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The aim of this paper is to present fixed point result of mappings satisfying a generalized rational contractive condition in the setup of multiplicative metric spaces. As an application, we obtain a common fixed point of a pair of weakly compatible mappings. Some common fixed point results of pair of rational contractive types mappings involved in cocyclic representation of a nonempty subset of a multiplicative metric space are also obtained. Some examples are presented to support the results proved herein. Our results generalize and extend various results in the existing literature.

We prove that functions analytic in the unit disk and continuous up to the boundary are dense in the de Branges–Rovnyak spaces induced by the extreme points of the unit ball of . Together with previous theorems, it follows that this class of functions is dense in any de Branges–Rovnyak space.

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift  is a contraction on the space. We present a model for this operator and use it to prove the surprising result that functions which extend continuously to the closure of the disk are dense in the space. This has several applications, for example we can answer a question regarding reverse Carleson embeddings for these spaces. We also identify a large class of spaces which are similar to the de Branges–Rovnyak spaces and prove some results which are new even in the classical case.

We study universality properties of the Epstein zeta function En(L,s) for lattices L of large dimension n and suitable regions of complex numbers s . Our main result is that, as n→∞ , En(L,s) is universal in the right half of the critical strip as L varies over all n -dimensional lattices L . The proof uses an approximation result for Dirichlet polynomials together with a recent result on the distribution of lengths of lattice vectors in a random lattice of large dimension and a strong uniform estimate for the error term in the generalized circle problem. Using the same basic approach we also prove that, as n→∞ , En(L1,s)−En(L2,s) is universal in the full half-plane to the right of the critical line as (L1,L2) varies over all pairs of n -dimensional lattices. Finally, we prove a more classical universality result for En(L,s) in the s -variable valid for almost all lattices L of dimension n . As part of the proof we obtain a strong bound of En(L,s) on the critical line that is subconvex for n≥5 and almost all n -dimensional lattices L.

As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions. (C) 2011 American Institute of Physics. [doi:10.1063/1.3653197]

In this paper, the generalizations of Gronwall’s type integral inequalities with singular kernels are established. In applications, theorems on stability estimates for the solutions of the nonliner integral equation and the integral-differential equation of the parabolic type are presented. Moreover, these inequalities can be used in the theory of fractional differential equations.

We study a nonlocal boundary value problem and a space-wise dependent source identification problem for one-dimensional hyperbolic-parabolic equation with involution and Neumann boundary condition. The stability estimates for the solutions of these two problems are established. The first order of accuracy stable difference schemes are constructed for the approximate solutions of the problems under consideration. Numerical results for two test problems are provided.

In the present paper, a space-dependent source identification problem for the hyperbolic-parabolic equation with unknown parameter p $$ \left\{ \begin{array}{l} \displaystyle u''(t) + Au(t) = p + f(t), ~ 0<t<1, \\ \displaystyle u'(t) + Au(t) = p + g(t), ~ -1<t<0, \\ \displaystyle u(0^{+})=u(0^{-}), ~ u'(0^{+})=u'(0^{-}), \\ \displaystyle u(-1)=\varphi, ~ \int \limits _{0}^{1} u(z)dz=\psi \end{array} \right. $${u′′(t)+Au(t)=p+f(t),0<t<1,u′(t)+Au(t)=p+g(t),-1<t<0,u(0+)=u(0-),u′(0+)=u′(0-),u(-1)=φ,∫01u(z)dz=ψ in a Hilbert space H with self-adjoint positive definite operator A is investigated. The stability estimates for the solution of this identification problem are established. In applications, the stability estimates for the solutions of four space-dependent source identification hyperbolic-parabolic problems are obtained.

In the present paper, we establish the well-posedness of an identification problem for determining the unknown space-dependent source term in the hyperbolic-parabolic equation with nonlocal conditions. The difference scheme is constructed for the approximate solution of this source identification problem. The stability estimates for the solution of the difference scheme are presented.

In the present paper, we study the first and second order of accuracy difference schemes for the approximate solution of the inverse problem for hyperbolic–parabolic equations with unknown time-independent source term. The unique solvability of constructed difference schemes and the stability estimates for their solutions are obtained. The proofs are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

In the present paper, a boundary value problem for a one-dimensional hyperbolic-parabolic equation with involution and the Dirichlet condition is studied. The stability estimates for the solution of the hyperbolic-parabolic problem are established. The first order of accuracy stable difference scheme for the approximate solution of the problem under consideration is constructed. Numerical algorithm for implementation of this scheme is presented. Numerical results are provided for a simple test problem.

In the present paper, we study a source identification problem for hyperbolic-parabolic equation with nonlocal conditions. The stability estimates for the solution of this source identification problem are established. Furthermore, we construct the second order of accuracy difference scheme for the approximate solution of the problem under consideration. The stability estimates for the solution of this difference scheme are presented.

In the present paper, a source identification problem for hyperbolic-parabolic equation with involution and Dirichlet condition is studied. The stability estimates for the solution of the source identification hyperbolic-parabolic problem are established. The first order of accuracy stable difference scheme is constructed for the approximate solution of the problem under consideration. Numerical results are given for a simple test problem.

An identification problem for an equation of mixed telegraph-parabolic type with an unknown parameter depending on spatial variables is considered. The unique solvability of this problem is proved, and stability inequalities for its solution are established. As applications, stability estimates are obtained for the solutions of four identification problems for telegraph-parabolic equations with an unknown source depending on spatial variables.

In the present study, a numerical study for source identification problems with the Neumann boundary condition for a one-dimensional hyperbolic-parabolic equation is presented. A first order of accuracy difference scheme for the numerical solution of the identification problems for hyperbolic-parabolic equations with the Neumann boundary condition is presented. This difference scheme is implemented for a simple test problem and the numerical results are presented.

The Escalator Boxcar Train (EBT) is a commonly used method for solving physiologically structured population models. The main goal of this paper is to overcome computational disadvantages of the EBT method. We prove convergence, for a general class of EBT models in which we modify the original EBT formulation, allowing merging of cohorts. We show that this modified EBT method induces a bounded number of cohorts, independent of the number of time steps. This in turn, improve the numerical algorithm from polynomial to linear time. An EBT simulation of the Daphnia model is used as an illustration of these findings.

Let M be an n-dimensional complex manifold. A holomorphic function f:M→C is said to be semi-Bloch if for every λ∈C the function (Formula presented.) is normal on M. We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate M in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.

   In the finance world, option pricing techniques have become an appealing topic among researchers, especially for pricing American options. Valuing this option involves more factors than pricing the European style one, which makes it more computationally challenging. This is mainly because the holder of American options has the right to exercise at any time up to maturity. There are several approaches that have been proved to be efficient and applicable for maximizing the price of this type of options. A common approach is the Least squares method proposed by Longstaff and Schwartz. The purpose of this thesis is to discuss and analyze the implementation of this approach under the Multiscale Stochastic Volatility model. Since most financial markets show randomly variety of volatility, pricing the option under this model is considered necessary. A numerical study is performed to present that the Least-squares approach is indeed effective and accurate for pricing American options.

We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson-type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors.

The stock market is a place in which numerous entities interact, operate, andchange state based on the decisions they make. Further, the stock market itselfevolves and changes its dynamics over time as a consequence of the individualactions taking place in it. In this sense, the stock market can be viewed andtreated as a complex adaptive system. In this study, an agent-based model,simulating the trading of a single asset has been constructed with the purposeof investigating how the collective behaviour affects the dynamics of the stockmarket. For this purpose, the agent-based modelling program NetLogo wasused. Lastly, the conclusion of the study revealed that the dynamics of thestock market are clearly dependent on some specific factors of the collectivebehaviour, such as the information source of the investors.

This thesis aims to provide a brief exposition of some chosen modes of convergence; namely uniform convergence, pointwise convergence and L¹ convergence. Theoretical discussion is complemented by simple applications to scientific computing. The latter include solving differential equations with various methods and estimating the convergence, as well as modelling problematic situations to investigate odd behaviors of usually convergent methods.

For dynamical systems defined by a covering map of a compact Hausdorff space and the corresponding transfer operator, the associated crossed product C ^*-algebras C(X)⋊ _α,ℒℕintroduced by Exel and Vershik are considered. An important property for homeomorphism dynamical systems is topological freeness. It can be extended in a natural way to in general non-invertible dynamical systems generated by covering maps. In this article, it is shown that the following four properties are equivalent: the dynamical system generated by a covering map is topologically free; the canonical embedding of C(X) into C(X)⋊ _α,ℒℕis a maximal abelian C ^*-subalgebra of C(X)⋊ _α,ℒN; any nontrivial two sided ideal of C(X)⋊ _α,ℒℕhas non-zero intersection with the embedded copy of C(X); a certain natural representation of C(X)⋊ _α,ℒℕis faithful. This result is a generalization to non-invertible dynamics of the corresponding results for crossed product C ^*-algebras of homeomorphism dynamical systems. 

One-sided shift spaces are a special kind of non-invertible topological dynamical system with which one can associate a C^*-algebra. We show how to construct the C^*-algebra associated with a one-sided shift space as the Cuntz-Pimsner C^*-algebra of a C^*-correspondence and use this to compute its K-theory.

In this article we study the Gleason problem locally. A new method for solving the Gleason A problem is presented. This is done by showing an equivalent statement to the Gleason A problem. In order to prove this statement, necessary and a sufficient conditions for a bounded domain to have the Gleason A property are found. Also an example of a bounded but not smoothly-bounded domain in Cⁿ is given, which satisfies the sufficient condition at the origin, and hence has the Gleason A property there.

The fractional derivative of the Dirichlet eta function is computed in order to investigate the behavior of the fractional derivative of the Riemann zeta function on the critical strip. Its convergence is studied. In particular, its half-plane of convergence gives the possibility to better understand the fractional derivative of the Riemann zeta function and its critical strip. As an application, two signal processing networks, corresponding to the fractional derivative of the eta function and to its Fourier transform, respectively, are shortly described.

It always exists different methods/models to build a yield curve from a set of observed market rates even when the curve completely reproduces the price of the given instruments. To create an accurate and smooth interest rate curve has been a challenging all the time. The purpose of this thesis is to use the real market data to construct the yield curves by the bootstrapping method and the Smith Wilson model in order to observe and compare the performance ability between the models. Furthermore, the extended Nelson Siegel model is introduced without implementation. Instead of implementation I compare the ENS model and the traditional bootstrapping method from a more theoretical perspective in order to perceive the performance capabilities of them.

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The first order and the second order of accuracy difference schemes approximately solving this problem are presented. The convergence estimates for the solutions of these difference schemes are obtained. Theoretical results are supported by numerical example.

This thesis is devoted to the construction and investigation of some properties of pairs of linear operators (A, B) (representations) satisfying commutation relations AB=BF(A), where F denotes certain polynomial. Commutation relations of this kind are called "covariance type" and finding their representations is equivalent to solving operator equations. This kind of commutation relations and operator representations on finite-dimensional or infinite-dimensional linear spaces are important in Physics, Engineering and many areas of Mathematics. The most original part of the present thesis contains the construction of representations of commutation relations AB=BF(A) by linear integral operators, multiplication operators and weighted composition operators defined on Banach spaces of continuous functions, on L_p spaces and l_p spaces. Conditions on kernels of integral operators and functions defining multiplication and composition operators are derived for these operators to satisfy the covariance type commutation relations, both for the general polynomials F and for important specific choices of F. These conditions are used for construction of various concrete pairs of operators satisfying such commutation relations. Many essential algebraic properties of commutation relations are studied here as well.

In this work, we present methods for constructing representations of polynomial covariance type commutation relations AB=BF(A) by linear integral operators in Banach spaces L_p. We derive necessary and sufficient conditions on the kernel functions for the integral operators to satisfy the covariance type commutation relation for general polynomials F, as well as for important cases, when F is arbitrary affine or quadratic polynomial, or arbitrary monomial of any degree. Using the obtained general conditions on the kernels, we construct concrete examples of representations of the covariance type commutation relations by integral operators on L_p. Also, we derive useful general reordering formulas for the integral operators representing the covariance type commutation relations, in terms of the kernel functions.

Representations of polynomial covariance type commutation relations by linear integral operators in Banach spaces L_p are constructed. Reordering formulas for this type of operators and commutation relations are obtained in terms of iterations of integrals and maps.

Representations of polynomial covariant type commutation relations by pairs of linear integral operators and multiplication operators on Banach spaces L_p are constructed.

Representations of polynomial covariance type commutation relations by linear integral operators on L_p over measures spaces are constructed. Conditions for such representations are described in terms of kernels of the corresponding integral operators. Representation by integral operators are studied both for general polynomial covariance commutation relations and for important classes of polynomial covariance commutation relations associated to arbitrary monomials and to affine functions. Examples of integral operators on L_p spaces representing the covariance commutation relations are constructed. Representations of commutation relations by integral operators with special classes of kernels such as separable kernels and convolution kernels are investigated. 

Representations of polynomial covariance type commutation relations by linear integral operators on L_p over measures spaces are investigated. Necessary and sufficient conditions for integral operators to satisfy polynomial covariance type commutation relations are obtained in terms of their kernels. For important classes of polynomial covariance commutation relations associated to arbitrary monomials and to affine functions, these conditions on the kernels are specified in terms of the coefficients of the monomials and affine functions. By applying these conditions, examples of integral operators on L_p spaces, with separable kernels representing covariance commutation relations associated to monomials, are constructed for the kernels involving multi-parameter trigonometric functions, polynomials and Laurent polynomials on bounded intervals. Commutators of these operators are computed and exact conditions for commutativity of these operators in terms of the parameters are obtained.

Representations of polynomial covariance type commutation relations are constructed on Banach spaces L_p and C[α, β], α, β∈ R. Representations involve operators with piecewise functions, multiplication operators and inner superposition operators.

In this work conditions for additivity property of representations of polynomial covariance commutation relations are derived for operator algebras. Some other properties that this kind of representations fulfill are described. A reduction degree of the polynomial property of representations of this kind of commutation relations is presented for operator algebras. Moreover, representations of polynomial covariance commutation relations are derived for linear operators acting on the space of bounded real infinite sequences l_p.

We study a decomposition problem for a class of unitary representations associated with wavelet analysis, wavelet representations, but our framework is wider and has applications to multi-scale expansions arising in dynamical systems theory for non-invertible endomorphisms. Our main results offer a direct integral decomposition for the general wavelet representation, and we solve a question posed by Judith Packer. This entails a direct integral decomposition of the general wavelet representation. We further give a detailed analysis of the measures contributing to the decomposition into irreducible representations. We prove results for associated Martin boundaries, relevant for the understanding of wavelet filters and induced random walks, as well as classes of harmonic functions. Published by Elsevier Inc.

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in [13], and we present several equivalent formulations of the problem.

This paper continues the study of orthonormal bases (ONB) of L²[0, 1] introduced in Dutkay et al. (J. Math. Anal. Appl. 409(2):1128-1139, 2014) by means of Cuntz algebra O_N representations on L²[0, 1]. For N = 2, one obtains the classic Walsh system. We show that the ONB property holds precisely because the O_N representations are irreducible. We prove an uncertainty principle related to these bases. As an application to discrete signal processing we find a fast generalized transform and compare this generalized transform with the classic one with respect to compression and sparse signal recovery.

We answer a question by Judith Packer about the irreducibility of the wavelet representation associated to the Cantor set. We prove that if the QMF filter does not have constant absolute value, then the wavelet representation is reducible.

We study the reducibility of the wavelet representation associated to various QMF filters, including those associated to Cantor sets. We show there are connections between this problem, the harmonic analysis of transfer operators and the ergodic properties of shifts on solenoids. We prove that if the QMF filter does not have constant absolute value, then the wavelet representations is reducible.

We derive an su(1, 1) coherent state path integral formula for a system of two one-dimensional anyons in a harmonic potential. By a change of variables we transform this integral into a coherent states path integral for a harmonic oscillator with a shifted energy. The shift is the same as the one obtained for anyons by other methods. We justify the procedure by showing that the change of variables corresponds to an su(1, 1) version of the Holstein-Primakoff transformation.

The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape and the physical performance.

In this paper a functional equation for the fractional derivative of the Riemann zeta function is presented. The fractional derivative of the zeta function is computed by a generalization of the Grunwald-Letnikov fractional operator, which satisfies the generalized Leibniz rule. It is applied to the asymmetric functional equation of the Rieman zeta function in order to obtain the result sought. Moreover, further properties of this fractional derivative are proposed and discussed.

The Caputo-Ortigueira fractional derivative provides the fractional derivativeof complex functions. This derivative plays an important role in the number theory, and has been shown suitable for the analysis of the Dirichlet series, Hurwitz zeta function and Riemann zeta function. An integral representation for the fractional derivative of the Riemann zeta function was discovered. Since the Riemann zeta function is widely used in Physics, the unilateral Fourier transform of its fractional derivative is computed to investigate its applications in Quantum Theory and Signal Processing.

In the following chapter we describe a wavelet expansion theory for positivedefinite distributions over the real line and define a fractional derivative operator for complex functions in the distribution sense. In order to obtain a characterisation of the complex fractional derivative through the distribution theory, the Ortigueira-Caputo fractional derivative operator is rewritten as a convolution product according to the fractional calculus of real distributions. In particular, the fractional derivative of the Gabor-Morlet wavelet is computed together with its plots and main properties.

In this paper, the Weierstrass-Mandelbrot function on the Cantor set is presented with emphasis on possible applications in science and engineering. An asymptotic estimation of its one-sided Fourier transform, in accordance with the simulation results, is analytically derived. Moreover, a time-frequency analysis of the Weierstrass-Mandelbrot function is provided by the numerical computation of its continuous wavelet transform.

Centrality measures in network analysis have become a popular measurement tool for identifying coherent nodes within a network. In the context of stock markets, the centrality measure helps to identify key performing ele- ments and strengths for specific stocks and determine their impact on disrupting market value and performance. Multiple studies presented practical implementations of centrality measures for determining trends and perform- ance of a particular market. However, fewer studies applied centrality measures to predict trends in the stock market.