Abstract
We consider the singular integral operator acting between two weighted Hölder spaces and give an estimate for its norm.
We consider the singular integral operator acting between two weighted Hölder spaces and give an estimate for its norm.
In this work, we delved into advanced modeling of economic relationships using the Cobb–Douglas production function as a theoretical foundation. Our primary goal was to develop an innovative multiple linear regression model by introducing innovations based on the (alpha)-stable distribution. By adapting the traditional multiple linear regression model, our approach incorporates the (alpha)-stable distribution to better represent the complexity of relationships between economic variables. This modification enables a better fit for asymmetric distributions and scenarios where data exhibit heavy tails. To assess the performance of our model, we applied it to real financial data. This practical step allowed us to evaluate the effectiveness and predictive capability of our approach in a real-world context, thus offering fresh perspectives for financial data analysis.
In this paper, we consider an impulsive homogeneous parabolic type partial integro-differential equation with degenerate kernel and involution. With respect to spatial variable (x) is used Dirichlet boundary value conditions and spectral problem is studied. The Fourier method of separation of variables is applied. The countable system of nonlinear functional equations is obtained with respect to the Fourier coefficients of unknown function. Theorem on a unique solvability of countable system of functional equations is proved. The method of successive approximations is used in combination with the method of contraction mapping. The unique solution of the impulsive mixed problem is obtained in the form of Fourier series. Absolutely and uniformly convergence of Fourier series is proved.
The article is devoted to construct a complete asymptotic expansion of the solution to the Cauchy problem for a linear analytical system of singularly perturbed ordinary differential equations of the first order. The peculiarities of the Cauchy problem are that a small parameter is present in front of the derivative, and the stability conditions are violated in the region under consideration. By modifying the method of boundary functions, a formal asymptotic expansion of the solution to the Cauchy problem is constructed. The remainder term of the expansion is estimated by the idea of L.S. Pontryagin entering the complex plane.
In this article, two dimensional inverse problem of determining convolution kernel in the fractional diffusion equation with the time-fractional Caputo derivative is studied. To represent the solution of the direct problem, the fundamental solution of the time-fractional diffusion equation with Riemann–Liouville derivative is constructed. Using the formulas of asymptotic expansions for the fundamental solution and its derivatives, an estimate for the solution of the direct problem is obtained in terms of the norm of the unknown kernel function, which was used for studying the inverse problem. The inverse problem is reduced to the equivalent integral equation of the Volterra type. The local existence and global uniqueness results are proven by the aid of fixed point argument in suitable functional classes. Also the stability estimate is obtained.
The article is devoted to study of controllability properties of a linear singularly perturbed discrete system with a small step. It is used the Gram operator, which transforms an infinite-dimensional space into a finite-dimensional one, and based on the separation of state variables of a linear discrete singularly perturbed system with a small step. The controllability criteria for this discrete system is derived.
We consider the remote state restoring and perfect transfer of the zero-order coherence matrix (PTZ) in a spin system governed by the XXZ-Hamiltonian conserving the excitation number. The restoring tool is represented by several nonzero Larmor frequencies in the Hamiltonian. To simplify the analysis we use two approximating models including either step-wise or pulse-type time-dependence of the Larmor frequencies. Restoring in spin chains with up to 20 nodes is studied. Studying PTZ, we consider the zigzag and rectangular configurations and optimize the transfer of the 0-order coherence matrix using geometrical parameters of the communication line as well as the special unitary transformation of the extended receiver. Overall observation is that XXZ-chains require longer time for state transfer than XX-chains, which is confirmed by the analytical study of the evolution under the nearest-neighbor approximation. We demonstrate the exponential increase of the state-transfer time with the spin chain length.
The problem of finding a solution, satisfying the non-local condition (u(xi_{0})=alpha u(+0)+varphi) in time for the Boussinesq type equation of the form (u_{tt}+Au_{tt}+Au=f) is studied in the article. Here (alpha) and (xi_{0}), (xi_{0}in(0,T],) are the given numbers, (A:Hrightarrow H) is the self-adjoint, unbounded, positive operator defined in the Hilbert separable space (H). By using the Fourier method, it was shown that the solution to the problem exists and is unique. The effect of parameter (alpha) on the existence and uniqueness of the solution is studied in the article. The inverse problem of determining the right-hand side of the equation is also considered.
In this article, we present some connections between the notation of D-stability, Strong D-stability, and structured singular values known as (mu)-values for square matrices.
In a sublinear space (left(Omega,mathcal{H},widehat{mathbb{E}}right)), we consider Mean Field stochastic differential equations ((G)-MFSDEs in short), called also (G)-McKean–Vlasov stochastic differential equations, which are SDEs where coefficients depend not only on the state of the unknown process but also on its law. We mean by law of a random variable (X) on (left(Omega,mathcal{H},widehat{mathbb{E}}right)), the set (left{P_{X}:Pinmathcal{P}right}), where (P_{X}) is the law of (X) with respect to (P) and (mathcal{P}) is the family of probabilities associated to the sublinear expectation (widehat{mathbb{E}}). In this paper, we study the existence and uniqueness of the solution of (G)-MFSDE by using the fixed point theorem. To this end, we introduce a new type Kantorovich metric between subsets of laws and adapted Lipchitz and linear growth conditions. Furthermore, we prove the validity of the averaging principle and obtain convergence theorem where the solution of the averaged (G)-MFSDE converges to that of the standard one in the mean square sense.
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