On the Oka principle in a Banach space,II

 Let X be one of the Banach spaces c ₀ , ℓ _p , 1≤p<∞; Ω⊂X pseudoconvex open, a holomorphic Banach vector bundle with a Banach Lie group G ^* for structure group. We show that a suitable Runge-type approximation hypothesis on X, G ^* (which we also prove for G ^* a solvable Lie group) implies the vanishing of the sheaf cohomology groups H ^q (Ω, 𝒪 ^E ), q≥1, with coefficients in the sheaf of germs of holomorphic sections of E. Further, letting 𝒪^Γ (𝒞^Γ) be the sheaf of germs of holomorphic (continuous) sections of a Banach Lie group bundle Γ→Ω with Banach Lie groups G, G ^* for fiber group and structure group, we show that a suitable Runge-type approximation hypothesis on X, G, G ^* (which we prove again for G, G ^* solvable Lie groups) implies the injectivity (and for X=ℓ₁ also the surjectivity) of the Grauert–Oka map H ¹ (Ω, 𝒪^Γ)→H ¹ (Ω, 𝒞^Γ) of multiplicative cohomology sets. Received: 1 March 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 32L20, 32L05, 46G20 RID="*" ID="*" Kedves Laci Móhan kisfiamnak. RID="*" ID="*" To my dear little Son     

On the Cohen–Lusk theorem

Let G be a finite group and X be a G-space. For a map f: X → ℝ ^m , the partial coincidence set A(f, k), k ≤ |G|, is the set of points x ∈ X such that there exist k elements g ₁,…, g _k of the group G, for which f(g ₁ x) = ⋅⋅⋅ = f(g _k x) holds. We prove that the partial coincidence set is nonempty for G = ℤ _p ⁿ under some additional assumptions. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 61–67, 2007.     

Mackey functors and bisets

For any finite group G, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.     

Order of a function on the Bruschlinsky group of a two-dimensional polyhedron

Homotopy classes of mappings of a compact polyhedron X to the circle T form an Abelian group B(X), which is called the Bruschlinsky group and is cananically isomorphic to H ¹ (X; ℤ), Let L be an Abelian group, and let f: B(X) → L be a function. One says that the order of f does not exceed r if for each mapping a: X → T the value f([a]) is ℤ-linearly expressed via the characteristic function I _r (a): (X × T) ^r → ℤ of (Γ _a ) ^r , where Γ _a ⊂ X × T is the graph of a. The (algebraic) degree of f is not greater than r if the finite differences of f of order r + 1 vanish. Conjecturally, the order of f is equal to the algebraic degree of f. The conjecture is proved in the case where dim X ≤ 2. Bibliography: 1 title.     

On geometric properties of functors of positive-homogenous and semiadditive functionals

We investigate the functor OH of positive-homogenous functionals and the functor OS of semiadditive functionals. We prove that OH(X) is an absolute retract if and only if X is an open-generated compactum, and OS(X) is an absolute retract if and only if X is an opengenerated compactum of weight ≤ ω₁. We investigate the softness of mappings of multiplication of monads generated by these functors.     

Pathwidth of Planar and Line Graphs

 We prove that for every 2-connected planar graph the pathwidth of its geometric dual is less than the pathwidth of its line graph. This implies that pathwidth(H)≤ pathwidth(H ^*)+1 for every planar triangulation H and leads us to a conjecture that pathwidth(G)≤pathwidth(G ^*)+1 for every 2-connected graph G. Received: May 8, 2001 Final version received: March 26, 2002 RID="*" ID="*" I acknowledge support by EC contract IST-1999-14186, Project ALCOM-FT (Algorithms and Complexity - Future Technologies) and support by the RFBR grant N01-01-00235. Acknowledgments. I am grateful to Petr Golovach, Roland Opfer and anonymous referee for their useful comments and suggestions.     

On the Oka principle in a Banach space,I

 Let X be a complex Banach space with a countable unconditional basis, Ω⊂X pseudoconvex open, G a complex Banach Lie group. We show that a Runge–type approximation hypothesis on X, G (which we also prove for G a solvable Lie group) implies that any holomorphic cocycle on Ω with values in G can be resolved holomorphically if it can be resolved continuously. Received: 1 March 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 32L05, 32E30, 46G20 RID="*" ID="*" Kedves Szímuskának. RID="*" ID="*" To my dear Wife.     

The functor <Emphasis Type="Italic">A</Emphasis><Superscript>min</Superscript> for (<Emphasis Type="Italic">p</Emphasis> − 1)-cell complexes and EHP sequences

Let X be a co-H-space of (p − 1)-cell complex with all cells in even dimensions. Then the loop space ΩX admits a retract Ā ^min(X) that is the evaluation of the functor Ā ^min on X. In this paper, we determine the homology H _*(Ā ^min(X)) and give the EHP sequence for the spaces Ā ^min(X).     

三角范畴和Abel范畴的Torsion理论

本文主要研究了三角范畴在Abel化过程中torsion理论的保持问题.利用三角范畴的coherent函子范畴是Abel范畴,证明了T的coherent函子范畴A(T)是A(D)的thick子范畴;若(X,Y)是D的torsion理论,且D=X*Y的扩张是可裂的,那么(A(X),A(Y))是A(D)的torsion理论.     

On the Schur multipliers of finite p-groups of given coclass

In this paper we obtain bounds for the order and exponent of the Schur multiplier of a p-group of given coclass. These are further improved for p-groups of maximal class. In particular, we prove that if G is p-group of maximal class, then |H ₂(G, ℤ)| < |G| and expH ₂(G, ℤ) ≤ expG. The bound for the order can be improved asymptotically.     

p-Group actions on homology spheres

When an arbitraryp-groupG acts on a _n -homologyn-sphereX, it is proved here that the dimension functionn:S(G)(S(G) is the set of subgroups ofG), defined byn(H)=dimX ^H, (dim here is cohomological dimension) is realised by a real representation ofG, and that there is an equivariant map fromX to the sphere of this representation. A converse is also established.     

Rational Mackey functors for compact lie groups I

A Mackey functor M is a structure analogous to the representationring functor H R(H) encoding good formal behaviour under inductionand restriction. More explicitly, M associates an abelian groupM(H) to each closed subgroup H of a fixed compact Lie groupG, and to each inclusion K H it associates a restriction map and an induction map . This paper gives an analysis of thecategory of Mackey functors M whose values are rational vectorspaces: such a Mackey functor may be specified by giving a suitablycontinuous family consisting of a Q ₀(W_G(H))-module V(H) foreach closed subgroup H with restriction maps V(K) V(K) wheneverK is normal in K and K/K is a torus (a ‘continuous Weyl-toralmodule’). We show that the category of rational Mackeyfunctors is equivalent to the category of rational continuousWeyl-toral modules. In Part II this will be used to give analgebraic analysis of the category of rational Mackey functors,showing in particular that it has homological dimension equalto the rank of the group. 1991 Mathematics Subject Classification:19A22, 20C99, 22E15, 55N91, 55P42, 55P91.     

Onsager-Machlup functional for the fractional Brownian motion

 Let X be the solution of the stochastic differential equation where B ^H is a fractional Brownian motion with Hurst parameter H. In this paper we compute the Onsager-Machlup functional of X for the supremum norm and H?lder norms of order β with in the case and for H?lder norms of order β with when . Received: 16 July 2001 / Revised version: 12 March 2002 / Published online: 10 September 2002     

Weighted Bergman spaces¶associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H _ℂ\G _ℂ. Then there exist certain G-invariant Stein subdomains Ξ of H _ℂ\G _ℂ. The Haar measure on H _ℂ\G _ℂ gives rise to a G-invariant measure on Ξ. With respect to this measure one can define the Bergman space B ²(Ξ) of square integrable holomorphic functions on Ξ. The group G acts unitarily on the Hilbert space B ²(Ξ) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B ²(Ξ), i.e., the disintegration formula for the left regular representation into irreducibles. Received: Received: 23 November 1998     

The Littlewood-Gowers problem

The paper has two main parts. To begin with, suppose that G is a compact abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions ƒ ∈ L ²(G). We prove an analogous result for functions ƒ ∈ A(G), where A(G) is the space endowed with the norm , and generalize this to the approximate Fourier transform on Bohr sets. As an application of the first part of the paper, we improve a recent result of Green and Konyagin. Suppose that p is a prime number and A ⊂ ℤ/pℤ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/3−ɛ; we improve this to ‖χ _A ‖ _A(ℤ/pℤ) ≫ ɛ (log p)^1/2−ɛ. To put this in context, it is easy to see that if A is an arithmetic progression, then ‖χ _A ‖ _A(ℤ/pℤ) ≪ log p.     

Maximal Abelian Subalgebras of the Hyperfinite Factor,Entropy and Ergodic Theory

Given a free ergodic action of a discrete abelian group G on a measure space (X, μ), the crossed product L ^∞ (X, μ)⋊ G contains two distinguished maximal abelian subalgebras. We discuss what kind of information about the action can be extracted from the positions of these two subalgebras inside the crossed product algebra. Received February 24, 2002, Accepted August 5, 2002     

On Vertex-Disjoint Complete Bipartite Subgraphs in a Bipartite Graph

 We show that, if G=(X,Y;E) is a bipartite graph with |X|=|Y|=4s and δ(G)≥4s−3 for any integer s≥2, then G contains four vertex-disjoint copies of K _s,s. This constitutes a partial answer to a conjecture in [4]. Received: March 27, 1996 Revised: March 7, 1997     

Duality for contravariant functors on Banach spaces

The following notion of duality is studied: IfG is a contravariant functor on Ban, then . We derive the following results: The^x-reflexive functors are exactly the maximal subfunctors ofH(.,A) with reflexiveA, G ^x is^x-reflexive if and only ifG(I) is a reflexive Banach space,G ^x=(G_e)^x, andG ^x(X)=G_e(X) ifX has the metric approximation property. the last result has the consequence that for the tensor product of functors holds, ifX has the m. A. P. Dedicated to Prof. Dr. E. Hlawka on the occasion of his 60th birthday     

Countable structures with a fixed group of automorphisms

We prove that, given a countable groupG, the set of countable structures (for a suitable languageL)U _G whose automorphism group is isomorphic toG is a complete coanalytic set and ifG ≄H thenU _G is Borel inseparable fromU _H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures forL whose automorphism group is isomorphic to ℤ _p ^ℕ ,p a prime number, is Π ₁ ¹ &σ ₁ ¹ -complete.