The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in N that does not contain x(1), x(2), y(1), y(2) with x(1 )+ x(2) = y(1 )+ y(2). Its growth and structure have remained...
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The greedy Sidon set, also known as the Mian-Chowla sequence, is the lexicographically first set in N that does not contain x(1), x(2), y(1), y(2) with x(1 )+ x(2) = y(1 )+ y(2). Its growth and structure have remained enigmatic for 80 years. In this work, we study a generalization from the form x(1)+x(2) to arbitrary linear forms c(1)x(1 )+ & mldr;+ c(h)x(h);these are called Sidon sets for linear forms. We explicitly describe the elements of the greedy Sidon sets for linear forms when c(i) = n(i-1) for some n >= 2, and also when h = 2, c(1 )= 2, c(2 )>= 4, the "structured" domain. We also contrast the "enigmatic" domain when h = 2, c(1) = 2, c(2 )= 3 with the "structured" domain, and give upper bounds on the growth rates in both cases. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
Let k >= 2 be an integer and let A be a set of nonnegative integers. For a k-tuple of positive integers lambda_=(lambda(1),& mldr;,lambda(k)) with 1 = 0.
Let k >= 2 be an integer and let A be a set of nonnegative integers. For a k-tuple of positive integers lambda_=(lambda(1),& mldr;,lambda(k)) with 1 <=lambda(1)= 0.
Instead of the axiom of choice, we assume that every set of reals has the Baire property. It is shown that under this condition the concept of slenderness known from the theory of abelian groups becomes meaningful for...
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Instead of the axiom of choice, we assume that every set of reals has the Baire property. It is shown that under this condition the concept of slenderness known from the theory of abelian groups becomes meaningful for vector spaces.
It is proved that if the conditional distribution of one linear form in two independent (not necessarily identically distributed) random variables given another is normal, then the variables are normal. The result com...
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It is proved that if the conditional distribution of one linear form in two independent (not necessarily identically distributed) random variables given another is normal, then the variables are normal. The result complements a series of characterizations of normal distribution via different properties of linear forms: independence, linearity of regression plus homoscedasticity, equidistribution, conditional symmetry and normality. The method is different from previous ones and is based on properties of densities, not characteristic functions.
This paper is the third in a series in which the author investigates the question of representation of forms by linear forms. Whereas in the first two treatments the proportion of forms F of degree 3 (resp. degree d) ...
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This paper is the third in a series in which the author investigates the question of representation of forms by linear forms. Whereas in the first two treatments the proportion of forms F of degree 3 (resp. degree d) which can be written as a sum of two cubes (resp. d-th powers) of linear forms with algebraic coefficients is determined, the generalization now consists in allowing more general expressions of degree d in two linear forms. The main result is thus to give an asymptotic formula, in terms of their height, for the number or decomposable forms that have a representation F(X) = f(L-1(X), L-2(X)), where f is some fixed homogeneous polynomial and L-1, L-2 are linear forms. This is achieved by analyzing some p-adic and archimedean absolute value inequalities combined methods of the geometry of numbers.
Convergence in probability of the linear forms Σ k =1 ∞ a nk X k is obtained in the space D [0, 1], where ( X k ) are random elements in D [0, 1] and ( a nk ) is an array of real numbers. These results are obtained ...
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Convergence in probability of the linear forms Σ k =1 ∞ a nk X k is obtained in the space D [0, 1], where ( X k ) are random elements in D [0, 1] and ( a nk ) is an array of real numbers. These results are obtained under varying hypotheses of boundedness conditions on the moments and conditions on the mean oscillation of the random elements ( X n ) on subintervals of a partition of [0, 1]. Since the hypotheses are in general much less restrictive than tightness (or convex tightness), these results represent significant improvements over existing weak laws of large numbers and convergence results for weighted sums of random elements in D [0, 1]. Finally, comparisons to classical hypotheses for Banach space and real-valued results are included.
Let X 1 , X 2 ,…, be independent, identically distributed random variables. Suppose that the linear forms L 1 = Σ j =1 ∞ a j X j and L 2 = Σ j =1 ∞ b j X j exist with probability one and are identically distribut...
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Let X 1 , X 2 ,…, be independent, identically distributed random variables. Suppose that the linear forms L 1 = Σ j =1 ∞ a j X j and L 2 = Σ j =1 ∞ b j X j exist with probability one and are identically distributed; necessary and sufficient conditions assuring that X 1 is normally distributed are presented. The result is an extension of a theorem of Linnik ( Ukrainian Math. J. 5 (1953) , 207–243, 247–290) concerning the case that the linear forms L 1 and L 2 have a finite number of nonvanishing components. This proof only makes use of elementary properties of characteristic functions and of meromorphic functions.
The main results of this paper are a generalization of the results of S. Fajtlowicz and J. Mycielski on convex linear forms. We show that if V-n is the variety generated by all possible algebras A = , where R denotes ...
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The main results of this paper are a generalization of the results of S. Fajtlowicz and J. Mycielski on convex linear forms. We show that if V-n is the variety generated by all possible algebras A = < R;f >, where R denotes the real numbers and f(x(1),...,x(n)) = p(1)x(1) + (...) + p(n)x(n), for some p(1), (...), p(n) is an element of R, then any basis for the set of all identities satisfied by Vn is infinite. But on the other hand, the identities satisfied by Vn are a consequence of gL and mu(n), where mu(n) is the n-ary medial law and the inference rule gL is an implication patterned after the classical rigidity lemma of algebraic geometry. We also prove that the identities satisfied by A = < R;f > are a consequence of gL and mu n iff {p(1),..., p(n)} is algebraically independent. We then prove analagous results for algebras A = (R;F) of arbitrary type T and in the final section of this paper, we show that analagous results hold for Abelian group hyperidentities.
Let L-1 = a(1)X(1) + (...) + a(n)X(n), L-2 = b(1)X(1) + (...) + b(n)X(n), L-3 =X-1 + (...) + Xn be three linear forms in independent random variables X-1, ... ,X-n. Assume that a(1) not equal b(1), ... , a(m) not equa...
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Let L-1 = a(1)X(1) + (...) + a(n)X(n), L-2 = b(1)X(1) + (...) + b(n)X(n), L-3 =X-1 + (...) + Xn be three linear forms in independent random variables X-1, ... ,X-n. Assume that a(1) not equal b(1), ... , a(m) not equal b(m), a(m+1) = b(m+1), ... , a(n) = b(n) for some m, 1 less than or equal to m less than or equal to n. It is proved that if E\X-j\(2m-1) < for all j = 1, ... , n then the relations E(L-1(j)\L-3) = E(L-2(j)/L-3), j = 1, 2, ... , 2m - 1 imply that all the so-called acyclic (defined in Section 2) among X-1, ... ,X-m are Gaussian. (C) 2001 Elsevier Science B.V, All rights reserved. MSc: primary 62E10.
Let R be a commutative ring R with 1(R) and with group of units R-x. Let Phi = Phi(t(1), ... ,t(h)) = Sigma(h)(i=0) phi(i)t(i) be an h-ary linear form with nonzero coefficients phi(1), ... ,phi(h) is an element of R. ...
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Let R be a commutative ring R with 1(R) and with group of units R-x. Let Phi = Phi(t(1), ... ,t(h)) = Sigma(h)(i=0) phi(i)t(i) be an h-ary linear form with nonzero coefficients phi(1), ... ,phi(h) is an element of R. Let M be an R-module. For every subset A of M, the image of A under Phi is Phi(A) = {Phi(a(1), ... ,a(h)) : (a(1), ... , a(h)) is an element of A(h)}. For every subset I of {1,2, ... , h}, there is the subset sum s(I) = Sigma(i is an element of I) phi(i). Let S(Phi) = {s(I) : emtey set not equal I subset of {1, 2, ... ,h}}. Theorem. Let (sic)(t(1), ... , t(g)) = Sigma(g)(i=1) v(i)t(i) and Phi(t(1), ... , t(h)) = Sigma(h)(i=1) phi(i)t(i) be linear forms with nonzero coefficients in the ring R. If {0, 1} subset of S((sic)) and S(Phi) is an element of R-x, then for every epsilon > 0 and c > 1 there exist a finite R-module M with vertical bar M vertical bar > c and a subset A of M such that (sic)(A boolean ORlinear forms) = M and vertical bar Phi(A)vertical bar < epsilon vertical bar M vertical bar. (C) 2017 Elsevier Inc. All rights reserved.
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