scholarly journals THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION

2009 ◽  
Vol 52 (1) ◽  
pp. 41-64 ◽  
Author(s):  
GIEDRIUS ALKAUSKAS
Keyword(s):  
Integral Equation ◽  
Generating Function ◽  
Period Function ◽  
Question Mark ◽  
Real Distribution ◽  
Exponential Generating Function ◽  
Wave Forms ◽  
Schmidt Operator ◽  
Definition Of ◽  
Minkowski Question Mark Function

AbstractThe Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern–Brocot tree. Surprisingly, the Eisenstein series G2(z) does manifest in both real and p-adic cases.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

scholarly journals IDENTITIES OF SYMMETRY FOR GENERALIZED TWISTED BERNOULLI POLYNOMIALS TWISTED BY RAMIFIED ROOTS OF UNITY

2014 ◽  
Vol 60 (1) ◽  
pp. 19-36
Author(s):  
Dae San Kim
Keyword(s):  
Generating Function ◽  
Bernoulli Polynomials ◽  
Roots Of Unity ◽  
Integral Expression ◽  
Power Sums ◽  
Exponential Generating Function

Abstract We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

On modified Green functions in exterior problems for the Helmholtz equation

1982 ◽  
Vol 383 (1785) ◽  
pp. 313-332 ◽  
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Helmholtz Equation ◽  
Least Squares ◽  
Present Note ◽  
Boundary Integral ◽  
Green Functions ◽  
Exterior Problems ◽  
The Green Function ◽  
Definition Of

The question of non-uniqueness in boundary integral equation formu­lations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of co­efficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

A generalized class of restricted Stirling and Lah numbers

2018 ◽  
Vol 68 (4) ◽  
pp. 727-740 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Combinatorial Properties ◽  
Exponential Generating Function ◽  
Cauchy Numbers ◽  
Restricted Stirling Numbers

Abstract In this paper, we consider a polynomial generalization, denoted by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) as well as for the associated Cauchy numbers.

Random Graphs from a Minor-Closed Class

2009 ◽  
Vol 18 (4) ◽  
pp. 583-599 ◽  
Author(s):  
COLIN McDIARMID
Keyword(s):  
Poisson Distribution ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Giant Component ◽  
Closed Class ◽  
Exponential Generating Function ◽  
Excluded Minor ◽  
A Minor ◽  
Labelled Graphs

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

scholarly journals On the affirmative solution to Salem’s problem

Analysis ◽  
2019 ◽  
Vol 39 (4) ◽  
pp. 135-149
Author(s):  
Semyon Yakubovich
Keyword(s):  
Linear Combination ◽  
Special Functions ◽  
Type Problem ◽  
Classical Analysis ◽  
Question Mark ◽  
Affirmative Solution ◽  
Minkowski Question Mark Function ◽  
Stieltjes Transforms

Abstract The Salem problem to verify whether Fourier–Stieltjes coefficients of the Minkowski question mark function vanish at infinity is solved recently affirmatively. In this paper by using methods of classical analysis and special functions we solve a Salem-type problem about the behavior at infinity of a linear combination of the Fourier–Stieltjes transforms. Moreover, as a consequence of the Salem problem, some asymptotic relations at infinity for the Fourier–Stieltjes coefficients of a power {m\in\mathbb{N}} of the Minkowski question mark function are derived.

scholarly journals On a New Family of Generalized Stirling and Bell Numbers

10.37236/564 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Recursion Relation ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Bell Numbers ◽  
New Family ◽  
Exponential Generating Function ◽  
Rook Numbers ◽  
Generalized Stirling Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

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