Analytic Arcs Research Articles

On the closed ideals inA(W)

scription of the closed ideals in the disc algebra is shown to apply to an ideal whose hull meets the boundary of the domain in a finite union of analytic arcs. The canonical factorization into inner and outer functions in the disc is replaced by a potential theoretic decomposition theorem, thus allowing essentially the same description to be carried over. The basically local nature of the problem is used to reduce it to the previously known ideal theory of a compact bordered Riemann surface. This reduction is facilitated by a factorization theorem that is proved by potential theoretic methods.

Maximum of the fourth diameter in the family of continua with prescribed capacity

We obtain a complete solution of the problem of the maximum of the fourth diameter in the family of continua with capacity 1. Let E(o, eiα, e−iα). 0<α<π/2, be a continuum of minimum capacity containing the points o, eiα, e−iα; H(α=cap E(o, eiα, e−iα). Let C(α) be the common point of three analytic arcs which form E(o, eiα, e−iα). One shows that the indicated maximum is realized by the continuum e={z:H(α0)z2∈E(o, eiα, e−iα)} whereα0, o<α0<π/2, is a solution of the equation . Any other extremal continuum of the gives problem is an image of e under the mappingz→ eiαz+C (γ is a real and C is a complex constant). One finds the value of the required maximum. The paper contains a brief exposition of the proof of this result.

The law of perceptual stability: Abstract foundations

Confronted with an object of perception, an individual will spontaneously try to identify unambiguously and consistently all its parts; except in rare instances of "illusory phenomena," he will immediately succeed. This elementary fact is formalized in a law of visual perception. It is used to define sets of stable states for a sensory mode of a biological system. As characterized, stable states are to perception as quantum states are to atomic structure: they represent natural states of physical systems. They are shown to be observable and to have an exact mathematical representation.A class of bounded open subsets of a two-dimensional Euclidean space, whose boundaries are piecewise compact analytic arcs, is used to construct a nontrivial mathematical model for stable states. The finitely many components of this mathematical model of a stable state (image) are mapped onto an object of perception (icon) by perceptual judgments. These judgments, which include the judgment of stability, have an exact interpretation in this model. They unify and make precise such traditional notions of psychology as "Gestalt," "figureground," and "(visual) boundary."Postulates for a general theory of perception are given. They are used to establish a formal relationship between biological and subjective studies of sensory phenomena and so provide a framework in which subjective studies can be used to analyze (their associated) biological processes. In applying these methods to cases, all icons are divided into two classes (the static and dynamic cases). The static case is treated.

Orthogonal polynomials and Padé approximants associated with a system of arcs

We study the convergence behavior of the diagonal sequence of the Padé table associated with a function with branch points. Given a set of even number, 2 l, of the branch points, a unique set S is constructed which consists of a number of analytic Jordan arcs ending at the branch points. We assume that these arcs are nonintersecting. Let σ( t) be a complex, never vanishing function defined on S, satisfying a Lipschitz type smoothness condition there, and let X( t) be the monic polynomial of degree 2 l with zeros at the branch points. We construct orthogonal polynomials with respect to the weight function X + −( 1 2 ) (t)σ(t) and study their asymptotic behavior. The orthogonal polynomials are defined without complex conjugation and the domain of integration is S. Some properties of these polynomials yield convergence in capacity of the diagonal sequence of Padé approximants to f(t) = ∫ sdt′X + −( 1 2 ) (t′)σ(t′ − t) −1 in any closed, bounded domain of the complex plain cut along S.

Regular analytic arcs and curves

Regular analytic arcs and curves

Closure of Similarity Orbits of Hilbert Space Operators, II: Normal Operators

Let L(H) be the algebra of operators on a complex separable Hilbert space H, let T ε L(H) and let ℓ(T) = {WTW1 : W is invertible in L(H)} be the “ similarity orbit ” of T. The norm closurey ℓ(N)− of ℓ(N) is completely characterized “up to compact perturbations” for N normal. A partial answer is also given for normal operators in non-separable Hilbert spaces. For the separable case, it is shown that if the spectrum of T is the union of a totally disconnected set and finitely many pairwise disjoint subsets of regular analytic Jordan arcs (or Jordan curves) and is perfect, then ℓ(T)− = ℓ(A)− for every A having the same spectrum as T. If T is a spectral operator, then ℓ(T)− contains the “scalar part” of T. The results also include some information about those operators T such that ℓ(T)− is maximal or minimal with respect to inclusion, in the family of the sets {ℓ(A)− : A ɛ L(H)}.

Toeplitz Matrices Generated by the Laurent Series Expansion of an Arbitrary Rational Function

Let ${T_n}(f) = ({a_{i - j}})_{i,j = 0}^n$ be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function. An identity is developed for $\det ({T_n}(f) - \lambda )$ which may be used to prove that the limit set of the eigenvalues of the ${T_n}(f)$ is a point or consists of a finite number of analytic arcs.

Toeplitz matrices generated by the Laurent series expansion of an arbitrary rational function

Let ${T_n}(f) = ({a_{i - j}})_{i,j = 0}^n$ be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function. An identity is developed for $\det ({T_n}(f) - \lambda )$ which may be used to prove that the limit set of the eigenvalues of the ${T_n}(f)$ is a point or consists of a finite number of analytic arcs.

Singularities of solutions to linear, second order, analytic elliptic equations in two independent variables. II

The analysis of Part I is extended to admit of a closed boundary composed of intersecting analytic arcs. The boundary data have branch point singularities for values of the arclength parameter corresponding to intersections of the boundary arcs. These singularities are associated with nonuniformity of the solution to the boundary value problem at the intersections. Complex characteristics and the generalized Huygens principle are used to relate singularities of the data to those of the solution. By way of illustration, the analytic continuation of the boundary data is performed for the Laplace equation whose solution satisfies a simple Neumann condition on a boundary consisting of a semicircle and its diameter

Addendum to "Analytic Functions and Jordan Arcs"

Addendum to "Analytic Functions and Jordan Arcs"

Addendum to “Analytic functions and Jordan arcs”

Professor Royden has kindly pointed out that in the theorem of [21 E may be replaced by any compact set K of positive two-dimensional measure and that f may be taken to be quasiconformal on all of S. Indeed, one has only to choose f as the normal solution of the Beltrami equation with ,u = 2XK, where XK is the characteristic function of K [1, p. 97]. Rothberger's example remains of considerable interest, however, since E has measure zero. On the other hand, f (E) has positive measure, so that f is not quasiconformal on S. In conclusion, we note that the arc J may also be taken to have zero measure; this follows from an elementary linearization argument suggested by Professor Rudin.

The Yang-Lee distribution of zeros for a classical one-dimensional fluid

The interaction potential is assumed to satisfy (γ) = + infinity if γ 2a, where a is a constant greater than zero, so that only nearest neighbours can interact. At any fixed temperature T let kTπ(z) be the thermodynamic pressure at fugacity z gt-or-equal, slanted 0, as calculated from the equation of state. Let Π (z) be the complete analytic function obtained by analytic continuation of π(z) into the complex z plane, and G be the set of values of z for which one branch of Π (z), say Π max(z), is regular and has a larger real part than all the others. It is proved that, in the limit where the length L of the system tends to infinity, the zeros of the grand partition function Ξ (z, L) approach a point set Z which consists of analytic arcs and is the complement of G. It is also proved that G is simply connected, that for all z in G, and that the limiting line density of zeros of Ξ along any arc of Z (each zero being given the weight L−1) is (2π)−1 times the discontinuity in Imπmax(z) across the arc. As an illustration, (a result of Hemmer et al.), that Z is - infinity < z less-than-or-eq, slant -1/ea for the hard-rod system is confirmed rigorously.

Shorter Notes: Analytic Functions and Jordan Arcs

Shorter Notes: Analytic Functions and Jordan Arcs

Sheets of Real Analytic Varieties

In a previous paper (4) the author worked out some results on the analytic connectivity properties of real algebraic varieties, that is to say, properties associated with the joining of points of the variety by analytic arcs lying on the variety. It is natural to ask whether these properties can be carried over to analytic varieties, since the proofs in the algebraic case depend mainly on local properties. But although this generalization can be carried out to a large extent, there are, nevertheless, difficulties in the analytic case, owing mainly to the fact (cf. 2, § 11) that a real analytic variety may not be definable by means of a set of global equations. Thus, although the general idea of the treatment given here is the same as in (4), some variation in the details of the method has proved to be necessary, and some of the final results are slightly weaker in form.

Analytic Coordinate Systems and Arcs in a Manifold

Analytic Coordinate Systems and Arcs in a Manifold

On Some Properties of the Medians of Closed Continuous Curves Formed by Analytic Arcs

On Some Properties of the Medians of Closed Continuous Curves Formed by Analytic Arcs

Conformal Classification of Analytic Arcs or Elements: Poincare's Local Problem of Conformal Geometry

Conformal Classification of Analytic Arcs or Elements: Poincare's Local Problem of Conformal Geometry

The Inscribed and Circumscribed Squares of a Quadrilateral and Their Significance in Kinematic Geometry

Professor Emch has shown that in any closed convex curve at least one square may be inscribed* and that about such a curve at least one square may be circumscribed. Moreover, the results may be extended to convex polygons formed by analytic arcs. This investigation leads to methods for actually constructing the inscribed squares of all regular polygons and any quadrilateral and the circumscribed square of any quadrangle. By inscribed square is meant one whose vertices lie on four sides of the figure and by circumscribed square is meant a square each of whose vertices passes through a vertex of the quadrangle. Because of the number of interesting propositions of kinematic geometry resulting from the solution of these problems for the quadrilateral and quadrangle, I shall confine myself to their discussion. THEOREM I. In every quadrilateral may be inscribed at least one square having a vertex on each of the four sides. If there is more than one such square in a given quadrilateral, there is an infinite number.