that leverages sparsity constraints on $x$. This approach significantly reduces the sampling complexity to the order of $L = O(\log n)$ when the non-zero elements of $x$ are sufficiently separated. Furthermore, we demonstrate that incorporating sparsity constraints yields a refined error bound compared to the traditional constrained least squares model. The proposed method results in more robust and precise signal recovery, as evidenced by both theoretical analysis and numerical simulations. These findings contribute to advancing the field of blind deconvolution and offer potential improvements in various applications requiring signal reconstruction from modulated inputs.
+
+## 1. INTRODUCTION
+
+1. 1. Problem Setup. Blind deconvolution is an inverse problem that aims to reconstruct two unknown signals, $ \boldsymbol{h}, \boldsymbol{x} \in \mathbb{C}^{n} $ , from their circular convolution $ \boldsymbol{y} \in \mathbb{C}^{n} $ , defined as $ \boldsymbol{y} := \boldsymbol{h} \circledast \boldsymbol{x} $ , where $ \circledast $ denotes the circular convolution operator. This operation can be equivalently expressed in matrix form as $ \boldsymbol{y} = \boldsymbol{h} \circledast \boldsymbol{x} = C_{\boldsymbol{h}}\boldsymbol{x} $ , where $ C_{\boldsymbol{h}} $ is the circulant matrix generated by $ \boldsymbol{h} = [h_{1},\dots,h_{n}]^{T} $ , defined as:
+
+$$
+C _ {h} = \left[ \begin{array}{c c c c} h _ {1} & h _ {n} & \dots & h _ {2} \\ h _ {2} & h _ {1} & \dots & h _ {3} \\ \vdots & \vdots & \ddots & \vdots \\ h _ {n} & h _ {n - 1} & \dots & h _ {1} \end{array} \right].
+$$
+
+This problem arises in numerous fields, including astronomy, optics, image processing, and communications engineering [17, 11, 21, 33]. The blind deconvolution problem is inherently ill-posed due to the presence of scaled-shift symmetry, which implies that there are infinitely many signal pairs that can yield the same convolution result. Consequently, incorporating prior information is crucial to overcoming this ill-posedness. For example, one might impose a subspace condition on $ \boldsymbol{x} $ [2], or enforce a short support condition on $ \boldsymbol{h} $ in combination with a sparsity constraint on $ \boldsymbol{x} $ [19, 31].
+
+In this work, we examine a related class of blind deconvolution problems, where the blur kernel $ \boldsymbol{h}\in\mathbb{C}^{n} $ is convolved with multiple modulated inputs. Specifically, the observations $ \boldsymbol{y}_{l}\in\mathbb{C}^{n} $

zai-org__GLM-OCR-api/arxiv_math/2503.03765_pg1_pg1_repeat3.md

@@ -0,0 +1,21 @@
+<div align="center">
+
+# STABLE RECOVERY GUARANTEES FOR BLIND DECONVOLUTION UNDER RANDOM MASK ASSUMPTION
+
+</div>
+
+SONG LI AND YU XIA
+
+ABSTRACT. This study addresses the blind deconvolution problem with modulated inputs, focusing on a measurement model where an unknown blurring kernel $h$ is convolved with multiple random modulations $\{d_{l}\}_{l=1}^{L}$ (coded masks) of a signal $x$, subject to $\ell_{2}$-bounded noise. We introduce a more generalized framework for coded masks, enhancing the versatility of our approach. Our work begins within a constrained least squares framework, where we establish a robust recovery bound for both $h$ and $x$, demonstrating its near-optimality up to a logarithmic factor. Additionally, we present a new recovery scheme that leverages sparsity constraints on $x$. This approach significantly reduces the sampling complexity to the order of $L = O(\log n)$ when the non-zero elements of $x$ are sufficiently separated. Furthermore, we demonstrate that incorporating sparsity constraints yields a refined error bound compared to the traditional constrained least squares model. The proposed method results in more robust and precise signal recovery, as evidenced by both theoretical analysis and numerical simulations. These findings contribute to advancing the field of blind deconvolution and offer potential improvements in various applications requiring signal reconstruction from modulated inputs.
+
+## 1. INTRODUCTION
+
+1. 1. Problem Setup. Blind deconvolution is an inverse problem that aims to reconstruct two unknown signals, $ \boldsymbol{h}, \boldsymbol{x} \in \mathbb{C}^{n} $ , from their circular convolution $ \boldsymbol{y} \in \mathbb{C}^{n} $ , defined as $ \boldsymbol{y} := \boldsymbol{h} \circledast \boldsymbol{x} $ , where $ \circledast $ denotes the circular convolution operator. This operation can be equivalently expressed in matrix form as $ \boldsymbol{y} = \boldsymbol{h} \circledast \boldsymbol{x} = C_{\boldsymbol{h}}\boldsymbol{x} $ , where $ C_{\boldsymbol{h}} $ is the circulant matrix generated by $ \boldsymbol{h} = [h_{1},\dots,h_{n}]^{T} $ , defined as:
+
+$$
+C _ {h} = \left[ \begin{array}{c c c c} h _ {1} & h _ {n} & \dots & h _ {2} \\ h _ {2} & h _ {1} & \dots & h _ {3} \\ \vdots & \vdots & \ddots & \vdots \\ h _ {n} & h _ {n - 1} & \dots & h _ {1} \end{array} \right].
+$$
+
+This problem arises in numerous fields, including astronomy, optics, image processing, and communications engineering [17, 11, 21, 33]. The blind deconvolution problem is inherently ill-posed due to the presence of scaled-shift symmetry, which implies that there are infinitely many signal pairs that can yield the same convolution result. Consequently, incorporating prior information is crucial to overcoming this ill-posedness. For example, one might impose a subspace condition on $ \boldsymbol{x} $ [2], or enforce a short support condition on $ \boldsymbol{h} $ in combination with a sparsity constraint on $ \boldsymbol{x} $ [19, 31].
+
+In this work, we examine a related class of blind deconvolution problems, where the blur kernel $ \boldsymbol{h}\in\mathbb{C}^{n} $ is convolved with multiple modulated inputs. Specifically, the observations $ \boldsymbol{y}_{l}\in\mathbb{C}^{n} $

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+Now, we prove that $\Psi_c\backslash \{(0,m):m\geq c\} \subset \Psi_c^*$. For any $0 < p\leq 1$, consider any $(p,m)\in \Psi_c$ and the following probability mass function for a random variable $T$:
+
+$$
+\Pr \{T = 0 \} = 1 - p \text {a n d} \Pr \{T = m / p \} = p.
+$$
+
+Since m $ \geq $ cp, we have m/p $ \geq c $ , and so
+
+$$
+\Pr \{T \geq c \} = \Pr \{T = m / p \} = p,
+$$
+
+and $ E[T] = p(m / p) = m $ . Therefore, $ (p,m) $ is achieved by the random variable $ T $ as constructed. Note that unless $ m = cp $ , $ (p,m) $ can always be achieved by more than one probability distribution.
+
+It remains to prove that every ordered pair $ (0,m) $ with $ 0\leq m<c $ is achievable. This can be done by noting that such an ordered pair can be achieved by any random variable $ T $ with $ \operatorname* {P r} \{ T=m \}=1. $ The theorem is proved.
+
+The region $ \Psi_{c} $ is defined by Markov's inequality together with the constraint $ 0\leq p\leq 1 $ which comes from the setup of the problem, and we have shown that for any fixed $ c>0 $ , every ordered pair in $ \Psi_{c} $ except for a region with Lebesgue measure 0 (namely the region $ \{(0,m):m\geq c\} $ ) is achievable by some random variable $ T $ . Specifically:
+
+- When $ \operatorname{Pr}\{T\geq c\} > 0 $ , Markov's inequality, namely
+
+$$
+E [ T ] \geq c \cdot \Pr \{T \geq c \},
+$$
+
+which gives a lower bound on $E[T]$, is the only constraint on $E[T]$ in terms of $\operatorname{Pr}\{T \geq c\}$.
+
+- When $ \operatorname{Pr}\{T\geq c\}=0 $ , Markov's inequality as in (10), which becomes $ E[T]\geq 0 $ , continues to be valid. However, we also have
+
+$$
+E [ T ] < c.
+$$
+
+Combining (10) and (11), we have
+
+$$
+0 \leq E [ T ] < c.
+$$

zai-org__GLM-OCR-api/arxiv_math/2503.03766_pg12_pg1_repeat2.md

@@ -0,0 +1,37 @@
+Now, we prove that $\Psi_c\backslash \{(0,m):m\geq c\}\subset \Psi_c^*$. For any $0 < p\leq 1$, consider any $(p,m)\in \Psi_c$ and the following probability mass function for a random variable $T$:
+
+$$
+\Pr \{T = 0 \} = 1 - p \text {a n d} \Pr \{T = m / p \} = p.
+$$
+
+Since m $ \geq $ cp, we have m/p $ \geq c $ , and so
+
+$$
+\Pr \{T \geq c \} = \Pr \{T = m / p \} = p,
+$$
+
+and $ E[T] = p(m / p) = m $ . Therefore, $ (p,m) $ is achieved by the random variable $ T $ as constructed. Note that unless $ m = cp $ , $ (p,m) $ can always be achieved by more than one probability distribution.
+
+It remains to prove that every ordered pair $ (0,m) $ with $ 0\leq m<c $ is achievable. This can be done by noting that such an ordered pair can be achieved by any random variable $ T $ with $ \operatorname* {P r} \{ T=m \}=1. $ The theorem is proved.
+
+The region $ \Psi_{c} $ is defined by Markov's inequality together with the constraint $ 0\leq p\leq 1 $ which comes from the setup of the problem, and we have shown that for any fixed $ c>0 $ , every ordered pair in $ \Psi_{c} $ except for a region with Lebesgue measure 0 (namely the region $ \{(0,m):m\geq c\} $ ) is achievable by some random variable $ T $ . Specifically:
+
+- When $ \operatorname{Pr}\{T\geq c\} > 0 $ , Markov's inequality, namely
+
+$$
+E [ T ] \geq c \cdot \Pr \{T \geq c \},
+$$
+
+which gives a lower bound on $E[T]$, is the only constraint on $E[T]$ in terms of $\operatorname{Pr}\{T \geq c\}$.
+
+- When $ \operatorname{Pr}\{T\geq c\}=0 $ , Markov's inequality as in (10), which becomes $ E[T]\geq 0 $ , continues to be valid. However, we also have
+
+$$
+E [ T ] < c.
+$$
+
+Combining (10) and (11), we have
+
+$$
+0 \leq E [ T ] < c.
+$$

zai-org__GLM-OCR-api/arxiv_math/2503.03766_pg12_pg1_repeat3.md

@@ -0,0 +1,37 @@
+Now, we prove that $\Psi_c\backslash \{(0,m):m\geq c\} \subset \Psi_c^*$. For any $0 < p \leq 1$, consider any $(p,m)\in \Psi_c$ and the following probability mass function for a random variable $T$:
+
+$$
+\Pr \{T = 0 \} = 1 - p \text {a n d} \Pr \{T = m / p \} = p.
+$$
+
+Since m $ \geq $ cp, we have m/p $ \geq c $ , and so
+
+$$
+\Pr \{T \geq c \} = \Pr \{T = m / p \} = p,
+$$
+
+and $ E[T] = p(m / p) = m $ . Therefore, $ (p,m) $ is achieved by the random variable $ T $ as constructed. Note that unless $ m = cp $ , $ (p,m) $ can always be achieved by more than one probability distribution.
+
+It remains to prove that every ordered pair $ (0,m) $ with $ 0\leq m<c $ is achievable. This can be done by noting that such an ordered pair can be achieved by any random variable $ T $ with $ \operatorname* {P r} \{ T=m \}=1. $ The theorem is proved.
+
+The region $ \Psi_{c} $ is defined by Markov's inequality together with the constraint $ 0\leq p\leq 1 $ which comes from the setup of the problem, and we have shown that for any fixed $ c>0 $ , every ordered pair in $ \Psi_{c} $ except for a region with Lebesgue measure 0 (namely the region $ \{(0,m):m\geq c\} $ ) is achievable by some random variable $ T $ . Specifically:
+
+- When $ \operatorname{Pr}\{T\geq c\} > 0 $ , Markov's inequality, namely
+
+$$
+E [ T ] \geq c \cdot \Pr \{T \geq c \},
+$$
+
+which gives a lower bound on $E[T]$, is the only constraint on $E[T]$ in terms of $\operatorname{Pr}\{T \geq c\}$.
+
+- When $ \operatorname{Pr}\{T\geq c\}=0 $ , Markov's inequality as in (10), which becomes $ E[T]\geq 0 $ , continues to be valid. However, we also have
+
+$$
+E [ T ] < c.
+$$
+
+Combining (10) and (11), we have
+
+$$
+0 \leq E [ T ] < c.
+$$

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@@ -0,0 +1,23 @@
+<div align="center">
+
+# Cardinalities in finite monoids of $ G $ -equivariant functions
+
+</div>
+
+Ramón H. Ruiz-Medina $ ^{*} $
+
+Centro Universitario de Ciencias Exactas e Ingenierías Universidad de Guadalajara, Guadalajara, México.
+
+## Abstract
+
+A set with a group action is referred to as a $G$-set, and the set of functions that commute with this action forms a monoid under function composition. This paper examines the case where the $G$-set is finite, which implies that the monoid of $G$-equivariant functions is also finite. The document provides formulas for calculating the cardinality of this monoid, its group of units, and explores special cases of $G$-equivariant functions, known as fixing elementary collapsings. All of these results are expressed in terms of specific properties of the $G$-set, including the number of orbits and certain indices of the subgroups acting as stabilizers.
+
+Keywords: Group actions, $G$-sets, $G$-equivariant function, cardinality.
+
+MSC 2020: 20B25, 20E22, 20M20.
+
+## 1 Introduction
+
+For any group $G$, a $G$-set is simply a set $X$ on which $G$ acts; that is, there exists a function $:\ G\times X\to X$ such that $e\cdot x = x$ for all $x\in X$ and $g\cdot(h\cdot x) = (gh)\cdot x$ for all $x\in X$, $g,h\in G$. In the context of semigroup theory, $G$-sets are also known as $G$-acts. A $G$-equivariant transformation of $X$, or a $G$-endomorphism of $X$, is a function $\tau : X\to X$ such that $\tau(g\cdot x) = g\cdot \tau(x)$ for all $g\in G$, $x\in X$. These maps are fundamental in the category of $G$-sets and find applications in various branches of mathematics such as equivariant topology, representation theory, and statistical inference.
+
+The set of all $ G $ - equivariant transformations of $ X $ , which are functions that commute with the group action, forms a monoid under function composition. We denote this monoid as $ \operatorname{E n d}_{G}(X) $ , and its group of units, consisting of all bijective $ G $ - equivariant transformations, as $ \operatorname{A u t}_{G}(X) $ . These objects have been extensively studied in various contexts (see [2], [3], [10], [16]). Many examples of objects with group actions and associated $ G $ - equivariant functions have been explored, such as cellular automata, which have been the motivation

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+<div align="center">
+
+# Cardinalities in finite monoids of $ G $ -equivariant functions
+
+</div>
+
+Ramón H. Ruiz-Medina $ ^{*} $
+
+Centro Universitario de Ciencias Exactas e Ingenierías Universidad de Guadalajara, Guadalajara, México.
+
+## Abstract
+
+A set with a group action is referred to as a $G$-set, and the set of functions that commute with this action forms a monoid under function composition. This paper examines the case where the $G$-set is finite, which implies that the monoid of $G$-equivariant functions is also finite. The document provides formulas for calculating the cardinality of this monoid, its group of units, and explores special cases of $G$-equivariant functions, known as fixing elementary collapsings. All of these results are expressed in terms of specific properties of the $G$-set, including the number of orbits and certain indices of the subgroups acting as stabilizers.
+
+Keywords: Group actions, $G$-sets, $G$-equivariant function, cardinality.
+
+MSC 2020: 20B25, 20E22, 20M20.
+
+## 1 Introduction
+
+For any group $G$, a $G$-set is simply a set $X$ on which $G$ acts; that is, there exists a function $:\ G\times X\to X$ such that $e\cdot x = x$ for all $x\in X$ and $g\cdot(h\cdot x) = (gh)\cdot x$ for all $x\in X$, $g,h\in G$ In the context of semigroup theory, $G$-sets are also known as $G$-acts. A $G$-equivariant transformation of $X$, or a $G$-endomorphism of $X$, is a function $\tau : X\to X$ such that $\tau(g\cdot x) = g\cdot \tau(x)$ for all $g\in G$, $x\in X$. These maps are fundamental in the category of $G$-sets and find applications in various branches of mathematics such as equivariant topology, representation theory, and statistical inference.
+
+The set of all $ G $ - equivariant transformations of $ X $ , which are functions that commute with the group action, forms a monoid under function composition. We denote this monoid as $ \operatorname{E n d}_{G}(X) $ , and its group of units, consisting of all bijective $ G $ - equivariant transformations, as $ \operatorname{A u t}_{G}(X) $ . These objects have been extensively studied in various contexts (see [2], [3], [10], [16]). Many examples of objects with group actions and associated $ G $ - equivariant functions have been explored, such as cellular automata, which have been the motivation

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@@ -0,0 +1,23 @@
+<div align="center">
+
+# Cardinalities in finite monoids of $ G $ -equivariant functions
+
+</div>
+
+Ramón H. Ruiz-Medina $ ^{*} $
+
+Centro Universitario de Ciencias Exactas e Ingenierías Universidad de Guadalajara, Guadalajara, México.
+
+## Abstract
+
+A set with a group action is referred to as a $G$-set, and the set of functions that commute with this action forms a monoid under function composition. This paper examines the case where the $G$-set is finite, which implies that the monoid of $G$-equivariant functions is also finite. The document provides formulas for calculating the cardinality of this monoid, its group of units, and explores special cases of $G$-equivariant functions, known as fixing elementary collapsings. All of these results are expressed in terms of specific properties of the $G$-set, including the number of orbits and certain indices of the subgroups acting as stabilizers.
+
+Keywords: Group actions, $G$-sets, $G$-equivariant function, cardinality.
+
+MSC 2020: 20B25, 20E22, 20M20.
+
+## 1 Introduction
+
+For any group $G$, a $G$-set is simply a set $X$ on which $G$ acts; that is, there exists a function $:\ G\times X\to X$ such that $e\cdot x = x$ for all $x\in X$ and $g\cdot(h\cdot x) = (gh)\cdot x$ for all $x\in X$, $g,h\in G$ In the context of semigroup theory, $G$-sets are also known as $G$-acts. A $G$-equivariant transformation of $X$, or a $G$-endomorphism of $X$, is a function $\tau : X\to X$ such that $\tau(g\cdot x) = g\cdot \tau(x)$ for all $g\in G$, $x\in X$. These maps are fundamental in the category of $G$-sets and find applications in various branches of mathematics such as equivariant topology, representation theory, and statistical inference.
+
+The set of all $ G $ - equivariant transformations of $ X $ , which are functions that commute with the group action, forms a monoid under function composition. We denote this monoid as $ \operatorname{E n d}_{G}(X) $ , and its group of units, consisting of all bijective $ G $ - equivariant transformations, as $ \operatorname{A u t}_{G}(X) $ . These objects have been extensively studied in various contexts (see [2], [3], [10], [16]). Many examples of objects with group actions and associated $ G $ - equivariant functions have been explored, such as cellular automata, which have been the motivation

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+This code was first proposed in Ref. [18] using the polynomials
+
+$$
+\begin{array}{l} f (x, y) = x ^ {2 5} + x ^ {2 6} + y ^ {3}, \\ g (x, y) = y + y ^ {2} + x ^ {9}, \\ \end{array}
+$$
+
+and implemented on an untwisted $ 30\times 6 $ torus. The stabilizers have a range of 9 in the $ x $ - direction (since $ x^{25} $ and $ x^{26} $ can be equivalently treated as $ x^{-5} $ and $ x^{-4} $ ).
+
+In contrast, our $[[360,12,24]]$ code is simply the (3,3)-BB code (Example 3), specified by the following polynomials:
+
+$$
+\begin{array}{l} f (x, y) = x + x ^ {2} + y ^ {3}, \\ g (x, y) = y + y ^ {2} + x ^ {3}, \\ \end{array}
+$$
+
+placed on a twisted $6\times 30$ torus with lattice vectors $\vec{a}_{1} = (0,30)$ and $\vec{a}_{2} = (6,6)$. For physical realization, once we have the architecture for the stabilizers of the (3,3)-BB code, we can generate optimal generalized toric codes on various lattices, as listed in Eq. (59).
+
+Similarly, the physical construction of the (3, -3) BB code (Example 4) can generate the quantum LDPC codes listed in Eq. (58). By comparing the stabilizers in Tables I, II, III, and IV with those in the literature, we observe that twisted tori generally reduce the range of stabilizers, making experimental realization more feasible.
+
+## C. Relation to one-dimensional generalized bicycle codes
+
+We present another example from Table III, the $[[254,14,16]]$ code, which achieves $ k d^{2} / n=14.11 $ . This code is defined on a twisted $ 1\times 127 $ torus with lattice vectors $ \vec{a}_{1}=(0,127) $ and $ \vec{a}_{2}=(1,25) $ . The associated polynomials are:
+
+$$
+\begin{array}{l} f (x, y) = 1 + x + x ^ {- 1} y ^ {- 3}, \\ g (x, y) = 1 + y + y ^ {- 6}. \\ \end{array}
+$$
+
+The code is local on the twisted torus, as the range of each stabilizer is small relative to the total system size n. Since the twisted torus is narrow in the x-direction, we can remove the x-direction periodicity by using the polynomial $ x y^{2 5} - 1 $ to cancel the x-dependence. Therefore, we can reduce the code to a non-local one-dimensional quantum code. This transformation yields the following polynomials:
+
+$$
+\begin{array}{l} f (y) = 1 + y ^ {2 2} + y ^ {1 0 2}, \\ g (y) = 1 + y + y ^ {1 2 1}, \\ \end{array}
+$$
+
+with a periodic boundary condition $ y^{127}-1=0 $ . In these one-dimensional codes, the Gröbner basis in Theorem 3 reduces to the gcd (greatest common divisor) for univariate polynomials, simplifying to the following expression:
+
+$$
+\begin{array}{l} k = 2 \dim \left(\frac {\mathbb {Z} _ {2} [ y , y ^ {- 1} ]}{\langle f (y) , g (y) , y ^ {l} - 1 \rangle}\right) \\ = 2 \deg \left(\gcd \left(f (y), g (y), y ^ {l} - 1\right)\right), \\ \end{array}
+$$
+
+which precisely matches Proposition 1 in Ref. [124], which computes the logical dimension of the generalized bicycle (GB) codes [125]. For each value of n, we can apply the same procedure to the generalized toric codes on the twisted $ 1 \times \frac{n}{2} $ tori:
+
+$$
+\vec {a} _ {1} = \left(0, \frac {n}{2}\right), \quad \vec {a} _ {2} = (1, \gamma), \quad \text {w i t h} 0 \leq \gamma < \frac {n}{2},
+$$
+
+to induce the corresponding one-dimensional generalized bicycle codes. The results are summarized in Table V, VI, and VII in Appendix A.
+
+For comparison, consider the GB code described in Ref. [124]. The polynomials for this GB code are:
+
+$$
+\begin{array}{l} f (y) = 1 + y ^ {1 5} + y ^ {2 0} + y ^ {2 8} + y ^ {6 6}, \\ f (y) = 1 + y ^ {5 8} + y ^ {5 9} + y ^ {1 0 0} + y ^ {1 2 1}, \\ \end{array}
+$$
+
+defined on a cycle of length $l = 127$. This GB code uses weight-10 stabilizers to achieve better code parameters $[[254,28,14\leq d\leq 20]]$.
+
+## IV. DISCUSSION AND FUTURE DIRECTIONS
+
+We have introduced a topological order perspective to studying quantum error-correcting codes on tori. From the algebraic structure of anyons, the logical dimension $k$ can be determined by counting independent anyon types. We showed that this corresponds to the dimension of the quotient ring $R / I$, where the ideal $I = \langle f(x,y),g(x,y)\rangle$ is generated by the stabilizers. This provides a systematic approach to characterizing the code space. Our framework naturally incorporates (twisted) periodic boundary conditions, enabling the construction and characterization of new quantum LDPC codes. To ensure computational feasibility, we employed Gröbner basis techniques, enabling a systematic analysis of generalized toric codes up to $n\leq 400$ physical qubits. The versatility of our method is reflected in the discovery of novel qLDPC codes listed in Tables I, II, III, and IV. These results illustrate the power of a ring-theoretic approach in advancing the understanding of topological quantum codes, paving the way for future explorations in both theory and practical implementation.
+
+Future work could extend this investigation to larger system sizes (higher $n$), as these may yield improved codes. Given that our search algorithm is fully parallelizable, supercomputers or computer clusters could be employed to examine all generalized toric codes within $n \leq 500$ or higher—scales that are comparable to the number of physical qubits in state-of-the-art experimental platforms [126-131]. The primary bottleneck, however, is the computation of code distances. When $n$ reaches a few hundred and $d$ exceeds 20, the probabilistic algorithm for computing the code distance may not be reliable and could only yield an upper bound for $d$.
+
+Alternatively, one could explore different forms of

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@@ -0,0 +1,61 @@
+This code was first proposed in Ref. [18] using the polynomials
+
+$$
+\begin{array}{l} f (x, y) = x ^ {2 5} + x ^ {2 6} + y ^ {3}, \\ g (x, y) = y + y ^ {2} + x ^ {9}, \\ \end{array}
+$$
+
+and implemented on an untwisted $ 30\times 6 $ torus. The stabilizers have a range of 9 in the $ x $ - direction (since $ x^{25} $ and $ x^{26} $ can be equivalently treated as $ x^{-5} $ and $ x^{-4} $ ).
+
+In contrast, our $[[360,12,24]]$ code is simply the (3,3)-BB code (Example 3), specified by the following polynomials:
+
+$$
+\begin{array}{l} f (x, y) = x + x ^ {2} + y ^ {3}, \\ g (x, y) = y + y ^ {2} + x ^ {3}, \\ \end{array}
+$$
+
+placed on a twisted $6\times 30$ torus with lattice vectors $\vec{a}_{1} = (0,30)$ and $\vec{a}_{2} = (6,6)$. For physical realization, once we have the architecture for the stabilizers of the (3,3)-BB code, we can generate optimal generalized toric codes on various lattices, as listed in Eq. (59).
+
+Similarly, the physical construction of the (3, -3) BB code (Example 4) can generate the quantum LDPC codes listed in Eq. (58). By comparing the stabilizers in Tables I, II, III, and IV with those in the literature, we observe that twisted tori generally reduce the range of stabilizers, making experimental realization more feasible.
+
+## C. Relation to one-dimensional generalized bicycle codes
+
+We present another example from Table III, the $[[254,14,16]]$ code, which achieves $ k d^{2} / n=14.11 $ . This code is defined on a twisted $ 1\times 127 $ torus with lattice vectors $ \vec{a}_{1}=(0,127) $ and $ \vec{a}_{2}=(1,25) $ . The associated polynomials are:
+
+$$
+\begin{array}{l} f (x, y) = 1 + x + x ^ {- 1} y ^ {- 3}, \\ g (x, y) = 1 + y + y ^ {- 6}. \\ \end{array}
+$$
+
+The code is local on the twisted torus, as the range of each stabilizer is small relative to the total system size n. Since the twisted torus is narrow in the x-direction, we can remove the x-direction periodicity by using the polynomial $ x y^{2 5} - 1 $ to cancel the x-dependence. Therefore, we can reduce the code to a non-local one-dimensional quantum code. This transformation yields the following polynomials:
+
+$$
+\begin{array}{l} f (y) = 1 + y ^ {2 2} + y ^ {1 0 2}, \\ g (y) = 1 + y + y ^ {1 2 1}, \\ \end{array}
+$$
+
+with a periodic boundary condition $ y^{127}-1=0 $ . In these one-dimensional codes, the Gröbner basis in Theorem 3 reduces to the gcd (greatest common divisor) for univariate polynomials, simplifying to the following expression:
+
+$$
+\begin{array}{l} k = 2 \dim \left(\frac {\mathbb {Z} _ {2} [ y , y ^ {- 1} ]}{\langle f (y) , g (y) , y ^ {l} - 1 \rangle}\right) \\ = 2 \deg \left(\gcd \left(f (y), g (y), y ^ {l} - 1\right)\right), \\ \end{array}
+$$
+
+which precisely matches Proposition 1 in Ref. [124], which computes the logical dimension of the generalized bicycle (GB) codes [125]. For each value of n, we can apply the same procedure to the generalized toric codes on the twisted $ 1 \times \frac{n}{2} $ tori:
+
+$$
+\vec {a} _ {1} = \left(0, \frac {n}{2}\right), \quad \vec {a} _ {2} = (1, \gamma), \quad \text {w i t h} 0 \leq \gamma < \frac {n}{2},
+$$
+
+to induce the corresponding one-dimensional generalized bicycle codes. The results are summarized in Table V, VI, and VII in Appendix A.
+
+For comparison, consider the GB code described in Ref. [124]. The polynomials for this GB code are:
+
+$$
+\begin{array}{l} f (y) = 1 + y ^ {1 5} + y ^ {2 0} + y ^ {2 8} + y ^ {6 6}, \\ f (y) = 1 + y ^ {5 8} + y ^ {5 9} + y ^ {1 0 0} + y ^ {1 2 1}, \\ \end{array}
+$$
+
+defined on a cycle of length $l = 127$. This GB code uses weight-10 stabilizers to achieve better code parameters $[[254,28,14\leq d\leq 20]]$.
+
+## IV. DISCUSSION AND FUTURE DIRECTIONS
+
+We have introduced a topological order perspective to studying quantum error-correcting codes on tori. From the algebraic structure of anyons, the logical dimension $k$ can be determined by counting independent anyon types. We showed that this corresponds to the dimension of the quotient ring $R / I$, where the ideal $I = \langle f(x,y),g(x,y)\rangle$ is generated by the stabilizers. This provides a systematic approach to characterizing the code space. Our framework naturally incorporates (twisted) periodic boundary conditions, enabling the construction and characterization of new quantum LDPC codes. To ensure computational feasibility, we employed Gröbner basis techniques, enabling a systematic analysis of generalized toric codes up to $n\leq 400$ physical qubits. The versatility of our method is reflected in the discovery of novel qLDPC codes listed in Tables I, II, III, and IV. These results illustrate the power of a ring-theoretic approach in advancing the understanding of topological quantum codes, paving the way for future explorations in both theory and practical implementation.
+
+Future work could extend this investigation to larger system sizes (higher $n$), as these may yield improved codes. Given that our search algorithm is fully parallelizable, supercomputers or computer clusters could be employed to examine all generalized toric codes within $n \leq 500$ or higher—scales that are comparable to the number of physical qubits in state-of-the-art experimental platforms [126-131]. The primary bottleneck, however, is the computation of code distances. When $n$ reaches a few hundred and $d$ exceeds 20, the probabilistic algorithm for computing the code distance may not be reliable and could only yield an upper bound for $d$.
+
+Alternatively, one could explore different forms of

zai-org__GLM-OCR-api/arxiv_math/2503.03827_pg10_pg1_repeat3.md

@@ -0,0 +1,61 @@
+This code was first proposed in Ref. [18] using the polynomials
+
+$$
+\begin{array}{l} f (x, y) = x ^ {2 5} + x ^ {2 6} + y ^ {3}, \\ g (x, y) = y + y ^ {2} + x ^ {9}, \\ \end{array}
+$$
+
+and implemented on an untwisted $ 30\times 6 $ torus. The stabilizers have a range of 9 in the $ x $ - direction (since $ x^{25} $ and $ x^{26} $ can be equivalently treated as $ x^{-5} $ and $ x^{-4} $ ).
+
+In contrast, our $[[360,12,24]]$ code is simply the (3,3)-BB code (Example 3), specified by the following polynomials:
+
+$$
+\begin{array}{l} f (x, y) = x + x ^ {2} + y ^ {3}, \\ g (x, y) = y + y ^ {2} + x ^ {3}, \\ \end{array}
+$$
+
+placed on a twisted $6\times 30$ torus with lattice vectors $\vec{a}_{1} = (0,30)$ and $\vec{a}_{2} = (6,6)$. For physical realization, once we have the architecture for the stabilizers of the (3,3)-BB code, we can generate optimal generalized toric codes on various lattices, as listed in Eq. (59).
+
+Similarly, the physical construction of the (3, -3) BB code (Example 4) can generate the quantum LDPC codes listed in Eq. (58). By comparing the stabilizers in Tables I, II, III, and IV with those in the literature, we observe that twisted tori generally reduce the range of stabilizers, making experimental realization more feasible.
+
+## C. Relation to one-dimensional generalized bicycle codes
+
+We present another example from Table III, the $[[254,14,16]]$ code, which achieves $ k d^{2} / n=14.11 $ . This code is defined on a twisted $ 1\times 127 $ torus with lattice vectors $ \vec{a}_{1}=(0,127) $ and $ \vec{a}_{2}=(1,25) $ . The associated polynomials are:
+
+$$
+\begin{array}{l} f (x, y) = 1 + x + x ^ {- 1} y ^ {- 3}, \\ g (x, y) = 1 + y + y ^ {- 6}. \\ \end{array}
+$$
+
+The code is local on the twisted torus, as the range of each stabilizer is small relative to the total system size n. Since the twisted torus is narrow in the x-direction, we can remove the x-direction periodicity by using the polynomial $ x y^{2 5} - 1 $ to cancel the x-dependence. Therefore, we can reduce the code to a non-local one-dimensional quantum code. This transformation yields the following polynomials:
+
+$$
+\begin{array}{l} f (y) = 1 + y ^ {2 2} + y ^ {1 0 2}, \\ g (y) = 1 + y + y ^ {1 2 1}, \\ \end{array}
+$$
+
+with a periodic boundary condition $ y^{127}-1=0 $ . In these one-dimensional codes, the Gröbner basis in Theorem 3 reduces to the gcd (greatest common divisor) for univariate polynomials, simplifying to the following expression:
+
+$$
+\begin{array}{l} k = 2 \dim \left(\frac {\mathbb {Z} _ {2} [ y , y ^ {- 1} ]}{\langle f (y) , g (y) , y ^ {l} - 1 \rangle}\right) \\ = 2 \deg \left(\gcd \left(f (y), g (y), y ^ {l} - 1\right)\right), \\ \end{array}
+$$
+
+which precisely matches Proposition 1 in Ref. [124], which computes the logical dimension of the generalized bicycle (GB) codes [125]. For each value of n, we can apply the same procedure to the generalized toric codes on the twisted $ 1\times \frac{n}{2} $ tori:
+
+$$
+\vec {a} _ {1} = \left(0, \frac {n}{2}\right), \quad \vec {a} _ {2} = (1, \gamma), \quad \text {w i t h} 0 \leq \gamma < \frac {n}{2},
+$$
+
+to induce the corresponding one-dimensional generalized bicycle codes. The results are summarized in Table V, VI, and VII in Appendix A.
+
+For comparison, consider the GB code described in Ref. [124]. The polynomials for this GB code are:
+
+$$
+\begin{array}{l} f (y) = 1 + y ^ {1 5} + y ^ {2 0} + y ^ {2 8} + y ^ {6 6}, \\ f (y) = 1 + y ^ {5 8} + y ^ {5 9} + y ^ {1 0 0} + y ^ {1 2 1}, \\ \end{array}
+$$
+
+defined on a cycle of length $l = 127$. This GB code uses weight-10 stabilizers to achieve better code parameters $[[254,28,14\leq d\leq 20]]$.
+
+## IV. DISCUSSION AND FUTURE DIRECTIONS
+
+We have introduced a topological order perspective to studying quantum error-correcting codes on tori. From the algebraic structure of anyons, the logical dimension $k$ can be determined by counting independent anyon types. We showed that this corresponds to the dimension of the quotient ring $R / I$, where the ideal $I = \langle f(x,y),g(x,y)\rangle$ is generated by the stabilizers. This provides a systematic approach to characterizing the code space. Our framework naturally incorporates (twisted) periodic boundary conditions, enabling the construction and characterization of new quantum LDPC codes. To ensure computational feasibility, we employed Gröbner basis techniques, enabling a systematic analysis of generalized toric codes up to $n\leq 400$ physical qubits. The versatility of our method is reflected in the discovery of novel qLDPC codes listed in Tables I, II, III, and IV. These results illustrate the power of a ring-theoretic approach in advancing the understanding of topological quantum codes, paving the way for future explorations in both theory and practical implementation.
+
+Future work could extend this investigation to larger system sizes (higher $n$), as these may yield improved codes. Given that our search algorithm is fully parallelizable, supercomputers or computer clusters could be employed to examine all generalized toric codes within $n \leq 500$ or higher—scales that are comparable to the number of physical qubits in state-of-the-art experimental platforms [126-131]. The primary bottleneck, however, is the computation of code distances. When $n$ reaches a few hundred and $d$ exceeds 20, the probabilistic algorithm for computing the code distance may not be reliable and could only yield an upper bound for $d$.
+
+Alternatively, one could explore different forms of

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+The above observed decreasing trends of compressive properties with increasing mesostructural stochastics can be explained through the weakest link principle (see also [61, 62]). Introducing the stochastic variations of more mesostructural features gives rise to the emergence of more weak regions. Indeed, Figure 19 demonstrates that the cell wall buckling events (using eq. (36)) tend to get promoted as more mesostructural stochastics are included (from "CtCt" to "StSt"). Here, $ N_{\mathrm{c}} $ and $ N_{\mathrm{w}} $ denote the number of buckled cell walls and the total number of cell walls, respectively. The fractions $ N_{\mathrm{c}} / N_{\mathrm{w}} $ of buckled cell walls for different model sets eventually become comparable upon a large applied strain. As expected, the cell wall buckling events for H200 are postponed (see also Figure 17) and accompanied by a slower growth of $ N_{\mathrm{c}} / N_{\mathrm{w}} $ , compared with H100.
+
+<div align="center">
+
+(a)
+
+</div>
+
+![](page=0,bbox=[252, 436, 577, 696])
+
+<div align="center">
+
+(b)
+
+</div>
+
+![](page=0,bbox=[593, 437, 918, 695])
+
+<div align="center">
+
+Figure 19: Fractions of buckled cell walls versus applied strain of different tessellation-based model sets for two Divinycell foam grades, under uniaxial compression in the transverse $(\vec{e}_1 / \vec{e}_2)$ and foam rise $(\vec{e}_3)$ directions.
+
+</div>
+
+For the sake of reference, the experimental compressive properties of H100 [97] and H200 [98], are provided in Figure 18 (leftmost bars). $ \tilde{\nu}_{13}^{*} $ are not measured and $ \tilde{\nu}_{31}=\tilde{\nu}_{32} $ has been assumed in [97, 98]. As an indication, the experimental data from other literature are also provided in Figure 18 (black crosses), although these studies are lacking either welldefined strain measurements or complete stress-strain curves under uniaxial compression. A remarkable inconsistency between the experimental data reported in different literature can be noticed. This places a clear need of more attention to the experimental aspects, e.g. test method, sample shape, sample size and determination of compressive properties.
+
+In the following, the numerical model predictions are compared with the experimental data from [97, 98] only, given their reliability and relevance. The model set "StSt", with all the cell size, cell wall thickness and cell shape anisotropy stochastics incorporated, seems to deliver the closest predictions with respect to the experimental data. In particular for the compressive moduli and Poisson's ratios, an excellent agreement between the experimental data and "StSt" predictions can be observed. Relatively large deviations appear on the strengths, which are overestimated by $ \sim 15\% $ , likely due to the disregarded plasticity in the present numerical models.
+
+## 7.3. Mechanical anisotropy
+
+With the effective compressive properties in Figure 18, the mechanical anisotropy $ \mathcal{R}^{E} $ and $ \mathcal{R}^{\sigma} $ of different model sets are computed, and reported in Figures 20(a) and (b), respectively. Three model sets ("StSt", "StCt" and "CtCt") are found to deliver comparable predictions of both $ \mathcal{R}^{E} $ and $ \mathcal{R}^{\sigma} $ , with the relative difference in between $ < 10\% $ . This suggests that the cell wall thickness and cell size stochastics only weakly affect the resulting mechanical anisotropy.

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@@ -0,0 +1,31 @@
+The above observed decreasing trends of compressive properties with increasing mesostructural stochastics can be explained through the weakest link principle (see also [61, 62]). Introducing the stochastic variations of more mesostructural features gives rise to the emergence of more weak regions. Indeed, Figure 19 demonstrates that the cell wall buckling events (using eq. (36)) tend to get promoted as more mesostructural stochastics are included (from "CtCt" to "StSt"). Here, $ N_{\mathrm{c}} $ and $ N_{\mathrm{w}} $ denote the number of buckled cell walls and the total number of cell walls, respectively. The fractions $ N_{\mathrm{c}} / N_{\mathrm{w}} $ of buckled cell walls for different model sets eventually become comparable upon a large applied strain. As expected, the cell wall buckling events for H200 are postponed (see also Figure 17) and accompanied by a slower growth of $ N_{\mathrm{c}} / N_{\mathrm{w}} $ , compared with H100.
+
+<div align="center">
+
+(a)
+
+</div>
+
+![](page=0,bbox=[252, 436, 577, 696])
+
+<div align="center">
+
+(b)
+
+</div>
+
+![](page=0,bbox=[593, 437, 918, 695])
+
+<div align="center">
+
+Figure 19: Fractions of buckled cell walls versus applied strain of different tessellation-based model sets for two Divinycell foam grades, under uniaxial compression in the transverse $(\vec{e}_1 / \vec{e}_2)$ and foam rise $(\vec{e}_3)$ directions.
+
+</div>
+
+For the sake of reference, the experimental compressive properties of H100 [97] and H200 [98], are provided in Figure 18 (leftmost bars). $ \tilde{\nu}_{13}^{*} $ are not measured and $ \tilde{\nu}_{31}=\tilde{\nu}_{32} $ has been assumed in [97, 98]. As an indication, the experimental data from other literature are also provided in Figure 18 (black crosses), although these studies are lacking either welldefined strain measurements or complete stress-strain curves under uniaxial compression. A remarkable inconsistency between the experimental data reported in different literature can be noticed. This places a clear need of more attention to the experimental aspects, e.g. test method, sample shape, sample size and determination of compressive properties.
+
+In the following, the numerical model predictions are compared with the experimental data from [97, 98] only, given their reliability and relevance. The model set "StSt", with all the cell size, cell wall thickness and cell shape anisotropy stochastics incorporated, seems to deliver the closest predictions with respect to the experimental data. In particular for the compressive moduli and Poisson's ratios, an excellent agreement between the experimental data and "StSt" predictions can be observed. Relatively large deviations appear on the strengths, which are overestimated by $ \sim 15\% $ , likely due to the disregarded plasticity in the present numerical models.
+
+## 7.3. Mechanical anisotropy
+
+With the effective compressive properties in Figure 18, the mechanical anisotropy $ \mathcal{R}^{E} $ and $ \mathcal{R}^{\sigma} $ of different model sets are computed, and reported in Figures 20(a) and (b), respectively. Three model sets ("StSt", "StCt" and "CtCt") are found to deliver comparable predictions of both $ \mathcal{R}^{E} $ and $ \mathcal{R}^{\sigma} $ , with the relative difference in between $ < 10\% $ . This suggests that the cell wall thickness and cell size stochastics only weakly affect the resulting mechanical anisotropy.

zai-org__GLM-OCR-api/arxiv_math/2503.03847_pg30_pg1_repeat3.md

@@ -0,0 +1,31 @@
+The above observed decreasing trends of compressive properties with increasing mesostructural stochastics can be explained through the weakest link principle (see also [61, 62]). Introducing the stochastic variations of more mesostructural features gives rise to the emergence of more weak regions. Indeed, Figure 19 demonstrates that the cell wall buckling events (using eq. (36)) tend to get promoted as more mesostructural stochastics are included (from "CtCt" to "StSt"). Here, $ N_{\mathrm{c}} $ and $ N_{\mathrm{w}} $ denote the number of buckled cell walls and the total number of cell walls, respectively. The fractions $ N_{\mathrm{c}} / N_{\mathrm{w}} $ of buckled cell walls for different model sets eventually become comparable upon a large applied strain. As expected, the cell wall buckling events for H200 are postponed (see also Figure 17) and accompanied by a slower growth of $ N_{\mathrm{c}} / N_{\mathrm{w}} $ , compared with H100.
+
+<div align="center">
+
+(a)
+
+</div>
+
+![](page=0,bbox=[252, 436, 577, 696])
+
+<div align="center">
+
+(b)
+
+</div>
+
+![](page=0,bbox=[593, 437, 918, 695])
+
+<div align="center">
+
+Figure 19: Fractions of buckled cell walls versus applied strain of different tessellation-based model sets for two Divinycell foam grades, under uniaxial compression in the transverse $(\vec{e}_1 / \vec{e}_2)$ and foam rise $(\vec{e}_3)$ directions.
+
+</div>
+
+For the sake of reference, the experimental compressive properties of H100 [97] and H200 [98], are provided in Figure 18 (leftmost bars). $ \tilde{\nu}_{13}^{*} $ are not measured and $ \tilde{\nu}_{31}=\tilde{\nu}_{32} $ has been assumed in [97, 98]. As an indication, the experimental data from other literature are also provided in Figure 18 (black crosses), although these studies are lacking either welldefined strain measurements or complete stress-strain curves under uniaxial compression. A remarkable inconsistency between the experimental data reported in different literature can be noticed. This places a clear need of more attention to the experimental aspects, e.g. test method, sample shape, sample size and determination of compressive properties.
+
+In the following, the numerical model predictions are compared with the experimental data from [97, 98] only, given their reliability and relevance. The model set "StSt", with all the cell size, cell wall thickness and cell shape anisotropy stochastics incorporated, seems to deliver the closest predictions with respect to the experimental data. In particular for the compressive moduli and Poisson's ratios, an excellent agreement between the experimental data and "StSt" predictions can be observed. Relatively large deviations appear on the strengths, which are overestimated by $ \sim 15\% $ , likely due to the disregarded plasticity in the present numerical models.
+
+## 7.3. Mechanical anisotropy
+
+With the effective compressive properties in Figure 18, the mechanical anisotropy $ \mathcal{R}^{E} $ and $ \mathcal{R}^{\sigma} $ of different model sets are computed, and reported in Figures 20(a) and (b), respectively. Three model sets ("StSt", "StCt" and "CtCt") are found to deliver comparable predictions of both $ \mathcal{R}^{E} $ and $ \mathcal{R}^{\sigma} $ , with the relative difference in between $ < 10\% $ . This suggests that the cell wall thickness and cell size stochastics only weakly affect the resulting mechanical anisotropy.

zai-org__GLM-OCR-api/arxiv_math/2503.03855_pg5_pg1_repeat1.md

@@ -0,0 +1,37 @@
+for all $r\in \mathbb{R}$ and $ \alpha\in\Phi $ . We also have a filtration of the maximal torus $ T $ by setting
+
+$$
+T _ {0} := \left\{t \in T \mid \forall \chi \in X ^ {*} (T), \omega (\chi (t)) = 0 \right\}
+$$
+
+and
+
+$$
+T _ {r} := \left\{t \in T _ {0} \mid \forall \chi \in X ^ {*} (T), \omega (\chi (t) - 1) \geq r \right\}
+$$
+
+for all $ r\in\mathbb{R}. $
+
+2. 1. The apartment. Let $N \leq G$ be generated by $T$ and $ \zeta_{\alpha}\left( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \right)$ for all $ \alpha \in \Phi$. Then $N = Z_{G}(T)$ and the quotient $ N / T $ is isomorphic to the (finite) Weyl group $ W $ of $ G $ . We write $ V := X_{*}(T)\otimes_{\mathbb{Z}} \mathbb{R} $ . It is clear that there is an action of $ W $ on $ V $ by linear transformations. This action can be extended to an action of $ N $ on $ V $ by affine transformations. When viewing $ V $ as an affine space over itself, we denote it by $ \mathbb{A} $ . It is called the apartment of $ G $ corresponding to the torus $ T $ . We will find it convenient to fix an origin $ o $ for $ \mathbb{A} $ , which corresponds to the origin of $ V $ . The action of $ N $ on $ \mathbb{A} $ is via the quotient $ N / T_{0} $ , which is isomorphic to a group $ \widehat{W} $ called the affine Weyl group of $ G $ . The subset of $ \widehat{W} $ acting via affine reflections gives rise to a set of affine hyperplanes in $ \mathbb{A} $ and by taking the closed half-spaces on both sides of all those hyperplanes, we obtain an affine root system $ \Sigma $ in the sense of [3]. Given $ a \in \Sigma $ the hyperplane $ \partial a $ is called a wall. Every element $ a \in \Sigma $ is of the form
+
+$$
+a = \alpha + r := \{x \in \mathbb {A} \mid \alpha (x) + r \geq 0 \}
+$$
+
+for some $\alpha \in \Phi$ and $r \in \mathbb{Z}$ and every set of this form is in $\Sigma$. Here we are abusing notation by writing $\alpha + r$ both for the affine function $x \mapsto \alpha(x) + r$ on $\mathbb{A}$ and for the half-space it defines. We obtain a surjective map $\Sigma \rightarrow \Phi : a \mapsto \hat{a} = \alpha$. The action of $N$ on $\mathbb{A}$ induces an action of $N$ on $\Sigma$. Let $a, b \in \Sigma$, then the walls $\partial a$ and $\partial b$ are called parallel if $\hat{a} = \pm \hat{b}$, moreover we say that $\partial a$ is parallel to $\hat{a}$.
+
+An equivalence relation on $ \mathbb{A} $ is now obtained by specifying that two points are equivalent when the sets of affine roots they are contained in are the same. The closures of the equivalence classes under this relation are called facets and they give rise to a simplicial structure on $ \mathbb{A} $ , where the equivalence classes are the open cells. The simplices of maximal dimension are called alcoves and they are of dimension d. Any alcove is a fundamental domain for the action of N on $ \mathbb{A} $ . Because we have fixed a pinning of G and a base II for our root system, there is a canonical alcove C given by
+
+$$
+C := \{x \in \mathbb {A} \mid \text {f o r} 1 \leq i \leq d, \alpha_ {i} (x) \geq 0 \text {a n d} \alpha_ {0} (x) \leq 1 \}.
+$$
+
+The vertices of $C$ are $v_{0} = 0$ and $v_{1},\dots,v_{d}$. Here the $v_{i}$ with $i > 0$ are given by $\alpha_{j}(v_{i}) = 0$ for all $j\neq i,0$ and $\alpha_{0}(v_{i}) = 1$. We now expand $\alpha_{0}$ in the basis given by II and find that
+
+$$
+\alpha_ {0} = \sum_ {i = 1} ^ {d} c _ {i} \alpha_ {i}
+$$
+
+for some positive integers $c_{i}$. We define $\omega_{i} \coloneqq c_{i} v_{i}$ and then we note that $\alpha_{i}(\omega_{j}) = \delta_{ij}$. Thus the $\omega_{i}$ form a basis for $V$ dual to $\Pi$ and they are called the fundamental cowweights.
+
+Associated to $C$ there is a fundamental Weyl chamber $C^{+} := \mathbb{R}_{\geq 0} \cdot C$. Because $C$ is a fundamental domain for the action of $N$ on $\mathbb{A}$, every vertex $x$ in $\mathbb{A}$ is $G$-conjugate to exactly one of the $v_{i}$, which gives rise to a map $\lambda : \mathbb{A}_{0} \to \{0, \dots, d\}$

zai-org__GLM-OCR-api/arxiv_math/2503.03855_pg5_pg1_repeat2.md

@@ -0,0 +1,37 @@
+for all $r\in \mathbb{R}$ and $ \alpha\in\Phi $ . We also have a filtration of the maximal torus $ T $ by setting
+
+$$
+T _ {0} := \left\{t \in T \mid \forall \chi \in X ^ {*} (T), \omega (\chi (t)) = 0 \right\}
+$$
+
+and
+
+$$
+T _ {r} := \left\{t \in T _ {0} \mid \forall \chi \in X ^ {*} (T), \omega (\chi (t) - 1) \geq r \right\}
+$$
+
+for all $ r\in\mathbb{R}. $
+
+2. 1. The apartment. Let $N \leq G$ be generated by $T$ and $ \zeta_{\alpha}\left( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \right)$ for all $ \alpha \in \Phi$. Then $N = Z_{G}(T)$ and the quotient $ N/T $ is isomorphic to the (finite) Weyl group $ W $ of $ G $ . We write $ V := X_{*}(T) \otimes_{\mathbb{Z}} \mathbb{R} $ . It is clear that there is an action of $ W $ on $ V $ by linear transformations. This action can be extended to an action of $ N $ on $ V $ by affine transformations. When viewing $ V $ as an affine space over itself, we denote it by $ \mathbb{A} $ . It is called the apartment of $ G $ corresponding to the torus $ T $ . We will find it convenient to fix an origin $ o $ for $ \mathbb{A} $ , which corresponds to the origin of $ V $ . The action of $ N $ on $ \mathbb{A} $ is via the quotient $ N/T_{0} $ , which is isomorphic to a group $ \widehat{W} $ called the affine Weyl group of $ G $ . The subset of $ \widehat{W} $ acting via affine reflections gives rise to a set of affine hyperplanes in $ \mathbb{A} $ and by taking the closed half-spaces on both sides of all those hyperplanes, we obtain an affine root system $ \Sigma $ in the sense of [3]. Given $ a \in \Sigma $ the hyperplane $ \partial a $ is called a wall. Every element $ a \in \Sigma $ is of the form
+
+$$
+a = \alpha + r := \{x \in \mathbb {A} \mid \alpha (x) + r \geq 0 \}
+$$
+
+for some $\alpha \in \Phi$ and $r \in \mathbb{Z}$ and every set of this form is in $\Sigma$. Here we are abusing notation by writing $\alpha + r$ both for the affine function $x \mapsto \alpha(x) + r$ on $\mathbb{A}$ and for the half-space it defines. We obtain a surjective map $\Sigma \rightarrow \Phi : a \mapsto \hat{a} = \alpha$. The action of $N$ on $\mathbb{A}$ induces an action of $N$ on $\Sigma$. Let $a, b \in \Sigma$, then the walls $\partial a$ and $\partial b$ are called parallel if $\hat{a} = \pm \hat{b}$, moreover we say that $\partial a$ is parallel to $\hat{a}$.
+
+An equivalence relation on $ \mathbb{A} $ is now obtained by specifying that two points are equivalent when the sets of affine roots they are contained in are the same. The closures of the equivalence classes under this relation are called facets and they give rise to a simplicial structure on $ \mathbb{A} $ , where the equivalence classes are the open cells. The simplices of maximal dimension are called alcoves and they are of dimension d. Any alcove is a fundamental domain for the action of N on $ \mathbb{A} $ . Because we have fixed a pinning of G and a base II for our root system, there is a canonical alcove C given by
+
+$$
+C := \{x \in \mathbb {A} \mid \text {f o r} 1 \leq i \leq d, \alpha_ {i} (x) \geq 0 \text {a n d} \alpha_ {0} (x) \leq 1 \}.
+$$
+
+The vertices of $C$ are $v_{0} = 0$ and $v_{1},\dots,v_{d}$. Here the $v_{i}$ with $i > 0$ are given by $\alpha_{j}(v_{i}) = 0$ for all $j\neq i,0$ and $\alpha_{0}(v_{i}) = 1$. We now expand $\alpha_{0}$ in the basis given by II and find that
+
+$$
+\alpha_ {0} = \sum_ {i = 1} ^ {d} c _ {i} \alpha_ {i}
+$$
+
+for some positive integers $c_{i}$. We define $\omega_{i} \coloneqq c_{i} v_{i}$ and then we note that $\alpha_{i}(\omega_{j}) = \delta_{ij}$. Thus the $\omega_{i}$ form a basis for $V$ dual to $\Pi$ and they are called the fundamental cowweights.
+
+Associated to $C$ there is a fundamental Weyl chamber $C^{+} := \mathbb{R}_{\geq 0} \cdot C$. Because $C$ is a fundamental domain for the action of $N$ on $\mathbb{A}$, every vertex $x$ in $\mathbb{A}$ is $G$-conjugate to exactly one of the $v_{i}$, which gives rise to a map $\lambda : \mathbb{A}_{0} \rightarrow \{0, \dots, d\}$

zai-org__GLM-OCR-api/arxiv_math/2503.03855_pg5_pg1_repeat3.md

@@ -0,0 +1,37 @@
+for all $r\in \mathbb{R}$ and $ \alpha\in \Phi $ . We also have a filtration of the maximal torus $ T $ by setting
+
+$$
+T _ {0} := \left\{t \in T \mid \forall \chi \in X ^ {*} (T), \omega (\chi (t)) = 0 \right\}
+$$
+
+and
+
+$$
+T _ {r} := \left\{t \in T _ {0} \mid \forall \chi \in X ^ {*} (T), \omega (\chi (t) - 1) \geq r \right\}
+$$
+
+for all $ r\in\mathbb{R}. $
+
+2. 1. The apartment. Let $N \leq G$ be generated by $T$ and $ \zeta_{\alpha}\left( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \right)$ for all $ \alpha \in \Phi$. Then $N = Z_{G}(T)$ and the quotient $ N/T $ is isomorphic to the (finite) Weyl group $ W $ of $ G $ . We write $ V := X_{*}(T)\otimes_{\mathbb{Z}} \mathbb{R} $ . It is clear that there is an action of $ W $ on $ V $ by linear transformations. This action can be extended to an action of $ N $ on $ V $ by affine transformations. When viewing $ V $ as an affine space over itself, we denote it by $ \mathbb{A} $ . It is called the apartment of $ G $ corresponding to the torus $ T $ . We will find it convenient to fix an origin $ o $ for $ \mathbb{A} $ , which corresponds to the origin of $ V $ . The action of $ N $ on $ \mathbb{A} $ is via the quotient $ N/T_{0} $ , which is isomorphic to a group $ \widehat{W} $ called the affine Weyl group of $ G $ . The subset of $ \widehat{W} $ acting via affine reflections gives rise to a set of affine hyperplanes in $ \mathbb{A} $ and by taking the closed half-spaces on both sides of all those hyperplanes, we obtain an affine root system $ \Sigma $ in the sense of [3]. Given $ a \in \Sigma $ the hyperplane $ \partial a $ is called a wall. Every element $ a \in \Sigma $ is of the form
+
+$$
+a = \alpha + r := \{x \in \mathbb {A} \mid \alpha (x) + r \geq 0 \}
+$$
+
+for some $\alpha \in \Phi$ and $r \in \mathbb{Z}$ and every set of this form is in $\Sigma$. Here we are abusing notation by writing $\alpha + r$ both for the affine function $x \mapsto \alpha(x) + r$ on $\mathbb{A}$ and for the half-space it defines. We obtain a surjective map $\Sigma \rightarrow \Phi : a \mapsto \hat{a} = \alpha$. The action of $N$ on $\mathbb{A}$ induces an action of $N$ on $\Sigma$. Let $a, b \in \Sigma$, then the walls $\partial a$ and $\partial b$ are called parallel if $\hat{a} = \pm \hat{b}$, moreover we say that $\partial a$ is parallel to $\hat{a}$.
+
+An equivalence relation on $ \mathbb{A} $ is now obtained by specifying that two points are equivalent when the sets of affine roots they are contained in are the same. The closures of the equivalence classes under this relation are called facets and they give rise to a simplicial structure on $ \mathbb{A} $ , where the equivalence classes are the open cells. The simplices of maximal dimension are called alcoves and they are of dimension d. Any alcove is a fundamental domain for the action of N on $ \mathbb{A} $ . Because we have fixed a pinning of G and a base II for our root system, there is a canonical alcove C given by
+
+$$
+C := \{x \in \mathbb {A} \mid \text {f o r} 1 \leq i \leq d, \alpha_ {i} (x) \geq 0 \text {a n d} \alpha_ {0} (x) \leq 1 \}.
+$$
+
+The vertices of $C$ are $v_{0} = 0$ and $v_{1},\dots,v_{d}$. Here the $v_{i}$ with $i > 0$ are given by $\alpha_{j}(v_{i}) = 0$ for all $j\neq i,0$ and $\alpha_{0}(v_{i}) = 1$. We now expand $\alpha_{0}$ in the basis given by II and find that
+
+$$
+\alpha_ {0} = \sum_ {i = 1} ^ {d} c _ {i} \alpha_ {i}
+$$
+
+for some positive integers $c_{i}$. We define $\omega_{i} \coloneqq c_{i} v_{i}$ and then we note that $\alpha_{i}(\omega_{j}) = \delta_{ij}$. Thus the $\omega_{i}$ form a basis for $V$ dual to $\Pi$ and they are called the fundamental cowweights.
+
+Associated to $C$ there is a fundamental Weyl chamber $C^{+} := \mathbb{R}_{\geq 0} \cdot C$. Because $C$ is a fundamental domain for the action of $N$ on $\mathbb{A}$, every vertex $x$ in $\mathbb{A}$ is $G$-conjugate to exactly one of the $v_{i}$, which gives rise to a map $\lambda : \mathbb{A}_{0} \rightarrow \{0, \dots, d\}$

zai-org__GLM-OCR-api/arxiv_math/2503.03861_pg30_pg1_repeat1.md

@@ -0,0 +1,23 @@
+Since $ H^{2}(X^{\mathrm{an}},\mathcal{O}_{X^{\mathrm{an}}}) $ is torsion free, any torsion element of $ H^{2}(X; \mathbb{Z}) $ vanishes under $ \beta. $ Therefore, to conclude the proof, it suffices to show $ \alpha $ is an injection. Since $ H^{1}(X,\mathcal{O}_{X^{\mathrm{an}}}^{\times}) $ is identified with the Picard group, to prove the desired injection, we only need to show $ H^{1}(X^{\mathrm{an}},\mathcal{O}_{X^{\mathrm{an}}})=0. $ Using GAGA for Deligne-Mumford stacks, [Hal11, Proposition A.4], we have $ H^{1}(X^{\mathrm{an}},\mathcal{O}_{X^{\mathrm{an}}})=H^{1}(X,\mathcal{O}_{X}). $ Since $ H^{1}(X; \mathbb{Q})=0, $ we also have $ H^{1}(X; \mathbb{C})=0, $ and hence we conclude $ H^{1}(X,\mathcal{O}_{X})=0 $ using [Sat12, Corollary 1.7], which says that the Hodge de Rham spectral sequence degenerates for smooth proper Deligne-Mumford stacks.
+
+7. 3. Proving the stable Picard rank conjecture. We now aim to prove Theorem 7.1.1. To do this, we next compute the first two stable cohomology groups of $ \left[ \left[ \mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G \right] / \mathrm{PGL}_{2} \right] $ To do so, we need a basic lemma about the number of connected components of Hurwitz spaces.
+
+Lemma 7.3.1. Let G be a group and $ c\subset G $ a conjugacy class generating G. For n sufficiently large, the set of connected components of $ \mathrm{CHur}_{n}^{c} $ with boundary monodromy $ g\in G $ is either empty or forms a torsor under $ H_{2}(G,c) $ ; it is nonempty if and only if the image of n in $ G^{\mathrm{ab}} $ (under the map $ \mathbb{Z}\rightarrow G^{\mathrm{ab}} $ sending the positive generator to the image of any element of c) agrees with the image of g in $ G^{\mathrm{ab}}. $
+
+Rephrasing the statement above, there are $ H_{2}(G,c) $ many components if the image of n in $ G^{\mathrm{ab}} $ agrees with the image of g, and 0 components otherwise.
+
+Proof. This essentially follows from [Woo21] as we now explain. Indeed, using [Woo21, Theorem 2.5 and Theorem 3.1] we can identify the number of components of $ \mathrm{CHur}_{n}^{c} $ for n sufficiently large with the set of elements in a certain reduced Schur cover $ S_{c}\rightarrow G $ having the same image in $ G^{\mathrm{ab}} $ as n. Moreover, the boundary monodromy of these components is the same as their image in G under the map $ S_{c}\rightarrow G $ . The kernel of $ S_{c}\rightarrow G $ is identified with $ H_{2}(G,c) $ and so connected components with boundary monodromy g either form a torsor under $ H_{2}(G,c) $ when the image of n in $ G^{\mathrm{ab}} $ agrees with the image of g, or else there are no such connected components.
+
+For the next lemma and its proof, the reader may wish to recall notation from Notation 2.4.1.
+
+Lemma 7.3.2. Let $G$ be a group, $c \subset G$ be a conjugacy class generating $G$, and $R := \mathbb{Z}[1/2|G|]$ For $n$ sufficiently large depending on $c$ and for each component $ \widetilde{Z} \subset [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G]$, with corresponding component $ Z \subset [[\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] / \mathrm{PGL}_2]$, we have
+
+$$
+H ^ {1} (\widetilde {Z}; R) = H ^ {1} (Z; R) = 0,
+$$
+
+$$
+H ^ {2} \left(\widetilde {Z}; R\right) = H ^ {2} \left(Z; R\right) = \left(\left(\mathbb {Z} / (2 n - 2) \mathbb {Z}\right) \otimes R\right).
+$$
+
+Proof. Taking $ g = \mathrm{id} $ in Lemma 7.3.1, we obtain that for $ n $ sufficiently large, both $ [\mathrm{CHur}_{n}^{G,c,\delta\in\mathrm{id}} / G] $ and $ [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] $ have $ |H_{2}(G,c)| $ many connected components. Indeed, the statement for $ [\mathrm{CHur}_{n}^{G,c,\delta\in\mathrm{id}} / G] $ follows from Lemma 7.3.1 and the fact that $ G $ conjugation acts trivially on $ H_{2}(G,c) $ as it is identified with the central kernel of $ S_{c}\rightarrow G $ by definition. Since $ [\mathrm{CHur}_{n}^{G,c,\delta\in\mathrm{id}} / G] $ is dense open in $ [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] $ , we obtain $ [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] $ also has

zai-org__GLM-OCR-api/arxiv_math/2503.03861_pg30_pg1_repeat2.md

@@ -0,0 +1,23 @@
+Since $ H^{2}(X^{\mathrm{an}},\mathcal{O}_{X^{\mathrm{an}}}) $ is torsion free, any torsion element of $ H^{2}(X; \mathbb{Z}) $ vanishes under $ \beta. $ Therefore, to conclude the proof, it suffices to show $ \alpha $ is an injection. Since $ H^{1}(X,\mathcal{O}_{X^{\mathrm{an}}}^{\times}) $ is identified with the Picard group, to prove the desired injection, we only need to show $ H^{1}(X^{\mathrm{an}},\mathcal{O}_{X^{\mathrm{an}}})=0. $ Using GAGA for Deligne-Mumford stacks, [Hal11, Proposition A.4], we have $ H^{1}(X^{\mathrm{an}},\mathcal{O}_{X^{\mathrm{an}}})=H^{1}(X,\mathcal{O}_{X}). $ Since $ H^{1}(X; \mathbb{Q})=0, $ we also have $ H^{1}(X; \mathbb{C})=0, $ and hence we conclude $ H^{1}(X,\mathcal{O}_{X})=0 $ using [Sat12, Corollary 1.7], which says that the Hodge de Rham spectral sequence degenerates for smooth proper Deligne-Mumford stacks.
+
+7. 3. Proving the stable Picard rank conjecture. We now aim to prove Theorem 7.1.1. To do this, we next compute the first two stable cohomology groups of $ \left[ \left[ \mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G \right] / \mathrm{PGL}_{2} \right] $ To do so, we need a basic lemma about the number of connected components of Hurwitz spaces.
+
+Lemma 7.3.1. Let G be a group and $ c\subset G $ a conjugacy class generating G. For n sufficiently large, the set of connected components of $ \mathrm{CHur}_{n}^{c} $ with boundary monodromy $ g\in G $ is either empty or forms a torsor under $ H_{2}(G,c) $ ; it is nonempty if and only if the image of n in $ G^{\mathrm{ab}} $ (under the map $ \mathbb{Z}\rightarrow G^{\mathrm{ab}} $ sending the positive generator to the image of any element of c) agrees with the image of g in $ G^{\mathrm{ab}}. $
+
+Rephrasing the statement above, there are $ H_{2}(G,c) $ many components if the image of n in $ G^{\mathrm{ab}} $ agrees with the image of g, and 0 components otherwise.
+
+Proof. This essentially follows from [Woo21] as we now explain. Indeed, using [Woo21, Theorem 2.5 and Theorem 3.1] we can identify the number of components of $ \mathrm{CHur}_{n}^{c} $ for n sufficiently large with the set of elements in a certain reduced Schur cover $ S_{c}\rightarrow G $ having the same image in $ G^{\mathrm{ab}} $ as n. Moreover, the boundary monodromy of these components is the same as their image in G under the map $ S_{c}\rightarrow G $ . The kernel of $ S_{c}\rightarrow G $ is identified with $ H_{2}(G,c) $ and so connected components with boundary monodromy g either form a torsor under $ H_{2}(G,c) $ when the image of n in $ G^{\mathrm{ab}} $ agrees with the image of g, or else there are no such connected components.
+
+For the next lemma and its proof, the reader may wish to recall notation from Notation 2.4.1.
+
+Lemma 7.3.2. Let $G$ be a group, $c \subset G$ be a conjugacy class generating $G$, and $R := \mathbb{Z}[1/2|G|]$ For $n$ sufficiently large depending on $c$ and for each component $ \widetilde{Z} \subset [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G]$, with corresponding component $ Z \subset [[\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] / \mathrm{PGL}_2]$, we have
+
+$$
+H ^ {1} (\widetilde {Z}; R) = H ^ {1} (Z; R) = 0,
+$$
+
+$$
+H ^ {2} \left(\widetilde {Z}; R\right) = H ^ {2} \left(Z; R\right) = \left(\left(\mathbb {Z} / (2 n - 2) \mathbb {Z}\right) \otimes R\right).
+$$
+
+Proof. Taking $ g = \mathrm{id} $ in Lemma 7.3.1, we obtain that for $ n $ sufficiently large, both $ [\mathrm{CHur}_{n}^{G,c,\delta\in\mathrm{id}} / G] $ and $ [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] $ have $ |H_{2}(G,c)| $ many connected components. Indeed, the statement for $ [\mathrm{CHur}_{n}^{G,c,\delta\in\mathrm{id}} / G] $ follows from Lemma 7.3.1 and the fact that $ G $ conjugation acts trivially on $ H_{2}(G,c) $ as it is identified with the central kernel of $ S_{c}\rightarrow G $ by definition. Since $ [\mathrm{CHur}_{n}^{G,c,\delta\in\mathrm{id}} / G] $ is dense open in $ [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] $ , we obtain $ [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] $ also has

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@@ -0,0 +1,23 @@
+Since $ H^{2}(X^{\mathrm{an}},\mathcal{O}_{X^{\mathrm{an}}}) $ is torsion free, any torsion element of $ H^{2}(X; \mathbb{Z}) $ vanishes under $ \beta. $ Therefore, to conclude the proof, it suffices to show $ \alpha $ is an injection. Since $ H^{1}(X,\mathcal{O}_{X^{\mathrm{an}}}^{\times}) $ is identified with the Picard group, to prove the desired injection, we only need to show $ H^{1}(X^{\mathrm{an}},\mathcal{O}_{X^{\mathrm{an}}})=0. $ Using GAGA for Deligne-Mumford stacks, [Hal11, Proposition A.4], we have $ H^{1}(X^{\mathrm{an}},\mathcal{O}_{X^{\mathrm{an}}})=H^{1}(X,\mathcal{O}_{X}). $ Since $ H^{1}(X; \mathbb{Q})=0, $ we also have $ H^{1}(X; \mathbb{C})=0, $ and hence we conclude $ H^{1}(X,\mathcal{O}_{X})=0 $ using [Sat12, Corollary 1.7], which says that the Hodge de Rham spectral sequence degenerates for smooth proper Deligne-Mumford stacks.
+
+7. 3. Proving the stable Picard rank conjecture. We now aim to prove Theorem 7.1.1. To do this, we next compute the first two stable cohomology groups of $ \left[ \left[ \mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G \right] / \mathrm{PGL}_{2} \right] $ To do so, we need a basic lemma about the number of connected components of Hurwitz spaces.
+
+Lemma 7.3.1. Let G be a group and $ c\subset G $ a conjugacy class generating G. For n sufficiently large, the set of connected components of $ \mathrm{CHur}_{n}^{c} $ with boundary monodromy $ g\in G $ is either empty or forms a torsor under $ H_{2}(G,c) $ ; it is nonempty if and only if the image of n in $ G^{\mathrm{ab}} $ (under the map $ \mathbb{Z}\rightarrow G^{\mathrm{ab}} $ sending the positive generator to the image of any element of c) agrees with the image of g in $ G^{\mathrm{ab}}. $
+
+Rephrasing the statement above, there are $ H_{2}(G,c) $ many components if the image of n in $ G^{\mathrm{ab}} $ agrees with the image of g, and 0 components otherwise.
+
+Proof. This essentially follows from [Woo21] as we now explain. Indeed, using [Woo21, Theorem 2.5 and Theorem 3.1] we can identify the number of components of $ \mathrm{CHur}_{n}^{c} $ for n sufficiently large with the set of elements in a certain reduced Schur cover $ S_{c}\rightarrow G $ having the same image in $ G^{\mathrm{ab}} $ as n. Moreover, the boundary monodromy of these components is the same as their image in G under the map $ S_{c}\rightarrow G $ . The kernel of $ S_{c}\rightarrow G $ is identified with $ H_{2}(G,c) $ and so connected components with boundary monodromy g either form a torsor under $ H_{2}(G,c) $ when the image of n in $ G^{\mathrm{ab}} $ agrees with the image of g, or else there are no such connected components.
+
+For the next lemma and its proof, the reader may wish to recall notation from Notation 2.4.1.
+
+Lemma 7.3.2. Let $G$ be a group, $c \subset G$ be a conjugacy class generating $G$, and $R := \mathbb{Z}[1/2|G|]$ For $n$ sufficiently large depending on $c$ and for each component $ \widetilde{Z} \subset [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G]$, with corresponding component $ Z \subset [[\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] / \mathrm{PGL}_2]$, we have
+
+$$
+H ^ {1} (\widetilde {Z}; R) = H ^ {1} (Z; R) = 0,
+$$
+
+$$
+H ^ {2} \left(\widetilde {Z}; R\right) = H ^ {2} \left(Z; R\right) = \left(\left(\mathbb {Z} / (2 n - 2) \mathbb {Z}\right) \otimes R\right).
+$$
+
+Proof. Taking $ g = \mathrm{id} $ in Lemma 7.3.1, we obtain that for $ n $ sufficiently large, both $ [\mathrm{CHur}_{n}^{G,c,\delta\in\mathrm{id}} / G] $ and $ [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] $ have $ |H_{2}(G,c)| $ many connected components. Indeed, the statement for $ [\mathrm{CHur}_{n}^{G,c,\delta\in\mathrm{id}} / G] $ follows from Lemma 7.3.1 and the fact that $ G $ conjugation acts trivially on $ H_{2}(G,c) $ as it is identified with the central kernel of $ S_{c}\rightarrow G $ by definition. Since $ [\mathrm{CHur}_{n}^{G,c,\delta\in\mathrm{id}} / G] $ is dense open in $ [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] $ , we obtain $ [\mathrm{CHur}_{\mathbb{P}^{1},n}^{G,c} / G] $ also has

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@@ -0,0 +1,37 @@
+We want to study the relations between the dual group $ \widehat{A} $ of A and the dual groups $ \widehat{H} $ and $ \widehat{A / H} $ of the subgroup H and the quotient group A/H. In fact, one can identify $ \widehat{A / H} $ with the subgroup $ H^{\perp} $ , called the annihilator of H, defined as
+
+$$
+H ^ {\perp} := \left\{\chi \in \widehat {A}: \chi (x) = 1 \quad \forall x \in H \right\},
+$$
+
+so that if we take $ \phi\in L^{1}(A) $ and define $ \phi^{H}\in L^{1}(A/H) $ as $ \phi^{H}(xH)=\int_{H}\phi(xh)dh $ , then by the above identification, we get $ \mathcal{F}_{A/H}(\phi^{H})=\mathcal{F}_{A}(\phi)|_{H^{\perp}} $ , see the proof of the theorem below, which explains this technique.
+
+Theorem 2.1 (General Poisson summation formula). Let H be a closed subgroup of the locally compact Abelian group A. For $ \phi\in L^{1}(A) $ , if $ \mathcal{F}_{A}(\phi)|_{H^{\perp}}\in L^{1}(H^{\perp}) $ , then
+
+$$
+\int_ {H} \phi (x h) d h = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \chi (x) d \chi ,
+$$
+
+for all $ x\in A $ , where Haar measure on $ H^{\perp}\cong A / H $ is the Plancherel measure with respect to the chosen Haar measure on $ A / H $ .
+
+Proof. For $ \chi\in H^{\perp} $ we have $ \chi(xh)=\chi(x) $ for every $ x\in A $ and $ h\in H $ . We therefore get from the quotient integral formula (2.2) that
+
+$$
+\begin{array}{l} \mathcal {F} _ {A / H} \left(\phi^ {H}\right) (\chi) = \int_ {A / H} \phi^ {H} (x H) \overline {{\chi (x)}} d (x H) \\ = \int_ {A / H} \int_ {H} \phi (x h) \overline {{\chi (x h)}} d h d (x H) \\ = \int_ {A} \phi (x) \overline {{\chi (x)}} d x = \mathcal {F} _ {A} (\phi) (\chi) \\ \end{array}
+$$
+
+for every $ \chi\in H^{\perp} $ . Moreover, if $ \mathcal{F}_{A}(\phi)|_{H^{\perp}}\in L^{1}(H^{\perp})=L^{1}\left(\widetilde{A}/\widetilde{H}\right) $ , then the Fourier inversion formula implies that for all $ x\in A $
+
+$$
+\begin{array}{l} \int_ {H} \phi (x h) d h = \phi^ {H} (x H) = \mathcal {F} _ {H ^ {\perp}} \mathcal {F} _ {A / H} \left(\phi^ {H}\right) (- x H) \\ = \mathcal {F} _ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (- x H) = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \overline {{- x H (\chi)}} d \chi \\ = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \overline {{\chi (- x)}} d \chi = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \chi (x) d \chi . \\ \end{array}
+$$
+
+In all our applications, the group $A$ will be of the form $A = (\mathbb{Z} / N\mathbb{Z})^{d_1}\times \mathbb{R}^{d_2}$ for $N\in \mathbb{N}_{+}, d_{1}, d_{2}\in \mathbb{N}$. Recall we have defined the standard symmetric bilinear form on $\mathbb{R}^{d_2}$ as in the beginning of §1, and for any $N\in \mathbb{N}_{+}$, we still denote the symmetric bilinear form on $(\mathbb{Z} / N\mathbb{Z})^{d_1}$ by $Q$ as defined in the standard way. We then identify $A$ with its dual group $ \widehat{A} $ through the pairing
+
+$$
+A \cong \widehat {A}
+$$
+
+$$
+(s, t) \mapsto \chi_ {s, t} (l, \xi) = e ^ {2 \pi i \left(\frac {Q (s , l)}{N} + Q (t, \xi)\right)},
+$$

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@@ -0,0 +1,37 @@
+We want to study the relations between the dual group $ \widehat{A} $ of A and the dual groups $ \widehat{H} $ and $ \widehat{A / H} $ of the subgroup H and the quotient group A/H. In fact, one can identify $ \widehat{A / H} $ with the subgroup $ H^{\perp} $ , called the annihilator of H, defined as
+
+$$
+H ^ {\perp} := \left\{\chi \in \widehat {A}: \chi (x) = 1 \quad \forall x \in H \right\},
+$$
+
+so that if we take $ \phi\in L^{1}(A) $ and define $ \phi^{H}\in L^{1}(A/H) $ as $ \phi^{H}(xH)=\int_{H}\phi(xh)dh $ , then by the above identification, we get $ \mathcal{F}_{A/H}(\phi^{H})=\mathcal{F}_{A}(\phi)|_{H^{\perp}} $ , see the proof of the theorem below, which explains this technique.
+
+Theorem 2.1 (General Poisson summation formula). Let H be a closed subgroup of the locally compact Abelian group A. For $ \phi\in L^{1}(A) $ , if $ \mathcal{F}_{A}(\phi)|_{H^{\perp}}\in L^{1}(H^{\perp}) $ , then
+
+$$
+\int_ {H} \phi (x h) d h = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \chi (x) d \chi ,
+$$
+
+for all $ x\in A $ , where Haar measure on $ H^{\perp}\cong A / H $ is the Plancherel measure with respect to the chosen Haar measure on $ A / H $ .
+
+Proof. For $ \chi\in H^{\perp} $ we have $ \chi(xh)=\chi(x) $ for every $ x\in A $ and $ h\in H $ . We therefore get from the quotient integral formula (2.2) that
+
+$$
+\begin{array}{l} \mathcal {F} _ {A / H} \left(\phi^ {H}\right) (\chi) = \int_ {A / H} \phi^ {H} (x H) \overline {{\chi (x)}} d (x H) \\ = \int_ {A / H} \int_ {H} \phi (x h) \overline {{\chi (x h)}} d h d (x H) \\ = \int_ {A} \phi (x) \overline {{\chi (x)}} d x = \mathcal {F} _ {A} (\phi) (\chi) \\ \end{array}
+$$
+
+for every $ \chi\in H^{\perp} $ . Moreover, if $ \mathcal{F}_{A}(\phi)|_{H^{\perp}}\in L^{1}(H^{\perp})=L^{1}\left(\widetilde{A}/\widetilde{H}\right) $ , then the Fourier inversion formula implies that for all $ x\in A $
+
+$$
+\begin{array}{l} \int_ {H} \phi (x h) d h = \phi^ {H} (x H) = \mathcal {F} _ {H ^ {\perp}} \mathcal {F} _ {A / H} \left(\phi^ {H}\right) (- x H) \\ = \mathcal {F} _ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (- x H) = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \overline {{- x H (\chi)}} d \chi \\ = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \overline {{\chi (- x)}} d \chi = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \chi (x) d \chi . \\ \end{array}
+$$
+
+In all our applications, the group $A$ will be of the form $A = (\mathbb{Z} / N\mathbb{Z})^{d_1}\times \mathbb{R}^{d_2}$ for $N\in \mathbb{N}_{+}, d_{1}, d_{2}\in \mathbb{N}$. Recall we have defined the standard symmetric bilinear form on $\mathbb{R}^{d_2}$ as in the beginning of §1, and for any $N\in \mathbb{N}_{+}$, we still denote the symmetric bilinear form on $(\mathbb{Z} / N\mathbb{Z})^{d_1}$ by $Q$ as defined in the standard way. We then identify $A$ with its dual group $ \widehat{A} $ through the pairing
+
+$$
+A \cong \widehat {A}
+$$
+
+$$
+(s, t) \mapsto \chi_ {s, t} (l, \xi) = e ^ {2 \pi i \left(\frac {Q (s , l)}{N} + Q (t, \xi)\right)},
+$$

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@@ -0,0 +1,37 @@
+We want to study the relations between the dual group $ \widehat{A} $ of A and the dual groups $ \widehat{H} $ and $ \widehat{A / H} $ of the subgroup H and the quotient group A/H. In fact, one can identify $ \widehat{A / H} $ with the subgroup $ H^{\perp} $ , called the annihilator of H, defined as
+
+$$
+H ^ {\perp} := \left\{\chi \in \widehat {A}: \chi (x) = 1 \quad \forall x \in H \right\},
+$$
+
+so that if we take $ \phi\in L^{1}(A) $ and define $ \phi^{H}\in L^{1}(A/H) $ as $ \phi^{H}(xH)=\int_{H}\phi(xh)dh $ , then by the above identification, we get $ \mathcal{F}_{A/H}(\phi^{H})=\mathcal{F}_{A}(\phi)|_{H^{\perp}} $ , see the proof of the theorem below, which explains this technique.
+
+Theorem 2.1 (General Poisson summation formula). Let H be a closed subgroup of the locally compact Abelian group A. For $ \phi\in L^{1}(A) $ , if $ \mathcal{F}_{A}(\phi)|_{H^{\perp}}\in L^{1}(H^{\perp}) $ , then
+
+$$
+\int_ {H} \phi (x h) d h = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \chi (x) d \chi ,
+$$
+
+for all $ x\in A $ , where Haar measure on $ H^{\perp}\cong A / H $ is the Plancherel measure with respect to the chosen Haar measure on $ A / H $ .
+
+Proof. For $ \chi\in H^{\perp} $ we have $ \chi(xh)=\chi(x) $ for every $ x\in A $ and $ h\in H $ . We therefore get from the quotient integral formula (2.2) that
+
+$$
+\begin{array}{l} \mathcal {F} _ {A / H} \left(\phi^ {H}\right) (\chi) = \int_ {A / H} \phi^ {H} (x H) \overline {{\chi (x)}} d (x H) \\ = \int_ {A / H} \int_ {H} \phi (x h) \overline {{\chi (x h)}} d h d (x H) \\ = \int_ {A} \phi (x) \overline {{\chi (x)}} d x = \mathcal {F} _ {A} (\phi) (\chi) \\ \end{array}
+$$
+
+for every $ \chi\in H^{\perp} $ . Moreover, if $ \mathcal{F}_{A}(\phi)|_{H^{\perp}}\in L^{1}(H^{\perp})=L^{1}\left(\widetilde{A}/\widetilde{H}\right) $ , then the Fourier inversion formula implies that for all $ x\in A $
+
+$$
+\begin{array}{l} \int_ {H} \phi (x h) d h = \phi^ {H} (x H) = \mathcal {F} _ {H ^ {\perp}} \mathcal {F} _ {A / H} \left(\phi^ {H}\right) (- x H) \\ = \mathcal {F} _ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (- x H) = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \overline {{- x H (\chi)}} d \chi \\ = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \overline {{\chi (- x)}} d \chi = \int_ {H ^ {\perp}} \mathcal {F} _ {A} (\phi) (\chi) \chi (x) d \chi . \\ \end{array}
+$$
+
+In all our applications, the group $A$ will be of the form $A = (\mathbb{Z} / N\mathbb{Z})^{d_1}\times \mathbb{R}^{d_2}$ for $N\in \mathbb{N}_{+}, d_{1}, d_{2}\in \mathbb{N}$. Recall we have defined the standard symmetric bilinear form on $\mathbb{R}^{d_2}$ as in the beginning of §1, and for any $N\in \mathbb{N}_{+}$, we still denote the symmetric bilinear form on $(\mathbb{Z} / N\mathbb{Z})^{d_1}$ by $Q$ as defined in the standard way. We then identify $A$ with its dual group $ \widehat{A} $ through the pairing
+
+$$
+A \cong \widehat {A}
+$$
+
+$$
+(s, t) \mapsto \chi_ {s, t} (l, \xi) = e ^ {2 \pi i \left(\frac {Q (s , l)}{N} + Q (t, \xi)\right)},
+$$

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@@ -0,0 +1,95 @@
+![](page=0,bbox=[133, 128, 479, 379])
+
+<div align="center">
+
+Fig. 2. Location of switch point at the end of finite element
+
+</div>
+
+The proposed step-equilibration approach is based on the principle that only the finite element(s) with the switch point(s) have non-uniform discretization $[t_{l-1}, t_l]$ . The approach uses an indicator variable $(\eta)$ for the switch point(s), which is determined by calculating and multiplying the sum of the complementarity variables in two consecutive finite elements.
+
+The auxiliary variables for the sum of the complementarity variables at each finite element are defined as:
+
+$$
+\hat {\lambda} _ {l} = \sum_ {k = 1} ^ {K} \lambda_ {l, k}, \quad \hat {\nu} _ {l} = \sum_ {k = 1} ^ {K} \nu_ {l, k}
+$$
+
+Then, the Hadamard product of the forward and backward sum of the complementarity variables determine if they have switched from positive to zero (or vice-versa).
+
+$$
+\pi_ {l} ^ {\lambda} = \hat {\lambda} _ {l - 1} \odot \hat {\lambda} _ {l}, \quad \pi_ {l} ^ {\nu} = \hat {\nu} _ {l - 1} \odot \hat {\nu} _ {l}
+$$
+
+(Here. $ \odot $ represents pointwise or elementwise product of vectors.)
+
+Since at least one of the vectors $ \pi_{l}^{\lambda} $ or $ \pi_{l}^{\nu} $ is zero at each element, and they are exactly equal to zero at the element corresponding to the switching point, the sum of the two vectors is a good candidate for the indicator function
+
+$$
+\tau_ {l} = \pi_ {l} ^ {\lambda} + \pi_ {l} ^ {\nu}, \eta_ {l} = \prod_ {j = 1} ^ {n _ {f}} \tau_ {l, j}
+$$
+
+Since the indicator variable $ \eta_{l} $ is non-negative and only zero at the switching element, the relation between step size and indicator variable can be represented by the following complementarity constraints.
+
+For $ l=1,\dots,N-1 $
+
+$$
+0 \leq \left(\Delta h _ {l} ^ {+} + \Delta h _ {l} ^ {-}\right) \perp \eta_ {l} \geq 0
+$$
+
+where $h_{l-1} - h_l = \Delta h_l^+ - \Delta h_l^-$, $\Delta h_l^+$, $\Delta h_l^- \geq 0$, The finite element with switch detection (FESD) algorithm was implemented as a package NOSNOC in [19].
+
+As mentioned in (10), the Nurkanovic formulation augments an additional $[2N\{\hat{\lambda},\hat{\nu}\} + (2N - 2)\{\pi^{\lambda},\pi^{\nu}\} + (N - 1)\{\tau\} + (N - 1)\{\eta\}]$ variables and $[2N\{(10a)\} + (2N - 2)\{(10b)\} + 2(N - 1)\{(10c)\} + (N - 1)\{(10d)\}]$ constraints for each complementarity constraint. This effectively decreases the degrees of freedom by $N - 1$ to that of the original problem and avoids non-unique solutions for the step size variables $h_i$.
+
+## 2.4 Proposed Formulation
+
+Although the Nurkanovic formulation makes the problem consistent with respect to the degrees of freedom and ensures uniform grid discretization away from the switch point(s), the formulation may be numerically unstable (i.e. the derivatives have large condition number) and increases the size of the problem, making it difficult to implement on larger optimal control problems.
+
+Inspired by the Nurkanovic [21] formulation, we propose a modification of the approach in [1], in order to keep the degrees of the problem consistent. In our proposed approach, we first apply the formulation in [1] with cross complementarities (9f). This locates the switching point(s) $(t_s)$ at the end of the finite element(s).
+
+We define the set of finite elements which have the switching point at the end (i.e. right) as:
+
+$$
+\begin{array}{l} \chi_ {s} = \left\{l \left[ \left| t _ {l - 1}, t _ {l} \right|, \lambda_ {l, K} = \nu_ {l, K} = 0, \right. \\ \lambda_ {l - 1, K} + \nu_ {l - 1, K} > 0 \text {o r} \lambda_ {l + 1, K} + \nu_ {l + 1, K} > 0 \} \\ \end{array}
+$$
+
+In the next step, we add additional constraints to the formulation which forces the finite elements to be equally spaced away from the switching points.
+
+$$
+h _ {l} - h _ {l + 1} = 0 \quad \forall l \in \{1, \dots , N - 1 \} \backslash \chi_ {s}
+$$
+
+Also, we add constraints to force the switching to happe at the boundary of finite elements found in the first step
+
+$$
+\lambda_ {l, K} + \nu_ {l, K} = 0 \quad \forall l \in \chi_ {s}
+$$
+
+This formulation adds the necessary $ N - 1 $ linear constraints without any additional variables making the formulation much more adaptable and applicable for large optimal control problems.
+
+The main assumption in our approach is that the location of the switching points in the optimal solution is independent of the step-size variables and the formulations. Thus, the switching points in the Baumrucker formulation would be the same as in the Nurkanovic formulation. The only difference between their solutions is in the value of the step size variables away from the switching elements. Therefore, we implement uniform discretization between the switch points, start time and the final time using (11) instead.
+
+## 3. SOLUTION METHODS FOR MPCCS
+
+To develop the solution strategy for the MPCC derived in the previous section, we discretize and rewrite (7) in the more general form:
+
+$$
+\min \varphi (x)
+$$
+
+$$
+\mathrm {s . t .} g _ {I} (x) \leq 0; g _ {E} (x) = 0,
+$$
+
+$$
+0 \leq G (x) \perp H (x) \geq 0
+$$
+
+Here the complementarity constraints (12d) represent the cross-complementarity constraints (9f) and step equilibration constraints (10e). The NLP equivalent formulation of the complementarity constraints is
+
+$$
+G (x) \geq 0, H (x) \geq 0, G _ {i} (x) H _ {i} (x) = 0, \forall i = 1 \dots n _ {c}
+$$
+
+## 3.1 MPCC Basics and Stationary Points
+
+The following index sets are defined at every feasible point $ \bar{x} $ of the MPCC (12):

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@@ -0,0 +1,95 @@
+![](page=0,bbox=[133, 128, 479, 379])
+
+<div align="center">
+
+Fig. 2. Location of switch point at the end of finite element
+
+</div>
+
+The proposed step-equilibration approach is based on the principle that only the finite element(s) with the switch point(s) have non-uniform discretization $[t_{l-1}, t_l]$ . The approach uses an indicator variable $(\eta)$ for the switch point(s), which is determined by calculating and multiplying the sum of the complementarity variables in two consecutive finite elements.
+
+The auxiliary variables for the sum of the complementarity variables at each finite element are defined as:
+
+$$
+\hat {\lambda} _ {l} = \sum_ {k = 1} ^ {K} \lambda_ {l, k}, \quad \hat {\nu} _ {l} = \sum_ {k = 1} ^ {K} \nu_ {l, k}
+$$
+
+Then, the Hadamard product of the forward and backward sum of the complementarity variables determine if they have switched from positive to zero (or vice-versa).
+
+$$
+\pi_ {l} ^ {\lambda} = \hat {\lambda} _ {l - 1} \odot \hat {\lambda} _ {l}, \quad \pi_ {l} ^ {\nu} = \hat {\nu} _ {l - 1} \odot \hat {\nu} _ {l}
+$$
+
+(Here. $ \odot $ represents pointwise or elementwise product of vectors.)
+
+Since at least one of the vectors $ \pi_{l}^{\lambda} $ or $ \pi_{l}^{\nu} $ is zero at each element, and they are exactly equal to zero at the element corresponding to the switching point, the sum of the two vectors is a good candidate for the indicator function
+
+$$
+\tau_ {l} = \pi_ {l} ^ {\lambda} + \pi_ {l} ^ {\nu}, \eta_ {l} = \prod_ {j = 1} ^ {n _ {f}} \tau_ {l, j}
+$$
+
+Since the indicator variable $ \eta_{l} $ is non-negative and only zero at the switching element, the relation between step size and indicator variable can be represented by the following complementarity constraints.
+
+For $ l=1,\dots,N-1 $
+
+$$
+0 \leq \left(\Delta h _ {l} ^ {+} + \Delta h _ {l} ^ {-}\right) \perp \eta_ {l} \geq 0
+$$
+
+where $h_{l-1} - h_l = \Delta h_l^+ - \Delta h_l^-$, $\Delta h_l^+$, $\Delta h_l^- \geq 0$, The finite element with switch detection (FESD) algorithm was implemented as a package NOSNOC in [19].
+
+As mentioned in (10), the Nurkanovic formulation augments an additional $[2N\{\hat{\lambda},\hat{\nu}\} + (2N - 2)\{\pi^{\lambda},\pi^{\nu}\} + (N - 1)\{\tau\} + (N - 1)\{\eta\}]$ variables and $[2N\{(10a)\} + (2N - 2)\{(10b)\} + 2(N - 1)\{(10c)\} + (N - 1)\{(10d)\}]$ constraints for each complementarity constraint. This effectively decreases the degrees of freedom by $N - 1$ to that of the original problem and avoids non-unique solutions for the step size variables $h_i$.
+
+## 2.4 Proposed Formulation
+
+Although the Nurkanovic formulation makes the problem consistent with respect to the degrees of freedom and ensures uniform grid discretization away from the switch point(s), the formulation may be numerically unstable (i.e. the derivatives have large condition number) and increases the size of the problem, making it difficult to implement on larger optimal control problems.
+
+Inspired by the Nurkanovic [21] formulation, we propose a modification of the approach in [1], in order to keep the degrees of the problem consistent. In our proposed approach, we first apply the formulation in [1] with cross complementarities (9f). This locates the switching point(s) $(t_s)$ at the end of the finite element(s).
+
+We define the set of finite elements which have the switching point at the end (i.e. right) as:
+
+$$
+\begin{array}{l} \chi_ {s} = \left\{l \left[ \left| t _ {l - 1}, t _ {l} \right|, \lambda_ {l, K} = \nu_ {l, K} = 0, \right. \\ \lambda_ {l - 1, K} + \nu_ {l - 1, K} > 0 \text {o r} \lambda_ {l + 1, K} + \nu_ {l + 1, K} > 0 \} \\ \end{array}
+$$
+
+In the next step, we add additional constraints to the formulation which forces the finite elements to be equally spaced away from the switching points.
+
+$$
+h _ {l} - h _ {l + 1} = 0 \quad \forall l \in \{1, \dots , N - 1 \} \backslash \chi_ {s}
+$$
+
+Also, we add constraints to force the switching to happe at the boundary of finite elements found in the first step
+
+$$
+\lambda_ {l, K} + \nu_ {l, K} = 0 \quad \forall l \in \chi_ {s}
+$$
+
+This formulation adds the necessary $ N - 1 $ linear constraints without any additional variables making the formulation much more adaptable and applicable for large optimal control problems.
+
+The main assumption in our approach is that the location of the switching points in the optimal solution is independent of the step-size variables and the formulations. Thus, the switching points in the Baumrucker formulation would be the same as in the Nurkanovic formulation. The only difference between their solutions is in the value of the step size variables away from the switching elements. Therefore, we implement uniform discretization between the switch points, start time and the final time using (11) instead.
+
+## 3. SOLUTION METHODS FOR MPCCS
+
+To develop the solution strategy for the MPCC derived in the previous section, we discretize and rewrite (7) in the more general form:
+
+$$
+\min \varphi (x)
+$$
+
+$$
+\mathrm {s . t .} g _ {I} (x) \leq 0; g _ {E} (x) = 0,
+$$
+
+$$
+0 \leq G (x) \perp H (x) \geq 0
+$$
+
+Here the complementarity constraints (12d) represent the cross-complementarity constraints (9f) and step equilibration constraints (10e). The NLP equivalent formulation of the complementarity constraints is
+
+$$
+G (x) \geq 0, H (x) \geq 0, G _ {i} (x) H _ {i} (x) = 0, \forall i = 1 \dots n _ {c}
+$$
+
+## 3.1 MPCC Basics and Stationary Points
+
+The following index sets are defined at every feasible point $ \bar{x} $ of the MPCC (12):

zai-org__GLM-OCR-api/arxiv_math/2503.03879_pg4_pg1_repeat3.md

@@ -0,0 +1,95 @@
+![](page=0,bbox=[133, 128, 479, 379])
+
+<div align="center">
+
+Fig. 2. Location of switch point at the end of finite element
+
+</div>
+
+The proposed step-equilibration approach is based on the principle that only the finite element(s) with the switch point(s) have non-uniform discretization $[t_{l-1}, t_l]$ . The approach uses an indicator variable $(\eta)$ for the switch point(s), which is determined by calculating and multiplying the sum of the complementarity variables in two consecutive finite elements.
+
+The auxiliary variables for the sum of the complementarity variables at each finite element are defined as:
+
+$$
+\hat {\lambda} _ {l} = \sum_ {k = 1} ^ {K} \lambda_ {l, k}, \quad \hat {\nu} _ {l} = \sum_ {k = 1} ^ {K} \nu_ {l, k}
+$$
+
+Then, the Hadamard product of the forward and backward sum of the complementarity variables determine if they have switched from positive to zero (or vice-versa).
+
+$$
+\pi_ {l} ^ {\lambda} = \hat {\lambda} _ {l - 1} \odot \hat {\lambda} _ {l}, \quad \pi_ {l} ^ {\nu} = \hat {\nu} _ {l - 1} \odot \hat {\nu} _ {l}
+$$
+
+(Here. $ \odot $ represents pointwise or elementwise product of vectors.)
+
+Since at least one of the vectors $ \pi_{l}^{\lambda} $ or $ \pi_{l}^{\nu} $ is zero at each element, and they are exactly equal to zero at the element corresponding to the switching point, the sum of the two vectors is a good candidate for the indicator function
+
+$$
+\tau_ {l} = \pi_ {l} ^ {\lambda} + \pi_ {l} ^ {\nu}, \eta_ {l} = \prod_ {j = 1} ^ {n _ {f}} \tau_ {l, j}
+$$
+
+Since the indicator variable $ \eta_{l} $ is non-negative and only zero at the switching element, the relation between step size and indicator variable can be represented by the following complementarity constraints.
+
+For $ l=1,\dots,N-1 $
+
+$$
+0 \leq \left(\Delta h _ {l} ^ {+} + \Delta h _ {l} ^ {-}\right) \perp \eta_ {l} \geq 0
+$$
+
+where $h_{l-1} - h_l = \Delta h_l^+ - \Delta h_l^-$, $\Delta h_l^+$, $\Delta h_l^- \geq 0$, The finite element with switch detection (FESD) algorithm was implemented as a package NOSNOC in [19].
+
+As mentioned in (10), the Nurkanovic formulation augments an additional $[2N\{\hat{\lambda},\hat{\nu}\} + (2N - 2)\{\pi^{\lambda},\pi^{\nu}\} + (N - 1)\{\tau\} + (N - 1)\{\eta\}]$ variables and $[2N\{(10a)\} + (2N - 2)\{(10b)\} + 2(N - 1)\{(10c)\} + (N - 1)\{(10d)\}]$ constraints for each complementarity constraint. This effectively decreases the degrees of freedom by $N - 1$ to that of the original problem and avoids non-unique solutions for the step size variables $h_i$.
+
+## 2.4 Proposed Formulation
+
+Although the Nurkanovic formulation makes the problem consistent with respect to the degrees of freedom and ensures uniform grid discretization away from the switch point(s), the formulation may be numerically unstable (i.e. the derivatives have large condition number) and increases the size of the problem, making it difficult to implement on larger optimal control problems.
+
+Inspired by the Nurkanovic [21] formulation, we propose a modification of the approach in [1], in order to keep the degrees of the problem consistent. In our proposed approach, we first apply the formulation in [1] with cross complementarities (9f). This locates the switching point(s) $(t_s)$ at the end of the finite element(s).
+
+We define the set of finite elements which have the switching point at the end (i.e. right) as:
+
+$$
+\begin{array}{l} \chi_ {s} = \left\{l \left[ \left| t _ {l - 1}, t _ {l} \right|, \lambda_ {l, K} = \nu_ {l, K} = 0, \right. \\ \lambda_ {l - 1, K} + \nu_ {l - 1, K} > 0 \text {o r} \lambda_ {l + 1, K} + \nu_ {l + 1, K} > 0 \} \\ \end{array}
+$$
+
+In the next step, we add additional constraints to the formulation which forces the finite elements to be equally spaced away from the switching points.
+
+$$
+h _ {l} - h _ {l + 1} = 0 \quad \forall l \in \{1, \dots , N - 1 \} \backslash \chi_ {s}
+$$
+
+Also, we add constraints to force the switching to happe at the boundary of finite elements found in the first step
+
+$$
+\lambda_ {l, K} + \nu_ {l, K} = 0 \quad \forall l \in \chi_ {s}
+$$
+
+This formulation adds the necessary $ N - 1 $ linear constraints without any additional variables making the formulation much more adaptable and applicable for large optimal control problems.
+
+The main assumption in our approach is that the location of the switching points in the optimal solution is independent of the step-size variables and the formulations. Thus, the switching points in the Baumrucker formulation would be the same as in the Nurkanovic formulation. The only difference between their solutions is in the value of the step size variables away from the switching elements. Therefore, we implement uniform discretization between the switch points, start time and the final time using (11) instead.
+
+## 3. SOLUTION METHODS FOR MPCCS
+
+To develop the solution strategy for the MPCC derived in the previous section, we discretize and rewrite (7) in the more general form:
+
+$$
+\min \varphi (x)
+$$
+
+$$
+\mathrm {s . t .} g _ {I} (x) \leq 0; g _ {E} (x) = 0,
+$$
+
+$$
+0 \leq G (x) \perp H (x) \geq 0
+$$
+
+Here the complementarity constraints (12d) represent the cross-complementarity constraints (9f) and step equilibration constraints (10e). The NLP equivalent formulation of the complementarity constraints is
+
+$$
+G (x) \geq 0, H (x) \geq 0, G _ {i} (x) H _ {i} (x) = 0, \forall i = 1 \dots n _ {c}
+$$
+
+## 3.1 MPCC Basics and Stationary Points
+
+The following index sets are defined at every feasible point $ \bar{x} $ of the MPCC (12):

zai-org__GLM-OCR-api/arxiv_math/2503.03899_pg9_pg1_repeat1.md

@@ -0,0 +1,41 @@
+Proposition A.1. For fixed $ 0 < \delta < \frac{1}{4} $ , we have
+
+$$
+\limsup _ {n \rightarrow \infty} n ^ {- \delta} \log P _ {n} \left(\inf _ {t \geq 0} \left| \frac {1}{\sqrt {n}} \sum_ {k \leq t \sqrt {n}} X _ {k} - L (t) \right| > n ^ {- \frac {1}{4} + \delta}\right) < 0.
+$$
+
+Remark. In particular, we have
+
+$$
+\lim _ {n \rightarrow \infty} P _ {n} \left(\sup _ {t \geq 0} \left| \frac {1}{\sqrt {n}} \sum_ {k \leq t \sqrt {n}} X _ {k} - L (t) \right|\leq n ^ {- \frac {1}{4} + \delta}\right)=1.
+$$
+
+Proof. Let $ d(n) $ be the number of distinct parts partitions of n. We require only a weak form of the well-known asymptotic expansion of $ d(n) $ [11],
+
+$$
+d (n) = e ^ {\frac {2 \sqrt {n}}{A}} + O (\log n).
+$$
+
+Let $ a_{n}, b_{n}\in \mathbb{N}_{0} $ and define $ \alpha_{n},\beta_{n} $ by
+
+$$
+a _ {n} = \alpha_ {n} \sqrt {n}, \quad b _ {n} = \beta_ {n} \sqrt {n}.
+$$
+
+Assume that $ \alpha_{n},\beta_{n}\geq n^{-\frac{1}{4}}+\delta $ . We use the saddle point bound to write, for any $ x_{n}\in\mathbb{R} $
+
+$$
+\begin{array}{l} P _ {n} \left(\frac {1}{\sqrt {n}} \sum_ {k \leq a _ {n}} X _ {k} = b _ {n}\right) \\ = \frac {1}{d (n)} \left[ q ^ {n} \right] \left[ \zeta^ {b _ {n}} \right] \prod_ {k \leq a _ {n}} \left(1 + \zeta q ^ {k}\right) \prod_ {k > a _ {n}} \left(1 + q ^ {k}\right) \\ \leq \frac {1}{d (n)} q _ {n} ^ {- n} e ^ {- b _ {n} x _ {n}} \prod_ {k \leq a _ {n}} \left(1 + e ^ {x _ {n}} q _ {n} ^ {k}\right) \prod_ {k > a _ {n}} \left(1 + q _ {n} ^ {k}\right) \\ = \exp \left(\frac {\sqrt {n}}{A} - \log d (n) - \beta_ {n} x _ {n} \sqrt {n} + \sum_ {k \leq a _ {n}} \log \left(1 + e ^ {x _ {n} - \frac {k}{A \sqrt {n}}}\right) + \sum_ {k > a _ {n}} \log \left(1 + e ^ {- \frac {k}{A \sqrt {n}}}\right)\right). \\ \end{array}
+$$
+
+We will take $x_{n}\in \{\pm n^{-\frac{1}{4}}\}$, where the sign will depend on $b_{n}$. By Taylor's Theorem,
+
+$$
+\begin{array}{l} \left| \log \left(1 + e ^ {x _ {n} - \frac {k}{A \sqrt {n}}}\right) - \log \left(1 + e ^ {- \frac {k}{A \sqrt {n}}}\right) - x _ {n} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{1 + e ^ {- \frac {k}{A \sqrt {n}}}} \right| \leq \frac {x _ {n} ^ {2}}{2} \sup _ {| x | \leq n \frac {1}{4}} \frac {e ^ {x - \frac {k}{A \sqrt {n}}}}{\left(1 + e ^ {x - \frac {k}{A \sqrt {n}}}\right) ^ {2}} \\ \leq x _ {n} ^ {2} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{\left(1 + \frac {1}{2} e ^ {- \frac {k}{A \sqrt {n}}}\right) ^ {2}}. \\ \end{array}
+$$
+
+Thus,
+
+$$
+\begin{array}{l} P _ {n} \left(\frac {1}{\sqrt {n}} \sum_ {k \leq a _ {n}} X _ {k} = b _ {n}\right) \leq \exp \left(\frac {\sqrt {n}}{A} - \log d (n) + \sum_ {k \geq 1} \log \left(1 + e ^ {- \frac {k}{A \sqrt {n}}}\right) - \beta_ {n} x _ {n} \sqrt {n} \right. \\ + x _ {n} \sum_ {k \leq a _ {n}} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{1 + e ^ {- \frac {k}{A \sqrt {n}}}} + O \left(x _ {n} ^ {2} \sum_ {k \leq a _ {n}} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{\left(1 + \frac {1}{2} e ^ {- \frac {k}{A \sqrt {n}}}\right) ^ {2}}\right)). \\ \end{array}
+$$

zai-org__GLM-OCR-api/arxiv_math/2503.03899_pg9_pg1_repeat2.md

@@ -0,0 +1,41 @@
+Proposition A.1. For fixed $ 0 < \delta < \frac{1}{4} $ , we have
+
+$$
+\limsup _ {n \rightarrow \infty} n ^ {- \delta} \log P _ {n} \left(\inf _ {t \geq 0} \left| \frac {1}{\sqrt {n}} \sum_ {k \leq t \sqrt {n}} X _ {k} - L (t) \right| > n ^ {- \frac {1}{4} + \delta}\right) < 0.
+$$
+
+Remark. In particular, we have
+
+$$
+\lim _ {n \rightarrow \infty} P _ {n} \left(\sup _ {t \geq 0} \left| \frac {1}{\sqrt {n}} \sum_ {k \leq t \sqrt {n}} X _ {k} - L (t) \right|\leq n ^ {- \frac {1}{4} + \delta}\right)=1.
+$$
+
+Proof. Let $ d(n) $ be the number of distinct parts partitions of n. We require only a weak form of the well-known asymptotic expansion of $ d(n) $ [11],
+
+$$
+d (n) = e ^ {\frac {2 \sqrt {n}}{A}} + O (\log n).
+$$
+
+Let $ a_{n}, b_{n}\in \mathbb{N}_{0} $ and define $ \alpha_{n},\beta_{n} $ by
+
+$$
+a _ {n} = \alpha_ {n} \sqrt {n}, \quad b _ {n} = \beta_ {n} \sqrt {n}.
+$$
+
+Assume that $ \alpha_{n},\beta_{n}\geq n^{-\frac{1}{4}}+\delta $ . We use the saddle point bound to write, for any $ x_{n}\in\mathbb{R} $
+
+$$
+\begin{array}{l} P _ {n} \left(\frac {1}{\sqrt {n}} \sum_ {k \leq a _ {n}} X _ {k} = b _ {n}\right) \\ = \frac {1}{d (n)} \left[ q ^ {n} \right] \left[ \zeta^ {b _ {n}} \right] \prod_ {k \leq a _ {n}} \left(1 + \zeta q ^ {k}\right) \prod_ {k > a _ {n}} \left(1 + q ^ {k}\right) \\ \leq \frac {1}{d (n)} q _ {n} ^ {- n} e ^ {- b _ {n} x _ {n}} \prod_ {k \leq a _ {n}} \left(1 + e ^ {x _ {n}} q _ {n} ^ {k}\right) \prod_ {k > a _ {n}} \left(1 + q _ {n} ^ {k}\right) \\ = \exp \left(\frac {\sqrt {n}}{A} - \log d (n) - \beta_ {n} x _ {n} \sqrt {n} + \sum_ {k \leq a _ {n}} \log \left(1 + e ^ {x _ {n} - \frac {k}{A \sqrt {n}}}\right) + \sum_ {k > a _ {n}} \log \left(1 + e ^ {- \frac {k}{A \sqrt {n}}}\right)\right). \\ \end{array}
+$$
+
+We will take $x_{n}\in \{\pm n^{-\frac{1}{4}}\}$, where the sign will depend on $b_{n}$. By Taylor's Theorem,
+
+$$
+\begin{array}{l} \left| \log \left(1 + e ^ {x _ {n} - \frac {k}{A \sqrt {n}}}\right) - \log \left(1 + e ^ {- \frac {k}{A \sqrt {n}}}\right) - x _ {n} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{1 + e ^ {- \frac {k}{A \sqrt {n}}}} \right| \leq \frac {x _ {n} ^ {2}}{2} \sup _ {| x | \leq n \frac {1}{4}} \frac {e ^ {x - \frac {k}{A \sqrt {n}}}}{\left(1 + e ^ {x - \frac {k}{A \sqrt {n}}}\right) ^ {2}} \\ \leq x _ {n} ^ {2} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{\left(1 + \frac {1}{2} e ^ {- \frac {k}{A \sqrt {n}}}\right) ^ {2}}. \\ \end{array}
+$$
+
+Thus,
+
+$$
+\begin{array}{l} P _ {n} \left(\frac {1}{\sqrt {n}} \sum_ {k \leq a _ {n}} X _ {k} = b _ {n}\right) \leq \exp \left(\frac {\sqrt {n}}{A} - \log d (n) + \sum_ {k \geq 1} \log \left(1 + e ^ {- \frac {k}{A \sqrt {n}}}\right) - \beta_ {n} x _ {n} \sqrt {n} \right. \\ + x _ {n} \sum_ {k \leq a _ {n}} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{1 + e ^ {- \frac {k}{A \sqrt {n}}}} + O \left(x _ {n} ^ {2} \sum_ {k \leq a _ {n}} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{\left(1 + \frac {1}{2} e ^ {- \frac {k}{A \sqrt {n}}}\right) ^ {2}}\right)). \\ \end{array}
+$$

zai-org__GLM-OCR-api/arxiv_math/2503.03899_pg9_pg1_repeat3.md

@@ -0,0 +1,41 @@
+Proposition A.1. For fixed $ 0 < \delta < \frac{1}{4} $ , we have
+
+$$
+\limsup _ {n \rightarrow \infty} n ^ {- \delta} \log P _ {n} \left(\inf _ {t \geq 0} \left| \frac {1}{\sqrt {n}} \sum_ {k \leq t \sqrt {n}} X _ {k} - L (t) \right| > n ^ {- \frac {1}{4} + \delta}\right) < 0.
+$$
+
+Remark. In particular, we have
+
+$$
+\lim _ {n \rightarrow \infty} P _ {n} \left(\sup _ {t \geq 0} \left| \frac {1}{\sqrt {n}} \sum_ {k \leq t \sqrt {n}} X _ {k} - L (t) \right|\leq n ^ {- \frac {1}{4} + \delta}\right)=1.
+$$
+
+Proof. Let $ d(n) $ be the number of distinct parts partitions of n. We require only a weak form of the well-known asymptotic expansion of $ d(n) $ [11],
+
+$$
+d (n) = e ^ {\frac {2 \sqrt {n}}{A}} + O (\log n).
+$$
+
+Let $ a_{n}, b_{n}\in \mathbb{N}_{0} $ and define $ \alpha_{n},\beta_{n} $ by
+
+$$
+a _ {n} = \alpha_ {n} \sqrt {n}, \quad b _ {n} = \beta_ {n} \sqrt {n}.
+$$
+
+Assume that $ \alpha_{n},\beta_{n}\geq n^{-\frac{1}{4}}+\delta $ . We use the saddle point bound to write, for any $ x_{n}\in\mathbb{R} $
+
+$$
+\begin{array}{l} P _ {n} \left(\frac {1}{\sqrt {n}} \sum_ {k \leq a _ {n}} X _ {k} = b _ {n}\right) \\ = \frac {1}{d (n)} \left[ q ^ {n} \right] \left[ \zeta^ {b _ {n}} \right] \prod_ {k \leq a _ {n}} \left(1 + \zeta q ^ {k}\right) \prod_ {k > a _ {n}} \left(1 + q ^ {k}\right) \\ \leq \frac {1}{d (n)} q _ {n} ^ {- n} e ^ {- b _ {n} x _ {n}} \prod_ {k \leq a _ {n}} \left(1 + e ^ {x _ {n}} q _ {n} ^ {k}\right) \prod_ {k > a _ {n}} \left(1 + q _ {n} ^ {k}\right) \\ = \exp \left(\frac {\sqrt {n}}{A} - \log d (n) - \beta_ {n} x _ {n} \sqrt {n} + \sum_ {k \leq a _ {n}} \log \left(1 + e ^ {x _ {n} - \frac {k}{A \sqrt {n}}}\right) + \sum_ {k > a _ {n}} \log \left(1 + e ^ {- \frac {k}{A \sqrt {n}}}\right)\right). \\ \end{array}
+$$
+
+We will take $x_{n}\in \{\pm n^{-\frac{1}{4}}\}$, where the sign will depend on $b_{n}$. By Taylor's Theorem,
+
+$$
+\begin{array}{l} \left| \log \left(1 + e ^ {x _ {n} - \frac {k}{A \sqrt {n}}}\right) - \log \left(1 + e ^ {- \frac {k}{A \sqrt {n}}}\right) - x _ {n} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{1 + e ^ {- \frac {k}{A \sqrt {n}}}} \right| \leq \frac {x _ {n} ^ {2}}{2} \sup _ {| x | \leq n \frac {1}{4}} \frac {e ^ {x - \frac {k}{A \sqrt {n}}}}{\left(1 + e ^ {x - \frac {k}{A \sqrt {n}}}\right) ^ {2}} \\ \leq x _ {n} ^ {2} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{\left(1 + \frac {1}{2} e ^ {- \frac {k}{A \sqrt {n}}}\right) ^ {2}}. \\ \end{array}
+$$
+
+Thus,
+
+$$
+\begin{array}{l} P _ {n} \left(\frac {1}{\sqrt {n}} \sum_ {k \leq a _ {n}} X _ {k} = b _ {n}\right) \leq \exp \left(\frac {\sqrt {n}}{A} - \log d (n) + \sum_ {k \geq 1} \log \left(1 + e ^ {- \frac {k}{A \sqrt {n}}}\right) - \beta_ {n} x _ {n} \sqrt {n} \right. \\ + x _ {n} \sum_ {k \leq a _ {n}} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{1 + e ^ {- \frac {k}{A \sqrt {n}}}} + O \left(x _ {n} ^ {2} \sum_ {k \leq a _ {n}} \frac {e ^ {- \frac {k}{A \sqrt {n}}}}{\left(1 + \frac {1}{2} e ^ {- \frac {k}{A \sqrt {n}}}\right) ^ {2}}\right)). \\ \end{array}
+$$

zai-org__GLM-OCR-api/arxiv_math/2503.03903_pg9_pg1_repeat1.md

@@ -0,0 +1,27 @@
+![](page=0,bbox=[482, 156, 705, 377])
+
+<div align="center">
+
+FIGURE 5. Above is the bottom pipe dream for 1427356 (cross signs denote crossings, and empty boxes denote non-crossings). If we draw a diagonal line out from the first crossing in row 3, then this diagonal line never intersects or is directly to the right of another crossing. However, the diagonal line emitting from the first box in row 5 enters the square directly to the right of the last crossing of row 3. Indeed, the subsequence 473 is a 231 pattern.
+
+</div>
+
+a square containing a crossing in P, nor does any square immediately to the left of this line contain a crossing in P.
+
+See Figure 3.2 for an example.
+
+Proof. For contradiction, let i the largest row index such that the line emanating from the leftmost crossing of row i intersects or is directly to the right of a crossing in row j. Since $ L(i) > 0 $ and i was the largest such row, we must have $ L(i+1)=0 $ , and thus, since $ L(i) > L(i+1) $ , we have $ w(i) > w(i+1) $ . But then, we claim that $ w(j) > w(i+1) $ . Either $ w(j) > w(i) $ , in which this follows by transitivity, or $ w(j) < w(i+1) $ , in which case, $ L(i+1)\geq L(j)+(i-j)+1>0 $ . Either way, we have either a 231 or a 321 pattern in w given by the indices j,i,i+1. So, we have proved the claim.
+
+Lemma 3.6 allows us to prove that 321 and 231-avoidance are sufficient:
+
+Lemma 3.7. If $w$ avoids 231 and 321, then $ \mathfrak{S}_{w} $ is a single CHM. In particular, $ \mathfrak{S}_{w}=h_{L(w)} $
+
+Proof. We use again the fact that all pipe dreams are obtained from the bottom pipe dream by ladder moves. Here, again all of the ladder moves we can perform are simple ladder moves that just move crossings along their diagonals. Two crossings in the same row of the bottom pipe dream can never slide past each other, and any two rows can slide independently by Lemma 3.6. Thus, row i contributes a factor $ h_{L(i)}^{i} $ , and multiplying these factors gives us $ \mathfrak{S}_{w}=\prod_{i} h_{L(i)}^{i}. $
+
+Notice that, unlike the case of SEMs and analogously to the case for usual monomials, the maximal monomial in the CHM expansion of $ \mathfrak{S}_{w} $ is always $ h_{L(w)} $ .
+
+Finally, we prove that 321 and 231 avoidance are necessary conditions in order for $ \mathfrak{S}_{w} $ to be a CHM.
+
+Lemma 3.8. If $w$ contains a 321 pattern, then $\mathfrak{S}_w$ is not a CHM.
+
+Proof. We induct on the length of w. First, suppose that w contains a 321 pattern and $ \mathfrak{S}_{w} $ is a single CHM. Let $ i < j < k $ be indices such that $ w_{i} > w_{j} > w_{k} $ . First, we show how to reduce to the case where i,j,k=i,i+1,i+2.

zai-org__GLM-OCR-api/arxiv_math/2503.03903_pg9_pg1_repeat2.md

@@ -0,0 +1,27 @@
+![](page=0,bbox=[482, 156, 705, 377])
+
+<div align="center">
+
+FIGURE 5. Above is the bottom pipe dream for 1427356 (cross signs denote crossings, and empty boxes denote non-crossings). If we draw a diagonal line out from the first crossing in row 3, then this diagonal line never intersects or is directly to the right of another crossing. However, the diagonal line emitting from the first box in row 5 enters the square directly to the right of the last crossing of row 3. Indeed, the subsequence 473 is a 231 pattern.
+
+</div>
+
+a square containing a crossing in P, nor does any square immediately to the left of this line contain a crossing in P.
+
+See Figure 3.2 for an example.
+
+Proof. For contradiction, let i the largest row index such that the line emanating from the leftmost crossing of row i intersects or is directly to the right of a crossing in row j. Since $ L(i) > 0 $ and i was the largest such row, we must have $ L(i+1)=0 $ , and thus, since $ L(i) > L(i+1) $ , we have $ w(i) > w(i+1) $ . But then, we claim that $ w(j) > w(i+1) $ . Either $ w(j) > w(i) $ , in which this follows by transitivity, or $ w(j) < w(i+1) $ , in which case, $ L(i+1)\geq L(j)+(i-j)+1>0 $ . Either way, we have either a 231 or a 321 pattern in w given by the indices j,i,i+1. So, we have proved the claim.
+
+Lemma 3.6 allows us to prove that 321 and 231-avoidance are sufficient:
+
+Lemma 3.7. If $w$ avoids 231 and 321, then $ \mathfrak{S}_{w} $ is a single CHM. In particular, $ \mathfrak{S}_{w}=h_{L(w)} $
+
+Proof. We use again the fact that all pipe dreams are obtained from the bottom pipe dream by ladder moves. Here, again all of the ladder moves we can perform are simple ladder moves that just move crossings along their diagonals. Two crossings in the same row of the bottom pipe dream can never slide past each other, and any two rows can slide independently by Lemma 3.6. Thus, row i contributes a factor $ h_{L(i)}^{i} $ , and multiplying these factors gives us $ \mathfrak{S}_{w}=\prod_{i} h_{L(i)}^{i}. $
+
+Notice that, unlike the case of SEMs and analogously to the case for usual monomials, the maximal monomial in the CHM expansion of $ \mathfrak{S}_{w} $ is always $ h_{L(w)} $ .
+
+Finally, we prove that 321 and 231 avoidance are necessary conditions in order for $ \mathfrak{S}_{w} $ to be a CHM.
+
+Lemma 3.8. If $w$ contains a 321 pattern, then $\mathfrak{S}_w$ is not a CHM.
+
+Proof. We induct on the length of w. First, suppose that w contains a 321 pattern and $ \mathfrak{S}_{w} $ is a single CHM. Let $ i < j < k $ be indices such that $ w_{i} > w_{j} > w_{k} $ . First, we show how to reduce to the case where i,j,k=i,i+1,i+2.

zai-org__GLM-OCR-api/arxiv_math/2503.03903_pg9_pg1_repeat3.md

@@ -0,0 +1,27 @@
+![](page=0,bbox=[482, 156, 705, 377])
+
+<div align="center">
+
+FIGURE 5. Above is the bottom pipe dream for 1427356 (cross signs denote crossings, and empty boxes denote non-crossings). If we draw a diagonal line out from the first crossing in row 3, then this diagonal line never intersects or is directly to the right of another crossing. However, the diagonal line emitting from the first box in row 5 enters the square directly to the right of the last crossing of row 3. Indeed, the subsequence 473 is a 231 pattern.
+
+</div>
+
+a square containing a crossing in P, nor does any square immediately to the left of this line contain a crossing in P.
+
+See Figure 3.2 for an example.
+
+Proof. For contradiction, let i the largest row index such that the line emanating from the leftmost crossing of row i intersects or is directly to the right of a crossing in row j. Since $ L(i) > 0 $ and i was the largest such row, we must have $ L(i+1)=0 $ , and thus, since $ L(i) > L(i+1) $ , we have $ w(i) > w(i+1) $ . But then, we claim that $ w(j) > w(i+1) $ . Either $ w(j) > w(i) $ , in which this follows by transitivity, or $ w(j) < w(i+1) $ , in which case, $ L(i+1)\geq L(j)+(i-j)+1>0 $ . Either way, we have either a 231 or a 321 pattern in w given by the indices j,i,i+1. So, we have proved the claim.
+
+Lemma 3.6 allows us to prove that 321 and 231-avoidance are sufficient:
+
+Lemma 3.7. If $w$ avoids 231 and 321, then $ \mathfrak{S}_{w} $ is a single CHM. In particular, $ \mathfrak{S}_{w}=h_{L(w)} $
+
+Proof. We use again the fact that all pipe dreams are obtained from the bottom pipe dream by ladder moves. Here, again all of the ladder moves we can perform are simple ladder moves that just move crossings along their diagonals. Two crossings in the same row of the bottom pipe dream can never slide past each other, and any two rows can slide independently by Lemma 3.6. Thus, row i contributes a factor $ h_{L(i)}^{i} $ , and multiplying these factors gives us $ \mathfrak{S}_{w}=\prod_{i} h_{L(i)}^{i}. $
+
+Notice that, unlike the case of SEMs and analogously to the case for usual monomials, the maximal monomial in the CHM expansion of $ \mathfrak{S}_{w} $ is always $ h_{L(w)} $ .
+
+Finally, we prove that 321 and 231 avoidance are necessary conditions in order for $ \mathfrak{S}_{w} $ to be a CHM.
+
+Lemma 3.8. If $w$ contains a 321 pattern, then $\mathfrak{S}_w$ is not a CHM.
+
+Proof. We induct on the length of w. First, suppose that w contains a 321 pattern and $ \mathfrak{S}_{w} $ is a single CHM. Let $ i < j < k $ be indices such that $ w_{i} > w_{j} > w_{k} $ . First, we show how to reduce to the case where i,j,k=i,i+1,i+2.

zai-org__GLM-OCR-api/arxiv_math/2503.03905_pg7_pg1_repeat1.md

@@ -0,0 +1,39 @@
+We show that $ \sigma^{\prime} $ is bijective on $ \mathsf{GF}(2^{m}) $ . Let us fix $ u,y_{1},y_{2}\in\mathsf{GF}(2^{m}),y_{1}\neq y_{2} $ . There is an element $ v\in\mathsf{GF}(2^{t}) $ such that
+
+$$
+\operatorname {T r} _ {\mathrm {G F} \left(2 ^ {t}\right) / \mathrm {G F} (2)} (u v) \neq h ^ {\prime} \left(y _ {1}\right) + h ^ {\prime} \left(y _ {2}\right).
+$$
+
+As $ \tau $ is surjective, there is a $ z\in\mathsf{GF}(2^{m}) $ with $ \tau(z)=v $ . By Lemma 7, there is $ x\in\mathsf{GF}(2^{m}) $ with $ y_{1}\star x+y_{2}\star x=z $ . Then
+
+$$
+\begin{array}{l} \alpha_ {y _ {1}} (x) + \alpha_ {y _ {2}} (x) = \operatorname {T r} _ {\mathrm {G F} \left(2 ^ {t}\right) / \mathrm {G F} (2)} \left(u \tau \left(y _ {1} \star x + y _ {2} \star x\right) + u h \left(y _ {1}\right) + u h \left(y _ {2}\right)\right) \\ = \operatorname {T r} _ {\mathrm {G F} \left(2 ^ {t}\right) / \mathrm {G F} (2)} \left(u \tau (z)\right) + h ^ {\prime} \left(y _ {1}\right) + h ^ {\prime} \left(y _ {2}\right) \\ = \operatorname {T r} _ {\mathrm {G F} \left(2 ^ {t}\right) / \mathrm {G F} (2)} (u v) + h ^ {\prime} \left(y _ {1}\right) + h ^ {\prime} \left(y _ {2}\right) \\ \neq 0. \\ \end{array}
+$$
+
+This implies $ \alpha_{y_{1}}(x)\neq\alpha_{y_{2}}(x) $ and $ \sigma^{\prime}(y_{1})\neq\sigma^{\prime}(y_{2}) $ . It follows that all component functions of f are of Maiorana-McFarland type bent function. This finishes the proof of the theorem.
+
+In this paper, we do not study the question of EA-equivalence of the $ (2m,t) $ -bent functions defined above. In general, this is a very difficult question. We only remark that Weng, Feng and Qui [23] proved that most of the $ \mathcal{PS} $ type bent functions, obtained from a Desarguesian spread are not EA-equivalent to any Maiorana-McFarland bent function. This leads us to conclude that, typically, for $ t > 1 $ , the $ (2m,t) $ -bent functions described by (9) are generally not EA-equivalent to the other two classes, as specified in (6) and (8).
+
+3. 4. Proof of Theorem 1. Let us recall the Carlet-Ding-Yuan bound (2) for the distance between affine and $ ( n,m) $ -bent functions:
+
+$$
+\left(1 - \frac {1}{2 ^ {m}}\right) \left(2 ^ {n} - 2 ^ {n / 2}\right) \leq d _ {H} (f, \mathcal {A}) \leq \left(1 - \frac {1}{2 ^ {m}}\right) \left(2 ^ {n} + 2 ^ {n / 2}\right).
+$$
+
+For an $ (n,m) $ -bent function f, the Walsh coefficients are
+
+$$
+W _ {f} (a, b) = \left\{ \begin{array}{l l} \pm 2 ^ {n / 2} & \text {i f} b \neq 0, \\ 0 & \text {i f} a \neq 0, b = 0, \\ 2 ^ {n} & \text {i f} a = 0, b = 0. \end{array} \right.
+$$
+
+Hence, the Carlet-Ding-Yuan bound follows from Lemma 3 easily. The Liu-Mesnager-Chen Conjecture implies that the true value of $ d_{H} ( f,\mathcal{A}) $ is $ \left( 1-\frac{1}{2^{m}} \right)\left( 2^{n}-2^{n/2} \right). $ Theorem 1 claims that this holds for two classes of $ (n,m) $ -bent functions.
+
+Proof of Theorem 1. Let $E_{i}$ be the set of pairs $(x,y)\in \mathrm{GF}(2^{m})^{2}$ such that $f_{i}(x,y) = f_{i}(0,0)$ $i = 1,2$. We show that $|E_{i}| = 2^{2m - t} + 2^{m} - 2^{m - t}$, which implies that $f_{i}$ has Hamming distance $(1 - 2^{-t})(2^{2m} - 2^{m})$ from the constant function $f_{i}(0,0)$. Therefore, $d_H(f_i,\mathcal{A}) \leq (1 - 2^{-t})(2^{2m} - 2^{m})$, and the theorem follows from the Carlet-Ding-Yuan bound.
+
+Let $T$ be the set of elements $z \in \mathrm{GF}(q^m)$ with $\gamma(z) = f_1(0,0)$. Since $\gamma$ is balanced, $|T| = 2^{m-t}$, and $f_1(x,y) = f_1(0,0)$ if and only if $\star \frac{x}{y} \in T$. Moreover, since $\star \frac{0}{0} = 0$, we have $0 \in T$. The number of solutions of $\star \frac{x}{y} = 0$ is $2 \cdot 2^m - 1$, and the number of solutions of $\star \frac{x}{y} = t \in T \setminus \{0\}$ is $2^m - 1$. This implies
+
+$$
+\left| E _ {1} \right| = \left| f _ {1} ^ {- 1} \left(f _ {1} (0, 0)\right) \right| = 2 \cdot 2 ^ {m} - 1 + \left(2 ^ {m - t} - 1\right) \left(2 ^ {m} - 1\right).
+$$
+
+The same argument applies to $ f_{2}. $ In this case, $ f_{2}(0,0)=h(0), $ and the number of solutions of $ \sigma(y)\star x=t $ is $ 2\cdot 2^{m}-1 $ or $ 2^{m}-1 $ , depending on $ t=0 $ or $ t\neq0. $

zai-org__GLM-OCR-api/arxiv_math/2503.03905_pg7_pg1_repeat2.md

@@ -0,0 +1,39 @@
+We show that $ \sigma^{\prime} $ is bijective on $ \mathsf{GF}(2^{m}) $ . Let us fix $ u,y_{1},y_{2}\in\mathsf{GF}(2^{m}),y_{1}\neq y_{2} $ . There is an element $ v\in\mathsf{GF}(2^{t}) $ such that
+
+$$
+\operatorname {T r} _ {\mathrm {G F} \left(2 ^ {t}\right) / \mathrm {G F} (2)} (u v) \neq h ^ {\prime} \left(y _ {1}\right) + h ^ {\prime} \left(y _ {2}\right).
+$$
+
+As $ \tau $ is surjective, there is a $ z\in\mathsf{GF}(2^{m}) $ with $ \tau(z)=v $ . By Lemma 7, there is $ x\in\mathsf{GF}(2^{m}) $ with $ y_{1}\star x+y_{2}\star x=z $ . Then
+
+$$
+\begin{array}{l} \alpha_ {y _ {1}} (x) + \alpha_ {y _ {2}} (x) = \operatorname {T r} _ {\mathrm {G F} \left(2 ^ {t}\right) / \mathrm {G F} (2)} \left(u \tau \left(y _ {1} \star x + y _ {2} \star x\right) + u h \left(y _ {1}\right) + u h \left(y _ {2}\right)\right) \\ = \operatorname {T r} _ {\mathrm {G F} \left(2 ^ {t}\right) / \mathrm {G F} (2)} \left(u \tau (z)\right) + h ^ {\prime} \left(y _ {1}\right) + h ^ {\prime} \left(y _ {2}\right) \\ = \operatorname {T r} _ {\mathrm {G F} \left(2 ^ {t}\right) / \mathrm {G F} (2)} (u v) + h ^ {\prime} \left(y _ {1}\right) + h ^ {\prime} \left(y _ {2}\right) \\ \neq 0. \\ \end{array}
+$$
+
+This implies $ \alpha_{y_{1}}(x)\neq\alpha_{y_{2}}(x) $ and $ \sigma^{\prime}(y_{1})\neq\sigma^{\prime}(y_{2}) $ . It follows that all component functions of f are of Maiorana-McFarland type bent function. This finishes the proof of the theorem.
+
+In this paper, we do not study the question of EA-equivalence of the $ (2m,t) $ -bent functions defined above. In general, this is a very difficult question. We only remark that Weng, Feng and Qui [23] proved that most of the $ \mathcal{PS} $ type bent functions, obtained from a Desarguesian spread are not EA-equivalent to any Maiorana-McFarland bent function. This leads us to conclude that, typically, for $ t > 1 $ , the $ (2m,t) $ -bent functions described by (9) are generally not EA-equivalent to the other two classes, as specified in (6) and (8).
+
+3. 4. Proof of Theorem 1. Let us recall the Carlet-Ding-Yuan bound (2) for the distance between affine and $ ( n,m) $ -bent functions:
+
+$$
+\left(1 - \frac {1}{2 ^ {m}}\right) \left(2 ^ {n} - 2 ^ {n / 2}\right) \leq d _ {H} (f, \mathcal {A}) \leq \left(1 - \frac {1}{2 ^ {m}}\right) \left(2 ^ {n} + 2 ^ {n / 2}\right).
+$$
+
+For an $ (n,m) $ -bent function f, the Walsh coefficients are
+
+$$
+W _ {f} (a, b) = \left\{ \begin{array}{l l} \pm 2 ^ {n / 2} & \text {i f} b \neq 0, \\ 0 & \text {i f} a \neq 0, b = 0, \\ 2 ^ {n} & \text {i f} a = 0, b = 0. \end{array} \right.
+$$
+
+Hence, the Carlet-Ding-Yuan bound follows from Lemma 3 easily. The Liu-Mesnager-Chen Conjecture implies that the true value of $ d_{H} ( f,\mathcal{A}) $ is $ \left( 1-\frac{1}{2^{m}} \right)\left( 2^{n}-2^{n/2} \right). $ Theorem 1 claims that this holds for two classes of $ (n,m) $ -bent functions.
+
+Proof of Theorem 1. Let $E_{i}$ be the set of pairs $(x,y)\in \mathrm{GF}(2^{m})^{2}$ such that $f_{i}(x,y) = f_{i}(0,0)$ $i = 1,2$. We show that $|E_{i}| = 2^{2m - t} + 2^{m} - 2^{m - t}$, which implies that $f_{i}$ has Hamming distance $(1 - 2^{-t})(2^{2m} - 2^{m})$ from the constant function $f_{i}(0,0)$. Therefore, $d_H(f_i,\mathcal{A}) \leq (1 - 2^{-t})(2^{2m} - 2^{m})$, and the theorem follows from the Carlet-Ding-Yuan bound.
+
+Let $T$ be the set of elements $z \in \mathrm{GF}(q^m)$ with $\gamma(z) = f_1(0,0)$. Since $\gamma$ is balanced, $|T| = 2^{m-t}$, and $f_1(x,y) = f_1(0,0)$ if and only if $\star \frac{x}{y} \in T$. Moreover, since $\star \frac{0}{0} = 0$, we have $0 \in T$. The number of solutions of $\star \frac{x}{y} = 0$ is $2 \cdot 2^m - 1$, and the number of solutions of $\star \frac{x}{y} = t \in T \setminus \{0\}$ is $2^m - 1$. This implies
+
+$$
+\left| E _ {1} \right| = \left| f _ {1} ^ {- 1} \left(f _ {1} (0, 0)\right) \right| = 2 \cdot 2 ^ {m} - 1 + \left(2 ^ {m - t} - 1\right) \left(2 ^ {m} - 1\right).
+$$
+
+The same argument applies to $ f_{2} $ . In this case, $ f_{2}(0,0)=h(0) $ , and the number of solutions of $ \sigma(y)\star x=t $ is $ 2\cdot 2^{m}-1 $ or $ 2^{m}-1 $ , depending on $ t=0 $ or $ t\neq0 $ .