Papers

We develop a projection and fibering method for sets of small combinatorial doubling in (not necessarily abelian) discrete groups. As an application, in the abelian case we prove that, if $A$ is finite, the ambient group has exponent $r$, and $|A+A|\leq K|A|$, then \[ |\langle A\rangle|\leq r^{(2+o(1))K}|A|. \] This answers a question of Ruzsa with an optimal leading coefficient, independent of Fox–Pham. The main ingredient is a discrete version of a fiber spillover argument. For sets in $2$-step nilpotent groups of exponent $r$, we also prove that $|A^3|\leq K|A|$ implies $|\langle A\rangle|\leq r^{(2+o_K(1))K}|A|$. The proof combines the abelian theorem with a weighted averaging of central fibers and commutators.

We develop a unified framework for $c$-sample homogeneity testing on the circle. The proposed class of $c$-sample Sobolev tests generalizes the two-sample Sobolev tests based on uniform scores and encompasses the existing multisample tests as particular cases. We further embed this class into a broader aggregation framework, where the samplewise Sobolev components are combined through general merging functions, including average-type, maximum-type, and interpolating families. Within this class, we introduce the first Anderson-Darling-type homogeneity test for circular data, and we further propose two new tests designed to detect multimodal departures from homogeneity, constructed from softmax and Poisson kernels. We derive the asymptotic null distribution of the class, prove its consistency against a broad family of fixed alternatives, and obtain the asymptotic distribution under shift-type local alternatives. The tests are distribution-free and therefore do not require resampling. A comprehensive simulation study demonstrates the strong power of the Anderson-Darling-type test compared with Cramér-von Mises-type competitors across several scenarios, and the effectiveness of the softmax and Poisson tests under several alternatives. The testing toolbox is applied to analyze the nursing patterns of polar bears.

Recent orthodoxy, which we call Curvature Fundamentalism, posits that the field strength $F$ on spacetime exhausts the physically significant content of electromagnetism, at least on prosaic (contractible) spacetimes. We argue that it is gauge invariance that constrains physical significance. As a result, there are physically significant quantities that $F$ does not capture, even on contractible spacetimes. We show that this result is robust: in the presence of charged matter fields, there are interference experiments whose outcome cannot be predicted by $F$ or any other point-local gauge invariant quantities on the basis of initial Cauchy data. We prove this using the example of relative phase of a Klein-Gordon-Maxwell field, and use it to identify a context in which the widely-claimed equivalence between the Maxwell-Faraday formulation $EM_1$ and the gauge formulation $EM_2$ of electromagnetism is untenable, and that new strategies may be needed for the reductionist project.

For a finite group $G$ let $\omega(G)$ denote its spectrum, the set of orders of its elements, and set $\eta(G)=|\omega(G)|$, the number of distinct element orders. The set $\omega(G)$ has been studied intensively, but its cardinality has received little systematic attention as an invariant in its own right. We develop the theory of $\eta$. After describing its behaviour under subgroups, quotients, sections, direct products and Frobenius extensions, we prove the two-sided bound $1+\sum_{p\mid|G|}v_p(\exp G)\le\eta(G)\le\tau(\exp G)$, in which the lower equality characterises the CP-groups, those all of whose elements have prime-power order, and the upper equality the groups whose spectrum realises every divisor of the exponent; nilpotent groups lie at the upper end, giving $\eta(G)=\tau(\exp G)$. We also prove $\eta(G)\le\tau(|G|)$ with equality only for cyclic groups, and $\eta(G)\ge\pi(|G|)+1$ with equality only when every nontrivial element has prime order. We show that $\eta(G)\le3$ forces solvability by an argument resting only on Burnside's theorem, and, combining this with the recognition of $A_5$ by its spectrum, that $A_5$ is the unique non-solvable group with $\eta(G)=4$. For the symmetric and alternating groups we record self-contained proofs of closed criteria for membership in the spectrum through a single arithmetic function, placing the associated counting sequences in the present framework, and we determine the extreme values of $\eta$ over all groups of a fixed order.

In this article, we prove that the solution of the skew stochastic heat equation converges to the solution of the stochastic heat equation with reflection in the limit of infinite skewness. The result holds at equilibrium. This provides an SPDE analog of the link between skew and reflected Brownian motion. One of the intermediate results is the convergence in law of a penalised Brownian bridge to a 3-Bessel bridge.

An old result of Tutte states that any $d$-regular graph contains a spanning subgraph in which every vertex has degree $k$ or $k+1$, for every $1\leq k\leq d$. We generalize this statement to hypergraphs, showing, for example, that every $3$-uniform $d$-regular hypergraph contains a subgraph in which all degrees are $k, k+1$ or $k+2$, for every $1\leq k\leq d$. This statement is best possible in the sense that the corresponding statement with only two allowed consecutive values is not true. We provide generalizations of this statement to higher uniformities and discuss several open problems.

In this paper, We obtain dispersive estimates for solutions to the Schrödinger equation with a uniform magnetic field on the hyperbolic plane \(\mathbb{H}\). The key ingredient is an explicit representation formula for the kernel of the corresponding Schrödinger propagator. As a consequence, we prove the corresponding Strichartz estimates for all admissible pairs on \(\mathbb{H}\).

We introduce an adaptive multi-parameter variant of ADMM and prove that it exhibits local superlinear convergence once the set of active constraints has been identified. In simulations, the proposed algorithm consistently outperforms OSQP, a standard ADMM solver, in terms of iteration count. We then implement an MPC solver based on our method and compare its runtime against a broader selection of state-of-the-art algorithms. Evaluations on challenging benchmark problems reveal that our approach delivers competitive performance both in terms of average and worst-case solve times, without being limited to coarse tolerances, as is typically the case for standard ADMM implementations and first-order methods.

We prove pointwise decay and scattering for small-mass solutions to the focusing and defocusing continuum Calogero–Moser models under suitable decay assumptions on the initial data. Our proof is based on an explicit formula for solutions and the exact conservation of the Galilean vector field associated to the equation.

We show that the rowmotion operator from dynamical algebraic combinatorics behaves the same on all alt $\nu$-Tamari lattices for a fixed lattice path $\nu$. We use this invariance of rowmotion to establish cyclic sieving and homomesy results for rational Tamari lattices. We also conjecture that this invariance extends to the more general cross Tamari lattices.

We prove that the solution map of the classical bilateral parabolic obstacle problem is directionally differentiable when interpreted as a function between suitable Lebesgue and Sobolev spaces. To the best of our knowledge, this paper is the first to establish this kind of differentiability for a parabolic evolution variational inequality with two-sided fully distributed pointwise inequality constraints. Our analysis relies on known differentiability results for unilateral parabolic obstacle problems and a new theorem on the directional differentiability of solution operators of non-expansive parameterized fixed-point equations. The latter may also be of interest for other applications. Our results can be used, for example, to derive optimality conditions for optimal control problems governed by bilateral parabolic obstacle problems.

In this paper, we study an inverse boundary value problem for a generalized Jordan-Moore-Gibson-Thompson equation with Westervelt-type nonlinearity and space-dependent coefficients. This third-order (in time) hyperbolic equation models nonlinear ultrasound propagation in viscous and thermally relaxing media. Assuming that the diffusivity and the sound speed are known a priori, we investigate the simultaneous stable determination of the friction coefficient, the weak damping coefficient, the potential, and the nonlinear coefficient from the associated Dirichlet-to-Neumann map. The proof combines the finite-difference linearization method with the construction of Gaussian beam and geometric optics solutions. These arguments reduce the inverse problem to stability estimates for (attenuated) geodesic ray transforms under the foliation condition. As a consequence, we obtain Hölder-type stability estimates of recovering the linear and nonlinear coefficients.

We develop a matrix pseudodifference operator framework that places theAdler-Pfaff and Pfaff-Toda hierarchies in a common algebraic setting. The bi-infinite Adler-Pfaff hierarchy then becomes a $2\times2$ matrix pseudodifference Lax hierarchy. We introduce the Matrix Pfaff-Toda hierarchy by dressing the matrix Laurent algebra $M_{2N}\bigl(\mathbb{C}[\mathcal{S},\mathcal{S}^{-1}]\bigr)$ using a Pfaff-Toda splitting of the matrix pseudodifference algebra. In the one-component case, its diagonal sector recovers the continuous Pfaff-Toda hierarchy, while the off-diagonal directions supply the missing odd flows, and the powers of a single bare operator reconstruct the full Adler-Pfaff hierarchy. For general $N$, on the regular factorization locus, the multicomponent Pfaff-Toda tau-functions of Savchenko and Zabrodin realize the commuting diagonal sector, with its dressing equations derived directly from the fermionic bilinear identity. Reductions to the block-diagonal and scalar cases recover multicomponent and scalar $2D$ Toda, identifying the even Pfaff hierarchy with an anti-diagonal Toda subhierarchy, and yielding the Krichever-Zabrodin C-Toda hierarchy.

We study non-archimedean Monge-Ampère measures of toric metrics on closed subvarieties of toric varieties, after reducing, if necessary, to the minimal torus-orbit closure containing the subvariety. The measure vanishes on the toric boundary and is supported on the tropical skeleton. Our main result explicitly describes their local structure in terms of the tropical germ of the subvariety and the tropical Green function of the metric. More precisely, the tropicalized Monge-Ampère measure is the corresponding polyhedral Monge-Ampère measure, which extends the tropical intersection product, while the original measure is recovered locally by tropical pullback from explicit tropical and convex data. This provides a toric counterpart to analogous results for subvarieties of abelian varieties, with new local phenomena arising from tropicalization.

This paper presents two types of novel, qualitative and quantitative, estimates for positive solutions of general uniformly elliptic PDEs in divergence form. First, we prove a Morel-Oswald type of extension of the Hopf-Oleinik lemma, in which in addition we specify the sharp dependence of the constant in the data of the operator and the size of the domain. Second, we establish a new differential Harnack (logarithmic gradient) estimate for non-homogeneous equations, as well as an optimal global differential Harnack estimate for homogeneous equations.

This paper proves the integrability of a nonholonomic billiard defined by a knife-edge in a disk. The paper begins by parametrizing the impact space using coordinates that reveal the system's rotational symmetry. It then constructs a map on the space of post-impact states that encodes the billiard dynamics through a recurrence relating each post-impact state to the next. In addition, the paper shows that the knife-edge billiard admits a family of generalized caustics.

In this article, we investigate rigidity results for constantly curved minimal immersions of the two-sphere $S^2$ into the unitary group $\mathrm{U}(n)$. Using loop group methods for harmonic maps, we establish a correspondence between such immersions and a distinguished class of constantly curved holomorphic immersions of $S^2$ into finite-dimensional Grassmannians. In the case $\mathrm{U}(3)$, we classify the constantly curved minimal immersions of $S^2$ with uniton number one and prove that, under a natural unramifiedness condition, those of uniton number two are $S^1$-invariant; as a consequence, every constantly curved totally unramified minimal immersion $S^2\to \mathrm{U}(3)$ of uniton number two is unitarily congruent to the composition of the first Gauss map of the Veronese curve in $\mathbb{C}P^2$ with the Cartan embedding $\mathbb{C}P^2\hookrightarrow \mathrm{U}(3)$.

This paper studies the problem of optimal control of Caputo linear fractional-order systems of order $\alpha \in (0, 1]$ with continuous-time state constraints and input constraints. The resulting problems are semi-infinite, since the state constraints are imposed over the sampling intervals and not merely at the sampling instants. To address this, we derive a sampled-data representation of the inter-sample trajectory and use Bernstein polynomials to construct polytopic approximations of the continuous-time state constraints. Explicit bounds are obtained for the conservatism introduced by this approximation. By subdividing each sampling interval and applying the Bernstein-based construction on every subinterval, the approximation is made arbitrarily tight. Numerical examples illustrate the tightness of the proposed enclosures, the computational tractability of the proposed method, and the trade-off between approximation accuracy and computational complexity.

We prove the generalized Lax conjecture for cubics in five variables: their hyperbolicity cones are spectrahedral. Our approach constructs spectrahedral realizations from completely positive maps on Clifford algebras. The construction passes through quaternionic determinantal representations of certain extremal cubics. Convexity of the set of Clifford-realizable cubics then yields the result for all cubics in five variables. The approach also covers all previously known cubic cases of the generalized Lax conjecture: forms in at most four variables and symmetric forms in arbitrarily many variables. We further establish a sufficient criterion for Clifford realizability in arbitrary dimension and use it to obtain a neighborhood of the origin in the cubic norm ball consisting of Clifford-realizable cubics. Finally, we prove spectrahedrality for sparse hyperbolic quartics in four variables.

We prove the all-genus open mirror symmetry of a toric Calabi-Yau 3-orbifold where the boundary condition is given by multiple Aganagic-Vafa branes. Each brane could be either inner or outer, and admit a fractional framing. The B-model is given by the Chekhov-Eynard-Orantin topological recursion expanded in local coordinates specified by the brane data.

We study the fractional porous medium equation on complete noncompact Riemannian manifolds with nonnegative Ricci curvature for $s\in(0,1]$ and $n>2s$. Assuming that the manifold is $s$-nonparabolic, so that the fractional Laplacian admits a suitable positive minimal Green function, we use this Green function to introduce a natural weighted space of initial data, strictly larger than $L^1$. This leads to a weak dual, or potential, formulation of the equation, for which we prove existence for nonnegative initial data in the weighted space. We then establish quantitative local smoothing estimates for initial data in either $L^1$ or the Green-weighted space. Under additional noncollapsing and uniform volume-growth assumptions, we obtain global smoothing estimates. We also show that, even in $\mathbb{R}^n$, data in the Green-weighted space need not generate bounded solutions unless a suitable uniform weighted integrability condition is imposed. The short- and long-time behaviors predicted by our estimates are shown to be optimal in appropriate senses. When $s\in(0,1/2)$, the results require only $\operatorname{Ric}\geq0$, together with a noncollapsing assumption where needed. Our estimates also cover the case $s=1$, several of them being new in that setting. Finally, we extend the approach to more general filtration equations.

We prove that all Morita classes $\mu_k \in H_{4k}(Aut(F_{2k+2});\mathbb{Q})$ are nonzero using a novel description of the Hopf algebra structure on Steinberg homology. We prove analogous results with twisted coefficients. Our results disprove a case of a finiteness conjecture of Kontsevich.

We establish a priori interior $C^{1,\alpha}$ and $C^{2,\alpha}$ Schauder estimates for a class of $Q$-valued quasilinear elliptic systems which are perturbations of the Laplace system, for arbitrary multiplicity $Q$, domain dimension $n\geq2$, and target dimension $k$. The $C^{1,\alpha}$ estimate generalises the work of Simon and Wickramasekera in \cite{SW16} on $2$-valued solutions of linear systems and concerns weak solutions of divergence-form systems, while the $C^{2,\alpha}$ estimate concerns strong solutions. In dimension $n=2$, both estimates hold for every $0<\alpha<1/Q$. In dimensions $n\geq3$, there exists $\delta=\delta(n,k,Q)>0$ such that both estimates hold for every $0<\alpha<\delta$. As applications, we obtain a small-slope Schauder estimate and a small-slope Bernstein theorem for $Q$-valued maps whose graph varifolds are stationary. Combining the Schauder estimate with the recent branch-set stratification theory of Krummel–Minter–Wickramasekera in \cite{KMW26}, we further prove that, for every $\gamma>0$, the branch set $\mathcal B_u$ of any $C^{1,\gamma}$ $Q$-valued map $u$ whose graph varifold is stationary satisfies $\dim_{\mathcal H}\mathcal B_u\leq n-2$. This bound is sharp already in codimension one.

We extend reflection coupling for uniform-in-time propagation of chaos from Euclidean space to complete Riemannian manifolds. Intrinsic reflection along minimizing geodesics replaces the Euclidean difference process, and the associated endpoint-index term quantifies the competition between curvature, confinement, and interaction. Under an endpoint-index bound and a radial Laplacian bound, we establish global well-posedness, uniform moment estimates, and uniform-in-time propagation of chaos with rate $N^{-1/2}$ in a modified Wasserstein distance. We further obtain exponential contraction of the nonlinear McKean–Vlasov semigroup, uniqueness of its invariant probability measure among probability measures with finite second moment, and exponential convergence to equilibrium. Although a Ricci lower bound is an important special case, our framework also includes a curvature-spike class with $\inf_M\operatorname{Ric}=-\infty$.

Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which \[ \sum_{n\leq x}f(n) \geq \frac{\sqrt{x}(\log\log x)^{1/4}}{V(x)}. \] The corresponding negative fluctuation holds as well. In particular, this gives an affirmative answer to Erdős Problem~\#1144. As a byproduct, our result has a direct corollary giving new almost sure lower bounds $\log\log x$ on the number of sign changes of partial sums up to $x$ for all sufficiently large $x$.

We introduce a generalized Cauchy–Kowalevski extension associated with a biaxial decomposition of $\mathbb{R}^{m+1}$, allowing monogenic functions to be reconstructed from data prescribed on the subspace $\{X\in\mathbb{R}^{m+1}: z=0\}$ where $z=x_0+x_1e_1$. The extension admits a natural decomposition into monogenic layers, leading to operators $\mathrm{CK}_{z,k}$. Classes of biaxial monogenic functions generated by these operators are characterized by systems of partial differential equations, and explicit power series solutions are obtained. The action of the operators $\mathrm{CK}_{z,k}$ on homogeneous polynomials is then investigated, leading to a class of biaxial monogenic plane waves and the associated system of partial differential equations.

In this paper, we study two families of quasilinear equations: Hunter-Saxton type and Camassa-Hom type equations, with a paramerter $\lambda\in(0,1)$ whose solutions form cusp singularities. When $\lambda=1$, the first system becomes the scalar conservation law. The main result of this paper is to give regularity of two types of structurally stable singularities: Type I on the singular curve, Type II at the point where cusp singularity forms, for some $\lambda\in(0,1)$. When $\lambda\rightarrow 1$, our result indicts the $C^{1/3}$ regularity at the point where singularity forms, which agrees with the regularity of the generic pre-shock solution.

We study the momentum equation of the decoupled unregularised Hibler sea-ice model with non-negative ice mass and ice strength, both of which may vanish. The ocean drag provides coercivity, while the stress law and mixed boundary conditions are encoded by a convex dissipation functional. We prove existence, uniqueness and stability for the time-discrete problem and reconstruct an admissible stress. A Rothe approximation yields global weak variational solutions for time-dependent coefficients under a one-sided growth condition on the mass. Weak solutions are unique when the ice strength is independent of time. For spatially Lipschitz ice strength, finite dissipation also yields a local measure structure of the weighted deformation and a global bound on the negative part of the weighted divergence.

Let $R$ be a unital ring equipped with a Sylvester matrix rank function $\rk$. A harmonic function $\alpha$ on a Bratteli diagram $B$ defines a weighted matrix rank on the associated algebraic direct limit $A(B,R)$. We prove that, if $\alpha$ is extreme and the total weight of blocks of any fixed bounded size tends to zero, then the rank completion of $A(B,R)$ is isomorphic to $\mathcal M_{R,\rk}$, the rank completion of the direct system $\Mat_{2^k}(R)$ with connecting maps $x\mapsto\diag(x,x)$ and normalized ranks $2^{-k}\rk$. The isomorphism preserves the unital $R$-algebra structure and the ranks on all rectangular matrices. The coefficient ring need not be regular, and the specified rank need not be induced from a regular ring. We recover factor-sequence uniqueness and construct corners of every prescribed rank in $(0,1]$ that are isomorphic to $\mathcal M_{R,\rk}$ with their normalized ranks. Examples show that the coefficient rank can affect the isomorphism type and that the completion can be non-regular and non-simple. For complex coefficients, the trace determined by $\alpha$ gives a rank completion of the associated AF $C^*$-algebra canonically isomorphic to the affiliated-operator ring of its GNS closure. The rank completion of the algebraic direct limit can be a proper subring of this ring.

Let $n\ge 2$, and let $M$ be either the $n$-dimensional sphere $\mathbb{S}^n$ or the real projective space $\mathbb{RP}^n$. Given $\alpha\ge 1$, we construct a finitely generated group $G_{n,\alpha}$ that embeds into the group of $C^\alpha$-diffeomorphisms of $M$, but does not embed into the group of $C^\beta$-diffeomorphisms of $M$ for any $\beta>\alpha$.