منابع مشابه
The Number of 2× 2 Integer Matrices Having a Prescribed Integer Eigenvalue
Random matrices arise in many mathematical contexts, and it is natural to ask about the properties that such matrices satisfy. If we choose a matrix with integer entries at random, for example, what is the probability that it will have a particular integer as an eigenvalue, or an integer eigenvalue at all? If we choose a matrix with real entries at random, what is the probability that it will h...
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This paper introduces two fundamental families of ‘quasipolyhedra’ — polyhedra with a countably infinite number of facets — that arise in the context of integer quadratic programming. It is shown that any integer quadratic program can be reduced to the minimisation of a linear function over a quasi-polyhedron in the first family. Some fundamental properties of the quasi-polyhedra are derived, a...
متن کاملThe Quadratic Character Experiment
A fast new algorithm is used compute the zeros of 10 quadratic character L-functions for negative fundamental discriminants with absolute value d > 10. These are compared to the 1-level density, including various lower order terms. These terms come from, on the one hand the Explicit Formula, and on the other the L-functions Ratios Conjecture. The latter give a much better fit to the data, provi...
متن کاملOn the Quadratic Character of Quadratic Units
Let p ≡ 1 (mod 4) be a prime. Let a, b ∈ Z with p a(a2 + b2). In the paper we mainly determine ( b+ √ a2+b2 2 ) p−1 2 (mod p) by assuming p = c2 + d2 or p = Ax2 + 2Bxy + Cy2 with AC − B2 = a2 + b2. As an application we obtain simple criteria for εD to be a quadratic residue (mod p), where D > 1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the qu...
متن کاملGeneralized Nonaveraging Integer Sequences
Let the sequence Sm of nonnegative integers be generated by the following conditions. Set the first term a0 = 0, and for all k ≥ 0, let ak+1 be the least integer greater than ak such that no element of {a0, . . . , ak+1} is the average of m − 1 distinct other elements. Szekeres gave a closed-form description of S3 in 1936, and Layman provided a similar description for S4 in 1999. We first find ...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1970
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-1970-0271006-x