Axioms

170 KiB

Open AccessArticle

Golden Ratio and a Ramanujan-Type Integral

by Hei-Chi Chan

Axioms 2013, 2(1), 58-66; https://doi.org/10.3390/axioms2010058 - 20 Mar 2013

Cited by 1 | Viewed by 6344

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

(This article belongs to the Special Issue q-Series and Related Topics in Special Functions and Analytic Number Theory)

499 KiB

Open AccessArticle

Signal Estimation Using Wavelet Analysis of Solution Monitoring Data for Nuclear Safeguards

by Tom Burr and Claire Longo

Axioms 2013, 2(1), 44-57; https://doi.org/10.3390/axioms2010044 - 20 Mar 2013

Cited by 1 | Viewed by 4439

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 [...] Read more.

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)

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291 KiB

Open AccessArticle

Some Modular Relations Analogues to the Ramanujan’s Forty Identities with Its Applications to Partitions

by Chandrashekar Adiga and Nasser Abdo Saeed Bulkhali

Axioms 2013, 2(1), 20-43; https://doi.org/10.3390/axioms2010020 - 18 Feb 2013

Cited by 2 | Viewed by 4603

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 [...] Read more.

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

(This article belongs to the Special Issue q-Series and Related Topics in Special Functions and Analytic Number Theory)

164 KiB

Open AccessArticle

Generalized q-Stirling Numbers and Their Interpolation Functions

by Hacer Ozden, Ismail Naci Cangul and Yilmaz Simsek

Axioms 2013, 2(1), 10-19; https://doi.org/10.3390/axioms2010010 - 08 Feb 2013

Cited by 3 | Viewed by 4981

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 [...] Read more.

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

(This article belongs to the Special Issue q-Series and Related Topics in Special Functions and Analytic Number Theory)

172 KiB

Open AccessArticle

On the Content Bound for Real Quadratic Field Extensions

by Robert G. Underwood

Axioms 2013, 2(1), 1-9; https://doi.org/10.3390/axioms2010001 - 28 Dec 2012

Viewed by 4217

Abstract

Let K be a finite extension of

ℚ

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 [...] Read more.

Let K be a finite extension of

ℚ

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}^{+} \to ℝ_{\geq 0}

denote the content map defined as

c ({a_{v}}) = \prod_{v \in s} v (a_{v})

for

{a_{v}} \in 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}} \in V_{K}^{+}

with

c ({a_{v}}) > c

, then there exists a non-zero element b ∈ K for which

v (b) \leq v (a_{v}), \forall v \in S

. Let c_K 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 c_K due to S. Lang. The purpose of this paper is to construct a new upper bound for c_K 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)

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Axioms, Volume 2, Issue 1 (March 2013) – 5 articles , Pages 1-66

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