scholarly journals Sequential Regular Variation: Extensions of Kendall’s Theorem

2020 ◽  
Vol 71 (4) ◽  
pp. 1171-1200
Author(s):  
Nicholas H Bingham ◽  
Adam J Ostaszewski
Keyword(s):  
Regular Variation ◽  
General Setting ◽  
Continuous Parameter ◽  
Special Cases ◽  
Parameter Theory

Abstract Regular variation is a continuous-parameter theory; we work in a general setting, containing the existing Karamata, Bojanic–Karamata/de Haan and Beurling theories as special cases. We give sequential versions of the main theorems, that is, with sequential rather than continuous limits. This extends the main result, a theorem of Kendall’s (which builds on earlier work of Kingman and Croft), to the general setting.

scholarly journals A perturbed algorithm for strongly nonlinear variational-like inclusions

2000 ◽  
Vol 62 (3) ◽  
pp. 417-426 ◽  
Author(s):  
C.-H. Lee ◽  
Q. H. Ansari ◽  
J.-C. Yao
Keyword(s):  
Exact Solution ◽  
General Setting ◽  
Approximate Solutions ◽  
Strongly Nonlinear ◽  
Special Cases ◽  
The One

In this paper, we define the concept of η- subdifferential in a more general setting than the one used by Yang and Craven in 1991. By using η-subdifferentiability, we suggest a perturbed algorithm for finding the approximate solutions of strongly nonlinear variational-like inclusions and prove that these approximate solutions converge to the exact solution. Several special cases are also discussed.

scholarly journals Unified Approach to Fractional Calculus Images of Special Functions—A Survey

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2260 ◽  
Author(s):  
Virginia Kiryakova
Keyword(s):  
Fractional Calculus ◽  
Hypergeometric Function ◽  
Special Functions ◽  
Great Part ◽  
General Setting ◽  
Generalized Hypergeometric Function ◽  
Unified Approach ◽  
Generalized Fractional Calculus ◽  
Special Cases ◽  
Google Search

Evaluation of images of special functions under operators of fractional calculus has become a hot topic with hundreds of recently published papers. These are growing daily and we are able to comment here only on a few of them, including also some of the latest of 2019–2020, just for the purpose of illustrating our unified approach. Many authors are producing a flood of results for various operators of fractional order integration and differentiation and their generalizations of different special (and elementary) functions. This effect is natural because there are great varieties of special functions, respectively, of operators of (classical and generalized) fractional calculus, and thus, their combinations amount to a large number. As examples, we mentioned only two such operators from thousands of results found by a Google search. Most of the mentioned works use the same formal and standard procedures. Furthermore, in such results, often the originals and the images are special functions of different kinds, or the images are not recognized as known special functions, and thus are not easy to use. In this survey we present a unified approach to fulfill the mentioned task at once in a general setting and in a well visible form: for the operators of generalized fractional calculus (including also the classical operators of fractional calculus); and for all generalized hypergeometric functions such as pΨq and pFq, Fox H- and Meijer G-functions, thus incorporating wide classes of special functions. In this way, a great part of the results in the mentioned publications are well predicted and appear as very special cases of ours. The proposed general scheme is based on a few basic classical results (from the Bateman Project and works by Askey, Lavoie–Osler–Tremblay, etc.) combined with ideas and developments from more than 30 years of author’s research, and reflected in the cited recent works. The main idea is as follows: From one side, the operators considered by other authors are cases of generalized fractional calculus and so, are shown to be (m-times) compositions of weighted Riemann–Lioville, i.e., Erdélyi–Kober operators. On the other side, from each generalized hypergeometric function pΨq or pFq (p≤q or p=q+1) we can reach, from the final number of applications of such operators, one of the simplest cases where the classical results are known, for example: to 0Fq−p (hyper-Bessel functions, in particular trigonometric functions of order (q−p)), 0F0 (exponential function), or 1F0 (beta-distribution of form (1−z)αzβ). The final result, written explicitly, is that any GFC operator (of multiplicity m≥1) transforms a generalized hypergeometric function into the same kind of special function with indices p and q increased by m.

General optimal stopping theorems for semi-Markov processes

1993 ◽  
Vol 25 (4) ◽  
pp. 825-846 ◽  
Author(s):  
Frans A. Boshuizen ◽  
José M. Gouweleeuw
Keyword(s):  
Markov Processes ◽  
Optimal Stopping ◽  
Stopping Time ◽  
General Setting ◽  
The State ◽  
Shock Models ◽  
Special Cases ◽  
Optimal Stopping Problems ◽  
The Times ◽  
Semi Markov Processes

In this paper, optimal stopping problems for semi-Markov processes are studied in a fairly general setting. In such a process transitions are made from state to state in accordance with a Markov chain, but the amount of time spent in each state is random. The times spent in each state follow a general renewal process. They may depend on the present state as well as on the state into which the next transition is made.Our goal is to maximize the expected net return, which is given as a function of the state at time t minus some cost function. Discounting may or may not be considered. The main theorems (Theorems 3.5 and 3.11) are expressions for the optimal stopping time in the undiscounted and discounted case. These theorems generalize results of Zuckerman [16] and Boshuizen and Gouweleeuw [3]. Applications are given in various special cases.The results developed in this paper can also be applied to semi-Markov shock models, as considered in Taylor [13], Feldman [6] and Zuckerman [15].

scholarly journals Endpoints of Multivalued Contraction Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Bhagwati Prasad ◽  
Ritu Sahni
Keyword(s):  
General Setting ◽  
Multivalued Contraction ◽  
Special Cases

The existence of the endpoints and approximate endpoints are studied in a general setting for the operators satisfying various contractive conditions. Some recent results are also derived as special cases.

scholarly journals On inequalities for integral operators

1970 ◽  
Vol 11 (2) ◽  
pp. 126-133 ◽  
Author(s):  
G. O. Okikiolu
Keyword(s):  
General Setting ◽  
Integral Operators ◽  
The Other ◽  
Special Cases ◽  
Measure Spaces ◽  
Schur Inequality ◽  
Abstract Measure ◽  
Schur Theorem ◽  
Image Position ◽  
Similar Extension

In two papers [3] and [4], the author has extended the inequality of Schur (Theorem 319 of [2]) to cases involving kernels which satisfy identities of the formThe purpose of this paper is to prove a general inequality, which includes the above and also the inequality of Young (Theorem 281 of [2]) as special cases. We shall give the results a general setting by considering functions defined on abstract measure spaces. From this we shall deduce an extension to n dimensions of the results given in [3], which also generalises a similar extension of the Schur inequality given by Stein and Weiss. In fact some cases of the other results given in [5] will follow directly from our theorem.

Large-Deviation Approximations for General Occupancy Models

2008 ◽  
Vol 17 (3) ◽  
pp. 437-470 ◽  
Author(s):  
JIM X. ZHANG ◽  
PAUL DUPUIS
Keyword(s):  
Large Deviation ◽  
Empirical Distribution ◽  
General Setting ◽  
Rate Function ◽  
Occupancy Models ◽  
Deviation Analysis ◽  
Finite Dimensional ◽  
Special Cases ◽  
Occupancy Problems ◽  
Bose Einstein

We obtain large-deviation approximations for the empirical distribution for a general family of occupancy problems. In the general setting, balls are allowed to fall in a given urn depending on the urn's contents prior to the throw. We discuss a parametric family of statistical models that includes Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics as special cases. A process-level large-deviation analysis is conducted and the rate function for the original problem is then characterized, via the contraction principle, by the solution to a calculus of variations problem. The solution to this variational problem is shown to coincide with that of a simple finite-dimensional minimization problem. As a consequence, the large-deviation approximations and related qualitative information are available in more-or-less explicit form.

scholarly journals Two-block configuration formulae for BTD's with parameters (V,B,R, 3, Λ)

2000 ◽  
Vol 61 (3) ◽  
pp. 463-471
Author(s):  
M. A. Francel
Keyword(s):  
General Setting ◽  
Block Designs ◽  
Special Cases

In this paper, we count two-block configurations in a general setting. In particular, no restriction is put on the pair repetition factor (that is, the parameter Λ) of the block designs being considered. Besides giving formulae for the counts of the two-block BTD (V, B, R, 3, Λ) configuration classes, a basis for the formulae is given and shown to have size four. Details of the special cases where Λ equals two and three are also presented, along with 2-block BIBD (v, b, r, 3, λ) configuration results.

scholarly journals Regular variation for measures on metric spaces

2006 ◽  
Vol 80 (94) ◽  
pp. 121-140 ◽  
Author(s):  
Henrik Hult ◽  
Filip Lindskog
Keyword(s):  
Regular Variation ◽  
Metric Spaces ◽  
General Setting ◽  
Continuous Functions ◽  
Separable Space ◽  
Real Numbers ◽  
Space Of Continuous Functions ◽  
Relative Compactness ◽  
Borel Measures

The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regular variation is defined in this general setting and several statements that are equivalent to this definition are presented. This extends the notion of regular variation for Borel measures on the Euclidean space Rd to more general metric spaces. Some examples, including regular variation for Borel measures on Rd, the space of continuous functions C and the Skorohod space D, are provided.

General optimal stopping theorems for semi-Markov processes

1993 ◽  
Vol 25 (04) ◽  
pp. 825-846 ◽  
Author(s):  
Frans A. Boshuizen ◽  
José M. Gouweleeuw
Keyword(s):  
Markov Processes ◽  
Optimal Stopping ◽  
Stopping Time ◽  
General Setting ◽  
The State ◽  
Shock Models ◽  
Special Cases ◽  
Optimal Stopping Problems ◽  
The Times ◽  
Semi Markov Processes

In this paper, optimal stopping problems for semi-Markov processes are studied in a fairly general setting. In such a process transitions are made from state to state in accordance with a Markov chain, but the amount of time spent in each state is random. The times spent in each state follow a general renewal process. They may depend on the present state as well as on the state into which the next transition is made. Our goal is to maximize the expected net return, which is given as a function of the state at time t minus some cost function. Discounting may or may not be considered. The main theorems (Theorems 3.5 and 3.11) are expressions for the optimal stopping time in the undiscounted and discounted case. These theorems generalize results of Zuckerman [16] and Boshuizen and Gouweleeuw [3]. Applications are given in various special cases. The results developed in this paper can also be applied to semi-Markov shock models, as considered in Taylor [13], Feldman [6] and Zuckerman [15].

scholarly journals GENERALIZED β-CONFORMAL CHANGE OF FINSLER METRICS

2010 ◽  
Vol 07 (04) ◽  
pp. 565-582 ◽  
Author(s):  
NABIL L. YOUSSEF ◽  
S. H. ABED ◽  
S. G. ELGENDI
Keyword(s):  
General Setting ◽  
Finsler Metrics ◽  
Finsler Spaces ◽  
General Transformation ◽  
Conformal Change ◽  
Special Cases ◽  
Geometric Objects

In this paper, we introduce and investigate a general transformation or change of Finsler metrics, which is referred to as a generalized β-conformal change: [Formula: see text] This transformation combines both β-change and conformal change in a general setting. The change, under this transformation, of the fundamental Finsler connections, together with their associated geometric objects, are obtained. Some invariants and various special Finsler spaces are investigated under this change. The most important changes of Finsler metrics existing in the literature are deduced from the generalized β-conformal change as special cases.

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