Consider the Cauchy problem for the ordinary 3+1 wave equation. Reduction of the spatial domain to a half-space involves an exact radiation boundary condition enforced on a planar boundary. This boundary condition is most easily formulated in terms of the tangential-Fourier and time-Laplace transform of the solution. Using the Schwartz theory of distributions, we examine two other formulations: (i) the nonlocal spacetime form and (ii) its three-dimensional (tangential/time) Fourier transform. The spacetime form features a convolution between two tempered distributions.

In this paper, we consider an inhomogeneous fifth-order nonlinear Schrodinger equation from Heisenberg ferromagnetism, which describes the dynamics of a site-dependent Hisenberg ferromagnetic spin chain. Based on its Lax pair, we study the determinant representation of the $n$-fold Darboux transformation (DT). Furthermore, by using the $n$-fold DT, we obtain the higher-order solitary wave, breather wave and rogue wave solutions of the equation, respectively. Finally, the dynamic characteristics of these exact solutions are discussed.

We discuss the regularity of extremal functions in certain weighted Bergman and Fock type spaces. Given an appropriate analytic function $k$, the corresponding extremal function is the function with unit norm maximizing $Re \int _\Omega f(z) \overline {k(z)}\, \nu (z) \, dA(z)$ over all functions $f$ of unit norm, where $\nu $ is the weight function and $\Omega $ is the domain of the functions in the space. We consider the case where $\nu (z)$ is a decreasing radial function satisfying some additional assumptions, and where $\Omega $ is either a disc centered at the origin or the entire complex plane. We show that, if $k$ grows slowly in a certain sense, then $f$ must grow slowly in a related sense. We also discuss a relation between the integrability and growth of certain log-convex functions and apply the result to obtain information about the growth of integral means of extremal functions in Fock type spaces.

In this work, we prove that, if $\mathfrak{R} $ is an arbitrary alternative division ring, then, under a mild condition, every semi-automorphism $\varphi : \mathfrak{R} \rightarrow \mathfrak{R} $ is either an automorphism or an anti-automorphism. We extend Hua's result L.K. Hua for an alternative division ring.

In this article, we introduce the abstract notion of Galois wavelet groups over finite fields as the finite group of Galois dilations, and translations. We then present a unified theoretical linear algebra approach to the theory of Galois wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as a finite coherent sum of Galois wavelet coefficients as well.

We study a variant of continuous Hochschild cohomology of a Banach algebra in connection with a higher-dimensional analogue of the approximate amenability of the algebra. Some results on higher-dimensional amenability have natural analogues in our context. Alternating cocycles, due to Johnson \cite {johnson}, are studied, and a previous result of the author B. E. Johnson on Lipschitz algebras over compact metric spaces is improved.

In this paper, we study the existence of multiple positive solutions to problem \[\left \{\begin{aligned} &\bigg (a+b \int _\Omega (|\nabla u|^2+|u|^2)\,dx\bigg )(-\Delta u+u)=|u|^{4}u &&\mbox {in } \Omega, \\ &\frac {\partial u}{\partial \nu }=\lambda |u|^{q-2}u &&\mbox {on } \partial \Omega,\end{aligned} \right . \] where $\Omega \subset \mathbb {R}^{3}$ is a smooth bounded domain, $a, b \gt 0$, $\lambda \gt 0$ and $1\lt q\lt 2$. Based on the Nehari manifold and variational methods, we prove that the problem has at least two positive solutions, and one of the solutions is a positive ground state solution.

In this paper, an autonomous Choquard equation with doubly critical exponents is studied. By using the Pohozaev constraint and the perturbed method, a positive and radially symmetric ground state solution in $H^1(\mathbb {R}^N)$ is obtained. The result here extends and complements the earlier theorems obtained by Seok and Moroz and Schaftingen.

We find the limiting proportion of periodic points in towers of finite fields for polynomial maps associated to algebraic groups, namely, pure power maps $\phi (z) = z^d$ and Chebyshev polynomials.

In this paper, we prove, for $\lambda $ and $\mu $ large, the existence of a positive solution for the semi-positone elliptic system \[ {\mathrm (P)} \qquad \left \{\begin{aligned} & - \Delta _p u = \lambda \omega (x) f(v) &&\mbox {in } \Omega , \\ &- \Delta _q v = \mu \rho (x) g(u) &&\mbox {in } \Omega , \\ &(u,v) = (0,0) &&\mbox {on } \partial \Omega , \end{aligned} \right . \] where $\Omega = B_1 (0) = \{ x\in \mathbb {R}^N: |x|\leq 1 \} $, and, for $m>1$, $\Delta _m$ denotes the $m$-Laplacian operator $p,q>1$. The weight functions $\omega $, $\rho \colon \overline {\Omega } \rightarrow \mathbb {R}$ are radial, continuous, nonnegative and not identically null, and the non-linearities $f,g\colon [0,\infty ) \rightarrow \mathbb {R}$ are continuous functions such that $f(t)$, $g(t)\geq -\sigma $. The result presented extends, for the radial case, some results in the literature [D. D. Hai and R. Shivaji]. In particular, we do not impose any monotonic condition on $f$ or $g$. The result is obtained as an application of the Schauder fixed point theorem and the maximum principle.

In this paper, a new approach for studying the practical stability and boundedness with respect to a manifold of the solutions of a class of fractional differential equations is applied. The technique is based on the recently defined ``fractional-like derivative'' of Lyapunov-type functions. Sufficient conditions using vector Lyapunov functions are established. Examples are also presented to illustrate the theory.

In this paper, a system of equations describing the motion of a one-dimensional, cylindrically symmetrical, inviscid, non-ideal gas with a constant axial magnetic field is considered and the singular surface theory is used to study different modes of wave propagation and its culmination into a shock wave. The transport equation for the jump in the velocity gradient is derived and solved numerically to study the affects of the ratio of specific heat, non ideal parameter and initial magnetic fields on jump in the velocity gradient.

We introduce the notion of localization operators associated with the Riemann-Liouville-Wigner transform, and we give a trace formula for the localization operators associated with the Riemann-Liouville-Wigner transform as a bounded linear operator in the trace class from $L^{2}(d\nu _{\alpha })$ into $L^{2}(d\nu _{\alpha })$ in terms of the symbol and the two admissible wavelets. Next, we give results on the boundedness and compactness of localization operators associated with the Riemann-Liouville-Wigner transform on $L^{p}(d\nu _{\alpha })$, $1 \leq p \leq \infty $.

A commuting $d$-tuple $T=(T_{1}, \ldots , T_{d})$ of bounded linear operators on a Hilbert space $\mathcal {H}$ is called a spherical $m$-isometry if $\sum _{j=0}^{m}(-1)^{j}\binom {m}{j}Q_{T}^{j}(I)=0$, where $I$ denotes the identity operator and $Q_{T}(A)=\sum _{i=1}^{d}T_{i}^{*}AT_{i}$ for every bounded linear operator $A$ on $\mathcal {H}$. Also, $T$ is called a toral $m$-isometry if $\sum _{p\in \mathbb {N}^{d},\, 0\leq p\leq n}(-1)^{\vert p\vert }\binom {n}{p}{T^{\ast }}^{p}T^{p}=0 $ for all $n\in \mathbb {N}^{d}$ with $\vert n\vert = m$. The present paper mainly focuses on the convex-cyclicity of the $d$-tuples of operators on a separable infinite-dimensional Hilbert space $\mathcal {H}$. In particular, we prove that spherical $m$-isometries are not convex-cyclic. Also, we show that toral and spherical $m$-isometric operators are never supercyclic.

In this paper, for iterated function systems, we define the classic concept of the dynamical systems: topological conjugacy of diffeomorphisms. We generalize the Hartman-Grobman theorem for one-dimensional iterated function systems on $\mathbb {R}$ Also, we introduce the basic concept of structural stability for an iterated function system, and therefore, we investigate the necessary condition for structural stability of an iterated function system on $\mathbb {R}$

We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorphism by using the Hamilton vector fields.