The scalar initial value problem \[ u_t = \rho Du + f(u), \] is a model for dispersal. Here $u$ represents the density at point $x$ of a compact spatial region $\Omega \in \mathbb{R}^n$ and time $t$, and $u(\cdot)$ is a function of $t$ with values in some function space $B$. $D$ is a bounded linear operator and $f(u)$ is a bistable nonlinearity for the associated ODE $u_t = f(u)$. Problems of this type arise in mathematical ecology and materials science where the simple diffusion model with $D=\Delta$ is not sufficiently general. The study of the dynamics of the equation presents a difficult problem which crucially differs from the diffusion case in that the semiflow generated is not compactifying. We study the asymptotic behaviour of solutions and ask under what conditions each positive semi-orbit converges to an equilibrium (as in the case $D=\Delta$). We develop a technique for proving that indeed convergence does hold for small $\rho$ and show by constructing a counter-example that this result does not hold in general for all $\rho$.

We set up the electromagnetic system and its plane-wave solutions with the associated slowness and wave surfaces. We treat the Cauchy initial-value problem for the electric vector and make explicit the quantities necessary for numerical evaluation. We use the Herglotz-Petrovskii representation as an integral around loops which, for each position and time form the intersection of a plane in the space of slownesses with the slowness surface. The field and especially its singularities are strongly dependent on the varying geometry of these loops; we use a level set numerical technique to compute those real loops which essentially gives us second order accuracy. We give the static term corresponding to the mode with zero wave speed. Numerical evaluation of the solution is presented graphically followed by some concluding remarks.

The exact numerical simulation of piezoelectric transducers requires the knowledge of all material tensors that occur in the piezoelectric constitutive relations. To account for mechanical, dielectric and piezoelectric losses, the material parameters are assumed to be complex. The issue of material tensor identification is formulated as an inverse problem: As input measured impedance values for different frequency points are used, the searched-for output is the complete set of material parameters. Hence, the forward operator F mapping from the set of parameters to the set of measurements, involves solutions of the system of partial differential equations arising from application of Newton's and Gauss' law to the piezoelectric constitutive relations. This, via two or three dimensional finite element discretisation, leads to an indefinite system of equations for solving the forward problem. Well-posedness of the infinite dimensional forward problem is proven and efficient solution strategies for its discretized version are presented. Since unique solvability of the inverse problem may hardly be verified, the system of equations we have to solve for recovering the material tensor entries can be rank deficient and therefore requires application of appropriate regularisation strategies. Consequently, inversion of the (nonlinear) parameter-to-measurement map F is performed using regularised versions of Newton's method. Numerical results for different piezoelectric specimens conclude this paper.

For a finite extension $F/\mathbb{Q}_p$ we construct and study a class of locally analytic representations of GL$_2(F)$ and related groups such as the quaternion algebra over $F$. The construction is based on inducing a locally analytic character of a maximal torus. We show that for a generic character the resulting representation is topologically irreducible, and not isomorphic to a locally analytic principal series, when the torus is non-split.

We study the existence of planar flames, in the case of a single-step chemical reaction with volumetric heat losses, with a general reaction term. We prove that for all positive Lewis numbers, and for small values of the heat loss rate parameter, two distinct solutions exist. We also give upper bounds for the flame speed and for the heat loss rate parameter. Moreover, we explicitly compute a lower bound for the unburned gases after reaction.

Soient $p$ un nombre premier et $\mathcal{C}$ la courbe de Hecke $p$-adique de $\mathrm{GL}_2$ de niveau moderé $1$ introduite par Coleman et Mazur. Nous montrons que $\mathcal{C}$ est lisse en ses points Eisenstein critiques et nous donnons une condition nécessaire et suffisante pour que l’application poids soit étale en un tel point en terme de certaines valeurs de la fonction $\zeta$$p$-adique. Un point important est la détermination en ces points du lieu de réductibilité schématique du pseudo-caractère Galoisien porté par $\mathcal{C}$, et ce restreint à un groupe de décomposition en $p$. L’assertion de lissité découle alors de ce que les fonctions $L$ de Dirichlet ont au plus des zéros simples aux entiers.

Let $p$ be a prime number and $\mathcal{C}$ be the $p$-adic tame level $1$ eigencurve introduced by Coleman and Mazur. We prove that $\mathcal{C}$ is smooth at the evil Eisenstein points and we give necessary and sufficient conditions for etaleness of the map to the weight space at these points in terms of $p$-adic zeta values. A key step is the determination at these points of the schematic reducibility locus of the pseudo-character carried by $\mathcal{C}$ restricted to a decomposition group at $p$. Then, the smoothness appears to be a consequence of the fact that the Dirichlet $L$-functions only have simple zeros at integers.

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 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.

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 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.

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.

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.

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.

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 $A_1^\pm$.

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.

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’.

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.