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arXiv:2605.21769v3 Announce Type: replace Abstract: We investigate the finite-time blow-up of solutions to a Tricomi-type equation with scale-invariant potential and power nonlinearities in the oscillatory regime. For smooth, compactly supported, nonnegative initial data, we prove nonexistence of global-in-time solutions when the power nonlinearity lies below the positive root of an explicit Strauss-type polynomial naturally associated wit....
arXiv:2605.22652v3 Announce Type: replace Abstract: We study inequalities between integer-valued knot invariants arising from classical knot theory, four-dimensional topology, knot homologies, and knot polynomials. We present a directed graph consisting of 46 inequalities between 33 knot invariants. Using these inequalities together with parity constraints, we construct and propagate a database NewDB, for knots up to 13 crossings, extendin..
arXiv:2605.22745v2 Announce Type: replace Abstract: We develop a first and second fundamental theorem for $n$--tuples of bosonic and fermionic matrices, by developing graded analogues of the classical case.
arXiv:2605.24148v2 Announce Type: replace Abstract: We address a new case of the Harris-Viehmann conjecture, which establishes a parabolic induction formula on the cohomology groups associated to non-basic local Shimura data. It follows that all supercuspidal representations on a Shimura variety are concentrated along the basic locus, making the conjecture relevant to the Langlands program. Historically, many cases of the Harris-Viehmann c..
arXiv:2605.24349v2 Announce Type: replace Abstract: This paper studies generalized P\'olya conversion problems for the $q$-permanent. We establish a sharp threshold governing the transition from low-dimensional algebraic flexibility to higher-dimensional combinatorial rigidity. For $n \ge 3$ and $q \ne \pm 1$, we prove that the $q$-permanent is not linearly convertible to the determinant or permanent. Conversely, we completely classify the....
arXiv:2605.24511v2 Announce Type: replace Abstract: Pipedreams and bumpless pipedreams are two combinatorial models that compute double Grothendieck polynomials. While studying matrix Schubert varieties, Pechen Pipedreams and marked bumpless pipedreams are two combinatorial models that compute double Grothendieck polynomials. While studying matrix Schubert varieties, Pechenik, Speyer, and Weigandt defined the Rajchgot code, denoted by $\ma..
arXiv:2605.25037v2 Announce Type: replace Abstract: Given $s\in(1,2]$, define $$H_s[0,1]=\{f\in C[0,1]:{\dim}_HG_f([0,1])=s\}$$ and $$\overline{B}_s[0,1]=\{f\in C[0,1]:\overline{{\dim}}_BG_f([0,1])=s\}.$$ The main goal of this paper is to study the $(\alpha,\beta)$-lineability/spaceability of the sets $H_s[0,1]$ and $\overline{B}_s[0,1]$. As a principal result, we prove that $H_s[0,1]$ is $(p,\mathfrak{c})$-spaceable for $p=1,2$ and also $..
arXiv:2605.25097v2 Announce Type: replace Abstract: In this paper, we prove that weak solutions to the 2D viscous and resistive magnetohydrodynamic (MHD) equations are non-unique in $L^2_t L^p(\mathbb{R}^2) \cap L^1_t W^{1,p}(\mathbb{R}^2)$ for given any $1\le p<\infty$, showing the sharpness of the Ladyzhenskaya--Prodi--Serrin condition at the endpoint $(2,\infty)$ and the solutions live on the borderline of the Beale--Kato--Majda criteri..
arXiv:2605.25637v2 Announce Type: replace Abstract: We study sharp weighted Sobolev-type inequalities of the form\[ \int_{0}^{1}|u(x)|\rho(x) \,\mathrm{d}x \leqslant \Lambda \Bigl(\int_{0}^{1}|u^{(k)}(x)|^2 \,\mathrm{d}x \Bigr)^{1/2}, \qquad u\in H_0^k(0,1), \]where $\rho$ is a non-negative weight. We characterize the minimizers and identify the optimal constant $\Lambda(k,\rho)$ by showing that every minimizer has a constant sign and..
arXiv:2605.25666v2 Announce Type: replace Abstract: Motivated by the recent approach of Milman, Shabelman, and Yehudayoff \cite{MilmanShabelmanYehudayoff2025}, we establish, for $p>1$, a complete characterization of the fixed points of the composition of the $L_p$-centroid operator and the polar $L_p$-projection operator. More precisely, for $p>1$, we prove that if a convex body $K \in \mathcal{K}_o^n$ satisfies \[ \Gamma_p \Pi_p^* K = cK ....
arXiv:2605.26439v2 Announce Type: replace Abstract: This note studies the 1D stochastic heat equation driven by a one-dimensional Brownian motion. We prove that the associated Markov process satisfies the strong Feller property under mild non-degeneracy conditions. The approach combines Malliavin calculus with the moment method from PDE control theory.
arXiv:2605.26775v2 Announce Type: replace Abstract: We generalize and prove the recursive relation \[ S_{\lambda}(V) = \sum_{L\subseteq V\text{ line}} S_{\lambda}(V \mathbin{/\mkern-5mu/} L) \] conjectured by I. G. Macdonald for his "7th variation" of the Schur functions. This variation is a family of polynomials over a finite field that mimic the (straight and skew) Schur polynomials using powers of the Frobenius.
arXiv:2605.26846v3 Announce Type: replace Abstract: We begin with the observation that the signed generalized Stirling polynomials $P_k(m,x)$, which occur in a generalization of Malmsten's integral, reduce to the falling factorials when $k=m$. The structure of these generalized Stirling polynomials is then used to obtain recurrence relations, gamma--polygamma formulas for the polynomials $P_{m-s}(m,x)$, a more transparent proof of a vanish....
arXiv:2605.26963v2 Announce Type: replace Abstract: Absolutely indecomposable vector bundle and parabolic vector bundles are well-studied via quiver representations. In this paper, we study absolutely indecomposable quasi-parabolic $G$-bundles over $\mathbb{P}^1$ with generic additive character varieties. Furthermore, we give a geometric interpretation of the multiplicity of the tensor product of irreducible characters of finite reductive ..
arXiv:2605.27302v2 Announce Type: replace Abstract: Our aim is to study matrix polynomials over max-algebras and their growth in terms of max-induced seminorms. In particular, we compare the set growth of a bounded family $\Psi$ of matrix polynomials, measured in terms of the seminorms $\eta_{\|\cdot\|}$ and $\hat{\eta}_{\|\cdot\|}$ with the induced joint spectral radius of the coefficient pool $\Psi_0$ of the matrix polynomials. Dynamics ..
arXiv:2605.28002v2 Announce Type: replace Abstract: Although irregular vectors for the Virasoro algebra are widely used in modern mathematical physics, a rigorous existence and uniqueness theorem in arbitrary rank has not been available in the literature. In this paper, we develop an algebraic framework, based on Virasoro differential operators on the parameter space, which gives such a theorem for arbitrary integer and half-integer ranks.....
arXiv:2605.28503v2 Announce Type: replace Abstract: We consider interpolation-based derivative-free optimization in settings where only some derivatives are available. Such situations arise in scientific computing applications involving simulations, adjoint-enabled components, legacy software, or partially differentiable models. We introduce a Birkhoff interpolation framework that permits arbitrary patterns of derivative availability and e....
arXiv:2605.28970v2 Announce Type: replace Abstract: In this article, we study mixed Killing vector fields, defined by the condition $L_V L_V g = f\,L_V g$, on Cigar Ricci--Bourguignon solitons. While conformal vector fields are always mixed Killing, the converse fails in flat and open cylinders with base manifold geometries, where the mixed Killing class is infinite-dimensional. We establish a rigidity phenomenon for Cigar Ricci--Bourguign....
arXiv:2605.29553v2 Announce Type: replace Abstract: We determine the sharp threshold for Hamilton cycles in randomly perturbed sparse graphs. For any $\alpha=\alpha(n)=o(1)$, let $G_{\alpha}$ be an $n$-vertex graph with minimum degree $\delta(G_{\alpha})\ge\alpha n$. We prove that if $$p\ge(1+\varepsilon)\frac{\log(1/\alpha)}{n},$$ then the union $G_{\alpha}\cup G(n,p)$ is Hamiltonian asymptotically almost surely. This significantly streng..
arXiv:2605.30285v2 Announce Type: replace Abstract: Given a finite abelian group $G$ and a Sylow $p$-subgroup $N_p$, we prove that the $KU_G/p$-local sphere spectrum is equivalent to the homotopy fixed points of a $p$-complete $KO_{N_p}$-module spectrum. Then we compute the $\mathbb{Z}$-graded homotopy Mackey functors of the $KU_G$-local sphere spectrum. This result generalizes the computation of arXiv:2303.12271 for finite $p$-groups, whe..
arXiv:2605.30379v2 Announce Type: replace Abstract: We study the convolution product of matrix-valued sequences and its role in the computation of Markov renewal equations. Explicit representations and recursive formulae for the convolutional inverse are derived and used to construct FFT-accelerated convolution and Newton-type inversion schemes, together with a Gauss--Jordan alternative in truncated power-series rings. The proposed framewo..
arXiv:2605.30823v2 Announce Type: replace Abstract: We prove interior $C^{2}$ regularity result for convex viscosity solutions of the quadratic Hessian equation $\sigma_2(D^2u) = f(x)$, under the assumption that $f\in C^{0,1}$ with $\inf f>0$. The result is almost sharp: if $f$ are merely continuous, there exist convex viscosity solutions that fail to be $C^{1,1}$. When $f\in C^{\alpha}$ for some $\alpha\in (0,1)$, the corresponding interi..
arXiv:2605.30845v2 Announce Type: replace Abstract: Model-based derivative-free trust-region methods build local interpolation models and restrict trial steps to regions where those models are reliable. This paper studies the shape of that region. When an objective is poorly scaled or locally anisotropic, a Euclidean ball can be governed by the steepest local direction and can restrict progress along directions of slow variation. We propos....
arXiv:2605.30921v2 Announce Type: replace Abstract: We consider an incompressible Navier--Stokes system nonlinearly coupled with a conserved Allen--Cahn equation with a singular potential (e.g., of Flory--Huggins type). This model describes a mass-conserving two-phase flow with constant density and non-constant viscosity. First, in three spatial dimensions, we prove the existence and uniqueness of local-in-time strong solutions to the asso....
Beyond Outerplanarity - arxiv.org - 3 months ago - eng
arXiv:1708.08723v4 Announce Type: replace-cross Abstract: We study straight-line drawings of graphs where the vertices are placed in convex position in the plane, i.e., \emph{convex drawings}. We consider two families of graph classes with convex drawings: \emph{outer $k$-planar} graphs, where each edge is crossed by at most $k$ other edges; and \emph{outer $k$-quasi-planar} graphs, where no $k$ edges can mutually cross. We show that the ....
arXiv:2211.04697v5 Announce Type: replace-cross Abstract: Sensitivity analysis for the unconfoundedness assumption is crucial in observational studies. For this purpose, the marginal sensitivity model gained popularity recently due to good interpretability and mathematical properties. However, most existing models only consider a worst-case parameter that bounds the logit difference between the observed and full data propensity scores, whi....
arXiv:2212.07944v4 Announce Type: replace-cross Abstract: We study a multi-factor block model for variable clustering and connect it to regularized subspace clustering through a distributionally robust version of nodewise regression. To solve the latter problem, we derive a convex relaxation, provide a data-driven approach for selecting the size of the robust region, and develop an ADMM algorithm for efficient implementation. We validate o..
arXiv:2307.03391v4 Announce Type: replace-cross Abstract: This paper proposes a unified adaptive portfolio-management framework that combines factor-based view generation, Black-Litterman (BL) posterior estimation, EWMA covariance estimation, and mean-variance optimization. The key mechanism is a dynamic sliding window that adjusts the estimation horizon according to realized portfolio volatility, thereby updating factor estimates, BL post..
arXiv:2310.20545v3 Announce Type: replace-cross Abstract: We present a multi-task optimization approach based on a deep learning architecture for time series forecasting. We leverage large collections of time series to identify the weights of forecasting models that can be combined to produce forecasts for each series. This method jointly addresses two tasks: the selection of different forecasting models, and their effective combination. I....
arXiv:2403.06588v3 Announce Type: replace-cross Abstract: Recently it was shown that the response time of First-Come-First-Served (FCFS) scheduling can be stochastically and asymptotically improved upon by the {\it Nudge} scheduling algorithm in case of light-tailed job size distributions. Such improvements are feasible even when the jobs are partitioned into two types and the scheduler only has information about the type of incoming jobs ....
arXiv:2408.11266v5 Announce Type: replace-cross Abstract: Deep learning is now common across many scientific fields, including the study of partial differential equations. This article provides a brief, accessible introduction to core deep learning concepts, including neural networks, backpropagation, and the universal approximation theorem. It mainly covers how to use deep learning in solving differential equations. The article aims to he....
arXiv:2410.08150v5 Announce Type: replace-cross Abstract: We use simple martingale methods to construct a large deviation theory of spin systems with pairwise interactions. As an application, we show that the fully connected case obeys a universal scaling limit that is just a product of magnetisation eigenstates.
arXiv:2412.15378v3 Announce Type: replace-cross Abstract: The functioning of complex systems depends on the coordination of diverse components, often supported by regulatory structures that incur costs. In human organizations, such costs manifest as administrative burden, which has been rising despite often reducing efficiency. Classic explanations point to bureaucrat self-interest or regulation, yet they do not explain variation across or....
arXiv:2412.19444v2 Announce Type: replace-cross Abstract: Optimization algorithms such as AdaGrad and Adam have significantly advanced the training of deep models by dynamically adjusting the learning rate during the optimization process. However, ad-hoc tuning of learning rates poses a challenge and leads to inefficiencies in practice. To address this issue, recent research has focused on developing ``parameter-free'' algorithms that oper....
arXiv:2501.18798v3 Announce Type: replace-cross Abstract: The Antibody Mediated Prevention (AMP) trials opened a new scientific frontier by showing that passively administered monoclonal broadly neutralizing antibodies (bnAbs) could prevent HIV-1 acquisition. Conducted across multiple geographic regions, including the United States, Brazil, Peru, Switzerland, and sub-Saharan Africa, the AMP trials revealed substantial regional heterogeneit....
arXiv:2504.04856v2 Announce Type: replace-cross Abstract: We approach the problem of constructing a quantum analogue of the immensely fruitful classical transport cost theory of Monge from a new angle. Going back to the original motivations, by which the transport is a bilinear function of a mass distribution (without loss of generality a probability density) and a transport plan (a stochastic kernel), we explore the quantum version where ....
arXiv:2505.18113v2 Announce Type: replace-cross Abstract: Training quantized neural networks requires addressing the non-differentiable and discrete nature of the underlying optimization problem. To tackle this challenge, the straight-through estimator (STE) has become the most widely adopted heuristic, allowing backpropagation through discrete operations by introducing biased yet valid surrogate gradients. However, its theoretical propert....
arXiv:2506.21278v3 Announce Type: replace-cross Abstract: We propose spherical Cauchy (spCauchy) latent variables for variational autoencoders on hyperspherical latent spaces. The spCauchy family has heavy-tailed global behavior and admits an exact differentiable reparameterization by applying a M\"obius transformation to uniform samples on the sphere. We show that, in the high-concentration limit, spCauchy recovers the local tangent-space....
arXiv:2507.15567v3 Announce Type: replace-cross Abstract: In this second work in a series, we prove the local-in-time well-posedness of the IBVP for the vacuum Einstein equations in general relativity with twisted Dirichlet boundary conditions on a finite timelike boundary. The boundary conditions consist of specification of the pointwise conformal class of the boundary metric, together with a scalar density involving a combination of the ..
arXiv:2508.14836v4 Announce Type: replace-cross Abstract: The space discreteness hypothesis asserts that the nature of space at short distances is radically different from that at large distances. Based on the Bronstein inequality, here, we use a totally disconnected topological space $\mathcal{X}$ as a model for the space. However, we consider the time as a real variable. In this framework, the formalism of Dirac-von Neumann can be used. ....
arXiv:2509.02293v2 Announce Type: replace-cross Abstract: In quantum gravity, the gravitational path integral involves a sum over topologies, representing the joining and splitting of multiple universes. To account for topology change, one is led to allow the creation and annihilation of closed and open universes in a framework often called third quantization or universe field theory. We argue that since topology change in gravity is a rar....
arXiv:2509.04631v2 Announce Type: replace-cross Abstract: Transductive conformal prediction addresses the simultaneous prediction for multiple data points. Given a desired confidence level, the objective is to construct a prediction set that includes the true outcomes with the prescribed confidence. We demonstrate a fundamental trade-off between confidence and efficiency in transductive methods, where efficiency is measured by the size of ....
arXiv:2509.05563v2 Announce Type: replace-cross Abstract: High-dimensional compositional data pose unique statistical challenges due to the simplex constraint and excess zeros. While dimension reduction is indispensable for analyzing such data, conventional approaches often rely on log-ratio transformations that compromise interpretability and distort the data through ad hoc zero replacements. To address these issues, we introduce a geomet....
arXiv:2510.03745v2 Announce Type: replace-cross Abstract: Low-discrepancy points are designed to efficiently fill the space in a uniform manner. This uniformity is highly advantageous in many problems in science and engineering, including in numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. Whereas most previous low-discrepancy constructions rely on abstract algebra and number ....
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