Existence and nonexistence of global positive solutions for the evolution P-Laplacian equations in exterior domains

This paper deals with the existence and nonexistence of global positive solutions for two evolution P-Laplacian equations in exterior domains with inhomogeneous boundary conditions. We demonstrate that _qc=n(p−1)/(n−p)$q_c=n(p-1)/(n-p)$ is its critical exponent provided 2n/(n+1)<p<n$2n/(n+1)<p<n$. Furthermore, we prove that if max{1,p−1}<q≤_qc$max\{1,p-1\}<q\le q_c$, then every positive solution of the equations blows up in finite time; whereas for q>_qc$q>q_c$, the equations admit the global positive solutions for some boundary value f(x)$f\left(x\right)$ and some initial data _u0(x)$u_0\left(x\right)$. We also demonstrate that every positive solution of the equations blows up in finite time provided n≤p$n\le p$.     

A method for solving differential equations of fractional order

In this paper, we consider Caputo type fractional differential equations of order 0<α<1$0<\alpha <1$ with initial condition x(0)=_x0$x\left(0\right)=x_0$. We introduce a technique to find the exact solutions of fractional differential equations by using the solutions of integer order differential equations. Generalization of the technique to finite systems is also given. Finally, we give some examples to illustrate the applications of our results.     

Characterization of soliton solutions in 2D nonlinear Schrödinger lattices by using the spatial disorder

In this paper, the pattern of the soliton solutions to the discrete nonlinear Schrödinger (DNLS) equations in a 2D   lattice is studied by the construction of horseshoes in _l∞$l_{\infty }$-spaces. The spatial disorder of the DNLS equations is the result of the strong amplitudes and stiffness of the nonlinearities. The complexity of this disorder is log(N+1)$\log (N+1)$ where N   is the number of turning points of the nonlinearities. For the case N=1$N=1$, there exist disjoint intervals _I0$I_0$ and _I1$I_1$, for which the state _um,n$u_{m,n}$ at site (m,n)$(m,n)$ can be either dark (_um,n∈_I0$u_{m,n}\in I_0$) or bright (_um,n∈_I1$u_{m,n}\in I_1$) that depends on the configuration _km,n=0$k_{m,n}=0$ or 1, respectively. Bright soliton solutions of the DNLS equations with a cubic nonlinearity are also discussed.     

Stability and Hopf bifurcation for a virus infection model with delayed humoral immunity response

In this paper, we investigate the dynamical behavior of a virus infection model with delayed humoral immunity. By using suitable Lyapunov functional and the LaSalle?s invariance principle, we establish the global stabilities of the two boundary equilibria. If _R0<1$R_0<1$, the uninfected equilibrium _E0$E_0$ is globally asymptotically stable; if _R1<1<_R0$R_1<1<R_0$, the infected equilibrium without immunity _E1$E_1$ is globally asymptotically stable. When _R1>1$R_1>1$, we obtain the sufficient conditions to the local stability of the infected equilibrium with immunity _E2$E_2$. The time delay can change the stability of _E2$E_2$ and lead to the existence of Hopf bifurcations. The stabilities of bifurcating periodic solutions is also studied. We check our theorems with numerical simulations in the end.     

The relaxed Newton method derivative: Its dynamics and non-linear properties

The dynamic behaviour of the one-dimensional family of maps f(x)=_c2[(a−1)x+_c1]^{−λ/(α−1)}$f\left(x\right)=c_2{[(a-1)x+c_1]}^{-\lambda /(\alpha -1)}$ is examined, for representative values of the control parameters a,_c1$a,c_1$, _c2$c_2$ and λ$\lambda$. The maps under consideration are of special interest, since they are solutions of the relaxed Newton method derivative being equal to a constant a$a$. The maps f(x)$f\left(x\right)$ are also proved to be solutions of a non-linear differential equation with outstanding applications in the field of power electronics. The recurrent form of these maps, after excessive iterations, shows, in an _xn$x_n$ versus λ$\lambda$ plot, an initial exponential decay followed by a bifurcation. The value of λ$\lambda$ at which this bifurcation takes place depends on the values of the parameters a,_c1$a,c_1$ and _c2$c_2$. This corresponds to a switch to an oscillatory behaviour with amplitudes of f(x)$f\left(x\right)$ undergoing a period doubling. For values of a$a$ higher than 1 and at higher values of λ$\lambda$ a reverse bifurcation occurs. The corresponding branches converge and a bleb is formed for values of the parameter _c1$c_1$ between 1 and 1.20. This behaviour is confirmed by calculating the corresponding Lyapunov exponents.     

The transition from ergodic to explosive behavior in a family of stochastic differential equations

We study a family of quadratic, possibly degenerate, stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, Hörmander’s hypoellipticity theorem, and geometric control theory, we find a critical parameter value _α1=_α2${\alpha }_1={\alpha }_2$ such that when _α2>_α1${\alpha }_2>{\alpha }_1$ the system is ergodic and when _α2<_α1${\alpha }_2<{\alpha }_1$ solutions are not defined for all times.     

Globally generated vector bundles of rank 2 on a smooth quadric threefold

We investigate the existence of globally generated vector bundles of rank 2 with _c1≤3$c_1\le 3$ on a smooth quadric threefold and determine their Chern classes. As an automatic consequence, every rank 2 globally generated vector bundle on Q$Q$ with _c1=3$c_1=3$ is an odd instanton up to twist.     

Traveling wave solutions of advection–diffusion equations with nonlinear diffusion

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c∈]_c?,+∞[$c\in ]c_{?},+\infty [$, where _c?>0$c_{?}>0$ is explicitly computed but may not be optimal. We also prove that a free boundary hypersurface separates a region where u=0$u=0$ and a region where u>0$u>0$, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u>0$u>0$, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.     

The exact analytic solutions of a nonlinear differential iterative equation

This paper is concerned with a second-order nonlinear iterated differential equation of the form _c0x^″(z)+_c1x^′(z)+_c2x(z)=x(p(z)+bx(z))+h(z)$c_0{x}^{″}\left(z\right)+c_1{x}^{\prime }\left(z\right)+c_{2}x\left(z\right)=x(p\left(z\right)+bx\left(z\right))+h\left(z\right)$. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only the general case |β|≠1$\left|\beta \right|\ne 1$, but also the critical case |β|=1$\left|\beta \right|=1$, especially when β$\beta$ is a root of unity. Furthermore, the exact and explicit analytic solution of the original equation is investigated for the first time. Such equations are important in both applications and the theory of iterations.     

On strictly singular nonlinear centralizers

We show that _c0$c_0$ is the only Banach space with unconditional basis that satisfies the equation Ext(X,X)=0$Ext(X,X)=0$. This partially improves an old result by Kalton and Peck. We prove that the Kalton–Peck maps are strictly singular on a number of sequence spaces, including _?p${?}_p$ for 0<p<∞$0<p<\infty$, Tsirelson and Schlumprecht spaces and their duals, as well as certain super-reflexive variations of these spaces. In the last section, we give estimates of the projection constants of certain finite-dimensional twisted sums of Kalton–Peck type.     

On the approximation by three consecutive continued fraction convergents

Denote by _pn/_qn,n=1,2,3,…$p_n/q_n,n=1,2,3,\dots$, the sequence of continued fraction convergents of the real irrational number x$x$. Define the sequence of approximation coefficients by _θn:=_qn|_qnx−_pn|,n=1,2,3,…${\theta }_n:=q_n|q_{n}x-p_n|,n=1,2,3,\dots$. A laborious way of determining the mean value of the sequence |_θn+1−_θn−1|,n=2,3,…$|{\theta }_{n+1}-{\theta }_{n-1}|,n=2,3,\dots$, is simplified. The method involved also serves for showing that for almost all x$x$ the pattern _θn−1<_θn<_θn+1${\theta }_{n-1}<{\theta }_n<{\theta }_{n+1}$ occurs with the same asymptotic frequency as the pattern _θn+1<_θn<_θn−1${\theta }_{n+1}<{\theta }_n<{\theta }_{n-1}$, namely 0.12109?$0.12109?$. All the four other patterns have the same asymptotic frequency 0.18945?$0.18945?$. The constants are explicitly given.     

Periodic traveling wave train and point-to-periodic traveling wave for a diffusive predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type scheme is investigated in this article. The existences of a small amplitude periodic traveling wave train _Γp${\Gamma }_p$ and the traveling wave solution connecting the boundary equilibrium _Eu(1,0)$E_u(1,0)$ to the periodic traveling wave _Γp${\Gamma }_p$ are obtained. The existence of this point-to-periodic solution reveals that the predator invasion leads to the periodic population densities in the coexistence domain, and thus plays a mild role in the evolution of predator–prey communities. The techniques used here are the Hopf bifurcation theorem, the improved shooting method combining with the geometric singular perturbation method.     

The inverse problem on subset sums

For a set A$A$, let P(A)$P\left(A\right)$ be the set of all finite subset sums of A$A$. We prove that if a sequence B={_b1<_b2<?}$B=\{b_1<b_2<?\}$ of integers satisfies _b1≥11,_b2≥3_b1+5,_b3≥3_b2+3$b_1\ge 11,b_2\ge 3b_1+5,b_3\ge 3b_2+3$ and _bn+1>3_bn−_bn−2$b_{n+1}>3b_n-b_{n-2}$  (n≥3)$(n\ge 3)$, then there exists a sequence of positive integers A={_a1<_a2<?}$A=\{a_1<a_2<?\}$ such that P(A)=N?B$P\left(A\right)=\mathbb{N}?B$. These lower bounds are optimal in a sense. We pose a problem for further research.     

Cumulants of the maximum of the Gaussian random walk

Let _X1,_X2,…$X_1,X_2,\dots$ be independent variables, each having a normal distribution with negative mean −β<0$-\beta <0$ and variance 1. We consider the partial sums _Sn=_X1+?+_Xn$S_n=X_1+?+X_n$, with _S0=0$S_0=0$, and refer to the process {_Sn:n≥0}$\{S_n:n\ge 0\}$ as the Gaussian random walk. This paper is concerned with the cumulants of the maximum _Mβ=max{_Sn:n≥0}$M_{\beta }=max\{S_n:n\ge 0\}$.     

Kolmogorov's problem on the class of multiply monotone functions

In this paper we give necessary and sufficient conditions for the system of positive numbers _Mk1,_Mk2,…,_Mkd$M_{k_1},M_{k_2},\dots ,M_{k_d}$, 0≤_k1<…<_kd≤r$0\le k_1<\dots <k_d\le r$, to guarantee the existence of an r  -monotone function defined on the negative half-line _R−${\mathbb{R}}_{-}$ and such that _{‖x(ki)‖∞}=_Mki${\Vert x^{(k_i)}\Vert }_{\infty }=M_{k_i}$, i=1,2,…,d$i=1,2,\dots ,d$. We also discuss some applications of the obtained results and connections with other problems.     

Uniform hypergraphs containing no grids

A hypergraph is called an r×r$r\times r$grid   if it is isomorphic to a pattern of r$r$ horizontal and r$r$ vertical lines, i.e., a family of sets {_A1,…,_Ar,_B1,…,_Br}$\{A_1,\dots ,A_r,B_1,\dots ,B_r\}$ such that _Ai∩_Aj=_Bi∩_Bj=0?$A_i\cap A_j=B_i\cap B_j=0?$ for 1≤i<j≤r$1\le i<j\le r$ and |_Ai∩_Bj|=1$|A_i\cap B_j|=1$ for 1≤i,j≤r$1\le i,j\le r$. Three sets _C1,_C2,_C3$C_1,C_2,C_3$ form a triangle   if they pairwise intersect in three distinct singletons, |_C1∩_C2|=|_C2∩_C3|=|_C3∩_C1|=1$|C_1\cap C_2|=|C_2\cap C_3|=|C_3\cap C_1|=1$, _C1∩_C2≠_C1∩_C3$C_1\cap C_2\ne C_1\cap C_3$. A hypergraph is linear  , if |E∩F|≤1$|E\cap F|\le 1$ holds for every pair of edges E≠F$E\ne F$.