Papers

In explicit dynamics, the critical time step is governed by the maximum eigenfrequency of the semi-discrete system. In Isogeometric Analysis (IGA), open knot vectors introduce a characteristic spectral boundary effect that can lead to high-frequency outliers and restrict the admissible time step. In the row-sum-lumped setting considered here, extending the computational patch and trimming away the exterior boundary functions mitigates this time-step penalty, but does not necessarily prevent boundary-adjacent basis functions from governing the maximum eigenfrequency. We show that reduced-support basis functions adjacent to the trimming boundary can remain critical spectral contributors even for knot-exact trimming, i.e., in the absence of small cut cells. Hence, small cut cells are not required for trimming-induced time-step penalties. Knot-exact and arbitrary trimming represent different degrees of support reduction within the same underlying mechanism. To control this effect, we propose the Boundary-Level-Constrained Refinement (BLCR) strategy, a local refinement constraint for LR- and THB-splines. BLCR constrains the refinement level of basis functions whose support intersects the trimming boundary relative to that of the refined interior. In all configurations investigated in this work, this constraint ensures that the maximum eigenfrequency is governed by untrimmed interior basis functions rather than by refined trimmed functions. Consequently, the trimming-induced high-frequency outliers governing the critical time step are suppressed, yielding a larger admissible time step than globally refined trimmed B-splines with the same interior resolution.[...]

We construct a smooth complete metric on $\R^3$ with strictly positive Ricci curvature and $\displaystyle \lim_{s\rightarrow\infty}s^{-1}\int_{B_p(s)}\Sc\dd\mu= \infty$. This gives a negative answer to Shing-Tung Yau's question on the asymptotic scalar-curvature integral. We also give counterexamples to Naber's local Shing-Tung Yau conjecture on compact manifolds. The noncompact example has zero asymptotic volume ratio, and the compact counterexamples have volumes tending to zero.

In this paper, we prove several results concerning the function theory of complete Hermitian manifolds with vanishing Chern curvature, which we refer to as complete Chern-flat manifolds. First, we prove that any complete simply connected Chern-flat manifold is elliptic in the sense of Gromov. Moreover, it is biholomorphic to a complex Lie group if the torsion has polynomial growth. Next, we obtain sharp dimension estimates for holomorphic functions of polynomial growth on complete Chern-flat manifolds, with no growth assumption on the torsion. Finally, we prove that any complete Chern-flat manifold admitting polynomial-growth holomorphic functions that form a local holomorphic coordinate system at some point is biholomorphic to complex Euclidean space. These results generalize our previous work, in which sublinear growth of the torsion was assumed. The new analytic tool is a version of Cauchy estimates along orbits of holomorphic flows generated by parallel unitary frames on such manifolds.

We provide a categorification for the hybrid Kazhdan-Lusztig bases of the Hecke algebra by constructing and investigating a series of new stratified structures on the principal block of BGG category $\mathcal{O}$.

We develop and analyse a new arbitrary-order method for the Kirchhoff–Love plate problem on general polygonal meshes. The construction relies on a discrete Hessian complex obtained through the Bernstein–Gelfand–Gelfand construction, which stacks a Stokes complex on top of a tensorised de Rham complex. A key ingredient of the analysis is a detailed study of the former. We specifically establish primal and adjoint consistency estimates, as well as uniform Poincaré inequalities, and use these results to derive the analytical properties of the resulting discrete Hessian complex. The proposed method is then shown to be coercive and to converge with order $k+1$ with respect to the mesh size, where $k\ge 0$ is the polynomial degree of the complex. Numerical experiments on polygonal meshes confirm the theoretical convergence rates.

Let $p$ be a prime and let $A,B$ be nonempty subsets of the cyclic group $\mathbb{Z}_p$ with $|A|\neq |B|$. The Alon–Nathanson–Ruzsa theorem gives the lower bound $|A\rplus B|\ge \min\{p,\,|A|+|B|-2\},$ where $A\rplus B=\{a+b:a\in A,\ b\in B,\ a\neq b\}$ is the restricted sumset. The inverse problem of characterizing all critical pairs $(A,B)$ attaining equality was posed by Alon, Nathanson, and Ruzsa in 1996 and remains open. Recently, Liu and Qian solved the inverse problem under the assumption that at least one of the sets is an arithmetic progression, and proposed a conjecture for the general case. The main purpose of this paper is to show that this conjecture fails in the boundary case $|A|+|B|=p$. More precisely, we construct infinite families of non-arithmetic critical pairs $(A,B)$ with $|A|+|B|=p$ and $|A\rplus B|=p-2$ for every prime $p\ge 11$. These families show that the boundary case is fundamentally different from the non-boundary case. Motivated by this, we formulate and prove a refined inverse theorem under the natural hypothesis $|A|+|B|\le p-1$, which excludes the boundary.

This document describes two developments. The first is a Multi-stage Decision Framework that permits the formulation of various sequential decision-making problems in a generic manner. The second concerns a robust stochastic solution procedure designed to handle the binary nonlinear optimization problem stemming from the proposed decision framework. The method adopts Monte-Carlo search with incremental probabilistic learning with reinforcement. The two developments are described herein along with a demonstrative application example. Key words: Multi-stage, decisions, binary, nonlinear systems, probabilistic and reinforcement learning

We consider Fisher-KPP equations with nearest-neighbor diffusion on $\mathbb Z^d$, $d\geq2$, with nonzero finitely supported initial data. We prove a logarithmic delay of the front along each signed coordinate axis and show that the transition region has uniformly bounded width. On every fixed-width half-tube around an axis, the solution converges to translates of the minimal-speed lattice traveling wave, with a bounded phase. We also obtain an upper bound with a logarithmic correction in every direction. The proof uses weighted estimates, bounds for tilted random walks, and a product lower solution. Concavity of the reaction is not assumed.

We prove a duality theorem for lax equalizers of stable $\infty$-categories, and we apply it to Frobenius and Cartier modules on a Noetherian $F$-finite scheme $X$ of characteristic $p$ with a unit dualizing complex. When $X$ has affine diagonal, comparison with the derived categories of coherent modules and passage to homotopy categories recovers Baudin's derived Hom duality. The duality of coherent modules exchanges the standard $t$-structure on one side with the perverse $t$-structure determined by the dualizing complex on the other. In particular, it identifies the abelian category of coherent Cartier modules with a perverse heart of coherent Frobenius modules, and conversely.

We study the one-corotational harmonic map heat flow from the disk to the sphere with a Dirichlet boundary control. We introduce a control-adapted gluing strategy to prevent blow-up. More precisely, we construct explicit upper profiles and well-prepared boundary traces such that every nonnegative datum below one of these profiles and having boundary value above $\pi$ blows up under the corresponding constant trace, whereas the corresponding solution is global under every control below the associated well-prepared boundary trace. Controls with uniformly bounded time derivative also prevent infinite-time concentration. We also show that boundary control can drive any nonnegative datum to blow-up before any prescribed time, whereas bounded controls cannot prevent suitably fast concentration.

We extend Cuenin's compact-manifold spectral bounds for Schrödinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators \(P+V\), where \(P\) is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and \(V\) is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for \(P\) imply \(L^p\)-\(L^{p'}\) resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

We present computer-assisted proofs of partial hyperbolicity, existence of a blender and robust transitivity for diffeomorphisms on closed manifolds. These proofs are implemented for a family of derived-from-Anosov systems on the 3-torus.

We study the dynamics of self-consistent transfer operators associated with mean-field coupled systems whose unit dynamics is given by an interval map with finitely many singularities of cusp type. We show that, when the coupling strength is sufficiently weak, such an operator has a unique fixed point in a suitable cone of the associated Banach space, and that this fixed point attracts the orbit of every density in the cone at an exponential rate in the corresponding norm.

We study three naturally associated notions of regularity for a unital inclusion of von Neumann algebras $\mathcal{B} \subseteq \mathcal{M}$: regularity, groupoid regularity, and unitary regularity. We first observe that regularity and groupoid regularity are unconditionally equivalent by showing that the standard normaliser and the groupoid normaliser of $\mathcal{B}$ in $\mathcal M$ generate the same linear span. The central question addressed in this work is whether regularity implies unitary regularity. For arbitrary von Neumann subalgebras, we demonstrate that this implication fails dramatically; we construct explicit examples to show that every factor (of type $\mathrm{I}_\infty$, $\mathrm{II}_1$, $\mathrm{II}_\infty$, and $\mathrm{III}$) contains a regular von Neumann subalgebra that is not unitarily regular. In stark contrast, we prove that the behaviour changes completely in the setting of subfactors. For subfactor inclusions across all these types, regularity always implies unitary regularity, and thus all three notions of regularity coincide.

In this paper, we study composition operators that act between different de Branges-Rovnyak spaces. Our main results are suggested by a paper of Mashreghi and Shabankhah concerning composition operators between model spaces. We also answer several related open questions posed by Dellepiane and Seco. To prove these results, we apply reproducing kernel Hilbert space methods, Sarason's approach to composition operators as integral operators, and Aleksandrov-Clark measures.

We study universal derived equivalences between diagram categories indexed by finite posets. Starting from a construction of Ladkani, we give an intrinsic criterion for determining when a finite poset admits a decomposition to which this construction can be applied. This leads to the notion of an admissible cut, formulated entirely in terms of the order structure of the poset. Our main result therefore provides a method for producing, from a given finite poset admitting such a cut, a new poset that is universally derived equivalent to it. The construction is reversible once the partition is retained, so that the original mixed order relations can be recovered from the transformed poset. As applications, we show that every finite poset of height at most one is universally derived equivalent to its opposite, give a criterion characterizing source-to-sink transformations at minimal elements, and recover the universal derived equivalence of all orientations of a finite tree through sequences of such local transformations. These results are also applied to persistence modules indexed by finite posets and to extensions obtained by adjoining further finite parameters.

In this paper, we generalize the concept of packing total coloring by introducing a new concept called the $S$-packing total coloring. For a graph $G$ and a non-decreasing sequence $S=(a_1,a_2,\ldots)$ of positive integers, an $S$-packing total coloring of $G$ is a mapping $c: V(G)\cup E(G)\rightarrow \{1,2,\ldots\}$ such that for any two distinct elements $A,B\in V(G)\cup E(G)$ with $c(A)=c(B)=i$, the distance between $A$ and $B$ is at least $a_i+1$. The smallest integer $k$ such that $G$ admits an $S$-packing total coloring using $k$ colors is called the $S$-packing total chromatic number of $G$, denoted by $\chi_S^{''}(G)$. For any sequence $S$, we establish general lower and upper bounds for $\chi_S^{''}(G)$, and characterize all graphs $G$ with $\chi_S^{''}(G)\in\{1,2,3\}$. Furthermore, we investigate $S$-packing total chromatic numbers of complete bipartite graphs, as well as infinite and finite paths and cycles.

We study additive observables of a non-confined Feller process characterized by a non- normalizable stationary probability density and return times with infinite mean. The process is equivalent, after a deterministic rescaling, to a squared Bessel process, but we focus here on a question: the spatial structure of its local-time field. We show that the local time admits a fac- torization in the long time limit. Its spatial profile is governed by the non-normalizable stationary density, whereas its temporal fluctuations are controlled by a single Mittag-Leffler random ampli- tude. As a consequence, normalized spatial correlations of the local time converge to a universal constant independent of the two observation levels. Occupation times of finite intervals follow as corollaries and display Darling-Kac fluctuations. In contrast, non-integrable power observables exhibit self-similar squared-Bessel-type limits rather than Mittag-Leffler statistics. This spatial- field perspective provides a unifying framework for infinite ergodic theory under state-dependent multiplicative noise.

We study axisymmetric stability of rigidly rotating viscous stars modeled by the free-boundary Navier–Stokes–Poisson (NSP) system. The unstable index of the linearized NSP generator, counted with Riesz algebraic multiplicity, equals the Morse index of the augmented energy at fixed mass and total angular momentum. We prove an abstract Kelvin–Tait–Chetaev theorem for damped gyroscopic equations with finite negative stiffness index, requiring no compactness assumptions and no spectral gap at zero; the stiffness kernel may be infinite-dimensional. In the NSP application, the viscous dissipation is degenerate: its axisymmetric kernel, generated by rigid axial translation and rigid rotation, is projected out before applying the abstract theorem, while conservation laws and a Routh reduction transfer the resulting instability index back to the full NSP generator. For slowly rotating branches with fixed total angular momentum, viscous instability begins at the continuation of a first nondegenerate spherical mass maximum. By contrast, for every sufficiently small nonzero angular momentum, the corresponding Euler–Poisson star remains axisymmetrically spectrally stable on an interval beyond this maximum. Thus dissipation can destabilize a rotating star before the corresponding inviscid star becomes unstable. This separation of the stability thresholds results from viscous redistribution of angular momentum, which removes the inviscid constraints on its material distribution while preserving its total value.

We consider the observability and controllability for the Schrödinger equation on polyhedra. We show that for the Schrödinger equation on convex polyhedra in $\mathbb{R}^d\,(d\geq2)$, when the initial data possess $H^s\,(s>d/2)$ regularity, observability and controllability hold, with the control region being an arbitrary nonempty open neighborhood of the singular set of the boundary, which is the same as the control region for eigenfunctions. Our proof of observability is a concrete adaptation of the method in the Burq–Zworski black box control work, where the observability inequality is proved via a resolvent estimate. Hence, before proving observability, we present an observability resolvent estimate on polyhedra. The idea of the proof of this resolvent estimate comes from the case of eigenfunction concentration on polyhedra considered by Cekić–Georgiev–Mukherjee, which uses semiclassical measures and relies closely on the dynamical properties of the billiard flow in polyhedra. Finally, by a standard HUM argument, we show that observability implies controllability.

We establish a Loomis-Whitney inequality for the Reiter-Heisenberg groups $\mathbb{G}_{qp}$, a family of step-two Carnot groups that includes the Heisenberg groups when $q=1$. The proof is based on the duality between Brascamp-Lieb inequalities and entropy subadditivity: we first derive the result for $\mathbb{G}_{q1}$ from the known inequality on the first Heisenberg group, using conditional entropy and the invariance of differential entropy under volume-preserving diffeomorphisms; then we pass from $\mathbb{G}_{q1}$ to $\mathbb{G}_{qp}$ via a stability principle for Loomis-Whitney inequalities under finite central sums, which generalizes the argument in (Zhang, 2024 arXiv:2402.02749v2). As consequences, we obtain the associated geometric projection inequality, a Gagliardo-Nirenberg-Sobolev inequality, and an isoperimetric inequality.

We use the full multiplication formula to study syzygy and suspension of cluster characters. In stably 2-Calabi–Yau Frobenius categories, syzygy and cosyzygy induce inverse automorphisms of localized character algebras; in Hom-finite 2-Calabi–Yau triangulated categories, the same uniqueness principle transports all cluster characters under suspension. The latter gives our Auslander–Reiten $F$-polynomial identity. No mutation reachability is required. We extend positroid twist identities to all objects, recover the Grassmannian and unipotent-cell formulas, identify partition-function algebras before boundary localization, and derive tropical covariance when the relevant tropical coordinate map is a homeomorphism.

A shell finite-element model of a stiffened aluminium wingbox panel, a laboratory structure shared by several experimental studies, is built with its plate-to-stiffener joints at the $78$ physical fastener positions and calibrated against measured modal data, a six-parameter sensitivity update of the substructure moduli within $\pm20\%$ reproducing the first five measured modes to $1.1$–$4.3\%$ with modal-assurance values of $0.88$–$1.00$. Making the fasteners nonlinear, at equal mesh, mass and damping, shows that joint nonlinearity enters the response strongly asymmetrically. Hardening is almost invisible, at most $+1\%$ in frequency, whereas softening or slipping the joints moves the first three resonances by $-2.1$, $-4.9$ and $-9.2\%$, and friction removes up to $80\%$ of the resonant peak before it recovers. Newmark integration of all $18{,}804$ degrees of freedom and a harmonic balance condensed exactly onto the joints and continued in arclength agree to $5.5\times10^{-4}$. A projection-based nonlinear model order reduction then locates the criterion that a jointed structure imposes on a reduced basis. The binding quantity is not the linear response, which eleven vectors reproduce to better than $0.01\%$, but the receptance of the structure between the joints, of which $126$ global eigenvectors carry about $2\%$, and without which the reduced model overpredicts the hardening shift of the fundamental by a factor of thirty and returns a value beyond the rigid-joint limit of the panel.

In a tracial von Neumann algebra M, two subsets X and Y are said to be N-independent if alternating products of length at most N of trace zero words from X and Y have trace zero. We show that two approximately 2N-independent finite tuples can be perturbed, in operator norm, to be exactly N-independent. Consequently, we are able to generalize Theorem 5.1 and Theorem H of Houdayer-Ioana and resolve Problem 31 of Kunnawalkam Elayavalli in the positive.

In this study, we use a constrained Schrödinger–induction system to investigate kinetic-energy fluctuations induced by a fluid singularity. Assuming the cited Navier–Stokes construction with zero initial velocity and its exact exterior profile, we construct a periodic trajectory from a constant normalized wave function and zero connection. The prescribed body force admits a smooth space–time extension across the normalized singular time $t=1$. In this construction, the probability density remains constant and the mean matter kinetic energy stays bounded, whereas the gauge-invariant kinetic-energy variance diverges. We prove a lower growth bound of order $(1-t)^{-1/2-5h}$, with $h$ fixed by the source construction, by integrating the fourth power of the complete velocity's magnitude over an exact exterior region where all oscillatory corrections vanish. After fixing a gauge with constant wave function, we identify a unique weak $L^2$ connection limit outside $L^4$ and construct its positive self-adjoint kinetic operator through the associated magnetic form. We further show that the normalized endpoint state belongs to the form domain but not the operator domain: its first kinetic spectral moment is finite, whereas its second is infinite. These results connect fluid concentration to a loss of kinetic-domain regularity and provide a dynamical benchmark for assessing the limits of mean-energy control in hydrodynamic wave formulations.

Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In particular, this resolves a question of E.M. Stein from the mid 1990s. The principal ingredient in our proof is that, when $t\lesssim d$ with $t$ sufficiently large, the associated multiplier $\mathfrak{m}_{\sqrt{\lfloor t^2\rfloor}}(\xi)$ admits an asymptotic expansion of arbitrary prescribed order, uniform in $\xi$, whose resulting maximal operators can be controlled by the discrete normalized Gaussian maximal function studied by Mirek–Szarek–Wróbel \cite{MSW25}.

Applying a matrix function to a vector is a common operation in large-scale scientific computing, such as Bayesian inference. When the matrix is symmetric positive definite and the function is its inverse, the Lanczos algorithm gives a matrix-free approximation from matrix-vector products alone. Stopping after a fixed number of steps keeps the approximation on the explored Krylov subspace but assigns nothing to its orthogonal complement, which holds the smallest eigenvalues and therefore the largest contributions to the inverse. This truncation is usually treated as a silent numerical error rather than as measurable uncertainty. We show that a single Lanczos run yields both a matrix-free approximation of the inverse and a Gaussian distribution over the exact result, whose covariance measures what the truncation discarded and concentrates as the number of steps grows. The covariance restores the complement in two parts: a rank-one coupling at the truncation boundary, given exactly by the Lanczos residual, and an isotropic term on the remaining bulk, whose size we estimate by Projected Stochastic Lanczos Quadrature, reusing one Krylov basis across all probes. We prove consistency, with bias decaying exponentially in the quadrature depth and variance at the Monte Carlo rate

Two sharp yes/no questions are mutually unbiased if and only if their involutions anticommute. A set of unbiased questions is therefore a Clifford algebra, whose maximal size is odd, meaning four anticommuting questions typically imply a fifth: the two-level systems of real, complex and quaternionic quantum theory allow two, three and five questions, while four is skipped. Without this operator product restriction, a two-level system with four unbiased questions exists as the four-dimensional Bloch ball, the hyperbit. The single-system postulates that pick out the Jordan state spaces allow such a model, and only energy observability rules it out. We show what it lacks, and realise it inside two qubits: its Kirkwood-Dirac imaginary parts are hidden $\mathfrak{so}(4)$ generators, its composites allow entanglement but no interaction, and its 24-cell of states stays Spekkens preparation contextual down to depolarising strength $2/3$. In fermionic terms, the four questions are the Majorana operators of two modes while the fifth is parity: the four-question system is therefore the sector removed by parity superselection.

In this paper we present a method for constructing directed regular graphs from a transitive permutation group. This method is a generalization of a construction method for transitive 1-designs from a finite group, described in \cite{dean1}. Using this construction, we prove the existence of directed strongly regular graphs with parameters $(72,25,15,6,10)$, $(96,11,4,3,1)$, $(96,22,16,8,4)$, $(96,26,11,7,7)$, $(96,29,23,10,8)$, $(96,42,32,16,20)$, $(96,45,35,22,20)$ and $(165,60,36,23,21)$. Finally, we classify quasi-strongly regular digraphs arising from transitive permutation groups of degree at most $30$ and from primitive permutation groups of degrees from $31$ to $110$ and rank at most $30$.

We study Haar-Bayesian prediction of one unmeasured copy of an unknown finite-dimensional pure quantum state after an arbitrary collective measurement on $n$ observed copies. Performance is evaluated by quantum relative entropy. For a fixed measurement, the Bayes predictive state is the posterior mean and the optimized conditional loss is its entropy. We then optimize the measurement over all POVMs on the symmetric subspace. For every nonzero positive effect $E$, the corresponding posterior predictive state is $\mu_E=(I+n\rho_E)/(n+d)$, where $\rho_E$ is the normalized one-particle marginal of $E$. Since a pure spectrum majorizes every density-operator spectrum, this identity gives an outcome-wise entropy lower bound. Coherent rank-one effects attain the bound, and their Haar orbit yields the highest-weight covariant POVM. Hence this POVM is globally Bayes optimal over all collective measurements and, by covariance, globally minimax. Its exact risk is $h_d((n+1)/(n+d))$, where $h_d(r)=-r\log r-(1-r)\log((1-r)/(d-1))$. The same arbitrary-effect reduction shows that the highest-weight POVM also maximizes the joint overlap between the latent pure state and its posterior predictive state, equivalently the mean posterior purity, with optimum $((n+1)^2+d-1)/(n+d)^2$.