Operator analogue of the Krein-Milman theorem in the generalized state spaces

We discuss the Krein-Milman-type problems in the C* -convexity theory for the generalized state space of C*-algebraA. The main results are that every BW-compact, C*-convex subset of possesses a C*-extreme point and every BW-compact, C* -convex subset of is the C^*-convex hull of its C*-extreme points.     

Symmetric centres of braided monoidal categories

This paper introduces the concept of ‘symmetric centres’ of braided monoidal categories. LetH be a Hopf algebra with bijective antipode over a fieldk. We address the symmetric centre of the Yetter-Drinfel’d module category: and show that a left Yetter-Drinfel’d moduleM belongs to the symmetric centre of and only ifM is trivial. We also study the symmetric centres of categories of representations of quasitriangular Hopf algebras and give a sufficient and necessary condition for the braid of, _Hℳ to induce the braid of , or equivalently, the braid of , whereA is a quantum commutativeH-module algebra     

The convergence rate of Euler means sequence

Let denote the Euler means of the Fourier series of the 2π-perodic function f(x). For an integer q>0, setting ℰ _nq (ω)=sup‖f−ℰ _n (q,f)‖, the precise value of is obtained.     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that and where ω(f,t)_p is the 1st-order modulus of continuity of f in L^p-metric which is defined in a way different than in the classical case of n=2. In Memory of Professor M. T. Cheng Project supported by the NSF of China under the grans # 19771009.     

Sull’unione di classi filtrali

Riassunto In questo lavoro si cercano condizioni perchè una classe di algebre simili filtrali sia essa stessa filtrale. Si trova che se è una famiglia finita di algebre simili e filtrali, che siano indipendenti (cfr.A. L. Foster [2]), allora anche è filtrale. Introdotto poi il concetto disemi-indipendenza, si arriva a un risultato analogo per le algebre semifiltrali e a una condizione sufficiente per la semifiltralità di un'algebra finita. Infine si osserva che i risultati trovati possono essere estesi alle classi ideali.

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Summary This paper refers to the concept offiltrale class of similar algebras, which has been introduced byR. Magari in [6]. First I observe that a class of similarfiltrale algebras is not generallyfiltrale, not even if is a finite family of finite algebras. However I find that if is a finite family of similarfiltrale algebras, which are also independent (using this term in the sense ofA. L. Foster [2]), then also isfiltrale. Then introducing the notion of independent varieties, one arrives at a generalization of the above mentioned result. Given then a family of similar algebras , we say that these aresemi-indipendenti if for everyJ⊂I there exists an algebraic functionf _J defined in II _i so thatf _J(x, y)=z wherez _i=x_i ifi∈J andz _i=y_i ifi∉J for every . I prove that if is a finite family ofsemifiltrali andsemi-indipendenti algebras, then also issemifiltrale. I also find that a finite algebra, which is simple andsemi-indipendente of itself (in the sense that there exists in a binary algebraic functionf withf(x, y)=<x ₀,y ₁>) issemifiltrale. Then I give the notion ofsemi-indipendente variety, which leads to some further results. Finally I observe that all these results can be extended to ideal classes of similar algebras.

     

Riesz frames and approximation of the frame coefficients

A frame is a fmaily {f _i } _i=1 ^∞ of elements in a Hilbert space with the property that every element in can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coefficients. From the mathematical point of view this is gratifying, but for applications it is a problem that the calculation requires inversion of an operator on . The projection method is introduced in order to avoid this problem. The basic idea is to consider finite subfamilies {f _i } _i=1 ⁿ of the frame and the orthogonal projection P_n onto its span. For has a representation as a linear combination of f_i, i=1,2,..., n and the corresponding coefficients can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n→∞, in which case we have avoided the inversion problem. In the same spirit we approximate the solution to a moment problem. It turns out, that the class of “well-behaving frames” are identical for the two problems we consider. The first named author is partially supported by NSF DMS 9201357, Danish NSRC grant 9401958, Missouri Research Board grant C-3-41743, and a Missouri Research Council Summer Fellowship.     

Strong mean approximation on HP(Tn) for the Riesz means at the critical index

In this note we extend the results in [1] to high dimensions. Let f∈H^p (Tⁿ), 0<p<1, n≥1 andσ ^δ , f denote the Riesz means of f at the critical index δ=n/p−(n+1)/2. We have the following estimate: were 0<s≤2 and , is the K-functional in H^p(Tⁿ). Supported by NSFC     

Non-Strebel points and variability sets

This paper studies the subset of the non-Strebel points in the universal Teichmuller space T. Let Z0 ∈ △be a fixed point. Then we prove that for every non-Strebel point h, there is a holomorphic curve γ : [0, 1]→ T with h as its initial point satisfying the following conditions.(1) The curve γ is on a sphere centered at the base-point of T, i.e. dT(id, γ(t)) = dT(id, h), (t ∈ [0, 1]).(2) For every t ∈ (0,1], the variability set Vγ(t)[Z0] of γ(t) has non-empty interior, i.e. Vγ(t) [Z0] ≠ .     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that      

The central Campanato spaces and its application

Let 1<p<∞, α≥0 andK be a local field. In this paper, the author introduces the spaces J _p ^α (K) and the central Campanato spaces as well as the S_p,α- and S _p,α ⁺ -type singular integrals. Then, the author investigates the behavior of these singular integrals and their altered operators on these spaces. The research was supported by the NNSF of China.     

Una dimostrazione del fatto che ogni varietà ammette algebre semplici

Riassunto Si dimostra che se è una struttura algebrica priva di quozienti semplici, allora un’opportuna sottoalgebra di ammette quozienti semplici. Ne segue il risultato del titolo.

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Summary Let be an algebraic structure with no simple quotients. Then there is a subalgebra of which possess simple quotients. Therefore every variety of algebras (=algebraic structures) has simple algebras.

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Lavoro eseguito nell’ambito dell’attività del Comitato Nazionale per le Scienze Matematiche del C.N.R. (anno ’68–’69, programma 8).     

μ-average N-widths on the Wiener space

In this paper we investigate the asymptotic behaviour of μ-averagen-widths of integral operatorK on the Wiener space, whereK is the inverse operator of an ordinary linear differential operatorL of orderm. For 1≤p.q<∞ . and forp∈[1, ∞),q∈[2, ∞) . Supported by the Fund. of Dooctoral program of NECC.     

Interpolation in Logiken monotoner systeme

A monotone structure ( ;μ) consists of a structure and a monotone systemμ over the domain of .L(Q ⁿ ) is , enlarged by a newn-ary quantifierQ ⁿ . says in ( ;μ) that there isU ∈μ such thatϕ[ā] is valid in ( ;μ) for allā ∈U ⁿ . If is a class of monotone structures, means thatϕ is valid in all expansions of monotone structures in . We show for the class of all ultrafilters that interpolation with respect to holds forL(Q ⁿ ) exactly in casen=1. Then we prove for a large class of (e.g. the class of topological groups) thatL(Q ⁿ ) satisfies interpolation with respect to for alln ≧ 1. Counterexamples indicate that the class of is sharp in some sense. Finally the results are carried over to certain topological structures and the interior quantifiersI ⁿ instead ofQ ⁿ , thereby generalizing results of Makowsky/Ziegler and Sgro, and to a multidimensional type of monotone structures including uniform spaces.     

Sistemi di assiomi che definiscono un piano affine di ordinen

Riassunto Un insieme di proprietà soddisfatte da una struttura è detto essere un sistema completo di assiomi se è sufficiente a definire come un piano affine. Un sistema completo è minimo se non possiede alcun sottosistema proprio completo. In questo lavoro sono determinati i possibili sistemi minimi completi di assiomi che definiscono un piano affine di ordinen.

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Summary A set of properties which are satisfied by an incidence structure is said to be a complete system of axioms if such a system is sufficient to define as an affine plane. A complete system is minimal if it does not possess any proper complete subsystem. In this work we find all possible minimal complete systems of axioms which define an affine plane of ordern.

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Lavoro eseguito nell’ambito dell’attività dei contratti di ricerca del Comitato per le Scienze Matematiche del C.N.R.     

Inductive definitions,models of comprehension and invariant definability

The connections between inductive definability and models of comprehension are studied. Let = 〈A, R _l, ...,R _n 〉 be an infinite structure and letI _φ be a set inductively defined by a formulaφ of the second order language . We prove that if is a model of Δ ₁ ¹ -Comprehension relativized toφ, andφ is -absolute, then for everyη smaller than the height of (h( )),I _φ is in . If is aβ-structure which satisfies Σ ₁ ¹ -Comprehension relativized toφ and WF(X), and φ is -absolute, thenI _φ is in and ‖φ| <h ( ). These results imply that Barwise-Grilliot theorem is false in the case of uncountable acceptable structures. We also study the notion of invariant definability over models₁ of Δ ₁ ¹ -Comprehension. This paper is registered as Report ZW 69/76 of the Mathematical Centre.     

Minimal presentations for certain group extensions

IfG is a finite group thend(G) denotes the minimal number of generators ofG. IfH andK are groups then the extension, 1 →H →G →K → 1, is called an outer extension ofK byH ifd(G)=d(H)+d(K). Let be the class of groups containing all finitep-groupsG which has a presentation withd(G) = dimH ¹(G,z _p ) generators andr(G)=dimH ² (G,Z _p ) relations: in this article it is shown that ifK is a non cyclic group belonging to andH is a finite abelian p-group then any outer extension ofK byH belongs to .     

Enumeration ofQ-acyclic simplicial complexes

Let (n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely ( _k ⁿ⁻¹ )k-faces, (3)H _k (C)=0. We prove that for ,H _k−1(C) is a finite group, and our main result is: . This formula extends to high dimensions Cayley’s formula for the number of trees onn labelled vertices. Its proof is based on a generalization of the matrix tree theorem.     

Constructions Using Conics

Conclusion  After recalling definitions and results about the constructibility of a geometric object, we have shown by more and more efficient methods how the works of Gauss, computer algebra systems (Maple), and dynamic geometry software (Cabri-Geometry, distributed by Texas Instruments) could be used together to construct regular polygons, using ruler, compass, and simple conics. In particular, we have given the list of small ₂-constructible polygons, and presented new ₂-constructions of the regular polygons with 19, 37, 73, and 97 sides. The ancient Greeks gave precedence to constructions using only ruler and compass, not because they did not know about the other curves (they invented a number of mechanical devices drawing some algebraic curves of degrees 2, 3, 4, and more), but for the neatness, perfection of reasoning, and the simplicity of the shapes involved (circle and straight line). Today’s tools such as Cabri-Geometry enlarge the notion of geometric simplicity by allowing the manipulation of algebraic expressions (the sequences defined by Gauss) and complex geometric objects (the conic sections). Some generalizations of the questions treated here may be considered:

Numerical evaluation of a 2-D Cauchy principal value integral based on quasi-interpolating splines

In this paper the author presents a method for the numerical solution of a 2-D Cauchy principal value of the form where S is a domain with a continuous boundary. By using polar coordinates, the integral is reduced to the form where the finite-part of the integral. We construct the relative product rule based on quasi-inter polating splines. Convergence results are proved and numerical examples are given.     

Large sample theory for distributions on the hypersphere with rotational symmetries

Summary A distribution on the unit sphere inℝ ^q with a densityf(‖x _v ‖) is considered where is ans(<q) dimensional subspace andx _v is the part ofx in . For a large sample the estimation of , a test that and a test for rotational symmetry within is given. For several samples with possibly different subspaces but the samef, a test that is given. For all tests power functions for contiguous alternatives are given. For the special density proportional to expk‖x _v ‖ ², additional results are given. Research supported in part by a Contract with the Office of Naval Research N00014-81-K-0146 awarded to Princeton University, Princeton, New Jersey 08544.