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SIAM Review, 1968
Abstract : It is shown that if A is an integral matrix having linearly independent rows, then the extreme points of the set of nonnegative solutions to Ax = b are integral for all integral b if and only if the determinant of every basis matrix is plus or minus 1.
Veinott, Arthur F. jun., Dantzig, G. B.
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Abstract : It is shown that if A is an integral matrix having linearly independent rows, then the extreme points of the set of nonnegative solutions to Ax = b are integral for all integral b if and only if the determinant of every basis matrix is plus or minus 1.
Veinott, Arthur F. jun., Dantzig, G. B.
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Convex Hulls of Integral Points
Journal of Mathematical Sciences, 2003After reviewing some basic facts about convex sets and polyhedrons in \(\mathbb{R}^d\), the author studies convex hulls of lattice points \(x\in \mathbb{Z}^d\). He shows by an example that such a convex hull need not be closed. He then gives conditions on a set \(C\in \mathbb{R}^d\) which guarantee that the convex hull of \(C\cap \mathbb{Z}^d\) is ...
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Point Group Symmetries and Gaussian Integration
Journal of Computational Physics, 1994The authors show how the coefficients for the symmetrized lattice harmonic functions that are used in solid state physics can be computed from sums of integrals of polynomials. They then use Gaussian integration rules to determine exact values for the polynomial integrals.
Fernando, G. W. +3 more
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Blockchain Points of Integration
2020At this stage of the book, we will discover the external and internal interactions of blockchains. Interactions can vary from sourcing of metadata, to onboarding of users, to connecting all the dots of the ecosystem surrounding the blockchain application.
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2010
In this chapter we begin to study the problem of counting the number of integer points in a convex polytope, or the equivalent problem of computing a partition function. We start with the simplest case of numbers. We continue with the theorems of Brion and Ehrhart and leave the general discussion to the next chapters.
Corrado De Concini, Claudio Procesi
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In this chapter we begin to study the problem of counting the number of integer points in a convex polytope, or the equivalent problem of computing a partition function. We start with the simplest case of numbers. We continue with the theorems of Brion and Ehrhart and leave the general discussion to the next chapters.
Corrado De Concini, Claudio Procesi
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S-INTEGRAL POINTS ON HYPERELLIPTIC CURVES
International Journal of Number Theory, 2011Let C : Y2 = an Xn + ⋯ + a0 be a hyperelliptic curve with the ai rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. Let S be a finite set of rational primes. We give a completely explicit upper bound for the size of the S-integral points on the model C, provided we know at least one rational point on C and a ...
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Computing S-integral points on elliptic curves
Acta Arithmetica, 1994A method is developed for computing explicitly all integral points on a Weierstrass model of an elliptic curve over \(\mathbb{Q}\), based on estimates of linear forms in elliptic logarithms. In a few words, we can describe the notion of elliptic logarithm of an elliptic curve as follows: Let \(y^2= x^3+ ax+b\) be a short Weierstrass model of an ...
Gebel, J., Pethö, A., Zimmer, H. G.
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INTEGRAL POINTS ON CONGRUENT NUMBER CURVES
International Journal of Number Theory, 2013We provide a precise description of the integer points on elliptic curves of the shape y2 = x3 - N2x, where N = 2apb for prime p. By way of example, if p ≡ ±3 (mod 8) and p > 29, we show that all such points necessarily have y = 0. Our proofs rely upon lower bounds for linear forms in logarithms, a variety of old and new results on quartic and ...
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