Papers

We establish a double-logarithmic stability estimate for the recovery of a time-dependent potential in the parabolic equation $(c^{-1}\partial_t-\Delta_g+q)u=0$ on a conformally transversally anisotropic manifold. The partial input–output map associates an initial state and lateral Dirichlet data supported in a prescribed open set with the terminal state and the Neumann trace on another open set. These sets contain the positive and negative boundary parts determined by a limiting Carleman weight, respectively. The measurement discrepancy is the operator norm of the difference of two such maps, whose range has improved Sobolev regularity. We prove stability under a uniform stability assumption for the attenuated geodesic ray transform on a family of non-tangential geodesics. This assumption is verified when the transversal manifold is simple and for regular families on certain non-simple manifolds. The proof uses Gaussian beam quasimodes with a uniform $O(h^{1/2})$ concentration estimate, geometric optics solutions obtained from boundary Carleman estimates, and stability of the geodesic ray transform for small attenuation. Quantitative analytic continuation for Hilbert-space-valued Fourier transforms and Sobolev interpolation yield the double-logarithmic modulus.

Grünbaum asked whether every planar convex body admits, for every $0\leq t\leq 1/4$, two orthogonal lines cutting it into pieces with cyclically ordered areas $t,t,1/2-t,1/2-t$. Bárány posed the analogous question for well-behaved planar measures and conjectured that the answer there is negative. We confirm Bárány's conjecture in a particularly robust form: for every fixed $0<t<1/4$ we construct smooth, strictly positive, centrally symmetric, strongly log-concave measures arbitrarily close to the standard Gaussian for which the prescribed partition does not exist. In contrast, we prove that the partition exists for every $t$ whenever the measure is invariant under an orientation-reversing affine involution. We also exhibit a $96$-point counterexample for which no pair of perpendicular lines produces cyclic counts $8,8,40,40$.

In this article, we study $2$-$(v,k,\lambda)$ designs $\mathcal{D}$ admitting a flag-transitive almost simple automorphism group $G$ with socle one of the Mathieu groups. In conclusion, we obtain all such $2$-designs with explicit constructions with appropriate references to known designs. To our knowledge, we obtain $101$ newly-constructed flag-transitive $2$-designs.

We study prime-power congruences for the prime-to-\(p\) Hecke eigensystems of normalized cuspidal level-one eigenforms as the weight varies, and we address the problem of classifying such systems for small prime powers. We construct Hecke-equivariant transfer maps between level-one modular-symbol modules via multiplication by suitable lifts of Dickson invariants; these maps reduce the problem to finite computation and propagate Hecke operator relations through all weights. For \(p=2,3,5,7\), we obtain all-weight classifications modulo \(2^8\), \(3^4\), \(5^3\), and \(7^2\), respectively.

The recently established duality between tensor, symmetric and exterior products of uniform matroids on the one hand, and abstract (bi-)rigidity matroids on the other hand, connects theorems and open questions from both areas, previously thought unrelated. In particular, a 1981 question of Mason asks if any two uniform matroids admit a freest product. Via duality, its reformulation due to Cruickshank, Jackson, Jordán and Tanigawa asks if the generic birigidity matroid is the freest abstract birigidity matroid for its parameters. A related conjecture of Jackson and Tanigawa asks if the generic $2$-hyperconnectivity matroid is the freest $\{K_4,K_{3,3}\}$-matroid. We show that these problems can be addressed effectively, and often solved completely for the class of linearly representable matroids. To this end, we introduce star-basis normal forms that allow to compare representations of abstract (bi-)rigidity matroids over the same field, resulting in a refinement of the weak order relation on the underlying matroids. Utilizing it, we prove that the generic birigidity matroid is the freest linearly representable abstract $(a,b)$-birigidity matroid. This confirms Mason's conjecture for linear matroids. We prove that every representable abstract $2$-rigidity matroid admits a rigidity matrix representation. This is a strengthening of the maximality property of the generic $2$-rigidity matroid. We also prove that every representable abstract $2$-rigidity matroid in which every copy of $K_{3,3}$ is a circuit admits a hyperconnectivity matrix representation. It follows that the generic rigidity and hyperconnectivity families $\mathcal{R}_2$ and $\mathcal{H}_2$ are the only linearly representable $2$-rigidity families.

We prove an asymptotic expansion of a weighted heat trace of the Dirichlet Laplacian on a bounded Lipschitz domain. The weight is given by a power of the distance to the boundary times a bounded function that is continuous near the boundary. Depending on the exponent of the weight, we obtain one-term or two-term asymptotics. We present two applications, namely a version of one-term asymptotics of weighted Riesz means for arbitrary orders, as well as convergence results for a family of measures that describes localization of Laplace eigenfunctions.

A multiplicative set $S$ of a commutative ring $R$ is strongly multiplicative if every family $(s_i)_{i \in I}$ of elements of $S$ admits a common multiple in $S \cap \bigcap_{i \in I} s_iR$. We combine the structural results on strongly multiplicative sets with their prime-theoretic and module-theoretic applications. We prove that $S$ is strongly multiplicative if and only if localization at $S$ commutes with arbitrary intersections of ideals, if and only if $R_S$ is the localization at an idempotent, and if and only if $D(S)$ is clopen in $\Spec(R)$. We then develop permanence results under homomorphisms, products, factor rings, trivial extensions, and amalgamations; describe the associated split torsion theory; show that almost multiplicative sets contribute no new cases beyond their multiplicative hull; and record a Mittag–Leffler refinement for intersections of submodules in finitely generated modules. On the prime-theoretic side, we relate strongly multiplicative sets to strongly prime ideals and strongly zero-dimensional rings, answer the Hamed–Malek question on the role of strong multiplicativity for chains of $S$-prime ideals, prove a strong Krull separation lemma together with a maximal-ideal correspondence for $R_S$, and connect the theory with the regular $m$-complement operator. In particular, every nontrivial strongly multiplicative localization must invert a zero divisor.

We prove that the closed subspace of $L^1[0,1]$ spanned by the even (respectively, odd) levels of the Haar system is not isomorphic to $L^1[0,1]$.

For any volume-collapsing finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface, we show the flow satisfies a Type I curvature bound and classify the corresponding tangent flows. Combined with previous results, this shows that any finite-time singularity of a Kähler-Ricci flow on a compact Kähler surface is of Type I.

We study the free-boundary Euler–Monge–Ampère system as a conservative relaxation of ideal incompressible fluid motion. The force is determined on the actual fluid domain by a Monge–Ampère equation coupled to a nonlinear boundary condition. For each fixed relaxation parameter, we establish local well-posedness for uniformly convex initial domains of class $C^{2,\alpha}$, densities in $C^{1,\alpha}$, and arbitrary initial velocities in $C^{2,\alpha}$. The solution retains these spatial regularities, and the force potential belongs to $C^{3,\alpha}$ up to the moving boundary. Within uniformly bounded data families, the solution depends continuously on the initial data at every lower Hölder exponent.

Given a positive integer $M$ with reduced residue system $\Phi_M$, we partition the set of squarefree integers according to some congruence conditions on the number of prime factors in each class $\phi \in \Phi_M$, and prove two relevant equidsitribution results in this setting. We discover the bias phenomenon in some special cases, and formulate a conjecture with numerical support.

Determining the momentum missing on a Poincaré section of a two-degree-of-freedom Hamiltonian is usually posed as the root-finding problem $H(0,q_2,p_1,p_2)=E$, which degenerates at the boundary of the energetically allowed region. We argue that the degeneration is the visible symptom of a geometric object and reorganize the problem around it. The section is a coisotropic hypersurface, the missing momentum is its characteristic direction, and recovering it is a reconstruction through the associated coisotropic reduction. Its carrier is not the fibrewise least-energy function, which we show is informationally incomplete, but the correspondence $\Sigma_E^{\mathrm{sec}}=\Sigma_E\cap C$ together with the reducing projection $\pi_E$, a branched reconstruction over the admissible region whose discriminant is identified with the Hill boundary under a base-side regularity condition. We prove, isolating for each conclusion its minimal hypothesis: that $\pi_E$ is proper and a covering away from the discriminant (coercivity); that the fibres have exactly two points over the interior (unimodality); that along the discriminant $\pi_E$ is a fold (fibrewise nondegeneracy and transversality); and that the discriminant coincides with the topological boundary of the admissible region precisely under a regularity of the least-energy function on the base, which no fibrewise hypothesis supplies. Strict convexity of $H$ in the momenta is shown to be a sufficient but non-fundamental bundle of these conditions. The boundary is a discriminant, not a Lagrangian caustic. Only afterwards does the squared fibrewise energy residual $F_b=(h_b-E)^2$ appear, as the analytic resolution of the fold; inside the admissible region it is a double well with two global minimizers, so branch selection is a separate rule.

Let $p\geq 11$ and let $\bar\rho$ be a très ramifié representation. For a primitive projective-augmented module of the corresponding type, we prove the très ramifié analogues of the local results of Liu-Truong-Xiao-Zhao and thereby establish the local ghost conjecture in this case. The main difference here is that the Iwahori dimension increases by one between consecutive relevant classical weights and may therefore be odd. Accordingly, the proof of the near-Steinberg results must treat both integral and half-integral centres.

Regularization choice changes the recovery phase diagrams of annealing-based binary compressed sensing. We develop a regularization-selection method that combines systematic parameter search with random forest regression. Under noiseless Gaussian measurements with known sparsity, reference parameters are selected from a candidate grid by minimizing mean squared reconstruction error over repeated simulated annealing (SA) trials. The fitted model predicts these reference values from signal dimension, sampling ratio, and sparsity. With predicted regularization, the SA recovery transition broadly follows the asymptotic reference boundary for box-constrained $\ell_1$ recovery at the larger signal dimensions examined. Without retraining, the same predictor supplies identical regularization values to SA and a quantum–classical hybrid solver. On matched problem instances, the hybrid solver yields smaller mean squared reconstruction errors than SA in parts of the evaluated parameter space. The resulting rule reuses the searched information for subsequent reconstruction without repeating candidate searches at each setting. The results quantify empirical performance under the stated finite candidate grid and solver settings; they do not constitute a solver-independent recovery guarantee or a time-to-solution comparison.

In this paper, we introduce and study a new non Hermitian matrix-valued process built from truncations of Brownian motion on complex positive-definite matrices and of its inverse. We prove that its eigenvalue process is valued in $(1,+\infty)$ and that it forms a non-compact Jacobi particle system. Surprisingly, the same particle system occurs in \cite{BDW} in connection with Brownian motion on a non-compact complex Grassmann manifold. In this respect, we supply evidences showing that this occurrence holds true only at the spectral level and not on the corresponding matrix diffusions. We then study the large-size limit of our matrix model and prove results parallel to those obtained by the second author for the free Jacobi process.

We study the dynamics of a single point vortex in an unbounded planar channel whose interfaces are quasi-periodic in the longitudinal direction. The motion is governed by the Robin function. We introduce a hull formulation that lifts the quasi-periodic geometry to a periodic problem on a finite-dimensional torus and yields an exact quasi-periodic representation of the Robin function. Under a natural transversality condition, we construct global quasi-periodic invariant graphs describing transporting vortex trajectories and show that a full neighborhood of each boundary component is foliated by such graphs. Under a Diophantine condition on the spatial frequencies, the dynamics along each graph can be straightened to a constant drift, so that the vortex motion is quasi-periodic modulo translation. A central contribution concerns the distribution of vortex equilibria. In the quasi-periodic setting, where no fundamental spatial cell exists, we identify critical points of the Robin function with crossings of a hypersurface by a Kronecker flow on the hull torus. Exploiting unique ergodicity, we establish a general zero-counting theorem, allowing finite-order tangencies, which yields an explicit geometric flux formula for the asymptotic density of critical points and their limiting phase distribution. The result is nonperturbative once the critical hull is constructed. Explicit periodic and quasi-periodic models reveal bifurcations and phase transitions in the critical-point distribution, while numerical computations provide quantitative validation of the analytical results.

This paper presents a stability analysis of the generalized multi-source Weber problems under the influence of data perturbation within the framework of the Minkowski function. First, we establish explicit Lipschitz constants for the objective functions and optimal value functions. Then, we prove several properties of the global optimal solution sets. Furthermore, we provide sufficient conditions for the upper semicontinuity of global solution mappings and the inner semicontinuity of local solution mappings. Three illustrative examples are constructed. Our results give indirect answers to several open questions regarding the stability of local optimal solution mappings of the optimization problems discussed.

In this article, we prove the Defect Conjecture for Cayley graphs of abelian groups, dihedral groups, and groups of the form $\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/(p-1)\mathbb{Z}$ where $p\geq 3$ is a prime. We also compute the Iwasawa invariants of the Bowen–Franks groups associated with these graphs in several cases.

An $A$-module $E$ is annihilator multiplication if the annihilator of each element equals that of $IE$ for a finitely generated ideal $I$. We characterize rings for which every faithful module has this property as the commutative quasi-Frobenius rings, and rings for which every module has the property as the Artinian principal ideal rings. Both characterizations reduce to two-generated modules. Over a reduced ring with finitely many minimal primes, faithful annihilator multiplication modules are exactly the regular-torsion-free modules, and principal ideals suffice in the definition. An ascending chain condition on annihilator ideals gives finite detection of the module annihilator, yielding localization and graph rigidity results. We also prove equality of associated primes with those of the faithful quotient and establish projective tensor and trace characterizations. Over local square-zero rings, the property is equivalent to nonsingularity of a bilinear multiplication map. A projective dimension argument gives a sharp length bound, attained by explicit faithful indecomposable nonprojective modules whose endomorphism rings are computed. Every cyclic submodule of these examples embeds in the ring, although the whole module is not torsionless. Support, hereditary torsion, and amalgamation criteria connect these structural results with further module constructions.

We study the equidistribution in arithmetic progressions modulo powers of a fixed odd prime of Hecke eigenvalues of noncuspidal $\mathrm{GL}_3$ automorphic forms, with a focus on the convolution coefficients $\lambda_{1\boxplus f}(n)=(\lambda_f\star 1)(n)$,attached to a level $1$ holomorphic primitive cusp form $f$. Utilizing techniques introduced by Kowalski–Lin–Michel and Milićević, we prove the exponent $\vartheta=1/2+7/458-\varepsilon$ is admissible.

We study the Laurent coefficients of the Barnes multiple zeta function with complex parameters in a common open half-plane and a fixed holomorphic determination of the logarithm. At the highest pole, we derive explicit limit formulae for every regular Laurent coefficient in terms of finite multiple sums and logarithmic correction terms. At the lower possible poles, we establish all-order relations with the Taylor coefficients at the origin and their parameter derivatives. We also determine the large-order behavior by subtracting all principal parts. The remaining function is entire, so Cauchy's estimate separates explicit residue contributions from a remainder that decays faster than any fixed geometric rate. The neighboring residues yield the limiting even and odd subsequences, with vanishing residues and cancellations accounted for. The constant-term cases recover known finite-part representations; the main focus is their extension to higher Laurent coefficients.

We introduce a matrix-valued finite difference formalism for the genus zero BPS resummed local Gromov-Witten potential of a threefold flop. Factoring the central difference as \[ \Delta_\eta=\nabla_\eta^2, \qquad \nabla_\eta=T_{h\eta/2}-T_{-h\eta/2}, \] we prove that the mixed differences \( \mathbf H=(\nabla_{\eta_i}\nabla_{\eta_j}F)_{i,j} \) admit an exact rank one signed \(q\)-Gram decomposition. Primitive BPS classes are therefore detected by rank one coefficient matrices, and a matrix-valued Möbius inversion reconstructs all multiplicities on a primitive ray. Under a simple threefold flop the Gram forcing is covariant away from the flopped ray and acquires the universal logarithmic anomaly \[ \kappa_C(C^+\otimes C^+)\log r, \qquad \kappa_C=\sum_{d\ge1}d^3n_{dC}. \] After a natural renormalization it becomes flop covariant, while one classical logarithmic derivative recovers the cubic correction in the crepant transformation formula. We interpret \(\kappa_C\) as the third moment of the GV width distribution: Toda's noncommutative width is the second moment, and contraction algebra BPS invariants give a finite cohomological width series whose numerical realization contains all these moments. For root supported BPS theories, simple root Gram coefficients reconstruct the signed Cartan matrix. Applied to the standard ADE foldings, the theory distinguishes \(B_n\) from \(C_n\), detects the \(F_4\) double bond and \(D_4\) triality, and yields explicit global \(q\)-Gram discriminants.

We construct an infinite family of imaginary biquadratic fields with large class numbers. The construction of this family is done by composing two distinct families of imaginary quadratic fields: the first one is the family Q(sqrt(k^2-p^l)) introduced by Banerjee and Hoque [1], subject to specific constraints on the integer parameters, l, k and p, and the second one is the parametrized family of the form Q(sqrt(-F_(50s+25))) based on Fibonacci numbers, introduced by Kishi [2]. One of the main tools that is used is Kuroda's class number formula of imaginary biquadratic fields. We choose these two families with the sole aim to get high order elements in the class group of the resultant biquadratic fields.

The Kantowski–Sachs interior of Grumiller's two-dimensional dilaton gravity model reduces to $\dot H=\tfrac12(\Lambda-3H^2-u^2)+Qu$, $\dot u=-Hu$, where $H$ is the expansion rate of the orbit two-spheres, $u$ their inverse areal radius, and $Q$ is minus twice the Rindler acceleration. For $Q\neq0$ it has two distinct rank-one nilpotent equilibria with complementary Bogdanov–Takens (BT) degeneracies: $(a,b)=(0,\neq0)$ with multiplicity $\mu=3$, and $(a,b)=(\neq0,0)$ with $\mu=2$. We show that this complementarity is forced by a single divisor-organized structure. The system is Darboux integrable, $X=\tfrac12u^4X_I$, where $I$ is the mass function and $u^4$ is an inverse integrating factor whose zero divisor is the invariant axis $\{u=0\}$. Off the divisor, every nilpotent equilibrium of $RX_I$ with $R(p)\neq0$ has $b=0$, from $b=D_{q_0}\operatorname{tr}DX$. On the divisor, invariance of the coordinate axis forces $a=0$ because $\det DX|_{u=0}=\Phi_xG$ and both factors vanish at the corner. Hence $ab=0$ throughout the class, so a versal two-parameter BT unfolding is impossible. The Bernstein–Kushnirenko bound fails at both points; instead, the exact local identity $\mu_0=m_1\ell_y+m_2\ell_x$ gives $\mu=3,2$ without a nondegeneracy hypothesis. Both multiplicities have a mixed-covolume interpretation, with $\operatorname{Covol}=\sum_{k,i}\min(p_kq_i',p_i'q_k)$ on convenient diagrams; the corresponding general theorem for the non-convenient diagrams arising here remains open. For $\dot H=\tfrac{1-m}{2}H^2+\psi(u)$, $\dot u=-Hu$, with $\psi(0)=0$, $\psi'(0)\neq0$, $m\neq1$, the vacuum always has $\mu=3$, $a=0$, $b=-m$. Its Dumortier–Llibre–Artés discriminant is $b_1^2+8a_3=(m-2)^2$, so for every $m>1$ it has one hyperbolic and one elliptic sector, independently of $\psi$.

A classical line of research characterizes analytic continuation of power series through analytic interpolation of their coefficients. Under suitable hypotheses, the growth of the interpolating function reflects the geometry of the continuation domain, and conversely. We survey the development of these results from Leau and Le Roy through Lindelöf, Carlson, Dufresnoy$\unicode{x2013}$Pisot and Arakelian. We formulate the main results in a unified notation using compactifications of the complex plane. In particular, we give a comprehensive treatment of Carlson's second continuation theorem, which has received little attention, and prove a refinement based on the Laplace transform.

We prove that sets in a certain class do not admit Riesz bases of exponentials. In particular, this class contains disks and triangles in the plane.

A well-known hypersurface classification theorem states that any isoparametric hypersurface in $S^{13}$ with six principal curvatures is homogeneous. This landmark result was first proved in her Annals paper in 2013 by R. Miyaoka with 58 pages, and with errata in Annals in 2016 with 15 pages. The main purpose of this paper is to provide a concise proof of this theorem through an interplay between topological and geometric insights into the Euler class of an oriented vector bundle.

We obtain decay estimates at infinity for Yang-Mills connections and Yang-Mills-Higgs pairs on higher-dimensional manifolds. The main tool is an abstract decay theorem for manifolds with suitable positive harmonic functions. Combining this theorem with refined Kato inequalities, we prove that a natural quadratic decay assumption implies an improved decay rate on certain asymptotically conical and product manifolds.

This work addresses a longstanding gap in the statistical foundations of marginal maximum likelihood estimation for high-dimensional latent variable models. Marginal maximum likelihood estimation is widely used to fit latent variable models across the social sciences, ecology, and machine learning. Despite its broad use, rigorous asymptotic theory for nonlinear models remains limited when both the sample size and the number of observed variables diverge. The gap arises largely from the fact that integration over the latent variables creates a nonlinear objective that tightly couples the high-dimensional model parameters. To address this issue, we first show that the marginal likelihood exhibits multiple nearly flat directions even at the true parameter, in contrast to the behavior in the fixed-dimensional case. Building on this geometric characterization, we develop new techniques to establish consistency and asymptotic normality for the marginal estimator of the high-dimensional parameters. For the latent variables, we provide frequentist and Bayesian uncertainty quantification, proving asymptotic normality of the maximum a posteriori estimator and a Bernstein-von Mises-type result for the plug-in posterior. Together, these results provide rigorous foundations for marginal estimation and latent-variable inference in high-dimensional models.

We present a systematic study of $a$-Davis–Wielandt radius estimates for elements in a unital $C^*$-algebra through an Orlicz function approach. We derive novel bounds for the algebraic $a$-Davis–Wielandt radius and further obtain estimates for the classical algebraic Davis–Wielandt radius via polar decomposition and the Moore–Penrose inverse. The results obtained in this work extend, generalize, and unify a variety of established inequalities in the literature as special cases.