Papers

Discrete Forman-Ricci curvature is a quantity associated to each edge of a graph that describes its local geometry. It has proven to be a useful tool in network analysis in a variety of applications. Recent work by Roost et al.\ (2024) proposed the use of Markov bases to sample from the space of graphs with prescribed vertex degrees and curvatures. In the present work, we further develop the algebraic and combinatorial theory of these Markov bases. We show that the degree of an indispensable Markov move grows at least quadratically in the maximum degree of the graph. In light of this result, a compact description of all Markov basis elements seems unattainable at present. Instead, we find a lattice basis for this problem using only degree three Markov moves, which allows us to employ recently-developed reinforcement learning methods for finding Markov moves that can be applied to a specific graph.

The \emph{$k$-color Ramsey number} $R_k(C_{2\ell+1})$ is the least integer $n$ such that any $k$-edge-coloring of a complete graph $K_n$ has a monochromatic odd cycle $C_{2\ell+1}$. Axenovich, Cames van Batenburg, Janzer, Michel, and Rundström~(JCT-B, 2026) recently proved \[ R_k(C_{2\ell+1})\le (4\ell-2)^k k^{k/\ell}+1, \] and Miyazaki, Mulrenin, Pohoata, and Zheng further improved the factor $k^{k/\ell}$ to $(k!)^{1/\ell}$. As Jenssen and Skokan (AM, 2021) determined $R_k(C_{2\ell+1})$ for fixed $k$ and sufficiently large $\ell$, it becomes even more interesting to seek better bound for fixed $\ell$ and sufficiently large $k$. In this paper, we show \[ R_k(C_{2\ell+1}) \le \frac{2\ell}{2\ell-1}(2\ell-1)^k(k!)^{1/\ell} \exp\!\left(k^{1-1/\ell}+O_\ell\!\left(k^{1-2/\ell}+\log k\right)\right)+1 \] for every fixed $\ell\ge 2$ and sufficiently large $k$, which improves the bound of Miyazaki et al. by a factor $2^{k-o(k)}$, and the bound of Axenovich et al. by a factor $(2\e^{1/\ell})^{k-o(k)}$.

The extensional quotient completion of relational doctrines provides a common generalization of both the exact completion of categories with weak finite limits and the elementary quotient completion of existential elementary doctrines. In this paper, we study projective objects in relational doctrines with quotients, characterizing those obtained through the extensional quotient completion as those admitting a projective cover. We apply this result to doctrines of algebras for monads on relational doctrines with quotients, describing in which cases these arise as the extensional quotient completion of their restriction to (appropriate subcategories of) free algebras. This extends a similar result for monadic categories over exact ones, covering also more examples such as monads over the category of metric spaces giving rise to variants of quantitative algebras.

For solutions to the Minkowski problem of the even dual curvature measures $\widetilde{C}_q$ in $\mathbb{R}^n$, $n\ge 2$, we prove the non-uniqueness for $q>n$ and $n\ge 2$, and the uniqueness for $0<q<n$ and $n=2$. These results are governed by our established logarithmic Brunn-Minkowski inequality for dual quermassintegrals $\widetilde V_q$.

This paper studies smoothed matrix-polar spectral gradient flows for unconstrained matrix-valued optimization.The canonical polar-factor map loses smoothness at rank-deficient matrices and becomes ill-conditioned as singular values approach zero, creating analytical difficulties.We therefore introduce a spectral feedback law generated by a smooth spectral potential and establish the regularity, monotonicity, boundedness, and dissipation properties of the feedback.Based on this feedback law, we propose a smoothed spectral gradient flow and prove well-posedness and global convergence of the flow.We derive convergence-rate results for the spectral gradient flow in nonconvex, convex, and Polyak–Lojasiewicz (PL) settings and analyze the Lyapunov structure and convergence of a momentum-augmented system under the same spectral feedback law. Furthermore, we provide a local descent-rate comparison between the smoothed spectral-gradient direction and the standard Frobenius-gradient direction using a general Hessian-based quadratic model. This analysis yields a verifiable normalized descent-rate advantage condition, showing that the local benefit of the spectral direction depends on both first-order alignment with the gradient matrix and the directional curvature induced by the Hessian.

We prove that the entanglement of formation of every two-mode Gaussian state coincides with its Gaussian restriction, solving a longstanding open problem in continuous variable quantum information theory. The key result is a sharp affine relation between entanglement and a generalized Einstein–Podolsky–Rosen observable, valid for arbitrary pure two-mode states, including non-Gaussian ones. We further prove that the entanglement of formation is additive on arbitrary tensor products of two-mode Gaussian states, and hence coincides with the entanglement cost, settling the latter for all two-mode Gaussian states. As a consequence, we obtain in closed form the entanglement cost of thermal-loss, thermal-amplifier and additive-noise bosonic channels. Our results, which also extend to bisymmetric multimode states, establish that no two-mode state is less entangled than the Gaussian state with the same covariance matrix, yielding measurable lower bounds on both the entanglement of formation and the entanglement cost of arbitrary non-Gaussian states.

AAn old conjecture in non-Kähler geometry states that any compact Hermitian manifold with constant Chern holomorphic sectional curvature must be either Kähler or Chern flat. The conjecture is known to be true in dimension 2 but still open in dimensions 3 or higher, except for several special classes of Hermitian manifolds. For the important class of locally conformally Kähler manifolds, the conjecture was proved by H. Chen, L. Chen, and Nie in 2021 when the constant holomorphic sectional curvature is non-positive and the remaining case was solved recently by Huang and Wan using the result of Kamishima on Bochner-Kähler manifolds. In this article, we use their technique to answer similar questions for locally conformally Kähler manifolds with constant Levi-Civita or Bismut holomorphic sectional curvature.

For every odd prime $p$, we construct a finite left brace $A_p$ of order $p^{2p+1}$ whose additive group is elementary abelian and whose multiplicative group $G_p$ is a powerful $p$-group, but such that $A_p$ is not right nilpotent. More precisely, $G_p'=G_p^p\cong C_p^2, \operatorname{cl}(G_p)=2, \exp(G_p)=p^2,$ and $\operatorname{Soc}(A_p)=0$. Thus powerfulness does not force right nilpotence even for multiplicative groups of class two with derived subgroup of order $p^2$. The obstruction is explicit: $A_p$ contains a three-dimensional trivial ideal $T$ satisfying $T*A_p=T$, whereas its left series is the ideal-power filtration of a nilpotent commutative algebra. Both $T$ and $A_p/T$ are right nilpotent, so right nilpotence of finite left braces is not closed under extensions. The construction is uniform and arises from a finite local commutative algebra, a square-zero derivation, and an invariant group homomorphism, yielding a regular affine subgroup as the kernel of a homomorphism on a semidirect product. It disproves the Shalev–Smoktunowicz conjecture in every odd characteristic and yields finite non-degenerate irretractable involutive set-theoretic solutions of the Yang–Baxter equation whose permutation groups are powerful $p$-groups of class two.

We present a novel approach to the superintegrability of collective Hamiltonians invariant under a Hamiltonian action of a connected semisimple compact Lie group, $G$, on a symplectic manifold, $M$. By exploiting a Hamiltonian torus action that goes back to Guillemin and Sternberg [GS,1983], we demonstrate that the functional dimensions of $\mathfrak{H} := \mathcal{J}^*(C^\infty(\mathfrak{g}^*)^G)$, where $\mathcal{J}: M \to \mathfrak{g}^*$ is the momentum map of the $G$ action, and its centralizer $\mathfrak{F}$ in $C^\infty(M)$ satisfy the equality $\mathrm{ddim}(\mathfrak{H}) + \mathrm{ddim}(\mathfrak{F}) = \mathrm{dim}(M)$. Together with a non-triviality condition, this ensures that the Abelian Poisson algebra $\mathfrak{H}\subset C^\infty(M)$ represents a superintegrable system, and it also follows that the momentum map of the GS torus action yields action variables for the system. Our work provides a new insight into collective superintegrability complementing earlier results of Bolsinov and Jovanović.

Let $\Gamma=(G,\sigma)$ be a signed graph, where $G$ is the underlying graph with vertex set $V(G)$ and edge set $E(G)$ such that $\sigma: E(G)\to \{-1,1\}$ is the sign function. For $U\subset V(G)$, the operation that changes the sign of all edges between $U$ and $V(G)\setminus U$ is called switching. Two signed graphs with the same underlying graph are switching equivalent if one is obtainable from the other one by switching a subset. Two signed graphs are switching isomorphic if one is isomorphic to a switching equivalent signed graph of the other one. A signed cycle is called negative if it contains an odd number of negative edges. A signed graph is balanced if none of its cycles is negative; otherwise it is unbalanced. The adjacency matrix $A(\Gamma)$ of $\Gamma$ is obtained from the standard $(0,1)$-adjacency matrix of $G$ by reversing the sign of all $1$s which correspond to negative edges. The index of $\Gamma$ is the largest eigenvalue of $A(\Gamma)$ and the spectral radius of $\Gamma$ is the largest absolute value of the eigenvalue of $A(\Gamma)$. The least eigenvalue of $\Gamma$ is the least eigenvalue of $A(\Gamma)$. We study the extremal problems of the index and the spectral radius among unbalanced signed multipartite graphs. More precisely, we determine the unbalanced signed $t$-partite graphs with fixed $t\ge 2$ and partite sizes (order, respectively) that maximizes the index and the spectral radius respectively, up to switching isomorphism. To determine the unbalanced signed multipartite graphs with fixed partite sizes (order, respectively) with maximum spectral radius, we also determine those with minimum least eigenvalue.

The bipartite Turán number of a graph $H$, denoted $\text{ex}(m, n; H)$, is the maximum number of edges in any $H$-free bipartite graph $G = (A, B; E)$ with parts of size $|A| = m$ and $|B| = n$. We study this problem for two families. For a tree $T = T(r, s)$ with parts $R$ and $S$ of sizes $|R| = r \le s = |S|$, we prove \[ (r - 1) n \;\le\; \text{ex}(m, n; T(r, s)) \;\le\; (r - 1) n + C(m,r,s) \] for $n$ sufficiently large compared to $m$, $r$, and $s$, where $C(m,r,s)$ does not depend on $n$, determining the leading-order term exactly (with the star case $r = 1$ solved with an exact formula). For a star forest $F = \bigcup_{i=1}^k S_{d_i}$ with $d_1 \ge \cdots \ge d_k$, we determine the exact value $\text{ex}(m, n; F) = (k - 1) n + (d_k - 1)(m - k + 1)$ for $n$ sufficiently large, and characterize the unique extremal graph.

We study the Euler–Riesz system on the torus $\mathbb T^d$, $d=2,3$: the compressible Euler equations with barotropic pressure $p(\varrho)=a\varrho^\gamma$ ($\gamma>1$, $a>0$), coupled to a repulsive nonlocal force ($\approx \varrho \nabla_x K \ast \varrho$) with Riesz kernel $K(x)\propto|x|^{\beta-d}$ of order $\beta\in(0,2)$. Since $K$ is the kernel of the inverse fractional Laplacian $(-\Delta)^{-\beta/2}$, we recast the force through the Caffarelli–Silvestre extension as the trace of a local stress tensor, replacing the nonlocal interaction by a local identity in one extra variable. For a repulsive kernel the total energy is coercive, and we use this to introduce a notion of global-in-time dissipative solution for arbitrarily large finite-energy data. Our main result is weak (measure-valued)–strong uniqueness, for every order $\beta\in(0,2)$ and every $\gamma>1$ independently: on any interval on which a strong solution exists, every dissipative solution with the same initial data coincides with it and all defects vanish. The proof rests on a suitable adaptation of relative energy.

In portfolio optimization, a cardinality constraint, which limits the number of assets held, plays a key role in cutting down monitoring and transaction costs. However, the resulting problem is NP-hard and becomes computationally difficult to solve globally as the number of candidate assets grows. Safe screening addresses this difficulty by fixing decision variables before optimization without excluding any globally optimal solution, thereby reducing the problem size while preserving optimality guarantees. We propose safe screening rules for cardinality-constrained portfolio optimization with a convex quadratic objective function and linear constraints. Using a perspective relaxation of the L2-regularization term and Fenchel duality, we derive asset-specific scores that incorporate the Lagrange multipliers of the linear constraints. Combined with a relaxation-based lower bound and a feasible-solution upper bound, these scores safely fix binary asset-selection variables to zero or one. Experiments on S&P 500 and Russell 2000 datasets show substantial computational improvements on challenging cases, particularly under moderate or strong regularization and less stringent return requirements. These results demonstrate the effectiveness of safe screening as an optimality-preserving preprocessing technique that greatly boosts computational efficiency in large-scale cardinality-constrained portfolio optimization.

We present a simple construction of a family of $0/1$-polytopes whose edge expansion decreases exponentially with the dimension, which disproves the Mihail-Vazirani conjecture that the graph of every $0/1$-polytope has edge expansion at least one.

We establish exponential mixing for the randomly forced and weakly damped KdV equation in $L^2(\mathbb{T})$. The noise is bounded, localized, and degenerate in high frequencies. Our proof relies on a general probabilistic framework in [11,33], nonlinear smoothing for KdV and its linearization via normal form transformation, and stabilization of the system by localized force. This paper continues a series of works connecting asymptotic compactness, control theory, and ergodicity and mixing for randomly forced dispersive PDEs.

We derive explicit characterizations of Gibbs-measures related to the Aviles-Giga functional. We build on the variational approach presented by Barashkov and Gubinelli (2020) in the context of the $\Phi^4_3$-model.

We construct a family of $\mathbb{R}/\mathbb{Z}$-valued CR invariants for compact strictly pseudoconvex CR manifolds admitting pseudo-Einstein contact forms. We define these invariants by applying the theory of Cheeger–Simons differential characters to a globally defined modification of the normal tractor connection. When the CR holomorphic tangent bundle is trivial, their natural $\mathbb{R}$-valued lifts agree with the generalized Burns–Epstein invariants. For CR manifolds arising as boundaries of relatively compact strictly pseudoconvex domains, we identify these differential characters with those determined by the renormalized connection and derive bulk–boundary formulas involving renormalized characteristic numbers and residues of Chern classes. These formulas recover and extend the results of Burns–Epstein and Marugame. We also obtain obstructions to CR embeddings into complex Euclidean space.

To address the demand for inherently interpretable optimization methods, we introduce novel linear surrogates for budget allocation problems. These surrogates consist of sparse linear rules that map instances to feature-based representations of solutions. We present an exact approach based on mixed-integer programming as well as a heuristic for their computation. The performance of both approaches is analyzed through computational experiments.

This letter investigates an integrated sensing and communication (ISAC) system aided by movable subarrays (MSAs) using a hybrid near-far field channel model. The sensing target and communication users are assumed to lie in the near field of the overall MSA aperture but in the far-field region of each subarray. Accordingly, a hybrid near-far field channel model is established, and the equivalent Fisher information matrix and Cramér-Rao bound (CRB) for joint range, elevation, and azimuth estimation are derived. The transmit beamforming matrix and subarray positions are jointly optimized to minimize the trace CRB subject to minimum communication signal-to-interference-plus-noise ratio (SINR), maximum transmit power and subarray movement constraints. An alternating optimization algorithm is developed combining iterative rank-one-penalized semidefinite relaxation with projected finite-difference block descent and backtracking. Numerical results show that the hybrid-field model closely matches the spherical-wave model, while MSAs substantially reduce the CRB.

Transcranial focused ultrasound (tFUS) requires accurate estimation of the intracranial acoustic field, which is distorted by skull-induced aberrations. Numerical solvers are accurate but computationally expensive for digital twins, where the field must be re-estimated repeatedly as treatment conditions change. Existing deep-learning surrogates are fast but typically use voxel-to-voxel regression on a fixed grid, with no mechanism reflecting how acoustic energy propagates through the skull. We instead cast tFUS simulation as an operator learning problem and propose tFUSOperator, a coordinate-aware neural operator that maps the free-field pressure, skull anatomy, and treatment parameters to the intracranial field within a shared physical coordinate frame. To our knowledge, this is the first operator-based formulation of tFUS field prediction. On both seen and unseen skulls, the model localizes the acoustic focus accurately-reaching about 90% and 72% Dice, respectively-and it performs nearly as well from magnetic resonance (MR) as from computed tomography (CT) input while running $5.6 \times 10^4$ times faster than numerical simulation. These results suggest a fast, radiation-free route to safe and practical digital twins for patient-specific tFUS treatment. The code is available at: https://github.com/CMME-Lab/tFUSOperator.git.

We study null controllability of backward stochastic heat systems with constant couplings in both the state and the martingale integrand, common scalar diffusion, and a localized drift control. We prove that null controllability is equivalent to a noncommutative word-rank condition on the two coupling matrices and the control matrix. The proof combines a factorization of the dual dynamics, positivity and small-time coercivity of a finite-dimensional stochastic Gramian, and a Lebeau–Robbiano spectral argument. Failure of the rank condition yields an invariant unobservable subspace.

We propose a quantum dominant energy condition (QDEC) for the stress tensor in the context of quantum field theory in curved spacetimes. A rigorous proof is given for the case of Rindler wedges, and a heuristic discussion about possible generalizations to more general geometric setups, including curved spacetime, is provided. In Minkowski spacetime, we establish a connection with state recovery bounds. We illustrate the QDEC for coherent states of the free scalar field, where it turns out to be related to the ordinary DEC for the stress tensor of the classical solution that defines the coherent state.

In this paper, we show existence and uniqueness of solutions to the viscous Burgers' equation in $\mathbb{R}^d$, when the initial condition $u_0$ is in Hermite-Sobolev space of index $p$, for suitable non-negative integers $p$. Our solutions are local in time. We also have a regularity result, viz. if $u_0$ belongs to Schwartz space, then so does the solution.

In this paper, we obtain the structure of a class of contractive projections, known as generalized tri-circular projections, on some functional Banach spaces on the open unit disk $\mathbb{D}$, which includes Hardy, Bergman and Bloch spaces. We then consider the space $S^p_{\mathcal{K}}$ of $\mathcal{K}$-valued analytic functions on $\mathbb{D}$ such that $f' \in H^p(\mathcal{K})$ and give complete description of generalized tri-circular projections on this space. Here, $\mathcal{K}$ is a complex separable Hilbert space, and $H^p(\mathcal{K})$ denotes the Hardy space of $\mathcal{K}$-valued analytic functions on $\mathbb{D}$.

In this paper, we develop the extension theory of commutative Poisson algebras. Specifically, we define the second cohomology of a commutative Poisson algebra $K$ with coefficients in an abelian Poisson algebra $H$. We then establish a natural bijective correspondence between this cohomology group and the set of equivalence classes of a special type of extensions.

Why does (-)x(-)=+? Although every student learns this rule, many accept it as a convention rather than as a mathematical necessity. This article develops a conceptual explanation by tracing the evolution of number from counting objects to representing opposites. Beginning with the emergence of numerals, place value, zero, and negative numbers, it shows how each stage extended the meaning of number and the operations defined on it. The discussion argues that multiplication of signed numbers is most naturally understood in terms of preserving or reversing direction, so that the familiar rule (-)x(-)=+ follows as a logical consequence of a coherent arithmetic rather than as an arbitrary convention. By combining historical development with mathematical reasoning, the article offers an accessible framework for teachers and students seeking a deeper understanding of signed multiplication and illustrates how the evolution of mathematical ideas can enrich classroom learning.

The regularity of solutions to elliptic partial differential equations degrades severely at mixed Dirichlet-Neumann boundary junctions, characterized classically by an $\mathcal{O}(r^{1/2})$ leading singular function. While this linear behavior is well documented, the introduction of gradient-dependent semilinear perturbations alters the local asymptotic profile. This article proves that gradient penalties scaling linearly with $|\nabla u|$ induce a highly localized $\mathcal{O}(r^{-1/2})$ source term that resonates with the half-integer spectrum of the principal homogeneous differential operator. This non-orthogonal resonance causes standard separable polynomial assumptions to fail at order $\mathcal{O}(r^{3/2})$. We establish a generalized resonance theorem that forces the emergence of a logarithmic anomaly, providing the exact analytical formulation of the resulting $r^{3/2} \ln(r)$ profile alongside rigorous local Sobolev regularity bounds for the remainder. By calculating the exact logarithmic coefficients and angular offsets for $\ell_1$, $\ell_2$, $\ell_\infty$, and arbitrary $\ell_q$-norm penalties, we demonstrate the universality of this obstruction. Finally, we formalize the corresponding enriched continuous Galerkin space (XFEM), establish its quasi-optimality, and present finite element experiments that confirm the predicted localized pollution, recover the predicted logarithmic coefficients across the $\ell_1$, $\ell_2$, and $\ell_\infty$ penalties on a curvature-free flat junction, and show that resolving the junction recovers optimal degree-of-freedom efficiency in standard finite element solvers; we close by outlining the targeted software architectures required for a fully enriched implementation.

For a prime \(p\), let \(A_p=\{k!\pmod p:1\leq k<p\}\). We prove \(|A_p|\gg p^{8/15}\), improving the general lower bound \((\sqrt{2}-o(1))p^{1/2}\). The proof begins with the identity \((n+2)!=(n+1)!+((n+1)!)^2/n!\) in \(\mathbb{F}_p\), which produces many incidences for a family of fractional-linear maps. After Cauchy–Schwarz, the transition maps between two members of this family become affine lines, with multiplicity at most two. The Cartesian-product point-line incidence theorem of Stevens and de Zeeuw then yields the exponent \(8/15\).

Fix a generic weight vector and a fake exponent of a homogeneous $A$-hypergeometric system. Using all corresponding standard pairs, including embedded ones, we construct an Artinian quotient of the Stanley–Reisner ring of the link of the negative support. Its Hilbert series gives the graded dimensions of the orthogonal complement of the local fake indicial ideal and, under the Okuyama–Saito Frobenius condition, those of the leading logarithmic coefficient space of actual series solutions. The construction requires no Cohen–Macaulay hypothesis. When a top-dimensional standard pair occurs and the link is Cohen–Macaulay, the Hilbert series specializes to the $h$-polynomial of the link.

A Gaussian mixture density can have more modes than components. It has been conjectured that the maximum number of modes of a $d$-variate $k$-component Gaussian mixture density is $\binom{d+k-1}{d}$, which equals six for $(d,k)=(2,3)$. We construct an explicit family of equally weighted heteroscedastic three-component bivariate Gaussian mixture densities with at least seven distinct nondegenerate modes, showing that this conjectured upper bound fails for $(d,k)=(2,3)$. To the best of our knowledge, this provides the first counterexample to the conjecture across all pairs $(d,k)$.