We prove Ulam stability of the hyperfinite II$_1$-factor with respect to the trace norm on the operator-norm unit ball. More precisely, every sufficiently additive, multiplicative, unital, $*$-preserving map from the hyperfinite II$_1$-factor-factor into a II$_1$-factor-factor von Neumann algebra is uniformly close, after passing to a small amplification of the target, to a genuine unital $*$-homomorphism. As a key finite-dimensional ingredient, we establish a dimension-free stability theorem for matrix algebras in the same trace-norm setting. As an application, we show that the hyperfinite II$_1$-factor is isolated among II$_1$-factors with respect to sufficiently accurate approximate $*$-isomorphisms.

We study the Dirichlet problem for the fully non-linear elliptic equation of second order traducing the prescription of the scalar curvature of a spacelike hypersurface in Anti-de Sitter space. The problem is solved if the datas are strictly convex.

In this note we present how the almost-orthogonality methods based on $TT^*$ arguments can be employed to study boundedness of discrete operators of Radon type. Almost-orthogonality methods have particular significance when the classical Fourier methods are not available. However here, to avoid technicalities and present the key ideas behind the discrete almost-orthogonality methods, we give a new proof of the $\ell^2(\mathbb{Z}^d)$-boundedness of Bourgain's maximal inequality for Radon polynomial averages.

For fixed integers $g\ge 2$ and $t\ge 1$, and every $\varepsilon>0$, we prove that there exists a constant $ρ>0$ such that every $n$-vertex graph $G$ with $δ(G)\ge (2/(2g+1)+\varepsilon)n$ either contains $C_{2g-1}[t]$, or can be made bipartite by deleting $O(n^{2-ρ})$ edges. This gives an affirmative answer to a question of Illingworth in [Minimum degree stability of $H$-free graphs, Combinatorica, 43(1):129-147, 2023.]

Recently studied dependence measures, such as Chatterjee's rank correlation, that characterize both independence and perfect functional dependence, provide a powerful framework for detecting nonlinear dependencies. However, these measures cannot be weakly continuous, which limits the applicability of classical plug-in estimators based on empirical distributions. This obstruction is natural, as such measures are defined via conditional distributions and not through their joint law alone. In this paper, we introduce an optimal transport-based mode of convergence that captures weak convergence of conditional distributions and restores continuity for a broad class of dependence measures. We relate this mode of convergence to the adapted Wasserstein distance, the Knothe-Rosenblatt distance and the d1-metric on copulas. Building on this perspective, we propose a copula estimator based on the adapted empirical measure and compare it with the classical rank-based checkerboard estimator. For both estimators, we derive O(N^{-1/3})-rates of convergence with respect to metrics that capture conditional weak continuity. As a consequence, we obtain the same rates for plug-in estimators of several classes of dependence measures, including rank-based and rearranged dependence measures.

We study the KPZ equation on a circle with an additive spatial perturbation $\partial_t h=\tfrac12Δh+\tfrac12|\nabla h|^2+ξ+ V$, where $ξ$ is a spacetime white noise and $V$ is a smooth spatial function. When $V=0$, it is well-known that the unique invariant measure is the Brownian bridge. In the presence of the perturbation, we show that the equation admits a unique invariant measure that is absolutely continuous with respect to the Brownian bridge. We further prove the measure has a finite relative entropy with respect to the law of the bridge and that, for any $p\in(1,\infty)$, the corresponding Radon-Nikodym derivative belongs to $L^p$, provided that $\int V^2$ is sufficiently small. The proof uses the discretization and mollification scheme of \cite{FQ}, together with an application of the log-Sobolev and spectral gap inequalities for the underlying Gaussian measure.

For tasks of likelihood families invariant under the action of a lie group, the quotient is the minimal sufficient invariant representation. On compact homogeneous spaces, this quotient representation admits a harmonic realization through spherical Fourier coefficients; for finite-band harmonic exponential families, the empirical harmonic coefficients are minimal sufficient statistics. The partition function can be expressed algebraically by extracting the trivial representation component through Clebsch-Gordan decomposition.

Momentum-based accelerated gradient methods are widely adopted to expedite convergence in nonconvex optimization, but are prone to overshooting and oscillatory behavior. A class of PID-damped accelerated gradient methods mitigates this issue by augmenting classical momentum methods with a discrete-time derivative damping term. However, the coupling among the step size, momentum, and derivative gain renders the explicit characterization of their convergence regions analytically intractable, leaving explicit theoretical convergence boundaries unexplored. In this paper, we model this class of algorithms as a third-order nonlinear feedback dynamical system and establish explicit three-dimensional convergence regions for the step size, momentum, and derivative gain via a robust control-theoretic analysis based on the Kalman-Yakubovich-Popov (KYP) lemma, formally guaranteeing linear convergence under the Regularity Condition. Furthermore, we reveal a strict geometric upper bound on the derivative gain dictated by the nonconvex curvature, beyond which over-damping severely contracts the feasible step-size region, providing a rigorous theoretical explanation for the overdamped stagnation phenomenon. Numerical experiments corroborate the theoretical boundaries and illustrate practical parameter selection guidelines for the derivative gain.

We compute the twisted first and second moments of the shifted central values of the Rankin-Selberg $L$-functions given by $L\left(\frac12+ω, f\otimes g\right)$ as $f$ varies over primitive forms of prime power level $p^ν$ with $ν\geq 3$. Here $ω$ is a bounded shift and $g$ is a fixed primitive form of level relatively prime to $p$.

We present a novel S-pair selection strategy called Homogeneity Entropy, for deciding the sequence of S-polynomials to construct in Buchberger's algorithm to compute a Groebner basis. The strategy uses an information theoretic measure derived from the distribution of degrees among the monomials of the S-polynomial: a very different approach to the classical heuristics such as Degree, Normal and Sugar, or indeed the more recent machine learning approaches to the problem. We implement this strategy and evaluate it on two different datasets: (1) variations of randomly generated polynomial systems with controlled numbers of variables, degrees, and densities; and (2) the PHCpack benchmark dataset sourced from real world problems. The Homogeneity Entropy strategy significantly outperforms classical strategies on random polynomial datasets, but on the PHCpack dataset the classical strategies perform better. This suggests the right strategy varies with the shape of the data and we explore this in several experiments. The new strategy offers practically meaningful gains on certain distributions, and represents the first use of such information-theoretic guidance in the optimisation of symbolic computation algorithms.

We introduce and study several conditions related to the factorization problem of composite numbers. For this purpose, we employ cyclotomic polynomials, Sylvester resultants, and the Fermat equation. We show that the existence of a specific solution to the Fermat type equation in positive characteristic $p$ implies polynomial-time factorization of a composite natural number that is a multiple of $p$. We also show that such solutions do not exist for many semi-prime integers.

Heisenberg curves are cyclic covers of Fermat curves that also arise as non-abelian covers of the projective line, branched over three points by the discrete Heisenberg group modulo an integer. As normal Belyi covers, these are curves with many automorphisms in the sense of Oort, who questioned whether such curves have CM Jacobians. In 1986, Ihara proposed using towers of curves to study the pro-$\ell$ Galois representation associated with the thrice-punctured projective line. To study the kernel of this representation, he suggested using Heisenberg curves, but it was unknown to him at the time whether their Jacobians lacked complex multiplication. In this paper, for any odd prime $\ell$, we prove that Heisenberg curves of level $\ell^n\neq 3$ do not have CM Jacobians. Thus, we resolve the missing part of Ihara's original argument and in doing so we provide an infinite family of new counterexamples to Oort's question.

Bohr pioneered the study of the statistical behavior of the Riemann zeta-function. A classical result by Bohr and Jessen revealed that the values of the Riemann zeta-function to the right of the critical line behave like a random variable. We now propose to extend Bohr's theory to the stage of hyperfunctions. In this paper, we introduce two random hyperfunctions: one is associated with the values of the Riemann zeta-function on the critical line, and the other is associated with the characteristic polynomial of a random matrix from the circular unitary ensemble. We then derive a relationship between these random hyperfunctions which is consistent with the Keating-Snaith conjecture on the moments of the Riemann zeta-function.

We study growth rates of infinite and finite sumset patterns in sets of positive density. In the infinite setting, we show that no such rate exists, answering a question of Kra, Moreira, Ritcher, and Robertson. Namely, for any proposed growth rate $\mathcal{H}: \mathbb{N} \to \mathbb{N}$ tending to infinity, we construct a set $A$ of lower density $1$ such that whenever $B,C \subseteq \mathbb{N}$ are infinite and $B+C \subseteq A$ we have the minimum of $|B\cap [N]|$ and $|C \cap [N]|$ is less than $\mathcal{H}(N)$ for infinitely many $N$. In the finitary setting, we prove that for all $δ\in (0,1)$, for all sufficiently large $N$, for all subsets $A$ of $\{1,\dots,N\}$ of proportion $δ$, one can always find sumset patterns $B+C\subseteq A$ with $|B|$ and $|C|$ of order $\log N$, partially resolving a conjecture of Kra, Moreira, Richter, and Robertson. Moreover, we generalize our second result to the case of the $k$-fold sum $B_1 + B_2 + \ldots + B_k \subseteq A$.

This paper is concerned with a study of a natural generalization of a classical Frank-Kamenetskii model of thermal explosion in the presence of a vortical flow in a two dimensional setting. This model describes possible stationary temperature distributions in a combustion vessel which boundary is maintained at a constant temperature. The model constitutes a Dirichlet boundary value problem for a certain semi-linear elliptic equation that depends on a parameter $λ,$ called Frank-Kamenetskii parameter. A remarkable property of this problem is that it admits a classical minimal solution when the Frank-Kamenetskii parameter does not exceed some critical value $λ^*$ and no classical solutions for $λ>λ^*$. The absence of a classical solution, in the framework of Frank-Kamenetskii theory, is associated with the thermal explosion event. Consequently, in the context of combustion, $λ^*,$ commonly called an explosion threshold, is a maximal value of the Frank-Kamenetskii parameter which allows to attain a thermal equilibrium within a combustion vessel and thus provides a sharp characterization of the thermal explosion. A critical temperature distribution corresponding to $λ^*$ is called an extremal solution. In this paper, we show that, under an assumption of sufficiently fast growth of the reaction term, there exists a regular vortical flow that allows to adjust an explosion threshold by reversing its direction, provided a combustion vessel is not a disk.We also give rather detailed description of extremal solutions. In particular, we show that extremal solutions are always classical.

A sharp lower bound for the first Dirichlet eigenvalue of the $p$-laplacian is derived for sets with prescribed $p$-torsional rigidity. The result provides an extension of the classical spectral inequality due to Kohler-Jobin. The proof is based on a careful analysis of the generalized $p$-torsional rigidity and on a sharp mass comparison result.

In this work, we study the boundary behaviour of the Volterra square- root process on $\mathbb{R}_+$. For regular Volterra kernels, we establish a time-dependent Feller condition that guarantees that the process does not hit zero on $[0, T]$, and prove finiteness of negative $p$-moments. For rough kernels that are regularly varying at zero, we show that the process necessarily hits zero with positive probability, and that its law has an atom at the boundary. Finally, for the limit distribution, we show that finiteness of negative moments is determined by the long-time asymptotics of the associated resolvent. In particular, while in the rough case the process has an atom at zero, its limit distribution has finite negative exponential moments. Our proofs are based on comparison principles for Volterra integral equations and generalized Riemann-Liouville fractional equations. The latter provide us with upper and lower bounds for the solution of the associated Volterra Riccati equation, and hence also on the asymptotics of the Laplace transform. As an application, we study the structure of equivalent martingale measures in the Volterra Heston model. For the rough case, we show that equivalent martingale measures exist only under very restrictive assumptions on the drift under the real-world measure.

We prove Fourier positivity for spherical functions on a semisimple linear algebraic group $G$ over a local field restricted to its split tori $A$ for unitary principal series parameters of $G$. For ${\rm SL}_n(F)$, where $F$ is a local field, we obtain an explicit recursive formula for the Fourier transform on the diagonal split torus in terms of local Rankin--Selberg factors for ${\rm GL}_n\times {\rm GL}_{n-1}$, together with uniform exponential lower bounds in the spectral parameters. The main input is a Plancherel expansion for the restriction of a ${\rm GL}_n(F)$-spherical function to ${\rm GL}_{n-1}(F)$. Its coefficients are spherical periods computed by Rankin--Selberg theory. Positivity of the Fourier transform for general semisimple groups with unitary principal series parameters is obtained by reduction to full-rank subgroups of type A. The results are motivated by variance non-vanishing problems for mixing abelian actions on homogeneous spaces.

This paper develops a natural decomposition method (NDM)for imposing essential boundary conditions in noninterpolatory meshfree Galerkin spaces without boundary parameter tuning or auxiliary constraint construction. In such spaces, algebraic coefficients generally do not coincide with boundary values; hence coefficient assignment or nodal boundary prescription is not equivalent to imposing the continuous trace required by the variational problem. NDM introduces boundary data before discretization through a natural transfer mechanism: a source subproblem accounts for the forcing term, a weighted curl correction transfers the remaining trace mismatch, and a scalar recovery step reconstructs the solution from the corrected weighted gradient. For topologically trivial single domains with connected boundary, the reconstructed solution is equivalent, at the continuous level, to the solution satisfying the prescribed essential boundary data. The discrete analysis separates the approximation defect of the recovery space from the upstream transfer error visible to that space. Numerical experiments on benchmark problems evaluate the proposed transfer mechanism and report the associated conditioning, computational cost, and boundary perturbation behavior.

The balance property is an important property of fitted statistical models deployed for insurance pricing. It guarantees that the total actuarial price in the fitted model is equal to the totally observed loss used to fit the model. This can be seen as an in-sample global unbiasedness property. Maximum likelihood fitted generalized linear models (GLMs) with canonical links automatically fulfill the balance property. Lindholm-Wüthrich (Scandinavian Actuarial Journal, 2026) discussed two popular balance correction methods in case the balance property fails to hold. This note extends this discussion with a third method, constrained GLM fitting, that turns out to be superior over the two previously discussed ones. Moreover, we highlight the connection between the balance property and ex-post risk sharing rules.

The coupling of the evolution of a surface with evolution equations defined on that surface is of relevance in many applications and has been in the focus of interest in the analysis of parabolic PDEs in recent years. In applications the evolution of two-phase vesicles and biomembranes is governed by flows decreasing an energy which involves Canham-Helfrich-type curvature energies coupled to a Ginzburg-Landau energy. We derive a new Cahn-Hilliard-Canham-Helfrich system for the evolution of two-phase membranes. The resulting system is highly non-linear and we use the theory of quasi-linear parabolic evolution equations in weighted $L_p$-spaces to show the existence of a strong local-in-time solution and hence demonstrate that the derived system is well-posed.

The study of set families with restricted Hamming distances is a classical topic of extremal combinatorics and coding theory. Let \(H=\{A\subseteq[n]: |A|\text{ is even}\}\) be the even subcube. Let \(\ell_1,\dots,\ell_t\) be distinct positive integers and set \(\mathcal L=\{2\ell_1,\dots,2\ell_t\}\). We prove that, for all sufficiently large \(n\), every family \(\mathcal F\subseteq H\) satisfying \( |A\triangle B|\in\mathcal L \) for all \(A\ne B\in\mathcal F\) has \[ |\mathcal F|\le \sum_{i=0}^{t}\binom{n-1}{i}. \] This is best possible as a universal bound, with equality attained at the distance set \(\mathcal L=\{2,4,\dots,2t\}\). Our proof uses a Fourier-inertia argument based on a universal low/high boundary-layer sign pattern for the Fourier transform of the distance-polynomial kernel on the even subcube: the prescribed distances enter only through lower-order Fourier terms, while the leading boundary-layer signs depend solely on \(t\). This even-subcube result immediately yields an odd-subcube analogue and, through parity reductions, provides bounds for arbitrary distance sets. In particular, this approach recovers the classical interval bounds of Kleitman and the corresponding interval bounds of Huang--Klurman--Pohoata, while offering a direct spectral proof of the maximality of \(\{2,4,\dots,2t\}\) among all fixed \(t\)-distance sets.

We study transport functions as a Morse-theoretical way of describing principal bundles. Transport functions are maps from the spaces of broken gradient flow lines to a topological group and they encode the transition functions of the principal bundle. We describe and extend a construction by Voigt that yields such transport functions and show that one can recover the principal bundle from the transport function. Using transport functions with values in a topological group $G$ and a differential graded module over the chains of $G$ we define a chain complex in the style of Barraud-Damian-Humilière-Oancea's Morse homology with differential graded coefficients. We prove that in many cases the homology of this complex is the homology of an associated bundle. In the case of smooth bundles transport functions arise also from parallel transport with respect to a connection and the corresponding DG Morse complex turns out to be isomorphic to a complex defined in the style of Barraud-Damian-Humilière-Oancea. We eventually consider certain aspects of the functoriality of our constructions.

Let $G$ be a finite solvable group on which a nilpotent group $A$ acts by automorphisms. The fixing size $\mathbf{c}(G;A)$ of $A$ on $G$ is the number of $A$-composition factors on which $A$ acts trivially in an $A$-composition series of $G$. In this paper we obtain a linear bound for the Fitting height of $G$ in terms of $\mathbf{c}(G;A)$ and $\ell(A)$ where $\ell(A)$ denotes the number of prime divisors (counted with multiplicities) of $A$, under some additional hypotheses.

We establish a scaling-critical continuation criterion for the three-dimensional incompressible magnetohydrodynamic equations in $\mathbb{R}^3$ with arbitrary positive viscosity and magnetic diffusivity. Let $β(t)$ be a unit vector that is piecewise $H^1$ in time and has only finitely many jumps. For $3\le p<\infty$, set $γ_p=2p/(2p-3)$. We prove that an $H^1$ strong solution can be continued beyond $T_*$ if \[ \int_0^{T_*}\left( \|u(t)\cdotβ(t)\|_{\dot H^{3/2}}^2+ \|b(t)\cdotβ(t)\|_{\dot H^{3/2}}^2+ \|β(t)\cdot\operatorname{curl}b(t)\|_{L^p}^{γ_p} \right)\,dt<\infty . \] Thus the observed component may vary in time, and the magnetic assumption is reduced to one moving magnetic component together with one Serrin-type current-density component. The proof is based on a moving-frame formulation of the vorticity-current system, an anisotropic product estimate adapted to the moving frame, and a horizontal Hodge decomposition that controls the current-Jacobian residual.

In this article, we examine the mean-field limit of the non-exchangeable Cucker--Dong model. This model corresponds to a biologically more realistic version of the classic Cucker-Smale model, which is used to describe the alignment phenomenon in large animal groups. In addition to alignment forces, the non-exchangeable Cucker--Dong model integrates attraction/repulsion forces and network-structured interactions. In order to enable convergence towards a flocking profile, the attraction/repulsion forces are weighted by a second-order coefficient called the alignment measure, which is smaller when individuals are more aligned overall. Deriving the mean-field limit of this model relies on a new stability result that is in agreement with with both the second-order nature of the alignment measure and the non-exchangeability induced by the graph-dependent interactions.

We study homogenization of a regularized viscoelastic Oldroyd-type model in a periodically perforated bounded domain. The system describes an incompressible non-Newtonian fluid coupled to an elastic extra stress tensor and includes both nonlinear viscosity and nonlinear stress diffusion effects. The governing model, introduced by Kreml, Pokorný, and Šalom (2015), covers Oldroyd-A- and Oldroyd-B-type constitutive laws. We establish qualitative and quantitative homogenization results in suitable scaling regimes and show convergence toward an effective Darcy law on the macroscopic domain. In particular, we prove that, under appropriate assumptions on the scaling parameters, the polymeric stress does not contribute to the effective limit equation. The analysis combines uniform estimates, oscillating test-function techniques, and a relative energy method, and additionally yields a weak-strong uniqueness principle for the viscoelastic system.

We present a novel adaptive isogeometric method for solving the biharmonic equation over planar multi-patch domains with possibly extraordinary vertices, parametrized by analysis-suitable G^1 multi-patch geometries. The proposed technique relies on the concept of Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP), which enforces the required C^1-smoothness of the solution across a common edge of two neighboring patches by imposing appropriate continuity conditions by means of Lagrange multipliers. The resulting saddle point problem is solved using a dual-primal formulation, first by a small linear problem for the Lagrange multipliers and then by local, parallelizable linear problems on the individual patches for the coefficients of the numerical solution. While for the local problems on the single patches standard diagonally scaling is used as preconditioner, a suitable preconditioner for the problem of finding the Lagrange multipliers is introduced. To perform adaptive refinement, the solution of the biharmonic equation on the single patches of the multi-patch domain is discretized by employing (truncated) hierarchical B-splines, and an appropriate refinement strategy of the underlying mesh is presented. Finally, the potential of the developed adaptive IETI-DP method for solving the biharmonic equation over planar multi-patch geometries is numerically tested on the basis of several numerical examples. Thereby, the numerical results show on the one hand optimal convergence behavior with respect to adaptive refinement, and on the other hand a good performance of the proposed preconditioner for the linear problem of determining the Lagrange multipliers.

We introduce and study the synthesis-sequencing channel, a two-stage model for DNA-based data storage that jointly captures synthesis and sequencing effects. The synthesis-sequencing channel provides a more nuanced and realistic model of the DNA storage process compared to prior work, as it distinguishes between physical coverage after synthesis and sequencing coverage after readout, relaxes the assumption of independent errors across reads, and naturally induces coverage bias through the composition of synthesis and sequencing stages. We establish the information-theoretic capacity of this channel by deriving matching converse and achievability bounds for the case where synthesis and sequencing errors are modeled by binary symmetric channels with possibly different error probabilities, under mild assumptions on the channel parameters. Our results reveal multiple trade-offs between physical coverage, synthesis errors, sequencing coverage, and sequencing errors that influence the maximum achievable rate for reliable data storage.

Let $B_n$ denote the book graph consisting of $n$ triangles sharing a common edge. Few exact values of $R(B_n,B_n)$ have been obtained since Rousseau and Sheehan (1978) proved, using Paley graphs, $R(B_n, B_n) = 4n + 2$ whenever $4n+1$ is a prime power. In this paper, we obtain $R(B_n,B_n)=4n+1$ for infinitely many $n$ by constructing new families of strongly regular graphs. Moreover, we prove that $R(B_{n-2},B_n)\le 4n-3$ for every $n\ge 3$ with $n\ne 6$, removing the original condition $n\equiv 2\pmod 3$ due to Rousseau and Sheehan. In particular, if there exists a symmetric Hadamard matrix of order $2n-2$ with all diagonal entries equal to $1$, then $R(B_{n-2},B_n)=4n-3$. As an application, we show that this equality holds for every $n=2^{2\ell-1}+1$ with $\ell\ge 1$.