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44680a8f8be39caa2bb704009ad3b2a73fb4efc4564127d9722b6446ce3e3620
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2026-01-16T00:00:00-05:00
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On the solutions of the generalized Fermat equation over totally real number fields
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arXiv:2404.09171v3 Announce Type: replace Abstract: Let $K$ be a totally real number field and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ over $K$ with prime exponent $p$, where $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even. For certain class of fields $K$, we prove that the equation $Ax^p+By^p+Cz^p=0$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2|abc$. Then, under some assumptions on $A,B,C$, we also prove that $Ax^p+By^p+Cz^p=0$ has no asymptotic solution in $K^3$. Finally, we give several purely local criteria of $K$ such that $Ax^p+By^p+Cz^p=0$ has no asymptotic solutions in $K^3$, and calculate the density of such fields $K$ when $K$ is a real quadratic field.
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https://arxiv.org/abs/2404.09171
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9be4f654410b0d75c17a871a2e7ec031d243be15e4baa475676f301a8b764e0b
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2026-01-16T00:00:00-05:00
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On the asymptotics of Kempner-Irwin sums
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arXiv:2404.13763v4 Announce Type: replace Abstract: Let $I(b,d,k)$ be the subseries of the harmonic series keeping the integers having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic expansion to all orders in descending powers of $b$, for fixed $d$ and $k$, of $I(b,d,k)-b\log(b)$. We explicitly give, depending on cases, either four or five terms. The coefficients involve the values of the zeta function at the integers.
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https://arxiv.org/abs/2404.13763
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06a7fbd9b70ef1adc014a9c5e93ae8bc491aff2be6c03082b74dac888e3ccf2c
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2026-01-16T00:00:00-05:00
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Preservation under Reduced Products in Continuous Logic
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arXiv:2405.12720v2 Announce Type: replace Abstract: We introduce a fragment of continuous first-order logic, analogue of Palyutin formulas (or h-formulas) in classical model theory, which is preserved under reduced products in both directions. We use it to extend classical results on complete theories which are preserved under reduced product and their stability. We also characterize the set of Palyutin sentences, Palyutin theories and other related fragments in terms of their preservation properties, both in the classical setting and the metric one.
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https://arxiv.org/abs/2405.12720
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8eb409b19247923978a436a3c7b2c12c4f39b0d4baef0d27b5837991b970b434
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2026-01-16T00:00:00-05:00
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Analytic Extended Dynamic Mode Decomposition
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arXiv:2405.15945v4 Announce Type: replace Abstract: We develop a novel EDMD-type algorithm that captures the spectrum of the Koopman operator defined on a reproducing kernel Hilbert space of analytic functions. This method, which we call analytic EDMD, relies on an orthogonal projection on polynomial subspaces, which is equivalent to a data-driven Taylor approximation. In the case of dynamics with a hyperbolic equilibrium, analytic EDMD demonstrates excellent performance to capture the lattice-structured Koopman spectrum based on the eigenvalues of the linearized system at the equilibrium. Moreover, it yields the Taylor approximation of associated principal eigenfunctions. Since the method preserves the triangular structure of the operator, it does not suffer from spectral pollution and, moreover, arbitrary accuracy on the spectrum can be reached with a fixed finite dimension of the approximation and with a (possibly non-uniform) sampling over an arbitrary set of nonzero measure. The performance of analytic EDMD is illustrated with numerical examples and is assessed through a comparative study with related methods. Finally, the method is complemented with theoretical results, proving strong convergence of the eigenfunctions and providing error bounds on the spectrum estimation.
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https://arxiv.org/abs/2405.15945
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ac45bc78e88b50faf49dce5c3fb6386548e757b79e8e86208b43481d05cbceea
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2026-01-16T00:00:00-05:00
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Moduli of rank two semistable sheaves on rational Fano threefolds of the main series
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arXiv:2405.20460v2 Announce Type: replace Abstract: In this paper we investigate the moduli spaces of semistable coherent sheaves of rank two on the projective space $\mathbb{P}^3$ and the following rational Fano manifolds of the main series - the three-dimensional quadric $X_2$, the intersection of two 4-dimensional quadrics $X_4$ and the Fano manifold $X_5$ of degree 5. For the quadric $X_2$, the boundedness of the third Chern class $c_3$ of rank two semistable objects in $\mathrm{D}^b(X_2)$, including sheaves, is proved. An explicit description is given of all the moduli spaces of semistable sheaves of rank two on $X_2$, including reflexive ones, with a maximal third class $c_3\ge0$. These spaces turn out to be irreducible smooth rational manifolds in all cases, except for the following two: $(c_1,c_2,c_3)=(0,2,2)$ or (0,4,8). Several new infinite series of rational components of the moduli spaces of semistable sheaves of rank two on $\mathbb{P}^3$, $X_2$, $X_4$ and $X_5$ are constructed, as well as a new infinite series of irrational components on $X_4$. The boundedness of the class $c_3$ is proved for $c_1=0$ and any $c_2>0$ for stable reflexive sheaves of general type on manifolds $X_4$ and $X_5$.
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https://arxiv.org/abs/2405.20460
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0da7c05344b929b512f46683e824e9901424f7e7867346deef30298508c6fc44
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2026-01-16T00:00:00-05:00
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Cross-Dimensional Mathematics: A Foundation For STP/STA
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arXiv:2406.12920v5 Announce Type: replace Abstract: A new mathematical structure, called the cross-dimensional mathematics (CDM), is proposed. The CDM considered in this paper consists of three parts: hyper algebra, hyper geometry, and hyper Lie group/Lie algebra. Hyper algebra proposes some new algebraic structures such as hyper group, hyper ring, and hyper module over matrices and vectors with mixed dimensions (MVMDs). They have sets of classical groups, rings, and modules as their components and cross-dimensional connections among their components. Their basic properties are investigated. Hyper geometry starts from mixed dimensional Euclidian space, and hyper vector space. Then the hyper topological vector space, hyper inner product space, and hyper manifold are constructed. They have a joined cross-dimensional geometric structure. Finally, hyper metric space, topological hyper group and hyper Lie algebra are built gradually, and finally, the corresponding hyper Lie group is introduced. All these concepts are built over MVMDs, and to reach our purpose in addition to existing semi-tensor products (STPs) and semi-tensor additions (STAs), a couple of most general STP and STA are introduced. Some existing structures/results about STPs/STAs have also been resumed and integrated into this CDM.
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https://arxiv.org/abs/2406.12920
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52e47abc26420eed5880d842dc8017df8943d4c09af8d80e0e68769befc63baf
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2026-01-16T00:00:00-05:00
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Polynomial convergence rate at infinity for the cusp winding spectrum of generalized Schottky groups
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arXiv:2407.12398v2 Announce Type: replace Abstract: We show that the convergence rate of the cusp winding spectrum to the Hausdorff dimension of the limit set of a generalized Schottky group with one parabolic generator is polynomial. Our main theorem provides the new phenomenon in which differences in the Hausdorff dimension of the limit set generated by a Markov system cause essentially different results on multifractal analysis. This paper also provides a new characterization of the geodesic flow on the Poinca\'re disc model of two-dimensional hyperbolic space and the limit set of a generalized Schottky group. To prove our main theorem we use thermodynamic formalism on a countable Markov shift, gamma function, and zeta function.
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https://arxiv.org/abs/2407.12398
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8ee13feeea4f51c61763e4263a244beb53c4a1a5718f5ad6a2614d5d7630118c
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2026-01-16T00:00:00-05:00
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Nearly-linear solution to the word problem for 3-manifold groups
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arXiv:2407.18029v2 Announce Type: replace Abstract: We show that the word problem for any 3-manifold group is solvable in time $O(n\log^3 n)$. Our main contribution is the proof that the word problem for admissible graphs of groups, in the sense of Croke and Kleiner, is solvable in $O(n\log n)$; this covers fundamental groups of non-geometric graph manifolds. Similar methods also give that the word problem for free products can be solved ``almost as quickly'' as the word problem in the factors.
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https://arxiv.org/abs/2407.18029
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d97ea10092611a746c16b0fddd49959cebd799c2c7d963481a6df6ef242669c0
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2026-01-16T00:00:00-05:00
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Littlewood-Offord problems for Ising models
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arXiv:2408.05720v2 Announce Type: replace Abstract: We consider the one-dimensional Littlewood-Offord problem for general Ising models. More precisely, we consider the concentration function \[Q_n(x,v)=P\left(\sum_{i=1}^{n}\varepsilon_iv_i\in(x-1,x+1)\right),\] where $x\in\mathbb{R}$, $v_1,v_2,\ldots,v_n$ are real numbers such that $|v_1|\geq 1, |v_2|\geq 1,\ldots, |v_n|\geq 1$, and $(\varepsilon_i)_{i=1,2,\ldots,n}\in\{-1,1\}^{n}$ are random spins of some Ising model. Let $Q_n=\sup_{x,v}Q_n(x,v)$. Under natural assumptions, we show that there exists a universal constant $C$ such that for all $n\geq 1$, \[\binom{n}{[n/2]}2^{-n}\leq Q_n\leq Cn^{-\frac{1}{2}}.\] As an application of the method, under the same assumption, we give a lower bound on the smallest eigenvalue of the truncated correlation matrix of the Ising model.
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https://arxiv.org/abs/2408.05720
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66602cdacf99da7c5f5d8b5a59bd2e8b3f1441b81e22b07c2adff0bc3402fda4
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2026-01-16T00:00:00-05:00
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Generalized Fruit Diophantine equation over number fields
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arXiv:2408.12278v3 Announce Type: replace Abstract: Let $K$ be a number field and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the solutions of the generalized fruit Diophantine equation $ax^d-y^2-z^2 +xyz-c=0$ over $K$, where $d \geq 3$ is an integer and $a,c\in \mathcal{O}_K\setminus \{0\}$. Subsequently, we provide explicit values of square-free integers $t$ such that the equation $ax^d-y^2-z^2 +xyz-c=0$ has no solution $(x_0, y_0, z_0) \in \mathcal{O}_{\mathbb{Q}(\sqrt{t})}^3$ with $2 | x_0$, and demonstrate that the set of all such square-free integers $t$ with $t \geq 2$ has density exactly $\frac{1}{6}$. As an application, we construct infinitely many elliptic curves $E$ defined over number fields $K$ having no integral point $(x_0,y_0) \in \mathcal{O}_K^2$ with $2|x_0$.
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https://arxiv.org/abs/2408.12278
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e78a7837a69ff71bb005d5140f305b32402598474f2b1a9c7d6b51db09669c55
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2026-01-16T00:00:00-05:00
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Graphs missing a connected partition
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arXiv:2409.12934v2 Announce Type: replace Abstract: We prove that a graph with a cut vertex whose deletion produces at least five connected components must be missing a connected partition of some type. We prove that this also holds if there are four connected components that each have at least two vertices. In particular, the chromatic symmetric function of such a graph cannot be $e$-positive. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-$e$-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an $e$-positive chromatic symmetric function.
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https://arxiv.org/abs/2409.12934
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c5372c7af6056ae6f025d74ff5a5e96922cb20af29b41d083c3bed91417f77b1
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2026-01-16T00:00:00-05:00
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K\"ahler metrics of negative holomorphic (bi)sectional curvature on a compact relative K\"ahler fibration
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arXiv:2409.14650v2 Announce Type: replace Abstract: For a compact relative K\"ahler fibration over a compact K\"ahler manifold with negative holomorphic sectional curvature, if the relative K\"ahler form on each fiber also exhibits negative holomorphic sectional curvature, we can construct K\"ahler metrics with negative holomorphic sectional curvature on the total space. Additionally, if this form induces a Griffiths negative Hermitian metric on the relative tangent bundle, and the base admits a K\"ahler metric with negative holomorphic bisectional curvature, we can also construct K\"ahler metrics with negative holomorphic bisectional curvature on the total space. As an application, for a non-trivial fibration where both the fibers and base have K\"ahler metrics with negative holomorphic bisectional curvature, and the fibers are one-dimensional, we can explicitly construct K\"ahler metrics of negative holomorphic bisectional curvature on the total space, thus resolving a question posed by To and Yeung for the case where the fibers have dimension one.
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https://arxiv.org/abs/2409.14650
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d158f208f20e99376a98e33ed22be0fd0b875a3d6808dee15b0fa928447f2484
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2026-01-16T00:00:00-05:00
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The five-color hypercube Adinkra and the Jacobian of a generalized Fermat curve
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arXiv:2410.11137v2 Announce Type: replace Abstract: Adinkras are highly structured graphs developed to study 1-dimensional supersymmetry algebras. A cyclic ordering of the edge colors of an Adinkra, or rainbow, determines a Riemann surface and a height function on the vertices of the Adinkra determines a divisor on this surface. We study the induced map from height functions to divisors on the Jacobian of the Riemann surface. In the first nontrivial case, a 5-dimensional hypercube corresponding to a Jacobian given by a product of 5 elliptic curves each with $j$-invariant 2048, we develop and characterize a purely combinatorial algorithm to compute height function images. We show that when restricted to a single elliptic curve, every height function is a multiple of a specified generating divisor, and raising and lowering vertices corresponds to adding or subtracting this generator. We also give strict bounds on the coefficients of this generator that appear in the collection of all divisors of height functions.
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https://arxiv.org/abs/2410.11137
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789b4fc6279e8e0f6a5e7144e93ba188244ca05265c0c2b414e4e14bb95efa3f
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2026-01-16T00:00:00-05:00
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A $\Lambda$-adic Kudla lift
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arXiv:2410.19992v2 Announce Type: replace Abstract: The Kudla lift studied in this article is a classical version for Picard modular forms of the automorphic theta lift between $\text{GU}(2)$ and $\text{GU}(3)$. We construct an explicit $p$-adic analytic family of Picard modular forms varying with respect to the weight and level, which interpolates a so-called $p$-modification of the lift at arithmetic weights, by exploiting a formula of Finis for the Fourier-Jacobi coefficients of a lifted form.
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https://arxiv.org/abs/2410.19992
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f102ef62e4876114eeb61fb12c0877001ecd42a2e304b72687d880248c3eb4a5
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2026-01-16T00:00:00-05:00
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Complexity and curvature of (pairs of) Cohen-Macaulay modules, and their applications
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arXiv:2411.17622v2 Announce Type: replace Abstract: The complexity and curvature of a module, introduced by Avramov, measure the growth of Betti and Bass numbers of a module, and distinguish the modules of infinite homological dimension. The notion of complexity was extended by Avramov-Buchweitz to pairs of modules that measure the growth of Ext modules. The related notion of Tor complexity was first studied by Dao. Inspired by these notions, we define Ext and Tor curvature of pairs of modules. The aim of this article is to study (Ext and Tor) complexity and curvature of pairs of certain CM (Cohen-Macaulay) modules, and establish lower bounds of complexity and curvature of pairs of modules in terms of that of a single module. It is known that among all modules, the residue field has maximal complexity and curvature, moreover they characterize complete intersection local rings. As applications of our results, we provide some upper bounds of the curvature of the residue field in terms of curvature and multiplicity of any nonzero CM module. As a final upshot, these allow us to characterize complete intersection local rings (including hypersurfaces and regular rings) in terms of complexity and curvature of pairs of certain CM modules. In particular, under some additional hypotheses, we characterize complete intersection and regular local rings via injective curvature of the ring and that of the module of K\"{a}hler differentials respectively. Thus, we make partial progress towards a question of Christensen-Striuli-Veliche, as well as another by Vasconcelos.
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https://arxiv.org/abs/2411.17622
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b9878900db0b67ae4f14947711b064d877c47ca6b8300dd7990ee028df184a76
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2026-01-16T00:00:00-05:00
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A note on the $L_p$-Brunn-Minkowski inequality for intrinsic volumes and the $L_p$-Christoffel-Minkowski problem
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arXiv:2411.17896v4 Announce Type: replace Abstract: The first goal of this paper is to improve some of the results in \cite{BCPR}. Namely, we establish the $L_p$-Brunn-Minkwoski inequality for intrinsic volumes for origin-symmetric convex bodies that are close to the ball in the $C^2$ sense for a certain range of $p<1$ (including negative values) and we prove that this inequality does not hold true in the entire class of origin-symmetric convex bodies for any $p<1$. The second goal is to establish a uniqueness result for the (closely related) $L_p$-Christoffel-Minkowski problem. More specifically, we show uniqueness in the symmetric case when $p\in[0,1)$ and the data function $g$ in the right hand side is sufficiently close to the constant 1. One of the main ingredients of the proof is the existence of upper and lower bounds for the (convex) solution, that depend only $\|\log g\|_{L^\infty}$, a fact that might be of independent interest.
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https://arxiv.org/abs/2411.17896
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cc781a61a75a6d8557e606916897fe4a0c725893b52f74e55690a3bfe25ee964
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2026-01-16T00:00:00-05:00
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Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor
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arXiv:2412.01636v2 Announce Type: replace Abstract: In this article, we show test properties, in the sense of finitely many vanishing of Ext or Tor, of CM (Cohen-Macaulay) modules whose multiplicity and number of generators (resp., type) are related by certain inequalities. We apply these test behaviour, along with other results, to characterize various kinds of local rings, including hypersurface rings of multiplicity at most two, surprisingly requiring only finitely many vanishing of Ext or Tor involving such CM modules. As further applications, we verify the long-standing (Generalized) Auslander-Reiten Conjecture for every CM module of minimal multiplicity over a Noetherian local ring, thus vastly extending a result of Huneke-\c{S}ega-Vraciu.
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https://arxiv.org/abs/2412.01636
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7216eebcc7b30802a8feb2dce0a65bd6536bd3806e57f6a57e00ee783f354077
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2026-01-16T00:00:00-05:00
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The John inclusion for log-concave functions
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arXiv:2412.18444v2 Announce Type: replace Abstract: John's inclusion states that a convex body in $\mathbb{R}^d$ can be covered by the $d$-dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions. As a byproduct of our approach, we establish the following asymptotically tight inequality: \\ \noindent For any log-concave function $f$ with finite, positive integral, there exist a positive definite matrix $A$, a point $a \in \mathbb{R}^d$, and a positive constant $\alpha$ such that \[ \chi_{\mathbf{B}^{d}}(x) \leq \alpha f\!\!\left(A(x-a)\right) \leq \sqrt{d+1} \cdot e^{-\frac{\left|x\right|}{d+2} + (d+1)}, \] where $\chi_{\mathbf{B}^{d}}$ is the indicator function of the unit ball $\mathbf{B}^{d}$.
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https://arxiv.org/abs/2412.18444
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4c156d543aed80ab660277c38af0f8242e751de0c207ab90c6a51ce34f5438f8
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2026-01-16T00:00:00-05:00
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Distributionally Robust Fault Detection Trade-off Design with Prior Fault Information
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arXiv:2412.20237v5 Announce Type: replace Abstract: The robustness of fault detection algorithms against uncertainty is crucial in the real-world industrial environment. Recently, a new probabilistic design scheme called distributionally robust fault detection (DRFD) has emerged and received immense interest. Despite its robustness against unknown distributions in practice, current DRFD focuses on the overall detectability of all possible faults rather than the detectability of critical faults that are a priori known. Henceforth, a new DRFD trade-off design scheme is put forward in this work by utilizing prior fault information. The key contribution includes a novel distributional robustness metric of detecting a known fault and a new relaxed distributionally robust chance constraint that ensures robust detectability. Then, a new DRFD design problem of fault detection under unknown probability distributions is proposed, and this offers a flexible balance between the robustness of detecting known critical faults and the overall detectability against all possible faults. To address the resulting semi-infinite chance-constrained problem, we first reformulate it to a finite-dimensional problem characterized by bilinear matrix inequalities. Subsequently, a tailored heuristic solution algorithm is developed, which includes a sequential minimization procedure and an initialization strategy. Finally, case studies on a simulated three-tank system and a real-world battery cell are carried out to showcase the effectiveness of the proposed heuristic algorithm and the advantages of our DRFD method.
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https://arxiv.org/abs/2412.20237
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eca2bef0719ad5d52bd8676b3b50c8a8c7beafc26929266f4285fd5b99808f56
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2026-01-16T00:00:00-05:00
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Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I
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arXiv:2501.04026v3 Announce Type: replace Abstract: In this first article of a multi-part series, we inspect a surprising relationship between the set of fixed points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}$ and the coefficient $c$, where $d > 2$ is an integer. Inspired greatly by the elegant counting problems along with the very striking results of Bhargava-Shankar-Tsimerman and their collaborators in arithmetic statistics, and also by interesting point-counting result of Narkiewicz on rational periodic points of any odd degree map $\varphi_{d, c}$ in arithmetic dynamics, we then first prove that for any prime $p\geq 3$, the average number of distinct integral fixed points of any $\varphi_{p, c}$ modulo $p$ is $3$ or $0$ as $c$ tends to infinity. Inspired further by a conjecture of Hutz on rational periodic points of $\varphi_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that the average number of distinct integral fixed points of any $\varphi_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density and number field-counting results from arithmetic statistics, and as a result obtain counting and statistical results on the irreducible integer polynomials and number fields arising naturally in our polynomial discrete dynamical settings.
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https://arxiv.org/abs/2501.04026
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fb7080ad8ddf7e2387dd4725a4eefa27b64ecf0882aecd54d07205367d5f7105
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2026-01-16T00:00:00-05:00
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Hypercyclicity of Toeplitz operators with smooth symbols
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arXiv:2502.03303v3 Announce Type: replace Abstract: This paper is devoted to the study of the dynamics of Toeplitz operators $T_F$ with smooth symbols $F$ on the Hardy spaces of the unit disk $H^p$, $p>1$. Building on a model theory for Toeplitz operators on $H^2$ developed by Yakubovich in the 90's, we carry out an in-depth study of hypercyclicity properties of such operators. Under some rather general smoothness assumptions on the symbol, we provide some necessary/sufficient/necessary and sufficient conditions for $T_F$ to be hypercyclic on $H^p$. In particular, we extend previous results on the subject by Baranov-Lishanskii and Abakumov-Baranov-Charpentier-Lishanskii. We also study some other dynamical properties for this class of operators.
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https://arxiv.org/abs/2502.03303
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ad98dfb06b86e8308ffbda44429ea75c954af83f734dbbfec2125a55e0c26f0e
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2026-01-16T00:00:00-05:00
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Improved regularity estimates for degenerate or singular fully nonlinear dead-core systems and H\'{e}non-type equations
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arXiv:2502.10099v3 Announce Type: replace Abstract: In this paper, we study the degenerate or singular fully nonlinear dead-core systems coupled with strong absorption terms. We establish several properties, including improved regularity of viscosity solutions along the free boundary, non-degeneracy, a measure estimate of the free boundary, Liouville-type results, and the behavior of blow-up solution. We also derive sharp regularity estimates for viscosity solutions to H\'{e}non-type equations with a degenerate weight and strong absorption, governed by a degenerate fully nonlinear operator. Our results are new even for the model equations involving degenerate Laplacian operators.
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https://arxiv.org/abs/2502.10099
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81d36cbfa02d7313d2fde652c623c14f529200cfdf4847a66583d0251e55cc45
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2026-01-16T00:00:00-05:00
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Isometries of spacetimes without observer horizons
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arXiv:2502.13904v3 Announce Type: replace Abstract: We study the isometry groups of (non-compact) Lorentzian manifolds with well-behaved causal structure, aka causal spacetimes satisfying the ``no observer horizons'' condition. Our main result is that the group of time orientation-preserving isometries acts properly on the spacetime. As corollaries, we obtain the existence of an invariant Cauchy temporal function, and a splitting of the isometry group into a compact subgroup and a subgroup roughly corresponding to time translations. The latter can only be the trivial group, $\mathbb{Z}$, or $\mathbb{R}$.
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https://arxiv.org/abs/2502.13904
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73cbfed74e9c8248e60018e54c4497a37c45ed2c62ec825ae5678a2b7cc0803e
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2026-01-16T00:00:00-05:00
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Abelian congruences and similarity in varieties with a weak difference term
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arXiv:2502.20517v2 Announce Type: replace Abstract: This is the first of three papers motivated by the author's desire to understand and explain "algebraically" one aspect of Dmitriy Zhuk's proof of the CSP Dichotomy Theorem. In this paper we study abelian congruences in varieties having a weak difference term. Each class of the congruence supports an abelian group structure; if the congruence is minimal, each class supports the structure of a vector space over a division ring determined by the congruence. A construction due to J. Hagemann, C. Herrmann and R. Freese in the congruence modular setting extends to varieties with a weak difference term, and provides a "universal domain" for the abelian groups or vector spaces that arise from the classes of the congruence within a single class of the annihilator of the congruence. The construction also supports an extension of Freese's similarity relation (between subdirectly irreducible algebras) from the congruence modular setting to varieties with a weak difference term.
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https://arxiv.org/abs/2502.20517
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a5f6d69246a8dcd578b68f7c02d94daddff2cda508324ac7083226921a854299
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2026-01-16T00:00:00-05:00
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Global solutions for supersonic flow of a Chaplygin gas past a conical wing with a shock wave detached from the leading edges
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arXiv:2503.03406v4 Announce Type: replace Abstract: In this paper, we first investigate the mathematical aspects of supersonic flow of a Chaplygin gas past a conical wing with diamond-shaped cross sections in the case of a shock wave detached from the leading edges. The flow under consideration is governed by the three-dimensional steady compressible Euler equations. For the Chaplygin gas, all characteristics are linearly degenerate, and shocks are reversible and characteristic. Using these properties, we can determine the location of the shock in advance and reformulate our problem as an oblique derivative problem for a nonlinear degenerate elliptic equation in conical coordinates. By establishing a Lipschitz estimate, we show that the equation is uniformly elliptic in any subdomain strictly away from the degenerate boundary, and then further prove the existence of a solution to the problem via the continuity method and vanishing viscosity method.
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https://arxiv.org/abs/2503.03406
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613f4382949017632d38e942deccebdf3555c8f4d2af9fdf9d03cc0f0c53f192
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2026-01-16T00:00:00-05:00
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Counting the number of $\mathcal{O}_{K}$-fixed points of a discrete dynamical system with applications from arithmetic statistics, II
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arXiv:2503.11393v3 Announce Type: replace Abstract: In this follow-up paper, we again inspect a surprising connection between the set of fixed points of a polynomial map $\varphi_{d,c}$ defined by $\varphi_{d,c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n > 1$ and $d > 2$ is an integer. As before, we wish to study counting problems which are inspired by exciting advances in arithmetic statistics, and again partly by point-counting result of Narkiewicz on real $K$-rational periodic points of any odd degree map $\varphi_{d,c}$ in arithmetic dynamics. In doing so, we then first prove that for any real algebraic number field $K$ of degree $n \geq 2$, and for any prime $p \geq 3$ and integer $\ell \geq 1$, the average number of distinct integral fixed points of any $\varphi_{p^{\ell},c}$ modulo prime ideal $p\mathcal{O}_{K}$ is $3$ or $0$ as $c\to \infty$. Motivated further by $K$-rational periodic point-counting result of Benedetto on any $\varphi_{(p-1)^{\ell},c}$ for any prime $p \geq 5$ and integer $\ell \in \mathbb{Z}_{\geq 1}$ in arithmetic dynamics, we then also prove unconditionally that for any number field (not necessarily real) $K$ of degree $n \geq 2$, the average number of distinct integral fixed points of any $\varphi_{(p-1)^{\ell},c}$ modulo prime $p\mathcal{O}_{K}$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density and number field-counting results from arithmetic statistics, and as a result obtain counting and statistical results on irreducible polynomials and number fields arising naturally in our polynomial discrete dynamical settings.
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https://arxiv.org/abs/2503.11393
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44abe2c33708fb9d358f1988bb2ee4783a86f0b8cd3cd66a1b4bd923fdc95835
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2026-01-16T00:00:00-05:00
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On a polygon version of Wiegmann-Zabrodin formula
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arXiv:2503.13718v2 Announce Type: replace Abstract: Let $P$ be a convex polygon in ${\mathbb C}$ and let $\Delta_{D, P}$ be the operator of the Dirichlet boundary value problem for the Lapalcian $\Delta=-4\partial_z\partial_{\bar z}$ in $P$. We derive a variational formula for the logarithm of the $\zeta$-regularized determinant of $\Delta_{D, P}$ for arbitrary infinitesimal deformations of the polygon $P$ in the class of polygons (with the same number of vertices). For a simply connected domain with smooth boundary such a formula was recently discovered by Wiegmann and Zabrodin as a non obvious corollary of the Alvarez variational formula, for domains with corners this approach is unavailable (at least for those deformations that do not preserve the corner angles) and we have to develop another one.
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https://arxiv.org/abs/2503.13718
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f4d571bc76d6f610dcf3f0a4d6f13b2c56658764e4c6e0c0d0d055dea72f8752
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2026-01-16T00:00:00-05:00
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Quasi-convex Splittings of Acylindrical Graphs of Locally Finite-Height Groups
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arXiv:2503.15459v2 Announce Type: replace Abstract: We find a condition on the acylindrical action of a finitely presented group on a simplicial tree which guarantees that this action will be dominated by an acylindrical action with finitely generated edge stabilisers, and find the first example of an action of a finitely presented group where there is no such dominating action. As a consequence, we show that any finitely presented group that admits a decomposition as an acylindrical graph of (possibly infinitely generated) free groups is virtually compact special, and that every finitely generated subgroup of a one-relator group with an acylindrical Magnus hierarchy is virtually compact special.
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https://arxiv.org/abs/2503.15459
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6fc2a8fdfcf332f0ed864de70255ec71c04cc4108f7a53b4599fc3fa8b9b9b92
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2026-01-16T00:00:00-05:00
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Talenti comparison results for solutions to $p$-Laplace equation on multiply connected domains
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arXiv:2504.06103v2 Announce Type: replace Abstract: In the last years comparison results of Talenti type for Elliptic Problems have been widely investigated. In this paper we obtain a comparison result for the $p$-Laplace operator in multiply connected domains with Robin boundary condition on the exterior boundary and non-homogeneous Dirichlet boundary conditions on the interior one, generalizing the results obtained in \cite{ANT, AGM} to this type of domains. This will be a generalization to Robin boundary condition of the results obtained in \cite{B, B2}, with an improvement of the $L^2$ comparison in the case $p=2$. As a consequence, we obtain a Bossel-Daners and Saint-Venant type inequalities for multiply connected domains.
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https://arxiv.org/abs/2504.06103
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908a95e49f35f3b7a5af3ac8d004e624b131e53fbe40be008b5fde23dae51a56
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2026-01-16T00:00:00-05:00
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On the uniqueness of a generalized quadrangle of order (4,16)
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arXiv:2504.09372v4 Announce Type: replace Abstract: In the manuscript [v4], we prove the uniqueness of a generalized quadrangle of order (4,16).
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https://arxiv.org/abs/2504.09372
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935f8963b7107d5e584876a025cd724950bc22cacd7e730da1ed3127ef867787
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2026-01-16T00:00:00-05:00
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Magnetic Thomas-Fermi theory for 2D abelian anyons
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arXiv:2504.13481v2 Announce Type: replace Abstract: Two-dimensional abelian anyons are, in the magnetic gauge picture, represented as fermions coupled to magnetic flux tubes. For the ground state of such a system in a trapping potential, we theoretically and numerically investigate a Hartree approximate model, obtained by restricting trial states to Slater determinants and introducing a self-consistent magnetic field, locally proportional to matter density. This leads to a fermionic variant of the Chern-Simons-Schr{\"o}dinger system. We find that for dense systems, a semi-classical approximation yields qualitatively good results. Namely, we derive a density functional theory of magnetic Thomas-Fermi type, which correctly captures the trends of our numerical results. In particular, we explore the subtle dependence of the ground state with respect to the fraction of magnetic flux units attached to particles.
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https://arxiv.org/abs/2504.13481
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ebc46ab21248cbc04f351582d6b1252452815534125fa4d5eb56f3d11d8effe9
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2026-01-16T00:00:00-05:00
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Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks
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arXiv:2504.15456v2 Announce Type: replace Abstract: A mixed equation in a group $G$ is given by a non-trivial element $w (x)$ of the free product $G \ast \mathbb{Z}$, and a solution is some $g\in G$ such that $w(g)$ is the identity. For $G$ acylindrically hyperbolic with trivial finite radical (e.g. torsion-free) we show that any mixed equation of length $n$ has a non-solution of length comparable to $\log(n)$, which is the best possible bound. Similarly, we show that there is a common non-solution of length $O(n)$ to all mixed equations of length $n$, again the best possible bound. In fact, in both cases we show that a random walk of appropriate length yields a non-solution with positive probability.
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https://arxiv.org/abs/2504.15456
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a54659e1a737167d376d7a951968502f08b3be134bf035789cd13907be6cf3b9
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2026-01-16T00:00:00-05:00
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Topological triviality and link-constancy in deformations of inner Khovanskii non-degenerate maps
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arXiv:2504.18816v5 Announce Type: replace Abstract: For real and mixed polynomial maps $f=(f^1,\dots,f^p)$ satisfying $f(0)=0$, we introduce the notion of Inner Khovanskii Non-Degeneracy (IKND), that generalizes a previous non-degeneracy condition introduced by Wall for complex polynomial functions (J. Reine Angew. Math. 509 (1999), 1-19). We prove that IKND is a sufficient condition ensuring that the link of the singularity of $f$ at the origin is smooth and well-defined. We study one-parameter deformations of an IKND map $f$, given by $F(\boldsymbol{x},\varepsilon)=f(\boldsymbol{x})+\theta(\boldsymbol{x},\varepsilon)$, with $ F(0,\varepsilon)=0$. We prove that the deformation is \textit{link-constant} under suitable conditions on $f$ and $\theta$, meaning that the ambient isotopy type of the link remains unchanged along the deformation. Furthermore, by employing a strong version of this non-degeneracy, Strong Inner Khovanskii Non-Degeneracy (SIKND), we obtain results on topological triviality. In the final section, we present link-constant deformations for IKND mixed polynomial functions of two variables. We also explore several applications motivated by the recent findings of Ara\'ujo dos Santos, Bode, and Sanchez Quiceno (Bull. Braz. Math. Soc. (N.S.) 55 (2024), no. 3, Paper No. 34).
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https://arxiv.org/abs/2504.18816
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d9d3018e0c33600817f1e9136ec3c7bcd79d01522ac07bc36c7d68062e7dfe2e
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2026-01-16T00:00:00-05:00
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Monoidal Relative Categories Model Monoidal $\infty$-Categories
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arXiv:2504.20606v3 Announce Type: replace Abstract: We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal $\infty$-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal $\infty$-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus--Sagave.
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https://arxiv.org/abs/2504.20606
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4b67f7b9e9b968e6a118e05deb177b3676f7a94e7b59a8d412a4042b01bde83d
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2026-01-16T00:00:00-05:00
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A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial
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arXiv:2505.06539v3 Announce Type: replace Abstract: Let \( P(z) \) be a polynomial of degree \( n \) and $\alpha \in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_\alpha P(z) \) and is defined by $D_\alpha P(z): = nP(z) + (\alpha -z)P'(z)$. The polar derivative \( D_\alpha P(z) \) is a polynomial of degree at most \( n - 1 \) and it generalizes the ordinary derivative \( P'(z) \). In this paper, we establish some \( L_p \) inequalities for the polar derivative of a polynomial with all its zeros located within a prescribed disk. Our results refine and generalize previously known findings.
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https://arxiv.org/abs/2505.06539
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0c2bd263ca795b14457ad55842e23bdd6ca2bf39bdd0dd1b90b475518a549609
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2026-01-16T00:00:00-05:00
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The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics
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arXiv:2505.07506v3 Announce Type: replace Abstract: We study a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~$\mathbf{Q}$-tensor for the liquid crystal component and a magnetisation vector field~$\mathbf{M}$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~$\mathbf{Q}$ and~$\mathbf{M}$. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\varepsilon$ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the~$\mathbf{Q}$-component concentrates, to leading order, on a finite number of singular points, while the energy density for the~$\mathbf{M}$-component concentrate along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~$\Q$-component.
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https://arxiv.org/abs/2505.07506
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c425a2b64c15b0d4de6aae1e3f158f520febf9e8ffd62c8a44685c20ac102337
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2026-01-16T00:00:00-05:00
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On the extremal length of the hyperbolic metric
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arXiv:2505.12400v2 Announce Type: replace Abstract: For any closed hyperbolic Riemann surface $X$, we show that the extremal length of the Liouville current is determined solely by the topology of \(X\). This confirms a conjecture of Mart\'inez-Granado and Thurston. We also obtain an upper bound, depending only on $X$, for the diameter of extremal metrics on $X$ with area one.
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https://arxiv.org/abs/2505.12400
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bd156592c7faae0b7c05a7d043364df24f747c9b2c4ac8df6d8285a591396232
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2026-01-16T00:00:00-05:00
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Invariant random subgroups on certain orbits
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arXiv:2506.09723v2 Announce Type: replace Abstract: Let $G$ be a connected Lie group and $\text{Sub}_G$ be the space of closed subgroups of $G$ equipped with the Chabauty topology. In this article, we investigate the existence of invariant random subgroups of $G$ supported on various orbits of the conjugation action of $G$ on $\text{Sub}_G$.
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https://arxiv.org/abs/2506.09723
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9bef1b093ec6978624040b3e3e9a407d57d03d01420d660387b66263ce0b1467
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2026-01-16T00:00:00-05:00
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The rational homotopy groups of virtual spheres for rank 1 compact Lie groups
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arXiv:2506.18085v4 Announce Type: replace Abstract: We calculate the rational representation-ring-graded stable stems for rank 1 groups, SU(2), SO(3), Pin (2), O(2), Spin(2) and SO(2), in the same spirit as the calculations for finite groups in arXiv:2205.02382 with J.D.Quigley. This illustrates the effectiveness of the algebraic models for these categories of G-spectra, and the way tom Dieck splitting fails for desuspensions. [v4: typos and tweaks in wording]
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https://arxiv.org/abs/2506.18085
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8be255dadd1ca1191c6ded4a789f5acd7eeab5475688a48ab7cb2b0b464b0afb
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2026-01-16T00:00:00-05:00
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Local H\"older Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights
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arXiv:2507.01899v2 Announce Type: replace Abstract: We establish local boundedness and local H\"older continuity of weak solutions to the following prototype problem: $$ -\operatorname{div}\left(|x|^{-2 \beta}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-\Delta)_{p(\cdot, \cdot), \beta}^{s(\cdot, \cdot)} u=0 \quad \text { in } \quad \Omega, $$ where $\Omega \subset \mathbb{R}^n, n \geq 2$, is a bounded domain. The nonlocal operator is defined by $$ (-\Delta)_{p(\cdot, \cdot), \beta}^{s(\cdot, \cdot)} u(x):=\mathrm{P} . \mathrm{V} . \int_{\Omega} \frac{|u(x)-u(y)|^{p(x, y)-2}(u(x)-u(y))}{|x-y|^{n+s(x, y) p(x, y)}} \frac{1}{|x|^\beta|y|^\beta} \mathrm{d} y $$ Here, $p: \Omega \times \Omega \rightarrow(1, \infty)$ and $s: \Omega \times \Omega \rightarrow(0,1)$ are measurable functions, $\mathbf{q}:=\operatorname{ess}_{\Omega \times \Omega} p$, and $0 \leq \beta<n$. Our approach is analytic and relies on an adaptation of the De Giorgi-Nash-Moser theory to a mixed local-nonlocal framework with variable exponents and weights.
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https://arxiv.org/abs/2507.01899
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07f0808adfc5b9ae59f26e85277e7c0e5a24ba9fd07fb7555e504c9bdcac31ff
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2026-01-16T00:00:00-05:00
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P\'olya's conjecture up to $\epsilon$-loss and quantitative estimates for the remainder of Weyl's law
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arXiv:2507.04307v3 Announce Type: replace Abstract: Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For any $\epsilon\in (0,1)$ we show that for any Dirichlet eigenvalue $\lambda_k(\Omega)>\Lambda(\epsilon,\Omega)$, it holds \begin{align*} k&\le (1+\epsilon)\frac{|\Omega|\omega(n)}{(2\pi)^n}\lambda_k(\Omega)^{n/2}, \end{align*} where $\Lambda(\epsilon,\Omega)$ is given explicitly. This reduces the $\epsilon$-loss version of P\'olya's conjecture to a computational problem. This estimate is based on quantitative estimates on the remainder of the Weyl law with explicit constants, which we give a new proof without using Neumann eigenvalues. Our arguments in deriving such uniform estimates yield also, in all dimensions $n\ge 2$, classes of domains that may even have rather irregular shapes or boundaries but satisfy P\'olya's conjecture. Another key observation is that on strip-tiling domains (and therefore any triangles for instance) one actually has better eigenvalue estimates than P\'olya conjectured.
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https://arxiv.org/abs/2507.04307
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0ee6442e216a8d970022117830fe0e3407191c27c28766cc988e14e23cb5afd3
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2026-01-16T00:00:00-05:00
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Cohen-Macaulay approximations and the $\text{SC}_r$-condition
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arXiv:2507.14424v3 Announce Type: replace Abstract: We study the relation between MCM approximations and FID hulls of modules over a Cohen-Macaulay local ring $R$ with canonical module, specifically when $R$ is generically Gorenstein. We then generalize a result of Kato, who proved that a Gorenstein complete local ring $R$ satisfies the $\text{SC}_{2}$-condition if and only if $R$ is a UFD. For $r \geq 3$, we prove a criterion for when an MCM $R$-module $M$ satisfies the $\text{SC}_{r}$-condition, assuming that its first syzygy $\Omega_{R}^{1}(M)$ satisfies the $\text{SC}_{r-1}$-condition.
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https://arxiv.org/abs/2507.14424
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0988dfdb23a01f7501c1c4ba554c86f5232936fe6525e73572ea586bdf803483
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2026-01-16T00:00:00-05:00
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Some reverse inequalities for scalar Birkhoff weak integrable functions
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arXiv:2507.16332v2 Announce Type: replace Abstract: Some inequalities and reverses of classic H\"{o}lder and Minkowski types are obtained for scalar Birkhoff weak integrable functions with respect to a non-additive measure.
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https://arxiv.org/abs/2507.16332
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f5f6677968b4ba71f387eb026942f9d109318e53ac4b3cf83e8e20565ab601a1
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2026-01-16T00:00:00-05:00
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Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra
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arXiv:2507.20325v2 Announce Type: replace Abstract: We show that the free spectrahedron determined by universal anticommuting self-adjoint unitaries is not equal to the minimal matrix convex set over the ball in dimension three or higher. This example, as well as other matrix convex sets over the ball, then provides context for structure results on the extreme points of coordinate projections. In particular, we show that the free polar dual of a real free spectrahedron is rarely the projection of a real free spectrahedron, contrasting a prior result of Helton, Klep, and McCullough over the complexes. We use this to show that spanning results for free spectrahedra that are closed under complex conjugation do not extend to free spectrahedrops that meet the same assumption. These results further clarify the role of the coefficient field.
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https://arxiv.org/abs/2507.20325
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d8ae34dea5d20cbcd8797f14ca2feb4a712564e51036570c26aba3e2c09497ea
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2026-01-16T00:00:00-05:00
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On the characteristic form of $\mathfrak{g}$-valued zero-curvature representations
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arXiv:2508.01224v2 Announce Type: replace Abstract: We study $\mathfrak{g}$-valued zero-curvature representations (ZCRs) for partial differential equations in two independent variables from the perspective of their extension to the entire infinite jet space, focusing on their characteristic elements. Since conservation laws -- more precisely, conserved currents -- and their generating functions for a given equation are precisely the $\mathbb{R}$-valued ZCRs and their characteristic elements, a natural question arises: to what extent can results known for conservation laws be extended to general $\mathfrak{g}$-valued ZCRs. For a fixed matrix Lie algebra $\mathfrak{g} \subset \mathfrak{gl}(n)$, we formulate ZCRs as equivalence classes of $\mathfrak{g}$-valued function pairs on the infinite jet space that satisfy the Maurer--Cartan condition. Our main result establishes that every such ZCR admits a characteristic representative -- i.e., a representative in which the Maurer--Cartan condition takes a characteristic form -- generalizing the characteristic form known for conservation laws. This form is preserved under gauge transformations and can thus be regarded as a kind of normal form for ZCRs. We derive a new necessary condition, independent of the Maurer--Cartan equation, that must be satisfied by any characteristic representative. This condition is trivial in the abelian case but nontrivial whenever $\mathfrak{g}$ is nonabelian. These findings not only confirm structural assumptions used in previous works but also suggest potential applications in the classification and computation of ZCRs.
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https://arxiv.org/abs/2508.01224
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efb521ff05fd567b396786e4d27e7d832fc8564b176983c2eea47fc5eef52407
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2026-01-16T00:00:00-05:00
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Canonical Frames for Bracket Generating Rank 2 Distributions which are not Goursat
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arXiv:2508.09307v5 Announce Type: replace Abstract: We complete a uniform construction of canonical absolute parallelism for bracket generating rank $2$ distributions with $5$-dimensional cube on $n$-dimensional manifold with $n\geq 5$ by showing that the condition of maximality of class that was assumed previously by Doubrov-Zelenko for such a construction holds automatically at generic points. This also gives analogous constructions in the case when the cube is not $5$-dimensional but the distribution is not Goursat through the procedure of iterative Cartan deprolongation. This together with the classical theory of Goursat distributions covers in principle the local geometry of all bracket generating rank 2 distributions in a neighborhood of generic points. As a byproduct, for any $n\geq 5$ we describe the maximally symmetric germs among bracket generating rank $2$ distributions with $5$-dimensional cube, as well as among those which reduce to such a distribution under a fixed number of Cartan deprolongations. Another consequence of our results on maximality of class is for optimal control problems with constraint given by a rank $2$ distribution with $5$-dimensional cube: it implies that for a generic point $q_0$ of $M$, there are plenty abnormal extremal trajectories of corank $1$ (which is the minimal possible corank) starting at $q_0$. The set of such points contains all points where the distribution is equiregular.
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https://arxiv.org/abs/2508.09307
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7be55acc8f5028d049f8040068087ef52227349eadee753850fc9b2cb6598e55
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2026-01-16T00:00:00-05:00
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Geometric inequalities for electrostatic systems with boundary
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arXiv:2508.10258v3 Announce Type: replace Abstract: In this article, we investigate electrostatic systems with a nonzero cosmological constant on compact manifolds with boundary. We establish new geometric properties for electrostatic manifolds in higher dimensions, extending previous results in the literature. Moreover, we prove sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds, as well as volume and boundary inequalities involving the Brown-York and Hawking masses.
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https://arxiv.org/abs/2508.10258
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d2b7d6ec3e563b0fe932b05e4bf289a1e3c6f6c0f955224ab0678a28321049b7
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2026-01-16T00:00:00-05:00
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False Data-Injection Attack Detection in Cyber-Physical Systems: A Wasserstein Distributionally Robust Reachability Optimization Approach
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arXiv:2508.12402v2 Announce Type: replace Abstract: Cyber-physical system (CPS) is the foundational backbone of modern critical infrastructures, so ensuring its security and resilience against cyber-attacks is of pivotal importance. This paper addresses the challenge of designing anomaly detectors for CPS under false-data injection (FDI) attacks and stochastic disturbances governed by unknown probability distribution. By using the Wasserstein ambiguity set, a prevalent data-driven tool in distributionally robust optimization (DRO), we first propose a new security metric to deal with the absence of disturbance distribution. This metric is designed by asymptotic reachability analysis of state deviations caused by stealthy FDI attacks and disturbance in a distributionally robust confidence set. We then formulate the detector design as a DRO problem that optimizes this security metric while controlling the false alarm rate robustly under a set of distributions. This yields a trade-off between robustness to disturbance and performance degradation under stealthy attacks. The resulting design problem turns out to be a challenging semi-infinite program due to the existence of distributionally robust chance constraints. We derive its exact albeit non-convex reformulation and develop an effective solution algorithm based on sequential minimization. Finally, a case study on a simulated three-tank is illustrated to demonstrate the efficiency of our design in robustifying against unknown disturbance distribution.
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https://arxiv.org/abs/2508.12402
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3b90c8492b986a3442270f4d11c08a6950d2a7822a715ffe68c25a434c66e584
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2026-01-16T00:00:00-05:00
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New weighted Riesz-type pointwise inequalities and applications to generalized Sobolev estimates
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arXiv:2508.12771v2 Announce Type: replace Abstract: In this article we study some new pointwise inequalities between rough singular integral operators, weighted maximal functions of the gradient and weighted Morrey spaces. These pointwise estimates will naturally lead us to a new class of weighted Sobolev-type inequalities.
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https://arxiv.org/abs/2508.12771
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eacc7eff51982b91661511c28789ebbeda5cc98e11b0308863214ef99f37b3fc
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2026-01-16T00:00:00-05:00
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Voter Model stability with respect to conservative noises
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arXiv:2509.02717v2 Announce Type: replace Abstract: The notions of noise sensitivity and stability were recently extended for the voter model. In this model, the vertices of a graph have opinions that are updated by uniformly selecting edges. We further extend stability results to different classes of perturbations. We consider two different types of noise: in the first one, an exclusion process is performed on the edge selections, while in the second, independent Brownian motions are applied to such a sequence. In both cases, we prove stability of the consensus opinion provided the noise is run for a short amount of time, depending on the underlying graph structure. This is done by analyzing the expected size of the pivotal set, whose definition differs from the usual one in order to reflect the change associated with these noises.
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https://arxiv.org/abs/2509.02717
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985676debacd619c6f8509e39a182f7f2efdb73676d86ccfac62b531d49b41d5
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2026-01-16T00:00:00-05:00
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Sequential Change Detection with Differential Privacy
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arXiv:2509.02768v2 Announce Type: replace Abstract: Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false-alarm rate when pre- and post-change distributions are fully known. However, releasing CUSUM statistics and the corresponding stopping time directly can compromise individual data privacy. We therefore introduce a differentially private (DP) variant, called DP-CUSUM, that injects calibrated Laplace noise into both the vanilla CUSUM statistics and the detection threshold, preserving the recursive simplicity of the classical CUSUM statistics while ensuring per-sample differential privacy. We derive closed-form bounds on the average run length to false alarm and on the worst-case average detection delay, explicitly characterizing the trade-off among privacy level, false-alarm rate, and detection efficiency. Our theoretical results imply that under a weak privacy constraint, our proposed DP-CUSUM procedure achieves the same first-order asymptotic optimality as the classical, non-private CUSUM procedure. Numerical simulations are conducted to demonstrate the detection efficiency of our proposed DP-CUSUM under different privacy constraints, and the results are consistent with our theoretical findings.
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https://arxiv.org/abs/2509.02768
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21d4fb6700e06e5420453eb4cee1037c624facb450a0e7f8cb030430207d517f
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2026-01-16T00:00:00-05:00
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Chebyshev's bias for modular forms
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arXiv:2509.04187v2 Announce Type: replace Abstract: We study Chebyshev's bias for the signs of Fourier coefficients of cuspidal newforms on $\Gamma_0(N)$. Our main result shows that the bias towards either sign is completely determined by the order of vanishing of the $L$-function $L(s, f)$ at the central point of the critical strip. We then give several examples of modular forms where we explicitly compute the order of vanishing of $L(s, f)$ at the central point and as a by-product, verify the super-positivity property, in the sense of Yun--Zhang (2017), for these examples.
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https://arxiv.org/abs/2509.04187
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3ccfa20a68aaafdf984baa8f70c9a5c07000c127855f6f91f53357924bd71bfb
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2026-01-16T00:00:00-05:00
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Polynomial Stability of Non-Linearly Damped Contraction Semigroups
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arXiv:2509.04275v2 Announce Type: replace Abstract: We investigate the stability properties of an abstract class of semi-linear systems. Our main result establishes rational rates of decay for classical solutions assuming a certain non-uniform observability estimate for the linear part and suitable conditions on the non-linearity. We illustrate the strength of our abstract results by applying them to a one-dimensional wave equation with weak non-linear damping and to an Euler-Bernoulli beam with a tip mass subject to non-linear damping.
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https://arxiv.org/abs/2509.04275
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11f062a95c2ad74dff68debc3ff6b213a930e670f6d30e5e7b22803ef07d8eda
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2026-01-16T00:00:00-05:00
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On the geometry of punctual Hilbert schemes on singular curves and their motivic zeta functions
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arXiv:2509.06761v4 Announce Type: replace Abstract: Inspired by the work of Soma and Watari, we define a tree structure on certain subsemimodules of the semigroup $\Gamma$ associated with an irreducible plane curve singularity $(C,O)$. Building on results of Oblomkov, Rasmussen, and Shende, we show that for specific classes of singularities, this tree encodes key aspects of the geometry of the punctual Hilbert schemes of $(C,O)$. As an application, we compute the motivic Hilbert zeta function for a family of singular curves. \vskip 0.1cm A point in the Hilbert scheme corresponds to an ideal in the local ring $\mathcal{O}_{C,O}$ of the singularity. We study the stratification of these Hilbert schemes induced by constraints on the minimal number of generators of the defining ideals, and we describe geometric properties of these strata, including their dimension and closure relations.\vskip 0.1cm More importantly, we study their motivic zeta functions, particularly the motivic Hilbert zeta function, which encodes the classes of all punctual Hilbert schemes in the Grothendieck ring of varieties.
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https://arxiv.org/abs/2509.06761
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686d150e53b2ce484b8a43f94b32d738c16b85a59d4554ef9bd38d4d4b18483d
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2026-01-16T00:00:00-05:00
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Timelike conjugate points in Lorentzian length spaces
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arXiv:2509.12855v3 Announce Type: replace Abstract: We study notions of conjugate points along timelike geodesics in the synthetic setting of Lorentzian (pre-)length spaces, inspired by earlier work for metric spaces by Shankar--Sormani. After preliminary considerations on convergence of timelike and causal geodesics, we introduce and compare one-sided, symmetric, unreachable and ultimate conjugate points along timelike geodesics. We show that all such notions are compatible with the usual one in the smooth (strongly causal) spacetime setting. As applications, we prove a timelike Rauch comparison theorem, as well as a result closely related to the recently established Lorentzian Cartan--Hadamard theorem by Er\"{o}s--Gieger. In the appendix, we give a detailed treatment of the Fr\'{e}chet distance on the space of non-stopping curves up to reparametrization, a technical tool used throughout the paper.
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https://arxiv.org/abs/2509.12855
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1188d45c49b03418899e6f5186f522374787a8616b4e111c7186b81df0475912
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2026-01-16T00:00:00-05:00
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The strange story of an almost unknown prime number counter: The Rafael Barrett formula
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arXiv:2509.19324v5 Announce Type: replace Abstract: In this brief article, we present the formula created by Rafael Barrett in 1903 in a note to Henri Poincar\'e, which remained unknown for decades. Discovered in the 1930s by a Uruguayan mathematician, this formula was published and analyzed in a journal published in Montevideo in 1935. In this study, we present Barrett's formula and analyze a challenge it could pose.
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https://arxiv.org/abs/2509.19324
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e43ce9f5663023c3e5d3359076b55bc300b6ce39d2e5fd4ec556a103ec3e615c
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2026-01-16T00:00:00-05:00
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On Rapid mixing for random walks on nilmanifolds
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arXiv:2510.00398v3 Announce Type: replace Abstract: We prove rapid mixing for almost all random walks generated by $m$ translations on an arbitrary nilmanifold under mild assumptions on the size of $m$. For several classical classes of nilmanifolds, we show $m=2$ suffices. This provides a partial answer to the question raised in \cite{D02} about the prevalence of rapid mixing for random walks on homogeneous spaces.
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https://arxiv.org/abs/2510.00398
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00f172b204bec669bcb3f2adcd2941f8a518845b8539d5dbe2b9783f6b923ec5
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2026-01-16T00:00:00-05:00
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A flux-based approach for analyzing the disguised toric locus of reaction networks
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arXiv:2510.03621v2 Announce Type: replace Abstract: Dynamical systems with polynomial right-hand sides are very important in various applications, e.g., in biochemistry and population dynamics. The mathematical study of these dynamical systems is challenging due to the possibility of multistability, oscillations, and chaotic dynamics. One important tool for this study is the concept of reaction systems, which are dynamical systems generated by reaction networks for some choices of parameter values. Among these, disguised toric systems are remarkably stable: they have a unique attracting fixed point, and cannot give rise to oscillations or chaotic dynamics. The computation of the set of parameter values for which a network gives rise to disguised toric systems (i.e., the disguised toric locus of the network) is an important but difficult task. We introduce new ideas based on network fluxes for studying the disguised toric locus. We prove that the disguised toric locus of any network $G$ is a contractible manifold with boundary, and introduce an associated graph $G^{\max}$ that characterizes its interior. These theoretical tools allow us, for the first time, to compute the full disguised toric locus for many networks of interest.
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https://arxiv.org/abs/2510.03621
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2026-01-16T00:00:00-05:00
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Diameter and mixing time of the giant component in the percolated hypercube
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arXiv:2510.13348v3 Announce Type: replace Abstract: We consider bond percolation on the $d$-dimensional binary hypercube with $p=c/d$ for fixed $c>1$. We prove that the typical diameter of the giant component $L_1$ is of order $\Theta(d)$, and the typical mixing time of the lazy random walk on $L_1$ is of order $\Theta(d^2)$. This resolves long-standing open problems of Bollob\'as, Kohayakawa and {\L}uczak from 1994, and of Benjamini and Mossel from 2003. A key component in our approach is a new tight large deviation estimate on the number of vertices in $L_1$ whose proof includes several novel ingredients: a structural description of the residue outside the giant component after sprinkling, a tight quantitative estimate on the spread of the giant in the hypercube, and a stability principle which rules out the disintegration of large connected sets under thinning. This toolkit further allows us to obtain optimal bounds on the expansion in $L_1$.
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https://arxiv.org/abs/2510.13348
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8af10c10c31900c8f4cec4718fd1661f9cd2f685bb6f813b93335e787878db90
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2026-01-16T00:00:00-05:00
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INTHOP: A Second-Order Globally Convergent Method for Nonconvex Optimization
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arXiv:2510.22342v3 Announce Type: replace Abstract: Second-order Newton-type algorithms that leverage the exact Hessian or its approximation are central to solve nonlinear optimization problems. However, their applications in solving large-scale nonconvex problems are hindered by three primary challenges: (1) the high computational cost associated with Hessian evaluations, (2) its inversion, and (3) ensuring descent direction at points where the Hessian becomes indefinite. We propose INTHOP, an interval Hessian-based optimization algorithm for nonconvex problems to deal with these primary challenges. The proposed search direction is based on approximating the original Hessian matrix by a positive definite matrix. The novelty of the proposed method is that the proposed search direction is guaranteed to be descent and requires approximation of Hessian and its inversion only at specific iterations. We prove that the difference between the calculated approximate and exact Hessian is bounded within an interval. Accordingly, the approximate Hessian matrix is reused if the iterates are in that chosen interval while computing the gradients at each iteration. We develop various algorithm variants based on the interval size updating methods and minimum eigenvalue computation methods. We also prove the global convergence of the proposed algorithm. Further, we apply the algorithm to an extensive set of test problems and compare its performance with the existing methods such as steepest descent, quasi-Newton, and Newton method. We show empirically that the proposed method solves more problems in fewer function and gradient evaluations than steepest descent and the quasi-Newton method. While in the comparison to the Newton method, we illustrate that for nonconvex optimization problems, we require substantially less $O(n^3)$ operations.
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https://arxiv.org/abs/2510.22342
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91b0c703ea37adfa68a217e1def0ed2830b655c64e4af2b5c90f0a963c140669
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2026-01-16T00:00:00-05:00
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Monotone Sobolev extensions in metric surfaces and applications to uniformization
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arXiv:2510.26458v2 Announce Type: replace Abstract: We prove a monotone Sobolev extension theorem for maps to Jordan domains with rectifiable boundary in metric surfaces of locally finite Hausdorff 2-measure. This is then used to prove a uniformization result for compact metric surfaces by minimizing energy in the class of monotone Sobolev maps.
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https://arxiv.org/abs/2510.26458
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43fe5d3a8cde5b8c6d995dceced450ec5d6ab13a9b8425f16e283c047c9c47c0
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2026-01-16T00:00:00-05:00
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Genus two embedded minimal surfaces in $\mathbb{S}^3$ with bidihedral symmetry
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arXiv:2511.16295v2 Announce Type: replace Abstract: The isometry group of the classical Lawson embedded minimal surface $\xi_{2,1}\subset \mathbb{S}^3$ of genus 2 is isomorphic to the product $S_3\times D_4$ of the permutation group of three elements and the dihedral group of order 8 (symmetries of a square). $S_3\times D_4$ has a subgroup of index 3 isomorphic to the bidihedral group $D_{4h}=\mathbb{Z}_2\times D_4$, where $D_4$ is the dihedral group of order 8. We prove that $\xi_{2,1}$ is the unique closed embedded minimal surface of genus 2 in $\mathbb{S}^3$ whose isometry group contains $D_{4h}$.
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https://arxiv.org/abs/2511.16295
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2026-01-16T00:00:00-05:00
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Exceptions to the Erd\H os--Straus--Schinzel conjecture
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arXiv:2511.16817v2 Announce Type: replace Abstract: A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel conjectured that for every integer $m\ge4$ there is a bound $n_m$ such that the fraction $m/n$ is the sum of 3 unit fractions for all integers $n\ge n_m$. Leveraging and generalizing work of Elsholtz and Tao, we show that if $n_m$ exists it must be at least $\exp(m^{1/3+o(1)})$; that is, there are numbers $n$ this large for which $m/n$ is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for $m\ge 6.52\times10^9$ there is a prime $p\in(m^2,2m^2)$ with $m/p$ not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound $m\ge20$. A result of Vaughan is that for each $m$, most $n$'s have $m/n$ representable; we make the dependence on $m$ in this result explicit. In addition, we prove a result generalizing the problem to the sum of $j$ unit fractions.
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https://arxiv.org/abs/2511.16817
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273ff989e90c196fdb9f01e06d02b630b934754080c80ac42e138b22b47081aa
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2026-01-16T00:00:00-05:00
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Skew-symmetrizable cluster algebras from surfaces and symmetric quivers
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arXiv:2512.12247v2 Announce Type: replace Abstract: We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $\sigma$. We give a geometric realization of such cluster algebras by showing that cluster variables of $\mathcal{A}$ correspond to $\sigma$-orbits of arcs of $\tilde{\mathbf{S}}$, while clusters are given by admissible $\sigma$-invariant triangulations. We establish a ring homomorphism from $\mathcal{A}$ to a skew-symmetric cluster algebra of the same rank, which is combinatorially derived from $\mathcal{A}$. We use this result to provide a cluster expansion formula for any $\sigma$-orbit $[\gamma]$ in terms of perfect matchings of some labeled modified snake graphs constructed from the arcs of $[\gamma]$. Then, we associate a symmetric finite-dimensional algebra $A$ to any seed of $\mathcal{A}$, such that non-initial cluster variables bijectively correspond to orthogonal indecomposable $A$-modules. Finally, we exhibit a purely representation-theoretic map from the category of orthogonal $A$-modules to $\mathcal{A}$, providing a Caldero-Chapoton map in this setting.
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https://arxiv.org/abs/2512.12247
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ba8c6e1caef0a242ce739b2923c9f6785db6b1821a625ec82b7a7b990f004f96
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2026-01-16T00:00:00-05:00
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Relative center construction for $G$-graded C$^*$-tensor categories and Longo-Rehren inclusions
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arXiv:2512.21485v2 Announce Type: replace Abstract: Gelaki-Naidu-Nikshych and Turaev-Virelizier showed the existence of $G$-braiding on the relative Drinfeld center of a $G$-graded tensor category. We will explain this concept from the viewpoint of Longo-Rehren inclusions.
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https://arxiv.org/abs/2512.21485
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2026-01-16T00:00:00-05:00
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Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Random-Walk Laplacian Formulation)
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arXiv:2512.21770v2 Announce Type: replace Abstract: The operator-theoretic dichotomy underlying diffusion on directed networks is \emph{symmetry versus non-self-adjointness} of the Markov transition operator. In the reversible (detailed-balance) regime, a directed random walk $P$ is self-adjoint in a stationary $\pi$-weighted inner product and admits orthogonal spectral coordinates; outside reversibility, $P$ is genuinely non-self-adjoint (often non-normal), and stability is governed by biorthogonal geometry and eigenvector conditioning. In this paper we develop a harmonic-analysis framework for directed graphs anchored on the random-walk transition matrix $P=D_{\mathrm{out}}^{-1}A$ and the random-walk Laplacian $L_{\mathrm{rw}}=I-P$. Using biorthogonal left/right eigenvectors we define a \emph{Biorthogonal Graph Fourier Transform} (BGFT) adapted to directed diffusion, propose a diffusion-consistent frequency ordering based on decay rates $\Re(1-\lambda)$, and derive operator-norm stability bounds for iterated diffusion and for BGFT spectral filters. We prove sampling and reconstruction theorems for $P$-bandlimited (equivalently $L_{\mathrm{rw}}$-bandlimited) signals and quantify noise amplification through the conditioning of the biorthogonal eigenbasis. A simulation protocol on directed cycles and perturbed non-normal digraphs demonstrates that asymmetry alone does not dictate instability; rather, non-normality and eigenvector ill-conditioning drive reconstruction sensitivity, making BGFT a natural analytical language for directed diffusion processes.
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https://arxiv.org/abs/2512.21770
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1a3e05253c7a8bbcdbfe1701435315d514fe38791d9e9adb4ea185027b6dbd08
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2026-01-16T00:00:00-05:00
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On the number of words of $N=3 \,n$ letters with a three-letter alphabet
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arXiv:2512.22362v2 Announce Type: replace Abstract: In this paper we address the well-known problem of counting the number of $3n$-letter words that can be formed from a three-letter alphabet by decomposing it into four possible cases based on its remainder when divided by three. The solution to the problem also gives us some sums of trinomial coefficients.
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https://arxiv.org/abs/2512.22362
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300b05e53aeb4a32353c4d48c0e643da2910fda5862952aa08af6ea22cadeeac
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2026-01-16T00:00:00-05:00
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A Proximal-Gradient Method for Solving Regularized Optimization Problems with General Constraints
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arXiv:2512.23166v2 Announce Type: replace Abstract: We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step acceptance or rejection. Under various assumptions, we establish a worst-case iteration complexity result, prove that limit points are first-order KKT points, and show that manifold identification and active-set identification properties hold. Preliminary numerical experiments on a subset of the CUTEst test problems and sparse canonical correlation analysis problems demonstrate the promising performance of our approach.
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https://arxiv.org/abs/2512.23166
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7bd4dc9a710a9b87eab7caebe8fc5b3bf1b9b6829a9e6e6dbfffb03eda587262
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2026-01-16T00:00:00-05:00
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Chamber zeta function and closed galleries in the standard non-uniform complex from $\operatorname{PGL}_3$
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arXiv:2512.23276v2 Announce Type: replace Abstract: We introduce the \emph{chamber zeta function} for a complex of groups, defined via an Euler product over primitive tailless chamber galleries, extending the Ihara--Bass framework from weighted graphs to higher-rank settings. Let $\mathcal{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_{3}(F)$ for a non-archimedean local field $F$ with residue field $\mathbb{F}_{q}$. For the standard arithmetic quotient $\Gamma\backslash\mathcal{B}$ with $\Gamma=\mathrm{PGL}_{3}(\mathbb{F}_{q}[t])$, we prove an Ihara--Bass type \emph{determinant formula} expressing the chamber zeta function as the reciprocal of a characteristic polynomial of a naturally defined chamber transfer operator. In particular, the chamber zeta function is \emph{rational} in its complex parameter. As an application of the determinant formula, we obtain explicit counting results for closed gallery classes arising from tailless galleries in $\mathcal{B}$, including exact identities and spectral asymptotics governed by the chamber operator.
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https://arxiv.org/abs/2512.23276
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5b8970d0412f294302973fcb06ef830bc7526bc9c46bb22ed5e74da6f227555a
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2026-01-16T00:00:00-05:00
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Polygons in Polygons with a Twist
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arXiv:2601.00899v3 Announce Type: replace Abstract: This is a study of the construction of particular regular sub-n-gons T in regular n-gons P using a special system of chords of P. In particular, some of these sub-n-gons have areas which are integer divisors of the area of the given n-gon P. Initially, the study will concentrate on chords which are from a vertex to special points of one of the opposite sides of P. Several examples are explored. However, it will become apparent that a much more general situation exists. A Dynamic Geometry software, such as Sketchpad, or GeoGebra, is the key to investigating this new relationship.
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https://arxiv.org/abs/2601.00899
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b2cf53a9caebe3c91f5e35be510d2f5af3e6a284b2df41290cf7a2bf04105eb3
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2026-01-16T00:00:00-05:00
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On the forward self-similar solutions to the two-dimensional Navier-Stokes equations
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arXiv:2601.03833v2 Announce Type: replace Abstract: We establish the global existence of forward self-similar solutions to the two-dimensional incompressible Navier-Stokes equations for any divergence-free initial velocity that is homogeneous of degree $-1$ and locally H\"older continuous. This result requires no smallness assumption on the initial data. In sharp contrast to the three-dimensional case, where $(-1)$-homogeneous vector fields are locally square-integrable, the major difficulty for the 2D problem is the criticality in the sense that the initial kinetic energy is locally infinite at the origin, and the initial vorticity fails to be locally integrable, so that the classical local energy estimates are not available. Our key ideas are to decompose the solution into a linear part solving the heat equation and a finite-energy perturbation part, and to exploit a kind of inherent cancellation relation between the linear part and the perturbation part. These, together with suitable choices of multipliers, enable us to control the interaction terms and to establish the $H^1$-estimates for the perturbation part. Furthermore, we can get an optimal pointwise estimate via investigating the corresponding Leray equations in weighted Sobolev spaces.This gives the faster decay of the perturbation part at infinity and compactness, which play important roles in proving the existence of global-in-time self-similar solutions.
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https://arxiv.org/abs/2601.03833
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89fbee612b1b9d91e8c47621a3968a3bd56c4e44fcbe62cbd342005af09f4708
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2026-01-16T00:00:00-05:00
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Ergodic Theorems and Equivalence of Green's Kernel for Random Walks in Random Environments
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arXiv:2601.04161v2 Announce Type: replace Abstract: We study the Ergodic Properties of Random Walks in stationary ergodic environments without uniform ellipticity under a minimal assumption. There are two main components in our work. The first step is to adopt the arguments of Lawler to first prove a uniqueness principle. We use a more general definition of environments using~\textit{Environment Functions}. As a corollary, we can deduce an invariance principle under these assumptions for balanced environments under some assumptions. We also use the uniqueness principle to show that any balanced, elliptic random walk must have the same transience behaviour as the simple symmetric random walk. The second is to transfer the results we deduce in balanced environments to general ergodic environments(under some assumptions) using a control technique to derive a measure under which the \textit{local process} is stationary and ergodic. As a consequence of our results, we deduce the Law of Large Numbers for the Random Walk and an Invariance Principle under our assumptions.
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https://arxiv.org/abs/2601.04161
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b1c3e4156cd1507012d9974229578f8508e700a147f2c2a883c9f0dc25d87291
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2026-01-16T00:00:00-05:00
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Linear identities for partition pairs with $5$-cores
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arXiv:2601.04743v2 Announce Type: replace Abstract: We prove an infinite family of linear identities for the number $A_5(n)$ of partition pairs of $n$ with $5$-cores by using certain theta function identities involving the Ramanujan's parameter $k(q)$ due to Cooper, and Lee and Park. Consequently, we deduce an infinite family of congruences for $A_5(n)$ using these linear identities.
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https://arxiv.org/abs/2601.04743
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629802001761d47d42d799c9573f591f19e7430fca9f94b9987e2e5a55b4c827
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2026-01-16T00:00:00-05:00
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A Unified Spectral Framework for Aging, Heterogeneous, and Distributed Order Systems via Weighted Weyl-Sonine Operators
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arXiv:2601.05423v2 Announce Type: replace Abstract: While General Fractional Calculus has successfully expanded the scope of memory operators beyond power-laws, standard formulations remain predominantly restricted to the half-line via Riemann-Liouville or Caputo definitions. This constraint artificially truncates the system's history, limiting the thermodynamic consistency required for modeling processes on unbounded domains. To overcome these barriers, we construct the \textbf{Weighted Weyl-Sonine Framework}, a generalized formalism that extends non-local theory to the entire real line without history truncation. Unlike recent algebraic approaches based on conjugation for finite intervals, we develop a rigorous harmonic analysis framework. Our central contribution is the \textbf{Generalized Spectral Mapping Theorem}, which establishes the Weighted Fourier Transform as a unitary diagonalization map for these operators. This result allows us to rigorously classify and solve distinct physical regimes under a single algebraic structure. We explicitly derive exact solutions for \textit{diffusive relaxation} (governed by Complete Bernstein Functions), \textit{inertial wave propagation} (exhibiting oscillatory dynamics), and \textit{retarded aging} (via distributed order), proving that our framework unifies the description of anomalous transport and wave mechanics in complex, time-deformed media.
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https://arxiv.org/abs/2601.05423
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Composition Ax-Kochen/Ershov principles and tame fields of mixed characteristic
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arXiv:2601.05790v2 Announce Type: replace Abstract: We study in which settings we have a composition AKE principle, i.e. when the theory of the coarsening $(K,w)$ and the theory of the induced valuation $(Kw,\overline{v})$ determine the theory of the composition $(K,v)$. We show that this is the case when $(K,w)$ is tame of equal characteristic, and provide counterexamples in mixed characteristic. We further show that, for a tame field of mixed characteristic, the theory of the valued field cannot, in general, be determined solely by the theories of its underlying field, its residue field, and its value group.
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https://arxiv.org/abs/2601.05790
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01785b4cf994ce1459231eb5a0242410cfa8848304d9555a03532814727bc196
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2026-01-16T00:00:00-05:00
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Global Optimization for Combinatorial Geometry Problems Revisited in the Era of LLMs
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arXiv:2601.05943v2 Announce Type: replace Abstract: Recent progress in LLM-driven algorithm discovery, exemplified by DeepMind's AlphaEvolve, has produced new best-known solutions for a range of hard geometric and combinatorial problems. This raises a natural question: to what extent can modern off-the-shelf global optimization solvers match such results when the problems are formulated directly as nonlinear optimization problems (NLPs)? We revisit a subset of problems from the AlphaEvolve benchmark suite and evaluate straightforward NLP formulations with two state-of-the-art solvers, the commercial FICO Xpress and the open-source SCIP. Without any solver modifications, both solvers reproduce, and in several cases improve upon, the best solutions previously reported in the literature, including the recent LLM-driven discoveries. Our results not only highlight the maturity of generic NLP technology and its ability to tackle nonlinear mathematical problems that were out of reach for general-purpose solvers only a decade ago, but also position global NLP solvers as powerful tools that may be exploited within LLM-driven algorithm discovery.
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https://arxiv.org/abs/2601.05943
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090fd11f9d96b512c0c3e86a591d54815736c7fc2867d8b4d9916d43baa6a122
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2026-01-16T00:00:00-05:00
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A Non-Renormalization Theorem for Local Functionals in Ghost-Free Vector Field Theories Coupled to Dynamical Geometry
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arXiv:2601.08081v2 Announce Type: replace Abstract: We establish a non-renormalization theorem for a class of ghost-free local functionals describing massive vector field theories coupled to dynamical geometry. Under the assumptions of locality, Lorentz invariance, and validity of the effective field theory expansion below a fixed cutoff, we show that quantum corrections do not generate local operators that renormalize the classical derivative self-interactions responsible for the constraint structure of the theory. The proof combines an operator-level analysis of the space of allowed local counterterms with a systematic decoupling-limit argument, which isolates the leading contributions to the effective action at each order in the derivative expansion. As a consequence, all radiatively induced local functionals necessarily involve additional derivatives per field and are suppressed by the intrinsic strong-coupling scales of the theory. In particular, the classical interactions defining ghost-free vector field theories are stable under renormalization, and any additional degrees of freedom arising from quantum corrections appear only above the effective field theory cutoff. This result extends known non-renormalization properties of flat-space vector theories to the case of dynamical geometry and provides a structural explanation for their perturbative stability to all loop orders.
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https://arxiv.org/abs/2601.08081
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b1851e64fa89033f52f808929b91206415130201be45c32a185224fb5b56b707
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2026-01-16T00:00:00-05:00
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The signless Laplacian spectral Tur\'an problems for hypergraphs
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arXiv:2601.08595v2 Announce Type: replace Abstract: Let $\mathcal{H}=(V, E)$ be an $r$-uniform hypergraph on $n$ vertices. The signless Laplacian spectral radius of $\mathcal{H}$ is defined as the maximum modulus of the eigenvalues of the tensor $\mathcal{Q}(\mathcal{H})=\mathcal{D}(\mathcal{H})+\mathcal{A}(\mathcal{H})$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the degree diagonal tensor and the adjacency tensor of $\mathcal{H}$, respectively. In this paper, we establish a general theorem that extends the spectral Tur\'an result of Keevash, Lenz and Mubayi [SIAM J. Discrete Math., 28 (4) (2014)] to the setting of signless Laplacian spectral Tur\'an problems. We prove that if a family $\mathcal{F}$ of $r$-uniform hypergraphs is degree-stable with respect to a family $\mathcal{H}_n$ of $r$-uniform hypergraphs and its extremal constructions satisfy certain natural assumptions, then the signless Laplacian spectral Tur\'an problem for $\mathcal{F}$ can be effectively reduced to the corresponding problem restricted to the family $\mathcal{H}_n$. As a concrete application, we completely determine the extremal hypergraph that maximizes the signless Laplacian spectral radius among all Fano plane-free $3$-uniform hypergraphs, showing that the unique extremal hypergraph is the balanced complete bipartite $3$-uniform hypergraph.
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https://arxiv.org/abs/2601.08595
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95f7c8183a7ef5ce19cb0cdc16e724f85e3c0ebebcbdc723b297a66b38d40e5b
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2026-01-16T00:00:00-05:00
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The Spectral Geometry of Ternary Gamma Schemes:Sheaf-Theoretic Foundations and Laplacian Clustering
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arXiv:2601.09268v2 Announce Type: replace Abstract: This article develops a self-contained affine $\Gamma$-scheme theory for a class of commutative ternary $\Gamma$-semirings. By establishing all geometric and spectral results internally, the work provides a unified framework for triadic symmetry and spectral analysis. The central thesis is that a triadic $\Gamma$-algebra canonically induces two primary structures: (i) an intrinsic triadic symmetry in the sense of a Nambu--Filippov-type fundamental identity on the structure sheaf, and (ii) a canonical Laplacian on the finite $\Gamma$-spectrum whose spectral decomposition detects the clopen (connected-component) decomposition of the underlying space. We define $\Gamma$-ideals and prime $\Gamma$-ideals, endow $\SpecG(T)$ with a $\Gamma$-Zariski topology, construct localizations and the structure sheaf on the basis of principal opens, and prove the affine anti-equivalence between commutative ternary $\Gamma$-semirings and affine $\Gamma$-schemes. Furthermore, we demonstrate that the triadic bracket on sections is invariant under $\Gamma$-automorphisms and compatible with localization. The main spectral theorem establishes the block-diagonalization of the Laplacian under topological decompositions and provides an algebraic-connectivity criterion. The theory is verified through explicit computations of finite $\Gamma$-spectra and their corresponding Laplacian spectra
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https://arxiv.org/abs/2601.09268
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7b0ea30f8975b94e570757e231bb232de25b6a51ff253c24d65be189836bd4bb
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2026-01-16T00:00:00-05:00
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Proof of a Conjecture on Young Tableaux with Walls
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arXiv:2601.09551v2 Announce Type: replace Abstract: Banderier, Marchal, and Wallner considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. In this work, we prove a conjecture of Fuchs and Yu concerning the enumeration of two classes of three-row Young tableaux with walls. Combining with the work by Chang, Fuchs, Liu, Wallner, and Yu leads to the verification of a conjecture on tree-child networks proposed by Pons and Batle. This conjecture was regarded as a specific and challenging problem in the Phylogenetics community until it was finally resolved by the present work.
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https://arxiv.org/abs/2601.09551
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8195abede5e92e689f079ade5720440583004b39273b903ee0bd000f9ee29ba1
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2026-01-16T00:00:00-05:00
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The Addition Theorem for the Algebraic Entropy of Torsion Nilpotent Groups
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arXiv:2601.09643v2 Announce Type: replace Abstract: The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved by Dikranjan, Goldsmith, Salce and Zanardo. It was later extended by Shlossberg to torsion nilpotent groups of class 2. As our main result, we prove the Addition Theorem for endomorphisms of torsion nilpotent groups of arbitrary nilpotency class. As an application, we show that if $G$ is a torsion nilpotent group, then for every $\phi\in \mathrm{End}(G)$ either the entropy $h(\phi)$ is infinite or $h(\phi)=\log(\alpha)$ for some $\alpha\in\mathbb N$. We further obtain, for automorphisms of locally finite groups, the Addition Theorem with respect to all terms of the upper central series; in particular, the Addition Theorem holds for automorphisms of $\omega$-hypercentral groups. Finally, we establish a reduction principle: if $\mathfrak X$ is a variety of locally finite groups, then the Addition Theorem for endomorphisms holds in $\mathfrak X$ if and only if it holds for locally finite groups generated by bounded sets.
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https://arxiv.org/abs/2601.09643
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5b4e27fab6d4f0ca2a4443a559842ebcb7ddc19b8c594d20e33b779d0ad536ac
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2026-01-16T00:00:00-05:00
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A GMM approach to estimate the roughness of stochastic volatility
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arXiv:2010.04610v5 Announce Type: replace-cross Abstract: We develop a GMM approach for estimation of log-normal stochastic volatility models driven by a fractional Brownian motion with unrestricted Hurst exponent. We show that a parameter estimator based on the integrated variance is consistent and, under stronger conditions, asymptotically normally distributed. We inspect the behavior of our procedure when integrated variance is replaced with a noisy measure of volatility calculated from discrete high-frequency data. The realized estimator contains sampling error, which skews the fractal coefficient toward "illusive roughness." We construct an analytical approach to control the impact of measurement error without introducing nuisance parameters. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show the bias correction attenuates any systematic deviance in the parameter estimates. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in stochastic volatility.
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https://arxiv.org/abs/2010.04610
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e641f0c8b9119b8280d097c913b8293282aeb7f05f8766ad95745a9569ab9d05
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2026-01-16T00:00:00-05:00
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A nonparametric test for diurnal variation in spot correlation processes
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arXiv:2408.02757v2 Announce Type: replace-cross Abstract: The association between log-price increments of exchange-traded equities, as measured by their spot correlation estimated from high-frequency data, exhibits a pronounced upward-sloping and almost piecewise linear relationship at the intraday horizon. There is notably lower-on average less positive-correlation in the morning than in the afternoon. We develop a nonparametric testing procedure to detect such variation in a correlation process. The test statistic has a known distribution under the null hypothesis, whereas it diverges under the alternative. We run a Monte Carlo simulation to discover the finite sample properties of the test statistic, which are close to the large sample predictions, even for small sample sizes and realistic levels of diurnal variation. In an application, we implement the test on a high-frequency dataset covering the stock market over an extended period. The test leads to rejection of the null most of the time. This suggests diurnal variation in the correlation process is a nontrivial effect in practice. We show how conditioning information about macroeconomic news and corporate earnings announcements affect the intraday correlation curve.
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https://arxiv.org/abs/2408.02757
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0eb5868e070a16396441924c8dbc6f0e515333da07a2c120f4d986977197575d
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2026-01-16T00:00:00-05:00
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An unbounded intensity model for point processes
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arXiv:2408.06519v2 Announce Type: replace-cross Abstract: We develop a model for point processes on the real line, where the intensity can be locally unbounded without inducing an explosion. In contrast to an orderly point process, for which the probability of observing more than one event over a short time interval is negligible, the bursting intensity causes an extreme clustering of events around the singularity. We propose a nonparametric approach to detect such bursts in the intensity. It relies on a heavy traffic condition, which admits inference for point processes over a finite time interval. With Monte Carlo evidence, we show that our testing procedure exhibits size control under the null, whereas it has high rejection rates under the alternative. We implement our approach on high-frequency data for the EUR/USD spot exchange rate, where the test statistic captures abnormal surges in trading activity. We detect a nontrivial amount of intensity bursts in these data and describe their basic properties. Trading activity during an intensity burst is positively related to volatility, illiquidity, and the probability of observing a drift burst. The latter effect is reinforced if the order flow is imbalanced or the price elasticity of the limit order book is large.
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https://arxiv.org/abs/2408.06519
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6865a2fe3e9f58cc38009551a3ce243fe5bc08effe4ad6ef14bf58e121df2e33
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2026-01-16T00:00:00-05:00
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Global dynamical structures from infinitesimal data
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arXiv:2410.02111v2 Announce Type: replace-cross Abstract: Scientists and engineers alike target modeling of complex, high dimensional, and nonlinear dynamical systems as a central goal. Machine learning breakthroughs alongside mounting computation and data advance the efficacy of learning from trajectory measurements. However scientifically interpreting data-driven models, e.g., localizing attracting sets and their basins, remains elusive. Such limitations particularly afflict identification of system-level regulatory mechanisms characteristic of living systems, e.g., stabilizing control for whole-body locomotion, where discontinuous, transient, and multiscale phenomena are common and prior models are rare. As a next step towards theory-grounded discovery of behavioral mechanisms in biology and beyond, we introduce VERT, a framework for discovering attracting sets from trajectories without recourse to any global model. Our infinitesimal-local-global (ILG) pipeline estimates the proximity of any sampled state to an attracting set, if one exists, with formal accuracy guarantees. We demonstrate our approach on phenomenological and physical oscillators with hierarchical and impulsive dynamics, finding sensitivity to both global and intermediate attractors composed in sequence and parallel. Application of VERT to human running kinematics data reveals insight into control modules that stabilize task-level dynamics, supporting a longstanding neuromechanical control hypothesis. The VERT framework promotes rigorous inference of underlying dynamical structure even for systems where learning a global dynamics model is impractical or impossible.
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https://arxiv.org/abs/2410.02111
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be5b0d77d1c17db49efc199594496756353d308a0f378716659e0eb7958695ad
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2026-01-16T00:00:00-05:00
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Nonlinear stability of extremal Reissner-Nordstr\"om black holes in spherical symmetry
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arXiv:2410.16234v2 Announce Type: replace-cross Abstract: In this paper, we prove the codimension-one nonlinear asymptotic stability of the extremal Reissner-Nordstr\"om family of black holes in the spherically symmetric Einstein-Maxwell-neutral scalar field model, up to and including the event horizon. More precisely, we show that there exists a teleologically defined, codimension-one "submanifold" $\mathfrak M_\mathrm{stab}$ of the moduli space of spherically symmetric characteristic data for the Einstein-Maxwell-scalar field system lying close to the extremal Reissner-Nordstr\"om family, such that any data in $\mathfrak M_\mathrm{stab}$ evolve into a solution with the following properties as time goes to infinity: (i) the metric decays to a member of the extremal Reissner-Nordstr\"om family uniformly up to the event horizon, (ii) the scalar field decays to zero pointwise and in an appropriate energy norm, (iii) the first translation-invariant ingoing null derivative of the scalar field is approximately constant on the event horizon $\mathcal H^+$, (iv) for "generic" data, the second translation-invariant ingoing null derivative of the scalar field grows linearly along the event horizon. Due to the coupling of the scalar field to the geometry via the Einstein equations, suitable components of the Ricci tensor exhibit non-decay and growth phenomena along the event horizon. Points (i) and (ii) above reflect the "stability" of the extremal Reissner-Nordstr\"om family and points (iii) and (iv) verify the presence of the celebrated "Aretakis instability" for the linear wave equation on extremal Reissner-Nordstr\"om black holes in the full nonlinear Einstein-Maxwell-scalar field model.
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https://arxiv.org/abs/2410.16234
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087e2aad6becb5ea9c48c6f7faef64e2ec2a78fd998b8e4a6f243e62b805b41a
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2026-01-16T00:00:00-05:00
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Characterising memory in quantum channel discrimination via constrained separability problems
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arXiv:2411.08110v2 Announce Type: replace-cross Abstract: Quantum memories are a crucial precondition in many protocols for processing quantum information. A fundamental problem that illustrates this statement is given by the task of channel discrimination, in which an unknown channel drawn from a known random ensemble should be determined by applying it for a single time. In this paper, we characterise the quality of channel discrimination protocols when the quantum memory, quantified by the auxiliary dimension, is limited. This is achieved by formulating the problem in terms of separable quantum states with additional affine constraints that all of their factors in each separable decomposition obey. We discuss the computation of upper and lower bounds to the solutions of such problems which allow for new insights into the role of memory in channel discrimination. In addition to the single-copy scenario, this methodological insight allows to systematically characterise quantum and classical memories in adaptive channel discrimination protocols. Especially, our methods enabled us to identify channel discrimination scenarios where classical or quantum memory is required, and to identify the hierarchical and non-hierarchical relationships within adaptive channel discrimination protocols.
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https://arxiv.org/abs/2411.08110
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8db6a33f38b3e729115fbae540918cccb06c120f2013e26ae4cdf3fbfb4536b1
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2026-01-16T00:00:00-05:00
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Rational RG flow, extension, and Witt class
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arXiv:2412.08935v3 Announce Type: replace-cross Abstract: Consider a renormalization group flow preserving a pre-modular fusion category $\mathcal S_1$. If it flows to a rational conformal field theory, the surviving symmetry $\mathcal S_1$ flows to a pre-modular fusion category $\mathcal S_2$ with monoidal functor $F:\mathcal S_1\to\mathcal S_2$. By clarifying mathematical (especially category theoretical) meaning of renormalization group domain wall/interface or boundary condition, we find the hidden extended vertex operator (super)algebra gives a unique (up to braided equivalence) completely $(\mathcal S_1\boxtimes\mathcal S_2)'$-anisotropic representative of the Witt equivalence class $[\mathcal S_1\boxtimes\mathcal S_2]$. The mathematical conjecture is supported physically, and passes various tests in concrete examples including non/unitary minimal models, and Wess-Zumino-Witten models. In particular, the conjecture holds beyond diagonal cosets. The picture also establishes the conjectured half-integer condition, which fixes infrared conformal dimensions mod $\frac12$. It further leads to the double braiding relation, namely braiding structures jump at conformal fixed points. As an application, we solve the flow from the $E$-type minimal model $(A_{10},E_6)\to M(4,3)$.
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https://arxiv.org/abs/2412.08935
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35e4d94d6f46dfb044b9b9270a06718a1a3975325e3d222b9ba79de74c43443e
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2026-01-16T00:00:00-05:00
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Encrypted Qubits can be Cloned
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arXiv:2501.02757v3 Announce Type: replace-cross Abstract: We show that encrypted cloning of unknown quantum states is possible. Any number of encrypted clones of a qubit can be created through a unitary transformation, and each of the encrypted clones can be decrypted through a unitary transformation. The decryption of an encrypted clone consumes the decryption key, i.e., only one decryption is possible, in agreement with the no-cloning theorem. Encrypted cloning represents a new paradigm that provides a form of redundancy, parallelism or scalability where direct duplication is forbidden by the no-cloning theorem. For example, a possible application of encrypted cloning is to enable encrypted quantum multi-cloud storage.
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https://arxiv.org/abs/2501.02757
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eb29c20d8ba51e749550e59c4a0cb07062fda7e0a5a6405d59295cd54e4b5225
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2026-01-16T00:00:00-05:00
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Bosonization of Noise Effects in Nonlocal Quantum Dynamics
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arXiv:2504.20891v2 Announce Type: replace-cross Abstract: Quantum systems that interact non-locally with an environment are paradigms for exploring collective phenomena. They naturally emerge in various physical contexts involving long-range, many-body interactions. We consider a general class of such open systems characterized by a coupling to the environment which is inversely proportional to the square root of the environment size. We show that the induced system dynamics has a universal bosonic nature: the same evolution arises from coupling the system to a collection of noninteracting bosonic modes, independently of the microscopic structure of the original environment. This emergent "bosonization" of the environment's influence results from the scaling of the coupling in the thermodynamic limit and is a manifestation of the quantum central limit theorem. While the effect has been observed in specific models before, we show that it is, in fact, a universal feature.
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https://arxiv.org/abs/2504.20891
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7f691fc6913213ae21f3559380e7a7a82775318be041956d5e39e878f52ed18c
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2026-01-16T00:00:00-05:00
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Non-uniqueness of the steady state for run-and-tumble particles with a double-well interaction potential
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arXiv:2510.07212v3 Announce Type: replace-cross Abstract: We study $N$ run-and-tumble particles (RTPs) in one dimension interacting via a double-well potential $W(r)=-k_0 \, r^2/2+g \, r^4/4$, which is repulsive at short interparticle distance $r$ and attractive at large distance. At large time, the system forms a bound state where the density of particles has a finite support. We focus on the determination of the total density of particles in the stationary state $\rho_s(x)$, in the limit $N\to+\infty$. We obtain an explicit expression for $\rho_s(x)$ as a function of the ''renormalized" interaction parameter $k=k_0-3m_2$ where $m_2$ is the second moment of $\rho_s(x)$. Interestingly, this stationary solution exhibits a transition between a connected and a disconnected support for a certain value of $k$, which has no equivalent in the case of Brownian particles. Analyzing in detail the expression of the stationary density in the two cases, we find a variety of regimes characterized by different behaviors near the edges of the support and around $x=0$. Furthermore, we find that the mapping $k_0\to k$ becomes multi-valued below a certain value of the tumbling rate $\gamma$ of the RTPs for some range of values of $k_0$ near the transition, implying the existence of two stable solutions. Finally, we show that in the case of a disconnected support, it is possible to observe steady states where the density $\rho_s(x)$ is not symmetric. All our analytical predictions are in good agreement with numerical simulations already for systems of $N = 100$ particles. The non-uniqueness of the stationary state is a particular feature of this model in the presence of active (RTP) noise, which contrasts with the uniqueness of the Gibbs equilibrium for Brownian particles. We argue that these results are also relevant for a class of more realistic interactions with both an attractive and a repulsive part, but which decay at infinity.
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https://arxiv.org/abs/2510.07212
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5072ba010aa12a1dbb87d15391aaa422e00ddbca7425005bfb5817cb0c1975ac
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2026-01-16T00:00:00-05:00
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Uniqueness of invariant measures as a structural property of markov kernels
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arXiv:2601.04900v2 Announce Type: replace-cross Abstract: We identify indecomposability as a key measure-theoretic underlying uniqueness of invariant probability measures for discrete-time Markov kernels on general state spaces. The argument relies on the mutual singularity of distinct invariant ergodic measures and on the observation that uniqueness follows whenever all invariant probability measures are forced to charge a common reference measure. Once existence of invariant probability measures is known, indecomposability alone is sufficient to rule out multiplicity. On standard Borel spaces, this viewpoint is consistent with the classical theory: irreducibility appears as a convenient sufficient condition ensuring indecomposability, rather than as a structural requirement for uniqueness. The resulting proofs are purely measure-theoretic and do not rely on recurrence, regeneration, return-time estimates, or regularity assumptions on the transition kernel.
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https://arxiv.org/abs/2601.04900
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17dcb4b790c1b878383553d9a0f657f49cf9590d1e9ad33daeec319ab1c205dd
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2026-01-16T00:00:00-05:00
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A Nonlinear Mechanism for Transient Anomalous Diffusion
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arXiv:2601.08083v2 Announce Type: replace-cross Abstract: Diffusion is a fundamental physical phenomenon with critical applications in fields such as metallurgy, cell biology, and population dynamics. While standard diffusion is well-understood, anomalous diffusion often requires complex non-local models. This paper investigates a nonlinear diffusion equation where the diffusion coefficient is linearly dependent on concentration. We demonstrate through a perturbative analysis that this physically-grounded model exhibits transient anomalous diffusion. The system displays a clear crossover from an initial subdiffusive regime to standard Fickian behavior at long times. This result establishes an important mechanism for trasient anomalous diffusion that arises purely from local interactions, providing an intuitive alternative to models based on fractional calculus or non-local memory effects.
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https://arxiv.org/abs/2601.08083
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6da3704aa9c1887720e23d341dbbb03d8671e0ae8644d1e3c4875d7cd30873af
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2026-01-16T00:00:00-05:00
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Bogomol'nyi Equations in Two-Species Born--Infeld Theories Governing Vortices and Antivortices
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arXiv:2601.09091v2 Announce Type: replace-cross Abstract: We derive several new Bogomol'nyi (self-dual) equations in two-species $U(1)\times U(1)$ gauge theories governed by the Born--Infeld nonlinear electrodynamics. By identifying appropriate Born--Infeld type Higgs potentials, we show that the highly nonlinear energy functionals admit exact topological lower bounds saturated by coupled first-order equations. The resulting models accommodate both vortex-vortex and vortex-antivortex configurations and generalize previously known single-species Born--Infeld systems to interacting multi-component settings. Beyond the derivation of the Bogomol'nyi equations, we develop an exact thermodynamic theory for pinned multivortex configurations in both the full plane and compact doubly periodic domains. Owing to the linear dependence of the Bogomol'nyi energy spectrum on topological charges, we obtain closed-form expressions for the canonical partition function, internal energy, heat capacity, and magnetization. In compact domains, the Bradlow type geometric bounds constrain admissible vortex numbers and lead to qualitatively new high-temperature behavior. In particular, vortex-only systems exhibit spontaneous magnetization, while vortex-antivortex systems do not, reflecting the underlying symmetry between opposite topological charges. These results provide a rare analytically solvable framework for studying thermodynamics in nonlinear multi-component gauge theories regulated by the Born--Infeld electrodynamics.
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https://arxiv.org/abs/2601.09091
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8b8da83d6f26e68203b6cf3f02e34237a0f926426f240008f2fe426fbfd48798
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2026-01-16T00:00:00-05:00
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Habitable Worlds Observatory (HWO): Living Worlds Community Working Group: The Search for Life on Potentially Habitable Exoplanets
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arXiv:2601.09766v1 Announce Type: new Abstract: The discovery of a biosphere on another planet would transform how we view ourselves, and our planet Earth, in relation to the rest of the cosmos. We now know Earth is one planet among eight circling our sun; our sun is part of a swirling galaxy of over one hundred billion other suns; and our galaxy is one of untold billions in the universe. While we do not yet know how many, if any, other biospheres exist on the countless worlds orbiting countless other suns, we stand at the precipice of a new era of discovery, enabled by powerful new facilities able to peer across the light years into the atmospheres of planets similar to our own. This article is an adaptation of a science case document (SCDD) developed for the NASA Astrophysics Flagship mission the Habitable Worlds Observatory (HWO) Science, Technology, and Architecture Review Team (START) Living Worlds Community Working Group.
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https://arxiv.org/abs/2601.09766
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87cb72efd794d5d6ebc4eb73f2717cb1a06403cc61ff08dd10417e380358e0d4
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2026-01-16T00:00:00-05:00
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The X-Ray Dot: Exotic Dust or a Late-Stage Little Red Dot?
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arXiv:2601.09778v1 Announce Type: new Abstract: JWST's "Little Red Dots" (LRDs) are increasingly interpreted as active galactic nuclei (AGN) obscured by dense thermalized gas rather than dust as evidenced by their X-ray weakness, blackbody-like continua, and Balmer line profiles. A key question is how LRDs connect to standard UV-luminous AGN and whether transitional phases exist and if they are observable. We present the "X-Ray Dot" (XRD), a compact source at $z=3.28$ observed by the NIRSpec WIDE GTO survey. The XRD exhibits LRD hallmarks: a blackbody-like ($T_{\rm eff} \simeq 6400\,$K) red continuum, a faint but blue rest-UV excess, falling mid-IR emission, and broad Balmer lines ($\rm FWHM \sim 2700-3200\,km\,s^{-1}$). Unlike LRDs, however, it is remarkably X-ray luminous ($L_\textrm{2$-$10$\,$keV} = 10^{44.18}\,$erg$\,$s$^{-1}$) and has a continuum inflection that is bluewards of the Balmer limit. We find that the red rest-optical and blue mid-IR continuum cannot be reproduced by standard dust-attenuated AGN models without invoking extremely steep extinction curves, nor can the weak mid-IR emission be reconciled with well-established X-ray--torus scaling relations. We therefore consider an alternative scenario: the XRD may be an LRD in transition, where the gas envelope dominates the optical continuum but optically thin sightlines allow X-rays to escape. The XRD may thus provide a physical link between LRDs and standard AGN, offering direct evidence that LRDs are powered by supermassive black holes and providing insight into their accretion properties.
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https://arxiv.org/abs/2601.09778
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4d89f0c1ac099f6b8c01b86ac78ad1a22051a970b852c4db08ab78d3938c6ed7
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2026-01-16T00:00:00-05:00
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Breathless BEARS: [O$_{\rm \,III}$] 88$\mu$m Emission of Dusty Star-Forming Galaxies at $z = 3-4$
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arXiv:2601.09780v1 Announce Type: new Abstract: We present [O$_{\rm \,III}$] 88$\mu$m observations towards four ${\it Herschel}$-selected dusty star-forming galaxies (DSFGs; log$_{10}$ $\mu$L$_{\rm IR}$/L$_{\odot}$ = 13.5 - 14 at $z = 2.9 - 4$) using the Atacama Compact Array (ACA) in Bands 9 and 10. We detect [O$_{\rm \,III}$] emission in all four targets at >3$\sigma$, finding line luminosity ratios ($L_{\rm [O_{\rm \,III}]}$ / L$_{\rm IR}$ = 10$^{-4.2}$ to 10$^{-3}$) similar to local spiral galaxies, and an order of magnitude lower when compared with local dwarf galaxies as well as high-redshift Lyman-break galaxies. Using the short-wavelength capabilities of the ACA, these observations bridge the populations of galaxies with [O$_{\rm \,III}$] emission at low redshift from space missions and at high redshift from ground-based studies. The difference in [O$_{\rm \,III}$] emission between these DSFGs and other high-redshift galaxies reflects their more evolved stellar populations (> 10 Myr), larger dust reservoirs (M$_{\rm dust}$ $\sim$ 10$^{9 - 11}$ M$_{\odot}$), metal-rich interstellar medium ($Z \sim 0.5 - 2$ Z$_{\odot}$), and likely weaker ionization radiation fields. Ancillary [C$_{\rm \,II}$] emission on two targets provide $L_{[{\rm O}_{\rm \,III}]} / L_{[{\rm C}_{\rm \,II}]}$ ratios at 0.3 - 0.9, suggesting that ionized gas represents a smaller fraction of the total gas reservoir in DSFGs, consistent with theoretical models of DSFGs as transitional systems between gas-rich, turbulent disks and more evolved, gas-poor galaxies. Expanding samples of DSFGs with [O$_{\rm \,III}$] emission will be key to place this heterogeneous, poorly-understood galactic phase in its astrophysical context.
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https://arxiv.org/abs/2601.09780
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Academic Papers
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d75443754d8bd26d70d2a9e0113bf1118aa678df134415598b06c745cf6db6a9
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2026-01-16T00:00:00-05:00
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New Hard X-Ray and Multiwavelength Study of the PeVatron Candidate PWN G0.9+0.1 in the Galactic Center Region
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arXiv:2601.09788v1 Announce Type: new Abstract: We present a new X-ray study and multiwavelength spectral energy distribution (SED) modeling of the young pulsar wind nebula (PWN) powered by the energetic pulsar PSR J1747-2809, inside the composite supernova remnant (SNR) G0.9+0.1, located in the Galactic Center region. The source is detected by NuSTAR up to 30 keV with evidence for the synchrotron burnoff effect in the changing spatial morphology with increasing energy. The broadband 2-30 keV spectrum of PWN G0.9+0.1 is modeled by a single power law with photon index $\Gamma=2.11\pm0.07$. We combined the new X-ray data with the multiwavelength observations in radio, GeV, and TeV gamma rays and modeled the SED, applying a one-zone and a multi-zone leptonic model. The comparison of the models is successful, as we obtained physically compatible results in the two cases. Through the one-zone model, we constrain the age of the system to $\sim2.2$ kyr, as well as reproduce the observed PWN and SNR radio sizes. In both the one-zone and multi-zone leptonic models, the electron injection spectrum is well-described by a single power law with spectral index $p \sim 2.6$ and a maximum electron energy of $\sim2$ PeV, suggesting the source could be a leptonic PeVatron candidate. We estimate the average magnetic field to be $B_{\rm PWN} \sim 20\ \mu$G. We also report the serendipitous NuSTAR detection of renewed X-ray activity from the very faint X-ray transient XMMU J174716.1-281048 and characterize its spectrum.
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https://arxiv.org/abs/2601.09788
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Academic Papers
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bb3243313a9ee130e0e065ef9acb5eca628647150f5afdde96403813e8834bbe
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2026-01-16T00:00:00-05:00
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Growing in number, passive in nature: tracing the evolution of the most massive quiescent galaxies since z ~ 0.8 with BOSS and DESI
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arXiv:2601.09789v1 Announce Type: new Abstract: Luminous Red Galaxies (LRGs) are among the most massive galaxies at any epoch, and lack ongoing star formation. As systems hosting most of the baryonic mass in the local Universe, they preserve imprints of the quenching mechanisms in the early Universe. We exploited the large BOSS and DESI spectroscopic datasets to perform the first homogeneous and continuous mapping of the evolution of stellar population properties of a complete sample of the most massive LRGs ($\log (M_*/\mathrm{M_\odot})> 11.5$) at 0.15 < z < 0.8. By consistently fitting the same spectral indices at all redshifts, we measured trends of [Fe/H], [alpha/Fe], and light-weighted age as a function of redshift. These galaxies exhibit a passive light-weighted age evolution and flat [Fe/H] and [alpha/Fe] trends towards lower redshift, indicating genuinely passive evolution. These trends are robust against the choice of stellar population models and analysis assumptions, and they support the predictions from IllustrisTNG, which predict negligible chemical evolution for the most massive quenched systems at z < 0.8. Our results suggest that, despite nearly 5 Gyr of cosmic time and a 3-4x increase in number density, the stellar population properties of massive quiescent galaxies remain essentially unchanged since z ~ 0.8. This shows a negligible progenitor bias below z ~ 0.8, and a genuinely passive evolution. Newly added systems after $z \sim 0.8$ were already largely quenched and chemically mature, while subsequent evolution was dominated by dry mergers without altering the bulk of the stellar populations.
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https://arxiv.org/abs/2601.09789
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Academic Papers
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svg
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f3f0a4bc1a24967186512aa9ff1746d6757dbb7ff4d41b2203a963b6fbd2c73e
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2026-01-16T00:00:00-05:00
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The influence of magnetic fields in Cloud-Cloud Collisions
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arXiv:2601.09794v1 Announce Type: new Abstract: Cloud-cloud collisions are expected to trigger star formation by compressing gas into dense, gravitationally unstable regions. However, the role of magnetic fields in this process is unclear. We use SPH to model head-on collisions between two uniform density clouds, each with mass $500 \,$M$_{\odot}$, initial radius 2 pc, and embedded in a uniform magnetic field parallel to the collision velocity. As in the nonmagnetic case, the resulting shock-compressed layer fragments into a network of filaments. If the collision is sufficiently slow, the filaments are dragged into radial orientations by non-homologous gravitational contraction, resulting in a $\textit{Hub Filament}$ morphology, which spawns a centrally concentrated monolithic cluster with a broad mass function shaped by competitive accretion and dynamical ejections. If the collision is faster, a $\textit{Spiders Web}$ of intersecting filaments forms, and star-systems condense out in small subclusters, often at the filament intersections; due to their smaller mass reservoirs, and the lower probability of dynamical ejection, the mass function of star-systems formed in these subclusters is narrower. Magnetic fields affect this dichotomy quantitatively by delaying collapse and fragmentation. As a result, the velocity threshold separating $\textit{Hub Filament}$ and $\textit{Spiders Web}$ morphologies is shifted upward in magnetised runs, thereby enlarging the parameter space in which $\textit{Hub Filament}$ morphologies form, and enhancing the likelihood of producing centrally concentrated clusters. Consequently, magnetic fields regulate both the morphology and timing of star formation in cloud-cloud collisions: they broaden filaments, delay the onset of star formation, and promote the formation of $\textit{Hub Filament}$ morphologies, monolithic clusters and high-mass star-systems.
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https://arxiv.org/abs/2601.09794
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Academic Papers
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