Negative number

In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced "minus three" or "negative

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Electric Field Eect on Skyrmion phase in Chiral lattice Ferrimagnet Cu2OSeO3

Naman Agarwal

Insulator materials are the recent research interest in Spintronics because of the absence of joule heating. There has been continued interest in insulator material Cu2OSeO3 because it possesses the novel Skyrmion phase [1], specific type of spin vortices in the lattice, characterized by quantized topological number, near the paramagnetic phase boundary [2]. This Skyrmion phase can be harnessed for the switching applications in the data storage devices [3–6]. We study the effect of positive or negative DC electric field on the Skyrmion phase boundary of this material by magnetoelectric susceptibility(dM/dE)measurements, which is a sensitive technique for scanning phase boundary [7,8]. We scan out the phase diagram for positive, negative and zero DC electric fields and find that electric field effect is maximum at the phase boundaries and decreases inside the phase There is no cut-off for the electric field effect. Also, the electric field effect is maximum at upper skyrmion boundary compared to other phase boundaries. We also show the small expansion(contraction) of Skyrmion phase along temperature axis with the application of positive(negative) DC electric field.

2016

Sign-Changing Solutions for a Class of Zero Mass Nonlocal Schrodinger Equations

We consider the following class of fractional Schrodinger equations: (-Delta)(alpha)u + V(x)u = K(x)f(u) in R-N, where alpha is an element of (0, 1), N > 2 alpha, (-Delta)(alpha) is the fractional Laplacian, V and K are positive continuous functions which vanish at infinity, and f is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when f is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.

2019

A note on the integrality gap of the configuration LP for restricted Santa Claus

Klaus Jansen, Lars Rohwedder

In the restricted Santa Claus problem we are given resources R and players P. Every resource j is an element of R. has a value v(j) and every player i is an element of P desires a set of resources R(i). We are interested in distributing the resources to players that desire them. The quality of a solution is measured by the least happy player, i.e., the lowest sum of resource values. This value should be maximized. The local search algorithm by Asadpour et al. [1] and its connection to the configuration LP has proved itself to be a very influential technique for this and related problems. Asadpour et al. used a local search to obtain a bound of 4 for the ratio of the fractional to the integral optimum of the configuration LP (the integrality gap). This bound is non-constructive, since the local search has not been shown to terminate in polynomial time. On the negative side, the worst instance known has an integrality gap of 2. Although much progress was made in this area, neither bound has been improved since. We present a better analysis that shows the integrality gap is not worse than 3 + 5/6 approximate to 3.8333. (C) 2020 Elsevier B.V. All rights reserved.

ELSEVIER2020

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values c

Algebra

Algebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation

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Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers r

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