Mathematics

Under appropiate hypotheses on the external force acting on an incompressible flow diffusing through a porous medium, we show that there is a unique stationary solution to the diffusive porous media equation. Moreover, we show that this…

Rado's Conjecture (RC) is a compactness principle for a certain class of partial orders, namely trees $T$ of height $\omega_1$ without cofinal branches, postulating that a partial order $P$ from this class can be decomposed into at most…

This work is concerned with the nonlinear information in the linear sampling method for the inverse medium scattering problem. In addition to the well-known capability in shape characterization, we demonstrate that the imaging indicator…

We construct a form of the $D_4^-$-singularity of fronts in $\R^3$ which uses coordinate transformation on the source and isometry on the target. As an application, we compute differential geometric invariants near the $D_4^-$-singularity,…

This paper characterizes the stabilized second James-Hopf invariant by means of three axioms. Specifically, we show that it is the unique natural transformation satisfying the Cartan formula, vanishing on suspensions, and a metastable EHP…

Let $G/\mathbb{Q}_p$ be a connected, split, reductive group over $\mathbb{Q}_p$. In this paper I show that if $K$ and $L$ are anabelomorphic $p$-adic fields i.e. $K$ and $L$ have topologically isomorphic absolute Galois groups, then the…

This paper develops an a posteriori error analysis framework for decoupled neural approximations of fully coupled forward--backward stochastic differential equations (FBSDEs). It provides an a posteriori error-analysis for the idealized…

In the present note, certain scenarios of potential Type II blowups of solutions to the Navier-Stokes equations are considered on the local level. They generalise particular scenarios described in the previous papers of the author. The main…

We study weak convergence rates of numerical approximations for stochastic Volterra integral equations (SVIEs), a class of non-Markovian models that arises naturally in stochastic volatility modeling and other fields. The intrinsic…

This paper investigates a novel quasi-singularity formation phenomenon in the isentropic compressible Euler equations in $\mathbb{R}^d$ for $d = 2, 3$. For any prescribed finite set of points and any sufficiently large parameter…

We establish handle attachment formulas for the Khovanov skein lasagna module with 1-dimensional inputs over $\mathbb{Q}$, defined recently by Ren, Wedrich, Willis, Zhang, and the second author. For a $4$-manifold built out of $1$- and…

The KKL Theorem, a seminal result in boolean function analysis, characterizes the structure of low-influence (non-expanding) functions on the hypercube. While recent years have seen breakthrough results across a variety of areas relying on…

Let $\mathcal{P}$ be a set of $n$ points in $\mathbb{R}^2$, with a convex hull of size $O(n/\log n)$. We prove that $\Omega(12.24^n)$ plane graphs can be drawn on $\mathcal{P}$, the first non-trivial bound for this problem. We also show…

In this paper, we develop a fractional stochastic neural network with residual dynamics driven by fractional Brownian motion. By introducing a discrete stochastic maximum principle for the network, we construct the corresponding adjoint…

We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of…

A graph homomorphism is an integer-valued function on the vertex set of a graph that assigns values differing by exactly one to adjacent vertices. We consider uniformly random homomorphisms on general finite trees, conditioned to take the…

The main objective of this paper is to give a positive answer to the natural question proposed by Ashot Minasyan: Is the fundamental group of finite graph of conjugacy separable groups with finite edge groups conjugacy separable?

Endo-Pajitnov manifolds are compact non-K\"ahler manifolds which generalize the Inoue surfaces $S_M$ to higher dimensions. We compute their Dolbeault cohomology and show that they satisfy the Hodge decomposition at the level of dimensions.

We give an explicit, cochain-level algebraic model for the pronilpotent completion of a group with finitely generated first cohomology. To each binomial $\cup_1$-dga $(A,d_A)$ over $R=\mathbb{Z}$ or $\mathbb{F}_p$ ($p$ prime) -- a…

We study standard spherical growth rates of right-angled Coxeter groups through the clique polynomial of the defining graph. We prove that every even degree at least four occurs as the degree of a strongly primitive Salem growth rate: for…