Symmetric inverse topological semigroups of finite rank ≤ <Emphasis Type="Italic">n</Emphasis>

We establish topological properties of the symmetric inverse topological semigroup of finite transformations of the rank ≤ n. We show that the topological inverse semigroup is algebraically h -closed in the class of topological inverse semigroups. Also we prove that a topological semigroup S with countably compact square S×S does not contain the semigroup for infinite cardinal λ and show that the Bohr compactification of an infinite topological symmetric inverse semigroup of finite transformations of the rank ≤ n is the trivial semigroup.     

Cohomology of a flag variety as a Bethe algebra

We interpret the equivariant cohomology H_GLn ^*H_{GL_n }^* (ℱ _λ ,ℂ) of a partial flag variety ℱ _λ parametrizing chains of subspaces 0 = F ₀ ⊂ F ₁ ⊂ … ⊂ F _N = ℂ ⁿ , dimF _i /F _i−1 = λ _i , as the Bethe algebra of the -weight subspace of a [t]-module .     

On Attractors for Multivalued Semigroups Defined by Generalized Semiflows

In this paper we extend results from Semigroup Theory on existence and characterization of attractors in order to include multivalued semigroups T(t) defined by generalized semiflows . In particular we show that, if is continuous, possesses a Lyapunov function, and has a global attractor which is maximal compact invariant, then  =  W ^u (Z()), where Z() is the stationary solutions set and W ^u (Z()) is the unstable set of Z(). We introduce the -attractor concept which does not enjoy any uniformity on time of attraction and we prove, under suitable conditions, that the global -attractor is the set of asymptotic states described by Z(). Jacson Simsen is supported by CAPES-Brazil.     

Two methods for avoiding hereditary completeness

A family of vectors of a Hubert space H is said to be hereditarily complete if it posses a biorthogonal family {x_n′;n≥1}((x_n,x_k′)=δ_nk) and if any elementx, xε H can be reconstructed in terms of the component of its Fourier series, i.e., x∈V((x,x′_n)x_n:n≥1),∀x∈H. In the paper we indicate two simple methods for constructing nonhereditary complete minimal families having a total biorthogonal family, which just not long ago has caused well-known difficulties (see Ref. Zh. Mat., 1975, 7B802). The first method consists in the fact that a given pair of biorthogonal families Y, Y′ of the space H′,H′⊂H is represented as the projection of the families of the same type but already complete in H.. Clearly, in this case cannot be hereditarily complete. The second method consists in considering linear deformation _n :n⩾1 of the orthogonal bases_n: n⩾1; here A is an unbounded operator of a special type. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 183–188, 1976.     

Solutions of the cauchy problem for the polyharmonic equation (uniqueness and approximation)

One finds conditions which ensure the possibility of weighted mean-square approximation of a vector-function defined on the boundary of an n-dimensional domain by vector-functions of the form , where u is, the solution of the equation Δ^m _u=0 in while∂/∂v denotes differentiation along the normal. The weight function is continuous and positive everywhere on with the point whose relative neighborhood is contained in some (n-1)-dimensional plane. The solution of this approximation problem is closely related with a certain uniqueness theorem for the solution of the Cauchy problem for the polyharmonic equation, also proved in the paper. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 65, pp. 164–171, 1976.     

On the regularity of domains satisfying a uniform hour–glass condition and a sharp version of the Hopf–Oleinik boundary point principle

We prove that an open, proper, nonempty subset of _boxclose^n{\mathbb{R}}^n is a locally Lyapunov domain if and only if it satisfies a uniform hour-glass condition. The limiting cases are as follows: Lipschitz domains may be characterized by a uniform double cone condition, and domains of class may be characterized by a uniform two-sided ball condition. We discuss a sharp generalization of the Hopf–Oleinik boundary point principle for domains satisfying an interior pseudoball condition, for semi-elliptic operators with singular drift and obtain a sharp version of the Hopf strong maximum principle for second order, nondivergence form differential operators with singular drift. Bibliography: 66 titles. Illustrations: 7 figures.     

Gap, Excess and Bornological Convergence

Let be the nonempty subsets of a metric space 〈X, d〉. Some classical convergences in - such as convergence in Hausdorff distance, Attouch-Wets convergence and Wijsman convergence - have been shown to be compatible with the weak topology on induced by all gap and excess functionals with fixed left argument ranging in some bornology. Here we consider an arbitrary ideal of subsets of X and compare the gap and excess topology so generated with the corresponding convergence defined in terms of truncations by elements of the ideal. Dedicated to the memory of Flora Daniel.     

Hankel-Schur multipliers and multipliers of the space H1

In this paper there is given a sufficient condition for a Hankel matrix _F to belong to the space of Schur multipliers of all bounded operators in ² (or, what is the same, to the tensor algebra V²). It is shown that ifw is a nonnegative function on T, such that is a sequence of integers, {F_i}_j1 is a sequence of polynomials,) and, then _FV². It follows from this that under these conditions F is a multiplier of the space H¹, i.e.,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 113–119, 1984     

Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

On estimation and detection of a function of infinitely many variables

We observe an unknown function of infinitely many variables f = f(t), t = (t₁, ..., t_n, ... ) ∈, [0, 1]^∞, in the Gaussian white noise of level ε > 0. We suppose that in each variable there exists a 1-periodical σ-smooth extension of the function f(t) to IR ^∞. Taking a quantity σ > 0 and a positive sequence a = {a_k}, we consider the set that consists of functions f such that . We consider the cases a_k = k^α and a_k = exp(λk), α > 0, λ > 0. We would like to estimate a function f ∈ or to test the null hypothesis H₀: f = 0 against the alternatives f ∈ , where the set consists of functions f ∈ such that ∥f∥₂ ≥ r. In the estimation problem, we obtain the asymptotics (as ε → 0) of the minimax quadratic risk. In the detection problem, we study the sharp asymptotics of minimax separation rates f _ɛ ^* that provide distiguishability in the problems. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 91–113.     

Estimates for multilinear singular integral operators with nonsmooth kernels

Let (, d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we consider the behavior on × ⋯ × for the m-linear singular integral operators with nonsmooth kernels which were first introduced by Duong, Grafakos and Yan.     

Normal Subgroups of B u Aut(Ω)

Aut(Ω) denotes the group of all order preserving permutations of the totally ordered set Ω, and if e ≤ u ∈ Aut(Ω), then B _u Aut(Ω) denotes the subgroup of all those permutations bounded pointwise by a power of u. It is known that if Aut(Ω) is highly transitive, then Aut(Ω) has just five normal subgroups. We show that if Aut(Ω) is highly transitive and u has just one interval of support, then B _u Aut(Ω) has normal subgroups, and there is a certain ideal of the lattice of subsets of (), the power set of the integers, such that the lattice of normal subgroups of every such Aut(Ω) is isomorphic to . To Bernhard Banaschewski on the occasion of his 80th birthday.     

Defining the integers in large rings of a number field using one universal quantifier

Julia Robinson gave a first-order definition of the set of integers in the rational numbers by a formula (∀∃∀∃)(F = 0) where the ∀-quantifiers run over a total of 8 variables and F is polynomial. We show that for a large class of number fields, not including , for every ε > 0 there exist a set of primes of natural density exceeding 1 − ε such that can be defined as a subset of the “large” subring

Venkov's reduction theory of positive-quadratic forms

We prove the absolute finiteness of the number of faces (independent of the parameter) of Venkov's reduction domain (Izv. Akad. Nauk SSSR, Ser. Mat.4, 37–52 (1940)) ofn-ary positive quadratic forms. The casen=3 is given special consideration. We study the change of the reduction domain when changes along a line segment in the space of coefficients.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 108–116, 1983.     

Nielsen equivalence classes of free abelianized extensions of groups

Let be a presentation of a group G. We obtain information about the Nielsen equivalence classes of F _n /R′ in terms of the group K ₁(ℤG) that arises in algebraic K-theory. We deduce some results about Nielsen equivalence classes and T-systems of polycyclic groups.     

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.     

Gauge group and gauge transformation in the continual theory of defects

A representation of the algebra (3)=t(3) S0(3, ) by differential Schaefer's operators is proposed, and an external algebra of (3)-valued differential forms is constructed. The requirement of local gauge invariance is formulated in the model of the (3)-valued field, which enables a group of gauge transformations of the continual theory of defects to be obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 173–184, 1991.I wish to thank V. N. Popov for his interest.     

A note on the degree conjecture for the Selberg class

The degree conjecture for the Selberg class of L-functions states that the degree d _F of every F ∈ is an integer. Moreover, it is expected that every F ∈ has polynomial Euler product, and that the degree ∂ _F of such an Euler product coincides with d _F . In this note we prove that a suitable continuity assumption on the degree d _F implies that ∂ _F = d _F for all F ∈ with polynomial Euler product.      

∞-jets of diffeomorphisms preserving orbits of vector fields

Let F be a C ^∞ vector field defined near the origin O ∈ ℝ ⁿ , F(O) = 0, and (F _t ) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝ ⁿ → ℝ ⁿ at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney W ^r topology. Then contains a subset consisting of maps of the form F _α(x)(x), where α: ℝ ⁿ → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O. We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.      

Edge offset meshes in Laguerre geometry

A mesh with planar faces is called an edge offset (EO) mesh if there exists a combinatorially equivalent mesh such that corresponding edges of and lie on parallel lines of constant distance d. The edges emanating from a vertex of lie on a right circular cone. Viewing as set of these vertex cones, we show that the image of under any Laguerre transformation is again an EO mesh. As a generalization of this result, it is proved that the cyclographic mapping transforms any EO mesh in a hyperplane of Minkowksi 4-space into a pair of Euclidean EO meshes. This result leads to a derivation of EO meshes which are discrete versions of Laguerre minimal surfaces. Laguerre minimal EO meshes can also be constructed directly from certain pairs of Koebe meshes with help of a discrete Laguerre geometric counterpart of the classical Christoffel duality.