In this paper, we study the inverse spectral problem for the diffusion operator on a finite interval with the Robin-Dirichlet boundary conditions, and prove that a particular set of eigenvalues can uniquely determine the diffusion operator, and give the reconstruction algorithm.

In this paper, a new generalized Bernstein-Bézier type operators is constructed. The estimates of the moments of these operators are investigated. The rate of convergence in terms of modulus of continuity is given. Then, the equivalent theorem of these operators is studied.

We characterize the similarity of $n$-shift plus certain weighted Volterra operator denoted by $T_{1}$ on the Dirichlet space ${\cal D}$, and prove that $T_{1}$ acting on ${\cal D}$ is similar to the multiplication operator $M_{p}$ acting on $S({\Bbb D})$ by using some techniques in operator theory. Furthermore, we consider the case of $p(z)=z^{n}$ corresponding to the operator $T_{2}$, and prove that the operator $T_{2}$ has exactly $2^{n}$ reducing subspaces.

In this paper, we study the exact solitary wave, periodic wave solutions and their evolutionary relationship with Hamilton energy for generalized symmetric regularized long wave equation. By the planar dynamical system method, we make detailed qualitative analysis on this equation. Then we use the first integral method to obtain the exact solutions for the equation. Furthermore, by establishing the corresponding relationships between the solitary wave, periodic wave solutions and Hamilton energy h, we reveal that the value of Hamilton energy h plays an important role in the appearance of solitary wave and periodic wave solutions. The evolution processes from the periodic wave solution to the solitary wave solution and the kink wave solution are also given in this paper.

In this paper, we consider the Cauchy problem to the tri-dimensional compressible Navier-Stokes-Korteweg system with a specific choice on the Korteweg tensor in $\mathbb{R}^3$, and construct the global solutions to the tri-dimensional Navier-Stokes-Korteweg equations with a class of of large initial data, where the $L^{\infty}$ norm of the initial velocity and the third component of initial vorticity could be arbitrarily large.

To consider the initial problem for one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity. By using the energy estimates, the lower and upper bounds for the density is derived, that is, nether vacuum states nor concentration states can occur. Finally, the approximate solution is constructed by transforming the viscous coefficient, the existence of global strong solution is obtained by using the local existence conclusion of the strong solution and combining a prior estimates of the density function and the velocity function.

Our purpose of this paper is to study weak solutions of nonlinear Helmholtz equation (0.1) $-\Delta u-u=Q|u{|^{p-2}}u+\sum\limits_{i=1}^N{{k_i}}{\delta_{{A_i}}}$ where $p>1$, $k_i\in$$\mathbb{R}$\{0} with i=1, …, N, Q: $\mathbb{R}^2$→[0, +∞) is a Hölder continuous function and ${\delta_{{A_i}}}$ is the Dirac mass concentrated at ${{A_i}}$.We obtain two solutions of (0.1) if $k=\sum\limits_{i=1}^N{|{k_i}}| < {k^*}$ for some $k^*$>0 when $Q$ decays as $|x|^{α}$ at infinity with $α ≤ $ 0 and $p$>max{2, 3(2+$α$)}. These two sequences of solutions of (0.1) are sign-changing real-valued solutions with isotropic singularity at ${{A_i}}$ by applying Mountain Pass Theorem to an related integral equation. By using the iteration technique, we obtain the decays of solution of (0.1) controlled by $|x|^{-\frac12}$ at infinity when $p>\max\{2, 4(2+\alpha)\}$.

In this paper, we consider the following Kirchhoff type problem (3.36) $ \begin{equation}\left\{ \begin{array}{l}-\Big(a+b\int_{\mathbb{R}^3}|\nabla u|^2{\rm d}x\Big)\triangle u+V(x)u=|u|^{p-2}u+\varepsilon|u|^4u, \, \, \, x\in\mathbb{R}^3, \\ u\in H^{1}(\mathbb{R}^3), \end{array} \right. \end{equation}$ where $a>0$, $b>0$, $4< p < 6$ and $V (x)\in L_{\rm loc}^{\frac{3}{2}}(\mathbb{R}^3)$ is a given nonnegative function such that $\lim\limits_{|x|\rightarrow\infty}V(x): =V_{\infty}$. Under suitable conditions on $V(x)$, we prove that the existence of ground state solutions for small $\varepsilon$.

In this paper, we study the existence of positive periodic solutions for a singular Liénard equation $x''(t)+f(x(t))x'(t)-\varphi(t)x^\delta(t)+\frac{\alpha(t)}{x^{\mu}(t)}=0, $ where $f: (0, +\infty)\rightarrow \mathbb{R} $ is continuous which may have a singularity at $x=0$, $\alpha$ and $\varphi$ are $T$ -periodic functions with $\alpha, \varphi\in L([0, T], \mathbb{R})$, $\mu\in(0, +\infty)$ and $\delta\in(0, 1]$ are constants. The signs of weight functions $\alpha(t)$ and $\varphi(t)$ are allowed to change on $[0, T]$. We prove that the given equation has at least one positive $T$ -periodic solution. The method of proof relies on a continuation theorem of coincidence degree principle.

We consider the following fractional Choquard equation with critical or supercritical growth $(-\triangle)^s u+u=f(u)+\lambda(|x|^{-\mu} \ast|u|^q\big)|u|^{q-2}u, x\in \Omega, $ where $s \in (0, 1)$, $\mu\in (0, N)$, $N>2s$, $q\geq 2_{\mu, s}^\ast$, $f$ is a continuous function. It is well-known that $2_{\mu, s}^\ast=\frac{2N-\mu}{N-2s}$ and $2_{\mu, s}=\frac{2N-\mu}{N}$ are critical exponents for the above equation in the sense of Hardy-Littlewood-Sobolev inequality. Many existence results have been established for $q \in[2_{\mu, s}, 2_{\mu, s}^\ast]$ in recent years. Here we are interested in critical or supercritical case for the above equation. Under some assumptions of $f$, the existence and multiplicity of solutions for the above equation can be obtained by applying some analytical techniques.

In this paper, we study the following elliptic system $\begin{eqnarray*} \left\{\begin{array}{ll} -\Delta u+\mu_1 u=\frac{p}{p+q}|x|^\alpha u^{p-1}v^q, \ &x\in\Omega, \[3mm] -\Delta v+\mu_2 v=\frac{q}{p+q}|x|^\alpha u^pv^{q-1}, \ &x\in\Omega, \[2mm] u, v>0, ~x\in\Omega, ~~u=v=0, ~~&x\in\partial\Omega, \end{array}\right. \end{eqnarray*} $ where $\Omega\subset\mathbb{R}^N$ is an annulus $N\geq 4$, $\mu_1, \mu_2>0$, $p, q>1$ and $p+q<\frac{2N-2}{N-3}$. By variational method and rescaling method, we prove that the system has many non-radial solutions.

In this paper, we study the n-dimensional plane wave solution of the Landau-Lifshtz equation on $\mathbb{S}^2$. Based on the Hasimoto transformation, the equivalent plane wave type Schrödinger equation is obtained. By the Strichartz estimation and energy method under Fourier transform, the global existence of this solution is proved under a small initial value. The global solution obtained here is smooth and the norm falls in any order Hilbert space. The results in this paper improve the regularity of the solution in the paper^[3].

This paper concerns an inverse problem of determining two spatially varying transport coefficients simultaneously in a linearly coupled Korteweg-de Vries (KdV) system with the first order terms. To obtain the stability result for the inverse problem with only one internal measurement data, we first prove a Carleman estimate including only one local integral for this coupled KdV system. Based on this Carleman estimate, we then obtain Lipschitz stability for the inverse problem under some priori information.

By a new energy approach involved in the high frequency and low frequency decomposition in the Besov spaces, we obtain the optimal decay for the incompressible Oldroyd-B model without damping mechanism in ${\mathbb R}^n$ ($n\ge 2$). More precisely, let $(u, \tau)$ be the global small solutions constructed in^[18], we prove for any $(u_0, \tau_0)\in{\dot{B}_{2, 1}^{-s}}({\mathbb R}^n)$ that $ \begin{eqnarray*} \big\|\Lambda^{\alpha}(u, \Lambda^{-1}{\mathbb P}{\rm div}\tau)\big\|_{L^q} \le C (1+t)^{-\frac n4-\frac {(\alpha+s)q-n}{2q}}, \quad\Lambda\stackrel{{\rm def}}{=}\sqrt{-\Delta}, \end{eqnarray*}$ with -\frac n2 < s < \frac np, $ \leq p \leq \min(4, {2n}/({n-2})), \ p\not=4\ \hbox{ if }\ n=2, $ and $p\leq q\leq\infty$, $\frac nq-\frac np-s<\alpha \leq\frac nq-1$. The proof relies heavily on the special dissipative structure of the equations and some commutator estimates and various interpolations between Besov type spaces. The method also works for other parabolic-hyperbolic systems in which the Fourier splitting technique is invalid.

The probability that three independent random subspaces in $\mathbb{R}^n$ intersecting a convex body K have their common point intersecting K is found by using of the mean curvature integral of convex sets. Then we focus on the particular case of hyperplanes. On the base of this, we state the geometric probability of hyperplanes that intersect a ball, a cube or a right parallelepiped having an intersection inside the same ball, the cube or the right parallelepiped respectively. Finally, the monotonicity, convergence, and size relationship of the geometric probabilistic sequence are discussed.

In this paper, the differential geometry technique is used to study the complexity of the system. Jacobi stability of the three-dimensional Rabinovich system is analyzed from any point of trajectory of the system. Based on KCC-theory, the Jacobi stable conditions of all equilibrium points of the system are obtained. On the basis of obtaining the time evolution of the deviation vector and its components near the equilibrium points of the system, the instability exponent and curvature are introduced, and the chaos mechanism of the system is analyzed tentatively by combining numerical simulation. Numerical results validate the existing theoretical analysis results.

The bending problem of a uniformly loaded orthotropic rectangular thin plate with a corner point-supported and the other opposite edge clamped is studied. First, we divide the bending problem into two sub-problems with two opposite edges slidingly clamped and another sub-problem with one edge slidingly clamped and its opposite edge simply supported. Then we obtain the eigenvalues and eigenfunctions of the Hamiltonian operator corresponding to the three sub-problems, respectively. Then the solutions of the above three sub-problems are solved by the method of symplectic eigenfunction expansion respectively. Finally, we obtain the symplectic superposition solution of the original bending problem by superposition of the solutions of the three sub-problems. In addition, we calculate the deflection and bending moment values at some points of the bending problems of isotropic rectangular plate and orthotropic rectangular plate by using the symplectic superposition solution obtained in this paper.

In this paper, we study the oscillation of second order generalized Emden-Fowler delay differential equations with a sub-linear neutral term. Under the irregularity condition, by using Riccati transformation and the inequalities technique, several simple new oscillation criteria of this kind of equations to ensure that every solution oscillates are established. These oscillation criteria generalize and improve the classical research results including those adapted to Euler equations established in previous literatures. Finally, some examples to verify the wide application of these oscillation criteria are given in this paper.

This paper deals with some new Farkas lemmas for a class of constraint fractional optimization with DC functions(the difference of convex functions). Following the idea due to Dinkelbach, we first associate the fractional optimization with a DC optimization problem. Then, by using the epigraph technique of the conjugate function, we introduce some new regularity conditions and establish the duality between the DC optimization problem and its Fenchel-Lagrange dual problem. Finally, we obtain some new Farkas lemmas for the fractional optimization problem. Furthermore, we also show that the results obtained in this paper extend and improve the corresponding results in the literature.

The conjugate gradient method is one of the most effective methods for solving large-scale unconstrained optimization. Combining the second inequality of the strong Wolfe line search, two new conjugate parameters are constructed. Under usual assumptions, it is proved that the improved PRP and HS conjugate gradient methods satisfy sufficient descent condition with the greater range of parameter in the strong Wolfe line search and converge globally for unconstrained optimization. Finally, two group numerical experiments for the proposed methods and their comparisons are tested, the numerical results and their corresponding performance files are reported, which show that the proposed methods are promising.

In this paper, we consider a type of time-changed Markov process, where the time-change is an inverse killed subordinator. This can be seen as an extension of Chen (Chen Zhenqing. Time fractional equations and probabilistic representation. Chaos Solitons and Fractals, 2017, 102: 168-174). As a result, it constructs a relationship between general Bernstein functions and a class of generalized time-fractional partial differential equations.

In this paper, the uniform extraordinary recurrent set of continuous time Markov processes is defined, and the determination method of uniform extraordinary recurrent set of continuous time Markov processes is discussed. Some conclusions on the recurrence of continuous time Markov processes with Ψ irreducibility are obtained, and some recurrence results related to petite sets are also obtained.

In this paper, using large deviations for two-parameter Brownian motion and increments of two-parameter Brownian motion, we obtain local functional law of the iterated logarithm for increments of two-parameter Brownian motion.

As a new type of experimental designs, the extended designs have been paid more and more attention in recent years. The extended design includes two parts: initial design and follow-up design. In many follow-up designs, some extra factors with two or three levels may be added in the follow-up stage since they are quite important but may be neglected in the initial stage. In this paper, under the uniformity criterion, we present the analytical expressions and the corresponding lower bounds on mixture-discrepancy of the mixed-level column augmented designs. In the sense of mixed discrepancy, mixed-level column augmented nearly uniform designs are proposed.

Based on nonstationarity measure, we design an electronic seal for matching by the random encryption method. The electronic seal contains time information, equipment information, content information and specific information. The electronic seal combines four parts of information into a sequence and is masked by strong noise. It is indeed an information sequence (Information plus strong noise sequences) with length of 1024. Since the nonstationarity measure is not affected by the noise distribution, the electronic seal can match with the electronic seal information security under the noise. Simulations indicate that the electronic seal achieves good performance in security, avoids key management and high match accuracy. It may have potential application in the field of information security such as authentication.

Lipschitz equivalence of self-similar sets is a kernel problem on geometric measure theory and fractal geometric. Rao-Ruan-Xi^[10] proved the problem of {1, 3, 5}-{1, 4, 5} by graph-directed sets. This paper gives another proof via the method of neighbor automaton.