scholarly journals Unbounded Fredholm Modules and Spectral Flow

1998 ◽  
Vol 50 (4) ◽  
pp. 673-718 ◽  
Author(s):  
Alan Carey ◽  
John Phillips
Keyword(s):  
Homology Class ◽  
Adjoint Operator ◽  
Spectral Flow ◽  
Continuous Path ◽  
Fredholm Operators ◽  
Positive Direction ◽  
Smooth Path ◽  
Dense Subalgebra ◽  
Image Position ◽  
The Right

AbstractAn odd unbounded (respectively, p-summable) Fredholm module for a unital Banach *-algebra, A, is a pair (H,D) where A is represented on the Hilbert space, H, and D is an unbounded self-adjoint operator on H satisfying:(1) (1 + D2)-1 is compact (respectively, Trace_(1 + D2)-(p/2)_∞), and(2) ﹛a ∈ A | [D, a] is bounded﹜ is a dense *- subalgebra of A.If u is a unitary in the dense *-subalgebra mentioned in (2) thenuDu* = D + u[D, u*] = D + Bwhere B is a bounded self-adjoint operator. The pathis a “continuous” path of unbounded self-adjoint “Fredholm” operators. More precisely, we show thatis a norm-continuous path of (bounded) self-adjoint Fredholm operators. The spectral flow of this path is roughly speaking the net number of eigenvalues that pass through 0 in the positive direction as t runs from 0 to 1. This integer,recovers the pairing of the K-homology class [D] with the K-theory class [u].We use I.M. Singer's idea (as did E. Getzler in the θ-summable case) to consider the operator B as a parameter in the Banach manifold, Bsa(H), so that spectral flow can be exhibited as the integral of a closed 1-formon this manifold. Now, for B in ourmanifold, any X ∈ TB_Bsa(H)_ is given by an X in Bsa(H) as the derivative at B along the curve t→ B + tX in the manifold. Then we show that for m a sufficiently large half-integer:is a closed 1-form. For any piecewise smooth path {Dt = D + Bt} with D0 and D1 unitarily equivalent we show thatthe integral of the 1-form ã. If D0 and D1 are not unitarily equivalent, wemust add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form:for an integer. The unbounded case is proved by reducing to the bounded case via the map . We prove simultaneously a type II version of our results.

Self-Adjoint Fredholm Operators And Spectral Flow

1996 ◽  
Vol 39 (4) ◽  
pp. 460-467 ◽  
Author(s):  
John Phillips
Keyword(s):  
Hilbert Space ◽  
Functional Calculus ◽  
Separable Hilbert Space ◽  
Spectral Flow ◽  
Intersection Numbers ◽  
Fredholm Operators ◽  
Positive Direction ◽  
The Family ◽  
Pass Through ◽  
Homotopy Groupoid

AbstractWe study the topology of the nontrivial component, , of self-adjoint Fredholm operators on a separable Hilbert space. In particular, if {Bt} is a path of such operators, we can associate to {Bt} an integer, sf({Bt}), called the spectral flow of the path. This notion, due to M. Atiyah and G. Lusztig, assigns to the path {Bt} the net number of eigenvalues (counted with multiplicities) which pass through 0 in the positive direction. There are difficulties in making this precise — the usual argument involves looking at the graph of the spectrum of the family (after a suitable perturbation) and then counting intersection numbers with y = 0.We present a completely different approach using the functional calculus to obtain continuous paths of eigenprojections (at least locally) of the form . The spectral flow is then defined as the dimension of the nonnegative eigenspace at the end of this path minus the dimension of the nonnegative eigenspace at the beginning. This leads to an easy proof that spectral flow is a well-defined homomorphism from the homotopy groupoid of onto Z. For the sake of completeness we also outline the seldom-mentioned proof that the restriction of spectral flow to is an isomorphism onto Z.

Residual Gas Reactions in the Electron Microscope: I. Qualitative Observations on the Water Gas Reaction

1968 ◽  
Vol 26 ◽  
pp. 292-293
Author(s):  
Richard E. Hartman ◽  
Roberta S. Hartman ◽  
Peter L. Ramos
Keyword(s):  
Electron Beam ◽  
Electron Microscope ◽  
Gas Phase ◽  
Absolute Temperature ◽  
Residual Gas ◽  
Gas Reactions ◽  
Image Position ◽  
The Right ◽  
Water Gas ◽  
Thinning Rate

The action of water and the electron beam on organic specimens in the electron microscope results in the removal of oxidizable material (primarily hydrogen and carbon) by reactions similar to the water gas reaction .which has the form:The energy required to force the reaction to the right is supplied by the interaction of the electron beam with the specimen.The mass of water striking the specimen is given by:where u = gH2O/cm2 sec, PH2O = partial pressure of water in Torr, & T = absolute temperature of the gas phase. If it is assumed that mass is removed from the specimen by a reaction approximated by (1) and that the specimen is uniformly thinned by the reaction, then the thinning rate in A/ min iswhere x = thickness of the specimen in A, t = time in minutes, & E = efficiency (the fraction of the water striking the specimen which reacts with it).

The Effect of Set on the Two-Point Tactual Threshold

1966 ◽  
Vol 18 (2) ◽  
pp. 169-174 ◽  
Author(s):  
Anima Sen
Keyword(s):  
University Students ◽  
Test Stimulus ◽  
Opposite Direction ◽  
Adaptation Level ◽  
Critical Stimulus ◽  
Positive Direction ◽  
Control Series ◽  
Longitudinal Line ◽  
Adaptation Level Theory ◽  
The Right

This experiment studies the influence of set on the two-point tactual threshold. The two-point limen (critical stimulus) was determined along the mid-longitudinal line of both volar and dorsal surfaces of the right forearms of seven university students. Test-stimuli were selected at 3 mm. steps both up and down from each of the critical stimuli. Each of these test-stimuli was presented separately, the critical stimulus being interpolated 30 times in each test of the series. The proportions of two-point and one-point responses to the critical stimulus were then determined again in a control series. It was found that, as the test-stimulus deviated in the positive direction from the critical stimulus, the proportion of two-point responses to the critical stimulus increased, reached a maximum and then began to decline. A similar rise and fall in one-point responses were found in the opposite direction. The results were explained by Adaptation Level Theory.

Register machine proof of the theorem on exponential diophantine representation of enumerable sets

1984 ◽  
Vol 49 (3) ◽  
pp. 818-829 ◽  
Author(s):  
J. P. Jones ◽  
Y. V. Matijasevič
Keyword(s):  
Natural Number ◽  
Polynomial Time ◽  
Simple Proof ◽  
Diophantine Equations ◽  
Polynomial Equation ◽  
Exponential Diophantine Equations ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Exponential Relation

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A(a1, …, an) is exponential diophantine, i.e. can be represented in the formwhere a1 …, an, x1, …, xm range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B, multiplication, AB, and exponentiation, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns.Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.

scholarly journals The Chaetotaxy of the Pupa of Stegomyia fasciata

1920 ◽  
Vol 10 (2) ◽  
pp. 161-169 ◽  
Author(s):  
J. W. S. Macfie
Keyword(s):  
The Body ◽  
Posterior Angle ◽  
Image Position ◽  
The Right ◽  
Two Sides

The pupa is bilaterally symmetrical, that is, setae occur in similar situations on each side of the body, so that it will suffice to describe the arrangement on one side only. The setae on the two sides of the same pupa, however, often vary as regards their sub-divisions, and similar variations occur between different individuals; as an example, in Table I are shown some of the variations that were found in ten pupae taken at random. An examination of a larger number would have revealed a wider range. As a rule, a seta which is sometimes single, sometimes divided, is longer when single. For example, in one pupa the seta at the posterior angle ofthe seventh segment was single on the right side, double on the left; the former measuring 266μ, and the latter only 159μ in length. This fact is not specifically mentioned in the descriptions which follow, but should be understood.

On the Non-Existence of Certain Euler Products

1980 ◽  
Vol 23 (3) ◽  
pp. 371-372
Author(s):  
M. V. Subbarao
Keyword(s):  
Euler Product ◽  
Product Decomposition ◽  
Euler Products ◽  
Image Position ◽  
The Right

In a paper with the above title, T. M. Apostol and S. Chowla [1] proved the following result:Theorem 1.For relatively prime integers h and k, l ≤ h ≤ k, the seriesdoes not admit of an Euler product decomposition, that is, an identity of the form1except when h = k = l; fc = 1, fc = 2. The series on the right is extended over all primes p and is assumed to be absolutely convergent forR(s)>1.

The rate of convergence of extremes of stationary normal sequences

1983 ◽  
Vol 15 (01) ◽  
pp. 54-80 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Joint Distribution ◽  
Point Processes ◽  
Rates Of Convergence ◽  
Joint Distribution Function ◽  
Maximal Correlation ◽  
Standard Normal ◽  
Image Position ◽  
The Right ◽  
Right Order

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r 1 , r 2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

A Segal-Langevin Type Stochastic Differential Equation on a Space Of Generalized Functionals

1992 ◽  
Vol 44 (3) ◽  
pp. 524-552 ◽  
Author(s):  
Gopinath Kallianpur ◽  
Itaru Mitoma
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Linear Functional ◽  
Adjoint Operator ◽  
Solution Process ◽  
Weak Form ◽  
Diffusion Operator ◽  
The Central Limit Theorem ◽  
Spatially Extended ◽  
Image Position

AbstractLet E′ be the dual of a nuclear Fréchet space E and L*(t) the adjoint operator of a diffusion operator L(t) of infinitely many variables, which has a formal expression:A weak form of the stochastic differential equationdX(t) = dW(t) + L*(t)X(t)dtis introduced and the existence of a unique solution is established. The solution process is a random linear functional (in the sense of I. E. Segal) on a space of generalized functionals on E′. The above is an appropriate model for the central limit theorem for an interacting system of spatially extended neurons. Applications to the latter problem are discussed.

scholarly journals Boundary problems for Riccati and Lyapunov equations

1986 ◽  
Vol 29 (1) ◽  
pp. 15-21 ◽  
Author(s):  
Lucas Jódar
Keyword(s):  
Boundary Condition ◽  
Boundary Problem ◽  
Transport Theory ◽  
Adjoint Operator ◽  
Linear Operators ◽  
Dimensional Case ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Image Position

The resolution problem of the systemwhere U(t), A, B, D and Uo are bounded linear operators on H and B* denotes the adjoint operator of B, arises in control theory, [9], transport theory, [12], and filtering problems, [3]. The finite-dimensional case has been introduced in [6,7], and several authors have studied the infinite-dimensional case, [4], [13], [18]. A recent paper, [17],studies the finite dimensional boundary problemwhere t ∈[0,b].In this paper we consider the more general boundary problemwhere all operators which appear in (1.2) are bounded linear operators on a separable Hilbert space H. Note that we do not suppose C = −B* and the boundary condition in (1.2) is more general than the boundary condition in (1.1).

scholarly journals Unbounded Fredholm Operators and Spectral Flow

2005 ◽  
Vol 57 (2) ◽  
pp. 225-250 ◽  
Author(s):  
Bernhelm Booss-Bavnbek ◽  
Matthias Lesch ◽  
John Phillips
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Direct Methods ◽  
Spectral Flow ◽  
Surprising Result ◽  
Graph Distance ◽  
Manifolds With Boundary ◽  
Fredholm Operators ◽  
Homotopy Invariance ◽  
Definition Of

AbstractWe study the gap (= “projection norm” = “graph distance”) topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transformand direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

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