We investigate the extinction, non-extinction and decay estimates of the non-negative nontrivial weak solutions of the initial-boundary value problem for the quasilinear parabolic equation with nonlinear gradient source.

In this paper the nonlinear matrix equation $X-\sum\limits_{i=1}^{m}A_{i}^*X^{-p_{i}}A_{i}=Q $ with $p_{i}\gt 0$ is investigated. Necessary and sufficient conditions for the existence of Hermitian positive definite solutions are obtained. An effective iterative method to obtain the unique solution is established. A perturbation bound and the backward error of an approximate solution to this solution is evaluated. Moreover, an explicit expression of the condition number for the positive definite solution is given. The theoretical results are illustrated by numerical examples.

The purpose of this paper is to present some fixed point results for multivalued operators in b-metric spaces endowed with a graph, as well for multivalued fractals in $b$-metric spaces, ussing contractive conditions of Ćirić type with respect to the functional $H$.

This paper is concerned with the study of controllability for a class of multi-valued functional differential systems with impulses and delayed control. By making use of fixed point theorem for multi-valued maps, we prove that our system is completely controllable. The result is obtained in the sense of Carathéodory, and without the requirement that the linear part is controllable.

Some characterizations of the multivariate wave packet systems in terms of the Fourier transforms of the wave packet systems' generating functions are discussed in this paper. First, the characterizations of the orthogonal wave packet systems are provided. Secondly, a sufficient condition of the completeness of wave packet system in some special cases is established. Finally, the necessary conditions and sufficient conditions for the wave packet systems to be wave packet Parseval frames with the very general lattices are obtained. Thus, the corresponding known results in Gabor systems and wavelet systems are obtained as some corollaries.

We consider the Timoshenko system in boundeddomain with a delay term in the nonlinear internal feedback $$\begin{cases} \rho_{1} \varphi_{tt}(x,t) - K(\varphi_{x}+\psi)_{x}(x,t) = 0, \\ \rho_{2} \psi_{tt}(x,t) - b \psi_{xx}(x,t) + K(\varphi_{x}+\psi)(x,t) \\ \qquad\qquad + \mu_1 g_1(\psi_{t}(x,t)) + \mu_2 g_2(\psi_{t}(x,t-\tau)) = 0, \end{cases}$$ and prove the global existence of its solutions in Sobolev spaces by means of the energy method combined with the Faedo-Galerkin procedureunder a condition between the weight of the delay term in the feedback and the weight of the term without delay. Furthermore, we establish a decay rate estimate for the energy by introducing suitable Lyapunov functionals.

On a compact $n$-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1984 by Besse, but has yet to be proved. In this paper, we prove that if the manifold with the critical point metric has harmonic curvature, then it is isometric to a standard sphere.

This paper devotes to establish complex dual Brunn-Minkowski theory. At first, we introduce the concepts of complex radial combination and complex radial-Blaschke combination, and obtain the relations between those two combinations and dual mixed volumes. Then, we extend the properties of real intersection body to the complex case. Finally, we prove some complex geometric inequalities about complex intersection bodies and complex mixed intersection bodies, such as dual Brunn-Minkowski type, dual Aleksandrov-Fenchel type and dual Minkowski type inequality. Moreover, as applications, we get some corollaries including an isoperimetric type inequality and a uniqueness theorem.

In this paper we introduce a new Cantor manifold theorem and then apply it to one new type of one-dimensional ($1d$) beam equations $$ u_{tt}+u_{xxxx}+mu-2u^2u_{xx}-2uu_x^2=0, m\gt 0,$$ with periodic boundary conditions. We show that the above equation admits small-amplitude linearly stable quasi-periodic solutions corresponding to finite dimensional invaraint tori of an associated infinite dimensional dynamical system. The proof is based on a partial Birkhoff normal form and an infinite dimensional KAM theorem for Hamiltonians with symmetry (cf. [19]).

In this paper, we discuss the closedness of set of efficient solutions for generalized Ky Fan inequality problems in topological vector spaces. We introduce a concept of section mapping of a bifunction. By using the lower semicontinuity of the section mapping, we present sufficient conditions for the closedness of set of efficient solutions to the generalized Ky Fan inequality problems. We give conditions to guarantee the lower semicontinuity of the section mapping. We give also an example to illustrate that the condition of the lower semicontinuity of the section mapping is essential for the closedness of set of efficient solutions for generalized Ky Fan inequality problems. As an application, we give results of closedness of set of efficient solutions for vector optimization problems and for Lipschitz vector variational inequalities.

In this paper, the author introduces the inhomogeneous Hausdorff type Besov and Triebel-Lizorkin spaces, which are the predual spaces of Besov-Morrey and Triebel-Lizorkin-Morrey spaces. By calculating the Calderón product of the sequence spaces related to Hausdorff type Besov and Triebel-Lizorkin spaces, the author obtains the complex interpolation of these spaces. In particular, the complex interpolation for the predual spaces of Morrey spaces is also obtained.

We classify pseudo parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one. With this result, the non-existence of recurrent as well as semi parallel proper CR-submanifolds in a non-flat complex space form with CR-dimension greater than one can also be obtained.

Let $K_n$ be a complete graph with $n$ vertices, $C_k$ denote a cycle of length $k$, and $S_k$ denote a star with $k$ edges. If $k=3$, then we call $C_3$ a triangle and $S_3$ a claw. In this paper, we show that for any nonnegative integers $p$ and $q$ and any positive integer $n$, there exists a decomposition of $K_n$ into $p$ copies of $C_3$ and $q$ copies of $S_3$ if and only if $3(p + q) = {n\choose 2}$, $q \ne 1, 2$ if $n$ is odd, $q = 1$ if $n=4$, and $q\ge \max \{3, \lceil\frac{n}{4}\rceil \}$ if $n$ is even and $n \geq 6$.

In this paper, we consider the strong solutions of the compressible Boussinesq equations in $\mathbb{R}^{3}$ and prove the existence of unique local strong solutions for all initial data satisfying some compatibility conditions. The initial density need not be positive and may vanish in an open set. We use the Lax-Milgram theorem and contraction mapping argument to get the result. Moreover, we establish a blow-up criterion for possible breakdown of strong solutions at a finite time in terms of the gradient of velocity.

In this paper we establish the existence and uniqueness of a weak solution of the strongly coupled chemotaxis model with Dirichlet boundary conditions.

Let $B$ be a subset of positive integers, and $\mathcal{P}$ the set of all positive primes. For a subset $A$ of positive integers, $A(x)$ denotes the number of integers in $A$ not exceeding $x$. Let $\mathcal{S}$ denote the set of integers of the form $p+b$ with $p\in \mathcal{P}$ and $b\in B$. In this paper, we prove that if $B(x)\gg \log x/\log\log x$ and $B(cx)\gg B(x)$ for some positive constant $c\lt 1$, then $\mathcal{S}(x)\gg x/\log \log x$. This result is best possible in a sense: For any positive integer $m$, we construct an explicit subset $B$ of positive integers with $B(x)\gg (\log x)^m$ and $B(cx)\gg B(x)$ for any positive constant $c\lt 1$ such that $\mathcal{S}(x)\ll x/\log\log x$. We also give an application to the integers of the form $p+2^{a^2}+2^{b^2}$, where $p\in \mathcal{P}$ and $a,b$ are integers. Two open problems are posed for further research.

In this paper we investigate a class of learning algorithms for classification generated by regularization schemes with polynomial kernels and $l^1-$regularizer. The novelty of our analysis lies in the estimation of the hypothesis error. A Bernstein-Kantorovich polynomial is introduced as a regularizing function. Although the hypothesis spaces and the regularizers in the schemes are sample dependent, we prove the hypothesis error can be removed from the error decomposition with confidence. As a result, we derive some explicit learning rates for the produced classifiers under some assumptions.

Ennola gives an example of a relation among the cyclotomic units which is not a combination of elementary relations. He also proves that twice any relation among the cyclotomic units is a consequence of elementary relations. In the sense of the distribution, the torsion part of the universal even punctured distribution $ \big(A_n ^0 \big)^+ $ is a 2-torsion group. In particular, when $n$ has three distinct prime divisors, $ \big(A_n ^0 \big)^+ $ has a unique 2-torsion element. The aim of this paper is to find an algorithm to produce the unique 2-torsion element when $n$ has three distinct odd prime divisors.

Let $L$ be a non-negative self-adjoint operator satisfying a pointwise Guassian estimate for its heat kernel. Let $w$ be some $A_s$ weight on $\mathbb{R}^n$. In this paper, we obtain a weighted $(p,q)-$atomic decomposition with $q\geq s$ for the weighted Hardy spaces $H^p_{L,w}(\mathbb{R}^n)$, $0\lt p\leq 1$. We also introduce the suitable weighted BMO spaces $BMO^p_{L,w}$. Then the duality between $H^1_{L,w}(\mathbb{R}^n)$ and $BMO_{L,w}$ is established.