Quaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space = Sp(n+1)/Sp(1) × Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry on modelled on . The integrability structure is shown to be geometrically encoded by a Poisson–Nijenhuis structure and a symplectic operator.

We characterize in terms of Darboux transformations the spaces in the Segal–Wilson rational Grassmannian, which lead to commutative rings of differential operators having coefficients which are rational functions of ex. The resulting subgrassmannian is parametrized in terms of trigonometric Calogero–Moser matrices.

Let G be a finite group. The symmetric genus σ(G) is the minimum genus of any Riemann surface on which G acts faithfully. We show that if G is a group of order 2m that has symmetric genus congruent to 3 (mod 4), then either G has exponent 2m−3 and a dihedral subgroup of index 4 or else the exponent of G is 2m−2. We then prove that there are at most 52 isomorphism types of these 2-groups; this bound is independent of the size of the 2-group G. A consequence of this bound is that almost all positive integers that are the symmetric genus of a 2-group are congruent to 1 (mod 4).

In [18] Shioda proved that the space of holomorphic 2-forms on a certain type of elliptic surface is canonically isomorphic to the space of modular forms of weight three for the associated Fuchsian group. Later, Hunt and Meyer [6] made an observation that the holomorphic 2-forms on a more general elliptic surface should in fact be identified with mixed automorphic forms associated to an automorphy factor of the form

for z in the Poincaré upper half plane ℋ, g = and χ(g) = , where g is an element of the fundamental group Γ⊂PSL(2, R) of the base space of the elliptic fibration, χ-Γ→SL(2, R) the monodromy representation, and w: ℋ→ℋ the lifting of the period map of the elliptic surface.

We denote by ‖…‖ the distance to the nearest integer. Let α and β be real. W. M. Schmidt [5] proved that for ε > 0 and N>c1(ε) there is a natural number n such that

This extends a theorem of H. Heilbronn [4] and also sharpens a theorem of H. Davenport [3].

We characterize real hypersurfaces of type (A) and ruled real hypersurfaces in a complex projective space in terms of two φ-invariances of their shape operators, and give geometric meanings of these real hypersurfaces by observing their some geodesics.

The main problem investigated in this paper is the following. Assume that we are given a convergent projective system of topological measure spaces ordered by ordinals. When does there exist a consistent system of liftings (densities, linear liftings) on the projective system converging to a lifting (density, linear lifting) on the limit space. We look mainly for strong or strong completion Baire liftings. We reduce the problem to the question about the existence of strong liftings being inverse images of other strong liftings under measure preserving mappings (Proposition 2.4) and then we adapt a condition applied earlier by A. and C. Ionescu Tulcea [14] to get a strong lifting for an arbitrary measure on a product space (Theorem 2.7). In this way we get some results (see Theorems 2.7, 5.3, 5.7, 6.4 and 6.5) extending the well known achievements of A. and C. Ionescu Tulcea [14] and Fremlin [9].

The application of projective limits allows us to carry over results obtained earlier only for product spaces (see e.g. [23], [18], [19], [20], [21]) to more general classes of topological probability spaces. In particular, we can extend the class of spaces for which there is a positive answer to a problem of J. Kupka [17] concerning the permanence of the strong lifting property under the formation of products (see Theorem 6.5).

For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p. We give a corollary that settles Iizuka’s conjecture for the case n=1 and p>2.

Let G be a finite group. The symmetric crosscap number \tilde \sigma (G) is the minimum topological genus of any compact non-orientable surface (with empty boundary) on which G acts effectively. We first survey some of the basic facts about the symmetric crosscap number; this includes relationships between this parameter and others. We obtain formulas for the symmetric crosscap number for three families of groups, the dicyclic groups, the abelian groups with most factors in the canonical form isomorphic to Z_(2), and the hamiltonian groups with no odd order part. We also determine \tilde \sigma (G) for each group G with order less than 16. The groups with symmetric crosscap numbers 1 and 2 have been classified. We show here that there are no groups with \tilde\sigma=3; this affirms a conjecture of Tucker.

In the study of connected partially ordered spaces a problem of fundamental interest is to determine sufficient conditions to ensure the existence of chains (i.e., simply ordered subsets) which are connected. Recently [5] R. J. Koch proved that, if X is a compact Hausdorff space with continuous partial order (i.e., the partial order has a closed graph), if L(x) = {y: y ≦ x} is connected for each x ∈ X, and if X has a zero (i.e., an element 0 such that 0 ≦ x for all x ∈ X), then each element of X lies in a connected chain containing zero. It is easy to find simple examples which show that this result is false if X is assumed only to be locally compact. However, if it is assumed that the partial order is that of a topological lattice then the existence of such chains can be shown by elementary methods. This solves a problem which was proposed in [3].

Let A be the product of an abelian variety and a torus over a number field K, and let $$m \ge 2$$ be a square-free integer. If $\alpha \in A(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of K such that the reduction $(\alpha \bmod \mathfrak p)$ is well defined and has order coprime to m. This set admits a natural density, which we are able to express as a finite sum of products of $\ell$ -adic integrals, where $\ell$ varies in the set of prime divisors of m. We deduce that the density is a rational number, whose denominator is bounded (up to powers of m) in a very strong sense. This extends the results of the paper Reductions of points on algebraic groups by Davide Lombardo and the second author, where the case m prime is established.

The Hilbert space methods in the theory of biholomorphic mappings were applied and developed by S. Bergman [1, 2]. In this approach the central role is played by the Hilbert space L2H(D) consisting of all functions which are square integrable and holomorphic in a domain D ⊂ ℂN. A biholomorphic mapping φ:D ⃗ G induces the unitary mapping Uφ:L2H(G) ⃗ L2H(D) defined by the formula

Here ∂φ/∂z denotes the complex Jacobian of φ. The mapping Uϕ is useful, since it permits to replace a problem for D by a problem for its biholomorphic image G (see for example [11], [13]). When ϕ is an automorphism of D we obtain a unitary operator Uϕ on L2H(D).

Let {Gr,i} be a sequence of r-generator subgroups of U(1,n; ℂ) and Gr be its algebraic limit group. In this paper, two algebraic convergence theorems concerning {Gr,i} and Gr are obtained. Our results are generalisations of their counterparts in the n-dimensional sense-preserving Möbius group.

A generalization is given of a theorem of F. Brafman [1] on the equivalence of generating relations for a certain sequence of functions. The main result, contained in Theorem 2 below, may be applied to several special functions including the classical orthogonal polynomials such as Hermite, Jacobi (and, of course, Legendre and ultraspherical), and Laguerre polynomials.

Let G be a finite abelian group of rank m, M an oriented compact connected surface, and F(G, M) the set of all orientation preserving free G-actions on M. Two actions φ1, φ2εF(G, M) are equivalent if there exists an orientation preserving homeomorphism h of M such that

hφ1(f) for all f ε G.

Since Azumaya introduced the notion of A-injectivity in 1974, several generalizations have been investigated by a number of authors. We introduce some more generalizations and discuss their connection to the previous ones.

We construct three p-adic L-functions attached to the symmetric square of a modular elliptic curve. Following a calculation of Perrin-Riou for one of these functions, we compute the derivative of the p-adic L-function associated to the square of the non-unit root of Frobenius at p. This generalises Greenberg's notion of [Lscr ]-invariant to these three-dimensional Galois representions.

Let be an n+ 1-dimensional, complete simply connected Riemannian manifold of constant sectional curvature c and We consider the function r(·) = d(·, P0) where d stands for the distance function in and we denote by grad r the gradient of The position vector (see [1]) with origin P0 is defined as where ϕ(r)equals

r, if c = 0, c< 0 or c <0 respectively.

In this paper we prove a theorem in Operational Calculus and use it to evaluate a few infinite integrals involving Legendre, Bessel and E-functions. We write

when

and

when

(2) is a generalisation of (1) as given by Meijer [2] and it reduces to (1) when v = ±½ by virtue of the relation

For a given (d−1)-dimensional simplicial complex Γ, we denote its h-vector by h(Γ)=(h0(Γ),h1(Γ),. . .,hd(Γ)) and set h−1(Γ)=0. The known Swartz equality implies that if Δ is a (d−1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ i ≤ d, the inequality ihi(Δ)+(d−i+1)hi−1(Δ) ≥ 0 holds true. In this paper, by using these inequalities, we give a simple proof for a result of Terai (N. Terai, On h-vectors of Buchsbaum Stanley–Reisner rings, Hokkaido Math. J. 25(1) (1996), 137–148) on the h-vectors of Buchsbaum simplicial complexes. We then generalize the Swartz equality (E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18(3) (2004/05), 647–661), which in turn leads to a generalization of the above-mentioned inequalities for Cohen–Macaulay simplicial complexes in co-dimension t.