On oriented graphs whose skew spectral radii do not exceed 2

Let S(G^σ)$S(G^{\sigma })$ be the skew adjacency matrix of the oriented graph G^σ$G^{\sigma }$ of order n   and _λ1,_λ2,…,_λn${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be all eigenvalues of S(G^σ)$S(G^{\sigma })$. The skew spectral radius _ρs(G^σ)${\rho }_s(G^{\sigma })$ of G^σ$G^{\sigma }$ is defined as max{|_λ1|,|_λ2|,…,|_λn|}$\max \{|{\lambda }_1|,|{\lambda }_2|,\dots ,|{\lambda }_n|\}$. In this paper, we investigate oriented graphs whose skew spectral radii do not exceed 2.     

Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type

We establish symmetrization results for the solutions of the linear fractional diffusion equation _∂tu+(−Δ)^σ/2u=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}u=f$ and its elliptic counterpart hv+(−Δ)^σ/2v=f$hv+{(-\mathrm{\Delta })}^{\sigma /2}v=f$, h>0$h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, _∂tu+(−Δ)^σ/2A(u)=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}A(u)=f$, but only when the nondecreasing function A:_R+→_R+$A:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is concave. In the elliptic case, complete symmetrization results are proved for B(v)+(−Δ)^σ/2v=f$B(v)+{(-\mathrm{\Delta })}^{\sigma /2}v=f$ when B(v)$B(v)$ is a convex nonnegative function for v>0$v>0$ with B(0)=0$B(0)=0$, and partial results hold when B is concave. Remarkable counterexamples are constructed for the parabolic equation when A is convex, resp. for the elliptic equation when B   is concave. Such counterexamples do not exist in the standard diffusion case σ=2$\sigma =2$.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Unilateral problem for the Navier–Stokes operators with variable viscosity in noncylindrical domain

We study a unilateral problem for the operator L perturbed of Navier–Stokes operator in a noncylindrical case, where

Lu=u^′-(_ν0+_ν1∥u(t)∥²)Δu+(u.∇)u-f+∇p.$\mathit{Lu}=u^{\prime }-({\nu }_0+{\nu }_1\parallel u(t){\parallel }^2)\mathrm{\Delta }u+(u.\nabla )u-f+\nabla p\text{.}$

Alternative approaches to asymptotic behaviour of eigenvalues of some unbounded Jacobi matrices

In this article we calculate the asymptotic behaviour of the point spectrum for some special self-adjoint unbounded Jacobi operators J   acting in the Hilbert space l²=l²(N)$l^2=l^2(\mathbb{N})$. For given sequences of positive numbers _λn${\lambda }_n$ and real _qn$q_n$ the Jacobi operator is given by J=SW+WS^*+Q$J=\text{<italic>SW</italic>}+{\mathit{WS}}^{*}+Q$, where Q=diag(_qn)$Q=\mathrm{diag}(q_n)$ and W=diag(_λn)$W=\mathrm{diag}({\lambda }_n)$ are diagonal operators, S is the shift operator and the operator J   acts on the maximal domain. We consider a few types of the sequences {_qn}$\{q_n\}$ and {_λn}$\{{\lambda }_n\}$ and present three different approaches to the problem of the asymptotics of eigenvalues of various classes of J's. In the first approach to asymptotic behaviour of eigenvalues we use a method called successive diagonalization, the second approach is based on analytical models that can be found for some special J's and the third method is based on an abstract theorem of Rozenbljum.     

Periodic solutions for Liénard type p-Laplacian equation with a deviating argument

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard type p-Laplacian equation with a deviating argument of the form:

(_?p(x^′(t)))^′+f(x(t))x^′(t)+g(t,x(t-τ(t)))=e(t).$({?}_p(x^{\prime }(t)){)}^{\prime }+f(x(t))x^{\prime }(t)+g(t,x(t-\tau (t)))=e(t)\text{.}$

The Schwarz genus of the Stiefel manifold

In this paper we compute: the Schwarz genus of the Stiefel manifold _Vk(Rⁿ)$V_k({\mathbb{R}}^n)$ with respect to the action of the Weyl group _Wk:=(Z/2)^k?_Sk$W_k:={(\mathbb{Z}/2)}^k?{\mathfrak{S}}_k$, and the Lusternik–Schnirelmann category of the quotient space _Vk(Rⁿ)/_Wk$V_k({\mathbb{R}}^n)/W_k$. Furthermore, these results are used in estimating the number of critically outscribed parallelotopes around a strictly convex body, and Birkhoff–James orthogonal bases of a normed finite dimensional vector space.     

Type III factors with unique Cartan decomposition

We prove that for any free ergodic nonsingular nonamenable action Γ?(X,μ)$\Gamma ?(X,\mu )$ of all Γ   in a large class of groups including all hyperbolic groups, the associated group measure space von Neumann algebra L^∞(X)?Γ${\mathrm{L}}^{\infty }(X)?\Gamma$ has L^∞(X)${\mathrm{L}}^{\infty }(X)$ as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in Popa and Vaes (in press) [38]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products _M1?BM2$M_1{?}_B{M}_2$ over a subalgebra B of type I.     

Global existence and nonexistence of solutions for second-order nonlinear differential equations

This paper deals with the global existence and nonexistence of solutions of the second-order nonlinear differential equation (φ(x^′))^′+λφ(x)=0${(\phi (x^{\prime }))}^{\prime }+\lambda \phi (x)=0$ satisfying x(0)=_x0$x(0)=x_0$ and x^′(0)=_x1$x^{\prime }(0)=x_1$, where λ   is a positive parameter and φ:(−ρ,ρ)→(−σ,σ)$\phi :(-\rho ,\rho )\to (-\sigma ,\sigma )$ with 0<ρ?∞$0<\rho ?\infty$ and 0<σ?∞$0<\sigma ?\infty$ is strictly increasing odd bijective and continuous on (−ρ,ρ)$(-\rho ,\rho )$. Necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic. Examples are given to illustrate our main result. Finally, a nonexistence result for the equation with a damping term is discussed as an application to our result.     

A spectral radius type formula for approximation numbers of composition operators

For approximation numbers _an(_Cφ)$a_n(C_{\phi })$ of composition operators _Cφ$C_{\phi }$ on weighted analytic Hilbert spaces, including the Hardy, Bergman and Dirichlet cases, with symbol φ   of uniform norm <1, we prove that _limn→∞?[_an(_Cφ)]^1/n=e^{−1/Cap[φ(D)]}${\lim }_{n\to \infty }?{[a_n(C_{\phi })]}^{1/n}={\mathrm{e}}^{-1/\mathrm{Cap}[\phi (\mathbb{D})]}$, where Cap[φ(D)]$\mathrm{Cap}[\phi (\mathbb{D})]$ is the Green capacity of φ(D)$\phi (\mathbb{D})$ in D$\mathbb{D}$. This formula holds also for H^p$H^p$ with 1≤p<∞$1\le p<\infty$.     

Layered viscosity solutions of nonautonomous Hamilton–Jacobi equations: Semiconvexity and relations to characteristics

We construct an explicit representation of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation (H,σ)$(H,\sigma )$ on a given domain Ω=(0,T)×Rⁿ$\Omega =(0,T)\times {\mathbb{R}}^n$. It is known that, if the Hamiltonian H=H(t,p)$H=H(t,p)$ is not a convex (or concave) function in p  , or H(⋅,p)$H(\cdot ,p)$ may change its sign on (0,T)$(0,T)$, then the Hopf-type formula does not define a viscosity solution on Ω  . Under some assumptions for H(t,p)$H(t,p)$ on the subdomains (_ti,_ti+1)×Rⁿ⊂Ω$(t_i,t_{i+1})\times {\mathbb{R}}^n\subset \Omega$, we are able to arrange “partial solutions” given by the Hopf-type formula to get a viscosity solution on Ω. Then we study the semiconvexity of the solution as well as its relations to characteristics.