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f524048ef83466119652723a17197084a1ac35085465c4bd11314e7808952828
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2026-01-21T00:00:00-05:00
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One-variable equations over the lamplighter group
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arXiv:2601.12112v1 Announce Type: new Abstract: We study one-variable equations over the lamplighter group $\MZ_2 \wr \MZ$. While the decidability of arbitrary equations over $L_2$ remains open, we prove that the Diophantine problem for single equations in one variable is decidable. Our approach reduces the problem to a divisibility question for families of parametric Laurent polynomials over $\MZ_2$, whose coefficients depend linearly on an integer parameter. We develop an automaton-theoretic framework to analyze divisibility of such polynomials, exploiting eventual periodicity phenomena arising from polynomial division over finite fields. This yields an explicit decision procedure, which is super-exponential in the worst case. On the other hand, we show that for a generic class of equations, solvability can be decided in nearly quadratic time. These results establish a sharp contrast between worst-case and typical computational behavior and provide new tools for the study of equations over wreath products.
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https://arxiv.org/abs/2601.12112
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8b9c38fd96b5468274f0461c8e48d6840370f0e58043de249b9fc90a95f9ef71
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2026-01-21T00:00:00-05:00
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Hodge decomposition for Kato manifolds
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arXiv:2601.12113v1 Announce Type: new Abstract: We prove that any Kato manifold satisfies the Hodge decomposition, in the sense that $b_k=\sum_{p+q=k}h^{p, q}$, by relating its cohomology to the corresponding cohomology of its modification data. We give, therefore, more evidence supporting a conjecture of Ornea--Verbitsky stating that compact locally conformally K\"ahler manifolds satisfy the Hodge decomposition. We further study Bott--Chern and Aeppli cohomology of Kato manifolds, showing that in certain degrees they coincide with Dolbeault cohomology.
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https://arxiv.org/abs/2601.12113
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c72b23b6200d042b4ef3029aed87d965ab4521a0ca19a69f0ded5eccf772975d
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2026-01-21T00:00:00-05:00
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Offline Policy Learning with Weight Clipping and Heaviside Composite Optimization
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arXiv:2601.12117v1 Announce Type: new Abstract: Offline policy learning aims to use historical data to learn an optimal personalized decision rule. In the standard estimate-then-optimize framework, reweighting-based methods (e.g., inverse propensity weighting or doubly robust estimators) are widely used to produce unbiased estimates of policy values. However, when the propensity scores of some treatments are small, these reweighting-based methods suffer from high variance in policy value estimation, which may mislead the downstream policy optimization and yield a learned policy with inferior value. In this paper, we systematically develop an offline policy learning algorithm based on a weight-clipping estimator that truncates small propensity scores via a clipping threshold chosen to minimize the mean squared error (MSE) in policy value estimation. Focusing on linear policies, we address the bilevel and discontinuous objective induced by weight-clipping-based policy optimization by reformulating the problem as a Heaviside composite optimization problem, which provides a rigorous computational framework. The reformulated policy optimization problem is then solved efficiently using the progressive integer programming method, making practical policy learning tractable. We establish an upper bound for the suboptimality of the proposed algorithm, which reveals how the reduction in MSE of policy value estimation, enabled by our proposed weight-clipping estimator, leads to improved policy learning performance.
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https://arxiv.org/abs/2601.12117
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Academic Papers
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07299667979cf79758659ba1ffb6979b465e2f718fa1f0fcd5a0ba2a55e696d5
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2026-01-21T00:00:00-05:00
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On the Hausdorff Dimension of weighted exactly Approximable Vectors
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arXiv:2601.12121v1 Announce Type: new Abstract: We show that the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-exactly approximable vectors in $\mathbb R^d$ coincides with the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-approximable vectors, generalizing a result of the first named author and De Saxc\'e.
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https://arxiv.org/abs/2601.12121
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9375bf990231fe36d68bb9ac9da5251649dc4af19a1cb7d6e04fab9bf85a5199
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2026-01-21T00:00:00-05:00
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Navier slip effects in micropolar thin-film flow: a rigorous derivation of Reynolds-type models
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arXiv:2601.12125v1 Announce Type: new Abstract: We study the stationary flow of incompressible micropolar fluid in a thin three-dimensional domain under Navier slip boundary condition for the velocity and no-spin condition for microrotation. After rescaling the governing equations, we perform a rigorous asymptotic analysis as the film thickness tends to zero, considering a friction coefficient dependent on the small parameter. According to the scaling of the slip coefficient, we identify three distinct regimes: perfect slip, partial slip, and no-slip. For each regime, we derive the corresponding reduced micropolar system and obtain explicit expressions for the velocity and microrotation fields. This leads to a generalized Reynolds-type equation for the pressure, highlighting the impact of slip effects on the micropolar thin-film flow.
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https://arxiv.org/abs/2601.12125
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5d0a0a01d0de6a1c3f3ab3deeefe97093451f6d379856e40d0ca45c940c04c7a
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2026-01-21T00:00:00-05:00
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Spectral Analysis of the $D_{\log}^{(\lambda, N)}$ Operators
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arXiv:2601.12133v1 Announce Type: new Abstract: This paper investigates the recent Connes-Consani-Moscovici $D_{\log}^{(\lambda, N)}$ operators, whose spectra are currently hypothesized to approach the zeros of $\zeta\left(\frac{1}{2} +is\right)$ as $\lambda, N \rightarrow \infty$. It turns out that when considering different standard notions of error, the dissonance between the spectra and Riemann $\zeta$ zeros either appears to or can be proven to be inverse logarithmic in nature, which elegantly fits the distribution of prime numbers.
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https://arxiv.org/abs/2601.12133
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1f0dd737baa88cb0c71bb05c309b64b7f86cc8e69010295fd6bf76a86c43c00d
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2026-01-21T00:00:00-05:00
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Fractional Semilinear Equations on Hyperbolic Spaces
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arXiv:2601.12140v1 Announce Type: new Abstract: We study a semilinear equation involving the fractional Laplacian on the hyperbolic space $\mathbb{H}^n$. Unlike in conformally compact Einstein manifolds, the fractional Laplacian on $\mathbb{H}^n$ does not enjoy conformal covariance. By employing Helgason-Fourier analysis, we explicitly derive the Green's function of the fractional Laplacian on $\mathbb{H}^n$ and study its asymptotic behaviors. We then apply a direct method of moving planes to the integral form of the equation, establishing symmetry of solutions and nonexistence of positive solutions in the critical and subcritical cases, respectively. In addition, we develop several maximum principles on hyperbolic space.
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https://arxiv.org/abs/2601.12140
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2552362a4295929676dcfd1c52a45b56cb056d364f239025f1c1d083460edbc0
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2026-01-21T00:00:00-05:00
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On dihedral invariants of the free associative algebra of rank two
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arXiv:2601.12144v1 Announce Type: new Abstract: Let $K\langle X_d\rangle$ denote the free associative algebra of rank $d \geq 2$ over a field $K$. By results of Lane (1976) and Kharchenko (1978), the algebra of invariants $K\langle X_d\rangle ^G$ is free for any subgroup $G \leq \GL_d(K)$ and any field $K$. Koryukin (1984) introduced an additional action of the symmetric group $Sym(n)$ on the homogeneous component of degree $n$ of $K\langle X_d\rangle$, given by permuting the positions of the variables. This endows $K\langle X_d\rangle $ with the structure of a $(K\langle X_d\rangle,\circ)$-$S$-algebra. With respect to this action, Koryukin proved that the invariant algebra $K\langle X_d\rangle ^G$ is finitely generated for every reductive group $G$. In this paper we study the algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ of invariants under the action of the dihedral group D_{2n} $ on the free associative algebra ${\mathbb C} \langle u,v\rangle$ of rank $2$. We compute the Hilbert series of ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ and construct an explicit set of generators for ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ as a free algebra. Furthermore, we describe a finite generating set for the $S$-algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$.
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https://arxiv.org/abs/2601.12144
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9a08a31123ec1f7e2925f9b79735776c7226d60651676a96761a8e13e1755a4a
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2026-01-21T00:00:00-05:00
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A $p$-adic cohomological approach to congruences of meromorphic modular forms
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arXiv:2601.12157v1 Announce Type: new Abstract: We study congruences relating Fourier coefficients of meromorphic modular forms and Frobenius eigenvalues of elliptic curves corresponding to their poles. We develop a $p$-adic cohomological framework that interprets these congruences via the interaction between the rigid cohomology of modular curves and the crystalline structure of the associated elliptic curves. Using comparison theorems and the Gysin sequence, we relate the Frobenius actions in cohomology to the $U_p$-operator acting on spaces of overconvergent modular forms. Our approach applies uniformly to both modular curves and Shimura curves admitting smooth integral models over $\mathbb{Z}_p$.
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https://arxiv.org/abs/2601.12157
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877bf27cfed03ea68b13355006a78f94e5f06ba277f8e93011c15cb27c680df6
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2026-01-21T00:00:00-05:00
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Locating critical points attracted to p-adic attracting cycles
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arXiv:2601.12163v1 Announce Type: new Abstract: In complex dynamics, a fundamental result of Fatou and Julia asserts that every attracting cycle of a rational map attracts a critical point. The analogous statement fails in non-Archimedean dynamics. For a non-Archimedean rational map, this paper establishes a sharp condition on the multiplier of an attracting cycle ensuring it attracts a critical point.
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https://arxiv.org/abs/2601.12163
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75a4b38d0afa975b685e5148918b40e754159ed34c3de8f4cb0f731005719b6b
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2026-01-21T00:00:00-05:00
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Fractional Quantum Hall States: Infinite Matrix Product Representation and its Implications
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arXiv:2601.12165v1 Announce Type: new Abstract: We present a novel matrix product representation of the Laughlin and related fractional quantum Hall wavefunctions based on a rigorous version of the correlators of a chiral quantum field theory. This representation enables the quantitative control of the coefficients of the Laughlin wavefunction times an arbitrary monomial symmetric polynomial when expanded in a Slater determinant or permanent basis. It renders the properties, such as factorization and the renewal structure, inherent in such fractional quantum Hall wavefunctions transparent. We prove bounds on the correlators of the chiral quantum field theory and utilize this representation to demonstrate the exponential decay of connected correlations and a gap in the entanglement spectrum on a thin cylinder.
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https://arxiv.org/abs/2601.12165
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3be303117f292a0674e5483dc7625cd3f040ea96167637c6d209dfb6f9ff1f1d
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2026-01-21T00:00:00-05:00
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Balancing adaptability and predictability: K-revision multistage stochastic programming
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arXiv:2601.12166v1 Announce Type: new Abstract: A standard assumption in multistage stochastic programming is that decisions are made after observing the uncertainty from the prior stage. The resulting solutions can be difficult to implement in practice, as they leave practitioners ill-prepared for future stages. To provide better foresight, we introduce the K-revision approach. This new framework requires plans to be specified in advance. To maintain flexibility, we allow plans to be revised a maximum of K times as new information becomes available. We analyze the complexity of K-revision problems, showing NP-hardness even in a simple setting. We examine, both theoretically and computationally, the impact of the K-revision approach on the objective compared with classical multistage stochastic programming models and the partially adaptive approach introduced in [1, 2]. We develop two MIP formulations, one directly from our definition and the other based on a combinatorial characterization. We analyze the tightness of these formulations and propose several methods to strengthen them. Computational experiments on synthetic problems and practical applications demonstrate that our approach is both computationally tractable and effective in reaching near-optimal performance while increasing the predictability of the solutions produced.
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https://arxiv.org/abs/2601.12166
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5e848749001382f2a36118249a8f4ea81d29426356fac12e5fbe4b9f38d5fa81
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2026-01-21T00:00:00-05:00
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The P\'olya Web
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arXiv:2601.12172v1 Announce Type: new Abstract: We introduce the P\'olya Web, a system of coalescing random walks based on the classic P\'olya urn model. This construction serves as an analogue to the web of coalescing random walks studied by T\'oth and Werner (1998), replacing simple symmetric random walks with P\'olya walks as primary constituents. First, we study the general web of up-right oriented coalescing random walks. We investigate its geometric properties and prove that certain indicator random variables satisfy negative association. Notably, the proof involves a non-trivial application of the van den Berg-Kesten-Reimer (BKR) inequality. Based on this property, we derive a strong law for the number of connected components generated by walks starting at the same time. Subsequently, we focus on the specific properties of the P\'olya Web. It is well-known that the normalized coordinates of a single P\'olya Walk converge almost surely to a beta-distributed random variable. We determine the joint distribution of these limiting variables in the coalescing framework. Using these joint densities, we provide exact calculations regarding the almost sure convergence of the number of components. Finally, by applying a local scaling to the P\'olya Web at the edges, we introduce the Yule Web, a web of coalescing Yule processes. We demonstrate that the fundamental properties and results derived for the P\'olya Web can be extended to this limiting case.
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https://arxiv.org/abs/2601.12172
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5f26a9305f32be833e9cff01f7f3952dcf28c75dc1fdb65cea881bb225b58b55
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2026-01-21T00:00:00-05:00
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Kato's Ramification filtration via de Rham-Witt complex and applications
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arXiv:2601.12177v1 Announce Type: new Abstract: Given an $F$-finite regular scheme $X$ of positive characteristic and a simple normal crossing divisor $E$ on $X$, we introduce a filtration on the de Rham-Witt complex $W_m\Omega^\bullet_{X\setminus E}$. When $X$ is the spectrum of a henselian discrete valuation ring $A$ with quotient field $K$, this extends the classical filtration on $W_m(K)$ due to Brylinski. We show that Kato's ramification filtration on $H^q_\et(X \setminus E, {\Q}/{\Z}(q-1))$ for $q \ge 1$ admits an explicit description in terms of the above filtration of the de Rham-Witt complex of $X \setminus E$. When $q =1$, this specializes to the results of Kato and Kerz-Saito. As applications, we prove refinements of the duality theorem of Jannsen-Saito-Zhao for smooth projective schemes over finite fields and the duality theorem of Zhao for semi-stable schemes over henselian discrete valuation rings of positive characteristic with finiteresidue fields. We also prove a modulus version of the duality theorem of Ekedahl. As another application, we prove Lefschetz theorems for Kato's ramification filtrations for smooth projective varieties over $F$-finite fields. This extends a result of Kerz-Saito for $H^1$ to higher cohomology. Similar results are proven for the Brauer group.
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https://arxiv.org/abs/2601.12177
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d5a9f3d4faef8307f324a2dbb8b0412bdeb49ad3c5c72b81f951e658ac5cda99
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2026-01-21T00:00:00-05:00
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On a theorem of Artin and the dimension of the space spanned by the rational valued characters of a group
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arXiv:2601.12185v1 Announce Type: new Abstract: In this paper, we sharpen a theorem of Artin to show that for a finite group, the dimension of the subspace of class functions spanned by the rational valued characters equals the number of conjugacy classes of cyclic subgroups.
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https://arxiv.org/abs/2601.12185
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5961122a48c3f59e28c389df2e0bdcc2ec790d8e8dcb5d3aa3d5f85c57497ab7
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2026-01-21T00:00:00-05:00
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Sets of Ramsey-limit points and IP-limit points
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arXiv:2601.12187v1 Announce Type: new Abstract: Let $X$ be an uncountable Polish space and let $\mathcal{H}$ be the Hindman ideal, that is, the family of all $S\subseteq \omega$ which are not $IP$-sets. For each sequence $x=(x_n)_{n \in \omega}$ taking values in $X$, let $\Lambda_{x}(FS)$ be the set of $IP$-limit points of $x$. Also, let $\Lambda_{x}(\mathcal{H})$ be the set of $\mathcal{H}$-limit points of $x$, that is, the set of ordinary limits of subsequences $(x_n)_{n \in S}$ with $S\notin \mathcal{H}$. After proving that these two notions do not coincide in general, we show that both families of nonempty sets of the type $\Lambda_{x}(FS)$ and of the type $\Lambda_{x}(\mathcal{H})$ are precisely the class of nonempty analytic subsets of $X$. An analogous result holds also for Ramsey convergence. In the proofs, we use the concept of partition regular functions introduced in J. Symb. Log. (2024) [doi:10.1017/jsl.2024.8], which provide a unified approach to these types of convergence.
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https://arxiv.org/abs/2601.12187
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73d6f4a160a295d6eb89dcc8f55afd9e152a35297888affb70a48c31815cab92
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2026-01-21T00:00:00-05:00
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Optimal Leveraging of Smoothness and Strong Convexity for Peaceman--Rachford Splitting
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arXiv:2601.12190v1 Announce Type: new Abstract: In this paper, we introduce a simple methodology to leverage strong convexity and smoothness in order to obtain an optimal linear convergence rate for the Peaceman--Rachford splitting (PRS) scheme applied to optimization problems involving two smooth strongly convex functions. The approach consists of adding and subtracting suitable quadratic terms from one function to the other so as to redistribute strong convexity in the primal formulation and smoothness in the dual formulation. This yields an equivalent modified optimization problem in which each term has adjustable levels of strong convexity and smoothness. In this setting, the Peaceman--Rachford splitting method converges linearly to the solution of the modified problem with a convergence rate that can be optimized with respect to the introduced parameters. Upon returning to the original formulation, this procedure gives rise to a modified variant of PRS. The optimal linear rate established in this work is strictly better than the best rates previously available in the general setting. The practical performance of the method is illustrated through an academic example and applications in image processing.
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https://arxiv.org/abs/2601.12190
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6229033734c4594611361e293404c38e691bc1be155d24e478acf2324d132098
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2026-01-21T00:00:00-05:00
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Sobolev inequalities for nonlinear Dirichlet forms
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arXiv:2601.12192v1 Announce Type: new Abstract: In this short note we show an equivalence between Sobolev type inequalities and so called isocapacitary inequalities in the context of a large class of nonlinear Dirichlet forms, their associated Dirichlet spaces and their associated capacities.
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https://arxiv.org/abs/2601.12192
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bc218f4ccde83cee733462ae521bae989f1159a30a62d4aef2dccb1c4f660866
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2026-01-21T00:00:00-05:00
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Bruhat Intervals in the Infinite Symmetric Group are Cohen-Macaulay
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arXiv:2601.12195v1 Announce Type: new Abstract: We show that the (non-Noetherian) Stanley-Reisner ring of the order complex of certain intervals in the Bruhat order on the infinite symmetric group $S_\infty$ of all auto-bijections of $\mathbb{N}$ is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. This gives an infinite-dimensional version of results due to Edelman, Bj\"{o}rner, and Kind and Kleinschmidt for finite symmetric groups $S_n$.
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https://arxiv.org/abs/2601.12195
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f3cc04d9e91f1f400c4af89fbdfef0f06e1afdafc5340b0c9e2ed9ae1b8e0b05
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2026-01-21T00:00:00-05:00
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Higher-Order Approximations of Sojourn Times in M/G/1 Queues via Stein's Method
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arXiv:2601.12197v1 Announce Type: new Abstract: We study the stationary sojourn time distribution in an M/G/1 queue operating under heavy traffic. It is known that the sojourn time converges to an exponential distribution in the limit. Our focus is on obtaining pre-asymptotic, higher-order approximations that go beyond the classical exponential limit. Using Stein's method, we develop an approach based on higher-order expansions of the generator of the underlying Markov process. The key technical step is to represent higher-order derivatives in terms of lower-order ones and control the resulting error via derivative bounds of the Stein equation. Under suitable moment-matching conditions on the service distribution, we show that the approximation error decays as a high-order power of the slack parameter $\varepsilon=1-\rho$. Error bounds are established in the Zolotarev metric, which further imply bounds on the Wasserstein distance as well as the moments. Our results demonstrate that the accuracy of the exponential approximation can be systematically improved by matching progressively more moments of the service distribution.
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https://arxiv.org/abs/2601.12197
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2c76d5d8bb1bddd7e5f5af3b1eef41b23ed8b830cbe80a80180bddfb14437b49
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2026-01-21T00:00:00-05:00
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A curvature-regularized variational problem with an area constraint
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arXiv:2601.12201v1 Announce Type: new Abstract: Interlocking interfaces are commonly employed to mitigate relative sliding under shear.Indeed, Their geometry is typically selected on grounds of fabrication convenience rather than analytical optimality. There is no reason to suppose that circular or polygonal profiles minimize localized stress concentration under fixed geometric constraints. We propose a variational model in which the interface is represented by a planar curve $y=f(x)$, and localized stress amplification is quantified by a curvature-sensitive functional \[ J[f] = \int_{-a}^{a} \bigl(1+\gamma \kappa^2\bigr) \sqrt{1+f'(x)^2}\,dx, \] defined on the Sobolev space $W^{2,2}([-a,a])$. The functional is motivated by elasticity-theoretic considerations in which curvature enters the leading-order stress field near a singular interface.Indeed, any profile possessing discontinuous tangents yields a divergent integral, thereby rendering it energetically inadmissible within the Sobolev space $W^{2,2}$. An area constraint $\int_{-a}^{a} f(x)\,dx = A_0$ is imposed to model fixed material volume. Using the direct method of the calculus of variations, we establish the existence of a minimizer and derive the associated Euler--Lagrange equation, a nonlinear fourth-order boundary value problem. Note, however, that constant-curvature and piecewise-linear profiles fail to satisfy the necessary optimality conditions under the imposed constraint. Indeed, we are thus forced to conclude that analytical optimality necessitates a more complex variation in the local tangent angle The analysis indicates that commonly employed interlock geometries are not variationally optimal for minimizing localized shear stress within this class of admissible interfaces.
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https://arxiv.org/abs/2601.12201
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9766b0af0726db37acb3c3977baa038dfb12e103fa0aa2652d3d6071b96fcf16
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2026-01-21T00:00:00-05:00
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Characterizations of Lorentz Type Sobolev Multiplier Spaces and Their Preduals
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arXiv:2601.12206v1 Announce Type: new Abstract: We provide several characterizations of Sobolev multiplier spaces of Lorentz type and their preduals. Block decomposition and K\"othe dual of such preduals are discussed. As an application, the boundedness of local Hardy-Littlewood maximal function on these spaces will be justified.
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https://arxiv.org/abs/2601.12206
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dc0fae7e68345b7bb609abb44d42464698090e10163572d5e5dbd7d12fbc4bb3
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2026-01-21T00:00:00-05:00
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Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws with Cattaneo's law
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arXiv:2601.12216v1 Announce Type: new Abstract: This paper examines the large-time behavior of solutions to a one-dimensional conservation law featuring a non-convex flux and an artificial heat flux term regulated by Cattaneo's law, forming a 2$\times$2 system of hyperbolic equations. Under the conditions of small wave strength and sufficiently small initial perturbations, we demonstrate the time-asymptotic stability of a composite wave that combines a degenerate Oleinik shock and a rarefaction wave. The proof utilizes the Oleinik entropy condition, the a-contraction method with time-dependent shifts, and weighted energy estimates.
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https://arxiv.org/abs/2601.12216
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a5c7cfe34dbf3706bc26c400ddc30c32550307b384073c38fd7bd0a342a25831
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2026-01-21T00:00:00-05:00
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Interval B-Tensors and Interval Double B-Tensors
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arXiv:2601.12217v1 Announce Type: new Abstract: This paper systematically investigates the properties and characterization of interval B-tensors and interval double B-tensors. We propose verifiable necessary and sufficient conditions that allow for determining whether an entire interval tensor family belongs to these classes based solely on its extreme point tensors. The study elucidates profound connections between these interval tensors and other structured ones such as interval Z-tensors and P-tensors, while also providing simplified criteria for special cases like circulant structures. Furthermore, under the condition of even order and symmetry, we prove that interval B-tensors (double B-tensors) ensure the property of being an interval P-tensor. This work extends interval matrix theory to tensors, offering new analytical tools for fields such as polynomial optimization and complementarity problems involving uncertainty.
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https://arxiv.org/abs/2601.12217
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0a088b68bc4dddf4189356f3cdab3d0de8df9a06041448a2cdc567d728e33d57
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2026-01-21T00:00:00-05:00
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Stabilization of arbitrary structures in a three-dimensional doubly degenerate nutrient taxis system
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arXiv:2601.12218v1 Announce Type: new Abstract: The doubly degenerate nutrient taxis system \begin{equation}\label {0.1} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u^{\alpha}v\nabla v)+\ell uv,&x\in \Omega,\, t>0,\\ & v_{t}=\Delta v-uv,&x\in \Omega,\, t>0,\\ \end{aligned} \right. \end{equation} is considered under zero-flux boundary conditions in a smoothly bounded domain $\Omega\subset\mathbb{R}^3$ where $\alpha>0,\chi>0$ and $\ell> 0$. By developing a novel class of functional inequalities to address the challenges posed by the doubly degenerate diffusion mechanism in \eqref{0.1}, it is shown that for $\alpha\in(\frac{3}{2},\frac{19}{12})$, the associated initial-boundary value problem admits a global continuous weak solution for sufficiently regular initial data. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty, 0)$ as $t\rightarrow \infty$. Notably, the limiting profile $u_{\infty}$ is non-homogeneous when the initial signal concentration $v_0$ is sufficiently small, provided the initial data $u_0$ is not identically constant.
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https://arxiv.org/abs/2601.12218
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637d44f689272ffb17d419bd19326c7a3573155f0d71166e3ceaf75a8a78d5d7
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2026-01-21T00:00:00-05:00
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Persistent Sheaf Laplacian Analysis of Protein Stability and Solubility Changes upon Mutation
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arXiv:2601.12219v1 Announce Type: new Abstract: Genetic mutations frequently disrupt protein structure, stability, and solubility, acting as primary drivers for a wide spectrum of diseases. Despite the critical importance of these molecular alterations, existing computational models often lack interpretability, and fail to integrate essential physicochemical interaction. To overcome these limitations, we propose SheafLapNet, a unified predictive framework grounded in the mathematical theory of Topological Deep Learning (TDL) and Persistent Sheaf Laplacian (PSL). Unlike standard Topological Data Analysis (TDA) tools such as persistent homology, which are often insensitive to heterogeneous information, PSL explicitly encodes specific physical and chemical information such as partial charges directly into the topological analysis. SheafLapNet synergizes these sheaf-theoretic invariants with advanced protein transformer features and auxiliary physical descriptors to capture intrinsic molecular interactions in a multiscale and mechanistic manner. To validate our framework, we employ rigorous benchmarks for both regression and classification tasks. For stability prediction, we utilize the comprehensive S2648 and S350 datasets. For solubility prediction, we employ the PON-Sol2 dataset, which provides annotations for increased, decreased, or neutral solubility changes. By integrating these multi-perspective features, SheafLapNet achieves state-of-the-art performance across these diverse benchmarks, demonstrating that sheaf-theoretic modeling significantly enhances both interpretability and generalizability in predicting mutation-induced structural and functional changes.
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https://arxiv.org/abs/2601.12219
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9193c290da9ce8c9125ffb17c9adf2ead1658800ad4def38ee3bf3c74865b557
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2026-01-21T00:00:00-05:00
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Mean-Field Games Under Model Uncertainty
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arXiv:2601.12226v1 Announce Type: new Abstract: We study discrete-time, finite-state mean-field games (MFGs) under model uncertainty, where agents face ambiguity about the state transition probabilities. Each agent maximizes its expected payoff against the worst-case transitions within an uncertainty set. Unlike in classical MFGs, model uncertainty renders the population distribution flow stochastic. This leads us to consider strategies that depend on both individual states and the realized distribution of the population. Our main results establish the asymptotic relationship between $N$-agent games and MFGs: every MFG equilibrium constitutes an $\varepsilon$-Nash equilibrium for sufficiently large populations, and conversely, limits of $N$-agent equilibria are MFG equilibria. We also prove the existence of equilibria for finite-agent games and construct a solvable mean-field example with closed-form solutions.
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https://arxiv.org/abs/2601.12226
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3f6b6cdc1cb21fcb00ef2f0d79f7043a11001dc7333ddd964e00e09f227f0503
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2026-01-21T00:00:00-05:00
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An optimal boundary control approach to the Cherrier-Escobar problem
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arXiv:2601.12232v1 Announce Type: new Abstract: We study an optimal boundary control problem associated to the boundary obstacle problem for the couple conformal Laplacian and conformal Robin operator on n-dimensional compact Riemannian manifolds with boundary and with n\geq 3. When the Cherrier-Escobar invariant of the compact Riemannian manifold with boundary is positive, we show that the optimal controls are equal to their associated optimal states. Moreover, we show that the optimal controls are minimizers of the Cherrier-Escobar functional, and hence induce conformal metrics with zero scalar curvature and constant mean curvature. Furthermore, we show the existence of an optimal control under an Aubin type assumption. For the standard unit ball, we derive a sharp Sobolev trace type inequality and prove that the standard bubbles-namely conformal factor of metrics conformal to the standard one with zero scalar curvature and constant mean curvature -- are the only optimal controls and hence equal to their associated optimal states.
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https://arxiv.org/abs/2601.12232
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e79978d3c6f4119f452b69d2eca2123a7d142822e85e149f009b8a66b9505308
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2026-01-21T00:00:00-05:00
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An alternative construction of the $G_2(2)$-graph
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arXiv:2601.12235v1 Announce Type: new Abstract: In this note, we give an alternative construction of the $G_2(2)$-graph from a $U_3(2)$-geometry.
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https://arxiv.org/abs/2601.12235
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004442f0cf8dfa61a0c63c713e056db61c3339682c96b4adee1af5dacbbb98c7
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2026-01-21T00:00:00-05:00
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Paley-type matrices and $1$-factorizations of complete graphs
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arXiv:2601.12250v1 Announce Type: new Abstract: Ball, Ortega--Moreno, and Prodromou asked whether, for every odd prime $p$, one can find a $1$-factor of the complete graph $K_{p+1}$ with some arithmetic restrictions related to quadratic residues. This problem is motivated by $1$-factorizations that are compatible with the sign pattern of certain Paley-type matrices. Recently, Afifurrahman et al. made some partial progress. In this paper, we completely resolve the problem.
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https://arxiv.org/abs/2601.12250
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d8f0697d79381e47a69253ca5174324cb5d349ee366eb7388bcbc36e502d9d45
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2026-01-21T00:00:00-05:00
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The generalized Lax conjecture is true for topological reasons related to compactness, convexity and determinantal deformations of increasing products of pointwise approximating linear forms
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arXiv:2601.12267v1 Announce Type: new Abstract: We develop a topological approach to prove the generalized Lax conjecture using the fact that determinants of sufficiently big symmetric linear pencils are able to express the rigidly convex sets of RZ polynomials of any degree $d$. Monicity of the representation is assessed through a topological argument that allows us to perturbate a sufficiently close linear approximation into a suitable nice determinantal multiple of the initial RZ polynomial with the same rigidly convex set. The perturbation can be smoothly performed. This fact is what will allow us to determine that the multiple obtained respects the initial rigidly convex sets. This argument provides thus a full proof of the generalized Lax conjecture. However, an effective proof providing the representation in nice terms seems far from reachable at this moment.
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https://arxiv.org/abs/2601.12267
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7ba65a24fad44f2a5837a2ca80d5c21a9b7bc5af8478c99637f3d4d1a60511f9
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2026-01-21T00:00:00-05:00
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The relative GAGA Theorem and an application to the analytic mapping stacks
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arXiv:2601.12299v1 Announce Type: new Abstract: We prove a relative GAGA theorem for perfect and pseudo-coherent complexes in non-archimedean analytic geometry, allowing bases given by Fredholm analytic rings, including those associated from affinoid perfectoid spaces. This answers a question raised in \cite{heuer2024padicnonabelianhodgetheory}. As an application, we show that for a proper scheme \(X\) and an Artin stack \(Y\) with suitable conditions, the analytification of the algebraic mapping stack \(\mathrm{Map}(X,Y)\) agrees with the intrinsic analytic mapping stack \(\mathrm{Map}(X^{\mathrm{an}},Y^{\mathrm{an}})\).
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https://arxiv.org/abs/2601.12299
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ef2ebb7dd2c1181b2193dde2f7f4318a7ab0ae74fe5673ec518e085afaa673fe
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2026-01-21T00:00:00-05:00
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Level of Faces for Exponential Sequence of Arrangements
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arXiv:2601.12328v1 Announce Type: new Abstract: In this paper, we introduce the bivariate exponential generating function $F_l(x,y)$ for the number of level-$l$ faces of an exponential sequence of arrangements (ESA), and establish the formula $F_l(x,y)=\big(F_1(x,y)\big)^l$ with a combinatorial interpretation. Its specialization at $x=0$ recovers a result first obtained by Chen et al. [3,4] for certain classic ESAs and later generalized to all ESAs by Southerland et al. [8]. As a byproduct, we obtain that an alternating sum of the number of level-$l$ faces is invariant with respect to the choice of ESA, and is exactly the Stirling number of the second kind. We also extend the binomial-basis expansion theorem [3,4,14] and Stanley's formula on ESAs [9] from characteristic polynomials to Whitney polynomials.
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https://arxiv.org/abs/2601.12328
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44e0eeb4da37dc5f6ca68e8e5ded50008f21e621a683dddfcee411ec4b8b9e5f
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2026-01-21T00:00:00-05:00
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A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szeg\H{o} Theorems
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arXiv:2601.12332v1 Announce Type: new Abstract: This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $d\mu = w(\theta) \frac{d\theta}{2\pi} + d\mu_s$ with Verblunsky coefficients $\alpha=\{\alpha_n\}_{n=0}^\infty$, distinct singular points $(\theta_k)_{k=1}^{\ell}$, and multiplicities $(m_k)_{k=1}^{\ell}$, we establish the equivalence between the entropy condition \[ \int_0^{2\pi} \prod_{k=1}^{\ell} [1 - \cos(\theta - \theta_k)]^{m_k} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \] and the decomposition condition \[ \exists \beta^{(1)}, \ldots, \beta^{(\ell)} : \alpha = \sum_{k=1}^\ell \beta^{(k)} \,\, \text{with} \,\, (S - e^{-i\theta_k})^{m_k} \beta^{(k)} \in \ell^2, \,\, \beta^{(k)} \in \ell^{2m_k + 2}. \] The proof synthesizes unitary transformations, discrete Sobolev-type inequalities, higher-order Szeg\H{o} expansions, and a novel algebraic decomposition technique. Our resolution affirms that spectral theory is fundamentally local-global behavior emerges from the superposition of local resonances, each governed by its intrinsic scale.
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https://arxiv.org/abs/2601.12332
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41f3492704dd969c22105a11d8e1ec49c6eb67950538fd398a245174b4c714b3
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2026-01-21T00:00:00-05:00
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Representation theorems for nonvariational solutions of the Helmholtz equation
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arXiv:2601.12335v1 Announce Type: new Abstract: We consider a possibly multiply connected bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{\max\{1,m\},\alpha}$ for some $m\in {\mathbb{N}}$, $\alpha\in]0,1[$ and we plan to solve both the Dirichlet and the Neumann problem for the Helmholtz equation in $\Omega$ and in the exterior of $\Omega$ in terms of acoustic layer potentials. Then we turn to prove an integral representation theorem solutions of the Helmholtz equation in terms of a single layer acoustic potential. The main focus of the paper is on $\alpha$-H\"{o}lder continuous solutions which may not have a classical normal derivative at the boundary points of $\Omega$ and that may have an infinite Dirichlet integral around the boundary of $\Omega$\, \textit{i.e.}, case $m=0$. Namely for solutions that do not belong to the classical variational setting.
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https://arxiv.org/abs/2601.12335
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ee1c5f03e024932dafe547be07ffc57437fdc838f96594af1ab00accee60f398
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2026-01-21T00:00:00-05:00
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Asymptotic Behavior of the Principal Eigenvalue Problems with Large Divergence-Free Drifts
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arXiv:2601.12342v1 Announce Type: new Abstract: In this paper, we consider the following principal eigenvalue problem with a large divergence-free drift: \begin{equation}\label{0.1} -\varepsilon\Delta \phi-2\alpha\nabla m(x)\cdot\nabla \phi+V(x)\phi=\lambda_\alpha \phi\ \,\ \text{in}\, \ H_0^1(\Omega),\tag{0.1} \end{equation} where the domain $\Omega\subset \mathbb{R}^N (N\ge 1)$ is bounded with smooth boundary $\partial\Omega$, the constants $\varepsilon>0$ and $\alpha>0$ are the diffusion and drift coefficients, respectively, and $m(x)\in C^{2}(\bar{\Omega})$, $V (x)\in C^{\gamma}(\bar{\Omega})~(0<\gamma<1)$ are given functions. For a class of divergence-free drifts where $m$ is a harmonic function in $\Omega$ and has no first integral in $H_{0}^{1}(\Omega)$, we prove the convergence of the principal eigenpair $(\lambda_\alpha, \phi)$ for (0.1) as $\alpha\rightarrow+\infty$, which addresses a special case of the open question proposed in [H. Berestycki, F. Hamel and N. Nadirashvili, CMP, 2005]. Moreover, we further investigate the refined limiting profiles of the principal eigenpair $(\lambda_\alpha, \phi)$ for (0.1) as $\alpha\rightarrow+\infty$, which display the visible effects of the large divergence-free drifts on the principal eigenpair $(\lambda_\alpha, \phi)$.
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https://arxiv.org/abs/2601.12342
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31380625ec73a8c3bb3607bd2903fdf1a11bd5361527ff680979172601cc3f1c
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2026-01-21T00:00:00-05:00
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Classification of the structures of stable radial solutions for semilinear elliptic equations in $\bf R^N$
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arXiv:2601.12350v1 Announce Type: new Abstract: We study the stability of radial solutions of the semilinear elliptic equation $\Delta u +f(u)=0$ in ${\bf R^N}$, where $N \geq 3$ and $f$ is a general superciritical nonlinearity. We give a classification of the solution structures with respect to the stability of radial solutions, and establish criteria for the existence and nonexistence of stable radial solutions in terms of the limits of $f'(u)F(u)$ as $u \to 0$ or $\infty$, where $F(u) = \int^{\infty}_u 1/f(t)dt$. Furthermore, we show the relation between the existence of singular stable solutions and the solution structure.
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https://arxiv.org/abs/2601.12350
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3f09a7bdc3daa22c163560a9d330655ffbfb35e0ad2004c67ce97434551c7a28
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2026-01-21T00:00:00-05:00
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Time-fractional nonlinear evolution equations with time-dependent constraints
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arXiv:2601.12352v1 Announce Type: new Abstract: This article is devoted to presenting an abstract theory of time-fractional gradient flow equations for time-dependent convex functionals in real Hilbert spaces. The main results are concerned with the existence of strong solutions to time-fractional abstract evolution equations governed by time-dependent subdifferential operators. To prove these results, Gronwall-type lemmas for nonlinear Volterra integral inequalities and fractional chain-rule formulae are developed. Moreover, the obtained abstract results are applied to the initial-boundary value problem for time-fractional nonlinear parabolic equations on moving domains.
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https://arxiv.org/abs/2601.12352
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cdfbde0fec6814bccd68994879c260cadc3c992e37dd35bb377cdfab9e7b29c7
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2026-01-21T00:00:00-05:00
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Skew brace extensions, second cohomology and complements
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arXiv:2601.12371v1 Announce Type: new Abstract: We study extensions and second cohomology of skew left braces via the natural semi-direct products associated with the skew left braces. Let $0 \to I \to E \to H \to 0$ be a skew brace extension and $\Lambda_H$ denote the natural semi-direct products associated with the skew left brace $H$. We establish a group homomorphism from ${\rm H}_{Sb}^2(H, I)$ into ${\rm H}_{Gp}^2(\Lambda_H, I \times I)$, which turns out to be an embedding when $I \le {\rm Soc}(E)$. In particular the Schur multiplier of a skew left braces $H$ embeds into the Schur multiplier of the group $\Lambda_H$. Analog of the Schur-Zassenhaus theorem is established for skew left braces in several specific cases. We introduce a concept called minimal extensions (which stay at the extreme end of split extensions) of skew left braces and derive many fundamental results. Several reduction results for split extensions of finite skew left braces by abelian groups (viewed as trivial left braces) are obtained.
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https://arxiv.org/abs/2601.12371
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5c52f033bb50afb359638a7a25cab5db332e6e2ef6b0f745476f383789e96028
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2026-01-21T00:00:00-05:00
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On a Modification of the Twistor Space
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arXiv:2601.12372v1 Announce Type: new Abstract: In the paper we construct a modification $S(M)$ of the twistor space of a K\"ahler scalar flat surface $M$ and study its complex-geometric and metric properties. In particular, we construct complete balanced metrics on $S(M)$ and show that $S(M)$ can not be K\"ahler when $M$ is a compact simple hyperk\"ahler manifold.
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https://arxiv.org/abs/2601.12372
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1bd5b22ce73e0a21acc1f269e4883bff8a78aa58e22c3f736a1ebf3a7f9c0437
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2026-01-21T00:00:00-05:00
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An efficient penalty decomposition algorithm for minimization over sparse symmetric sets
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arXiv:2601.12383v1 Announce Type: new Abstract: This paper proposes an improved quasi-Newton penalty decomposition algorithm for the minimization of continuously differentiable functions, possibly nonconvex, over sparse symmetric sets. The method solves a sequence of penalty subproblems approximately via a two-block decomposition scheme: the first subproblem admits a closed-form solution without sparsity constraints, while the second subproblem is handled through an efficient sparse projection over the symmetric feasible set. Under a new assumption on the gradient of the objective function, weaker than global Lipschitz continuity from the origin, we establish that accumulation points of the outer iterates are basic feasible and cardinality-constrained Mordukhovich stationarity points. To ensure robustness and efficiency in finite-precision arithmetic, the algorithm incorporates several practical enhancements, including an enhanced line search strategy based on either backtracking or extrapolation, and four inexpensive diagonal Hessian approximations derived from differences of previous iterates and gradients or from eigenvalue-distribution information. Numerical experiments on a diverse benchmark of $30$ synthetic and data-driven test problems, including machine-learning datasets from the UCI repository and sparse symmetric instances with dimensions ranging from $10$ to $500$, demonstrate that the proposed algorithm is competitive with several state-of-the-art methods in terms of efficiency, robustness, and strong stationarity.
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https://arxiv.org/abs/2601.12383
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9f5bb5ddb10b0a892044766833605e44d5a037652b1d9a2ab638669bcc6ec103
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2026-01-21T00:00:00-05:00
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Strong Hollowness in Commutative Rings
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arXiv:2601.12388v1 Announce Type: new Abstract: In this paper we study strongly hollow ideals and completely strongly hollow ideals in commutative rings without finiteness assumptions. We establish basic structural properties, including maximality phenomena and permanence under quotients and surjective homomorphisms. We obtain several characterizations of completely strongly hollow ideals in terms of extremal ideals avoiding a given ideal, and we show that a strongly hollow ideal which is not contained in the Jacobson radical is necessarily completely strongly hollow. As applications, we derive strong restrictions in integral domains and consequences for principal ideal domains, including a discrete valuation ring criterion. We develop the connection between complete hollowness and complete irreducibility and obtain a correspondence between completely strongly hollow ideals and completely strongly irreducible ideals. Finally, we develop a condition related to greatest common divisors which is equivalent to strongly hollowness under mild finiteness conditions.
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https://arxiv.org/abs/2601.12388
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226c36918ee700dbddacf75a1bf17f24e66f3da813b9a61842ec590bf7b218f9
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2026-01-21T00:00:00-05:00
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Anderson Acceleration for Distributed Constrained Optimization over Time-varying Networks
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arXiv:2601.12398v1 Announce Type: new Abstract: This paper applies the Anderson Acceleration (AA) technique to accelerate the Fenchel dual gradient method (FDGM) to solve constrained optimization problems over time-varying networks. AA is originally designed for accelerating fixed-point iterations, and its direct application to FDGM faces two challenges: 1) FDGM in time-varying networks cannot be formulated as a standard fixed-point update; 2) even if the network is fixed so that FDGM can be expressed as a fixed-point iteration, the direct application of AA is not distributively implementable. To overcome these challenges, we first rewrite each update of FDGM as inexactly solving several \emph{local} problems where each local problem involves two neighboring nodes only, and then incorporate AA to solve each local problem with higher accuracy, resulting in the Fenchel Dual Gradient Method with Anderson Acceleration (FDGM-AA). To guarantee global convergence of FDGM-AA, we equip it with a newly designed safe-guard scheme. Under mild conditions, our algorithm converges at a rate of \(O(1/\sqrt{k})\) for the primal sequence and \(O(1/k)\) for the dual sequence. The competitive performance of our algorithm is validated through numerical experiments.
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https://arxiv.org/abs/2601.12398
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6f490ed84ea1521aed74a86f941a878c98d6b499d64003475b91909629ca5afc
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2026-01-21T00:00:00-05:00
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BiCoLoR: Communication-Efficient Optimization with Bidirectional Compression and Local Training
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arXiv:2601.12400v1 Announce Type: new Abstract: Slow and costly communication is often the main bottleneck in distributed optimization, especially in federated learning where it occurs over wireless networks. We introduce BiCoLoR, a communication-efficient optimization algorithm that combines two widely used and effective strategies: local training, which increases computation between communication rounds, and compression, which encodes high-dimensional vectors into short bitstreams. While these mechanisms have been combined before, compression has typically been applied only to uplink (client-to-server) communication, leaving the downlink (server-to-client) side unaddressed. In practice, however, both directions are costly. We propose BiCoLoR, the first algorithm to combine local training with bidirectional compression using arbitrary unbiased compressors. This joint design achieves accelerated complexity guarantees in both convex and strongly convex heterogeneous settings. Empirically, BiCoLoR outperforms existing algorithms and establishes a new standard in communication efficiency.
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https://arxiv.org/abs/2601.12400
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4078ddb238021363632a336b6656551a3484947fc51ab7a3eabfedec0ee70f82
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2026-01-21T00:00:00-05:00
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Weak quantum hypergroups from finite index C*-inclusions
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arXiv:2601.12406v1 Announce Type: new Abstract: We study a finite index inclusion of simple unital C*-algebras and construct a canonical completely positive coproduct on the second relative commutant, thereby endowing it with a natural coalgebra structure. Motivated by this construction, we introduce the notion of a weak quantum hypergroup, a generalization of the quantum hypergroups of Chapovsky and Vainerman. We show that every finite index inclusion gives rise to such a weak quantum hypergroup, and that the corresponding weak quantum hypergroup possesses a Haar integral. In the irreducible case, this structure satisfies the axioms of a quantum hypergroup in the sense of Chapovsky and Vainerman, while in the depth 2 setting our framework yields the associated weak Hopf algebra constructed by Nikshych and Vainerman. These results provide a unified and intrinsically C*-algebraic framework for generalized quantum symmetries associated with finite index inclusions.
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https://arxiv.org/abs/2601.12406
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2482ca2ce60439396ec29ce2549af91ba279f39d523fc43cc2b1fb22606c24ae
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2026-01-21T00:00:00-05:00
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Majorization between symplectic spectra of positive semidefinite matrices
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arXiv:2601.12408v1 Announce Type: new Abstract: Given $2n \times 2n$ real symmetric positive semidefinite matrix $A$ with symplectic kernel, there exists a real $2n \times 2n$ \emph{symplectic matrix} $M$ such that $M^TAM= D \oplus D$, where $D$ is an $n \times n$ non-negative diagonal matrix which is unique up to permutation of its diagonal entries. The diagonal entries of $D$ are called the \emph{symplectic eigenvalues} or symplectic spectrum of $A$. In this work, we investigate some majorization and weak supermajorization relations between the symplectic spectra of two positive semidefinite matrices. More explicitly, suppose $A$ and $B$ are $2n \times 2n$ real symmetric positive semidefinite matrices with symplectic kernels. We show that if the symplectic spectrum of $A$ is majorized by the symplectic spectrum of $B$, then $A$ lies in the convex hull of the symplectic orbit of $B$. We also establish that only a weak converse of this statement holds; i.e., if $A$ lies in the convex hull of the symplectic orbit of $B$ then the symplectic spectrum of $A$ is \emph{weakly supermajorized} by the symplectic spectrum of $B$. Several consequences of our results are also presented. Our methods make use of well-known connections between the theory of majorization, doubly stochastic, doubly superstochastic, and symplectic matrices.
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https://arxiv.org/abs/2601.12408
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9b8f6f0332f3e838ee2553c1c2eabd17e16f2362b665b282d2801d82aa3fe451
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2026-01-21T00:00:00-05:00
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Dynamic resource allocation in eukaryotic Resource Balance Analysis
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arXiv:2601.12411v1 Announce Type: new Abstract: Resource Balance Analysis (RBA) is a framework for predicting steady-state cellular growth under resource constraints. However, classical RBA formulations are static and do not capture the dynamic regulation of biosynthetic resources or macromolecular turnover, which is particularly important in eukaryotic cells. In this work, we propose a dynamic extension of eukaryotic RBA based on an optimal control formulation. Cellular growth is modeled as the result of a time-dependent allocation of translational capacity between metabolic enzymes and macromolecular machinery, aimed at maximizing biomass accumulation over a finite time horizon. Using Pontryagin's Maximum Principle, we characterize optimal allocation strategies and show that steady-state RBA solutions arise as limiting regimes of the dynamic problem.
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https://arxiv.org/abs/2601.12411
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d564fe7fa8361c37ce55b22b55e3dcdd4310163b5bd11c3ad6c4cb9329904b23
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2026-01-21T00:00:00-05:00
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Localization and interpolation of parabolic $L^p$ Neumann problems
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arXiv:2601.12429v1 Announce Type: new Abstract: We show a localization estimate for local solutions to the parabolic equation $-\partial_t u+\mbox{div} (A\nabla u)=0$ with zero Neumann data, assuming that the $L^p$ Neumann problem and $L^{p'}$ Dirichlet problem for the adjoint operator are solvable in a Lipschitz cylinder for some $p\in(1,\infty)$. Using this result, we establish the solvability of the Neumann problem in the atomic Hardy space for parabolic operators with bounded, measurable, time-dependent coefficients, and hence obtain the interpolation of solvability of the $L^p$ Neumann problem.
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https://arxiv.org/abs/2601.12429
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ed3b6e8f9c805b49a0f4f0aff54cfc7b1db44f66a7783374ac6cd10819f27770
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2026-01-21T00:00:00-05:00
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Periodic families in the homology of $GL_n(F_2)$
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arXiv:2601.12431v1 Announce Type: new Abstract: We construct infinite families of nonzero classes in $H_d(GL_n(F_2);F_2)$ along lines of the form $d =\frac{2}{3}n +$(constant), thereby showing that the known slope $\frac{2}{3}$-stability for these homology groups are optimal. Using the new stability Hopf algebra perspective of Randal-Williams, our computations in addition recover the slope-$\frac{2}{3}$ stability for $GL_n(Z)$ with coefficients in $F_2$, improve that for $Aut(F_n)$ to $\frac{2}{3}$, and demonstrate that those slopes are optimal. Perhaps of independent interest, we also provide a manual for computing stability Hopf algebras over $F_2$.
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https://arxiv.org/abs/2601.12431
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ec434fdde61c30a0b44f84b737b617ea68e7a0ef5059feca10154726963cdb8b
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2026-01-21T00:00:00-05:00
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Disjoint non-forking amalgamation in stable AECs
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arXiv:2601.12439v1 Announce Type: new Abstract: The \emph{disjoint amalgamation property} (DAP), which asserts that all spans of a class of models can be amalgamated with minimal intersection, is an important property in the context of abstract elementary classes, with connections to both Grossberg's question and Shelah's categoricity conjecture. We prove that, in a nice AEC $\mathbf{K}$ stable in $\lambda \geq \operatorname{LS}(\mathbf{K})$ with a strong enough independence relation, all high cofinality $\lambda$-limit models are disjoint (non-forking) amalgamation bases. $\textbf{Theorem.}$ Let $\mathbf{K}$ be an AEC stable in $\lambda$, where $\mathbf{K}_\lambda$ has AP, JEP, and NMM, and let $\mathbf{K}'$ be some AC where $\mathbf{K}_{(\lambda,\geq\kappa)} \subseteq \mathbf{K}' \subseteq \mathbf{K}_\lambda$. Suppose there is an independence relation on $\mathbf{K}'$ satisfying uniqueness, existence, non-forking amalgamation, $\mathbf{K}_{(\lambda,\geq\kappa)}$-universal continuity* in $\mathbf{K}_\lambda$, and $(\geq \kappa)$-local character. Assume $M_0, M_1, M_2 \in \mathbf{K}_{(\lambda,\geq\kappa)}$, and that $M_0 \leq_{\mathbf{K}} M_l$ and $a_l \in M_l$ for $l = 1, 2$. Then there exist $N \in \mathbf{K}_{(\lambda,\geq\kappa)}$ and $f_l : M_l \rightarrow N$ fixing $M_0$ for $l = 1, 2$ such that $\operatorname{gtp}(f_l(a_l)/f_{3-l}[M_{3-l}], N)$ does not fork over $M_0$ and $f_1[M_1] \cap f_2[M_2] = M_0$. That is, our independence relation has disjoint non-forking amalgamation. In particular, every $M_0 \in \mathbf{K}_{(\lambda,\geq\kappa)}$ is a disjoint amalgamation base in $\mathbf{K}_\lambda$. The hypotheses on the independence relation can be weakened (closer to $\lambda$-non-splitting in $\lambda$-stable AECs) if we are willing to give up the `non-forking' conditions of the amalgamation.
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https://arxiv.org/abs/2601.12439
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f32e48cc3683ad1021f53670baed61946a2f005bb31d4d8d464512b7b51d8958
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2026-01-21T00:00:00-05:00
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Schr\"odinger Operators, Integral Curvature, and the Euler Characteristic of Riemannian Manifolds
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arXiv:2601.12440v1 Announce Type: new Abstract: We establish new connections between integral curvature bounds and the Euler characteristic of closed Riemannian manifolds through the perspective of Schr\"odinger-type operators. Central to our approach is the twisted Dirac operator \(\mathcal{D}_{\theta}\), whose index equals \(\chi(M)\). Under integral smallness conditions on the negative part of a potential \(V\) and a Sobolev--Poincar\'e inequality, we show that a suitable scaling of \(\theta\) forces the kernel of \(\mathcal{D}_{t\theta}\) to vanish, thereby implying \(\chi(M)=0\). Applying this framework to geometrically natural potentials yields several topological consequences. In even dimensions, sufficiently small integral bounds on partial sums of curvature operator eigenvalues force \(\chi(M)\) either to vanish or to have a sign determined by the middle dimension. For four-manifolds, a small \(L^{p}\)-norm of the negative Ricci curvature relative to the diameter guarantees \(\chi(M)\ge 0\). Moreover, when \(\chi(M)\neq 0\) we obtain a Li--Yau type lower bound for the first eigenvalue of the rough Laplacian on \(1\)-forms in terms of the diameter and an integral curvature quantity. Subsequently, we provide an explicit lower bound for the first eigenvalue of the Laplacian on $1$-forms under almost nonnegative curvature conditions, thereby giving an affirmative answer to Yau's Problem 79.
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https://arxiv.org/abs/2601.12440
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686b9bd9f29cf196e700a9007e2038d1ea72c137869abb0254af87d9f994f40c
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2026-01-21T00:00:00-05:00
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Distinct permutation dot products
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arXiv:2601.12445v1 Announce Type: new Abstract: We show that for any two sets of reals numbers $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_n\}$, the sums of the form $\sum_{i=1}^n a_i\,b_{\pi(i)}$ always take on $\Omega(n^{3})$ distinct values, as we range over all permutations $\pi \in S_n$. An important ingredient is a ``supportive'' version of Hal\'asz's anticoncentration theorem from Littlewood-Offord theory, which may be of independent interest.
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https://arxiv.org/abs/2601.12445
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ec62813e71e735fe71b1686ee787e2a18342740b6e131bfb84a8e8d840440974
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2026-01-21T00:00:00-05:00
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On spaces of embeddings of circles in surfaces
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arXiv:2601.12450v1 Announce Type: new Abstract: We consider the space of embeddings of finitely many circles that bound disks in non-positively curved surfaces. We index the connected components of this space with finite rooted trees and show that the connected components are classifying spaces of the ``braided" automorphism groups of the associated trees. An intermediate step to proving these results is to construct a strong deformation retract onto the subspace of geometric circles; moreover, this strong deformation retraction is equivariant with respect to transformations of the surface.
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https://arxiv.org/abs/2601.12450
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62a4afbfde1d128d71d658e65cd8faf7e0d6e659bb66b66bd7a08fd82a333dd1
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2026-01-21T00:00:00-05:00
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Even Sets and Dual Projective Geometric Codes: A Tale of Cylinders
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arXiv:2601.12451v1 Announce Type: new Abstract: In this paper, we prove that the smallest even sets in ${\rm PG}(n,q)$, i.e. sets that intersect every line in an even number of points, are cylinders with a hyperoval as base. This fits into a more general study of dual projective geometric codes. Let $q$ be a prime power, and define $\mathcal C_k(n,q)^\perp$ as the kernel of the $k$-space vs. point incidence matrix of ${\rm PG}(n,q)$, seen as a matrix over the prime order subfield of $\mathbb F_q$. Determining the minimum weight of this linear code is still an open problem in general, but has been reduced to the case $k=1$. There is a known construction that constructs small weight codewords of $\mathcal C_1(n,q)^\perp$ from minimum weight codewords of $\mathcal C_1(2,q)^\perp$. We call such codewords cylinder codewords. We pose the conjecture that all minimum weight codewords of $\mathcal C_1(n,q)^\perp$ are cylinder codewords. This conjecture is known to be true if $q$ is prime. We take three steps towards proving that the conjecture is true in general: (1) We prove that the conjecture is true if $q$ is even. This is equivalent to our classification of the smallest even sets. (2) We prove that the minimum weight of $\mathcal C_1(n,q)^\perp$ is $q^{n-2}$ times the minimum weight of $\mathcal C_1(2,q)^\perp$, which matches the weight of cylinder codewords. Thus, we completely reduce the problem of determining the minimum weight of $\mathcal C_1(n,q)^\perp$ to the case $n=2$. (3) We prove that if the conjecture is true for $n=3$, it is true in general.
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https://arxiv.org/abs/2601.12451
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c91019f20a1bcac73c149d8da898de1208d8f5eddcb6ba891d5d967f9375b1a3
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2026-01-21T00:00:00-05:00
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Unbounded banded matrices with positive bidiagonal factorization and mixed-type multiple orthogonal polynomials
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arXiv:2601.12453v1 Announce Type: new Abstract: A spectral Favard theorem is proved for semi-infinite banded matrices admitting a positive bidiagonal factorization, without assuming boundedness of the associated operator, thus covering both the bounded and unbounded settings. The result yields a matrix-valued spectral measure and an explicit spectral representation of the matrix powers in terms of the associated mixed-type multiple orthogonal polynomials. The argument follows the constructive truncation scheme: principal truncations are oscillatory, hence have simple positive spectra, and a suitable choice of initial conditions ensures positivity of the Christoffel coefficients and of the resulting discrete matrix-valued measures supported at the truncation eigenvalues. The main difficulty is the passage to the limit of these discrete measures beyond the bounded case. This is resolved by combining the available Gaussian quadrature structure with a Helly-type compactness argument, leading to a limiting matrix-valued measure and completing the spectral theorem. The role of normality (maximal degree pattern) for the mixed-type families is also addressed.
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https://arxiv.org/abs/2601.12453
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4055eb21a16056113d71e66bdb163a10e80f6ccc7c3b40dd34b874f6ac4d1f47
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2026-01-21T00:00:00-05:00
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Hirzebruch-Riemann-Roch for complex analytic infinity-prestacks
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arXiv:2601.12454v1 Announce Type: new Abstract: We provide a cocycle-level Hirzebruch-Riemann-Roch (HRR) identity for arbitrary complex analytic infinity-prestacks. We view this work as the natural setting for Toledo and Tong's HRR philosophy and technical machinery.
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https://arxiv.org/abs/2601.12454
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13bb1edb914c16a4cdea0c9c31cd5ea516b450e4380229c41b1b253a43bf0d3c
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2026-01-21T00:00:00-05:00
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Structure theory of set addition with two operations
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arXiv:2601.12457v1 Announce Type: new Abstract: We take the first step toward a structure theory that includes both operations of a ring $\mathcal{R}$. More precisely, we prove a series of inverse results for the structure of sets $A\subseteq \mathbf{F}_p$ such that, under certain conditions on integers $r_1, \dots, r_k$, one has $|A^{r_1} + \dots + A^{r_k}| \ll \sqrt[k]{p^{k-1} |A|}$.
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https://arxiv.org/abs/2601.12457
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688af3e7aae74d743edd6d618ac7dbb8bd00ee6f50c5b4a328bf3afa766ea7c9
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2026-01-21T00:00:00-05:00
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Symmetric preparation of systems
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arXiv:2601.12458v1 Announce Type: new Abstract: In this paper we generalize the Weierstrass and Malgrange preparation theorems to the symmetric matrix valued case, proving symmetric preparation of analytic and smooth symmetric systems that vanish of first order.
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https://arxiv.org/abs/2601.12458
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f5b02721e73dc3ba18d9705659dca7c7e4b250c1755a3587911fd2e27895ee86
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2026-01-21T00:00:00-05:00
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Homological $(n-2)$-systole in $n$-manifolds with positive triRic curvature
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arXiv:2601.12461v1 Announce Type: new Abstract: In this paper, we prove an optimal systolic inequality and characterize the case of equality on closed Riemannian manifolds with positive triRic curvature. This extends prior work of Bray-Brendle-Neves \cite{BrayBrenleNevesrigidity} and Chu-Lee-Zhu \cite{chuleezhu_n_systole} to higher codimensions. The proof relies on the notion of stable weighted $k$-slicing, a weighted volume comparison theorem and metric-deformation.
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https://arxiv.org/abs/2601.12461
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3e1e5317349a9cb6d00289fec279cf582e599b3b215ee55c49afcc43b5c114d9
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2026-01-21T00:00:00-05:00
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Non-intersecting Squared Bessel Process: Spectral Moments and Dynamical Entanglement Entropy
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arXiv:2601.12484v1 Announce Type: new Abstract: Statistical ensembles of reduced density matrices of bipartite quantum systems play a central role in entanglement estimation, but do not capture the non-stationary nature of entanglement relevant to realistic quantum information processing. To address this limitation, we propose a dynamical extension of the Hilbert-Schmidt ensemble, a baseline statistical model for entanglement estimation, arising from non-intersecting squared Bessel processes and perform entanglement estimation via average entanglement entropy and quantum purity. The investigation is enabled by finding spectral moments of the proposed dynamical ensemble, which serves as a new approach for systematic computation of entanglement metrics. Along the way, we also obtain new results for the underlying multiple orthogonal polynomials of modified Bessel weights, including structure and recurrence relations, and a Christoffel-Darboux formula for the correlation kernels.
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https://arxiv.org/abs/2601.12484
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27a9246dc97b3e3d97b92d69abc73aad51ba10f025ca07a10aaf3f984800f0c6
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2026-01-21T00:00:00-05:00
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Fast Computing Formulas for some Dirichlet L-Series
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arXiv:2601.12495v1 Announce Type: new Abstract: For $\chi_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,\chi_k)$, expressed as linear combinations of Hurwitz $\zeta$ functions, are found for $s=2,3$ and some selected values of $k$. By using a merged approach between the Wilf$-$Zeilberger method and a Dougall$'$s $_5H_5$ technique, new proven accelerated series of hypergeometric$-$type are derived for specific Hurwitz $\zeta$ function values. These fast series that are computed by means of the binary splitting algorithm, enter into the reduced identities found producing very efficient formulas to compute these selected $L-$functions. The new algorithms include $\zeta(3):=L_1(3)$, (Apery$'$s constant), $G:=L_\text{-4}(2)$ (Catalan$'$s constant) as well as $\text{}L_\text{k}(2)\text{}$ for $k=-7, -8, -15, -20, -24$ together with $L_k(3)$ for $k=5, 8, 12$. Formulas were tested and verified up to 100 million decimal places for each $L-$value.
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https://arxiv.org/abs/2601.12495
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3ee1f2207a90aa421d98be7132745c17deff1a2324fda466c511f5d72c90abe4
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2026-01-21T00:00:00-05:00
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The $\ell$-modular local theta correspondence in type II and partial permutations
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arXiv:2601.12497v1 Announce Type: new Abstract: In this paper we compute the multiplicities appearing in the ${\overline{\mathbb{F}}_\ell}$-modular theta correspondence in type II over a non-archimedean field $\mathrm{F}$, where $\ell$ is a prime not dividing the residue cardinality of $\mathrm{F}$. Unlike for representations with complex coefficients, highly non-trivial multiplicities can emerge. We show that these multiplicities are precisely governed by the action of symmetric groups on the set of partial permutations, and the ${\overline{\mathbb{F}}_\ell}$-representation of symmetric groups these give rise to. The problem is thus reduced to certain branching problems in the modular representation theory of symmetric groups. In particular, if $d$ is the order of the residue cardinality of $\mathrm{F}$ in ${\overline{\mathbb{F}}_\ell}$, and the rank of the involved general linear groups is bounded above by $ d\ell$, the behavior of the theta correspondence can be predicted via explicit algorithms coming from Pieri's Formula.
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https://arxiv.org/abs/2601.12497
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c6251843ca541868af7eaee560cb0da28a015f309197aa02685c477b1bb6c2d7
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2026-01-21T00:00:00-05:00
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Heun-function analysis of the Dirac spinor spectrum in a sine-Gordon soliton background
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arXiv:2601.12504v1 Announce Type: new Abstract: We study the Dirac spectrum in a sine-Gordon soliton background, where the induced position-dependent mass reduces the spectral problem to a Heun-type differential equation. Bound and scattering sectors are treated within a unified framework, with spectral data encoded in Wronskians matching local Heun solutions and exhibiting explicit dependence on the soliton parameters and the bare fermion mass. This formulation enables a systematic analysis of spinor bound and scattering states, supported by analytic and numerical verification of wave function matching across the soliton domain. The present work is related to arXiv:2512.07658 and emphasizes a pedagogical treatment of scattering states within the Heun-equation formalism.
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https://arxiv.org/abs/2601.12504
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244be761145582637d5c0a2caee43fb0d1b62e2f73116325fd4a38de434e7da9
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2026-01-21T00:00:00-05:00
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Approximability for Lagrangian submanifolds
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arXiv:2601.12506v1 Announce Type: new Abstract: This paper introduces a notion of categorical approximability for metric spaces that can be viewed as a categorification of approximability for metric groups, as defined by Turing in 1938. Approximability as introduced here is a property of metric spaces that is more general than precompactness. It is shown that several classes of Lagrangian submanifolds - closed Lagrangian submanifolds in a cotangent disk bundle; equators on the sphere; weakly exact Lagrangians on the torus-endowed with the spectral metric are approximable in this sense. Among other geometric applications, we show that there are such examples of spaces of Lagrangians that are approximable but are not precompact.
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https://arxiv.org/abs/2601.12506
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06a20021055b7e977be59ebf1d76cb669696c778f06f311c365792afd516b3d3
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2026-01-21T00:00:00-05:00
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Extending graph total colorings to cell complexes
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arXiv:2601.12514v1 Announce Type: new Abstract: Let $2\le k\in\mathbb{Z}$. A total coloring of a simple connected regular graph via color set $ \{0,1,\ldots, k\}$ is said to be {\it efficient} if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. In this work, focus is set upon 2-cell complexes whose 1-skeletons, namely their induced 1-cell complexes, are toroidal graphs. Each such 2-cell complex is said to cover its induced 1-skeleton. An efficient total coloring of one such skeleton induces an efficient total cell coloring of its covering 2-cell complex if it assigns a vertex-and-edge $k$-color set to the border skeleton of each of its 2-cells, with the consequently missing color in $\{0,1,\ldots,k\}$ assigned to the 2-cell itself, so that the two adjacent 2-cells along any 1-cell are assigned different colors. Applications are given for plane tilings, cycle products, toroidal triangulations, honeycombs and star-of-David tilings.
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https://arxiv.org/abs/2601.12514
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26d6b6266e14a982e1d581dfbda6eda26a2b61baf839b5161c6701489dce4d71
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2026-01-21T00:00:00-05:00
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Rigidity results in multi-bubble dynamics for non-radial energy-critical heat equation
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arXiv:2601.12517v1 Announce Type: new Abstract: This paper concerns the classification of asymptotic behaviors in multi-bubble dynamics for the energy-critical nonlinear heat equations in large dimensions $N\geq7$ without symmetry. This multi-bubble dynamics appears naturally at least for a sequence of times in view of soliton resolution. We assume each bubble is given by the scalings and translations of $\pm W$ with (localized) non-colliding conditions for a sequence of times, where $W$ is the ground state. The case of one soliton was previously established and in particular there is no blow-up. We consider the case of $J\geq2$ solitons, where we expect only infinite-time blow-up. We are able to identify three different scenarios, where we have a continuous-in-time resolution with an unexpected universal blow-up speed. The first one is when one scaling is much larger than the others. In this case, one bubble does not concentrate (hence stabilize) and the other bubbles concentrate with the universal blow-up speed $t^{-2/(N-6)}$ together with strong sign constraints. Next, assuming we are not in the first scenario, we establish a non-degenerate condition on the positions of bubbles to obtain that all bubbles concentrate with the universal blow-up speed $t^{-1/(N-4)}$. The last case we consider is a degenerate, but not too much degenerate, scenario. Here again, we obtain that all bubbles concentrate with the universal blow-up speed $t^{-1/(N-3)}$. This last rate has not been discovered before. Our theorem covers the case of four or less bubbles and we provide the construction of examples. To our knowledge, this is the first classification result in the non-radial multi-bubble dynamics, where both the scales, positions, and signs enter the dynamics nontrivially.
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https://arxiv.org/abs/2601.12517
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444ada904f3173c774d221c14d9025dfd0003e293f3151b5cd4d402f049fb75b
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2026-01-21T00:00:00-05:00
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Quasigeodesic languages are not context-free in some non-hyperbolic groups
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arXiv:2601.12520v1 Announce Type: new Abstract: We study the full language of quasigeodesics in Cayley graphs, with fixed error constants. We show that, given a non-virtually-cyclic nilpotent group or Baumslag--Solitar group, and any finite generating set, such languages fail to be context-free for sufficiently large error constants. In fact, this conclusion holds for any finitely generated group which contains one of these groups as an undistorted subgroup. This strengthens a recent theorem of Hughes, Nairne, and Spriano, who showed that such languages fail to be regular in any non-hyperbolic group, for sufficiently large error constants.
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https://arxiv.org/abs/2601.12520
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cf3a00e2d18149779c5aff42b262a1a6448aebe5d7057468235f80502d983901
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2026-01-21T00:00:00-05:00
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Derived equivalences via Tate resolutions
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arXiv:2601.12531v1 Announce Type: new Abstract: For any finite sequence of elements $s_1, \ldots , s_d$ in a commutative noetherian ring $R$, we show that for $n \gg 0$, the natural map from the Koszul complex $K(s_1^n, \ldots , s_d^n)$ to the Koszul complex $K(s_1, \ldots , s_d)$ factors through the Tate resolution on $s_1^n, \ldots , s_d^n$. Using this, for any resolving subcategory $\mathcal A$ of mod($R$) and any ideal $I$ such that it has a filtration $\{ I_n \}$ which is equivalent to the $I$-adic filtration and $\textrm{dim}_{\mathcal A}(R/I_n) < \infty$, we show a derived equivalence between the bounded derived category of finitely generated modules supported on $V(I)$ having finite $\mathcal A$-dimension and the bounded derived category of $\mathcal A$ with homologies supported on $V(I)$. As a special case, when $R$ is of prime characteristic and $I$ is of finite projective dimension, we obtain a derived equivalence between the bounded derived category of finite projective dimension modules supported on $V(I)$ and the bounded derived category of projective modules with homologies supported on $V(I)$.
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https://arxiv.org/abs/2601.12531
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05003ae61e640e6c167603ade153323d504806b462d7b153b66dcfcf2a2881ea
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2026-01-21T00:00:00-05:00
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Examples and counterexamples of injective types
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arXiv:2601.12536v1 Announce Type: new Abstract: It is known that, in univalent mathematics, type universes, the type of $n$-types in a universe, reflective subuniverses, and the underlying type of any algebra of the lifting monad are all (algebraically) injective. Here, we further show that the type of ordinals, the type of iterative (multi)sets, the underlying type of any pointed directed complete poset, as well as the types of (small) $\infty$-magmas, monoids, and groups are all injective, among other examples. Not all types of mathematical structures are injective in general. For example, the type of inhabited types is injective if and only if all propositions are projective. In contrast, the type of pointed types and the type of non-empty types are always injective. The injectivity of the type of two-element types implies Fourman and \v{S}\v{c}edrov's world's simplest axiom of choice. We also show that there are no nontrivial small injective types unless a weak propositional resizing principle holds. Other counterexamples include the type of booleans, the simple types, the type of Dedekind reals, and the type of conatural numbers, whose injectivity implies weak excluded middle. More generally, any type with an apartness relation and two points apart cannot be injective unless weak excluded middle holds. Finally, we show that injective types have no non-trivial decidable properties, unless weak excluded middle holds, which amounts to a Rice-like theorem for injective types.
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https://arxiv.org/abs/2601.12536
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b328f6af4da2004ea9f9b9a4c97eb2f1ffcef33848125bb1661a0fd11f3cd132
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2026-01-21T00:00:00-05:00
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Connections and na\"{i}ve lifting of DG modules
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arXiv:2601.12550v1 Announce Type: new Abstract: In this paper, we generalize the notion of connections, which was introduced by Alain Connes in noncommutative differential geometry, to the differential graded (DG) homological algebra setting. Then, along a DG algebra homomorphism $A \to B$, where $B$ is assumed to be projective as an underlying graded $A$-module, we give necessary and sufficient conditions for a semifree DG $B$-module to be na\"{i}vely liftable to $A$ in terms of connections.
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https://arxiv.org/abs/2601.12550
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b51458a49350abd9047e37e55941ce4f4df0d66a8f7b4f4cbf92cca174865bbd
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2026-01-21T00:00:00-05:00
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Remarks on the second Chern class of a foliation
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arXiv:2601.12558v1 Announce Type: new Abstract: We bound the second Chern class of the tangent sheaf of a codimension-one foliation. Equivalently, we bound the degree of the pure codimension-two part of the singular scheme. In particular, for a degree-$d$ foliation on the projective space, the codimension-two part of its singular scheme must have degree at least $d+1$. Moreover, equality holds only for rational foliations of type $(1,d+1)$. These bounds involve counting an invariant related to first-order unfoldings of 2-dimensional foliated singularities.
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https://arxiv.org/abs/2601.12558
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d977fbe82bbd25fbef31790b3d1078c3335247c881eed0dbe86a918cc875026e
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2026-01-21T00:00:00-05:00
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Sheared displays and $p$-divisible groups
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arXiv:2601.12565v1 Announce Type: new Abstract: We develop a Dieudonn\'e theory for $p$-divisible groups using sheared Witt vectors.
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https://arxiv.org/abs/2601.12565
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e0f1baa78aed1038eb68d93b5295d14e6f35a0c961760a71426515ff9cffcc1c
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2026-01-21T00:00:00-05:00
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Self-avoiding walk, connective constant, cubic graph, Fisher transformation, quasi-transitive graph
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arXiv:2601.12571v1 Announce Type: new Abstract: We study self-avoiding walks (SAWs) on infinite quasi-transitive cubic graphs under \emph{local transformations} that replace each degree-$3$ vertex by a finite, symmetric three-port gadget. To each gadget we associate a two-port SAW generating function $g(x)$, defined by counting SAWs that enter and exit the gadget through prescribed ports. Our first main result shows that, if $G$ is cubic and $G_1=\phi(G)$ is obtained by applying the local transformation at every vertex, then the connective constants $\mu(G)$ and $\mu(G_1)$ satisfy the functional relation \[ \mu(G)^{-1}=g\bigl(\mu(G_1)^{-1}\bigr). \] We next consider critical exponents defined via susceptibility-type series that do not rely on an ambient Euclidean dimension, and prove that the exponents $\gamma$ and $\eta$ are invariant under local transformations; moreover $\nu$ is invariant under a standard regularity hypothesis on SAW counts (a common slowly varying function). Our second set of results concerns bipartite graphs, where the local transformation is applied to one colour class (or to both classes, possibly with different gadgets). In this setting we obtain an analogous relation \[ \mu(G)^{-2}=h\bigl(\mu(G_{\mathrm e})^{-1}\bigr), \] with $h(x)=xg(x)$ when only one class is transformed and $h(x)=g_{\phi_1}(x)\,g_{\phi_2}(x)$ when both are transformed. We further present explicit families of examples, including replacing each degree-3 vertex by a complete-graph gadget $K_N$.
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https://arxiv.org/abs/2601.12571
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99bf850bebcb0cb8a08944419d75f4d4df55c912bc0fd63a182b5ba7a4d3950a
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2026-01-21T00:00:00-05:00
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A Functorial Approach to Multi-Space Interpolation with Function Parameters
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arXiv:2601.12572v1 Announce Type: new Abstract: We introduce an extension of interpolation theory to more than two spaces by employing a functional parameter, while retaining a fully functorial and systematic framework. This approach allows for the construction of generalized intermediate spaces and ensures stability under natural operations such as powers and convex combinations. As a significant application, we demonstrate that the interpolation of multiple generalized Sobolev spaces yields a generalized Besov space. Our framework provides explicit tools for handling multi-parameter interpolation, highlighting both its theoretical robustness and practical relevance.
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https://arxiv.org/abs/2601.12572
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475d28dccf99e5f9d502ff8531092143ca2a934ae266b94313934fc3be71d828
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2026-01-21T00:00:00-05:00
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L(3,2,1)-labelings of three classes of 4-valent circulants
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arXiv:2601.12574v1 Announce Type: new Abstract: An $L(3,2,1)$-labeling of a graph $G$ is an assignment $f$ of nonnegative integers to vertices such that $\vert f(x)-f(y)\vert > 3-\mbox{dist}_G(x,y)$ for every pair $x,y$ of vertices of $G$, where $\mbox{dist}_G(x,y)$ denotes the distance between $x$ and $y$ in $G$. The minimum span (i.e., the difference between the largest and the smallest value) among all $L(3,2,1)$-labelings of $G$ is denoted by $\lambda_{(3,2,1)}(G)$. In this paper, we study $L(3,2,1)$-labelings of three classes of circulant graphs. Namely, we investigate $\lambda_{(3,2,1)}$ of $C_n(\{1,s_2,n-s_2,n-1\})$, where $s_2\in\{3,4,5\}$. This paper is a continuation of a recent publication of T. Calamoneri who studied the square of cycles, i.e., circulants $C_n(\{1,2,n-2,n-1\})$.
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https://arxiv.org/abs/2601.12574
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d012b69b52648d4e9c82965cb31e9caebbb308a40cb069cb7b5ce040ced52bc9
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2026-01-21T00:00:00-05:00
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A semigroup approach to iterated binomial transforms
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arXiv:2601.12579v1 Announce Type: new Abstract: We study a one-parameter family of binomial-convolution operators acting on sequences. These operators form an additive semigroup with an explicit inverse, and they subsume iterated classical binomial transforms as a special case. We describe the action in terms of ordinary and exponential generating functions, interpret the transform in the Riordan-array framework, and prove a general root-shift principle for constant-coefficient linear recurrences: applying the transform shifts the characteristic roots by a fixed amount. Several classical families (Fibonacci, Lucas, Pell, Jacobsthal, Mersenne) are treated uniformly as illustrative examples.
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https://arxiv.org/abs/2601.12579
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bbf5148beb9993119d10e4fd48803f5625dd9a5b84d7d8e2c4de3f95e6bf5e9b
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2026-01-21T00:00:00-05:00
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Integrals of products of four modified Bessel functions
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arXiv:2601.12590v1 Announce Type: new Abstract: We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized hypergeometric and Lauricella $F_C$ functions, and study a number of special cases in which the integrals can be evaluated in terms of simpler special functions or indeed take an elementary form. As a consequence, we deduce some new formulas for definite integrals of products of four Airy functions.
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https://arxiv.org/abs/2601.12590
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ac98de0762ac3c9ecca52299225ebcc6b73710eb4ec15dbac362bbf3193e2454
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2026-01-21T00:00:00-05:00
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Ehrhart quasi-polynomials via Barnes polynomials and discrete moments of parallelepipeds
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arXiv:2601.12596v1 Announce Type: new Abstract: We give novel and explicit formulas for the Ehrhart quasi-polynomials of rational simple polytopes, in terms of Barnes polynomials and discrete moments of half-open parallelepipeds. These formulas also hold for all positive dilations of a rational polytope. There is an interesting appearance of an extra complex z-parameter, which seems to allow for more compact formulations. We also give similar formulas for discrete moments of rational polytopes, and their positive dilates, objects known in the literature as sums of polynomials over a polytope. The appearance of the Barnes polynomials and the Barnes numbers allow for explicit computations. From this work, it is clear that the complexity of computing Ehrhart quasi-polynomials lies mainly in the computation of various discrete moments of parallelepipeds. These discrete moments are in general summed over a particular lattice flow on a closed torus, defined in this paper. Some of the consequences involve novel vanishing identities for rational polytopes, novel formulations of Ehrhart polynomials of unimodular polytopes, and a differential equation that extends the work of Eva Linke.
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https://arxiv.org/abs/2601.12596
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7d5eb2bbdc3aba95c282cd0ba3d16fa37df59cb77903571fdaa24ed21e9be943
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2026-01-21T00:00:00-05:00
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Conjugating full cycles by adjacent transpositions: diameter and sorting time
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arXiv:2601.12597v1 Announce Type: new Abstract: We establish upper and lower bounds on the maximal number of steps needed to transform a cyclic permutation to the canonical cyclic permutation using conjugation by adjacent transpositions, and on the diameter of the underlying Schreier graph.
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https://arxiv.org/abs/2601.12597
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ed842edb82331cda32390431d40d5eff86c99ac7220fc3fee86bf366a49d0b15
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2026-01-21T00:00:00-05:00
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Elementary proofs of ring commutativity theorems
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arXiv:2601.12599v1 Announce Type: new Abstract: Jacobson's commutativity theorem says that a ring is commutative if, for each $x$, $x^n = x$ for some $n > 1$. Herstein's generalization says that the condition can be weakened to $x^n-x$ being central. In both theorems, $n$ may depend on $x$. In this paper, in certain cases where $n$ is a fixed constant, we find equational proofs of each theorem. For the odd exponent cases $n = 2k+1$ of Jacobson's theorem, our main tool is a lemma stating that for each $x$, $x^k$ is central. For Herstein's theorem, we consider the cases $n=4$ and $n=8$, obtaining proofs with the assistance of the automated theorem prover Prover9.
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https://arxiv.org/abs/2601.12599
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1f8567c613c9dfca95bc11203f8d0efa27fa9a7b35bfe674a096ac909db9a794
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2026-01-21T00:00:00-05:00
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Improved Averaged Distribution of $d_3(n)$ in Prime Arithmetic Progressions
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arXiv:2601.12601v1 Announce Type: new Abstract: We say that $d_3(n)$ has exponent of distribution $\theta$ if, for all $\varepsilon>0$, the expected asymptotic holds uniformly for all moduli $q \le x^{\theta-\varepsilon}$. Nguyen proved that, after averaging over reduced residue classes $a \bmod q$, the function $d_3(n)$ has exponent of distribution $2/3$, following earlier work of Banks et al. Using the Petrow--Young subconvexity bound for Dirichlet $L$-functions, we improve this to an exponent of distribution $8/11$ when averaging over residue classes modulo a prime $q$.
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https://arxiv.org/abs/2601.12601
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044bc9d3930b42ca2b0ed514bdeb23c51c576c541575dd6862f420f302287c96
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2026-01-21T00:00:00-05:00
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An unbounded number of canard limit cycles in linear regularizations of piecewise linear systems
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arXiv:2601.12602v1 Announce Type: new Abstract: The purpose of this paper is to study the number of limit cycles of canard type in linear regularizations of piecewise linear systems with non-monotonic transition functions. Using the notion of slow divergence integral and elementary breaking mechanisms, we construct systems with an arbitrary finite number of hyperbolic limit cycles. The Hopf breaking mechanism deals with transition functions with precisely one critical point in the interval $(-1,1)$. On the other hand, the jump breaking mechanism produces any number of limit cycles using transition functions with precisely three critical points in $(-1,1)$.
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https://arxiv.org/abs/2601.12602
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3805071344f55e5502a0d317bbe8b5206c96812446489251ab219cded07f37e1
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2026-01-21T00:00:00-05:00
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On the second homology of the genus 3 hyperelliptic Torelli group
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arXiv:2601.12605v1 Announce Type: new Abstract: Let $s$ be a fixed hyperelliptic involution of the closed, oriented genus $g$ surface $\Sigma_g$. The hyperelliptic Torelli group $\mathcal{SI}_g$ is the subgroup of the mapping class group $\mathrm{Mod}(\Sigma_g)$ consisting of elements that act trivially on $\mathrm{H}_1(\Sigma_g;\mathbb{Z})$ and commute with $s$. It is generated by Dehn twists about $s$-invariant separating curves, and its cohomological dimension is $g-1$. In this paper we study the top homology group $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$. For each pair of disjoint $s$-invariant separating curves there is a naturally associated abelian cycle in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$; we call such cycles \emph{simple}. We show that simple abelian cycles are in bijection with orthogonal (with respect to the intersection form) splittings of $\mathrm{H}_1(\Sigma_3;\mathbb{Z})$ satisfying a simple algebraic condition, and prove that these abelian cycles are linearly independent in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$.
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https://arxiv.org/abs/2601.12605
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af5a64ef5c66e48bcf179a7c5ab58be2748b4affc4a1709b34be6c0e6cad875a
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2026-01-21T00:00:00-05:00
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On the second Bohr radius for vector valued pluriharmonic functions
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arXiv:2601.12608v1 Announce Type: new Abstract: In this paper, we introduce the notion of the second Bohr radius for vector valued pluriharmonic functions on complete Reinhardt domains in $\mathbb{C}^n$. This investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 1147-1155], where the corresponding problem was studied for complex valued holomorphic functions. We show that the second Bohr radius constant for pluriharmonic functions is strictly positive under suitable condition. In addition, we obtain its asymptotic behavior in the finite-dimensional settings using invariants from local Banach space theory. Asymptotic estimates for this constant are obtained on both convex and non-convex complete Reinhardt domains. Our results also apply to a broad class of Banach sequence spaces, including symmetric and convex Banach spaces. The framework developed here also includes the second Bohr radius problem for vector valued holomorphic functions. As an application of our results, we derive several consequences that extend known results in the scalar valued setting as well as existing results in the literature.
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https://arxiv.org/abs/2601.12608
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9fcb40439a645c38d22454978590ce7b87bf9d8322ae1027e973d56899f80a59
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2026-01-21T00:00:00-05:00
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A class of non-cylindrical domains for parabolic equations
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arXiv:2601.12609v1 Announce Type: new Abstract: We present a class of non-cylindrical domains where Dirichlet-type problems for parabolic equations, such as the heat equation, can be posed and solved. The regularity for the boundary of this class of domains is a mixed Lipschitz condition, as described in the bulk of the paper. The main tool is an adequate version of the implicit function theorem for functions with this kind of regularity. It is proved that the class introduced herein is of the same type as domains previously considered by several authors.
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https://arxiv.org/abs/2601.12609
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7928debcf53ce1550da5885fe88102d0cb650b33616e256eb6bdebf4ec006cd4
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2026-01-21T00:00:00-05:00
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An Eventown Result for Permutations
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arXiv:2601.12613v1 Announce Type: new Abstract: A family of permutations $\mathcal{F} \subseteq S_n$ is even-cycle-intersecting if $\sigma \pi^{-1}$ has an even cycle for all $\sigma,\pi \in \mathcal{F}$. We show that if $\mathcal{F} \subseteq S_n$ is an even-cycle-intersecting family of permutations, then $|\mathcal{F}| \leq 2^{n-1}$, and that equality holds when $n$ is a power of 2 and $\mathcal{F}$ is a double-translate of a Sylow 2-subgroup of $S_n$. This result can be seen as an analogue of the classical eventown problem for subsets and it confirms a conjecture of J\'anos K\"orner on maximum reversing families of the symmetric group. Along the way, we show that the canonically intersecting families of $S_n$ are also the extremal odd-cycle-intersecting families of $S_n$ for all even $n$. While the latter result has less combinatorial significance, its proof uses an interesting new character-theoretic identity that might be of independent interest in algebraic combinatorics.
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https://arxiv.org/abs/2601.12613
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7ca6511145a62b7b3f48ed2fd47e2afdda4494f95655da7dec9483f44713da42
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2026-01-21T00:00:00-05:00
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Density of growth rates of subgroups of a free group -- an alternative proof
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arXiv:2601.12620v1 Announce Type: new Abstract: We give an alternative proof to the theorem recently proved by Louvaris, Wise and Yehuda, that the growth rates of finitely generated subgroups of $F_r$ are dense in $[1,2r-1]$.
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https://arxiv.org/abs/2601.12620
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e6d7f9319232d0f6e564b047be9b2bb93b97d9d7dd853464834f946f5e4fde2f
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2026-01-21T00:00:00-05:00
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New Trends in the Stability of Sinkhorn Semigroups
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arXiv:2601.12633v1 Announce Type: new Abstract: Entropic optimal transport problems play an increasingly important role in machine learning and generative modelling. In contrast with optimal transport maps which often have limited applicability in high dimensions, Schrodinger bridges can be solved using the celebrated Sinkhorn's algorithm, a.k.a. the iterative proportional fitting procedure. The stability properties of Sinkhorn bridges when the number of iterations tends to infinity is a very active research area in applied probability and machine learning. Traditional proofs of convergence are mainly based on nonlinear versions of Perron-Frobenius theory and related Hilbert projective metric techniques, gradient descent, Bregman divergence techniques and Hamilton-Jacobi-Bellman equations, including propagation of convexity profiles based on coupling diffusions by reflection methods. The objective of this review article is to present, in a self-contained manner, recently developed Sinkhorn/Gibbs-type semigroup analysis based upon contraction coefficients and Lyapunov-type operator-theoretic techniques. These powerful, off-the-shelf semigroup methods are based upon transportation cost inequalities (e.g. log-Sobolev, Talagrand quadratic inequality, curvature estimates), $\phi$-divergences, Kantorovich-type criteria and Dobrushin contraction-type coefficients on weighted Banach spaces as well as Wasserstein distances. This novel semigroup analysis allows one to unify and simplify many arguments in the stability of Sinkhorn algorithm. It also yields new contraction estimates w.r.t. generalized $\phi$-entropies, as well as weighted total variation norms, Kantorovich criteria and Wasserstein distances.
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https://arxiv.org/abs/2601.12633
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0327428abd61fc43ef128495f6f32220db2a73a81b636cd4b629af16a998f6ee
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2026-01-21T00:00:00-05:00
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Torsion points of small order on cyclic covers of $\mathbb{P}^1$. III
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arXiv:2601.12643v1 Announce Type: new Abstract: Let $d>1$ be an integer and $K_0$ a perfect field such that $char(K_0)$ does not divide $d$. Let $n>d$ be an integer that is prime to $d$. Let $f(x)\in K_0[x]$ be a degree $n$ monic polynomial without repeated roots, and $\mathcal{C}_{f,d}$ a smooth projective model of the affine curve $y^d=f(x)$. Let $J(\mathcal{C}_{f,d})$ be the Jacobian of the $K_0$-curve $\mathcal{C}_{f,d} $. As usual, we identify $\mathcal{C}_{f,d}$ with its canonical image in $J(\mathcal{C}_{f,d})$ (such that the only ``infinite point'' of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$). We say that an integer $m>1$ is $(n,d)$-reachable over $K_0$ if there exists a polynomial $f(x)$ as above such that $\mathcal{C}_{f,d}(K_0)$ contains a torsion point of order $m$. Let us put $\ell_0:=[(n+d)/d], \ m_0:=\ell_0 d$. Earlier we proved that if $m$ is $(n,d)$-reachable, then either $m=d$ or $m = n$ or $m \ge m_0$ (in addition, both $d$ and $n$ are $(n,d)$-reachable over every $K_0$). We also proved that if $m_0$ is $(n,d)$-reachable over some $K_0$ then $n-m_0+\ell_0\ge 0$. In the present paper we discuss the $(n,d)$-reachability of $m_0$ when $n-m_0+\ell_0=0$ or $1$.
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https://arxiv.org/abs/2601.12643
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d4ef9a4a06a64fa64db75d7dfe32540d8f4213d0ad2dd7eff65ec6bb2680e596
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2026-01-21T00:00:00-05:00
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On Some Properties of Matrices with Entries Defined by Products of $k$-Fibonacci and $k$-Lucas Numbers
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arXiv:2601.12644v1 Announce Type: new Abstract: In this paper, we employ combinatorial and algebraic tools to derive closed-form expressions for several classical matrix invariants, including the determinant, inverse, trace, and powers, for a family of matrices whose entries are given by products of $k$-Fibonacci and $k$-Lucas numbers. Moreover, we compute the spectral radius and the energy of the graphs associated with this family of matrices. Finally, we investigate connections between the obtained formulas and certain integer sequences listed in the On-Line Encyclopedia of Integer Sequences (OEIS).
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https://arxiv.org/abs/2601.12644
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dcf510ae475c748e91a612fca9944f0a151fb7241e9a25e23fb4380fd22829eb
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2026-01-21T00:00:00-05:00
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A Landau-de Gennes Type Theory for Cholesteric-Helical Smectic-Smectic C* Liquid Crystal Phase Transitions
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arXiv:2601.12653v1 Announce Type: new Abstract: We present a rigorous mathematical analysis of a modified Landau-de Gennes (LdG) theory modeling temperature-driven phase transitions between cholesteric, helical smectic, and smectic C* phases. This model couples a tensor-valued order parameter (nematic orientational order) with a real-valued order parameter (smectic layer modulation). We establish the existence of energy minimizers of the modified LdG energy in three dimensions, subject to Dirichlet conditions, and rigorously analyze the energy minimizers in two asymptotic limits. First, in the Oseen--Frank limit, we show that the global minimizer strongly converges to a minimizer of the Landau-de Gennes bulk energy. Second, in the limit of dominant elastic constants, we prove that the global minimizers converge to a classical helical director profile. Finally, through stability analysis and bifurcation theory, we derive the complete sequence of symmetry-breaking transitions with decreasing temperature-from the cholesteric phase (with in-plane twist and no layering) to an intermediate helical smectic phase (with in-plane twist and layering), and ultimately to the smectic C* phase (with out-of-plane twist and layering). These theoretical results are supported by numerical simulations.
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https://arxiv.org/abs/2601.12653
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3446c24303a02757ee7ab6edcc56090e2f977727abb766efd4caca4990abe9b8
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2026-01-21T00:00:00-05:00
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Stable and Fr\'echet limit theorem for subgraph functionals in the hyperbolic random geometric graph
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arXiv:2601.12677v1 Announce Type: new Abstract: We study the fluctuations of subgraph counts in hyperbolic random geometric graphs on the $d$-dimensional Poincar\'e ball in the heterogeneous, heavy-tailed degree regime. In a hyperbolic random geometric graph whose vertices are given by a Poisson point process on a growing hyperbolic ball, we consider two basic families of subgraphs: star shape counts and clique counts, and we analyze their global counts and maxima over the vertex set. Working in the parameter regime where a small number of vertices close to the center of the Poincar\'e ball carry very large degrees and act as hubs, we establish joint functional limit theorems for suitably normalized star shape and clique count processes together with the associated maxima processes. The limits are given by a two-dimensional dependent process whose components are a stable L\'evy process and an extremal Fr\'echet process, reflecting the fact that a small number of hubs dominates both the total number of local subgraphs and their extremes. As an application, we derive fluctuation results for the global clustering coefficient, showing that its asymptotic behavior is described by the ratio of the components of a bivariate L\'evy process with perfectly dependent stable jumps.
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https://arxiv.org/abs/2601.12677
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700db4310fa98cd65b8a3655f65d1b12a5cd29027fe465278772157918dc9a9e
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2026-01-21T00:00:00-05:00
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Non-parabolic Spatial Hybrid Framed Curves and Their Applications in the Spatial Hybrid Number Space
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arXiv:2601.12679v1 Announce Type: new Abstract: In this paper, we define non-parabolic spatial hybrid framed curves in the spatial hybrid number space, which may have singularities, and prove the existence and uniqueness theorem for non-parabolic spatial hybrid framed curves. As appliciations, we define evolutes, involutes, pedal and contrapedal curves of non-parabolic spatial hybrid framed curves and discuss their relations.
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https://arxiv.org/abs/2601.12679
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124b6149d577265d9aea2b5c872a892f3298ae4dd458fb1d12a50f5f02ff8f17
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2026-01-21T00:00:00-05:00
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Igusa-Todorov properties of recollements of abelian categories
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arXiv:2601.12702v1 Announce Type: new Abstract: In this paper, we investigate the behavior of Igusa-Todorov properties under recollements of abelian categories. In particular, we study how the Igusa-Todorov distances of the categories involved in a recollement are related. Applications are given to Artin algebras, especially to Morita context rings.
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https://arxiv.org/abs/2601.12702
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504d7c68a8c5b28ddf7d320e1e65ca3c061121e1ec34408c3e959848bb12676b
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2026-01-21T00:00:00-05:00
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On a class of logarithmic Schr\"odinger equations via perturbation method
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arXiv:2601.12732v1 Announce Type: new Abstract: In this paper, we consider the following logarithmic Schr\"odinger equation \[ -\Delta u + V(x)u = u \log u^{2} \quad \text{in }\ \mathbb{R}^{N}. \] Assuming that \(V(x)\in C(\mathbb{R}^{N})\) and \(V(x)\to+\infty\) as \(|x|\to\infty\), we develop a new perturbative variational approach to overcome the lack of \(C^{1}\)-smoothness of the associated functional and prove the existence and multiplicity of nontrivial weak solutions.
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https://arxiv.org/abs/2601.12732
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525fef17ec11c5a7b2fec6264746eeb152b9e9e22bf1efbc5b13050c974fbcff
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2026-01-21T00:00:00-05:00
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A Sharp Global Boundedness Result for Keller--Segel--(Navier--)Stokes Systems with Rapid Diffusion and Saturated Sensitivities
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arXiv:2601.12733v1 Announce Type: new Abstract: We investigate the Keller--Segel--(Navier--)Stokes system posed in a smooth bounded domain \(\Omega \subset \mathbb{R}^N\) with \(N = 2,3\): \begin{equation*} \begin{cases} n_t + u \cdot \nabla n = \Delta n - \nabla \cdot \big( n S(n)\nabla c \big), \\[2mm] u \cdot \nabla c = \Delta c - c + n, \\[2mm] u_t + \kappa (u \cdot \nabla) u = \Delta u - \nabla P + n \nabla \phi, \\[2mm] \nabla \cdot u = 0, \end{cases} \end{equation*} where \(\kappa \in \left \{0,1 \right \} \), the given gravitational potential \(\phi \in W^{2, \infty}(\Omega)\), and the chemotactic sensitivity function \(S \in C^2([0,\infty))\). Under no-flux boundary conditions for \(n\) and \(c\), together with the Dirichlet boundary condition for \(u\), we show that, provided the initial data satisfy suitable regularity assumptions, the following results hold: \begin{itemize} \item If \(N = 2\), \(\kappa = 1\), and the sensitivity function satisfies \(\lim_{\xi \to \infty} S(\xi) = 0\), then the Keller--Segel--Navier--Stokes system admits a global classical solution that remains uniformly bounded in time. \item If \(N = 3\), \(\kappa = 0\), and \(S\) satisfies \[ |S(\xi)| \le K_S (\xi + 1)^{-\alpha} \quad \text{for all } \xi \ge 0, \] with some constants \(K_S > 0\) and \(\alpha > \frac{1}{3}\), then the Keller--Segel--Stokes system possesses a global bounded classical solution. \end{itemize} Our results are optimal, since it is well established that, in the absence of fluid effects, blow-up can occur when $S \equiv \mathrm{const}$ in two dimensions, or when $\alpha < \tfrac{1}{3}$ in three dimensions.
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https://arxiv.org/abs/2601.12733
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af68ae7c728c1211f41424c2d0526d5e114bfc64a0b567bfd4ff7ef09c5ae355
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2026-01-21T00:00:00-05:00
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Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions
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arXiv:2601.12737v1 Announce Type: new Abstract: Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question is whether it contains arbitrarily long arithmetic progressions. In this paper we study the fractal structure of irrational numbers whose sequences of partial quotients are strictly increasing and contain arbitrarily long, quantified arithmetic progressions.
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https://arxiv.org/abs/2601.12737
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5f2cbb26293dfbb94af38d2edf84ab98d3118dbdb95e9d124ae163ce2d392b2e
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2026-01-21T00:00:00-05:00
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Monotonicity of Pairs of Operators and Generalized Inertial Proximal Method
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arXiv:2601.12738v1 Announce Type: new Abstract: Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to obtain the monotone pairs. In this paper, we solve this problem and propose a Generalized Inertial Proximal Point Algorithm (GIPPA) using warped resolvents under the monotonicity of pairs. The weak, strong and linear convergence of the algorithm under some mild assumptions are established. We also provide numerical examples illustrating the implementability and effectiveness of the proposed method.
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https://arxiv.org/abs/2601.12738
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d6514fe8813a7081b1efc5a3d18e048118d756daddc28ebbd66cc18c98127d10
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2026-01-21T00:00:00-05:00
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When all directed cycles have the same weight
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arXiv:2601.12746v1 Announce Type: new Abstract: A digraph $G$ is weightable if its edges can be weighted with real numbers such that the total weight in each directed cycle equals 1. There are several equivalent conditions: that $G$ admits a 0/1-weighting with the same property, or that $G$ contains no subdivided "double-cycle" as a subdigraph, or that for every triple of vertices, all directed cycles containing all three pass through them in the same cyclic order. And there is quite a rich supply of such digraphs: for instance, any digraph drawn in the plane such that each of its directed cycles rotates clockwise around the origin is weightable (let us call such digraphs "circular"), and there are weightable planar digraphs with much more complicated structure than this. Until now the general structure of weightable digraphs was not known, and that is our objective in this paper. We will show that: - there is a construction that builds every planar weightable digraph from circular digraphs; and - there is a (different) construction that builds every weightable digraph from planar ones. We derive a poly-time algorithm to test if a digraph is weightable.
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https://arxiv.org/abs/2601.12746
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Academic Papers
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03739e989e380b812d1f79957531cdf1759f915b8517dfabaf4ed3213baf8042
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2026-01-21T00:00:00-05:00
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Non-Wieferich property of prime ideals and a conjecture of Erd\"os
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arXiv:2601.12753v1 Announce Type: new Abstract: Let $K$ be a number field with ring of integers $\mathcal{O}$ and $\alpha\in\mathcal{O}$. For any prime ideal $\mathfrak{p}$ of $\mathcal{O}$, we obtain its higher $\alpha$-Wieferich property, which implies a nonexistence theorem for higher Wieferich unramified prime ideals. If $\beta\in\mathcal{O}$ is relatively prime to $\alpha$ and all prime ideal factors of $(\beta)$ are unramified and have residue degree $1$, we apply our higher $\alpha$-Wieferich property to establish the asymptotic equidistribution of digits in $\beta$-adic expansions of $\alpha^n$, which is a generalization of the Dupuy-Weirich theorem. When $(\beta)$ have ramified prime ideal factors, we also obtain a result on the block complexity of $\beta$-adic expansions of $\alpha^n$.
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https://arxiv.org/abs/2601.12753
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Academic Papers
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