Papers

We investigate the compressible Navier-Stokes equations with heat conduction subject to inflow/outflow boundary conditions, establishing the existence of a strong solution. The primary contribution of this work lies in the derivation of a new regularity criterion for three-dimensional compressible flows, which serves as an analogue to the Beale-Kato-Majda criterion for incompressible fluids. Notably, our findings refine both the strong solution results established by Valli and Zajaczkowski and the conditional regularity framework proposed by Basaric, Feireisl, and Mizerova. Our analysis extends the framework of Basaric, Feireisl, and Mizerova, allowing us to successfully derive the required uniform estimates.

Lee and O'Clarey recently conjectured that every integer $n>24$ can be written as the sum of a prime and a positive integer that is not square-free. They proved this for odd $n$, and for all $n>24$ assuming the generalised Riemann hypothesis for Dirichlet $L$-functions. We prove their conjecture unconditionally for every integer $n>24$ that is not divisible by $997\#$, where $997\#=\prod_{p\leq 997}p$. In particular, any counterexample must be divisible by every prime up to $997$, and hence exceeds $10^{415}$. The proof uses an elementary congruence argument together with finite computation.

Let $X$ be a compact Kähler manifold and $(\alpha,\beta)$ be a pair of Kähler classes. We show that when $(\alpha,\beta)$ is $J$-semistable, the collection of positive-dimensional irreducible $J$-null subvarieties is finite. We also obtain a uniform positive lower bound for the normalized $J$-slope of every irreducible non-$J$-null subvariety. This confirms the finiteness and uniform-gap expectation of Khalid-Sjöström Dyrefelt \cite[Section 1.2, Theorem 1.5; Remark 1.8]{KSD26}. As a corollary, we show that the collection of optimal destabilizers is finite when the stability threshold is nonpositive.

An important problem in quantum technologies is the generation of a target evolution of an open quantum system. A core component of this task is the ability to determine whether the actual evolution of the system coincides with the target evolution. For the generation of unitary quantum channels in open quantum systems, it was shown by Goerz, Reich and Koch [New J. Phys. 16 055012 (2014)] that for determining whether the actual evolution coincides with a desired unitary it is sufficient to compare their action on three special density matrices. In this work, we consider controlled generation and unique determination of non-unitary quantum channels in open quantum systems with particular emphasis on rank two single-qubit quantum channels. We prove that to uniquely determine such quantum channels it is sufficient to consider their action on only three suitably chosen density matrices. Based on this theoretical result, we numerically investigate generation of various non-unitary target single-qubit quantum channels in open quantum systems using coherent and incoherent controls.

A subgroup $H$ of Thompson's group $F$ is closed if every piecewise-$H$ function in $F$ belongs to $H$. Prominent examples of closed subgroups are the maximal subgroups of infinite index, stabilizers and pointwise stabilizers of sets of points, and many of Jones' subgroups. A closed subgroup is finitely generated if and only if its Stallings $2$-core is finite, and it is then isomorphic to a diagram group over the core, in the sense of Guba and Sapir. We study the finiteness properties of closed finitely generated subgroups of $F$ with finitely many orbits on the dyadic rationals. For such a subgroup $H$ the following are equivalent: $H$ is of type $\mathrm{FP}_2$; $H$ is finitely presented; $H$ is of type $F_\infty$; $H$ has finitely many orbits on pairs of dyadic rationals; the action of $H$ on the dyadic rationals is oligomorphic. If moreover the action of $H$ on $(0,1)$ is minimal, these conditions hold if and only if the image of $H$ in the abelianization $\mathbb Z^2$ of $F$ has rank two. All these conditions can be decided from the core of $H$, and when they hold a finite presentation of $H$ can be computed. As a consequence, every finitely generated maximal subgroup of $F$ is of type $F_\infty$.

We prove that the SU-rank of a supersimple theory is no greater than that of any of its supersimple expansions. A proof for finite rank theories already exists in the literature, but non-continuity presents a problem at limit ranks. We prove the non-decreasing of SU-rank in expansions type-by-type by reducing to finding, in the base theory, certain trees of forking extensions in which the formulas and dividing numbers which witness dividing have a sufficiently uniform pattern. In the case of countable ranks, we can give an especially nice description of this uniformity: for a countable ordinal $\alpha$, any type of SU-rank at least $\alpha$ in a supersimple theory has a chain of forking extensions in the reverse order type of $\alpha$.

We prove that the blow-up of the Hirzebruch surface $\mathbb F_n$ at $n+6$ general points admits a phantom subcategory for every $n\geqslant 3$.

In this paper, we prove the $C^{1,1}$ regularity of Green functions associated with the complex $k$-Hessian operator for $1\leq k<n$, and give a new proof of the corresponding regularity of pluricomplex Green functions. We establish real Hessian estimates for approximating problems associated with homogeneous complex Hessian equations on punctured domains. The main idea is to introduce new auxiliary functions generated by complex linear vector fields to reduce the global real Hessian estimate to the boundary. We also treat the complex Monge-Ampère case. Similar real Hessian estimates also works for the exterior problem.

In this paper, we investigate the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity formulated on a three dimensional Riemannian manifold. We derive the gap equation in the path integral picture by employing a Hubbard-Stratonovich transformation followed by a saddle point approximation. Under the assumption that the length scale set by the metric is much larger than the healing length, we solve the gap equation to find a slowly varying gap function that exhibits the effects of curvature. To rigorously address the ultraviolet divergences inherent in the gap equation, we utilize heat kernel expansion techniques, which provide a systematic and mathematically transparent renormalization scheme. We explicitly evaluate the renormalized gap equation at both zero and finite temperatures, computing the relevant integrals asymptotically in the weak-coupling limit ($\Delta \ll \mu$). Furthermore, we establish a universal, regularization-independent relation connecting the finite-temperature gap to the zero-temperature gap and the temperature itself. Based on this fundamental relation, we analytically prove the monotonic decrease of the energy gap with increasing temperature ($\partial \Delta / \partial T < 0$), successfully recovering the standard universal BCS behavior near the critical temperature within a curved space framework.

We prove uniqueness of solutions to the Navier-Stokes equations in the space \[ L^p\bigl([0,T],\widetilde{L}^{3,\infty}(\mathbb{R}^3)\bigr) \cap C\bigl([0,T],B^{-1,\infty}_{\infty}(\mathbb{R}^3)\bigr), \qquad p>2, \] where $\widetilde{L}^{3,\infty}(\mathbb{R}^3)$ denotes the closure of the Schwartz space $\mathcal{S}(\mathbb{R}^3)$ in the Lorentz space $L^{3,\infty}(\mathbb{R}^3)$. Moreover, we prove that any solution $u$ in this class, if it exists, is smooth for positive time, namely, $u\in C^\infty((0,T]\times \mathbb{R}^3)$. In addition, we establish precise estimates for the behavior of $u(t)$ as $t\to0^+ $ in various Banach spaces.

We develop a certified finite-horizon quantum optimal-control framework based on fixed-rank Hierarchical Tucker (HT) surrogates. Under a uniform rank-dependent HT truncation-accuracy condition, we establish exponentially decaying trajectory, cost, and value-function errors and near-optimality of surrogate-generated controls. We further derive a logarithmic rank–performance relation, an a posteriori cost certificate from realized truncation perturbations, and guarantees for inexact surrogate optimization and control convergence under additional regularity conditions. Numerical experiments on controlled XXZ spin chains provide finite-sample evidence consistent with the truncation condition and illustrate the trade-off between HT rank, representation size, surrogate accuracy, and control performance.

We prove a dimension-free comparison between the expectation threshold $q(\mathcal F)$ and the fractional expectation threshold $q_f(\mathcal F)$ for any nontrivial increasing family $\mathcal F$ on a finite ground set. Specifically, we show that there is a universal constant $K>0$ such that \[ q_f(\mathcal F)\le Kq(\mathcal F)\max\{1,\log\log(1/q(\mathcal F))\}. \] Combining this comparison with a recent result of Li [arXiv:2609.08967] on the fractional version of Talagrand's discrete Convexity Conjecture, we obtain a dimension-independent bound toward the conjecture.

A communication system may convey urgent information through a fast physical stream and more specific information through a slower material stream. In several biological and engineered settings, however, the fast process also changes the transport law of the slow one. We study this architecture under a shared resource constraint, with a strictly increasing concave fast-channel capacity–cost function and a deadline-constrained slow molecular channel. We first characterize the capacity region under separated message routing and message-independent operating-point schedules, and identify the marginal criterion for complementarity rather than competition between the streams. For one-dimensional drift diffusion, we prove that arrival probability before a deadline is strictly log-concave in Péclet number. For a distinguishable-token deadline-erasure channel, any increasing concave transport-actuation law then yields an exact single-crossing theorem: complementarity exists if and only if an initial transport-assistance elasticity exceeds one, the transition is unique when it exists, and the decreasing allocation branch remains the Pareto boundary after convexification. For positive baseline drift and sufficiently strong coupling, a unique critical normalized deadline determines when complementarity disappears. Short- and long-deadline limits clarify the associated temporal regimes. Numerical examples for a finite-frame LTI-Poisson slow channel exhibit analogous allocation behavior with counting noise and intersymbol interference.

We study a free boundary problem for a diffusive Lotka–Volterra competition system describing the invasion of a new species into the habitat of a native competitor, where habitat suitability shifts from unfavourable to favourable at a constant speed $c>0$ due to climate change. Only the invader is affected by the shifting environment and only its range is governed by a Stefan-type free boundary, while the native species occupies the whole half line. We work throughout in the weak competition regime, in which the two species may coexist. We prove a spreading–vanishing dichotomy: either the invader spreads and the pair converges to the coexistence steady state $(u^*,v^*)$, or the invader vanishes and the native species recovers its carrying capacity. In the vanishing case we obtain the explicit bound $\lim_{t\to\infty}h(t)\le\frac{\pi}{2}\sqrt{d_1c_2/(a_1c_2-a_2c_1)}$, and we give criteria guaranteeing each alternative. When spreading occurs, we determine the exact asymptotic spreading speed: $\lim_{t\to\infty}h(t)/t=\min\{c,c_0\}$, where $c_0$ is the spreading speed of the corresponding homogeneous weak competition system. In particular the invasion is slowed down both by the competitor and by the climate shift, and the slower of the two mechanisms is the one that determines the speed. Numerical simulations illustrate the results.

A Grothendieck topology on a category determines both a subform of covering sieves and a quotient form obtained by identifying locally equivalent sieves. We place these two constructions in a single short exact sequence. For this purpose we introduce distributivity forms: indexed meet-semilattices equipped with distinguished indexed joins over which finite indexed meets distribute, and a strong version in which these joins also satisfy Beck–Chevalley. Over a fixed base, the resulting categories have all small limits and have kernels and cokernels relative to the closed ideal of fibrewise constant-top morphisms. Their monomorphisms are the fibrewise injective morphisms, and their relative cokernels, together with the top-reflecting morphisms, form an orthogonal factorization system. Cokernels compose but need not be stable under pullback, whereas kernels need not compose. We call a short exact sequence with prescribed middle term an extensional presentation, and prove that Grothendieck topologies on a category are precisely the extensional presentations of its maximally distributive form of sieves. Further applications recover universal productive closure operators and Lawvere–Tierney topologies, functorial non-Archimedean group topologies, and functorial linear topologies on commutative rings.

We study two-point fractional boundary value problems with uncertain input data, where randomness may enter through the coefficients and boundary conditions. To quantify the resulting uncertainty in the solution, we employ the generalized polynomial chaos (gPC) framework and develop a stochastic Galerkin formulation of the problem. A particular focus of this work is the convergence analysis of the resulting approximation. Rather than imposing assumptions directly on the stochastic coefficients appearing in the gPC representation, we introduce minimal regularity assumptions on the input data and use them to establish the properties required for the convergence analysis. Based on these results, we prove the convergence of the stochastic Galerkin approximation to the corresponding gPC solution. Numerical experiments are presented to illustrate the theoretical findings and to investigate the influence of random coefficients and boundary conditions on the statistical behavior of the solution.

Finite-dimensional zigzag modules decompose into interval modules. For an ordinary persistence module, every fully supported interval submodule of a restriction splits as a direct summand, whereas this need not hold for a zigzag module. We study this failure of splitting. A breaker pair is a local configuration of two overlapping interval summands with a backward arrow at the left boundary of the second interval and a forward arrow at the right boundary of the first. A single breaker pair gives a local nonsplitting interval submodule, but it need not control a prescribed larger restriction. We therefore introduce breaker chains. Our main result shows that a restriction $V[p,q]$ contains a nonsplitting submodule isomorphic to $I[p,q]$ if and only if its interval decomposition contains a breaker chain spanning $[p,q]$. Equivalently, the proper interval summands $J$ for which $\operatorname{Hom}(I[p,q],J)\ne0$ have supports covering $[p,q]$. This cover formulation gives a constructive description of nonsplitting embeddings, a trichotomy for their splitting behavior, and an $O(m\log m)$ algorithm from an oriented barcode with $m$ intervals.

In this article, we show that every norm-increasing $3$-concave weighted shift on a directed tree admits Wold-type decomposition. We then classify the $4$-isometries within a class of weighted shifts on a rootless directed tree introduced by S. Chavan and S. Trivedi, and use this classification to construct analytic norm-increasing strict $4$-isometries without the wandering subspace property. This answers a question of S. Shimorin in the negative for every $m\in \mathbb{Z}_{\geqslant 4}$.

We consider the Nernst–Planck–Darcy system describing the electrodiffusion of $N$ ionic species with arbitrary valences and different diffusivities, transported by a Darcy flow driven by the electric force, in the whole space $\mathbb{R}^d$, $d \ge 3$. We prove the existence of global solutions for initial ionic concentrations that are small in the critical Lebesgue space $L^{d/2}(\mathbb{R}^d)$. Moreover, we establish uniqueness for initial data in $L^{d+\epsilon}(\mathbb{R}^d)$ when $3 \le d \le 4$ and in $L^{2d-1}(\mathbb{R}^d)$ when $d \ge 5$, improving on results in the literature, which require Sobolev regularity of the data. The proofs rely on energy estimates exploiting cancellations of the highest-order nonlinear terms, and on the resulting dissipative structures, which yield an instantaneous gain of regularity used to control the doubly nonlinear velocity term in the uniqueness argument.

We prove that topological conjugacy and flip conjugacy of minimal homeomorphisms of a Cantor space are complete analytic relations and hence are not Borel. We also prove that mutual reducibility by injective continuous graph homomorphisms and topological graph isomorphism are complete analytic relations on the space of nonempty compact graphs on a fixed Cantor vertex space with continuous chromatic number two. Both relations remain complete analytic on the larger space of nonempty compact graphs on the same Cantor vertex space with continuous chromatic number two or three.

An oriented graph $H$ is said to be tournament anti-Sidorenko (TAS) if a uniformly random tournament asymptotically maximizes the homomorphism density of $H$ among all tournaments. For an oriented path $P$, a direction flip is a non-leaf source or sink. Sah, Sawhney and Zhao proved that consistently directed paths (paths with no direction flips) are TAS. He, Mani, Nie, Tung and Wei proved that for $3\le k\le 7$, every oriented path of length $k$ with exactly one direction flip is TAS, and Chen, Clemen and Noel recently extended this to every $k\ge 3$. In this paper, we extend these results by proving that for every integer $r\ge 0$, every oriented path of length at least $1665r+1454$ with $r$ direction flips is tournament anti-Sidorenko.

Motivated by the role of the Wigner caustic in semiclassical phase-space analysis, we extend it, together with the centre symmetry set, from regular planar curves to cooriented frontals. For an angularly regular parallel pair, the signed speed of the Wigner caustic is one half of the difference of the extended signed radii of curvature, while singular points of the finite centre symmetry set are the critical points of their projective ratio. These formulas yield criteria and explicit invariants for ordinary and higher cusps, an order-lowering relation between the two constructions, and a multiplicity-weighted extension of the classical cusp-count inequality for strictly convex ovals. We also analyse parallel pairs containing a singular frontal which is not a front. A $5/2$-cusp produces either a $5/2$- or, at a signed-radius resonance, a $5/3$-cusp on the Wigner caustic. The opposite resonance sends the centre symmetry set to infinity. We obtain the corresponding transfer results for a $5/3$-cusp and give a projective completion which resolves vanishing denominators and simultaneous finite-order zeros of the two signed radii.

The Sinkhorn Riemannian metric characterizes the local behavior of the Sinkhorn divergence, a debiased version of entropy-regularized optimal transport. Although the Sinkhorn divergence is known to approximate the squared Wasserstein-2 distance as the regularization parameter $\varepsilon$ goes to $0$, the behavior of the Sinkhorn Riemannian metric as $\varepsilon\to 0$ remains largely open. To the best of our knowledge, the only existing argument is for measures with a density and is still a formal computation. In this work, we prove the convergence of the Sinkhorn Riemannian metric as $\varepsilon\to 0$ in the case of finitely supported measures undergoing horizontal perturbations. Under our assumptions, our techniques additionally allow us to prove the convergence of higher order derivatives of the Sinkhorn Riemannian metric, thereby ensuring the convergence of the associated Riemann curvature tensor and Christoffel symbols as $\varepsilon\to 0$. We also compare with the absolutely continuous setting and show where our approach fails for measures with a density.

The infinitesimal Stoker conjecture states that an infinitesimal deformation of a convex polytope that preserves all dihedral angles also preserves all normal directions up to isometry. We give a short polyhedral proof, self-contained up to assuming spectral and geometric properties of the Izmestiev matrix.

We study almost Lipschitz embeddings in the quantitative geometry of unit spheres of classical \(L_p\)-spaces. Our main result is a sphere version of the classical Banach–Kadec–Paley theorem: for \(1\le p,q<\infty\), the unit sphere \(S_{L_q}\) admits an almost Lipschitz embedding into \(L_p\) if and only if \(L_q\) admits a linear embedding into \(L_p\). Consequently, the almost Lipschitz geometry of \(L_p\)-spheres completely determines the underlying linear structure of \(L_p\)-spaces. The proof relies on quantitative estimates for moduli of continuity of sphere mappings. These estimates are derived from Kalton–Randrianarivony's concentration inequalities, Mendel–Naor's metric cotype inequality, and Naor's sharp metric \(X_p\) inequality. They also yield several sharp results on the Hölder geometry of \(L_p\)-spheres.

By constructing an appropriate quadratic form, we prove that if an analytic Toeplitz operator commutes with a co-analytic Toeplitz operator on the harmonic Bergman space, then at least one of the symbols is constant. This resolves an open question posed by Choe and Lee in 1999.

We prove that the quantum monad in dimension $2n$ admits no natural transformation to the polymorphism clone of linear equations modulo $n$. Consequently, for every $n\geq 2$, there exists an unsatisfiable system of linear equations over $\mathbb{Z}_n$ whose constraint system game admits a perfect finite-dimensional quantum strategy. As a corollary, we completely characterise pseudo-telepathic constraint languages in finite dimension. The proof combines a result of Harding, Jager, and Smith on group-valued measures on subspaces of Hilbert spaces with the polymorphism-minion characterisation of quantum pseudo-telepathy.

Let $G$ be a countable locally finite abelian group, and let \[ G_1\leq G_2\leq\cdots, \qquad \bigcup_{i\geq1}G_i=G, \] be any filtration of $G$ by finite subgroups. For $A\subseteq G$, let $\mathcal P(A)$ denote the set of all finite subset sums of elements of $A$, and let $2G:=\{2g:g\in G\}$. We prove that $|2G|=\infty$ if and only if for every filtration and every interval $[\alpha,\beta]\subseteq[0,1]$, there exists $A\subseteq G$ such that the set of limit points of \[ \left( \frac{|\mathcal P(A)\cap G_i|}{|G_i|} \right)_{i\geq1} \] is exactly $[\alpha,\beta]$. We also prove a positive-density result that does not require $|2G|=\infty$: if $G$ is infinite and $\mathcal P(A)$ has positive upper density along the given filtration, then there is an infinite set $B\subseteq \mathcal P(A)$ such that $B+B\subseteq \mathcal P(A)$. Combining this with the realization theorem, we show that, when $|2G|=\infty$, every interval $[\alpha,\beta]$ with $0\leq\alpha\leq\beta\leq1$ and $\beta>0$ can be realized by a set $A$ with this additional property.

We study homogenization of norm-optimal and time-optimal controls for a one-dimensional heat equation with rapidly oscillating periodic diffusion coefficients. The controls act on a fixed subinterval and are subject to pointwise constraints in space and time. We first derive final-state observability and null-controllability estimates that are uniform with respect to the oscillation scale. Combining these estimates with periodic homogenization and an abstract theory of optimal controls for parabolic systems, we prove convergence of the norm-optimal values and controls to their homogenized counterparts. We then deduce convergence of the optimal times and time-optimal controls. The controls converge weakly-star in the space of essentially bounded functions and strongly in every Lebesgue space with finite exponent, after extension to a common time interval when necessary.

The classical capacity of a quantum channel generally requires regularization because input codewords may be entangled across channel uses. We remove this regularization for generalized amplitude-damping channels (GADCs) and determine their exact strong-converse exponent throughout the full parameter range. The exponent is a single-letter transform of the sandwiched Rényi Holevo information and is attained by product input codewords with collective decoding. The key ingredient is a decomposition of the joint output into positive operators whose triangular factors have identical scalar Gram matrices after norm compression. A sharp triangular Schatten-norm compression inequality then establishes maximum output norm multiplicativity for diagonally sandwiched GADCs tensored with arbitrary completely positive maps. This yields additivity of the sandwiched Rényi Holevo information at every order above one. Consequently, the classical capacity equals the known one-copy Holevo information and admits a scalar optimization. Thus, entangled input codewords improve neither the capacity nor the optimal exponential decay of decoding success above capacity.