scholarly journals On Convergence Rates of Some Limits

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 634
Author(s):  
Edward Omey ◽  
Meitner Cadena
Keyword(s):  
General Condition ◽  
Representation Theorem ◽  
Convergence Rates ◽  
Convergence Condition ◽  
Class Of Functions ◽  
General Convergence

In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each y > 0 , x L ( x + y ) L ( x ) − 1 → 0 as x → ∞ . Very recently, we have extended this result by considering a wider class of functions U related to the following more general condition. For each y > 0 , r ( x ) U ( x + y g ( x ) ) U ( x ) − 1 → 0 as x → ∞ , for some functions r and g. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided.

scholarly journals Convergence rates for estimators of geodesic distances and Frèchet expectations

2018 ◽  
Vol 55 (4) ◽  
pp. 1001-1013
Author(s):  
Catherine Aaron ◽  
Olivier Bodart
Keyword(s):  
Probability Distribution ◽  
Convergence Rate ◽  
Compact Manifold ◽  
Convergence Rates ◽  
Geodesic Distance ◽  
Convergence Result ◽  
Regularity Conditions ◽  
Compact Set ◽  
Geodesic Distances ◽  
General Convergence

Abstract Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

25.—On the Eigenfunction Expansion associated with a Singular Complex-valued Fourth-order Differential Equation

1976 ◽  
Vol 75 (4) ◽  
pp. 325-332
Author(s):  
Jyoti Chaudhuri ◽  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Representation Theorem ◽  
Eigenfunction Expansion ◽  
Order Differential Equation ◽  
Convergence Theory ◽  
Infinite Intervals ◽  
Complex Valued ◽  
Class Of Functions ◽  
Fourth Order Differential Equation

SynopsisThe direct convergence theory of eigenfunction expansions associated with boundry value problems, not necessarily self-adjoint, generated from complex-valued fourth-order symmetric ordinary differential expressions on semi-infinite intervals, is discussed. An admissible class of functions for the expansion is characterised. Also a generalisation of Stieltjes representation theorem for analytic functions discussed in [13, §§ 22.23 and 24] is obtained.

Characterization of convergence rates for the approximation of the stationary distribution of infinite monotone stochastic matrices

1996 ◽  
Vol 33 (04) ◽  
pp. 974-985 ◽  
Author(s):  
F. Simonot ◽  
Y. Q. Song
Keyword(s):  
Convergence Rate ◽  
Generating Function ◽  
Convergence Rates ◽  
Transition Probability ◽  
Stochastic Matrix ◽  
Input Sequence ◽  
Transition Probability Matrix ◽  
Stochastic Matrices ◽  
Preceding Result

Let P be an infinite irreducible stochastic matrix, recurrent positive and stochastically monotone and Pn be any n × n stochastic matrix with Pn ≧ Tn , where Tn denotes the n × n northwest corner truncation of P. These assumptions imply the existence of limit distributions π and π n for P and Pn respectively. We show that if the Markov chain with transition probability matrix P meets the further condition of geometric recurrence then the exact convergence rate of π n to π can be expressed in terms of the radius of convergence of the generating function of π. As an application of the preceding result, we deal with the random walk on a half line and prove that the assumption of geometric recurrence can be relaxed. We also show that if the i.i.d. input sequence (A(m)) is such that we can find a real number r 0 > 1 with , then the exact convergence rate of π n to π is characterized by r 0. Moreover, when the generating function of A is not defined for |z| > 1, we derive an upper bound for the distance between π n and π based on the moments of A.

scholarly journals Geometric applications of critical point theory to submanifolds of complex projective space

1974 ◽  
Vol 55 ◽  
pp. 5-31 ◽  
Author(s):  
Thomas E. Cecil
Keyword(s):  
Projective Space ◽  
Critical Point ◽  
Distance Function ◽  
Complex Projective Space ◽  
Euclidean Distance ◽  
Point Theory ◽  
Euclidean Distance Function ◽  
Complex Projective ◽  
Class Of Functions

In a recent paper, [6], Nomizu and Rodriguez found a geometric characterization of umbilical submanifolds Mn ⊂ Rn+p in terms of the critical point behavior of a certain class of functions Lp, p ⊂ Rn+p, on Mn. In that case, if p ⊂ Rn+p, x ⊂ Mn, then Lp(x) = (d(x,p))2, where d is the Euclidean distance function.

scholarly journals Convergence Rates in the Implicit Renewal Theorem on Trees

2013 ◽  
Vol 50 (4) ◽  
pp. 1077-1088
Author(s):  
Predrag R. Jelenković ◽  
Mariana Olvera-Cravioto
Keyword(s):  
Fixed Point ◽  
Rate Of Convergence ◽  
Convergence Rates ◽  
Tail Asymptotics ◽  
Renewal Theorem ◽  
Branching Trees

We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

scholarly journals What is invexity?

1986 ◽  
Vol 28 (1) ◽  
pp. 1-9 ◽  
Author(s):  
A. Ben-Israel ◽  
B. Mond
Keyword(s):  
Mathematical Programming ◽  
Convex Functions ◽  
Constraint Qualification ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Slater Constraint Qualification ◽  
The Relationship ◽  
Unconstrained Problems ◽  
Class Of Functions

AbstractRecently it was shown that many results in Mathematical Programming involving convex functions actually hold for a wider class of functions, called invex. Here a simple characterization of invexity is given for both constrained and unconstrained problems. The relationship between invexity and other generalizations of convexity is illustrated. Finally, it is shown that invexity can be substituted for convexity in the saddle point problem and in the Slater constraint qualification.

scholarly journals Estudio de las fachadas de inmuebles protegidos del casco urbano de Cieza (Murcia) = Study of the facades of protected buildings in the urban area of Cieza (Murcia)

2018 ◽  
Vol 4 (1) ◽  
pp. 56
Author(s):  
Eloisa González Ponce ◽  
Nuria Rosa Roca ◽  
Silvia Spairani Berrio ◽  
Borja Perez Pardos
Keyword(s):  
Urban Area ◽  
General Condition ◽  
Research Work ◽  
The State ◽  
Architectural Heritage ◽  
Good General Condition ◽  
Sustainable Materials ◽  
The City

ResumenEl propósito principal de este trabajo de investigación se centra en el estudio del estado de conservación de las fachadas de 34 inmuebles pertenecientes al Catálogo de Bienes Inmuebles y Elementos Protegidos del Plan General del Ayuntamiento de Cieza (Murcia), concretamente de aquellos situados en el casco urbano, para aportar directrices que garanticen una intervención con materiales sostenibles coherente y respetuosa con el patrimonio arquitectónico de la ciudad. Evidentemente, un desarrollo correcto del estudio diagnóstico constructivo de las fachadas de inmuebles protegidos debe llevarse a cabo en 2 fases: la caracterización constructiva de la arquitectura de sus fachadas y el estudio de lesiones o estado actual que presentan las mismas. Tras analizar los 34 inmuebles se establece una metodología y un modelo sistemático para la documentación del grado de deterioro que presentan las fachadas de los inmuebles del Catálogo de la ciudad. Destacamos que los resultados de materiales propuestos en las fachadas dan unos niveles de deterioro “bajo” o “muy bajo” en el 70 % de los casos analizados, corroborándose el buen estado general de las mismas.AbstractThe main intention of this research work focus interest on the 34 building’s facade from the state of preservation belonging a Municipalities Heritage Catalogue of Cieza's Town (Murcia), it specifically those that located in the urban area from to comply with the guidelines established by sustainable materials coherent and respectful with the architectural heritage of the city. Evidently, a correct development of the diagnostic constructive study of the 34 building’s facade must be carried out in 2 phases: the constructive characterization of the architecture of his building’s facade and the study of the state of conservation or current condition that the same ones present. After analyzing 34 building’s facade methodology and a systematic model it establish for the documentation of a Municipalities Heritage Catalogue of Cieza's Town of the city. We emphasize that the results of proposed materials building’s facade give a few levels of low or very low deterioration in 70 % of the analyzed cases, there being corroborated the good general condition of the same ones.

scholarly journals Chemotaxis on networks: Analysis and numerical approximation

2020 ◽  
Vol 54 (4) ◽  
pp. 1339-1372
Author(s):  
Herbert Egger ◽  
Lukas Schöbel-Kröhn
Keyword(s):  
Convergence Rates ◽  
Energy Estimates ◽  
Implicit Time ◽  
Discrete Approximations ◽  
Uniform Bounds ◽  
Time Stepping ◽  
Characterization Of Solutions ◽  
Networks Analysis ◽  
Theoretical Results

We consider the Keller–Segel model of chemotaxis on one-dimensional networks. Using a variational characterization of solutions, positivity preservation, conservation of mass, and energy estimates, we establish global existence of weak solutions and uniform bounds. This extends related results of Osaki and Yagi to the network context. We then analyze the discretization of the system by finite elements and an implicit time-stepping scheme. Mass lumping and upwinding are used to guarantee the positivity of the solutions on the discrete level. This allows us to deduce uniform bounds for the numerical approximations and to establish order optimal convergence of the discrete approximations to the continuous solution without artificial smoothness requirements. In addition, we prove convergence rates under reasonable assumptions. Some numerical tests are presented to illustrate the theoretical results.

scholarly journals The class of functions convex in the negative direction of the imaginary axis of order(α,β)

2002 ◽  
Vol 29 (11) ◽  
pp. 641-650
Author(s):  
Adam Lecko
Keyword(s):  
Imaginary Axis ◽  
Negative Direction ◽  
Alpha Beta ◽  
Class Of Functions

The aim of this paper is to present an analytic characterization of the class of functions convex in the negative direction of the imaginary axis of order(α,β). The method of the proof is based on Julia's lemma.

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