Related papers: Upper bound for isometric embeddings \ell_2^m\to\e…
Yudin's lower bound for the spherical designs is generalized to the cubature formulas on the projective spaces over a field K, where K can be R, C, or H (the field of quaternions), and thus to isometric embeddings of l_2 into l_p with p an…
In this paper, we study existence of isometric embedding of $S_q^m$ into $S_p^n,$ where $1\leq p\neq q\leq \infty$ and $n\geq m\geq 2.$ We show that for all $n\geq m\geq 2$ if there exists a linear isometry from $S_q^m$ into $S_p^n$, where…
The main result is that a finite dimensional normed space embeds isometrically in $\ell_p$ if and only if it has a discrete Levy $p$-representation. This provides an alternative answer to a question raised by Pietch, and as a corollary, a…
For some centrally symmetric convex bodies $K\subset \mathbb R^n$, we study the energy integral $$ \sup \int_{K} \int_{K} \|x - y\|_r^{p}\, d\mu(x) d\mu(y), $$ where the supremum runs over all finite signed Borel measures $\mu$ on $K$ of…
We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible…
We construct symplectic embeddings of ellipsoids of dimension $2n \ge 6$ into the product of a 4-ball or 4-dimensional cube with Euclidean space. A sequence of these embeddings can be shown to be optimal.
Given parameters k, l, and d, we give new lower bounds on the dimensions N such that there are maps from R^d to R^N that are k-regular, l-skew embeddings, or k-regular-l-skew embeddings. This extends and sharpens results due to Chisholm…
We give upper bounds for the number $\Phi_\ell(G)$ of matchings of size $\ell$ in (i) bipartite graphs $G=(X\cup Y, E)$ with specified degrees $d_x$ ($x\in X$), and (ii) general graphs $G=(V,E)$ with all degrees specified. In particular,…
The length function $\ell_2(r,R)$ is the smallest length of a binary linear code with codimension (redundancy) $r$ and covering radius $R$. We obtain the following new upper bounds on $\ell_2(r,R)$, which yield a decrease $\Delta(r,R)$…
We prove an exponential upper bound for the number $f(m,n)$ of all maximal triangulations of the $m\times n$ grid: \[ f(m,n) < 2^{3mn}. \] In particular, this improves a result of S. Yu. Orevkov (1999).
We study general properties of holomorphic isometric embeddings of complex unit balls $\mathbb B^n$ into bounded symmetric domains of rank $\ge 2$. In the first part, we study holomorphic isometries from $(\mathbb B^n,kg_{\mathbb B^n})$ to…
It has been known since 1996 that a lower bound for the measure, $\mu(B(x,r))\geq br^s$, implies Sobolev embedding theorems for Sobolev spaces $M^{1,p}$ defined on metric-measure spaces. We prove that, in fact Sobolev embeddings for…
We study embedding a subset $K$ of the unit sphere to the Hamming cube $\{-1,+1\}^m$. We characterize the tradeoff between distortion and sample complexity $m$ in terms of the Gaussian width $\omega(K)$ of the set. For subspaces and several…
In this paper, we provide an method to obtain the lower bound on the number of the distinct maximum genus embedding of the complete bipartite graph Kn;n (n be an odd number), which, in some sense, improves the results of S. Stahl and H.…
The lower bound W(K_{2n})>=3n-2 is proved for the greatest possible number of colors in an interval edge coloring of the complete graph K_{2n}.
Let $\mathcal{M}$ be a smooth submanifold of $\mathbb{R}^n$ equipped with the Euclidean (chordal) metric. This note considers the smallest dimension $m$ for which there exists a bi-Lipschitz function $f: \mathcal{M} \mapsto \mathbb{R}^m$…
We study existence of linear isometric embedding of $\ell_p^m$ into $S_\infty,$ for $1\leq p< \infty$ and unique operator space structure on two dimensional Banach spaces. For $p\in(2,\infty)\cup\{1\},$ we show that indeed $\ell_p^2$ does…
The best known upper estimates for the constants of the Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}$ spaces are of the form $\left(\sqrt{2}\right) ^{m-1}.$ We present better estimates which depend on $p$ and $m$. An…
In this paper, isometric embedding results of Greene, Gromov and Rokhlin are extended to what are called "indefinite metric polyhedra". Two definitions for an indefinite metric polyhedron are given, an intuitive definition and a more useful…
In this paper, we establish several asymptotical bounds for the complete elliptic integrals of the second kind $\mathcal{E}(r)$, and improve the well-known conjecture $\mathcal{E}(r)>\pi[(1+(1-r^2)^{3/4})/2]^{2/3}/2$ for all $r\in(0,1)$…