On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Number of quasimodes of “bouncing-ball” type

A new two-scale expansion is proposed for eigenfunctions of bouncing-ball type and the corresponding eigenvalues of the Laplace operator with the Dirichlet condition in a domain of the plane. The eigenfunctions are concentrated in a neighborhood of a stable diameter of the domain and are numbered with two indices (p, q) where p is the number of longitudinal nodes and q the number of nodes in a direction orthogonal to the diameter. The validity of the asymptotic expansions is guaranteed for 0 q const p^–1 and >0 as p +.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 172–182, 1981.     

Lie bialgebras of generalized Witt type

In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W(?)W) is trivial.     

Theorems of Denjoy–Wolff type

Using the Kobayashi distance, we first establish a version of the Denjoy–Wolff theorem for a bounded and strictly convex domain in ${{\mathbb{C}}^k}$ . Next, we prove analogous results for semigroups of holomorphic mappings and the resolvents of their generators. Finally, we obtain theorems of Denjoy–Wolff type for families of holomorphic retracts of the open unit ball in a complex, reflexive, and strictly convex Banach space.     

Poincaré series of Drinfeld type

Let K be the rational function field $\mathbb{F}_q (t)$ . We construct Poincaré series on the Bruhat-Tits tree of GL₂ over K _∞ and show that they generate the space of automorphic cusp forms of Drinfeld type.     

Rigidity of H–type groups

On H–type groups N the left invariant horizontal vector fields span a subbundle of the tangent bundle, called the horizontal bundle HN. Generalized contact mappings f on N are smooth mappings which preserve HN. The question is: how many such mappings exist? In the case of the Heisenberg group these are the contact mappings in the classical sense and they exist in abundance. In this paper it is shown that if the dimension of the center of N is at least three, then the generalized contact mappings are in the automorphism group of a finite dimensional Lie algebra g. The elements in are the infinitesimal generators of local one parameter subgroups of generalized contact transformations. Rigidity is defined as the property that is finite dimensional. For the case of the complexified Heisenberg group, i.e. the case when the dimension of the center of N is two, it has been shown [RR] that g is infinite dimensional. Received January 4, 2000; in final form March 20, 2000 / Published online April 12, 2001     

Geometry of D’Atri spaces of type k

A Riemannian n-dimensional manifold M is a D’Atri space of type k (or k-D’Atri space), 1 ≤ k ≤ n ? 1, if the geodesic symmetries preserve the k-th elementary symmetric functions of the eigenvalues of the shape operators of all small geodesic spheres in M. Symmetric spaces are k-D’Atri spaces for all possible k ≥ 1 and the property 1-D’Atri is the D’Atri condition in the usual sense. In this article we study some aspects of the geometry of k-D’Atri spaces, in particular those related to properties of Jacobi operators along geodesics. We show that k-D’Atri spaces for all k = 1, . . ., l satisfy that ${{\rm{tr}}(R_{v}^{k})}$ , v a unit vector in TM, is invariant under the geodesic flow for all k = 1, . . ., l. Further, if M is k-D’Atri for all k = 1, . . ., n ? 1, then the eigenvalues of Jacobi operators are constant functions along geodesics. In the case of spaces of Iwasawa type, we show that k-D’Atri spaces for all k = 1, . . ., n ? 1 are exactly the symmetric spaces of noncompact type. Moreover, in the class of Damek-Ricci spaces, the symmetric spaces of rank one are characterized as those that are 3-D’Atri.     

The κ ring of the moduli of curves of compact type

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the κ classes is studied in all genera. Relations, constructed via the virtual geometry of the moduli of stable quotients, are used to obtain minimal sets of generators. Bases and Betti numbers of the κ rings are computed. A universality property relating the higher genus κ rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps. The λ_g-formula for Hodge integrals arises as the simplest consequence.     

Convergence of rational multistep methods of Adams-Padé type

Rational generalizations of multistep schemes, where the linear stiff part of a given problem is treated by an A-stable rational approximation, have been proposed by several authors, but a reasonable convergence analysis for stiff problems has not been provided so far. In this paper we directly relate this approach to exponential multistep methods, a subclass of the increasingly popular class of exponential integrators. This natural, but new interpretation of rational multistep methods enables us to prove a convergence result of the same quality as for the exponential version. In particular, we consider schemes of rational Adams type based on A-acceptable Padé approximations to the matrix exponential. A numerical example is also provided.     

Analysis of degenerate elliptic operators of Grušin type

We analyse degenerate, second-order, elliptic operators H in divergence form on L ₂(R ⁿ  × R ^m ). We assume the coefficients are real symmetric and a ₁ H _δ  ≥ H ≥ a ₂ H _δ for some a ₁, a ₂ > 0 where

Fréchet algebras of finite type

The main objects of study in this paper are Fréchet algebras having an Arens Michael representation in which every Banach algebra is finite dimensional. We shall classify these algebras using a theorem which ensures that the image of any continuous linear map of a Fréchet space of finite type (i.e., for which the defining seminorms have a finite dimensional cokernel) into any Fréchet space is in fact closed.This work is part of the research project of the European Research Training Network Analysis and Operators, contract HPRN-CT 2000 00116, funded by the European Commission.     

Universal Padé approximants of Seleznev type

We use the chordal metric in order to approximate all meromorphic functions on \({\mathbb{C} \backslash \{0\}}\) by Padé approximants of formal power series. This is a generic universality of Seleznev type which implies Menchov type almost everywhere approximation with respect to any σ-finite Borel measure on \({\mathbb{C} \backslash \{0\}}\).     

On a model of thermoviscoelasticity of Jeffreys–Oldroyd type

In the plane case, the initial–boundary value problem for a thermoelastic medium model with a rheological relation determined by the Jeffreys–Oldroyd model is shown to be nonlocally weakly solvable. The study is based on separating the system, reducing it to an operator equation, and performing an iterative process.     

Normal forms of functional-differential equations of Beardon type and Briot–Bouquet type equations in the complex domain

We study local analytic solutions of the functional-differential equation of the form \({h(\psi(z)) = b(z) h(z) h^\prime(z) + d(z)h(z)^{2}}\) which are called Beardon type functional-differential equations. All functions involved are supposed to be holomorphic in a neighbourhood of zero. Special cases are the equations f(kz) =  kf(z) f′(z) where k is a complex number, \({k \neq 0}\), and \({f(\varphi(z)) = a(z) f(z) f'(z)}\) with given \({\varphi}\) and a. The class of these equations is invariant under transformations \({h \to \alpha h, \alpha(z) \neq 0}\) for all z in a neighbourhood of zero, of the unknown function and \({z \to T(z)}\) of the argument z. In particular, we are interested to know under which conditions a Beardon type functional-differential equation can be transformed to the simplified (normal form) \({h(kz) = k h(z) h'(z) + c(z) h(z)^2}\) where \({k \in \mathbb {C} \backslash\left\{0\right\}}\). We solve this normal form by another transfomation to a so-called Briot–Bouquet type functional-differential equation.     

An average of special values of Dirichlet series of Rankin–Selberg type

The Ramanujan Journal - For an integer n and a Dirichlet character $$\xi $$ modulo N, we denote by $$\mathcal {S}_n(N,\xi )$$ the space of cusp forms of weight n with respect to $$\varGamma _0(N)$$...