We consider a natural resource allocation problem in which we are given a set of items, where each item has a list of pairs associated with it. Each pair is a configuration of an allowed size for this item, together with a nonnegative penalty, and an item can be packed using any configuration in its list. The goal is to select a configuration for each item so that the number of unit bins needed...

Dynamic resource allocation is an important means to increase the sum capacity of fading multiple access channels. In this paper we consider vector multiaccess channels (channels where each user has multiple degrees of freedom) and study the eeect of power allocation as a function of the channel state on the sum capacity (or spectral eeciency) deened as the maximum sum of rates of users per uni...

It is shown that the set {1, 2, . . . , n} contains at most 2n/2−2n maximal sum-free subsets, provided n is large enough. A set A ⊆ [n] = {1, 2, . . . , n} is sum-free if for any two elements a, b ∈ A we have a + b / ∈ A. A sum-free set A ⊆ [n] is maximal if it is not contained in any other sum-free subset of [n]. Let s(n) and smax(n) denote the number of sum-free and maximal sum-free subsets o...

A graph G is called a sum graph if there exists a labelling of the vertices of G by distinct positive integers such that the vertices labelled u and v are adjacent if and only if there exists a vertex labelled u + v. If G is not a sum graph, adding a nite number of isolated vertices to it will always yield a sum graph, and the sum number (G) of G is the smallest number of isolated vertices that...

A sequence S of terms from an abelian group is zero-sum if the sum of the terms of S is zero. In 1961 Erdős, Ginzburg and Ziv proved that any sequence of 2m− 1 terms from an abelian group of order m contains an m-term zero-sum subsequence [10]. This sparked a flurry of generalizations, variations and extensions [1] [3] [7] [8] [11] [13] [14] [15] [16] [17] [18] [22] [26] [27] [28] [37]. Since a...

A graph G=(V; E) is said to be an integral sum graph (sum graph) if its vertices can be given a labeling with distinct integers (positive integers), so that uv ∈ E if and only if u+ v ∈ V . The integral sum number (sum number) of a given graph G, denoted by (G) ( (G)), was de:ned as the smallest number of isolated vertices which when added to G result in an integral sum graph (sum graph). In th...

Let G be an additive finite abelian p-group. For a given (long) sequence S in G and some element g ∈ G, we investigate the number of subsequences of S which have sum g. This refines some classical results of J.E. Olson and recent results of I. Koutis.

In this paper, we consider a wireless communication system where a large intelligent surface (LIS) is deployed comprising a number of small and distributed LIS-Units. Each LISUnit has a separate signal process unit (SPU) and is connected to a central process unit (CPU) that coordinates the behaviors of all the LIS-Units. With such a LIS system, we consider the user assignments both for sum-rate...

The prefix sum is the sum of all values in preceding locations in the sequence: in this case those to the left of the current location. In the case of the radix sort this means that the prefix sum computes the total count of all values less than the current value. For example, the prefix sum of location, and hence value, 2 is o2 = 3. This means there are 3 entries for 0s and 1s in the sequence....

Dynamic resource allocation is an important means to increase the sum capacity of fading multi access channels. In this paper we consider vector multiaccess channels (channels where each user has multiple degrees of freedom) and study the eeect of power allocation as a function of the channel state on the sum capacity (or spectral eeciency) deened as the maximum sum of rates of users per unit p...