A less formal approach to kaluza-klein formalism

The action integrals (a) and , corresponding respectively to gravitational and gravitational-electromagnetic phenomena, are shown to be related under continuous groups of null translations. This relation motivates a modified Kaluza—Klein formalism for which the classical cylindrical metric preserving transformations (c)y ⁵ = =x ⁵ +f ⁵(x ^j ),y ⁱ =f ⁱ (x ^j ) fori = 1, 2, 3, 4 are replaced by (d)y ⁵ =x ⁵,y ⁱ =f ⁱ (x ^j ,x ⁵). The cylindrical metric of V⁵ is nevertheless preserved under (d), since it is assumed thatV ⁵ admits a metric of the form (corresponding to (a)) and that (d) defines a continuous group of null translations in theV ⁴ metric defined byg _ij whenx ⁵ is considered the group parameter. Application of (d) leads to the cylindrical metric corresponding to (b). The resulting electromagnetic fieldsF _ij =B _i,j –B _j,i are then related to the curvatures of theV ⁴ corresponding tog _ij andh _ij ; in particular it is shown that and . When it is shown thatF _ij is a null electromagnetic field which is generally non-trivial. Some physical and geometric interpretations of the mathematical results are also presented.Dedicated to Professor A. Ostrowski on the occasion of his 75th birthday     

On the number of faces of centrally-symmetric simplicial polytopes

I. Bárány and L. Lovász [Acta Math. Acad. Sci. Hung.40, 323–329 (1982)] showed that ad-dimensional centrally-symmetric simplicial polytopeP has at least 2 ^d facets, and conjectured a lower bound for the numberf _i ofi-dimensional faces ofP in terms ofd and the numberf ₀ =2n of vertices. Define integers A. Björner conjectured (unpublished) that (which generalizes the result of Bárány-Lovász sincef _d–1 = h _i ), and more strongly that , which is easily seen to imply the conjecture of Bárány-Lovász. In this paper the conjectures of Björner are proved.Partially supported by NSF grant MCS-8104855. The research was performed when the author was a Sherman Fairchild Distinguished Scholar at Caltech.     

The dirichlet problem for sub-elliptic second order equations

The C-regularity up to the boundary of solutions to the Dirichlet problem: is proved, using a comparison principle of L with a Hörmander's type operator X _j ^* X_j, where is a smooth bounded open subset of Rⁿ, and is a second-order degenerate elliptic operator with smooth coefficients, satisfying the so-called Fefferman-Phong's condition.     

Easy transportation-like problems onK-dimensional arrays

TheK-dimensional version of two transportation-like problems is posed and efficiently solved. The caseK=2 goes as follows: Given an (m,n)-matrixA of reals, a realm-vectoru, and a realn-vectorv, find a real (m,n)-matrixX minimizing

K0 of multiplier algebras of C*-algebras with real rank zero

For a large class of -unital C ^*-algebras A with real rank zero and stable rank one, the structure of the Grothendieck group k ₀ of the multiplier algebra (A) is investigated. The ordered group K ₀( (A)) is shown to be an unperforated Riesz group, and its additive structure is completely determined, as is — in important cases — its order structure. These structures, and the attendant consequences for the ideal structure of (A), are richer than previously anticipated. In particular, it is shown that the corona algebra (A)/A can have very large stably finite quotient algebras. For example, there exist simple, separable, approximately finite-dimensional C ^*-algebras A such that the maximal stably finite quotient algebra of (A)/A has uncountably many maximal ideals modulo which a W ^*-factor of Type II₁ results. The analysis of the additive structure of K ₀( (A)) yields as a byproduct that if A is a -unital approximately finite-dimensional C ^*-algebra without nonzero unital quotient algebras, then all quasitraces on (A) are traces.This research was partially supported by a grant from the National Science Foundation.     

Analytic semigroups generated on hölder spaces by second order elliptic systems under Dirichlet boundary conditions

Sunto Si studia il problema della generazione di semigruppi analitici, nella topologia hölderiana, per un operatore ellittico del tipo con dati al bordo di Dirichlet e supponendo i coefficienti A_ij continui in .

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This work is partially supported by the Research Funds of the Ministero della Pubblica Istruzione.     

Majorization of Gaussian processes and geometric applications

Summary Let (x _i ^* ) _i=1 ⁿ denote the decreasing rearrangement of a sequence of real numbers (x _i ) _i=1 ⁿ . Then for everyij, and every 1kn, the 2-nd order partial distributional derivatives satisfy the inequality, . As a consequence we generalize the theorem due to Fernique and Sudakov. A generalization of Slepian's lemma is also a consequence of another differential inequality. We also derive a new proof and generalizations to volume estimates of intersecting spheres and balls in ⁿ proved by Gromov.Supported by NSF grant # DMS 8909745, and the USA-Israel Binational Science Foundation grant # 86-00074, and grant for the Promotion of Research at the Technion     

Limits of indeterminacy of sequences obtained from a given sequence using a regular transformation

The problem considered is how there can be a set of weak accumulation points of the subsequences of a sequence obtained from a given sequence by using a regular transformation of the class T(C, C) when the terms of the sequences are elements of a reflexive Banach space. T(C, C) denotes the class of complex regular matrices (c_mn) (c_mn=a _mn+ib_mn, wherea _mn and b_mn are real numbers) satisfying the conditions and Translated from Matematicheskie Zametki, Vol. 16, No. 6, pp. 887–897, December, 1974.In conclusion the author thanks D. E. Men'shov for his help and interest, and S. B. Stechkin for valuable advice.     

The closed ideal of (c o ,p,q)-summing operators

For 1/p+1/q1, we study the closed ideal formed by the (c _o ,p,q)-summing operators. It turns out thatT:XY does not belong to if and only if it factors the mapId:l _p ^*l _q . By localization, we get the ideal that consists of those operatorsT for which all ultrapowersT ^u are contained in . Operators in the complement of are characterized by the property that they factor the mapsId:l _p ^*n l _q ⁿ uniformly. Our main tools are ideal norms.Supported by DFG grant PI 322/1-2     

Positive Periodic Solutions of N-Species Neutral Delay Systems

In this paper, we employ some new techniques to study the existence of positive periodic solution of n-species neutral delay system N _i = N _i (t)[ _i (t) — _ij (t)N _j (t) — b _ij (t)N _j (t— _ij (t))— c _ij (t)N _j (t— _ij (t))].As a corollary, we answer an open problem proposed by Y. Kuang.     

Asymptotic Distribution of Quadratic Forms and Applications

We consider the quadratic formsQ X _j X _k+ (X _j ² -E X _j ² )where X _j are i.i.d. random variables with finite sixth moment. For a large class of matrices (a _jk ) the distribution of Q can be approximated by the distribution of a second order polynomial in Gaussian random variables. We provide optimal bounds for the Kolmogorov distance between these distributions, extending previous results for matrices with zero diagonals to the general case. Furthermore, applications to quadratic forms of ARMA-processes, goodness-of-fit as well as spacing statistics are included.     

On Brualdi's Theorem

This paper studies irreducible matrices , satisfying Brualdi's conditions , or, shortly, Brualdi matrices. Here, is the set of circuits of length in the directed graph of A, and is the support of . Among the results obtained are a characterization of Brualdi matrices, implying, in particular, that they are generalized diagonally dominant; the necessary and sufficient conditions of singularity of Brualdi matrices; explicit expressions for the absolute values of the components of right null-vectors of a singular Brualdi matrix, and conditions necessary and sufficient for a boundary point of Brualdi's inclusion region to be an eigenvalue of an irreducible matrix. Bibliography: 8 titles.     

Patterns of exponential decay for solutions to second order elliptic equations in a sector ofR 2

We consider solutions ψ to equations of the form in a sector Ω ofR ². The basic assumptions are that the limitsa _ij(x)→δ_ij,b _i(x)→0,c _i→E at infinity are achieved at certain rates and thatg decays faster than ψ. We then discuss the possible patterns of exponential decay for ψ in Ω. NSERC University Research Fellow. Research partially supported by USNEF grant MCS-83-01159.     

The wavelet transform on Sobolev spaces and its approximation properties

Summary We extend the continuous wavelet transform to Sobolev spacesH ^s() for arbitrary reals and show that the transformed distribution lies in the fiber spaces . This generalisation of the wavelet transform naturally leads to a unitary operator between these spaces.Further the asymptotic behaviour of the transforms ofL ₂-functions for small scaling parameters is examined. In special cases the wevelet transform converges to a generalized derivative of its argument. We also discuss the consequences for the discrete wavelet transform arising from this property. Numerical examples illustrate the main result.Supported by the Deutsche Forschungsgemeinschaft under grant Lo 310/2-4     

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

The C*-algebra generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities we reduce the study to simpler C*-algebras associated with points and pairs We construct a symbol calculus for unital C*-algebras generated by n orthogonal projections sum of which equals the unit and by m one-dimensional orthogonal projections. Such algebras are models of local algebras at points z ∈∂Π being the discontinuity points of coefficients. A symbol calculus for the C*- algebra and a Fredholm criterion for the operators are obtained. Finally, a C*-algebra isomorphism between the quotient algebra where is the ideal of compact operators, and its analogue for the unit disk is constructed.     

A constructive method of solving the Liapounov equation for complex matrices

Summary This paper describes a method of solving the Liapounov equation (1)HM+M ^* H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M ^* A ^* =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH _j andD _j respectively for whichH _j M+M ^* H _j =2D _j is valid. Then one can solve the real system to obtain the solution of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom.     

Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C^*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors – real K-theory, complex K-theory, and self-conjugate K-theory – and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, nonsplitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C^*-algebras A and B which holds as long as the complexification of A is in the bootstrap category . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C^*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product is not determined solely by the greatest common divisor of K and l. Hence, we have examples of nonisomorphic, simple, purely infinite, real C^*-algebras whose complexifications are isomorphic.     

An Algebra Associated with a Spin Model

To each symmetric n × n matrix W with non-zero complex entries, we associate a vector space N, consisting of certain symmetric n × n matrices. If W satisfies then N becomes a commutative algebra under both ordinary matrix product and Hadamard product (entry-wise product), so that N is the Bose-Mesner algebra of some association scheme. If W satisfies the star-triangle equation: then W belongs to N. This gives an algebraic proof of Jaeger's result which asserts that every spin model which defines a link invariant comes from some association scheme.     

Aspects of Number Theory in Terms of Potential Theory

We develop the Potential Theory on the set N^* of all natural numbers 2 associated with the kernel V (resp. V ^*) given by

Completely positive completion of partial matrices whose entries are completely bounded maps

We study completion problems of partial matrices associated with a graph where entries are completely bounded maps on aC ^*-algebra. We characterize a graph for which every -partial completely positive matrix has a completely positive completion. As a special case we study -partial functional matrices. We give a necessary and sufficient condition for a -partial functional matrix to have a positive completion and a representation for such matrices. These generalize some results on inflated Schur product maps due to Paulsen, Power and Smith. As an application, we study completely positive completions of partial matrices whose entries are completely bounded multipliers of the Fourier algebra of a locally compact group.