Original Theorem Research Articles

Bound excited states in density-functional theory

Using recent modifications of the original Hohenberg-Kohn theorem due to Levy [Proc. Natl. Acad. Sci. USA 76, 6062 (1979)] and Valone [J. Chem. Phys. 73, 1344 (1980)] and the modified Ritz variational principles [J.K.L. MacDonald, Phys. Rev. 46, 828 (1934)] alternative density functionals are exhibited which respect the bounds of the modified principles. Excited-state energies and electron densities may be calculated by direct minimization of the new functionals. The density functional ${R}_{2}[\ensuremath{\rho},U]$ obeys the bound ${R}_{2}[\ensuremath{\rho},U]\ensuremath{\ge}{({E}_{b}\ensuremath{-}U)}^{2}$, where $U$ is a fixed constant energy and ${E}_{b}$ is the bound-state energy of the system closest to $U$. At present, it appears that the functional depends nontrivially on the external potential. Some properties of reduced density-matrix functionals are presented. The nature of the 2 matrix functional, ${R}_{2}[\ensuremath{\rho},U]$, provides clues to the nature of the density functional. The evident dependence of ${R}_{2}[\ensuremath{\rho},U]$ on the external potential indicates that the presence of excited states in the Levy density functional is very unlikely. The point of view in this paper is found to complement the recently proposed Theophilou density functional for excited states.

Iterative methods for quasilinear hyperbolic systems in the first canonic form

Quasilinger hyberbolic equation and system in the “first canonic” (diagonal) from areconsidered, with both Cauchy data and boundary data “a la Cesari”, and an arbitrary number of independent variables. The aim of the paper is to develop iterative methods converging uniformly to the a.e. solution of the Cauchy or boundary value problem, based on a modified version of the proof of Cesari's original existence and uniqueness theorem and on some general results concerning contractive Lipschitz maps in the product of two Banach spaces, as derived in earlier papers. The existence proof given here is in fact founded on a new contraction map in the product of two (closed convex subsets of) Banach function spaces.

Preference strength and two kinds of ordinalism

In his paper Extended Sympathy and the Possibility of Social Choice, Professor Arrow first considers an approach to individual utilities which is and rules out interpersonal comparisons. This is the approach he took when he formulated his original impossibility theorem [1]. He then takes up an approach which allows interperso~al comparisons, but only of an ordinal kind. The strictly ordinal approach without interpersonal comparisons turns out to lead to a dead end, and the second approach the one that allows interpersonal comparisons of an ordinal kind alone leads either to lexical mz'ximax or to lexical maximin. What are we to make of ~11 this? The approach without interpersonal comparisons leads to a dead end in the sense that it is inconsistent with certain other conditions: in the version of the Arrow theorem in the paper I am discussing, with the conditions of Binary Relevancy, Anonymity, and the Weak Pareto Principle. A further condition is built into the formalism Arrow uses: that the just or optimal alternatives from among the feasible alternatives are determined by a weak ordering which is independent of the feasibility of alternatives. The ordinal approach without interpersonal comparisons, then, is discredited by the Arrow theorem only in so far as these other conditions are plausible. Just what, more precisely, is the approach that is supposed to lead to trouble? I can think of two positions the term 'a strictly ordinal approach' might designate; I shall call them 'Basic Ordinalism' and 'Pairwise Ordinalism'. One is this.

Strassen type limit points for moving averages of a wiener process

AbstractLet {W(s); 8 ≥ 0} be a standard Wiener process, and let βN = (2aN (log (N/aN) + log log N)‐1/2, 0 < αN ≤ N < ∞, where αN↑ and αN/N is a non‐increasing function of N, and define γN(t) = βN[W(nN + taN) − W(nN)), 0 ≥ t ≥ 1, with nN = N – aN. Let K = {x ϵ C[0,1]: x is absolutely continuous, x(0) = 0 and }. We prove that, with probability one, the sequence of functions {γN(t), t ϵ [0,1]} is relatively compact in C[0,1] with respect to the sup norm ||·||, and its set of limit points is K. With aN = N, our result reduces to Strassen's well‐known theorem. Our method of proof follows Strassen's original, direct approach. The latter, however, contains an oversight which, in turn, renders his proof invalid. Strassen's theorem is true, of course, and his proof can also be rectified. We do this in a somewhat more general context than that of his original theorem. Applications to partial sums of independent identically distributed random variables are also considered.

Rice Theorems for ∑n-1 Sets

In [3] Hay proves generalizations of Rice's Theorem and the Rice-Shapiro Theorem for differences of recursively enumerable sets (d.r.e. sets). The original Rice Theorem [5, p. 3G4, Corollary B] says that the index set of a class C of r.e. sets is recursive if and only if C is empty or C contains all r.e. sets. The Rice-Shapiro Theorem conjectured by Rice [5] and proved independently by McNaughton, Shapiro, and Myhill [4] says that the index set of a class C of r.e. sets is r.e. if and only if C is empty or C consists of all r.e. sets which extend some element of a canonically enumerable class of finite sets. Since a d.r.e. set is a difference of r.e. sets, a d.r.e. set has an index associated with it, namely, the pair of indices of the r.e. sets of which it is the difference.

Restricted Ramsey configurations

In recent years there has been great development in the area becoming known as Ramsey Theory. In this paper we explore some restricted Ramsey theorems, best illustrated by the following beautiful result of NeSetIil and RodI. We begin with the first nontrivial example of the original theorem of Ramsey: Given any two-coloring of the edges of KB there exists a monochromatic triangle. Let us use the notation H--f (G), if any c-coloring of the edges of a graph H yields a monochromatic G. In this instance we write K, -+ (K& . It was asked, by P. Erdos, what graphs H have the property H--f (K& , and whether such H exist if you restrict the clique number w(H). In an elegant, though complex, paper Folkman [4] showed that there exist H satisfying H---f (K& , where w(H) = 3. A full generalization, using a totally different method, was given by NeSetIil and Rod1 [6], who showed that for all G, c there exists a graph H so that H -+ (G), and w(H) = w(G).

On an extension of Hilbert's second ε-theorem

The aim of this paper is to correct an error in the proof in [2] of a strengthened version of Hilbert's second ε-theorem.Hilbert's original theorem says (effectively) that formulae of the form ∃xA → A(εxA) can be eliminated from deductions of ∈-free sentences B (i.e. those which do not contain the ε-symbol) from collections X of ε-free sentences. The extended version (Theorem III. 11 of [2]) says that, in addition, instances of Ackermann's schema can also be eliminated.The error in [2] occurs in the proof of Theorem III.7 and is described as follows. If A′ is the formula obtained from A by replacing every occurrence of ∃yB by B(εyB) (or ∀yC by C(εyB) if B is of the form ¬ C), and if A is a Q3 or Q4-axiom then, contrary to the claim on p. 73 of [2], A′ is not necessarily an axiom of the same form. For example, if A is the Q3-axiomwhere the rank of εx∃yD(x, y) is ≤ the rank of εyB (see p. 70) and where B is D(t,y), then A′ iswhich is not a Q3-axiom. In other words, the notion of rank as defined in [2] is inadequate for the proof of Theorem III.11.

A generalization of Ramsey's theorem for regular trees

An analog of Ramsey's theorem for regular trees is proved. The original theorem is a special case of it. It is shown that the denumerable case of Ramsey's theorem has no analog for binary trees.

Applications of Group Theory to Connecting Networks

Group theory impinges on the combinatorial study of connecting networks in a natural way: the stages, frames, and cross-connect fields from which many existing networks are built provide simple permutations out of which desired, complex assignments are built by composition. Some of the consequences of this interpretation are explored in this paper. In the group-theoretic setting, the action and role of the stages and fields become transparent, and many questions and results regarding networks can be regarded as problems about cosets, subgroups, factorizations, etc. This approach is particularly useful for the study of rearrangeable networks made of stages of square switches; such a network is rearrangeable if and only if the symmetric group of appropriate degree can be factored into products of certain subgroups associated with the network. Or again, the original Slepian-Duguid rearrangeability theorem corresponds to factoring a symmetric group into a product of double cosets of subgroups generated by stages.

On the Local Convergence of Update Methods

This paper concerns a class of update methods, or, as they are also called, quasi-Newton methods, variable metric methods, or modification methods. A general theory of rank-one and symmetric rank-two update formulas is presented which covers many of the special methods proposed in the literature. Recently, Broyden, Dennis and More found a local convergence theorem for a class of these methods. A new and unified proof of this theorem is given here which uses a geometrically more intuitive and also more general convergence condition than the original theorem. The proof utilizes elliptic-norm estimates to derive a majorizing system of difference inequalities. Then more careful estimates involving a generalized Frobenius norm show that under rather general conditions the methods under consideration are superlinearly convergent.

An abstract Church-Rosser theorem. II: Applications

This paper is a continuation of An abstract form of the Church-Rosser theorem. I (this Journal, vol. 35 (1969), pp. 545–560). In Part I, the Church-Rosser property was deduced from abstract premises (A1)–(A8). The original draft of Part II contained some applications of this result, and a fairly simple abstract result by which the Church-Rosser property could be extended from λβ-reduction to λβη-reduction (Curry's notation [3, Chapter 3]). But since this draft was written, these results have been obtained independently and improved by other workers, and a simple and natural new proof for λβ-reduction has been discovered by W. W. Tait and P. Martin-Löf (see §11 later, and [17, §2.4.3]).So the main purpose of the present Part II is merely to justify the claim in Part I that the abstract theorem does cover the case of λβ-reduction (and various modifications). I shall also include a summary of the main kinds of Church-Rosser proofs. The paragraph and theorem numbers in Part II will continue from those of Part I.In §§5 and 6 below, Theorem 1 will be specialised to reductions defined by replacements of parts of expressions by others (Theorems 2 and 2A). At the end of §6 an important subclass of such reductions will be treated (Theorem 3).In §7, Theorem 3 will be applied to prove the Church-Rosser property for combinatory weak reduction [10, §11B], with or without type-restrictions and extra “arithmetical” reduction-rules (Theorems 4 and 5). (In the original draft Theorem 5 was deduced directly from Theorem 2A; the present intervening Theorem 3 is an independent result of B. Rosen [7].)

Singular side approximations for $2$-spheres in $E^{3}$

Bing has proved that each $2$-sphere in $E^{3}$ can almost be mapped free of itself in the following very nice sense: Suppose that $S$ is a $2$-sphere in $E^{3}$ and $\varepsilon > 0$; then there is an $\varepsilon$-map $$f:S \rightarrow S \cup \mathrm{Int}\,S$$ such that $f(S)\cap S$ and $f^{-1}(f(S)\cap S)$ are $0$-dimensional and $$f|S - f^{-1} (S) \cap S$$ is a homeomorphism. This paper illustrates how Bing's theorem can be used advantageously as a substitute for Bing's original side approximation theorem. The following are the principal results. (1) A $2$-sphere $S$ is tame if it is (singularly) spanned or capped on tame sets.(2) A $2$-sphere $S$ is tame if each of its points is an inaccessible point of a Sierpiński curve in $S$ which can be pushed by a homotopy into each complementary domain of $S$.

The Existence of Optimal Price Vectors in the General Balanced-Growth Model of Gale

I N 1956 Gale [4] considered a general model of balanced growth and asserted the existence of price vectors which equate the economic growth with the technological growth rate. This model of Gale was an extension of fundamental results earlier proven by von Neumann for the case in which the production space was polyhedral. Recently Hulsmann and Steinmetz [6] demonstrated that Gale's theorem was not true by constructing a counterexample. In this paper we shall prove that Gale's theorem in a modified form is true and with a certain regularization the original theorem of Gale is valid. This regularization will be automatically satisfied by polyhedral production spaces so that as a corollary the proof for the polyhedral version of Gale's theorem will be attained. Finally we show that the counterexample of Hulsmann and Steinmetz [6] does not satisfy this regularization.

Bounds for Initial Value Problems

A new method has been developed for finding rigorous upper and lower bounds to the solution of a wide class of initial value problems. The method is applicable to initial value problems of the following type: x(¨t)+f(t,x,x)˙=0,x(0)=X0,x(˙0)=V0, where f is continuous with continuous first derivatives, Lipschitzian, and ∂f/∂x ≥ 0. An original bounding theorem has been formulated and proven and a numerical technique has been developed for finding the bounding functions in analytic form as linear combinations of Tchebyshev polynomials. The method has been applied to several problems of engineering interest.

Infinite exponent partition relations and well-ordered choice

The study of partition relations for cardinal numbers introduced by Erdös and his school in the 1950's has, for the past several years, had a profound impact in logic. Unfortunately, quite early in their development, it was noticed by Rado [1] that the potentially most fruitful class of such relations, infinite exponent partition relations, were always in contradiction with the axiom of choice (AC). As a result, such relations were overlooked. This turned out to be a mistake; for, as has been noticed recently, a close study of infinite exponent partition relations is both interesting and rewarding. For example, there are weakened versions of such relations which are provable in ZF and which have valuable applications in recursion theory and set theory. In addition, the pure theory of these relations, like that of the axiom of determinateness, is fruitful as well as elegant. For more background here one should refer to [3].At any rate, with a more detailed look at infinite exponent partition relations came a more refined version of Rado's original theorem. Specifically, Rado used the full axiom of choice to carefully construct partitions to violate any desired relation—a more sophisticated look at the actual theory of such relations indicated how one could put together some desired partitions using only well-ordered choice [3].The distinction between well-ordered choice and full choice is by no means vacuous in this context. For Mathias has shown [4] that the simplest infinite exponent partition relation, ω → (ω)ω, is consistent with countable choice (well-ordered choice of length ℵ0) and, in fact, is consistent with dependent choice.

The early days of information theory

Shannon's communication (information) theory cast about as much light on the problem of the communication engineer as can be shed. It reflected or encompassed earlier work, but it was so new that it was not really understood by many of those who first talked or wrote about it. Most of the papers published on information theory through 1950 are irrelevant to Shannon's work. Later work has given us useful information and encoding schemes as well as greater rigor, but the wisdom of Shannon's way of looking at things and his original theorems are of primary importance.

The Jahn–Teller Effect in Molecules and Crystals

R Englman Chichester: Wiley 1972 pp xx + 350 price £8 The Jahn–Teller effect has come to include almost all manifestations of the coupling of degenerate electron states to nuclear vibrations. The original theorem showed that there should be an instability, namely that for any orbitally degenerate molecule or complex (linear systems and Kramers' degeneracy excluded) there will be a distortion which lowered the symmetry and lowered the total energy.

The spectral theorem in Banach algebras

The concept of a hermitian element of a Banach algebra was first introduced by Vidav [21] who proved that, if a Banach algebra 𝒜 has “enough” hermitian elements, then 𝒜 can be renormed and given an involution to make it a stellar algebra. (Following Bourbaki [5] we shall use the expression “stellar algebra” in place of the term “C*-algebra”.) This theorem was improved by Berkson [2], Glickfeld [10] and Palmer [17]. The improvements consist of removing hypotheses from Vidav's original theorem and in showing that Vidav's new norm is in fact the original norm of the algebra. Lumer [13] gave a spatial definition of a hermitian operator on a Banach space E and proved it to be equivalent to Vidav's definition when one considers the Banach algebra 𝓛(E) of continuous linear mappings of E into E.

On Lerner's theorem of equal income distribution

In their paper "Distributional Equality and Aggregate Utility", Breit and Culbertson (1) distinguish two propsitions of A. Lerner on optimum division of income: a "meek" proposition that income equalization maximizes the excepted total satisfaction and a "bold" proposition, that income equalization increases total satisfaction with absolute certainty. While the "meek" proposition is Lerner's original theorem in his "The Economics of Control" (2), the "bold" proposition is the invention of Breit and Culbertson1. In this note we give two rigorous proofs of Lerner's original theorem. The first is on the line of Lerner's original exposition in his "The Economics of Control", the second gives also a new interpretation of the problem using the von Neumann-Morgenstern utility theory. The second proof also covers a generalization of Lerners' "meek" theorem proposed by Breit and Cubertson (1). Lerner's most recent formulation of his theorem is the following: ". . . if it is desired to maximize the total satisfaction in a society , the rational procedure , in the absence of the knowledge, that would enable us to equalize the marginal utilities, is to maximize the probable total satisfaction i. e. to divide income on an equalitarian basis" (3).

An extension of the Nagumo uniqueness theorem

The classical Nagumo uniqueness theorem is a best possible result in the sense that if the Nagumo constant is replaced by a number greater than one then the result is false. This classical result uses the continuity of the right-hand side, $f(x,y)$, of the first order ordinary differential equation, and there is no explicit connection shown between the constant and the continuity of $f$; the observation that one makes is that the counterexample originally given by Perron uses a discontinuity at the origin together with a constant greater than one to obtain an initial-value problem with many solutions. This paper shows explicitly that the original theorem remains true with Nagumo constant greater than one provided that $f$ is sufficiently small at the origin, this sufficiency being determined by the actual value of the constant. Moreover, it is shown that the Nagumo inequality need not be imposed on the entire right-hand side of the equation; it suffices that only a certain factor satisfy a weakened form of the inequality.