The prime, completely prime, maximal, and primitive spectra are classified for the universal enveloping algebra of the Schrödinger algebra. The explicit generators are given for all of these ideals. A counterexample is constructed to the conjecture of Cheng and Zhang about nonexistence of simple singular Whittaker modules for the Schrödinger algebra (and all such modules are classified). It is proved that the conjecture holds ‘generically’.

The Pólya-Vinogradov inequality states that for any non-principal character x modulo q and any N ≧ 1,

where c is an absolute constant. We show that (*) holds with c = 2/(3π2) + o(1) in the case x is primitive and x (— 1) =1 with c = 1/(3π) + o(l) in the case x is primitive and x(— 1) = — 1- This improves by a factor 2/3 the previously best-known values for these constants.

The well-known Falkner–Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to $\text{ }\!\!\lambda\!\!\text{ }\!\!\pi\!\!\text{ /2}$, where $\text{ }\!\!\lambda\!\!\text{ }\,\in \,\mathbb{R}$ is a parameter involved in the equation. It is known that there exists ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,<\,0$ such that the equation with suitable boundary conditions has at least one positive solution for each $\text{ }\!\!\lambda\!\!\text{ }\,\ge \,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ and has no positive solutions for $\text{ }\!\!\lambda\!\!\text{ }\,<\,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ . The known numerical result shows ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,=\,-0.1988$ . In this paper, ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,\in \,[-0.4,\,-0.12]$ is proved analytically by establishing a singular integral equation which is equivalent to the Falkner–Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner–Skan equation.

A presentation of a group G is an exact sequence of groups

where F is a free group. Let l→S ⊆ F→G→1 be another presentation of G involving the same free group F.

In [4], A. Sade defines the singular direct product for quasigroups. In this paper we use the singular direct product to construct quasigroups satisfying the identity x(xy)=yx. In particular, we show that the singular direct product preserves the identity x(xy)=yx.

We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

Regular C-fractions f(α) = 1 + a1α/1 + a2α/1 + . .. with an = an2 + bn + c + Vn, |Vn| sufficiently small are examined. In the case Vn = 0, exact expressions are obtained which reveal a two sheeted Riemann structure for f(α). If Vn ≠ 0 analytic properties are obtained by means of perturbation theory applied to the associated difference equation. A conjecture that f(α) is the ratio of two entire functions of for an even larger class of C-fractions is proved for the case .

An irreducible supercuspidal representation $\pi$ of $G\,=\,\text{GL}\left( n,\,F \right)$, where $F$ is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup $H$ of $G$ and a quasicharacter $\text{ }\!\!\chi\!\!\text{ }$ of $H$ if $\text{Ho}{{\text{m}}_{H}}\left( \pi ,\,\text{ }\!\!\chi\!\!\text{ } \right)\,\ne \,0$. There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to $\text{GL}\left( n \right)$. Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

We give a simple characterization of coefficient multipliers from the mixed norm space Hp,q,α, 2 ≤ p ≤ ∞, into Hu,v,β, 0 ≤ u ≤ 2, which includes the main results of Wojtaszczyk in [5]. We also calculate multipliers from the Hardy space Hp, 2 ≤ p ≤ ∞, into Hq, 0 < q ≤ 2.

Biharmonic maps are defined as critical points of the bienergy. Every harmonic map is a stable biharmonic map. In this article, the stability of nonharmonic biharmonic Legendrian submanifolds in Sasakian space forms is discussed.

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.

We introduce a new constant in Banach spaces which implies, in certain cases, the weak- or weak*-normal structure.

We prove the converse of a theorem of Zaanen about the duality problem of positive $\text{AM}$-compact operators.

Real hypersurfaces in a complex space form whose structure Jacobi operator is symmetric along the Reeb flow are studied. Among them, homogeneous real hypersurfaces of type $\left( A \right)$ in a complex projective or hyperbolic space are characterized as those whose structure Jacobi operator commutes with the shape operator.

Consider a real potential $V$ on ${{\text{R}}^{d}}$, $d\ge 2$, and the Schrödinger equation:

$$\left( \text{LS} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i{{\partial }_{t}}u+\Delta u-Vu=0,{{u}_{\upharpoonright }}_{t=0}={{u}_{0}}\in {{L}^{2}}$$

In this paper, we investigate the minimal local regularity of $V$ needed to get local in time dispersive estimates (such as local in time Strichartz estimates or local smoothing effect with gain of $1/2$ derivative) on solutions of $\left( \text{LS} \right)$. Prior works show some dispersive properties when $V$ (small at infinity) is in ${{L}^{d/2}}$ or in spaces just a little larger but with a smallness condition on $V$ (or at least on its negative part).

In this work, we prove the critical character of these results by constructing a positive potential $V$ which has compact support, bounded outside 0 and of the order ${{\left( \log \left| x \right| \right)}^{2}}/{{\left| x \right|}^{2}}$ near 0. The lack of dispersiveness comes from the existence of a sequence of quasimodes for the operator $P:=-\Delta +V$.

The elementary construction of $V$ consists in sticking together concentrated, truncated potential wells near 0. This yields a potential oscillating with infinite speed and amplitude at 0, such that the operator $P$ admits a sequence of quasi-modes of polynomial order whose support concentrates on the pole.

By a group ring we mean in this paper a ring defined by a finite group G and an integral domain K:

such that A contains G and is freely generated by G over K, so that

The ring A = KG has a co-multiplication

defined by

so that A is a Hopf algebra.

Let $L\left( X \right)$ be the free locally convex space over a Tychonoff space $X$. Then $L\left( X \right)$ is a $k$-space if and only if $X$ is a countable discrete space. We prove also that $L\left( D \right)$ has uncountable tightness for every uncountable discrete space $D$.

We derive a necessary and sufficient condition for $\text{HNN}$-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of $\text{HNN}$-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of $\text{HNN}$-extensions of nilpotent groups with cyclic associated subgroups.

We describe all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,\,y\,\in \,K$. When is $F\left[ x,\,y \right]\,=\,F\left[ \alpha x\,+\,\beta y \right]$ for some nonzero elements $\alpha ,\,\beta \,\in \,F?$

Let Ω be an index set and for each αϵΩ let fα:X → Xα be a function where X and Xα are sets. Assume that, for each α, a topology is given for Xα. Then, as is well-known, the functions fα and the topologies determine a topology for X. This is the so-called weak or initial topology, which is generated by .