相关论文: On the largest subsets avoiding the diameter of $(…

In the present paper, we study problems related to the classical Borsuk's problem. Recall that the Borsuk's problem consists in finding the smallest number $ f(n) $ of parts of smaller diameter into which an arbitrary set of diameter 1 in…

Multiple kernel learning (MKL) algorithms combine different base kernels to obtain a more efficient representation in the feature space. Focusing on discriminative tasks, MKL has been used successfully for feature selection and finding the…

In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin's conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph…

We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we…

We revisit the following problem (along with its higher dimensional variant): Given a set $S$ of $n$ points inside an axis-parallel rectangle $U$ in the plane, find a maximum-area axis-parallel sub-rectangle that is contained in $U$ but…

In this note, we give short proofs of three theorems concerning extremal problems in the Johnson scheme, or, in other terminology, on $(n,k,L)$-systems. The main result is a proof of the Aljohani--Bamberg--Cameron conjecture which claims…

Let $M(x)$ be the length of the largest subinterval of $[1,x]$ which does not contain any sums of two squareful numbers. We prove a lower bound \[ M(x)\gg \frac{\ln x}{(\ln\ln x)^2} \] for all $x\geq 3$. The proof relies on properties of…

Let $k$, $t$ and $m$ be positive integers. A $k$-multiset of $[m]$ is a collection of $k$ integers from the set $\{1,...,m\}$ in which the integers can appear more than once. We use graph homomorphisms and existing theorems for intersecting…

Finding a largest Euclidean ball in a given convex body $K \subset \mathbb{R}^d$ and finding a largest volume ellipsoid in $K$ are two problems of fundamentally different nature. The first is a purely Euclidean problem, where we consider…

Under which conditions and with which distortions can we preserve the pairwise-distances of low-complexity vectors, e.g., for structured sets such as the set of sparse vectors or the one of low-rank matrices, when these are mapped in a…

Let $V$ denote an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. For an $m$-dimensional $\mathbb{F}_q$-subspace $U$ of $V$ assume that $\dim_q \left(\langle {\bf v}\rangle_{\mathbb{F}_{q^n}} \cap U\right) \geq 2$ for each non zero vector…

Let $G$ be an abelian group. A set $A \subset G$ is a \emph{$B_k^+$-set} if whenever $a_1 + \dots + a_k = b_1 + \dots + b_k$ with $a_i, b_j \in A$ there is an $i$ and a $j$ such that $a_i = b_j$. If $A$ is a $B_k$-set then it is also a…

The metric sketching problem is defined as follows. Given a metric on $n$ points, and $\epsilon>0$, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up…

Let $\mathcal{F}$ be a family of subsets of $[n]=\{1,\ldots,n\}$ and let $L$ be a set of nonnegative integers. The family $\mathcal{F}$ is \emph{$L$-intersecting} if $|F\cap F'|\in L$ for every two distinct members $F,F'\in\mathcal{F}$; and…

In [Sau11,SPW13], Saunderson, Parrilo and Willsky asked the following elegant geometric question: what is the largest $m= m(d)$ such that there is an ellipsoid in $\mathbb{R}^d$ that passes through $v_1, v_2, \ldots, v_m$ with high…

The weighted Euclidean norm $\|x\|_w$ of a vector $x\in \mathbb{R}^d$ with weights $w\in \mathbb{R}^d$ is the Euclidean norm where the contribution of each dimension is scaled by a given weight. Approaches to dimensionality reduction that…

Extending a classical theorem of Sperner, we characterize the integers $m$ such that there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$, that is, the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion. As an…

We consider the problem of determining the maximum cardinality of a subset containing no arithmetic progressions of length $k$ in a given set of size $n$. It is proved that it is sufficient, in a certain sense, to consider the interval…

The deletion distance between two binary words $u,v \in \{0,1\}^n$ is the smallest $k$ such that $u$ and $v$ share a common subsequence of length $n-k$. A set $C$ of binary words of length $n$ is called a $k$-deletion code if every pair of…

The symmetry algebras of certain families of quantum spin chains are considered in detail. The simplest examples possess m states per site (m\geq2), with nearest-neighbor interactions with U(m) symmetry, under which the sites transform…