Positivity of quadratic base change -functions
Hervé Jacquet, Chen Nan (2001)
Bulletin de la Société Mathématique de France
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We show that certain quadratic base change -functions for are non-negative at their center of symmetry.
Hervé Jacquet, Chen Nan (2001)
Bulletin de la Société Mathématique de France
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We show that certain quadratic base change -functions for are non-negative at their center of symmetry.
Noam D. Elkies, Daniel M. Kane, Scott Duke Kominers (2013)
Journal de Théorie des Nombres de Bordeaux
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In this note, we give simple examples of sets of quadratic forms that have minimal -universality criteria of multiple cardinalities. This answers a question of Kim, Kim, and Oh [KKO05] in the negative.
I. D. Shkredov (2014)
Acta Arithmetica
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We describe all sets which represent the quadratic residues in the sense that R = A + A or R = A ⨣ A. Also, we consider the case of an approximate equality R ≈ A + A and R ≈ A ⨣ A and prove that A is then close to a perfect difference set.
Zbigniew Leszczyński, Daniel Simson (2015)
Colloquium Mathematicae
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The incidence coalgebras of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form , where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category of finite-dimensional left -modules is equivalent to the tameness of the category of finitely copresented left -modules. Hence, the tame-wild dichotomy for the coalgebras is deduced. Moreover, we prove that for an interval finite...
G. L. Watson
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CONTENTSIntroduction.......................................................................................61. Definition of certain special forms...........................................62. Statement of results...................................................................83. Proof of Theorem 2.....................................................................94. Preliminaries for Theorem 1.....................................................105. Further preliminaries for Theorem...
Gianfranco Cimmino (1989)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
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For every positive definite quadratic form in variables the reciprocal of the square root of the discriminant is equal to the arithmetic mean of the values assumed by the form on the sphere centered at and with radius raised to the ()-th. power. Various consequences are deduced from this, in particular a simplification of some calculations from which one obtains the possibility of solving linear systems using spherical means rather than determinants.
Kamila Kliś-Garlicka (2016)
Czechoslovak Mathematical Journal
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The notion of a bilattice was introduced by Shulman. A bilattice is a subspace analogue for a lattice. In this work the definition of hyperreflexivity for bilattices is given and studied. We give some general results concerning this notion. To a given lattice we can construct the bilattice . Similarly, having a bilattice we may consider the lattice . In this paper we study the relationship between hyperreflexivity of subspace lattices and of their associated bilattices. Some examples...
Clemens Heuberger, Daniel Krenn (2013)
Journal de Théorie des Nombres de Bordeaux
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We consider digit expansions with an endomorphism of an Abelian group. In such a numeral system, the -NAF condition (each block of consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight admits an optimal -NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...
Debopam Chakraborty, Anupam Saikia (2014)
Acta Arithmetica
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The relative class number of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of and , where denotes the ring of integers of K and is the order of conductor f given by . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the...
Julian Ławrynowicz, Massimo Vaccaro (2011)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions and , respectively, but the procedure fails for in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are . Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower...
Shriram K. Nimbhorkar, Rupal C. Shroff (2017)
Mathematica Bohemica
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The concept of a Goldie extending module is generalized to a Goldie extending element in a lattice. An element of a lattice with is said to be a Goldie extending element if and only if for every there exists a direct summand of such that is essential in both and . Some properties of such elements are obtained in the context of modular lattices. We give a necessary condition for the direct sum of Goldie extending elements to be Goldie extending. Some characterizations...
Brandon Hanson (2013)
Acta Arithmetica
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An open problem of arithmetic Ramsey theory asks if given an r-colouring c:ℕ → 1,...,r of the natural numbers, there exist x,y ∈ ℕ such that c(xy) = c(x+y) apart from the trivial solution x = y = 2. More generally, one could replace x+y with a binary linear form and xy with a binary quadratic form. In this paper we examine the analogous problem in a finite field . Specifically, given a linear form L and a quadratic form Q in two variables, we provide estimates on the necessary size...