A partial differential operator associated with natural oscillations of an incompressible fluid in the neighbourhood of an elliptical flow is considered. The differentiation is only taken with respect to the angular variable, and thus the operator becomes a family of ordinary differential operators parametrized by the radial variable. It is shown that the spectra of these ordinary differential operators completely determine the spectrum of the given operator which turns out to have a kind of skeleton structure.

The paper deals with the well posedness of a class of ordinary differential equations. The vector field depends on the solution to a scalar conservation law, whose flux function is assumed to have a single inflection point (from whence ‘nonconvex’ is derived). Filippov solutions to the ordinary differential equations are considered, and Hölder continuous dependence on the initial data is proved. The motivation for the problem is a model of traffic flow.

A model (M, <, …) is κ-like if M has cardinality κ but, for all α ∈ M, the cardinality of {x ∈ M [ratio ] x < a} is strictly less than κ. In this paper we shall give constructions of κ-like models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The main results are: (1) for each countable nonstandard M [models ] Π2−Th(PA) with arbitrarily large initial segments satisfying PA and each uncountable κ of cofinality ω there is a cofinal extension K of M which is κ-like; also hierarchical variants of this result for Πn−Th(PA); and (2) for every n [ges ] 1, every singular κ and every M [models ] B[sum ]n+ exp+¬ I[sum ]n there is a κ-like model K elementarily equivalent to M.

It is shown that the topological classification and the smooth classification are generically the same for certain families of plane curves in a semi-local case (that is, the double local case). The normal form of transversely joined two families of plane curves with second-order contact at the envelope is also given.

Let G be a group endowed with its profinite topology, then G is called product separable if the profinite topology of G is Hausdorff and, whenever H1, H2, [ctdot ], Hn are finitely generated subgroups of G, then the product subset H1H2 [ctdot ] Hn is closed in G. In this paper, we prove that if G=F×Z is the direct product of a free group and an infinite cyclic group, then G is product separable. As a consequence, we obtain the result that if G is a generalized free product of two cyclic groups amalgamating a common subgroup, then G is also product separable. These results generalize the theorems of M. Hall Jr. (who proved the conclusion in the case of n=1, [3]), and L. Ribes and P. Zalesskii (who proved the conclusion in the case of that G is a finite extension of a free group, [6]).

The j-invariants of the quadratic Q-curves without complex multiplication are studied. Some properties of the norms of these invariants are shown and a relationship between the field Q(j) and the degree of an isogeny of the Q-curve to its Galois conjugate is found. In the case when the degree of the isogeny is a prime p, some properties of the primes of potentially multiplicative reduction for the Q-curve and of the reduction of j modulo a prime [Pfr ] in Q(j) over p when the Q-curve has potentially good reduction at [Pfr ] are found.

Hall's theorem for bipartite graphs gives a necessary and sufficient condition for the existence of a matching in a given bipartite graph. Aharoni and Ziv have generalized the notion of matchability to a pair of possibly infinite matroids on the same set and given a condition that is sufficient for the matchability of a given pair $(\mathcal{M},\mathcal{W})$ of finitary matroids, where the matroid $\mathcal{M}$ is SCF (a sum of countably many matroids of finite rank). The condition of Aharoni and Ziv is not necessary for matchability. The paper gives a condition that is necessary for the existence of a matching for any pair of matroids (not necessarily finitary) and is sufficient for any pair $(\mathcal{M},\mathcal{W})$ of finitary matroids, where the matroid $\mathcal{M}$ is SCF.

A functorial and categorical defined cyclotomic trace is given, extending the usual one for rings to ring spectra. There are two ingredients to this: first a cyclotomic trace is needed that accepts a categorical input with few restrictive assumptions. This is important in its own right, since this allows one to transport rich structures through the cyclotomic trace. Secondly, a sufficiently nice model is needed for the category of finitely generated free modules which is functorial in the ring spectrum.

The notion of stationary reflection is one of the most important notions of combinatorial set theory. Weak reflection, which is, as its name suggests, a weak version of stationary reflection, is investigated. The main result is that modulo a large cardinal assumption close to 2-hugeness, there can be a regular cardinal $\kappa$ such that the first cardinal weakly reflecting at $\kappa$ is the successor of a singular cardinal. This answers a question of Cummings, Džamonja and Shelah.

We characterize the minimal isometric dilation of a non-commutative contractive sequence of operators as a universal object for certain diagrams of completely positive maps. A non-spatial construction of the minimal isometric dilation is also given, using Hilbert modules over C*-algebras.

It is shown that the non-commutative disc algebras [Ascr ]n (n[ges ]2) are the universal algebras generated by contractive sequences of operators and the identity, and C*(S1, …, Sn) (n[ges ]2), the extension through compact operators of the Cuntz algebra [Oscr ]n, is the universal C*-algebra generated by a contractive sequence of isometries. It is also shown that the algebras [Ascr ]n and C*(S1, …, Sn) are completely isometrically isomorphic to some free operator algebras considered by D. Blecher. In particular, the universal operator algebra of a row (respectively column) contraction is identified with a subalgebra of C*(S1, …, Sn). The internal characterization of the matrix norm on a universal algebra leads to some factorization theorems.

Working over an algebraically closed, complete, non-Archimedean, nontrivially valued field of characteristic zero, it is shown that (i) any subanalytic subset of a one-dimensional semialgebraic set is semialgebraic, (ii) any one-dimensional subanalytic set is semianalytic, and (iii) any subanalytic subset of a two-dimensional semianalytic set is semianalytic.

On page 45 of his lost notebook, Ramanujan recorded two asymptotic formulas for two continued fractions involving the Riemann zeta-function and Dirichlet $L$ -functions. The paper proves a more general theorem and derives Ramanujan's claims as a corollary of the theorem.

Dans cet article, nous nous intéressons aux problèmes de structure galoisienne associés aux couples (E/L, a), où E désigne une courbe elliptique définie sur le corps de nombres L et a un endomorphisme de E. Si E est à multiplication complexe, nous notons k le corps quadratique imaginaire dont l'anneau des entiers décrit End(E) et L une extension de k. Ainsi a est-il selon les cas un entier algébrique de k ou un élément de ℤ.

Equivariant and cocompact retractions of certain symmetric spaces are constructed. These retractions are defined using the natural geometry of symmetric spaces and in relation to the theory of lattices of euclidean space. The following cases are considered: the symmetric space corresponding to lattices endowed with a finite group action, from which is obtained some information relating to the classification problem of these lattices, and the Siegel space Sp2g(R)/Ug, for which a natural Sp2g(Z)-equivariant cocompact retract of codimension 1 is obtained.

Weierstrass points are special points on a Riemann surface that carry a lot of information. Ogg studied such points on $X_0(pM)$ (for $M$ such that $X_0(M)$ has genus zero and $p$ prime with $p\nmid M$), and he proved that if $Q$ is a $\mathbb{Q}$-rational Weierstrass point on $X_0(pM)$, then its reduction modulo $p$ is supersingular. The paper shows that, for square-free $M$ on the list, all supersingular $j$-invariants are obtained in this way. Furthermore, for most cases where $M$ is prime, the explicit correspondence between Weierstrass points and supersingular $j$-invariants in characteristic $p$ is described. Along the way, a useful formula of Rohrlich for computing a certain Wronskian of modular forms modulo $p$ is generalized.

The paper shows that the distinction between tangential sequences and tangential curves, which is known to be relevant for the boundary behaviour of harmonic functions in Euclidean half spaces, is also meaningful and relevant in the discrete setting of the potential theory on a tree associated to a very regular, nearest neighbour random walk. In particular, the paper first defines the class of tangential curves on a tree, and then proves that, for any family of tangential curves which have the same order of tangency and are associated (and converging) to points in the boundary, there is a bounded harmonic function which, for (almost) every point in the boundary, has positive oscillation along the corresponding curve. This result does not hold if, in place of tangential curves, tangential sequences are admitted.

The tree is not assumed to be homogeneous. In particular, the results do not depend on the existence of isometries.

The study of reductive group actions on a normal surface singularity X is facilitated by the fact that the group Aut X of automorphisms of X has a maximal reductive algebraic subgroup G which contains every reductive algebraic subgroup of Aut X up to conjugation. If X is not weighted homogeneous then this maximal group G is finite (Scheja, Wiebe). It has been determined for cusp singularities by Wall. On the other hand, if X is weighted homogeneous but not a cyclic quotient singularity then the connected component G1 of the unit coincides with the [Copf ]* defining the weighted homogeneous structure (Scheja, Wiebe, Wahl). Thus the main interest lies in the finite group G/G1. Not much is known about G/G1. Ganter has given a bound on its order valid for Gorenstein singularities which are not log-canonical. Aumann-Körber has determined G/G1 for all quotient singularities.

We propose to study G/G1 through the action of G on the minimal good resolution X˜ of X. If X is weighted homogeneous but not a cyclic quotient singularity, let E0 be the central curve of the exceptional divisor of X˜. We show that the natural homomorphism G→Aut E0 has kernel [Copf ]* and finite image. In particular, this re-proves the rest of Scheja, Wiebe and Wahl mentioned above. Moreover, it allows us to view G/G1 as a subgroup of Aut E0. For simple elliptic singularities it equals (ℤb×ℤb)[rtimes ]Aut0E0 where −b is the self-intersection number of E0, ℤb×ℤb is the group of b-torsion points of the elliptic curve E0 acting by translations, and Aut0E0 is the group of automorphisms fixing the zero element of E0. If E0 is rational then G/G1 is the group of automorphisms of E0 which permute the intersection points with the branches of the exceptional divisor while preserving the Seifert invariants of these branches. When there are exactly three branches we conclude that G/G1 is isomorphic to the group of automorphisms of the weighted resolution graph. This applies to all non-cyclic quotient singularities as well as to triangle singularities. We also investigate whether the maximal reductive automorphism group is a direct product G≃G1×G/G1. This is the case, for instance, if the central curve E0 is rational of even self-intersection number or if X is Gorenstein such that its nowhere-zero 2-form ω has degree ±1. In the latter case there is a ‘natural’ section G/G1[rarrhk ]G of G[rarrhk ]G/G1 given by the group of automorphisms in G which fix ω. For a simple elliptic singularity one has G≃G1×G/G1 if and only if −E0 · E0 = 1.

The paper shows that for every positive integer $p > 2$, there exists a compact non-orientable surface of genus $p$ with at least $4p$ automorphisms if $p$ is odd, or at least $8\,(p-2)$ automorphisms if $p$ is even, with improvements for odd $p\not\equiv 3$ mod 12. Further, these bounds are shown to be sharp (in that no larger group of automorphisms exists with genus $p$) for infinitely many values of $p$ in each congruence class modulo 12, with the possible (but unlikely) exception of 3 mod 12.

Assume that $G$ is a group covered by countably many sets $X_n,\,n\,{<}\,\omega$. It is proved that $G$ is generated in two or three steps by a small number of the sets $X_n$. These results are generalized for countable coverings of types and countable colourings of graphs.

The paper studies the dynamics of rational maps with indifferent parabolic points by comparing their dynamical properties with those of their ‘jump transformation’ which is uniformly expanding on a non-compact set with infinite Markov partition. It establishes the spectral properties of a two-variables operator-valued function associated to the jump transformation and exploits their dynamical relevance to allow the analytic properties of the pressure, the escape rate from a neighbourhood of the Julia set and the asymptotic distribution of pre-images to be studied.