Hypersurfaces of restricted type in Minkowski space

A submanifold M _n ^r of Minkowski space is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of to the tangent space of M _n ^r at every point of M _n ^r . In this paper we completely classify hypersurfaces of restricted type in . More precisely, we prove that a hypersurface of is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S ^k × , S _k ¹ × , H ^k × , S _n ¹ , H ⁿ , with 1kn–1, or an open part of a cylinder on a plane curve of restricted type.This work was done when the first and fourth authors were visiting Michigan State University.Aangesteld Navorser N.F.W.O., Belgium.     

Reducibility of lacunary polynomials,VII

Letk>1 and let be non-zero algebraic numbers contained in the field . It is shown that for almost all, in the sense of density integer vectorsn ₁,...,n _k the polynomial becomes irreducible over on dividing by the product of all factorsx–, where is a root of unity.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

Positive elements in the algebra of the quantum moment problem

Summary Let denote the extended Weyl algebra, , the Weyl algebra. It is well known that every element of of the formA=B _k ^* B _k is positive. We prove that the converse implication also holds: Every positive elementA in has a quadratic sum factorization for some finite set of elements (B _k ) in . The corresponding result is not true for the subalgebra . We identify states on which do not extend to states on . It follows from a result of Powers (and Arveson) that such states on cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on , and this property is not in general shared by representations generated by states defined only on the subalgebra .Work supported in part by the NSF     

Über die Riemannsche Einbettungsgeometrie der Veronesemannigfaltigkeiten

Let E be a n-dimensional euclidean vector space. The subset V _k ⁿ ={x ... x | x E} of ^kE is called a Veronesemanifold. The scalar product of E induces a euclidean structure on ^kE. Passing to the corresponding projective space , one may consider as a riemannian submanifold of the space form . In this paper we study properties of the pair of riemannian manifolds.     

On Bergman-Toeplitz operators with commutative symbol algebras

Let be the unit disk in, be the Bergman space, consisting of all analytic functions from , and be the Bergman projection of onto . We constructC ^*-algebras , for functions of which the commutator of Toeplitz operators [T _a ,T _b ]=T _a T _b –T _b T _a is compact, and, at the same time, the semi-commutator [T _a ,T _b )=T _a T _b –T _ab is not compact.It is proved, that for each finite set =n ₀,n ₁, ...,n _m , where 1=n _{0 1} <... _m , andn _k {}, there are algebras of the above type, such that the symbol algebras Sym of Toeplitz operator algebras arecommutative, while the symbol algebras Sym of the algebras , generated by multiplication operators and , haveirreducible representations exactly of dimensions n ₀,n ₁,..., n _m .This work was partially supported by CONACYT Project 3114P-E9607, México.     

A note on certain fields of finite transcendence type

Let be a finitely generated extension field of ℚ, andα _i,β_j(1⩽i⩽m,1⩽j⩽n) be some complex numbers. Let (k=1,2,3) be fields obtained by adjoining to the numbers {α _i,β_j exp(α_iβ_j)}, {α_i, exp(α_iβ_j)}, and {exp(α_iβ_j)}, respectively. In the present note the relation between the transcendental degree of over and the transcendence type of over ℚ is given. This work was completed in Dpt. Math., Univ. of Southern Mississippi, Hattiesburg, USA.     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

On a theorem of Seidel and Walsh

Given a sequence ( _n ) _n in with there are functions such that , is a dense subset of , and the set of functions with this property is residual in . We will show that in and some related Banach spaceX there are functionsf with is dense in , and we will give a sufficient condition when the set of such functions is residual inX.     

Gray codes from antimatroids

We show three main results concerning Hamiltonicity of graphs derived from antimatroids. These results provide Gray codes for the feasible sets and basic words of antimatroids.For antimatroid (E, ), letJ( ) denote the graph whose vertices are the sets of , where two vertices are adjacent if the corresponding sets differ by one element. DefineJ( ;k) to be the subgraph ofJ( )² induced by the sets in with exactlyk elements. Both graphsJ( ) andJ( ;k) are connected, and the former is bipartite.We show that there is a Hamiltonian cycle inJ( )×K ₂. As a consequence, the ideals of any poset % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFpepuaaa!414C!\[\mathcal{P}\] may be listed in such a way that successive ideals differ by at most two elements. We also show thatJ( ;k) has a Hamilton path if (E, ) is the poset antimatroid of a series-parallel poset.Similarly, we show thatG( )×K ₂ is Hamiltonian, whereG( ) is the basic word graph of a language antimatroid (E, ). This result was known previously for poset antimatroids.Research supported in part by NSERC.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant A3379.     

Subcoercivity and subelliptic operators on Lie groups II: The general case

Let (,G, U) be a continuous representation of a Lie groupG by bounded operatorsg U (g) on the Banach space and let (, ,dU) denote the representation of the Lie algebra obtained by differentiation. Ifa ₁, ...,a _d is a Lie algebra basis of ,A _i =dU (a _i ) and whenever =(i ₁, ...,i _k ) we reconsider the operators

Towards a model theory for 2-hyponormal operators

We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension ofT we only require that hold forf ; in this case we call a partially normal extension ofT. After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let { _n } _n=0 be a weight sequence and letW denote the associated unilateral weighted shift on . IfW is 2-hyponormal thenW is weakly subnormal. Moreover, there exists a partially normal extension on such that (i) is hyponormal; (ii) ; and (iii) . In particular, if is strictly increasing then can be obtained as

On dualities between function spaces

By [6], the dualities between and , whereX andW are two sets and (i.e., the mappings satisfying for all and all index setsI), can be represented with the aid of functions . Here we show that they can be also represented with the aid of functions , whereR = (–, +). As an application, we show that every duality is completely determined by a suitable duality between 2 ^{X ×R} and 2 ^{W ×R} (i.e., a mapping 2 ^{X ×R} 2 ^{W ×R} satisfying for all {M _i} _iI 2 ^{X ×R} and all index setsI), applied to the epigraphs of the functions .     

Prime ideal decomposition inF({}^m\sqrt \mu )

LetF be an algebraic number field and F such thatx ^m– is irreducible, wherem is an integer. Let be a prime ideal inF with . The prime decomposition of in is explicitly obtained in the following cases. Case 1: , (a,m) = 1 (where means , 0 ). Case 2:m lt, wherel is a prime andl 0 . Case 3:m 0 and every prime that dividesm also dividespf–1. It is not assumed that thev ^th roots of unity are inF for anyv 2.     

One-step prediction forP n-weakly stationary processes

The one-step prediction problem is studied in the context ofP _n-weakly stationary stochastic processes , where is an orthogonal polynomial sequence defining a polynomial hypergroup on . This kind of stochastic processes appears when estimating the mean of classical weakly stationary processes. In particular, it is investigated whether these processes are asymptoticP _n-deterministic, i.e. the prediction mean-squared error tends to zero. Sufficient conditions on the covariance function or the spectral measure are given for being asymptoticP _n-deterministic. For Jacobi polynomialsP _n(x) the problem of being asymptoticP _n-deterministic is completely solved.     

A geometric approach to the cascade approximation operator for wavelets

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically thecascade algorithm in wavelet theory. Let be a Hilbert space, and let be a representation ofL ( ) on . LetR be a positive operator inL ( ) such thatR(1) =1, where1 denotes the constant function 1. We study operatorsM on (bounded, but noncontractive) such that

K-theory of twisted differential operators on flag varieties

Let be a semisimple Lie algebra overk, an algebraically closed field of characteristic zero, and let be a Cartan subalgebra inside a Borel subalgebra of . LetU be the enveloping algebra of . For letM() denote the corresponding Verma modúle and letU _u=U/AnnM(). LetW be the Weyl group and letW ⁰ be the stabiliser of inW. We prove the following theorem, which affirms a conjecture of T.J. Hodges.Oblatum 16-XII-1994     

Tensor ideals in the category of tilting modules

We study the tensor category of tilting modules over a quantum groupU _q with divided powers. The setX ₊ of dominant weights is a union of closed alcoves numbered by the elementswW ^f of a certain subset of affine Weyl groupW. G. Lusztig and N. Xi defined a partition ofW ^f into canonical right cells and the right order _R on the set of cells. For a cellAW ^f we consider a full subcategory formed by direct sums of tilting modulesQ() with highest weights . We prove that is a tensor ideal in , generalizing H. Andersen's theorem about the ideal of negligible modules which in our notations is nothing else then . The proof is an application of a recent result by W. Soergel who has computed the characters of tilting modules.This material is based upon work supported by the U.S. Civilian Research and Development Foundation under Award No. RM1-265.