For two finite sets of integers A and B their additive energy E(A, B) is the number of solutions to a + b = a + b, where a, a ∈ A and b, b ∈ B. Given finite sets A, B ⊆ Z with additive energy E(A, B) = |A||B| + E, we investigate the sizes of largest subsets A ⊆ A and B ⊆ B with all |A||B| sums a + b, a ∈ A, b ∈ B, being different (we call such subsets A, B co-Sidon). In particular, for |A| = |B...

In this paper we show that the following problem is NP -complete: Given an alphabet Σ and two strings over Σ, the question is whether there exists a permutation of Σ which is a subsequence of both of the given strings.

Let G = (V,E) be a finite (non-empty) graph, where V and E are the sets of vertices and edges of G. An edge magic total labeling is a bijection α from V ∪E to the integers 1,2, . . . , n+e, with the property that for every xy ∈E, α(x)+α(y)+α(xy)= k, for some constant k. Such a labeling is called an a-vertex consecutive edge magic total labeling if α(V )= {a + 1, . . . , a + n} and a b-edge cons...

We show that for any two convex curves C1 and C2 in Rd parametrized by [0, 1] with opposite orientations, there exists a hyperplane H with the following property: For any t ∈ [0, 1] the points C1(t) and C2(t) are never in the same open halfspace bounded by H. This will be deduced from a more general result on equipartitions of ordered point sets by hyperplanes.

We ask, for which n does there exists a k, 1 ≤ k < n and (k, n) = 1, so that k/n has a continued fraction whose partial quotients are bounded in average by a constant B? This question is intimately connected with several other well-known problems, and we provide a lower bound in the case of B = 2.

Let X be a finite set of points in R. The Tukey depth of a point q with respect to X is the minimum number τX(q) of points of X in a halfspace containing q. In this paper we prove a depth version of Carathéodory’s theorem. In particular, we prove that there exists a constant c (that depends only on d and τX(q)) and pairwise disjoint sets X1, . . . , Xd+1 ⊂ X such that the following holds. Each ...

A (v, k, t) directed trade (or simply a (v, k, t)DT) of volume s consists of two disjoint collections Tl and each containing ordered k-tuples of distinct elements of a v-set called blocks, such that the number of blocks containing any t-tuple of V is the same in Tl as in T2 . Our study shows that the volume of a (v, k, t)DT is at least 2Lt/ 2J and that directed trades with minimum volume and mi...

In this paper, we prove Dirichlet’s theorem that, given any pair h, k with (h, k) = 1, there are infinitely many prime numbers congruent to h (mod k). Although this theorem lies strictly within the realm of number theory, its proof employs a range of tools from other branches of mathematics, most notably characters from group theory and holomorphic functions from complex analysis.

sup t>0 ‖η( · )m(t · )‖W 2,β(Rd) ≤ Cη, where ‖ · ‖W 2,β(Rd) is the standard Sobolev norm on R, then the multiplier operator f 7→ F−1(mFf), initially defined on L(R) ∩ L(R), is bounded on L(R) for 1 < p < ∞, and is of weak-type (1,1). Here F denotes the Fourier transform. Let (Ω, d(x, y)) be a metric space equipped with a positive measure μ. We assume that (Ω, d, μ) is a space of homogeneous typ...