The multidimensional Fourier-Bessel transform is a generalization of Fourier-Bessel transform that obeys the same uncertainty principles as the classical Fourier transform. In this paper, we establish the following uncertainty principles; an -version of Morgan’s theorem, the Donoho-Stark uncertainty principles and bandlimited principles of concentration type for the multidimensional Fourier-Bessel transform.

We construct explicit strictly ascending chains of dense subalgebras of length 𝔠 in every separable infinite-dimensional complex Banach algebra. For large classes of commutative C*-algebras we also construct strictly descending chains of the same length. The constructions rely on algebraic independence, Stone–Weierstrass arguments, and transfinite recursion.

In this paper, we give the definitions of s-convex set and s-convex function on Heisenberg group. And some inequalities of Jensen’s type for this class of mappings are pointed out.

We develop and analyze an adaptive spacetime finite element method for nonlinear parabolic equations of p–Laplace type. The model problem is governed by a strongly nonlinear diffusion operator that may be degenerate or singular depending on the exponent p, which typically leads to limited regularity of weak solutions. To address these challenges, we formulate the problem in a unified spacetime variational framework and discretize it using conforming finite element spaces defined on adaptive spacetime meshes. We prove the well-posedness of both the continuous problem and the spacetime discrete formulation, and establish a discrete energy stability estimate that is uniform with respect to the mesh size. Based on residuals in the spacetime domain, we construct a posteriori error estimators and prove their reliability and local efficiency. These results form the foundation for an adaptive spacetime refinement strategy, for which we prove global convergence and quasi-optimal convergence rates without assuming additional regularity of the exact solution. Numerical experiments confirm the theoretical findings and demonstrate that the adaptive spacetime finite element method significantly outperforms uniform refinement and classical time-stepping finite element approaches, particularly for problems exhibiting localized spatial and temporal singularities.

<p>This article concerns the problem on the growth and the oscillation of some differential polynomials generated by solutions of the second order non-homogeneous linear differential equation <span class="math display">\[\begin{equation*} f^{\prime \prime }+P\left( z\right) e^{a_{n}z^{n}}f^{\prime }+B\left( z\right) e^{b_{n}z^{n}}f=F\left( z\right) e^{a_{n}z^{n}}, \end{equation*}\]</span> where <span class="math inline">\(a_{n}\)</span>, <span class="math inline">\(b_{n}\)</span> are complex numbers, <span class="math inline">\(P\left( z\right)\)</span> <span class="math inline">\(\left( \not\equiv 0\right)\)</span> is a polynomial, <span class="math inline">\(B\left( z\right)\)</span> <span class="math inline">\(\left( \not\equiv 0\right)\)</span> and <span class="math inline">\(F\left( z\right)\)</span> <span class="math inline">\(\left( \not\equiv 0\right)\)</span> are entire functions with order less than <span class="math inline">\(n\)</span>. Because of the control of differential equation, we can obtain some estimates of their hyper-order and fixed points.</p>

This paper studies a natural one-parameter extension of the Hardy-Hilbert integral inequality. The proposed generalization introduces a parameter that interpolates between different forms. This allows us to establish a hierarchy among a family of related double integrals. We provide sharp upper bounds expressed in terms of the integral norms of the functions involved. In doing so, we extend a classical result while maintaining the optimality of the constant in the original inequality.

The purpose of this paper is to give sufficient conditions for the existence and uniqueness of positive solutions to a rather general type of elliptic system of the Dirichlet problem on a bounded domain \(\Omega\) in \(R^{n}\). Also considered are the effects of perturbations on the coexistence state and uniqueness. The techniques used in this paper are super-sub solutions method, eigenvalues of operators, maximum principles, spectrum estimates, inverse function theory, and general elliptic theory. The arguments also rely on some detailed properties for the solution of logistic equations. These results yield an algebraically computable criterion for the positive coexistence of species of animals with predator-prey relation in many biological models.

<p>This article concerns the existence and multiplicity of homoclinic solutions for the following fourth-order differential equation with <span class="math inline">\(p-\)</span>Laplacian <span class="math display">\[\Big(\left|u''(t)\right|^{p-2}u''(t)\Big)''-\omega\Big(\left|u'(t)\right|^{p-2}u'(t)\Big)'+V(t)\left|u(t)\right|^{p-2}u(t)=f(t,u(t)),\]</span> where <span class="math inline">\(p>1\)</span>, <span class="math inline">\(\omega\)</span> is a constant, <span class="math inline">\(V\in C(\mathbb{R},\mathbb{R})\)</span> is noncoercive and <span class="math inline">\(f\in C(\mathbb{R}^{2},\mathbb{R})\)</span> is of subquadratic growth at infinity. Some results are proved using variational methods, the minimization theorem and the generalized Clark’s theorem. Recent results in the literature are extended and improved.</p>

We consider the unsteady problem for the general planar Broadwell model with fourh velocities in a rectangular spatial domain over a finite time interval. We impose a class of non-negative initial and Dirichlet boundary data that are bounded and continuous, along with their first-order partial derivatives. We then prove the existence and uniqueness of a non-negative continuous solution, bounded together with its first-order partial derivatives, to the initial-boundary value problem.