Axioms

222 KiB

Open AccessArticle

Some Aspects of Extended Kinetic Equation

by Dilip Kumar

Axioms 2015, 4(3), 412-422; https://doi.org/10.3390/axioms4030412 - 18 Sep 2015

Viewed by 4559

Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the [...] Read more.

Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

771 KiB

Open AccessArticle

POVMs and the Two Theorems of Naimark and Sz.-Nagy

by James D. Malley and Anthony R. Fletcher

Axioms 2015, 4(3), 400-411; https://doi.org/10.3390/axioms4030400 - 1 Sep 2015

Viewed by 3407

Abstract

In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be [...] Read more.

In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be so resolved, with the resolution respecting all linear sums. Crucially, both resolutions return the correct Born probabilities for the original observables. Here, an alternative proof of the Sz.-Nagy result is given using elementary inner product spaces. A version of the resolution is then shown to respect all products of observables on the base space. Practical and theoretical consequences are indicated. For example, quantum statistical inference problems that involve any algebraic functionals can now be studied using classical statistical methods over commuting observables. The estimation of quantum states is a problem of this type. Further, as theoretical objects, classical and quantum systems are now distinguished by only more or less degrees of freedom. Full article

567 KiB

Open AccessArticle

Limiting Approach to Generalized Gamma Bessel Model via Fractional Calculus and Its Applications in Various Disciplines

by Nicy Sebastian

Axioms 2015, 4(3), 385-399; https://doi.org/10.3390/axioms4030385 - 26 Aug 2015

Cited by 1 | Viewed by 4299

Abstract

The essentials of fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes are established. In addition to this, from this fractional integral, one can list out almost all of the extended [...] Read more.

The essentials of fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes are established. In addition to this, from this fractional integral, one can list out almost all of the extended densities for the pathway parameter q < 1 and q → 1. Here, we bring out the idea of thicker- or thinner-tailed models associated with a gamma-type distribution as a limiting case of the pathway operator. Applications of this extended gamma model in statistical mechanics, input-output models, solar spectral irradiance modeling, etc., are established. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

► Show Figures

Figure 1

1303 KiB

Open AccessArticle

An Overview of Generalized Gamma Mittag–Leffler Model and Its Applications

by Seema S. Nair

Axioms 2015, 4(3), 365-384; https://doi.org/10.3390/axioms4030365 - 26 Aug 2015

Cited by 3 | Viewed by 5261

Abstract

Recently, probability models with thicker or thinner tails have gained more importance among statisticians and physicists because of their vast applications in random walks, Lévi flights, financial modeling, etc. In this connection, we introduce here a new family of generalized probability distributions associated [...] Read more.

Recently, probability models with thicker or thinner tails have gained more importance among statisticians and physicists because of their vast applications in random walks, Lévi flights, financial modeling, etc. In this connection, we introduce here a new family of generalized probability distributions associated with the Mittag–Leffler function. This family gives an extension to the generalized gamma family, opens up a vast area of potential applications and establishes connections to the topics of fractional calculus, nonextensive statistical mechanics, Tsallis statistics, superstatistics, the Mittag–Leffler stochastic process, the Lévi process and time series. Apart from examining the properties, the matrix-variate analogue and the connection to fractional calculus are also explained. By using the pathway model of Mathai, the model is further generalized. Connections to Mittag–Leffler distributions and corresponding autoregressive processes are also discussed. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

► Show Figures

Figure 1

274 KiB

Open AccessArticle

Almost Periodic Solutions of Nonlinear Volterra Difference Equations with Unbounded Delay

by Yoshihiro Hamaya, Tomomi Itokazu and Kaori Saito

Axioms 2015, 4(3), 345-364; https://doi.org/10.3390/axioms4030345 - 24 Aug 2015

Cited by 1 | Viewed by 3957

Abstract

In order to obtain the conditions for the existence of periodic and almost periodic solutions of Volterra difference equations, \( x(n+1)=f(n,x(n))+\sum_{s=-\infty}^{n}F(n,s, {x(n+s)},x(n)) \), we consider certain stability properties, which are referred to as (K, \( \rho \))-weakly uniformly-asymptotic stability and (K, \( \rho [...] Read more.

In order to obtain the conditions for the existence of periodic and almost periodic solutions of Volterra difference equations, \( x(n+1)=f(n,x(n))+\sum_{s=-\infty}^{n}F(n,s, {x(n+s)},x(n)) \), we consider certain stability properties, which are referred to as (K, \( \rho \))-weakly uniformly-asymptotic stability and (K, \( \rho \))-uniformly asymptotic stability. Moreover, we discuss the relationship between the \( \rho \)-separation condition and the uniformly-asymptotic stability property in the \( \rho \) sense. Full article

(This article belongs to the Special Issue Functional Differential Equations)

425 KiB

Open AccessArticle

On the Fractional Poisson Process and the Discretized Stable Subordinator

by Rudolf Gorenflo and Francesco Mainardi

Axioms 2015, 4(3), 321-344; https://doi.org/10.3390/axioms4030321 - 4 Aug 2015

Cited by 19 | Viewed by 5331

Abstract

We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe [...] Read more.

We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe that for physical time likewise we have t ≥ 0. Thus we apply the Laplace transform with respect to both variables x and t. Applying then a modification of the Montroll-Weiss-Cox formalism of continuous time random walk we obtain the essential characteristics of a renewal process in the transform domain and, if we are lucky, also in the physical domain. The process t = t(N) of accumulation of waiting times is inverse to the counting number process, in honour of the Danish mathematician and telecommunication engineer A.K. Erlang we call it the Erlang process. It yields the probability of exactly n renewal events in the interval (0; t]. We apply our Laplace-Laplace formalism to the fractional Poisson process whose waiting times are of Mittag-Leffler type and to a renewal process whose waiting times are of Wright type. The process of Mittag-Leffler type includes as a limiting case the classical Poisson process, the process of Wright type represents the discretized stable subordinator and a re-scaled version of it was used in our method of parametric subordination of time-space fractional diffusion processes. Properly rescaling the counting number process N(t) and the Erlang process t(N) yields as diffusion limits the inverse stable and the stable subordinator, respectively. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

► Show Figures

Figure 1

192 KiB

Open AccessArticle

Fixed Points of Local Actions of Lie Groups on Real and Complex 2-Manifolds

by Morris W. Hirsch

Axioms 2015, 4(3), 313-320; https://doi.org/10.3390/axioms4030313 - 27 Jul 2015

Cited by 2 | Viewed by 4539

Abstract

I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

263 KiB

Open AccessArticle

Pro-Lie Groups: A Survey with Open Problems

by Karl H. Hofmann and Sidney A. Morris

Axioms 2015, 4(3), 294-312; https://doi.org/10.3390/axioms4030294 - 24 Jul 2015

Cited by 16 | Viewed by 5988

Abstract

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete [...] Read more.

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

1015 KiB

Open AccessArticle

Heat Kernel Embeddings, Differential Geometry and Graph Structure

by Hewayda ElGhawalby and Edwin R. Hancock

Axioms 2015, 4(3), 275-293; https://doi.org/10.3390/axioms4030275 - 21 Jul 2015

Cited by 7 | Viewed by 5071

Abstract

In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian [...] Read more.

In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian eigen-system over time. A Young–Householder decomposition is performed on the heat kernel to obtain the matrix of the embedded coordinates for the nodes of the graph. With the embeddings at hand, we establish a graph characterization based on differential geometry by computing sets of curvatures associated with the graph edges and triangular faces. A sectional curvature computed from the difference between geodesic and Euclidean distances between nodes is associated with the edges of the graph. Furthermore, we use the Gauss–Bonnet theorem to compute the Gaussian curvatures associated with triangular faces of the graph. Full article

(This article belongs to the Special Issue Discrete Differential Geometry and Its Applications to Imaging and Graphics)

► Show Figures

Figure 1

217 KiB

Open AccessArticle

Closed-Form Representations of the Density Function and Integer Moments of the Sample Correlation Coefficient

by Serge B. Provost

Axioms 2015, 4(3), 268-274; https://doi.org/10.3390/axioms4030268 - 20 Jul 2015

Cited by 1 | Viewed by 4172

Abstract

This paper provides a simplified representation of the exact density function of R, the sample correlation coefficient. The odd and even moments of R are also obtained in closed forms. Being expressed in terms of generalized hypergeometric functions, the resulting representations are [...] Read more.

This paper provides a simplified representation of the exact density function of R, the sample correlation coefficient. The odd and even moments of R are also obtained in closed forms. Being expressed in terms of generalized hypergeometric functions, the resulting representations are readily computable. Some numerical examples corroborate the validity of the results derived herein. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

235 KiB

Open AccessArticle

Lindelöf Σ-Spaces and R-Factorizable Paratopological Groups

by Mikhail Tkachenko

Axioms 2015, 4(3), 254-267; https://doi.org/10.3390/axioms4030254 - 10 Jul 2015

Cited by 3 | Viewed by 4458

Abstract

We prove that if a paratopological group G is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is R-factorizable and has countable cellularity. If in addition, G is regular, then it is totally w-narrow and satisfies cel_w [...] Read more.

We prove that if a paratopological group G is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is R-factorizable and has countable cellularity. If in addition, G is regular, then it is totally w-narrow and satisfies cel_w(G) ≤ w, and the Hewitt–Nachbin completion of G is again an R-factorizable paratopological group. Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

235 KiB

Open AccessArticle

On Elliptic and Hyperbolic Modular Functions and the Corresponding Gudermann Peeta Functions

by Thomas Ernst

Axioms 2015, 4(3), 235-253; https://doi.org/10.3390/axioms4030235 - 8 Jul 2015

Viewed by 4149

Abstract

In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations [...] Read more.

In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann’s formulas in Carlson’s modern notation. The transformations between squares of elliptic functions can be expressed as general Möbius transformations, and a conjecture of twelve formulas, extending a Gudermannian formula, is presented. In the second part of the paper, we consider the corresponding formulas for hyperbolic modular functions, and show that these Möbius transformations can be used to prove integral formulas for the inverses of hyperbolic modular functions, which are in fact Schwarz-Christoffel transformations. Finally, we present the simplest formulas for the Gudermann Peeta functions, variations of the Jacobi theta functions. 2010 Mathematics Subject Classification: Primary 33E05; Secondary 33D15. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

259 KiB

Open AccessEditorial

Scientific Endeavors of A.M. Mathai: An Appraisal on the Occasion of his Eightieth Birthday, 28 April 2015

by Hans J. Haubold and Arak M. Mathai

Axioms 2015, 4(3), 213-234; https://doi.org/10.3390/axioms4030213 - 3 Jul 2015

Viewed by 4279

Abstract

A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill University, Canada. He is currently the Director of the Centre for Mathematical and Statistical Sciences India. His research contributions cover a wide spectrum of topics in mathematics, statistics, physics, astronomy, and biology. [...] Read more.

A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill University, Canada. He is currently the Director of the Centre for Mathematical and Statistical Sciences India. His research contributions cover a wide spectrum of topics in mathematics, statistics, physics, astronomy, and biology. He is a Fellow of the Institute of Mathematical Statistics, National Academy of Sciences of India, and a member of the International Statistical Institute. He is a founder of the Canadian Journal of Statistics and the Statistical Society of Canada. He was instrumental in the implementation of the United Nations Basic Space Science Initiative (1991–2012). This paper highlights research results of A.M. Mathai in the period of time from 1962 to 2015. He published over 300 research papers and over 25 books. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

Journal Menu

Journal Browser

Axioms, Volume 4, Issue 3 (September 2015) – 13 articles , Pages 213-422

Further Information

Guidelines

MDPI Initiatives

Follow MDPI