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Axioms, Volume 2, Issue 3 (September 2013), Pages 286-476

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Editorial

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Open AccessEditorial Special Issue: “q-Series and Related Topics in Special Functions and Analytic Number Theory”—Foreword
Axioms 2013, 2(3), 435-436; doi:10.3390/axioms2030435
Received: 30 August 2013 / Accepted: 30 August 2013 / Published: 3 September 2013
Cited by 1 | PDF Full-text (19 KB) | HTML Full-text | XML Full-text
Abstract
It is indeed a fairly common practice for scientific research journals and scientific research periodicals to publish special issues as well as conference proceedings. Quite frequently, these special issues are devoted exclusively to specific topics and/or are dedicated respectfully to commemorate the [...] Read more.
It is indeed a fairly common practice for scientific research journals and scientific research periodicals to publish special issues as well as conference proceedings. Quite frequently, these special issues are devoted exclusively to specific topics and/or are dedicated respectfully to commemorate the celebrated works of renowned research scientists. The following Special Issue: “q-Series and Related Topics in Special Functions and Analytic Number Theory” (see [1–8] below) is an outcome of the ongoing importance and popularity of such topics as Basic (or q-) Series and Basic (or q-) Polynomials. [...] Full article

Research

Jump to: Editorial

Open AccessArticle Some Notes on the Use of the Windowed Fourier Transform for Spectral Analysis of Discretely Sampled Data
Axioms 2013, 2(3), 286-310; doi:10.3390/axioms2030286
Received: 24 April 2013 / Revised: 20 May 2013 / Accepted: 21 May 2013 / Published: 24 June 2013
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Abstract
The properties of the Gabor and Morlet transforms are examined with respect to the Fourier analysis of discretely sampled data. Forward and inverse transform pairs based on a fixed window with uniform sampling of the frequency axis can satisfy numerically the energy [...] Read more.
The properties of the Gabor and Morlet transforms are examined with respect to the Fourier analysis of discretely sampled data. Forward and inverse transform pairs based on a fixed window with uniform sampling of the frequency axis can satisfy numerically the energy and reconstruction theorems; however, transform pairs based on a variable window or nonuniform frequency sampling in general do not. Instead of selecting the shape of the window as some function of the central frequency, we propose constructing a single window with unit energy from an arbitrary set of windows that is applied over the entire frequency axis. By virtue of using a fixed window with uniform frequency sampling, such a transform satisfies the energy and reconstruction theorems. The shape of the window can be tailored to meet the requirements of the investigator in terms of time/frequency resolution. The algorithm extends naturally to the case of nonuniform signal sampling without modification beyond identification of the Nyquist interval. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle Complexity L0-Penalized M-Estimation: Consistency in More Dimensions
Axioms 2013, 2(3), 311-344; doi:10.3390/axioms2030311
Received: 7 April 2013 / Revised: 15 May 2013 / Accepted: 4 June 2013 / Published: 9 July 2013
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Abstract
We study the asymptotics in L2 for complexity penalized least squares regression for the discrete approximation of finite-dimensional signals on continuous domains—e.g., images—by piecewise smooth functions. We introduce a fairly general setting, which comprises most of the presently popular partitions of signal [...] Read more.
We study the asymptotics in L2 for complexity penalized least squares regression for the discrete approximation of finite-dimensional signals on continuous domains—e.g., images—by piecewise smooth functions. We introduce a fairly general setting, which comprises most of the presently popular partitions of signal or image domains, like interval, wedgelet or related partitions, as well as Delaunay triangulations. Then, we prove consistency and derive convergence rates. Finally, we illustrate by way of relevant examples that the abstract results are useful for many applications. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle Wavelet-Based Monitoring for Biosurveillance
Axioms 2013, 2(3), 345-370; doi:10.3390/axioms2030345
Received: 5 June 2013 / Revised: 18 June 2013 / Accepted: 19 June 2013 / Published: 9 July 2013
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Abstract
Biosurveillance, focused on the early detection of disease outbreaks, relies on classical statistical control charts for detecting disease outbreaks. However, such methods are not always suitable in this context. Assumptions of normality, independence and stationarity are typically violated in syndromic data. Furthermore, [...] Read more.
Biosurveillance, focused on the early detection of disease outbreaks, relies on classical statistical control charts for detecting disease outbreaks. However, such methods are not always suitable in this context. Assumptions of normality, independence and stationarity are typically violated in syndromic data. Furthermore, outbreak signatures are typically of unknown patterns and, therefore, call for general detectors. We propose wavelet-based methods, which make less assumptions and are suitable for detecting abnormalities of unknown form. Wavelets have been widely used for data denoising and compression, but little work has been published on using them for monitoring. We discuss monitoring-based issues and illustrate them using data on military clinic visits in the USA. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle Nonnegative Scaling Vectors on the Interval
Axioms 2013, 2(3), 371-389; doi:10.3390/axioms2030371
Received: 17 April 2013 / Revised: 28 June 2013 / Accepted: 1 July 2013 / Published: 9 July 2013
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Abstract
In this paper, we outline a method for constructing nonnegative scaling vectors on the interval. Scaling vectors for the interval have been constructed in [1–3]. The approach here is different in that the we start with an existing scaling vector ϕ that [...] Read more.
In this paper, we outline a method for constructing nonnegative scaling vectors on the interval. Scaling vectors for the interval have been constructed in [1–3]. The approach here is different in that the we start with an existing scaling vector ϕ that generates a multi-resolution analysis for L2(R) to create a scaling vector for the interval. If desired, the scaling vector can be constructed so that its components are nonnegative. Our construction uses ideas from [4,5] and we give results for scaling vectors satisfying certain support and continuity properties. These results also show that less edge functions are required to build multi-resolution analyses for L2 ([a; b]) than the methods described in [5,6]. Full article
(This article belongs to the Special Issue Wavelets and Applications)
Open AccessArticle Discrete Integrals Based on Comonotonic Modularity
Axioms 2013, 2(3), 390-403; doi:10.3390/axioms2030390
Received: 16 April 2013 / Revised: 31 May 2013 / Accepted: 19 June 2013 / Published: 23 July 2013
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Abstract
It is known that several discrete integrals, including the Choquet and Sugeno integrals, as well as some of their generalizations, are comonotonically modular functions. Based on a recent description of the class of comonotonically modular functions, we axiomatically identify more general families [...] Read more.
It is known that several discrete integrals, including the Choquet and Sugeno integrals, as well as some of their generalizations, are comonotonically modular functions. Based on a recent description of the class of comonotonically modular functions, we axiomatically identify more general families of discrete integrals that are comonotonically modular, including signed Choquet integrals and symmetric signed Choquet integrals, as well as natural extensions of Sugeno integrals. Full article
(This article belongs to the Special Issue Axiomatic Approach to Monotone Measures and Integrals)
Open AccessArticle On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices
Axioms 2013, 2(3), 404-434; doi:10.3390/axioms2030404
Received: 28 May 2013 / Revised: 4 June 2013 / Accepted: 3 July 2013 / Published: 23 July 2013
Cited by 4 | PDF Full-text (368 KB) | HTML Full-text | XML Full-text
Abstract
The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the [...] Read more.
The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author. Full article
Open AccessCommunication Yang-Baxter Systems, Algebra Factorizations and Braided Categories
Axioms 2013, 2(3), 437-442; doi:10.3390/axioms2030437
Received: 13 August 2013 / Revised: 30 August 2013 / Accepted: 30 August 2013 / Published: 3 September 2013
Cited by 3 | PDF Full-text (143 KB) | HTML Full-text | XML Full-text
Abstract
The Yang-Baxter equation first appeared in a paper by the Nobel laureate, C.N. Yang, and in R.J. Baxter’s work. Later, Vladimir Drinfeld, Vaughan F. R. Jones and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equation. After [...] Read more.
The Yang-Baxter equation first appeared in a paper by the Nobel laureate, C.N. Yang, and in R.J. Baxter’s work. Later, Vladimir Drinfeld, Vaughan F. R. Jones and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equation. After a short review on this equation and the Yang-Baxter systems, we consider the problem of constructing algebra factorizations from Yang-Baxter systems. Our sketch of proof uses braided categories. Other problems are also proposed. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2013)
Open AccessArticle R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation
Axioms 2013, 2(3), 443-476; doi:10.3390/axioms2030443
Received: 14 August 2013 / Revised: 28 August 2013 / Accepted: 30 August 2013 / Published: 5 September 2013
Cited by 3 | PDF Full-text (402 KB)
Abstract
In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel0d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general [...] Read more.
In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel0d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the “braided” aspects of YD module structure, exhibiting a braided system encoding all the axioms from the definition of YD modules. The functoriality and several generalizations of this construction are studied using the original machinery of YD systems. As consequences, we get a conceptual interpretation of the tensor product structures for YD modules, and a generalization of the deformation cohomology of YD modules. This homology theory is thus included into the unifying framework of braided homologies, which contains among others Hochschild, Chevalley-Eilenberg, Gerstenhaber-Schack and quandle homologies. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2013)

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