scholarly journals Convexity of certain integrals of the calculus of variations

1987 ◽  
Vol 107 (1-2) ◽  
pp. 15-26 ◽  
Author(s):  
B. Dacorogna
Keyword(s):  
Calculus Of Variations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Necessary And Sufficient

SynopsisIn this paper we study the convexity of the integral over the space . We isolate a necessary condition on f and we find necessary and sufficient conditions in the case where f(x, u, u′) = a(u)u′2n or g(u) + h(u′).

scholarly journals On circulant codes with prescribed distances

1977 ◽  
Vol 16 (3) ◽  
pp. 361-369
Author(s):  
M. Deza ◽  
Peter Eades
Keyword(s):  
Sufficient Conditions ◽  
Difference Sets ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Weighing Matrices ◽  
Cyclic Difference Sets ◽  
The Matrix ◽  
Circulant Codes ◽  
Necessary And Sufficient ◽  
Square Matrix

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

scholarly journals Square roots in finite full transformation semigroups

1982 ◽  
Vol 23 (2) ◽  
pp. 137-149 ◽  
Author(s):  
Mary Snowden ◽  
J. M. Howie
Keyword(s):  
Transformation Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Square Root ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Square Roots ◽  
Finite Set ◽  
Necessary And Sufficient

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root.

scholarly journals Buying Supermajorities in Finite Legislatures

2000 ◽  
Vol 94 (3) ◽  
pp. 677-681 ◽  
Author(s):  
Jeffrey S. Banks
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Vote Buying ◽  
Necessary And Sufficient ◽  
Coalition Size

I analyze the finite-voter version of the Groseclose and Snyder vote-buying model. I identify how the optimal coalition size varies with the underlying preference parameters; derive necessary and sufficient conditions for minimal majority and universal coalitions to form; and show that the necessary condition for minimal majorities found in Groseclose and Snyder is incorrect.

scholarly journals The necessary and sufficient conditions for wavelet frames in Sobolev space over local fields

2021 ◽  
Vol 39 (3) ◽  
pp. 81-92
Author(s):  
Ashish Pathak ◽  
Dileep Kumar ◽  
Guru P. Singh
Keyword(s):  
Sobolev Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Local Fields ◽  
Necessary Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Necessary And Sufficient

In this paper we construct wavelet frame on Sobolev space. A necessary condition and sufficient conditions for wavelet frames in Sobolev space are given.

scholarly journals Generalising the Horodecki criterion to nonprojective qubit observables

2021 ◽  
Author(s):  
Michael J W Hall ◽  
Shuming Cheng
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Weak Measurements ◽  
Chsh Inequality ◽  
Bell Nonlocality ◽  
Necessary And Sufficient ◽  
Projective Measurements

Abstract The Horodecki criterion provides a necessary and sufficient condition for a two-qubit state to be able to manifest Bell nonlocality via violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. It requires, however, the assumption that suitable projective measurements can be made on each qubit, and is not sufficient for scenarios in which noisy or weak measurements are either desirable or unavoidable. By characterising two-valued qubit observables in terms of strength, bias, and directional parameters, we address such scenarios by providing necessary and sufficient conditions for arbitrary qubit measurements having fixed strengths and relative angles for each observer. In particular, we find the achievable maximal values of the CHSH parameter for unbiased measurements on arbitrary states, and, alternatively, for arbitrary measurements on states with maximally-mixed marginals, and determine the optimal angles in some cases. We also show that for certain ranges of measurement strengths it is only possible to violate the CHSH inequality via biased measurements. Finally, we use the CHSH inequality to obtain a simple necessary condition for the compatibility of two qubit observables.

Sufficiency and the Jacobi Condition in the Calculus of Variations

1986 ◽  
Vol 38 (5) ◽  
pp. 1199-1209 ◽  
Author(s):  
Frank H. Clarke ◽  
Vera Zeidan
Keyword(s):  
Calculus Of Variations ◽  
Basic Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jacobi Condition ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Differentiable Mappings

Besides stating the problem and the results, we shall give in this section a brief overview of the classical necessary and sufficient conditions in the calculus of variations, in order to clearly situate the contribution of this article.1.1 The problem. We are given an interval [a, b], two points xa, xb in Rn, and a function L (the Lagrangian) mapping [a, b] × Rn × Rn to R. The basic problem in the calculus of variations, labeled (P), is that of minimizing the functionalover some class X of functions x and subject to the constraints x(a) = xa, x(b) = xb. Let us take for now the class X of functions to be the continuously differentiable mappings from [a, b] to Rn; we call such functions smooth arcs.

scholarly journals Equality of Dedekind sums mod ℤ, 2ℤ and 4ℤ

2015 ◽  
Vol 11 (06) ◽  
pp. 1735-1738 ◽  
Author(s):  
Emmanuel Tsukerman
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dedekind Sums ◽  
Necessary Condition ◽  
Necessary And Sufficient ◽  
Condition B

In [K. Girstmair, A criterion for the equality of Dedekind sums mod ℤ, Int. J. Number Theory10 (2014) 565–568], it was shown that the necessary condition b ∣ (a1a2- 1) × (a1- a2) for equality of two Dedekind sums s(a1, b) and s(a2, b) given in [S. Jabuka, S. Robins and X. Wang, When are two Dedekind sums equal? Int. J. Number Theory7 (2011) 2197–2202] is equivalent to 12s(a1, b) - 12s(a2, b) ∈ ℤ. In this paper, we give a new proof of this result and then find two additional necessary and sufficient conditions for 12s(a1, b) - 12s(a2, b) ∈ 2ℤ, 4ℤ. These give new necessary conditions on equality of Dedekind sums.

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