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Axioms, Volume 2, Issue 1 (March 2013), Pages 1-66

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Research

Open AccessArticle On the Content Bound for Real Quadratic Field Extensions
Axioms 2013, 2(1), 1-9; doi:10.3390/axioms2010001
Received: 31 October 2012 / Revised: 18 December 2012 / Accepted: 20 December 2012 / Published: 28 December 2012
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Abstract
Let K be a finite extension of and let S = {ν} denote the collection of K normalized absolute values on K. Let VK+ denote the additive group of adeles over K and let c:V
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Let K be a finite extension of and let S = {ν} denote the collection of K normalized absolute values on K. Let V K + denote the additive group of adeles over K and let c : V K + 0 denote the content map defined as c( { a v } ) = vs v( a v ) for { a v } V K + . A classical result of J. W. S. Cassels states that there is a constant c > 0 depending only on the field K  with the following property: if { a v } V K + with c( { a v } ) > c , then there exists a non-zero element b  ∈ K for which v(b)v( a v ), v S . Let cK be the greatest lower bound of the set of all c that satisfy this property. In the case that K is a real quadratic extension there is a known upper bound for cK due to S. Lang. The purpose of this paper is to construct a new upper bound for cK in the case that K has class number one. We compare our new bound with Lang’s bound for various real quadratic extensions and find that our new bound is better than Lang’s in many instances. Full article
(This article belongs to the Special Issue Axioms in Number Theory)
Open AccessArticle Generalized q-Stirling Numbers and Their Interpolation Functions
Axioms 2013, 2(1), 10-19; doi:10.3390/axioms2010010
Received: 21 November 2012 / Revised: 28 January 2013 / Accepted: 31 January 2013 / Published: 8 February 2013
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Abstract
In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to
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In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to q-Bernoulli numbers and polynomials and q-Stirling numbers of the second kind. Full article
Open AccessArticle Some Modular Relations Analogues to the Ramanujan’s Forty Identities with Its Applications to Partitions
Axioms 2013, 2(1), 20-43; doi:10.3390/axioms2010020
Received: 1 November 2012 / Revised: 22 January 2013 / Accepted: 28 January 2013 / Published: 18 February 2013
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Abstract
Recently, the authors have established a large class of modular relations involving the Rogers-Ramanujan type functions J(q) and K(q) of order ten. In this paper, we establish further modular relations connecting these two functions with Rogers-Ramanujan functions, Göllnitz-Gordon functions and cubic functions, which
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Recently, the authors have established a large class of modular relations involving the Rogers-Ramanujan type functions J(q) and K(q) of order ten. In this paper, we establish further modular relations connecting these two functions with Rogers-Ramanujan functions, Göllnitz-Gordon functions and cubic functions, which are analogues to the Ramanujan’s forty identities for Rogers-Ramanujan functions. Furthermore, we give partition theoretic interpretations of some of our modular relations. Full article
Open AccessArticle Signal Estimation Using Wavelet Analysis of Solution Monitoring Data for Nuclear Safeguards
Axioms 2013, 2(1), 44-57; doi:10.3390/axioms2010044
Received: 31 January 2013 / Revised: 4 March 2013 / Accepted: 7 March 2013 / Published: 20 March 2013
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Abstract
Wavelets are explored as a data smoothing (or de-noising) option for solution monitoring data in nuclear safeguards. In wavelet-smoothed data, the Gibbs phenomenon can obscure important data features that may be of interest. This paper compares wavelet smoothing to piecewise linear smoothing and
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Wavelets are explored as a data smoothing (or de-noising) option for solution monitoring data in nuclear safeguards. In wavelet-smoothed data, the Gibbs phenomenon can obscure important data features that may be of interest. This paper compares wavelet smoothing to piecewise linear smoothing and local kernel smoothing, and illustrates that the Haar wavelet basis is effective for reducing the Gibbs phenomenon. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle Golden Ratio and a Ramanujan-Type Integral
Axioms 2013, 2(1), 58-66; doi:10.3390/axioms2010058
Received: 1 November 2012 / Revised: 2 March 2013 / Accepted: 5 March 2013 / Published: 20 March 2013
Cited by 1 | PDF Full-text (170 KB) | HTML Full-text | XML Full-text
Abstract In this paper, we give a pedagogical introduction to several beautiful formulas discovered by Ramanujan. Using these results, we evaluate a Ramanujan-type integral formula. The result can be expressed in terms of the Golden Ratio. Full article

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