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dots.mocr-4bit/arxiv_math/2502.15977_pg21_pg1_repeat0.md

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+FIGURE 3. The decorated fan of an action of $Q(1)^2$ on $\mathbb{P}^{2|2}$
+
+**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 $i = 1, \dots, n$ 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
+
+$$
+\mathbb{C}[t_1, \dots, t_n, t_1\xi_1, \dots, t_n\xi_n]
+$$
+
+and
+
+$$
+\mathbb{C}[t_i^{-1}t_1, \dots, t_i^{-1}, \dots, t_i^{-1}t_n\xi_1, \dots, \xi_i, \dots, t_i^{-1}t_n\xi_n].
+$$
+
+Figure 3 depicts the corresponding decorated fan for $n = 2$,
+
+If instead we change the decoration of $\rho_0$ to $\theta_1 + \dots + \theta_n$, then the resulting affine charts have coordinate superalgebras
+
+$$
+\mathbb{C}[t_1, \dots, t_n, t_1\xi_1, \dots, t_n\xi_n]
+$$
+
+and
+
+$$
+\mathbb{C}[t_i^{-1}t_1(1 + \xi_i\xi_1), \dots, t_i^{-1}, \dots, t_i^{-1}t_n(1 + \xi_i\xi_n), t_i^{-1}t_1(\xi_i - \xi_1), \dots, t_i^{-1}\xi_i, \dots, t_i^{-1}t_n(\xi_i - \xi_n)],
+$$
+
+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 $\rho = \mathbb{R}_+(a_1x_1 + \dots + a_nx_n)$, there are $2^{d-1}$ many possible “square root subspaces” $\mathbb{C}(\sqrt{a_1}\theta_1 \pm \dots \pm \sqrt{a_n}\theta_n)$, where $d$ is the number of indices $i = 1, \dots, n$ 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 $\rho_1 = \mathbb{R}_+(a_1x_1 + \dots + a_nx_n)$ and $\rho_2 = \mathbb{R}_+(b_1x_1 + \dots + b_nx_n)$, compatibility is less common if the linear matroid on the $n$ vectors
+
+$$
+\begin{pmatrix} a_1 \\ b_1 \end{pmatrix}, \dots, \begin{pmatrix} a_n \\ b_n \end{pmatrix}
+$$

dots.mocr-4bit/arxiv_math/2503.02004_pg9_pg1_repeat0.md

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+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 $O(2^M \cdot K)$. 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:** $\mathbf{G}, N_r, N_{\text{init}}, \Delta n$
+**Ensure:** $\mathcal{I}_s$
+1: Initialize: $l = 0, T_{(0)} = \{\emptyset\}$
+2: **while** $l \le N_{\text{init}}$ **do**
+3:     $i = \arg\max_{i \in T_{(l)}^c} \bar{g}_i$, s.t. $D_1(\{i\}, T_{(l)}) \le \Delta n$
+4:     $l = l + 1$
+5:     $T_{(l)} = T_{(l)} \cup \{i\}$
+6: **end while**
+7: **while** $l \le N_r$ **do**
+8:     $i = \arg\max_{i \in T_{(l)}^c} \min_k \|\mathbf{h}_k\|_2$, s.t. $\mathbf{h}_k \in \text{Col}(\mathbf{H}_{T_{(l)} \cup \{i\}})$
+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 $O(MKN_r)$. The algorithm starts by choosing $N_{\text{init}}$ 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 points.
+
+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.
+
+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 algorithms (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.

dots.mocr-4bit/arxiv_math/2503.03754_pg10_pg1_repeat0.md

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+[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
+
+$$
+P_{XY} = \begin{pmatrix} 1-s & 0 \\ sd & s(1-d) \end{pmatrix}
+$$
+
+where $s, d \in [0, 1]$. Let $u = f_X(0)$ and $v = f_X(1)$ for some $u, v \ge 0$. Assume that $\mathbb{E}[f] = m = (1-s)u+sv$. Then $u = \frac{m-sv}{1-s}$. One can verify directly that
+
+$$
+g(v, m) := \frac{H_{\Phi}(\mathbb{E}[f|Y])}{H_{\Phi}(f)} = \frac{(1-s(1-d))\Phi\left(\frac{1-s}{1-s(1-d)}u + \frac{sd}{1-s(1-d)}v\right) + s(1-d)\Phi(v) - \Phi((1-s)u + sv)}{(1-s)\Phi(u) + s\Phi(v) - \Phi((1-s)u + sv)} \quad (13)
+$$
+
+$$
+= \frac{(1-s(1-d))\Phi\left(\frac{m-s(1-d)v}{1-s(1-d)}\right) + s(1-d)\Phi(v) - \Phi(m)}{(1-s)\Phi\left(\frac{m-sv}{1-s}\right) + s\Phi(v) - \Phi(m)}. \quad (14)
+$$
+
+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
+
+$$
+\frac{s\Phi'(v) - s\Phi'\left(\frac{m-sv}{1-s}\right)}{(1-s)\Phi\left(\frac{m-sv}{1-s}\right) + s\Phi(v) - \Phi(m)} \ge \frac{s(1-d)\Phi'(v) - s(1-d)\Phi'\left(\frac{m-s(1-d)v}{1-s(1-d)}\right)}{(1-s(1-d))\Phi\left(\frac{m-s(1-d)v}{1-s(1-d)}\right) + s(1-d)\Phi(v) - \Phi(m)}. \quad (15)
+$$
+
+For any $t \in [0, 1]$, define
+
+$$
+k(t) = \frac{st\Phi'(v) - st\Phi'\left(\frac{m-svt}{1-st}\right)}{(1-st)\Phi\left(\frac{m-svt}{1-st}\right) + st\Phi(v) - \Phi(m)} \quad (16)
+$$
+
+$$
+= \frac{\Phi'(v) - \Phi'\left(\frac{m-svt}{1-st}\right)}{\left(\frac{1}{st} - 1\right)\Phi\left(\frac{m-svt}{1-st}\right) + \Phi(v) - \frac{1}{st}\Phi(m)}. \quad (17)
+$$
+
+Then, (15) can be written as $k(1) \ge k(1-d)$. We would be done if we can show that $k(t)$ is an increasing function. Showing $k'(t) \ge 0$ is equivalent with
+
+$$
+C(t) = -\frac{1}{st^2} \left( \Phi'(v) - \Phi' \left( \frac{m-svt}{1-st} \right) \right) \left( -\Phi \left( \frac{m-svt}{1-st} \right) + \frac{(m-v)st}{1-st} \Phi' \left( \frac{m-svt}{1-st} \right) + \Phi(m) \right) \\ - \frac{s(m-v)}{(1-st)^2} \Phi'' \left( \frac{m-svt}{1-st} \right) \left( \Phi(v) + \frac{1-st}{st} \Phi \left( \frac{m-svt}{1-st} \right) - \frac{1}{st} \Phi(m) \right) \ge 0.
+$$
+
+Let $x_1 = v$, $x_2 = \frac{m-svt}{1-st}$. Then we can compute $s$ from $x_1$ and $x_2$ as follows:
+
+$$
+s = \frac{m - x_2}{t(x_1 - x_2)}.
+$$

dots.mocr-4bit/arxiv_math/2503.03759_pg9_pg1_repeat0.md

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+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, non-negative 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
+
+$$
+H(Z) = \mathbb{E}[I(Z)] = \sum_z P(z)I(z) = -\sum_z P(z) \log P(z), \quad (10)
+$$
+
+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
+
+$$
+\log P(z) = \log |P(z)| + i\theta(P(z)), \quad (11)
+$$
+
+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:
+
+$$
+\begin{aligned} H(Z) &= -\sum_z P(z) \log P(z) \\ &= -\sum_z P(z) (\log |P(z)| + i\theta(P(z))) \\ &= -\sum_z P(z) \log |P(z)| - i \sum_z P(z) \theta(P(z)). \end{aligned} \quad (12)
+$$
+
+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.

dots.mocr-4bit/arxiv_math/2503.03762_pg1_pg1_repeat0.md

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+arXiv:2503.03762v1 [cs.IT] 24 Feb 2025
+
+DISPROVING SOME THEOREMS IN SHARMA AND CHAUHAN *et al.*
+(2018, 2021)*
+
+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 Solé
+
+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 $\ell \ge 1$, let $0 \ne \lambda_i \in \mathbb{F}_q$ and $m_i \ge 1$ for $1 \le i \le \ell$. If $\Lambda = (\lambda_1, \lambda_2, \dots, \lambda_\ell)$, then a $\Lambda$-MT code $\mathcal{C}$ 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
+
+$$
+T_{\Lambda} : (c_{1,0}, c_{1,1}, \dots, c_{1,m_1-1}; c_{2,0}, c_{2,1}, \dots, c_{2,m_2-1}; \dots; c_{\ell,0}, c_{\ell,1}, \dots, c_{\ell,m_\ell-1}) \mapsto \\ (\lambda_1 c_{1,m_1-1}, c_{1,0}, \dots, c_{1,m_1-2}; \lambda_2 c_{2,m_2-1}, c_{2,0}, \dots, c_{2,m_2-2}; \dots; \lambda_\ell c_{\ell,m_\ell-1}, c_{\ell,0}, \dots, c_{\ell,m_\ell-2}).
+$$
+
+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
+
+$$
+V = \bigoplus_{i=1}^{\ell} \frac{\mathbb{F}_q[x]}{\langle x^{m_i} - \lambda_i \rangle} = \frac{\mathbb{F}_q[x]}{\langle x^{m_1} - \lambda_1 \rangle} \oplus \frac{\mathbb{F}_q[x]}{\langle x^{m_2} - \lambda_2 \rangle} \oplus \dots \oplus \frac{\mathbb{F}_q[x]}{\langle x^{m_\ell} - \lambda_\ell \rangle}.
+$$
+
+*This research was conducted at Université d'Artois, La Faculté Jean Perrin in Lens, and was funded by the Science, Technology & Innovation Funding Authority (STDF); International Cooperation Grants, project number 49294. Ramy Takieldin would like to express his deepest gratitude to Professor André Leroy for his invaluable guidance throughout this project.

dots.mocr-4bit/arxiv_math/2503.03765_pg1_pg1_repeat0.md

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+arXiv:2503.03765v1 [cs.IT] 27 Feb 2025
+
+STABLE RECOVERY GUARANTEES FOR BLIND DECONVOLUTION UNDER RANDOM MASK ASSUMPTION
+
+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 $\boldsymbol{h}$ is convolved with multiple random modulations $\{\boldsymbol{d}_l\}_{l=1}^L$ (coded masks) of a signal $\boldsymbol{x}$, 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 $\boldsymbol{h}$ and $\boldsymbol{x}$, demonstrating its near-optimality up to a logarithmic factor. Additionally, we present a new recovery scheme that leverages sparsity constraints on $\boldsymbol{x}$. This approach significantly reduces the sampling complexity to the order of $L = O(\log n)$ when the non-zero elements of $\boldsymbol{x}$ 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 $\boldsymbol{y} = \boldsymbol{h} \circledast \boldsymbol{x} = \boldsymbol{C}_\boldsymbol{h} \boldsymbol{x}$, where $\boldsymbol{C}_\boldsymbol{h}$ is the circulant matrix generated by $\boldsymbol{h} = [h_1, \cdots, h_n]^T$, defined as:
+
+$$
+\boldsymbol{C}_\boldsymbol{h} = \begin{bmatrix} h_1 & h_n & \cdots & h_2 \\ h_2 & h_1 & \cdots & h_3 \\ \vdots & \vdots & \ddots & \vdots \\ h_n & h_{n-1} & \cdots & h_1 \end{bmatrix}.
+$$
+
+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 $\boldsymbol{y}_l \in \mathbb{C}^n$,
+
+2020 Mathematics Subject Classification. Primary 94A15, 46C05; Secondary 94A12, 49N45.
+Key words and phrases. Self calibration, General random mask, Optimal complexity, Alternating minimization.
+Song Li is supported by NSFC grant (U21A20426, 12071426).
+Yu Xia was supported by NSFC grant (12271133, 11901143) and the key project of Zhejiang Provincial Natural Science Foundation grant (LZ23A010002).

dots.mocr-4bit/arxiv_math/2503.03766_pg12_pg1_repeat0.md

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+Now, we prove that $\Psi_c \setminus \{(0, m) : m \ge c\} \subset \Psi_c^*$. For any $0 < p \le 1$, consider any $(p, m) \in \Psi_c$ and the following probability mass function for a random variable $T$:
+
+$$
+\Pr\{T = 0\} = 1 - p \quad \text{and} \quad \Pr\{T = m/p\} = p.
+$$
+
+Since $m \ge cp$, we have $m/p \ge c$, and so
+
+$$
+\Pr\{T \ge c\} = \Pr\{T = m/p\} = p,
+$$
+
+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 $0 \le m < c$ is achievable. This can be done by noting that such an ordered pair can be achieved by any random variable $T$ with $\Pr\{T = m\} = 1$. The theorem is proved. $\square$
+
+The region $\Psi_c$ is defined by Markov's inequality together with the constraint $0 \le p \le 1$ 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 $\{(0, m) : m \ge c\}$) is achievable by some random variable $T$. Specifically:
+
+* When $\Pr\{T \ge c\} > 0$, Markov's inequality, namely
+
+$$
+E[T] \ge c \cdot \Pr\{T \ge c\}, \qquad (10)
+$$
+
+which gives a lower bound on $E[T]$, is the only constraint on $E[T]$ in terms of $\Pr\{T \ge c\}$.
+
+* When $\Pr\{T \ge c\} = 0$, Markov's inequality as in (10), which becomes $E[T] \ge 0$, continues to be valid. However, we also have
+
+$$
+E[T] < c. \qquad (11)
+$$
+
+Combining (10) and (11), we have
+
+$$
+0 \le E[T] < c.
+$$

dots.mocr-4bit/arxiv_math/2503.03772_pg1_pg1_repeat0.md

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+arXiv:2503.03772v1 [math.GR] 4 Mar 2025
+
+Cardinalities in finite monoids of $G$-equivariant functions
+
+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 $\cdot : G \times X \to X$ 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 $\mathrm{End}_G(X)$, and its group of units, consisting of all bijective $G$-equivariant transformations, as $\mathrm{Aut}_G(X)$. 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
+
+*Email: harath.ruiz@academicos.udg.mx

dots.mocr-4bit/arxiv_math/2503.03827_pg10_pg1_repeat0.md

@@ -0,0 +1,85 @@
+This code was first proposed in Ref. [18] using the polynomials
+
+$$
+\begin{aligned}
+f(x, y) &= x^{25} + x^{26} + y^3, \\
+g(x, y) &= y + y^2 + x^9,
+\end{aligned} 
+\qquad (69)
+$$
+
+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:
+
+$$
+\begin{aligned}
+f(x, y) &= x + x^2 + y^3, \\
+g(x, y) &= y + y^2 + x^3,
+\end{aligned} 
+\qquad (70)
+$$
+
+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 $kd^2/n = 14.11$. 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:
+
+$$
+\begin{aligned}
+f(x, y) &= 1 + x + x^{-1}y^{-3}, \\
+g(x, y) &= 1 + y + y^{-6}.
+\end{aligned} 
+\qquad (71)
+$$
+
+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 $xy^{25} - 1$ 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:
+
+$$
+\begin{aligned}
+f(y) &= 1 + y^{22} + y^{102}, \\
+g(y) &= 1 + y + y^{121},
+\end{aligned} 
+\qquad (72)
+$$
+
+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:
+
+$$
+\begin{aligned}
+k &= 2 \dim \left( \frac{\mathbb{Z}_2[y, y^{-1}]}{\langle f(y), g(y), y^l - 1 \rangle} \right) \\
+&= 2 \deg \left( \gcd(f(y), g(y), y^l - 1) \right),
+\end{aligned} 
+\qquad (73)
+$$
+
+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:
+
+$$
+\vec{a}_1 = (0, \frac{n}{2}), \quad \vec{a}_2 = (1, \gamma), \quad \text{with } 0 \le \gamma < \frac{n}{2}, \qquad (74)
+$$
+
+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:
+
+$$
+\begin{aligned}
+f(y) &= 1 + y^{15} + y^{20} + y^{28} + y^{66}, \\
+f(y) &= 1 + y^{58} + y^{59} + y^{100} + y^{121},
+\end{aligned} 
+\qquad (75)
+$$
+
+defined on a cycle of length $l = 127$. This GB code uses weight-10 stabilizers to achieve better code parameters [[254, 28, 14 $\le d \le 20$]].
+
+## 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 $n \le 400$ 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 $n \le 500$ 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

dots.mocr-4bit/arxiv_math/2503.03847_pg30_pg1_repeat0.md

@@ -0,0 +1,11 @@
+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, $N_c$ and $N_w$ denote the number of buckled cell walls and the total number of cell walls, respectively. The fractions $N_c/N_w$ 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 $N_c/N_w$, compared with H100.
+
+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.
+
+For the sake of reference, the experimental compressive properties of H100 [97] and H200 [98], are provided in Figure 18 (leftmost bars). $\hat{\nu}_{13}^*$ are not measured and $\hat{\nu}_{31} = \hat{\nu}_{32}$ 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 well-defined 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 ~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.

dots.mocr-4bit/arxiv_math/2503.03855_pg5_pg1_repeat0.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
+
+$$
+T_0 := \{t \in T \mid \forall \chi \in X^*(T), \omega(\chi(t)) = 0\}
+$$
+
+and
+
+$$
+T_r := \{t \in T_0 \mid \forall \chi \in X^*(T), \omega(\chi(t) - 1) \ge r\}
+$$
+
+for all $r \in \mathbb{R}$.
+
+2.1. **The apartment.** Let $N \le G$ be generated by $T$ and $\zeta_\alpha \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ 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
+
+$$
+a = \alpha + r := \{x \in \mathbb{A} \mid \alpha(x) + r \ge 0\}
+$$
+
+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 $\Sigma \to \Phi : a \mapsto \hat{a} = \alpha$. 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 $\Pi$ for our root system, there is a canonical alcove $C$ given by
+
+$$
+C := \{x \in \mathbb{A} \mid \text{for } 1 \le i \le d, \alpha_i(x) \ge 0 \text{ and } \alpha_0(x) \le 1\}.
+$$
+
+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 $j \ne i$, $0$ and $\alpha_0(v_i) = 1$. We now expand $\alpha_0$ in the basis given by $\Pi$ and find that
+
+$$
+\alpha_0 = \sum_{i=1}^{d} c_i \alpha_i
+$$
+
+for some positive integers $c_i$. We define $\omega_i := c_i v_i$ 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 coweights*.
+
+Associated to $C$ there is a *fundamental Weyl chamber* $C^+ := \mathbb{R}_{\ge 0} \cdot C$. 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 $\lambda : \mathbb{A}_0 \to \{0, \dots, d\}$

dots.mocr-4bit/arxiv_math/2503.03861_pg30_pg1_repeat0.md

@@ -0,0 +1,23 @@
+Since $H^2(X^{\text{an}}, \mathcal{O}_{X^{\text{an}}})$ 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 $H^1(X, \mathcal{O}_{X^{\text{an}}})$ is identified with the Picard group, to prove the desired injection, we only need to show $H^1(X^{\text{an}}, \mathcal{O}_{X^{\text{an}}}) = 0$. Using GAGA for Deligne-Mumford stacks, [Hal11, Proposition A.4], we have $H^1(X^{\text{an}}, \mathcal{O}_{X^{\text{an}}}) = H^1(X, \mathcal{O}_X)$. 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. $\square$
+
+### 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 $[[\mathrm{CHur}_{\mathbb{P}^1,n}^{G,c} / G] / \mathrm{PGL}_2]$. 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 $G^{\text{ab}}$ (under the map $\mathbb{Z} \to G^{\text{ab}}$ sending the positive generator to the image of any element of $c$) agrees with the image of $g$ in $G^{\text{ab}}$.
+
+Rephrasing the statement above, there are $H_2(G, c)$ many components if the image of $n$ in $G^{\text{ab}}$ 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 $S_c \to G$ having the same image in $G^{\text{ab}}$ as $n$. Moreover, the boundary monodromy of these components is the same as their image in $G$ under the map $S_c \to G$. The kernel of $S_c \to G$ 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 $G^{\text{ab}}$ agrees with the image of $g$, or else there are no such connected components. $\square$
+
+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 $\tilde{Z} \subset [\mathrm{CHur}_{\mathbb{P}^1,n}^{G,c} / G]$, with corresponding component $Z \subset [[\mathrm{CHur}_{\mathbb{P}^1,n}^{G,c} / G] / \mathrm{PGL}_2]$, we have
+
+$$
+\begin{aligned}
+H^1(\tilde{Z}; R) &= H^1(Z; R) = 0, \\
+H^2(\tilde{Z}; R) &= H^2(Z; R) = ((\mathbb{Z}/(2n-2)\mathbb{Z}) \otimes R).
+\end{aligned}
+$$
+
+*Proof.* Taking $g = \mathrm{id}$ in Lemma 7.3.1, we obtain that for $n$ sufficiently large, both $[\mathrm{CHur}_n^{G,c,\partial \in \mathrm{id}} / G]$ and $[\mathrm{CHur}_{\mathbb{P}^1,n}^{G,c} / G]$ have $|H_2(G, c)|$ many connected components. Indeed, the statement for $[\mathrm{CHur}_n^{G,c,\partial \in \mathrm{id}} / G]$ 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 $S_c \to G$ by definition. Since $[\mathrm{CHur}_n^{G,c,\partial \in \mathrm{id}} / G]$ 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

dots.mocr-4bit/arxiv_math/2503.03873_pg5_pg1_repeat0.md

@@ -0,0 +1,35 @@
+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
+
+$$
+H^{\perp} := \{ \chi \in \widehat{A} : \chi(x) = 1 \quad \forall x \in H \},
+$$
+
+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
+
+$$
+\int_H \phi(xh)dh = \int_{H^{\perp}} \mathcal{F}_A(\phi)(\chi)\chi(x)d\chi, \quad (2.3)
+$$
+
+for all $x \in A$, where Haar measure on $H^{\perp} \cong \widehat{A/H}$ 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
+
+$$
+\begin{align*} \mathcal{F}_{A/H}(\phi^H)(\chi) &= \int_{A/H} \phi^H(xH)\overline{\chi(x)}d(xH) \\ &= \int_{A/H} \int_H \phi(xh)\overline{\chi(xh)}dhd(xH) \\ &= \int_A \phi(x)\overline{\chi(x)}dx = \mathcal{F}_A(\phi)(\chi) \end{align*}
+$$
+
+for every $\chi \in H^{\perp}$. Moreover, if $\mathcal{F}_A(\phi)|_{H^{\perp}} \in L^1(H^{\perp}) = L^1(\widehat{A/H})$, then the Fourier inversion formula implies that for all $x \in A$
+
+$$
+\begin{align*} \int_H \phi(xh)dh &= \phi^H(xH) = \mathcal{F}_{H^{\perp}} \mathcal{F}_{A/H}(\phi^H)(-xH) \\ &= \mathcal{F}_{H^{\perp}} \mathcal{F}_A(\phi)(-xH) = \int_{H^{\perp}} \mathcal{F}_A(\phi)(\chi) \overline{-xH(\chi)} d\chi \\ &= \int_{H^{\perp}} \mathcal{F}_A(\phi)(\chi) \overline{\chi(-x)} d\chi = \int_{H^{\perp}} \mathcal{F}_A(\phi)(\chi) \chi(x) d\chi. \end{align*}
+$$
+
+□
+
+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
+
+$$
+\begin{align*} A &\cong \widehat{A} \\ (s, t) &\mapsto \chi_{s,t}(l, \xi) = e^{2\pi i \left( \frac{Q(s,l)}{N} + Q(t,\xi) \right)}, \end{align*}
+$$

dots.mocr-4bit/arxiv_math/2503.03879_pg4_pg1_repeat0.md

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+Fig. 2. Location of switch point at the end of finite element
+
+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:
+
+$$
+\hat{\lambda}_l = \sum_{k=1}^{K} \lambda_{l,k}, \quad \hat{\nu}_l = \sum_{k=1}^{K} \nu_{l,k} \quad (10a)
+$$
+
+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).
+
+$$
+\pi_l^\lambda = \hat{\lambda}_{l-1} \odot \hat{\lambda}_l, \quad \pi_l^\nu = \hat{\nu}_{l-1} \odot \hat{\nu}_l \quad (10b)
+$$
+
+(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
+
+$$
+\tau_l = \pi_l^\lambda + \pi_l^\nu, \quad \eta_l = \prod_{j=1}^{n_f} \tau_{l,j} \quad (10c)
+$$
+
+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 $l = 1, \dots, N-1$
+
+$$
+0 \le (\Delta h_l^+ + \Delta h_l^-) \perp \eta_l \ge 0 \quad (10d)
+$$
+
+where $h_{l-1} - h_l = \Delta h_l^+ - \Delta h_l^-$, $\Delta h_l^+, \Delta h_l^- \ge 0$. 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:
+
+$$
+\chi_s = \{l|[t_{l-1}, t_l], \lambda_{l,K} = \nu_{l,K} = 0, \lambda_{l-1,K} + \nu_{l-1,K} > 0 \text{ or } \lambda_{l+1,K} + \nu_{l+1,K} > 0\} \quad (11a)
+$$
+
+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.
+
+$$
+h_l - h_{l+1} = 0 \quad \forall l \in \{1, \dots, N-1\} \setminus \chi_s \quad (11b)
+$$
+
+Also, we add constraints to force the switching to happen at the boundary of finite elements found in the first step.
+
+$$
+\lambda_{l,K} + \nu_{l,K} = 0 \quad \forall l \in \chi_s \quad (11c)
+$$
+
+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:
+
+$$
+\min \varphi(x) \quad (12a)
+$$
+
+$$
+\text{s.t. } g_I(x) \le 0; \quad g_E(x) = 0, \quad (12b)
+$$
+
+$$
+0 \le G(x) \perp H(x) \ge 0 \quad (12c)
+$$
+
+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
+
+$$
+G(x) \ge 0, \quad H(x) \ge 0, \quad G_i(x)H_i(x) = 0, \quad \forall i = 1 \dots n_c
+$$
+
+### 3.1 MPCC Basics and Stationary Points
+
+The following index sets are defined at every feasible point $\bar{x}$ of the MPCC (12):

dots.mocr-4bit/arxiv_math/2503.03899_pg9_pg1_repeat0.md

@@ -0,0 +1,52 @@
+**Proposition A.1.** For fixed $0 < \delta < \frac{1}{4}$, we have
+
+$$
+\limsup_{n \to \infty} n^{-\delta} \log P_n \left( \inf_{t \ge 0} \left| \frac{1}{\sqrt{n}} \sum_{k \le t \sqrt{n}} X_k - L(t) \right| > n^{-\frac{1}{4} + \delta} \right) < 0.
+$$
+
+**Remark.** In particular, we have
+
+$$
+\lim_{n \to \infty} P_n \left( \sup_{t \ge 0} \left| \frac{1}{\sqrt{n}} \sum_{k \le t \sqrt{n}} X_k - L(t) \right| \le n^{-\frac{1}{4} + \delta} \right) = 1.
+$$
+
+*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 $\alpha_n, \beta_n \ge n^{-\frac{1}{4}+\delta}$. We use the saddle point bound to write, for any $x_n \in \mathbb{R}$,
+
+$$
+\begin{align*}
+P_n \left( \frac{1}{\sqrt{n}} \sum_{k \le a_n} X_k = b_n \right) \\
+&= \frac{1}{d(n)} [q^n] [\zeta^{b_n}] \prod_{k \le a_n} (1 + \zeta q^k) \prod_{k > a_n} (1 + q^k) \\
+&\le \frac{1}{d(n)} q_n^{-n} e^{-b_n x_n} \prod_{k \le a_n} (1 + e^{x_n} q_n^k) \prod_{k > a_n} (1 + q_n^k) \\
+&= \exp \left( \frac{\sqrt{n}}{A} - \log d(n) - \beta_n x_n \sqrt{n} + \sum_{k \le a_n} \log \left( 1 + e^{x_n - \frac{k}{A \sqrt{n}}} \right) + \sum_{k > a_n} \log \left( 1 + e^{-\frac{k}{A \sqrt{n}}} \right) \right).
+\end{align*}
+$$
+
+We will take $x_n \in \{\pm n^{-\frac{1}{4}}\}$, where the sign will depend on $b_n$. By Taylor's Theorem,
+
+$$
+\begin{align*}
+\left| \log \left( 1 + e^{x_n - \frac{k}{A \sqrt{n}}} \right) - \log \left( 1 + e^{-\frac{k}{A \sqrt{n}}} \right) - x_n \frac{e^{-\frac{k}{A \sqrt{n}}} }{1 + e^{-\frac{k}{A \sqrt{n}}} } \right| &\le \frac{x_n^2}{2} \sup_{|x| \le n^{\frac{1}{4}}} \frac{e^{x - \frac{k}{A \sqrt{n}}}}{\left(1 + e^{x - \frac{k}{A \sqrt{n}}} \right)^2} \\
+&\le x_n^2 \frac{e^{-\frac{k}{A \sqrt{n}}}}{\left(1 + \frac{1}{2} e^{-\frac{k}{A \sqrt{n}}} \right)^2}.
+\end{align*}
+$$
+
+Thus,
+
+$$
+\begin{align*}
+P_n \left( \frac{1}{\sqrt{n}} \sum_{k \le a_n} X_k = b_n \right) \le \exp \left( \frac{\sqrt{n}}{A} - \log d(n) + \sum_{k \ge 1} \log \left( 1 + e^{-\frac{k}{A \sqrt{n}}} \right) - \beta_n x_n \sqrt{n} \right. \\
+\left. + x_n \sum_{k \le a_n} \frac{e^{-\frac{k}{A \sqrt{n}}} }{1 + e^{-\frac{k}{A \sqrt{n}}} } + O \left( x_n^2 \sum_{k \le a_n} \frac{e^{-\frac{k}{A \sqrt{n}}} }{\left(1 + \frac{1}{2} e^{-\frac{k}{A \sqrt{n}}} \right)^2} \right) \right).
+\end{align*}
+$$

dots.mocr-4bit/arxiv_math/2503.03903_pg9_pg1_repeat0.md

@@ -0,0 +1,21 @@
+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.
+
+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 $i$ be 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, $L(i + 1) \ge L(j) + (i - j) + 1 > 0$. 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. $\square$
+
+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 $\mathfrak{S}_w = \prod_i h_{L(i)}^i$. $\square$
+
+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 $i, j, k = i, i + 1, i + 2$.

dots.mocr-4bit/arxiv_math/2503.03905_pg7_pg1_repeat0.md

@@ -0,0 +1,44 @@
+We show that $\sigma'$ is bijective on $\text{GF}(2^m)$. Let us fix $u, y_1, y_2 \in \text{GF}(2^m)$, $y_1 \neq y_2$. There is an element $v \in \text{GF}(2^t)$ such that
+
+$$
+\text{Tr}_{\text{GF}(2^t)/\text{GF}(2)}(uv) \neq h'(y_1) + h'(y_2).
+$$
+
+As $\tau$ is surjective, there is a $z \in \text{GF}(2^m)$ with $\tau(z) = v$. By Lemma 7, there is $x \in \text{GF}(2^m)$ with $y_1 \star x + y_2 \star x = z$. Then
+
+$$
+\begin{align*}
+\alpha_{y_1}(x) + \alpha_{y_2}(x) &= \text{Tr}_{\text{GF}(2^t)/\text{GF}(2)}(u\tau(y_1 \star x + y_2 \star x) + uh(y_1) + uh(y_2)) \\
+&= \text{Tr}_{\text{GF}(2^t)/\text{GF}(2)}(u\tau(z)) + h'(y_1) + h'(y_2) \\
+&= \text{Tr}_{\text{GF}(2^t)/\text{GF}(2)}(uv) + h'(y_1) + h'(y_2) \\
+&\neq 0.
+\end{align*}
+$$
+
+This implies $\alpha_{y_1}(x) \neq \alpha_{y_2}(x)$ and $\sigma'(y_1) \neq \sigma'(y_2)$. It follows that all component functions of $f$ are of Maiorana-McFarland type bent function. This finishes the proof of the theorem. $\square$
+
+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
+
+$$
+W_f(a, b) = \begin{cases} \pm 2^{n/2} & \text{if } b \neq 0, \\ 0 & \text{if } a \neq 0, b = 0, \\ 2^n & \text{if } a = 0, b = 0. \end{cases}
+$$
+
+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 $(x, y) \in \text{GF}(2^m)^2$ 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, $d_H(f_i, \mathcal{A}) \le (1-2^{-t})(2^{2m} - 2^m)$, and the theorem follows from the Carlet-Ding-Yuan bound.
+
+Let $T$ be the set of elements $z \in \text{GF}(2^m)$ 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
+
+$$
+|E_1| = |f_1^{-1}(f_1(0, 0))| = 2 \cdot 2^m - 1 + (2^{m-t} - 1)(2^m - 1).
+$$
+
+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 $t \neq 0$. $\square$

dots.mocr-4bit/arxiv_math/2503.03909_pg14_pg1_repeat0.md

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+### 4.2.2. Bratu problem.
+We use lrAA to solve the non-linear Bratu problem
+
+$$
+u_{xx} + u_{yy} + \lambda e^u = 0, \quad (x, y) \in [0, 1] \times [0, 1],
+$$
+
+with $\lambda = 1$ and homogeneous Dirichlet boundary conditions. To find the approximate solution we discretize this equation using standard second order finite difference approximations for the $x$ and $y$ derivatives. Given an approximation $X(i, j) \approx u(x_i, y_j)$ this results in a function $G_B(i, j; X)$ describing the equation
+
+$$
+(4.2) \qquad \begin{aligned} G_B(i, j; X) = \frac{1}{h_x^2} (X(i+1, j) - 2X(i, j) + X(i-1, j)) \\ + \frac{1}{h_x^2} (X(i, j+1) - 2X(i, j) + X(i, j-1)) + \lambda e^{X(i,j)}. \end{aligned}
+$$
+
+Near the boundaries some of the terms in this expression will be set to zero to account for the homogeneous Dirichlet boundary conditions. In the numerical examples below we take the mesh to be $x_i = ih_x$, $h_x = \frac{1}{m+1}$ and $y_j = jh_y$, $h_y = \frac{1}{n+1}$ with $m = n = 200$, making the setup the same as in [52].
+
+The fixed point function $G(i, j)$ is obtained by applying the preconditioned Richardson iteration. We have
+
+$$
+X^{k+1}(i, j) = G(i, j; X^k, \alpha) \equiv X^k(i, j) + \alpha M(G_B(i, j; X^k)).
+$$
+
+We test lrAA with no preconditioner and with the ES preconditioner (corresponding to `Rel1_x_n10.1E10`) described above. The lrAA parameters used are the following TOL = $10^{-6}$, $\hat{m} = 5$, $\theta = 0.9$, $\alpha = 0.125h_x^2$ (un-preconditioned case) and $\alpha = 0.1$ (preconditioned case, no scheduling is used). In all experiments we use a rank 1 matrix with Frobenius norm around 1.
+
+The results in Figure 11 display the numerical solutions obtained by lrAA methods with and without the ES preconditioner. In particular, both methods obtain visually similar numerical results in terms of solution contours and column and row index section for the final iterates. The un-preconditioned lrAA gives a monotonically increasing intermediate ranks. The ES-preconditioned lrAA converges very rapidly in 8 iterations.
+
+FIG. 11. Bratu problem solved by lrAA. The top left figure displays the contour levels of the converged solution along with markers at the intersection points of the final index sets $\mathcal{I}$ and $\mathcal{J}$ (for both lrAA with and without ES preconditioner). The top right figure displays the singular values from lrAA solutions with and without the ES preconditioner. The bottom left figure and right figures display the rank evolution and the decay of the residual throughout the lrAA iterations for lrAA with and without the ES preconditioner.

dots.mocr-4bit/arxiv_math/2503.03948_pg3_pg1_repeat0.md

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+FIG. 1. (a) Bulk band structure for the $C_2T$-symmetric TBG model. Inset diagrams this lattice model. (b) The local gap $\mu_{(x,y,E)}$ for $x$ and $E$ at $y = 0$. Position $x$ is scaled in terms of the lattice constant $a$ and $x_m$ denotes the length of the edge from the origin. (c) $\mu_{(x,y,E)}$ for $E$ at the center of rotation denoted by blue dotted line in panel (b). The energy-resolved invariant $\zeta_E$ is shown by the green dots, where $\kappa = 0.1t/a$ for all calculations.
+
+Specifically, when perturbing the system $H \to H + \delta H$, $\zeta_E$ is guaranteed by Weyl's inequality to be preserved so long as $\|\delta H\| < \mu_{(0,0,E)}(X, Y, H)$ [60, 61]. In addition, as locations where $\mu_{(x,y,E)} = 0$ are associated with the locations of a system's states [62], changes in a system's topological marker necessarily imply changes in the structure of its states. A detailed mathematical discussion of Eq. (9), its essential properties, and its relation to atomic limits is given in Supplementary Secs. SI.E-G. In particular, Examples SI.11 and SI.12 show the form of the two different classes of atomic limits distinguished by $\zeta_E$.
+
+Summarizing our derivation, we first used a system's physical symmetries to define a transpose-like matrix operation that translates the physical symmetries to matrix symmetries of the system's $H$, $X$, and $Y$. Then, we found a change of basis that transformed this matrix operation into the standard matrix transpose, such that in this atypical basis the system's operators were either symmetric or skew-symmetric. Finally, by tensoring these operators using the Pauli matrices, we formed a single skew-symmetric matrix whose Pfaffian's sign discriminates between which atomic limits a given system can be connected to without closing the system's local gap. Altogether, by using results from matrix homotopy, this argument yields an energy-resolved invariant for classifying fragile topology as well as a quantitative measure of topological protection.
+
+Having derived a classification framework applicable to finite systems, we demonstrate its use in a four-band model that is a low-energy approximation of TBG and exhibits fragile topology [18, 21]. This model con-
+
+sists of a bilayer honeycomb lattice, as schematically shown in Fig. 1(a), where $t_1$ and $t_2$ represent the intra- and inter-layer hopping amplitudes, respectively. The blue lines spirally connecting inter-layer sites represent next-nearest neighbor (NNN) hoppings with the hopping phase $\pm\phi$, which can induce a nontrivial fragile band gap. We have provided the expressions of the Hamiltonian in both position and momentum space the End Matter and further information in Supplementary Sec. SII.
+
+Comparison of the bulk band structure of the infinite four-band TBG model with the local gap of a finite system confirms that the locations in $(x, y, E)$-space with $\mu_{(x,y,E)} \approx 0$ indicate the presence of states at the specified energies and positions, see Figs. 1(a), (b). For choices of $E$ residing within the spectral extent of the bulk bands, extended Bloch states are distributed throughout the system, whereas within the bulk band gap, only localized states exist at the system's boundaries. Note that the fluctuation pattern of $\mu_{(x,0,E)}$ only intermittently touching zero near the band gap inherently suggests the weak topological nature of this fragile system; strong topological phases instead exhibit a spheroid of appropriate dimension where $\mu_{(x,E)} = 0$ (see Supplementary Sec. SI.G).
+
+Within the bulk band gap, the energy-resolved marker $\zeta_E$ confirms the finite system's fragile topology, while the large local gap at the rotation center indicates this phase's strong topological protection, see Fig. 1(c). For energies above and below the bulk gap, $\zeta_E$ maintains a nontrivial value of $-1$ until the first closing points of $\mu_{(0,0,E)}$, beyond which it switches to $+1$. Although the exact energy where $\mu_{(0,0,E)} = 0$ may vary with the parameter $\kappa$ (see Supplementary Sec. SIII), these spectral regions with small local gaps are not topologically robust, as very small system perturbations are able to change the topology.
+
+To confirm that the local fragile marker $\zeta_E$ captures phase transitions, we uniformly vary $\phi$ between all NNN sites from $-\pi$ to $\pi$. As can be seen in Fig. 2(a), the width of the bulk spectral gap under this alteration is symmetric about $\phi = 0$ and touches zero twice at $\phi = \pi/3$ and $\phi = 2\pi/3$. Similarly, the local gap closes at precisely the same points where $E_{\text{gap}} = 0$ and $\zeta_E$ changes across these values of $\phi$, indicating a change in the material's fragile topological phase.
+
+As our framework works directly with a finite system expressed in position-space, it can inherently be applied to disordered and aperiodic systems. To illustrate this capability, we consider an ensemble of disordered variants of the four-band TBG model where the hopping phases $\phi_{jk}$ between each pair of NNN sites $j$ and $k$ are randomly assigned a value within an angle range of $2S$ from $\pm i$ from a uniform distribution while preserving $C_2T$-symmetry. Therefore, $S$ represents the median value of the disorder strength, allowing us to investigate the phase diagram of the disordered system based on this variable. In Figs. 2(d), (e), and (f), we present $E_{\text{gap}}$, $\mu_{(0,0,E)}$, and $\zeta_E$ as functions of $S$ for 10 different disorder realizations. We numerically observe that the disordered samples exhibit

dots.mocr-4bit/arxiv_math/2503.03949_pg1_pg1_repeat0.md

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+arXiv:2503.03949v1 [math.AG] 5 Mar 2025
+
+Volumes in Calabi-Yau Complete Intersection of Products of Projective Space
+
+Yi-Heng Tsai
+
+**Abstract.** We prove that the birational automorphism group of a general Calabi-Yau complete intersection $X$ given by ample divisors in $\mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_l}$ is always Lorentzian. Applying the Kawamata-Morrison cone theorem on such $X$, we compute $\mathrm{vol}_X(D + sA)$ for any divisor $D \in \partial\overline{\mathrm{Eff}}(X)$ and ample divisor $A$ when $s$ is small. We also provide examples of volumes of certain Cartier divisors that involve the digamma function.
+
+# 1 Introduction
+
+It is conjectured in [Mor93] and [Kaw97] that the movable effective cone of a Calabi-Yau manifold has a rational fundamental domain under the action of the birational automorphism group.
+
+**Conjecture 1.0.1** (Kawamata-Morrison cone conjecture) Let $X$ be a Calabi-Yau manifold. Then there exists a rational polyhedral fundamental domain $\Pi$ for the action of the birational automorphism group $\mathrm{Bir}(X)$ on the movable effective cone $\overline{\mathrm{Mov}}^e(X)$ in the sense that
+
+$$
+(i) \overline{\mathrm{Mov}}^e(X) = \bigcup_{g \in \mathrm{Bir}(X)} g^* \Pi,
+$$
+
+$$
+(ii) \mathrm{int}(\Pi) \cap \mathrm{int}(g^* \Pi) = \emptyset \text{ if } g^* \neq \mathrm{id}.
+$$
+
+The conjecture has been proven in the case when $X$ is a general Wehler $N$-fold i.e. a general hypersurface of multidegree $(2, \cdots, 2)$ in $(\mathbb{P}^1)^{N+1}$ for $N \ge 3$ ([CO15, Theorem 1.3]). In [FLT23], the structure of the boundary of the pseudoeffective cone $\overline{\mathrm{Eff}}(X)$ has been studied. The divergent-recurrent decomposition of $\partial\overline{\mathrm{Eff}}(X)$ is proven in [FLT23, Theorem 2.4.2], and the following result on the asymptotic behavior of volume function near $\partial\overline{\mathrm{Eff}}(X)$ is derived:
+
+**Theorem 1.0.2** ([FLT23, Theorem 1.3.7]) Suppose $X$ is a general Wehler Calabi-Yau $N$-fold. Then, for every pseudoeffective $\mathbb{R}$-divisor $D$ in $\partial\overline{\mathrm{Eff}}(X)$ and sufficiently ample divisor $A$, there exists an integer $s(D) \in \{0, \cdots, N-2\}$ and a real number $\delta(D) \in [1, \frac{1}{2}(N-s(D))]$ such that
+
+$$
+\liminf_{s \downarrow 0} \frac{\log \mathrm{vol}(D + sA)}{\log s} = \delta(D),
+$$
+
+$$
+\limsup_{s \downarrow 0} \frac{\log \mathrm{vol}(D + sA)}{\log s} = \frac{N - s(D)}{2}.
+$$
+
+The fact that the birational automorphism group of a general Wehler $N$-fold is Lorentzian provides a hyperbolic subspace in $\mathbb{P}N^1(X)$, and plays an important role when investigating the structure of $\partial\overline{\mathrm{Eff}}(X)$ (see [FLT23, §2.3]). Let $\mathbf{n} = (n_1, \cdots, n_l) \in \mathbb{N}$ such that $|\mathbf{n}| \ge 4$ and $\mathbf{n} \ne (2, 2)$. In [Yá22], the author generalized the results in [CO15], and proved the Kawamata-Morrison cone conjecture when $X$ is a general Calabi-Yau complete intersection in $\mathbb{P}^\mathbf{n} := \mathbb{P}^{n_1} \times \cdots \times \mathbb{P}^{n_l}$ given by the intersection of $n$ ample divisors, with $n \le \min\{n_i\}$. He also conjectured that the

dots.mocr-4bit/arxiv_math/2503.03952_pg5_pg1_repeat0.md

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+Also taking the limit $u \to 0$, the single letter index is given by
+
+$$
+f(t, 0; q) = \frac{t - q}{1 - q}. \qquad (2.8)
+$$
+
+In this case, the corresponding index is known as the “half-index” (of Neumann boundary condition) [14]. Due to an obvious symmetry between $t$ and $u$, the limit $t \to 0$ is essentially same as $u \to 0$. However, in the analysis in the next subsection, these two limits look different, and lead to the equivalent result non-trivially.
+
+Finally, if we set $u = q/t$, the resulting index is known as the flavored Schur index. The single letter index is now given by
+
+$$
+f(t, q/t; q) = \frac{t + q/t - 2q}{1 - q}. \qquad (2.9)
+$$
+
+For the further specialization to $t = q^{1/2}$ (i.e., $u = q^{1/2}$), the index $I_N(q^{1/2}, q^{1/2}; q)$ is nothing but the original Schur index [3, 4].
+
+We stress that all of them are obtained from the index (2.10) as special limits. The reduced index $I_N(t, u; q)$ is regarded as a two-parameter deformation of the Schur index $I_N(q^{1/2}, q^{1/2}; q)$. As mentioned in the introductory section, we refer to $I_N(t, u; q)$ as the deformed Schur index.
+
+## 2.2 Exact evaluation of deformed Schur indices
+
+In this subsection, we evaluate the matrix integral of the deformed Schur index exactly. When $v = p = 0$, we can rewrite the integral representation (2.4) as a more convenient form in terms of the $q$-Pochhammer symbol:
+
+$$
+I_N(t, u; q) = \frac{1}{N!} \frac{(q; q)_\infty^N (tu; q)_\infty^N}{(t; q)_\infty^N (u; q)_\infty^N} \oint_{\mathbb{T}^N} \prod_{i=1}^N \frac{dx_i}{2\pi i x_i} \prod_{1 \le i \ne j \le N} \frac{(x_i/x_j; q)_\infty (tux_i/x_j; q)_\infty}{(tx_i/x_j; q)_\infty (ux_i/x_j; q)_\infty}. \quad (2.10)
+$$
+
+This is a starting point of our analysis. The $q$-Pochhammer symbol is defined by
+
+$$
+(x; q)_\infty = \prod_{k=0}^{\infty} (1 - xq^k), \quad (x; q)_n = \prod_{k=0}^{n-1} (1 - xq^k), \quad (x; q)_0 = 1, \qquad (2.11)
+$$
+
+and we have used an identity,
+
+$$
+(qx; q)_\infty = \frac{(x; q)_\infty}{1 - x}, \qquad (2.12)
+$$
+
+to derive (2.10). A method to perform the integral (2.10) is simple. The computation consists of three steps.
+
+In the first step, we recognize that the integrand of (2.10) includes a weight function of Macdonald polynomials of type $A_{N-1}$. In Appendix A, we review basics on the Macdonald polynomials of type A, based on [15, 16], for the reader's convenience. The weight function of the Macdonald polynomials of type $A_{N-1}$ is given by
+
+$$
+w(x) = \prod_{1 \le i \ne j \le N} \frac{(x_i/x_j; q)_\infty}{(tx_i/x_j; q)_\infty}. \qquad (2.13)
+$$

dots.mocr-4bit/arxiv_math/2503.03994_pg108_pg1_repeat0.md

@@ -0,0 +1,45 @@
+Note that one can choose any $\vec{\Theta}$ satisfying
+
+$$
+\max\{-1, -1 - v_p(x_1)\} \le v_p(\Theta_0) = 1 - v_p(\Theta_1) \le \min\{0, v_p(x_0)\}.
+$$
+
+We will divide $R(\vec{r}; \vec{k}')$ into following four areas, and in each area we will determine the mod-$p$ reduction:
+
+(i) $\vec{t} \in R_{\text{int}}(\vec{r}; \vec{k}')$: Note that we have either $v_p(\Theta_0) \notin \{0, v_p(x_0)\}$ or $v_p(\Theta_0) \notin \{-1, -1 - v_p(x_1)\}$, in this case. Consider two $\overline{S_F}$-submodules $\mathcal{M}' = \overline{S_F}(\overline{E_1^{(0)}}, \overline{E_2^{(1)}})$ and $\mathcal{M}'' = \overline{S_F}(\overline{E_2^{(0)}}, \overline{E_1^{(1)}})$ of $\mathcal{M}$. Then one can observe that
+
+○ if $v_p(\Theta_0) \notin \{0, v_p(x_0)\}$, then $\mathcal{M}'$ is a Breuil submodule of $\mathcal{M}$ such that $\mathcal{M}' \cong \mathcal{M}(1, 1; -\beta_0, \alpha_1)$ and $\mathcal{M}/\mathcal{M}' \cong \mathcal{M}(1, 1; \alpha_0, -\beta_1)$;
+
+○ if $v_p(\Theta_0) \notin \{-1, -1 - v_p(x_1)\}$, then $\mathcal{M}''$ is a Breuil submodule of $\mathcal{M}$ such that $\mathcal{M}'' \cong \mathcal{M}(1, 1; \alpha_0, -\beta_1)$ and $\mathcal{M}/\mathcal{M}'' \cong \mathcal{M}(1, 1; -\beta_0, \alpha_1)$.
+
+In particular, if $v_p(\Theta_0) \notin \{-1, 0, v_p(x_0), -1 - v_p(x_1)\}$, then $\mathcal{M} = \mathcal{M}' \oplus \mathcal{M}''$ as Breuil modules. Otherwise, It is not difficult to show that the short exact sequence determined by $\mathcal{M}' \hookrightarrow \mathcal{M}$ or $\mathcal{M}'' \hookrightarrow \mathcal{M}$ is non-split. So, $\overline{\rho}^{ss}|_{I_{\mathcal{Q}_{p^2}}} \cong \omega_2^{p+1} \oplus \omega_2^{p+1}$, by Lemma 2.3.6 (i). Furthermore, the monodromy type is given as follow.
+
+(a) If $v_p(\Theta_0) = 0$, then $t_0 \ge 0$, and the monodromy type is (1, 0).
+
+(b) If $v_p(\Theta_0) = -1$, then $t_1 \ge 0$, and the monodromy type is (0, 1).
+
+(c) Otherwise, the monodromy type is (0, 0).
+
+(ii) $t_1 = -1, t_0 \ge 0$: We have $v_p(\Theta_0) = 0 = -1 - v_p(x_1)$, and so $\frac{1}{\Theta_0}, \frac{x_1}{\Theta_1} \in \mathbf{F}^\times$ and $\frac{1}{\Theta_1} = 0$ in $\mathbf{F}$. In particular, the monodromy type is (1, 0). Put $\mathcal{M}' = \overline{S_F}(\overline{E_2^{(0)}}, \overline{E_1^{(1)}})$. It is easy to see that $\mathcal{M}' \cong \mathcal{M}(1, 2; \alpha_0, -px_1\beta_1)$ and $\mathcal{M}/\mathcal{M}' \cong \mathcal{M}(1, 0; -\beta_0, \frac{\alpha_1}{px_1})$, and so we conclude that $\overline{\rho}^{ss}|_{I_{\mathcal{Q}_{p^2}}} \cong \omega_2^p \oplus \omega_2^{p+2}$, by Lemma 2.3.6 (i), and $\overline{\rho}$ is non-split as the monodromy type is nonzero.
+
+(iii) $t_0 + t_1 = -1, -1 < t_0 < 0$: We have $v_p(\Theta_0) = v_p(x_0) = -1 - v_p(x_1) \notin \{-1, 0\}$, and so $\frac{x_0}{\Theta_0}, \frac{x_1}{\Theta_1} \in \mathbf{F}^\times$ and $\frac{1}{\Theta_0} = \frac{1}{\Theta_1} = 0$ in $\mathbf{F}$. In particular, the monodromy type is (0, 0). Put
+
+$$
+A := \begin{bmatrix} -\frac{x_1}{\Theta_1} \alpha_1 & \alpha_1 \\ -\beta_1 & 0 \end{bmatrix} \begin{bmatrix} -\frac{x_0}{\Theta_0} \alpha_0 & \alpha_0 \\ -\beta_0 & 0 \end{bmatrix} = \begin{bmatrix} \frac{x_0 x_1}{\Theta_0 \Theta_1} \alpha_0 \alpha_1 - \beta_0 \alpha_1 & -\frac{x_1}{\Theta_1} \alpha_0 \alpha_1 \\ \frac{x_0}{\Theta_0} \alpha_0 \beta_1 & -\alpha_0 \beta_1 \end{bmatrix}.
+$$
+
+Under the base change
+
+$$
+\underline{E}'^{(1)} = \underline{E}^{(1)} \begin{bmatrix} 0 & -\frac{1}{\beta_1} \\ \frac{1}{\alpha_1} & -\frac{x_1}{\beta_1 \Theta_1} \end{bmatrix}, \underline{F}'^{(0)} = \underline{F}^{(0)} \begin{bmatrix} 1 & 0 \\ -\frac{x_0}{\Theta_0} & 1 \end{bmatrix}, \underline{F}'^{(1)} = \underline{F}^{(1)} \begin{bmatrix} 0 & -\frac{1}{\beta_1} \\ \frac{1}{\alpha_1} & 0 \end{bmatrix},
+$$
+
+we have
+
+○ $\mathrm{Mat}_{\underline{E}^{(0)}, \underline{F}'^{(0)}}(\mathrm{Fil}^r \mathcal{M}^{(0)}) = \mathrm{Mat}_{\underline{E}'^{(1)}, \underline{F}'^{(1)}}(\mathrm{Fil}^r \mathcal{M}^{(1)}) = uI_2$;
+
+○ $\mathrm{Mat}_{\underline{E}'^{(1)}, \underline{F}'^{(0)}}(\phi_2^{(0)}) = A \quad \& \quad \mathrm{Mat}_{\underline{E}^{(0)}, \underline{F}'^{(1)}}(\phi_2^{(1)}) = I_2$.
+
+Note that we have $\frac{\alpha_0}{\beta_0} = \frac{\alpha_1}{\beta_1} = p\Theta_0\Theta_1$ and $\alpha_0\alpha_1 = \lambda^2\Theta_0\Theta_1$, and so one can describe the mod-$p$ reduction as follow:
+
+○ if $px_0x_1 \ne 4$ in $\mathbf{F}$, then $A$ is diagonalizable. So, extending $\mathbf{F}$ large enough, we have $\overline{\rho}|_{I_{\mathcal{Q}_{p^2}}} \cong \omega_2^{p+1} \oplus \omega_2^{p+1}$ by Lemma 2.3.6 (i).

dots.mocr-4bit/arxiv_math/2503.04024_pg4_pg1_repeat0.md

@@ -0,0 +1,43 @@
+we introduce the PG-VarMiON emulating the optimal Petrov-Galerkin formulation, describe the training procedure, and present an analysis of the generalization error. Numerical results are presented in Section 4 for the diffusion and advection-diffusion equations in one dimension, and the advection-diffusion problem in two dimensions. We end with concluding remarks in Section 5.
+
+## 2 Problem Formulation
+
+Let $\Omega \in \mathbb{R}^d$ be an open, bounded domain with piecewise smooth boundary $\Gamma$. The boundary is further split into the Dirichlet boundary $\Gamma_D$ and natural boundary $\Gamma_\eta$, with $\Gamma = \Gamma_D \cup \Gamma_\eta$. Define the space $H_D^r(\Omega) = \{u \in H^r(\Omega) : u|_{\Gamma_D} = 0\}$. We consider the following scalar elliptic boundary value problem
+
+$$
+\begin{align}
+\mathcal{L}(u(\mathbf{x}); \mathbf{g}(\mathbf{x})) &= f(\mathbf{x}) && \forall \mathbf{x} \in \Omega, \nonumber \\
+\mathcal{B}(u(\mathbf{x}); \mathbf{g}(\mathbf{x})) &= \eta(\mathbf{x}) && \forall \mathbf{x} \in \Gamma_\eta, \tag{2.1} \\
+u(\mathbf{x}) &= 0 && \forall \mathbf{x} \in \Gamma_D, \nonumber
+\end{align}
+$$
+
+where $\mathcal{L}$ is a linear elliptic PDE operator and $\mathcal{B}$ is the natural boundary operator, both parametrized by a set of functions $\mathbf{g} \in \mathcal{G}$. Also, $f \in \mathcal{F} \subseteq L^2(\Omega)$ is the source term, and $\eta \in \mathcal{H} \subseteq L^2(\Gamma_\eta)$. The solution $u \in \mathcal{V} := H_D^r(\Omega)$, where $r$ depends on the order of the operator $\mathcal{L}$.
+
+A particular example of (2.1) is the steady advection-diffusion equation with
+
+$$
+\begin{align}
+-\nabla \cdot (\kappa(\mathbf{x})\nabla u(\mathbf{x})) + \mathbf{c}(\mathbf{x}) \cdot \nabla u(\mathbf{x}) &= f(\mathbf{x}) && \forall \mathbf{x} \in \Omega, \nonumber \\
+\kappa(\mathbf{x})\nabla u(\mathbf{x}) \cdot \mathbf{n} &= \eta(\mathbf{x}) && \forall \mathbf{x} \in \Gamma_\eta, \tag{2.2} \\
+u(\mathbf{x}) &= 0 && \forall \mathbf{x} \in \Gamma_D, \nonumber
+\end{align}
+$$
+
+where $\mathcal{V} = H_D^1(\Omega)$, $\mathbf{n}$ is the unit outward normal on $\Gamma_\eta$ and the set of parametrizing functions are $\mathbf{g} = [\kappa, \mathbf{c}]$. Here $\kappa \in L^\infty(\Omega) \cup \{\kappa \mid \kappa(\mathbf{x}) \ge \kappa_{\min} \text{ a.e. } \mathbf{x} \in \Omega\}$ for some (fixed) scalar $\kappa_{\min} > 0$ is the diffusion coefficient, while $\mathbf{c} \in H_{\text{div}}^1(\Omega) = \{\mathbf{c} \in [L^2(\Omega)]^2 \mid \nabla \cdot \mathbf{c} \in L^2(\Omega)\}$ is the velocity field. We will use (2.2) as a canonical example for the numerical results in Section 4.
+
+### 2.1 Variational form and symmetrization
+
+The variational formulation of (2.1) is given by: find $u \in \mathcal{V}$ such that
+
+$$
+a(u, w; \mathbf{g}) = (f, w) + (\eta, w)_{\Gamma_{\eta}} \quad \forall w \in \mathcal{V}, \tag{2.3}
+$$
+
+where $(\cdot, \cdot)$ is the $L^2(\Omega)$ inner-product, $(\cdot, \cdot)_{\Gamma_\eta}$ is the $L^2(\Gamma_\eta)$ inner-product, while $a(u, w; \mathbf{g})$ is the associated bilinear form parameterized by $\mathbf{g}$. We also assume that the bilinear form is coercive, which requires additional conditions on $\mathbf{g}$. With this assumption, a unique solution of (2.3) exists, as guaranteed by the Lax-Milgram theorem [10, 9].
+
+For the particular case of the advection-diffusion equation, we have
+
+$$
+a(u, w; \kappa, \mathbf{c}) := (\kappa \nabla u, \nabla w) + (\mathbf{c} \cdot \nabla u, w) \tag{2.4}
+$$

dots.mocr-4bit/arxiv_math/2503.04026_pg2_pg1_repeat0.md

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+Euclidean and non-Euclidean 3-dimensional spaces is a fundamental area of research in geometry and physics. Over time, researchers have developed various results, theorems, and insights concerning these surfaces. In [8], the authors investigated the Bertrand offset for ruled surfaces and showed that, similarly to planar curves, a ruled surface can have an infinite number of Bertrand offsets. The relationship between the Bertrand offsets of trajectory ruled surfaces and their projections on spherical areas has been explored in [9, 10], along with their corresponding invariants. The Bertrand offsets of ruled surfaces in Minkowski 3-space were considered in [11, 12]. Additionally, the study of Mannheim offsets for timelike ruled and developable surfaces, particularly their invariants, was addressed in [13, 14]. In [15], Senturk and Yuce analyzed involute-evolute offsets of ruled surfaces and developed associated invariants using the geodesic Frenet frame. More recently, research on evolute offsets of ruled surfaces, especially those with constant Gaussian and mean curvatures, in both Euclidean and Minkowski 3-spaces has been presented in [16, 17].
+
+The characteristics of ruled surfaces and their offset surfaces have been extensively studied in both Euclidean and non-Euclidean spaces (see [15-21]). However, existing literature lacks a detailed approach to constructing evolute offsets of slant $\mathfrak{T}\mathfrak{L}$-ruled surfaces in terms of the striction curve. In this study, we explore the geometric properties of slant $\mathfrak{T}\mathfrak{L}$-ruled surfaces and their evolute offsets within three-dimensional Minkowski space $\mathbb{E}_1^3$. By establishing a bijective correspondence through their rulings, we derive conditions under which an evolute offset $\mathfrak{M}^*$ maintains a coaxial relationship with the central normal of $\mathfrak{M}$. Furthermore, we formulate expressions governing curvature behavior and classify $\mathfrak{M}$ and $\mathfrak{M}^*$ based on specific functional parameters. Special cases, including $\mathfrak{T}\mathfrak{L}$-developable, $\mathfrak{T}\mathfrak{L}$-binormal, and $\mathfrak{T}\mathfrak{L}$-cone surfaces, are also examined, with graphical visualizations provided to support the analysis.
+
+This paper investigates slant timelike ($\mathfrak{T}\mathfrak{L}$) ruled surfaces and their evolute offsets in Minkowski 3-space $\mathbb{E}_1^3$. We present a parametric formulation of skew $\mathfrak{T}\mathfrak{L}$-ruled surfaces and establish conditions ensuring the coaxial alignment of the central normal with the ruling direction of the corresponding offset surface. The study explores the geometric properties using the Blaschke and Darboux frames, deriving curvature characteristics and fundamental invariants. Special cases, including $\mathfrak{T}\mathfrak{L}$-developable and $\mathfrak{T}\mathfrak{L}$-binormal surfaces, are analyzed with illustrative examples. The findings provide a deeper understanding of the differential geometry of $\mathfrak{T}\mathfrak{L}$-ruled surfaces and their evolute offsets in Lorentzian space.
+
+## 2 Basic concepts
+
+Let $\mathbb{E}_1^3$ denote Minkowski 3-space [22, 23]. For vectors $\mathfrak{e} = (e_1, e_2, e_3)$ and $\mathfrak{v} = (\nu_1, \nu_2, \nu_3)$ in $\mathbb{E}_1^3$, the inner product is defined as:
+
+$$
+\langle \mathfrak{e}, \mathfrak{v} \rangle = e_1 \nu_1 + e_2 \nu_2 - e_3 \nu_3.
+$$

dots.mocr-4bit/arxiv_math/2503.04033_pg9_pg1_repeat0.md

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+$$
+\mathbf{A}\mathbf{u} = \mathbf{f} \equiv \begin{bmatrix} \mathbf{A}_{11} & & \mathbf{A}_{15} \\ & \mathbf{A}_{22} & \mathbf{A}_{25} \\ \mathbf{A}_{51} & \mathbf{A}_{52} & \mathbf{A}_{55} \end{bmatrix} \begin{bmatrix} \mathbf{u}_1 \\ \mathbf{u}_2 \\ \mathbf{u}_5 \end{bmatrix} = \begin{bmatrix} \mathbf{f}_1 \\ \mathbf{f}_2 \\ \mathbf{f}_5 \end{bmatrix}. \quad (8)
+$$
+
+Here $\mathbf{u}$ is the solution and $\mathbf{f}$ is the body load on the interior of $\Omega$, split into the two subdomains and shared boundary. Because an elliptic PDO is a local operator there is no interaction through $\mathbf{A}$ on $\mathbf{f}_1$ by $\mathbf{u}_2$ or $\mathbf{f}_2$ by $\mathbf{u}_1$, so $\mathbf{A} = 0$ in those submatrices. However, both $\mathbf{u}_1$ and $\mathbf{u}_2$, are local to the shared boundary where $\mathbf{u}_5$ lies, so they both interact with the body load there, $\mathbf{f}_5$, through $\mathbf{A}$. To interpret the first two block rows, note that
+
+$$
+\mathbf{A}_{11}\mathbf{u}_1 + \mathbf{A}_{15}\mathbf{u}_5 = \mathbf{f}_1 \iff \mathbf{u}_1 = -\mathbf{A}_{11}^{-1}\mathbf{A}_{15}\mathbf{u}_5 + \mathbf{A}_{11}^{-1}\mathbf{f}_1 \quad (9)
+$$
+
+This is similar to our solution operator formulation shown in (4), with the part of the subdomain boundary on $\partial\Omega$ now folded into $\mathbf{u}_1$. Next, let $\tilde{\mathbf{u}}_1 = \mathbf{A}_{11}^{-1}\mathbf{f}_1$ and $\tilde{\mathbf{u}}_2 = \mathbf{A}_{22}^{-1}\mathbf{f}_2$. If we supply these vectors in place of $\mathbf{u}_1$, $\mathbf{u}_2$ in (8) and set $\mathbf{u}_5 = \mathbf{0}$, then the equation still holds with the same body loads $\mathbf{f}_1$ and $\mathbf{f}_2$. Thus $\tilde{\mathbf{u}}_1$ and $\tilde{\mathbf{u}}_2$ are the particular solutions to our PDE on the interiors of $\Omega^{(1)}$, and $\Omega^{(2)}$ with $\mathbf{u}_5 = \mathbf{0}$. Now consider an upper triangular matrix $\mathbf{U}$ that satisfies
+
+$$
+\underbrace{\begin{bmatrix} \mathbf{I} & \mathbf{A}_{11}^{-1}\mathbf{A}_{15} \\ \mathbf{I} & \mathbf{A}_{22}^{-1}\mathbf{A}_{25} \\ \mathbf{I} \end{bmatrix}}_{\mathbf{U}} \begin{bmatrix} \mathbf{u}_1 \\ \mathbf{u}_2 \\ \mathbf{u}_5 \end{bmatrix} = \begin{bmatrix} \tilde{\mathbf{u}}_1 \\ \tilde{\mathbf{u}}_2 \\ \mathbf{u}_5 \end{bmatrix}. \quad (10)
+$$
+
+This linear system can be confirmed with (9). $\mathbf{U}$ decouples our solutions $\mathbf{u}_1$, $\mathbf{u}_2$ into the particular solutions $\tilde{\mathbf{u}}_1$, $\tilde{\mathbf{u}}_2$ which are derived from our global Dirichlet BC without the subdomain-only (shared face) Dirichlet BC, and $\mathbf{u}_5$ which is that subdomain-only boundary condition. Thus $\mathbf{U}^{-1}$ collects both components of our solutions - in effect it represents the end of solving the merged system, where we use $\mathbf{u}_3$ and $\mathbf{u}_5$ to get $\mathbf{u}_1$, and $\mathbf{u}_4$ and $\mathbf{u}_5$ to get $\mathbf{u}_2$. Next, let $\mathbf{L}$ be a lower triangular matrix defined such that
+
+$$
+\underbrace{\begin{bmatrix} \mathbf{I} & & \\ & \mathbf{I} & \\ \mathbf{A}_{51}\mathbf{A}_{11}^{-1} & \mathbf{A}_{52}\mathbf{A}_{22}^{-1} & \mathbf{I} \end{bmatrix}}_{\mathbf{L}} \begin{bmatrix} \mathbf{f}_1 \\ \mathbf{f}_2 \\ \mathbf{f}_5 - \mathbf{A}_{51}\mathbf{A}_{11}^{-1}\mathbf{f}_1 - \mathbf{A}_{52}\mathbf{A}_{22}^{-1}\mathbf{f}_2 \end{bmatrix} = \begin{bmatrix} \mathbf{f}_1 \\ \mathbf{f}_2 \\ \mathbf{f}_5 \end{bmatrix}. \quad (11)
+$$

dots.mocr-4bit/arxiv_math/2503.04040_pg2_pg1_repeat0.md

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+proach cannot be directly applied to this joint optimization problem. To address this challenge, the authors of [25] studied a secure rate maximization problem with joint beamforming, FA position control, and artificial noise optimization problem. Then, they proposed the minimum mean square error (MMSE) method to decouple the optimization variables, demonstrating that the joint optimization significantly improves performance. The FAS was further extended to multiuser systems by [12], where the authors considered the transmit power minimization problem in an FA-assisted multiuser system with joint beamforming and FA position optimization. They decoupled the variables using zero-forcing (ZF) and MMSE methods. Then, they optimized FA positions via a multi-directional descent (MDD) framework. It was demonstrated that FAS can effectively cancel MUI, thereby significantly reducing transmit power while ensuring the data rate. More recently, [26] evaluated the system capacity gain of the FAS by solving the weighted sum rate (WSR) maximization problem in an FA-assisted downlink MU-multiple-input single-output (MISO) system. The authors first applied the weighted minimum mean square error (WMMSE) algorithm to decouple beamforming vectors and FA positions. Then, they leveraged the majorization maximization (MM) framework to solve the resulting non-concave FA position optimization problem. Simulation results demonstrated that FAS enhances overall system capacity in MU-MISO networks.
+
+However, the MM framework in [26] has unaffordable complexity when the number of FAs at the BS is large. Specifically, the MM framework optimizes one FA position at a time while keeping the others fixed in each iteration, leading to the increased computational time as the number of FAs grows.
+
+## B. Contributions
+
+In this paper, we propose a novel block coordinate ascent (BCA)-based algorithm for joint beamforming and FA position optimization in the FA-assisted downlink MU-MIMO system, as well as provide a decentralized implementation of the proposed algorithm. Our contributions are summarized as follows.
+
+* Different from [26] that focuses on MU-MISO networks, we consider the more general FA-assisted MU-MIMO networks. We formulate the joint beamforming and antenna position optimization as a WSR maximization problem, where the objective function is non-concave and the optimization variables are highly coupled. To decouple the beamforming matrices and FA positions, we utilize two matrix fractional programming (FP) techniques, i.e., the quadratic transform and the Lagrangian dual transform [27], [28]. These FP techniques enable us to solve the decoupled subproblems by a BCA-based algorithm.
+* Unlike existing FA position optimization algorithms that sequentially update FA positions, we propose a novel optimization algorithm based on the MM framework that optimizes all FA positions simultaneously. The proposed algorithm reduces computational time by exploiting parallelism and can be easily extended to a decentralized implementation. Additionally, we introduce a box-constrained movement mode to simplify both the formulated problem and the engineering implementation of FAS. This movement mode provides closed-form so-
+
+lutions to reduce complexity and enables decentralized implementation by allowing the independent movement of each FA.
+
+* To further reduce computational, storage, and interconnection costs, we propose a decentralized implementation of our algorithm by utilizing the decentralized baseband processing (DBP) architecture [29], which partitions the transmit FA array into several clusters. The DBP architecture can decompose the optimization problem into smaller subproblems, and enable decentralized units (DUs) to solve them in parallel. To enable the decentralized implementation of the proposed BCA-based algorithm, we employ the non-homogeneous transform and Nesterov's extrapolation [30], [31] to avoid the matrix inversion in the optimization of beamforming matrices. The decentralized implementation can significantly alleviate computational, storage, and interconnection costs with negligible performance loss.
+* We conduct various numerical experiments to verify the performance of our proposed algorithm. It is shown that the proposed algorithm can improve the WSR of FA-assisted MU-MIMO networks by 47% compared with conventional MIMO networks using FPAs. Moreover, the decentralized implementation reduces computation time by approximately 70% and has similar performance compared with the centralized implementation.
+
+The remainder of this paper is organized as follows. Section II presents the channel model of the FA-assisted MU-MIMO system and formulates the WSR problem. Section III reformulates the problem using FP techniques and solves it using BCA and MM. The decentralized implementation of the proposed algorithm is introduced in Section IV. Simulation results are provided in Section V, and conclusions are drawn in Section VI.
+
+*Notation:* Italic letters, boldface lowercase letters, and boldface uppercase letters denote scalars, vectors, and matrices, respectively. The imaginary unit is denoted by $j$. For a complex number $a$, its amplitude and phase are given by $|a|$ and $\angle a$, respectively. The $\ell_2$ norm of a vector $\mathbf{a}$ is $\|\mathbf{a}\|_2$. $[\mathbf{A}]_m$, $[\mathbf{A}]_{mn}$, $\mathbf{A}^T$, $\mathbf{A}^H$, $\det(\mathbf{A})$, $\operatorname{tr}(\mathbf{A})$, $\operatorname{vec}(\mathbf{A})$, $\|\mathbf{A}\|_\infty$, and $\|\mathbf{A}\|_F$ denote the $m$-th row, the $(m, n)$-th element, transpose, conjugate transpose, determinant, trace, vectorization, the infinity norm, and the Frobenius norm of matrix $\mathbf{A}$, respectively. $\mathbf{A} \succeq \mathbf{0}$ and $\mathbf{A} \succ \mathbf{0}$ indicate that $\mathbf{A}$ is positive semi-definite and positive definite, respectively. $\mathbb{C}^{M \times N}$, $\mathbb{R}^{M \times N}$, and $\mathbb{R}_+^{M \times N}$ denote the sets of $M \times N$ complex, real, and non-negative real matrices, respectively. The circularly symmetric complex Gaussian (CSCG) distribution with zero mean and covariance $\sigma^2 \mathbf{I}$ is represented as $\mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I})$, and the uniform distribution over $[a, b]$ is denoted by $\mathcal{U}[a, b]$. Operator $\partial(\cdot)$ denotes the partial differential. $\nabla_x f(\mathbf{x})$ and $\nabla_x^2 f(\mathbf{x})$ denote the gradient vector and Hessian matrix of $f(\mathbf{x})$ with respect to (w.r.t.) $\mathbf{x}$ respectively.
+
+## II. SYSTEM MODEL AND PROBLEM FORMULATION
+
+As shown in Fig. 1, we consider a downlink MU-MIMO system where a BS with $M$ FAs serves $K$ users, each equipped with $N$ FAs. A three-dimensional (3D) Cartesian coordinate system is established to describe the positions of the transmit FAs at the BS and receive FAs at the users.

dots.mocr-4bit/arxiv_math/2503.04041_pg20_pg1_repeat0.md

@@ -0,0 +1,16 @@
+due to the loss of orthogonality of basis vectors. Thus we set the maximum iteration steps to $10n$. We have observed that for the test problems in our experiments, $10n$ is enough to ensure convergence. We perform full reorthogonalization on $V'_k$, $U_k$ and $\hat{U}_k$ to ensure that they are numerically orthonormal to the working precision $\epsilon$. By default, the convergence of line 2 in Algorithm 6.1 is judged by (3.18) with the stopping tolerance $tol = 10^{-8}$. The starting vector $u_1$ is randomly generated by the standard normal distribution function `randn` in MATLAB and then normalized.
+
+<table><thead><tr><th>Parameters</th><th>Default values</th><th>Description</th></tr></thead><tbody><tr><td>target</td><td>5</td><td>$|target| = l$ is the number of desired GSVD components, where $target > 0$ or $target < 0$ means that the largest or smallest GSVD components are required.</td></tr><tr><td>$k_{\max}$</td><td></td><td>Maximum subspace dimension</td></tr><tr><td>adjust</td><td>3</td><td>Integer added to $l$ to speed up convergence</td></tr><tr><td>$tol$</td><td>$10^{-8}$</td><td>Stopping tolerance</td></tr><tr><td>$maxit$</td><td>1000</td><td>Maximum number of restarts</td></tr><tr><td>$lsqrtol$</td><td>$10\epsilon$</td><td>Stopping tolerance of lsqr</td></tr><tr><td>$lsqrmaxit$</td><td>$10n$</td><td>Maximum number of iterations of lsqr</td></tr><tr><td>$u_1$</td><td>generated by randn and then normalized</td><td>The unit length starting vector</td></tr></tbody></table>
+
+TABLE 1
+Parameters of IRJBD
+
+7. Numerical experiments. We now present a number of numerical experiments to demonstrate the performance of Algorithm 6.1, written in MATLAB language, and compare it with the public C language code of TRJBD in [1]. The experiments will illustrate that Algorithm 6.1 is at least competitive with and can outperform TRJBD considerably in terms of restarts. Since section 4.4 has shown that each restart of IRJBD and TRJBD basically costs the same for the same $l$ and $k_{\max}$, fewer restarts mean higher efficiency. The experiments were performed on an AMD Ryzen 7 5800X CPU with 78 GB RAM and 8 cores using the MATLAB R2024a with the machine precision $\epsilon = 2.22 \times 10^{-16}$ under the Ubuntu 20.04.3 LTS 64-bit system.
+
+We use some matrices from the SuiteSparse Matrix Collection [6] as $A$, and $L$ is taken as (cf. [11]):
+
+$$
+L = \begin{pmatrix} 3 & 1 & & & \\ 1 & \ddots & \ddots & & \\ \ddots & \ddots & \ddots & 1 & \\ 1 & 3 \end{pmatrix},
+$$
+
+which is well conditioned. Table 2 lists the test matrices $A$ and some of their properties, where $nnz$ is the total number of nonzero entries of $\{A, L\}$ and $\kappa(A) = \frac{\sigma_{\max}(A)}{\sigma_{\min}(A)}$, available from the SuiteSparse Matrix Collection, is the condition number of $A$ and indicates the level of ill-conditioning of $A$ and reflects magnitudes of generalized singular values of $\{A, L\}$. In order to ensure that all the $A$ are flat or square, i.e., $m \le n$, we transpose some matrices, denoted by superscript T. Note that the smallest singular value of $\text{knese\_8\_3\_1}^T$ is tiny, which causes that the matrix is numerically row

dots.mocr-4bit/arxiv_math/2503.04045_pg2_pg1_repeat0.md

@@ -0,0 +1,61 @@
+**Theorem 1.2** (Chen). Every sufficiently large even number $N$ can be expressed as
+
+$$
+N = p + \eta, \qquad (1.2)
+$$
+
+where $p$ is prime and $\eta > 0$ has at most 2 prime factors.
+
+**Theorem 1.3** (Li). Every sufficiently large odd number $N$ can be expressed as
+
+$$
+N = p + 2\eta, \qquad (1.3)
+$$
+
+where $p$ is prime and $\eta > 0$ has at most 2 prime factors.
+
+Here, the factor of 2 in (1.3) is to force $p + 2\eta$ to be odd; provided of course $p \neq 2$.
+
+Combining Theorems 1.2 and 1.3, we have that every large $N \ge 1$ can be represented as the sum of a prime and a number with at most 3 prime factors. Any further lowering of the number of prime factors in Chen or Li's results appears out of reach, so we do not attempt this here.
+
+Now, representing $N$ of the form (1.1) when $k \ge 2$ is naturally more difficult, owing to sparseness of the set of $k$th prime powers, and additional “forced” prime factors of $N - p^k$. Thus in this setting, much less is known. However, the following classical result, due to Erdős [7] and Rao [24], can be viewed as partial progress towards representing large $N$ in the form (1.1).
+
+**Theorem 1.4** (Erdős–Rao). Let $k \ge 2$. Then every sufficiently large integer $N$ such that
+
+$$
+N \not\equiv 1 \begin{cases} (\text{mod } 16), & \text{when } k = 4, \\ (\text{mod } 4), & \text{when } k = 2, \end{cases} \qquad (1.4)
+$$
+
+can be expressed as
+
+$$
+N = p^k + \eta
+$$
+
+where $p$ is prime and $\eta > 0$ is $k$th power-free.
+
+Here, the condition that $\eta$ is $k$th power-free is much simpler than $\eta$ having a bounded number of prime factors. In particular, most integers ($> 60\%$) are square-free (and thus $k$th power-free), whereas the set of integers with a bounded number of prime factors has a natural density of 0.
+
+1.2 Statement of results
+
+Our main result is as follows.
+
+**Theorem 1.5.** Let $k \ge 2$. There exists an integer $M(k)$ such that every sufficiently large integer $N$ can be expressed as
+
+$$
+N = p^k + \eta \qquad (1.5)
+$$
+
+where $p$ is prime and $\eta > 0$ has at most $M(k)$ prime factors. Here, $M(k) = 6k$ is admissible for all even $k \ge 2$, and $M(k) = 4k$ is admissible for all odd $k \ge 3$. In addition, for sufficiently large $k \ge k_\varepsilon$ one can set
+
+$$
+M(k) = (2 + \varepsilon)k, \qquad (1.6)
+$$
+
+for any $\varepsilon > 0$. Or, under assumption of the Elliott–Halberstam conjecture¹,
+
+$$
+M(k) = (1 + \varepsilon)k. \qquad (1.7)
+$$
+
+¹More precisely, a standard variant of the Elliott–Halberstam conjecture (see Conjecture 2.7).

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@@ -0,0 +1,35 @@
+The efficiency of simulated annealing around $T_c$ has been reported in numerous studies. Kirkpatrick et al. (1983) demonstrated that during phase transition, the search becomes more efficient. Strobl & Barker (2016) showed that when using simulated annealing to solve phylogeny reconstruction, the search is constrained to a small valley of the search space when the temperature is below $T_c$. The search efficiency around $T_c$ can also be seen in Figure 2 of (Cai & Ma, 2010b).
+
+While some algorithms (Basu & Frazer, 1990; Cai & Ma, 2010a) have used critical temperature $T_c$ to design the initial and final temperatures, its use for reheating is less common. Abramson et al. (1999) proposes a reheating strategy tied to function cost that implicitly involves $T_c$. However, this method lacks theoretical support and is sensitive to hyperparameters. Instead, our approach directly reheats to the critical temperature, which injects just the right amount of stochastic energy back into the system, enabling it to escape “wandering in contours” without incurring excessive randomness of a high-temperature regime.
+
+### 5.2.2 Specific Heat
+
+Determining the critical temperature in simulated annealing requires identifying the phase transition point during optimization, which is characterized by peaks in specific heat (Kirkpatrick et al., 1983; Strobl & Barker, 2016). In statistical physics, the system's energy at temperature $T$, denoted as $E(T)$, adheres to the Boltzmann distribution. Thus, the expected energy of a system can be seen as a function of the temperature, $E[E(T)]$. The specific heat at temperature $T$, denoted as $C_T$, is traditionally defined in thermodynamics as the rate of change of the expected energy $E[E(T)]$ to temperature (Aarts et al., 1987), given by $C_T = \frac{\partial E[E(T)]}{\partial T}$. Integrating over the Boltzmann distribution allows deriving specific heat in terms of energy variance:
+
+$$
+C_T = \frac{\sigma^2(E(T))}{T^2} \qquad (9)
+$$
+
+where $\sigma^2(E(T))$ is the variance of the system energy at $T$.
+
+### 5.2.3 Determination of Critical Temperature
+
+To determine the critical temperature based on specific heat, we first denote $T(t)$ as the temperature at step $t$, $C(t)$ as the specific heat at temperature $T(t)$, and $x_t$ as the solution sampled at step $t$, then by selecting an appropriate sample size $M$, we define the approximation of $C(t)$ as :
+
+$$
+\hat{C}(t) = \frac{\sigma^2(\{f(x_{t-M+1}), \dots, f(x_t)\})}{T(t)^2}, \quad t \ge M \qquad (10)
+$$
+
+where $\sigma^2(\{f(x_{t-M+1}), \dots, f(x_t)\})$ represents the variance in objective values over the $M$ most recent steps. The critical temperature $T_c$ can be determined as $T(t^*)$, where $t^* = \arg\max_{t \ge M} \hat{C}(t)$. However, SA with gradient-based discrete samplers convergences rapidly in the initial stage, resulting in an *abnormal* initial peak in specific heat (due to the high variance). As the annealing progresses, the specific heat quickly decreases, eventually stabilizing at a level more typical of a critical temperature. This behavior is shown in Figure 4a&4b.
+
+To address the abnormal peak, we introduce a “skip step” threshold, denoted as $t_{skip}$, ensuring that the initial transient behavior is excluded from the analysis. Thus, the critical temperature is identified as $T(\tilde{t}^*)$, where
+
+$$
+\tilde{t}^* = \arg\max_{t \ge t_{skip}} \hat{C}(t). \qquad (11)
+$$
+
+Our method diverges from traditional SA reheat strategies (Abramson et al., 1999) by accounting for inhomogeneous chains and addressing gradient-based methods' unique abnormal peaks.
+
+## 5.3 Reheated Gradient-based Discrete Sampling for Combinatorial Optimization
+
+Combining results in Section 5.1 and 5.2, we obtain the *Reheated Sampling for Combinatorial Optimization* algorithm (ReSCO), which is summarized in Algorithm 1. ReSCO is compatible with any gradient-based

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@@ -0,0 +1,23 @@
+Figure 9:
+
+which implies that:
+
+$$
+x_m = \frac{r_1 \tan \theta_1 - r_2 \tan \theta_2 - (u_1 - u_2)}{\tan \theta_1 + \tan \theta_2} \quad (130)
+$$
+
+Then we use $x_m$ and $\psi_m$ to express the total volume. By using geometric computation, we have:
+
+$$
+\begin{aligned} \mathcal{V}(\psi_m) = & (r_1 u_1 - \int_{\psi_1}^{\psi_m} \int_{\psi}^{\psi_m} \frac{\cos \gamma}{g \sqrt{\frac{2\sigma}{g}(\cos \psi_m - \cos \gamma)}} d\gamma \frac{-\sin \psi}{g \sqrt{\frac{2\sigma}{g}(\cos \psi_m - \cos \psi)}} d\psi) \\ & + \int_{\psi_2}^{\psi_m} \int_{\psi}^{\psi_m} \frac{\cos \gamma}{g \sqrt{\frac{2\sigma}{g}(\cos \psi_m - \cos \gamma)}} d\gamma \frac{-\sin \psi}{g \sqrt{\frac{2\sigma}{g}(\cos \psi_m - \cos \psi)}} d\psi \\ & + \frac{1}{2}(u_1 + u_2)^2 \frac{1}{-\tan \theta_1} + (-u_2)r_1 + \frac{1}{2}(u_2 - u_1 - x_1 \tan \theta_1)^2 \left(\frac{1}{\tan \theta_1} + \frac{1}{\tan \theta_2}\right) \end{aligned} \quad (131)
+$$
+
+Then we need to prove the following theorem about the total volume:
+
+**Theorem 3.12.** When $\psi_m \to 0$, we have $\mathcal{V}(\psi_m) \to +\infty$.
+
+*Proof.* Using the computation in Theorem 3.7, we know that:
+
+$$
+\int_{\psi}^{\psi_m} \frac{\cos \gamma}{g \sqrt{\frac{2\sigma}{g}(\cos \psi_m - \cos \gamma)}} d\gamma \to +\infty
+$$

dots.mocr-4bit/arxiv_math/2503.04056_pg25_pg1_repeat0.md

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+Figure 17: (Color online) Data-driven discovery of the nonlinear coefficient $v(t) = \cos(t)$ in the variable coefficient KP equation: the upper part of the figure displays the 3D predicted solution at these three cross-sections. The red dashed lines represent the predicted solution, while the blue lines represent the true solution; the lower part displays the errors between the predicted and true solutions at the same cross-sections.
+
+Figure 18: (Color online) The data-driven variable coefficient $V(t)$ discovery of the variable coefficient KP equation by R-gPINN related to solution (21): the comparison of the error between the learning and exact variable coefficient $V(t)$ over the interval $[-6, 6]$.

dots.mocr-4bit/arxiv_math/2503.04068_pg2_pg1_repeat0.md

@@ -0,0 +1,21 @@
+The time-one map $\Psi$ is defined by setting $\Psi(x_0) = x(1)$ for each $x_0 \in \mathbb{R}^d$. Classical results on approximation properties of shallow neural networks do not immediately apply since $A(t)$ is fixed to be a square matrix of input dimension. For this reason, the neural ODE (1) is referred to as *narrow* following the definition in [2].
+
+In [11], it has been shown that flows of this ODE can be used to approximate continuous maps on $\mathbb{R}^d$ in the uniform norm by considering a NODEs on $\mathbb{R}^{2d+1}$, provided the activation function satisfies a quadratic equation. In [9], it has been shown that any $L_2$ map can be approximated using a narrow NODE. The capability NODEs for transporting probability densities have been explored in [5, 9, 2]. For $L_p$ maps, universal approximations have been shown in [4].
+
+The strategy used in this paper departs from the techniques of the other cited works, where either the arguments are constructive or control theoretic arguments are used. Firstly, we note that it can be shown that flows of (1) can approximate flows of the wide NODE
+
+$$
+\dot{x}(t) = \sum_{i=1}^{m} A_i \Sigma(W_i x + b_i) \quad x(0) = x_0 \quad (3)
+$$
+
+This, in turn, can be used to show that flow maps of (1) can be used to approximate flows of any dynamical system of the form
+
+$$
+\dot{x}(t) = V(x, t) \quad x(0) = x_0 \quad (4)
+$$
+
+where $V(x, t)$ is a time dependent vector field and hence, $V(x, t)$ can be approximated by an arbitrarily wide network $\sum_{i=1}^{m} A_i \Sigma(W_i x + b_i)$ for any $t$. Therefore, narrow neural ODEs inherit the approximation properties of their shallow but wide counterparts. This strategy is used in [5] and in [6] to study flow approximation properties using narrow NODEs. In [6] it was in fact shown that the dimension of the weight parameters used to approximate maps can be taken to be $m \times d$, with $m$ less than the input dimension $d$, by additionally exploiting geometric non-commutative properties of some chosen $m$ basis vector fields, owing to diffeomorphism controllability results due to [1].
+
+Our goal in this paper is to derive quantitative rates of approximation of (1). Existing quantitative rates are either established to approximate homoemorphism in the $L_2$ norm [9] or for the purposes of generative modeling [9, 2]. In contrast, we are interested in quantitative rates for approximation of flow maps in the uniform norm. The problem we address is how many switches are needed in $A(t)$, $W(t)$, $b(t)$, to approximate the flow of (3). In this way, our result complements [9, 2]. While we restrict to reference flows of the differential equations of the form (3), an extension to general flows is straightforward since shallow but wide neural networks are known to be dense in the set of continuous functions. Hence, quantitative approximation properties of shallow neural networks would immediately translate to quantitative approximation properties of narrow NODEs.
+
+Before we present our analysis, we give some brief intuition behind our proof strategy. The idea behind our analysis is the following. Given a differential equation of the

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+Periodic Inscription of Isosceles Trapezoids
+
+Ali Naseri Sadr
+
+Abstract
+
+We prove that a pair of continuous disjoint periodic curves in $\mathbb{C}$ inscribes an isosceles trapezoid with any similarity type. The case of smooth curves can be identified with a Lagrangian intersection problem for a pair of Lagrangian cylinders in $\mathbb{R} \times S^1 \times \mathbb{C}$, and the continuous case follows from the smooth one by a standard convergence argument.
+
+1 Introduction
+
+Let $\gamma_1, \gamma_2: \mathbb{R} \to \mathbb{C}$ be two continuous embeddings of the real line into $\mathbb{C}$ that satisfy the periodicity condition
+
+$$
+\gamma_i(t + 1) = \gamma_i(t) + \sqrt{-1}
+$$
+
+for every $t$ and $i = 1, 2$. Furthermore, assume the images of $\gamma_1$ and $\gamma_2$ are disjoint. Tao conjectured in [10] that there exist four points in $\gamma_1(\mathbb{R}) \cup \gamma_2(\mathbb{R})$ which are vertices of a square; this is a variation of the Toeplitz square peg problem for periodic curves, and Hugelmeyer proved it in [5].
+
+For any given isosceles trapezoid $Q$, we show there are four points in $\gamma_1(\mathbb{R}) \cup \gamma_2(\mathbb{R})$ that are vertices of a quadrilateral similar to $Q$. The approach of [5] does not directly generalize even to the case of rectangles. By contrast, in this article, we use a different approach to prove not only that every pair of periodic curves inscribes every similarity type of rectangles, but also every similarity type of isosceles trapezoids.
+
+**Definition 1.1.** Assume $Q$ is an isosceles trapezoid. We say that the pair $(\gamma_1, \gamma_2)$ admits a balanced inscription of $Q$ if there exist $p_1, p_2 \in \gamma_1(\mathbb{R})$ and $p_3, p_4 \in \gamma_2(\mathbb{R})$ such that the quadrilateral formed by $p_1, p_2, p_3, p_4$ is similar to $Q$, the line segments $\overline{p_1p_2}$ and $\overline{p_3p_4}$ are parallel, and $|\overline{p_1p_2}| \le |\overline{p_3p_4}|$.
+
+Note that our definition depends on the order of the pair $(\gamma_1, \gamma_2)$ unless $Q$ is a rectangle.
+
+**Theorem 1.2.** Suppose $\gamma_1$ and $\gamma_2$ are two continuous disjoint periodic embeddings of the real line into the plane, and suppose $Q$ is an isosceles trapezoid. Then $(\gamma_1, \gamma_2)$ admits a balanced inscription of $Q$. Furthermore, there is a generic subset of smooth disjoint periodic pairs such that each pair in this set admits at least two balanced inscriptions of $Q$ that are not related under translation by $\sqrt{-1}$.
+
+**Corollary 1.3.** Let $\theta \in (0, \frac{\pi}{2}]$; then every pair of continuous disjoint periodic curves in the plane inscribes a rectangle with angle $\theta$ between its two diagonals.
+
+We conjecture that Theorem 1.2 is optimal, in the following sense.
+
+**Conjecture 1.4.** Let $Q$ be a quadrilateral that admits an inscription in any pair of disjoint periodic curves in $\mathbb{C}$. Then $Q$ is an isosceles trapezoid.
+
+arXiv:2503.04077v1 [math.SG] 6 Mar 2025

dots.mocr-4bit/arxiv_math/2503.04086_pg3_pg1_repeat0.md

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+recall that a positive integer $n$ and two integers $a, b$, $\gcd(a, n) = \gcd(b, n)$ if and only if $a$ and $b$ generate the same ideal in the ring $\mathbb{Z}/n$. This is also equivalent to the condition that $a \equiv ub \pmod{n}$ for some $u \in (\mathbb{Z}/n)^\times$. This property generalizes well for Artinian rings, and in particular, to finite rings. More precisely
+
+**Lemma 2.2.** (See [5, Lemma 2.1] Let $R$ be an Artinian ring. If $Ra = Rb$, then there exists $u \in R^\times$ such that $b = ua$.
+
+From this perspective, a crucial observation here is that a generating set $S$ in a finite ring $R$ such as $\mathbb{Z}/n$ or $\mathbb{F}_q[x]/f$ gives rise to a gcd-graph if and only if $S$ is stable under the action of $R^\times$; that is, if $s \in S$, then $us \in S$ for all $u \in R^\times$. Motivated by this observation, we introduce the following definition.
+
+**Definition 2.3.** We say that $\Gamma(R, S)$ is a gcd-graph if $S$ is stable under the action of $R^\times$.
+
+**Remark 2.4.** When $S = R^\times$, the graph $\Gamma(R, R^\times)$ is known as a unitary Cayley graph (see [1, 6]). This shows that the unitary Cayley graph over $R$ is a special case of gcd-graphs. Other examples, as explained previously, include the gcd-graphs defined in [6] for the ring $\mathbb{Z}/n$ and the gcd-graphs defined in [9] for the ring $\mathbb{F}_q[x]/f$ where $\mathbb{F}_q$ is a finite field with $q$ elements and $f$ is a non-zero polynomial in $\mathbb{F}_q[x]$.
+
+We provide below the necessary and sufficient conditions for a Cayley graph over $R$ to be a gcd-graph. These conditions are somewhat more explicit than Definition 2.3. Furthermore, together with Corollary 2.7, this description will be important later on when we study the spectrum of these gcd-graphs.
+
+**Proposition 2.5.** Let $R$ be a finite commutative ring and $S$ a subset of $R$. Then the following are equivalent.
+
+(1) $\Gamma(R, S)$ is a gcd-graph.
+(2) There exist distinct nonzero principal ideals $\mathcal{I}_1, \mathcal{I}_2, \dots, \mathcal{I}_k$ such that for each $r \in R$, $r \in S$ if and only if there exists $1 \le i \le k$ such that $\mathcal{I}_i = Rr$.
+
+*Proof.* First, we claim that (2) $\implies$ (1). Clearly, $S$ is symmetric and $0 \notin S$. By definition, we need to show that if $s \in S$ and $u \in R^\times$ then $us \in S$. Since $s \in S$, there exists an ideal $\mathcal{I}_i$ with $1 \le i \le k$ such that $Rs = \mathcal{I}_i$. Since $u \in R^\times$, $\mathcal{I}_i = Rs = Rus$. By (2), this shows that $us \in S$.
+
+Let us show that (1) implies (2) as well. For each $s \in S$, let $\mathcal{I}_s = Rs$ be the ideal generated by $s$. Let $\{\mathcal{I}_1, \mathcal{I}_2, \dots, \mathcal{I}_k\}$ be the set of all $\mathcal{I}_s$. We conclude that for each $s \in S$, $\mathcal{I}_s = \mathcal{I}_i$ for some $1 \le i \le k$. Conversely, if $r \in R$ such that $Rr \in \{\mathcal{I}_1, \mathcal{I}_2, \dots, \mathcal{I}_k\}$, then $Rr = Rs$ for some $s \in S$. By Lemma 2.2, we know that $r = us$ for some $u \in R^\times$. Since $\Gamma(R, S)$ is a gcd-graph, $S$ is stable under the action of $R^\times$. This shows that $r \in S$ as well. $\square$

dots.mocr-4bit/arxiv_math/2503.04092_pg10_pg1_repeat0.md

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+with $T_c = 0.8$ and $T = 0.36$.
+
+The physical parameters are set as seen in Table 1. The forward problem is solved using
+
+<table><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>ρ (gr · cm<sup>3</sup>)</td><td>1.2</td></tr><tr><td>μ (P)</td><td>0.035</td></tr><tr><td>U (cm · s<sup>-1</sup>)</td><td>75</td></tr><tr><td>T<sub>c</sub> (s)</td><td>0.80</td></tr><tr><td>T (s)</td><td>0.36</td></tr><tr><td>κ (s<sup>-1</sup>)</td><td>70</td></tr></tbody></table>
+
+<table><thead><tr><th></th><th>Γ<sub>1</sub></th><th>Γ<sub>2</sub></th><th>Γ<sub>3</sub></th><th>Γ<sub>4</sub></th></tr></thead><tbody><tr><td>R<sub>p</sub> (dyn · s · cm<sup>-5</sup>)</td><td>480</td><td>520</td><td>520</td><td>200</td></tr><tr><td>R<sub>d</sub> (dyn · s · cm<sup>-5</sup>)</td><td>7200</td><td>11520</td><td>11520</td><td>4800</td></tr><tr><td>C (dyn<sup>-1</sup> · cm<sup>5</sup>)</td><td>4 · 10<sup>-4</sup></td><td>3 · 10<sup>-4</sup></td><td>3 · 10<sup>-4</sup></td><td>4 · 10<sup>-4</sup></td></tr></tbody></table>
+
+Table 1: Physical parameters and numerical values of the three-element Windkessel parameters for every outlet.
+
+a semi-implicit 3D-0D coupling scheme as seen in [9]. The full algorithm is detailed in the appendix.
+
+### 3.1.2 Synthetic measurements
+
+The forward solution is generated with a time step of $dt = 1ms$ and undersampled in time to $dt_{meas} = 15ms$, leading to a total of 56 measurements. From the solution of the forward problem, we simulate a PC-MRI acquisition by subsampling into a rectangular measurement mesh with a resolution of [2mm, 2mm, 2mm] and then applying the process described in Section 2.1 with a $venc$ of double the maximal velocity. The magnitude is modelled as
+
+$$
+M(\boldsymbol{x}) = \begin{cases} 1.0 & \text{if } \boldsymbol{x} \text{ is in the lumen of the vessel} \\ 0.5 & \text{otherwise.} \end{cases} \qquad (16)
+$$
+
+Finally a complex Gaussian noise $\boldsymbol{\epsilon} \in \mathbb{C}^N$ is added with a signal-to-noise ratio (SNR) of 15. Fifty independent realizations of the noise were generated.
+
+For comparison, we reconstructed velocity measurements from these synthetic measurements using the Berkeley Advanced Reconstruction Toolbox (BART)[11]. BART is a command-line-based software that provides a flexible framework of compressed sensing methods, as well as tools for simulation, pre-processing, and image reconstruction, providing a multitude of different regularization options. In this work, we have used this toolbox for compressed sensing reconstructions of the velocities, using total variation in time as for the regularization.
+
+Next, the sampling mask is applied to these simulated frequency space measurements. We take a 2D subsampled mask in the $x-y$-plane and sample fully in the $z$-direction as in [4]. We consider different subsampling rates $R = \frac{N_{sampled}}{N_{total}} = 8, 16, 32$, with two different masks: the pseudo-spiral mask and the pseudo-random Gaussian mask, which is sampled according to a Gaussian probability distribution, as shown in Figure 2. For the pseudo-spiral mask, the points are placed evenly on a cartesian grid along a spiral with six turns and a final radius reaching the edge of the mask.

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+Thanks to this technique, we have been able to reveal that the hidden symmetry algebra emerging from the embedding chain $\mathfrak{su}(4) \supset \mathfrak{su}(2) \times \mathfrak{su}(2)$ gives rise to a quartic polynomial Poisson algebra. As a byproduct, this paper has also demonstrated that the grading method not only simplifies complex expansions in the Poisson bracket relations of a given degree but is, in fact, indispensable for deriving these relations when the degree becomes so high that standard “brute-force” methods are no longer applicable.
+
+This ansatz can be generalized and applied to any other reduction chain $\mathfrak{g} \supset \mathfrak{g}'$ involving reductive Lie algebras where the subalgebra embedding is not regular, and for which the usual argumentation using the root system is no longer applicable [10]. However, as shown in [39], there are several alternatives to define non-canonical gradings in Lie algebras, that can be conveniently adapted to various non-regular reduction chains (see [40, 41] and references therein). Following this argumentation, the reduction chains of interest in physical applications (see e.g. [31, Chapter 12], [42]) examined in the context of the missing label problem can be reevaluated from the perspective of polynomial algebras, allowing us to describe the complete algebraic structure, and potentially allowing us to make more effective of labeling operators to separate degenerate states. For applications in superintegrable systems [43, 44], the precise knowledge of the associated polynomial algebra can be useful for the systematic construction of Hamiltonians admitting constants of the motion of degrees higher than two, providing new hierarchies of systems for which the classical criteria for the separation of variables no longer necessarily apply [45, 46]. It is worthy to be mentioned in this context that the observed relations between polynomial algebras and quasi-exactly solvable problems (see e.g. [47]) could be systematized by means of a detailed analysis of the (hidden) symmetry algebras and their associated polynomial structures. Work in these directions is currently in progress.
+
+ACKNOWLEDGEMENTS
+
+This work was partially supported by the Future Fellowship FT180100099 and the Discovery Project DP190101529 from the Australian Research Council. RCS acknowledges financial support by the Agencia Estatal de Investigación (Spain) under the grant PID2023-148373NB-I00 funded by MCIN/AEI/10.13039/501100011033/FEDER, UE. The research of DL is partially funded by MUR - Dipartimento di Eccellenza 2023-2027, codice CUP G43C22004580005 - codice progetto DECC23_012_DIP and partially supported by INFN-CSN4 (Commissione Scientifica Nazionale 4 - Fisica Teorica), MMNLP project. DL is a member of GNFM, INdAM.

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+FIGURE 9. $\Gamma_5$
+
+FIGURE 10. $\Gamma_6$
+
+$$
+(3) \quad \sigma_1(\vec{e}_i) = \vec{e}_i \text{ and } \sigma_1(\vec{f}_j) = \overleftarrow{\vec{f}}_j; 1 \le i \le 3 \text{ and } 1 \le j \le 7.
+$$
+
+$$
+(4) \quad \sigma_0 = C_1 \dots C_5, \text{ where}
+$$
+
+$$
+\begin{align*}
+C_1 &= (\vec{e}_1, \vec{f}_1, \vec{e}_3, \vec{f}_1), \\
+C_2 &= (\vec{e}_2, \vec{f}_3, \vec{e}_1, \vec{f}_2), \\
+C_3 &= (\vec{e}_3, \vec{f}_6, \vec{e}_2, \vec{f}_5), \\
+C_4 &= (\vec{f}_2, \vec{f}_4, \vec{f}_3, \vec{f}_4), \\
+C_5 &= (\vec{f}_5, \vec{f}_7, \vec{f}_6, \vec{f}_7).
+\end{align*}
+$$
+
+Now, we count the boundary components of $\Gamma_5$ using the permutation $\sigma_\infty$. The boundary components of the graph $\Gamma_5$ given by:
+
+$$
+\partial_1 = \vec{e}_1 \vec{f}_2 \vec{f}_4 \vec{f}_2 \vec{e}_2 \vec{f}_5 \vec{f}_7 \vec{f}_5 \vec{e}_3 \vec{f}_1 \vec{e}_3 \vec{f}_6 \vec{f}_7 \vec{f}_6 \vec{e}_2 \vec{f}_3 \vec{f}_4 \vec{f}_3 \vec{e}_1 \vec{f}_1.
+$$
+
+So, there is only one boundary component.
+
+4. UPPER BOUND ON MAPPING CLASS ORBITS OF MAXIMUM SIZE FILLING
+
+Let $I(S_g)$ represent the set of all isotopy classes of simple closed curves on $S_g$. $I(S_g)^n$ denotes the Cartesian product of $n \in \mathbb{N}$ copies of $I(S_g)$. Define an equivalence relation $\sim$ on $I(S_g)^n$ by

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+whose center vertex is $x$, by adding an edge $vx$ and a pendent 4-cycle at vertex $v$. In that case, by Proposition 2.6, there is a hop dominating set $S$ of $H$, such that $x \in S$ and $|S| \le \frac{2(n-4)}{5}$, which implies that $S \cup \{v\}$ is a hop dominating set $S$ of $G$, a contradiction. Thus such a path $P$ exists. Note that $k \ge 2$.
+
+**Claim 3.10.** $k \in \{2, 3\}$
+
+*Proof.* Suppose that $k \ge 4$. Let $G' = (G - xx_1) + x_1 x_4$. Note that $x_1 x_2 x_3 x_4 v_1$ is a pendent 4-cycle at $x_4$ in $G'$. Note that $|V(G')| = n$, $|E(G')| = |E(G)|$, and $G'$ has more pendent 4-cycles than $G$.
+
+Suppose that $\deg_G(x) \ge 3$. Then $\delta(G') \ge 2$ and $G'$ has no connected component in $\mathcal{B}$. There is a minimum hop dominating set $S'$ of $G'$ such that $x_4, v, x \in S'$ and $N_G(x_4) \cap S' \ne \emptyset$ by Proposition 2.7. By the choice of $G$, $|S'| \le \frac{2n}{5}$. Thus $S'$ is a hop dominating set of $G$, which is a contradiction.
+
+Suppose that $\deg_G(x) = 2$. Then let $H$ be the connected component of $G'$ containing $x_4$. Note that $|V(H)| \le n - 5$. There is a minimum hop dominating set $S'$ of $G'$ such that $x_4 \in S'$, $N_{G'}(x_4) \cap S' \ne \emptyset$ by Proposition 2.7. If $S'$ contains $x_1$, then we replace $x_1$ with $x_3$ so that $S'$ does not have $x_1$. Then $S' \cup \{v, x\}$ is a minimum hop dominating set of $G$, a contradiction. $\square$
+
+Let $G' = G - \{x_1, \dots, x_{k-1}\}$. Suppose that $\deg_G(x) \ge 3$. Then $\delta(G') \ge 2$. If $G'$ has no connected component in $\mathcal{B}$, then take a minimum hop dominating set $S'$ of $G'$ such that $v, x \in S'$ by Proposition 2.7, which implies that $S'$ is a minimum hop dominating set of $G$, a contradiction. Thus, $G'$ has two connected components $D_1$ and $D_2$. We may assume that $D_1 \notin \mathcal{B}$ and $D_2 \in \mathcal{B}$. There is a hop dominating set $S_1$ of $D_1$ such that $v, x \in S_1$ and $|S_1| \le \frac{2|V(D_1)|}{5}$ by Proposition 2.7. If $k = 3$ or $D_2 \ne C_8$, then by (3.1), there is a minimum hop dominating set $S_2$ of $D_2$ such that $|S_2| \le \frac{2|V(D_2)|+2(k-1)}{5}$. If $k = 2$ and $D_2 = C_8$, then we take a minimum hop dominating set $S_2$ of a path $D_2 - x_2$ such that $|S_2| = 3 \le \frac{2|V(D_2)|+2(k-1)}{5}$. Note that $S_1 \cup S_2$ is a hop dominating set of $G$. Then
+
+$$
+|S_1 \cup S_2| \le \frac{2|V(D_1)| + 2|V(D_2)| + 2(k-1)}{5} = \frac{2n}{5},
+$$
+
+which is a contradiction.
+
+Suppose that $\deg_G(x) = 2$. Let $H$ be the connected component containing $x_k$. Then $|V(H)| \le n - 6$. If $H \ne C_8$, then for a minimum hop dominating set $S$ of $H$, we have $|S| \le \frac{2(n-6)+2}{5}$ by (3.1), and so $S \cup \{v, x\}$ is a hop dominating set of $G$ whose size is at most $\frac{2n}{5}$, a contradiction. Thus $H = C_8$. Then $G$ is one of the last two graphs in Figure 6, which shows that $\gamma_h(G) \le \frac{2n}{5}$. This completes the proof of our main theorem.
+
+# 4 An open problem
+
+It would be a natural extension to consider giving a sharp upper bound on the hop domination number for a graph with a large girth. So we propose the following open problem.

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+Fig. 1. System model.
+
+finally, our paper is concluded in Section VI.
+
+*Notations:* In this paper, boldface upper letters, boldface lower letters, and lower letters denote matrices, vectors, and scalars, respectively. $\mathbb{C}^{x \times y}$ and $\mathbb{R}^{x \times y}$ represent the set of complex matrices and real matrices with the dimension of $x \times y$, respectively. $|x|$ represents the amplitude of a complex number $x$. $\|x\|$ and $\|x\|_1$ denote the 2-norm and 1-norm of a vector $x$, respectively. The notation $(\cdot)^T$ and $(\cdot)^H$ refer to the transpose and the conjugate transpose of a vector or matrix, while $\mathbf{E}$ represents the expectation operator.
+
+## II. SYSTEM MODEL
+
+As shown in Fig. 1, we consider a multiple FAs assisted IDET system, which is comprised of a transmitter, a DR and an ER. The transmitter is equipped with $N_t$ FAs, each of which has the same size and parameters without loss of generality. The DR and ER are both equipped with a single traditional antenna due to the limited hardware space. We assume that the transmit antennas are far apart in space and the channels of different antennas are fully independent, while the channels among different ports in the same antenna are strongly spatially correlated. The set of FAs and FA's port are denoted by $\mathcal{N}_t = \{1, \dots, N_t\}$ and $\mathcal{N} = \{1, \dots, N\}$, respectively.
+
+### A. Fluid antenna model
+
+In this paper, we consider the practical architecture of the liquid metal-based fluid antenna. The length of fluid antenna is $W\lambda$, where $\lambda$ is the wavelength of RF signal and $W$ is a scaling constant. The MEMS is adopted in each fluid antenna to drive the metal droplet to move. Specifically, due to the electrocapillary effect, the motion of metal droplet is achieved by applying a voltage gradient along the FA, resulting in a phenomenon known as continuous electrowetting [24]. According to [5], the average velocity of the metal droplet motion is given by
+
+$$
+u = \frac{q}{6\mu} \frac{D}{L} \Delta\phi, \qquad (1)
+$$
+
+where $q$ is the initial charge in the electrical double layer for EGIn, $\mu$ denotes the viscosity of EGIn, $D$ and $L$ are the thickness and length of metal droplet, respectively. $\Delta\phi$ represents the voltage differential resulting from the flow of current through the narrow electrolyte layer between EGIn and the wall, which is much smaller than the voltage $U$ applied
+
+between two ends of the FA [25]. Therefore, the moving delay from $i$-th port to $j$-th port can be expressed as
+
+$$
+\tau = \frac{W\lambda |i-j|}{u (N-1)}. \qquad (2)
+$$
+
+### B. Wireless channel model
+
+The block fading channel model is assumed between the transmitter and receivers, implying that the channel remains constant during a coherence time $T$ but varies from one frame to another. Since the ports are located much closer in the same FA, strong correlation exists between the channels of different ports. Following Jake's two dimensional correlation model [26], the correlation between any two ports can be represented by a matrix $\mathbf{J}$, which is given by
+
+$$
+\mathbf{J} = \begin{bmatrix} J_{1,1} & J_{1,2} & \cdots & J_{1,N} \\ J_{2,1} & J_{2,2} & \cdots & J_{2,N} \\ \vdots & \vdots & \ddots & \vdots \\ J_{N,1} & J_{N,2} & \cdots & J_{N,N} \end{bmatrix}, \qquad (3)
+$$
+
+where $J_{i,j} = J_0(\frac{2\pi(i-j)}{N-1}W)$ and $J_0(\cdot)$ is the zeroth-order Bessel function of the first kind. The channels between $i$-th FA and DR and ER in the $t$-th block are denoted as $\mathbf{h}_i(t)$ and $\mathbf{g}_i(t)$, which can be modeled as
+
+$$
+\begin{aligned} \mathbf{h}_i(t) &= \sqrt{\frac{1}{PL_{h,i}}} \mathbf{U} \sqrt{\mathbf{\Lambda}} \mathbf{z}_h(t), \\ \mathbf{g}_i(t) &= \sqrt{\frac{1}{PL_{g,i}}} \mathbf{U} \sqrt{\mathbf{\Lambda}} \mathbf{z}_g(t), \end{aligned} \qquad (4)
+$$
+
+where $PL_{h,i}$ and $PL_{g,i}$ are the path-loss from $i$-th FA to DR and ER, respectively. $\mathbf{U}$ and $\mathbf{\Lambda}$ are a unitary matrix composed of the eigenvectors of $\mathbf{J}$ and a diagonal matrix consisting of the eigenvalues of $\mathbf{J}$, which satisfy $\mathbf{J} = \mathbf{U}\mathbf{\Lambda}\mathbf{U}^T$. In addition, $\mathbf{z}_h(t) \in \mathbb{C}^{N \times 1}$ and $\mathbf{z}_g(t) \in \mathbb{C}^{N \times 1}$ are complex Gaussian random vector with each element having zero mean and variance of 1.
+
+On the other hand, the channels of different transmission frames are temporally correlated, which can be expressed as
+
+$$
+\begin{aligned} \mathbf{h}_i(t) &= \rho \mathbf{h}_i(t-1) + \sqrt{\frac{1-\rho^2}{PL_{h,i}}} \mathbf{\delta}, \\ \mathbf{g}_i(t) &= \rho \mathbf{g}_i(t-1) + \sqrt{\frac{1-\rho^2}{PL_{g,i}}} \mathbf{\delta} \end{aligned} \qquad (5)
+$$
+
+where $\rho$ is the temporal correlation factor and $\mathbf{\delta} \sim \mathcal{CN}(0, \mathbf{I}_N)$ is the random difference with an identity matrix $\mathbf{I}_N$.
+
+### C. Signal model
+
+Before the transmitter delivering wireless signal to the DR and ER, an appropriate antenna port should be selected on each FA. Assuming that the initial port of $i$-th FA is $k_{i,0}(t)$ in the $t$-th block, the port switching delay on the $i$-th antenna from the initial port to the selected port $k_i(t)$ is expressed as
+
+$$
+\tau_i(t) = \frac{W\lambda |k_i(t) - k_{i,0}(t)|}{u (N-1)}. \qquad (6)
+$$

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+Prior to proceed let us consider the product $D_p x$. Since $D_p$ is an $n \times n$ matrix with entries in $\mathbb{Q}_p$ and $x$ is an $n$-tuple with entries in $\mathbb{Q}_p$, the result $D_p x$ is an $n$-tuple with entries in $\mathbb{Q}_p$. Formally, if $D_p = (d_{ij})$ and $x = (x_1, x_2, \dots, x_n)$, then the $i$-th component of $D_p x$ is given by:
+
+$$
+(D_p x)_i = \sum_{j=1}^{n} d_{ij} x_j \in \mathbb{Q}_p
+$$
+
+The p-adic norm $|\cdot|_p$ is applied component-wise to the resulting $n$-tuple $D_p x$. For each component $(D_p x)_i$ of $D_p x$, we have:
+
+$$
+|(D_p x)_i|_p = \frac{1}{p^{\text{ord}_p((D_p x)_i)}}
+$$
+
+where $\text{ord}_p((D_p x)_i)$ is the highest power of $p$ dividing $(D_p x)_i$.
+
+Since the p-adic norm $|\cdot|_p$ maps elements of $\mathbb{Q}_p$ to $\mathbb{Q}_p$, applying it component-wise to an $n$-tuple will result in another $n$-tuple with entries in $\mathbb{Q}_p$. Therefore, $|D_p x|_p$ is an $n$-tuple with entries in $\mathbb{Q}_p$.
+
+Thus, $\delta_p(x) = |D_p x|_p \in \mathbb{Q}_p^n$, hence, the map $\delta_p$ is well-defined.
+
+As we have seen in previous sections we can define the $p$-adic Ducci sequence the sequence
+
+**Definition 0.4** ($p$-adic Ducci sequence).
+
+$$
+x, \delta_p(x), \delta_p^2 = \delta_p(\delta_p(x)) = \delta_p \circ \delta_p, \dots, \delta_p^n(x), \dots
+$$
+
+with initial seed $x \in \mathbb{Q}_p^n$ respect to $D_p \in \mathbb{M}_{n \times n}(\mathbb{Q}_p)$. In this $p$-adic context, we define a $p$-adic Ducci map associated with a $p$-adic matrix $D_p$ as a function that maps a $p$-adic vector $x$ to $|D_p x|_p$, where $|\cdot|_p$ is applied elementwise.
+
+Without loss of generality we can define the following For convergence to zero, we analyze when $x_k \to 0$ as $k \to \infty$.
+
+Prior to move forward we recall the following
+
+**Remark 1.** In the field of $p$-adic numbers $\mathbb{Q}_p$, the $p$-adic absolute value $|x|_p$ of a nonzero element $x$ is defined as:
+
+$$
+|x|_p = p^{-v_p(x)},
+$$
+
+where $v_p(x)$ is the $p$-adic valuation of $x$. The valuation $v_p(x)$ is the highest power of $p$ that divides $x$ in $\mathbb{Q}_p$.
+
+If $|x|_p < 1$ and $x \neq 0$, this means that $v_p(x) > 0$, so $x$ is divisible by $p$. The possible values of $|x|_p$ in this case are:
+
+$$
+|x|_p = p^{-k}, \quad \text{where } k \in \mathbb{N} \text{ and } k \ge 1.
+$$
+
+Thus, the possible values of $|x|_p$ when $|x|_p < 1$ are:
+
+$$
+\{p^{-1}, p^{-2}, p^{-3}, \dots\}.
+$$

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+with
+
+$$
+\rho(y, x, \eta') = \varphi(y, \eta') - \varphi(x, \eta') - (y - x)\varphi_x(x, \eta'),
+$$
+
+$\rho$ vanishes at order $\ge 2$ in $y = x$. For checking (3.10), we see
+
+$$
+(3.11) \qquad Q(ae^{i\frac{\varphi}{h}}) = \iint e^{i\frac{1}{h}((x'-y')\theta' + \varphi(y, \eta'))} \frac{q(x, \theta')}{(2\pi h)^{n-1}} a(y, \eta') dy'd\theta'.
+$$
+
+a having compact support and $\varphi$ being a symbol, one has $|\varphi'_y| \le C'$ on $\text{supp}(qa)$, so for $|\theta'| \ge C > 0$, large, the phase of (3.11) $H = (x' - y')\theta' + \varphi(y, \eta')$ satisfies
+
+$$
+|H'_{y'}| \ge c(1 + |\theta'|), \quad \text{for } |\theta'| \ge C > 0, \text{ large}
+$$
+
+and if we split the integrand of (3.11) in $qa = \chi(\theta')qa + (1 - \chi(\theta'))qa$, we integrate the second term by parts and obtain
+
+$$
+(3.12) \qquad Q(ae^{i\frac{\varphi}{h}}) = Q'_{\varphi}(a)e^{i\frac{\varphi}{h}} + R(a),
+$$
+
+where $Q'_{\varphi}$ is a $\mathcal{G}^s$ symbol of order $\tilde{S}_s^{m,k+1}$ having the expansion (3.10) by the stationary phase lemma as $R(a)$ is an $\mathcal{O}_s(h^\infty)$ remainder. It is easy to see in view of these arguments that $a_1 \in \tilde{S}_s^{m-1,k}$. Moreover, it is to be observed that the above expansion is only a formal Gevrey $2s-1$ symbol.
+
+The microlocal invertibility of FIO reduces to the PDO case. We refer to [5] for a proof of the Gevrey elliptic result in classes $S_s^m$.
+
+For proving Theorem 2, we rewrite (3.8) in the form
+
+$$
+(hD_{x_1} + Q(x, hD_x; h))Fu = F(hD_{x_1} + Q')u,
+$$
+
+close to $(x_0, \xi_0; x_0, \xi_0)$ for some PDO $Q'(x, hD_x; h)$ of bi-order $(-1, 0)$ in using a left microlocal inverse of $F$ close to $(x_0, \xi'_0)$. Indeed, we compute $FF^*$ and $F^*F$, and one has writing $y = (x_1, y')$.
+
+One has, following Eskin [8],
+
+$$
+FF^*u(x, h) = \frac{1}{(2\pi h)^{n-1}} \iint e^{i\frac{1}{h}(\varphi(x, \xi') - \varphi(y, \xi'))} a(x, \xi') \overline{a(y, \xi')} u(x_1, y') dy' d\xi',
+$$
+
+$\varphi(x, \xi')$ having been obtained in (3.6). We split the integral above into two terms. The first is a $h$-PDO, the second is a smoothing operator. First, we note that the map:
+
+$$
+(x, y', \xi') \to (x, y', \Sigma(x, y', \xi')),
+$$
+
+with
+
+$$
+(3.13) \qquad \Sigma(x, y', \xi') = \int_0^1 \varphi'_{x'}(x_1, y' + t(x' - y'), \xi') dt
+$$
+
+is a $\mathcal{G}^s$-diffeo in a neighbourhood of $(x_0, y'_0, \eta'_0)$ with $|x_1| \le \delta$, $|x' - y'| \le \delta$, $0 < \delta$ small, close to the identity.
+
+Let $(x, y', \eta') \to (x, y', \Sigma^{-1}(x, y', \eta'))$ be an inverse map.
+
+One has obviously $\varphi(x, \xi') - \varphi(y, \xi') = \Sigma(x, y', \xi')(x' - y')$, and
+
+$$
+(3.14) \qquad FF^*u(x, h) = K_1 u(x, h) + K_2 u(x, h),
+$$

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+(resp. dual) closedness criterion. To this aim, let us recall the concept of closedness regarding a set [1].
+
+Given two subsets $U$ and $V$ of a topological space $Z$, one says that $U$ is closed regarding $V$ if $\overline{U} \cap V = U \cap V$, where $\overline{U}$ is the closure of $U$ in $Z$. In particular, Given $z \in Z$, $U$ is closed regarding $\{z\}$ if and only if either $z \in U$ or $z \notin \overline{U}$, $U$ is closed regarding $V$ if and only if $U$ is closed regarding $\{z\}$ for all $z \in V$, and $U$ is closed if and only if it is closed regarding $Z$.
+
+The term *characterization of Farkas' lemma* was introduced by V. Jeyakumar, S. Kum, and G. M. Lee [13] and that of *characterization of stable Farkas' lemma* by V. Jeyakumar and G. M. Lee [14], both papers published in 2008. Their characterization of stable Farkas' lemma consisted in the closedness of certain cone associated with the given conic system. After the introduction of the concept of closedness regarding a given set by Boğ in 2010, successive characterizations of (stable) Farkas' lemma in terms of closedness of certain set regarding another set were obtained for different types of systems, e.g., systems involving convex, nonconvex composite functions [8], [9], [10], systems involving vector-valued functions ([4] and [6]), systems involving a family of finite subsets of the given index set [7], etc. The characterization of stable Farkas' lemma for nonconvex composite semi-infinite programming has been considered in [16] and [17].
+
+Considering feasible sets of the form $C \cap \mathbb{A}^{-1}(D)$, with the convex set $D$ not being necessarily a cone, is the main novelty of this paper, which is organized as follows. Section 2 characterizes the equivalence $(\mathcal{A}) \iff (\mathcal{B})$ in terms of the closedness regarding $\{(0_X, 0_Y, -1)\}$ and $\{(0_Y, -1)\}$ of certain subsets of the primal spaces $X \times Y \times \mathbb{R}$ and $Y \times \mathbb{R}$, respectively (see Theorems 1 and 2). Section 3, in turn, characterizes the equivalence $(\mathcal{A}) \iff (\mathcal{B})$ in terms of the closedness regarding $\{(0_{X'}, 0)\}$ of certain subset of the dual space $X' \times \mathbb{R}$ (Theorem 11) as well as the non-emptiness of the feasible set $C \cap \mathbb{A}^{-1}(D)$ (Proposition 9) and the stable Farkas' lemma (Proposition 16). Section 4 is devoted to linear infinite systems. Section 5 characterizes Farkas-type lemmas oriented to identify the minima of convex (resp. concave) functions on $C \cap \mathbb{A}^{-1}(D)$, reason why they are known in the literature as convex (resp. concave) Farkas' lemmas. More in detail, the convex Farkas' lemmas provided in Section 3 determine when a convex feasible set of the form $C \cap \mathbb{A}^{-1}(D)$ is contained in the reverse-convex set $\{x \in X : f(x) \ge 0\}$, where $f$ is convex. Similarly, Section 5 characterizes the containment of $C \cap \mathbb{A}^{-1}(D)$ in the sublevel set of a convex function $f$ (Theorem 23). Finally, Section 6 shows applications to constrained convex minimization problems (optimization, strong duality and stable strong duality theorems) and to functional approximation by polynomials (Farkas-type lemmas).
+
+## 2 Primal closedness characterizations
+
+Recall that $f: X \to \mathbb{R}$ ($f \in \mathbb{R}^X$ in short) is a proper convex function. We consider the vector-valued mapping $H: X \times Y \times \text{dom } f \to X \times Y \times \mathbb{R}$ such that
+
+$$
+H(x, y, v) = (x - v, \mathbb{A}x - y, f(v)),
+$$

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+## 2 Simple Proof
+
+This section presents a simple proof of the Apex Minor Theorem. The next lemma is the key. A *centre* of a graph $G$ is a vertex $\alpha \in V(G)$ such that $\max\{\text{dist}(\alpha, v) : v \in V(G)\}$ is equal to the radius of $G$.
+
+**Lemma 10.** Let $A$ be a planar graph such that $A - z$ is a minor of the $k \times k$ grid for some $z \in V(A)$. Let $G$ be an $A$-minor-free graph with radius $r$ and centre $\alpha$. Then $G - \alpha$ has no $k^r \times k^r$ grid minor.
+
+*Proof.* We proceed by induction on $r$. In the $r = 0$ case, $V(G - \alpha) = \emptyset$ and the result holds. Now consider the radius $r$ case, and assume the radius $r - 1$ case holds. Let $V_i := \{x \in V(G) : \text{dist}_G(x, \alpha) = i\}$ for $i \in \{0, \dots, r\}$. So $V_0 = \{\alpha\}$ and $V_0 \cup \dots \cup V_r = V(G)$.
+
+Suppose for the sake of contradiction that $G - \alpha$ has a $k^r \times k^r$ grid minor. Partition this $k^r \times k^r$ grid into $k \times k$ subgrids, so that contracting each subgrid to a vertex gives a $k^{r-1} \times k^{r-1}$ grid.
+
+First suppose that some $k \times k$ subgrid is contained in $V_r$. Thus $H$ is a minor of $G[V_r]$. By construction, $G[V_0 \cup \dots \cup V_{r-1}]$ is connected. Let $G'$ be obtained from $G$ by contracting $V_0 \cup \dots \cup V_{r-1}$ to a vertex $w$. Since every vertex in $V_r$ has a neighbour in $V_{r-1}$, every vertex in $G'$ is at distance at most 1 from $w$. Since $H$ is a minor of $G[V_r]$, $A$ is a minor of $G'$ and thus of $G$, which is a contradiction.
+
+Now assume that every $k \times k$ subgrid intersects $V_1 \cup \dots \cup V_{r-1}$. Let $G'$ be obtained from $G$ by contracting each subgrid to a vertex, and deleting any vertices in $V_r$ not in one of the subgrids. So every vertex in $G'$ is at distance at most $r-1$ from $\alpha$, and $G' - \alpha$ contains a $k^{r-1} \times k^{r-1}$ grid minor. Since $G'$ is a minor of $G$, $G'$ is $A$-minor-free. Hence $G'$ contradicts the assumed truth of the $r-1$ case.
+
+Therefore, $G - \alpha$ has no $k^r \times k^r$ grid minor.
+
+□
+
+Lemmas 2 and 10 imply that for every apex graph $A$ with $t$ vertices, every $A$-minor-free graph $G$ with radius $r$ has no $(2t-2)^r \times (2t-2)^r$ grid minor. Equation (1) implies that $\text{tw}(G) \in O^*((2t)^{9r})$. This completes our simple proof of the Apex Minor Theorem by Eppstein [18] (Theorem 4).
+
+## 3 Polynomial Upper Bound
+
+This section proves Theorem 5 from Section 1. We need the following straightforward lemma.
+
+**Lemma 11.** If a graph $G$ is a minor of the $k \times \ell$ grid, then there is an $H$-model ($B_u : u \in V(H)$) in the $2k \times 2\ell$ grid $J$, such that for each vertex $u \in V(H)$ there is a vertex $h_u$ in $B_u$ incident to no edge of $J$ representing an edge of $H$.

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+for the Dirichlet form associated to sticky Brownian motion. Denoting $a_0 = \frac{1}{2+\omega L}$ and $b_0 = \omega a_0$, for all $s > 0$ we have
+
+$$
+\begin{align*}
+\int f^2 d\mu &= a_0(f(0)^2 + f(L)^2) + b_0 \int_0^L f^2 dx, \\
+&\le a_0(f(0)^2 + f(L)^2) + b_0 s \int_0^L (f')^2 dx + b_0 \beta(s) \left( \int_0^L |f| dx \right)^2, \\
+&\le b_0 s \int_0^L (f')^2 dx + b_0 \max(b_0^{-2}, a_0^{-1}) \beta(s) \left( \int |f| d\mu \right)^2.
+\end{align*}
+$$
+
+Therefore the sticky Brownian satisfies a super Poincaré inequality. Then by [Wan00, Th. 5.1], it has an empty essential spectrum. Now, by [BGL14, Th. A.6.4], the resolvent is compact and thus the generator has discrete spectrum. $\square$
+
+**Corollary 19.** Choosing $T = m^{-1/2}$, the transition semigroup of the RTP process is exponentially contractive in T-average with rate
+
+$$
+\nu = \Omega \left( \frac{\omega}{1 + (\omega L)^2} \right).
+$$
+
+Note that the relaxation time corresponding to this decay rate is of the same order as the mixing time obtained in [GHM24]. It reveals the existence of two regimes controlled by the parameter $\omega L$. In the ballistic regime $\omega L \ll 1$, velocity flips are rare, leading to a fast exploration of the position space $\mathcal{S}$ and a comparatively slow exploration of the velocity space $\mathcal{V}$. This results in the scaling $\nu \propto \omega$. On the contrary, in the diffusive regime $\omega L \gg 1$, the high frequency of velocity flips makes the exploration of $\mathcal{V}$ faster than the exploration of $\mathcal{S}$. This leads to the scaling $\nu \propto \omega^{-1} L^{-2}$.
+
+*Proof.* We begin by verifying Assumption (A). Recall that $\text{Dom}(\mathcal{L}_{C^0})$ is a core of $\mathcal{L}$ by Theorem 7. For all $f \in \text{Dom}(\mathcal{L}_{C^0})$ we have $\hat{\mathcal{L}}_v(f \circ \pi) = 0$ hence $\hat{\mathcal{L}}_{\text{tr}}$ is a lift of $\mathcal{L}$ by Remark 8. Furthermore, for $f \in \text{Dom}(\mathcal{L}_{C^0})$ one has
+
+$$
+\hat{\mathcal{L}}_{\text{tr}}^*(f \circ \pi)(x, v) = -v 1_{\{0<x<L\}} f'(x) = -\hat{\mathcal{L}}_{\text{tr}}(f \circ \pi)(x, v).
+$$
+
+A straightforward computation yields
+
+$$
+\int_{\mathcal{V}} \hat{\mathcal{L}}_v f(x, v) d\kappa_x(v) = 0 \text{ for all } x \in \mathcal{S} \text{ and } f \in \text{Dom}(\hat{\mathcal{L}}).
+$$
+
+Finally, we prove $\|f - \Pi_v f\|_{L^2(\hat{\mu})}^2 \le \frac{1}{m_v} \mathcal{E}_v(f)$ with $m_v = 2$. Define the matrices
+
+$$
+S = \begin{pmatrix} 1/4 & 0 & 0 \\ 0 & 1/2 & 0 \\ 0 & 0 & 1/4 \end{pmatrix}, \quad \mathcal{Q} = \begin{pmatrix} -2 & 2 & 0 \\ 1 & -2 & 1 \\ 0 & 2 & -2 \end{pmatrix},
+$$
+
+as well as the scalar product $\langle x, y \rangle_S = x^\top S y$ and let $\Pi$ be the orthogonal projection on the kernel of $\mathcal{Q}$ with respect to $\langle \cdot, \cdot \rangle_S$. The matrix $\mathcal{Q}$ is symmetric w.r.t. the scalar

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+admits a finite index subgroup with positive first Betti number. Since the quotient of the unit ball in $\mathbf{C}^n$ by a discrete cocompact subgroup has nonzero Euler characteristic, the proposition 3.1 can be applied.
+
+*Remark 3.5.* — Llosa Isenrich and Py showed in fact the existence of infinitely many, pairwise not commensurable, word hyperbolic groups admitting subgroups of type $\mathcal{F}_n$ and not of type $\mathcal{F}_{n+1}$.
+
+# 4. Construction from right-angled polytopes
+
+Hereafter the article Italiano, Martelli, and Migliorini (2023) will be mentioned as IMM5 and the article Italiano, Martelli, and Migliorini (2022) will be mentioned as IMM8.
+
+The approach developed by IMM5 for the proof of theorem 1.2 is more combinatorial in nature. It starts by constructing a finite volume hyperbolic 5-manifold from a right-angled polytope in the hyperbolic space $\mathbb{H}_5$, then a fibration $f: M \to S^1$ is constructed using a natural cubulation of the manifold $M$. In order to produce a compact object (and hence a word hyperbolic group) one needs to cap the boundary components of $M$ to obtain a metric space $M^\vee$; this can be done maintaining the negatively curved metric on $M^\vee$ and an extension of the fibration exists.
+
+## 4.1. The polytope and the manifold
+
+The chosen model for hyperbolic space $\mathbb{H}_5$ is the Klein model: the unit ball in $\mathbf{R}^5$ with geodesic given by Euclidean segments.
+
+The polytope $P_5$ in $\mathbb{H}_5$ is described as the intersections of the half-spaces
+
+$$
+\underline{\varepsilon} \cdot \underline{x} = \sum_{i=1}^{5} \varepsilon_i x_i \le 1, \quad \underline{x} \in \mathbb{H}_5
+$$
+
+where $\underline{\varepsilon}$ varies in the subgroup of $\{\pm 1\}^5$ defined by $\prod \varepsilon_i = 1$, i.e. an even number of the $\varepsilon_i$ are equal to $-1$. We refer to IMM5 (section 1.1) for a complete description, and to IMM8 for further details on that polytope as well as a related series of right-angled polytopes in dimensions 3, ..., 8. These polytopes were previously studied by Potyagailo and Vinberg (2005) who explained them starting from certain hyperbolic simplices. They are related (by duality) to a series of semiregular polytopes discovered by Gosset (1899).
+
+The polytope $P_5$ has finite volume, is right-angled, and has 16 facets given by the hyperplanes where equality is achieved in the equation above. It has a big group of symmetries: the permutation of coordinates as well as the coordinate-wise pluttifkation by $\underline{\varepsilon}$ (cf. Lindgren, 1945, for this classical operation); this produces a group of symmetries of type $D_4$ and of order $2^4 \times 5! = 1920$. The hyperbolic reflections through the 16 hyperplanes bounding $P_5$ generate a discrete subgroup $\Gamma$ of $\text{Isom}(\mathbb{H}_5)$ that is known to be isomorphic to the congruence two subgroup of the group of integral matrices in

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+and cutting the polygon into quadrilateral regions. After Baryhsnikov, one can define a compatibility relation between quadrangulations. On the set $R_Q$ of quadrangulations compatible with $Q$, there is a mutation process, that defines a mutation graph. Moreover, this mutation process is oriented and defines a partial order. The mutation graph is also the skeleton of a polytope, called the Stokes polytope for the quadrangulation [MP19, Man18, BMP17].
+
+There is a simple way to map the planar-embedded arbor $t$ into a quadrangulation. There is also a way to define a gentle quiver from any quadrangulation. The composite map recovers the definition of $G_t$ from $t$. By this correspondence, the $\tau$-tilting theory exactly matches the combinatorial description using mutations of compatible quadrangulations.
+
+It follows that the conjectural statements above could be rephrased entirely using the structures (mutation graph, poset, polytope) coming from this context instead of those coming from $\tau$-tilting theory.
+
+## 6 Arbors of type $\mathbb{A}$
+
+In this section, we consider the case of linear arbors in which each vertex has one element and at most one sub-tree, so that their shape is a simple chain. This is the intersection of the set of unary-binary arbors considered before with the set of linear arbors. In order to compute the Zeta polynomial and the $M$-triangle for the poset $P_t$ in this case, it is convenient to study a more general family of posets.
+
+In this section, we will use the following convention: for a product $\beta_\ell = \prod_{r=2}^\ell \alpha_r$, the value at $\ell = 1$ is 1 and the value at $\ell = 0$ is $1/\alpha_1$.
+
+Let $m \ge 1$ be an integer, used as a slope parameter. Let $R_m$ be the region in the plane of coordinates $(x, y)$ defined by the conditions
+
+$$
+0 \le x, \quad 0 \le y, \quad my < x. \qquad (28)
+$$
+
+Let now $(x, y)$ be nonnegative integers such that
+
+$$
+my < x. \qquad (29)
+$$
+
+We consider the set of lattice paths in $R_m$ that go from $(0, 0)$ to the fixed point $(x, y)$, by steps $(+1, 0)$ (denoted by $\mathsf{X}$) or $(0, +1)$ (denoted by $\mathsf{Y}$). The point $(0, 0)$ is the unique allowed point on the line $my = x$ at the boundary of $R_m$.
+
+Figure 2: Example of lattice path $\mathsf{X}^3\mathsf{Y}^2\mathsf{Y}^3\mathsf{Y}$ with $m = 2$, $x = 8$ and $y = 3$. The word $\mathsf{A}$ associated with vertical steps is $(0, 0, 1, 0, 1, 0, 0)$.

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+(2) For all objects $x$ and $y$ of $\mathcal{B}$, the torsion groups $T_\bullet = \mathrm{Tor}_\bullet^\mathcal{A}(\phi^*h_{\mathcal{B}^{\mathrm{op}}}^x, \phi^*h_{\mathcal{B}}^y)$ (calculated in the category of additive functors from $\mathcal{A}$ to abelian groups) satisfy
+
+$$
+k \otimes_{\mathbb{Z}} T_i = 0 = \mathrm{Tor}_1^\mathbb{Z}(k, T_j) = 0, \text{ for } 0 < i < e \text{ and } 0 < j < e - 1.
+$$
+
+The remainder of the section is devoted to the proof of theorem 3.4. This proof depends on the use of simplicial techniques, and in particular on variants of the Hurewicz theorem. For the convenience of the reader, the simplicial notions and results that we need are recalled in the appendix (section 6). The first step of the proof is the following general lemma, which is of independent interest. It is well-known to experts, but we do not know any written reference for it.
+
+**Lemma 3.5.** Let $A : \mathcal{A}^{\mathrm{op}} \to \mathbb{Z}\text{-Mod}$ and $B : \mathcal{A} \to \mathbb{Z}\text{-Mod}$ be two additive functors, and let $k[A]$ and $k[B]$ denote the composition of these functors with the $k$-linearization functor $k[-]$. There is an isomorphism of $k$-modules, natural with respect to $A$ and $B$:
+
+$$
+k[A] \otimes_{k[\mathcal{A}]} k[B] \simeq k[A \otimes_{\mathbb{Z}[\mathcal{A}]} B].
+$$
+
+*Proof.* We first recall concrete formulas for tensor products. For all commutative rings $K$, the tensor product $F \otimes_{K[\mathcal{A}]} G$ can be concretely computed as the quotient of the direct sum $\bigoplus_x F(x) \otimes_K G(x)$ indexed by a set of representatives of the isomorphism classes of objects of $\mathcal{A}$, modulo the relations $F(f)(s) \otimes t = s \otimes G(f)(t)$ for all morphisms $f : x \to y$ and for all elements $s \in F(y)$ and $t \in G(x)$. We will denote by $\llbracket s \otimes t \rrbracket \in F \otimes_{K[\mathcal{A}]} G$ the class of an element $s \otimes t \in F(x) \otimes_K G(x)$. When $G = P^c$ is a standard projective there is a ‘Yoneda isomorphism’
+
+$$
+\Upsilon : F \otimes_{K[\mathcal{A}]} P^c \simeq F(c)
+$$
+
+given by sending the class $\llbracket s \otimes f \rrbracket$ with $s \in F(x)$ and $f \in \mathcal{A}(c, x)$ to $F(f)(s)$ (the inverse isomorphism sends $u \in F(c)$ to $\llbracket u \otimes \mathrm{id}_c \rrbracket$). Similarly, if $F$ is additive and $G = h^c$ is a standard additive projective, there is an ‘additive Yoneda isomorphism isomorphism’
+
+$$
+\Upsilon_{\mathrm{add}} : F \otimes_{K[\mathcal{A}]} h^c \simeq F(c).
+$$
+
+When $k = \mathbb{Z}$, the isomorphism $\Upsilon_{\mathrm{add}}$ sends class $\llbracket s \otimes f \rrbracket$ with $s \in F(x)$ and $f \in \mathcal{A}(c, x)$ to $F(f)(s)$.
+
+We are now ready to construct the isomorphism of lemma 3.5. For all objects $x$ of $\mathcal{A}$, we let $\theta_{A,B,x} : k[A(x)] \otimes k[B(x)] \to k[A \otimes_{\mathbb{Z}[\mathcal{A}]} B]$ be the $k$-linear map such that $\theta_{A,B,x}(s \otimes t) = \llbracket s \otimes t \rrbracket$ for all $s$ in $A(x)$ and all $t \in B(x)$. The maps $\theta_{A,B,x}$ induce a $k$-linear map, natural in $A$ and $B$:
+
+$$
+\Theta_{A,B} : k[A] \otimes_{k[\mathcal{A}]} k[B] \to k[A \otimes_{\mathbb{Z}[\mathcal{A}]} B].
+$$
+
+If $B = h^c$ is a standard projective additive, the composition $k[\Upsilon_{\mathrm{add}}] \circ \Theta_{A,B}$ is equal to $\Upsilon$, hence $\Theta_{A,B}$ is an isomorphism in this case. Finite direct sums of standard projective additives are isomorphic to standard projective additives, hence $\Theta_{A,B}$ is also an isomorphism if $B$ is a finite direct sum of standard additive projectives. Now the source and the target of $\Theta_{A,B}$, viewed as functors of $B$ preserve filtered colimits of monomorphisms, which implies in turn that $\Theta_{A,B}$ is an isomorphism if $B$ is an arbitrary direct sum of standard projective additives.
+
+Now let $B$ be arbitrary and let $\epsilon : P \to B$ be a projective simplicial resolution of $B$ in $\mathcal{A}\text{-Mod}$ by direct sums of standard projective additives. Then we have a commutative square of simplicial $k$-modules in which the top row is an isomorphism,

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+A Constructive Approach for Building Wavelet Bases in $L^2(\mathbb{R}^d, \mathbb{R}^m)$ with Optimal Properties
+
+Hicham TARIF, Nadir MAAROUFI
+
+March 2024
+
+Abstract
+
+The main contribution of this paper is a constructive method for building separable multivariate vector-valued wavelet bases in the general framework of $L^2(\mathbb{R}^d, \mathbb{R}^m)$ for any $d, m \ge 1$. While separable wavelet bases in $L^2(\mathbb{R}^d, \mathbb{R})$ are well-established and widely applied, the explicit construction of truly vector-valued wavelet bases remains an open problem, even in the simplest case of $L^2(\mathbb{R}, \mathbb{R}^2)$, let alone in $L^2(\mathbb{R}^2, \mathbb{R}^2)$. In practice, the conventional approach applies standard separable wavelet bases of $L^2(\mathbb{R}^2, \mathbb{R})$ independently to each component of vector-valued signals in $L^2(\mathbb{R}^2, \mathbb{R}^2)$. However, this approach fails to capture the intrinsic vectorial structure of the signals. To address this limitation, we propose a constructive approach within the vector-valued wavelet framework, providing a systematic method for constructing such bases in the general case of $L^2(\mathbb{R}^d, \mathbb{R}^m)$. By linking $m$-multiwavelets to vector-valued wavelets, our approach not only enables the systematic construction of separable multivariate bases in $L^2(\mathbb{R}^d, \mathbb{R}^m)$ that satisfy the vector-valued multiresolution analysis but also ensures that these bases inherit key structural properties, making them well-suited for practical applications.
+
+**Key words :** Multivariate wavelet bases; Vector-valued wavelet bases; Vector-valued multiresolution decomposition; $m$-multiwavelets; Separable wavelets.
+
+# 1 Introduction
+
+Since the 1990s, wavelets have gained significant attention in signal processing and data analysis due to their ability to capture both frequency and time-domain information simultaneously. This dual capability makes them particularly well-suited for analyzing non-stationary signals. Wavelets have become a standard theoretical framework for analyzing signals [2], processes [21], and multifractal functions [22]. A fundamental concept in this framework is multiresolution decomposition (MRA), which enables the representation of a function at different levels of resolution. By capturing local and global information, wavelets provide a flexible tool for tasks ranging from function approximation to data compression and reconstruction from samples [1]. This versatility has revolutionized the analysis of functions and signals in theoretical mathematics and engineering applications. The origins of wavelets date back to 1909 when Alfred Haar introduced the first wavelet basis [30], long before the term "wavelet" was coined. The Haar wavelet, while simple and orthogonal, is discontinuous, limiting its applicability in some practical contexts. However, Haar wavelets have proven useful in various theoretical contexts [29].
+
+Significant advances in wavelet theory were achieved in the 1980s, notably by Yves Meyer [31] and Ingrid Daubechies [16]. Meyer's wavelets are constructed using a band-limited function, ensuring smoothness and regularity in the time domain. Although they lack compact support, their smoothness and orthogonality make them suitable for applications requiring continuously differentiable wavelets. In contrast, Daubechies wavelets are constructed to possess compact support and a specified number of vanishing moments. These properties enhance localization and facilitate the approximation of smooth functions. A key advantage of Daubechies wavelets is their ability to combine compact support with adjustable vanishing moments, making them particularly effective for smooth function approximation, signal analysis, and image processing [16, 39, 38]. The vanishing moments of the Daubechies wavelets play a crucial role in the multifractal formalism. More precisely, they make the wavelets orthogonal to low-order polynomials, enabling the removal of such trends. Sequentially, wavelet transforms based on Daubechies wavelets characterize Hölder spaces and pointwise regularity. They allow estimating the Hölder exponent through decay conditions in wavelet leaders [3].
+
+Yves Meyer's pioneering work on wavelets laid the foundation for Stéphane Mallat's multiresolution analysis (MRA) in the scalar case. MRA provides a systematic and theoretical framework for constructing wavelet bases for spaces such as $L^2(\mathbb{R})$. A key starting point in MRA is to find a scaling function (father wavelet) that satisfies specific properties, from which the corresponding wavelet function (mother wavelet) is derived [31]. This framework enables the decomposition of signals across multiple scales, facilitating analysis at various levels of resolution, and permitting the construction of a wavelet basis. MRA also provides a theoretical framework for constructing multivariate scalar wavelet bases in spaces such as $L^2(\mathbb{R}^d, \mathbb{R})$. However, despite the general scheme of MRA, building such multivariate wavelet bases in practice remains challenging [27]. The primary difficulty lies in finding a suitable multivariate scaling
+
+arXiv:2503.04255v1 [math.FA] 6 Mar 2025

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+so $[\phi]$ is a class in $\mathcal{F}^k H_{-1}(\mathrm{gr}_\mathcal{L}^i \mathfrak{h})$. We now consider the maps
+
+$$
+\mathcal{F}^k H_{-1} (\mathrm{gr}_\mathcal{L}^i \mathfrak{h}) \to \mathcal{F}^k H_{-1} \left( \frac{\mathfrak{h}}{\mathcal{L}^{i+1} \mathfrak{h}} \right) \to \mathcal{F}^k H_{-1} \left( \frac{\mathfrak{h}}{\mathcal{L}^{i+1} \mathfrak{h} + \mathcal{F}^{k+1} \mathfrak{h}} \right) \xleftarrow{f} \mathcal{F}^k H_{-1} \left( \frac{\mathfrak{h}}{\mathcal{L}^{i+1} \mathcal{F}^k \mathfrak{h} + \mathcal{F}^{k+1} \mathfrak{h}} \right)
+$$
+
+The last group is where the $k$-th intermediate gauge triviality class $\vartheta_k^i$ lives. But the image of this class in $\mathcal{F}^k H_{-1}(\mathfrak{h}/\mathcal{L}^{i+1}\mathfrak{h} + \mathcal{F}^{k+1}\mathfrak{h})$ is equal to the image of $[\phi] \in \mathcal{F}^k H_{-1}(\mathrm{gr}_\mathcal{L}^i \mathfrak{h})$, which vanishes by assumption. Therefore, to show that all classes $\vartheta_k^i$ vanish, it is sufficient to show that the map $f$ is injective. Suppose we have $[\alpha] \in \ker(f)$. Then we can find $\lambda$ such that
+
+$$
+d^\psi \lambda \equiv \alpha \pmod{\mathcal{L}^{i+1} \mathfrak{h} + \mathcal{F}^{k+1} \mathfrak{h}}.
+$$
+
+Let us set $\lambda' = \lambda^{(k-1)} + \lambda^{(k)} + \cdots$. Since $d^\psi$ is homogeneous of weight one, we must have
+
+$$
+d^\psi (\lambda') \equiv \alpha \pmod{\mathcal{L}^{i+1} \mathfrak{h} + \mathcal{F}^{k+1} \mathfrak{h}},
+$$
+
+since $\alpha$ is in $\mathcal{F}^k \mathfrak{h}$. Thus $\lambda'$ is a primitive of $\alpha$ in $\mathfrak{h}/\mathcal{L}^{i+1} \mathcal{F}^k \mathfrak{h} + \mathcal{F}^{k+1} \mathfrak{h}$. $\square$
+
+2.3. **Application to formality of properadic algebras.** The aim of the present article is to study the (intrinsic) coformality properties of Definition 1.33. To do so, we will use the obstruction theories developed in Section 2.1. Beforehand, let us recall the approach of formality as a deformation problem. In all this section, the ring $R$ is a $\mathbb{Q}$-algebra. Let $\mathcal{C}$ be a reduced weight-graded dg coproperad, e.g. $\mathcal{C} = \mathcal{Y}^{(n)i}$. Given any chain complex $(A, d_A)$, we have a convolution dg Lie admissible algebra
+
+$$
+\mathfrak{g}_A = (\mathrm{Hom}_S (\bar{\mathcal{C}}, \mathrm{End}_A), \partial, \star),
+$$
+
+whose Maurer–Cartan elements are in bijection with $\Omega\mathcal{C}$-structures on $A$. Recall that an $\infty$-morphism between two $\Omega\mathcal{C}$-algebra structures is an $\infty$-isotopy if its first component is the identity. We denote by $\Gamma_A$ the set of all $\infty$-isotopies. The existence of gauge equivalences between Maurer–Cartan elements in $\mathfrak{g}_A$ corresponds to existence of $\infty$-isotopies between the corresponding $\Omega\mathcal{C}$-structures thanks to the following theorem.
+
+**Theorem 2.14** ([3, Theorem 2.16]). If $R$ is a $\mathbb{Q}$-algebra, the set of all the $\infty$-isotopies between $\Omega\mathcal{C}$-algebra structures forms a group which is isomorphic though the graph exponential/logarithm maps to the gauge group of $\mathfrak{g}_A$
+
+$$
+\exp : ((\mathfrak{g}_A)_0, \mathrm{BCH}, 0) \cong (\Gamma_A, \odot, 1) : \log .
+$$
+
+Suppose that $\mathcal{C}$ is a reduced weight-graded coproperad (with no differential) and let $H$ be a graded $R$-module. The associated convolution dg Lie admissible algebra $\mathfrak{g}_H$ is weight-graded Lie algebra in the sense of Assumptions 1, with the weight grading coming from that of $\mathcal{C}$. It also has an extra filtration where $\mathcal{L}^i \mathfrak{g}_A$ is all the operations with $(i+1)$ or more inputs. More precisely, the $(\mathbb{S}^{\mathrm{op}} \times \mathbb{S})$-module $\mathcal{C}$ has a direct sum decomposition under a *second* grading
+
+$$
+\mathcal{C} = I \oplus \mathcal{C}_{(1)} \oplus \mathcal{C}_{(2)} \oplus \mathcal{C}_{(3)} \oplus \cdots
+$$
+
+where $\mathcal{C}_{(i)}$ is spanned by the operations with $i$ outputs. This gives a weight decomposition
+
+$$
+\mathfrak{g}_A = (\mathfrak{g}_A)_{(1)} \times (\mathfrak{g}_A)_{(2)} \times (\mathfrak{g}_A)_{(3)} \times \cdots
+$$
+
+and we can then define the extra filtration by
+
+$$
+\mathcal{L}^i \mathfrak{g}_A = \prod_{j \ge i+1} (\mathfrak{g}_A)_{(j)}.
+$$
+
+Assuming $\mathcal{L}$ is relatively bounded, that is bounded with respect to $\mathcal{F}$, we can apply all the methods of Section 2.1, giving the following applications of Theorem 2.10 and Theorem 2.13.

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+Observe that $\mathbf{I} : \mathcal{L}^2 \to \mathcal{L}^2$ is a monotone operator as soon as $h$ is a nondecreasing map, since
+
+$$
+\begin{align*}
+\langle \mathbf{I}(u) - \mathbf{I}(v), u - v \rangle &= \frac{1}{k(T)} \mathbb{E} \left[ (h(Z_T^u) - h(Z_T^v)) \int_0^T G(T-t)(u_t - v_t) \mathrm{d}t \right] \\
+&= \frac{1}{k(T)} \mathbb{E} \left[ (h(Z_T^u) - h(Z_T^v)) (Z_T^u - Z_T^v) \right] \ge 0, \quad u, v \in \mathcal{L}^2.
+\end{align*}
+$$
+
+Then, applying Fubini's Theorem we get
+
+$$
+\langle \mathbf{III}(u) - \mathbf{III}(v), u - v \rangle = -\mathbb{E} \left[ \int_0^T (h'(Z_t^u) - h'(Z_t^v))(Z_t^u - Z_t^v) \frac{\mathrm{d}g_t}{k(t)} \right], \quad u, v \in \mathcal{L}^2,
+$$
+
+so that the monotonicity of the operator $\mathbf{III}$ may not be true in general and depends on the dynamics of the endogenous signal $g$. The particular case where $\mathrm{d}g_t = g'(t)\mathrm{d}t$ with $g' \ge 0$ (i.e., $g$ is a nondecreasing input curve) and $h'$ is nonincreasing (i.e., the impact function $h$ is concave on the real line) would yield the monotonicity of $\mathbf{III}$, but such assumptions may be too restrictive.
+
+Furthermore, verifying the monotonicity property of the operators $\mathbf{II}$ and $\mathbf{IV}$ in general is not obvious and depends on the form of the kernel $G$ and its first-argument derivative $\partial_x G$.
+
+**Case of one exponential for the impact decay.** For this reason, in what follows we assume (i) $h$ to be nondecreasing and we restrict our attention to the exponential Volterra kernel, i.e.,
+
+$$
+G(t, s) = \mathbb{1}_{\{t \ge s\}} a e^{-b(t-s)}, \quad t, s \in [0, T], \quad a > 0, b \ge 0,
+$$
+
+to study the monotonicity property of the operator $\mathbf{V} := \mathbf{II} + \mathbf{IV}$. In this way, we conveniently use the relation $\partial_x G(t, s) = -bG(t, s)$, for $0 \le s < t \le T$, and the fact that $k \equiv a$. Specifically, by Fubini's Theorem and straightforward calculus, we get for any $u, v \in \mathcal{L}^2$
+
+$$
+\begin{align*}
+\langle \mathbf{V}(u) - \mathbf{V}(v), u - v \rangle &= \frac{b}{a} \langle h(Z^u) - h(Z^v), Z^u - Z^v \rangle \\
+&\quad + \frac{b}{a} \langle Z^u h'(Z^u) - Z^v h'(Z^v), Z^u - Z^v \rangle \\
+&\quad - \frac{b}{a} \mathbb{E} \left[ \int_0^T g_t (h'(Z_t^u) - h'(Z_t^v)) (Z_t^u - Z_t^v) \mathrm{d}t \right].
+\end{align*}
+$$
+
+Concluding on the nonnegativity of the above quantity in general depends on the endogenous signal $g$ as was the case for $\mathbf{III}$. But if we additionally assume that (ii) $g \equiv 0$ and (iii) $x \mapsto xh'(x)$ is nondecreasing, then clearly $\mathbf{V}$ is monotone, which proves Proposition 2.8.
+
+**Case of a sum of exponential time scales for the impact decay.** More generally, one may wonder whether we could derive in a similar way sufficient conditions to get the monotonicity property of the operator $\mathbf{V} : \mathcal{L}^2 \to \mathcal{L}^2$ in the case of a Volterra kernel given by a finite sum of exponentials, i.e.,
+
+$$
+G(t, s) = \mathbb{1}_{\{t \ge s\}} \sum_{i=1}^{n} a_i e^{-b_i(t-s)}, \quad t, s \in [0, T], \quad a_i > 0, b_i \ge 0, \quad i \in \{1, \dots, n\}, \quad n \in \mathbb{N}.
+$$
+
+The answer turns out to be negative. Indeed, assume without loss of generality that the mean-reversion speeds $(b_i)_i$ differ from one another. Then, in this case, $k \equiv \sum_{i=1}^n a_i =: A$ and
+
+$$
+\partial_x G(t, s) = - \sum_{i=1}^{n} a_i b_i \exp\{-b_i(t-s)\}, \quad 0 \le s < t \le T.
+$$
+
+For any $i = 1, \dots, n$, we also define the Volterra kernels $G_i(t, s) = \mathbb{1}_{\{t \ge s\}} a_i e^{-b_i(t-s)}$, so that $G(t, s) = \sum_{i=1}^n G_i(t, s)$, $s, t \in [0, T]$, and $\partial_x G(t, s) = -\sum_{i=1}^n b_i G_i(t, s)$, $0 < s < t < T$. Considering the initial input curve $g \equiv 0$, we can write
+
+$$
+Z^u = \sum_{i=1}^{n} \mathbf{G}_i u =: \sum_{i=1}^{n} Z^{i,u}, \quad u \in \mathcal{L}^2.
+$$