We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using Riemann sums. In our axiomatic approach even the proof of the existence of the definite integral (which does use Riemann sums) becomes slightly more elegant ...

Let A be a nite set of integers. For h 1, let S h (A) denote the set of all sums of h distinct elements of A. Let S(A) denote the set of all nonempty sums of distinct elements of A. The direct problem for subset sums is to nd lower bounds for jS h (A)j and jS(A)j in terms of jAj. The inverse problem for subset sums is to determine the structure of the extremal sets A of integers for which jS h ...

Recent advances in the technology of transmission electron microscopy have allowed for a more precise visualization of materials and physical processes, such as metal oxidation. Nevertheless, the quality of information is limited by the damage caused by an electron beam, movement of the specimen or other environmental factors. A novel registration method has been proposed to remove those limita...

‎For a coloring $c$ of a graph $G$‎, ‎the edge-difference coloring sum and edge-sum coloring sum with respect to the coloring $c$ are respectively‎ ‎$sum_c D(G)=sum |c(a)-c(b)|$ and $sum_s S(G)=sum (c(a)+c(b))$‎, ‎where the summations are taken over all edges $abin E(G)$‎. ‎The edge-difference chromatic sum‎, ‎denoted by $sum D(G)$‎, ‎and the edge-sum chromatic sum‎, ‎denoted by $sum S(G)$‎, ‎a...

In this paper we present an one-and-a-half-line proof, involving Cogalois Theory, of a folklore result asking when is an irrational number a sum of radicals of positive rational numbers. Some of the main ingredients of Cogalois Theory like G-Kneser extension, G-Cogalois extension, etc., used in the proof are briefly explained, so that the paper is selfcontained. We also discuss some older and n...

A generalization of the Davenport constant is investigated. For a finite abelian group G and a positive integer k, let D k (G) denote the smallest ℓ such that each sequence over G of length at least ℓ has k disjoint non-empty zero-sum subsequences. For general G, expanding on known results, upper and lower bounds on these invariants are investigated and it is proved that the sequence (D k (G)) ...

The present paper reports on the findings and conclusions of an experimental study that investigated the perception of Scottish English /i/-/I/ by Spanish speakers. The conclusions suggest that 1) L2 speakers can learn to identify and discriminate a new contrast, 2) these speakers may also learn to signal a new contrast using an acoustic cue that is secondary for the L1 perception of the same c...

We examined how variations in color and brightness are used by the visual system in distinguishing textured surfaces that differed in their first- or second-order statistics. Observers viewed a 32 x 32 array containing two types of square elements differing in chromaticity or luminance or both. The spatial distributions of the two kinds of elements were varied within the array until observers c...

We present an algorithm for minimizing a sum of functions that combines the computational efficiency of stochastic gradient descent (SGD) with the second order curvature information leveraged by quasi-Newton methods. We unify these disparate approaches by maintaining an independent Hessian approximation for each contributing function in the sum. We maintain computational tractability and limit ...

Recent work has revealed multiple pathways for cross-orientation suppression in cat and human vision. In particular, ipsiocular and interocular pathways appear to assert their influence before binocular summation in human but have different (1) spatial tuning, (2) temporal dependencies, and (3) adaptation after-effects. Here we use mask components that fall outside the excitatory passband of th...