Bundle-connection pairs and loop group representations

LetM be a connected differentiable manifold. Denote by Ω _m (M) the space ofH ¹-loops based at a fixed pointm∈M. Associated to Ω _m (M) one has , the group of unparameterized loops. Given a bundle-connection pair (E,∇) overM with fiber the finite-dimensional vector spaceV and structure groupG⊂GL(V) we get (up to equivalence) a smooth representation of inG given by the parallel transport operatorP ^∇. It is possible to find in the literature several versions of the converse theorem, namely: all (smooth) representations of arise in the above described way from a bundle-connection pair. It is shown in the present paper that the correct setting for this theorem is the theory of induced representations for groupoids. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 503–518, April, 1997.     

Rational curves on moduli spaces of vector bundles

We identify the spaces Hom_i(ℙ¹,M) fori = 1, 2, whereM is the moduli space of vector bundles of rank 2 and determinant isomorphic to ,x ₀ ∈X, on a compact Riemann surface of genusg ≥ 2.     

On a characterization of analytic compactifications for ℂ* × ℂ*

Let M be a minimal compact surface, let Γ ⊂ M be a compact analytic sub-variety. Assume that X:= M \ Γ is Stein. Then we will show that X admits algebraic compactifications M _i (resp. non algebraic compactifications $ \mathbb{M}_i $ \mathbb{M}_i ) which are not birationally equivalent (resp. not bimeromorphically equivalent) iff X is biholomorphic to      

Packing,covering and decomposing of a complete uniform hypergraph into delta-systems

Anh-uniform hypergraph generated by a set of edges {E ₁,...,E _c} is said to be a delta-system Δ(p,h,c) if there is ap-element setF such that ∇F|=p andE _i⌢E _j=F,∀i≠j. The main result of this paper says that givenp, h andc, there isn ₀ such that forn≥n ₀ the set of edges of a completeh-uniform hypergraphK _n ^h can be partitioned into subsets generating isomorphic delta-systems Δ(p, h, c) if and only if . This result is derived from a more general theorem in which the maximum number of delta-systems Δ(p, h, c) that can be packed intoK _n ^h and the minimum number of delta-systems Δ(p, h, c) that can cover the edges ofK _n ^h are determined for largen. Moreover, we prove a theorem on partitioning of the edge set ofK _n ^h into subsets generating small but not necessarily isomorphic delta-systems.     

Euler operator and homogeneous hida distributions

Let (S)⊄L ²(S′(∔),μ)⊄(S)^* be the Gel'fand triple over the white noise space (S′(∔),μ). Let (e _n ,n>-0) be the ONB ofL ²(∔) consisting of the eigenfunctions of the s.a. operator . In this paper the Euler operator Δ _E is defined as the sum , where ∂ _i stands for the differential operatorD _{e i}. It is shown that Δ _E is the infinitesimal generator of the semigroup (T _t ), where (T _t ϕ)(x)=ϕ(e ^t x) for ϕ∈(S). Similarly to the finite dimensional case, the λ-order homogeneous test functionals are characterized by the Euler equation: Δ _Eϕ =λ_ϕ. Via this characterization the λ-order homogeneous Hida distributions are defined and their properties are worked out. Supported by the National Natural Science Foundation of China.     

On spaces of p-adic vector measures

LetX be a Hausdorff zero-dimensional topological space,K(X) the algebra of all clopen subsets of X, E a Hausdorff locally convex space over a non-Archimedean valued field and C _b (X) the space of all bounded continuous -valued functions on X. The space M(K(X),E), of all bounded finitely-additive measures m: K(X) → E, is investigated. If we equip C _b (X) with the topologies β _o , β, β _u , τ _b or β _ob , it is shown that, for E (compete, the corresponding spaces of continuous linear operators from C _b (X) to E (are algebraically isomorphic to certain subspaces of M(K(X),E). The text was submitted by the author in English.     

Compactness of the congruence group of measurable functions in several variables

We solve a problem, which appears in functional analysis and geometry, on the group of symmetries of functions of several arguments. Let be a measurable function defined on the product of finitely many standard probability spaces (X_i, , μ_i), 1 ≤ i ≤ n, that takes values in any standard Borel space Z. We consider the Borel group of all n-tuples (g₁, ..., g_n) of measure preserving automorphisms of the respective spaces (X_i, , μ_i) such that f(g₁ x ₁, ..., g_nx_n) = f(x₁, ..., x_n) almost everywhere and prove that this group is compact, provided that its “trivial” symmetries are factored out. As a consequence, we are able to characterize all groups that result in such a way. This problem appears with the question of classifying measurable functions in several variables, which was solved by the first author but is interesting in itself. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 57–67.     

Banach manifolds of algebraic elements in the algebra $$\mathcal{L}$$ ( H) of bounded linear operatorsof bounded linear operators

Given a complex Hilbert space H, we study the manifold of algebraic elements in . We represent as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C^*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C^*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M, and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection. Supported by Ministerio de Educación y Cultura of Spain, Research Project BFM2002-01529.     

Amalgamated free products of weakly rigid factors and calculation of their symmetry groups

We consider amalgamated free product II₁ factors M = M _1*B M _2*B … and use “deformation/rigidity” and “intertwining” techniques to prove that any relatively rigid von Neumann subalgebra Q ⊂ M can be unitarily conjugated into one of the M _i ’s. We apply this to the case where the M _i ’s are w-rigid II₁ factors, with B equal to either C, to a Cartan subalgebra A in M _i , or to a regular hyperfinite II₁ subfactor R in M _i , to obtain the following type of unique decomposition results, àla Bass–Serre: If M = (N _1 * CN_2*C …) ^t , for some t > 0 and some other similar inclusions of algebras C ⊂ N _i then, after a permutation of indices, (B ⊂ M _i ) is inner conjugate to (C ⊂ N _i ) ^t , for all i. Taking B = C and , with {t _i } _i⩾1 = S a given countable subgroup of R ₊ ^*, we obtain continuously many non-stably isomorphic factors M with fundamental group equal to S. For B = A, we obtain a new class of factors M with unique Cartan subalgebra decomposition, with a large subclass satisfying and Out(M) abelian and calculable. Taking B = R, we get examples of factors with , Out(M) = K, for any given separable compact abelian group K.     

On the Matlis duals of local cohomology modules

Let ( ) be a commutative Noetherian local ring with non-zero identity, an ideal of R and M a finitely generated R-module with . Let D(–) := Hom _R (–, E) be the Matlis dual functor, where is the injective hull of the residue field . We show that, for a positive integer n, if there exists a regular sequence and the i-th local cohomology module H ⁱ _a (M) of M with respect to is zero for all i with i > n then The author was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 85130023). Received: 9 August 2006     

Comparing the relative volume with a revolution manifold as a model

Given a pair (P, M), whereM is ann-dimensional connected compact Riemannian manifold andP is a connected compact hypersurface ofM, the relative volume of (P, M) is the quotient volume(P)/volume(M). In this paper we give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature ofM and the mean curvature ofP, with respect to that of a model pair where ℳ is a revolution manifold and a “parallel” of ℳ. Work partially supported by a DGICYT Grant No. PB91-0324.     

Sumsets in difference sets

We study some properties of sets of differences of dense sets in ℤ² and ℤ³ and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results.

Generators of the unoriented bordism ring which are fibered over products of projective spacesRP(2 r −1)

For each positive integerk≦∞ we construct a family {M _k ⁿ } of generators of the unoriented bordims ring. The manifoldsM _k ⁿ are total spaces of fiber bundles whose base spaces are high-dimensional products of projective spaces wherer _i≦k. The fibers are themselves iterated projective bundles with maximal fiber dimension two. In the special casek=3 we obtain generatorsM ₃ ⁿ which admit approximately 7/8·n pointwise linearly independent vector fields. This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag     

The Hardy-Littlewood theorem for trigonometric series with generalized monotone coefficients

Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ ( $ \tfrac{1} {2} Earlier we introduced a continuous scale of monotony for sequences (classes M _α, α ≥ 0), where, for example, M ₀ is the set of all nonnegative vanishing sequences, M ₁ is the class of all nonincreasing sequences, tending to zero, etc. In addition, we extended several results obtained for trigonometric series with monotone convex coefficients onto more general classes. The main result of this paper is a generalization of the well-known Hardy—Littlewood theorem for trigonometric series, whose coefficients belong to classes M _α, where α ∈ (, 1). Namely, the following assertion is true. Let α ∈ (, 1), < p < 2, a sequence a ∈ M _α, and . Then the series cos nx converges on (0,2π) to a finite function f(x) and f(x) ∈ L _p (0,2π). Original Russian Text ? M.I. D’yachenko, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, 2008, No. 5, pp. 38–47.     

Change point estimation by local linear smoothing under a weak dependence condition

We consider a change-point problem in regression estimation. Observations (X _i , Y _i ), i = 1, ..., n, are governed by the model Y _i = m(X _i ) + σ(X _i )ɛ _i , where the (ɛ _i ) _i∈ℤ are independent and identically distributed and independent of (X _i ) _i∈ℤ. The latter sequence satisfies a weak dependence condition proposed by Dedecker and Prieur [4]. We essentially study the basic situation, where the regression function has a unique change point. The construction of the jump estimate process, t → (t), is based on local linear regression. Under a positivity condition regarding the asymmetric kernel involved, we prove the convergence of a local dilated-rescaled version of (t) to a compound Poisson process with an additional drift. We also derive asymptotic normality results.      

Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field F of characteristic zero, a Lie algebra B（G） of Block type is defined with basis {Lα,i｜ α∈G, i∈Z＋} and relations [Lα,i, Lβ,j] = （β-α）Lα＋β,i＋j＋（αj-βi）Lα＋β,Lα＋β,i＋j-1.It is proved that an irreducible highest weight B（Z）-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B（G）0^＊（the dual space of B（G）0 = span{L0,i｜i∈Z＋}）, a Verma B（G）-module M（∧,λ） is defined, and the irreducibility of M（A,λ） is completely determined.     

Everywhere H?lder continuity of the spatial gradient of weak solutions to nonlinear parabolic systems

We consider weak solutions to the parabolic system ∂u ⁱ∂t−D _α A _i ^α (∇u)=B _i(∇u) in (i=1,...,) (Q=Ω×(0,T), R ⁿ a domain), where the functionsB _i may have a quadratic growth. Under the assumptionsn≤2 and ∇u ɛL _loc ^4+δ (Q; R ^nN ) (δ>0) we prove that ∇u is locally H?lder continuous inQ.     

ℓ<Superscript>1</Superscript>-spreading models in mixed Tsirelson spacemodels in mixed Tsirelson space

Suppose that (F _n ) _n=1 ^∞ is a sequence of regular families of finite subsets of ℝ and (θ _n ) _n=1 ^∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson spaceT[(θ _n ,F _n ) _n=1 ^∞ ] is the completion ofc ₀₀ with respect to the implicitly defined norm , where the last supremum is taken over all sequences (E _i ) _i=1 ^k in [ℕ]^<∞ such that maxE _i<minE _i +1 and . Necessary and sufficient conditions are obtained for the existence of higher order ℓ¹-spreading models in every subspace generated by a subsequence of the unit vector basis ofT[(θ _n ,F _n ) _n=1 ^∞ ].     

Some properties and applications of weakly equicompact sets

Let X and Y be Banach spaces. A set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n) in X, there exists a subsequence (x _k(n)) so that (Tx_k(n)) is uniformly weakly convergent for T ∈ M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X ↩̸ ℓ₁, of spaces X such that B _X* is weak* sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Received: 5 July 2006