Papers

Motivated by the importance of the purification principle in quantum theory and generalized probabilistic theories, we study purifications using only the geometry of a finite-dimensional proper convex cone. We prove an existence theorem for indecomposable cones and intermediate tensor cones containing the maximally entangled state; in particular, every interior point of an indecomposable homogeneous cone admits a purification. This applies to Lorentz cones, for example. We also give a criterion for uniqueness up to local automorphisms. On the boundary, we show that if every proper face of $C$ is simplicial, then only pure points can admit purifications, and we demonstrate that this conclusion fails in the presence of non-simplicial faces. Examples involving positive semidefinite cones, Lorentz cones, $k$-positive maps, PPT tensors, and polyhedral cones illustrate both existence and non-uniqueness phenomena.

This paper addresses the approximation of entropy solutions to a one-dimensional scalar conservation law via the so-called Follow-the-Leader particle scheme in case the velocity map is non monotone. The result is based on a discrete maximum principle, on $BV$ estimates of the approximated density, and on the characterization of the limit as entropy solution in the sense of Kru\v zkov.

We study the one-dimensional focusing mass-critical half-wave equation \[ i\partial_tu=|D|u-|u|^2u. \] For even initial data with negative energy and mass slightly above the ground-state mass, we prove finite-time blow-up and obtain the upper bound \[ \|u(t)\|_{\dot H^{1/2}} \lesssim \frac{|\log(T-t)|^{1/4}}{\sqrt{T-t}} \qquad\text{as }t\uparrow T. \] This is the first finite-time blow-up result for negative-energy solutions to the mass-critical half-wave equation in the near-ground-state regime. The proof uses the modulation analysis developed by Merle–Raphaël \cite{MerleRaphael2005AnnMath}. For the half-wave equation, the key missing ingredient is a coercivity estimate for a nonlocal quadratic form generated by the scaling direction. Unlike the local NLS, the half-wave equation does not admit the ODE methods used to establish the corresponding coercivity estimate. Instead, we prove the coercivity estimate by an analytic reduction followed by a rigorous computer-assisted proof. The spectral analysis part reduces the coercivity problem to a finite collection of spectral inequalities by combining constrained Morse index arguments with the Birman–Schwinger principle. The computer-assisted part certifies these inequalities by interval arithmetic using a validated approximation of the ground state obtained by compactifying the real line. Together with the modulation analysis, this coercivity theorem gives the finite-time blow-up result.

We study finite-time blow-up for the one-dimensional focusing mass-critical half-wave equation \begin{equation*} i\partial_tu=|D|u-|u|^2u. \end{equation*} For even initial data with negative energy and mass slightly above the ground-state mass, we prove the log-log upper bound \begin{equation*} \|u(t)\|_{\dot H^{1/2}}\lesssim \left(\frac{\log|\log(T-t)|}{T-t}\right)^{1/2} \quad \text{as}\quad t\uparrow T. \end{equation*} This gives, for the half-wave equation, the same log-log law upper bound as in the mass-critical nonlinear Schrödinger equation. The proof follows a similar strategy developed by Merle and Raphaël, but requires a new construction of the blow-up profile. Main difficulty arises from the nonlocal operator $|D|$ and the absence of pseudo-conformal symmetry. We construct an almost self-similar profile with exponentially small error by combining tail computations carried out to arbitrary order, depending on a dynamical parameter, with Borel integral summation in $\Lambda$-analytic spaces. Then, in the modulation analysis, we use a local-virial spectral property proved in the companion paper \cite{Park2026arXiv}.

Oversmoothing is a fundamental limitation of deep graph neural networks (GNNs), where repeated message passing causes node representations to become increasingly similar, eventually collapsing toward a low-dimensional subspace. This phenomenon limits the effective depth of message-passing architectures and motivates the search for mechanisms that preserve representation diversity. In this paper, we study a recurrent graph neural network in which independent Gaussian noise is injected after every propagation step and analyze the resulting architecture as a stochastic dynamical system. Under a standard global contraction assumption on the deterministic update, we prove that the hidden representations form a geometrically ergodic Markov chain admitting a unique invariant probability measure. Our main theoretical result establishes an explicit positive lower bound on the expected stationary Dirichlet energy, proportional to both the noise variance and the spectral gap of the underlying graph. Consequently, the stationary representations cannot collapse onto the constant manifold, providing a rigorous guarantee that asymptotic oversmoothing is prevented in the sense of non-vanishing Dirichlet energy. Our analysis reveals persistent stochastic perturbations as a fundamentally different mechanism for combating oversmoothing, complementing existing deterministic approaches based on residual connections, normalization, and graph rewiring. Finally, numerical experiments on both linear and nonlinear recurrent graph neural networks closely match the theoretical predictions, illustrating the emergence of a stationary distribution and the predicted dependence of the limiting Dirichlet energy on the noise intensity.

Operators invariant under a symmetry group are also invariant under any finite-index subgroup. The equivariant indices of such operators must be consistent under symmetry breaking. We use this principle to establish a canonical weak/strong dichotomy for equivariant indices, building on an idea originating in the theory of topological insulators in solid-state physics. We also study the relationship to coarse-geometric, or macroscopic, indices.

We study a finite-horizon nonstationary workforce planning problem in which staff turnover in each period follows a binomial distribution whose parameters depend on the post-hiring workforce level. The model incorporates a fixed hiring cost that is incurred whenever recruitment occurs, regardless of the number of employees hired; this represents explicit (e.g. administration costs) or implicit (e.g. team disruption) factors that are commonplace in workforce planning. The objective is to minimise the expected total cost, including recruitment, salary, and shortage costs, where deviations below period-specific staffing requirements are penalised. To analyse this stochastic dynamic program with decision-dependent transition probabilities, we establish the discrete convexity of the variable single-period cost (the sum of expected salary and penalty costs) and the K-convexity of the expected total cost in presence of endogenous uncertainty. We then show that the optimal hiring policy exhibits an (s, S)-type structure: when the workforce level in a given period falls below a threshold s, staff are hired up to level S; otherwise, no hiring occurs. To compute approximate policy parameters, we develop a mixed-integer linear programming heuristic based on a piecewise approximation. In our computational study, we demonstrate that our approach achieves fast computation and small optimality gaps on the test instances.

The inverse synthesis problem of selecting a minimal leader set to guarantee strong structural controllability (SSC) in matrix-weighted networks remains an unresolved NP-hard challenge. This paper proposes a rigorous mathematical framework to solve this. We prove that structural uncontrollability stems exclusively from dimension-specific reachability isolation and topological symmetry equivalence. To overcome these bottlenecks, we formulate a two-phase synthesis: a reachability prerequisite to identify structural roots, followed by three distinct symmetry-breaking algorithms (Greedy Weisfeiler-Lehman Selection, Submodular Bound Maximization, and Partition Entropy Maximization). Mathematical proofs guarantee immunity to invariant subspaces and structural dilation, validated by extensive numerical evaluations across diverse topologies.

This paper investigates the nonlinear stochastic fractional heat equation driven by a Gaussian noise that is white in time and fractional in space with a Hurst parameter $H \in \big(\frac{3-\alpha}{4}, \frac{1}{2}\big)$. Specifically, the driving operator is the fractional Laplacian of order $\alpha/2 \in (1/2, 1)$. We characterize the asymptotic behavior of the temporal increment $u(t+\varepsilon,x)-u(t,x)$ for fixed $t\ge 0$ and $x\in\mathbb{R}$ as $\varepsilon\downarrow 0$. Utilizing these precise asymptotic estimates, we establish Khinchin's and Chung's laws of the iterated logarithm for the temporal process $t \mapsto u(t,x)$.

In this note, we prove that for every Lie series $\psi$ with no terms of degree less than 3, the relation $$[\psi(x,y),x]+[\psi(-x-y,y),-x-y]=0$$ is equivalent to $S_*(\psi_*)=-\psi_*$, where $\psi_*$ denotes the regularization of $\psi$ and $S_*$ is the harmonic antipode. The proof relies on the calculations of the harmonic antipode $S_*$ and the shuffle antipode $S$. As a consequence, we prove that every $\psi$ in Racinet's double shuffle Lie algebra $\mathfrak{dmr}_0$ satisfies the relation $[\psi(x,y),x]+[\psi(-x-y,y),-x-y]=0$. We further prove that $\mathfrak{dmr}_0$ injects into the symmetric Kashiwara–Vergne Lie algebra $\mathfrak{krv}^{\mathrm{sym}}_2$ of Alekseev and Torossian.

Motivated by a conjecture of Amari and Kobayashi, we study the problem of identifying pairs of Boolean functions that maximize the Kullback-Leibler divergence between two distributions obtained by separately compressing two correlated sources. When the reference distribution corresponds to independent sources, this problem reduces to the problem of maximizing mutual information, for which the optimality of dictator functions has been proved by Pichler, Piantanida, and Matz. For the problem of maximizing Fisher information, which can be viewed as a local version of the problem studied in this paper, Amari and Kobayashi conjectured that parity functions are optimal. For unbiased pairs of Boolean functions, and for identical pairs in the nonnegative correlation regime, we prove that both the divergence and the Fisher information are maximized by level-$k$ functions, namely, functions whose Fourier coefficients are supported only on level $k$. Since level-$k$ functions include parity functions, this gives a partial resolution of the conjecture of Amari and Kobayashi. Furthermore, in the framework of Bayesian distributed one-bit hypothesis testing, we prove that level-$k$ functions are optimal among all pairs of functions. Finally, we also discuss the one function version of the problem studied in this paper, which can be regarded as the divergence analogue of the Courtade and Kumar conjecture.

Many modern outcomes, including images, point clouds, networks, and spatial fields, are structured objects for which \(Y^1-Y^0\) may be undefined or scientifically inadequate. We introduce Topological Causal Data Analysis (TCDA), a framework separating the observation space, causal-model class, topological representation, and causal query. Topology does not define interventions; it supplies stable, shape-sensitive summaries after causal assumptions have been specified. We distinguish outcome-level TCDA, which transforms individual potential outcomes, from distribution-level TCDA, which transforms interventional outcome laws, and characterize when outcome and distribution level contrasts agree. Building on recent outcome-level theory, we formulate identification and doubly robust representations for Banach-space-valued summaries. At the distribution level, we identify targets through the standard causal \(g\)-formula and derive stability-transfer bounds and plug-in consistency. We also place target-specific topological ignorability within the framework, clarifying when a covariate-standardized coarse effect can be identified without identifying the full interventional laws. Finally, we delimit the role of observational topology in causal discovery: it can assist diagnosis on restricted model classes but cannot by itself identify causal structure.

Let $\mathbf{x}_{n \times n}$ be an $n \times n$ matrix of variables and let $S = \mathbb{F}[\mathbf{x}_{n \times n}]$ be the polynomial ring over these variables where $\mathbb{F}$ is a field of characteristic zero. Regard $S$ as the coordinate ring of the affine space $\mathbb{F}^{n \times n}$ of $n \times n$ $\mathbb{F}$-matrices. Let $\mathfrak{D}_n \subseteq \mathbb{F}^{n \times n}$ be the locus of derangement permutation matrices. We study the orbit harmonics quotient ring ${\bf R}(\mathfrak{D}_n) = S/\mathrm{gr} \, \mathbf{I}(\mathfrak{D}_n)$ where $\mathrm{gr} \, \mathbf{I}(\mathfrak{D}_n)$ is the associated graded ideal of the vanishing ideal $\mathbf{I}(\mathfrak{D}_n) \subseteq S$. We give an explicit generating set of $\mathrm{gr} \, \mathbf{I}(\mathfrak{D}_n),$ relate the Hilbert series of $\mathbf{R}(\mathfrak{D}_n)$ to the Foata transformation and the longest increasing subsequence statistic on $\mathfrak{S}_n$, and give an alternating sum formula for the graded $\mathfrak{S}_n$-character of $\mathbf{R}(\mathfrak{D}_n)$. Our proofs make heavy use of the mapping cone construction of homological algebra.

The weight distribution bound gives a lower bound on the weight of eigenvectors for eigenvalues of distance-regular graphs. In this note, we study when the weight distribution bound is tight for the negative eigenvalue of collinearity graphs of finite embedded polar spaces, and elliptic and hyperbolic affine polar graphs. We give a complete classification of eigenvectors meeting the weight distribution bound.

We prove that every Diophantine quadruple $\{a,b,c,d\}$ over $\mathbb{Q}(i)[X]$ that contains at least one non-constant polynomial is regular; that is, $$(a+b-c-d)^2=4(ab+1)(cd+1).$$ This result is consistent with the corresponding results over $\mathbb{Z}[i][X]$ and $\mathbb{R}[X]$, but contrasts with the situation over $\mathbb{C}[X]$.

The Bier sphere of a simplicial complex $K$ is defined as the deleted join of $K$ and its combinatorial Alexander dual. We focus on the class of Bier spheres of the skeleta of a simplex. Since these Bier spheres are known to be polytopal, they give rise to small covers. We classify small covers over these Bier spheres up to Davis–Januszkiewicz equivalence. As applications, for all $m \geq 4$, we determine the homeomorphism types of small covers over the Bier spheres of the $0$-skeleton and the $(m-3)$-skeleton of an $(m-1)$-simplex. For the remaining cases $0<r<m-3$, we compute their rational Betti numbers.

We show that if an Anosov flow on a 3-dimensional manifold has orientable stable and unstable foliations, then the complement of any filling periodic orbit is a hyperbolic manifold. This generalizes the known case of the complement of a closed, filling geodesic orbit in the unit tangent bundle of a hyperbolic surface. Furthermore, we show that the orientability condition on invariant foliations is necessary, by constructing a counterexample in the absence of this property. In the case of the suspension flows we obtain that the complement of every collection of periodic orbits is hyperbolic, while for the geodesic flow (regardless of orientability of the invariant foliations) the complement of every filling and anannular collection of periodic orbits is hyperbolic.

The game of Hex is one of the most celebrated connection games in combinatorics. Although it is known that the first player always has a winning strategy, very little is understood about the complexity of optimal play. Following Campbell's introduction of the parameters $\lambda(n)$ and $\delta(n)$, measuring respectively the length of the shortest guaranteed winning path and the minimum number of stones required by the first player to force a win, we establish new structural properties of these quantities. Our main result determines the exact value \[ \lambda(5)=7, \] thereby confirming one of Campbell's conjectures. We also show that the corresponding problem on the infinite strip $5\times\infty$ has a markedly different behavior. In this setting we prove that \[ \lambda(5\times\infty)=5, \] which is strictly smaller than $\lambda(5)$. To the best of our knowledge, this paper provides the first solution to one of Campbell's 2004 conjectures and answers a question posed by Stromquist (2006) in the affirmative. Furthermore, it uncovers a fundamental distinction between finite and infinite boards, offering new insights into the nature of optimal winning strategies in Hex.

We prove structure theorems for multipersistence modules indexed by finite posets that are not totally ordered. Specifically, we consider pointwise finite-dimensional modules over the opposite of the poset of non-empty subsets of a finite set, and give sufficient conditions, expressed through transition morphisms, for such modules to split as direct sums of interval modules. In the general case, the interval summands and multiplicities are explicitly determined by dimensions at finitely many indices. Although the assumptions may look algebraically restrictive, we show that they have a natural geometric origin in a robustness theory of cellular sheaf data over simplicial complexes, where one studies how algebraic information, compatibility constraints, and cohomological obstructions persist under structural failures. We extend thickness and cohesion from simplicial cohomology to cellular sheaves: thickness detects the dependence of cohomology classes on high-dimensional support, while cohesion captures the influence of higher-order adjacencies on the cohomological features. We leverage our abstract structure theorems to obtain interval decompositions for the resulting geometric cohesion modules. Finally, we introduce biparameter persistence constructions for sheaf resilience, tracking whether global sections and cohomological obstructions remain detectable on thick or cohesive substructures during topological degradation.

In this note we extend the Ambrosio-DiPerna-Lions theory on the well-posedness of the transport equation to the case in which the vector field satisfies an $L^p$ Osgood condition. In particular we show that bounded distributional solutions to the transport equation are unique and thus renormalized. As opposed to the classical proof, we do not rely on the vanishing of a commutator, but we show that a weighted energy built out of the Littlewood-Paley decomposition of the solution is conserved.

We propose a notion of homotopy limit in the category of complexes over an abelian category with products by totalizing the classic construction of the Roos' complex that computes derived inverse limits. For complexes of modules over a non-necessarily commutative ring, we show that our construction of homotopy limits computes the derived limit complex, and under an acyclicity hypothesis on the inverse system, we prove that it is quasi-isomorphic to the limit. We further show that, in general, the construction is appropriately dual of the previous construction of homotopy colimits of complexes from [Alonso, Jeremías and Souto: Localization in categories of complexes and unbounded resolutions. Canad. J. Math. (2000)], and that there also is a dual behavior between derived limits and colimits in derived categories of modules. Finally, we show that colocalizing subcategories are stable for homotopy limits.

Carlson and Simpson proved that for every finite coloring of the 1-variable words over a finite alphabet~$A$, there is an infinite $\omega$-variable word on which all the 1-variable words are monochromatic. This statement for $\ell$-colorings, written $\mathsf{CSL}^1_\ell$, is known to be strictly weaker than $\mathsf{ACA}_0$. We prove that $\mathsf{RCA}_0 + \mathsf{CSL}^1_2$ is a $\forall \Pi^0_4$-conservative extension of $\mathsf{RCA}_0 + \mathsf{B}Sigma_2$. Among its consequences, it implies that neither the indivisibility of the universal triangle-free Henson graph for 2-colorings, nor the tree theorem for pairs and two colors, imply $\Sigma^0_2$-induction. This answers a question of Chong, Li, Wang and Yang.

For a positive integer $n$, let $A_n=\mathbb{C}[t_1^{\pm1},\ldots,t_n^{\pm1},x_1,\ldots,x_n]$ and $\mathfrak{g}_n=\bigoplus_{i=1}^n A_nd_i$, where $d_i=t_i\frac{\partial}{\partial t_i} +\frac{\partial}{\partial x_i}$. We first determine when the tensor module $T(P,V)=P\otimes V$ is simple, where $P$ is a simple module over the Weyl type algebra $D_n$ and $V$ is a simple $\mathfrak{gl}_n$-module. We then prove a canonical algebra isomorphism $A_n\#U(\mathfrak{g}_n)\cong D_n\otimes U(\mathfrak{m}_{\mathbf{1},\mathbf{0}}\Delta)$, and use it to show that every simple cuspidal $\mathfrak{g}_n$-module is isomorphic to a simple quotient of some $T(A_n(\lambda),V)$, where $V$ is a finite-dimensional simple $\mathfrak{gl}_n$-module.

We introduce the convolution between abstractly given anisotropic weight matrices and investigate properties of this new construction. Further, we apply the knowledge to the particular but interesting case when the (isotropic) matrices are associated with weight functions in the sense of Braun-Meise-Taylor and show how the convolution of associated weight matrices modifies the underlying weight functions. Finally, we give a concrete application of the convolution when studying the concept of local solvability for a certain hyperbolic PDE. Indeed, the convolution allows to treat in a natural way a mixed setting; i.e. having a controlled loss of regularity expressed in terms of two, in general different, weight sequences.

This work presents a new adaptive finite-element framework for level-set transport, achieving high accuracy in kinematic interface transport problems relevant to interface-capturing methods for two-phase flows while reducing computational cost. This framework accommodates dynamic refinement and coarsening, and is compatible with standard finite-element data structures. The algorithm is applicable to both structured and unstructured discretizations in two and three dimensions. The method uses a tree-based hierarchical mesh with dynamic local refinement that tracks the evolving interface, concentrating resolution in a narrow band around the zero level set while preserving a coarse discretization elsewhere. The level-set field is updated through marker transport on adaptively refined meshes, enabling interface prediction and the construction of an evolving adaptive hierarchy. Concurrently, backward characteristic tracing provides accurate evaluation of the advected level-set field. Both operations employ an efficient multilevel marker and point-location algorithm to identify containing elements across refinement levels. Numerical experiments on two- and three-dimensional structured and unstructured grids, including quadrilateral, simplicial, wedge, and hexahedral meshes, subjected to severe interface deformations, demonstrate that the adaptive strategy achieves accuracy comparable to uniform discretizations at a substantially lower cost, while maintaining good conservation properties under aggressive adaptivity. Consequently, the proposed approach provides an efficient and flexible framework that can be naturally integrated into level-set-based multiphase flow solvers.

We construct noetherian rings admitting local cohomology modules with nonclosed support, or equivalently with infinitely many minimal primes; this answers a question of Huneke–Lyubeznik.

We prove that every polylogarithmically dense subset of $[N]$ contains a nontrivial configuration $x+b_1m,\ldots,x+b_km$ for almost all choices of the coefficient vector $(b_1,\ldots, b_k)$ in a wide range of scales. We prove the same statement for polylogarithmically relatively dense subsets of the primes, in a shorter range of scales. The main ingredients are a new quantitative generalised von Neumann theorem, degree lowering to the $U^{1+}$ norm, and densification arguments that transfer the result to the primes.

We establish strong existence and pathwise uniqueness for McKean-Vlasov stochastic differential equations with coefficients satisfying a distribution-dependent Lyapunov condition. Under a hybrid Perron-Nagumo condition that permits a non-integrable singularity at the initial time, pathwise uniqueness holds within the class of strong solutions satisfying the corresponding Lyapunov estimate. For existence, we truncate the coefficients on nested bounded domains, construct absorbed local weak solutions, pass to a weak solution via tightness arguments, and then apply a restricted Yamada-Watanabe theorem to obtain a strong solution. Our existence proof, different from the classical truncation-patching method, is interesting in its own right. We also provide an explicit example to which our criterion applies, while none of the Lipschitz, Osgood, or monotonicity conditions is satisfied.

Let $F$ denote the Fano plane, the $3$-uniform hypergraph with $7$ vertices and $7$ edges. Frankl and Füredi, and independently Keevash and Sudakov, proved that the largest $F$-free subhypergraph of $K_n^{(3)}$ is bipartite. In this paper, we determine the sharp threshold for this property in the random setting. We show that for $\hat{p} = \Theta_F \cdot n^{-2/3} \left(\log n\right)^{1/6}$, where $\Theta_F$ is an explicit constant depending on $F$, we have: (i) if $(1+\epsilon) \hat{p} \le p = o(1)$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is bipartite; and (ii) if $\frac{1}{n^2} \ll p \le (1-\epsilon) \hat{p}$, then with high probability every largest $F$-free subhypergraph of $G_{n,p}^{(3)}$ is not bipartite. To the best of our knowledge, this work provides the first sharp threshold result obtained for a Turán-type problem in random hypergraphs.

We present a unification of Kendall's landmark shape spaces, where rigid motions are factored out and scale fixed on landmark configurations equipped with Euclidean geometry, with landmark configuration spaces carrying Riemannian metrics descending from right-invariant Sobolev metrics on the diffeomorphism group. The resulting new landmark shape spaces achieve the defining properties of both approaches: The regularity of the descending metric prevents landmarks from colliding, the metric is defined in the ambient space independent of the number of landmarks, local rigid transformations are preserved, global rigid motions are removed, and scale fixed. To achieve this, we define a particular Sobolev-type operator, the screened elasticity operator, whose null-space consists exactly of the rigid motions, we show how this operator descends to achieve the desired geometry, and we present approaches to solving matching problems and computing geodesics numerically. The resulting construction allows the use of landmark configuration spaces with sufficiently regular metrics in applications while retaining the shape invariances that are a hallmark of Kendall's shape spaces.