Mathematics
We consider the following quasilinear critical problem involving the mixed local-nonlocal operator: \begin{equation}\label{main_prob_abstract_1}\tag{$\mathcal{P}_p$} -\Delta_p u+(-\Delta_p)^s u=|u|^{p^*-2}u+f(x)\text{ in }\mathbb{R}^N,…
Recently, Escobar, Harada, and Manon introduced the theory of polyptych lattices. This theory gives a general framework for constructing projective varieties from polytopes in a polyptych lattice. When all the mutations of the polyptych…
Bayesian inverse problems use data to update a prior probability distribution on uncertain parameter values to a posterior distribution. Such problems arise in many structural engineering applications, but computational solution of Bayesian…
We investigate planar piecewise-smooth vector fields with a discontinuity line, focusing on the bifurcation of crossing limit cycles that arise when one of the vector fields is translated along the discontinuity set. We establish…
This paper develops a mean field game framework for dynamic two-sided matching markets, extending existing matching theory by integrating micro-macro dynamics in two-sided environments. Unlike traditional matching models focusing on static…
J. C. C. Nitsche proved that any minimal disk satisfying the free boundary condition in the unit ball $B^3$ must be an equatorial flat disk. Later, Fraser and Schoen extended this rigidity theorem to higher dimensions and to ambient spaces…
A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…
We extend the Euler Poincare formalism from Lie groups to Lie groupoids for optimal control problems. While Lie algebroids provide the standard infinitesimal framework, the groupoid formulation enables global trajectory reconstruction and…
Sustainable irrigation planning requires balancing economic benefits with environmental flow requirements under increasing climatic and resource constraints. Building on the irrigation optimization framework developed by Ullah and Nehring,…
Nonlinear PDEs give rise to complex dynamics that are often difficult to analyze in state space due to their relatively large numbers of degrees of freedom, ill-conditioned operators, and changing spatial and parameter resolution…
Understanding how a system loses memory of its initial state is a central problem in probability and statistics. In this manuscript, we introduce the notion of abrupt decorrelation, which explicitly characterises a sharp and sudden loss of…
We propose and analyze a randomization scheme for a general class of impulse control problems. The solution to this randomized problem is characterized as the fixed point of a compound operator which consists of a regularized nonlocal…
The Tomas-Stein inequality for a compact subset $\Gamma$ of the sphere $S^d$ states that the mapping $f\mapsto \widehat{f\sigma}$ is bounded from $L^2(\Gamma,\sigma)$ to $L^{2+4/d}(\R^{d+1})$. Then conditional on a strict comparison between…
We first prove a Boundary Schwarz lemma for holomorphic disks on the unit ball in $\mathbb{C}^n$. Further by using a Schwarz lemma for minimal conformal disks of Forstneri\v c and Kalaj (F.~Forstneri{\v{c}} and D.~Kalaj. \newblock…
Conjugate gradient (CG) methods are widely acknowledged as efficient for minimizing continuously differentiable functions in Euclidean spaces. In recent years, various CG methods have been extended to Riemannian manifold optimization, but…
We prove that the Hardy--Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(X,d,\mu)$, with $1<p_-\le p_+<\infty$, over an unbounded space of homogeneous type $(X,d,\mu)$ with a Borel-semiregular measure…
The conjugate function method is an algorithm for numerical computation of conformal mappings for simply and multiply connected domains on surfaces. In this paper the conjugate function method, earlier used for simply connected domains, is…
In this preprint we discuss a definition of a $q$-ary graph. Furthermore, we describe how to make an incidence matrix for it, with an eye on the corresponding $q$-matroid.
George Andrews and Mohamed El Bachraoui recently explored identities for two-color partitions. In particular, they studied the connection between two-colored partitions and overpartitions. Their proofs were analytical, but they conjectured…
The Butcher theory provides a powerful tool for analyzing order conditions of Runge-Kutta schemes for ordinary differential equations (ODEs); however, such a theory has not yet been well established for backward stochastic differential…