In this paper, we use trace methods to study the algebraic $K$-theory of rings form $R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2$. We compute relative $p$-adic $K$ groups for $R$ a perfectoid ring. particular, get integral when is finite field, and $K_*(R[x_1,\ldots, x_d)^2, (x_1,\ldots, x_d))$ perfect $\mathbb{F}_p$-algebra. conclude paper with some other notable computations, including which are n...

Let $p$ be a prime. $R$ regular local ring of dimension $d\ge 2$ whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ Cohen the same residue field $k$ as and $h\in C(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo does not belong ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ $k[[x_1,\ldots,x_d]]$. We extend result Vasiu-Zink (for $d=2$) show each Barsotti-Tate...

A combinatorial proof of the Gaussian product inequality (GPI) is given under assumption that each component a centered random vector $\boldsymbol{X} = (X_1, \ldots, X_d)$ arbitrary length can be written as linear combination, with coefficients identical sign, components standard vector. This condition on $\boldsymbol{X}$ shown to strictly weaker than density $(|X_1|, |X_d|)$ multivariate total...

We consider the Dirichlet problem for a class of elliptic and parabolic equations in upper-half space $\mathbb{R}^d_+$, where coefficients are product $x_d^\alpha, \alpha \in (-\infty, 1),$ bounded uniformly matrix coefficients. Thus, singular or degenerate near boundary $\{x_d =0\}$ they may not locally integrable. The novelty work is that we find proper weights under which existence, uniquene...

Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study an set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is density isotropic stable process (or more generally, a pure unimodal process) $\mathbb...

We study a class of second-order degenerate linear parabolic equations in divergence form ( <mml:mo...

Given any diagonal cyclic subgroup $\Lambda \subset GL(n+1,k)$ of order $d$, let $I_d\subset k[x_0,\ldots, x_n]$ be the ideal generated by all monomials $\{m_{1},\ldots, m_{r}\}$ degree $d$ which are invariants $\Lambda$. $I_d$ is a monomial Togliatti system, provided $r \leq \binom{d+n-1}{n-1}$, and in this case projective toric variety $X_d$ parameterized $(m_{1},\ldots, m_{r})$ called $GT$-v...

For a topological dynamical system (X, T), $$l\in \mathbb {N}$$ and $$x\in X$$ , let $$N_l(X)$$ $$L_x^l(X)$$ be the orbit closures of diagonal point $$(x,\ldots ,x)$$ (l times) under actions $$\mathscr {G}_{l}$$ $$\tau _l $$ respectively, where is generated by $$T\times \ldots \times T$$ _l=T\times T^l$$ . In this paper, we show that for minimal T) $$d,l\in maximal d-step pro-nilfactor $$(N_l(X...