We study a continuous time growth process on the $d$-dimensional hypercubic lattice ${𝒵}^d$, which admits a phenomenological interpretation as the combustion reaction $A+B\to 2A$, where $A$ represents heat particles and $B$ inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site...

Take a centered random walk $S_n$ and consider the sequence of its partial sums $A_n:={\sum }_{i=1}^n{S}_i$. Suppose $S_1$ is in the domain of normal attraction of an $\alpha$-stable law with $1lt;\alpha \le 2$. Assuming that $S_1$ is either right-exponential (i.e. $\mathbb{P}(S_{1}gt;x|S_{1}gt;0)={\mathrm{e}}^{-ax}$ for some $agt;0$ and all $xgt;0$) or right-continuous (skip free), we prove that $$\mathbb{P}\{A_{1}gt;0,\cdots ,A_{N}gt;0\}\sim C_{\alpha }{N}^{1/\left(2\alpha \right)-1/2}$$ as $N\to \infty$, where $C_{\alpha }gt;0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

Suppose that $𝒢$ is a finite, connected graph and $X$ is a lazy random walk on $𝒢$. The lamplighter chain $X^{\diamond }$ associated with $X$ is the random walk on the wreath product ${𝒢}^{\diamond }={𝐙}_2\wr 𝒢$, the graph whose vertices consist of pairs $(\underline{f},x)$ where $f$ is a labeling of the vertices of $𝒢$ by elements of ${𝐙}_2=\{0,1\}$ and $x$ is a vertex in $𝒢$. There is an edge between $(\underline{f},x)$ and $(\underline{g},y)$ in ${𝒢}^{\diamond }$ if and only if $x$ is adjacent to $y$ in $𝒢$ and $f_z=g_z$ for all $z\ne x,y$. In each step, $X^{\diamond }$ moves from a configuration $(\underline{f},x)$ by updating $x$ to $y$ using the transition rule of $X$ and then...

The author proved in 2018 that if $G$ is an open subset of a Hilbert space, $f_1,f_2:G\to ℝ$ continuous functions and $\omega$ a nontrivial modulus such that $f_1\le f_2$, $f_1$ is locally semiconvex with modulus $\omega$ and $f_2$ is locally semiconcave with modulus $\omega$, then there exists $f\in C_{\text{loc}}^{1,\omega }\left(G\right)$ such that $f_1\le f\le f_2$. This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of ${ℝ}^n$). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to $L^p$ spaces, $p\in [2,\infty )$. We...

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k\ge 2$ and every set $A$ of words over $k$ satisfying $\lim {\sup }_{n\to \infty }|A\cap {\left[k\right]}^n|/k^n>0$ there exist a word $c$ over $k$ and a sequence $\left(w_n\right)$ of left variable words over $k$ such that the set $c\cup \{c^{}{w}_0{\left(a_0\right)}^{}..{.}^{}{w}_n\left(a_n\right):n\in \mathbb{N}\text{and}{a}_0,...,a_n\in \left[k\right]\}$ is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

Let $X$ be a proper, smooth, geometrically connected curve over a $p$-adic field $k$. Lichtenbaum proved that there exists a perfect duality: $$\text{Br}\left(X\right)\times \text{Pic}\left(X\right)\to \mathbb{Q}/\mathbb{Z}$$ between the Brauer and the Picard group of $X$, from which he deduced the existence of an injection of $\text{Br}\left(X\right)$ in ${\displaystyle \prod _{P\in X}\text{Br}\left(k_P\right)}$ where $P\in X$ and $k_P$ denotes the residual field of the point $P$. The aim of this paper is to prove that if $G=\tilde{G}$ is an $X_{et}$- scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results...

We consider random walk on a discrete torus $E$ of side-length $N$, in sufficiently high dimension $d$. We investigate the percolative properties of the vacant set corresponding to the collection of sites which have not been visited by the walk up to time $u{N}^d$. We show that when $u$ is chosen small, as $N$ tends to infinity, there is with overwhelming probability a unique connected component in the vacant set which contains segments of length const $logN$. Moreover, this connected component occupies a...

In this paper, we study the size of the giant component $C_G$ in the random geometric graph $G=G(n,r_n,f)$ of $n$ nodes independently distributed each according to a certain density $f(·)$ in ${[0,1]}^2$ satisfying ${inf}_{x\in {[0,1]}^2}f\left(x\right)gt;$ $0$. If $\frac{c_1}{n}\le r_n^2\le c_2\frac{logn}{n}$ for some positive constants $c_1$, $c_2$ and $n{r}_n^2⟶\infty$ as $n\to \infty$, we show that the giant component of $G$ contains at least $n-\mathrm{o}\left(n\right)$ nodes with probability at least $1-{\mathrm{e}}^{-\beta n{r}_n^2}$ for all $n$ and for some positive constant $\beta$. We also obtain estimates on the diameter and number of the non-giant components of $G$.

Let $\mathrm{T}$ be the family of all typically real functions, i.e. functions that are analytic in the unit disk $\Delta :=\{z\in ℂ:|z|<1\}$, normalized by $f\left(0\right)=f^{\text{'}}\left(0\right)-1=0$ and such that Im $z$ Im $f\left(z\right)$ $\ge 0$ for $z\in \Delta$. Moreover, let us denote: ${\mathrm{T}}^{\left(2\right)}:=\{f\in \mathrm{T}:f\left(z\right)=-f(-z)\text{for}z\in \Delta \}$ and ${\mathrm{T}}^{M,g}:=\{f\in \mathrm{T}:f\prec Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}\cap \mathrm{S}$ and $\mathrm{S}$ consists of all analytic functions, normalized and univalent in $\Delta$.We investigate  classes in which the subordination is replaced with the majorization and the function $g$ is typically real but does not necessarily univalent, i.e. classes $\{f\in \mathrm{T}:f\ll Mg\text{in}\Delta \}$, where $M>1$, $g\in \mathrm{T}$, which we denote...

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring $R$. Our focus is on homological properties of contracting endomorphisms of $R$, e.g., the Frobenius endomorphism when $R$ contains a field of positive characteristic. For instance, in this case, when $R$ is $F$-finite and $C$ is a semidualizing $R$-complex, we prove that the following conditions are equivalent: (i) $C$ is a dualizing $R$-complex; (ii) $C\sim 𝐑{\mathrm{Hom}}_R{(}^{n}R,C)$ for some $n>0$; (iii) ${\mathrm{G}}_C{\text{-dim}}^{n}R<\infty$ and $C$ is derived...

Let $f^j={\sum }_k{a}_{k}f(2^{j+1}x-2k)$, where the sum is taken over the lattice of all points k in ${ℝ}^n$ having integer-valued components, j∈ℕ and $a_k\in ℂ$. Let $A_{pq}^s$ be either $B_{pq}^s$ or $F_{pq}^s$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on ${ℝ}^n.$ The aim of the paper is to clarify under what conditions $\parallel f^j|A_{pq}^s\parallel$ is equivalent to $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}\left(V\right):=dim{H}^0(V,12K_V)\&gt;0$ and $P_{m_0}\left(V\right)\&gt;1$ for some positive integer $m_0\le 24$. A direct consequence is the birationality of the pluricanonical map ${\varphi }_m$ for all $m\ge 126$. Besides, the canonical volume $\text{Vol}\left(V\right)$ has a universal lower bound $\nu \left(3\right)\ge \frac{1}{63·{126}^2}$.

Let $\Omega \in L^s\left({\mathrm{S}}^{n-1}\right)$ for $s\ge 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral ${\mu }_{\Omega }$ and $b$ is defined by $${\displaystyle [b,{\mu }_{\Omega }]\left(f\right)\left(x\right)=({\int }_0^{\infty }{\left|{\int }_{|x-y|\le t}\frac{\Omega (x-y)}{{|x-y|}^{n-1}}[b\left(x\right)-b\left(y\right)]f\left(y\right)\mathrm{d}y{|}^2\frac{\mathrm{d}t}{t^3}\right)}^{1/2}.}$$ In this paper, the author proves the $(L^{p(·)}\left({ℝ}^n\right),L^{p(·)}\left({ℝ}^n\right))$-boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ and its commutator $[b,{\mu }_{\Omega }]$ when $p(·)$ satisfies some conditions. Moreover, the author obtains the corresponding result about ${\mu }_{\Omega }$ and $[b,{\mu }_{\Omega }]$ on Herz spaces with variable exponent.