Papers

This paper proposes and analyzes a numerical method for coupled wave-heat systems that arise, for example, in visco-elasticity with memory, thermo-elastic wave propagation with finite thermal speed, and Maxwell's equations coupled to dispersive material laws. Motivated by the widespread use of explicit leapfrog schemes for wave problems and the parabolic time-step restriction for heat equations, we design a second-order scheme that combines leapfrog time integration for the wave part with a Crank–Nicolson step for the heat part. The coupling is arranged in a Strang-splitting–type fashion so that the overall scheme remains explicit in the coupling and is subject only to a CFL condition of hyperbolic type, rather than the more restrictive parabolic constraint. We introduce an abstract space discretization that covers a broad class of wave–heat systems and accommodates both conforming and stabilized non-conforming discretizations. Using a general error decomposition, we derive error estimates for the fully discrete scheme under the desired CFL condition. Numerical experiments for our model applications confirm the theoretical results.

The full shadow of a complex polynomial $P$ of degree $d\ge2$ is the closure of all zeros of $(P^n)^{(m)}$, $n\ge1$, $0\le m<dn$. We prove that it consists of a compact connected limiting core containing all roots and critical points of $P$, together with isolated transient zeros, each occurring at only finitely many powers. The core is the Hausdorff limit of the zero sets pooled over all derivative orders as $n\to\infty$ and the support of their limiting normalized root-counting measure. For $m/n\to\alpha\in(0,d)$, we establish weak convergence of normalized root-counting measures and Hausdorff convergence of zero sets to the limiting supports, both stable under coefficient perturbations. We exhibit quartics with infinitely many transients. A bounded operator constructed from polynomial differentiators realizes the full shadow, core, and transients as its spectrum, Fredholm essential spectrum, and discrete spectrum, respectively.

We study graph transformations and their effects on the eigenvectors of three fundamental graph matrices: the adjacency matrix $A$, the signless Laplacian $Q$, and the Laplacian $L$. We extend the Laplacian eigenvector edge and matching principles introduced by Merris to $Q$ and $A$. We also extend some of the known $L$ graph transformations to $Q$ and $A$ and prove that graphs admitting a common eigenvector $\mathbf{x}$ for $A, L$ and $Q$ are such that vertices corresponding to nonzero components of $\mathbf{x}$ have same degree. Following our previous work for $L$, we investigate bivalent graphs for $A$ and $Q$, that is graphs admitting an eigenvector with all components in $\{-1, 1\}$. In particular, for trees, we characterize $Q$-bivalence and establish that $A$-bivalence forces all vertex degrees to be odd.

We establish the free-boundary version of Schoen-Simon's regularity and compactness theory for stable minimal hypersurfaces in a complete $(n+1)$-dimensional Riemannian manifold-with-boundary, following the intrinsic PDE scheme introduced by the first-named author. The key step is to prove sheeting theorems in the two possible scenarios in which local weak flatness can occur near a point on $\partial M$: flatness with respect to $\partial M$, and flatness with respect to a reference hypersurface-with-boundary meeting $\partial M$ orthogonally. We obtain these theorems via $\varepsilon$-regularity results for geometrically adapted tilt functions that quantify flatness while keeping track of the ambient geometry. These tilt functions satisfy nonlinear elliptic PDEs on the hypersurface which are amenable to an intrinsic analysis via De Giorgi iteration, made possible by the weak Caccioppoli-type inequalities obtained from stability. Both results apply to immersed hypersurfaces with a singular set of locally finite $(n-2)$-dimensional Hausdorff measure. When specialised to embeddings, they yield compactness and regularity: a sequence of stable free-boundary embedded minimal hypersurfaces with uniformly bounded area and singular sets of locally finite $(n-2)$-measure admits a subsequence converging to a free-boundary minimal hypersurface that is smoothly embedded away from a singular set of Hausdorff dimension at most $n-7$. Convergence is smooth and graphical, possibly with multiplicity, away from this singular set. As an application, we extend to arbitrary dimension the free-boundary Almgren-Pitts existence theory developed by the second-named author with X. Zhou and prove that every compact Riemannian manifold-with-boundary of dimension $n+1$ contains a free-boundary minimal hypersurface smoothly embedded outside a singular set of Hausdorff dimension at most $n-7$.

This paper is concerned with the existence and local gradient regularity of solutions to a quasilinear elliptic Dirichlet problem with finite Radon measure and a degenerate matrix-valued weight. Under a small log-BMO condition on $\mathbb M$, we first prove the existence of a SOLA (solution obtained by a limit of approximations) and then establish a local Calderón-Zygmund estimate for $|\mathbb{M}|^{\frac{1}{p-1}}\mathbb{M} Du$ in terms of the fractional maximal function $\mathcal{M}_1(\mu)^{\frac{1}{p-1}}$.

In this paper, we establish the absence of absolutely continuous spectrum for scalar one-dimensional Laurent operators with multiplicative disorder and critical long-range hoppings. In particular, the requirement of at least quartic off-diagonal decay rate in the existing theory is bypassed by exploiting the finite-difference bound of the convolution kernel, which is available even when the interaction decays only as $|n|^{-1}\log^{-\beta}|n|$ with $\beta>1$. We then apply the abstract main result to a concrete physical model, the capacitance operator in subwavelength physics. For a chain of three-dimensional resonators with one resonator per period, we verify the required decay and finite-difference estimates and conclude the absence of absolutely continuous spectrum. These results leave open the distinction between pure point and singular continuous spectrum, but constitute a key step in proving Anderson localization in wave systems with critical long-range interactions.

The purpose of this paper is to study Faber–Krahn inequality of Dirichlet eigenvalues $\{ \lambda^{\ln,\mu}_{k}(\Omega)\}_{k\in \N}$ for the logarithmic Hardy operator \[ \mathcal{L}_{\mu}=(-\Delta)^{\ln}+2\mu\log\frac{1}{|x|} \] on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^{N}$ containing the origin, with $\mu>-1$. We establish a Faber–Krahn inequality for $\mu\in(-1,0)$, the ball centered at the origin minimizes the first eigenvalue at fixed volume, with equality only for that ball. Finally, we show this inequality fails for $\mu>0$ via an explicit two-ball-and-tube construction.

Let an Itô semimartingale be observed at times satisfying $\tau_{i+1}^n-\tau_i^n=(n\theta_{\tau_i^n})^{-1}$, where $\theta$ is adapted and càdlàg, and $\theta$ and its reciprocal are bounded on compact intervals. Under the usual hypothesis (H) on the semimartingale, we prove fixed-time stable convergence for test functions whose Hessian is $o(\norm{x})$ at zero. The limit is a sum of jump contributions with sampling scale $\theta_{T-}^{-1}$ and separate pre-jump and post-jump volatilities. The proof establishes joint stable convergence of the local Poisson positions and Brownian increments on the endogenous grid, and controls infinitely many small jumps by a uniform Itô estimate. Three examples identify the left-limit sampling scale and show why positivity and càdlàg regularity alone cannot replace local control of the reciprocal sampling intensity.

We prove the Gowers uniformity of the von Mangoldt function minus its Cramér model in all short intervals $[X,X+X^{3/5+\varepsilon}]$, improving on the work of the first and fourth authors with Shao and Tao, where the exponent was $5/8$. This also implies a local-to-global theorem for linear equations in primes in such short intervals. We also show that the Möbius function has cancellation in all intervals $[X,X+X^{19/35+\varepsilon}]$, improving on the work of the first and fourth authors, where the exponent was $11/20$. Both improvements are based on improved treatment of trilinear sums over short intervals which naturally reduces to estimating mean values of products of three Dirichlet polynomials. In a general setting, we reduce the task of estimating mean values of products of Dirichlet polynomials with given large value bounds to the task of showing that a certain piecewise linear function is positive. This reduction allows recent large value estimates to be incorporated. Combining this general framework with the most recent large value estimates stemming from the work of Guth and Maynard, we prove a Heath-Brown–Iwaniec type estimate for type I/II sums in intervals of length $X^{19/35+\varepsilon}$ and a Baker–Harman–Pintz parallelogram type estimate for general trilinear sums in intervals of length $X^{3/5+\varepsilon}$.

An $r$-dynamic coloring of a graph $G$ is a proper vertex coloring in which every vertex $v$ sees at least $\min\{r,d(v)\}$ distinct colors in its neighborhood. The minimum number of colors in such a coloring is the $r$-dynamic chromatic number $\chi_r^d(G)$. We study dynamic colorings of regular graphs. A straighforward observation shows that $\chi_r^d(G)=\chi(G^2)$, for any $r$-regular graph $G$. We prove that $\chi_3^d(G)=\chi(G^2)\le6$ for every claw-free cubic graph $G$, and the bound is sharp. For Hamiltonian claw-free cubic graphs, we improve the bound to $5$ apart from four explicit exceptions. We also determine exactly the $r$-dynamic chromatic number of the $4$-regular circulant graph $C_p(1,3)$ for each $r\in\{2,3,4\}$. In particular, the case $r=4$ determines $\chi(C^2_p(1,3))$.

We study the projection outer volume ratio ${\rm povr}_k(K)$ introduced by D. Galicer, A.E. Litvak, M. Merzbacher, and D. Pinasco. When $k\leq n/2$, we provide sharp bounds in the general case, namely \[ {\rm povr}_k(K)\leq C \max\Big\{1, \sqrt{\frac{k}{\ln(2n/k)}}\Big\}, \] where the maximal order is obtained for $K=B_1^n$. Moreover, in the important particular case $K=B_2^n$ we show that ${\rm povr}_k(B_2^n)$ is bounded by an absolute constant whenever $k\leq n/2$ and \[ {\rm povr}_k(B_2^n)\leq C\sqrt{\frac{n}{n-k}}\, \ln\left(1+\frac{n}{n-k}\right) \] for $k>n/2$. The latter bound is sharp up to a logarithmic term.

Let $A$ be the fixed point set of a semifree action of a compact (metric) group $G$ on a Keller space $X$ (an infinite-dimensional compact convex subset of a separable Fréchet space). All Keller spaces are homeomorphic and all nontrivial compact Lie groups act on all $X$'s semifreely with unique fixed points (see Section 3). We prove (1) if $A$ is a single point, then it is the fixed point set of a semifree action of $G$ with uniformly small orbits, (2) if $A$ has Property Z (e.g., is of infinite codimension) in $X$, then the action of $G$ may be replaced by one with fixed point set $A$ and uniformly small orbits, and (3) if $A$ is an \ANR, then $G$ acts on $X$ semifreely with fixed point set homeomorphic to any compact (metric) space that is Shape equivalent to $A$. Corollaries are (a) every Cell-like set (i.e., continuum of trivial Shape) embedded in a Keller space $X$ as a Z-set is the fixed point set of semifree actions on $X$ with uniformly small orbits of all nontrivial compact Lie groups (Corollary \ref{CE}) and (b) (a partial converse to a Theorem of P.\ A.\ Smith) if a compact (metric) space $Y$ has the Shape of a compact \ANR, then $Y$ embeds in $X$ as the fixed point set of a homeomorphism of $X$ of period $n$ if it is acyclic in \v Cech homology with $\mathbb{Z}_{n}$ coefficients. Finally, we (4) apply recent results of Cappell, Weinberger, and Yan \cite{cwy} to give for a compact manifold $M$ modeled on a Keller space homological and algebraic K-theoretic conditions equivalent to the existence of semifree actions of finite groups $G$ on $M$ with prescribed \ANR\ fixed point sets and (5) generalize (3) to prove that if $A$ is an \ANR\ that is the fixed point set of a semifree action of $G$ on $M$, then every compact (metric) space that is Shape equivalent to $A$ is homeomorphic to the fixed point set of a semifree action of $G$ on $M$.

We provide an alternative and independent proof of the analogue of the Gage-Hamilton-Grayson Theorem for embedded free boundary curve shortening flows in a convex planar domain established by Langford-Zhu. Our proof is based on blowup arguments introduced by B. White in his work on mean-convex mean curvature flow.

We study global and variational properties of $\mathbb R$-equivariant minimal Lagrangian surfaces in the complex quadric $Q_2$ and the complex hyperbolic quadric $Q_2^*$ arising from loop group potentials. In the $Q_2^*$ case, we work under the timelike-axis condition throughout the associated $\mathbb{S}^1$-family. Excluding the flat case in $Q_2$, we give a necessary and sufficient criterion for closing in the $y$-direction: a member $f^{\lambda_0}$ closes if and only if the spectral ratio $\mu_2/\mu_1$ is a positive rational number. The closing parameters form a dense countable subset of $\mathbb{S}^1$, while the non-closing parameters are dense and have full arc-length measure. The closing members yield cylindrical quotients in $Q_2$ and annular quotients in $Q_2^*$. Apart from the flat case in $Q_2$, the surfaces under consideration have infinite absolute total curvature. The equivariant surfaces considered here in $Q_2$ are unstable, whereas minimal Lagrangian surfaces in $Q_2^*$ are stable. Finally, for compact cylindrical domains of the $y$-closing surfaces in $Q_2$, we obtain explicit sufficient width conditions for Hamiltonian stability and instability under variations fixed near the boundary. We also establish the existence of a finite geometric critical width and determine it explicitly in the flat case.

The iteration of $z\mapsto z^2+c$ over the octonions has been drawn for more than thirty years, and it is known that the octonionic Julia sets are bodies of revolution about the imaginary part of the constant. Real eight-dimensional composition algebras, however, are much richer than the octonions: beside the Hurwitz algebras there are the symmetric composition algebras, the para-octonions and the Okubo algebras, which have no unit and whose automorphism groups are $G_2$ and $PSU(3)$ respectively. In this work we study, to our knowledge for the first time, the quadratic dynamics over these algebras, and we do it in an experimental spirit: an atlas of 134 four-dimensional renderings, measured quarter by quarter, suggests the statements, which we then prove and test against new, predicted pictures. We show that the symmetries of the Julia sets of the Okubo algebra are governed by the root decomposition of $\mathfrak{su}(3)$ with respect to the plane of $1$ and $i$: for a constant in that plane the continuous symmetry is a maximal torus, which acts by rotations on three root planes and grows to $U(2)$ on the Weyl walls; the automorphism group of the symmetrized product is $PSU(3)\rtimes\mathbb{Z}_2$. We then prove that the Mandelbrot set with seed $0$ of every symmetric composition algebra is the pull-back of a single planar set through the invariants $n(c)$ and $n(c,c\diamond c)$: the tricorn in the compact case, the tricorn and a hyperbolic tricorn in the split case. The result is quite surprising since the para-octonionic and the Okubonic Mandelbrot sets turn out to be the same planar figure saturated by two different groups. Finally, we show that every bilateral map over a symmetric composition algebra is an octonionic quadratic map in disguise, and that at the Petersson angle $\theta=90^\circ$ the dynamics acquires symmetries that do not come from automorphisms.

For any regular projective surface $X$ over an arbitrary field $k$ of characteristic $p>0$ with $K_X\equiv0$, we prove that \[ 60K_X\sim0 . \] This uniform bound is optimal. To show that the factor $5$ is necessary, we construct an example in characteristic $5$ for which $K_X$ has exact order five. If $X$ is geometrically normal, then $12K_X\sim0$. Our main focus is the geometrically non-normal case, where we obtain more precise bounds according to the Albanese morphism. If it is nontrivial, then $p=2$ or $3$, and we prove $12K_X\sim0$ and $6K_X\sim0$, respectively; both bounds are optimal. If it is trivial, then $K_X\sim0$ for $p\ge7$, while $5K_X\sim0$, $3K_X\sim0$, and $4K_X\sim0$ for $p=5,3,2$, respectively.

We formulate the shortest-path problem in a domain containing a crack as a Hamilton–Jacobi equation carrying a state constraint on the crack, which on the binding part of the crack takes the form of a variational inequality: the crack is a two-sided barrier that may be touched and followed but not traversed, so that it acts as a lower-dimensional state constraint, carried separately by each of its two lips, which admits tangential propagation but forbids crossing. We establish a comparison principle and uniqueness of the value function through trajectory-based arguments that circumvent the failure of the classical doubling technique of Crandall, Ishii, and Lions at the crack. The variational inequality induces a free boundary on the crack separating a sliding regime, where optimal trajectories run along the crack over a finite segment, from an illuminated regime, where the crack is reached transversally by direct rays; we give a complete geometric characterization of both regimes, including grazing (tangential) contact, the absence of reflection at regular crack points, and a diffraction law at crack tips, all derived from optimality alone. A simulation of the forward problem on an exactly solvable configuration reproduces the three regimes and the tip-diffraction law to the accuracy of the closed-form solution.

In this paper, we establish quantitative Darcy limits for a three-dimensional inhomogeneous incompressible fluid with temperature-dependent viscosity and quasi-static thermal feedback. The domain is either the flat torus or a bounded domain with $C^{3,\beta}$ boundary, $0<\beta<1$. The holes have size $\varepsilon^\alpha$ and separation of order $\varepsilon$, with $1<\alpha<3$. Before normalization, the fluid and solid conductivities are $\sigma_\varepsilon^2\kappa_f$ and $\sigma_\varepsilon^2\kappa_s$, where $\sigma_\varepsilon^2\sim\varepsilon^{3-\alpha}$. This scaling preserves the thermal feedback in the limit. For a sufficiently regular reference solution and under an explicit absorption condition, the squared density, velocity and temperature errors are bounded by the weighted initial discrepancy together with $\varepsilon^{\alpha-1}+\varepsilon^{3-\alpha}$ on the torus and $\varepsilon^{\alpha-1}+\varepsilon^{(3-\alpha)/2}$ in a bounded domain. An interface-independent temperature estimate controls the viscosity discrepancy, and a weighted Stokes residual retains the coefficient and cell-pressure commutators. The bounded comparison uses a solenoidal wall correction and cellwise divergence repair. We also construct fixed-parameter microscopic weak solutions. The local strong effective solutions require a separate contraction condition in both settings and stronger boundary regularity in the bounded case.

We obtain new results on Lane-Emden conjecture to prove a Liouville theorem for the Lane-Emden system $$\begin{cases} -\Delta u=v^p,& x\in \mathbb{R}^N,\\ -\Delta v=u^q,& x\in \mathbb{R}^N,\end{cases}$$ with $N\ge3$, $p,q>0$, $pq>1$, and \[ \left(p-\frac2{N-2}\right)^2+ \left(q-\frac2{N-2}\right)^2\le\frac{2N^2}{(N-2)^2}, \] the system has no positive classical solution as $(p,q)$ in the disk unless $p=q=(N+2)/(N-2)$, no growth or decay assumptions are imposed, and the disk is tangent to the critical hyperbola. Especially, for $N\ge5$, the disk contains a nonempty open set not covered by the criteria of Busca-Manásevich~\cite{BM} and Souplet~\cite{Souplet}, or by the explicit conditions of Li-Li-Wei~\cite{LLW}. After reducing the exponent range by known Liouville criteria, we first construct vector fields to derive a divergence inequality, then apply Sobolev embedding theorem, Gagliardo-Nirenberg interpolation inequality, and Young's inequality to give a uniform local integral bound $\int_{B_1}(u^rv^{2p}+u^{2q}v^s)dx\le C(N,p,q)$ for suitable weights for any positive solution $(u,v)$, finally the nonexistence of positive classical solutions follows by scaling.

We give an algebraic criterion for certain sequences $(G_n)_{n\in\mathbb{N}}$ of groups to have Kazhdan's property $(T)$ eventually. As a consequence, we prove that the Brin–Thompson groups $nV$ have Kazhdan's property $(T)$ for large $n$. The algebraic criterion provides a uniform proof that $nV$, $\mathrm{Aut}(F_n)$ and $\mathrm{EL}_n(R)$ for a finitely generated associative unital ring $R$ have property $(T)$ for large $n$. In an appendix, written by Francesco Fournier-Facio, it is shown that the group $nV$ for large $n$ is neither hyperlinear, in particular not sofic, nor $\mathrm{MF}$. Further, it is not Schatten $p$-approximable for any $1\le p<\infty$.

Let $F$ be an algebraically closed field of characteristic different from $2$, and let $V$ be a nonnegatively integer-graded vertex operator algebra over $F$. We consider higher Zhu algebras obtained by taking a quotient of $V$ by a relation space containing the vectors $D_V^{(h)}a-\binom{-\mathrm{wt}(a)}{h}a$ for all homogeneous $a\in V$ and integers $h\geq1$, where $D_V^{(h)}a=a_{-h-1}\mathbf1$. For $n\geq1$ and a nonzero unital module $U$ over the corresponding higher Zhu algebra, we study the quotients $L_n(U)$ of induced modules and determine the kernel of the canonical map $U\to\Omega_n(L_n(U))/\Omega_{n-1}(L_n(U))$. This kernel is the largest submodule of $U$ that factors through the adjacent lower-level algebra. When this submodule is zero, the canonical map is an isomorphism, and each $\Omega_r$ is the direct sum of the first $r+1$ homogeneous subspaces. These results are proved using an expansion of zero modes with integer coefficients and the grading of the standard induced modules. Moreover, the construction is natural with respect to module homomorphisms, and $L_n(U)$ is indecomposable in the graded module category whenever $U$ is indecomposable. From nonsplit extensions of modules for higher Zhu algebras satisfying the stated hypotheses, we construct nonsplit short exact sequences of weak $V$-modules.

First order Langevin algorithms for constrained sampling in machine learning, such as projected Langevin Monte Carlo which are based on discretizations of reflected Langevin dynamics, require differentiable log densities that limits their applicability. This paper introduces reflected anchored Langevin dynamics (RALD), a reflected diffusion that converges to non-differentiable targets on constrained domains. The method uses a smooth anchored reference potential and multiplies the drift and noise covariance of its reflected Langevin dynamics by the same state dependent scaling factor. Its Euler-Maruyama discretization with projection gives reflected anchored Langevin Monte Carlo (RALMC) algorithm. We prove explicit convergence bounds and iteration complexity for RALMC in the 2-Wasserstein distance to the target distribution. Numerical experiments are provided to illustrate the theoretical predictions and the empirical performance of the method.

We study a strongly damped stochastic wave equation on the three-dimensional torus driven by finite-rank pure-jump Lévy noise obtained from subordinated Brownian motion. Lyapunov estimates and an asymptotic compactness decomposition yield invariant measures, and a small-noise event gives weak irreducibility. To prove uniqueness, we combine conditional Malliavin calculus with terminal observability and estimates for the full variational flow starting from high modes. A control constructed on alternating stopping blocks yields the e-property under a fixed-rank balance condition involving high-mode decay, spectral leakage, and the random clock. Consequently, the invariant measure is unique and ergodic, and the Cesàro averages of the transition laws from every initial state converge weakly to it. We verify this condition for sufficiently small positive cubic coefficients at each fixed forcing operator of sufficiently large rank. Under a separate explicit smallness condition, synchronous coupling proves the e-property and weak convergence of the transition laws, without a lower bound on the forcing rank.

The paper contains the study of the weighted $L^{p_1}\times L^{p_2}\times \ldots \times L^{p_m}\to L^p$ estimates for the multilinear maximal operator, in the context of abstract probability spaces equipped with a tree-like structure. Using the Bellman function method, we identify the associated optimal constants for weights satisfying the multilinear Muckenhoupt condition.

Optimal transport (OT) has found broad applications in machine learning, economics, and management. Classical OT models, however, are not directly suited to allocation problems with graph-structured connectivity and unbalanced mass. Distributed OT formulations on bipartite graphs address these requirements but are typically solved by ADMM, whose slow convergence limits communication efficiency. We adapt the accelerated preconditioned ADMM (AP-ADMM) to the quadratically regularized distributed OT model and develop DAP-ADMM, a fully distributed algorithm in which each node communicates only with its graph neighbors. For fixed local penalty parameters, we establish global convergence and non-ergodic $O(1/k)$ bounds for both the KKT-residual norm and the primal objective gap. We derive an exact solver-free routine for the interval-constrained projections arising in the local ADMM subproblems. We propose a distributed self-adaptive penalty mechanism that updates node-specific penalties using locally computable KKT-residual components without global aggregation. Numerical experiments confirm the efficiency of the solver-free routine and the practical acceleration of DAP-ADMM over standard distributed ADMM.

Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly matched to PDE solutions with strongly heterogeneous complexity across space and space–time, leaving capacity insufficient where the physics is difficult and redundant where it is simple. We introduce physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics. We instantiate this principle with Representation Capacity Adaptation for PDEs (ReCAP), a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging. ReCAP uses responsibility-weighted error indicators and the geometry of residual energy to determine where and how to refine. To limit the disturbance introduced by splitting, we introduce quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation. We further establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Across five challenging 3D and 4D PDE benchmarks against 11 physics-informed solvers, ReCAP achieves the lowest relative $L^2$ error on every problem, reducing error by $10.7\%$–$27.5\%$ relative to the strongest competing result. These results suggest that physics-informed solvers need not merely learn their parameters—they can learn how their representational capacity should be organized.

The switch chain is a Markov chain defined on the set of labelled realizations of a given graphical degree sequence. At each step, a pair of vertex-disjoint edges is chosen at random and the process attempts to replace them with a uniformly chosen perfect matching of the same four vertices, rejecting any proposal that would create a multiple edge. The resulting process is reversible with respect to the uniform distribution on all realizations. We investigate the mixing time of this chain by viewing realizations as the facets of a simplicial complex and studying a variant of the original process called the simplicial switch chain, which we analyze using tools from the theory of high-dimensional expansion. Our technical contributions include a proof that links of faces of sufficiently high codimension are strong spectral expanders and a comparison between the Dirichlet energies of large block updates and two-edge updates. Our main result is an $O(\Delta^{2}m\log m)$ bound on the mixing time of both simplicial and classical switch chains whenever $m\ge C\Delta^{8}$, where $m$ is the number of edges, $\Delta$ is the maximum prescribed degree, and $C>0$ is an absolute constant. For sequences on $n$ vertices with fixed maximum degree, this proves that the chain mixes in $O(n\log n)$ steps, resolving a longstanding conjecture of Cooper, Dyer, and Greenhill and extending its conclusion to irregular degree sequences.

We provide a simple ansatz for finding integer spin $SO(3)\times\Z_2^T\times\Z_2^R$-invariant matrix product states representing a phase for each triple of $\Z_2$ topological indices studied in the literature by Tasaki \cite{tasakiheistop}\cite{tasakitop} and Ogata (corresponding to time-reversal and reflection symmetries) \cite{ogatatime}\cite{ogatareflection}. Every combination of trivial and non-trivial indices has a spin-$1$ $SU(2)$, time-reversal, and site-reflection symmetric MPS representative with injectivity length $l\leq 8$. We provide some exact examples for spin-$1$ and a Fortran program which will find representatives of any triple for any integer spin, as well as some data from random sampling for small spin. The existence of such states agrees with the classification of Chen, Gu, and Wen \cite{chen}, and with the recent results of Tasaki \cite{tasakinew} equating the hidden-order and one of Ogata's indices, wherein the Ogata index is that associated to the dihedral $D_2$ symmetry.

We establish uniform logarithmic Hessian estimates for solutions of non-uniformly elliptic equations in non-divergence form. The proof is based on a measure estimate from convex analysis without localization or covering arguments. Moreover, we construct a counterexample that rules out uniform power-type Hessian estimates and shows the sharpness of the logarithmic exponent in the plane. Finally, we extend our estimates to strong solutions and apply them to obtain regularity results for Grushin-type equations.

Function-correcting codes (FCCs) protect specified function evaluations of messages against errors while reducing the redundancy required for reliable communication. We study FCCs for symbol-pair read channels, motivated by high-density storage systems that read overlapping symbol pairs. Phase-shift keying (PSK) modulation is well-suited to such systems due to its bandwidth efficiency and noise robustness. While FCCs for symbol-pair read channels have been studied under the Hamming metric, the Lee metric is a more appropriate error model for $q$-ary PSK and, hence for $q$-ary symbol-pair read channels. We generalize the symbol-pair Lee distance, previously defined only over $\mathbb{Z}_4$, to $\mathbb{Z}_q$, $q\ge2$, and introduce function-correcting symbol-pair Lee-distance codes (FCSPLCs) over $\mathbb{Z}_q$, specializing to $q=2^m$, $m\ge1$, for $2^m$-ary PSK constellations. We investigate their redundancy requirements by introducing irregular-pair Lee-distance codes and relating the optimal redundancy of FCSPLCs to the shortest length of such codes. We derive Plotkin-type and Gilbert–Varshamov-type bounds on the optimal redundancy through lower and upper bounds on the shortest length of irregular-pair Lee-distance codes. For bijective functions, we obtain corresponding Plotkin-type and Gilbert–Varshamov-type bounds for classical Lee metric codes for symbol-pair read channels over $\mathbb{Z}_q$, which, to the best of our knowledge, are the first such bounds for the Lee metric symbol-pair setting. We then specialize the FCSPLC framework to pair-locally bounded functions, the Pair-Lee weight function, and the Pair-Lee weight distribution function, giving explicit constructions and corresponding bounds on the optimal redundancy. Finally, for linear functions, we derive a Plotkin-type lower bound on the optimal redundancy.