Papers

We introduce magnetic cotangent groupoids on the cotangent bundle of a Lie groupoid using normalized cocycles, and characterize when magnetic symplectic forms are multiplicative on these groupoids. We show that $\mathbb{S}^1$-central extensions of Lie groupoids yield such structures via symplectic reduction. For a Kac–Moody extension, the resulting deformation connects the coadjoint and gauge action groupoids, which are not Morita equivalent.

In this work, we compare the De Rham cohomology and the Koszul-Vinberg cohomology groups on the Lie groups SO(2), H3(R) and SGal(3). We model their interactions by constructing a three vertex directed graph connecting associative algebras, KV-cohomology, and Lie groups. By computing the exact dimensions of these complexes, we evaluate their algebraic quotient, which measures the gap separating global topological invariants from left-invariant flat affine structures. Extending this geometric framework to the coadjoint orbits of the Heisenberg group H3(R) and the Galilei group SGal(3), we investigate their properties under an invariant Lagrangian foliation inherited from a constant rank Nijenhuis endomorphism preserving the Boyom complex. Finally, we establish a vanishing theorem for the second KV-cohomology group. We demonstrate that any infinitesimal deformation of the affine structure governed by the polarized Maurer Cartan equation is trivial, thereby proving the structural rigidity of these orbits.

We study finite-time blow-up for the nonlinear wave equation \begin{equation*} v_{tt}-\Delta v=|\nabla_x v|^2 \end{equation*} in dimensions $n\geq2$, under radial symmetry. For every prescribed radius $r_0>0$, we construct solutions which blow up in finite time $T>0$ on the sphere $\{|x|=r_0\}$ with logarithmic Type-I rate. The leading singular dynamics are governed by “generalised self-similar” profiles of the associated one-dimensional equation, while the radial geometry generates a curvature correction of size $\mathcal{O}((\frac{T}{r_0})^2)$. A key simplification in our approach is a logarithmic radial correction which removes the first-order radial drift and reduces the geometry to a decaying inverse-square forcing. We further prove asymptotic stability of the resulting family under radial perturbations. A new feature compared with the one-dimensional theory is that the stable blow-up family is not fully explicit. To overcome this, we develop spectral and semigroup estimates on an extended light cone, together with Lipschitz dependence on the modulation parameters for the spectral projections, the stable flow, and the non-explicit correction.

This paper considers a single-input multiple-output (SIMO) setup with a fluid antenna system (FAS) at the transmitter side and multiple fixed antennas at the receiver, which is referred to as a Tx-SIMO-FAS. We investigate the use of spatial modulation (SM) utilizing the FAS on a single radio-frequency (RF) chain while the receiver side performs maximum-likelihood detection. Unlike conventional antenna arrays, however, the large number of fluid antenna ports accommodated within a limited aperture introduces strong spatial correlation, which reduces the distinguishability of port indices and degrades the reliability of index detection. To address this challenge, three correlation-aware port-selection schemes are proposed: successive fluid Euclidean-distance-optimized selection (SF-EDAS), successive orthogonal port selection (SOPS), and correlation-constrained orthogonal array selection (CC-COAS). These schemes focus on enhancing received-constellation separation, improving channel-basis conditioning, and jointly optimizing channel gain and inter-port decorrelation, respectively. To understand the performance limits of FAS-SM, a reliability analysis is developed by decomposing the channel into an energy-based degree of freedom (DoF), and an extreme-value DoF. High signal-to-noise ratio (SNR) analysis reveals an effective diversity order determined by the number of selected ports, the number of receive antennas, and the energy-based spatial DoF. Furthermore, the aperture-limited array gain is characterized through a scalar equivalent independent-look approximation involving the Digamma function. Numerical results demonstrate that the proposed schemes significantly outperform conventional SM and grouping-based benchmarks. Among them, CC-COAS achieves the most favorable tradeoff between error performance and computational complexity.

We study the following question of Farb: For which positive integers n is it possible to choose n distinct points continuously on every smooth cubic plane curve? This paper is a continuation of the work of Banerjee-Chen. We prove that such choice is unique up to homotopy for n=27 (smallest open case), and does not exist for n=36k+18.

We disprove Conjecture 5.1 of Lin concerning a multivariable extension of Ando's matrix Young inequality for positive semidefinite matrices. For every integer $m\geq4$, the conjectured eigenvalue inequality fails at the largest eigenvalue for $2\times2$ real rank-one orthogonal projections. We also give a $3\times3$ real positive semidefinite counterexample for $m=3$, where the failure occurs at the second eigenvalue.

This note discusses our formalisation in Lean 4 of the classification of groups of order $p^3$ for a prime number $p$, using mathlib4. We present the five isomorphism classes and give a detailed account of the formalisation, with particular emphasis on the non-abelian case, which requiring the most substantial formal development. For odd~$p$, the non-abelian groups are the Heisenberg group $\Heis(\Z/p\Z)$ and the semidirect product $\Z/p^2\Z\rtimes\Z/p\Z$; for $p=2$, they are $D_4$ and $Q_8$. We describe the construction of these concrete groups, the structural lemmas about centers, commutators, and exponents, and the explicit isomorphism constructions that classify an arbitrary non-abelian $p^3$-group.

Isohedral or face-transitive polyhedra are polyhedra with identical faces, such that any face can be mapped onto any other via a symmetry of the polyhedron. Examples include the Platonic solids, Catalan solids, bipyramids, and trapezohedra. We use the orbit-counting theorem to count the number of ways of tiling the faces of such polyhedra up to their isometries given any arbitrary set of tile designs. This approach also counts tilings fixed under each isometry of a given polyhedron and provides explicit enumeration formulas. We use this framework to recover sixteen sequences from the On-Line Encyclopedia of Integer Sequences and fill in the gaps by contributing twelve new sequences.

We examine a Riccati–Gamma approach to the logarithmic derivative of the completed Riemann zeta function. The first part proves, in full local detail, that a naive two-sided vertical concavity criterion for $\Xi'/\Xi$ cannot be a proof of the Riemann Hypothesis, because every zero produces opposite vertical curvatures on the two horizontal sides of the pole of the logarithmic derivative. The second part replaces this obstruction by a rigorously formulated finite spectral averaging framework. We prove cancellation at the critical line, positivity of the off-critical paired contribution on the left of the critical line under a concrete low-frequency kernel condition, a conditional zero-density consequence, and a precise conditional theorem showing which additional localisation hypotheses would imply the Riemann Hypothesis. The results are therefore not presented as an unconditional proof of RH. They give a partial resolution of the Riccati–Gamma question: one natural route is ruled out unconditionally, a second symmetric mechanism is proved at the finite spectral level, and the remaining step is isolated as explicit analytic hypotheses. Reproducible Python routines and numerical figures accompany the analytic discussion.

In recent work on the Fourier coefficients of light-cone Eisenstein series arising in the problem of counting rational points on spheres, Kelmer and Yu conjectured that the leading coefficient in the asymptotic formula for the counting function is rational up to a volume factor. We prove this conjecture by showing that every cusp volume appearing in the corresponding light-cone Eisenstein series is rational.

In this paper, we investigate two recent conjectures posed by Yang concerning the periodicity of entire functions. A portion of these problems was recently addressed by Qiong and Peichu [Acta Math. Sci. Ser. B (Engl. Ed.) 38 (2018) 209-214] and Liu [Bull. Aust. Math. Soc. 101 (2020) 290-296], who provided partial solutions within the class of entire functions. The purpose of this work is to establish several new periodicity theorems that not only significantly improve the results of Qiong-Peichu and Liu, but also extend them to a much broader setting. Furthermore, we provide a series of illustrative examples to demonstrate the sharpness and validity of the hypotheses in our main results.

Panmagic permutations are permutations whose matrices are panmagic squares, better known as solutions to the modular n-queens problem, configurations of n non-attacking queens on a toroidal nxn chessboard. Some of them, affine panmagic permutations, can be conveniently described by linear formulas of modular arithmetic, and we show that their sets are a generalization of groups with N-ary multiplication instead of binary one. With the help of the Post coset theorem, we identify panmagic N-ary groups as cosets of the dihedral subgroup and its extensions in the group of all affine permutations. We also investigate decomposition of panmagic permutations into disjoint cycles and find many connections with classical topics of number theory and combinatorics: square-free numbers, 4k+1 primes, quadratic residues, cycle indices from Polya counting, and linear congruential generators.

A continuous action $\mathbb{G}\circlearrowright X$ of a topological group is principal if its isotropy groups are all conjugate to $\mathbb{H}\le \mathbb{G}$ and the quotient map $X\to X/\mathbb{G}$ is a locally trivial $\mathbb{G}/\mathbb{H}$-fiber bundle. We prove that compact groups whose identity component has metrizable abelianization are Lie provided their free actions on Tychonoff (equivalently, compact Hausdorff) spaces are all principal; this is a converse to Gleason's theorem. A variant confirms the conclusion for Tychonoff or compact Hausdorff actions with constant central isotropy by compact connected groups.

We study the Samuelson area condition for planar Lagrangian \(2\)-webs formed by tangent lines. For the tangent line family \(L_t:y=tx+h(t)\), where \(h\) is smooth and \(h''\neq0\), let \(\Psi(u,v)=L_u\cap L_v\) for \(u\neq v\), and let \(J_\Psi\) be its Jacobian. Domitrz and Zubilewicz (2024) proved that these webs do not satisfy the Samuelson condition. We study the Jacobian as the two tangents coalesce. In web coordinates, the Samuelson condition is equivalent to \(\partial_u\partial_v\log|J_\Psi|=0\). We show that \(J_\Psi\) has a simple zero along \(u=v\) and that \[ \partial_u\partial_v\log|J_\Psi(u,v)| =\frac{1}{(v-u)^2}+R_h(u,v), \] where \(R_h\) extends smoothly across the diagonal. The double pole therefore prevents the Samuelson condition from holding near the envelope. We also give a direct local proof of this failure for webs formed by two separated smooth line families with varying slopes.

This paper investigates convergence properties of regularized Nyström subsampling applied to the unsupervised domain adaptation problem under covariate shift. We focus on the low-smoothness (misspecified) case where the target function lies outside the reproducing kernel Hilbert space. By combining Tikhonov regularization with Nyström projection onto a subsampled subspace, we obtain upper bounds on the excess risk that hold with high probability and are expressed in terms of the source condition, the effective dimension, and the sample sizes. We further extend the analysis to the setting where the Radon-Nikodym derivative between the target and source marginal distributions is unknown and must be approximated, and we identify the minimal additional sample sizes required to maintain the same convergence rate as in the oracle case.

We investigate the minimum non-zero number of perfect matchings in planar graphs. We prove that this is a constant for 2-connected planar graphs of minimum degree 3 and 3-connected planar graphs. In the former case, the constant is 4 and this is best possible. In the 3-connected case, we give several infinite families, including nearly 3-regular graphs and triangulations with a constant number of perfect matchings. In contrast, it was known that in the 4-connected case the minimum non-zero number of perfect matchings is at least linear. For 5-connected triangulations, it follows from a result of Alahmadi, Aldred, and Thomassen that there must be exponentially many perfect matchings. The families of planar graphs we investigate here are classified by connectivity. We conclude the article with an infinite family of counterexamples to a conjecture published by Zaks; these are not planar, but very much concern connectivity constraints.

We initiate the study of transmission zero forcing, a variant of the well-studied zero forcing graph parameter. In this variant, a subset of vertices is assigned an initial unit weight, and these vertices can increase the weight of a neighbor subject to the zero forcing color change rule at a rate determined by the transmission proportion. A vertex is considered filled when its weight exceeds the transmission threshold, at which point the process can continue. The transmission zero forcing number of a graph is the minimum cardinality of the initial set that results in all vertices exceeding the transmission threshold. This iterative graph coloring process is a generalization of zero forcing that allows for a vertex to be forced by multiple neighbors. We develop tools for studying this graph parameter, determine its value on some common classes of graphs, and investigate its behavior under various graph operations.

The generalized Lyapunov equation has a natural splitting that leads to the standard fixed-point iteration (sFPI) for its numerical solution. sFPI is convergent, for an arbitrarily given initial guess, if and only if the spectral radius of the associated linear operator is smaller than $1$. This means that sFPI may diverge if the spectral radius is $1$ or bigger. In this paper, we propose a compensated splitting scheme that aims to reduce the spectral radius so that the resulting compensated fixed-point iteration (cFPI) has a better convergence property, namely, cFPI may still converge even if sFPI does not or cFPI converges faster than sFPI does when the latter is also convergent. Numerical results are presented to demonstrate the superiority of cFPI to sFPI.

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct sizes. In this paper, we show that if G is an ah-group and \pi(G) \subseteq {2,3,5}, where \pi(G) denotes the set of prime divisors of |G|, then G \cong S_3.

This paper studies the definability of natural language generalized quantifiers. The semantics of generalized quantifiers are provided by a collection of subsets of the underlying domain. However, the generalized quantifiers appearing in natural language are definable either by first-order quantification or by cardinality notions. This paper provides an explanation for this observed phenomenon. The explanation is that the famous constraints of domain independence and conservativity, when extended to Henkin models, suffice to ensure low-level definability, namely $\Delta^1_1$-definability or at least $\Sigma^1_1$-definability; and in most cases this definability can be made to be bounded. This is basically a consequence of Feferman's Preservation Theorem, which Marker has provided a short model-theoretic proof of. Further, we verify that the paradigmatic cardinality quantifiers are indeed $\Delta^1_1$-definable for a reasonable choice of background theory. Finally, in many other cases, we show that this definability can be lowered to first-order definability.

In this note we solve a problem posed by Lagarias and Shallit concerning Lagrange constants under linear fractional transformations $Mx=\frac{ax+b}{cx+d}$. For an integer matrix $M$ with nonzero determinant and relatively prime entries, define the ratio spectrum \begin{equation*} \mathcal{V}(M)=\left\{\frac{k(Mx)}{k(x)}:x\in\mathrm{Bad}\right\}, \end{equation*} where $k(x)$ denotes the Lagrange constant of the irrational number $x$ and $\mathrm{Bad}$ is the set of badly approximable numbers. Lagarias and Shallit proved that \begin{equation*} \mathcal{V}(M)\subseteq\left[\frac{1}{|\det M|},|\det M|\right], \end{equation*} and asked for the determination of $\mathcal{V}(M)$. We prove that \begin{equation*} \mathcal{V}(M)=\left[\frac{1}{|\det M|},|\det M|\right]. \end{equation*}

We consider Gaussian free field (GFF) on simply connected domains with piecewise constant Dirichlet boundary data. We show that the crossing probabilities for its level lines are determined by conformal blocks of primary fields in a conformal field theory (CFT) with central charge $c = 1$ which are degenerate at each insertion. Alternatively, the crossing probabilities are ratios of explicit partition functions of fused multiple $\mathrm{SLE}_4$ curves, which can be written in terms of fused Specht polynomials introduced recently by Lafay, Peltola, & Roussillon in a representation-theoretic context. We also prove that for the metric graph GFF introduced by Lupu, with appropriate boundary conditions, the crossing probabilities for its level sets converge in the scaling limit to our formulas. In particular, the geometry of the level-set percolation for both the continuum GFF and the metric graph GFF has a CFT description in terms of the aforementioned $c=1$ conformal blocks, which are linearly independent and solve the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs of arbitrary orders. Interestingly, not all combinatorial boundary conditions are amenable for the GFF models – while the CFT contains conformal blocks of primary fields labeled by generalized Dyck paths (i.e., semi-standard Young tableaux), the ones appearing in the above models satisfy specific monotonicity constraints.

Misalignment can change how information moves from an AI agent to a human user. We model this as an information advantage: the AI agent observes the world state, while the human receiver only knows a prior and must act after seeing the agent's signal. A strategic AI sender may withhold evidence or garble information in order to steer the human's decision. We ask how much useful information can still reach the human when the AI optimizes a misaligned objective. We study a Bayesian persuasion model in which the world state is a bit string, the human receiver wants to guess the bits correctly, and a single AI sender wants the receiver to guess as many bits as possible as $1$. For a prior $\mu$, let $R_0(\mu)$ be the receiver's utility from using only the prior, and let $R_{\max}(\mu)$ be the largest receiver utility among signaling schemes that are optimal for the sender. We prove $R_{\max}(\mu)/R_0(\mu)\leq 3/2$. This bound improves for priors close to the independent product prior with the same marginals: if $\mu(x)\geq (1-\eta)\pi_\mu(x)$ for every state $x$, then $R_{\max}(\mu)\leq R_0(\mu)+\eta n$. We also give a six-bit prior for which $R_{\max}(\mu)/R_0(\mu)=39/31>5/4$, so no universal $5/4$ bound is possible.

Panmagic permutations are permutations whose matrices are panmagic squares. Positions of 1-s in the latter describe maximal configurations of non-attacking queens on a toroidal chessboard. Some of them, affine panmagic permutations, can be conveniently described by linear formulas of modular arithmetic, and we show that their sets have remarkable algebraic properties when one multiplies three or more of them rather than just two. In group-theoretic terms, they are special cosets of the dihedral group in the group of all affine permutations. We also investigate decomposition of panmagic permutations into disjoint cycles and find many connections with classical topics of number theory: multiplicative orders, 4k+1 primes, primitive roots and quadratic residues.

A central goal of quantum optimal control is to achieve high-fidelity and low-energy control pulses. When quantum optimal control methods optimize every point of a pulse discretized over small time steps independently this can yield high fidelity control but also results in broadband and energy-hungry waveforms. We extend MAGICARP, a shooting method inspired by Pontryagin's maximum principle on energy that generates an entire pulse from a small set of parameters, making it smooth and energy-efficient by construction, from driftless systems to closed systems with the constant drift Hamiltonian of two exchange-coupled spins in an external magnetic field. The optimization proceeds in stages: the dressed states of the drift Hamiltonian structure the target, an initial shooting optimization is performed in the rotating-wave frame, and an exact laboratory-frame refinement follows. Benchmarked against Krotov and GRAPE at matched gate infidelity, MAGICARP consistently achieves the lowest energy and a conserved pulse area, concentrates its spectral weight on the gate-relevant transitions, and is the most robust to fluctuations in the exchange coupling; GRAPE independently converges to essentially the same pulse while black-box Krotov meets the same error at an order-of-magnitude energy premium. This method-independence is what qualifies the bounded solver as a measurement instrument, and the central result follows: a large statistical survey of unselected optimization runs resolves a weak-driving quantum speed limit for two exchange-coupled electron spins: low-amplitude realizations of the two-qubit quantum Fourier transform cease to exist below a critical gate time, and the minimum control energy diverges on approach to this limit. The divergence obeys a simple two-parameter area–pole law, $E_2^{\mathrm{law}}(T)=A/T+B/(T-T^*)$.

We investigate the transport and fluctuation properties of self-propelled particles whose motion is governed by an age-dependent phase-switching mechanism. The dynamics alternate between a Markovian downstream phase with a constant switching probability $r$ and a semi-Markovian upstream phase in which the age-dependent hazard probability $a/(b+c)$ decays with the internal clock $c$, representing persistent orientation. The time-averaged velocity, as an order parameter, shows a continuous transition at $a=1$ which separates an upstream-dominated ballistic regime ($a<1$) from an ergodic diffusive regime ($a>1$). Through generating-function methods and discrete-time moment recurrences, we derive exact expressions for the propagator and determine the long-time asymptotics of the mean displacement and variance. At the critical point $a=1$, the system exhibits giant fluctuations, with the variance scaling ballistically up to a logarithmic correction, $\mathrm{Var}(x_T) \propto T^2 / \log T$. These results demonstrate how slowly decaying reorientation probabilities lead to a marginal breakdown of the Central Limit Theorem, enabling unusually high-variance exploratory dynamics in biased environments.

We prove a converse mean-value theorem for harmonic functions associated with the invariant Laplace–Beltrami operator on the real unit ball. Under suitable integrability and topological assumptions on the domain and its Green potential, we show that the invariant volume mean-value property characterizes hyperbolic balls centered at the origin. As a consequence, we also answer a question left open by Bruna and Détraz in the setting of invariantly harmonic functions.

We develop methods for computing framed motives associated with motivic Thom spectra. Our main tool is a motivic Atiyah–Hirzebruch spectral sequence relating framed motives to framed motivic cohomology. As a consequence, after inverting a finite set of primes, the bigraded homotopy sheaves of motivic Thom spectra are computed in terms of framed motivic cohomology. We further analyze the symmetric-group actions inherent in framed correspondences and introduce a theory of torsion framed motivic cohomology that yields new computational descriptions of framed motivic cohomology groups. These constructions lead to a category of permutation-free framed correspondences from which we reconstruct rational stable motivic homotopy theory.

We consider feedback stabilization for one-parameter families of finite-dimensional linear systems over compact parameter sets in the complex field. Classical ensemble feedback induces compact control operators and therefore cannot modify the essential spectrum of the associated multiplication operators describing the free motion of the system. This precludes stabilization in many infinite-dimensional settings. To address this issue, multiplication feedback operators are introduced. For systems with constant Hermite indices, an analogue of Heymann's lemma is proved, as well as a pole placement theorem, and stabilization results. The relation of pointwise and ensemble controllability is investigated. For systems with nonconstant Hermite indices, corresponding results are obtained under additional assumptions on the structure of the parameter set.

Tempered time-fractional coupled phase-field (tTFCP) systems are used to model interfacial phenomena involving memory-dependent transport and relaxation mechanisms. Numerical solutions to these systems are challenging due to the simultaneous presence of nonlocal temporal operators, weak initial singularities, moving diffuse interfaces, and strongly coupled multiphysics dynamics. In this work, we introduce FM-tfPINN (fractional-memory physics-informed neural network), which is used for forward simulation and inverse parameter identification in tempered fractional coupled phase-field systems. Unlike conventional fractional PINNs, which enforce memory effects solely through residual constraints, our framework incorporates tempered fractional memory directly into the neural representation via latent memory-source functions and a tempered fractional integral operator. We develop a fast shifted residual formulation based on graded temporal meshes and sum-of-exponentials (SOE) history compression to efficiently evaluate the tempered fractional operators. This framework combines interface-aware and residual-adaptive collocation strategies, improving resolution near evolving diffuse interfaces. A unified, physics-informed loss formulation allows for the forward prediction and inverse recovery of unknown physical parameters from sparse observations. We assess the proposed method on a class of tempered fractional corrosion phase-field models, including one-dimensional corrosion-front propagation, activation- and diffusion-controlled regimes, two-dimensional pitting corrosion, and inverse mobility identification problems. The numerical results demonstrate the accurate recovery of coupled phase and concentration fields, the robust prediction of physically relevant interface diagnostics, and the reliable estimation of parameters from limited data.