math updates on arXiv.org

On a square packing conjecture of Erd\H{o}s

Anshul Raj Singh — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22163v1 Announce Type: new Abstract: Let $f(n)$ be the maximum sum of the sides of non-overlapping squares (or equilateral triangles) packed inside a unit square or (unit equilateral triangle). In this paper, we explore some properties of $f$ and examine how the square and triangle cases are similar. We prove that a conjecture of Erd\H{o}s, which says that $f(k^2+1) = k$ for all $k$, is equivalent to the convergence of the series $\sum_{k\geqslant 1}(f(k^2+1)-k)$. We also explore the case of parallelograms and discuss how that is similar to the case of unit square and triangle.

Some new results on the Seidel energy of graphs with self-loops

Kalpesh M. Popat, Irena M. Jovanovic — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22165v1 Announce Type: new Abstract: Harshitha et al. recently introduced Seidel energy of graphs with self loops. In this paper, we extend some of their results by giving a necessary and sufficient condition for the Seidel energy of a looped graph to be equal to the Seidel energy of its underlying graph. We also consider Seidel energy of the union of certain graphs, and show that graph operations complement and Seidel switching preserve Seidel energy in the looped setting.

Large Language Models: A Mathematical Formulation

Ricardo Baptista, Andrew Stuart, Son Tran — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22170v1 Announce Type: new Abstract: Large language models (LLMs) process and predict sequences containing text to answer questions, and address tasks including document summarization, providing recommendations, writing software and solving quantitative problems. We provide a mathematical framework for LLMs by describing the encoding of text sequences into sequences of tokens, defining the architecture for next-token prediction models, explaining how these models are learned from data, and demonstrating how they are deployed to address a variety of tasks. The mathematical sophistication required to understand this material is not high, and relies on straightforward ideas from information theory, probability and optimization. Nonetheless, the combination of ideas resting on these different components from the mathematical sciences yields a complex algorithmic structure; and this algorithmic structure has demonstrated remarkable empirical successes. The mathematical framework established here provides a platform from which it is possible to formulate and address questions concerning the accuracy, efficiency and robustness of the algorithms that constitute LLMs. The framework also suggests directions for development of modified and new methodologies.

Pseudo-Riemannian Spectral Triples for $\mathrm{SU}(1,1)$

Jort de Groot — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22171v1 Announce Type: new Abstract: We use the harmonic analysis of $\mathrm{SU}(1,1)$ to show that the triple $(\mathcal{A},\mathcal{H},D)$, with $D$ (the closure of) Kostant's cubic Dirac operator acting on the Hilbert space $\mathcal{H}=L^2(\mathrm{SU}(1,1))\otimes\mathbb{C}^2$, and with $*$-algebra $\mathcal{A}=C^\infty_c(\mathrm{SU}(1,1))\otimes 1$, forms both a pseudo-Riemannian spectral triple in the sense of Van den Dungen, Paschke and Rennie, and an indefinite spectral triple in the sense of Van den Dungen and Rennie.

On the $L^p$-Convergence and Denoising Performance of Durrmeyer-Type Max-Min Neural Network Operators

Berke \c{S}ahin, \.Ismail Aslan — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22174v1 Announce Type: new Abstract: In this paper, we investigate Durrmeyer-type generalizations of maximum-minimum neural network operators. The primary objective of this study is to establish the convergence of these operators in the $L^{p}$ norm for functions $f\in L^{p}([a,b],[0,1])$ with $1\leq p<\infty$. To this end, we analyze the properties of sigmoidal functions and maximum-minimum operations, subsequently establishing the convergence of the proposed operator in pointwise, supremum, and $L^{p}$ norms. Furthermore, we derive quantitative estimates for the rates of convergence. In the applications section, numerical and graphical examples demonstrate that the proposed Durrmeyer-type operators provide smoother approximations compared to Kantorovich-type and standard max-min operators. Finally, we highlight the superior filtering performance of these operators in signal analysis, validating their effectiveness in both approximation and data processing tasks.

Proliferating series by Jean Barraqu\'e: a study and classification in mathematical terms

Isabel Tard\'on, Pablo Mart\'in-Santamar\'ia — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22176v1 Announce Type: new Abstract: Barraqu\'e's proliferating series give an interesting turn on the concept of classic serialism by creating a new invariant when it comes to constructing the series: rather than the intervals between consecutive notes, what remains unaltered during the construction of the proliferations of the given base series is the permutation of the notes which happens between two consecutive series, that is to say, the transformation of the order of the notes in the series. This presents new possibilities for composers interested in the serial method, given the fact that the variety of intervals obtained by this method is far greater than that of classic serialism. In this manuscript, we will study some unexplored possibilities that the proliferating series offer from a mathematical point of view, which will allow composers to gain much more familiarity with them and potentially result in the creation of pieces that take serialism to the next level.

A root finding method with arbitrary order of convergence

Alois Schiessl — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22187v1 Announce Type: new Abstract: Let $a\in \mathbb{R}^{+}\backslash\left\{0\right\}$ and $M\in\mathbb{N}$. We consider the equation $t^M-a=0$, which is equivalent to $1-\frac{t^M}{a}=0\,.$ The real solution is $\sqrt[M]{a}$. In this publication, we present a method that enables the calculation of $\sqrt[M]{a}$ with arbitrary order of convergence using only polynomials. We define the fixed point function \[ F\left(x\right) =\prod_{\ell=1}^{P}\left(1+\frac{1}{\ell\cdot M}\right) \int\limits_{0}^{x}\!\left(1-{\frac{{t}^{M}}{a}}\right)^{P}{\rm d}t =\sum\limits_{k=0}^{P}\frac{\left(-1\right)^{\,k}}{a^{\,k}}\cdot\binom{P}{k}\cdot\frac{x^{\,k\,\cdot M+1}}{k\,\cdot M+1} \] This is a polynomial of degree $\left(P\cdot M+1\right)$ with $\left(P+1\right)$ terms. The calculation of $\sqrt[M]{a}$ is thus reduced to a polynomial evaluation. The computational tests we performed demonstrate the efficiency of the method. -- Es sei $a\in \mathbb{R}^{+}\backslash\left\{0\right\}$ und $M\in\mathbb{N}$. Vorgelegt ist die Gleichung $t^M-a=0$, die \"aquivalent zu $1-\frac{t^M}{a}=0$ ist. Die reelle L\"osung hiervon ist $\sqrt[M]{a}$. In dieser Ver\"offentlichung stellen wir ein Verfahren vor, das die Berechnung von $\sqrt[M]{a}$ mit beliebiger Konvergenzordnung erm\"oglicht und nur Polynome verwendet. Wir definieren die Fixpunktfunktion \[F\left(x\right) =\prod_{\ell=1}^{P}\left(1+\frac{1}{\ell\cdot M}\right) \int\limits_{0}^{x}\!\left(1-{\frac{{t}^{M}}{a}}\right)^{P}{\rm d}t =\sum\limits_{k=0}^{P}\frac{\left(-1\right)^{\,k}}{a^{\,k}}\cdot\binom{P}{k}\cdot\frac{x^{\,k\,\cdot M+1}}{k\,\cdot M+1} \] Das ist ein Polynom vom Grad $\left(P\cdot M+1\right)$ mit $\left(P+1\right)$ Summanden. Anhand ausgew\"ahlter Beispiele von Wurzelberechnungen zeigen wir die Effizienz des Verfahrens.

When is the convolution a t-norm on normal, convex and upper semicontinuous fuzzy truth values?

Jie Sun — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22190v1 Announce Type: new Abstract: In Type-2 rule-based fuzzy systems (T2 RFSs), triangular norms on complete lattice $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$ can be used to model the compositional rule of inference, where $\textbf{L}$ is the set of all convex normal fuzzy truth values, $\mathbf{L_u}$ is the set of all convex normal and upper semicontinuous fuzzy truth values, and $\sqsubseteq$ is the so-called convolution order. Hence, the choice of t-norms on $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$ may influence the performance of T2 RFSs, and thus, it is significant to broad the set of t-norms among which domain experts can choose most suitable one. To construct t-norms on $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$, the mainstream method is based on convolution $\ast_\vartriangle$ induced by two operators $\ast$ and $\vartriangle$ on the unit interval $[0,1]$. Recently, we have complete solve the question when convolution $\ast_\vartriangle$ is a t-norm on $(\mathbf{L},\sqsubseteq)$. This paper aim to provide the necessary and sufficient conditions under which convolution $\ast_\vartriangle$ is a t-norm on $(\mathbf{L_u}, \sqsubseteq)$.

Transitive Sets of Mutually Orthogonal Latin Squares

Amadou Keita, Ilya Shapiro — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22205v1 Announce Type: new Abstract: We investigate MacNeish's conjecture (known to be false in general) in the setting of what we call "transitive" Mutually Orthogonal Latin Squares (MOLS). When we restrict our attention to "simply transitive" MOLS, we find that the conjecture holds. We provide some partial results towards the transitive case, as well as the outcome of a computer search, which introduces a new construction of MOLS. In particular, we were unable to find any transitive large (conjecture-violating) sets of MOLS in the literature.

Zero-level integrable modules over twisted affine Lie superalgebras

Hajar Kiamehr, Senapathi Eswara Rao, Malihe Yousofzadeh — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22210v1 Announce Type: new Abstract: The main result of this paper is the characterization of zero-level integrable finite weight modules, over twisted affine Lie superalgebras. We prove that such a module is parabolically induced from a module which is obtained, in a prescribed way, from a module over a Lie algebra $\mathscr{L}$ which is either a $\bbbz$-graded abelian Lie algebra or a direct sum of a $\bbbz$-graded abelian Lie algebra and the so-called quadratic Lie superalgebra $\mathcal{Q}$. We give also a complete characterization of both finite dimensional simple $\mathcal{Q}$-modules as well as bounded finite weight $\bbbz$-graded simple $\mathcal{Q}$-modules.

The metaplectic semigroup and its applications to time-frequency analysis and evolution operators

Gianluca Giacchi, Luigi Rodino, Davide Tramontana — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22252v1 Announce Type: new Abstract: We develop a systematic analysis of the metaplectic semigroup $\mathrm{Mp}_+(d,\mathbb{C})$ associated with positive complex symplectic matrices, a notion introduced almost simultaneously and independently by H\"ormander, Brunet, Kramer, and Howe, thereby extending the classical metaplectic theory beyond the unitary setting. While the existing literature has largely focused on propagators of quadratic evolution equations, for which results are typically obtained via Mehler formulas, our approach is operator-theoretic and symplectic in spirit and adapts techniques from the standard metaplectic group $\mathrm{Mp}(d,\mathbb{R})$ to a substantially broader framework that is not driven by differential problems or particular propagators. This point of view provides deeper insight into the structure of the metaplectic semigroup, and allows us to investigate its generators, polar decomposition, and intertwining relations with complex conjugation and with the Wigner distribution. We then exploit these structural results to characterize, from a metaplectic perspective, classes of time-frequency representations satisfying prescribed structural properties. Finally, we discuss further implications for parabolic equations with complex quadratic Hamiltonians, we study the boundedness of their propagators on modulation spaces, we obtain estimates in time of their operator norms. Finally, we apply our theory to the study of propagation of Wigner singularities.

Smooth correspondences between quiver varieties

Nicolle Gonz\'alez, Eugene Gorsky, Jos\'e Simental — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22287v1 Announce Type: new Abstract: We introduce a new class of smooth correspondences between Nakajima quiver varieties called split parabolic quiver varieties, and study their properties. We use these correspondences to construct an explicit resolution of singularities of quiver Brill--Noether loci and prove that the latter are irreducible and Cohen-Macaulay of expected dimension (if non-empty). This generalizes the results of Nakajima--Yoshioka and Bayer--Chen--Jiang for Hilbert schemes of points on surfaces.

Forecasting in the presence of scale-free noise

Serhii Kryhin, Tatiana Mouzykantskii, Vivishek Sudhir — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22294v1 Announce Type: new Abstract: The extraction of signals from noise is a common problem in all areas of science and engineering. A particularly useful version is that of forecasting: determining a causal filter that estimates a future value of a hidden process from past observations. Current techniques for deriving the filter require that the noise be well described by rational power spectra. However, scale-free noises, whose spectra scale as a non-integer power of frequency, are ubiquitous in practice. We establish a method, together with performance guarantees, that solves the forecasting problem in the presence of scale-free noise. Via the duality between estimation and control, our technique can be used to design control for distributed systems. These results will have wide-ranging applications in neuroscience, finance, fluid dynamics, and quantum measurements.

Operating Imperfect AI: Reliability Drift and Human Congestion

Ziyao Wang, Svetlozar T Rachev — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22295v1 Announce Type: new Abstract: The deployment of machine learning in high-stakes services relies on ``human-in-the-loop'' architectures to mitigate algorithmic uncertainty. However, existing static policies fail to address a fundamental tension: algorithms suffer from stochastic ``reliability drift,'' while human override capacity is scarce and congestible. We formulate the management of such systems as a dynamic queueing control problem. The system state is defined by the tuple (queue backlog, reliability regime), and the control variable is a state-dependent risk threshold. We prove that the optimal escalation policy is driven by the endogenous ``Shadow Price of Capacity.'' We establish two key structural monotonicity results: (i) Congestion Shedding, where the threshold rises with backlog to sacrifice marginal accuracy for responsiveness; and (ii) Safety Buffering, where the threshold lowers during drift to use the queue as a ``risk capacitor.'' Furthermore, we identify a critical ``Capacity Phase Transition'' in the arrival-drift parameter space, beyond which no policy can maintain safety standards without causing structural system failure (infinite queues). Our results provide rigorous operational rules for managing the interface between imperfect algorithms and congested experts.

The Homology of Complex Equivariant Bordism

Julius Groenjes — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22303v1 Announce Type: new Abstract: Let $A$ be an abelian compact Lie group and let $E$ be an oriented $A$-spectrum. We compute the $E$-homology of tom Dieck's homotopical $A$-equivariant complex bordism spectrum $MU_A$ in two ways, correcting an error in Cole-Greenlees-Kriz (2002). Additionally, we calculate the $E$-homology of the geometric $A$-equivariant complex bordism spectrum $mU_A$.

Geometric configuration of integrally closed Noetherian domains

Gyu Whan Chang, Giulio Peruginelli — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22314v1 Announce Type: new Abstract: In this paper, we completely describe the family of integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. We accomplish this result by classifying the Krull domains between these two polynomial rings. To this end, we first describe the DVRs of $\mathbb{Q}(X)$ lying over $\mathbb{Z}_{(p)}$ for some prime $p \in \mathbb{Z}$, by distinguishing them according to whether the extension of the residue fields is algebraic or transcendental. We unify the known descriptions of such valuations by considering ultrametric balls in $\mathbb{C}_p$, the completion of the algebraic closure of the field $\mathbb{Q}_p$ of $p$-adic numbers. We then study when the intersection $R$ of such DVRs with $\mathbb{Q}[X]$ is of finite character, so that $R$ is a Krull domain, and we finally compute the divisor class group of $R$. It turns out that such a ring is formed by those polynomials which simultaneously map a finite union of ultrametric balls of $\mathbb{C}_p$ to its valuation domain $\mathbb{O}_p$, as $p\in\mathbb{Z}$ ranges through the set of primes. By a result of Heinzer, the Krull domains of this class are precisely the integrally closed Noetherian domains between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$. This novel approach provides a geometric understanding of this class of integrally closed domains. Furthermore, we also describe the UFDs between $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.

A Generalized Analytical Heat Transfer Model for Enhanced Geothermal Systems: Capturing Fracture Interactions and Correcting Classical Optimistic Predictions

Nelson Barros-Galvis or Christine Ehlig-Economides or Cristi Darley Guevara — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22316v1 Announce Type: new Abstract: Numerical analytical heat transfer models play a critical role in geothermal design and feasibility studies. Classical solutions, such as those proposed by Gringarten et al. 1975, rely on simplified assumptions and systematically overestimate thermal performance, which can lead to unrealistic engineering decisions. This study presents a generalized analytical model for enhanced geothermal systems that explicitly captures thermal interactions between fractures while preserving analytical tractability. The formulation is based on Green\'s functions and reproduces realistic thermal behavior under conditions representative of fractured geothermal reservoirs. The resulting solution is computationally efficient and sufficiently simple to be implemented directly in standard spreadsheets, without requiring Laplace space transformations or numerical inversion algorithms. The model is validated against numerical simulations performed using CMG STARS and Volsung software, showing close agreement in temperature evolution, including the effects of interacting fractures. Compared with classical analytical approaches, the proposed model corrects optimistic bias and provides more reliable predictions of production temperature and energy recovery. These results have direct implications for geothermal feasibility studies, well design, and power forecasting, effectively bridging the gap between legacy analytical models and numerical or commercial engineering tools. Building on the analytical framework originally introduced by Gringarten et al. 1975, the proposed formulation generalizes classical heat transfer solutions to account for fracture interaction while retaining analytical simplicity and practical applicability.

Quaternionic Perfect Sequences and Hadamard Matrices

Aidan Bennett, Curtis Bright, Paul Colinot, Ashwin Nayak — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22337v1 Announce Type: new Abstract: A finite sequence of numbers is perfect if it has zero periodic autocorrelation after a nontrivial cyclic shift. In this work, we study quaternionic perfect sequences having a one-to-one correspondence with the binary sequences arising in Williamson's construction of quaternion-type Hadamard matrices. Using this correspondence, we devise an enumeration algorithm that is significantly faster than previously used algorithms and does not require the sequences to be symmetric. We implement our algorithm and use it to enumerate all circulant and possibly non-symmetric Williamson-type matrices of orders up to 21; previously, the largest order exhaustively enumerated was 13. We prove that when the blocks of a quaternion-type Hadamard matrix are circulant, the blocks are necessarily pairwise amicable. This dramatically improves the filtering power of our algorithm: in order 20, the number of block pairs needing consideration is reduced by a factor of over 25,000. We use our results to construct quaternionic Hadamard matrices of interest in quantum communication and prove they are not equivalent to those constructed by other means. We also study the properties of quaternionic Hadamard matrices analytically, and demonstrate the feasibility of characterizing quaternionic Hadamard matrices with a fixed pattern of entries. These results indicate a richer set of properties and suggest an abundance of quaternionic Hadamard matrices for sufficiently large orders.

Convergence Analysis of the Discrete Constrained Saddle Dynamics and Their Momentum Variants

Qiang Du, Baoming Shi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22341v1 Announce Type: new Abstract: We study the discrete constrained saddle dynamics and their momentum variants for locating saddle points on manifolds. Under the assumption of exact unstable eigenvectors, we establish a local linear convergence of the discrete constrained saddle dynamics and show that the convergence rate depends on the condition number of the Riemannian Hessian. To mitigate this dependence, we introduce a momentum-based constrained saddle dynamics and prove local convergence of the continuous-time dynamics as well as the corresponding discrete scheme, which further demonstrates that momentum accelerates convergence, particularly in ill-conditioned settings. In addition, we show that a single-step eigenvector update is sufficient to guarantee local convergence; thus, the assumption of exact unstable eigenvectors is not necessary, which substantially reduces the computational cost. Finally, numerical experiments, including applications to the Thomson problem, the Rayleigh quotient on the Stiefel manifold, and the energy functional of Bose-Einstein condensates, are presented to complement the theoretical analysis.

Low-Rank Approximation by Randomly Pivoted LU

Marc Aur\`ele Gilles, Heather Wilber — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22344v1 Announce Type: new Abstract: The low-rank approximation properties of Randomly Pivoted LU (RPLU), a variant of Gaussian elimination where pivots are sampled proportional to the squared entries of the Schur complement, are analyzed. It is shown that the RPLU iterates converge geometrically in expectation for matrices with rapidly decaying singular values. RPLU outperforms existing low-rank approximation algorithms in two settings: first, when memory is limited, RPLU can be implemented with $\mathcal{O}(k^2 + m + n)$ storage and $\mathcal{O}( k(m + n)+ k\mathcal{M}(\mat{A}) + k^3)$ operations, where $\mathcal{M}(\mat{A})$ is the cost of a matvec with $\mat{A}\in\mathbb{C}^{n\times m}$ or its adjoint, for a rank-$k$ approximation. Second, when the matrix and its Schur complements share exploitable structure, such as for Cauchy-like matrices. The efficacy of RPLU is illustrated with several examples, including applications in rational approximation and solving large linear systems on GPUs.

Square Root-Factorized Covariance Steering

Naoya Kumagai, Kenshiro Oguri — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22348v1 Announce Type: new Abstract: Covariance steering (CS) synthesizes a control policy which drives the state's mean and covariance matrix towards desired values. Offering tractable computation of a closed-loop policy which can obey chance constraints in uncertain environments, application to many real-world control problems have been proposed. We consider the chance-constrained, discrete-time, linear time-varying CS with Gaussian noise. The contribution of this paper is a novel solution method for this problem, explicitly writing the propagation equations of the Cholesky factor of the state covariance matrix by using the QR decomposition. The use of the square-root form of covariance matrices brings two key benefits over other existing methods: (i) computational scalability and (ii) numerical reliability. (i) Compared to solution methods that require large block matrix formulations, the proposed method scales better with the growth in horizon length, shows better optimality, and uses memoryless state feedback. (ii) Compared to another class of methods that explicitly define the covariance matrix as variables, the proposed method allows flexible cost formulations and shows better numerical reliability when uncertainty terms are smaller than the mean. On the other hand, these benefits come with a minor drawback: the propagation equation of covariance square roots is non-convex, necessitating sequential convex programming to solve. However, this paper proves the global optimality of the proposed approach for CS without chance constraints. When chance constraints are present, the existing optimal CS formulation is also non-convex, and we prove that the proposed approach shares the same local minima. We verify the mathematical arguments via extensive numerical simulations.

Forward-KL Convergence of Time-Inhomogeneous Langevin Diffusions

Andreas Habring, Martin Zach — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22349v1 Announce Type: new Abstract: Many practical samplers rely on time-dependent drifts -- often induced by annealing or tempering schedules -- to improve exploration and stability. This motivates a unified non-asymptotic analysis of the corresponding Langevin diffusions and their discretizations. We provide a convergence analysis that includes non-asymptotic bounds for the continuous-time diffusion and its Euler--Maruyama discretization in the forward-Kullback--Leibler divergence under a single set of abstract conditions on the time-dependent drift. The results apply to many practically-relevant annealing schemes, including geometric tempering and annealed Langevin sampling. In addition, we provide numerical experiments comparing the annealing schemes covered by our theory in low- and high-dimensional settings.

Capacity of Two-User Wireless Systems Aided by Movable Signals

Matteo Nerini, Bruno Clerckx — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22358v1 Announce Type: new Abstract: Movable signals have emerged as a third approach to enable smart radio environments (SREs), complementing reconfigurable intelligent surfaces (RISs) and flexible antennas. This paper investigates their potential to enhance multi-user wireless systems. Focusing on two-user systems, we characterize the capacity regions of the multiple access channel (MAC) and broadcast channel (BC). Interestingly, movable signals can dynamically adjust the operating frequency to orthogonalize the user channels, thereby significantly expanding the capacity regions. We also study frequency optimization, constraining it in a limited frequency range, and show that movable signals provide up to 45% sum rate gain over fixed signals.

Hermitian indices and factorization of selfadjoint operators on a Kre\u{i}n space

Michael A. Dritschel, Alejandra Maestripieri, James Rovnyak — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22366v1 Announce Type: new Abstract: The hermitian indices of a selfadjoint operator $C$ on a Kre\u{i}n space $\mathcal H$ are defined as geometric measures of positivity and negativity of the operator. A different pair of indices arises in the Bogn\'ar-Kr\'amli factorization of $C$, which writes $C$ as a product $AA^*$ where $A$ acts on a Kre\u{i}n space $\mathcal A$ into $\mathcal H$ and has zero kernel; the new indices are the positive and negative indices of $\mathcal A$. Such factorizations are far from unique. When $\mathcal H$ is separable, it is known that the two notions of indices always coincide, and this has applications to index formulas in the theory of Julia operators and completion problems for operator matrices. A new proof of the equality of indices that does not require separability is given in this work.

Dynamical stability of various convex graphical translators

Junyoung Park — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22368v1 Announce Type: new Abstract: In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of graphical translators to mean curvature flow, namely the grim reaper, two dimensional graphical translators, and asymptotically cylindrical translators.

Operator Splitting with Hamilton-Jacobi-based Proximals

Nicholas Di, Eric C. Chi, Samy Wu Fung — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22370v1 Announce Type: new Abstract: Operator splitting algorithms are a cornerstone of modern first-order optimization, decomposing complex problems into simpler subproblems solved via proximal operators. However, most functions lack closed-form proximal operators, which has long restricted these methods to a narrow set of problems. Hamilton-Jacobi-based proximal operator (HJ-Prox) is a recent derivative-free Monte Carlo technique based on Hamilton-Jacobi PDE theory, that approximates proximal operators numerically. In this work, we introduce a unified framework for operator splitting via HJ-Prox, which allows for deployment of operator splitting even when functions are not proximable. We prove that replacing exact proximal steps with HJ-Prox in algorithms such as proximal point, proximal gradient descent, Douglas-Rachford splitting, Davis-Yin splitting, and primal-dual hybrid gradient preserves convergence guarantees under mild assumptions. Numerical experiments demonstrate HJ-Prox is competitive and effective on a wide variety of statistical learning tasks.

Visibility in Polygonal Environments with Holes: Finding Best Spots for Hiding and Surveillance

Neilabh Banzal, Jorge Cort\'es, Sonia Mart\'inez — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22405v1 Announce Type: new Abstract: Visibility plays an important role for decision making in cluttered, uncertain environments. This paper considers the problem of identifying optimal hiding spots for an agent against line-of-sight detection by an adversary whose location is unknown. We consider environments modeled as polygons with holes. We develop a set of mathematical tools for reasoning about visibility as a function of position and rely on non-smooth analysis to formally characterize the regularity properties of various visibility-based metrics. These metrics are non-smooth and non-convex, so off-the-shelf optimization algorithms can only guarantee convergence to Clarke critical points. To address this, the proposed Normalized Descent algorithm leverages the structure of non-smooth points in visibility problems and introduces randomness to escape saddle points. Our technical analysis allows for the non-monotonic decrease in the visibility metric and strengthens the algorithm guarantees, ensuring convergence to local minima with high probability. Simulations on two hide-and-seek scenarios showcase the effectiveness of the proposed approach.

Quasihomomorphisms to real algebraic groups

Sami Douba, Francesco Fournier-Facio, Sam Hughes, Simon Machado — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22411v1 Announce Type: new Abstract: A quasihomomorphism is a map that satisfies the homomorphism relation up to bounded error. Fujiwara and Kapovich proved a rigidity result for quasihomomorphisms taking values in discrete groups, showing that all quasihomomorphisms can be built from homomorphisms and sections of bounded central extensions. We study quasihomomorphisms with values in real linear algebraic groups, and prove an analogous rigidity theorem.

The Riemann Hypothesis in Oaxaca

Carlos Segovia — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22413v1 Announce Type: new Abstract: An equivalence of the Riemann Hypothesis (RH) enables a direct bridge to the Young lattice. In specific, the classical threshold $\lim_{n\to\infty} \sigma(n)/(n \log\log n) = e^{\gamma} \approx 1.78107$, derived from the asymptotic behavior of the sum-of-divisors function, can be realized combinatorially via limiting proportions associated to specific families of integer partitions.

Leader-Follower Linear-Quadratic Stochastic Graphon Games

Weijia Chen, Jingtao Shi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22429v1 Announce Type: new Abstract: This paper investigates leader-follower linear-quadratic stochastic graphon games, which consist of a single leader and a continuum of followers. The state equations of the followers interact through graphon coupling terms, with their diffusion coefficients depending on the state, the graphon aggregation term, and the control variables. The diffusion term of the leader's state equation depends on its state and control variables. Within this framework, a hierarchical decision-making structure is established: for any strategy adopted by the leader, the followers compete to attain a Nash equilibrium, while the leader optimizes its own cost functional by anticipating the followers' equilibrium response. This work develops a rigorous mathematical model for the game, proves the existence and uniqueness of solutions to the system's state equations under admissible control sets, and constructs a Stackelberg-Nash equilibrium for the continuum follower game. By employing the continuity method, we establish the existence, uniqueness, and stability of solutions to the associated forward-backward stochastic differential equation with a graphon aggregation term.

Selective Adaptation of Beliefs and Communication on Cellular Sheaves

Vicente Bosca, Robert Ghrist — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22431v1 Announce Type: new Abstract: We extend opinion dynamics on discourse sheaves to incorporate "directional stubbornness": agents may hold fixed positions in specified directions of their opinion stalk while remaining flexible in others. This converts the equilibrium problem from harmonic extension to a forced sheaf equation: the free-opinion component satisfies a sheaf Poisson equation with forcing induced by the clamped directions. We develop a parallel theory for "selective learning" of expression policies. When only a designated subset of incidence maps may adapt, the resulting gradient flow is sheaf diffusion on an auxiliary structure sheaf whose global sections correspond to sheaf structures making a fixed opinion profile publicly consistent. For joint evolution of beliefs and expressions, we give conditions (and regularized variants) guaranteeing convergence to nondegenerate equilibria, excluding spurious agreement via vanishing opinions or trivialized communication maps. Finally, we derive stagnation bounds in terms of the rate ratio between opinion updating and structural adaptation, quantifying when rapid rhetorical accommodation masks nearly unchanged beliefs, and conversely when flexible beliefs conform to rigid communication norms.

On the computability of cofinal Fra\"iss\'e limits

Nathanael Ackerman, Cameron Freer, Mostafa Mirabi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22435v1 Announce Type: new Abstract: For any collection of finite structures closed under isomorphism (i.e., an age) which has the Hereditary Property (HP), the Joint Embedding Property (JEP), and the Cofinal Amalgamation Property (CAP), there is a unique (up to isomorphism) countable structure which is cofinally ultrahomogeneous with the given age. Such a structure is called the cofinal Fra\"iss\'e limit of the age. In this paper, we consider the computational strength needed to construct the cofinal Fra\"iss\'e limit of a computable age. We show that this construction can always be done using the oracle 0''', and that there are ages that require 0''. In contrast, we show that if one assumes the strengthening of (CAP) known as the Amalgamation Property (AP), then the resulting limit, called the Fra\"iss\'e limit, can be constructed from the age using 0'. Our results therefore show that the more general case of cofinal Fra\"iss\'e limits requires greater computational strength than Fra\"iss\'e limits.

Divergence Identity for the scalar curvature and Rigidity of Codazzi Tensors

Xu Cheng, Andr\'es Lipa, Detang Zhou — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22437v1 Announce Type: new Abstract: We introduce a local vector field on an $n$-dimensional Riemannian manifold, defined as the sum of the covariant derivatives of a local orthonormal frame, and derive an explicit identity for its divergence, decomposed into a scalar curvature term and an auxiliary term involving connection coefficients. This result is applied to rigidity problems for Codazzi symmetric tensors. In particular, we give a new proof of a Tang-Yan theorem, which states that on a closed $n$-dimensional manifold with nonnegative scalar curvature, a smooth Codazzi symmetric tensor whose trace invariants up to order $n-1$ are constant must have constant eigenvalues. We also obtain further rigidity results under assumptions on elementary symmetric functions of the eigenvalues, with applications to the isoparametric rigidity of closed hypersurfaces in the unit sphere.

On Monogeneity of reciprocal polynomials

Rupam Barman, Anuj Narode, Vinay Wagh — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22453v1 Announce Type: new Abstract: Let $\mathbb{Z}_K$ denote the ring of integers of the number field $K = \mathbb{Q}(\theta)$, where $\theta$ is a root of the monic irreducible polynomial $f(x) \in \mathbb{Z}[x]$. We say that $f(x)$ is monogenic if $\mathbb{Z}_K = \mathbb{Z}[\theta]$. A polynomial $f(x) \in \mathbb{Z}[x]$ is called reciprocal if $f(x) = x^{\operatorname{deg}(f)} f(1/x)$. In this article, we derive sufficient conditions for the monogeneity of even degree reciprocal polynomials. By employing properties of the discriminant of reciprocal polynomials, we partially prove a conjecture proposed by Jones in $2021$. Furthermore, we establish a lower bound on the number of certain sextic monogenic reciprocal polynomials.

Compactification of Reductive Group Schemes

Ayan Nath — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22462v1 Announce Type: new Abstract: Let $\mathrm G$ be an isotrivial reductive group over a scheme $S$. We construct a smooth projective $S$-scheme containing $\mathrm G$ as a fiberwise-dense open subscheme equipped with left and right actions of $\mathrm G$ which extend the translation actions of $\mathrm G$ on itself. This verifies a conjecture of \v{C}esnavi\v{c}ius (arXiv:2201.06424). When $\mathrm G$ is adjoint, we recover fiberwise the wonderful compactification. Finally, we give an example of a non-isotrivial torus admitting no equivariant compactification.

The isomorphism problem for reduced finitary power monoids

Balint Rago — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22469v1 Announce Type: new Abstract: Let $H$ be a multiplicatively written monoid with identity $1_H$ and let $\mathcal{P}_{\text{fin},1}(H)$ denote the reduced finitary power monoid of $H$, that is, the monoid consisting of all finite subsets of $H$ containing $1_H$ with set multiplication as operation. Building on work of Tringali and Yan, we give a complete description of pairs of commutative and cancellative monoids $H,K$ for which $\mathcal{P}_{\text{fin},1}(H)$ and $\mathcal{P}_{\text{fin},1}(K)$ are isomorphic.

5G LDPC Codes as Root LDPC Codes via Diversity Alignment

Hyuntae Ahn, Inki Kim, Hee-Youl Kwak, Yongjune Kim, Chanki Kim, Sang-Hyo Kim — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22470v1 Announce Type: new Abstract: This paper studies the diversity of protographbased quasi-cyclic low-density parity-check (QC-LDPC) codes over nonergodic block-fading channels under iterative beliefpropagation decoding. We introduce diversity evolution (DivE), a Boolean-function-based analysis method that tracks how the fading dependence of belief-propagation messages evolves across decoding iterations. Under a Boolean approximation of block fading, DivE derives a Boolean fading function for each variable node (VN) output (i.e., the a-posteriori reliability after iterative decoding), from which the VN diversity order can be directly determined. Building on this insight, we develop a greedy blockmapping search that assigns protograph VNs to fading blocks so that all information VNs achieve full diversity, while including the minimum additional parity VNs when full diversity is infeasible at the nominal rate. Numerical results on the 5G New Radio LDPC codes show that the proposed search finds block mappings that guarantee full diversity for all information bits without modifying the base-graph structure, yielding a markedly steeper high-SNR slope and lower BLER than random mappings.

Tangents to Lipschitz and Sobolev images

Matthew Badger, Jared Krandel, Vyron Vellis — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22473v1 Announce Type: new Abstract: We develop geometric versions of Rademacher and Calderon type differentiability theorems in two categories. A special case of our results is that for any Lipschitz or continuous $W^{1,p}$ Sobolev map $f$ from $[0,1]^n$ into a Euclidean space with $p>n$, the image $f([0,1]^n)$ has a unique tangent set (Attouch-Wets convergence) at almost every point with respect to the $n$-dimensional Hausdorff measure. In the analogous case when $f$ is a continuous $N^{1,p}$ map from $[0,1]^n$ into a metric space, we show that the image $f([0,1]^n)$ has a unique metric tangent (Gromov-Hausdorff convergence) almost everywhere. These results complement, but are distinct from Federer's theorem on existence and uniqueness of approximate tangents of $n$-rectifiable sets in $\mathbb{R}^d$. We show that approximate tangents to Sobolev images can be upgraded to Attouch-Wets or Gromov-Hausdorff tangents by first proving that the $n$-packing content of Sobolev images is finite, then proving that the inability to upgrade on a set of positive measure implies infinite packing content.

Grothendieck rigidity and virtual retraction of higher-rank GBS groups

Daxun Wang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22477v1 Announce Type: new Abstract: A rank $n$ generalized Baumslag-Solitar group ($GBS_n$ group) is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to $\mathbb{Z}^n$. This paper investigates Grothendieck rigidity and virtual retraction properties of $GBS_n$ groups. We show that every residually finite $GBS_n$ group is Grothendieck rigid. Further, we characterize when a $GBS_n$ group satisfies property (VRC), showing that it holds precisely when the monodromy is trivial.

Successive Cancellation List Decoding of Extended Reed-Solomon Codes

Xiaoqian Ye, Jingyu Lin, Junjie Huang, Li Chen, Chang-An Zhao — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22482v1 Announce Type: new Abstract: Reed-Solomon (RS) codes are an important class of non-binary error-correction codes. They are particularly competent in correcting burst errors, being widely applied in modern communications and data storage systems. This also thanks to their distance property of reaching the Singleton bound, being the maximum distance separable (MDS) codes. This paper proposes a new list decoding for extended RS (eRS) codes defined over a finite field of characteristic two, i.e., F_{2^n}. It is developed based on transforming an eRS code into n binary polar codes. Consequently, it can be decoded by the successive cancellation (SC) decoding and further their list decoding, i.e., the SCL decoding. A pre-transformed matrix is required for reinterpretating the eRS codes, which also determines their SC and SCL decoding performances. Its column linear independence property is studied, leading to theoretical characterization of their SC decoding performance. Our proposed decoding and analysis are validated numerically.

Note on Euler characteristic of a toric vector bundle

Suhyon Chong, Shaoyu Huang, Kiumars Kaveh — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22514v1 Announce Type: new Abstract: A convex chain is a finite integer linear combination of indicator functions of convex polytopes. Khovanskii-Pukhlikov extend the Ehrhart theory of convex lattice polytopes to the setting of convex chains. Extending the relationship between equivariant line bundles on projective toric varieties and virtual lattice polytopes, we associate a lattice convex chain to a torus equivariant vector bundle on a toric variety and show that sum of values of this convex chain on lattice points gives the Euler characteristic of the bundle.

Local existence and nonexistence of solutions to the Hardy parabolic equation with general nonlinearity

Yo Tsusaka — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22520v1 Announce Type: new Abstract: In this paper, we consider the Cauchy problem for the Hardy parabolic equation with general nonlinearity and establish the local existence and nonexistence results. Our results provide the optimal integrability conditions on initial function for the existence of a local-in-time nonnegative solution. The proof of the existence result is based on the supersolution method.

Flexible FTN-OTFS for High-Mobility LEO Satellite-to-Ground Communication

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22526v1 Announce Type: new Abstract: In this paper, a lightweight LEO satellite-assisted flexible faster-than-Nyquist (FTN)-orthogonal time frequency space (OTFS) (LEO-FFTN-OTFS) scheme is proposed to address the stringent constraints on onboard power consumption and the severe impact of fast time-varying channels in non-terrestrial networks. A rigorous system framework incorporating realistic 3GPP Tapped Delay Line (TDL) channel models is established to accurately capture high-mobility propagation characteristics. To counteract channel aging effects while maintaining low computational complexity, an SNR-aware flexible FTN strategy is introduced, wherein a low-complexity Look-Up Table (LUT) is utilized to adaptively optimize the time-domain compression factor based on instantaneous channel responses. Through this mechanism, the trade-off between rate acceleration and interference penalty is effectively resolved, ensuring that spectral efficiency is maximized while strict reliability constraints are satisfied with minimal processing overhead. Moreover, a comprehensive theoretical analysis is provided, in which analytical expressions for effective throughput, energy efficiency, and bit error rate are derived. Finally, it is demonstrated by extensive simulations that the proposed scheme significantly outperforms static FTN benchmarks, offering a superior balance of high throughput and robustness for next-generation LEO communications.

Transmission and Reflection coefficients for Schr\"odinger Operators with Truncated Periodic Potentials that support defect states

Joseph C. Stellman, Jeremy L. Marzuola — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22544v1 Announce Type: new Abstract: We consider scattering waves through truncated periodic potentials with perturbations that support localized gap eigenstates. In a small complex neighborhood around an assumed positive bound state of the model operator, we prove the existence of a distinct zero reflection state, or transmission resonance. We compare its location to a previously found scattering resonance and use the properties of solutions near these interesting points to analyze the behavior of transmission and reflection coefficients of scattering solutions near the assumed bound state. By example, we also discuss the truncated simple harmonic oscillator and compare the analysis to the crystalline case.

Corrigendum: Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups (J. Topol. 16 (2023), no. 2, 634--649.)

Chetan Balwe, Amit Hogadi, Anand Sawant — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22555v1 Announce Type: new Abstract: The proof of Lemma 5.1 in the paper Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups (J. Topol. 16 (2023), no. 2, 634--649) is incomplete as it relies on some results of Choudhury-Hagadi, the proof of which contains a gap. The goal of this note is to give a complete and self-contained proof of this lemma.

Generalized Zalcman Conjecture for Starlike Mappings in Several Complex Variables

Surya Giri — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22558v1 Announce Type: new Abstract: Generalizing the Zalcman conjecture given by $\vert a_n^2 - a_{2n-1}\vert \leq (n-1)^2$, Ma proposed and proved that the inequality $$\vert a_n a_m-a_{n+m-1}\vert \leq (n-1)(m-1), \quad m,n \in \mathbb{N},$$ holds for functions $f(z)=z+a_2z^2 +a_3 z^3 +\cdots\in \mathcal{S}^*$, the class of starlike functions in the open unit disk. In this work, we extend this problem to several complex variables for $m=2$ and $n=3$, considering the class of starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$.

Inverse acoustic scattering for random obstacles with multi-frequency data

Zhiqi Sun, Xiang Xu, Yiwen Lin — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22560v1 Announce Type: new Abstract: We study an inverse random obstacle scattering problems in $\mathbb{R}^2$ where the scatterer is formulated by a Gaussian process defined on the angular parameter domain. Equipped with a modified covariance function which is mathematically well-defined and physically consistent, the Gaussian process admits a parameterization via Karhunen--Lo\`eve (KL) expansion. Based on observed multi-frequency data, we develop a two-stage inversion method: the first stage reconstructs the baseline shape of the random scatterer and the second stage estimates the statistical characteristics of the boundary fluctuations, including KL eigenvalues and covariance hyperparameters. We further provide theoretical justifications for the modeling and inversion pipeline, covering well-definedness of the Gaussian-process model, convergence for the two-stage procedure and a brief discussion on uniqueness. Numerical experiments demonstrate stable recovery of both geometric and statistical information for obstacles with simple and more complex shapes.

Quantum $(r,\delta)$-Locally Recoverable BCH and Homothetic-BCH Codes

Carlos Galindo, Fernando Hernando, Ryutaroh Matsumoto — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22567v1 Announce Type: new Abstract: Quantum $(r,\delta)$-locally recoverable codes ($(r,\delta)$-LRCs) are the quantum version of classical $(r,\delta)$-LRCs designed to recover multiple failures in large-scale distributed and cloud storage systems. A quantum $(r,\delta)$-LRC, $Q(C)$, can be constructed from an $(r,\delta)$-LRC, $C$, which is Euclidean or Hermitian dual-containing. This article is devoted to studying how to get quantum $(r,\delta)$-LRCs from BCH and homothetic-BCH codes. As a consequence, we give pure quantum $(r,\delta)$-LRCs which are optimal for the Singleton-like bound.

An ultra-weak three-field finite element formulation for the biharmonic and extended Fisher--Kolmogorov equations

Rekha Khot, Bishnu P. Lamichhane, Ricardo Ruiz-Baier — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22587v1 Announce Type: new Abstract: This paper discusses a so-called ultra-weak three-field formulation of the biharmonic problem where the solution, its gradient, and an additional Lagrange multiplier are the three unknowns. We establish the well-posedness of the problem using the abstract theory for saddle-point problems, and develop a conforming finite element scheme based on Raviart--Thomas discretisations of the two auxiliary variables. The well-posedness of the discrete formulation and the corresponding a priori error estimate are proved using a discrete inf-sup condition. We further extend the analysis to the time-dependent semilinear equation, namely extended Fisher--Kolmogorov equation. We present a few numerical examples to demonstrate the performance of our approach.

Kernels of Arithmetic Jet Spaces and Frobenius Morphism

Rajat Kumar Mishra, Arnab Saha — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22591v1 Announce Type: new Abstract: For any $\pi$-formal group scheme $G$, the Frobenius morphism between arithmetic jet spaces restricts to generalized kernels of the projection map. Using the functorial properties of such kernels of arithmetic jet spaces, we show that this morphism is indeed induced by a natural ring map between shifted $\pi$-typical Witt vectors. In the special case when $G = \hat{\mathbb{G}}_a$, the arithmetic jet space, as well as the generalized kernels are affine $\pi$-formal planes with Witt vector addition as the group law. In that case the above morphism is the multiplication by $\pi$ map on Witt vector schemes. In fact, the system of arithmetic jet spaces and generalized kernels of any $\pi$-formal group scheme $G$ along with their maps and identitites satisfied among them are a generalization of the case of the Witt vector scheme with the system of maps such as the Frobenius, Verschiebung and multiplication by $\pi$.

A spectral approach for online covariance change point detection

Zhigang Bao, Kha Man Cheong, Yuji Li, Jiaxin Qiu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22602v1 Announce Type: new Abstract: Change point detection in covariance structures is a fundamental and crucial problem for sequential data. Under the high-dimensional setting, most of the existing research has focused on identifying change points in historical data. However, there is a significant lack of studies on the practically relevant online change point problem, which means promptly detecting change points as they occur. In this paper, applying the limiting theory of linear spectral statistics for random matrices, we propose a class of spectrum based CUSUM-type statistic. We first construct a martingale from the difference of linear spectral statistics of sequential sample Fisher matrices, which converges to a Brownian motion. Our CUSUM-type statistic is then defined as the maximum of a variant of this process. Finally, we develop our detection procedure based on the invariance principle. Simulation results show that our detection method is highly sensitive to the occurrence of change point and is able to identify it shortly after they arise, outperforming the existing approaches.

Spectral properties and bound states of the Dirac equation on periodic quantum graphs

Zhipeng Yang, Ling Zhu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22603v1 Announce Type: new Abstract: We investigate nonlinear Dirac equations on a periodic quantum graph $G$ and develop a variational approach to the existence and multiplicity of bound states. After introducing the Dirac operator on $G$ with a $\mathbb Z^{d}$-periodic potential, we describe its spectral decomposition and work in the natural energy space. Under asymptotically linear or superquadratic assumptions on the nonlinearity, we establish the required linking geometry and a Cerami-type compactness property modulo $\mathbb Z^{d}$-translations. As a consequence, we prove the existence of at least one bound state and, when the nonlinearity is even, infinitely many geometrically distinct bound states.

Weighted estimates for Hodge-Maxwell systems

Rohit Mahato, Swarnendu Sil — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22604v1 Announce Type: new Abstract: We establish up to the boundary regularity estimates in weighted $L^{p}$ spaces with Muckenhoupt weights $A_{p}$ for weak solutions to the Hodge systems \begin{align*} d^{\ast}\left(Ad\omega\right) + B^{\intercal}dd^{\ast}\left(B\omega\right) = \lambda B\omega + f \quad \text{ in } \Omega \end{align*} with either $\nu \wedge \omega $ and $\nu \wedge d^{\ast}\left(B\omega\right)$ or $\nu \lrcorner B\omega$ and $\nu \lrcorner Ad\omega$ prescribed on $\partial\Omega.$ As a consequence, we prove the solvability of Hodge-Maxwell systems and derive Hodge decomposition theorems in weighted Lebesgue spaces. Our proof avoids potential theory, does not rely on representation formulas and instead uses decay estimates in the spirit of `Campanato method' to establish weighted $L^{p}$ estimates.

An inertial minimal-deformation-rate framework for shape optimization

Falai Chen, Buyang Li, Jiajie Li, Rong Tang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22605v1 Announce Type: new Abstract: We propose a robust numerical framework for PDE-constrained shape optimization and Willmore-driven surface hole filling. To address two central challenges -- slow progress in flat energy landscapes, which can trigger premature stagnation at suboptimal configurations, and mesh deterioration during geometric evolution -- we couple a second-order inertial flow with a minimal-deformation-rate (MDR) mesh motion strategy. This coupling accelerates convergence while preserving mesh quality and thus avoids remeshing. To further enhance robustness for non-smooth or non-convex initial geometries, we incorporate surface-diffusion regularization within the Barrett-Garcke-N"urnberg (BGN) framework. Moreover, we extend the inertial MDR methodology to Willmore-type surface hole filling, enabling high-order smooth reconstructions even from incompatible initial data. Numerical experiments demonstrate markedly faster convergence to lower original objective values, together with consistently superior mesh preservation throughout the evolution.

Local controllability of the Cahn-Hilliard-Burgers' equation around certain steady states

Manika Bag, Sheetal Dharmatti, Subrata Majumdar, Debanjana Mitra — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22611v1 Announce Type: new Abstract: In this article we study the local controllability of the one-dimensional Cahn-Hilliard-Navier-Stokes equation, that is Cahn-Hilliard-Burgers' equation, around a certain steady state using a localized interior control acting only in the concentration equation. To do it, we first linearize the nonlinear equation around the steady state. The linearized system turns out to be a system coupled between second order and fourth order parabolic equations and the control acts in the fourth order parabolic equation. The null controllability of the linearized system is obtained by a duality argument proving an observability inequality. To prove the observability inequality, a new Carleman inequality for the coupled system is derived. Next, using the source term method, it is shown that the null controllability of the linearized system with non-homogeneous terms persists provided the non-homogeneous terms satisfy certain estimates in a suitable weighted space. Finally, using a Banach fixed point theorem in a suitable weighted space, the local controllability of the nonlinear system is obtained.

Sequence entropy of rank one systems

Shigenori Takeda — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22626v1 Announce Type: new Abstract: We study the sequence entropy of rank one measure-preserving systems along subexponential sequences. We prove that the sequence entropy along a large class of sequences can be infinite using Ornstein's probabilistic constructions. Moreover, we show that sequence entropy necessarily vanishes for subexponential sequences if the growth of tower heights remains below certain growth rates, and obtain a flexibility result for polynomial sequences at this critical threshold.

Maximal Prikry Sequences

Ernest Schimmerling, Jiaming Zhang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22643v1 Announce Type: new Abstract: In this paper we investigate the covering machinery of the Mitchell-Steel core model $K$, under the hypothesis that there is no inner model with a Woodin cardinal. In an earlier work, Mitchell and the first author showed that if $\nu>\omega_2$ is a regular cardinal in $K$ but a singular ordinal in $V$, then $\nu$ is a measurable cardinal in $K$. In this article, we further show that under certain circumstances, there exists a maximal Prikry sequence $C$ for a measure on $\nu$ in $K$. The first author shows that the anti-large cardinal hypothesis is necessary. In a more restrictive setting, we prove that every subset of $\nu$ with size $<|\nu|$ can be covered by a set in $K[C]$ with size $<|\nu|$. Benhemou and the first author show that the result is optimal.

On the BSE and BED properties of the Beurling algebra $L^1(G,\omega)$

Jekwin J. Dabhi, Prakash A. Dabhi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22646v1 Announce Type: new Abstract: Let $G$ be a locally compact abelian group, and let $\omega:G \to [1,\infty)$ be a weight, i.e., $\omega$ is measurable, $\omega$ is locally bounded and $\omega(s+t)\leq \omega(s)\omega(t)$ for all $s, t \in G$. If $\omega^{-1}$ is vanishing at infinity, then we show that the Beurling algebra $L^1(G,\omega)$ is both BSE- algebra and BED- algebra.

Multisets of finite intervals and a universal category of poset representations

Henning Krause, Balduin Stoye — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22649v1 Announce Type: new Abstract: For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some cases new integer sequences arise. The formulation of this counting problem leads to a universal construction which assigns to any poset a finitely cocomplete additive category; it is abelian when the poset is finite and does not depend on the choice of any ring of coefficients. For a general poset the universal category of representations is abelian if and only if for the lattice of ideals the meet of two compact elements is again compact.

Sharp thresholds for Escobar and Gagliardo-Nirenberg functionals: the Escobar-Willmore mass, geometric selection, and compactness trichotomy

Mayukh Mukherjee, Utsab Sarkar — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22665v1 Announce Type: new Abstract: We develop a unified quantitative framework for sharp threshold phenomena in boundary-critical variational problems on compact Riemannian manifolds, covering the Escobar quotient and Gagliardo-Nirenberg inequalities. Via transfer-stability-reduction, we obtain attainment-versus-bubbling alternatives, $H^1$-compactness, and finite-dimensional reductions. Geometric selection is governed by mean curvature $H_g$ and a Willmore-type anisotropy from $|\mathring{\mathrm{II}}|^2$. At hemisphere threshold $S_\ast=C^*_{\mathrm{Esc}}(\mathbb S^n_+)$ for $n\ge5$ on $H_g\equiv0$, we identify a renormalized boundary mass $\mathfrak R_g=\kappa_1(n)\,\mathrm{Ric}_g(\nu,\nu)+\kappa_2(n)\,\mathrm{Scal}_{g|\partial M}+\kappa_3(n)\,|\mathring{\mathrm{II}}|^2$, $\kappa_3(n)<0$, yielding one-bubble expansions and energy-only estimators. Threshold dichotomy: if the first nonvanishing coefficient among $\{\rho_n^{\mathrm{conf}}H_g,\mathfrak R_g,\Theta_g\}$ is negative somewhere, then $C^*_{\mathrm{Esc}}(M,g)0$ at all critical points, no bubbling occurs. In multi-bubble regime ($n\ge5$), dynamics governed by $\mathcal W_k=\sum_{i=1}^k\mathfrak R_g(x_i)$ produce $k$-bubble critical points at levels $k^{1/(n-1)}S_\ast$. In the degenerate case we obtain conformal hemispherical rigidity. The GN track yields analogous dichotomies and resolves a question of Christianson et al.: the sharp constant with small Dirichlet windows diverges at optimal capacitary rate, relating threshold to spectral/isoperimetric invariants. Applications include entropy inequalities for fast diffusion, curvature-driven NLS ground states, and (in $n=2$) Euler characteristic recovery from GN measurements.

Classification of horospherical invariant measures in higher rank: The Full Story

Inhyeok Choi, Dongryul M. Kim — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22668v1 Announce Type: new Abstract: In this paper, we classify horospherical invariant Radon measures for Anosov subgroups of arbitrary semisimple real algebraic groups. This generalizes the works of Burger and Roblin in rank one to higher ranks. At the same time, this extends the works of Furstenberg, Veech, and Dani, and a special case of Ratner's theorem for finite-volume homogeneous spaces to infinite-volume Anosov homogeneous spaces. Especially, this resolves the open problems proposed by Landesberg--Lee--Lindenstrauss--Oh and by Oh. Our measure classification is in fact for a more general class of discrete subgroups, including relatively Anosov subgroups with respect to any parabolic subgroups, not necessarily minimal. Our method is rather geometric, not relying on continuous flows or ergodic theorems.

Wall singularity of spaces with an upper curvature bound

Koichi Nagano — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22673v1 Announce Type: new Abstract: We study typical wall singularity of codimension one for locally compact geodesically complete metric spaces with an upper curvature bound. We provide a geometric structure theorem of codimension one singularity, and a geometric characterization of codimension two regularity. These give us necessary and sufficient conditions for singular sets to be of codimension at least two.

Anisotropic Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Sets

Filip Fry\v{s} — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22681v1 Announce Type: new Abstract: This paper investigates the existence of the anisotropic lower-dimensional Minkowski content. We establish that the $C$-anisotropic $k$-dimensional Minkowski content of a $k$-rectifiable compact set always exists and coincides with a specific functional that depends naturally on $C$. We further show that the same conclusion holds for countably $\mathcal{H}^k$-rectifiable compact sets, provided that the so-called \emph{AFP-condition} is satisfied. In addition, we discuss how the existence of the $C$-anisotropic $k$-dimensional Minkowski content for a countably $\mathcal{H}^k$-rectifiable compact set depends on the choice of $C$.

SUN-DSBO: A Structured Unified Framework for Nonconvex Decentralized Stochastic Bilevel Optimization

Yaoshuai Ma, Xiao Wang, Wei Yao, Jin Zhang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22682v1 Announce Type: new Abstract: Decentralized stochastic bilevel optimization (DSBO) is a powerful tool for various machine learning tasks, including decentralized meta-learning and hyperparameter tuning. Existing DSBO methods primarily address problems with strongly convex lower-level objective functions. However, nonconvex objective functions are increasingly prevalent in modern deep learning. In this work, we introduce SUN-DSBO, a Structured Unified framework for Nonconvex DSBO, in which both the upper- and lower-level objective functions may be nonconvex. Notably, SUN-DSBO offers the flexibility to incorporate decentralized stochastic gradient descent or various techniques for mitigating data heterogeneity, such as gradient tracking (GT). We demonstrate that SUN-DSBO-GT, an adaptation of the GT technique within our framework, achieves a linear speedup with respect to the number of agents. This is accomplished without relying on restrictive assumptions, such as gradient boundedness or any specific assumptions regarding gradient heterogeneity. Numerical experiments validate the effectiveness of our method.

A Mathematical Analysis of a Smooth-Convex-Concave Splitting Scheme for the Swift--Hohenberg Equation

Yuki Yonekura, Daiki Iwade, Shun Sato, Takayasu Matsuo — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22687v1 Announce Type: new Abstract: The Swift--Hohenberg equation is a widely studied fourth-order model, originally proposed to describe hydrodynamic fluctuations. It admits an energy-dissipation law and, under suitable assumptions, bounded solutions. Many structure-preserving numerical schemes have been proposed to retain such properties; however, existing approaches are often fully implicit and therefore computationally expensive. We introduce a simple design principle for constructing dissipation-preserving finite difference schemes and apply it to the Swift--Hohenberg equation in three spatial dimensions. Our analysis relies on discrete inequalities for the underlying energy, assuming a Lipschitz continuous gradient and either convexity or $\mu$-strong convexity of the relevant terms. The resulting method is linearly implicit, yet it preserves the original energy-dissipation law, guarantees unique solvability, ensures boundedness of numerical solutions, and admits an a priori error estimate, provided that the time step is sufficiently small. To the best of our knowledge, this is the first linearly implicit finite difference scheme for the Swift--Hohenberg equation for which all of these properties are established.

Multi-target DoA estimation with a single Rydberg atomic receiver by spectral analysis of spatially-resolved fluorescence

Liangcheng Han, Haifan Yin, M\'erouane Debbah — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22704v1 Announce Type: new Abstract: Rydberg-based Direction-of-Arrival (DoA) estimation has been hampered by the complexity of receiver arrays and the single-target, narrow-band limitations of existing single-receiver methods. This paper introduces a novel approach that addresses these limitations. We demonstrate that by spatially resolving the fluorescence profile along the vapor cell, the multi-target problem can be effectively solved. Our approach hinges on the insight that by superimposing incoming signals with a strong local oscillator (LO), the complex atomic absorption pattern is linearized into a simple superposition of sinusoids. In this new representation, each spatial frequency uniquely and directly maps to the DoA of a target. This reduces the multi-target challenge into a spectral estimation problem, which we address using Prony's method. Our approach, termed Imaging-based Spectral Estimation (ISE), inherently supports multi-target detection and restores the full broadband capability of the sensor by removing the restrictive cell-length dependency. This development also shows potential for realizing multi-channel Rydberg receivers and the continuous-aperture sensing required for holographic multiple-input multiple-output (MIMO). We develop a comprehensive theoretical model, derive the Cramer-Rao Lower Bound (CRLB) as a performance benchmark, and present simulations validating the effectiveness of the approach to resolve multiple targets.

Profunctorial algebras

Quentin Aristote, Umberto Tarantino — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22721v1 Announce Type: new Abstract: We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first characterize, in terms of exact squares, when pseudomonads on a bicategory extend to its bicategory of two-sided discrete fibrations. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories satisfying our criterion and thus extending to profunctors. Among these, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces.

Enhancing Exploration in Global Optimization by Noise Injection in the Probability Measures Space

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22753v1 Announce Type: new Abstract: McKean-Vlasov (MKV) systems provide a unifying framework for recent state-of-the-art particlebased methods for global optimization. While individual particles follow stochastic trajectories, the probability law evolves deterministically in the mean-field limit, potentially limiting exploration in multimodal landscapes. We introduce two principled approaches to inject noise directly into the probability law dynamics: a perturbative method based on conditional MKV theory, and a geometric approach leveraging tangent space structure. While these approaches are of independent interest, the aim of this work is to apply them to global optimization. Our framework applies generically to any method that can be formulated as a MKV system. Extensive experiments on multimodal objective functions demonstrate that both our noise injection strategies enhance consistently the exploration and convergence across different configurations of dynamics, such as Langevin, Consensus-Based Optimization, and Stein Boltzmann Sampling, providing a versatile toolkit for global optimization.

Numerical Differentiation of Functions of Two Variables Using Chebyshev Polynomials

Maksym Kyselov, Sergiy G. Solodky — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22762v1 Announce Type: new Abstract: We investigate the problem of numerical differentiation of bivariate functions from weighted Wiener classes using Chebyshev polynomial expansions. We develop and analyze a new version of the truncation method based on Chebyshev polynomials and the idea of hyperbolic cross to reconstruct partial derivatives of arbitrary order. The method exploits the approximation properties of Chebyshev polynomials and their natural connection to weighted spaces through the Chebyshev weight function. We derive a choice rule for the truncation parameter as a function of the noise level, smoothness parameters of the function class, and the order of differentiation. This approach allows us to establish explicit error estimates in both weighted integral norms and uniform metric.

Characterization of $n$-Lie Derivations on Generalized Matrix Algebras

Xinfeng Liang, Minghao Wang, Feng Wei — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22774v1 Announce Type: new Abstract: The principal objective of this paper is to determine the structure of $n$-Lie derivations ($n\geq 3$) on generalized matrix algebras.It is shown that under certain mild assumptions, every $n$-Lie derivation can be decomposed into the sum of an extremal $n$-derivation and an $n$-linear centrally-valued mapping. As direct applications, we provide complete characterizations of $n$-Lie derivations on both full matrix algebras and triangular algebras.

The Symplectic-to-Contact Dictionary

Fabrizio Pugliese, Giovanni Sparano, Luca Vitagliano — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22775v1 Announce Type: new Abstract: Contact Geometry is an odd dimensional analogue of Symplectic Geometry. This vague idea can actually be formalized in a rather precise way by means of a Symplectic-to-Contact Dictionary. The aim of this review paper is discussing the basic entries in this dictionary. Surprisingly, the dictionary can also be applied to apparently far away situations like complex and $G$-structures, to get old and new interesting geometries.

Generators for automorphisms of special groups

Elia Fioravanti — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22789v1 Announce Type: new Abstract: Let $G$ be a (compact) special group in the sense of Haglund and Wise. We show that ${\rm Out}(G)$ is finitely generated, and provide a virtual generating set consisting of Dehn twists and ``pseudo-twists''. We exhibit instances where Dehn twists alone do not suffice and completely characterise this phenomenon: it is caused by certain abelian subgroups of $G$, called ``poison subgroups'', which can be removed by replacing $G$ with a finite-index subgroup. Similar results hold for coarse-median preserving automorphisms, without the pathologies: For every special group $G$, the coarse-median preserving subgroups ${\rm Out}(G,[\mu])\leq{\rm Out}(G)$ are virtually generated by finitely many Dehn twists with respect to splittings of $G$ over centralisers. Proofs are based on a novel, hierarchical version of Rips and Sela's shortening argument.

Convergence of Multi-Level Markov Chain Monte Carlo Adaptive Stochastic Gradient Algorithms

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22799v1 Announce Type: new Abstract: Stochastic optimization in learning and inference often relies on Markov chain Monte Carlo (MCMC) to approximate gradients when exact computation is intractable. However, finite-time MCMC estimators are biased, and reducing this bias typically comes at a higher computational cost. We propose a multilevel Monte Carlo gradient estimator whose bias decays as $O(T_{n}^{-1} )$ while its expected computational cost grows only as $O(log T_n )$, where $T_n$ is the maximal truncation level at iteration n. Building on this approach, we introduce a multilevel MCMC framework for adaptive stochastic gradient methods, leading to new multilevel variants of Adagrad and AMSGrad algorithms. Under conditions controlling the estimator bias and its second and third moments, we establish a convergence rate of order $O(n^{-1/2} )$ up to logarithmic factors. Finally, we illustrate these results on Importance-Weighted Autoencoders trained with the proposed multilevel adaptive methods.

Rapid stabilizability of delayed infinite-dimensional control systems

Yaxing Ma, Lijuan Wang, Huaiqiang Yu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22819v1 Announce Type: new Abstract: In this paper, the rapid stabilizability of linear infinite-dimensional control system with constant-valued delay is studied. Under assumptions that the state operator generates an immediately compact semigroup and the coefficient of the delay term is constant, we mainly prove the following two results: (i) the delay does not affect rapid stabilizability of the control system; (ii) from the perspective of observation-feedback, it is not necessary to use historical information to stabilize the control system when the system is rapidly stabilizable. Some applications are given.

On the average number of representations of an integer as a sum of polynomials computed at prime values

Alessandra Migliaccio, Alessandro Zaccagnini — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22822v1 Announce Type: new Abstract: We study the average number of representations of an integer $n$ as $n = \phi(n_{1}) + \dots + \phi(n_{j})$, for polynomials $\phi \in \mathbb{Z}[n]$ with $\partial\phi = k\ge 1$, $\operatorname{lead}(\phi) = 1$, $j \ge k$, where $n_{i}$ is a prime power for each $i \in \{1, \dots, j\}$. We extend the results of Languasco and Zaccagnini (2019), for $k=3$ and $j=4$, and of Cantarini, Gambini and Zaccagnini (2020), where they focused on monomials $\phi(n) = n^k$, $k\ge 2$ and $j=k, k + 1$.

Approximation of PDE solution manifolds: Sparse-grid interpolation and quadrature

Dinh D\~ung, Van Kien Nguyen, Duong Thanh Pham, Christoph Schwab — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22825v1 Announce Type: new Abstract: We study fully-discrete approximations and quadratures of infinite-variate functions in abstract Bochner spaces associated with a Hilbert space $X$ and an infinite-tensor-product Jacobi measure. For target infinite-variate functions taking values in $X$ which admit absolutely convergent Jacobi generalized polynomial chaos expansions, with suitable weighted summability conditions for the coefficient sequences, we generalize and improve prior results on construction of sequences of finite sparse-grid tensor-product polynomial interpolation approximations and quadratures, based on the univariate Chebyshev points. For a generic stable discretization of $X$ in terms of a dense sequence $(V_m)_{m \in \mathbb{N}_0}$ of finite-dimensional subspaces, we obtain fully-discrete, linear approximations in terms of so-called sparse-grid tensor-product projectors, with convergence rates of approximations as well as of sparse-grid tensor-product quadratures of the target functions. We verify the abstract assumptions in two fundamental application settings: first, a linear elliptic diffusion equation with affine-parametric coefficients and second, abstract holomorphic maps between separable Hilbert spaces with affine-parametric input data encoding. For these settings, as in [37,20], cancellation of anti-symmetric terms in ultra-spherical Jacobi generalized polynomial chaos expansion coefficients implies crucially improved convergence rates of sparse-grid tensor-product quadrature with respect to the infinite-tensor-product Jacobi weight, free from the ``curse-of-dimension". Largely self-contained proofs of all results are developed. Approximation convergence rate results in the present setting which are based on construction of neural network surrogates, for unbounded parameter ranges with Gaussian measures, will be developed in extensions of the present work.

Simplicity of eigenvalues for elliptic problems with mixed Steklov-Robin boundary condition

Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22829v1 Announce Type: new Abstract: This paper investigates the spectral properties of two classes of elliptic problems characterized by mixed Steklov-Robin boundary conditions. Our main objective is to prove that, for a generic domain, all the eigenvalues are simple. This result is established by employing domain perturbation techniques and analyzing the transversality of the associated operators.

Asymmetric conformal prediction with penalized kernel sum-of-squares

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22834v1 Announce Type: new Abstract: Conformal prediction (CP) is a distribution-free method to construct reliable prediction intervals that has gained significant attention in recent years. Despite its success and various proposed extensions, a significant practical feature which has been overlooked in previous research is the potential skewed nature of the noise, or of the residuals when the predictive model exhibits bias. In this work, we leverage recent developments in CP to propose a new asymmetric procedure that bridges the gap between skewed and non-skewed noise distributions, while still maintaining adaptivity of the prediction intervals. We introduce a new statistical learning problem to construct adaptive and asymmetric prediction bands, with a unique feature based on a penalty which promotes symmetry: when its intensity varies, the intervals smoothly change from symmetric to asymmetric ones. This learning problem is based on reproducing kernel Hilbert spaces and the recently introduced kernel sum-of-squares framework. First, we establish representer theorems to make our problem tractable in practice, and derive dual formulations which are essential for scalability to larger datasets. Second, the intensity of the penalty is chosen using a novel data-driven method which automatically identifies the symmetric nature of the noise. We show that consenting to some asymmetry can let the learned prediction bands better adapt to small sample regimes or biased predictive models.

Vidinli algebras

Alberto Elduque, Javier R\'andez-Ib\'a\~nez — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22839v1 Announce Type: new Abstract: A new class of nonassociative algebras, Vidinli algebras, is defined based on recent work of Co\c{s}kun and Eden. These algebras are conic (or quadratic) algebras with the extra restriction that the commutator of any two elements is a scalar multiple of the unity. Over fields of characteristic not 2, Vidinli algebras may be considered as generalizations of the Jordan algebras of Clifford type. However, in characteristic 2, the class of Vidinli algebras is much larger and include the unitizations of anticommutative algebras.

Distance Optimization in the Grassmannian of Lines

Hannah Friedman, Andrea Rosana, Bernd Sturmfels — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22843v1 Announce Type: new Abstract: The square of a skew-symmetric matrix is a symmetric matrix whose eigenvalues have even multiplicities. When the matrices have rank two, they represent the Grassmannian of lines, and the squaring operation takes Pl\"ucker coordinates to projection coordinates. We develop metric algebraic geometry for varieties of lines in this linear algebra setting. The Grassmann distance (GD) degree is introduced as a new invariant for subvarieties of a Grassmannian. We study the GD degree for Schubert varieties and other models.

Unconditional well-posedness of the master equation for monotone mean field games of controls

Joe Jackson, Alp\'ar R. M\'esz\'aros — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22845v1 Announce Type: new Abstract: We establish the first unconditional well-posedness result for the master equation associated with a general class of mean field games of controls. Our analysis covers games with displacement monotone or Lasry--Lions monotone data, as well as those with a small time horizon. By unconditional, we mean that all assumptions are imposed solely at the level of the Lagrangian and the terminal cost. In particular, we do not require any a priori regularity or structural assumptions on the additional fixed-point mappings arising from the control interactions; instead we show that these fixed-point mappings are well-behaved as a consequence of the regularity and the monotonicity of the data. Our approach is bottom-up in nature, unlike most previous results which rely on a generalized method of characteristics. In particular, we build a classical solution of the master equation by showing that the solutions of the corresponding $N$-player Nash systems are compact, in an appropriate sense, and that their subsequential limit points must be solutions to the master equation. Compactness is obtained via uniform-in-$N$ decay estimates for derivatives of the $N$-player value functions. The underlying games are driven by non-degenerate idiosyncratic Brownian noise, and our results allow for the presence of common noise with constant intensity.

Relative Kazhdan Lusztig isomorphism for $GL_{2n}/Sp_{2n}$

Guy Shtotland — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22846v1 Announce Type: new Abstract: The Kazhdan Lusztig isomorphism, relating the affine Hecke algebra of a $p$-adic group to the equivariant $K$ theory of the Steinberg variety of its Langlands dual, played a key role in the proof of the Deligne Langlands conjectures concerning the classification of smooth irreducible representations with an Iwahori fixed vector. In this work we state and prove a relative version of the Kazhdan Lusztig isomorphism for the symmetric pair $(GL_{2n},Sp_{2n})$. The relative isomorphism is an isomorphism between the module of compactly supported Iwahori invariant functions on $X=GL_{2n}/Sp_{2n}$ and another module over the affine Hecke algebra constructed using equivariant $K$ theory and the relative Langlands duality. We use this isomorphism to give a new proof of a condition on $X$ distinguished representations.

Existence of a solution of the TV Wasserstein gradient flow

Kexin Lin, Filippo Santambrogio — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22847v1 Announce Type: new Abstract: On the flat torus in any dimension we prove existence of a solution to the TV Wasserstein gradient flow equation, only assuming that the initial density $\rho_0$ is bounded from below and above by strictly positive constants. This solution preserves upper and lower bounds of the densities, and shows a certain decay of the BV norm (of the order of $t^{-1/3}$ for $t\to 0$ -- if $\rho_0\notin BV$, otherwise the BV norm is of course bounded -- and of the order of $t^{-1}$ as $t\to\infty$). This generalizes a previous result by Carlier and Poon, who only gave a full proof in one dimension of space and did not consider the case $\rho_0\notin BV$. The main tool consists in considering an approximated TV-JKO scheme which artificially imposes a lower bound on the density and allows to find a continuous-in-time solution regular enough to prove that the lower bounds of the initial datum propagates in time, and study on this approximated equation the decay of the BV norm.

Convergence Rates for the Alternating Minimization Algorithm in Structured Nonsmooth and Nonconvex Optimization

Glaydston C. Bento, Boris S. Mordukhovich, Tiago S. Mota, Antoine Soubeyran — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22850v1 Announce Type: new Abstract: This paper is devoted to developing the alternating minimization algorithm for problems of structured nonconvex optimization proposed by Attouch, Bolt\'e, Redont, and Soubeyran in 2010. Our main result provides significant improvements of the convergence rate of the algorithm, especially under the low exponent Polyak-{\L}ojasiewicz-Kurdyka condition when we establish either finite termination of this algorithm or its superlinear convergence rate instead of the previously known linear convergence. We also investigate the PLK exponent calculus and discuss applications to noncooperative games and behavioral science.

On the convergence and efficiency of splitting schemes for the Cahn-Hilliard-Biot model

Cedric Riethm\"uller, Erlend Storvik — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22854v1 Announce Type: new Abstract: In this paper, we present a novel solution strategy for the Cahn-Hilliard-Biot model, a three-way coupled system that features the interplay of solid phase separation, fluid dynamics, and elastic deformations in porous media. It is a phase-field model that combines the Cahn-Hilliard regularized interface equation and Biot's equations of poroelasticity. Solving the system poses significant challenges due to its coupled, nonlinear, and non-convex nature. The main goal of this work is to provide a consistent and efficient solution strategy. With this in mind, we introduce a semi-implicit time discretization such that the resulting discrete system is equivalent to a convex minimization problem. Then, using abstract theory for convex problems, we prove the convergence of an alternating minimization method to the time-discrete system. The solution strategy is relatively flexible in terms of spatial discretization, although we require standard inverse inequalities for the guaranteed convergence of the alternating minimization method. Finally, we perform some numerical experiments that show the promise of the proposed solution strategy, both in terms of efficiency and robustness.

The two-nest ants process on triangle-series-parallel graphs

C\'ecile Mailler, Zo\'e Varin — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22855v1 Announce Type: new Abstract: The ants process is a stochastic process introduced by Kious, Mailler and Schapira as a model for the phenomenon of ants finding shortest paths between their nest and a source of food (seen as two marked nodes in a finite graph), with no other means of communications besides the pheromones they lay behind them as they explore their environment. The ants process relies on a reinforcement learning mechanism. In this paper, we modify the ants process by having more than one ants nest (and still one source of food). For technical reasons, we restrict ourselves to the case when there are two nests, and when the graph is a triangle between the two nests and the source of food, whose edges have been replaced by series-parallel graphs. In this setting, using stochastic approximation techniques, comparison with P\'olya urns, and combinatorial arguments, we are able to prove that the ants process converges and to describe its limit.

Bayesian Interpolating Neural Network (B-INN): a scalable and reliable Bayesian model for large-scale physical systems

Chanwook Park, Brian Kim, Jiachen Guo, Wing Kam Liu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22860v1 Announce Type: new Abstract: Neural networks and machine learning models for uncertainty quantification suffer from limited scalability and poor reliability compared to their deterministic counterparts. In industry-scale active learning settings, where generating a single high-fidelity simulation may require days or weeks of computation and produce data volumes on the order of gigabytes, they quickly become impractical. This paper proposes a scalable and reliable Bayesian surrogate model, termed the Bayesian Interpolating Neural Network (B-INN). The B-INN combines high-order interpolation theory with tensor decomposition and alternating direction algorithm to enable effective dimensionality reduction without compromising predictive accuracy. We theoretically show that the function space of a B-INN is a subset of that of Gaussian processes, while its Bayesian inference exhibits linear complexity, $\mathcal{O}(N)$, with respect to the number of training samples. Numerical experiments demonstrate that B-INNs can be from 20 times to 10,000 times faster with a robust uncertainty estimation compared to Bayesian neural networks and Gaussian processes. These capabilities make B-INN a practical foundation for uncertainty-driven active learning in large-scale industrial simulations, where computational efficiency and robust uncertainty calibration are paramount.

Randomized Methods for Kernelized DMD

Peter Oehme — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22867v1 Announce Type: new Abstract: Dynamic Mode Decomposition (DMD) is a data-driven method related to Koopman operator theory that extracts information about dominant dynamics from data snapshots. In this paper we examine techniques to accelerate the application of DMD to large-scale data sets with an eye on randomized techniques. Randomized techniques exploit low-rank matrix approximations at a much smaller computational cost, therefore permitting the use of increased data set sizes. In particular, we propose the application of the RPCholesky algorithm in the setting of kernelized DMD (KDMD). This algorithm relies on adaptive randomized sampling to approximate positive semidefinite kernel matrices and provides better stability guarantees than previously implemented randomized methods for KDMD. Differences between existing competitive randomized techniques and our proposed implementation are discussed with a focus on numerical stability and tradeoff between exploration and exploitation of information obtained from data. The efficacy of this new combination of algorithms is demonstrated on well-established benchmark problems from DMD literature increasing in problem dimension.

Global Well-posedness of Strong Solutions to the Cauchy Problem of 2D Nonhomogeneous Navier-Stokes Equations with Density-Dependent Viscosity and Vacuum

Bing Yuan, Rong Zhang, Peng Zhou — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22877v1 Announce Type: new Abstract: This paper is concerned with the Cauchy problem for the modified two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity. By fully using the structure of the system, we can obtain the key estimates of $\|\nabla \rho\|_{L_t^\infty L_x^q},q>2$ without any smallness asuumption on the initial data, and thus establish the global existence of the strong solutions with the far-field density being either vacuum or nonvacuum. Notably, the initial data can be arbitrarily large and the initial density is allowed to vanish. Furthermore, the large-time asymptotic behavior of the gradients of the velocity and the pressure is also established.

Arbitrary harmonic functions as Bose--Einstein condensates

Michiel De Wilde, Robert Seiringer — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22883v1 Announce Type: new Abstract: We show that a suitable choice of boundary conditions for the Laplacian allows for the appearance of an an arbitrary number of condensates, described by arbitrary harmonic functions, in the thermodynamic limit of an ideal Bose gas.

Uncoupled Dirac-Yang-Mills Pairs on Closed Riemannian Spin Manifolds

Adam Lindstr\"om — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22886v1 Announce Type: new Abstract: We study the Dirac-Yang-Mills equations on closed spin manifolds with a focus on uncoupled solutions, i.e. solutions for which the connection form satisfies the Yang-Mills equation. Such solutions require the Dirac current, a quadratic form on the spinor bundle, to vanish. We study the condition that this current vanishes on all harmonic spinors using perturbation theory and obtain a classification of the connection forms for which this holds, which we show contains an open and dense subset of connections. This has several implications for the generic dimension of the kernel of the Dirac operator. We further establish existence results for uncoupled solutions, in particular in dimension $4$ using the index theorem. Finally we generalize a construction method for twisted harmonic spinors to construct explicit uncoupled solutions on $4$-manifolds admitting twistor spinors and on spin manifolds of any dimension admitting parallel spinors.

Grassmannian Geometry and Global Convergence of Variable Projection for Neural Networks

Mathias Dus (IRMA) — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22897v1 Announce Type: new Abstract: Training deep neural networks and Physics-Informed Neural Networks (PINNs) often leads to ill-conditioned and stiff optimization problems. A key structural feature of these models is that they are linear in the output-layer parameters and nonlinear in the hiddenlayer parameters, yielding a separable nonlinear least-squares formulation. In this work, we study the classical variable projection (VarPro) method for such problems in the context of deep neural networks. We provide a geometric formulation on the Grassmannian and analyze the structure of critical points and convergence properties of the reduced problem. When the feature map is parametrized by a neural network, we show that these properties persist except in rank-deficient regimes, which we address via a regularized Grassmannian framework. Numerical experiments for regression and PINNs, including an efficient solver for the heat equation, illustrate the practical effectiveness of the approach.

Status Updating via Integrated Sensing and Communication: Freshness Optimisation

Touraj Soleymani, Mohamad Assaad, John S. Baras — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22901v1 Announce Type: new Abstract: This paper studies strategic design in an integrated sensing and communication (ISAC) architecture for status updating of remotely navigating agents. We consider an ISAC-enabled base station that can sense the state of a remote source and communicate this information back to the source. Both sensing and communication succeed with given probabilities and incur distinct costs. The objective is to optimise a long-term cost that captures information freshness, measured by the age of information (AoI), at the source together with sensing and communication overheads. The resulting sequential decision problem is formulated as a discounted infinite-horizon Markov decision process with a two-dimensional AoI state, representing information freshness at the source and at the base station. We prove that the optimal stationary policy admits a monotone threshold structure characterised by a nondecreasing switching curve in the AoI state space. Our numerical analysis illustrates the structures of the value function and the optimal decision map. These results demonstrate that freshness-based objectives can be naturally integrated into ISAC design, while yielding interpretable and implementable strategies.

Rigidity of circle polyhedra and hyperideal polyhedra: the tangency case

John C. Bowers, Philip L. Bowers, Carl O. R. Lutz — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22903v1 Announce Type: new Abstract: We prove the global rigidity of proper triangulated convex hyperbolic circle polyhedra on the sphere $\mathbb{S}^2$. These circle polyhedra correspond to proper triangulated convex hyperbolic polyhedra in the Beltrami-Klein model $\mathbb{B}^{3}$ of hyperbolic space with hyperideal vertices whose faces meet $\mathbb{B}^{3}$. Although the vertices of these polyhedra lie outside $\mathbb{B}^{3} \cup \mathbb{S}^{2}$ and the faces meet $\mathbb{B}^{3}$, the edges may miss $\mathbb{B}^{3}$ entirely, meet $\mathbb{B}^{3}$, or, more importantly, lie tangent to $\mathbb{B}^{3}$ at ideal points on the boundary $\partial \mathbb{B}^{3} = \mathbb{S}^{2}$. The latter case is new and generalizes the global rigidity results of both Bao-Bonahon and arXiv:1703.09338. This result also generalizes the uniqueness part of the celebrated Koebe-Andre'ev-Thurston theorem to the case where adjacent circles need not touch.

Examples of finitely presented groups with strong fixed point properties and property (T)

Indira Chatterji, Martin Kassabov — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22907v1 Announce Type: new Abstract: We construct a finitely presented group with property (T) which can not act on on reasonable spaces. Such group is constructed using an generalization of Hall embedding theorem, where property (T) is added at the expense of weakening the simplicity requirement.

Reducibility of self-maps in monoid and its related invariants

Gopal Chandra Dutta — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22908v1 Announce Type: new Abstract: Given a positive integer $k$, we investigate the $k$-redcibility of self-maps in the monoid $\AA^k(X\vee Y)$, consisting of self-maps that induce isomorphisms on homology groups up to degree $k$. In general, verifying $k$-reducibility is a subtle problem. We show that the $k$-reducibility of a self-map is determine through its induced endomorphisms on homology or cohomology groups. Moreover, under the k-reducibility assumption, the computation of the homology self-closeness number of the wedge sum of spaces essentially reduces to the computation of the homology self-closeness numbers of the individual wedge summands. We generalize the notion of an atomic space to that of an $n$-atomic space and establish some of its fundamental properties. We show that the $k$-reducibility criteria for self-maps in a monoid $\AA^k(X)$ is satisfied when the space $X$ decomposes as a wedge sum of distinct $n$-atomic spaces. Finally, we determine the homology self-closeness numbers of wedge sums of distinct $n$-atomic spaces.

Feedback Control via Integrated Sensing and Communication: Uncertainty Optimisation

Touraj Soleymani, Mohamad Assaad, John S. Baras — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22912v1 Announce Type: new Abstract: This paper studies strategic design in an integrated sensing and communication (ISAC) architecture for feedback control of cyber-physical systems. We focus on a setting in which the regulation of a physical process (i.e., remote source) is performed via an ISAC-enabled base station. The base station can alternate between tracking the state of the source and delivering control-relevant information back to the source. For a Gauss-Markov source subject to i.i.d. Bernoulli sensing and communication links, under a finite-horizon linear-quadratic-Gaussian cost, we rigorously characterise the optimal policies through an uncertainty-aware synthesis. We establish that the optimal switching policy, for the ISAC system at the base station, is threshold-based in terms of the source and base-station estimation covariances, while the optimal control policy, for the actuator at the source, is linear in the source state estimate. We show that the threshold region$\unicode{x2014}$defined as the set of estimation covariance pairs for which communication is preferred over sensing$\unicode{x2014}$expands with increasing source uncertainty and contracts with increasing base-station uncertainty.

Left Ehresmann monoids with a proper basis

Gracinda Gomes, Victoria Gould, Yanhui Wang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22923v1 Announce Type: new Abstract: This article gives an abstract characterisation of a class of left Ehresmann monoids possessing certain universal properties. It is known that every left Ehresmann monoid has a cover, that is, a projection separating preimage, of the form $\mathcal{P}_{\ell}(T,X)$, where $\mathcal{P}_{\ell}(T,X)$ is a left Ehresmann monoid constructed from a monoid $T$ and an order-preserving action of $T$ on a semilattice $X$ with identity. We introduce the class of $*$-left Ehresmann monoids and show that each $\mathcal{P}_{\ell}(T,X)$ belongs to this class; in particular so does any free left Ehresmann monoid. Further, we present the notion of a proper basis, and show that $\mathcal{P}_{\ell}(T,X)$ possesses a proper basis. Next, we exhibit a class of subsemigroups $\mathcal{Q}_{\ell}(T,X,Y)$ (properly, biunary monoid subsemigroups) of the monoids $\mathcal{P}_{\ell}(T,X)$ which are also $*$-left Ehresmann with a proper basis, and prove that up to isomorphism they form exactly the class of all such monoids. Our results can be regarded as being analogous to those for proper inverse semigroups, due to McAlister and O'Carroll, the $\mathcal{Q}_{\ell}(T,X,Y)$ playing the role of the $P$-semigroups and the $\mathcal{P}_{\ell}(T,X)$ the role of the semidirect products of a semilattice by a group.

Poset modules of the $0$-Hecke algebras of type $B$

Young-Hun Kim, Dominic Searles — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22926v1 Announce Type: new Abstract: In 2001, Chow developed the theory of the $B_n$ posets $P$ and the type $B$ $P$-partition enumerators $K^B_P$. To provide a representation-theoretic interpretation of $K^B_P$, we define the poset modules $M^B_P$ of the 0-Hecke algebra $H_n^B(0)$ of type $B$ by endowing the set of type-$B$ linear extensions of $P$ with an $H_n^B(0)$-action. We then show that the Grothendieck group of the category associated to type-$B$ poset modules is isomorphic to the space of type $B$ quasisymmetric functions as both a $\mathrm{QSym}$-module and comodule, where $\mathrm{QSym}$ denotes the Hopf algebra of quasisymmetric functions. Considering an equivalence relation on $B_n$ posets, where two posets are equivalent if they share the same set of type-$B$ linear extensions, we identify a natural representative of each equivalence class, which we call a distinguished poset. We further characterize the distinguished posets whose sets of type-$B$ linear extensions form intervals in the right weak Bruhat order on the the hyperoctahedral groups. Finally, we discuss the relationship among the categories associated to type-$B$ weak Bruhat interval modules, $B_n$ poset modules, and finite-dimensional $H_n^B(0)$-modules.

Prescribed $T$-curvature flow on the four-dimensional unit ball

Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun, Heming Wang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22934v1 Announce Type: new Abstract: In this paper, we study the prescribed $T$-curvature problem on the unit ball $\mathbb{B}^4$ of $\mathbb{R} ^4$ via the $T$-curvature flow approach. By combining Ache-Chang's inequality with the Morse-theoretic approach of Malchiodi-Struwe, we establish existence results under strong Morse-type inequalities at infinity. As a byproduct of our argument, we also prove the exponential convergence of the $T$-curvature flow on $\mathbb{B}^4$, starting from a $Q$-flat and minimal metric conformal to the standard Euclidean metric, to an extremal metric of Ache-Chang's inequality whose explicit expression was derived by Ndiaye-Sun.

Local Well-posedness and Blow-up for the Restricted Fourth-Order Prandtl Equation

Ik Hyun Choi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22940v1 Announce Type: new Abstract: We prove local well-posedness and finite-time blow-up for a restricted fourth-order Prandtl equation posed on the half-line with clamped boundary conditions. The equation arises from a two-dimensional fourth-order Prandtl system via an ansatz reduction, and its nonlinearity involves a nonlocal integral term. To close a Duhamel fixed-point argument, we need uniform $L^1$ bounds for the associated half-line biharmonic heat kernel. We establish uniform $L^1$ estimates for the kernel and its derivatives, and we show that the semigroup preserves spatial regularity under appropriate compatibility conditions, using an alternative representation derived by integration by parts. These kernel estimates yield local existence and uniqueness for the restricted model and allow us to construct solutions that blow-up in finite time.

Compact group Rohlin actions and $G$-kernels on von Neumann algebras

Takumi Nishihara — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22941v1 Announce Type: new Abstract: We provide a new construction of a topological group model for the string group of a compact, simple, and simply-connected Lie group, by solving the obstruction realization problem for compact group $G$-kernels on full factors. Furthermore, we introduce the Rohlin property for actions and cocycle actions of compact groups in order to establish cohomology vanishing theorems.

FNWoS: Fractional Neural Walk-on-Spheres Methods for High-Dimensional PDEs Driven by $\alpha$-stable L\'{e}vy Process on Irregular Domains

Ling Guo, Mingxin Qin, Changtao Sheng, Hao Wu, Fanhai Zeng — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22942v1 Announce Type: new Abstract: In this paper, we develop a highly parallel and derivative-free fractional neural walk-on-spheres method (FNWoS) for solving high-dimensional fractional Poisson equations on irregular domains. We first propose a simplified fractional walk-on-spheres (FWoS) scheme that replaces the high-dimensional normalized weight integral with a constant weight and adopts a correspondingly simpler sampling density, substantially reducing per-trajectory cost. To mitigate the slow convergence of standard Monte Carlo sampling, FNWoS is then proposed via integrating this simplified FWoS estimator, derived from the Feynman-Kac representation, with a neural network surrogate. By amortizing sampling effort over the entire domain during training, FNWoS achieves more accurate evaluation at arbitrary query points with dramatically fewer trajectories than classical FWoS. To further enhance efficiency in regimes where the fractional order $\alpha$ is close to 2 and trajectories become excessively long, we introduce a truncated path strategy with a prescribed maximum step count. Building on this, we propose a buffered supervision mechanism that caches training pairs and progressively refines their Monte Carlo targets during training, removing the need to precompute a highly accurate training set and yielding the buffered fractional neural walk-on-spheres method (BFNWoS). Extensive numerical experiments, including tests on irregular domains and problems with dimensions up to $1000$, demonstrate the accuracy, scalability, and computational efficiency of the proposed methods.

Persuasive Privacy

Joshua J Bon, James Bailie, Judith Rousseau, Christian P Robert — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22945v1 Announce Type: new Abstract: We propose a novel framework for measuring privacy from a Bayesian game-theoretic perspective. This framework enables the creation of new, purpose-driven privacy definitions that are rigorously justified, while also allowing for the assessment of existing privacy guarantees through game theory. We show that pure and probabilistic differential privacy are special cases of our framework, and provide new interpretations of the post-processing inequality in this setting. Further, we demonstrate that privacy guarantees can be established for deterministic algorithms, which are overlooked by current privacy standards.

On polynomial functors and polynomial comonads over infinity groupoids

Kun Chen — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22968v1 Announce Type: new Abstract: We show that single-variable polynomial functors over the category $\mathcal{S}$ of infinity groupoids, as defined by Gepner-Haugseng-Kock, are exactly colimits of representable copresheaves indexed by infinity groupoid. This allows us to establish certain categorical properties of the $\infty$-category $Poly_{\mathcal{S}}$, in parallel with the case of the ordinary category $Poly$. We define the notion of polynomial comonad under the monoidal structure of $Poly_{\mathcal{S}}$ induced by composition of polynomials, and describe a construction toward exploring the connection between polynomial comonads and complete Segal spaces. This construction partially generalizes the classical one given in the proof of a theorem of Ahman-Uustalu.

Periods of Ehrhart coefficients of rational polytopes

Tyrrell B. McAllister, H\'el\`ene O. Rochais — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22992v1 Announce Type: new Abstract: Let $\mathcal{P} \subseteq \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a quasi-polynomial function of $k$ -- that is, a "polynomial" in which the coefficients are themselves periodic functions of $k$. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values.

A Remark on Stability Conditions on Smooth Projective Varieties

Chunyi Li — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22994v1 Announce Type: new Abstract: Let $X$ be a smooth projective variety over $\mathbb C$. In this paper, we prove that $\mathrm{D}^b(X)$, the bounded derived category of coherent sheaves on $X$, always admits stability conditions in the sense of Bridgeland.

Baire-type properties of topological vector spaces

Saak Gabriyelyan, Alexander Osipov, Evgenii Reznichenko — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23008v1 Announce Type: new Abstract: Burzyk, Kli\'{s} and Lipecki proved that every topological vector space (tvs) $E$ with the property $(K)$ is a Baire space. K\c{a}kol and S\'{a}nchez Ruiz proved that every sequentially complete Fr\'{e}chet--Urysohn locally convex space (lcs) is Baire. Being motivated by the property $(K)$ and the notion of a Mackey null sequence we introduce a property $(MK)$ which is strictly weaker than the property $(K)$, and show that any locally complete lcs has the property $(MK)$. We prove that any $\kappa$-Fr\'{e}chet--Urysohn tvs with the property $(MK)$ is a Baire space; consequently, each locally complete $\kappa$-Fr\'{e}chet--Urysohn lcs is a Baire space. This generalizes both the aforementioned results. We construct a feral Baire space $E$ with the property $(K)$ and which is not $\kappa$-Fr\'{e}chet--Urysohn. Although a $\kappa$-Fr\'{e}chet--Urysohn lcs $E$ can be not a Baire space, we show that $E$ is always $b$-Baire-like in the sense of Ruess. Applications to spaces of Baire functions and $C_k$-spaces are given.

On the finiteness of prime trees and their relation to modular forms

Yusuke Fujiyoshi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23016v1 Announce Type: new Abstract: In this paper, we introduce the prime trees associated with a finite subset $P$ of the set of all prime numbers, and provide conditions under which the tree is of finite type. Moreover, we compute the density of finite-type subsets $P$. As an application, we show that for weight $k \ge 2$ and levels $N = N'\prod_{p \in P} p^{a_p}$, where $N'$ is squarefree and $a_{p} \geq 2$, every cusp form $f \in \mathcal{S}_k(\Gamma_0(N))$ can be expressed as a linear combination of products of two specific Eisenstein series whenever $P$ is of finite type.

The uniqueness theorem for Kasparov theory

G\'abor Szab\'o — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23029v1 Announce Type: new Abstract: Answering a question of Carri\'on et al in their recent landmark paper on C*-algebra classification, we prove a general uniqueness theorem for $KK$-theory. Given arbitrary separable C*-algebras $A$ and $B$ and a Cuntz pair consisting of two absorbing representations $\varphi,\psi: A\to\mathcal{M}(B\otimes\mathcal{K})$, the induced element of $KK(A,B)$ vanishes if and only if $\varphi$ and $\psi$ are strongly asymptotically unitarily equivalent. This improves upon the Lin-Dadarlat-Eilers stable uniqueness theorem. The conclusion is deduced by first showing the $K_1$-injectivity of an auxiliary C*-algebra associated to the C*-pair $(A,B)$, which is sometimes called the Paschke dual algebra in the literature. Most of the article is concerned with the treatment of an umbrella theorem, which yields such a uniqueness theorem for other variants of $KK$-theory. This encompasses nuclear $KK$-theory, ideal-related $KK$-theory, equivariant $KK$-theory, or any combinations thereof.

Breaking the Stochasticity Barrier: An Adaptive Variance-Reduced Method for Variational Inequalities

Yungi Jeong, Takumi Otsuka — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23034v1 Announce Type: new Abstract: Stochastic non-convex non-concave optimization, formally characterized as Stochastic Variational Inequalities (SVIs), presents unique challenges due to rotational dynamics and the absence of a global merit function. While adaptive step-size methods (like Armijo line-search) have revolutionized convex minimization, their application to this setting is hindered by the Stochasticity Barrier: the noise in gradient estimation masks the true operator curvature, triggering erroneously large steps that destabilize convergence. In this work, we propose VR-SDA-A (Variance-Reduced Stochastic Descent-Ascent with Armijo), a novel algorithm that integrates recursive momentum (STORM) with a rigorous Same-Batch Curvature Verification mechanism. We introduce a theoretical framework based on a Lyapunov potential tracking the Operator Norm, proving that VR- SDA-A achieves an oracle complexity of O(epsilon -3) for finding an epsilon-stationary point in general Lipschitz continuous operators. This matches the optimal rate for non-convex minimization while uniquely enabling automated step-size adaptation in the saddle-point setting. We validate our approach on canonical rotational benchmarks and non-convex robust regression tasks, demonstrating that our method effectively suppresses limit cycles and accelerates convergence with reduced dependence on manual learning rate scheduling.

Accelerated Inertial Gradient Algorithms with Vanishing Tikhonov Regularization

Samir Adly, Vinh Thanh Ho, Huu Nhan Nguyen — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23035v1 Announce Type: new Abstract: In this paper, we study an explicit Tikhonov-regularized inertial gradient algorithm for smooth convex minimization with Lipschitz continuous gradient. The method is derived via an explicit time discretization of a damped inertial system with vanishing Tikhonov regularization. Under appropriate control of the decay rate of the Tikhonov term, we establish accelerated convergence of the objective values to the minimum together with strong convergence of the iterates to the minimum-norm minimizer. In particular, for polynomial schedules $\varepsilon_k = k^{-p}$ with $0

Stationary Mean-Field singular control of an Ornstein-Uhlenbeck process

Federico Cannerozzi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23036v1 Announce Type: new Abstract: Motivated by continuous-time optimal inventory management, we study a class of stationary mean-field control problems with singular controls. The dynamics are modeled by a mean-reverting Ornstein-Uhlenbeck process, and the performance criterion is given by a quadratic long-time average expected cost functional. The mean-field dependence is through the stationary mean of the controlled process itself, which enters the ergodic cost functional. We characterize the solution to the stationary mean-field control problem in terms of the equilibria of an associated stationary mean-field game, showing that solutions of the control problem are in bijection with the equilibria of this mean-field game. Finally, we solve the stationary mean-field game explicitly, thereby providing a solution to the original stationary mean-field control problem.

Instability of two-dimensional Taylor-Green Vortices

Gonzalo Cao-Labora, Maria Colombo, Michele Dolce, Paolo Ventura — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23040v1 Announce Type: new Abstract: For a wide class of linear Hamiltonian operators we develop a general criterion that characterizes the unstable eigenvalues as the zeros of a holomorphic function given by the determinant of a finite-dimensional matrix. We apply the latter result to prove the spectral instability of the Taylor-Green vortex in two-dimensional ideal fluids. The linearized Euler operator at this steady state possesses different invariant subspaces, within which we apply our criterion to rule out or detect instabilities. We show linear stability of odd perturbations, for which the unstable spectrum can appear only on the real axis. We exclude this possibility by applying our stability criterion. Real instabilities, instead, exist and can be detected with the same criterion if we consider suitable rescalings of the Taylor-Green vortex. In the subspace of functions even in both variables, the problem is reduced to finding a single complex root of our stability function. We successfully locate this value by combining our general criterion with a rigorous computer-assisted argument. As a consequence, we fully characterize the unstable spectrum of the Taylor-Green vortex.

On two-dimensional Dirac operators with critical delta-shell interactions

William Borrelli, Pietro Carimati, Davide Fermi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23053v1 Announce Type: new Abstract: We study two-dimensional Dirac operators with singular interactions of electrostatic and Lorentzscalar type, supported either on a straight line or a circle. For certain critical values of the interaction strengths, the essential spectrum of such operators comprises an isolated point lying within the mass gap. We clarify the nature of this point in both geometries. For the straight line model, this point is known to be an eigenvalue of infinite multiplicity, and we provide a detailed analysis of the corresponding eigenfunctions. By contrast, in the case of a circle, we show that the said point is not itself an eigenvalue, but rather an accumulation point of a double sequence of simple eigenvalues. In view of the high degree of symmetry of the configurations under analysis, this behavior is unexpected and our findings lead us to formulate some conjectures concerning critical singular interactions supported on generic smooth curves.

Some elementary amenable subgroups of interval exchange transformations

Nancy Guelman, Isabelle Liousse — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23054v1 Announce Type: new Abstract: In this paper, we study a family of finitely generated elementary amenable iet-groups. These groups are generated by finitely many rationals iets and rotations. For them, we state criteria for not virtual nilpotency or solvability, and we give conditions to ensure that they are not virtually solvable. We precise their abelianizations, we determine when they are isomorphic to certain lamplighter groups and we provide non isomorphic cases among them. As consequences, in the class of infinite finitely generated subgroups of iets up to isomorphism, we exhibit infinitely many non virtually solvable and non linear groups, and infinitely many solvable groups of arbitrary derived length.

Spectrum of bidual uniform algebras

Marek Kosiek, Krzysztof Rudol — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23055v1 Announce Type: new Abstract: We obtain a description of the spectrum of bidual algebra $A^{**}$ of a uniform algebra $A$. This spectrum turns out to be a quotient space of the hyper-Stonean envelope of the spectrum of $A$.

Some notes on plump ordinals

Shuwei Wang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23070v1 Announce Type: new Abstract: In this exposition, we attempt to formalise a treatment of Paul Taylor's notion of plump ordinals in weak intuitionistic axiomatic set theories such as IKP. We will explore basic properties of plump ordinals, especially in relation to G\"odel's constructible universe $L$ and incomparable codings. As a quick application, we explain at the end how plump ordinals can be used to build a Heyting-valued model $V^\mathbb{H}$ from a classical $V \vDash \mathrm{ZFC}$ such that for some arbitrary, fixed $x \in V$ we have $V^\mathbb{H} \vDash \mathcal{P}{\left(\check{x}\right)} \in L$.

The $L^p$- regularity problem for the Bergman projection of two-dimensional Rudin ball quotients

Debraj Chakrabarti, Alessandro Monguzzi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23074v1 Announce Type: new Abstract: We solve the $L^p$-regularity problem of the Bergman projection of two-dimensional Rudin ball quotients.

Mermin-Wagner theorems for quantum systems with multipole symmetries

Timo Feistl, Severin Schraven, Simone Warzel — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23078v1 Announce Type: new Abstract: We prove Mermin-Wagner-type theorems for quantum lattice systems in the presence of multipole symmetries. These theorems show that the presence of higher-order symmetries protects against the breaking of lower-order ones. In particular, we prove that the critical dimension in which the charge symmetry can be broken increases if the system admits higher multipole symmetries, e.g. $ d = 4 $ on the regular lattice $ \mathbb{Z}^d $ in the presence of dipole symmetry.

Lifting property for finite groups

Chandrashekhar B. Khare, Alexander Merkurjev — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23089v1 Announce Type: new Abstract: We classify all finite groups that have lifting property of mod $p$ representations to mod $p^2$ representations for all prime $p$.

Existence of Traveling Waves in Infinite Range FPUT Lattices

Michael Herrmann, Karsten Matthies, Jan-Patrick Meyer — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23091v1 Announce Type: new Abstract: We prove the existence of solitary waves in a lattice where all particles interact with each other by pair-wise repulsive forces that decay with distance. The variational existence proof is based on constrained optimization and provides a one-parameter family of unimodal solutions. We also describe the asymptotic behavior of large, fast, high-energy waves.

Seminoetherian Modules over Non-Primitive HNP rings

Askar Tuganbaev — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23099v1 Announce Type: new Abstract: We study the structure of seminoetherian modules. Seminoetherian modules over non-primitive hereditary noetherian prime rings are completely described.

Bipartite Graphs Are Not Well-Ordered by Bipartite Minors

Therese Biedl, Dinis Vitorino — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23101v1 Announce Type: new Abstract: In "Bipartite minors," Chudnovsky etal. introduced the bipartite minor relation, a partial order on the set of bipartite graphs somewhat analogous the minor relation on general graphs and asked whether it is a well-order. We answer this question negatively by giving an infinite set of $2$-connected bipartite graphs that are pairwise incomparable with respect to the bipartite minor relation. We additionally give two sets of infinitely many pairs of bipartite graphs: one set of pairs $G,H$ such that $H$ is a bipartite minor, but not a minor, of $G$, and one set of pairs $G,H$ such that $H$ is a minor, but not a bipartite minor, of $G$.

Series-Parallel and Planar Graphs for Efficient Broadcasting

David Evangelista, Hovhannes A. Harutyunyan, Aram Khanlari — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23104v1 Announce Type: new Abstract: The broadcasting problem concerns the efficient dissemination of information in graphs. In classical broadcasting, a single originator vertex initially has a message to be transmitted to all vertices. Every vertex which has received the message informs at most one uninformed neighbor at each discrete time unit. In this paper, we introduce infinite families of series-parallel graphs with efficient broadcast times: graphs on $n$ vertices with broadcast time at most $\lceil\log_2 n \rceil + 1$ for any $n$, graphs on $n$ vertices with broadcast time $\lfloor \frac{3 \lceil \log_2 n \rceil}{2} \rfloor$ and maximum degree $\lceil \log_2 n \rceil - 1$ for any $n$, and broadcast graphs on up to $2^{k-1} + 2^{\lfloor \frac{k}{2} \rfloor }$ vertices with broadcast time $k$ for any $k$. We also introduce an infinite family of planar broadcast graphs on up to $2^{k-1} + 2^{\lfloor \frac{3k}{4} \rfloor - 1}$ vertices with broadcast time $k$ for any $k$, which improves the known lower bound on the maximum number of vertices in a planar broadcast graph.

Lifts of endomorphisms of Weyl algebras modulo $p^2$

Niels Lauritzen, Jesper Funch Thomsen — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23110v1 Announce Type: new Abstract: Let $\varphi$ denote a $k$-algebra endomorphism of the $n$-th Weyl algebra $A_n(k)$ over a perfect field $k$ of positive characteristic $p$. We prove that $\varphi$ can be lifted to an endomorphism of the Weyl algebra $A_n(W_2(k))$ over the Witt vectors $W_2(k)$ of length two over $k$ if and only if $\varphi$ induces a Poisson morphism of the center of $A_n(k)$. Furthermore, we improve a result of Tsuchimoto, which enables us to conclude that these equivalent statements hold at least when ${\rm deg}(\varphi) < p$. In particular, we conclude that $\varphi$ is injective if ${\rm deg}(\varphi) < p$.

The Coxeter Flag Variety

Nantel Bergeron, Lucas Gagnon, Hunter Spink, Vasu Tewari — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23111v1 Announce Type: new Abstract: For a Coxeter element $c$ in a Weyl group $W$, we define the $c$-Coxeter flag variety $\operatorname{CFl}_c\subset G/B$ as the union of left-translated Richardson varieties $w^{-1}X^{wc}_w$. This is a complex of toric varieties whose geometry is governed by the lattice $\operatorname{NC}(W,c)$ of $c$-noncrossing partitions. We show that $\operatorname{CFl}_c$ is the common vanishing locus of the generalized Pl\"ucker coordinates indexed by $W\setminus\operatorname{NC}(W,c)$. We also construct an explicit affine paving of $\operatorname{CFl}_c$ and identify the $T$-weights of each cell in terms of $c$-clusters. This paving gives a GKM description of $H^\bullet(\operatorname{CFl}_c)$ and $H^\bullet_{T_{ad}}(\operatorname{CFl}_c)$ in terms of the induced Cayley subgraph on $\operatorname{NC}(W,c)$, and we show these rings are naturally isomorphic for different choices of $c$. In type $\mathrm{A}$, this recovers the quasisymmetric flag variety for a special $c$, and for general $c$ we show the cohomology ring has a presentation as permuted quasisymmetric coinvariants.

Graded Lie superalgebras from embedding tensors

Sylvain Lavau, Jakob Palmkvist — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23113v1 Announce Type: new Abstract: We show how various constructions of $\mathbb{Z}$-graded Lie superalgebras are related to each other. These Lie superalgebras have a Lie algebra $\mathfrak{g}$ as the subalgebra at degree 0, an odd $\mathfrak{g}$-module V as the subspace at degree 1, and an embedding tensor as an element at degree -1. This is a linear map from V to $\mathfrak{g}$ satisfying a quadratic constraint, which equips V with the structure of a Leibniz algebra.

Nonlinear Schr\"odinger Equation with magnetic potential on metric graphs

Riccardo Adami, Nicol\`o Cangiotti, Ivan Gallo, David Spitzkopf — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23115v1 Announce Type: new Abstract: In this manuscript, we shall investigate the Nonlinear Magnetic Schr\"odinger Equation on noncompact metric graphs, focusing on the existence of ground states. We prove that the magnetic Hamiltonian is variationally equivalent to a non-magnetic operator with additional repulsive potentials supported on the graph's cycles. This effective potential is strictly determined by the Aharonov-Bohm flux through the topological loops. Leveraging this reduction, we extend classical existence criteria to the magnetic setting. As a key application, we characterize the ground state structure on the tadpole graph, revealing a mass-dependent phase transition. The ground states exist for sufficiently small repulsion in an intermediate regime of masses while sufficiently strong flux prevents the formation of ground states.

Log canonical thresholds at infinity

Carles Bivi\`a-Ausina, Alexander Rashkovskii — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23118v1 Announce Type: new Abstract: The paper considers a global version of the notion of log canonical threshold for plurisubharmonic functions $u$ of logarithmic growth in $\mathbb{C}^n$, aiming at description of the range of all $p>0$ such that $e^{-u}\in L^p(\mathbb{C}^n)$. Explicit formulas are obtained in the toric case. By considering Bergman functions of corresponding weighted Hilbert spaces, a new polynomial approximation of plurisubharmonic functions of logarithmic growth with control over its singularities and behavior at infinity (a global version of Demailly's approximation theorem) is established. Some application to tame polynomial maps are given.

A General Tikhonov Regularized Second-Order Dynamical System for Convex-Concave Bilinear Saddle Point Problems

Bohan Zhang, Xiaojun Zhang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23120v1 Announce Type: new Abstract: In this paper, we propose a general Tikhonov regularized second-order dynamical system with viscous damping, time scaling and extrapolation coefficients for the convex-concave bilinear saddle point problem. By the Lyapunov function approach, we show that the convergence properties of the proposed dynamical system depend on the choice of the Tikhonov regularization parameter. Specifically, when the Tikhonov regularization parameter tends to zero rapidly, the convergence rate of the primal-dual gap along the generated trajectory is O(1 over t squared times beta(t)); when the Tikhonov regularization parameter tends to zero slowly, the convergence rate of the primal-dual gap is o(1 over beta(t)). We also prove the strong convergence property of the trajectory generated by the Tikhonov regularized dynamical system to the minimum-norm solution of the convex-concave bilinear saddle point problem, and derive several integral estimates. In addition, the effectiveness of the proposed dynamical system is verified through a series of numerical experiments.

Learning and Teaching Calculus Through Its History

Chamila Gamage — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23122v1 Announce Type: new Abstract: This paper frames calculus as a global, centuries-long development rather than a subject that began only with Newton and Leibniz. Drawing on ideas from Greek, Indian, Islamic, and later European mathematics, it highlights how concepts like infinity, area, motion, and continuous change slowly evolved through solving problems and cultural exchange. I argue that bringing this history into the classroom helps students see calculus as more than a set of procedures: it becomes a story of human creativity and persistence. By revisiting the questions early mathematicians struggled with, students can better appreciate and better understand the core ideas behind the formulas they use today.

Semi-knockoffs: a model-agnostic conditional independence testing method with finite-sample guarantees

Angel Reyero-Lobo, Bertrand Thirion, Pierre Neuvial — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23124v1 Announce Type: new Abstract: Conditional independence testing (CIT) is essential for reliable scientific discovery. It prevents spurious findings and enables controlled feature selection. Recent CIT methods have used machine learning (ML) models as surrogates of the underlying distribution. However, model-agnostic approaches require a train-test split, which reduces statistical power. We introduce Semi-knockoffs, a CIT method that can accommodate any pre-trained model, avoids this split, and provides valid p-values and false discovery rate (FDR) control for high-dimensional settings. Unlike methods that rely on the model-$X$ assumption (known input distribution), Semi-knockoffs only require conditional expectations for continuous variables. This makes the procedure less restrictive and more practical for machine learning integration. To ensure validity when estimating these expectations, we present two new theoretical results of independent interest: (i) stability for regularized models trained with a null feature and (ii) the double-robustness property.

On the b-function with respect to weights of annihilating ideals in the Weyl algebra

Helena Cobo — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23125v1 Announce Type: new Abstract: Given a polynomial $f\in\mathbb{C}[x_1,\ldots,x_n]$ and an integer $\ell\in\mathbb{Z}$, we study some properties of the b-function with respect to weights of the annihilating ideal Ann$(f^\ell)$. In some particular cases the expression of the b-function is given explicitly.

Hyperbolic partial differential equations with complex characteristics on Fourier Lebesgue spaces

Duv\'an Cardona, William Obeng-Denteh, Frederick Opoku — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23138v1 Announce Type: new Abstract: The aim of this paper is to establish well-posedness properties for hyperbolic PDEs on Fourier Lebesgue spaces. We consider hyperbolic operators with complex characteristics. Since our approach comes from harmonic analysis, we establish boundedness properties of Fourier integral operators with complex-valued phase functions on Fourier Lebesgue spaces, Besov spaces and Triebel-Lizorkin spaces. Indeed, these classes of operators serve as propagators of the considered PDE problems. In terms of the boundedness properties, we prove new results in the case where the canonical relation of the operator is assumed to satisfy the {\it spatial smooth factorization condition}

2-covering numbers of some finite solvable groups

Andrea Lucchini — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23144v1 Announce Type: new Abstract: A 2-covering for a finite group $G$ is a set of proper subgroups of $G$ such that every pair of elements of $G$ is contained in at least one subgroup in the set. The minimal number of subgroups needed to 2-cover a group $G$ is called the 2-covering number and denoted by $\sigma_2(G).$ In \cite{gk} it is conjectured that if $G$ is solvable and not 2-generated, then $\sigma_2(G)=1+q+q^2,$ where $q$ is a prime power. We disprove this conjecture.

Some series representing the zeta function for $\Re s>1$

Jean-Fran\c{c}ois Burnol — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23158v1 Announce Type: new Abstract: We present series converging to the Riemann zeta function in its half-plane of convergence, and possessing remainders whose sizes decrease geometrically. They are easy to implement numerically, using only polynomial and power functions, and are efficient for obtaining dozens or hundreds of digits (when the imaginary part is not too large). They may prove less suited to very high precision (tens of thousands of digits), due to a linear cost for each new term. One can express the coefficients as linear combinations of Bernoulli numbers, but this is not advantageous numerically. The method is a development of tools introduced by the author for the evaluation of harmonic series with restricted digits in a given radix.

Class choice and the surprising weakness of Kelley-Morse set theory

Victoria Gitman, Joel David Hamkins, Thomas A. Johnstone — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23165v1 Announce Type: new Abstract: Kelley-Morse set theory KM is weaker than generally supposed and fails to prove several principles that may be desirable in a foundational second-order set theory. Even though KM includes the global choice principle, for example, (i) KM does not prove the class choice scheme, asserting that whenever every set $x$ admits a class $X$ with $\varphi(x,X)$, then there is a class $Z\subseteq V\times V$ for which $\varphi(x,Z_x)$ on every section. This scheme can fail with KM even in low-complexity first-order instances $\varphi$ and even when only a set of indices $x$ are relevant. For closely related reasons, (ii) the theory KM does not prove the {\L}o\'s theorem scheme for internal second-order ultrapowers, even for large cardinal ultrapowers, such as the ultrapower by a normal measure on a measurable cardinal. Indeed, the theory KM itself is not generally preserved by internal ultrapowers. Finally, (iii) KM does not prove that the $\Sigma^1_n$ logical complexity is invariant under first-order quantifiers, even bounded first-order quantifiers. For example, $\forall \alpha{<}\delta\ \psi(\alpha,X)$ is not always provably equivalent to a $\Sigma^1_1$ assertion when $\psi$ is. Nevertheless, these various weaknesses in KM are addressed by augmenting it with the class choice scheme, thereby forming the theory KM+, which we propose as a robust KM alternative for the foundations of second-order set theory.

The Total Chromatic Quasisymmetric Functions of a Graph

Laura Colmenarejo, Ian Klein — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23170v1 Announce Type: new Abstract: In this paper, we introduce and study two variants of the chromatic quasisymmetric function of a graph: the total chromatic quasisymmetric function via vertex labeling and via acyclic orientations. The original definition of the chromatic quasisymmetric function of a graph by Shareshian and Wachs depends on a labeling of the vertices of the graph, which directly affects the properties of the coefficients appearing in the decomposition of the chromatic quasisymmetric function of a graph into different bases. Motivated by this, we construct the first variant of the chromatic quasisymmetric function of a graph by normalizing it with respect to all the labelings of the vertices. The second variant is motivated by the \emph{tree isomorphism conjecture} and is constructed in terms of acyclic orientations. We investigate the properties of the coefficients in the expansion in the monomial quasisymmetric basis for both variants and provide a comparative analysis. Furthermore, we derive explicit formulas for the coefficients in the monomial decomposition of the two variants for the star graph. For the labeling-based variant, these coefficients arise from a binomial identity for which we provide a combinatorial proof.

Interacting dynamical systems on networks and fractals: discrete and continuous models, mean-field limit, and convergence rates

Georgi S. Medvedev — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23175v1 Announce Type: new Abstract: We develop a continuum limit and mean-field theory for interacting particle systems (IPS) on self-similar networks, a new class of discrete models whose large-scale behavior gives rise to nonlocal evolution equations on fractal domains. This work extends the graphon-based framework for IPS, used to derive continuum and mean-field limits in the non-exchangeable setting, to situations where the spatial domain is fractal rather than Euclidean. The motivation arises from both physical models naturally formulated on fractals and real-world networks exhibiting hierarchical or quasi-self-similar structure. Our analysis relies on tools from fractal geometry, including Iterated Function Systems and self-similar measures. A central result is an explicit isomorphism between self-similar IPS and graphon IPS, which allows us to justify the continuum and mean-field limits in the self-similar setting. This connection reveals that macroscopic dynamics on fractal domains emerge naturally as limits of dynamics on appropriate discretizations of fractal sets. Another contribution of the paper is the derivation of optimal convergence rates for the discrete self-similar models. We introduce a scale of generalized Lipschitz spaces on fractals, extending the Nikolskii-Besov spaces used in the Euclidean setting, and obtain convergence estimates for discontinuous Galerkin approximations of nonlocal equations posed on fractal domains. These results apply to kernels with minimal regularity addressing models relevant in applications.

Preconditioning and Numerical Stability in Neural Network Training for Parametric PDEs

Markus Bachmayr, Wolfgang Dahmen, Chenguang Duan, Mathias Oster — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23185v1 Announce Type: new Abstract: In the context of training neural network-based approximations of solutions of parameter-dependent PDEs, we investigate the effect of preconditioning via well-conditioned frame representations of operators and demonstrate a significant improvement on the performance of standard training methods. We also observe that standard representations of preconditioned matrices are insufficient for obtaining numerical stability and propose a generally applicable form of stable representations that enables computations with single- and half-precision floating point numbers without loss of precision.

Noetherianity for powers of algebraic representations

Alessandro Danelon — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23186v1 Announce Type: new Abstract: Powers of a polynomial $\operatorname{GL}$-representation are topologically Noetherian under the action of $\operatorname{Sym} \times \operatorname{GL}$. We show that this result extends to powers of algebraic representations of the orthogonal and the symplectic groups. This work is a natural follow-up to arXiv:2212.05790 and to arXiv:1708.06420.

General Optimal Stopping without Time Consistency

Hanqing Jin, Yanzhao Yang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23187v1 Announce Type: new Abstract: In this paper, we propose a new framework for solving a general dynamic optimal stopping problem without time consistency. A sophisticated solution is proposed and is well-defined for any time setting with general flows of objectives. A backward iteration is proposed to find the solution. The iteration works with an additional condition, which holds in interesting cases including the time inconsistency arising from non-exponential discounting. Even if the iteration does not work, the equilibrium solution can still be studied by a forward definition.

Schopieray's Galois-modular extension conjecture

Theo Johnson-Freyd — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23192v1 Announce Type: new Abstract: Plavnik, Schopieray, Yu, and Zhang have drawn attention to those (automatically premodular) fusion subcategories of modular fusion categories which are submodules for the Galois action on the ambient category. In particular, they showed that a subcategory is a Galois submodule if and only if its centralizer is integral. In the other direction, Schopieray has conjectured that every premodular fusion category can be embedded as a Galois-closed subcategory of a modular category; Schopieray calls such an embedding a "Galois-modular extension." We prove Schopieray's conjecture for pseudounitary categories. Along the way we record some general comments about the minimal nondegenerate extension problem for braided fusion categories.

Non-uniformly elliptic variational problems on BV

Lisa Beck, Franz Gmeineder, Mathias Sch\"affner — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23195v1 Announce Type: new Abstract: We establish $\mathrm{W}^{1,1}$-regularity and higher gradient integrability for relaxed minimizers of convex integral functionals on $\mathrm{BV}$. Unlike classical examples such as the minimal surface integrand, we only require linear growth from below but not necessarily from above. This typically comes with a non-uniformly degenerate elliptic behaviour, for which our results extend the presently available bounds from the superlinear growth case in a sharp way.

Vector-valued Gelfand-Kazhdan criterion

Fulin Chen, Binyong Sun, Yixiang Weng — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23199v1 Announce Type: new Abstract: The Gelfand-Kazhdan criterion is a fundamental tool for studying multiplicity-one properties of local periods of representations. However, it does not apply to many cases arising in the relative Langlands program. Generalizing the usual Gelfand-Kazhdan criterion, we formulate and prove a vector-valued Gelfand-Kazhdan criterion that fits into the general framework of the relative Langlands program. As an illustration of its effectiveness, we establish the multiplicity-one property for the local Asai Rankin-Selberg periods.

A complete characterisation of conditional entropies

Roberto Rubboli, Erkka Haapasalo, Marco Tomamichel — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23213v1 Announce Type: new Abstract: Entropies are fundamental measures of uncertainty with central importance in information theory and statistics and applications across all the quantitative sciences. Under a natural set of operational axioms, the most general form of entropy is captured by the family of R\'enyi entropies, parameterized by a real number $\alpha$. Conditional entropy extends the notion of entropy by quantifying uncertainty from the viewpoint of an observer with access to potentially correlated side information. However, despite their significance and the emergence of various useful definitions, a complete characterization of measures of conditional entropy that satisfy a natural set of operational axioms has remained elusive. In this work, we provide a complete characterization of conditional entropy, defined through a set of axioms that are essential for any operationally meaningful definition: additivity for independent random variables, invariance under relabeling, and monotonicity under conditional mixing channels. We prove that the most general form of conditional entropy is captured by a family of measures that are exponential averages of R\'enyi entropies of the conditioned distribution and parameterized by a real parameter and a probability measure on the positive reals. Finally, we show that these quantities determine the rate of transformation under conditional mixing and provide a set of second laws of quantum thermodynamics with side information for states diagonal in the energy eigenbasis.

Secure Integrated Sensing and Communication against Communication and Sensing Eavesdropping

Sidong Guo, Matthieu R. Bloch — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23216v1 Announce Type: new Abstract: Sensing privacy and communication confidentiality play fundamentally different but interconnected roles in adversarial wireless environments. Capturing this interplay within a single physical-layer framework is particularly challenging in integrated sensing and communication (ISAC) systems, where the same waveform simultaneously serves dual purposes. We study a secure ISAC system in which a monostatic transmitter simultaneously sends a confidential message to a legitimate receiver and senses an environmental state, while a passive adversary attempts both message decoding and state estimation. We partially characterize the fundamental trade-offs among three performance measures: the transmitter's secrecy rate, its detection exponent, and the adversary's detection exponent. Beyond the joint input distribution that governs overall performance, the trade-offs are further shaped by the transmitter's ability to extract keys via feedback and hide both the content and structure of the codewords via wiretap and resolvability codes. We derive an achievable region, and illustrate the resulting design trade-offs through a numerical example.

Applications of QR-based Vector-Valued Rational Approximation

Simon Dirckx — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23237v1 Announce Type: new Abstract: Several applications of the QR-AAA algorithm, a greedy scheme for vector-valued rational approximation, are presented. The focus is on demonstrating the flexibility and practical effectiveness of QR-AAA in a variety of computational settings, including Stokes flow computation, multivariate rational approximation, function extension, the development of novel quadrature methods and near-field approximation in the boundary element method.

A Primal-Dual Level Set Method for Computing Geodesic Distances

Hailiang Liu, Laura Zinnel — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23244v1 Announce Type: new Abstract: The numerical computation of shortest paths or geodesics on surfaces, along with the associated geodesic distance, has a wide range of applications. Compared to Euclidean distance computation, these tasks are more complex due to the influence of surface geometry on the behavior of shortest paths. This paper introduces a primal-dual level set method for computing geodesic distances. A key insight is that the underlying surface can be implicitly represented as a zero level set, allowing us to formulate a constraint minimization problem. We employ the primal-dual methodology, along with regularization and acceleration techniques, to develop our algorithm. This approach is robust, efficient, and easy to implement. We establish a convergence result for the high-resolution PDE system, and numerical evidence suggests that the method converges to a geodesic in the limit of refinement.

Eigenweights for arithmetic Hirzebruch Proportionality

Tony Feng — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23245v1 Announce Type: new Abstract: Prior work of Feng--Yun--Zhang established a (Higher) Arithmetic Hirzebruch Proportionality Principle, expressing the arithmetic volumes of moduli stacks of shtukas in terms of differential operators applied to $L$-functions. This formula involves certain "eigenweights" which were calculated in simple cases by Feng--Yun--Zhang, but not in general. We document work of a (custom) AI Agent built upon Gemini Deep Think, which employs tools from algebraic combinatorics to connect these eigenweights to the representation theory of symmetric groups, and then determines them for all classical groups.

Radicals and Nilpotents in Equivariant Algebra

David Chan, Ben Spitz — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23247v1 Announce Type: new Abstract: Associated to each Tambara functor $T$ is its Nakaoka spectrum $\mathrm{Spec}(T)$, analogous to the Zariski spectrum of a commutative ring. We establish that this topological space is spectral. This result follows from an analysis of the notion of nilpotence in Tamabra functors. We prove that the nilradical of a Tambara functor $T$ (the intersection of all of its prime ideals) is computed levelwise, i.e. consists precisely of the nilpotent elements in $T$. In contrast to ordinary commutative algebra, the nilpotents of $T$ are not the same as the elements $x$ such that $T[1/x] = 0$; we therefore also give a classification of these elements. As a corollary, we observe that the set of these elements in $\pi_\star^s$ (the equivariant stable stems, viewed as an $\mathrm{RO}(G)$-graded Tambara functor) forms an ideal.

Theoretical Challenges in Learning for Branch-and-Cut

Hongyu Cheng, Amitabh Basu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23249v1 Announce Type: new Abstract: Machine learning is increasingly used to guide branch-and-cut (B&C) for mixed-integer linear programming by learning score-based policies for selecting branching variables and cutting planes. Many approaches train on local signals from lookahead heuristics such as strong branching, and linear programming (LP) bound improvement for cut selection. Training and evaluation of the learned models often focus on local score accuracy. We show that such local score-based methods can lead to search trees exponentially larger than optimal tree sizes, by identifying two sources of this gap. The first is that these widely used expert signals can be misaligned with overall tree size. LP bound improvement can select a root cut set that yields an exponentially larger strong branching tree than selecting cuts by a simple proxy score, and strong branching itself can be exponentially suboptimal (Dey et al., 2024). The second is that small discrepancies can be amplified by the branch-and-bound recursion. An arbitrarily small perturbation of the right-hand sides in a root cut set can change the minimum tree size from a single node to exponentially many. For branching, arbitrarily small score discrepancies, and differences only in tie-breaking, can produce trees of exponentially different sizes, and even a small number of decision differences along a trajectory can incur exponential growth. These results show that branch-and-cut policies trained and learned using local expert scores do not guarantee small trees, thus motivating the study of data-driven methods that produce policies better aligned with tree size rather than only accuracy on expert scores.

Geometric Quantization by Paths, Part III: The Metaplectic Anomaly

Patrick Iglesias-Zemmour — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23259v1 Announce Type: new Abstract: In the previous parts of this work, we established the Prequantum Groupoid $\mathbf{T}_\omega$ as the universal geometric container for quantum mechanics. This approach, which we call the "Geometric Quantization by Paths" (GQbP) framework, replaces the traditional construction of principal bundles with the distillation of the space of histories. In this third part, we cross the "Threshold of Analysis" by constructing the intrinsic observable algebra of the system. The harmonic oscillator is treated here as a validation case, demonstrating that the standard resolution via complex polarization and half-forms is naturally integrated into the GQbP framework. Starting from the complexified groupoid, we define the algebra using symplectic half-densities to ensure a canonical convolution product. We then show that the transition to a polarized representation forces a factorization of these densities. The action of the symmetry group on the polarized half-forms generates a divergence term, which we identify as the source of the zero-point energy of the harmonic oscillator, $E_0 = n\hbar/2$. This derivation resolves the "Metaplectic Anomaly" as a necessary geometric consequence of the intrinsic quantization process.

Rank Reduction AutoEncoders for Mechanical Design: Advancing Novel and Efficient Data-Driven Topology Optimization

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23269v1 Announce Type: new Abstract: This work presents a data-driven framework for fast forward and inverse analysis in topology optimization (TO) by combining Rank Reduction Autoencoders (RRAEs) with neural latent-space mappings. The methodology targets the efficient approximation of the relationship between optimized geometries and their corresponding mechanical responses or Quantity of Interest (QoI), with a particular focus on compliance-minimized linear elastic structures. High-dimensional TO results are first compressed using RRAEs, which encode the data into a low-rank approximation via Singular Value Decomposition (SVD), obtained in this sense the most important features that approximate the data. Separate RRAE models are trained for geometry and for different types of QoIs, including scalar metrics, one-dimensional stress fields, and full two-dimensional von Mises stress distributions. The resulting low-dimensional latent coefficients of the latent space are then related through multilayer perceptrons to address both direct problems -- predicting structural responses from geometry -- and inverse problems -- recovering geometries from prescribed performance targets. The proposed approach is demonstrated on a benchmark TO problem based on a half MBB beam, using datasets generated via density-based Solid Isotropic Material with Penalization (SIMP) optimization. Numerical results show that the framework enables accurate and computationally efficient surrogate models, with increasing robustness and fidelity as richer QoIs are considered. The methodology also provides a foundation for generative mechanical design by enabling the synthesis of new geometries and responses through latent-space exploration.

On graphs with girth at least five achieving Steffen's edge coloring bound

Guantao Chen, Alireza Fiujlaali, Anna Johnsen-Yu, Jessica McDonald — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23274v1 Announce Type: new Abstract: Vizing and Gupta showed that the chromatic index $\chi'(G)$ of a graph $G$ is bounded above by $\Delta(G) + \mu(G)$, where $\Delta(G)$ and $\mu(G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined this bound, proving that $\chi'(G) \leq \Delta(G) + \left\lceil \mu(G)/\left\lfloor g(G)/2 \right\rfloor \right\rceil$, where $g(G)$ is the girth of the graph $G$. A {\it ring graph} is a graph obtained from a cycle by duplicating some edges. The equality in Steffen's bound is achieved by ring graphs of the form $\mu C_g$, obtained from an odd cycle $C_g$ by duplicating each edge $\mu$ times. We answer two questions posed by Stiebitz et al. regarding the characterization of graphs which achieve Steffen's bound. In particular, we show that if $G$ is a critical graph which achieves Steffen's bound with $g(G)\geq 5$ and $\chi'(G)\geq \Delta+2$, then $G$ must be a ring graph of odd girth.

Variational Tail Bounds for Norms of Random Vectors and Matrices

Sohail Bahmani — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2503.17300v4 Announce Type: cross Abstract: We propose a variational tail bound for norms of random vectors under moment assumptions on their one-dimensional marginals. A simplified version of the bound that parametrizes the ``aggregating distribution'' using a certain pushforward of the Gaussian distribution is also provided. We apply the proposed method to reproduce some of the well-known bounds on norms of Gaussian random vectors, and also obtain dimension-free tail bounds for the Euclidean norm of random vectors with arbitrary moment profiles. Furthermore, we reproduce a dimension-free concentration inequality for sum of independent and identically distributed positive semidefinite matrices with sub-exponential marginals, and obtain a concentration inequality for the sample covariance matrix of sub-exponential random vectors. We also obtain a tail bound for the operator norm of a random matrix series whose random coefficients may have arbitrary moment profiles. Furthermore, we use coupling to formulate an abstraction of the proposed approach that applies more broadly.

Optimizing Shanghai's Household Waste Recycling Collection Program by Decision-Making based on Mathematical Modeling

Jiaxuan Chen, Ling Zhou Shen, Jinchen Liu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2507.03844v1 Announce Type: cross Abstract: In this article, we will discuss the optimization of Shanghai's recycling collection program, with the core of the task as making a decision among the choice of the alternatives. We will be showing a vivid and comprehensive application of the classical mathematical multi-criteria decision model: Analytical Hierarchy Process (AHP), using the eigenvector method. We will also seek the key criteria for the sustainability development of human society, by assessing the important elements of waste recycling.First, we considered the evaluation for a quantified score of the benefits and costs of recycling household glass wastes in Shanghai, respectively. In the evaluation of each score, we both adopted the AHP method to build a hierarchical structure of the problem we are facing. We first identified the key assessment criteria of the evaluation, on various perspectives including direct money costs and benefits, and further environmental and indirect considerations. Then, we distributed questionnaires to our school science teachers, taking the geometric mean, to build the pairwise comparison matrix of the criterion. After the theoretical modeling works are done, we began collecting the essential datasets for the evaluation of each score, by doing research on the official statistics, Internet information, market information and news reports. Sometimes, we proceed a logical pre-procession of the data from other data, if the data wanted isn't directly accessible. Then, we crucially considered the generalization of our mathematical model. We considered from several perspectives, including the extension of assessment criteria, and the consideration of the dynamic interdependency between the wastes, inside a limited transportation container.

Emergent spatial organization of competing species under environmental stress and cooperation

Ton Viet Ta — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22177v1 Announce Type: cross Abstract: Understanding how species persist under interacting stressors is a central challenge in ecology. We develop a spatially explicit reaction-diffusion framework to investigate competing species in landscapes shaped by climate variability, pollution, resource heterogeneity, and cooperation. Here, temperature follows low-frequency oscillations, while pollution and resources diffuse from localized sources. Growth is governed by a dynamic carrying capacity integrating abiotic stress with an endogenous, pollution-sensitive cooperation field. Numerical simulations reveal the spontaneous emergence of persistent spatial organization, including dominance segregation and stable competitive boundaries. Quantitative analyses-using boundary geometry, fractal dimension, and spatial entropy-demonstrate a transition from intermixed initial states to low-complexity, quasi-stationary configurations. Coexistence occurs through distinct strategies: one species occupies more area, while the other maintains higher local densities. Cooperation enhances resilience but collapses in polluted zones, creating heterogeneous "social buffering." We further introduce a hybrid inverse modeling framework using a Swin Transformer to infer high-dimensional parameters from only two temporal snapshots. Trained on synthetic data, the model accurately recovers demographic, diffusive, and environmental-sensitivity parameters. While it achieves reliable short-term spatial predictions, long-term forecasts diverge due to the intrinsic sensitivity of nonlinear systems. This unified framework links sparse observations to mechanistic dynamics, advancing biodiversity forecasting under accelerating global change.

Adaptive Benign Overfitting (ABO): Overparameterized RLS for Online Learning in Non-stationary Time-series

Luis Ontaneda Mijares, Nick Firoozye — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22200v1 Announce Type: cross Abstract: Overparameterized models have recently challenged conventional learning theory by exhibiting improved generalization beyond the interpolation limit, a phenomenon known as benign overfitting. This work introduces Adaptive Benign Overfitting (ABO), extending the recursive least-squares (RLS) framework to this regime through a numerically stable formulation based on orthogonal-triangular updates. A QR-based exponentially weighted RLS (QR-EWRLS) algorithm is introduced, combining random Fourier feature mappings with forgetting-factor regularization to enable online adaptation under non-stationary conditions. The orthogonal decomposition prevents the numerical divergence associated with covariance-form RLS while retaining adaptability to evolving data distributions. Experiments on nonlinear synthetic time series confirm that the proposed approach maintains bounded residuals and stable condition numbers while reproducing the double-descent behavior characteristic of overparameterized models. Applications to forecasting foreign exchange and electricity demand show that ABO is highly accurate (comparable to baseline kernel methods) while achieving speed improvements of between 20 and 40 percent. The results provide a unified view linking adaptive filtering, kernel approximation, and benign overfitting within a stable online learning framework.

Efficient learning of logical noise from syndrome data

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22286v1 Announce Type: cross Abstract: Characterizing errors in quantum circuits is essential for device calibration, yet detecting rare error events requires a large number of samples. This challenge is particularly severe in calibrating fault-tolerant, error-corrected circuits, where logical error probabilities are suppressed to higher order relative to physical noise and are therefore difficult to calibrate through direct logical measurements. Recently, Wagner et al. [PRL 130, 200601 (2023)] showed that, for phenomenological Pauli noise models, the logical channel can instead be inferred from syndrome measurement data generated during error correction. Here, we extend this framework to realistic circuit-level noise models. From a unified code-theoretic perspective and spacetime code formalism, we derive necessary and sufficient conditions for learning the logical channel from syndrome data alone and explicitly characterize the learnable degrees of freedom of circuit-level Pauli faults. Using Fourier analysis and compressed sensing, we develop efficient estimators with provable guarantees on sample complexity and computational cost. We further present an end-to-end protocol and demonstrate its performance on several syndrome-extraction circuits, achieving orders-of-magnitude sample-complexity savings over direct logical benchmarking. Our results establish syndrome-based learning as a practical approach to characterizing the logical channel in fault-tolerant quantum devices.

Exact closed-form Gaussian moments of residual layers

Simon Kuang, Xinfan Lin — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22307v1 Announce Type: cross Abstract: We study the problem of propagating the mean and covariance of a general multivariate Gaussian distribution through a deep (residual) neural network using layer-by-layer moment matching. We close a longstanding gap by deriving exact moment matching for the probit, GeLU, ReLU (as a limit of GeLU), Heaviside (as a limit of probit), and sine activation functions; for both feedforward and generalized residual layers. On random networks, we find orders-of-magnitude improvements in the KL divergence error metric, up to a millionfold, over popular alternatives. On real data, we find competitive statistical calibration for inference under epistemic uncertainty in the input. On a variational Bayes network, we show that our method attains hundredfold improvements in KL divergence from Monte Carlo ground truth over a state-of-the-art deterministic inference method. We also give an a priori error bound and a preliminary analysis of stochastic feedforward neurons, which have recently attracted general interest.

Quantum bootstrap product codes

Meng-Yuan Li — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22363v1 Announce Type: cross Abstract: Product constructions constitute a powerful method for generating quantum CSS codes, yielding celebrated examples such as toric codes and asymptotically good low-density parity check (LDPC) codes. Since a CSS code is fully described by a chain complex, existing product formalisms are predominantly homological, defined via the tensor product of the underlying chain complexes of input codes, thereby establishing a natural connection between quantum codes and topology. In this Letter, we introduce the \textit{quantum bootstrap product} (QBP), an approach that extends beyond this standard homological paradigm. Specifically, a QBP code is determined by solving a consistency condition termed the ``bootstrap equation''. We find that the QBP paradigm unifies a wide range of important codes, including general hypergraph product (HGP) codes of arbitrary dimensions and fracton codes typically represented by the X-cube code. Crucially, the solutions to the bootstrap equation yield chain complexes where the chain groups and associated boundary maps consist of multiple components. We term such structures \textit{fork complexes}. This structure elucidates the underlying topological structures of fracton codes, akin to foliated fracton order theories. Beyond conceptual insights, we demonstrate that the QBP paradigm can generate self-correcting quantum codes from input codes with constant energy barriers and surpass the code-rate upper bounds inherent to HGP codes. Our work thus substantially extends the scope of quantum product codes and provides a versatile framework for designing fault-tolerant quantum memories.

Proof Complexity of Linear Logics

Amirhossein Akbar Tabatabai, Raheleh Jalali — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22393v1 Announce Type: cross Abstract: Proving proof-size lower bounds for $\mathbf{LK}$, the sequent calculus for classical propositional logic, remains a major open problem in proof complexity. We shed new light on this challenge by isolating the power of structural rules, showing that their combination is extremely stronger than any single rule alone. We establish exponential (resp. sub-exponential) proof-size lower bounds for $\mathbf{LK}$ without contraction (resp. weakening) for formulas with short $\mathbf{LK}$-proofs. Concretely, we work with the Full Lambek calculus with exchange, $\mathbf{FL_e}$, and its contraction-extended variant, $\mathbf{FL_{ec}}$, substructural systems underlying linear logic. We construct families of $\mathbf{FL_e}$-provable (resp. $\mathbf{FL_{ec}}$-provable) formulas that require exponential-size (resp. sub-exponential-size) proofs in affine linear logic $\mathbf{ALL}$ (resp. relevant linear logic $\mathbf{RLL}$), but admit polynomial-size proofs once contraction (resp. weakening) is restored. This yields exponential lower bounds on proof-size of $\mathbf{FL_e}$-provable formulas in $\mathbf{ALL}$ and hence for $\mathbf{MALL}$, $\mathbf{AMALL}$, and full classical linear logic $\mathbf{CLL}$. Finally, we exhibit formulas with polynomial-size $\mathbf{FL_e}$-proofs that nevertheless require exponential-size proofs in cut-free $\mathbf{LK}$, establishing exponential speed-ups between various linear calculi and their cut-free counterparts.

Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erd\H{o}s Problems

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22401v1 Announce Type: cross Abstract: We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erd\H{o}s Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erd\H{o}s Problems.

On the undecidability of quantum channel capacities

Archishna Bhattacharyya, Arthur Mehta, Yuming Zhao — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22471v1 Announce Type: cross Abstract: An important distinction in our understanding of capacities of classical versus quantum channels is marked by the following question: is there an algorithm which can compute (or even efficiently compute) the capacity? While there is overwhelming evidence suggesting that quantum channel capacities may be uncomputable, a formal proof of any such statement is elusive. We initiate the study of the hardness of computing quantum channel capacities. We show that, for a general quantum channel, it is QMA-hard to compute its quantum capacity, and that the maximal-entanglement-assisted zero-error one-shot classical capacity is uncomputable.

Structural Conditions for Native CCZ Magic-State Fountains in qLDPC Codes

Mohammad Rowshan — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22489v1 Announce Type: cross Abstract: Quantum low-density parity-check (qLDPC) codes promise constant-rate, linear-distance families with bounded-weight checks, and recent work has realized transversal or constant-depth non-Clifford gates on various (often non-LDPC) codes. However, no explicit \emph{qubit} qLDPC family is known that simultaneously has constant rate, linear distance, bounded stabilizer weight, and a native \emph{magic-state fountain} that prepares many non-Clifford resource states in constant depth. We take a structural approach and identify coding-theoretic conditions under which a CSS qLDPC family necessarily supports a constant-depth $\CCZ$ magic-state fountain. The key ingredients are: (i) an algebraic notion of \emph{magic-friendly triples} of $X$-type logical operators, defined by pairwise orthogonality and a triple-overlap form controlling diagonal $\CCZ$ phases, and (ii) a 3-uniform hypergraph model of physical $\CCZ$ circuits combined with a packing lemma that turns large collections of such triples with bounded overlaps into bounded-degree hypergraphs. Our main theorem shows that if a CSS code family on $n$ qubits admits $\Omega(n^{1+\gamma})$ magic-friendly triples whose supports have bounded per-qubit participation, then there exists a constant-depth circuit of physical $\CCZ$ gates implementing $\Omega(n^{\gamma})$ logical $\CCZ$ gates in parallel while preserving distance up to a constant factor. For asymptotically good qLDPC families such as quantum Tanner codes, this reduces the existence of a native $\CCZ$ magic-state fountain to a concrete combinatorial problem about counting and distributing magic-friendly triples in the logical $X$ space.

Spin quantum Hall transition on random networks: exact critical exponents via quantum gravity

Esteban Mac\'ias, Ilya Gruzberg, Eldad Bettelheim — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22639v1 Announce Type: cross Abstract: We solve the problem of the spin quantum Hall transition on random networks using a mapping to classical percolation that focuses on the boundary of percolating clusters. Using tools of two-dimensional quantum gravity, we compute critical exponents that characterize this transition and confirm that these are related to the exponents for the regular (square) network through the KPZ relation. Our results demonstrate the relevance of the geometric randomness of the networks and support conclusions of numerical simulations of random networks for the integer quantum Hall transition.

Parametric vector flows for registration fields in bounded domains with applications to nonlinear interpolation of shock-dominated flows

Jon Labatut, Jean-Baptiste Chapelier, Angelo Iollo, Tommaso Taddei — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22712v1 Announce Type: cross Abstract: We present a registration procedure for parametric model order reduction (MOR) in two- and three-dimensional bounded domains. In the MOR framework, registration methods exploit solution snapshots to identify a parametric coordinate transformation that improves the approximation of the solution set through linear subspaces. For each training parameter, optimization-based (or variational) registration methods minimize a target function that measures the alignment of the coherent structures of interest (e.g., shocks, shear layers, cracks) for different parameter values, over a family of bijections of the computational domain $\Omega$. We consider diffeomorphisms $\Phi$ that are vector flows of given velocity fields $v$ with vanishing normal component on $\partial \Omega$; we rely on a sensor to extract appropriate point clouds from the solution snapshots and we develop an expectation-maximization procedure to simultaneously solve the point cloud matching problem and to determine the velocity $v$ (and thus the bijection $\Phi$); finally, we combine our registration method with the nonlinear interpolation technique of [Iollo, Taddei, J. Comput. Phys., 2022] to perform accurate interpolations of fluid dynamic fields in the presence of shocks. Numerical results for a two-dimensional inviscid transonic flow past a NACA airfoil and a three-dimensional viscous transonic flow past an ONERA M6 wing illustrate the many elements of the methodology and demonstrate the effectiveness of nonlinear interpolation for shock-dominated fields.

Geometric Selection Rules for Singularity Formation in Modified Gravity

Soumya Chakrabarti — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22739v1 Announce Type: cross Abstract: We argue that the polynomial degeneracies of curvature invariants can act as geometric selection rules for spacetime singularities in modified theories of gravity. The degeneracies arise purely from the algebraic structure of Riemannian geometry and impose non-trivial constraints on the effective energy-momentum tensor. We derive these constraints for metric $f(R)$ gravity and a wide class of scalar-tensor theories to show that a singularity formation is generally occluded by curvature and/or scalar-induced anisotropies. Therefore, formation of a singularity in modified theories of gravity is not always a generic outcome but can occur only along algebraically admissible branches selected by Riemannian curvature invariants.

Discovering Scaling Exponents with Physics-Informed M\"untz-Sz\'asz Networks

Gnankan Landry Regis N'guessan, Bum Jun Kim — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22751v1 Announce Type: cross Abstract: Physical systems near singularities, interfaces, and critical points exhibit power-law scaling, yet standard neural networks leave the governing exponents implicit. We introduce physics-informed M"untz-Sz'asz Networks (MSN-PINN), a power-law basis network that treats scaling exponents as trainable parameters. The model outputs both the solution and its scaling structure. We prove identifiability, or unique recovery, and show that, under these conditions, the squared error between learned and true exponents scales as $O(|\mu - \alpha|^2)$. Across experiments, MSN-PINN achieves single-exponent recovery with 1--5% error under noise and sparse sampling. It recovers corner singularity exponents for the two-dimensional Laplace equation with 0.009% error, matches the classical result of Kondrat'ev (1967), and recovers forcing-induced exponents in singular Poisson problems with 0.03% and 0.05% errors. On a 40-configuration wedge benchmark, it reaches a 100% success rate with 0.022% mean error. Constraint-aware training encodes physical requirements such as boundary condition compatibility and improves accuracy by three orders of magnitude over naive training. By combining the expressiveness of neural networks with the interpretability of asymptotic analysis, MSN-PINN produces learned parameters with direct physical meaning.

Conditional Performance Guarantee for Large Reasoning Models

Jianguo Huang, Hao Zeng, Bingyi Jing, Hongxin Wei, Bo An — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22790v1 Announce Type: cross Abstract: Large reasoning models have shown strong performance through extended chain-of-thought reasoning, yet their computational cost remains significant. Probably approximately correct (PAC) reasoning provides statistical guarantees for efficient reasoning by adaptively switching between thinking and non-thinking models, but the guarantee holds only in the marginal case and does not provide exact conditional coverage. We propose G-PAC reasoning, a practical framework that provides PAC-style guarantees at the group level by partitioning the input space. We develop two instantiations: Group PAC (G-PAC) reasoning for known group structures and Clustered PAC (C-PAC) reasoning for unknown groupings. We prove that both G-PAC and C-PAC achieve group-conditional risk control, and that grouping can strictly improve efficiency over marginal PAC reasoning in heterogeneous settings. Our experiments on diverse reasoning benchmarks demonstrate that G-PAC and C-PAC successfully achieve group-conditional risk control while maintaining substantial computational savings.

Aligning the Unseen in Attributed Graphs: Interplay between Graph Geometry and Node Attributes Manifold

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22806v1 Announce Type: cross Abstract: The standard approach to representation learning on attributed graphs -- i.e., simultaneously reconstructing node attributes and graph structure -- is geometrically flawed, as it merges two potentially incompatible metric spaces. This forces a destructive alignment that erodes information about the graph's underlying generative process. To recover this lost signal, we introduce a custom variational autoencoder that separates manifold learning from structural alignment. By quantifying the metric distortion needed to map the attribute manifold onto the graph's Heat Kernel, we transform geometric conflict into an interpretable structural descriptor. Experiments show our method uncovers connectivity patterns and anomalies undetectable by conventional approaches, proving both their theoretical inadequacy and practical limitations.

Robust Rigid Body Assembly via Contact-Implicit Optimal Control with Exact Second-Order Derivatives

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22849v1 Announce Type: cross Abstract: Efficient planning of assembly motions is a long standing challenge in the field of robotics that has been primarily tackled with reinforcement learning and sampling-based methods by using extensive physics simulations. This paper proposes a sample-efficient robust optimal control approach for the determination of assembly motions, which requires significantly less physics simulation steps during planning through the efficient use of derivative information. To this end, a differentiable physics simulation is constructed that provides second-order analytic derivatives to the numerical solver and allows one to traverse seamlessly from informative derivatives to accurate contact simulation. The solution of the physics simulation problem is made differentiable by using smoothing inspired by interior-point methods applied to both the collision detection as well as the contact resolution problem. We propose a modified variant of an optimization-based formulation of collision detection formulated as a linear program and present an efficient implementation for the nominal evaluation and corresponding first- and second-order derivatives. Moreover, a multi-scenario-based trajectory optimization problem that ensures robustness with respect to sim-to-real mismatches is derived. The capability of the considered formulation is illustrated by results where over 99\% successful executions are achieved in real-world experiments. Thereby, we carefully investigate the effect of smooth approximations of the contact dynamics and robust modeling on the success rates. Furthermore, the method's capability is tested on different peg-in-hole problems in simulation to show the benefit of using exact Hessians over commonly used Hessian approximations.

A Framework for the Bayesian Calibration of Complex and Data-Scarce Models in Applied Sciences

Christina Schenk, Ignacio Romero — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22890v1 Announce Type: cross Abstract: In this work, we review the theory involved in the Bayesian calibration of complex computer models, with particular emphasis on their use for applications involving computationally expensive simulations and scarce experimental data. In the article, we present a unified framework that incorporates various Bayesian calibration methods, including well-established approaches. Furthermore, we describe their implementation and use with a new, open-source Python library, ACBICI (A Configurable BayesIan Calibration and Inference Package). All algorithms are implemented with an object-oriented structure designed to be both easy to use and readily extensible. In particular, single-output and multiple-output calibration are addressed in a consistent manner. The article completes the theory and its implementation with practical recommendations for calibrating the problems of interest. These guidelines -- currently unavailable in a unified form elsewhere -- together with the open-source Python library, are intended to support the reliable calibration of computational codes and models commonly used in engineering and related fields. Overall, this work aims to serve both as a comprehensive review of the statistical foundations and (computational) tools required to perform such calculations, and as a practical guide to Bayesian calibration with modern software tools.

A categorical account of the Metropolis-Hastings algorithm

Rob Cornish, Andi Q. Wang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22911v1 Announce Type: cross Abstract: Metropolis-Hastings (MH) is a foundational Markov chain Monte Carlo (MCMC) algorithm. In this paper, we ask whether it is possible to formulate and analyse MH in terms of categorical probability, using a recent involutive framework for MH-type procedures as a concrete case study. We show how basic MCMC concepts such as invariance and reversibility can be formulated in Markov categories, and how one part of the MH kernel can be analysed using standard CD categories. To go further, we then study enrichments of CD categories over commutative monoids. This gives an expressive setting for reasoning abstractly about a range of important probabilistic concepts, including substochastic kernels, finite and $\sigma$-finite measures, absolute continuity, singular measures, and Lebesgue decompositions. Using these tools, we give synthetic necessary and sufficient conditions for a general MH-type sampler to be reversible with respect to a given target distribution.

Open strings on knot complements

Sachin Chauhan, Tobias Ekholm, Pietro Longhi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.22922v1 Announce Type: cross Abstract: Using skein valued holomorphic curve counting techniques, we give a flow loop formula for the skein valued partition function of the Lagrangian knot complement of a fibered knot (of the $A$-model open topological strings with Lagrangian $A$-branes wrapping the complement) in the cotangent bundle of the three-sphere and in the resolved conifold. For torus knots we show that the partition function in the cotangent bundle localizes on two or three holomorphic annuli and give a corresponding generalized quiver structure for the partition function in the resolved conifold. We connect the formula to the augmentation curve, the representation variety of the knot contact homology algebra of the knot, generated by Reeb chords of its Legendrian conormal and with differential given by holomorphic disks interpolating between words of Reeb chords. The curve admits a quantization as a $q$-difference equation for the generating function of symmetrically colored HOMFLYPT-polynomials of the knot or, geometrically, for the $U(1)$-partition function of the knot conormal. For $(2,2p+1)$-torus knots we show that, after a change of variables, the partition function of the knot complement also satisfies this $q$-difference equation. This gives another geometrically defined coordinate chart for the $D$-module defined by the quantized augmentation polynomial.

Robust Control of Constrained Linear Systems using Online Convex Optimization and a Reference Governor

Marko Nonhoff, Mohammad Taher Al Torshan, Matthias A. M\"uller — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23160v1 Announce Type: cross Abstract: This article develops a control method for linear time-invariant systems subject to time-varying and a priori unknown cost functions, that satisfies state and input constraints, and is robust to exogenous disturbances. To this end, we combine the online convex optimization framework with a reference governor and a constraint tightening approach. The proposed framework guarantees recursive feasibility and robust constraint satisfaction. Its closed-loop performance is studied in terms of its dynamic regret, which is bounded linearly by the variation of the cost functions and the magnitude of the disturbances. The proposed method is illustrated by a numerical case study of a tracking control problem.

Causal spinfoam vertex for 4d Lorentzian quantum gravity

Eugenio Bianchi, Chaosong Chen, Mauricio Gamonal — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23162v1 Announce Type: cross Abstract: We introduce a new causal spinfoam vertex for $4$d Lorentzian quantum gravity. The causal data are encoded in Toller $T$-matrices, which add to Wigner $D$-matrices $T^{(+)}+T^{(-)}=D$, and for which we provide a Feynman $\mathrm{i}\varepsilon$ representation. We discuss how the Toller poles cancel in the EPRL vertex, how the Livine-Oriti model is obtained in the Barrett-Crane limit, and how spinfoam causal data are distinct from Regge causal data. In the large-spin limit, we show that only Lorentzian Regge geometries with causal data compatible with the spinfoam data are selected, resulting in a single exponential $\exp(+\mathrm{i}\, S_{\mathrm{Regge}}/\hbar)$ and a new form of causal rigidity.

A unified theory of order flow, market impact, and volatility

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23172v1 Announce Type: cross Abstract: We propose a microstructural model for the order flow in financial markets that distinguishes between {\it core orders} and {\it reaction flow}, both modeled as Hawkes processes. This model has a natural scaling limit that reconciles a number of salient empirical properties: persistent signed order flow, rough trading volume and volatility, and power-law market impact. In our framework, all these quantities are pinned down by a single statistic $H_0$, which measures the persistence of the core flow. Specifically, the signed flow converges to the sum of a fractional process with Hurst index $H_0$ and a martingale, while the limiting traded volume is a rough process with Hurst index $H_0-1/2$. No-arbitrage constraints imply that volatility is rough, with Hurst parameter $2H_0-3/2$, and that the price impact of trades follows a power law with exponent $2-2H_0$. The analysis of signed order flow data yields an estimate $H_0 \approx 3/4$. This is not only consistent with the square-root law of market impact, but also turns out to match estimates for the roughness of traded volumes and volatilities remarkably well.

YuriiFormer: A Suite of Nesterov-Accelerated Transformers

Aleksandr Zimin, Yury Polyanskiy, Philippe Rigollet — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23236v1 Announce Type: cross Abstract: We propose a variational framework that interprets transformer layers as iterations of an optimization algorithm acting on token embeddings. In this view, self-attention implements a gradient step of an interaction energy, while MLP layers correspond to gradient updates of a potential energy. Standard GPT-style transformers emerge as vanilla gradient descent on the resulting composite objective, implemented via Lie--Trotter splitting between these two energy functionals. This perspective enables principled architectural design using classical optimization ideas. As a proof of concept, we introduce a Nesterov-style accelerated transformer that preserves the same attention and MLP oracles. The resulting architecture consistently outperforms a nanoGPT baseline on TinyStories and OpenWebText, demonstrating that optimization-theoretic insights can translate into practical gains.

Graph Attention Network for Node Regression on Random Geometric Graphs with Erd\H{o}s--R\'enyi contamination

Somak Laha, Suqi Liu, Morgane Austern — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23239v1 Announce Type: cross Abstract: Graph attention networks (GATs) are widely used and often appear robust to noise in node covariates and edges, yet rigorous statistical guarantees demonstrating a provable advantage of GATs over non-attention graph neural networks~(GNNs) are scarce. We partially address this gap for node regression with graph-based errors-in-variables models under simultaneous covariate and edge corruption: responses are generated from latent node-level covariates, but only noise-perturbed versions of the latent covariates are observed; and the sample graph is a random geometric graph created from the node covariates but contaminated by independent Erd\H{o}s--R\'enyi edges. We propose and analyze a carefully designed, task-specific GAT that constructs denoised proxy features for regression. We prove that regressing the response variables on the proxies achieves lower error asymptotically in (a) estimating the regression coefficient compared to the ordinary least squares (OLS) estimator on the noisy node covariates, and (b) predicting the response for an unlabelled node compared to a vanilla graph convolutional network~(GCN) -- under mild growth conditions. Our analysis leverages high-dimensional geometric tail bounds and concentration for neighbourhood counts and sample covariances. We verify our theoretical findings through experiments on synthetically generated data. We also perform experiments on real-world graphs and demonstrate the effectiveness of the attention mechanism in several node regression tasks.

Decoupled Diffusion Sampling for Inverse Problems on Function Spaces

Thomas Y. L. Lin, Jiachen Yao, Lufang Chiang, Julius Berner, Anima Anandkumar — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.23280v1 Announce Type: cross Abstract: We propose a data-efficient, physics-aware generative framework in function space for inverse PDE problems. Existing plug-and-play diffusion posterior samplers represent physics implicitly through joint coefficient-solution modeling, requiring substantial paired supervision. In contrast, our Decoupled Diffusion Inverse Solver (DDIS) employs a decoupled design: an unconditional diffusion learns the coefficient prior, while a neural operator explicitly models the forward PDE for guidance. This decoupling enables superior data efficiency and effective physics-informed learning, while naturally supporting Decoupled Annealing Posterior Sampling (DAPS) to avoid over-smoothing in Diffusion Posterior Sampling (DPS). Theoretically, we prove that DDIS avoids the guidance attenuation failure of joint models when training data is scarce. Empirically, DDIS achieves state-of-the-art performance under sparse observation, improving $l_2$ error by 11% and spectral error by 54% on average; when data is limited to 1%, DDIS maintains accuracy with 40% advantage in $l_2$ error compared to joint models.

Cubical approximation for directed topology I

Sanjeevi Krishnan — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:1012.0509v3 Announce Type: replace Abstract: Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than classical homotopy invariants on underlying spaces because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with temporal structure encoding the orientations of simplices and 1-cubes. In an attempt to develop calculational tools for directed homotopy theory, we prove appropriate simplicial and cubical approximation theorems. We consequently show that geometric realization induces an equivalence between weak homotopy diagram categories of cubical sets and directed spaces and that its right adjoint satisfies an excision theorem. Along the way, we give criteria for two different homotopy relations on directed maps in the literature to coincide.

Chernoff bounds for branching random walks

Changqing Liu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:1604.00056v5 Announce Type: replace Abstract: Concentration inequalities, which have proved very useful in a variety of fields, provide fairly tight bounds on large deviation probabilities while central limit theorem (CLT) describes the asymptotic distribution around the mean (at the $\sqrt{n}$ scale). Harris (1963) conjectured that for a supercritical branching random walk (BRW) of i.i.d offspring and i.i.d displacements, positions of individuals in $nth$ generation approach to Gaussian distribution -- central limit theorem. This conjecture was later proved by Stam (1966) and Kaplan \& Asmussen (1976). Refinements and extensions followed. However, to the best of our knowledge, there is no corresponding existing work on concentration inequalities for BRWs. In this note, we propose a new definition of BRW, providing a more general framework. Owing to this definition, a Chernoff-type (subgaussian) bound for BRWs follows directly from the Chernoff bound for random walk. The relation between RW (random walk) and BRW is discussed.

The Partition-Frequency Enumeration Matrix

Hartosh Singh Bal, Gaurav Bhatnagar — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2102.04191v3 Announce Type: replace Abstract: We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan's $\tau$ function, sums of squares and triangular numbers, and for $\zeta(2n)$, where $n$ is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer and others. As one application, we embed Ramanujan's famous congruences $p(5n+4)\equiv 0$ (mod $5)$ and $\tau(5n+5)\equiv 0$ (mod $5)$ into an infinite family of such congruences.

On quasi-isospectrality of potentials and Riemannian manifolds

Clara L. Aldana, Camilo Perez — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2202.06110v5 Announce Type: replace Abstract: In this article, we study quasi-isospectral operators as a generalization of isospectral operators. The paper contains both expository material and original results. We begin by reviewing known results on isospectral potentials on compact manifolds and finite intervals, and then introduce the notion of quasi-isospectrality. We next investigate the BMT method as a systematic approach to constructing quasi-isospectral Sturm-Liouville operators on a finite interval, and apply it to several boundary value problems. Our main result shows that any two quasi-isospectral closed manifolds of odd dimension are, in fact, isospectral. In addition, we extend classical compactness results for isospectral potentials on low-dimensional manifolds to the quasi-isospectral setting via heat trace asymptotics.

On classification of continuous first order theories

Karim Khanaki — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2205.12051v3 Announce Type: replace Abstract: We give several new characterizations of $IP$ (the independence property) and $SOP$ (the strict order property) for continuous first order logic and study their relations to the function theory and the Banach space theory. We suggest new dividing lines of unstable theories by the study of subclasses of Baire-1 functions and argue why one should not expect a perfect analog of Shelah's theorem, namely a theory is unstable iff it has $IP$ or $SOP$, for real-valued logics, especially for continuous logic.

Ramification theory from homotopical point of view, I

Tomoyuki Abe — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2206.02401v2 Announce Type: replace Abstract: We prove the compatibility of pushforward along a proper morphism of an \'{e}tale constructible sheaf and the pushforward of its characteristic cycle up to $p$-torsion. This was conjectured by Takeshi Saito. For this, we revisit the construction of the characteristic cycle, due to Saito and Beilinson, from more homotopical point of view. In particular, the language of $\infty$-categories is indispensable to carry this out.

Cardinal Optimizer (COPT) User Guide

Dongdong Ge, Qi Huangfu, Zizhuo Wang, Jian Wu, Yinyu Ye — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2208.14314v4 Announce Type: replace Abstract: Cardinal Optimizer is a high-performance mathematical programming solver for efficiently solving largescale optimization problem. This documentation provides basic introduction to the Cardinal Optimizer.

Conjectures on the reduced Kronecker coefficients

Tao Gui — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2210.14668v4 Announce Type: replace Abstract: We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of Okounkov's conjecture on the log-concavity of the Littlewood--Richardson coefficients and the Schur log-concavity theorem of Lam--Postnikov--Pylyavskyy. We prove our conjectures in some special cases and discuss some implications of these conjectures.

Optimal Stabilization of Periodic Orbits: A Symplectic Geometry Approach

Fabian Beck, Noboru Sakamoto — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2211.11955v3 Announce Type: replace Abstract: In this contribution, the optimal stabilization problem of periodic orbits is studied via invariant manifold theory and symplectic geometry. The stable manifold theory for the optimal point stabilization case is generalized to the case of periodic orbit stabilization, where a normally hyperbolic invariant manifold plays the role of a hyperbolic equilibrium point. A sufficient condition for the existence of an NHIM of an extended Hamiltonian system is derived in terms of a periodic Riccati differential equation. It is shown that the problem of optimal orbit stabilization has a solution if a linearized periodic system is stabilizable and detectable. A moving orthogonal coordinate system is employed along the periodic orbit, which is a natural framework for orbital stabilization and linearization along the orbit. Two illustrative examples are presented: the first involves stabilizing a spring-mass oscillator at a target energy level, and the second addresses an orbit transfer problem for a satellite-a classic scenario in orbital mechanics. In both cases, we show that the proposed nonlinear feedback controller outperforms traditional linear control.

A universal sheaf of algebras governing representations of vector fields on quasi-projective varieties

Yuly Billig, Colin Ingalls — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2302.07918v2 Announce Type: replace Abstract: We construct a quasi-coherent sheaf of associative algebras which controls a category of $AV$-modules over a smooth quasi-projective variety. We establish a local structure theorem, proving that in \'etale charts these associative algebras decompose into a tensor product of the algebra of differential operators and the universal enveloping algebra of the Lie algebra of power series vector fields vanishing at the origin.

On the finite time blow-ups for solutions of nonlinear differential equations

Luan Hoang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2303.10153v2 Announce Type: replace Abstract: We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\alpha$ for some $\alpha>0$. We prove that the asymptotic behavior of a solution $y(t)$ near a finite blow-up time $T_*$ is $(T_*-t)^{-1/\alpha}\xi_*$ for some nonzero vector $\xi_*$. Specific error estimates for $|(T_*-t)^{1/\alpha}y(t)-\xi_*|$ are provided. In some typical cases, they can be a positive power of $(T_*-t)$ or $1/|\ln(T_*-t)|$. This depends on whether the decaying rate of the lower order term, relative to the size of the dominant term, is of a power or logarithmic form. Similar results are obtained for a class of nonlinear differential inequalities with finite time blow-up solutions. Our results cover larger classes of nonlinear equations, differential inequalities and error estimates than those in the previous work.

Higher topological complexity of Seifert fibered manifolds

Navnath Daundkar, Rekha Santhanam, Soumyadip Thandar — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2304.01274v3 Announce Type: replace Abstract: In this article, we investigate the higher topological complexity of oriented Seifert fibered manifolds that are Eilenberg--MacLane spaces $K(G,1)$ with infinite fundamental group $G$. We first refine the cohomological lower bounds for higher topological complexity by introducing the notion of higher topological complexity weights. As an application, we show that the $r^{\text{th}}$ topological complexity of these manifolds lies in $\{3r-1, 3r, 3r+1\}$, and characterize large families where the value is $3r$ or $3r+1$. Additionally, we establish a sufficient condition for higher topological complexity to be exactly $3r$ when the base surface is orientable and aspherical. Finally, we show that the higher topological complexity of the wedge of finitely many closed, orientable, aspherical $3$-manifolds is exactly $3r+1$.

Amending the Lonely Runner Spectrum Conjecture

Ho Tin Fan, Alec Sun — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2306.10417v2 Announce Type: replace Abstract: Let $||x||$ be the absolute distance from $x$ to the nearest integer. For a set of distinct positive integral speeds $v_1, \ldots, v_n$, we define its maximum loneliness, also known as the gap $\delta$, to be $$ML(v_1,\ldots,v_n) = \max_{t \in \mathbb{R}}\min_{1 \leq i \leq n} || tv_i||.$$ The Loneliness Spectrum Conjecture, recently proposed by Kravitz (2021), asserts that $$\exists s \in \mathbb{N}, \text{ML}(v_1,\ldots,v_n) = \frac{s} {sn + 1} \text{ or } \text{ML}(v_1,\ldots,v_n) \geq \frac{1}{n}. $$ We disprove the Loneliness Spectrum Conjecture for $n = 4$ with an infinite family of counterexamples and propose an alternative conjecture. We confirm the amended conjecture for $n = 4$ whenever there exists a pair of speeds with a common factor of at least $3$ and also prove some related results.

Tilings in quasi-random $k$-partite hypergraphs

Shumin Sun — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2306.10539v2 Announce Type: replace Abstract: Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an $F$-factor in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. Lenz and Mubayi were first to study the $F$-factor problems in quasi-random $k$-graphs with a minimum degree condition. Recently, Ding, Han, Sun, Wang and Zhou gave the density threshold for having all $3$-partite $3$-graphs factors in quasi-random $3$-graphs with vanishing minimum codegree condition $\Omega(n)$. In this paper, we consider embedding factors when the host $k$-graph is $k$-partite and quasi-random with partite minimum codegree condition. We prove that if $p>1/2$ and $F$ is a $k$-partite $k$-graph with each part having $m$ vertices, then for $n$ large enough and $m\mid n$, any $p$-dense $k$-partite $k$-graph with each part having $n$ vertices and partite minimum codegree condition $\Omega(n)$ contains an $F$-factor. We also present a construction showing that $1/2$ is best possible. Furthermore, for $1\leq \ell \leq k-2$, by constructing a sequence of $p$-dense $k$-partite $k$-graphs with partite minimum $\ell$-degree $\Omega(n^{k-\ell})$ having no $K_k(m)$-factor, we show that the partite minimum codegree constraint can not be replaced by other partite minimum degree conditions. On the other hand, we prove that $n/2$ is the asymptotic partite minimum codegree threshold for having all fixed $k$-partite $k$-graph factors in sufficiently large host $k$-partite $k$-graphs even without quasi-randomness.

Quartic Gauss sums over primes and metaplectic theta functions

Chantal David, Alexander Dunn, Alia Hamieh, Hua Lin — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2306.11875v5 Announce Type: replace Abstract: We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over $\mathbb{Q}(i)$, and Suzuki's evaluation of the Fourier-Whittaker coefficients of quartic theta functions at squares. We also conjecture asymptotics for certain moments of quartic Gauss sums over primes.

The center of the asymptotic Hecke category and unipotent character sheaves

Liam Rogel, Ulrich Thiel — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2307.07276v3 Announce Type: replace Abstract: In 2015, Lusztig [Bull. Inst. Math. Acad. Sin. (N.S.)10(2015), no.1, 1-72] showed that for a connected reductive group over an algebraic closure of a finite field the associated (geometric) Hecke category admits a truncation in a two-sided Kazhdan--Lusztig cell, making it a categorification of the asymptotic algebra (J-ring), and that the categorical center of this "asymptotic Hecke category" is equivalent to the category of unipotent character sheaves supported in the cell. Subsequently, Lusztig noted that an asymptotic Hecke category can be constructed for any finite Coxeter group using Soergel bimodules. Lusztig conjectured that the centers of these categories are modular tensor categories (which was then proven by Elias and Williamson) and that for non-crystallographic finite Coxeter groups the S-matrices coincide with the Fourier matrices that were constructed in the 1990s by Lusztig, Malle, and Brou\'e--Malle. If the conjecture is true, the centers may be considered as categories of "unipotent character sheaves" for non-crystallographic finite Coxeter groups. In this paper, we show that the conjecture is true for dihedral groups and for some (we cannot resolve all) cells of H3 and H4. The key ingredient is the method of H-reduction and the identification of the (reduced) asymptotic Hecke category with known categories whose center is already known as well. We conclude by studying the asymptotic Hecke category and its center for some infinite Coxeter groups with a finite cell.

Logarithmic Asymptotic Relations Between $p$-Values and Mutual Information

Tsutomu Mori, Takashi Kawamura — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2308.14735v2 Announce Type: replace Abstract: We establish a precise connection between statistical significance in dependence testing and information-theoretic dependence as quantified by Shannon mutual information (MI). In the absence of prior distributional information, we consider a maximum-entropy model and show that the probability associated with the realization of a given magnitude of MI takes an exponential form, yielding a corresponding tail-probability interpretation of a $p$-value. In contingency tables with fixed marginal frequencies, we analyze Fisher's exact test and prove that its $p$-value $P_F$ satisfies a logarithmic asymptotic relation of the form $MI=-(1/N)\log P_F + O(\log(N+1)/N)$ as the sample size $N\to\infty$. These results clarify the role of MI as the exponential rate governing the asymptotic behavior of $p$-values in the settings studied here, and they enable principled comparisons of dependence across datasets with different sample sizes. We further discuss implications for combining evidence across studies via meta-analysis, allowing mutual information and its statistical significance to be integrated in a unified framework.

Frank--Wolfe algorithms for piecewise star-convex functions with a nonsmooth difference-of-convex structure

R. D\'iaz Mill\'an, O. P. Ferreira, J. Ugon — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2308.16444v4 Announce Type: replace Abstract: In the present paper, we formulate two versions of Frank--Wolfe algorithm or conditional gradient method to solve the DC optimization problem with an adaptive step size. The DC objective function consists of two components; the first is thought to be differentiable with a continuous Lipschitz gradient, while the second is only thought to be convex. The second version is based on the first and employs finite differences to approximate the gradient of the first component of the objective function. In contrast to past formulations that used the curvature/Lipschitz-type constant of the objective function, the step size computed does not require any constant associated with the components. For the first version, we established that the algorithm is well-defined of the algorithm and that every limit point of the generated sequence is a stationary point of the problem. We also introduce the class of weak-star-convex functions and show that, despite the fact that these functions are non-convex in general, the rate of convergence of the first version of the algorithm to minimize these functions is ${\cal O}(1/k)$. The finite difference used to approximate the gradient in the second version of the Frank-Wolfe algorithm is computed with the step-size adaptively updated using two previous iterations. Unlike previous applications of finite difference in the Frank-Wolfe algorithm, which provided approximate gradients with absolute error, the one used here provides us with a relative error, simplifying the algorithm analysis. In this case, we show that all limit points of the generated sequence for the second version of the Frank-Wolfe algorithm are stationary points for the problem under consideration, and we establish that the rate of convergence for the duality gap is ${\cal O}(1/\sqrt{k})$.

A linearization map for genuine equivariant algebraic $K$-theory

Maxine Calle, David Chan, Andres Mejia — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2309.08025v4 Announce Type: replace Abstract: We introduce a version of algebraic $K$-theory for coefficient systems of rings which is valued in genuine $G$-spectra for a finite group $G$. We use this construction to build a genuine $G$-spectrum $K_G(\mathbb{Z}[\underline{\pi_1(X)}])$ associated to a $G$-space $X$, which provides a home for equivariant versions of classical invariants like the Wall finiteness obstruction and Whitehead torsion. We provide a comparison between our $K$-theory spectrum and the equivariant $A$-theory of Malkiewich--Merling via a genuine equivariant linearization map.

High-Dimensional Bernstein Von-Mises Theorems for Covariance and Precision Matrices

Partha Sarkar, Kshitij Khare, Malay Ghosh, Matt P. Wand — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2309.08556v3 Announce Type: replace Abstract: This paper aims to examine the characteristics of the posterior distribution of covariance/precision matrices in a "large $p$, large $n$" scenario, where $p$ represents the number of variables and $n$ is the sample size. Our analysis focuses on establishing asymptotic normality of the posterior distribution of the entire covariance/precision matrices under specific growth restrictions on $p_n$ and other mild assumptions. In particular, the limiting distribution turns out to be a symmetric matrix variate normal distribution whose parameters depend on the maximum likelihood estimate. Our results hold for a wide class of prior distributions which includes standard choices used by practitioners. Next, we consider Gaussian graphical models which induce sparsity in the precision matrix. Asymptotic normality of the corresponding posterior distribution is established under mild assumptions on the prior and true data-generating mechanism.

Imaginaries, products and the adele ring

Jamshid Derakhshan, Ehud Hrushovski — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2309.11678v3 Announce Type: replace Abstract: We describe the imaginary sorts of infinite products in terms of imaginary sorts of the factors. We extend the result to certain reduced powers and then to infinite products $\prod_{i\in I} M_i$ enriched with a predicate for the ideal of finite subsets of $I$. As a special case, using the Hils-Rideau-Kikuchi uniform $p$-adic elimination of imaginaries, we find the imaginary sorts of the ring of rational adeles. Our methods include the use of the Harrington-Kechris-Louveau Glimm-Efros dichotomy both for transitioning from monadic second order imaginaries to first-order reducts, and for proving a certain ``one-way'' model-theoretic orthogonality within the adelic imaginaries.

Cubical Approximation for Directed Topology II

Sanjeevi Krishnan — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2309.16619v4 Announce Type: replace Abstract: The paper establishes an equivalence between directed homotopy categories of (diagrams of) cubical sets and (diagrams of) directed topological spaces. This equivalence both lifts and extends an equivalence between classical homotopy categories of cubical sets and topological spaces. Some simple applications include combinatorial descriptions and subsequent calculations of directed homotopy monoids and directed singular 1-cohomology monoids. Another application is a characterization of isomorphisms between small categories up to zig-zags of natural transformations as directed homotopy equivalences between directed classifying spaces. Cubical sets throughout the paper are taken to mean presheaves over the minimal symmetric monoidal variant of the cube category. Along the way, the paper characterizes morphisms in this variant as the interval-preserving lattice homomorphisms between finite Boolean lattices.

Balanced metrics, Zoll deformations and isosystolic inequalities in $\mathbb{C}P^n$

Luciano L. Junior — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2310.10877v2 Announce Type: replace Abstract: The k-systole of a Riemannian manifold is the infimum of the volume over all homologically non-trivial k-cycles. In this paper we discuss the behavior of the dimension two and co-dimension two systole of the complex projective space for distinguished classes of metrics, namely the homogeneous metrics and the balanced metrics. In particular, we argue that every homogeneous metric maximizes the systole in its volume-normalized conformal class, as well as that each K\"ahler metric locally minimizes the systole on the set of volume-normalized balanced metrics. The proof demands the implementation of integral geometric techniques, and a careful analysis of the second variation of the systole functional. As an application, we characterize the systolic behavior of almost-Hermitian 1-parameter Zoll-like deformations of the Fubini-Study metric.

Symmetry-Enforced Quadratic Degradability Beyond Low Dimensions

Yun-Feng Lo, Yen-Chi Lee, Min-Hsiu Hsieh — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2401.16312v5 Announce Type: replace Abstract: Approximate degradability provides a powerful framework for bounding the quantum and private capacities of noisy quantum channels in regimes where exact degradability fails. While generic low-noise channels exhibit a non-degradability parameter that decays as a fractional power of the noise strength, certain symmetric channels are known to display an enhanced quadratic suppression. In this work, we investigate the structural origin of this phenomenon through a family of high-dimensional, rotationally symmetric noise models constructed from angular momentum operators. We first establish that the pure noise component of these channels is maximally distinguishable from the identity channel in diamond norm, revealing a geometric orthogonality between signal and noise. Building on this structure, we construct an explicit symmetric degrading map and prove that the approximate degradability parameter scales quadratically with the noise parameter for all system dimensions. To clarify the mechanism behind this behavior, we identify algebraic conditions on the noise operators that guarantee the cancellation of leading-order non-degradability terms. These conditions apply not only to the rotationally symmetric model studied here, but also to a distinct family of high-dimensional depolarizing channels based on discrete unitary operator bases. Numerical evaluations of capacity lower bounds further illustrate the practical impact of the quadratic suppression. Together, these results demonstrate that enhanced approximate degradability arises from symmetry-induced orthogonality and invariance properties, rather than from low-dimensional or model-specific effects.

Optimal sampling for stochastic and natural gradient descent

Robert Gruhlke, Anthony Nouy, Philipp Trunschke — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2402.03113v2 Announce Type: replace Abstract: We consider the problem of optimising the expected value of a loss functional over a nonlinear model class of functions, assuming that we have only access to realisations of the gradient of the loss. This is a classical task in statistics, machine learning and physics-informed machine learning. A straightforward solution is to replace the exact objective with a Monte Carlo estimate before employing standard first-order methods like gradient descent, which yields the classical stochastic gradient descent method. But replacing the true objective with an estimate ensues a generalisation error. Rigorous bounds for this error typically require strong compactness and Lipschitz continuity assumptions while providing a very slow decay with sample size. To alleviate these issues, we propose a version of natural gradient descent that is based on optimal sampling methods. Under classical assumptions on the loss and the nonlinear model class, we prove that this scheme converges almost surely monotonically to a stationary point of the true objective. Under Polyak-Lojasiewicz-type conditions, this provides bounds for the generalisation error. As a remarkable result, we show that our stochastic optimisation scheme achieves the linear or exponential convergence rates of deterministic first order descent methods under suitable conditions.

Cyclotomic Factors and LRS-Degeneracy

John Abbott, Nico Mexis — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2403.08751v4 Announce Type: replace Abstract: We present three new, practical algorithms for polynomials in $\mathbb{Z}[x]$: one to test if a polynomial is cyclotomic, one to determine which cyclotomic polynomials are factors, and one to determine whether the given polynomial is LRS-degenerate. A polynomial is "LRS-degenerate" iff it has two distinct roots $\alpha, \beta$ such that $\beta = \zeta \alpha$ for some root of unity $\zeta$. All three algorithms are based on "intelligent brute force". The first two produce the indexes of the cyclotomic polynomials; the third produces a list of degeneracy orders. The algorithms are implemented in CoCoALib.

Quasi-invariant lifts of completely positive maps for groupoid actions

Suvrajit Bhattacharjee, Marzieh Forough — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2405.07859v2 Announce Type: replace Abstract: Let $G$ be a locally compact, Hausdorff, second countable groupoid and $A$ be a separable, $C_0(G^{(0)})$-nuclear, $G$-$C^*$-algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from $A$ into a separable, quotient $C^*$-algebra. Along the way, we construct the Busby invariant for $G$-actions.

Asymptotic vanishing of cohomology in triangulated categories

Petter Andreas Bergh, David A. Jorgensen, Peder Thompson — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2405.12763v2 Announce Type: replace Abstract: Given a graded-commutative ring acting centrally on a triangulated category, our main result shows that if cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.

SLE and its partition function in multiply connected domains via the Gaussian Free Field and restriction measures

Juhan Aru, Phil\'emon Bordereau — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2405.20148v2 Announce Type: replace Abstract: One way to uniquely define Schramm-Loewner Evolution (SLE) in multiply connected domains is to use the restriction property. This gives an implicit definition of a $\sigma$-finite measure on curves; yet it is in general not clear how to construct such measures nor whether the mass of these measures, called the partition function, is finite. We provide an explicit construction of the such conformal restriction SLEs in multiply connected domains when $\kappa = 4$ using the Gaussian Free Field (GFF). In particular, both when the target points of the curve are on the same or on distinct boundary components, we show that there is a mixture of laws of level lines of GFFs that satisfies the restriction property. This allows us to give an expression for the partition function of $\mathrm{SLE}_4$ on multiply connected domains and shows that the partition function is finite, answering the question raised in [Lawler, J. Stat. Phys. 2009]. In a second part, we provide a second construction of $\mathrm{SLE}_\kappa$ in multiply-connected domains for the whole range $\kappa \in (8/3,4]$; specific, however, to the case of the two target points belonging to the same boundary components. This is inspired by [Werner, Wu, Electron. J. Probab. 2013] and consists of a mixture of laws on curves obtained by following $\mathrm{CLE}_\kappa$ loops and restriction hulls attached to parts of the boundary of the domain. In this case as well, we obtain as a corollary the finiteness of the partition function for this type of $\mathrm{SLE}_\kappa$.

Enumeration of minimal transversals of hypergraphs of bounded VC-dimension

Arnaud Mary — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2407.00694v4 Announce Type: replace Abstract: We consider the problem of enumerating all minimal transversals (also called minimal hitting sets) of a hypergraph $\mathcal{H}$. An equivalent formulation of this problem known as the \emph{transversal hypergraph} problem (or \emph{hypergraph dualization} problem) is to decide, given two hypergraphs, whether one corresponds to the set of minimal transversals of the other. The existence of a polynomial time algorithm to solve this problem is a long standing open question. In \cite{fredman_complexity_1996}, the authors present the first sub-exponential algorithm to solve the transversal hypergraph problem which runs in quasi-polynomial time, making it unlikely that the problem is (co)NP-complete. In this paper, we show that when one of the two hypergraphs is of bounded VC-dimension, the transversal hypergraph problem can be solved in polynomial time, or equivalently that if $\mathcal{H}$ is a hypergraph of bounded VC-dimension, then there exists an incremental polynomial time algorithm to enumerate its minimal transversals. This result generalizes most of the previously known polynomial cases in the literature since they almost all consider classes of hypergraphs of bounded VC-dimension. As a consequence, the hypergraph transversal problem is solvable in polynomial time for any class of hypergraphs closed under partial subhypergraphs. We also show that the proposed algorithm runs in quasi-polynomial time in general hypergraphs and runs in polynomial time if the conformality of the hypergraph is bounded, which is one of the few known polynomial cases where the VC-dimension is unbounded.

A unified theory of regular functions of a hypercomplex variable

Riccardo Ghiloni, Caterina Stoppato — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2408.01523v2 Announce Type: replace Abstract: This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises Fueter-regular functions, slice-regular functions and a recently-discovered function class. In the special case of Clifford-valued functions of one paravector variable, it encompasses monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and a variety of function classes not yet considered in literature. For $T$-regular functions over an associative $*$-algebra, this work provides integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula. It also proves some foundational results about $T$-regular functions over an alternative but nonassociative $*$-algebra, such as the real algebra of octonions.

Homogenization of Poisson-Nernst-Planck equations for multiple species in a porous medium

Apratim Bhattacharya — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2408.08831v3 Announce Type: replace Abstract: We rigorously derive a homogenized model for the Poisson--Nernst--Planck (PNP) equations for the case of multiple species defined on a periodic porous medium in spatial dimensions two and three. This extends the previous homogenization results for the PNP equations concerning two species. Here, the main difficulty is that the microscopic concentrations remain uniformly bounded in a space with relatively weak regularity. Therefore, the standard Aubin-Lions-Simon type compactness results for porous media, which give strong convergence of the microscopic solutions, become inapplicable in our weak setting. We overcome this problem by constructing suitable cut-off functions. The cut-off function, together with the application of a previously known energy functional, yields strong convergence of the microscopic concentrations in $L^1_t L^r_x$, for some $r>2$, enabling us to pass to the limit in the nonlinear drift term. Finally, we derive the homogenized equations by means of two-scale convergence in $L^p_t L^q_x$ setting.

Lannes' $T$-functor and mod-$p$ cohomology of profinite groups

Marco Boggi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2408.12488v2 Announce Type: replace Abstract: The Lannes-Quillen theorem relates the mod-$p$ cohomology of a finite group $G$ with the mod-$p$ cohomology of centralizers of abelian elementary $p$-subgroups of $G$, for $p>0$ a prime number. This theorem was extended to profinite groups whose mod-$p$ cohomology algebra is finitely generated by Henn. In a weaker form, the Lannes-Quillen theorem was then extended by Symonds to arbitrary profinite groups. Building on Symonds' result, we formulate and prove a full version of this theorem for all profinite groups. For this purpose, we develop a theory of products for families of discrete torsion modules, parameterized by a profinite space, which is dual, in a very precise sense, to the theory of coproducts for families of profinite modules, parameterized by a profinite space, developed by Haran, Melnikov and Ribes. In the last section, we give applications to the problem of conjugacy separability of $p$-torsion elements and finite $p$-subgroups.

Multivariate functorial difference

Robert Par\'e — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2409.09494v2 Announce Type: replace Abstract: Partial difference operators for a large class of functors between presheaf categories are introduced, extending our difference operator from \cite{Par24} to the multivariable case. These combine into the Jacobian profunctor which provides the setting for a lax chain rule. We introduce a functorial version of multivariable Newton series whose aim is to recover a functor from its iterated differences. Not all functors are recovered but we get a best approximation in the form of a left adjoint, and the induced comonad is idempotent. Its fixed points are what we call soft analytic functors, a generalization of the multivariable analytic functors of Fiore et al.~\cite{FioGamHylWin08}.

Positively closed $Sh(B)$-valued models

Krist\'of Kanalas — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2409.11231v3 Announce Type: replace Abstract: We study positively closed and strongly positively closed topos-valued models of coherent theories. Positively closed is a global notion (it is defined in terms of all possible outgoing homomorphisms), while strongly positively closed is a local notion (it only concerns the definable sets inside the model). For $\mathbf{Set}$-valued models of coherent theories they coincide. We prove that if $\mathcal{E}=Sh(B,\tau _{coh})$ for a complete Boolean algebra, then positively closed but not strongly positively closed $\mathcal{E}$-valued models of coherent theories exist, yet, there is an alternative local property which characterizes positively closed $\mathcal{E}$-valued models. A large part of our discussion is given in the context of infinite quantifier geometric logic, dealing with the fragment $L^g_{\kappa \kappa }$ where $\kappa $ is weakly compact.

On the tails of log-concave density estimators

Didier B. Ryter, Lutz Duembgen — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2409.17910v3 Announce Type: replace Abstract: It is shown that the nonparametric maximum likelihood estimator of a univariate log-concave probability density satisfies desirable consistency properties in the tail regions. Specifically, let $P$ and $f$ denote the true underlying distribution and density, respectively. If $\hat{f}_n$ is the estimated log-concave density, and $\hat{\varphi}_n = \log \hat{f}_n$, then we specify sequences $(b_n)_{n\in \mathbb{N}}$ such that $P([b_n,\infty)) \to 0$ at a specific speed, ensuring that the absolute errors or absolute relative errors of $\hat{f}_n, \ \hat{\varphi}_n$ and $\hat{\varphi}_n'$ converge to zero uniformly on sets $[a, b_n]$. The main tools, besides characterizations of $\hat{f}_n$, are exponential and maximal inequalities for truncated moments of log-concave distributions, which are of independent interest.

On the Uniqueness of the Norton-Sullivan Quasiconformal extension

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2409.19805v2 Announce Type: replace Abstract: We show that the extension map \[ \mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\mbox{ for all }z=x+iy\in\mathbb{H}\,, \] defined by Norton and Sullivan in '96, is the only locally linear extension map taking bi-Lipschitz functions on $\mathbb{R}$ to quasiconformal functions on $\mathbb{H}$, modulo the action of a group isomorphic to the linear group. In fact, we discovered many other extension like this one (lying in the orbit of such group action), such as: $f(x)\mapsto f(x)+i(f(x)-f(x-y))$.

Large Deviations of Mean-Field Jump-Markov Processes on Structured Sparse Disordered Graphs

James MacLaurin — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2410.13682v2 Announce Type: replace Abstract: We prove a Large Deviation Principle for {\color{blue} jump-Markov } Processes on sparse large disordered network with disordered connectivity. The network is embedded in a geometric space, with the probability of a connection a (scaled) function of the spatial positions of the nodes. This type of model has numerous applications, including neuroscience, epidemiology and social networks. We prove that the rate function (that indicates the asymptotic likelihood of state transitions) is the same as for a network with all-to-all connectivity. We apply our results to a stochastic $SIS$ epidemiological model on a disordered networks, and determine Euler-Lagrange equations that dictate the most likely transition path between different states of the network.

Zarankiewicz bounds from distal regularity lemma

Mervyn Tong — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2410.13695v3 Announce Type: replace Abstract: Since K\H{o}v\'ari, S\'os, and Tur\'an proved upper bounds for the Zarankiewicz problem in 1954, much work has been undertaken to improve these bounds, and some have done so by restricting to particular classes of graphs. In 2017, Fox, Pach, Sheffer, Suk, and Zahl proved better bounds for semialgebraic binary relations, and this work was extended by Do in the following year to arbitrary semialgebraic relations. In this paper, we show that Zarankiewicz bounds in the shape of Do's are enjoyed by all relations satisfying the distal regularity lemma, an improved version of the Szemer\'edi regularity lemma satisfied by relations definable in distal structures (a vast generalisation of o-minimal structures).

State Estimation Using Sparse DEIM and Recurrent Neural Networks

Mohammad Farazmand — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2410.15982v3 Announce Type: replace Abstract: Sparse Discrete Empirical Interpolation Method (S-DEIM) was recently proposed for state estimation in dynamical systems when only a sparse subset of the state variables can be observed. The S-DEIM estimate involves a kernel vector whose optimal value is inferred through a data assimilation algorithm. This data assimilation step suffers from two drawbacks: (i) It requires the knowledge of the governing equations of the dynamical system, and (ii) It is not generally guaranteed to converge to the optimal kernel vector. To address these issues, here we introduce an equation-free S-DEIM framework that estimates the optimal kernel vector from sparse observational time series using recurrent neural networks (RNNs). We show that the recurrent architecture is necessary since the kernel vector cannot be estimated from instantaneous observations. But RNNs, which incorporate the past history of the observations in the learning process, lead to nearly optimal estimations. We demonstrate the efficacy of our method on three numerical examples with increasing degree of spatiotemporal complexity: a conceptual model of atmospheric flow known as the Lorenz-96 system, the Kuramoto-Sivashinsky equation, and the Rayleigh-Benard convection. In each case, the resulting S-DEIM estimates are satisfactory even when a relatively simple RNN architecture, namely the reservoir computing network, is used. More specifically, our RNN-based S-DEIM state estimations reduce the relative error between 42% and 58% when compared to Q-DEIM which ignores the kernel vector by setting it equal to zero.

Gelfand-Fuks cohomology of vector fields on algebraic varieties

Yuly Billig, Kathlyn Dykes — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2410.20316v2 Announce Type: replace Abstract: For an affine algebraic variety, we introduce algebraic Gelfand-Fuks cohomology of polynomial vector fields with coefficients in differentiable $AV$-modules. Its complex is given by cochains that are differential operators in the sense of Grothendieck. Using the jets of vector fields, we compute this cohomology for varieties with uniformizing parameters. We prove that in this case, Gelfand-Fuks cohomology with coefficients in a tensor module decomposes as a tensor product of the de Rham cohomology of the variety and the cohomology of the Lie algebra of vector fields on affine space, vanishing at the origin. We explicitly compute this cohomology for affine space, the torus, and Krichever-Novikov algebras.

On uniqueness in structured model learning

Martin Holler, Erion Morina — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2410.22009v3 Announce Type: replace Abstract: This paper addresses the problem of uniqueness in learning physical laws for systems of partial differential equations (PDEs). Contrary to most existing approaches, it considers a framework of structured model learning, where existing, approximately correct physical models are augmented with components that are learned from data. The main results of the paper are a uniqueness and a convergence result that cover a large class of PDEs and a suitable class of neural networks used for approximating the unknown model components. The uniqueness result shows that, in the limit of full, noiseless measurements, a unique identification of the unknown model components as functions is possible as classical regularization-minimizing solutions of the PDE system. This result is complemented by a convergence result showing that model components learned as parameterized neural networks from incomplete, noisy measurements approximate the regularization-minimizing solutions of the PDE system in the limit. These results are possible under specific properties of the approximating neural networks and due to a dedicated choice of regularization. With this, a practical contribution of this analytic paper is to provide a class of model learning frameworks different to standard settings where uniqueness can be expected in the limit of full measurements.

Generalized random processes related to Hadamard operators and Le Roy measures

Luisa Beghin, Lorenzo Cristofaro, Federico Polito — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2410.22880v3 Announce Type: replace Abstract: The definition of generalized random processes in Gel'fand sense allows to extend well-known stochastic models, such as the fractional Brownian motion, and study the related fractional pde's, as well as stochastic differential equations in distributional sense. By analogy with the construction (in the infinite-dimensional white-noise space) of the latter, we introduce two processes defined by means of Hadamard-type fractional operators. When used to replace the time derivative in the governing p.d.e.'s, the Hadamard-type derivatives are usually associated with ultra-slow diffusions. On the other hand, in our construction, they directly determine the memory properties of the so-called Hadamard fractional Brownian motion (H-fBm) and its long-time behaviour. Still, for any finite time horizon, the H-fBm displays a standard diffusing feature. We then extend the definition of the H-fBm from the white noise space to an infinite dimensional grey-noise space built on the Le Roy measure, so that our model represents an alternative to the generalized grey Brownian motion. In this case, we prove that the one-dimensional distribution of the process satisfies a heat equation with non-constant coefficients and fractional Hadamard time-derivative. Finally, once proved the existence of the distributional derivative of the above defined processes and derived an integral formula for it, we construct an Ornstein-Uhlenbeck type process and evaluate its distribution.

Index estimates for constant mean curvature surfaces in three-manifolds by energy comparison

Luca Seemungal, Ben Sharp — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2411.02932v2 Announce Type: replace Abstract: We prove a linear upper bound on the Morse index of closed constant mean curvature (CMC) surfaces in orientable three-manifolds in terms of genus, number of branch points and a Willmore-type energy.

Numerical analysis of a constrained strain energy minimization problem

Tilman Aleman, Arnold Reusken — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2411.19089v2 Announce Type: replace Abstract: We consider a setting in which an evolving surface is implicitly characterized as the zero level of a level set function. Such an implicit surface does not encode any information about the path of a single point on the evolving surface. In the literature different approaches for determining a velocity that induces corresponding paths of points on the surface have been proposed. One of these is based on minimization of the strain energy functional. This then leads to a constrained minimization problem, which has a corresponding equivalent formulation as a saddle point problem. The main topic of this paper is a detailed analysis of this saddle point problem and of a finite element discretization of this problem. We derive well-posedness results for the continuous and discrete problems and optimal error estimates for a finite element discretization that uses standard $H^1$-conforming finite element spaces.

Well-Posedness of the Linear Regularized 13-Moment Equations Using Tensor-Valued Korn Inequalities

Peter Lewintan, Lambert Theisen, Manuel Torrilhon — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2501.14108v2 Announce Type: replace Abstract: In this paper, we finally prove the well-posedness of the linearized R13 moment model, which describes, e.g., rarefied gas flows. As an extension of the classical fluid equations, moment models are robust and have been frequently used, yet they are challenging to analyze due to their additional equations. By effectively grouping variables, we identify a 2-by-2 block structure, allowing us to analyze well-posedness within the abstract LBB framework for saddle point problems. Due to the unique tensorial structure of the equations, in addition to an interesting combination of tools from Stokes' and linear elasticity theory, we also need new coercivity estimates for tensor fields. These Korn-type inequalities are established by analyzing the symbol map of the symmetric and trace-free part of tensor derivative fields. Together with the corresponding right inverse of the tensorial divergence, we obtain the existence and uniqueness of weak solutions. This result also serves as the basis for future numerical analysis of corresponding discretization schemes.

Families of singular algebraic varieties that are rationally elliptic spaces

A. Libgober — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2501.17970v3 Announce Type: replace Abstract: We discuss families of hypersurfaces with isolated singularities in projective space with the property that the sum of the ranks of the rational homotopy and the homology groups is finite. They represent infinitely many distinct homotopy types with all hypersurfaces having a nef canonical or anti-canonical class. In the appendix we show that such an infinite family of smooth rationally elliptic 3-folds does not exist.

Sparsity-Guided Multi-Parameter Selection in $\ell_1$-Regularized Models via a Fixed-Point Proximity Approach

Qianru Liu, Rui Wang, Yuesheng Xu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2502.00655v2 Announce Type: replace Abstract: We study a regularization framework that combines a convex fidelity term with multiple $\ell_1$-based regularizers, each linked to a distinct linear transform. This multi-penalty model enhances flexibility in promoting structured sparsity. We analyze how the choice of regularization parameters governs the sparsity of solutions under the given transforms and derive a precise relationship between the parameters and resulting sparsity patterns. This insight enables the development of an iterative strategy for selecting parameters to achieve prescribed sparsity levels. A key computational challenge arises in practice: effective parameter tuning requires simultaneous access to the regularized solution and two auxiliary vectors derived from the sparsity analysis. To address this, we propose a fixed-point proximity algorithm that jointly computes all three vectors. Together with our theoretical characterization, this algorithm forms the basis of a practical multi-parameter selection scheme. Numerical experiments demonstrate that the proposed method reliably produces solutions with desired sparsity patterns and strong approximation accuracy.

Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions

Yankun Huang, Qihang Lin, Yangyang Xu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2502.19764v2 Announce Type: replace Abstract: In this paper, we investigate how structural properties of the constraint system impact the oracle complexity of smooth non-convex optimization problems with convex inequality constraints over a simple polytope. In particular, we show that, under a local error bound condition with exponent $d\in[1,2]$ on constraint functions, an inexact Moreau envelope Lagrangian method can attain an $\epsilon$-Karush--Kuhn--Tucker point with $\tilde O(\epsilon^{-2d})$ gradient oracle complexity. When $d=1$, this result matches the best-known complexity in literature up to logarithmic factors. Importantly, the assumed error bound condition with any $d\in[1,2]$ is strictly weaker than the local linear independence constraint qualification that is required to achieve the best-known complexity. Our results clarify the interplay between error bound conditions of constraints and algorithmic complexity, and extend complexity guarantees to a broader class of constrained non-convex problems.

Dynamic Programming in Ordered Vector Space

Nisha Peng, John Stachurski — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2503.06055v2 Announce Type: replace Abstract: New approaches to the theory of dynamic programming view dynamic programs as families of policy operators acting on partially ordered sets. In this paper, we extend these ideas by shifting from arbitrary partially ordered sets to ordered vector spaces. The integrated algebraic and order structure in such spaces leads to sharper fixed point results. These fixed point results can then be exploited to obtain optimality properties. We illustrate our results through applications ranging from firm management to data valuation. These applications include features from the recent literature on dynamic programming, including risk-sensitive preferences, nonlinear discounting, and state-dependent discounting. In all cases we establish existence of optimal policies, characterize them in terms of Bellman optimality relationships, and prove convergence of major algorithms.

Supersimplicity and arithmetic progressions

Amador Martin-Pizarro, Daniel Palac\'in — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2503.08258v3 Announce Type: replace Abstract: The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for definable groups in simple theories. In the last sections of this article, we apply our model-theoretic results to bound the number of initial points starting few arithmetic progression of length $3$ in the structure of the additive group of integers with a predicate for the prime integers, assuming Dickson's conjecture, or with a predicate for the square-free integers, as well as for asymptotic limits of finite fields. Our techniques yield similar results for the elements appearing as distances in skew-corners and for S\'ark\"ozy's theorem on the distance of distinct elements being perfect squares.

Some Geometric Aspects Related to Lim's Condition

Deepak Gothwal, T. S. S. R. K. Rao — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2504.09464v2 Announce Type: replace Abstract: In their seminal work, Lau and Mah (1986) study $w^*$-normal structure in the space of operators $\mathcal{L}(H)$, on a Hilbert space $H$, using a geometric property of the dual unit ball called Lim's condition. In this paper, we study a weaker form of Lim's condition, which we call property ($\ddagger$), for $C^\ast$-algebras, uniform algebras, and $L^1$-predual spaces. In the case of a $C^\ast$-algebra, we prove that property $(\ddagger)$ is equivalent to Lim's condition and consequently, we obtain a geometric characterization of $C^*$-algebras which are $c_0$-direct sum of finite-dimensional operator spaces. For a uniform algebra, we extend a result of Lau and Mah to show that property $(\ddagger)$ implies that the space is finite-dimensional. In the case of an $L^1$-predual space, we show that this condition implies $k$-smoothness of the norm in the sense considered in Lin and Rao (2007).

On anti-coproximinal and strongly anti-coproximinal subspaces of function spaces

Shamim Sohel, Souvik Ghosh, Debmalya Sain, Kallol Paul — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2504.13464v2 Announce Type: replace Abstract: The purpose of this article is to study the anti-coproximinal and strongly anti-coproximinal subspaces of the Banach space of all bounded (continuous) functions. We obtain a tractable necessary condition for a subspace to be stronsgly anti-coproximinal. We prove that for a subspace $\mathbb{Y}$ of a Banach space $\mathbb{X}$ to be strongly anti-coproximinal, $\mathbb Y$ must contain all w-ALUR points of $\mathbb{X}$ and intersect every maximal face of $B_{\mathbb{X}}.$ We also observe that the subspace $\mathbb{K}(\mathbb{X}, \mathbb{Y})$ of all compact operators between the Banach spaces $ \mathbb X $ and $ \mathbb Y$ is strongly anti-coproximinal in the space $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ of all bounded linear operators between $ \mathbb X $ and $ \mathbb Y$, whenever $\mathbb{K}(\mathbb{X}, \mathbb{Y})$ is a proper subset of $\mathbb{L}(\mathbb{X}, \mathbb{Y}),$ and the unit ball $B_{\mathbb{X}}$ is the closed convex hull of its strongly exposed points.

Geometry of regular semisimple Lusztig varieties

Patrick Brosnan, Jaehyun Hong, Donggun Lee — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2504.15868v2 Announce Type: replace Abstract: Lusztig varieties are subvarieties in flag manifolds $G/B$ associated to an element $w$ in the Weyl group $W$ and an element $x$ in $G$, introduced in Lusztig's papers on character sheaves. We study the geometry of these varieties when $x$ is regular semisimple. In the first part, we establish that they are normal, Cohen-Macaulay, of pure expected dimension and have rational singularities. We then show that the cohomology of ample line bundles vanishes in positive degrees, in arbitrary characteristic. This extends to nef line bundles when the base field has characteristic zero or sufficiently large characteristic. Along the way, we prove that Lusztig varieties are Frobenius split in positive characteristic and that their open cells are affine. We also prove that the open cells in Deligne-Lusztig varieties are affine, settling a question that has been open since the foundational paper of Deligne and Lusztig. In the second part, we explore their relationship with regular semisimple Hessenberg varieties. Both varieties admit Tymoczko's dot action of $W$ on their (intersection) cohomology. We associate to each element $w$ in $W$ a Hessenberg space using the tangent cone of the Schubert variety associated with $w$, and show that the cohomology of the associated regular semisimple Lusztig varieties and Hessenberg varieties is isomorphic as graded $W$-representations when they are smooth. This relationship extends to the level of varieties: we construct a flat degeneration of regular semisimple Lusztig varieties to regular semisimple Hessenberg varieties. In particular, this proves a conjecture of Abreu and Nigro on the homeomorphism types of regular semisimple Lusztig varieties in type $A$, and generalizes it to arbitrary Lie types.

Discrete analogues of second-order Riesz transforms

Rodrigo Ba\~nuelos, Daesung Kim — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2504.18739v2 Announce Type: replace Abstract: Discrete analogues of classical operators in harmonic analysis have been widely studied, revealing deep connections with areas such as ergodic theory and analytic number theory. This line of research is commonly known as \emph{Discrete Analogues in Harmonic Analysis (DAHA)}. In this paper, we study the $\ell^p$ norms of discrete analogues of second-order Riesz transforms. Using probabilistic methods, we construct a new class of second-order discrete Riesz transforms $\mathcal{R}^{(jk)}$ on the lattice $\mathbb{Z}^d$, $d \ge 2$. We show that for $1

Weakly Einstein curvature tensors

Andrzej Derdzinski, JeongHyeong Park, Wooseok Shin — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2504.18752v2 Announce Type: replace Abstract: We classify weakly Einstein algebraic curvature tensors in an oriented Euclidean 4-space, defined by requiring that the three-index contraction of the curvature tensor against itself be a multiple of the inner product. This algebraic formulation parallels the geometric notion of weakly Einstein Riemannian four-manifolds, which include conformally flat scalar-flat, and Einstein manifolds. Our main result provides a complete classification of non-Einstein weakly Einstein curvature tensors in dimension four, naturally dividing them into three disjoint five-dimensional families of algebraic types. These types are explicitly constructed using bases that simultaneously diagonalize both the Einstein tensor and the (anti)self-dual Weyl tensors, which consequently proves that such simultaneous diagonalizability follows from the weakly Einstein property. We also point out that our classification has immediate applications, and describe how some known geometric examples that are neither Einstein, nor conformally flat scalar-flat (namely, the EPS space and certain K\"ahler surfaces) fit within our classification framework.

Analysis and Elimination of Numerical Pressure Dependency in Coupled Stokes-Darcy Problem

Jiachuan Zhang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2504.19116v2 Announce Type: replace Abstract: This paper analyses the classical mixed finite element method (FEM) and a pressure-robust variant with divergence-free reconstruction operators for the coupled Stokes-Darcy problem. Its main contribution is to provide viscosity-explicit a priori error estimates that clearly distinguish the pressure dependence of the two discretizations: the velocity error of the classical scheme depends on both the exact pressure and the viscosity, whereas the pressure-robust method eliminates both entirely. Moreover, we derive pressure error estimates and quantify their dependence on the exact solution and model parameters. Two-dimensional numerical experiments validate the theoretical findings, including higher-order tests up to polynomial degree three and a lid-driven cavity benchmark with a piecewise linear interface. The implementation code is made publicly available to facilitate reproducibility.

Sharp asymptotics for $N$-point correlation functions of coalescing heavy-tailed random walk

Jinjiong Yu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2505.05000v2 Announce Type: replace Abstract: We study a system of coalescing continuous-time random walks starting from every site on $\mathbb{Z}$, where the jump increments lie in the domain of attraction of an $\alpha$-stable distribution with $\alpha\in(0,1]$. We establish sharp asymptotics for the $N$-point correlation function of the system. Our analysis relies on two precise tail estimates for the system density, as well as the non-collision probability of $N$ independent random walks with arbitrary fixed initial configurations. In addition, we derive refined estimates for heavy-tailed random walks, which may be of independent interest.

A Dantzig-Wolfe Decomposition Method for Quasi-Variational Inequalities

Manoel Jardim, Claudia Sagastiz\'abal, Mikhail Solodov — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2505.08108v2 Announce Type: replace Abstract: We propose an algorithm to solve quasi-variational inequality problems, based on the Dantzig-Wolfe decomposition paradigm. Our approach solves in the subproblems variational inequalities, which is a simpler problem, while restricting quasi-variational inequalities in the master subproblems, making them generally (much) smaller in size when the original problem is large-scale. We prove global convergence of our algorithm, assuming that the mapping of the quasi-variational inequality is either single-valued and continuous or it is set-valued maximally monotone. Quasi-variational inequalities serve as a framework for several equilibrium problems, and we apply our algorithm to an important example in the field of economics, namely the Walrasian equilibrium problem formulated as a generalized Nash equilibrium problem. Our numerical assessment demonstrates good performance and usefullness of the approach for the large-scale cases.

Riguet congruences, Generalized congruences and Free monoids

Juan Climent Vidal, Enric Cosme Ll\'opez, Ra\'ul Ruiz Mora — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2505.15767v4 Announce Type: replace Abstract: We examine Riguet congruences and generalized congruences on a category, giving particular attention to their interrelations from both lattice-theoretic and category-theoretic perspectives. This investigation constitutes the principal contribution of the paper. As an application of these results, starting from a category associated with the free monoid on a set $A$ of sorts, we obtain a skeletal category via the quotient of that category by a suitable Riguet congruence on it. Moreover, we prove that this quotient category is equivalent to the category of finite $A$-sorted sets, while being neither a subcategory of it nor identical to the category of finite $A$-sorted cardinal numbers.

Bias-Optimal Bounds for SGD: A Computer-Aided Lyapunov Analysis

Daniel Cortild, Lucas Ketels, Juan Peypouquet, Guillaume Garrigos — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2505.17965v2 Announce Type: replace Abstract: The non-asymptotic analysis of Stochastic Gradient Descent (SGD) typically yields bounds that decompose into a bias term and a variance term. In this work, we focus on the bias component and study the extent to which SGD can match the optimal convergence behavior of deterministic gradient descent. Assuming only (strong) convexity and smoothness of the objective, we derive new bounds that are bias-optimal, in the sense that the bias term coincides with the worst-case rate of gradient descent. Our results hold for the full range of constant step-sizes $\gamma L \in (0,2)$, including critical and large step-size regimes that were previously unexplored without additional variance assumptions. The bounds are obtained through the construction of a simple Lyapunov energy whose monotonicity yields sharp convergence guarantees. To design the parameters of this energy, we employ the Performance Estimation Problem framework, which we also use to provide numerical evidence for the optimality of the associated variance terms.

Calder\'{o}n-Zygmund estimates for double phase problems with matrix weights

Sun-Sig Byun, Yumi Cho, Seungjin Ryu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2505.20856v3 Announce Type: replace Abstract: We establish an optimal Calder\'{o}n-Zygmund theory for nonuniformly elliptic double phase problems with matrix weights. For $11$, $$ (|\M F|^p+a(x)|\M F|^q)\in L^\gamma_{\mathrm{loc}} \;\Longrightarrow\; (|\M Du|^p+a(x)|\M Du|^q)\in L^\gamma_{\mathrm{loc}}. $$ Our argument combines a freezing of the logarithm of the matrix field, $\log \M$, with a fractional maximal-operator method governed by the Muckenhoupt-Wheeden $\mathcal{A}_{p,s}$ classes (where $1/s=1/p-\alpha/(nq)$). This yields scale-invariant comparison and level-set estimates and precludes Lavrentiev gaps at the sharp threshold $q/p\le 1+\alpha/n$. Our result recovers the identity case $\,\M\equiv {\rm I}_n\,$, i.e., the classical (unweighted) Calder\'{o}n-Zygmund theory for double-phase problems.

Classification of exact structures using the Ziegler spectrum

Julia Sauter — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2506.02304v2 Announce Type: replace Abstract: Given an idempotent complete additive category, we show the there is an explicitly constructed topological space such that the lattice of exact substructures is anti-isomorphic to the lattice of closed subsets. In the special case that the additive category has weak cokernels, this topological space is an open subset of the Ziegler spectrum and this is a result of Kevin Schlegel. We also look at some module categories of rings where the Ziegler spectrum is known and calculate the global dimensions of the corresponding exact substructures. Second version contains minor changes to first version.

A hybrid isogeometric and finite element method: NURBS-enhanced finite element method for hexahedral meshes (NEFEM-HEX)

Duygu Sap — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2506.13694v3 Announce Type: replace Abstract: In this paper, we present a NURBS-enhanced finite element method that integrates the NURBS-based boundary representation of a geometric domain into a standard finite element framework for hexahedral meshes. We decompose an open, bounded, convex three-dimensional domain with a NURBS boundary into two parts, define NURBS-enhanced finite elements over the boundary layer, and use piecewise-linear Lagrange finite elements in the interior region. We introduce a special quadrature rule and a stable interpolation operator for the NURBS-enhanced elements. We discuss how the h-refinement in finite element analysis and the knot insertion in isogeometric analysis can be utilized in the refinement of the NURBS-enhanced elements. To illustrate an application of our methodology, we utilize a generic weak formulation of a second-order linear elliptic boundary value problem and derive a priori error estimates in the $H^{1}$ norm. In addition, we use the Poisson problem as a model problem and provide numerical results that support the theoretical results. The proposed methodology combines the efficiency of finite element analysis with the geometric precision of NURBS, and may enable more accurate and efficient simulations over complex geometries.

On some results of Harish-Chandra for representations of p-adic groups, extended to their central extensions

Volker Heiermann — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2506.21334v2 Announce Type: replace Abstract: The aim of this article is to give a complete proof of results of Harish-Chandra linking the irreducibility of parabolic induction of a supercuspidal representation of a p-adic group to the analytic behavior of the mu-function of Harish-Chandra and to show that the proof remains valid in the case of a central extension.M

The lightning method for the heat equation

Hunter La Croix, Alan E. Lindsay — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2506.22576v3 Announce Type: replace Abstract: This paper introduces a new method for solving the planar heat equation based on the Lightning Method. The lightning method is a recent development in the numerical solution of linear PDEs which expresses solutions using sums of polynomials and rational functions, or more generally as sums of fundamental solutions. The method is particularly well suited to handle domains with sharp corners where solution singularities are present. Boundary conditions are formed on a set of collocation points which is then solved as an overdetermined linear system. The approach of the present work is to utilize the Laplace transform to obtain a modified Helmholtz equation which is solved by an application of the lightning method. The numerical inversion of the Laplace transform is then performed by means of Talbot integration. Our validation of the method against existing results and multiple challenging test problems shows the method attains spectral accuracy with root-exponential convergence while being robust across a wide range of time intervals and adaptable to a variety of geometric scenarios.

Projective Transformations for Regularized Central-Force Dynamics: Hamiltonian Formulation

Joseph T. A. Peterson, Manoranjan Majji, John L. Junkins — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2506.22681v5 Announce Type: replace Abstract: This work introduces a Hamiltonian approach to regularization and linearization of central-force particle dynamics through a new canonical extension of the so-called "projective decomposition". The regularization scheme is formulated within the framework of classic analytical Hamiltonian dynamics as a redundant-dimensional canonical/symplectic coordinate transformation, combined with an evolution parameter transformation, on extended phase space. By considering a generalized version of the standard projective decomposition, we obtain a family of such canonical transformations which differ at the momentum level. From this family of transformations, a preferred coordinate set is chosen that possesses a simple and intuitive connection to the particle's local reference frame. Using this transformation, closed-form solutions are readily obtained for inverse-square and inverse-cubic radial forces, or any superposition thereof. Governing equations are numerically validated for the classic two-body problem incorporating the J2 gravitational perturbation.

On the rank weight hierarchy of $M$-codes

G. Berhuy, J. Molina — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2507.00609v3 Announce Type: replace Abstract: We study the rank weight hierarchy of linear codes which are stable under a linear endomorphism defined over the base field, in particular when the endomorphism is cyclic. In this last case, we give a necessary and sufficient condition for such a code to have first rank weight equal to $1$ in terms of its generator polynomial, as well as an explicit formula for its last rank weight.

Perturbed Toroidal Vortices Display Internal Simply Connected Topology

Taosif Ahsan, Samuel A. Cohen, Alan H. Glasser — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2507.04596v2 Announce Type: replace Abstract: This work shows that the interiors of perturbed zero-helicity vortices display simply connected topology with a crescent-shaped boundary. Flux surfaces in fluid and magnetic vortices were explored analytically, while particle trajectories in the context of plasma confinement were examined numerically, demonstrating the existence of both toroidal and simply connected topologies. This new topology appears for perturbations in a broad class, with amplitudes and spatial variance allowed to be arbitrarily small. This work proves the closedness of field lines under odd-parity perturbations of zero-helicity vortices.

Illumination number of 3-dimensional cap bodies

Andrii Arman, Jaskaran Singh Kaire, Andriy Prymak — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2507.08712v2 Announce Type: replace Abstract: The illumination conjecture asserts that any convex body in $n$-dimensional Euclidean space can be illuminated by at most $2^n$ external light sources or parallel beams of light. Despite recent progress on the illumination conjecture, it remains open in general, as well as for specific classes of bodies. Bezdek, Ivanov, and Strachan showed that the conjecture holds for symmetric cap bodies in sufficiently high dimensions. Further, Ivanov and Strachan calculated the illumination number for the class of 3-dimensional centrally symmetric cap bodies to be 6. In this paper, we show that even the broader class of all 3-dimensional cap bodies has the same illumination number 6, in particular, the illumination conjecture holds for this class. The illuminating directions can be taken to be vertices of a regular tetrahedron, together with two special directions depending on the body. The proof is based on probabilistic arguments and integer linear programming.

Twisted periods of modular forms

Tianyu Ni, Hui Xue — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2507.17041v2 Announce Type: replace Abstract: Let $S_k$ denote the space of cusp forms of weight $k$ and level one. For $0\leq t\leq k-2$ and primitive Dirichlet character $\chi$ mod $D$, we introduce twisted periods $r_{t,\chi}$ on $S_k$. We show that for a fixed natural number $n$, if $k$ is sufficiently large relative to $n$ and $D$, then any $n$ periods with the same twist but different indices are linearly independent. We also prove that if $k$ is sufficiently large relative to $D$ then any $n$ periods with the same index but different twists mod $D$ are linearly independent. These results are achieved by studying the trace of the products and Rankin-Cohen brackets of Eisenstein series of level $D$ with nebentypus. Moreover, we give two applications of our method. First, we prove certain identities that evaluate convolution sums of twisted divisor functions. Second, we show that Maeda's conjecture implies a non-vanishing result on twisted central $L$-values of normalized Hecke eigenforms.

Self-Similar Solutions to the Hele-Shaw Problem with Surface Tension

Siddhant Agrawal, Neel Patel — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2507.19443v2 Announce Type: replace Abstract: We consider the Hele-Shaw problem with surface tension in an infinite domain. We prove the existence of a family of self-similar solutions. At $t=0$, these solutions have a corner of angle $\theta$ with $ 0 < |\theta - \pi| \ll 1$, and for $t>0$, the solutions are smooth.

A Generalized Analytical Framework for the Nonlinear Best-Worst Method

Harshit M. Ratandhara, Mohit Kumar — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2508.06048v2 Announce Type: replace Abstract: The nonlinear model of the best-worst method frequently produces multiple optimal weight sets, which are conventionally determined through optimization software. While an analytical approach exists that provides both a closed-form expression for the optimal interval-weights and a secondary objective function to determine the best optimal weight set, we demonstrate that this approach is only valid when preferences are quantified using the Saaty scale and only a single decision-maker is involved. To tackle this issue, we propose a framework compatible with any scale and any number of decision-makers. We first derive an analytical expression for optimal interval-weights and then select the best optimal weight set. After demonstrating that the values of consistency index for the Saaty scale in the existing literature are not well-defined, we derive a formula of consistency index. We also obtain an analytical expression for the consistency ratio, enabling its use as an input-based consistency indicator. Furthermore, we establish that when multiple best/worst criteria are present, weights may vary among best criteria and among the worst criteria. To address this limitation, we modify the original optimization model for weight computation in such instances, solve it analytically to obtain optimal interval-weights and then select the best optimal weight set using a secondary objective function. Finally, we demonstrate and validate the proposed approach using numerical examples and a real-world case study of ranking barriers to energy efficiency in buildings.

A bottleneck model with shared autonomous vehicles: Scale economies and price regulations

Koki Satsukawa, Yuki Takayama — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2508.08848v2 Announce Type: replace Abstract: This study examines how scale economies in the operation of shared autonomous vehicles (SAVs) affect the efficiency of a transportation system where SAVs coexist with normal vehicles (NVs). We develop a bottleneck model where commuters choose their departure times and mode of travel between SAVs and NVs, and analyze equilibria under three SAV fare-setting scenarios: marginal cost pricing, average cost pricing, and unregulated monopoly pricing. Marginal cost pricing reduces commuting costs but results in financial deficits for the service provider. Average cost pricing ensures financial sustainability but has contrasting effects depending on the timing of implementation due to the existence of multiple equilibria: when implemented too early, it discourages adoption of SAVs and increases commuting costs; when introduced after SAV adoption reaches the monopoly equilibrium level, it promotes high adoption and achieves substantial cost reductions without a deficit. We also show that expanding road capacity may increase commuting costs under average cost pricing, demonstrating the Downs--Thomson paradox in transportation systems with SAVs. We next examine two optimal policies that improve social cost, including the operator's profit: the first-best policy that combines marginal cost pricing with congestion tolls, and the second-best policy that relies on fare regulation alone. Our analysis shows that these policies can limit excessive adoption by discouraging overuse of SAVs. This suggests that promoting SAV adoption does not always reduce social cost.

A Generalized Alternating Anderson Acceleration Method

Yunhui He, Santolo Leveque — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2508.10158v2 Announce Type: replace Abstract: In this work, we propose a generalized alternating Anderson acceleration method, a periodic scheme composed of $t$ fixed-point iteration steps, interleaved with $s$ steps of Anderson acceleration with window size $m$, to solve linear and nonlinear problems. This allows flexibility to use different combinations of fixed-point iteration and Anderson iteration. We present a convergence analysis of the proposed scheme for accelerating the Richardson iteration in the linear case, with a focus on specific parameter choices of interest. Specifically, we prove convergence of the proposed method under contractive fixed-point iteration and provide a sufficient condition for convergence when the Richardson iteration matrix is diagonalizable and noncontractive. To demonstrate the broader applicability of our proposed method, we use it to accelerate Jacobi iteration, Picard iteration, gradient descent, and the alternating direction method of multipliers in solving partial differential equations and nonlinear, nonsmooth optimization problems. The numerical results illustrate that the proposed scheme is more efficient than the existing windowed Anderson acceleration and alternating Anderson ($s=1$) in terms of iteration number and CPU time for careful choice of parameters $m, s, t$.

Relative braid group symmetries on modified iquantum groups and their modules

Weiqiang Wang, Weinan Zhang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2508.12041v2 Announce Type: replace Abstract: We present a comprehensive generalization of Lusztig's braid group symmetries for quasi-split iquantum groups. Specifically, we give 3 explicit rank one formulas for symmetries acting on integrable modules over a quasi-split iquantum group of arbitrary Kac-Moody type with general parameters. These symmetries are formulated in terms of idivided powers and iweights of the vectors being acted upon. We show that these symmetries are consistent with the relative braid group symmetries on iquantum groups, and they are also related to Lusztig's symmetries via quasi $K$-matrices. Furthermore, through appropriate rescaling, we obtain compatible symmetries for the integral forms of modified iquantum groups and their integrable modules. We verify that these symmetries satisfy the relative braid group relations.

Cycles of Length 4 or 8 in Graphs with Diameter 2 and Minimum Degree at Least 3

Avery Carr — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2508.19302v4 Announce Type: replace Abstract: In this short note it is shown that every graph of diameter 2 and minimum degree at least 3 contains a cycle of length 4 or 8. This result contributes to the study of the Erd\H{o}s-Gy\'arf\'as Conjecture by confirming it for the class of diameter-2 graphs.

The Tautochrone of Huygens and Abel: From Constructive Geometry to Fractional Calculus

Luiz Roberto Evangelista, Francesco Mainardi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.05308v2 Announce Type: replace Abstract: In this paper, we explore the connections between Christiaan Huygens and Niels Henrik Abel through the tautochrone problem. The problem -- determining the curve along which a particle descends under gravity in the same time, regardless of its starting point -- has been a central topic at the intersection of physics, geometry, and analysis. Though these two major figures are separated by nearly two centuries, they approached the problem in radically different ways. While Huygens proposed a physical solution based on geometric construction, Abel approached the problem within the analytic framework of integral equations, employing a procedure that can be seen as anticipating and paving the way for the development of differential calculus of arbitrary order. This contrast highlights a broader historical narrative: the transformation of mathematical thinking from constructive geometry to abstract analysis.

Carryless Pairing: Additive Pairing in the Fibonacci Basis

Milan Rosko — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.10382v4 Announce Type: replace Abstract: We define a pairing map $\pi : \mathbb{N}^2\to\mathbb{N}$ that encodes $x$ and $y$ into disjoint index bands inside the Zeckendorf support of a single integer. Evaluation and inversion use only addition, comparison, and bounded scans of supports; no multiplication, factorization, or digit interleaving is used. The device is carryless by construction: supports remain non-adjacent, so the output is already in Zeckendorf-normal form. The map is injective but not surjective; membership in its image is decidable by the same support machinery used for decoding. The core claims are mechanized in Rocq.

Rough stochastic filtering

Fabio Bugini, Peter K. Friz, Khoa L\^e, Huilin Zhang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.11825v2 Announce Type: replace Abstract: This article is concerned with the well-posedness of the "filtering equations", due to Zakai and Kushner-Stratonovich, arising in nonlinear stochastic filtering. In general situations, notably in correlated diffusion models and when signal coefficients depend on the observation process, the well-posedness is a difficult problem, mainly due to conflicting martingale structures of the involved forward and backward equations. Crisan-Pardoux (2024) address this classical problem with BSPDE techniques, Du et al. (2013), a Sobolev-based approach that however requires increasingly strong regularity assumptions in high dimensions. In this work, we take a new mixed rough stochastic perspective which allows us to derive well-posed rough counterparts of the filtering equations. Importantly, the rough filtering equations are seen, upon randomization, to coincide with the classical filtering equations. Our framework yields well-posedness (existence, uniqueness, stability) under dimension-independent regularity assumptions, providing a robust and conceptually unified solution to a longstanding problem in stochastic filtering theory. To illustrate the flexibility of the method, we also treat rough versions of the classical Kalman-Bucy filter, with characteristics described by a new class of RDEs of rough Riccati type.

Haussdorff consistency of MLE in folded normal and Gaussian mixtures

Koustav Mallik — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.12206v2 Announce Type: replace Abstract: We develop a constant-tracking likelihood theory for two nonregular models: the folded normal and finite Gaussian mixtures. For the folded normal, we prove boundary coercivity for the profiled likelihood, show that the profile path of the location parameter exists and is strictly decreasing by an implicit-function argument, and establish a unique profile maximizer in the scale parameter. Deterministic envelopes for the log-likelihood, the score, and the Hessian yield elementary uniform laws of large numbers with finite-sample bounds, avoiding covering numbers. Identification and Kullback-Leibler separation deliver consistency. A sixth-order expansion of the log hyperbolic cosine creates a quadratic-minus-quartic contrast around zero, leading to a nonstandard one-fourth-power rate for the location estimator at the kink and a standard square-root rate for the scale estimator, with a uniform remainder bound. For finite Gaussian mixtures with distinct components and positive weights, we give a short identifiability proof up to label permutations via Fourier and Vandermonde ideas, derive two-sided Gaussian envelopes and responsibility-based gradient bounds on compact sieves, and obtain almost-sure and high-probability uniform laws with explicit constants. Using a minimum-matching distance on permutation orbits, we prove Hausdorff consistency on fixed and growing sieves. We quantify variance-collapse spikes via an explicit spike-bonus bound and show that a quadratic penalty in location and log-scale dominates this bonus, making penalized likelihood coercive; when penalties shrink but sample size times penalty diverges, penalized estimators remain consistent. All proofs are constructive, track constants, verify measurability of maximizers, and provide practical guidance for tuning sieves, penalties, and EM-style optimization.

Linear Complexity Computation of Code Distance and Minimum Size of Trapping Sets for LDPC Codes with Bounded Treewidth

Qingqing Peng, Ke Liu, Guiying Yan, Guanghui Wang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.13040v2 Announce Type: replace Abstract: It is well known that, given $b\ge 0$, finding an $(a,b)$-trapping set with the minimum $a$ in a binary linear code is NP-hard. In this paper, we demonstrate that this problem can be solved with linear complexity with respect to the code length for codes with bounded treewidth. Furthermore, suppose a tree decomposition corresponding to the treewidth of the binary linear code is known. In that case, we also provide a specific algorithm to compute the minimum $a$ and the number of the corresponding $(a, b)$-trapping sets for a given $b$ with linear complexity. Simulation experiments are presented to verify the correctness of the proposed algorithm.

Error Analysis of Discrete Flow with Generator Matching

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.21906v2 Announce Type: replace Abstract: Discrete flow models offer a powerful framework for learning distributions over discrete state spaces and have demonstrated superior performance compared to the discrete diffusion models. However, their convergence properties and error analysis remain largely unexplored. In this work, we develop a unified framework grounded in stochastic calculus theory to systematically investigate the theoretical properties of discrete flow models. Specifically, by leveraging a Girsanov-type theorem for the path measures of two continuous-time Markov chains (CTMCs), we present a comprehensive error analysis that accounts for both transition rate estimation error and early stopping error. In fact, the estimation error of transition rates has received little attention in existing works. Unlike discrete diffusion models, discrete flow incurs no initialization error caused by truncating the time horizon in the noising process. Building on generator matching and uniformization, we establish non-asymptotic error bounds for distribution estimation without the boundedness condition on oracle transition rates. Furthermore, we derive a faster rate of total variation convergence for the estimated distribution with the boundedness condition, yielding a nearly optimal rate in terms of sample size. Our results provide the first error analysis for discrete flow models. We also investigate model performance under different settings based on simulation results.

Abstract Integration in Net Convergence Structures

Alexandre Reggiolli Teixeira — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.24430v2 Announce Type: replace Abstract: In this article, we propose a general theory of integration of the Riemann and Lebesgue types with respect to arbitrary measures and functions, connected by a continuous bilinear product, with values in abstract vector spaces endowed with a convergence structure given by nets. This covers both the topological and order based convergences in the literature. We then show that this integral satisfies most of the common properties of the objects that comprises integration theory. By establishing a generalized notion of summability on Riesz spaces and an integral built upon countable partitions of the base space, we then stablish some uniform, monotone and dominated convergence theorems for the refereed integrals, as well as a non-topological or order based Henstock Lemma and a general convergence theorem based on the notion of conjugated lattice seminorms. An application of these theorems is made to prove various equivalences concerning the Lebesgue, for which we give a brief survey, Saks and Riemann type integrals in partially ordered and topological vector spaces presented in the literature, for which we also make a thorough review. We finish the article with a possible way of classifying general integration procedures defined in abstract convergence structures, and pose some open problems based on them.

Multifractality in the Tree of Life: A Branching-Process RIFS Proof

Kevin Hudnall — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.26637v2 Announce Type: replace Abstract: We study a branching-process random iterated function system (RIFS) defined by a recursive replacement of leaves by finite subtrees at strictly smaller contraction scales. This construction yields a tree-valued, infinite-depth random geometry that unifies classical branching processes and random iterated function systems while remaining distinct from both. We prove rigorously that the resulting branching-process RIFS is multifractal under explicit and mild assumptions. Two variants are analyzed: a non-anchored case with a nontrivial compact attractor, and a biologically motivated anchored case in which the invariant geometric set collapses to a point, while tangent measures obey the same multifractal law. The construction formalizes the foundational principles of nestedness, duality, and randomness in the living tree of life (Hudnall & D'Souza, 2025), yielding a minimal-condition theorem that explains the ubiquity of multifractal signatures in biological data.

The Magmoid of Normalized Stochastic Kernels

Elena Di Lavore, Mario Rom\'an, M\'ark Sz\'eles — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.01131v2 Announce Type: replace Abstract: Normalization, $D(X + 1) \to D(X) + 1$, is almost a distributive law; but because one of the distributive law axioms only holds up-to-idempotent, it yields a non-associative composition of normalized kernels. We introduce the Markov magmoid of normalized stochastic kernels: a normalized-by-construction semantics for probabilistic inference, unifying exact Bayesian observations and interventions as two parenthesizations of the same composite. Front-door and back-door criteria follow from the axioms of Markov magmoids; we implement these with non-associative monadic notation.

Absolutely Abelian Hilbert Class Fields and $\ell-$torsion conjecture

Mahesh Kumar Ram, Prem Prakash Pandey, Nimish Kumar Mahapatra — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.10725v2 Announce Type: replace Abstract: There are several recent works where authors have shown that number fields $K$ with `sufficiently many' units and cyclic class group contain a Euclidean ideal class provided the Hilbert class field $H(K)$ is absolutely abelian. In this article, we explore the latter hypothesis: how often a number field $K$ has absolutely abelian Hilbert class field? For a number field $K$ to have absolutely abelian Hilbert class field, we obtain several criteria in terms of class number of $K$, P\'olya group of $K$, and genus number of $K$. We also show that for such number fields the $\ell-$torsion conjecture is true. Along with these, the article also reports some results on a theme to study class groups, developed by the authors, where primes of higher degree are used to study class groups.

On estimation of weighted cumulative residual Tsallis entropy for complete and censored samples

Siddhartha Chakraborty, Asok K. Nanda — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.12442v2 Announce Type: replace Abstract: Recently, weighted cumulative residual Tsallis entropy has been introduced in the literature as a generalization of weighted cumulative residual entropy. We study some new properties of weighted cumulative residual Tsallis entropy measure. Next, we propose some non-parametric estimators of this measure. Asymptotic properties of these estimators are discussed. Performance of these estimators are compared by mean squared error. Non-parametric estimators for weighted cumulative residual entropy measure are also discussed. Estimator for weighted cumulative residual Tsallis entropy for progressive type-II censored data is proposed and its performance is investigated by Monte-Carlo simulations for various censoring schemes. Two uniformity tests for complete samples are proposed based on an estimator of these two measures and power of the tests are compared with some popular tests. The tests perform reasonably well. Uniformity test under progressively type-II censored data is also developed. Some real datasets are analysed for illustration.

The spectrum of Dirichlet-to-Neumann maps for radial conductivities

Thierry Daud\'e, Fabricio Maci\`a, Crist\'obal Mero\~no, Fran\c{c}ois Nicoleau — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.22585v2 Announce Type: replace Abstract: The problem of characterizing sequences of real numbers that arise as spectra of Dirichlet-to-Neumann (DtN) maps for elliptic operators has attracted considerable attention over the past fifty years. In this article, we address this question in the simple setting of DtN maps associated with a rotation-invariant elliptic operator $\nabla \cdot (\gamma\nabla \centerdot )$ in the ball in Euclidean space. We show that the spectrum of such a DtN operator can be expressed as a universal term, determined solely by the boundary values of the conductivity $\gamma$, plus a sequence of Hausdorff moments of an integrable function, which we call the Born approximation of $\gamma$. We also show that this object is locally determined from the boundary by the corresponding values of the conductivity, a property that implies a local uniqueness result for the Calder\'on Problem in this setting. We also give a stability result: the functional mapping the Born approximation to its conductivity is H\"older stable in suitable Sobolev spaces. Finally, in order to refine the characterization of the Born approximation, we analyze its regularity properties and their dependence on the conductivity.

Prime and Semiprime Ideals in Commutative Ternary $\Gamma$-Semirings: Quotients, Radicals, Spectrum

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.23885v2 Announce Type: replace Abstract: The theory of ternary $\Gamma$-semirings extends classical ring and semiring frameworks by introducing a ternary product controlled by a parameter set $\Gamma$. Building on the foundational axioms recently established by Rao, Rani, and Kiran (2025), this paper develops the first systematic ideal-theoretic study within this setting. We define and characterize prime and semiprime ideals for commutative ternary $\Gamma$-semirings and prove a quotient characterization: an ideal $P$ is prime if and only if $T/P$ is free of nonzero zero-divisors under the induced ternary $\Gamma$-operation. Semiprime ideals are shown to be stable under arbitrary intersections and coincide with their radicals, providing a natural bridge to radical and Jacobson-type structures. A correspondence between prime ideals and prime congruences is established, leading to a Zariski-like spectral topology on $\mathrm{Spec}(T)$. Computational classification of all commutative ternary $\Gamma$-semirings of order $\leq 4$ confirms the theoretical predictions and reveals novel structural phenomena absent in binary semiring theory. The results lay a rigorous algebraic and computational foundation for subsequent categorical, geometric, and fuzzy extensions of ternary $\Gamma$-algebras.

Uniqueness of the non-commutative divergence cocycle

Pauline Baudat — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2511.06903v2 Announce Type: replace Abstract: We show that, for $n \geq 3 $, 1-cocycles of degree zero on the Lie algebra of derivations of the free associative algebra $T(A_n)$ with values in $ \rvert T(A_n) \rvert \otimes \rvert T(A_n) \rvert $ are linear combinations of the non-commutative divergence and its switch, when restricted to finite-degree quotients. Here, $ \rvert T(A_n) \rvert $ denotes the space of cyclic words. Furthermore, we study 1-cocycles of degree zero on the Lie algebra of symplectic derivations of the free Lie algebra $ \mathfrak{L_{2n}}$, and prove the uniqueness of the Enomoto-Satoh trace.

Model-oriented Graph Distances via Partially Ordered Sets

Armeen Taeb, F. Richard Guo, Leonard Henckel — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2511.10625v2 Announce Type: replace Abstract: A well-defined distance on the parameter space is key to evaluating estimators, ensuring consistency, and building confidence sets. While there are typically standard distances to adopt in a continuous space, this is not the case for combinatorial parameters such as graphs that represent statistical models. Defined on the graphs alone, existing proposals like the structural Hamming distance ignore the structure of the model space and can thus exhibit undesirable behaviors. We propose a model-oriented framework for defining the distance between graphs that is applicable across different graph classes. Our approach treats each graph as a statistical model and organizes the graphs in a partially ordered set based on model inclusion. This induces a neighborhood structure, from which we define the model-oriented distance as the length of a shortest path through neighbors, yielding a metric in the space of graphs. We apply this framework to probabilistic undirected graphs, causal directed acyclic graphs, probabilistic completed partially directed acyclic graphs, and causal maximally oriented partially directed acyclic graphs. We analyze theoretical and empirical behaviors of the model-oriented distance and draw comparison with existing distances. By exploiting the underlying poset structures, we develop algorithms for computing and bounding the proposed distance that scale to moderate-sized graphs.

Stable subgroups of graph products

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2511.11176v2 Announce Type: replace Abstract: We extend the characterization of stable subgroups of right-angled Artin groups of Koberda, Mangahas and Taylor to the case of graph products of infinite groups. Specifically, we show that the stable subgroups of such graph products are exactly the subgroups that quasi-isometrically embed in the associated contact graph. Equivalently, they are the subgroups that satisfy a condition arising from the defining graph: a stable subgroup is an almost join-free subgroup. In particular, we generalize the equivalence between stable and purely loxodromic subgroups from Koberda, Mangahas and Taylor in the case where all torsion subgroups of the vertex groups are finite, and the equivalence between stable and infinite index Morse subgroups from Tran in the case where the defining graph is connected.

Data-integrated neural networks for solving partial differential equations

Jiachun Zheng, Yunqing Huang, Nianyu Yi, Yunlei Yang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2511.12055v4 Announce Type: replace Abstract: In this work, we propose data-integrated neural networks (DataInNet) for solving partial differential equations (PDEs), offering a novel approach to leveraging data (e.g., source terms, initial conditions, and boundary conditions). The core of this work lies in the integration of data into a unified network framework. DataInNet comprises two subnetworks: a data integration neural network responsible for accommodating and fusing various types of data, and a fully connected neural network dedicated to learning the residual physical information not captured by the data integration neural network. This network architecture inherently excludes function classes that violate known physical constraints, thereby substantially narrowing the solution search space. Numerical experiments demonstrate that the proposed DataInNet delivers superior performance on challenging problems, such as the Helmholtz equation (relative $L^2$ error: O($10^{-6}$)) and PDEs with high frequency solutions (relative $L^2$ error: O($10^{-5}$)).

Generalized ovals, 2.5-dimensional additive codes, and multispreads

Denis S. Krotov, Sascha Kurz — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2511.15843v2 Announce Type: replace Abstract: We present constructions and bounds for additive codes over a finite field in terms of their geometric counterpart, i.e., projective systems. It is known that the maximum number of $(h-1)$-spaces in PG$(2,q)$, such that no hyperplane contains three, is given by $q^h+1$ if $q$ is odd. Those geometric objects are called generalized ovals. We show that cardinality $q^h+2$ is possible if we decrease the dimension a bit. We completely determine the minimum possible lengths of additive codes over GF$(9)$ of dimension $2.5$ and give improved constructions for other small parameters, including codes outperforming the best linear codes. As an application, we consider multispreads in PG$(4,q)$, in particular, completing the characterization of parameters of GF$(4)$-linear $64$-ary one-weight codes. Keywords: additive code, projective system, generalized oval, multispread, one-weight code, two-weight code

On finiteness properties of separating semigroup of real curve

Matthew Magin — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2511.18545v2 Announce Type: replace Abstract: A real morphism $f$ from a real algebraic curve $X$ to $\mathbb{P}^1$ is called separating if $f^{-1}(\mathbb{R} \mathbb{P}^1) = \mathbb{R} X$. A separating morphism defines a covering $\mathbb{R} X \to \mathbb{R} \mathbb{P}^1$. Let $X_1, \ldots, X_r$ denote the components of $\mathbb{R} X$. M. Kummer and K. Shaw~\cite{kummer_separating_2020} defined the separating semigroup of a curve $X$ as the set of all vectors $d(f) = (d_1(f), \ldots, d_r(f)) \in \mathbb{N}^{r}$ where $f$ is a separating morphism $X \to \mathbb{P}^1$ and $d_i(f)$ is the degree of the restriction of $f$ to $X_i$. In the present paper we prove that for a non-negative integer number $g$ the set of all separating semigroups of genus $g$ curves is finite.

Age Optimal Sampling and Routing under Intermittent Links and Energy Constraints

Adem Utku Atasayar, Aimin Li, \c{C}a\u{g}r{\i} Ar{\i}, Elif Uysal — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.00985v2 Announce Type: replace Abstract: Links in practical systems, such as satellite--terrestrial integrated networks, exhibit distinct delay distributions, intermittent availability, and heterogeneous energy costs. These characteristics pose significant challenges to maintaining timely and energy-efficient status updates. While link availability restricts feasible transmission routes, routing decisions determine the actual delay and energy expenditure. This paper tackles these challenges by jointly optimizing sampling and routing decisions to minimize monotonic, non-linear Age of Information (AoI). The proposed formulation incorporates key system features, including multiple routes with correlated random delays, stochastic link availability, and route-dependent energy consumption. We model the problem as an infinite-horizon Constrained Semi-Markov Decision Process (CSMDP) with a hybrid state--action space and develop an efficient nested algorithm, termed Bisec-\textsc{ReaVI}, to solve this problem. We analyze the structural properties of the solution and reveal a well-defined jointly optimal policy structure: (i) For general monotonic penalty functions, the optimal sampling policy is a piecewise linear waiting policy with at most $N$ breakpoints given $N$ routes; and (ii) under a derived Expected Penalty Ordering condition, the optimal routing policy is a monotonic threshold-based handover policy characterized by at most $\binom{N}{2}$ thresholds. Numerical experiments in a \textit{satellite--terrestrial} integrated routing scenario demonstrate that the proposed scheme efficiently balances energy usage and information freshness, and reveal a counter-intuitive insight: \textit{even routes with higher average delay, higher delay variance or lower availability can still play a critical role in minimizing monotonic functions of AoI}.

Covariance Estimation for Matrix-variate Data via Fixed-rank Core Covariance Geometry

Bongjung Sung — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.01070v3 Announce Type: replace Abstract: We study the geometry of the fixed-rank core covariance manifold and propose a novel covariance estimator for matrix-variate data leveraging this geometry. To generalize the separable covariance model, Hoff, McCormack, and Zhang (2023) showed that every covariance matrix $\Sigma$ of $p_1\times p_2$ matrix-variate data uniquely decomposes into a separable component $K$ and a core component $C$. Such a decomposition also exists for rank-$r$ $\Sigma$ if $p_1/p_2+p_2/p_1

Function-Correcting Codes for Insertion-Deletion Channel

Anamika Singh, Abhay Kumar Singh — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.07243v2 Announce Type: replace Abstract: In coding theory, handling errors that occur when symbols are inserted or deleted from a transmitted message is a long-standing challenge. Optimising redundancy for insertion and deletion channels remains a key open problem with significant importance for applications in DNA data storage and document exchange. Recently, a coding framework known as function-correcting codes has been proposed to address the challenge of minimising redundancy while preserving specific functions of the message. This framework has gained attention due to its potential applications in machine learning systems and long-term archival data storage. Motivated by the problem of redundancy optimisation for insertion and deletion channels, we propose a new framework called function-correcting codes for insdel channels. In this paper, we introduce the notions of function-correcting insertion codes, function-correcting deletion codes, and function-correcting insdel codes, and we show that these three formulations are equivalent. We then define insdel distance matrices and irregular insdel-distance codes, and derive lower and upper bounds on the optimal redundancy achievable by function-correcting codes for insdel channels. In addition, we establish Gilbert-Varshamov and Plotkin-like bounds on the length of irregular insdel-distance codes. Using the relation between optimal redundancy and the length of such codes, we obtain a simplified lower bound on optimal redundancy. Finally, we derive bounds on the optimal redundancy of function-correcting insdel codes for several classes of functions, including locally bounded functions, VT syndrome functions, the number-of-runs function, and the maximum-run-length function.

Generalizations of the Normalized Radon Cumulative Distribution Transform for Limited Data Recognition

Matthias Beckmann, Robert Beinert, Jonas Bresch — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.08099v2 Announce Type: replace Abstract: The Radon cumulative distribution transform (R-CDT) exploits one-dimensional Wasserstein transport and the Radon transform to represent prominent features in images. It is closely related to the sliced Wasserstein distance and facilitates classification tasks, especially in the small data regime, like the recognition of watermarks in filigranology. Here, a typical issue is that the given data may be subject to affine transformations caused by the measuring process. To make the R-CDT invariant under arbitrary affine transformations, a two-step normalization of the R-CDT has been proposed in our earlier works. The aim of this paper is twofold. First, we propose a family of generalized normalizations to enhance flexibility for applications. Second, we study multi-dimensional and non-Euclidean settings by making use of generalized Radon transforms. We prove that our novel feature representations are invariant under certain transformations and allow for linear separation in feature space. Our theoretical results are supported by numerical experiments based on 2d images, 3d shapes and 3d rotation matrices, showing near perfect classification accuracies and clustering results.

L-equivalence and Fourier--Mukai partners of cubic fourfolds

Reinder Meinsma, Riccardo Moschetti — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.08651v2 Announce Type: replace Abstract: We study L-equivalence in the Grothendieck ring of varieties and its interaction with categorical invariants of cubic fourfolds. Assuming a Derived Torelli-type criterion for Kuznetsov components and a mild condition on the discriminant of the transcendental lattice, we prove a counting formula for Fourier--Mukai partners of such cubic fourfolds. As an application, we exhibit cubic fourfolds with a fixed algebraic lattice admitting a unique non-trivial Fourier--Mukai partner, which is trivially L-equivalent to the original. Finally, we show that L-equivalence classes of cubic fourfolds are finite.

On the complex zeros and the computational complexity of approximating the reliability polynomial

Ferenc Bencs, Chiara Piombi, Guus Regts — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.11504v2 Announce Type: replace Abstract: In this paper we relate the location of the complex zeros of the reliability polynomial to parameters at which a certain family of rational functions derived from the reliability polynomial exhibits chaotic behaviour. We use this connection to prove new results about the location of reliability zeros. In particular we show that there are zeros with modulus larger than $1$ with essentially any possible argument. We moreover use this connection to show that approximately evaluating the reliability polynomial for planar graphs at a non-positive algebraic number in the unit disk is #P-hard.

Taylor polynomials on left-quotients of Carnot groups

Alessandro Ottazzi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.12239v2 Announce Type: replace Abstract: We prove classical Taylor polynomial theorems for sub-Riemannian manifolds that are obtained as the submetric image of a Carnot group. For these theorems we also prove a sufficient condition for real analyticity and a result on L-harmonicity of Taylor polynomials.

A Note on the Sum-Product Problem and the Convex Sumset Problem

Adam Cushman — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.13849v2 Announce Type: replace Abstract: We provide a new exponent for the Sum-Product conjecture on $\mathbb{R} $. Namely for $A \subset \mathbb{R}$ finite, \[ \max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{4}{3} + \frac{10}{4407} - \epsilon} .\] We also provide new exponents for $A \subset \mathbb{R} $ finite and convex, namely \[ \left\lvert A+A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{46}{29} - \epsilon}, \] and \[ \left\lvert A-A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{8}{5} + \frac{1}{3440} -\epsilon} .\]

Equidistribution of polynomial sequences in function fields: resolution of a conjecture

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.16118v2 Announce Type: replace Abstract: Let $\mathbb F_q$ be the finite field of $q$ elements having characteristic $p$, and denote by $\mathbb K_\infty=\mathbb F_q((1/t))$ the field of formal Laurent series in $1/t$. We consider the equidistribution in $\mathbb T=\mathbb K_\infty/\mathbb F_q[t]$ of the values of polynomials $f(u)\in \mathbb K_\infty [u]$ as $u$ varies over $\mathbb F_q[t]$. Let $\mathcal K$ be a finite set of positive integers, and suppose that $\alpha_r\in \mathbb K_\infty$ for $r\in \mathcal K\cup \{0\}$. We show that the polynomial $\sum_{r\in \mathcal K\cup\{0\}}\alpha_ru^r$ is equidistributed in $\mathbb T$ whenever $\alpha_k$ is irrational for some $k\in \mathcal K$ satisfying $p\nmid k$, and also $p^vk\not\in \mathcal K$ for any positive integer $v$. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.

Study of a TPFA scheme for the stochastic Allen-Cahn problem with constraint through numerical experiments

Niklas Sapountzoglou, Aleksandra Zimmermann — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.17712v3 Announce Type: replace Abstract: This contribution provides numerical experiments for a finite volume scheme for an approximation of the stochastic Allen-Cahn equation with homogeneous Neumann boundary conditions. The approximation is done by a Yosida approximation of the subdifferential operator. The problem is set on a polygonal bounded domain in two or three dimensions. The non-linear character of the projection term induces challenges to implement the scheme. To this end, we provide a splitting method for the finite volume scheme. We show that the splitting method is accurate. The computational error estimates induce that the squared $L^2$-error w.r.t. time is of order $1$ as long as the noise term is small enough. For larger noise terms the order of convergence w.r.t. time might become worse.

Higher Order Dualities between Prime Ideals

Sroyon Sengupta — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.22346v2 Announce Type: replace Abstract: Extending the works of Alladi and Sweeting and Woo, we state and prove the general higher order duality between prime ideals in number rings. We then use the second order duality to obtain the a new formula for the Chebotarev Density involving sums of the generalized M\"obius function and the prime ideal counting function. We also provide two estimates of such sums as an application of the duality identity. A discussion of the duality in a slightly more general setting is done at the end.

Tameness of actions on finite rank median algebras

Michael Megrelishvili — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.01681v2 Announce Type: replace Abstract: We prove that for (compact) finite-rank median algebras the geometric rank equals the independence number of all (continuous) median-preserving functions to $[0,1]$. Combined with Rosenthal's dichotomy, this yields a generalized Helly selection principle: for finite-rank median algebras, the space of all median-preserving functions to $[0,1]$ is sequentially compact in the pointwise topology. Generalizing joint results with E. Glasner on dendrons (rank-1), we establish that every continuous action of a topological group $G$ by median automorphisms on a finite-rank compact median algebra is Rosenthal representable, hence dynamically tame. As an application, the Roller-Fioravanti compactification of finite-rank topological median $G$-algebras with compact intervals is often a dynamically tame $G$-system.

Singular basins in multiscale systems: tunneling between stable states

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.02001v2 Announce Type: replace Abstract: Real-world systems often evolve on different timescales and possess multiple coexisting stable states. Whether or not a system returns to a given stable state after being perturbed away from it depends on the shape and extent of its basin of attraction. We show that basins of attraction in multiscale systems can exhibit special geometric properties in the form of singular funnels. Although singular funnels are narrow, they can extend to different regions of the phase space and, unexpectedly, impact the system's resilience to perturbations. Consequently, singular funnels may prevent common dimensionality reductions in the limit of large timescale separation, such as the quasi-static approximation, adiabatic elimination and time-averaging of the fast variables. We refer to basins of attraction with singular funnels as singular basins. We show that singular basins are universal and occur robustly in a range of multiscale systems: the normal form of a pitchfork bifurcation with a slowly adapting parameter, an adaptive active rotator, and an adaptive network of phase rotators.

Hyperconvexity in partial metric spaces: challenges and outlooks

Dariusz Bugajewski, Piotr Kasprzak, Olivier Olela-Otafudu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.02279v2 Announce Type: replace Abstract: In this article, we present several different ways to define hyperconvexity in partial metric spaces. In particular, we show that the analogue of the Aronszajn--Panitchpakdi notion of hyperconvexity fails to exhibit certain key properties present in the classical metric setting.

On $W^{2,\varepsilon}$-estimates for a class of singular-degenerate parabolic equations

Junyuan Fang, Tuoc Phan — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.04324v2 Announce Type: replace Abstract: We study a class of parabolic equations in non-divergence form with measurable coefficients that exhibit singular and/or degenerate behavior governed by weights in the $A_{1+\frac{1}{n}}$-Muckenhoupt class. Under a smallness assumption on a weighted mean oscillation of the weights, we establish weighted $W^{2,\varepsilon}$-estimates in the spirit of F.-H. Lin. Our results particularly holds for equations whose leading coefficients are of logistic-type singularities, as well as to those with polynomial blow-up or vanishing with sufficiently small exponents. A central component of our approach is the development of local quantitative lower estimates for solutions, which are interpreted as the mean sojourn time of sample paths, a stochastic-geometric perspective that generalizes the seminal work of L. C. Evans. We address the singular-degenerate nature of the operators by employing a class of intrinsic weighted parabolic cylinders, combined with a perturbation argument and parabolic Aleksandrov-Bakelman-Pucci (ABP) estimates. Furthermore, we conduct a rigorous analysis of weight regularization and truncation to ensure that the estimates are independent of the regularization and truncation parameters. The results extend classical regularity theory to a broad class of second-order parabolic equations and provide a functional analytic foundation for further study of fully nonlinear parabolic equations with singular-degenerate structure

Robust and Secure Blockage-Aware Pinching Antenna-assisted Wireless Communication

Ruotong Zhao, Shaokang Hu, Deepak Mishra, Derrick Wing Kwan Ng — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.06430v2 Announce Type: replace Abstract: In this work, we investigate a blockage-aware pinching antenna (PA) system designed for secure and robust wireless communication. The considered system comprises a base station equipped with multiple waveguides, each hosting multiple PAs, and serves multiple single-antenna legitimate users in the presence of multi-antenna eavesdroppers under imperfect channel state information (CSI). To safeguard confidential transmissions, artificial noise (AN) is deliberately injected to degrade the eavesdropping channels. Recognizing that conventional linear CSI-error bounds become overly conservative for spatially distributed PA architectures, we develop new geometry-aware uncertainty sets that jointly characterize eavesdroppers position and array-orientation errors. Building upon these sets, we formulate a robust joint optimization problem that determines per-waveguide beamforming and AN covariance, individual PA power-ratio allocation, and PA positions to maximize the system sum rate subject to secrecy constraints. The highly non-convex design problem is efficiently addressed via a low computational complexity iterative algorithm that capitalizes on block coordinate descent, penalty-based methods, majorization-minimization, the S-procedure, and Lipschitz-based surrogate functions. Simulation results demonstrate that sum rates for the proposed algorithm outperforms conventional fixed antenna systems by 4.7 dB, offering substantially improved rate and secrecy performance. In particular, (i) adaptive PA positioning preserves LoS to legitimate users while effectively exploiting waveguide geometry to disrupt eavesdropper channels, and (ii) neglecting blockage effects in the PA system significantly impacts the system design, leading to performance degradation and inadequate secrecy guarantees.

The wanted extension of Fujii and Tsurumaru's formula for the spectral radius of the Bell-CHSH operator

Albrecht B\"ottcher, Ilya M. Spitkovsky — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.09392v2 Announce Type: replace Abstract: This paper is motivated by a recent paper of Yuki Fujii and Toyohiro Tsurumaru in which they established a beautiful formula for the spectral radius of the Bell-CHSH operator on finite-dimensional Hilbert spaces. To tackle the operator on infinite-dimensional spaces, they elaborated a method based on appropriate approximation of commutators of infinite-dimensional orthogonal projections by commutators of orthogonal projections on finite-dimensional spaces. We here give a proof of Fujii and Tsurumaru's original formula that works in all dimensions. We also present an alternative approximation procedure, uncover the connection of the problem with block Toeplitz operators, and derive good estimates and explicit expressions for the spectral radius in concrete cases.

Kov\'acs' conjecture on characterization of projective space and hyperquadrics

Soham Ghosh — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.10055v2 Announce Type: replace Abstract: We prove Kov\'acs' conjecture that claims that if the $p^{th}$ exterior power of the tangent bundle of a smooth complex projective variety contains the $p^{th}$ exterior power of an ample vector bundle then the variety is either projective space or the $p$-dimensional quadric hypersurface. We also prove a similar characterization involving symmetric powers instead of exterior powers. This provides a common generalization of Mori, Wahl, Cho-Sato, Andreatta-Wi\'sniewski, Kobayashi-Ochiai, and Araujo-Druel-Kov\'acs type characterizations of such varieties.

An Ito Formula via Predictable Projection for Non-Semimartingale Processes

Ramiro Fontes — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.10359v3 Announce Type: replace Abstract: We derive an Ito-type change-of-variables formula for stochastic processes admitting a causal derivation-divergence representation. The result applies to a broad class of non-semimartingale and rough processes, including Gaussian models with irregular covariance structure. The Ito correction term is expressed explicitly through a product rule for divergences rather than through quadratic variation. This formulation provides a unified operator-theoretic representation of classical and generalized Ito formulas.

Ehrhart quasi-polynomials via Barnes polynomials and discrete moments of parallelepipeds

Sinai Robins — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.12596v2 Announce Type: replace Abstract: We give novel and explicit formulas for the Ehrhart quasi-polynomials of rational simple polytopes, in terms of Barnes polynomials and discrete moments of half-open parallelepipeds. These formulas also hold for all positive dilations of a rational polytope. There is an interesting appearance of an extra complex z-parameter, which seems to allow for more compact formulations. We also give similar formulas for discrete moments of rational polytopes, and their positive dilates, objects known in the literature as sums of polynomials over a polytope. The appearance of the Barnes polynomials and the Barnes numbers allow for explicit computations. From this work, it is clear that the complexity of computing Ehrhart quasi-polynomials lies mainly in the computation of various discrete moments of parallelepipeds. These discrete moments are in general summed over a particular lattice flow on a compact torus, defined in this paper. Some of the consequences involve novel vanishing identities for rational polytopes. As another consequence, we obtain a differential equation for discrete moments of rational polytopes, which extends the work of Eva Linke. For smooth polytopes, we obtain novel and much simpler formulations of Ehrhart polynomials, discrete moments, and vanishing identities that may be of independent interest from the perspective of Barnes polynomials and Barnes numbers. These formulations show the utility of Barnes polynomials in geometric combinatorics, due to their very rich structure that extends the 1-dimensional Bernoulli polynomials.

Volume polynomials

June Huh — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.13249v3 Announce Type: replace Abstract: Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss applications to the combinatorics of algebraic matroids. These notes are based on lectures given at the 2025 Summer Research Institute in Algebraic Geometry at Colorado State University.

On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture

Barinder S. Banwait, Xiaoyu Huang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.16044v2 Announce Type: replace Abstract: Recent work of Burungale-Skinner-Tian-Wan established the first infinite families of quadratic twists of non-CM elliptic curves over $\mathbb{Q}$ for which the strong Birch-Swinnerton-Dyer (BSD) conjecture holds. Building on their results, we encode the required hypotheses into an explicit algorithm and apply it to the database of elliptic curves in the $L$-functions and Modular Forms Database (LMFDB), identifying all elliptic curves $E$ of conductor at most $500{,}000$ that admit infinitely many quadratic twists satisfying the strong BSD conjecture. Our computations provide certain numerical evidence for a conjecture of Radziwi{\l}{\l} and Soundararajan predicting Gaussian behavior in the analytic order of the Shafarevich-Tate group, while also observing a systematic positive bias within the BSD-satisfying subfamily.

Some Families of Type $B$ Set Partitions Counted by the Dowling Numbers

Per Alexandersson, Fufa Beyene, Roberto Mantaci — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.17174v2 Announce Type: replace Abstract: In this paper, we study type $B$ set partitions without zero block. Certain classes of these partitions, such as merging-free and separated partitions (enumerated by the Dowling numbers), are investigated. We show that these classes are in bijection with type $B$ set partitions. The intersection of these two classes is also studied, and we prove that their block-generating polynomials are real-rooted. Finally, we study the descent statistics on the class of permutations obtained by flattening type $B$ merging-free partitions. Using the valley-hopping action, we prove the Gamma-positivity of the descent distribution and provide a combinatorial interpretation of the Gamma-coefficients. We also show that the descent statistic is homomesic under valley-hopping.

Asymptotics of the d'Arcais Numbers at Small $k$

Shannon Starr — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.18599v3 Announce Type: replace Abstract: The d'Arcais numbers are the triangular array $\{A(2,n,k)\, :\, n=0,1,\dots,\, k=0,\dots,n\}$, such that $\sum_{n=0}^{\infty} \sum_{k=0}^{n} A(2,n,k) x^k z^n/n! = ((z;z)_{\infty})^{-x}$. The infinite $q$-Pochhammer symbol is $(q;q)_{\infty} = \prod_{n=1}^{\infty} (1-q^n)$. Holding $k$ fixed and considering large $n$, we note that the ratio $k! A(2,n,k)/n!$ is asymptotic to $C(k) \sigma_{2k-1}(n)/n^k$ where the divisor sum function is $\sigma_p(n) = \sum_{d|n} d^p$ and $C(k) = (\zeta(2))^k/(\Gamma(k) \zeta(2k))$. This is a slightly generalized version of one of Ramanujan's formulas from his paper, ``On Certain Arithmetical Functions," and it is an immediate consequence of the more recent article of Oliver, Shreshta and Thorne. Heim and Neuhauser made a conjecture, that $A(2,n,k)/A(2,n,k-1)$ is greater than or equal to $A(2,n,k+1)/A(2,n,k)$, for $k=2,3,\dots$ and all $n$. The conjecture is false for $k=2$, and it is true for $k=3,4,\dots$ when $n$ is sufficiently large. We consider the Hardy-Ramanujan circle method as a heuristic step.

On the SOS Rank of Simple and Diagonal Biquadratic Forms

Yi Xu, Chufeng Cui, Liqun Qi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.19195v2 Announce Type: replace Abstract: We study the sum-of-squares (SOS) rank of simple and diagonal biquadratic forms. For simple biquadratic forms in $3 \times 3$ variables, we show that the maximum SOS rank is exactly $6$, attained by a specific six-term form. We further prove that for any $m \ge 3$, there exists an $m \times m$ simple biquadratic form whose SOS rank is exactly $2m$, providing a general lower bound that extends the $3\times3$ case. For diagonal biquadratic forms with nonnegative coefficients, we prove an SOS rank upper bound of $7$, improving the general bound of $8$ for $3 \times 3$ forms. In addition, we extend the techniques to a broader class of \textbf{sparse biquadratic forms}, obtaining combinatorial upper bounds and constructing explicit families whose SOS rank grows linearly with the number of terms. These results provide new lower and upper bounds on the worst-case SOS rank of biquadratic forms and highlight the role of structure in reducing the required number of squares.

Spectral stability of shock profiles for the Navier-Stokes-Poisson system

Wanyong Shim — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.20684v2 Announce Type: replace Abstract: We investigate the spectral stability of small-amplitude shock profiles for the one-dimensional isothermal Navier-Stokes-Poisson system, which describes ion dynamics in a collision-dominated plasma. Specifically, we establish (i) bounds on the essential spectrum, (ii) bounds on the point spectrum, and (iii) simplicity of the zero eigenvalue for the linearized operator about the profile in $L^2$. The result in (i) shows that the zero eigenvalue arising from translation invariance is embedded in the essential spectrum. Consequently, the standard Evans function approach cannot be applied directly to prove (iii). To resolve this, we employ an Evans-function framework that extends into regions of the essential spectrum, thereby enabling us to compute the derivative of the Evans function at the origin. Our result establishes that this derivative admits a factorization into two factors: one associated with transversality of the connecting profile and the other with hyperbolic stability of the corresponding shock of the quasi-neutral Euler system. We further show that both factors are nonzero, which implies simplicity of the zero eigenvalue.

A locking-free mixed virtual element discretization for the elasticity eigenvalue problem

Felipe Lepe, Gonzalo Rivera — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.20807v2 Announce Type: replace Abstract: In this paper, we introduce a mixed virtual element method to approximate the eigenvalues and eigenfunctions of the two-dimensional elasticity eigenvalue problem. Under standard assumptions on the meshes, we prove the convergence of the discrete solution operator to the continuous one as the mesh size tends to zero. Using the theory of compact operators, we analyze the convergence of the method and derive error estimates for both the eigenvalues and eigenfunctions. We validate our theoretical results with a series of numerical tests, in which we compute convergence orders and show that the method is locking-free and capable of accurately approximating the spectrum independently of the shape of the polygons on the meshes.

Identification of space-dependent coefficients in two competing terms of a nonlinear subdiffusion equation

Barbara Kaltenbacher, William Rundell — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.21018v2 Announce Type: replace Abstract: We consider a (sub)diffusion equation with a nonlinearity of the form $pf(u)-qu$, where $p$ and $q$ are space dependent functions. Prominent examples are the Fisher-KPP, the Frank-Kamenetskii-Zeldovich and the Allen-Cahn equations. We devise a fixed point scheme for reconstructing the spatially varying coefficients from interior observations a) at final time under two different excitations b) at two different time instances under a single excitation. Convergence of the scheme as well as local uniqueness of these coefficients is proven. Numerical experiments illustrate the performance of the reconstruction scheme.

Decay rates to equilibrium in a nonlinear subdiffusion equation with two counteracting terms

Barbara Kaltenbacher — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.21038v2 Announce Type: replace Abstract: In this paper we prove convergence to a steady state as $t\to\infty$ for solutions to the subdiffusion equation \[ \partial_t^\alpha u - \mathbb{L} u = q(x)u - p(x)f(u) + r \] with the exponential ($\alpha=1$) or power law ($\alpha\in[0,1)$) rates under mild conditions on the coefficients $p$, $q$, the nonlinearity $f$, the source $r$, and the elliptic operator $\mathbb{L}$.

Explicit Construction of Maass Wave Forms and Their Petersson Inner Products

Daichi Tanaka — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.21588v2 Announce Type: replace Abstract: In this paper, we explicitly construct Maass wave cusp forms associated to Hecke characters on arbitrary real quadratic fields. This result is a generalization of Maass (1949), who constructed Maass wave cusp forms under the assumption that narrow class number is one. We also compute its Petersson inner product explicitly and give a few examples involving dihedral Artin representation.

The sum-product problem for small sets II

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.21828v2 Announce Type: replace Abstract: We establish that every set of $k=10$ natural numbers determines at least $30$ distinct pairwise sums or at least $30$ distinct pairwise products, as well as the analogous result for $k=11$ and at least $34$ sums/products, with sharpness exhibited by $\{1, 2, 3, 4, 6, 8, 9, 12, 16, 18\}$ and $\{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24\}$, respectively. This extends previous work of the fifth author with Clevenger, Havard, Heard, Lott, and Wilson, which established the corresponding thresholds for $k\leq 9$. Included is a classification result for sets of $10$ real numbers (resp. positive real numbers) determining at most $29$ pairwise sums (resp. pairwise products) that do not contain $8$ elements of any single arithmetic progression (resp. geometric progression), as well as some observations controlling additive quadruples in small subsets of two-dimensional generalized geometric progressions.

Iterative execution of discrete and inverse discrete Fourier transforms with applications for signal denoising via sparsification

H. Robert Frost — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2211.09284v4 Announce Type: replace-cross Abstract: We describe a family of iterative algorithms that involve the repeated execution of discrete and inverse discrete Fourier transforms. One interesting member of this family is motivated by the discrete Fourier transform uncertainty principle and involves the application of a sparsification operation to both the real domain and frequency domain data with convergence obtained when real domain sparsity hits a stable pattern. This sparsification variant has practical utility for signal denoising, in particular the recovery of a periodic spike signal in the presence of Gaussian noise. General convergence properties and denoising performance relative to existing methods are demonstrated using simulation studies. An R package implementing this technique and related resources can be found at https://hrfrost.host.dartmouth.edu/IterativeFT.

The complexity of solving a system of equations of the same degree

Giulia Gaggero, Elisa Gorla — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2309.03855v3 Announce Type: replace-cross Abstract: Many systems of interest in cryptography consist of equations of the same degree. Under the assumption that the degree of regularity is finite, we prove upper bounds on the degree of regularity of a system of equations of the same degree, with or without adding the field equations to the system. The bounds translate into upper bounds on the solving degree of the systems, and hence on the complexity of solving them via Gr\"obner bases methods. Our bounds depend on the number of equations in the system, the number of variables, and the degree of the equations.

Quantum speedups for linear programming via interior point methods

Simon Apers, Sander Gribling — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2311.03215v3 Announce Type: replace-cross Abstract: We describe a quantum algorithm based on an interior point method for solving a linear program with $n$ inequality constraints on $d$ variables. The algorithm explicitly returns a feasible solution that is $\varepsilon$-close to optimal, and runs in time $\sqrt{n} \cdot \mathrm{poly}(d,\log(n),\log(1/\varepsilon))$ which is sublinear for tall linear programs (i.e., $n \gg d$). Our algorithm speeds up the Newton step in the state-of-the-art interior point method of Lee and Sidford [FOCS '14]. This requires us to efficiently approximate the Hessian and gradient of the barrier function, and these are our main contributions. To approximate the Hessian, we describe a quantum algorithm for the \emph{spectral approximation} of $A^T A$ for a tall matrix $A \in \mathbb R^{n \times d}$. The algorithm uses leverage score sampling in combination with Grover search, and returns a $\delta$-approximation by making $O(\sqrt{nd}/\delta)$ row queries to $A$. This generalizes an earlier quantum speedup for graph sparsification by Apers and de Wolf [FOCS '20]. To approximate the gradient, we use a recent quantum algorithm for multivariate mean estimation by Cornelissen, Hamoudi and Jerbi [STOC '22]. While a naive implementation introduces a dependence on the condition number of the Hessian, we avoid this by pre-conditioning our random variable using our quantum algorithm for spectral approximation.

Estimating the Decoding Failure Rate of Binary Regular Codes Using Iterative Decoding

Alessandro Annechini, Alessandro Barenghi, Gerardo Pelosi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2401.16919v4 Announce Type: replace-cross Abstract: Providing closed-form estimates of the decoding failure rate of iterative decoders for low- and moderate-density binary parity-check codes has attracted significant interest in the research community. Recently, interest in this topic has increased due to the use of iterative decoders in post-quantum cryptosystems, where the desired decoding failure rates (DFRs) are less than or equal to $2^{-128}$ and impossible to estimate via Monte Carlo simulations. We propose a new technique that provides accurate DFR estimates for a two-iteration (parallel) bit-flipping decoder that can be used for cryptographic purposes. We estimate the bit-flipping probabilities at the second decoder iteration and the syndrome weight distribution before and after the first iteration as a function of the code parameters and error weight. We validate our results numerically by comparing the modelled and simulated syndrome weights, the incorrectly guessed error bit distribution at the end of the first iteration, and the DFR after two iterations in both the floor and waterfall regimes. Finally, we apply our method to estimate the DFR of the LEDAcrypt cryptographic system, a post-quantum key encapsulation method that employs a two-iteration bit-flipping decoder. We show that the DFR estimate resulting from the chosen code parameters can be improved by a factor larger than $2^{70}$ with respect to previous estimation techniques, when $128$-bit security is required. This allows for a $20$% reduction in public key and ciphertext sizes at no security loss. We note that our results can be applied to the post-quantum cryptosystem known as Bit Flipping Key Encapsulation (BIKE) replacing the current ``BIKE-flip decoder'' with the two-iteration decoder and consequently endowing BIKE with the property of indistinguishability under an adaptive chosen-ciphertext attack (IND-CCA$2$), provably.

TorchCP: A Python Library for Conformal Prediction

Jianguo Huang, Jianqing Song, Xuanning Zhou, Bingyi Jing, Hongxin Wei — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2402.12683v5 Announce Type: replace-cross Abstract: Conformal prediction (CP) is a powerful statistical framework that generates prediction intervals or sets with guaranteed coverage probability. While CP algorithms have evolved beyond traditional classifiers and regressors to sophisticated deep learning models like deep neural networks (DNNs), graph neural networks (GNNs), and large language models (LLMs), existing CP libraries often lack the model support and scalability for large-scale deep learning (DL) scenarios. This paper introduces TorchCP, a PyTorch-native library designed to integrate state-of-the-art CP algorithms into DL techniques, including DNN-based classifiers/regressors, GNNs, and LLMs. Released under the LGPL-3.0 license, TorchCP comprises about 16k lines of code, validated with 100\% unit test coverage and detailed documentation. Notably, TorchCP enables CP-specific training algorithms, online prediction, and GPU-accelerated batch processing, achieving up to 90\% reduction in inference time on large datasets. With its low-coupling design, comprehensive suite of advanced methods, and full GPU scalability, TorchCP empowers researchers and practitioners to enhance uncertainty quantification across cutting-edge applications.

Extending Mean-Field Variational Inference via Entropic Regularization: Theory and Computation

Bohan Wu, David Blei — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2404.09113v4 Announce Type: replace-cross Abstract: Variational inference (VI) has emerged as a popular method for approximate inference for high-dimensional Bayesian models. In this paper, we propose a novel VI method that extends the naive mean field via entropic regularization, referred to as $\Xi$-variational inference ($\Xi$-VI). $\Xi$-VI has a close connection to the entropic optimal transport problem and benefits from the computationally efficient Sinkhorn algorithm. We show that $\Xi$-variational posteriors effectively recover the true posterior dependency, where the dependence is downweighted by the regularization parameter. We analyze the role of dimensionality of the parameter space on the accuracy of $\Xi$-variational approximation and how it affects computational considerations, providing a rough characterization of the statistical-computational trade-off in $\Xi$-VI. We also investigate the frequentist properties of $\Xi$-VI and establish results on consistency, asymptotic normality, high-dimensional asymptotics, and algorithmic stability. We provide sufficient criteria for achieving polynomial-time approximate inference using the method. Finally, we demonstrate the practical advantage of $\Xi$-VI over mean-field variational inference on simulated and real data.

On the statistical analysis of grouped data: when Pearson $\chi^2$ and other divisible statistics are not goodness-of-fit tests

Sara Algeri, Estate V. Khmaladze — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2406.09195v5 Announce Type: replace-cross Abstract: Thousands of experiments are analyzed and papers are published each year involving the statistical analysis of grouped data. While this area of statistics is often perceived -- somewhat naively -- as saturated, several misconceptions still affect everyday practice, and new frontiers have so far remained unexplored. Researchers must be aware of the limitations affecting their analyses and what are the new possibilities in their hands. Motivated by this need, the article introduces a unifying approach to the analysis of grouped data, which allows us to study the class of divisible statistics -- that includes Pearson's $\chi^2$, the likelihood ratio as special cases -- with a fresh perspective. The contributions collected in this manuscript span from modeling and estimation to distribution-free goodness-of-fit tests. Perhaps the most surprising result presented here is that, in a sparse regime, all tests proposed in the literature are dominated by members of the class of weighted linear statistics.

Modularity maximization and community detection in complex networks through recursive and hierarchical annealing in the D-Wave Advantage quantum processing units

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2410.07744v3 Announce Type: replace-cross Abstract: Quantum adiabatic optimization has long been expected to outperform classical methods in solving NP-type problems. While this has been proven in certain experiments, its main applications still reside in academic problems where the size of the system to be solved would not represent an obstacle to any modern desktop computer. Here we develop a systematic procedure to find the global optima of the modularity function to discover community structure in complex networks solely relying on pure annealers rather than hybrid solutions. We bypass the one-hot encoding constraints by hierarchically and recursively encoding binary instances of the problem that can be solved without the need to guess the exact penalties for the Lagrange multipliers. We study the variability, and robustness of the annealing process as a function of network size, directness of connections, topology, and the resolution of the communities. We show how our approach produces meaningful and at least equally optimal solutions to state-of-the-art community detection algorithms while maintaining tractable computing times. Lastly, due to its recursive nature, the annealing process returns intermediate subdivisions thus offering interpretable rather than black-box solutions. These \textit{dendrograms} can be used to unveil normal and pathological hidden hierarchies in brain networks hence opening the door to clinical workflows. Overall, this represents a first step towards an applicable practice-oriented usage of pure quantum annealing potentially bridging two segregated communities in modern science and engineering; that of network science and quantum computing.

Stein's method for marginals on large graphical models

Tiangang Cui, Shuigen Liu, Xin T. Tong — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2410.11771v3 Announce Type: replace-cross Abstract: Many spatial models exhibit locality structures that effectively reduce their intrinsic dimensionality, enabling efficient approximation and sampling of high-dimensional distributions. However, existing approximation techniques primarily focus on joint distributions and do not provide precise accuracy control for low-dimensional marginals, which are of primary interest in many practical scenarios. By leveraging the locality structures, we establish a dimension independent uniform error bound for the marginals of approximate distributions. Inspired by the Stein's method, we introduce a novel $\delta$-locality condition that quantifies the locality in distributions, and link it to the structural assumptions such as the sparse graphical models. The theoretical guarantee motivates the localization of existing sampling methods, as we illustrate through the localized likelihood-informed subspace method and localized score matching. We show that by leveraging the locality structure, these methods greatly reduce the sample complexity and computational cost via localized and parallel implementations.

Capacity-Achieving Entanglement Purification Protocol for Pauli Dephasing Channel

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2411.14573v2 Announce Type: replace-cross Abstract: Quantum communication enables secure information transmission and entanglement distribution, but these tasks are fundamentally limited by the capacities of quantum channels. While quantum repeaters can mitigate losses and noise, entanglement swapping via a central node is ineffective against the Pauli dephasing channel due to degradation from Bell-state measurements. This suggests that purifying distributed Bell states before entanglement swapping is necessary. Although one-way hashing codes are known to saturate the dephasing channel capacity, no explicit two-way purification protocol has previously been shown to achieve this bound. In this work, we present a two-way entanglement purification protocol with an explicit, scalable circuit that asymptotically achieves the dephasing channel capacity. With each iteration, the fidelity of Bell states increases. At the final round, the residual dephasing error is suppressed doubly-exponentially, scaling as $\mathcal{O}(p^{2^{n}})$, enabling near-perfect Bell pairs for any fixed number of purification rounds $n$. The explicit circuit we propose is versatile and applicable to any number of Bell pairs, offering a practical solution for mitigating decoherence in quantum networks and distributed.

Estimation of relative risk, odds ratio and their logarithms with guaranteed accuracy and controlled sample size ratio

Luis Mendo — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2503.04876v3 Announce Type: replace-cross Abstract: Given two populations from which independent binary observations are taken with parameters $p_1$ and $p_2$ respectively, estimators are proposed for the relative risk $p_1/p_2$, the odds ratio $p_1(1-p_2)/(p_2(1-p_1))$ and their logarithms. The sampling strategy used by the estimators is based on two-stage sequential sampling applied to each population, where the sample sizes of the second stage depend on the results observed in the first stage. The estimators guarantee that the relative mean-square error, or the mean-square error for the logarithmic versions, is less than a target value for any $p_1, p_2 \in (0,1)$, and the ratio of average sample sizes from the two populations is close to a prescribed value. The estimators can also be used with group sampling, whereby samples are taken in batches of fixed size from the two populations simultaneously, each batch containing samples from the two populations. The efficiency of the estimators with respect to the Cram\'er-Rao bound is good, and in particular it is close to $1$ for small values of the target error.

Fault Tolerant Quantum Simulation via Symplectic Transvections

Zhuangzhuang Chen, Jack Owen Weinberg, Narayanan Rengaswamy — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2504.11444v2 Announce Type: replace-cross Abstract: Conventional approaches to fault-tolerant quantum computing realize logical circuits gate-by-gate, synthesizing each gate independently on one or more code blocks. This incurs excess overhead and doesn't leverage common structures in quantum algorithms. In contrast, we propose a framework that enables the execution of entire logical circuit blocks at once, preserving their global structure. This whole-block approach allows for the direct implementation of logical Trotter circuits - of arbitrary rotation angles - on any stabilizer code, providing a powerful new method for fault tolerant Hamiltonian simulation within a single code block. At the heart of our approach lies a deep structural correspondence between symplectic transvections and Trotter circuits. This connection enables both logical and physical circuits to share the Trotter structure while preserving stabilizer centralization and circuit symmetry even in the presence of non-Clifford rotations. We discuss potential approaches to fault tolerance via biased noise and code concatenation. While we illustrate the key principles using a $[[8,3,3]]$ code, our simulations show that the framework applies to Hamiltonian simulation on even good quantum LDPC codes. These results open the door to new algorithm-tailored, block-level strategies for fault tolerant circuit design, especially in quantum simulation.

Generalization Dynamics of Linear Diffusion Models

Claudia Merger, Sebastian Goldt — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2505.24769v2 Announce Type: replace-cross Abstract: Diffusion models are powerful generative models that produce high-quality samples from complex data. While their infinite-data behavior is well understood, their generalization with finite data remains less clear. Classical learning theory predicts that generalization occurs at a sample complexity that is exponential in the dimension, far exceeding practical needs. We address this gap by analyzing diffusion models through the lens of data covariance spectra, which often follow power-law decays, reflecting the hierarchical structure of real data. To understand whether such a hierarchical structure can benefit learning in diffusion models, we develop a theoretical framework based on linear neural networks, congruent with a Gaussian hypothesis on the data. We quantify how the hierarchical organization of variance in the data and regularization impacts generalization. We find two regimes: When $N d$, we find that the sampling distributions of linear diffusion models approach their optimum (measured by the Kullback-Leibler divergence) linearly with $d/N$, independent of the specifics of the data distribution. Our work clarifies how sample complexity governs generalization in a simple model of diffusion-based generative models.

PPO in the Fisher-Rao geometry

Razvan-Andrei Lascu, David \v{S}i\v{s}ka, {\L}ukasz Szpruch — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2506.03757v2 Announce Type: replace-cross Abstract: Proximal Policy Optimization (PPO) is widely used in reinforcement learning due to its strong empirical performance, yet it lacks formal guarantees for policy improvement and convergence. PPO's clipped surrogate objective is motivated by a lower bound on linearization of the value function in flat geometry setting. We derive a tighter surrogate objective and introduce Fisher-Rao PPO (FR-PPO) by leveraging the Fisher-Rao (FR) geometry. Our scheme provides strong theoretical guarantees, including monotonic policy improvement. In the direct parametrization setting, we show that FR-PPO achieves sub-linear convergence with no dependence on action or state space dimensions, and for parametrized policies we further obtain sub-linear convergence up to the compatible function approximation error. Finally, although our primary focus is theoretical, we also demonstrate empirically that FR-PPO performs well across a range of standard reinforcement learning tasks.

Antithetic Noise in Diffusion Models

Jing Jia, Sifan Liu, Bowen Song, Wei Yuan, Liyue Shen, Guanyang Wang — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2506.06185v2 Announce Type: replace-cross Abstract: We systematically study antithetic initial noise in diffusion models, discovering that pairing each noise sample with its negation consistently produces strong negative correlation. This universal phenomenon holds across datasets, model architectures, conditional and unconditional sampling, and even other generative models such as VAEs and Normalizing Flows. To explain it, we combine experiments and theory and propose a \textit{symmetry conjecture} that the learned score function is approximately affine antisymmetric (odd symmetry up to a constant shift), supported by empirical evidence. This negative correlation leads to substantially more reliable uncertainty quantification with up to $90\%$ narrower confidence intervals. We demonstrate these gains on tasks including estimating pixel-wise statistics and evaluating diffusion inverse solvers. We also provide extensions with randomized quasi-Monte Carlo noise designs for uncertainty quantification, and explore additional applications of the antithetic noise design to improve image editing and generation diversity. Our framework is training-free, model-agnostic, and adds no runtime overhead. Code is available at https://github.com/jjia131/Antithetic-Noise-in-Diffusion-Models-page.

Matrix-Driven Identification and Reconstruction of LLM Weight Homology

Ruichong Zhang, Daniel Goldstein — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2508.06309v3 Announce Type: replace-cross Abstract: We propose Matrix-Driven Identification and Reconstruction (MDIR), a SOTA large language model homology method that accurately detects weight correspondences between models and provides rigorous $p$-value estimation of the statistical significance of these correspondences. Our method does not require model inference, and allows the detection of unattributed reuse or replication of model weights even on low-resource devices as it compares only a single pair of matrices at a time. We leverage matrix analysis, polar decomposition, and Large Deviation Theory (LDT) to achieve accurate reconstruction of weight relationships between models. Notably, MDIR is the first method to achieve perfect scores on both Area-Under-Curve (AUC) and accuracy metrics across different source models on LeaFBench.

Effective permeability conditions for diffusive transport through impermeable membranes with gaps

Molly Brennan, Edwina F. Yeo, Philip Pearce, Mohit P. Dalwadi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2508.10694v5 Announce Type: replace-cross Abstract: Membranes regulate transport in a wide variety of industrial and biological applications. The microscale geometry of the membrane can significantly affect overall transport through the membrane, but the precise nature of this multiscale coupling is not well characterised in general. Motivated by the application of transport across a bacterial membrane, in this paper we use formal multiscale analysis to derive explicit effective coupling conditions for macroscale transport across a two-dimensional impermeable membrane with periodically spaced gaps, and validate these with numerical simulations. We derive analytic expressions for effective macroscale quantities associated with the membrane, such as the permeability, in terms of the microscale geometry. Our results generalise the classic constitutive membrane coupling conditions to a wider range of membrane geometries and time-varying scenarios. Specifically, we demonstrate that if the exterior concentration varies in time, for membranes with long channels, the transport gains a memory property where the coupling conditions depend on the system history. By applying our effective conditions in the context of small molecule transport through gaps in bacterial membranes called porins, we predict that bacterial membrane permeability is primarily dominated by the thickness of the membrane. Furthermore, we predict how alterations to membrane microstructure, for example via changes to porin expression, might affect overall transport, including when external concentrations vary in time. These results will apply to a broad range of physical applications with similar membrane structures, from medical and industrial filtration to carbon capture.

First passage of a run-and-tumble particle with exponentially-distributed tumble duration in the presence of a drift

Pascal Grange, Linglong Yuan — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.11308v2 Announce Type: replace-cross Abstract: We consider a run-and-tumble particle on a finite interval $[a,b]$ with two absorbing end points. The particle has an internal velocity state that switches between three values $v,0,-v$ at exponential times, thus incorporating positive tumble times. Moreover, a constant drift is added to the run-and-tumble motion at all times. The combination of these two features constitutes the main novelty of our model. The densities of the first-passage time through $a$ (given the initial position and velocity states) satisfy certain forward Fokker--Planck equations, whose Laplace transforms induce evolution equations for the exit probabilities and mean first-passage times of the particle. We solve these equations explicitly for all possible initial states. We consider the limiting regimes of instantaneous tumble and/or the limit of large $b$ to confirm consistency with existing results in the literature. In particular, in the limit of a half-line (large $b$), the mean first-passage time conditioned on the exit through $a$ is an affine function of the initial position if the drift is positive, as in the case of instantaneous tumble.

Instability of the halocline at the North Pole

Christian Puntini — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.21378v2 Announce Type: replace-cross Abstract: In this paper we address the issue of stability for the near-inertial Pollard waves, as a model for the halocline in the region of the Arctic Ocean centered around the North Pole, derived in Puntini (2025a). Adopting the short-wavelength instability approach, the stability of such flows reduces to study the stability of a system of ODEs along fluid trajectories, leading to the result that, when the steepness of the near-inertial Pollard waves exceeds a specific threshold, those waves are linearly unstable. The explicit dispersion relation of the model allows to easily compute such threshold, knowing the physical properties of the water column.

Direct Bias-Correction Term Estimation for Average Treatment Effect Estimation

Masahiro Kato — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.22122v2 Announce Type: replace-cross Abstract: This study considers the estimation of the direct bias-correction term for estimating the average treatment effect (ATE). Let $\{(X_i, D_i, Y_i)\}_{i=1}^{n}$ be the observations, where $X_i$ denotes $K$-dimensional covariates, $D_i \in \{0, 1\}$ denotes a binary treatment assignment indicator, and $Y_i$ denotes an outcome. In ATE estimation, $h_0(D_i, X_i) = \frac{1[D_i = 1]}{e_0(X_i)} - \frac{1[D_i = 0]}{1 - e_0(X_i)}$ is called the bias-correction term, where $e_0(X_i)$ is the propensity score. The bias-correction term is also referred to as the Riesz representer or clever covariates, depending on the literature, and plays an important role in construction of efficient ATE estimators. In this study, we propose estimating $h_0$ by directly minimizing the Bregman divergence between its model and $h_0$, which includes squared error and Kullback--Leibler divergence as special cases. Our proposed method is inspired by direct density ratio estimation methods and generalizes existing bias-correction term estimation methods, such as covariate balancing weights, Riesz regression, and nearest neighbor matching. Importantly, under specific choices of bias-correction term models and Bregman divergence, we can automatically ensure the covariate balancing property. Thus, our study provides a practical modeling and estimation approach through a generalization of existing methods.

Antiferromagnetic domain walls under spin-orbit torque

George Theodorou, Stavros Komineas — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.22241v2 Announce Type: replace-cross Abstract: Domain walls in antiferromagnets under a spin-polarized current present dynamical behavior that is not observed in ferromagnets, and it is tunable by the current polarization. Precessional dynamics is obtained for perpendicular spin polarization. In-plane polarization gives propagating walls. We obtain the velocity as a function of current by a perturbation method for low velocities, and the wall profile is found to lack a definite parity. For high velocities, a power-law decay develops in the trailing tail of the wall. The main features of the wall profile are manifest in a direct solution of an equation that is valid in a limiting case. Oscillatory motion of domain walls is obtained for a spin polarization that has both perpendicular and in-plane components, and an analytical description is given. We discuss the modifications of the dynamics when a Dzyaloshinskii-Moriya interaction is present. Finally, we give the magnetization of the dynamical walls and find that this can become large, providing a potential method for observations.

On the Separability of Information in Diffusion Models

Akhil Premkumar — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.23937v4 Announce Type: replace-cross Abstract: Diffusion models transform noise into data by injecting information that was captured in their neural network during the training phase. In this paper, we ask: \textit{what} is this information? We find that, in pixel-space diffusion models, (1) a large fraction of the total information in the neural network is committed to reconstructing small-scale perceptual details of the image, and (2) the correlations between images and their class labels are informed by the semantic content of the images, and are largely agnostic to the low-level details. We argue that these properties are intrinsically tied to the manifold structure of the data itself. Finally, we show that these facts explain the efficacy of classifier-free guidance: the guidance vector amplifies the mutual information between images and conditioning signals early in the generative process, influencing semantic structure, but tapers out as perceptual details are filled in.

Think Less, Label Better: Multi-Stage Domain-Grounded Synthetic Data Generation for Fine-Tuning Large Language Models in Telecommunications

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.25736v2 Announce Type: replace-cross Abstract: The success of large language models (LLMs) depends heavily on large-scale, high-quality instruction-following and reinforcement datasets. However, generating such data through human annotation is prohibitively time-consuming particularly for domain-specific tasks like telecom network troubleshooting, where accurate responses require deep technical expertise and contextual understanding. In this paper, we present a fully automated, retrieval-augmented pipeline for generating synthetic question-answer (QA) pairs grounded in structured domain knowledge. Our multi-stage framework integrates a retriever, base generator, and refinement model to synthesize and enhance QA pairs using documents retrieved from a domain-specific knowledge graph. To ensure data quality, we employ customized RAGAS-based scoring to filter low-quality samples, producing a high-quality dataset suitable for reinforcement fine-tuning (RFT). We demonstrate our approach in a real-world telecom scenario focused on radio access network (RAN) troubleshooting. The resulting pipeline generates complex, context-rich troubleshooting solution plans without human intervention. This work offers a scalable solution for building instruction and reinforcement datasets in specialized domains, significantly reducing dependence on manual labeling while maintaining high technical fidelity.

A Generalized Information Bottleneck Theory of Deep Learning

Charles Westphal, Stephen Hailes, Mirco Musolesi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2509.26327v3 Announce Type: replace-cross Abstract: The Information Bottleneck (IB) principle offers a compelling theoretical framework to understand how neural networks (NNs) learn. However, its practical utility has been constrained by unresolved theoretical ambiguities and significant challenges in accurate estimation. In this paper, we present a \textit{Generalized Information Bottleneck (GIB)} framework that reformulates the original IB principle through the lens of synergy, i.e., the information obtainable only through joint processing of features. We provide theoretical and empirical evidence demonstrating that synergistic functions achieve superior generalization compared to their non-synergistic counterparts. Building on these foundations we re-formulate the IB using a computable definition of synergy based on the average interaction information (II) of each feature with those remaining. We demonstrate that the original IB objective is upper bounded by our GIB in the case of perfect estimation, ensuring compatibility with existing IB theory while addressing its limitations. Our experimental results demonstrate that GIB consistently exhibits compression phases across a wide range of architectures (including those with \textit{ReLU} activations where the standard IB fails), while yielding interpretable dynamics in both CNNs and Transformers and aligning more closely with our understanding of adversarial robustness.

Stable Evaluation of Lefschetz Thimble Intersection Numbers: Towards Real-Time Path Integrals

Yutaro Shoji, Katarina Trailovi\'c — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.06334v3 Announce Type: replace-cross Abstract: We introduce a robust numerical method for determining intersection numbers of Lefschetz thimbles in multivariable settings. Our approach employs the multiple shooting method to solve the upward flow equations from the saddle points to the original integration cycle, which also enables us to determine the signs of the intersection numbers. The method demonstrates stable and reliable performance, and has been tested for systems with up to $20$ variables, which can be further extended by adopting quadruple-precision arithmetic. We determine intersection numbers for several complex saddle points in a discretized path integral, providing new insights into the structure of real-time path integrals. The proposed method is broadly applicable to a wide range of problems involving oscillatory integrals in physics and mathematics.

On the Provable Performance Guarantee of Efficient Reasoning Models

Hao Zeng, Jianguo Huang, Bingyi Jing, Hongxin Wei, Bo An — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.09133v2 Announce Type: replace-cross Abstract: Large reasoning models (LRMs) have achieved remarkable progress in complex problem-solving tasks. Despite this success, LRMs typically suffer from high computational costs during deployment, highlighting a need for efficient inference. A practical direction of efficiency improvement is to switch the LRM between thinking and non-thinking modes dynamically. However, such approaches often introduce additional reasoning errors and lack statistical guarantees for the performance loss, which are critical for high-stakes applications. In this work, we propose Probably Approximately Correct (PAC) reasoning that controls the performance loss under the user-specified tolerance. Specifically, we construct an upper confidence bound on the performance loss and determine a threshold for switching to the non-thinking model. Theoretically, using the threshold to switch between the thinking and non-thinking modes ensures bounded performance loss in a distribution-free manner. Our comprehensive experiments on reasoning benchmarks show that the proposed method can save computational budgets and control the user-specified performance loss.

MARS-M: When Variance Reduction Meets Matrices

Yifeng Liu, Angela Yuan, Quanquan Gu — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.21800v3 Announce Type: replace-cross Abstract: Matrix-based preconditioned optimizers, such as Muon, have recently been shown to be more efficient than scalar-based optimizers for training large-scale neural networks, including large language models (LLMs). Recent benchmark studies of LLM pretraining optimizers have demonstrated that variance-reduction techniques such as MARS can substantially speed up training compared with standard optimizers that do not employ variance reduction. In this paper, we introduce MARS-M, a new optimizer that integrates MARS-style variance reduction with Muon. Under standard regularity conditions, we prove that MARS-M converges to a first-order stationary point at a rate of $\tilde{\mathcal{O}}(T^{-1/3})$, improving upon the $\tilde{\mathcal{O}}(T^{-1/4})$ rate attained by Muon. Empirical results on language modeling and computer vision tasks demonstrate that MARS-M consistently yields lower losses and improved performance across various downstream benchmarks. The implementation of MARS-M is available at https://github.com/AGI-Arena/MARS/tree/main/MARS_M.

Deep Ensembles for Epistemic Uncertainty: A Frequentist Perspective

Anchit Jain, Stephen Bates — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.22063v2 Announce Type: replace-cross Abstract: Decomposing prediction uncertainty into aleatoric (irreducible) and epistemic (reducible) components is critical for the reliable deployment of machine learning systems. While the mutual information between the response variable and model parameters is a principled measure for epistemic uncertainty, it requires access to the parameter posterior, which is computationally challenging to approximate. Consequently, practitioners often rely on probabilistic predictions from deep ensembles to quantify uncertainty, which have demonstrated strong empirical performance. However, a theoretical understanding of their success from a frequentist perspective remains limited. We address this gap by first considering a bootstrap-based estimator for epistemic uncertainty, which we prove is asymptotically correct. Next, we connect deep ensembles to the bootstrap estimator by decomposing it into data variability and training stochasticity; specifically, we show that deep ensembles capture the training stochasticity component. Through empirical studies, we show that this stochasticity component constitutes the majority of epistemic uncertainty, thereby explaining the effectiveness of deep ensembles.

Reactive capacitance of flat patches of arbitrary shape

Denis S. Grebenkov, Raphael Maurette — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2510.25288v2 Announce Type: replace-cross Abstract: We investigate the capacity of a flat partially reactive patch of arbitrary shape to trap independent particles that undergo steady-state diffusion in the three-dimensional space. We focus on the total flux of particles onto the patch that determines its reactive capacitance. To disentangle the respective roles of the reactivity and the shape of the patch, we employ a spectral expansion of the reactive capacitance over a suitable Steklov eigenvalue problem. We derive several bounds on the reactive capacitance to reveal its monotonicity with respect to the reactivity and the shape. Two probabilistic interpretations are presented as well. An efficient numerical tool is developed for solving the associated Steklov spectral problem for patches of arbitrary shape. We propose and validate, both theoretically and numerically, a simple, fully explicit approximation for the reactive capacitance that depends only on the surface area and the electrostatic capacitance of the patch. This approximation opens promising ways to access various characteristics of diffusion-controlled reactions in general domains with multiple small well-separated patches. Direct applications of these results in statistical physics and physical chemistry are discussed.

The Mean-Field Dynamics of Transformers

Philippe Rigollet — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.01868v4 Announce Type: replace-cross Abstract: We develop a mathematical framework that interprets Transformer attention as an interacting particle system and studies its continuum (mean-field) limits. By idealizing attention on the sphere, we connect Transformer dynamics to Wasserstein gradient flows, synchronization models (Kuramoto), and mean-shift clustering. Central to our results is a global clustering phenomenon whereby tokens cluster asymptotically after long metastable states where they are arranged into multiple clusters. We further analyze a tractable equiangular reduction to obtain exact clustering rates, show how commonly used normalization schemes alter contraction speeds, and identify a phase transition for long-context attention. The results highlight both the mechanisms that drive representation collapse and the regimes that preserve expressive, multi-cluster structure in deep attention architectures.

The Blueprints of Intelligence: A Functional-Topological Foundation for Perception and Representation

Eduardo Di Santi — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.05089v4 Announce Type: replace-cross Abstract: Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

Random-Bridges as Stochastic Transports for Generative Models

Stefano Goria, Levent A. Meng\"ut\"urk, Murat C. Meng\"ut\"urk, Berkan Sesen — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.14190v2 Announce Type: replace-cross Abstract: This paper motivates the use of random-bridges -- stochastic processes conditioned to take target distributions at fixed timepoints -- in the realm of generative modelling. Herein, random-bridges can act as stochastic transports between two probability distributions when appropriately initialized, and can display either Markovian or non-Markovian, and either continuous, discontinuous or hybrid patterns depending on the driving process. We show how one can start from general probabilistic statements and then branch out into specific representations for learning and simulation algorithms in terms of information processing. Our empirical results, built on Gaussian random bridges, produce high-quality samples in significantly fewer steps compared to traditional approaches, while achieving competitive Frechet inception distance scores. Our analysis provides evidence that the proposed framework is computationally cheap and suitable for high-speed generation tasks.

ScoreMatchingRiesz: Score Matching for Debiased Machine Learning and Policy Path Estimation

Masahiro Kato — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.20523v2 Announce Type: replace-cross Abstract: We propose ScoreMatchingRiesz, a family of Riesz representer estimators based on score matching. The Riesz representer is a key nuisance component in debiased machine learning, enabling $\sqrt{n}$-consistent and asymptotically efficient estimation of causal and structural targets via Neyman-orthogonal scores. We formulate Riesz representer estimation as a score estimation problem. This perspective stabilizes representer estimation by allowing us to leverage denoising score matching and telescoping density ratio estimation. We also introduce the policy path, a parameter that captures how policy effects evolve under continuous treatments. We show that the policy path can be estimated via score matching by smoothly connecting average marginal effect (AME) and average policy effect (APE) estimation, which improves the interpretability of policy effects.

Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources

Alejandro Borda, Julian Rincon, C\'esar Galindo — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2512.20787v2 Announce Type: replace-cross Abstract: The Clifford group is efficiently classically simulable, and universality is obtained by supplementing it with non-Clifford resources. We determine which single-qudit gates suffice to achieve universality. We show that the structure of such resources is governed by the prime factorization of the qudit dimension $d$. Using the adjoint action on the space of complex trace-zero matrices, we relate density to irreducibility together with an infiniteness criterion, yielding a trichotomy based on the factorization of $d$. When $d$ is prime, any non-Clifford gate generates a dense subgroup of the determinant-one unitaries. If $d$ is a prime power, the adjoint action is reducible, and universality requires gates that couple the resulting invariant subspaces. For composite $d$ with pairwise coprime factors, generalized intra-qudit controlled-NOT gates connecting the factors already suffice. These findings suggest that ``composite architectures'' -- hybrid registers combining incommensurate dimensions -- offer a route to bypass the standard overhead associated with magic-state injection.

Optimal Transport under Group Fairness Constraints

Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.07144v2 Announce Type: replace-cross Abstract: Ensuring fairness in matching algorithms is a key challenge in allocating scarce resources and positions. Focusing on Optimal Transport (OT), we introduce a novel notion of group fairness requiring that the probability of matching two individuals from any two given groups in the OT plan satisfies a predefined target. We first propose a modified Sinkhorn algorithm to compute perfectly fair transport plans efficiently. Since exact fairness can significantly degrade matching quality in practice, we then develop two relaxation strategies. The first one involves solving a penalized OT problem, for which we derive novel finite-sample complexity guarantees. Our second strategy leverages bilevel optimization to learn a ground cost that induces a fair OT solution, and we establish a bound on the deviation of fairness when matching unseen data. Finally, we present empirical results illustrating the performance of our approaches and the trade-off between fairness and transport cost.

A game-theoretic probability approach to loopholes in CHSH experiments

Takara Nomura, Koichi Yamagata, Akio Fujiwara — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.09339v2 Announce Type: replace-cross Abstract: We study the CHSH inequality from an informational, timing-sensitive viewpoint using game-theoretic probability, which avoids assuming an underlying probability space. The locality loophole and the measurement-dependence (``freedom-of-choice'') loophole are reformulated as structural constraints in a sequential hidden-variable game between Scientists and Nature. We construct a loopholes-closed game with capital processes that test (i) convergence of empirical conditional frequencies to the CHSH correlations and (ii) the absence of systematic correlations between measurement settings and Nature's hidden-variable assignments, and prove that Nature cannot satisfy both simultaneously: at least one capital process must diverge. This yields an operational winning strategy for Scientists and a game-theoretic probabilistic interpretation of experimentally observed CHSH violations.

Rewriting Systems on Arbitrary Monoids

Eduardo Magalh\~aes — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.10564v5 Announce Type: replace-cross Abstract: In this paper, we introduce monoidal rewriting systems (MRS), an abstraction of string rewriting in which reductions are defined over an arbitrary ambient monoid rather than a free monoid of words. This shift is partly motivated by logic: the class of free monoids is not first-order axiomatizable, so "working in the free setting" cannot be treated internally when applying first-order methods to rewriting presentations. To analyze these systems categorically, we define $\mathbf{NCRS_2}$ as the 2-category of Noetherian Confluent MRS. We then prove the existence of a canonical biadjunction between $\mathbf{NCRS_2}$ and $\mathbf{Mon}$. Finally, we classify all Noetherian Confluent MRS that present a given fixed monoid. For this, we introduce Generalized Elementary Tietze Transformations (GETTs) and prove that any two presentations of a monoid are connected by a (possibly infinite) sequence of these transformations, yielding a complete characterization of generating systems up to GETT-equivalence.

The Compound BSDE Method: A Fully Forward Method for Option Pricing and Optimal Stopping Problems in Finance

Zhipeng Huang, Cornelis W. Oosterlee — Mon, 02 Feb 2026 00:00:00 -0500

arXiv:2601.18634v2 Announce Type: replace-cross Abstract: We propose the Compound BSDE method, a fully forward, deep-learning-based approach for solving a broad class of problems in financial mathematics, including optimal stopping. The method is based on a reformulation of option pricing problems in terms of a system of backward stochastic differential equations (BSDEs), which offers a new perspective on the numerical treatment of compound options and optimal stopping problems such as Bermudan option pricing. Building on the classical deep BSDE method for a single BSDE, we develop an algorithm for compound BSDEs and establish its convergence properties. In particular, we derive an a posteriori error estimate for the proposed method. Numerical experiments demonstrate the accuracy and computational efficiency of the approach, and illustrate its effectiveness for high-dimensional option pricing and optimal stopping problems.