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)     

Über die Häufigkeit großer Differenzen konsekutiver Primzahlen

The following theorem is going to be proved. Letp _m be them-th prime and putd _m :=p _m+1–p _m . LetN(,T), 1/21,T3. denote the number of zeros =+i of the Riemann zeta function which fulfill and ||T. Letc2 andh0 be constants such thatN(,T)T ^c(1–) (logT) ^h holds true uniformly in 1/21. Let >0 be given. Then there is some constantK>0 such that      

On the circles of Apollonius associated with a triangle

Given a ratio , >>0, and a triangle ABC, on the sides and , using ratios , and , three circles of Apollonius are denned. In this paper, we will show that the three centers are collinear, the circles are coaxal and develop a necessary and sufficient condition that these circles intersect. J. A. Hoskins, W. D. Hoskins and R. G. Stanton obtained these results in a recent paper using algebraic computation. Our aim is to establish all these results using only results from elementary Euclidean geometry and thereby uncovering more geometric insights and avoid lengthy calculations.     

Mittelwert der Eulerschen φ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern

LetK be an algebraic number field, and for every integer K let () andd(), respectively, denote the number of relatively prime residue classes and the number of divisors of the principal ideal (). Asymptotic equalities are proved for the sums () and d ²(), where runs through certain finite sets of integers ofK.     

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.     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

Angular limits of holomorphic functions which satisfy an integrability condition

Let (–1,1), let 2/(1–)p<, letp denote the Hölder conjugate ofp, and let be an open arc of the unit circle. It is shown that, iff is a holomorphic function on the unit disc such that: (i) (1–|z|)log⁺|f(z)| isL ^p -integrable on the sector {r:0f has an infinite asymptotic value has -finite (2–(1+)p)-dimensional Hausdorff, measure, thenf has finite angular limits on a subset of of positive linear measure. In fact, a stronger conclusion will be established.     

Über geschlossene affine Zwangläufe in der Ebene

We consider integral coverings y:{1,2,..,} of an affine plane which occur when is moved under a continuous periodic affine motion(t):. One can distinguish normal points × , i.e. is constant in a certain neighborhood of x, and singular points. If (x) is the number of times x passes through its orbit (t)x all normal points x have (x)=1, and the set of all singular points consists of a number of isolated points and lines. If (x) is the tangent rotation number of the orbit of x all singular points lie on the moving pole curve.     

The ε c -Product of a Schwartz b-Space by a Quotient Banach Space and Applications

We define the -product of a -space by a quotient Banach space. We give conditions under which this -product will be monic. Finally, we define the _c -product of a Schwartz b-space by a quotient Banach space and we give some examples of applications.     

Independence in hypergraphs

Suppose an integral function (|A|)q₁ defined on the subsets of edges of a hypergraph (X,u,) satisfies the following two conditions: 1) any set W u such that |A|(|A|) for any AW is matroidally independent; 2) if W is an independent set, then there exists a unique partitionW=T₁+ T₂+...+T_v such that |T ⁱ |=(|T ⁱ |),i1:v, and for any AW, |A|(|A|) there exists a T_i such that AT_i. The form of such a function is found, in terms of parameters of generalized connected components, hypercycles, and hypertrees.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 196–204, 1982.     

Interfaces and typical Gibbs configurations for one-dimensional Kac potentials

Summary We consider a one dimensional Ising spin system with a ferromagnetic Kac potential J(|r|),J having compact support. We study the system in the limit, »0, below the Lebowitz-Penrose critical temperature, where there are two distinct thermodynamic phases with different magnetizations. We prove that the empirical spin average in blocks of size ^–1 (for any positive ) converges, as »0, to one of the two thermodynamic magnetizations, uniformly in the intervals of size ^–p , for any given positivep1. We then show that the intervals where the magnetization is approximately constant have lengths of the order of exp(c ^–1),c>0, and that, when normalized, they converge to independent variables with exponential distribution. We show this by proving large deviation estimates and applying the Ventsel and Friedlin methods to Gibbs random fields. Finally, if the temperature is low enough, we characterize the interface, namely the typical magnetization pattern in the region connecting the two phases.The research has been partially supported by CNR, GNFM, GNSM and by grant SC1CT91-0695 of the Commission of European Communities     

Some propositions related to a dilation theorem of W. Benz

Summary This paper begins with another proof of a theorem of W. Benz [2] concerning dilations in normed linear spaces. Our proof motivates several questions which are addressed thereafter. For instance it is shown that, ifI is an open interval in ,: I ⁿ , is continuously differentiable and there exista ₁,...,a _n I such that {(a ₁,...,(a _n )} is linearly independent, then {(t): t I} contains a Hamel basis for ⁿ over .     

Nonclassical analogs of solitons in quantum field theory

The possible existence of quantum copies of classical soliton solutions that cannot exist if the effective Planck constant of the theory tends to zero is discussed. In the framework of the standard expansion in , such nonclassical solitons have energy of orderO() and thus occupy an intermediate position between purely classical configurations and second-quantized degrees of freedom. Analytic calculations are made for the profile functions, masses, and stability of such configurations in the ⁴ kink model.Physics Faculty, Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 1, pp. 52–65, January, 1993.     

Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. Part II: The general case

We consider the abstract dynamical framework of [LT3, class (H.2)] which models a variety of mixed partial differential equation (PDE) problems in a smooth bounded domain ⁿ , arbitraryn, with boundaryL ₂-control functions. We then set and solve a min-max game theory problem in terms of an algebraic Riccati operator, to express the optimal quantities in pointwise feedback form. The theory obtained is sharp. It requires the usual Finite Cost Condition and Detectability Condition, the first for existence of the Riccati operator, the second for its uniqueness and for exponential decay of the optimal trajectory. It produces an intrinsically defined sharp value of the parameter, here called _c (critical), _c0, such that a complete theory is available for > _c, while the maximization problem does not have a finite solution if 0 < < _c. Mixed PDE problems, all on arbitrary dimensions, except where noted, where all the assumptions are satisfied, and to which, therefore, the theory is automatically applicable include: second-order hyperbolic equations with Dirichlet control, as well as with Neumann control, the latter in the one-dimensional case; Euler-Bernoulli and Kirchhoff equations under a variety of boundary controls involving boundary operators of order zero, one, and two; Schroedinger equations with Dirichlet control; first-order hyperbolic systems, etc., all on explicitly defined (optimal) spaces [LT3, Section 7]. Solution of the min-max problem implies solution of theH -robust stabilization problem with partial observation.The research of C. McMillan was partially supported by an IBM Graduate Student Fellowship and that of R. Triggiani was partially supported by the National Science Foundation under Grant NSF-DMS-8902811-01 and by the Air Force Office of Scientific Research under Grant AFOSR-87-0321.     

On the γ-Interior and γ-Closure of a Set

For a set X, let : exp X exp X satisfy A B whenever A B X. In [4], -open subsets of X, -interior iA and -closure cA of A X have been defined. The purpose of the present paper is to show that, under suitable conditions on , explicit formulas furnish iA and cA.     

On the radicals of structural matrix rings

The relationship between the radical of a ringR and a structural matrix ring overR has been determined for some radicals. We continue these investigations, amongst others, determining exactly which radicals have the property (M(,R))=M( _s ,(R))+M( _a ,⁺(R))for any structural matrix ringM(,R) and finding (M(,R)) for any hereditary subidempotent radical .     

A geometric property of extremal surfaces

Let the surface R³ be defined by the equation z = f(x, y), where f(x, y) is a function 3 times continuously differentiable in R². It is proved that if the total (Gaussian) curvature of the surface is nonzero almost everywhere on in the sense of Lebesgue measure in R²), then is extremal, i.e., for almost all (x,y) R² the inequality max (||qx||, ||qy, qf (x, y)) > q^–1/s–. holds for all integral q qo (f), where x is the distance from the real number x to the nearest integer and > 0 is arbitrarily small.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 177–181, February, 1978.In conclusion, the author thanks V. G. Sprindzhuk for suggesting the problem.     

An extension of the fenchel theorem

We give an extension of the Fenchel-Borsuk result by introducing the total absolute torsion-curvature KT() for regular curves whose tangent indicatrix is a piecewise regular curve (-closed curves). We prove that KT() is low bounded by 2 and we give a geometric characterization for the -closed curves whose KT() is minimal.     

Exponential integrability of sub-Gaussian vectors

Summary In this paper we define two classes of Banach space (B, ·)-valued random vectors called sub-Gaussian vectors and -sub-Gaussian vectors. The main purpose of this paper is to prove the exponential integrability of a sub-Gaussian vectorX, that is, for some >0, in the case whereB=L _p . On the other hand, using the arguments ofX. Fernique and M. Talagrand, we also show that the exponential integrability of a -sub-Gaussian vector in an arbitrary separable Banach space.These two definitions of sub-Gaussian vectors and -sub-Gaussian vectors are not comparable, and neither of these definitions is a necessary condition for the exponential integrability. We shall give illuminating examples.     

Minimal Surfaces with an Elastic Boundary

LetD:= { C ³ (