We present a class of higher-dimensional Hilbert-type inequalities on a fractal set $(\mathbb{R}_+^{\alpha n})^{k}$. The crucial step in establishing our results are higher-dimensional spherical coordinates on a fractal space. Further, we impose the corresponding conditions under which the constants appearing in the established Hilbert-type inequalities are the best possible. As an application, our results are compared with the previous results known from the literature.

The aim of this paper is to give an alternative technique for the derivation of a prior energy estimate. Consequently, this allows to define a natural energy norm of the long-term well-posedness result established by S. Israwi in [2] but for the original system, in which the partial operator ∇× is not involved.

The stability analysis of dynamic continuous structural system (DCSS) has often been investigated by discretizing it into several low-dimensional elements. The integrated results of all elements are employed to describe the whole dynamic behavior of DCSS. In this paper, DCSS is regarded as the complex dynamic network with the discretized elements as the dynamic nodes and the time-varying stiffness as the dynamic link relations between them, by which the DCSS can be regarded to be the large-scale system composed of the node subsystem (NS) and link subsystem (LS). Therefore, the dynamic model of DCSS is proposed as the combination of dynamic equations of NS and LS, in which their state variables are coupled mutually. By using the model, this paper investigates the stability of DCSS. The research results show that the state variables of NS and LS are uniformly ultimately bounded (UUB) associated with the synthesized coupling terms in LS. Finally, the simulation example is utilized to demonstrate the validity of method in this paper.

By the Lagrangian multiplier and constraint variational derivative, a relationship between conserved quantities and multi-Hamiltonian structures is built. Making using the relation a method is founded to prove the infinite-dimensional Liouville integrability of evolution equations with continuous variables. As the application, the conservation laws of the Kundu equation are firstly obtained. Its conserved quantities are deduced for comparing by Fokas' method different from the method used in the existed literature. The integrability of the equation is proved through taking the conservation laws as a starting point.

Very recently one has started to study Bergman and Szeg\“o kernels in the setting of octonionic monogenic functions. In particular, explicit formulas for the Bergman kernel for the octonionic unit ball and for the octonionic right half-space as well as a formula for the Szeg\”o kernel for the octonionic unit ball have been established. In this paper we extend this line of investigation by developing explicit formulas for the Szeg\“o kernel of strip domains of the form ${\cal{S}} := \{z \in \mathbb{O} \mid 0 < \Re(z) < d\}$ from which we derive by a limit argument considering $d \to \infty$ the Szeg\”o kernel of the octonionic right half-space. Additionally, we set up formulas for the Bergman kernel of such strip domains and relate both kernels with each other. In fact, these kernel functions can be expressed in terms of one-fold periodic octonionic monogenic generalizations of the cosecant function and the cotangent function, respectively.

In general, the basic reproduction number (R0) cannot be explicitly calculated for HIV(Human Immunodeficiency Virus) infection model with age-structured in a infinite dimensional spaces. To find R0, we need to transform the HIV model into a finite-dimensional space. In this paper, we are absorbed in numerical approximation of R0, which is the non-negative dominant eigenvalues of the positive irreducible matrices whose spectrum radius is defined as the next generation matrix. The linear operators generated by infected population are discretized into ordinary differential equations in a finite n-dimensional space. Thus, the abstract problem is transformed to find the positive dominant eigenvalues of the next generation matrix, we obtain a threshold R_0,n. Based on the spectral approximation theory, we show that R_0,n →R0 as n →+∞. Finally, by virtue of a numerical simulation, we demonstrate the results of the theorem.

In this article, the Cauchy problem of three-dimensional (3-D) incompressible magnetohydrodynamic system with infinite energy initial data is investigated. Via some elaborate analysis of the time evolution of both the vorticity ω=: ▽×u and the current density j=: ▽×b , the local-in-time well-posedness of mild solutions with arbitrarily large initial data in Morrey spaces is established.

In this paper, we establish existence and finite time blow up of weak solutions of a parabolic equation of p(x)-Laplacian type with the Dirichlet boundary condition. Moreover, we obtain upper and lower bounds for the blow up time of solutions, by employing concavity method and differential inequality technique respectively.

Pursuit-evasion games are the next logical stage in the exploring of powerful, intelligent, adaptive performance. In fact the optimal strategy is known for games in an infinitely sized playing field. The quality of the machine learning methods can thus be compared to the optimal performance possible. Therefore, we consider in this study a pursuit-evasion differential game in Hilbert space l2 with a hybrid system of dynamics. The game consists of a non-inertial pursuer and an inertial evader where controls of the pursuer and the evader are satisfied to the integral constraints. The duration of the game, φ, is fixed. The position of the evader at time φ satisfies to the phase constraint. We obtain attainability domains of the players and then we make a winning strategy for the pursuer which guarantees capturing the evader. We show that our constructed strategy is admissible as well.

The evolution of individual crystals of ellipsoidal shape in supercooled one-component and binary melts as well as in supersaturated solutions is studied theoretically. The crystal volume growth rate is derived using the prolate ellipsoidal coordinates. We show that this rate is a function of the current crystal volume and supercooling/supersaturation of the ambient liquid. Also, we demonstrate that the particle growth rate increases with increasing the volume of ellipsoidal crystals and supercooling.

Massless field equations are fundamental in particle physics. In Clifford analysis, the Euclidean version of these equations has been dealt with but it is not clear, even in dimension 4, what should be the right analogue of massless field equations for fields with values in a general irreducible Spin(4)-module. The main aim of the paper is to explain that a good possibility is to take the so-called generalized Cauchy-Riemann equations proposed a long time ago by E. Stein and G. Weiss. For this choice of the equations, we show that their polynomial solutions form different irreducible Spin(4)-modules. This is an important step in developing the corrresponding function theory.

In this paper, an avian influenza model with saturation and psychological effect on heterogeneous complex networks is proposed. Firstly, the basic reproduction number $\mathscr{R}_{0}$ is given through mathematical analysis, which is a threshold to determine whether or not the disease spreads. Secondly, the locally and globally asymptotical stability of the disease-free equilibrium point and the endemic equilibrium point are investigated by using Lyapunov functions and Kirchhoff’s matrix tree theorem. If $\mathscr{R}_{0}<1$, the disease-free equilibrium is globally asymptotically stable and the disease will die out. If $\mathscr{R}_{0}>1$, the endemic equilibrium is globally asymptotically stable. Thirdly, an optimal control problem is established by taking slaughter rate and cure rate as control variables. Finally, numerical simulations are given to demonstrate the main results.

The inverse problems of determining the energy-temperature relation α(t) and the heat conduction relation k(t) functions in the one-dimensional integro– differential heat equation are investigated. The direct problem is the initial-boundary problem for this equation. The integral terms have the time convolution form of unknown kernels and direct problem solution. As additional information for solving inverse problems, the solution of the direct problem for x = x₀ is given. At the beginning an auxiliary problem, which is equivalent to the original problem is introduced. Then the auxiliary problem is reduced to an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the continuous class of functions with weighted norms, we prove the main result of the article, which is a global existence and uniqueness theorem of inverse problem solutions.

This paper is devoted to the study of a nonlinear pseudoparabolic equation in an annular with Robin-Dirichlet conditions. At first, by applying the standard Faedo-Galerkin method, we prove existence and uniqueness results. Next, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative, then we also establish the lifespan for the equation via finding the upper bound and the lower bound for the blow-up times. Finally, we give a sufficient condition for the global existence and decay of weak solutions.

This note presents corrections to our paper

In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum $\Lambda=\left\{0, r/N\right\}+2\mathbb Z$, where $N\ge 1$ is an integer and $r$ is an odd integer with $1\le r\le 2N-1$ such that $r$ and $N$ are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families. Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

This note presents corrections to our paper [1]. We also give remarks and additions related to other papers [2]-[15].

In order to provide a significantly richer representation of non-stationary signals appearing in various disciplines of science and engineering, we introduce here a novel fractional nonuniform multiresolution analysis (FrNUMRA) on the spectrum $\Lambda$ given by $\Lambda = \left\{0,\frac{r}{N}\right\}+2\mathbb{Z}$, where $N \geqq 1$ is an integer and $r$ is an odd integer with $ 1 \leqq r \leqq 2N-1,$ such that $r$ and $N$ are relatively prime. The necessary and sufficient condition for the existence of nonuniform wavelets of fractional order is derived and an algorithm is also presented for the construction of fractional NUMRA starting from a fractional low-pass filter $h_{0}^{\alpha}$ with appropriate conditions. Moreover, we provide a complete characterization for the biorthogonality of the translates of the scaling functions of two fractional nonuniform multiresolution analyses and the associated fractional biorthogonal wavelet families.