The Parseval Equality and Expansion Formula for Singular Hahn-Dirac System

This work studies the singular Hahn-Dirac system given by Here 𝜇 is a complex spectral parameter, p(.) and r(.) are real-valued continuous functions at 𝜔0, defined on [𝜔0,∞) and q∈(0,1), , 𝜔>0, x∈[𝜔0,∞). The existence of a spectral function for this system is proved. Further, a Parseval equality and an expansion formula in eigenfunctions are proved in terms of the spectral function.

Cooperative Grey Games

In this chapter, the authors extend transportation situations under uncertainty by using grey numbers. Further, they try in this research building models for grey game problems on transportation situations proposing the ideas of grey solutions and their corresponding structures. They introduce cooperative grey games and grey solutions. They focus on the grey Shapley value and the grey core of the modeled game arising from transportation situations. Moreover, they prove the nonemptiness of the grey core for the transportation grey games, and some results on the relationship between the grey core.

COGARCH Models

In this chapter, the features of a continuous time GARCH (COGARCH) process is discussed since the process can be applied as an explicit solution for the stochastic differential equation which is defined for the volatility of unequally spaced time series. COGARCH process driven by a Lévy process is an analogue of discrete time GARCH process and is further generalized to solutions of Lévy driven stochastic differential equations. The Compound Poisson and Variance Gamma processes are defined and used to derive the increments for the COGARCH process. Although there are various parameter estimation methods introduced for COGARCH, this study is focused on two methods which are Pseudo Maximum Likelihood Method and General Methods of Moments. Furthermore, an example is given to illustrate the findings.

Application of the Exponential Rational Function Method to Some Fractional Soliton Equations

In this chapter, the authors study the exponential rational function method to find new exact solutions for the time-fractional fifth-order Sawada-Kotera equation, the space-time fractional Whitham-Broer-Kaup equations, and the space-time fractional generalized Hirota-Satsuma coupled KdV equations. These fractional differential equations are converted into ordinary differential equations by using the fractional complex transform. The fractional derivatives are defined in the sense of Jumarie's modified Riemann-Liouville. The proposed method is direct and effective for solving different kind of nonlinear fractional equations in mathematical physics.

Spectral Problem for a Polynomial Pencil of the Sturm-Liouville Equations

A boundary value problem for the second order differential equation -y′′+∑_{m=0}N−1λ^{m}q_{m}(x)y=λ2Ny with two boundary conditions a_{i1}y(0)+a_{i2}y′(0)+a_{i3}y(π)+a_{i4}y′(π)=0, i=1,2 is considered. Here n>1, λ is a complex parameter, q0(x),q1(x),...,q_{n-1}(x) are summable complex-valued functions, a_{ik} (i=1,2; k=1,2,3,4) are arbitrary complex numbers. It is proved that the system of eigenfunctions and associated eigenfunctions is complete in the space and using elementary asymptotical metods asymptotic formulas for the eigenvalues are obtained.

Stability Analysis of a Nonlinear Epidemic Model With Generalized Piecewise Constant Argument

The authors consider a nonlinear epidemic equation by modeling it with generalized piecewise constant argument (GPCA). The authors investigate invariance region for the considered model. Sufficient conditions guaranteeing the existence and uniqueness of the solutions of the model are given by creating integral equations. An important auxiliary result giving a relation between the values of the unknown function solutions at the deviation argument and at any time t is indicated. By using Lyapunov-Razumikhin method developed by Akhmet and Aruğaslan for the differential equations with generalized piecewise constant argument (EPCAG), the stability of the trivial equilibrium is investigated in addition to the stability examination of the positive equilibrium transformed into the trivial equilibrium. Then sufficient conditions for the uniform stability and the uniform asymptotic stability of trivial equilibrium and the positive equilibrium are given.

On the Orthogonality of the q-Derivatives of the Discrete q-Hermite I Polynomials

Discrete q-Hermite I polynomials are a member of the q-polynomials of the Hahn class. They are the polynomial solutions of a second order difference equation of hypergeometric type. These polynomials are one of the q-analogous of the Hermite polynomials. It is well known that the q-Hermite I polynomials approach the Hermite polynomials as q tends to 1. In this chapter, the orthogonality property of the discrete q-Hermite I polynomials is considered. Moreover, the orthogonality relation for the k-th order q-derivatives of the discrete q-Hermite I polynomials is obtained. Finally, it is shown that, under a suitable transformation, these relations give the corresponding relations for the Hermite polynomials in the limiting case as q goes to 1.

Auditors in the Economy and the Impact of Rent-Seeking Behaviour and Penalties

Society often relies on information disclosed by enterprises and verified by auditors to decide on an efficient allocation of capital. Auditing sector serves as a means of verification to protect investors from making decisions based on inaccurate information. However, auditors can use their superior information for extracting additional rents. This study explores an economy where entrepreneurs choose their financial reporting quality considering incentives imposed by the society, and rent-seeking auditors may manipulate their reports to extract gains in the expense of public interest. The analysis captures the dynamics of strategy changes among different actors by introducing a population game framework. The steady-state equilibrium analysis shows that there is a pure state and mixed states whose stability is affected by policy parameters such as subsidies, taxes, competitive auditor fee, and rate of adjustment of different behavioral dynamics. It appears that corruption in auditing sector and poor quality in financial reporting may arise as a temporally persistent outcome.

A Generalization of Groups

For a nonempty set G, the authors define an operation * reckoned with closeness property (i.e.,* is an operation which is not a binary operation). Then they define the partial group as a generalisation of a group. A partial group G which is a non empty set satisfies following conditions hold for all a,b and c∈d:(PG1) If ab,(ab)c, bc and a (bc) is defined, then,(ab) c=a (bc)(PG2) . For every,a∈G there exists an e∈G such that ae and ea are defined and, ae=ea=a (PG3) . For every,a∈G there exists an a∈G such that aa and aa are defined and aa=aa=e. The * operation effects and changes the properties of group axioms. So that lots of group theoretic theorems and conclusions do not work in partial groups. Thus, this description gives us some fundemental and important properties and analogous to group theory. Also the authors have some differences from group theory.

The Stability of an Epidemic Model With Piecewise Constant Argument by Lyapunov-Razumikhin Method

The authors propose a nonlinear epidemic model by developing it with generalized piecewise constant argument (GPCA) introduced by Akhmet. The authors investigate invariance region for the considered model. For the taken model into consideration, they obtain a useful inequality concerning relation between the values of the solutions at the deviation argument and at any time for the epidemic model. The authors reach sufficient conditions for the existence and uniqueness of the solutions. Then, based on Lyapunov-Razumikhin method developed by Akhmet and Aruğaslan for the differential equations with generalized piecewise constant argument (EPCAG), sufficient conditions for the stability of the trivial equilibrium and the positive equilibrium are investigated. Thus, the theoretical results concerning the uniform stability of the equilibriums are given.