A method is developed for obtaining Ramanujan's mock theta functions from ordinary theta functions by performing certain operations on their q-series expansions. The method is then used to construct several new mock theta functions, including the first ones of eighth order. Summation and transformation formulae for basic hypergeometric series are used to prove that the new functions actually have the mock theta property. The modular transformation formulae for these functions are obtained.

Commutative rings for which every ideal can be written as a finite product of invertible and prime ideals are characterized.

In the appendix to [20] Waldhausen discussed a trace map tr:K(R)&→HH(R), from the algebraic K-theory of a ring to its Hochschild homology, which can be used to obtain information about K(R) from HH(R). In [1] Bökstedt described a factorization of this trace map. The intermediate functor THH(HR) is called the topological Hochschild homology of the Eilenberg–MacLane spectrum HR associated with R, because it is constructed similarly to Hochschild homology with the tensor product replaced by the smash product of spectra.

The moduli problem for (algebraic completely) integrable systems is introduced. This problem consists in constructing a moduli space of affine algebraic varieties and explicitly describing a map which associates to a generic affine variety, which appears as a level set of the first integrals of the system (or, equivalently, a generic affine variety which is preserved by the flows of the integrable vector fields), a point in this moduli space. As an illustration, the example of an integrable geodesic flow on SO(4) is worked out. In this case, the generic invariant variety is an affine part of the Jacobian of a Riemann surface of genus 2. The construction relies heavily on the fact that these affine parts have the additional property of being 4:1 unramified covers of Abelian surfaces of type (1,4).

It is proved that if an algebra R over a field can be endowed with a pointed and finite-dimensional ℕn-filtration such that the associated ℕn-graded algebra T is semi-commutative, then R is left and right finitely partitive. In order to do this, a multi-variable Poincaré series for every finitely generated graded T-module is considered and it is shown that this Poincaré series is a rational function. The methods apply to some iterated Ore extensions such as quantum matrices and quantum Weyl algebras as well as to the quantized enveloping algebra of [sfr ][lfr ](ν+1).

A classical result of M. Gerstenhaber [5] states that finite-dimensional semisimple complex Lie algebras are rigid. One interpretation of this fact is as follows. Let [Lfr ] be a semisimple complex Lie algebra of dimension d and let [Bfr ] be a [Copf ]-basis of [Lfr ]. Let L(d) ⊂ [Copf ]d·(d2) denote the affine variety of all structure constants of [Copf ]-Lie algebras of dimension d and let c ∈ L(d) denote the point corresponding to the structure constants of [Lfr ] with respect to [Bfr ]. Then there exists an open neighbourhood in the metric topology U ⊂ L(d) of c ∈ L(d) such that [Lfr ]c* is isomorphic to [Lfr ] for all c* ∈ U, where [Lfr ]c* denotes the Lie algebra defined by the structure constants c* ∈ L(d). Our aim is to generalize this result to semisimple Lie algebras over discrete valuation domains and to apply these results to powerful pro-p-groups.

Some subnormal embeddings of groups into groups with additional properties are established, and in particular, embeddings of countable groups into two-generated groups with some extra properties. The results obtained are generalizations of theorems of P. Hall, R. Dark, B. Neumann, H. Neumann and G. Higman on embeddings of that type. Through the consideration of subnormal embeddings of finite groups into finite groups with additional properties, a result of H. Heineken and J. Lennox is illustrated.

All finite groups G are determined that admit a subgroup K of index pa, p a prime, under the assumption that G has an irreducible and faithful GF(q)-module in characteristic p whose dimension over GF(q) is at most a. As an application to the theory of permutation groups, the maximal transitive subgroups of the primitive affine permutation groups are determined. The above-mentioned classification is generalized by dropping the assumption that p|q. In both cases surprisingly nice results are obtained.

We are interested in Weil representations of Sp(2n, R), where R is the ring Z/plZ, p is an odd prime and l is a positive integer, or, more generally, R = [Oscr ]/[pfr ]l, where [Oscr ] is the ring of integers of a local field, [pfr ] is the maximal ideal of [Oscr ] and [Oscr ]/[pfr ] has odd characteristic. One reason for this interest is that a continuous finite-dimensional complex representation of Sp(2n, [Oscr ]) has to factor through a representation of Sp(2n, [Oscr ]/[pfr ]l) for some l.

We study here some foundational aspects of the classification problem for torsion-free abelian groups of finite rank. These are, up to isomorphism, the subgroups of the additive groups (ℚn, +), for some n = 1, 2, 3,…. The torsion-free abelian groups of rank [les ] n are the subgroups of (ℚn, +).

It is proved that a Tychonoff topological group is locally compact if and only if it is of pointwise countable type and its left uniformity is cofinally complete. From this result a characterization is derived of those T0 paratopological groups (X, τ) of pointwise countable type for which (X, τ ∨ τ−1) is locally compact and also a characterization is deduced of locally pseudocompact topological groups in terms of cofinal completeness. Also characterized are the Tychonoff topological groups of pointwise countable type for which their left uniformity has property U. Finally, cofinal completeness of the Hausdorff–Bourbaki uniformity of a topological group is studied.

A theorem due to Hardy states that, if f is a function on R such that |f(x)| [les ] C e−α|x|2 for all x in R and |fˆ(ξ)| [les ] C e−β|ξ|2 for all ξ in R, where α > 0, β > 0, and αβ > 1/4, then f = 0. A version of this celebrated theorem is proved for two classes of Lie groups: two-step nilpotent Lie groups and harmonic NA groups, the latter being a generalisation of noncompact rank-1 symmetric spaces. In the first case the group Fourier transformation is considered; in the second case an analogue of the Helgason–Fourier transformation for symmetric spaces is considered.

The paper is concerned with the uniqueness and existence problem for a multidimensional reacting and convection system with the vanishing viscosity method. The uniqueness theorem is obtained from the stability with respect to the initial data. To solve the existence problem, the uniqueness and existence of solutions to a viscous system are first proved, and then the L1-modulus of continuity of solutions independent of the small viscosity is obtained.

For a given integer n, all zero-mean cosine polynomials of order at most n which are non-negative on [0,(n/(n+1))π] are found, and it is shown that this is the longest interval [0,θ] on which such cosine polynomials exist. Also, the longest interval [0,θ] on which there is a non-negative zero-mean cosine polynomial with non-negative coefficients is found.

As an immediate consequence of these results, the corresponding problems of the longest intervals [θ,π] on which there are non-positive cosine polynomials of degree n are solved.

For both of these problems, all extremal polynomials are found. Applications of these polynomials to Diophantine approximation are suggested.

The paper provides a new look at compensated compactness and paracommutators. It is shown that there is a one-to-one correspondence between compensated quantities and paracommutators, that the H1-regularity of compensated quantities is equivalent to bounded mean oscillation boundedness of paracommutators, that the weak convergence of compensated quantities is equivalent to vanishing mean oscillation compactness of paracommutators, that the Schatten–von Neumann Sp-property is a natural generalization of vanishing mean oscillation compactness, that the theory of paracommutators provides a ready-made tool for Sp-regularity of compensated quantities, and that the cut-off phenomenon of paracommutators exactly coincides with the vanishing moment properties of the compensated quantities. Some further examples are given.

In this paper we study the second-order parabolic equation

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in a domain [0,T]×ℝd ⊂ ℝd+1, where a = (aij)di,j=1 is matrix of bounded measurable coefficients, b = (bj)dj=1, and bˆ = (bˆj)dj=1 are measurable (in general, singular) vector fields, V is a measurable potential, T is a fixed positive number, and ∂tu = ∂u/∂t, and we employ the notation

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We introduce a new class of coefficients in the lower-order terms for which we prove the existence and the uniqueness of the weak fundamental solution, and for this we derive Gaussian upper and lower bounds.

The compression theorem is used to prove results for equivariant configuration spaces that are analogous to the well-known non-equivariant results of May, Milgram and Segal.

Let [ ] be the open unit disk in the complex plane [Copf ]. The Bergman space L2a([ ]) is the Hilbert space of analytic functions f in [ ] such that

formula here

where dA is the normalized area measure on [ ]. If

formula here

are two functions in L2a([ ]), then the inner product of f and g is given by

formula here

We study multiplication operators on L2a([ ]) induced by analytic functions. Thus for φ ∈ H ∞([ ]), the space of bounded analytic functions in [ ], we define

formula here

by

formula here

It is easy to check that Mφ is a bounded linear operator on L2a([ ]) with

formula here

It is shown that if a plane of PG(3,q), q even, meets an ovoid in a conic, then the ovoid must be an elliptic quadric. This is proved by using the generalized quadrangles T2([Cscr ]) ([Cscr ] a conic), W(q) and the isomorphism between them to show that every secant plane section of the ovoid must be a conic. The result then follows from a well-known theorem of Barlotti.

The tangent developables of space curves are classified locally and topologically. It turns out that only seven different tangent developables arise, except for very degenerate cases. An extension of the known classification due to Cleave, Shcherbak and Mond is given.