scholarly journals q-Binomial Convolution and Transformations of q-Appell Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 70
Author(s):  
Alaa Mohammed Obad ◽  
Asif Khan ◽  
Kottakkaran Sooppy Nisar ◽  
Ahmed Morsy
Keyword(s):  
Abelian Group ◽  
Group Structure ◽  
Random Variable ◽  
Scale Transformation ◽  
Appell Polynomials ◽  
Quantum Calculus ◽  
Appell Sequences ◽  
Binomial Convolution ◽  
Definition Of ◽  
Determinantal Form

In this paper, binomial convolution in the frame of quantum calculus is studied for the set Aq of q-Appell sequences. It has been shown that the set Aq of q-Appell sequences forms an Abelian group under the operation of binomial convolution. Several properties for this Abelian group structure Aq have been studied. A new definition of the q-Appell polynomials associated with a random variable is proposed. Scale transformation as well as transformation based on expectation with respect to a random variable is used to present the determinantal form of q-Appell sequences.

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

scholarly journals Certain results of hybrid families of special polynomials associated with appell sequences

Filomat ◽  
2019 ◽  
Vol 33 (12) ◽  
pp. 3833-3844 ◽  
Author(s):  
Ghazala Yasmin ◽  
Abdulghani Muhyi
Keyword(s):  
Generating Functions ◽  
Mixed Type ◽  
Bernoulli Polynomials ◽  
Operational Method ◽  
Appell Polynomials ◽  
Special Polynomials ◽  
Appell Sequences

In this article, the Legendre-Gould-Hopper polynomials are combined with Appell sequences to introduce certain mixed type special polynomials by using operational method. The generating functions, determinant definitions and certain other properties of Legendre-Gould-Hopper based Appell polynomials are derived. Operational rules providing connections between these formulae and known special polynomials are established. The 2-variable Hermite Kamp? de F?riet based Bernoulli polynomials are considered as an member of Legendre-Gould-Hopper based Appell family and certain results for this member are also obtained.

scholarly journals Fractional negative binomial and Pólya processes

2018 ◽  
Vol 38 (1) ◽  
pp. 77-101
Author(s):  
Palaniappan Vellai Samy ◽  
Aditya Maheshwari
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Infinitely Divisible ◽  
One Dimensional ◽  
Fractional Poisson Process ◽  
Negative Binomial Process ◽  
The One ◽  
Definition Of ◽  
Binomial Process

In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

scholarly journals Formulation of scale transformation in a stochastic data assimilation framework

2017 ◽  
Vol 24 (2) ◽  
pp. 279-291 ◽  
Author(s):  
Feng Liu ◽  
Xin Li
Keyword(s):  
Data Assimilation ◽  
Spatial Scale ◽  
Measure Theory ◽  
Stochastic Calculus ◽  
Scale Transformation ◽  
Earth Observations ◽  
Model Unit ◽  
Stochastic Data ◽  
Definition Of ◽  
New Formulation

Abstract. Understanding the errors caused by spatial-scale transformation in Earth observations and simulations requires a rigorous definition of scale. These errors are also an important component of representativeness errors in data assimilation. Several relevant studies have been conducted, but the theory of the scale associated with representativeness errors is still not well developed. We addressed these problems by reformulating the data assimilation framework using measure theory and stochastic calculus. First, measure theory is used to propose that the spatial scale is a Lebesgue measure with respect to the observation footprint or model unit, and the Lebesgue integration by substitution is used to describe the scale transformation. Second, a scale-dependent geophysical variable is defined to consider the heterogeneities and dynamic processes. Finally, the structures of the scale-dependent errors are studied in the Bayesian framework of data assimilation based on stochastic calculus. All the results were presented on the condition that the scale is one-dimensional, and the variations in these errors depend on the differences between scales. This new formulation provides a more general framework to understand the representativeness error in a non-linear and stochastic sense and is a promising way to address the spatial-scale issue.

scholarly journals On Viscosity and Equivalent Notions of Solutions for Anisotropic Geometric Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Cecilia De Zan ◽  
Pierpaolo Soravia
Keyword(s):  
Mean Curvature Flow ◽  
Group Structure ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Carnot Groups ◽  
Singular Limits ◽  
Euclidian Space ◽  
Characteristic Points ◽  
Definition Of ◽  
Homogeneous Norm

We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the Euclidian space, and in Carnot groups, it is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the Euclidian setting. These results simplify the handling of the singularities of the equation, for instance, to study the asymptotic behaviour of singular limits of reaction diffusion equations. We provide examples of using the simplified definition, showing, for instance, that boundaries of strictly convex subsets in the Carnot group structure become extinct in finite time when subject to the horizontal mean curvature flow even if characteristic points are present.

scholarly journals Collective rituals in team sports: Implications for team resilience and communal coping

2019 ◽  
pp. 27-36
Author(s):  
Devin Bonk ◽  
Chloé Leprince ◽  
Katherine A. Tamminen ◽  
Julie Doron
Keyword(s):  
Group Dynamics ◽  
Cultural Anthropology ◽  
Group Structure ◽  
Team Sports ◽  
Future Research ◽  
Communal Coping ◽  
Sports Teams ◽  
Team Resilience ◽  
Human And Social Sciences ◽  
Definition Of

Many sports teams engage in collective rituals (e.g., the New Zealand All Blacks’ haka). While the concept has been studied extensively in other fields (e.g., social psychology and cultural anthropology), literature on collective rituals specific to sport is limited. Leveraging theoretical positions and empirical findings from across the human and social sciences, the application of an existing definition of collective ritual in team sports is explored. Complementary research is suggestive of a potential link between collective rituals and two growing topics of interest in group dynamics, namely, team resilience and communal coping. Collective rituals can bolster team resilience by strengthening the group structure and increasing a team’s social capital. They can also serve as communal coping strategies, helping to manage team stressors as they arise. However, at the extremes, collective rituals can become problematic. Over-reliance and abusive rites of passage (i.e., hazing) are considered. Potential applied implications and future research directions in sport psychology are then discussed.

scholarly journals A Generalization of Final Rank of Primary Abelian Groups

1970 ◽  
Vol 22 (6) ◽  
pp. 1118-1122 ◽  
Author(s):  
Doyle O. Cutler ◽  
Paul F. Dubois
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Basic Subgroup ◽  
Primary Abelian Group ◽  
Limit Ordinal ◽  
Final Rank ◽  
Definition Of ◽  
Group Recall

Let G be a p-primary Abelian group. Recall that the final rank of G is infn∈ω{r(pnG)}, where r(pnG) is the rank of pnG and ω is the first limit ordinal. Alternately, if Γ is the set of all basic subgroups of G, we may define the final rank of G by supB∈Γ {r(G/B)}. In fact, it is known that there exists a basic subgroup B of G such that r(G/B) is equal to the final rank of G. Since the final rank of G is equal to the final rank of a high subgroup of G plus the rank of pωG, one could obtain the same information if the definition of final rank were restricted to the class of p-primary Abelian groups of length ω.

A regression problem

1951 ◽  
Vol 47 (2) ◽  
pp. 337-346
Author(s):  
D. V. Lindley
Keyword(s):  
Complete Solution ◽  
Practical Importance ◽  
Random Variable ◽  
Partial Solution ◽  
Point Of View ◽  
General Definition ◽  
Regression Problem ◽  
Definition Of ◽  
Regression Theory ◽  
Complete Solutions

During the Oxford Conference of the Econometric Society in September 1936, Ragnar Frisch proposed a problem in regression theory. A partial solution was found in 1938 by Miss H. V. Allen (1). A more complete solution was given by C. R. Rao (6) in 1947, and in the same year the present author (5) obtained a solution as a particular case of a more general result. These last two papers contained a flaw, and a correct solution was provided by Miss E. Fix (2). This last solution still leaves a part of the problem unanswered, and in the present paper a result of P. Lévy's (4), is used to complete the solution. At the same time further generalizations of the problem are considered and, in the cases of most practical importance, complete solutions are obtained. It is advisable, both from the point of view of rigour and simplicity of analysis, to use a general definition of the conditional expectation of a random variable. Accordingly, the paper begins with a summary of the relevant definitions. These notions were introduced by Kolmogoroff (3). It has been thought worth while giving the definitions here, in forms which are slightly different from Kolmogoroff's and seem more suitable for applications, in order to explain the notation and nomenclature used. The relevant consequences of these definitions are also stated in the form in which they are used.

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