We consider the systems of hyperbolic equations ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S1) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$ ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+{h(t,x)\left|v\right|}^p$, t > 0, $x\in {ℝ}^N$, (S2) ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{l(t,x)\left|v\right|}^m+k(t,x){\left|u\right|}^q$, t > 0, $x\in {ℝ}^N$, (S3) ⎧$uₜₜ=\Delta \left(a(t,x)u\right)+\Delta \left(b(t,x)v\right)+{h(t,x)\left|u\right|}^p$, t > 0, $x\in {ℝ}^N$, ⎨ ⎩$vₜₜ=\Delta \left(c(t,x)v\right)+{k(t,x)\left|v\right|}^q$, t > 0, $x\in {ℝ}^N$, in $(0,\infty )\times {ℝ}^N$ with u(0,x) = u₀(x), v(0,x) = v₀(x), uₜ(0,x) = u₁(x), vₜ(0,x) = v₁(x). We show that, in each case, there exists a bound B on N such that for 1 ≤ N ≤ B solutions to the systems blow up in finite time.

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B(X^{\epsilon }\left(t\right)$, $𝛯\left(q_1\left(t\right)\right),𝛯\left(q_2\left(t\right)\right),...,𝛯\left(q_{\ell }\left(t\right)\right))$ where $𝛯\left(t\right)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while $q_j\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}lt;{\alpha }_{2}lt;\cdots lt;{\alpha }_k$ and $q_j$, $j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

Let $Y$ be a hyperbolic surface and let $\phi$ be a Laplacian eigenfunction having eigenvalue $-1/4-{\tau }^2$ with $\tau >0$. Let $N\left(\phi \right)$ be the set of nodal lines of $\phi$. For a fixed analytic curve $\gamma$ of finite length, we study the number of intersections between $N\left(\phi \right)$ and $\gamma$ in terms of $\tau$. When $Y$ is compact and $\gamma$ a geodesic circle, or when $Y$ has finite volume and $\gamma$ is a closed horocycle, we prove that $\gamma$ is “good” in the sense of [TZ]. As a result, we obtain that the number of intersections between $N\left(\phi \right)$ and $\gamma$ is $O\left(\tau \right)$. This bound is...

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...

Let $m$ and $n$ be positive integers, and let $R=(r_1,...,r_m)$ and $S=(s_1,...,s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m\times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F\left(R\right)$ be the $m\times n$ $(0,1)$-matrix, where for each $i$, the $i$th row of $F(R,S)$ consists of $r_i$ 1’s followed by $(n-r_i)$ 0’s. Let $A\in A(R,S)$. The discrepancy of A, $\mathrm{disc}\left(A\right)$, is the number of positions in which $F\left(R\right)$ has a 1 and $A$ has a 0. In this paper we investigate linear operators mapping $m\times n$ matrices over...

Let $X$ be a connected closed manifold and $f$ a self-map on $X$. We say that $f$ is almost quasi-unipotent if every eigenvalue $\lambda$ of the map $f_{*k}$ (the induced map on the $k$-th homology group of $X$) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ odd is equal to the sum of the multiplicities of $\lambda$ as eigenvalue of all the maps $f_{*k}$ with $k$ even. We prove that if $f$ is $C^1$ having finitely many periodic points all of them...

In a Tychonoff space $X$, the point $p\in X$ is called a $C^{*}$-point if every real-valued continuous function on $C\setminus \left\{p\right\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^{*}$-point. A classic example is the space ${𝐖}^{*}={\omega }_1+1$ consisting of the countable ordinals together with ${\omega }_1$. The point ${\omega }_1$ is known to be a $C^{*}$-point as well as a $P$-point. We supply a characterization of $C^{*}$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...

Let $S$ be a Riemann surface. Let ${ℍ}^3$ be the $3$-dimensional hyperbolic space and let ${\partial }_{\infty }{ℍ}^3$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $\varphi :S\to {\partial }_{\infty }{ℍ}^3=\widehat{ℂ}$. If $i:S\to {ℍ}^3$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\widehat{ı}$, of $i$ by $\widehat{ı}=N$. Let $\overrightarrow{n}:U{ℍ}^3\to {\partial }_{\infty }{ℍ}^3$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S,\varphi )$ is a convex immersion $i$ of constant Gaussian curvature equal to $k\in (0,1)$ such that the Gauss lifting $(S,\widehat{ı})$ is complete and $\overrightarrow{n}\circ \widehat{ı}=\varphi$. In this...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

For integers $m>r\ge 0$, Brietzke (2008) defined the $(m,r)$-central coefficients of an infinite lower triangular matrix $G=(d,h)={\left(d_{n,k}\right)}_{n,k\in \mathbb{N}}$ as $d_{mn+r,(m-1)n+r}$, with $n=0,1,2,\cdots$, and the $(m,r)$-central coefficient triangle of $G$ as $$G^{(m,r)}={\left(d_{mn+r,(m-1)n+k+r}\right)}_{n,k\in \mathbb{N}}.$$ It is known that the $(m,r)$-central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array $G=(d,h)$ with $h\left(0\right)=0$ and $d\left(0\right),h^{\text{'}}\left(0\right)\ne 0$, we obtain the generating function of its $(m,r)$-central coefficients and give an explicit representation for the $(m,r)$-central Riordan array $G^{(m,r)}$ in terms of the Riordan array $G$....

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 $2^{j(s-n/p)}({\sum }_k|a_k{|}^p{)}^{1/p}\parallel f|A_{pq}^s\parallel$.

For each n ≥ 1, let $v_{n,k},k\ge 1$ and $u_{n,k},k\ge 1$ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means v̅ₙ and u̅̅ₙ, respectively. Let $X{ₙ}^B\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(v_{n,j}-v̅ₙ)$, $X{ₙ}^A\left(t\right)=(1/cₙ){\sum }_{j=1}^{\left[nt\right]}(u_{n,j}-u̅̅ₙ)$, t ≥ 0, and $Xₙ=X{ₙ}^B-X{ₙ}^A$. The main result gives conditions under which the weak convergence $Xₙ\stackrel{}{\to }X$, where X is a Lévy process, implies $X{ₙ}^B\stackrel{}{\to }{X}^B$ and $X{ₙ}^A\stackrel{}{\to }{X}^A$, where $X^B$ and $X^A$ are mutually independent Lévy processes and $X=X^B-X^A$.

For $X,Y\in {𝐌}_{n,m}$ it is said that $X$ is gut-majorized by $Y$, and we write $X{\prec }_{\mathrm{gut}}Y$, if there exists an $n$-by-$n$ upper triangular g-row stochastic matrix $R$ such that $X=RY$. Define the relation ${\sim }_{\mathrm{gut}}$ as follows. $X{\sim }_{\mathrm{gut}}Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$. The (strong) linear preservers of ${\prec }_{\mathrm{gut}}$ on ${ℝ}^n$ and strong linear preservers of this relation on ${𝐌}_{n,m}$ have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ${\sim }_{\mathrm{gut}}$ on ${ℝ}^n$ and ${𝐌}_{n,m}$.

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...