Let $(X,d_X)$, $(\Omega ,d_{\Omega })$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_p$ the p-Wasserstein metric with some p ∈ [1,∞), and by ${}_p\left(X\right)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f:\Omega {\to }_p\left(X\right)$, $\omega \mapsto f_{\omega }$, be a Borel map such that f is a contraction from $(\Omega ,d_{\Omega })$ into ${(}_p\left(X\right),W_p)$. Let ν₁,ν₂ be probability measures on Ω with $W_p(\nu ₁,\nu ₂)$ finite. On X we consider the subordinated measures ${\mu }_i={\int }_{\Omega }{f}_{\omega }d{\nu }_i\left(\omega \right)$. Then $W_p(\mu ₁,\mu ₂)\le W_p(\nu ₁,\nu ₂)$. As an application we show that the solution measures ${\varrho }_{\alpha }\left(t\right)$...

Suppose that $\mu$ is an absolutely continuous probability measure on ${}^R$n, for large $n$. Then $\mu$ has low-dimensional marginals that are approximately spherically-symmetric. More precisely, if $n\ge {(C/\epsilon )}^{Cd}$, then there exist $d$-dimensional marginals of $\mu$ that are $\epsilon$-far from being sphericallysymmetric, in an appropriate sense. Here $C>0$ is a universal constant.

Let ${\alpha }_i,{\beta }_i>0$, 1 ≤ i ≤ n, and for t > 0 and x = (x₁,...,xₙ) ∈ ℝⁿ, let $t•x=(t^{\alpha ₁}x₁,...,t^{\alpha ₙ}xₙ)$, $t\circ x=(t^{\beta ₁}x₁,...,t^{\beta ₙ}xₙ)$ and $\left|\right|x\left|\right|={\sum }_{i=1}^n{\left|x_i\right|}^{1/{\alpha }_i}$. Let φ₁,...,φₙ be real functions in $C^{\infty }(ℝⁿ-0)$ such that φ = (φ₁,..., φₙ) satisfies φ(t • x) = t ∘ φ(x). Let γ > 0 and let μ be the Borel measure on ${ℝ}^{2n}$ given by $\mu \left(E\right)={\int }_{ℝⁿ}{\chi }_E{(x,\phi \left(x\right))\left|\right|x\left|\right|}^{\gamma -\alpha }dx$, where $\alpha ={\sum }_{i=1}^n{\alpha }_i$ and dx denotes the Lebesgue measure on ℝⁿ. Let $T_{\mu }f=\mu \ast f$ and let $\left|\right|T_{\mu }{\left|\right|}_{p,q}$ be the operator norm of $T_{\mu }$ from $L^p\left({ℝ}^{2n}\right)$ into $L^q\left({ℝ}^{2n}\right)$, where the $L^p$ spaces are taken with respect to the Lebesgue measure. The type set $E_{\mu }$ is defined by $E_{\mu }=(1/p,1/q):\left|\right|T_{\mu }{\left|\right|}_{p,q}<\infty ,1\le p,q\le \infty$. In the case ${\alpha }_i\ne {\beta }_k$ for 1 ≤ i,k ≤ n we characterize the...

We consider $C^2$ families $t\mapsto f_t$ of $C^4$ unimodal maps $f_t$ whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure ${\mu }_t$ of $f_t$ depends differentiably on $t$, as a distribution of order $1$. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of ${\mu }_t$ for a Benedicks-Carleson map $f_t$, in terms of a single smooth function and the...

We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space $K_{2n}$ such that $C\left(K_{2n}\right)$ has no uncountable (semi)biorthogonal sequence ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of at most 2n-1 points of $K_{2n}$, but has biorthogonal systems ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves...

Let $G_1$ and $G_2$ be two given graphs. The Ramsey number $R(G_1,G_2)$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_1$ or $\overline{G}$ contains a $G_2$. Parsons gave a recursive formula to determine the values of $R(P_n,K_{1,m})$, where $P_n$ is a path on $n$ vertices and $K_{1,m}$ is a star on $m+1$ vertices. In this note, we study the Ramsey numbers $R(P_n,K_1\vee F_m)$, where $F_m$ is a linear forest on $m$ vertices. We determine the exact values of $R(P_n,K_1\vee F_m)$ for the cases $m\le n$ and $m\ge 2n$, and for the case that $F_m$ has no odd component. Moreover, we...

The Ramsey number $R(G,H)$ for a pair of graphs $G$ and $H$ is defined as the smallest integer $n$ such that, for any graph $F$ on $n$ vertices, either $F$ contains $G$ or $\overline{F}$ contains $H$ as a subgraph, where $\overline{F}$ denotes the complement of $F$. We study Ramsey numbers for some subgraphs of generalized wheels versus cycles and paths and determine these numbers for some cases. We extend many known results studied in [5, 14, 18, 19, 20]. In particular we count the numbers $R(K_1+L_n,P_m)$ and $R(K_1+L_n,C_m)$ for some integers $m$, $n$, where $L_n$ is...

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type $L_{1}u=-{\mu }_{1}v$, $L_{2}v=-{\mu }_{2}u$, on a domain $D$ of ${ℝ}^d$, where ${\mu }_1$ and ${\mu }_2$ are suitable measures on $D$, and $L_1$, $L_2$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${𝒞}^{\infty }$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with $L_1$ and $L_2$, and...

Motivated by recent developments on calculus in metric measure spaces $(X,\mathrm{d},\mathrm{m})$, we prove a general duality principle between Fuglede’s notion [15] of $p$-modulus for families of finite Borel measures in $(X,\mathrm{d})$ and probability measures with barycenter in $L^q(X,\mathrm{m})$, with $q$ dual exponent of $p\in (1,\infty )$. We apply this general duality principle to study null sets for families of parametric and non-parametric curves in $X$. In the final part of the paper we provide a new proof, independent of optimal transportation, of the...

Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets ${}_{\nu }\left(X\right)$ and ${}_{\nu }\left(X\right)$ of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient...

Recently it was proved for 1 < p < ∞ that ${\omega }^m{(f,t)}_p$, a modulus of smoothness on the unit sphere, and $K̃ₘ{(f,t^m)}_p$, a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence ${\omega }^m{(f,t)}_p\approx K̃ₘ{(f,t^r)}_p$ does not hold either for p = ∞ or for p = 1.

A graph $G$ is a $k$-tree if either $G$ is the complete graph on $k+1$ vertices, or $G$ has a vertex $v$ whose neighborhood is a clique of order $k$ and the graph obtained by removing $v$ from $G$ is also a $k$-tree. Clearly, a $k$-tree has at least $k+1$ vertices, and $G$ is a 1-tree (usual tree) if and only if it is a $1$-connected graph and has no $K_3$-minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of $k$-trees as follows: if $G$ is a graph with at least $k+1$ vertices, then $G$ is...

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

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...