Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each $r$-th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian–Totik $k$th modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs $(k,\,r)$ we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grünbaum–Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes.

In this paper, first, we will investigate the Dirichlet problem for one type of vortex equation, which generalizes the well-known Hermitian Einstein equation. Secondly, we will give existence results for solutions of these vortex equations over various complete noncompact Kähler manifolds.

We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL\left( 2,\,k \right)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL\left( 2,\,k \right)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

Let $L(X)$ be the space of bounded linear operators on the Banach space $X$. We study the strict singularity and cosingularity of the two-sided multiplication operators $S\,\mapsto \,ASB$ on $L(X)$, where $A,\,B\,\in \,L(X)$ are fixed bounded operators and $X$ is a classical Banach space. Let $1\,<\,p\,<\,\infty $ and $p\,\ne \,2$. Our main result establishes that the multiplication $S\,\mapsto \,ASB$ is strictly singular on $L\left( {{L}^{p}}\left( 0,\,1 \right) \right)$ if and only if the non-zero operators $A,\,B\,\in \,L\left( {{L}^{p}}\left( 0,\,1 \right) \right)$ are strictly singular. We also discuss the case where $X$ is a ${\mathcal{L}^{1}}-$ or a ${{\mathcal{L}}^{\infty }}-$space, as well as several other relevant examples.

Let $\rho :\,{{G}_{Q}}\,\to \,\text{G}{{\text{L}}_{n}}\left( {{Q}_{\ell }} \right)$ be a motivic $\ell $-adic Galois representation. For fixed $m\,>\,1$ we initiate an investigation of the density of the set of primes $p$ such that the trace of the image of an arithmetic Frobenius at $p$ under $\rho $ is an $m$-th power residue modulo $p$. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals $1/m$ whenever the image of $\rho $ is open. We further conjecture that for such $\rho $ the set of these primes $p$ is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain $m$ in the complementary case of modular forms of $\text{CM}$-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the $\text{CM}$ case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at $p$ in abelian extensions of imaginary quadratic fields unramified away from $p$.

To every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the minimum number of wild primes. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes.

In this paper, we find configurations of points in $n$-dimensional projective space $\left( {{\mathbb{P}}^{n}} \right)$ which simultaneously generalize both $k$-configurations and reduced 0-dimensional complete intersections. Recall that $k$-configurations in ${{\mathbb{P}}^{2}}$ are disjoint unions of distinct points on lines and in ${{\mathbb{P}}^{n}}$ are inductively disjoint unions of $k$-configurations on hyperplanes, subject to certain conditions. Furthermore, the Hilbert function of a $k$-configuration is determined from those of the smaller $k$-configurations. We call our generalized constructions ${{k}_{D}}$-configurations, where $D\,=\,\left\{ {{d}_{1}},\ldots ,{{d}_{r}} \right\}$ (a set of $r$ positive integers with repetition allowed) is the type of a given complete intersection in ${{\mathbb{P}}^{n}}$. We show that the Hilbert function of any ${{k}_{D}}$-configuration can be obtained from those of smaller ${{k}_{D}}$-configurations. We then provide applications of this result in two different directions, both of which are motivated by corresponding results about $k$-configurations.

We investigate the problem of whether every immersed flat plane in a nonpositively curved square complex is the limit of periodic flat planes. Using a branched cover, we reduce the problem to the case of $\mathcal{V}\mathcal{H}$-complexes. We solve the problem for malnormal and cyclonormal $\mathcal{V}\mathcal{H}$-complexes. We also solve the problem for complete square complexes using a different approach. We give an application towards deciding whether the elements of fundamental groups of the spaces we study have commuting powers. We note a connection between the flat approximation problem and subgroup separability.

We study the semilinear equation

$$-{{\Delta }_{\mathbb{H}}}u(\eta )\,+\,u(\eta )\,=\,f(\eta ,\,\,u(\eta )),\,u\in \,\overset{\circ }{\mathop{S}}\,_{1}^{2}(\Omega ),$$

where $\Omega $ is an unbounded domain of the Heisenberg group ${{\mathbb{H}}^{N}},\,N\,\ge \,1$. The space $\overset{\circ }{\mathop{S}}\,_{1}^{2}(\Omega )$ is the Heisenberg analogue of the Sobolev space $W_{0}^{1,\,2}\,\left( \Omega \right)$. The function $f\,:\,\overset{-}{\mathop{\Omega }}\,\,\times \,\mathbb{R}\,\to \,\mathbb{R}$ is supposed to be odd in $u$, continuous and satisfy some (superlinear but subcritical) growth conditions. The operator ${{\Delta }_{\mathbb{H}}}$ is the subelliptic Laplacian on the Heisenberg group. We give a condition on $\Omega $ which implies the existence of infinitely many solutions of the above equation. In the proof we rewrite the equation as a variational problem, and show that the corresponding functional satisfies the Palais–Smale condition. This might be quite surprising since we deal with domains which are far frombounded. The technique we use rests on a compactness argument and the maximum principle.

A duality formula is found for coalescing Brownian motions on the real line. It is shown that the joint distribution of a coalescing Brownian motion can be determined by another coalescing Brownian motion running backward. This duality is used to study a measure-valued process arising as the high density limit of the empirical measures of coalescing Brownian motions.

We study Dupin hypersurfaces in ${{\mathbb{R}}^{5}}$ parametrized by lines of curvature, with four distinct principal curvatures. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. We show that these vector valued functions are invariant by inversions and homotheties.

In 1999 V. Arnol’d introduced the local contact algebra: studying the problem of classification of singular curves in a contact space, he showed the existence of the ghost of the contact structure (invariants which are not related to the induced structure on the curve). Our main result implies that the only reason for existence of the local contact algebra and the ghost is the difference between the geometric and (defined in this paper) algebraic restriction of a 1-form to a singular submanifold. We prove that a germ of any subset $N$ of a contact manifold is well defined, up to contactomorphisms, by the algebraic restriction to $N$ of the contact structure. This is a generalization of the Darboux-Givental’ theorem for smooth submanifolds of a contact manifold. Studying the difference between the geometric and the algebraic restrictions gives a powerful tool for classification of stratified submanifolds of a contact manifold. This is illustrated by complete solution of three classification problems, including a simple explanation of V. Arnold's results and further classification results for singular curves in a contact space. We also prove several results on the external geometry of a singular submanifold $N$ in terms of the algebraic restriction of the contact structure to $N$. In particular, the algebraic restriction is zero if and only if $N$ is contained in a smooth Legendrian submanifold of $M$.