Almost Fredholm operators in von Neumann algebras

LetA be a von Neumann algebra,J be the ideal of compact operators relative toA and letF ₊ be the left-Fredholm class ofA. We call almost left-Fredholm the class = {A A: if P A is a projection and AP J then P J}. Then and the inclusion is proper unlessA is semifinite and has a non-large center. satisfies all of the algebraic properties ofF ₊ but it is generally not open. IfA is semifinite then A iff there are central projectionsG with G = I such that AG F₊(AG). Let :A A/J. Then the left almost essential spectrum ofA A, , coincides with the set of eigenvalues of (A)     

Decompositions of complete graphs into isomorphic bipartite subgraphs

Let |E(G)|= andf, a 1-1 mapping ofV(G) into {0,1,...,}. Thenf is called a -valuation ofG if the induced function given by , for alluvE(G) is 1-1. A -valuationf is called an -valuation ofG if there exists a nonnegative number such that for everyuvE(G) withf(u)<f(v),f(u)<f(v). Let denote the graph of then-dimensionalG-cube. ForG=K _{3, 3},K _{4, 4}, andP _k ,it is shown that for any positive integern, then-dimensionalG-cube has an -valuation. This gives rise to decompositions of some complete graphs into certain bipartite graphs.     

Finitely Valued f-Modules, An Addendum

In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets ₁ and ₂ and the corresponding -value sets and . If R is a unital -ring, then each unital -module over R is an f-module and has exactly when R is an f-ring in which 1 is a strong order unit.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Differentiability properties of the minimal average action

Given aZ ⁿ⁺¹-periodic variational principle onR ⁿ⁺¹ we look for solutionsu:R ⁿ R minimizing the variational integral with respect to compactly supported variations. To every vector R ⁿ we consider a subset of solutions which have an average slope when averaging overR ⁿ. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R ⁿ{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the i^th component of is rational with denominator sⁱ N in lowest terms, we show: The difference of right- and left-sided derivative in the i^th standard unit direction is bounded by const · .     

On singular perturbation of a semilinear hyperbolic equation

We consider a hyperbolic version of Eells-Sampson's equation: . This equation is semilinear with respect to space derivative and time derivative. Letu (x) be the solution with initial data u(0) and (0), and putv (t,x)=u (t,x). We show that when the resistance ,V (t,x) converges to a solution of the original parabolic Eells-Sampson's equation: . Note thatv _t(0)= (0) diverges when . We show that this phenomena occurs in more general situations.This article was processed by the author using the Springer-Verlag Pjourlg macro package.     

Conformally Invariant Regularization and Skeleton Expansions in Gauge Theory

We consider a conformally invariant regularization of an Abelian gauge theory in an Euclidean space of even dimension D 4 and regularized skeleton expansions for vertices and higher Green's functions. We set the respective regularized fields and with the scaling dimensions and into correspondence to the gauge field A and Euclidean current j . We postulate special rules for the limiting transition 0. These rules are different for the transversal and longitudinal components of the field and the current . We show that in the limit 0, there appear conformally invariant fields A and j each of which is transformed by a direct sum of two irreducible representations of the conformal group. Removing the regularization, we obtain a well-defined skeleton theory constructed from conformal two- and three-point correlation functions. We consider skeleton equations on the transversal component of the vertex operator and of the spinor propagator in conformal quantum electrodynamics. For simplicity, we restrict the consideration to an Abelian gauge field A , but generalization to a non-Abelian theory is straightforward.     

Isolation theorem for forms corresponding to purely real algebraic fields

Let M be the complete module of a purely real algebraic field of degree n 3, let be a lattice in this module, and let F(X) be its form. We use to denote any lattice for which we have = , where is a nondiagonal matrix for which – I . With each lattice we can associate a factorizable formF (X) in a natural manner. We denote the complete set of forms corresponding to the set {} by {F (X)}. It is proved that for any > 0 there exists an > 0 such that for eachF (X) {F } we have |F (X₀₎| for some integer vector X₀ 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 5–12, 1990.In conclusion, the author would like to express his deep gratitude to B. F. Skubenko for stating the problem and for his constant attention.     

Estimates of the asphericity of sections of convex bodies

We define the function (n, k) to be the infimum of all such that any bounded centrally symmetric convex body inR ⁿ possesses an -asphericalk-dimensional central section. It is proved that (3, 2)=2–1 and (n, n-1)n-1-1. Several related functions are defined and their values on the pairs (n, n-1) are estimated.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 76–79.     

The Dvoretzky-Hanani Theorem for the Group "ax + b"

Let G be the group "ax + b" of affine transformations of the line and let U be a neighbourhood of 1 in G. It is proved that there is another neighborhood V of 1 such that to each finite sequence g1,...,gn V there corresponds a sequence of signs 1,...,n = ±1 with U for k = 1,...,n. This implies that G satisfies the following analogue of the Dvoretzky-Hanani theorem: to each sequence converging to 1 in G there corresponds a sequence of signs k = ±1 such that the infinite product is convergent.     

On the sums of a family of Kapteyn series

The series ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) arise in studying the emission and absorption of radiation by a charged particle on a Kepler orbit. For the first few even,r, the sums are obtained in closed form, and for oddr they are given in terms of a certain definite integral. The integral is expressed as a power series in for ||<1, and, near =1, an asymptotic expansion in powers of (1–²)^1/2 may be obtained.

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Résumé La série ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) se trouve par l'émission et l'absorption du rayonnement d'une particule chargée sur l'orbite Keplerien. Pour les plus petites valuers paires der, les sommes s'obtienment en forme compacte, et pour les valuers impaires, elles se déterminent d'après une intégrale definie. Pour ||<1, cette intégrale se développe dans une série de puissances de , et dans le voisinage de =1, on obtient une série asymptotique et puissances de (1–²)^1/2.

     

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a covering Y is introduced whereY is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations of which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.     

Sharp determinants

We introduce a sharp trace Tr^# and a sharp determinant Det^#(1-z) for an algebra of operators acting on functions of bounded variation on the real line. We show that the zeroes of the sharp determinant describe the discrete spectrum of . The relationship with weighted zeta functions of interval maps and Milnor-Thurston kneading determinants is explained. This yields a result on convergence of the discrete spectrum of approximated operators.Oblatum 8-V-1995 & IX-1995On leave from CNRS, UMR 128, ENS Lyon, France     

An isolation theorem for decomposable forms of purely real algebraic fields of degree n⩾3

Let M be a complete module of a purely algebraic field of degree n3, let be the lattice of this module and let F(X) be its form. By we denote any lattice for which we have = , where is a nondiagonal matrix satisfying the condition ¦-I¦ , I being the identity matrix. The complete collection of such lattices will be denoted by {}. To each lattice we associate in a natural manner the decomposable form F(X). The complete collection of forms, corresponding to the set {}, will be denoted by {F} It is shown that for any given arbitrarily small interval (N–, N+), one can select an such that for each F(X) from {F} there exists an integral vector X₀ such that N– < F(X₀) < N+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 167–171, 1981.     

The correctness problem for best approximations by trigonometric polynomials in the class W o r H[ω]C

Suppose thatk, rz₊, W _o ^r H[]_C= {ff is a 2-periodic function,f C^r [–, ], (f^(r), ) ()}, T_k is the space of trigonometric polynomials of order k, p_k(f)T_k is the polynomial of best uniform approximation to f, and E_k(f) is the error of the best approximation. It is shown that for an arbitrary > 0 we have,where for 0<&#x2A7D;(1),k > 0.R () is the root of the equation , and for k = 0 or > (1) we have R()=.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 85–101, July, 1977.The author thanks S. B. Stechkin for posing the problem and for his attention to this work.     

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

On Shikata's distance between differentiable structures

Shikata proved: there is a number (n) with the following property: If two compact homeomorphic n-dimensional manifolds have a distance less than (n), then they are diffeomorphic. We improve the known lower bound (n!)^–n for (n) to 1/3n ^–2.This work was done under the program Sonderforschungsbereich Theoretische Mathematik (SFB 40) at Bonn University while Shikata was SFB-guest at Bonn.     

Noether modules over abelian groups of finite free rank

We prove that if M is a Noether JG-module, where G is an abelian group of finite free rank, and either J=, or J=Ft, where F is a finite field and t is an infinite cyclic group, then the module M belongs to a class (J, ) for some finite set in the sense defined by P. Hall.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1042–1048, July–August, 1991.     

Rate of approximation by rectangular partial sums of double orthogonal series

( ) . .

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Dedicated to Professor K. Tandori on his seventieth birthday

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This research was supported in part by Grant # K41 100 of the Joint Fund of the Government of Ukraine and the International Science Foundation.     

Higher order methods for a singularly perturbed problem

Summary A finite element method using piecewise polynomials of degree k is used to approximate the problem u+u=f, >0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h ^k+1) are obtained uniformly in ,h the maximum stepsize, fork2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.Work performed while visiting the IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y.