Clean Single-Valued Polylogarithms

We define a variant of real-analytic polylogarithms that are single-valued and that satisfy ''clean'' functional relations that do not involve any products of lower weight functions. We discuss the basic properties of these functions and, for depths one and two, we present some explicit formulas and results. We also give explicit formulas for the single-valued and clean single-valued version attached to the Nielsen polylogarithms Sn,2(x), and we show how the clean single-valued functions give new evaluations of multiple polylogarithms at certain algebraic points.

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New family of Whitney numbers

Filomat ◽

10.2298/fil1702309e ◽

2017 ◽

Vol 31 (2) ◽

pp. 309-320 ◽

Cited By ~ 2

Author(s):

B.S. El-Desouky ◽

Nenad Cakic ◽

F.A. Shiha

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Stirling Numbers ◽

Explicit Formulas ◽

Basic Properties ◽

New Family ◽

Whitney Numbers ◽

Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

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Variations of Mixed Hodge Structures of Multiple Polylogarithms

Canadian Journal of Mathematics ◽

10.4153/cjm-2004-057-2 ◽

2004 ◽

Vol 56 (6) ◽

pp. 1308-1338 ◽

Cited By ~ 3

Author(s):

Jianqiang Zhao

Keyword(s):

Number Field ◽

Zeta Functions ◽

Explicit Construction ◽

Hodge Structures ◽

Special Values ◽

Multiple Polylogarithms ◽

Real Analytic ◽

Mixed Hodge Structures

AbstractIt is well known that multiple polylogarithms give rise to good unipotent variations of mixed Hodge-Tate structures. In this paper we shall explicitly determine these structures related to multiple logarithms and some other multiple polylogarithms of lower weights. The purpose of this explicit construction is to give some important applications. First we study the limit of mixed Hodge-Tate structures and make a conjecture relating the variations of mixed Hodge-Tate structures of multiple logarithms to those of general multiple polylogarithms. Then following Deligne and Beilinson we describe an approach to defining the single-valued real analytic version of the multiple polylogarithms which generalizes the well-known result of Zagier on classical polylogarithms. In the process we find some interesting identities relating single-valued multiple polylogarithms of the same weight k when k = 2 and 3. At the end of this paper, motivated by Zagier's conjecture we pose a problem which relates the special values of multiple Dedekind zeta functions of a number field to the single-valued version of multiple polylogarithms.

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A q-analogue of α־-Whitney numbers

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm1801178e ◽

2018 ◽

Vol 12 (1) ◽

pp. 178-191 ◽

Cited By ~ 1

Author(s):

B.S. El-Desouky ◽

F.A. Shiha

Keyword(s):

Generating Functions ◽

Recurrence Relations ◽

Explicit Formulas ◽

Basic Properties ◽

Whitney Numbers

We define the (q,??)-Whitney numbers which are reduced to the ??-Whitney numbers when q ? 1. Moreover, we obtain several properties of these numbers such as explicit formulas, recurrence relations, generating functions, orthogonality and inverse relations. Finally, we define the ??-Whitney-Lah numbers as a generalization of the r-Whitney-Lah numbers and we introduce their important basic properties.

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EXACT FORMULAS FOR COEFFICIENTS OF JACOBI FORMS

International Journal of Number Theory ◽

10.1142/s1793042111004617 ◽

2011 ◽

Vol 07 (03) ◽

pp. 825-833 ◽

Cited By ~ 6

Author(s):

KATHRIN BRINGMANN ◽

OLAV K. RICHTER

Keyword(s):

Fourier Coefficients ◽

Theta Functions ◽

Poincaré Series ◽

Jacobi Forms ◽

Mock Theta Functions ◽

Explicit Formulas ◽

Poincare Series ◽

Real Analytic

In previous work, we introduced harmonic Maass–Jacobi forms. The space of such forms includes the classical Jacobi forms and certain Maass–Jacobi–Poincaré series, as well as Zwegers' real-analytic Jacobi forms, which play an important role in the study of mock theta functions and related objects. Harmonic Maass–Jacobi forms decompose naturally into holomorphic and non-holomorphic parts. In this paper, we give exact formulas for the Fourier coefficients of the holomorphic parts of harmonic Maass–Jacobi forms and, in particular, we obtain explicit formulas for the Fourier coefficients of weak Jacobi forms.

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More Explicit Formulas for the Matrix Exponential

Linear Algebra and its Applications ◽

10.1016/s0024-3795(96)00478-8 ◽

1997 ◽

Vol 262 (1-3) ◽

pp. 131-163 ◽

Cited By ~ 13

Author(s):

H Cheng

Keyword(s):

Matrix Exponential ◽

Explicit Formulas ◽

The Matrix

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Case Formulation and Design of Behavioral Treatment Programs

European Journal of Psychological Assessment ◽

10.1027//1015-5759.19.3.164 ◽

2003 ◽

Vol 19 (3) ◽

pp. 164-174 ◽

Cited By ~ 27

Author(s):

Stephen N. Haynes ◽

Andrew E. Williams

Keyword(s):

Functional Analysis ◽

Behavior Problems ◽

Behavioral Treatment ◽

Clinical Case ◽

Relative Magnitude ◽

Mechanisms Of Action ◽

Treatment Programs ◽

Case Formulation ◽

Functional Relations ◽

Causal Variables

Summary: We review the rationale for behavioral clinical case formulations and emphasize the role of the functional analysis in the design of individualized treatments. Standardized treatments may not be optimally effective for clients who have multiple behavior problems. These problems can affect each other in complex ways and each behavior problem can be influenced by multiple, interacting causal variables. The mechanisms of action of standardized treatments may not always address the most important causal variables for a client's behavior problems. The functional analysis integrates judgments about the client's behavior problems, important causal variables, and functional relations among variables. The functional analysis aids treatment decisions by helping the clinician estimate the relative magnitude of effect of each causal variable on the client's behavior problems, so that the most effective treatments can be selected. The parameters of, and issues associated with, a functional analysis and Functional Analytic Clinical Case Models (FACCM) are illustrated with a clinical case. The task of selecting the best treatment for a client is complicated because treatments differ in their level of specificity and have unequally weighted mechanisms of action. Further, a treatment's mechanism of action is often unknown.

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On the Essence and Tipology of International Public Goods

Voprosy Ekonomiki ◽

10.32609/0042-8736-2005-3-131-141 ◽

2005 ◽

pp. 131-141

Author(s):

V. Mortikov

Keyword(s):

Economic Policy ◽

Social Construction ◽

Public Goods ◽

Public Good ◽

Global Public Goods ◽

Club Goods ◽

Basic Properties ◽

International Public Goods ◽

Regional Public Goods ◽

Joint Products

The basic properties of international public goods are analyzed in the paper. Special attention is paid to the typology of international public goods: pure and impure, excludable and nonexcludable, club goods, regional public goods, joint products. The author argues that social construction of international public good depends on many factors, for example, government economic policy. Aggregation technologies in the supply of global public goods are examined.

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