If k is a positive integer we say that a set S of real numbers is k-sum-free if there do not exist x, y, x is S such that x + y = kz. For k greater than or equal to 4 we find the essentially unique measurable k-sum-free subset of (0, 1] of maximum size.

A finite set A of integers is square-sum-free if there is no subset of A sums up to a square. In 1986, Erd˝ os posed the problem of determining the largest cardinality of a square-sum-free subset of {1,. .. , n}. Answering this question, we show that this maximum cardinality is of order n 1/3+o(1) .

If k is a positive real number, we say that a set S of real numbers is k-sum-free if there do not exist x, y, z in S such that x+y = kz. For k greater than or equal to 4 we find the essentially unique measurable k-sum-free subset of (0, 1] of maximum size.

We are given a sequence A of n real numbers which is to be preprocessed. In the Range Maximum-Sum Segment Query (RMSQ) problem, a query is comprised of two intervals [i, j] and [k, l] and our goal is to return the maximum-sum segment of A where the staring index of the segment lies in [i, j] and the ending index lies in [k, l]. We provide the ̄rst known optimal algorithm with O(n) preprocessing...

If k is a positive integer, we say that a set A of positive integers is k-sum-free if there do not exist a, b, c in A such that a + b = kc. In particular we give a precise characterization of the structure of maximum sized k-sum-free sets in {1, . . . , n} for k ≥ 4 and n large.

Utilizing the concept of Perron complement, a new estimate for the spectral radius of a nonnegative irreducible matrix is presented. A new matrix is derived that preserves the spectral radius while its minimum row sum increases and its maximum row sum decreases. Numerical examples are provided to illustrate the effectiveness of this approach.

Utilizing the concept of Perron complement, a new estimate for the spectral radius of a nonnegative irreducible matrix is presented. A new matrix is derived that preserves the spectral radius while its minimum row sum increases and its maximum row sum decreases. Numerical examples are provided to illustrate the effectiveness of this approach.

A high-speed capacitance difference-to-sum ratio measurement circuit is presented for differential capacitance transducers. It consists of a switched-capacitor input stage, two sample-and-hold (S/H) circuits followed by voltage-to-current (V/I) converters, and a current-ratio-controlled relaxation oscillator. This circuit offers a square-wave output whose oscillation period is directly proporti...

We consider single facility location problems with equity measures, de0ned on networks. The models discussed are, the variance, the sum of weighted absolute deviations, the maximum weighted absolute deviation, the sum of absolute weighted di2erences, the range, and the Lorenz measure. We review the known algorithmic results and present improved algorithms for some of these models. ? 2002 Elsevi...

For a planar graph on n vertices we determine the maximum values for the following: 1) the sum of the m largest vertex degrees. 2) the number of vertices of degree at least k. 3) the sum of the degrees of vertices with degree at least k.