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About 10,000 papers for “Partial function”

Partial sums of the cotangent function

Sandro Bettin, Sary Drappeau · Journal de Théorie des Nombres de Bordeaux · 2020

We prove the existence of reciprocity formulae for sums of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mfrac> <mml:mi>m</mml:mi> <mml:mi>k</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo form="prefix">cot</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>π</mml:mi> <mml:mfrac> <mml:mrow> <mml:mi>m</mml:mi> <mml:mi>h</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:mfrac> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> is a piecewise <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> function, featuring an alternating phenomenon not visible in the classical case where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> . We deduce bounds for these sums in terms of the continued fraction expansion of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>/</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:math> .

Cited by 3Open access

Green's function in partial subdivision networks

Ángeles Carmona Mejías, Margarida Mitjana Riera, Enrique P.J. Monsó Burgués · Linear and Multilinear Algebra · 2018

In the present work, we define a partial subdivision network ΓS of a given network Γ, by inserting a new vertex in some selected edges of Γ, so that each of these edges is replaced by two new edges with conductances that fulfil the Kirchhoff series law on the new network. Then, we obtain an expression for the Green kernel of the partial subdivision network in terms of the Green kernel of the base network. For that, we show the relation between Poisson problems on the partial subdivision network and Poisson problems on the base network. Moreover, we also obtain the effective resistance and the Kirchhoff index of the partial subdivision network in terms of the corresponding parameters on the base network. Finally, as an example, we carry out the computations in the case of a star network in which we have subdivided the even edges.

Cited by 2

Partial sums of the Rabotnov function

Sercan Kazımoğlu, Erhan Deni̇z · Acta Universitatis Sapientiae Mathematica · 2022

Abstract This article deals with the ratio of normalized Rabotnov function ℝ α,β ( z ) and its sequence of partial sums (ℝ α,β ) m ( z ). Several examples which illustrate the validity of our results are also given.

Cited by 1Open access

Partial sums of the cotangent function

Sandro Bettin, Sary Drappeau · CINECA IRIS Institutial Research Information System (University of Genoa) · 2020

We prove the existence of reciprocity formulae for sums of the form [Formula Presented] where f is a piecewise C1 function, featuring an alternating phenomenon not visible in the classical case where f(x) = x. We deduce bounds for these sums in terms of the continued fraction expansion of h/k.

Cited by 8

Partial differentials of Marcum's Q function

William K. Pratt · Proceedings of the IEEE · 1968

The partial derivatives of Marcum's Q function Q(a,b), with respect to a and b, are required for many communication system optimization problems. A simple derivation of the derivatives is given.

Cited by 35

Partial sums of Mittag-Leffler function

Deepak Bansal, Halit Orhan · Journal of Mathematical Inequalities · 2018

In the present investigation, Mittag-Leffler function with their normalization are considered. In this paper, we will study the ratio of a function of the form (??) to its sequence of par-

Cited by 5Open access

Variance function additive partial linear models

Yixin Fang, Heng Lian, Hua Liang et al. · Electronic Journal of Statistics · 2015

To model heteroscedasticity in a broad class of additive partial linear models, we allow the variance function to be an additive partial linear model as well and the parameters in the variance function to be different from those in the mean function. We develop a two-step estimation procedure, where in the first step initial estimates of the parameters in both the mean and variance functions are obtained and then in the second step the estimates are updated using the weights based on the initial estimates. We use polynomial splines to approximate the additive nonparametric components in both the mean and variation functions and derive their convergence rates. The resulting weighted estimators of the linear coefficients in both the mean and variance functions are shown to be asymptotically normal and more efficient than the initial un-weighted estimators. Simulation experiments are conducted to examine the numerical performance of the proposed procedure, which is also applied to analyze the dataset from a nutritional epidemiology study.

Cited by 5Open access

On Partial-Function Application in Z

David A. Duffy · Electronic workshops in computing · 1998

We discuss the application of partial functions to elements outside their domain in the context of the Z language and CADiZ tool. We illustrate some surprising results that can arise, and show that they may be readily justified, but also show that undesirable results can arise that are less readily resolved. We offer two possible resolutions, one that involves a simple change to, for example, free types, and another more contentious approach that involves, in particular, a new semantics for Z. We discuss the advantages and disadvantages of both approaches.

Cited by 1Open access

Function Spaces and Partial Differential Equations

Ali Taheri · 2015

Abstract This book presents a comprehensive treatment of aspects of classical and modern analysis relating to theory of ‘partial differential equations’ and the associated ‘function spaces’. It begins with a quick review of basic properties of harmonic functions and Poisson integrals and then moves into a detailed study of Hardy spaces. The classical Dirichlet problem is considered and a variety of methods for its resolution ranging from potential theoretic (Perron’s method of sub-harmonic functions and Wiener’s criterion, Green’s functions and Poisson integrals, the method of layered potentials or integral equations) to variational (Dirichlet principle) are presented. Parallel to this is the development of the necessary function spaces: Lorentz and Marcinkiewicz spaces, Sobolev spaces (integer as well as fractional order), Hardy spaces, the John-Nirenberg space BMO, Morrey and Campanato spaces, Besov spaces and Triebel-Lizorkin spaces. Harmonic analysis is deeply intertwined with the topics covered and the subjects of summability methods, Tauberian theorems, convolution algebras, Calderon-Zygmund theory of singular integrals and Littlewood-Paley theory that on the one hand connect to various PDE estimates (Calderon-Zygmund inequality, Strichartz estimates, Mihlin-Hormander multipliers, etc.) and on the other lead to a unified characterisation of various function spaces are discussed in great depth. The book ends by a discussion of regularity theory for second order elliptic equations in divergence form— first with continuous and next with measurable coefficients—and covers, in particular, De Giorgi’s theorem, Moser iteration, Harnack inequality and local boundedness of solutions. (The case of elliptic systems and related topics is discussed in the exercises.)

Cited by 18