An operator $T$ on a Banach space $X$ is said to be weakly supercyclic (respectively $N$-supercyclic) if there exists a one-dimensional (respectively $N$-dimensional) subspace of $X$ whose orbit under $T$ is weakly dense (respectively norm dense) in $X$. We show that a weakly supercyclic hyponormal operator is necessarily a multiple of a unitary operator, and we give an example of a weakly supercyclic unitary operator. On the other hand, we show that hyponormal operators are never $N$-supercyclic. Finally, we characterize $N$-supercyclic weighted shifts.

Given a Banach space $E$ and positive integers $k$ and $l$ we investigate the smallest constant $C$ that satisfies $\|P\|\hskip1pt\|Q\|\le C\|PQ\|$ for all $k$-homogeneous polynomials $P$ and $l$-homogeneous polynomials $Q$ on $E$. Our estimates are obtained using multilinear maps, the principle of local reflexivity and ideas from the geometry of Banach spaces (type and uniform convexity). We also examine the analogous problem for general polynomials on Banach spaces.

We construct a Legendrian version of envelope theory. A tangential family is a one-parameter family of rays emanating tangentially from a regular plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle $PT^*\mathbb{R}^2$. We study the singularities of Legendrian graphs and their stability under small tangential deformations. We also find normal forms of their projections into the plane. This allows us to interpret the beak-to-beak perestroika as the apparent contour of a deformation of the double Whitney umbrella singularity $A_1^\pm$.

In this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:

(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.

(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.

(iii) The inequalities

$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$

for every Banach space $E$.

(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.

(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.

(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.

Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.

In this paper we study the existence and nonexistence of multiple positive solutions for the Dirichlet problem:

$$ -\Delta{u}-\mu\frac{u}{|x|^2}=\lambda(1+u)^p,\quad u\gt0,\quad u\in H^1_0(\varOmega), \tag{*} $$

where $0\leq\mu\lt(\frac{1}{2}(N-2))^2$, $\lambda\gt0$, $1\ltp\leq(N+2)/(N-2)$, $N\geq3$. Using the sub–supersolution method and the variational approach, we prove that there exists a positive number $\lambda^*$ such that problem (*) possesses at least two positive solutions if $\lambda\in(0,\lambda^*)$, a unique positive solution if $\lambda=\lambda^*$, and no positive solution if $\lambda\in(\lambda^*,\infty)$.

We define the notion of diffractive geodesic for a polygonal billiard or, more generally, for a Euclidean surface with conical singularities. We study the local geometry of the set of such geodesics of given length and we relate it to a number that we call classical complexity. This classical complexity is then computed for any diffractive geodesic. As an application we describe the set of periodic diffractive geodesics as well as the symplectic aspects of the ‘diffracted flow’.

The well-posedness of the Ostrovsky equation is considered. Local well-posedness for data in $\tilde{H}^s(\mathbb{R})$ $(s\geq-\frac{1}{8})$ and global well-posedness for data in $\tilde{L}^{2}(\mathbb{R})$ are obtained.

We consider a simple self-similar sequence of graphs that does not satisfy the symmetry conditions that imply the existence of a spectral decimation property for the eigenvalues of the graph Laplacians. We show that, for this particular sequence, a very similar property to spectral decimation exists, and we obtain a complete description of the spectra of the graphs in the sequence.

We discuss the existence of breather solutions for a discrete nonlinear Schrödinger equation in an infinite $N$-dimensional lattice, involving site-dependent anharmonic parameters. We give a simple proof of the existence of (non-trivial) breather solutions based on a variational approach, assuming that the sequence of anharmonic parameters is in an appropriate sequence space (decays with an appropriate rate). We also give a proof of the non-existence of (non-trivial) breather solutions, and discuss a possible physical interpretation of the restrictions, in both the existence and non-existence cases.

A sharp norm estimate will be given to the pre-Schwarzian derivatives of close-to-convex functions of specified type. In order to show the sharpness, we introduce a kind of maximal operator which may be of independent interest. We also discuss a relation between the subclasses of close-to-convex functions and the Hardy spaces.

We investigate the multidimensional non-isentropic Euler–Poisson (or full hydrodynamic) model for semiconductors, which contain an energy-conserved equation with non-zero thermal conductivity coefficient. We first discuss existence and uniqueness of the non-constant stationary solutions to the corresponding drift–diffusion equations. Then we establish the global existence of smooth solutions to the Cauchy problem with initial data, which are close to the stationary solutions. We find that these smooth solutions tend to the stationary solutions exponentially fast as $t\rightarrow+\infty$.

We present methods for the computation of the Hochschild and cyclic continuous cohomology and homology of some locally convex topological algebras. Let $(A_{\alpha},T_{\alpha,\beta})_{(\varLambda,\le)}$ be a reduced projective system of complete Hausdorff locally convex algebras with jointly continuous multiplications, and let $A$ be the projective limit algebra $A=\lim_{\substack{\raisebox{-3pt}{\tiny$\leftarrow$}\\ \raisebox{2pt}{\tiny$\,\alpha$}}} A_\alpha$. We prove that, for the continuous cyclic cohomology $HC^*$ and continuous periodic cohomology $HP^*$ of $A$ and $A_\alpha$, $\alpha\in\varLambda$, for all $n\ge0$, $HC^n(A)=\lim_{\substack{\raisebox{-3pt}{\tiny$\rightarrow$}\\ \raisebox{2pt}{\tiny$\!\alpha\,$}}} HC^n(A_\alpha)$, the inductive limit of $HC^n(A_\alpha)$, and, for $k=0,1$, $HP^k(A)=\lim_{\substack{\raisebox{-3pt}{\tiny$\rightarrow$}\\ \raisebox{2pt}{\tiny$\!\alpha\,$}}} HP^k(A_\alpha)$. For a projective limit algebra $A=\lim_{\substack{\raisebox{-3pt}{\tiny$\leftarrow$}\\ \raisebox{2pt}{\tiny$\,m$}}}A_m$ of a countable reduced projective system $(A_m,T_{m,\ell})_{\mathbb{N}}$ of Fréchet algebras, we also establish relations between the cyclic-type continuous homology of $A$ and $A_m$, $m\in\mathbb{N}$. For example, we show the exactness of the following short sequence for all $n\ge0$:

$$ 0\rightarrow {\lim_{\substack{\leftarrow\\ \raisebox{2pt}{\scriptsize$\,m$}}}}^{1}HC_{n+1}(A_m)\rightarrow HC_n(A)\rightarrow \lim_{\substack{\leftarrow\\ \raisebox{2pt}{\scriptsize$\,m$}}} HC_{n}(A_m)\rightarrow0. $$

We present a class of Fréchet algebras $A$ for which the continuous periodic cohomology $HP^k(A)$, $k=0,1$, is isomorphic to the continuous cyclic cohomology $HC^{2\ell+k}(A)$ starting from some integer $\ell$. We apply the above results to calculate the continuous cyclic-type homology and cohomology of some Fréchet locally $m$-convex algebras.

We establish results concerning the detailed asymptotic structure of the eigenvalues of a class of regular and singular Sturm–Liouville problems as the domain size shrinks to zero. This is motivated by the recent paper of Bailey et al. and provides an extensive generalization of the results therein.

In this work, we estimate the blow-up time for the non-local hyperbolic equation of ohmic type, $u_t+u_{x}=\lambda f(u)/(\int_{0}^1f(u)\,\mathrm{d} x)^{2}$, together with initial and boundary conditions. It is known that, for $f(s)$, $-f'(s)$ positive and $\int_0^\infty f(s)\,\mathrm{d} s\lt\infty$, there exists a critical value of the parameter $\lambda>0$, say $\lambda^\ast$, such that for $\lambda>\lambda^\ast$ there is no stationary solution and the solution $u(x,t)$ blows up globally in finite time $t^\ast$, while for $\lambda\leq\lambda^\ast$ there exist stationary solutions. Moreover, the solution $u(x,t)$ also blows up for large enough initial data and $\lambda\leq\lambda^\ast$. Thus, estimates for $t^\ast$ were found either for $\lambda$ greater than the critical value $\lambda^\ast$ and fixed initial data $u_0(x)\geq0$, or for $u_0(x)$ greater than the greatest steady-state solution (denoted by $w_2\geq w^*$) and fixed $\lambda\leq\lambda^\ast$. The estimates are obtained by comparison, by asymptotic and by numerical methods. Finally, amongst the other results, for given $\lambda$, $\lambda^*$ and $0\lt\lambda-\lambda^*\ll1$, estimates of the following form were found: upper bound $\epsilon+c_1\ln[c_2(\lambda-\lambda^*)^{-1}]$; lower bound $c_3(\lambda-\lambda^*)^{-1/2}$; asymptotic estimate $t^\ast\sim c_4(\lambda-\lambda^\ast)^{-1/2}$ for $f(s)=\mathrm{e}^{-s}$. Moreover, for $0\lt\lambda\leq\lambda^*$ and given initial data $u_0(x)$ greater than the greatest steady-state solution $w_2(x)$, we have upper estimates: either $c_5\ln(c_6A^{-1}_0+1)$ or $\epsilon+c_7\ln(c_8\zeta^{-1})$, where $A_0$, $\zeta$ measure, in some sense, the difference $u_0-w_2$ (if $u_0\to w_2+$, then $A_0,\zeta\to0+$). $c_i\gt0$ are some constants and $0\lt\epsilon\ll1$, $0\ltA_0,\zeta$. Some numerical results are also given.

This paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an $n({\geq}\,2)$-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss–Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean $n$-sphere. We also show that an $n({\geq}\,2)$-dimensional complete connected orientable hypersurface immersed in a unit sphere $S^{n+1}$ whose Gauss image is contained in a closed geodesic ball of radius less than $\pi/2$ in $S^{n+1}$ is diffeomorphic to a sphere. Finally, we prove that an $n({\geq}\,2)$-dimensional connected closed orientable hypersurface in $S^{n+1}$ with constant scalar curvature greater than $n(n-1)$ and Gauss image contained in an open hemisphere is totally umbilic.