Spectral asymmetry and Riemannian geometry. III

1976 ◽  
Vol 79 (1) ◽  
pp. 71-99 ◽  
Author(s):  
M. F. Atiyah ◽  
V. K. Patodi ◽  
I. M. Singer
Keyword(s):  
Fundamental Group ◽  
Real Number ◽  
Analytic Continuation ◽  
Riemannian Geometry ◽  
Compact Manifold ◽  
Adjoint Operator ◽  
The Real ◽  
Spectral Asymmetry ◽  
Image Position ◽  
Adjoint Operators

In Parts I and II of this paper ((4), (5)) we studied the ‘spectral asymmetry’ of certain elliptic self-adjoint operators arising in Riemannian geometry. More precisely, for any elliptic self-adjoint operator A on a compact manifold we definedwhere λ runs over the eigenvalues of A. For the particular operators of interest in Riemannian geometry we showed that ηA(s) had an analytic continuation to the whole complex s-plane, with simple poles, and that s = 0 was not a pole. The real number ηA(0), which is a measure of ‘spectral asymmetry’, was studied in detail particularly in relation to representations of the fundamental group.

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

On the essential spectrum of self-adjoint operators

1980 ◽  
Vol 86 (3-4) ◽  
pp. 261-274
Author(s):  
B. J. Harris
Keyword(s):  
Differential Expression ◽  
Essential Spectrum ◽  
Adjoint Operator ◽  
Matrix Differential ◽  
Image Position ◽  
Adjoint Operators

SynopsisWe provide estimates of the formfor the length of gap centre μ in the essential spectrum of a self-adjoint operator generated by a matrix differential expression.

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

Banach games

1984 ◽  
Vol 49 (2) ◽  
pp. 343-375 ◽  
Author(s):  
Chris Freiling
Keyword(s):  
Real Number ◽  
Perfect Information ◽  
Real Numbers ◽  
The Real ◽  
Infinite Game ◽  
Image Position ◽  
The Relationship

Abstract.Banach introduced the following two-person, perfect information, infinite game on the real numbers and asked the question: For which sets A ⊆ R is the game determined?Rules: The two players alternate moves starting with player I. Each move an is legal iff it is a real number and 0 < an, and for n > 1, an < an−1. The first player to make an illegal move loses. Otherwise all moves are legal and I wins iff exists and .We will look at this game and some variations of it, called Banach games. In each case we attempt to find the relationship between Banach determinacy and the determinacy of other well-known and much-studied games.

6.—On the Distribution of the Eigenvalues and the Order of the Eigenfunctions of a Fourth-order Singular Boundary Value Problem

1972 ◽  
Vol 71 (1) ◽  
pp. 61-75
Author(s):  
Jyoti Chaudhuri ◽  
W. N. Everitt
Keyword(s):  
Boundary Value Problem ◽  
Real Number ◽  
Boundary Value ◽  
Asymptotic Properties ◽  
Singular Boundary ◽  
Positive Real ◽  
The Real ◽  
Asymptotic Formulae ◽  
Eigenvalue Parameter ◽  
Image Position

SynopsisThis paper is concerned with the asymptotic properties of the eigenvalues and eigenfunctions of the boundary value problemWith suitable restrictions placed on the real-valued coefficient q the spectrum of this problem, with respect to the eigenvalue parameter λ, is discrete; let {λn; n = 1, 2, …} and {ψn; n = 1, 2, …} be the eigenvalues and associated eigenfunctions. Asymptotic formulae are obtained for N(λ), the number of eigenvalues not exceeding the real number λ, and for ψn(x) as n→∞ where x is a fixed, positive real number.

scholarly journals Class Numbers and Biquadratic Reciprocity

1982 ◽  
Vol 34 (4) ◽  
pp. 969-988 ◽  
Author(s):  
Kenneth S. Williams ◽  
James D. Currie
Keyword(s):  
Real Number ◽  
Class Number ◽  
Quadratic Field ◽  
Class Numbers ◽  
Kronecker Symbol ◽  
The Real ◽  
Unique Manner ◽  
Image Position

0. Notation. Throughout this paper p denotes a prime congruent to 1 modulo 4. It is well known that such primes are expressible in an essentially unique manner as the sum of the squares of two integers, that is,(0.1)with |a| and |b| uniquely determined by (0.1). Since a is odd, replacing a by –a if necessary, we can specify a uniquely by(0.2)Further, as {[(p – l)/2]!}2 = – 1 (mod p), we can specify b uniquely by(0.3)These choices are assumed throughout.The following notation is also used throughout the paper: h(d) denotes the class number of the quadratic field of discriminant d, (d/n) is the Kronecker symbol of modulus |d|, [x] denotes the greatest integer less than or equal to the real number x, and {x} = x – [x].

scholarly journals Delange's characterization of the sine function

1974 ◽  
Vol 15 (1) ◽  
pp. 66-68 ◽  
Author(s):  
Chin-Hung Ching ◽  
Charles K. Chui
Keyword(s):  
Entire Function ◽  
Analytic Continuation ◽  
Complex Plane ◽  
Real Line ◽  
Sine Function ◽  
The Real ◽  
Image Position

In [2], H. Delange gives the following characterization of the sine function.Theorem A. f(x)=sin x is the only infinitely differentiable real-valued function on the real line such that f'(O)= 1 andfor all real x and n = 0,1,2,….It is clear that, if f satisfies (1), then the analytic continuation of f is an entire function satisfyingfor all z in the complex plane. Hence f is of at most order one and type one. In this note, we prove the following theorem.

On the almost sure approximation of self-adjoint operators in L2 (0, 1)

1996 ◽  
Vol 119 (3) ◽  
pp. 537-543
Author(s):  
L. J. Ciach ◽  
R. Jajte ◽  
A. Paszkiewicz
Keyword(s):  
Adjoint Operator ◽  
Almost Sure Convergence ◽  
The Other ◽  
Orthogonal Projections ◽  
Weak Operator Topology ◽  
Weak Operator ◽  
Image Position ◽  
Adjoint Operators ◽  
Orthogonal Systems ◽  
Martingale Convergence

There are several important theorems concerning the almost sure convergence of (monotone) sequences of orthogonal projections in L2-spaces. Let us mention here the martingale convergence theorems or the results on the developments of functions with respect to orthogonal systems. On the other hand every self-adjoint operator with the spectrum on the interval [0, 1] is a limit of some sequence of orthogonal projections in the weak operator topology (see [1]). This paper is devoted to a problem of approximation of a self-adjoint operator A acting in L2 (0, 1) by a sequence Pn of orthogonal projections in the sense that

scholarly journals Real parts of quasi-nilpotent operators

1979 ◽  
Vol 22 (3) ◽  
pp. 263-269 ◽  
Author(s):  
P. A. Fillmore ◽  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Dimensional Case ◽  
Nilpotent Operator ◽  
The Real ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Nilpotent Operators ◽  
Finite Dimensional Case ◽  
Adjoint Operators

The purpose of this paper is to answer the question: which self-adjoint operators on a separable Hilbert space are the real parts of quasi-nilpotent operators? In the finite-dimensional case the answer is: self-adjoint operators with trace zero. In the infinite dimensional case, we show that a self-adjoint operator is the real part of a quasi-nilpotent operator if and only if the convex hull of its essential spectrum contains zero. We begin by considering the finite dimensional case.

scholarly journals On Square Roots and Logarithms of Self-Adjoint Operators

1958 ◽  
Vol 4 (1) ◽  
pp. 1-2 ◽  
Author(s):  
C. R. Putnam
Keyword(s):  
Hilbert Space ◽  
Sufficient Condition ◽  
Square Root ◽  
The Real ◽  
Square Roots ◽  
Simple Sufficient Condition ◽  
Image Position ◽  
Adjoint Operators

All operators considered in this paper are bounded and linear (everywhere defined) on a Hilbert space. An operator A will be called a square root of an operator B ifA simple sufficient condition guaranteeing that any solution A of (1) be normal whenever B is normal was obtained in [1], namely: If B is normal and if there exists some real angle θ for which Re(Aeιθ)≥0, then (1) implies that A is normal. Here, Re (C) denotes the real part ½(C + C*) of an operator C.

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