Papers

We consider the semilinear parabolic equation \[ \partial_t u = \Delta u + 2\mathfrak{q}\,\delta_{\mathbb{S}}\,\nabla u + |u|^{p-1}u \qquad \text{in } (0,\infty)\times\mathbb{R}^N, \] where $|\mathfrak{q}|\le 1$, $p>1$, and $\mathbb{S}$ is a fixed interface hyperplane. Working in Lebesgue spaces, we first establish local well-posedness of mild solutions. This is achieved by combining Gaussian bounds for the associated fundamental solution with a contraction mapping argument adapted to the lack of spatial homogeneity induced by the interface term. We then prove a sharp Fujita-type dichotomy for nonnegative solutions. Specifically, we show that every nontrivial solution blows up in finite time when $1<p \le 1+\frac{2}{N}$, whereas for $p>1+\frac{2}{N}$ global solutions exist for sufficiently small initial data. The blow-up analysis relies on a suitably adapted test-function method that accounts for the presence of the interface. It is noteworthy that the critical exponent coincides with the classical Fujita exponent for the heat equation, indicating that the Fujita phenomenon remains stable under the presence of discontinuous diffusion effects and interface transmission conditions. To the best of our knowledge, this is the first result of this type for operators involving a singular drift supported on a hypersurface.

We consider solutions of uniformly elliptic equations with measurable coefficients. We assume that the lowest eigenvalue of the coefficient matrix is at least $K^{-1}$ and the largest eigenvalue is at most $K$. In three and higher dimensions we construct $\alpha$-Hölder continuous solutions with $\alpha = \exp(- c_n K)$. This disproves a long-standing conjecture by showing that, except for the two-dimensional case, the Hölder exponent obtained from the Bombieri–Giusti Harnack inequality has the optimal dependence on the ellipticity constant $K$. Using the same construction as a starting point, we disprove a conjecture of De Giorgi about continuity of solutions to non-uniformly elliptic equations.

We develop a risk-neutral option-pricing model where the activity scale of an infinite-activity jump process is endogenously driven by the asset's own realized price jumps. Jump sizes are governed by a normalized asymmetric tempered-stable Lévy shape, while a predictable activity scale controls the overall jump intensity and is normalized to coincide with the local jump-induced quadratic-variation rate. Endogenous feedback is introduced through the bounded excitation function $g(y)=1-e^{-ay^2}$, so that small realized jumps excite future activity approximately in proportion to squared jump size while the total average excitation remains finite. We construct the coupled log-price and activity-state dynamics by state-dependent thinning of a Poisson random measure, prove pathwise existence and uniqueness, derive the mean-subcriticality condition, and obtain both the risk-neutral drift restriction and a sufficient true-martingale condition. The resulting two-dimensional state process admits an affine transform representation. We derive the associated generalized Riccati system and prove real-axis well-posedness with forward invariance of the relevant complex half-plane. European options are priced by a Fourier-cosine (COS) method, which requires only the real-axis transform, and are benchmarked against a damped Carr–Madan (CM) inversion. Numerical experiments illustrate the model-implied volatility surface and show how current activity shifts near-term volatility levels, while endogenous feedback affects the persistence of jump-induced skew across maturities.

We study the realization of finite groups as automorphism groups of finite posets. Given a finite group $G$, let $\beta(G)$ denote the smallest number of elements in a poset $P$ with $\Aut(P)\cong G$. While $\beta(G)$ is known for several cyclic and small abelian groups, the non-cyclic abelian case is largely open. In this paper we prove that $\beta(\Z_{2}\times\Z_{2^{n}})=2^{\,n+1}+2$ for every $n\ge 3$.

In this paper, we study nonconvex equality-constrained optimization problems in which only stochastic first-order approximations of the objective and constraint functions are available. Owing to the stochasticity in both objective and constraints, most existing stochastic first-order methods incur relatively high oracle complexity, particularly in terms of stochastic constraint function evaluations. To address this issue, we develop a stochastic first-order method based on a decomposed stochastic search direction, and employ Fletcher's augmented Lagrangian as a smooth merit function for step-size selection. To cope with the possible loss of uniform nondegeneracy of the stochastic Jacobian, we introduce an event decomposition based on the smallest singular value, which enables us to control perturbations in the stochastic search direction. Under an additional Lipschitz continuity assumption on the second-order derivatives of the objective and constraint functions, we show that the proposed algorithm attains a stochastic \(\epsilon\)-KKT point with an expected total oracle complexity of \(\mathcal O(\epsilon^{-3})\) in terms of stochastic gradient and stochastic constraint function evaluations. Finally, we present numerical experiments to demonstrate the performance of the proposed method.

Let $P_s(n)$ denote the $n$-th $s$-gonal number. Consider the Diophantine equation $P_{s}(n) = t^{m}$ for integers $n, s, t$ and $m > 2$. All solutions to this equation are known for $m>2$ and $s\in\{3,5,6,8,10,20\}$. Here we extend these results to the cases $s = 2k+4$ (where $k = 4,6$ or $5 \leq k \leq 97$ is a prime number) and $s = k+4$ (where $k = 9,15$ or $3 \leq k \leq 97$ is a prime number). The proofs of our results use the modular and hypergeometric methods, linear forms in logarithms and extensive calculations. We were unable to completely solve the above Diophantine equations, but we expect (based on GRH and the weak effective $abc$ conjecture) that there will be no additional solutions beyond those explicitly shown in Theorems~1, 2 and 3.

Temporal link prediction (TLP) is typically evaluated by predictive performance on unseen edges, but this criterion can conflate predictive accuracy with recovery of the underlying causal mechanism. In stochastic models, Fisher information governs the Cramér–Rao (CR) bound on parameter estimation error: higher Fisher information permits more accurate parameter recovery. We show that, under comonotonicity conditions between Fisher information and entropy, binary logistic models exhibit an estimation–prediction tradeoff: regimes with higher Fisher information, and hence smaller CR bounds, also have higher irreducible predictive entropy. To study this tradeoff in TLP, we introduce a probabilistic causal generator for temporal graphs with transient edges and known ground-truth causal structure, and validate the phenomenon empirically.

This work presents the analysis and numerical simulation of a stationary drift-diffusion model for electrical discharge in micro-electro-mechanical systems (MEMS). The model couples Poisson's equation for the electrostatic potential with continuity equations for positive ions and electrons, incorporating a Townsend-type impact ionization source term that depends exponentially on the electric field magnitude. We prove the existence of weak solutions under physically relevant assumptions and establish uniform bounds on the carrier densities. The proof relies on a regularization-approximation scheme with truncated nonlinearities, monotone operator theory (Browder-Minty), iterative energy estimates, and Stampacchia-type truncation arguments. We further develop a robust finite element solver to simulate the carrier density and electrostatic potential profiles for several geometries, including two-dimensional domains and a three-dimensional axisymmetric geometry.

We establish weighted Gaussian approximations for the uniform empirical and quantile processes and for their increments ending at a fixed point t in (0,1). We first place the classical weighted approximations for the ordinary processes in a common framework and then show that the corresponding increment approximations remain valid uniformly down to the finite-count scale lambda/n, for every fixed lambda > 0. For the empirical increments, the proof splits the sample at t, couples the two resulting conditional empirical processes with independent Brownian bridges, and approximates the binomial fluctuation at t by a Gaussian variable. The three Gaussian components are then combined into a single standard Brownian bridge. For the quantile increments, the Renyi representation and a reversal of the relevant exponential spacings reduce the problem to the weighted approximation of an ordinary uniform quantile process. The resulting bounds hold for 0 <= nu < 1/4 in the empirical case and for 0 <= eta < 1/2 in the quantile case. As an application of the empirical increment approximation, we derive simultaneous weighted Gaussian approximations for the censored and uncensored empirical subdistribution-tail processes arising under random right censoring.

We study the coupon collector's problem in a generalized setting where each draw reveals a fixed number of coupons and the sampling mechanism is required to be fair, meaning that every coupon appears with the same frequency among the admissible draws. Grunbaum and Yaakobi conjectured that, among all fair mechanisms with fixed parameters, the fully random model maximizes the expected time to complete coverage. We disprove this conjecture by exhibiting explicit counterexamples arising from finite geometry. In particular, we show that the line set of the Fano plane yields a fair mechanism whose expected coverage time exceeds that of the full model. Further exact and computational results are obtained for projective planes of higher order. In addition, we analyze a simple infinite family of fair mechanisms, the star mechanism, for which the expected coverage time admits a closed form. Depending on the scaling regime, this mechanism can be asymptotically slower or faster than the full model, showing that no universal extremality principle holds for fair mechanisms without additional structural assumptions.

Let $(T_1,\ldots,T_d)$ be a commuting $d$-tuple of Ritt$_E$ operators on some UMD Banach space $X$. We show that $(T_1,\ldots,T_d)$ admits a bounded $H^\infty$-functional calculus if and only if $T_k$ is an $R$-Ritt$_E$ operator for every $k=1,\ldots,d$, and the $d$-tuple $(T_1,\ldots,T_d)$ admits an isometric dilation $(U_1,\ldots,U_d)$ on some UMD Banach space $Y$ such that $(U_1,\ldots,U_d)$ is polynomially bounded. In the case where $X$ further possesses property $(\alpha)$, we establish other characterizations of the $H^\infty$-functional calculus property for $(T_1,\ldots,T_d)$ in terms of isometric dilations.

We prove that convexity of the Boltzmann entropy at Wasserstein barycenters is strong enough to distinguish Hilbert spaces from general Banach spaces. Thus Wasserstein barycenters provide an intrinsic optimal-transport test for Hilbertian geometry. More precisely, we show that if a finite-dimensional normed vector space, equipped with Lebesgue measure, satisfies the Wasserstein Jensen's inequality for the entropy at barycenters of arbitrary finite families of probability measures, then its norm must be induced by an inner product. This contrasts sharply with a well-known result: every finite-dimensional normed vector space satisfies the nonnegative Ricci curvature condition in the sense of Lott–Sturm–Villani, whereas barycenter convexity excludes all non-Hilbertian norms. As a consequence, smooth reversible Finsler manifolds satisfying the corresponding barycentric curvature-dimension condition have Riemannian tangent norms. The proof does not assume strict convexity of the norm. Its two main ingredients are a rank-one polarization argument, which yields the dual parallelogram identity in the strictly convex case, and a maximal-face trapping argument, which rules out flat faces of the unit ball.

We investigate a variational problem for eigenvalues of the Laplace-Beltrami operator on smooth manifolds with respect to Radon measures belonging to a suitable class; we are motivated by conformal eigenvalues in dimension two. Our main result is a regularity result for stationary measures with respect to outer variations. More precisely, we prove that any sufficiently regular stationary measure is absolutely continuous with respect to the classical volume measure and that its density is induced by an harmonic map. Our result has some interesting applications to Steklov eigenvalues on subdomains.

For minimizers of a degenerate diffusion functional with a singular reaction term, we prove that the free boundary is $(n-1)$-rectifiable. The argument relies on a suitable integrability property, derived from a pointwise gradient estimate, combined with a Hausdorff dimension estimate for a portion of the zero set.

Non-linear dRGT massive and bimetric gravities are complicated theories constructed in terms of square roots of matrices. Apart from the technical issues of successfully working with such square roots, there is also a problem of their non-uniqueness. There are claims in the literature that one should better use the principal root. This is a very reasonable conclusion. However, the motivation they give for it is that otherwise there would be non-primary square roots violating the general covariance. In this paper, I would like to show that, if properly understood, the non-primary square roots are also perfectly covariant. At the same time, I recall the relatively old observation that the real problem with such square roots lies in perturbation theory around them. In terms of matrices, it simply does not exist. In terms of the elementary symmetric polynomials used in the Lagrangian density, it is not analytic. Moreover, the non-principal square roots are more prone to getting into the complex domain.

Construction of Earth-Moon transfers is the basis of missions to explore the Moon and cislunar space. The traditional grid search method suffers from a relatively low convergence rate and computational efficiency, mainly focusing on the distribution of transfer characteristic parameters. Moreover, when constructing transfers with different departure/arrival orbital altitudes, the process of grid search and trajectory correction should be repeated with a low convergence rate and computational efficiency. To address these limitations of the traditional grid search method, this paper is devoted to exploring an effective way to augment the grid search method. Bi-impulsive Earth-Moon transfers from a circular Earth parking orbit to a circular Moon target orbit in the Earth-Moon planar circular restricted three-body problem are considered in this paper. Firstly, the transfers are constructed, and the corresponding solution space is explored in terms of construction parameters, including departure phase angle at the Earth parking orbit, initial-to-circular velocity ratio, and time of flight. An interesting phenomenon about the discontinuous behavior of the time-of-flight distribution with respect to departure phase angle is identified. This phenomenon is further used to train a diffusion model, which aims to augment the traditional grid search method and generate high-quality initial guesses for transfers with different departure/arrival orbital altitudes. The construction results and the preliminary applicable range of the proposed method are presented and analyzed. The proposed diffusion model-augmented grid search method improves the convergence rate by 47.34-56.25$\%$ and saves wall-clock time by 39.39-40.52$\%$ with respect to the traditional grid search method, while ensuring comparable transfer characteristics.

We classify, up to conjugacy, the 3-dimensional subalgebras of the Lie algebras associated with the 4-dimensional Thurston geometries whose isometry groups have dimension 4. Since homogeneous hypersurfaces arise as orbits of subgroups of the isometry group acting transitively on the ambient space, we determine all such subgroups and describe their corresponding orbits, thereby obtaining a classification of the homogeneous hypersurfaces, up to ambient isometries, and we study the geometry of the orbit foliations in these geometries.

We investigate the existence of segregated rotating waves, arising in the singular limit of competition-diffusion systems of the type \[ \partial_t u_i -\partial_{xx} u_i = f(u_i)-\beta u_i \sum_{j \neq i} a_{ij} u_j,\qquad x\in\mathbb{S}^1,\ t>0, 1\le i,j\le k, \] as $\beta\to+\infty$. Here $k\ge3$, the reaction $f$ is of Fisher-KPP (logistic) type, and the competition coefficients $a_{ij}>0$ are not necessarily symmetric. Assuming that, for every $i$, \[ \dfrac{a_{i+1,i}}{a_{i,i+1}}=\lambda>0, \] we provide a complete characterization of the rotating waves enjoying an equivariant structure, where each density is a suitable rotation of any other one: such waves exist if and only if $\lambda$ belongs to an explicit range, in which case the angular velocity $\omega=\omega(\lambda)$ is uniquely prescribed, as is the rotating profile. In particular, stationary solutions (with $\omega=0$) exist only in the symmetric case $\lambda=1$. This marks a strong difference with the same problem with either Dirichlet or Neumann boundary conditions, where it is known that no periodic in time solution exists, also in the asymmetric case, sheding more light on some conjectures and open problems concerning the long time behavior of competition-diffusion systems.

A graph $G$ is $\tau_k$-maximal if $G$ contains no subgraph admitting $k+1$ edge-disjoint spanning trees, while the addition of any edge in the complement of $G$ yields a subgraph that admits $k+1$ edge-disjoint spanning trees. In this paper, we prove that for any integers $k\geq 1$ and $n\geq 2k+2$, every $\tau_k$-maximal graph of order $n$ satisfies $|E(G)|\leq (k+1)(n-1)-1$. Furthermore, we construct a family of $\tau_k$-maximal graphs on $n\ge 2k+2$ vertices that have exactly $(k+1)(n-1)-1$ edges, which establishes the tightness of the upper bound. Then we conjecture that every $\tau_k$-maximal graph on $n$ vertices has exactly $(k+1)(n-1)-1$ edges, and we verify the conjecture for the case $k=1$.

We present a fast, high-order algorithm for the free-space fractional Fokker-Planck equation (FFPE) in arbitrary spatial dimension. Its fundamental solution, corresponding to a Dirac-delta initial condition, is obtained from the explicit Fourier representation by applying a sum-of-Gaussians (SOG) approximation to the nonseparable stretched exponential, using its complete monotonicity as the Laplace transform of a one-sided $\alpha$-stable density. Each Gaussian term is an ordinary heat kernel and therefore factorizes across spatial coordinates. On a tensor-product grid, the separated form can be assembled in $O(MdN)$ work and storage, rather than forming all $O(N^d)$ grid values, where $M$ is the number of Gaussian terms and $N$ is the number of points per dimension. We prove an a~priori error estimate for the pure-fractional fundamental solution and give a parameter-selection procedure for prescribed accuracy over specified ranges of space and time. In numerical experiments the method achieves more than ten digits of relative accuracy, with $M$ growing only logarithmically in the inverse tolerance, and maintains this accuracy in dimensions up to $d=10^{5}$. This exceeds the dimensions reached in comparable radial-quadrature tests, where the integrand becomes increasingly oscillatory as the dimension grows. Because the method represents the fundamental solution as a separated sum of heat kernels, any initial datum given as a finite sum of tensor products can be evolved in closed form using only one-dimensional convolutions. This yields a computable class of high-dimensional solutions that is amenable to error analysis, and tensor neural networks provide one possible way to construct such separated representations for more general data.

Classical works of Hall and McLain show that solubility and local nilpotency play a key role in deriving finite generation in groups from maximal or minimal conditions on normal subgroups. In this work, brace-theoretical analogues of Hall's and McLain's results are analysed for skew braces satisfying the maximal or minimal condition on ideals. We also introduce finiteness and chain conditions on non-degenerate set-theoretic solutions of the Yang-Baxter equation, and their impact on associated structure and permutation skew braces of solutions is also described.

We identify necessary and sufficient conditions for a class of random mappings to send exchangeable $\{0,1\}$-sequences to other exchangeable $\{0,1\}$-sequences. We call this property the propagation of exchangeability, and show that any mapping that propagates exchangeability induces a pair of forward- and backward-in-time processes that are moment duals. This establishes a transparent, tractable, and applicable connection between moment duality and the propagation of exchangeability. We illustrate the usefulness of our results by constructing lookdown models for $\Xi$-Fleming–Viot processes with frequency-dependent selection.

The largest matching root $\mu(G)$ of a graph $G$ is that of its matching polynomial. In this paper, all limit points of the largest matching roots of graphs are determined. More precisely, we identify the limit points of the largest matching roots of graphs less than $\tau^{\frac{1}{2}}+\tau^{-\frac{1}{2}}$. For any $\gamma \geq \tau^{\frac{1}{2}}+\tau^{-\frac{1}{2}}$ with $\tau=\frac{\sqrt{5}+1}{2}$, there exists a graph sequence $\{G_i\, |\, i\in \mathbb{N}\}$ such that $\lim\limits_{i \rightarrow \infty}\mu(G_i)=\gamma$.

Physics-informed neural networks (PINNs) have recently emerged as a promising framework for addressing the Calderón inverse problem from limited boundary data. In this work, we revisit neural Calderón inversion by introducing multiscale boundary excitations based on randomized wavelet functions and investigating the role of Fourier-feature encoding (FFE) for representing sharp conductivity variations. We propose a physics-informed reconstruction framework that represents the unknown conductivity and the associated family of electric potentials with separate neural networks conditioned on the applied boundary excitations. The governing elliptic PDE is enforced through physics-informed residuals, while finite Dirichlet-to-Neumann (DtN) data are incorporated through boundary losses. Using synthetic data from a finite-difference forward solver, we evaluate the method on conductivity fields with inclusions, sharp interfaces, smooth profiles, and heterogeneous media. Results show that the framework recovers dominant conductivity structures from finite boundary measurements with relative errors between $3\%-12\%$ approximately. We show that FFE improves the reconstruction of localized sharp features, particularly for inclusions and interfaces, but are not universally optimal, with raw-coordinate networks performing competitively for smoother fields. These results highlight coordinate representations and boundary excitation design as key factors in neural Calderón inversion.

We prove that there exists a finite unit-distance graph in the plane with independence ratio strictly smaller than 1/4, answering a question of Erdős. Our proof closely follows the framework of Matolcsi, Ruzsa, Varga, and Zsámboki, based on the geometric fractional chromatic number, but adds a carefully chosen two-vertex augmentation that pushes their 27-vertex construction from geometric fractional chromatic number $4$ to a value strictly larger than 4. This disproves their Conjecture 1, and implies that the fractional chromatic number of the plane is strictly larger than 4. The proof can be made fully constructive, but the resulting finite graph has an enormous number of vertices.

We prove that for any matrix $A \in \mathbb{R}^{m \times n}$ and any $\varepsilon > 0$ there is a diagonal matrix $D \in \mathbb{R}_{\geq 0}^{m \times m}$ with at most $O(\frac{n}{\varepsilon^2})$ nonzero entries so that \[ (1-\varepsilon) \|Ax\|_1 \leq \|DAx\|_1 \leq (1+\varepsilon)\|Ax\|_1 \quad \forall x \in \mathbb{R}^n. \] In particular, for any zonotope $Z \subseteq \mathbb{R}^{n}$ there exists a zonotope $Z' \subseteq \mathbb{R}^{n}$ generated by at most $O(\frac{n}{\varepsilon^2})$ segments so that $(1-\varepsilon) Z \subseteq Z' \subseteq (1+\varepsilon) Z$. Previously, the best known bound was $O(\frac{n}{\varepsilon^2} \log n)$ due to Talagrand (1990).

We compute the transitive realizations for the low dimensional cases of real Lie algebras up to dimension four using Shirokov's method. First, the generic realizations are given, then, making use of the known list of subalgebras, nongeneric realizations are computed. The result is compared with the known classification of Popovych et al.

We prove global-in-time strong pathwise well-posedness for a stochastic fluid-structure interaction problem coupling a two-dimensional incompressible Navier-Stokes fluid to a one-dimensional damped Kirchhoff plate. The coupling is imposed on a fixed interface through continuity of velocities and balance of normal stresses, and stochastic forcing, modeled by a cylindrical Wiener process, acts on both the fluid and structure equations. We split the problem into a linear stochastic part and a nonlinear deterministic remainder. The linear stochastic problem is treated by proving that the associated fluid-structure operator admits a bounded \(\mathcal{H}^\infty\)-calculus, yielding stochastic maximal regularity. This requires a decoupling procedure for the non-diagonal operator domain, and pressure estimates via suitable lifting constructions. The deterministic remainder is solved locally by quasilinear methods, and the resulting blow-up criterion is ruled out by higher-order a priori estimates. This is the first global-in-time strong pathwise well-posedness result for a stochastically forced Navier-Stokes system interacting with a deformable elastic structure.

We study a class of nonconvex cardinality-constrained optimization problems arising in sparse learning. These problems are NP-hard due to the combinatorial nature of sparsity constraints. We introduce a Reservoir Zero-Coordinatewise Projected Subspace Search (RZCW-PSS) algorithm, a simplex-style method on sparse manifolds that integrates coordinatewise search, symmetry-aware swap-based support updates, randomized low-dimensional subspace exploration, and zero-coordinatewise reservoir injection. The proposed method augments classical coordinate and swap moves with sparse-compatible subspace searches constructed from a dynamically maintained reservoir of previously accepted feasible points. A key feature of the approach is a refined reservoir initialization strategy that embeds sparse projection directly into a uniform sampling procedure, preserving geometric diversity within the feasible set. The algorithm also includes an optional support-identification safeguard that enforces full-support stabilization under a fixed support-change decrease threshold. We establish that, under the stated regularity, sampling, and subproblem-accuracy assumptions, every full-support accumulation point of the RZCW-PSS iterates is Beck–Hallak zero-coordinatewise stationary almost surely; with the safeguard and full-support initialization, this conclusion applies to all accumulation points. We further prove a conditional local linear convergence rate after support stabilization and derive the corresponding logarithmic local iteration complexity. Numerical experiments on synthetic sparse learning problems demonstrate that RZCW-PSS improves robustness and solution quality while remaining computationally competitive with Partial Simplex Search, Basic Feasible Search, and Zero-Coordinatewise Search methods.

Last-iterate convergence and generalization guarantees in first-order convex learning hinge on the monotonicity of the update operator. While linear averaging preserves the monotonicity of gradient updates, this property is often violated when gradients are aggregated non-affinely, as in modern pipelines enforcing constraints like adaptivity, privacy, robustness or fairness. Whether it is possible to design non-affine aggregation rules that maintain monotonicity has remained an open question. We answer this question negatively: we prove that the monotonicity of aggregated gradients is preserved if and only if the aggregation rule is positively affine. Consequently, non-affine aggregation prevents steady convergence and substantially degrade algorithmic stability. We quantify these drawbacks and propose a path forward by identifying sufficient conditions under which monotonicity can be restored. Our results provide a unified theoretical framework explaining the disparate failure modes observed in modern learning systems.