Interesting Formula Research Articles

Generalised heat coefficients and associated spectral zeta functions on complex projective spaces Pn(ℂ)

ABSTRACTThe heat invariants or the Minakshisundaram-Pleijel heat coefficients describe the residues of the spectral zeta functions on any compact Riemannian manifold M. In this paper, we first give an explicit description of the heat coefficients via the Jacobi theta functions and their higher order derivatives, and then express these Minakshisundaram-Pleijel coefficients in terms of the residues of the associated zeta functions . Different interesting formulae are obtained for these zeta functions. A new class of heat coefficients, the Maclaurin heat coefficients () (i.e. the coefficients appearing in the Maclaurin expansion of the heat kernel ), are explicitly studied in terms of the classical and generalised Minakshisundaram-Pleijel coefficients and () respectively. Remarkable asymptotic expansions for the Maclaurin spectral functions are established. We also introduce and construct new zeta functions associated with these Maclaurin heat coefficients (generalised Minakshisundaram-Pleijel zeta functions), and it is interesting to see that these generalised zeta functions are explicitly understood in terms of the classical (Minakshisundaram-Pleijel) zeta functions.

A Study on Novel Extensions for the p-adic Gamma and p-adic Beta Functions

In this paper, we introduce the ρ , q -analog of the p-adic factorial function. By utilizing some properties of ρ , q -numbers, we obtain several new and interesting identities and formulas. We then construct the p-adic ρ , q -gamma function by means of the mentioned factorial function. We investigate several properties and relationships belonging to the foregoing gamma function, some of which are given for the case p = 2 . We also derive more representations of the p-adic ρ , q -gamma function in general case. Moreover, we consider the p-adic ρ , q -Euler constant derived from the derivation of p-adic ρ , q -gamma function at x = 1 . Furthermore, we provide a limit representation of aforementioned Euler constant based on ρ , q -numbers. Finally, we consider ρ , q -extension of the p-adic beta function via the p-adic ρ , q -gamma function and we then investigate various formulas and identities.

The Regulation on Village Governance in Indonesia: Efficient Contracting in Agency Theory

The application of efficient contracting concept in Village Fund management is an interesting idea and formula in the public governance perspective. The concept provides a potential solution so that the Village Government obey the Law No. 6 of 2014 and brings impact to public-oriented service. The philosophical concept of “Gotong Royong” or mutual assistance reinforces the efficient contracting essence. The aim of this research is to build a framework of public governance on Village Fund by implementing the efficient contracting, and to identify, illustrate and describe the variables in efficient contracting concept to Village Fund Management. The research used qualitative method, designed in narrative strategy and grounded theory approach, and formulated using constructivism paradigm. Results of the study have demonstrated that (1) Efficient contracting concept encompasses economic constitution, government policy, public finance management and governance, and the functional relationship in the government body (Central, Regional and Village); (2) the essence of Efficient Contracting is the community active participation, legal aspect, government attention, balance consensus, effectiveness and efficiency, justice and equality, transparency, responsibility and accountability. The descriptive findings were interpreted in textual and structural method to expand the Efficient Contracting concept and to bring implication in development administration and public management.

On Certain Transformation Formulas Involving Basic Hypergeometric Series

In this paper, making use of Bailey’s transform and certain summations of basic hypergeometric series, we have established interesting transformation formulae involving basic hypergeometric series.

More on a hyperbolic‐cotangent class of difference equations

We present a natural method for solving the difference equation urn:x-wiley:mma:media:mma5541:mma5541-math-0001 where , parameter a, and initial values x−j, , , are real or complex numbers. As a concrete example, the case k = 1, l = 2 is studied in detail, and several interesting formulas for solutions and objects used in the analysis of the equation are given. A useful remark on solvability of difference equations on complex domains is presented and used here.

The Generalized Quadratic Gauss Sum and Its Fourth Power Mean

In this article, our main purpose is to introduce a new and generalized quadratic Gauss sum. By using analytic methods, the properties of classical Gauss sums, and character sums, we consider the calculating problem of its fourth power mean and give two interesting computational formulae for it.

Phase-type distributions in population genetics

Probability modelling for DNA sequence evolution is well established and provides a rich framework for understanding genetic variation between samples of individuals from one or more populations. We show that both classical and more recent models for coalescence (with or without recombination) can be described in terms of the so-called phase-type theory, where complicated and tedious calculations are circumvented by the use of matrix manipulations. The application of phase-type theory in population genetics consists of describing the biological system as a Markov model by appropriately setting up a state space and calculating the corresponding intensity and reward matrices. Formulae of interest are then expressed in terms of these aforementioned matrices. We illustrate this procedure by a number of examples: (a) Calculating the mean, (co)variance and even higher order moments of the site frequency spectrum in multiple merger coalescent models, (b) Analysing a sample of DNA sequences from the Atlantic Cod using the Beta-coalescent, and (c) Determining the correlation of the number of segregating sites for multiple samples in the two-locus ancestral recombination graph. We believe that phase-type theory has great potential as a tool for analysing probability models in population genetics. The compact matrix notation is useful for clarification of current models, and in particular their formal manipulation and calculations, but also for further development or extensions.

Configurational stability of a crack propagating in a material with mode-dependent fracture energy - Part I: Mixed-mode I+III

In a previous paper (Leblond et al., 2011), we proposed a theoretical interpretation of the experimentally well-known instability of coplanar crack propagation in mode I+III. The interpretation relied on a stability analysis based on analytical expressions of the stress intensity factors for a crack slightly perturbed both within and out of its original plane, due to Gao and Rice (1986) and Movchan et al. (1998), coupled with a double propagation criterion combining Griffith (1920)’s energetic condition and Goldstein and Salganik (1974)’s principle of local symmetry. Under such assumptions instability modes were indeed evidenced for values of the mode mixity ratio - ratio of the mode III to mode I stress intensity factors applied remotely - larger than some threshold depending only on Poisson’s ratio. Unfortunately, the predicted thresholds were much larger than those generally observed for typical values of this material parameter. While the subcritical character of the nonlinear bifurcation from coplanar to fragmented fronts has been proposed as a possible explanation for this discrepancy (Chen et al., 2015), we propose here an alternative explanation based on the introduction of a constitutive relationship between the fracture energy and the mode mixity ratio, which is motivated by experimental observations. By re-examining the linear stability analysis of a planar propagating front, we show that such a relationship suffices, provided that it is strong enough, to lower significantly the threshold value of the mode mixity ratio for instability so as to bring it in a range more consistent with experiments. Interesting formulae are also derived for the distributions of the perturbed stress intensity factors and energy-release-rate, in the special case of perturbations of the crack surface and front obeying the principle of local symmetry and having reached a stationary state (corresponding to instability modes in near-threshold conditions).

Sums of weighted averages of gcd-sum functions

Let [Formula: see text] be the greatest common divisor of the integers [Formula: see text] and [Formula: see text]. In this paper, we give several interesting asymptotic formulas for weighted averages of the [Formula: see text]-sum function [Formula: see text] and the function [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text], namely [Formula: see text] with any fixed integer [Formula: see text] and any arithmetical function [Formula: see text]. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of [Formula: see text]-sum functions [Formula: see text]

On the classical Gauss sum and the recursive properties

Let p be a prime with pequiv1bmod8, ψ be an eighth character modp, and tau(psi) denote the classical Gauss sum modp. The main purpose of this paper is using the analytic method and the properties of the classical Gauss sum to study the computational problem of one kind rational polynomial of tau(psi). In the end, we prove an interesting second-order linear recursive formula for it.

On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums

Abstract The main purpose of this paper is to study the computational problem of one kind hybrid power mean involving two-term exponential sums and quartic Gauss sums using the analytic method and the properties of the classical Gauss sums, and to prove some interesting fourth-order linear recurrence formulae for this problem. As an application of our result, we can also obtain an exact computational formula for one kind congruence equation mod p, an odd prime.

Some Symmetric Identities Involving Fubini Polynomials and Euler Numbers

The aim of this paper is to use elementary methods and the recursive properties of a special sequence to study the computational problem of one kind symmetric sums involving Fubini polynomials and Euler numbers, and give an interesting computational formula for it. At the same time, we also give a recursive calculation method for the general case.

The hybrid power mean of the quartic Gauss sums and the two-term exponential sums

In this paper, we use the analytic method and the properties of classical Gauss sums to study the computational problems of one kind hybrid power mean of quartic Gauss sums and two-term exponential sums, and give an interesting fourth-order linear recurrence formula for it.

A spacetime derivation of the Lorentzian OPE inversion formula

Caron-Huot has recently given an interesting formula that determines OPE data in a conformal field theory in terms of a weighted integral of the four-point function over a Lorentzian region of cross-ratio space. We give a new derivation of this formula based on Wick rotation in spacetime rather than cross-ratio space. The derivation is simple in two dimensions but more involved in higher dimensions. We also derive a Lorentzian inversion formula in one dimension that sheds light on previous observations about the chaos regime in the SYK model.

Classical Magnetism and an Integral Formula Involving Modified Bessel Functions

Abstract We study an integral expression that is encountered in some classical spin models of magnetism. The idea is to calculate the key integral that represents the building block for the expression of the partition function of these models. The general calculation allows one to have a better look at the internal structure of the quantity of interest which, in turn, may lead to potentially new useful insights. We find out that application of two different approaches to solve the problem in a general-case scenario leads to an interesting integral formula involving modified Bessel functions of the first kind which appears to be new. We performed Monte Carlo simulations to verify the correctness of the integral formula obtained. Additional numerical integration tests lead to the same result as well. The approach under consideration, when generalized, leads to a linear integral equation that might be of interest to numerical studies of classical spin models of magnetism that rely on the well-established transfer-matrix formalism.

Some remarks regarding the $(p,q)-$Fibonacci and Lucas octonion polynomials

We investigate the $(p,q)-$Fibonacci and Lucas octonion polynomials. The main purpose of this paper is using of some properties of the $(p,q)-$ Fibonacci and Lucas polynomials. Also for present some results involving these octonion polynomials, we obtain some interesting computational formulas.

A Probabilistic Visit to Gauss’s Product Formula for Gamma Function

Some probabilistic methods are used to find a number of combinatorial identities in very elementary manners. One of the identities is for the Laplace transform of a suitable sequence of ordered random variables from an exponential distribution. This identity leads to Gauss’s product formula for gamma function, which itself provides many interesting formulas. Some asymptotics of the underlying distribution are described in connection with some of these identities and Gauss’s product formula.

Trade credit and replenishment decisions considering default risk

Abstract This paper determines the optimal credit period and replenishment decisions when credit period affects demand rate and default risk. This study uses the compounded interest formula to calculate the present value of expected profit and treats the rate of default risk as uncertainty. The problem is solved from both risk-neutral and risk-averse perspectives. Two risk-averse approaches, rate constraint and revenue constraint, are proposed, which limit the solution space to the set of credit periods which guarantee an upper bound of the expected rate of default risk and guarantee an expected revenue level when facing default risk. Numerical analyses are carried out to illustrate some management insights. We show that setting lower rate constraint or revenue constraint results in shorter credit period, longer replenishment cycle time and lower default risk. These approaches are useful for the case of a low-probability high-consequence contingency event.

Wallis formula from the harmonic oscillator

We show that the asymptotic formula for $\pi$, the Wallis formula, that was related with quantum mechanics and the hydrogen atom in \cite{HF}, can also be related to the harmonic oscillator using a quantum duality between these two systems. As a corollary we show that this very interesting asymptotic formula is not related with the hydrogen atom or quantum mechanics itself but with a clever choice of a trial function and a potential in the Schroedinger equation when we use the variational approach to calculate the ground state energy associated with the given potential function.

General solution to a higher-order linear difference equation and existence of bounded solutions

We present a closed-form formula for the general solution to the difference equation xn+k−qnxn=fn,n∈N0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$x_{n+k}-q_{n}x_{n}=f_{n},\\quad n\\in \\mathbb {N}_{0}, $$\\end{document} where kin mathbb {N}, (q_{n})_{nin mathbb {N}_{0}}, (f_{n})_{nin mathbb {N}_{0}}subset mathbb {C}, in the case q_{n}=q, nin mathbb {N}_{0}, qin mathbb {C}setminus{0}. Using the formula, we show the existence of a unique bounded solution to the equation when |q|>1 and sup_{nin mathbb {N}_{0}}|f_{n}|<infty by finding a solution in closed form. By using the formula for the bounded solution we introduce an operator that, together with the contraction mapping principle, helps in showing the existence of a unique bounded solution to the equation in the case where the sequence (q_{n})_{nin mathbb {N}_{0}} is real and nonconstant, which shows that, in this case, there is an elegant method of proving the result in a unified way. We also obtain some interesting formulas.