Papers

Let $G$ be an unramified group over a local function field of characteristic $p>0$. This article introduces an abstract norm map between the twisted conjugacy classes and conjugacy classes of $G$ in the positive characteristics setting, applies the process of stabilization of (twisted) trace formula and proves the corresponding base change fundamental lemma for regular semisimple elements.

We study maximum-sum red-blue matchings and matching equilibrium for finite planar point sets. For a red-blue perfect matching $M = \{(a_i,b_i) : 1 \le i \le n\}$, we define the gain of a directed red cycle as the change in total weight produced by cyclically shifting the corresponding blue partners. We prove that $M$ is maximum-sum if and only if every directed red cycle has nonpositive gain, and we derive a geometric sufficient condition for optimality from cyclic intersections of distance-difference regions. We then characterize balanced matchings, in which all red-blue perfect matchings have the same total weight. Equilibrium is shown to be equivalent to vanishing cycle gains, to an additive form of the distance matrix, and to a common level-set condition for distance-difference functions. In the squared Euclidean case this yields an orthogonality classification, while in the Euclidean case it yields a hyperbolic level-set description and a collinear-separation classification in the nondegenerate setting.

While autoregressive models optimize the exact data likelihood via the chain rule, diffusion models are typically trained with denoising objectives. We develop conservation laws based on generalized extrinsic information transfer (GEXIT) functions for a broad class of memoryless noise processes, showing that the data–model cross-entropy (CE) can be characterized exactly as an integral of local information-theoretic derivatives along the noise path. This yields a unified characterization of the likelihood for discrete and continuous diffusion, with the Gaussian case reducing to the well-known mutual information–minimum mean-square error (I-MMSE) relationship. An immediate implication is a locality property: one can compute the information-theoretic derivatives using only the marginal posteriors along the noise path. As a result, training reduces to learning the marginal posteriors by minimizing the negative log-likelihood. While the conservation law implies that the entropy does not depend on the noise path, finite-capacity denoisers approximate the posteriors with varying accuracy across noise types, leading to differences in performance. We validate these predictions on synthetic Markov sources and standard benchmarks, including text8 and CIFAR-10.

For an edge-weighted graph $G=(V,E)$ and a stretch parameter $t\geq 1$, a $t$-spanner is a subgraph $H\subseteq G$ such that the shortest path distances in $G$ and $H$ satisfy $\delta_H(u,v)\leq t\, \delta_G(u,v)$ for all $u,v\in V$. In metric spanners, $V$ is a finite metric space, and $G$ is the complete graph with edge weights corresponding to the distances between the endpoints. When $G$ is the complete graph on $n$ points in the plane, $O(n)$-size $t$-spanners are possible for any $t>1$: For every $\varepsilon>0$, there is an $(1+\varepsilon)$-spanner with $O(n/\varepsilon)$ edges (i.e., the stretch can be arbitrarily close to 1). When $G=K(R,B)$ is the complete bipartite graph on $n$ bichromatic points in the plane, in general, no spanner construction can guarantee stretch $t<3$ with $o(n^2)$ edges. Bose et al.~(SICOMP 2009) constructed a $(3+\varepsilon)$-spanner with $O(n\log n)$ edges for any constant $\varepsilon>0$. Our main result is a new construction for a $(3+\varepsilon)$-spanner with $O(\sqrt{1/\varepsilon}\cdot n)$ edges. Eliminating the $O(\log n)$ factor resolves a problem left open for more than 17 years, and raises a new research problem about optimizing the dependence on $\varepsilon$. We also study spanners for $G=K(R,B)$ on $n$ bichromatic points on the real line: In this case, we show that the MST of $K(R,B)$ is a 7-spanner, and we construct a 3-spanner with at most $2n-3$ edges.

Fix a prime $p$ and a constant $\frac{1}{p}<\alpha\leq 1$. Consider the random Erdős–Rényi bipartite graph $G_{\alpha}(n,u)$ with bipartition $(V_1,V_2)$ of sizes $|V_1|=n$ and $|V_2|=\lceil\alpha n\rceil$, and edge probability $0<u<1$. The authors of [1] and [8] conjectured a limiting distribution for the $p$-Sylow subgroup of the sandpile group of $G_{\alpha}(n,u)$ as $n\to\infty$. We prove this conjecture for odd primes $p$. Similar results have previously been proved by computing the expected number of surjections from the random abelian $p$-group to $H$, for each finite abelian $p$-group $H$. However, in our setting, these surjective moments often diverge to infinity, despite the conjectured limiting distribution having finite moments. We resolve this issue by discarding the graphs for which too many vertices have degrees divisible by $p$. Once we remove the contribution of this rare set of graphs, then the surjective moments converge to the expected values. When $p$ is odd, applying Wood's universality theorem yields the desired convergence in distribution. For $p=2$, our computed moments (after excluding the rare set of graphs) match those of the conjectured distribution. However, these moments do not uniquely determine a distribution.

We generalise a local $p$-adic monodromy theorem of Igusa that studies the monodromy representation associated to the universal elliptic curve around a supersingular point of the modular curve. Let $X$ be a smooth quasi-projective variety over $\mathbb{F}_p$. We set up and prove the analog of Igusa's theorem for overconvergent F-isocrystals on $X$ such that the action of the Frobenius is algebraic, $p$-plain, and semisimple at the closed points of $X$. We study the local monodromy of their unit root sub-objects around a point in $X$ with isoclinic Newton slopes. In particular, our result generalises Igusa's Theorem to overconvergent F-isocrystals arising from Shimura varieties, unconditionally for Shimura varieties of abelian type and conditional on Frobenius semisimplicity for exceptional Shimura varieties where this property is not known yet. In the particular case of Siegel Shimura varieties, we also prove an analogous result for the local monodromy of a point in the boundary of its compactification. As a consequence of our results, we prove a finiteness result for the reduction of the Hecke orbit of abelian varieties over a local field of equicharacteristic.

In this article, we refine the method of our earlier work with N. Palojärvi to obtain a sharper explicit bound for the error term $\Delta_{3}(x)$ associated with the summatory function of $d_{3}(n)$. We prove that \begin{equation*} |\Delta_3(x)| < \begin{cases} 0.6901\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & 3.682\cdot 10^{31}\le x < 4.133\cdot 10^{87},\\[4pt] 0.2067\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & 4.133\cdot 10^{87} \le x < 1.597\cdot 10^{98},\\[4pt] 0.1947\,x^{\frac{1}{2}}\log^{\frac{7}{2}}x, & x \ge 1.597\cdot 10^{98}. \end{cases} \end{equation*} These explicit results improve the exponent of $x$ from $2/3$, due to Tudzi, and $859/1400$, due to Palojärvi and Tudzi, to $1/2$, giving the best known bound for all $x\ge 3.682\cdot 10^{31}$.

It is well known that the study of the Kolmogorov widths of a function class, which is the image of the unit ball of the $L_q$ space of an integral operator $J_K$ with the kernel $K$, is closely connected with the study of sparse approximations of the kernel $K$ with respect to the classical bilinear dictionary. Recently, it was discovered that if instead of the Kolmogorov widths we study the errors of optimal linear sampling recovery of the same classes, then we need to study sparse approximations of the kernel $K$ with respect to an adaptive dictionary, which is determined by the kernel $K$. In this paper we study this important problem of nonlinear approximation with respect to an adaptive dictionary. Also, in this paper we continue to develop the following general approach, which is related to the above nonlinear approximation problem. We study asymptotic behavior of the errors of sampling recovery not for an individual smoothness class, how it is usually done, but for the collection of classes, which are defined by integral operators with kernels coming from a given class of functions. Earlier, such approach was realized for the Kolmogorov widths and very recently for the entropy numbers.

Green's popular difference theorem says that for every \(\varepsilon>0\), all sufficiently large primes \(p\), and every set \(A\subseteq\mathbb F_p\) of density \(\alpha\), there exists a nonzero \(d\in\mathbb F_p\) such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq \alpha^3-\varepsilon. \] We show that a stronger simultaneous popular difference phenomenon holds for polynomial configurations. Namely, if $\mathcal P=\{P_1,\dots,P_k\} \subset \mathbb Z[t]$ is a fixed collection of linearly independent polynomials with zero constant terms, we show that for every \(\varepsilon>0\), all sufficiently large primes \(p\), and every set \(A\subseteq\mathbb F_p\) of density \(\alpha\), there exists a nonzero \(d\in\mathbb F_p\) such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x) \prod_{i=1}^k 1_A\bigl(x+P_i(d)\bigr)^{\omega_i} \geq \alpha^{1+\sum_i\omega_i}-\varepsilon \] simultaneously for every \(\omega=(\omega_1,\dots,\omega_k)\in\{0,1\}^k\). We also show that such simultaneous popular difference phenomena have sharp limitations by proving that for every sufficiently large prime \(p\), there is a constant \(c>0\) such that, for all sufficiently large \(n\), one can find a set \(A\subseteq\mathbb F_p^n\) of density \(1/2+o_n(1)\) satisfying \[ \max_{d\neq 0} \min\left\{ \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+d)1_A(x+2d), \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+2d)1_A(x+4d) \right\} \leq \frac18-c. \] That is, the strengthening of Green's result, in this case over $\mathbb F_p^n$ for $p$ fixed and $n$ tending to infinity, requiring that both \(d\) and \(2d\) are simultaneously popular differences for three-term arithmetic progressions is false.

We prove a full Adam theorem for spectral heavy-tail onset in a closed Gaussian Stein-Hermite teacher-student state-evolution model. The theorem begins with the actual full-batch Adam recurrences, derives the population gradient by Stein-Hermite calculus, proves finite-width covariance concentration, converts multi-step Adam momentum into an exact non-centered Gaussian sign kernel, controls the diagonal Adam denominator by a basis-homogenization theorem, derives a regularly varying projected update response from a Hermite edge-transfer theorem, pushes the response through the exact Gram update, and proves approximate-target KL contraction with matching upper and lower hitting bounds. The final law is (\tau_\varepsilon=\Theta(\Delta_1^{-\gamma}d^\rho\log(\Psi_0/\varepsilon))), where (\Delta_1) is the first spike-bulk spectral gap. The result is full in the following precise sense: every step from Adam's momentum and denominator to the spectral hitting law is formalized inside the closed state-evolution model. We also prove that a stronger arbitrary-gradient Adam theorem is impossible, and that exact two-step linear-network loss dynamics do not identify factor spectra or heavy-tail hitting times.

This paper presents an analytical approach to constructing the Pareto set in multiobjective variational control problems, based on Noether's second theorem. A fundamental connection is established between the gauge symmetries of the dynamical system describing the plant and the structure of the Pareto set. It is shown that the Pareto front can be interpreted as a conservation law arising from the invariance of the system and the quality criteria with respect to a symmetry group. A gauge regularization concept is proposed, which allows eliminating variables that do not affect the criteria without changing the Pareto set, thereby reducing the dimension of the solution space. Numerical examples are provided for the linear-quadratic regulator with two conflicting criteria and for the standard ZDT1 test problem.

It is well-established that single-axis Transverse Ray Transform tomography data from high-energy X-ray diffraction is insufficient for general reconstruction of three-dimensional elastic strain. In this paper we show that, when combined with the constraint of mechanical equilibrium, this problem becomes uniquely solvable for isotropic elastic samples with known elastic constants and non-zero Poisson's ratio. Building on the reconstruction framework of Desai and Lionheart [N.M. Desai, W.R.B. Lionheart, An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data, Inverse Problems 32 (11) (2016) 115009], we formulate this constrained single-axis tomography problem in the Fourier domain and show that the resulting system is invertible everywhere outside a finite collection of measure zero characteristic planes. In the case of bounded samples, this is sufficient to establish uniqueness for reconstructions. We further derive conditional stability estimates that quantify the regularity requirements associated with reconstruction of the various strain components.

The Kaplansky–L'vov conjecture asserts that the image of a multilinear polynomial map on a full matrix algebra over a field is always a vector space. Although the conjecture remains open in general, substantial progress has been made for $2\times 2$ and $3\times 3$ matrix algebras over various fields. Recently, Panja, Saini, and Singh formulated a generalized Kaplansky–L'vov conjecture for polynomial maps with matrix coefficients over algebraically closed fields and verified it for $2\times 2$ matrices. In this work, we investigate an analogous problem for polynomial maps with constant matrix coefficients over an infinite division algebra. Specifically, we consider polynomials in the free algebra $M_2(\mathbb D)\langle x_1,\ldots,x_m\rangle$ of the form $\omega = A_1x_1^{k_1}+\cdots+A_mx_m^{k_m},$ where the $A_1,\ldots, A_m\in M_2(\mathbb D)$ are fixed matrices, $\mathbb D$ is an infinite division algebra, and $k_1,\ldots, k_m$ are positive integers. We prove that the corresponding generalized Kaplansky–L'vov conjecture holds for $2\times 2$ matrices over $\mathbb R$ and the quaternion division algebra $\mathbb H$. We also investigate the surjectivity of these polynomial maps. This can be viewed as a generalized Waring problem for matrix algebras.

Large-scale robotic swarms motivate the use of mean-field control (MFC). Classical partial differential equation (PDE)-based formulations provide a principled framework but can become computationally challenging in higher dimensions, whereas machine learning achieves scalability at the cost of approximation and guarantees. In this work, we establish an optimization-based framework that lifts the MFC problem into the space of occupation measures, resulting in a convex relaxation formulated as an optimization over measures. The resulting problem is solved using a Frank-Wolfe (FW) algorithm in the measure space, with each iteration reduced to a tractable optimal control problem. This approach retains the O(1/k) convergence rate of FW, avoids discretization of the state space, and naturally incorporates interaction and safety constraints. Numerical experiments demonstrate agreement with analytic and PDE-based baselines in two dimensions and show that the method scales to three-dimensional environments with multiple obstacles, where standard grid-based PDE solvers become impractical. A full 3D instance with ten obstacles is solved in minutes on a standard workstation, underscoring the practicality and scalability of the proposed framework.

The management of cash replenishment in Automated Teller Machine (ATM) networks requires scheduling recharges in order to minimize operational costs while maintaining high service levels and avoiding cash-outs, under uncertain and time-varying withdrawal demand. This work formulates the ATM cash replenishment problem through a Quadratic Unconstrained Binary Optimization (QUBO) model, which naturally captures nonlinear cost interactions, while incorporating operational constraints through penalty terms. The objective function combines fixed and variable replenishment costs with co-location discounts, as well as penalties for a late replenishment that could cause a service interruption. The resulting QUBO instances are solved using MegaQUBO, a GPU-accelerated QUBO solver. An empirical evaluation on a real dataset of 276 ATMs located in Italy, covering four representative months of 2022 (April, May, October, and November), benchmarks the proposed approach against a threshold-based operational policy. Results show consistent cost reductions of approximately 15%-18% while maintaining an excellent average service level (around 99.8%-99.9). Overall, the study demonstrates that QUBO-based optimization, coupled with GPU-based solving, can provide a practically deployable decision-support tool for large-scale ATM cash logistics.

The central question of this paper is the following: "If a topological space equipped with a weight function satisfies a certain property only approximately, can we recover the original structure without significantly altering the weights?” To each open set of a topological space, we assign a non-negative weight and study approximate versions of three natural properties: monotonicity, subadditivity, and convexity. Through this study, we develop Hyers–Ulam-type stability results in general topology. In addition, we investigate several topology-specific cases, as well as present minorant and sandwich-type results.

We study self-referential multiple-choice tests with the question: How many correct answer choices are there? The answer choices are positive integers. A value $a$ is called valid if it occurs exactly $a$ times among answer choices. The cost of a test is the sum of all answer choices, linking the problem to integer partitions. Using this framework, we define solvable and $k$-solvable tests and derive generating functions that enumerate them by cost, number of distinct valid values, and number of options. We also investigate extremal questions, including minimum costs and the maximum possible number of valid values. The paper was inspired by a puzzle from Mathematical Puzzles and Curiosities.

Let $\mathfrak{G}$ be a locally compact Lie group and \(d\mu\) a Haar measure defined on $\mathfrak{G}$. We consider the existence of a fuzzy analogue of $\mathfrak{G}$ denoted by $\mathfrak{G_f}$ called a fuzzy Lie group and develop a fuzzy Haar measure \(\mu_f\) on $\mathfrak{G_f}$. Also we construct a fuzzy Haar integral with respect to the corresponding fuzzy Haar measure on $\mathfrak{G_f}$. And finally we show that there exists a fuzzy Haar integral that is unique up to a multiplicative constant.

For squarefree $M>1$, we determine the quotient of the principal cuspidal lattice of $X_0(M)$ by the sum of its primitive Atkin-Lehner character sublattices. Its invariant factors are explicit at every level. The quotient is a 2-group, and with $k\ge3$ prime divisors its exponent is exactly $2^{k-2}$, the sharp universal power expressing every modular unit, modulo constants, through primitive sign units. We construct mixed generators and operator actions and relate the quotient to cuspidal class numbers and integral Eisenstein periods. At the Fricke point, singleton exponent sums determine the prime factors of every normalized unit value; the remaining factor is an algebraic unit. When the quadratic order has class group of exponent at most two, Kronecker's limit formula expresses the even sign values as explicit rational powers of real quadratic fundamental units. The Fricke-fixed coefficient identity for $1/\pi$ leads to a strictly monotone equation with rigorous truncation certificates. For reciprocal sign products we prove a colored-partition asymptotic and a congruence modulo 5. The classical eta representation and Kronecker-Pell results are explicitly identified as inputs.

We study a symbolic search space for the Collatz conjecture based on finite exponent codes of the accelerated map. Each code records the number of divisions by two after every 3n + 1 step and determines three quantities: real drift, a 2-adic start representative, and a 3-adic endpoint representative. Their combination defines the 2-3-infinity diagnostic. Counterexample-like codes should exhibit near-critical drift, small 2-adic start representatives, and endpoints compatible with growth on the scale of (3/2)^k. We prove that every infinite code generated by a fixed positive integer has asymptotically vanishing 2-adic and 3-adic residue rates. Experiments with random critical codes, mechanical critical codes, and adaptive evolutionary search at lengths 100, 200, and 400 show that adaptive search improves finite-length trade-offs, while all methods retain clearly positive residue rates. The proposed framework is not a verification method for the Collatz conjecture, but a symbolic diagnostic approach for investigating obstruction structures in exponent-code space.

For many years, pairwise comparison methods have been widely used for eliciting preferences and ranking alternatives in decision-making problems. These methods estimate priority weights from a pairwise comparison matrix and are now standard tools in multi-criteria decision analysis. In this paper, we present a statistical view of the pairwise comparison method using reference values and the geometric mean to calculate alternative priorities. Thanks to this approach, we can simultaneously capture the phenomenon of inconsistency in pairwise comparisons and the preference distance between different alternatives. In this paper, we define indicators that measure the quality of the obtained weight vector, which, thanks to the statistical approach, have an understandable interpretation.

In this note we determine the maximum number of arcs in a digraph on $n$ vertices that does not contain a length-$k$ directed path $\overrightarrow{P}_{k+1}$ for $n \geq 50k^6$. This improves a theorem of Zhou and Li [$\textit{Graphs Combin.}$ 39(3), 2023] who proved the result with a superexponential threshold on $n$.

Hypothesis Testing is the most contentious procedure in statistical Methodology. P values rejects Null Hypotheses far too easily, specially for large samples. On the other hand, Bayes Factors depends on assumptions, for example regarding Intrinsic Bayes Factors, which average? Arithmetic, Geometric, Median? Our bound is the infimum over all the averages. We develop a lower bound on Intrinsic Bayes Factors that adjust authomatically with the sample size. Furthermore, we introduce the new idea of {\it{Least Favorable Intrinsic Prior}}, which corresponds to the least favourable possible training samples. The bound sets a bridge between Intrinsic Bayes Factors and Adrian Smith and David Spiegelhalter methodology.

Examples of odd equivariant cohomology exist in the literature, but they are generally in degree one and used to show only that odd equivariant cohomology can exist. This paper produces a family of toric varieties which have non trivial odd equivariant cohomology which arise as torus orbits of generic points in the grassmannian in many odd degrees. We then show that is it is possible to compute the equivariant cohomology explicitly with \mathbb{Q}-coefficients in H^3_T for small examples.

This paper proposes a new notion of robust invertibility for nonlinear dynamical systems, and introduces constructive parameterizations of recurrent neural network which are robustly invertible by design. We define robust invertibility as the existence of a causal inverse system such that both the forward and inverse systems are contracting and have bounded incremental input-output gains (the system is bi-Lipschitz), implying that both forward prediction and input reconstruction are robust to signal perturbations and initial-state mismatch. We construct robustly invertible recurrent models via series composition of static orthogonal layers and dynamic layers satisfying a strong input-output monotonicity property, and provide a differentiable neural network parameterizations in the form of the bi-Lipschitz recurrent equilibrium network (BiLipREN). Additionally, composition with dynamic orthogonal layers yields a nonlinear minimum-phase/all-pass (a.k.a. inner–outer) factorization. We illustrate the utility of the framework through a series of application examples in data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows, illustrating its utility for robust control, trajectory optimization, and generative modeling of complex trajectory distributions.

We prove that, given any contact $3$-manifold and any computable function $f: \mathbb{N} \dashrightarrow \mathbb{N}$, there exists a defining contact form and a Poincaré section of its Reeb flow whose partially defined return map computes $f$.

We study various constructions of spanning solids for knotted surfaces in the 4-sphere. In particular, we consider a 4-dimensional analogue of Murasugi sum, which we use to define a notion of arborescent knotted surfaces. We give a variety of examples and applications to broken surface diagrams, such as deciding which resolutions of the crossings in these diagrams yield spanning solids.

Weights play an essential role in the classification of simple modules for affine Hecke algebras. In this paper we show the extent to which weights determine simple modules and relate this to induction and restriction along isogenies. We also describe the behaviour of hermitian duals and unitarity under induction and restriction along isogenies.

We study scalar and vector bosons in the Bonnor–Melvin–$\Lambda$ spacetime within the Duffin–Kemmer–Petiau (DKP) formalism. By employing Umezawa's projection operators, we separate the physical spin-0 and spin-1 sectors and derive the corresponding exact second-order equations in the full curved spacetime, without relying on the conical approximation. For the scalar sector, the radial equation reduces to a Schrödinger-like equation with a trigonometric Pöschl–Teller effective potential. In the vector sector, the longitudinal mode is governed by the same effective potential, whereas the transverse polarizations are described by generalized trigonometric Pöschl–Teller potentials. Because the me\-tric function vanishes at a discrete set of radial points, the radial dynamics is naturally formulated as a singular Sturm–Liouville problem on a fundamental interval, with the physical radial domain fixed by the Friedrichs self-adjoint extension of the corresponding singular radial operators. As a result, all physical sectors exhibit purely discrete radial spectra, and their eigenfunctions are obtained in closed form. These results provide a unified exact treatment of scalar and vector bosons in the Bonnor–Melvin–$\Lambda$ spacetime, complement previous analyses based on the conical approximation, and clarify the role of the global geometric structure of the background in shaping confinement and spectral properties.

We study the sedimentation of stationary random suspensions of rigid particles in Stokes flow. Batchelor's formula predicts the first dilute correction to the infinite-volume mean settling speed due to hydrodynamic interactions between suspended particles. A rigorous derivation has long been obstructed by the long-range nature of the Stokes flow, which gives rise to infrared divergences in the large-volume limit. In dimension $d>2$, for stationary suspensions satisfying quantitative decorrelation assumptions, we construct the infinite-volume mean settling speed and show that it governs the relative settling speed of particles in large containers, independently of the container shape. We then establish a renormalized cluster expansion of this mean settling speed in the dilute regime and compute it up to the two-particle term, thereby justifying Batchelor's formula. The proof is based on the infrared renormalization of hydrodynamic interactions. Infinite-volume observables are decomposed into an explicit singular part, carrying the non-integrable large-scale contribution, and a regular remainder controlled by elliptic estimates. The singular part is renormalized through counterterms that encode the diverging mean backflow generated by the suspension. At the level of the dilute cluster expansion, the renormalization is implemented cluster by cluster and the singular-regular decomposition is achieved through a finitary diagrammatic expansion of hydrodynamic interactions, inspired by the method of reflections, which isolates the leading divergent substructures and exposes the key cancellations.