The existence of periodic solutions for the nonlinear asymmetric oscillator $ x''{+}\alpha x^+{-}\beta x^- {=} h(t),\qquad (' = d/dt) $ is discussed, where $\alpha, \beta$ are positive constants satisfying $ {1}/{\sqrt{\alpha}} + {1}/{\sqrt{\beta}} = {2}/{n} $ for some positive integer $n\in {\bf \mathbb{N}}$ and $h(t)\in L^\infty(0,2\pi)$ is $2\pi$-periodic with $x^{\pm}=\max\{\pm x, 0\}$.

In the context of Lebesgue integration the Gauss–Green theorem is proved for bounded vector fields with substantial sets of singularities with respect to continuity and differentiability. The resulting integration by parts is applied to removable sets for the Cauchy–Riemann, Laplace, and minimal surface equations. A simple connection between the Gauss–Green theorem and distributional divergence is established.

it is proved in this paper that super-exponentially decaying, possibly non-selfadjoint perturbations of the free jacobi operator are uniquely determined by the location of all their eigenvalues and resonances.

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Fréchet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space $\varphi $ of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on $\varphi$). (2) The space of all test functions for distributions, which is also a complete direct sum of Fréchet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Fréchet space contains a dense hyperplane that admits no transitive operator.

Let B be a complete ccc Boolean algebra and let $\tau_s$ be the topology on B induced by the algebraic convergence of sequences in B.

1. Either there exists a Maharam submeasure on B or every nonempty open set in $(B,\tau_s)$ is topologically dense.

2. It is consistent that every weakly distributive complete ccc Boolean algebra carries a strictly positive Maharam submeasure.

3. The topological space $(B,\tau_s)$ is sequentially compact if and only if the generic extension by B does not add independent reals.

Examples are also given of ccc forcings adding a real but not independent reals.

There exists a set $\cal U$ in the plane, such that elements of $\cal U$ correspond to minimal stable solutions of a two-parameter elliptic system. For points on the boundary $\Upsilon$ of $\cal U$, there exist weak solutions to the elliptic system. The paper studies the properties of the curve $\Upsilon$.

Effective formulae are found for the complex geodesics in the symmetrized bidisc.

A class of operators is introduced on $L_1$ that is stable under taking sums of pointwise unconditionally convergent series, contains all compact operators and does not contain isomorphic embeddings. It follows that any operator from $L_1$ into a space with an unconditional basis belongs to this class.

in this paper, the space $\mathcal{a}_\psi(\mathbb{d})$ is considered, consisting of those holomorphic functions $f$ on the unit disk $\mathbb{d}$ such that $\|f\|_\psi=\sup_{z\in\mathbb{d}}|f(z)|\psi(|z|)<+\infty$, with $\psi(1)=0$. the sampling problem is studied for weights satisfying $\ln\psi(r)/\ln(1-r)\to0$. möbius stability of sampling is shown to fail in this space.

Smale's mean value conjecture asserts that $\min_\theta |P(\theta)/\theta| \le K|P'(0)|$ for every polynomial $P$ of degree $d$ satisfying $P(0)\,{=}\,0$, where $K\,{=}\,(d-1)/d$ and the minimum is taken over all critical points $\theta$ of $P$. A stronger conjecture due to Tischler asserts that \[ \min_\theta\left|\frac12-\frac{P(\theta)}{\theta\cdot P'(0)}\right| \le K_1 \] with $K_1=\frac12-1/d$. Tischler's conjecture is known to be true: (i) for local perturbations of the extremum $P_0(z)=z^d-dz$, and (ii) for all polynomials of degree $d\le 4$. In this paper, Tischler's conjecture is verified for all local perturbations of the extremum $P_1(z)=(z-1)^d-(-1)^d$, but counterexamples to the conjecture are given in each degree $d\ge 5$. In addition, estimates for certain weighted $L^1$- and $L^2$-averages of the quantities $\frac12-{P(\theta)}/{\theta\cdot P'(0)}$ are established, which lead to the best currently known value for $K_1$ in the case $d=5$.

This paper investigates the tightness property of the capacity induced by the reduction operator with respect to the resolvent of a right Markov process. Tightness is verified in two particular situations: under the ‘weak duality hypothesis’, and if a substitute for ‘the axiom of polarity for the dual theory’ holds. In the second context, the quasi-continuity property for the excessive functions is derived. These are extensions and improvements of results of Lyons and Röckner, Ma and Röckner, Le Jan, and Fitzsimmons, mainly obtained in the context of Dirichlet forms.

Consider the parabolic equation \[ \begin{array}{@{}l@{\qquad}l@{}} u_t=\Delta u+|u|^p& \text{in}\ R^d\times(0,T), u(x,0)=\phi(x)& \text{in}\ R^d, \end{array} \] up to a maximal time $T=T_\infty$, where $p>1$. Let $p^{\ast}=1+2/d$. It is a classical result that if $p\le p^{\ast}$, then there exist no non-negative, global solutions to the above equation for any choice of $\phi\geq0$; that is, necessarily $T_\infty<\infty$ and the solution blows up in some sense. On the other hand, if $p>p^{\ast}$, then there do exist non-negative, global solutions for appropriate choices of $\phi$. A recent paper by Zhang seems to be the first to consider the existence of global solutions for the above equation with sign-changing initial data. He proved that if $p\le p^{\ast}$ and $\int_{R^d}\phi(x)dx>0$, then the solution to the above equation is not global. In this paper it is shown that this result continues to hold if $\int_{R^d}\phi(x)dx=0$, and then it is shown that for each $p{>}1$ and each $l\in(-\infty,0)$, there exists a $\phi$ satisfying $\int_{R^d}\phi(x)dx=l$ and such that the corresponding solution to the above equation is global, and there exists a $\phi$ satisfying $\int_{R^d}\phi(x)dx=l$ and such that the corresponding solution is non-global.

an index of a collection of 1-forms on a complex isolated complete intersection singularity corresponding to a chern number is defined and – in the case when the 1-forms are complex analytic – expressed as the dimension of a certain algebra.

A norm on a Banach space $X$ is called maximal when no equivalent norm has a larger group of isometries. If, besides this, there is no equivalent norm with the same isometries (apart from its scalar multiples), the norm is said to be uniquely maximal, which is equivalent to the convex-transitivity of $X$: the convex hull of the orbits under the action of the isometry group on the unit sphere is dense in the unit ball of $X$. The main result of the paper is that the complex $C_0(\Omega)$ is convex-transitive in its natural supremum norm if $\Omega$ is a connected manifold (without boundary). As a complement, it is shown that if $\Omega$ is a connected manifold of dimension at least two, then the diameter norm is convex transitive on the corresponding space of real functions.

This paper concerns rigidity of the mapping class groups. It is shown that any homomorphism $\varphi\,{:}\,\mcg_g\,{\to}\,\mcg_h$ between mapping class groups of closed orientable surfaces with distinct genera $g\,{>}\,h$ is trivial if $g\,{\geq}\, 3$, and has finite cyclic image for all $g\,{\geq}\, 1$.

Some implications are drawn for more general homomorphs of these groups.

The theory of the spectral shift function is extended to the case where only the difference of some powers of the resolvents of self-adjoint operators belongs to the trace class. As an example, a pair of Dirac operators is considered.

The authors of this paper study positive supersolutions to the elliptic equation $-\Delta U=c|x|^{-s}u^p$ in Cone-like domains of $\mathbb{R}^N$ ($N\ge 2$), where $p,s\in\mathbb{R}$ and $c>0$. They prove that in the sublinear case $p<1$ there exists a critical exponent $p_\ast<1$ such that the equation has a positive supersolution if and only if $-\infty<p<p_\ast$. The value of $p_\ast$ is determined explicitly by $s$ and the geometry of the cone.

In this paper, a proof is given that certain boundedness properties of operators yield distributional estimates that have exponential decay at infinity. Such distributional estimates imply local exponential integrability, and apply to many operators such as $m$-linear Calderón–Zygmund operators and their maximal counterparts.

In this paper, the existence problem is studied for extremals of the Sobolev trace inequality $W^{1,p}(\Omega)\to L^{p_*}(\partial\Omega)$, where $\Omega$ is a bounded smooth domain in $\RR^N$, $p_*=p(N-1)/(N-p)$ is the critical Sobolev exponent, and $1<p<N$.

Using Morse theory and the truncation technique, a proof is given of the existence of at least three nontrivial solutions for a class of $p$-Laplacian equations. When $p=2$, the existence of four nontrivial solutions is also considered.