In this paper, we propose a new method, called the level-collapsing method, to construct branching Latin hypercube designs (BLHDs). The obtained design has a sliced structure in the third part, that is, the part for the shared factors, which is desirable for the qualitative branching factors. The construction method is easy to implement, and (near) orthogonality can be achieved in the obtained BLHDs. A simulation example is provided to illustrate the effectiveness of the new designs.

In this paper we consider a class of polynomial planar system with two small parameters, $\varepsilon$ and $\lambda$, satisfying $0<\varepsilon\ll\ lambda\ll1$. The corresponding first order Melnikov function $M_1$ with respect to $\varepsilon$ depends on $\lambda$ so that it has an expansion of the form $M_1(h,\lambda)=\sum\limits_{k=0}^\infty M_{1k}(h)\lambda^k.$ Assume that $M_{1k'}(h)$ is the first non-zero coefficient in the expansion. Then by estimating the number of zeros of $M_{1k'}(h)$, we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for $0<\varepsilon\ll\lambda\ll1$, when $k'=0$ or $1$. In addition, for each $k\in \mathbb{N}$, an upper bound of the maximal number of zeros of $M_{1k}(h)$, taking into account their multiplicities, is presented.

We establish a slow manifold for a fast-slow dynamical system with anomalous diffusion, where both fast and slow components are influenced by white noise. Furthermore, we verify the exponential tracking property for the established random slow manifold, which leads to a lower dimensional reduced system. Alongside this we consider a parameter estimation method for a nonlocal fast-slow stochastic dynamical system, where only the slow component is observable. In terms of quantifying parameters in stochastic evolutionary systems, the provided method offers the advantage of dimension reduction.

This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease. This model can be viewed as a nonlocal version of the free boundary problem studied by Kim et al. (An SIR epidemic model with free boundary. Nonlinear Anal RWA, 2013, 14:1992-2001). We first prove that this problem has a unique solution defined for all time, and then we give sufficient conditions for the disease vanishing and spreading. Our result shows that the disease will not spread if the basic reproduction number $R_0<1$, or the initial infected area $h_0$, expanding ability $\mu$, and the initial datum $S_0$ are all small enough when $1 < R_0 < 1+\frac{d}{\mu_2+\alpha}$. Furthermore, we show that if $1 < R_0 < 1+\frac{d}{\mu_2+\alpha}$, the disease will spread when $h_0$ is large enough or $h_0$ is small but $\mu$ is large enough. It is expected that the disease will always spread when $R_0\geq1+\frac{d}{\mu_2+\alpha}$, which is different from the local model.

This paper is concerned with a stability problem on perturbations near a physically important steady state solution of the 3D MHD system. We obtain three major results. The first assesses the existence of global solutions with small initial data. Second, we derive the temporal decay estimate of the solution in the L²-norm, where to prove the result, we need to overcome the difficulty caused by the presence of linear terms from perturbation. Finally, the decay rate in L² space for higher order derivatives of the solution is established.

In this paper, we construct k-valued random analytic algebroid functions for the first time. By combining the properties of random series, we study the growth and Borel points of random analytic algebroid functions in the unit disc and obtain some interesting theorems.

We are concerned with the shock diffraction configuration for isothermal gas modeled by the conservation laws of nonlinear wave system. We reformulate the shock diffraction problem into a linear degenerate elliptic equation in a fixed bounded domain. The degeneracy is of Keldysh type-the derivative of a solution blows up at the boundary. We establish the global existence of solutions and prove the $C^{0,\frac{1}{2}}$-regularity of solutions near the degenerate boundary. We also compare the difference of solutions between the isothermal gas and the polytropic gas.

In this paper, we study the existence of nontrivial solutions to the elliptic system \begin{equation*} \begin{cases} -\Delta u=\lambda v + F_u(x,u,v),& \ x\in\Omega,\\ -\Delta v=\lambda u + F_v(x,u,v),& \ x\in\Omega,\\ u=v=0,& \ x\in\partial\Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is bounded with a smooth boundary. By the Morse theory and the Gromoll-Meyer pair, we obtain multiple nontrivial vector solutions to this system.

The goal of this paper is to study a mathematical model of a nonlinear static frictional contact problem in elasticity with the mixed boundary conditions described by a combination of the Signorini unilateral frictionless contact condition, and nonmonotone multivalued contact, and friction laws of subdifferential form. First, under suitable assumptions, we deliver the weak formulation of the contact model, which is an elliptic system with Lagrange multipliers, and which consists of a hemivariational inequality and a variational inequality. Then, we prove the solvability of the contact problem. Finally, employing the notion of H-convergence of nonlinear elasticity tensors, we provide a result on the convergence of solutions under perturbations which appear in the elasticity operator, body forces, and surface tractions.

We investigate the asymptotic behavior of solutions to the initial boundary value problem for the micropolar fluid model in a half line $\mathbb{R}_{+}:=(0,\infty).$ Inspired by the relationship between a micropolar fluid model and Navier-Stokes equations, we prove that the composite wave consisting of the transonic boundary layer solution, the 1-rarefaction wave, the viscous 2-contact wave and the 3-rarefaction wave for the inflow problem on the micropolar fluid model is time-asymptotically stable under some smallness conditions. Meanwhile, we obtain the global existence of solutions based on the basic energy method.

In this paper, we introduce the $\Delta _{h}$-Gould-Hopper Appell polynomials $\mathcal{A}_{n}(x,y;h)$ via $h$-Gould-Hopper polynomials $ G_{n}^{h}(x,y)$. These polynomials reduces to $\Delta _{h}$-Appell polynomials in the case $y=0$, $\Delta $-Appell polynomials in the case $y=0$ and $ h=1 $, $2D$-Appell polynomials in the case $h\rightarrow 0$, $2D$ $ \Delta$-Appell polynomials in the case $h=1$ and Appell polynomials in the case $ h\rightarrow 0$ and $y=0$. We obtain some well known main properties and an explicit form, determinant representation, recurrence relation, shift operators, difference equation, integro-difference equation and partial difference equation satisfied by them. Determinants satisfied by $ \Delta _{h}$-Gould-Hopper Appell polynomials reduce to determinant of all subclass of the usual polynomials. Recurrence, shift operators and difference equation satisfied by these polynomials reduce to recurrence, shift operators and difference equation of $\Delta _{h}$-Appell polynomials, $\Delta $-Appell polynomials; recurrence, shift operators, differential and integro-differential equation of $2D$-Appell polynomials, recurrence, shift operators, integro-difference equation of $2D$ $\Delta$-Appell polynomials, recurrence, shift operators, differential equation of Appell polynomials in the corresponding cases. In the special cases of the determining functions, we present the explicit forms, determinants, recurrences, difference equations satisfied by the degenerate Gould-Hopper Carlitz Bernoulli polynomials, degenerate Gould-Hopper Carlitz Euler polynomials, degenerate Gould-Hopper Genocchi polynomials, $\Delta _{h}$-Gould-Hopper Boole polynomials and $\Delta _{h}$-Gould-Hopper Bernoulli polynomials of the second kind. In particular cases of the degenerate Gould-Hopper Carlitz Bernoulli polynomials, degenerate Gould-Hopper Genocchi polynomials, $\Delta _{h}$-Gould-Hopper Boole polynomials and $\Delta _{h}$-Gould-Hopper Bernoulli polynomials of the second kind, corresponding determinants, recurrences, shift operators and difference equations reduce to all subclass of degenerate so-called families except for Genocchi polynomials recurrence, shift operators, and differential equation. Degenerate Gould-Hopper Carlitz Euler polynomials do not satisfy the recurrences and differential equations of $2D$-Euler and Euler polynomials.

Given $n\geq2$ and $\alpha > \frac 12$, we obtained an improved upbound of Hausdorff's dimension of the fractional Schrödinger operator; that is, $$ \sup\limits_{f\in H^s(\mathbb{R}^n)}\dim _H\left\{x\in\mathbb{R}^n:\ \lim_{t\rightarrow0}e^{{\rm i}t(-\Delta)^\alpha}f(x)\neq f(x)\right\}\leq n+1-\frac{2(n+1)s}{n}%\ \ \text{under}\ \ \frac{n}{2(n+1)} < s\leq\frac{n}{2} $$ for $\frac{n}{2(n+1)} < s\leq\frac{n}{2}$.

In this article, we propose a new algorithm and prove that the sequence generalized by the algorithm converges strongly to a common element of the set of fixed points for a quasi-pseudo-contractive mapping and a demi-contraction mapping and the set of zeros of monotone inclusion problems on Hadamard manifolds. As applications, we use our results to study the minimization problems and equilibrium problems in Hadamard manifolds.

Boyd's interpolation theorem for quasilinear operators is generalized in this paper, which gives a generalization for both the Marcinkiewicz interpolation theorem and Boyd's interpolation theorem. By using this new interpolation theorem, we study the spherical fractional maximal functions and the fractional maximal commutators on rearrangementinvariant quasi-Banach function spaces. In particular, we obtain the mapping properties of the spherical fractional maximal functions and the fractional maximal commutators on generalized Lorentz spaces.

For entire or meromorphic function $f$, a value $\theta\in[0,2\pi)$ is called a Julia limiting direction if there is an unbounded sequence $\{z_n\}$ in the Julia set satisfying $\lim\limits_{n\rightarrow\infty}\arg z_n=\theta.$ Our main result is on the entire solution $f$ of $P(z,f)+F(z)f^s=0$, where $P(z,f)$ is a differential polynomial of $f$ with entire coefficients of growth smaller than that of the entire transcendental $F$, with the integer $s$ being no more than the minimum degree of all differential monomials in $P(z,f)$. We observe that Julia limiting directions of $f$ partly come from the directions in which $F$ grows quickly.

Some sufficient conditions of the energy conservation for weak solutions of incompressible viscoelastic flows are given in this paper. First, for a periodic domain in $\mathbb R^3$, and the coefficient of viscosity $\mu=0$, energy conservation is proved for $u$ and $F$ in certain Besov spaces. Furthermore, in the whole space $\mathbb R^3$, it is shown that the conditions on the velocity $u$ and the deformation tensor $F$ can be relaxed, that is, $u\in B^{\frac 13}_{3,c(\mathbb N)}$, and $F\in B^{\frac 13}_{3,\infty}$. Finally, when $\mu>0$, in a periodic domain in $\mathbb R^d$ again, a result independent of the spacial dimension is established. More precisely, it is shown that the energy is conserved for $u\in L^r(0,T;L^s(\Omega))$ for any $\frac 1r+\frac 1s\leqslant\frac 12$, with $s\geqslant4$, and $F\in L^m(0,T;L^n(\Omega))$ for any $\frac 1m+\frac 1n\leqslant\frac 12$, with $n\geqslant4$.

Let $0 < \alpha,\beta < n$ and $f,g \in C([0,\infty) \times[0,\infty))$ be two nonnegative functions. We study nonnegative classical solutions of the system \[\begin{cases} (-\Delta)^{\frac{\alpha}{2}} u=f(u,v) &\text{ in } \mathbb{R}^n,\\ (-\Delta)^{\frac{\beta}{2}} v=g(u,v) &\text{ in } \mathbb{R}^n, \end{cases} \] and the corresponding equivalent integral system. We classify all such solutions when $f(s,t)$ is nondecreasing in $s$ and increasing in $t$, $g(s,t)$ is increasing in $s$ and nondecreasing in $t$, and $\frac{f(\mu^{n-\alpha}s,\mu^{n-\beta}t)}{\mu^{n+\alpha}}$, $\frac{g(\mu^{n-\alpha}s,\mu^{n-\beta}t)}{\mu^{n+\beta}}$ are nonincreasing in $\mu > 0$ for all $s,t\ge0$. The main technique we use is the method of moving spheres in integral forms. Since our assumptions are more general than those in the previous literature, some new ideas are introduced to overcome this difficulty.

In this paper, we consider the following new Kirchhoff-type equations involving the fractional $p$-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:\begin{eqnarray*} && \left (a+ b\iint _{\mathbb{R}^{2N}} \frac{|u (x)-u (y)|^p}{|x-y|^{N + ps}} {\rm d}x {\rm d}y\right)^{p-1} (-\Delta)_p^s u + \lambda V(x)|u|^{p-2}u\\ &=& \bigg(\int_{\mathbb{R}^N} \frac{|u|^{p^*_{\mu,s}}}{|x-y|^\mu}{\rm d}y\bigg)|u|^{p^*_{\mu,s}-2}u, \; x \in \mathbb{R}^N, \end{eqnarray*} where $(-\Delta)_p^s$ is the fractional $p$-Laplacian with $0 < s < 1 < p$, $0 < \mu < N$, $N > ps$, $a,b>0$, $\lambda>0$ is a parameter, $V:\mathbb{R}^N \to \mathbb{R}^+$ is a potential function, $\theta \in[1, 2^*_{\mu,s})$ and $p^*_{\mu,s}=\frac{pN-p\frac{\mu}{2}}{N-ps}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii's genus theory. To the best of our knowledge, our result is new even in Choquard-Kirchhoff-type equations involving the $p$-Laplacian case.

We study the regularity of weak solutions to a class of second order parabolic system under only the assumption of continuous coefficients. We prove that the weak solution $u$ to such system is locally Hölder continuous with any exponent $\alpha\in(0,1)$ outside a singular set with zero parabolic measure. In particular, we prove that the regularity point in $Q_T$ is an open set with full measure, and we obtain a general criterion for the weak solution to be regular in the neighborhood of a given point. Finally, we deduce the fractional time and fractional space differentiability of $D u$, and at this stage, we obtain the Hausdorff dimension of a singular set of $u$.

By applying the standard fixed point theorems, we prove the existence and uniqueness results for a system of coupled differential equations involving both left Caputo and right Riemann-Liouville fractional derivatives and mixed fractional integrals, supplemented with nonlocal coupled fractional integral boundary conditions. An example is also constructed for the illustration of the obtained results.

In this paper, we consider a delayed diffusive SVEIR model with general incidence. We first establish the threshold dynamics of this model. Using a Nonstandard Finite Difference (NSFD) scheme, we then give the discretization of the continuous model. Applying Lyapunov functions, global stability of the equilibria are established. Numerical simulations are presented to validate the obtained results. The prolonged time delay can lead to the elimination of the infectiousness.