Necessity Part Research Articles

On stability of interval matrices

New sufficient, and sometimes necessary and sufficient conditions, are obtained for Schur- and Hurwitz-stability of interval matrices by relying on the concept of connective stability and M-matrices. The necessity part is broadened to include interval matrices with mixed signs of the off-diagonal elements, provided the sign patterns follow that of the Morishima matrix. The obtained results are extended to cover convex combinations of interval matrices.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The simple substitution property of the intermediate propositional logics on finite slices

The simple substitution property provides a systematic and easy method for proving a theorem by an axiomatic way. The notion of the property was introduced in Hosoi [4] but without a definite name and he showed three examples of the axioms with the property. Later, the property was given it's name as above in Sasaki [7]. Our main result here is that the necessary and sufficient condition for a logicL on a finite slice to have the simple substitution property is thatL is finite. Here the necessity part is essentially new, for the sufficiency part has been proved in Hosoi and Sasaki [5]. Also the proof of sufficiency part is improved here. For logics on the ω-th slice, the condition for them to have the simple substitution property is not yet known. We abbreviate the simple substitution property asSSP.

Two-Riccati formulae for the discrete-time H∞-control problem

A two-Riccati state-space approach to the discrete-time H∞ control problem is presented. The main theoretical tool is the Kalman-Szego-Popov-Yakubovich system written in adequate forms in order to reduce the problems to a standard form for which the solutions are easily obtained. The necessity part of our main results uses a 'signed' counterpart of the positivity theorem.

The Behnke-Stein theorem for open Riemann surfaces

Using the Riemann-Roch theorem and the set-topological part of Bishop's polyhedron lemma, we show that the usual Runge approximation theorem for compact subsets of the Riemann sphere is valid word-for-word on any compact Riemann surface X, with meromorphic functions on X playing the role of rational functions; this result is essentially equivalent to the Behnke-Stein approximation theorem. The Behnke-Stein generalization of the Runge approximation theorem [1], which is the basic tool for many existence questions on open Riemann surfaces, can be stated in several equivalent ways, for instance: Let X be a Riemann surface, and K a compact subset of X. Then, in order that every holomorphic function in a neighborhood of K be uniformly approximable on K by holomorphic functions on X, it is necessary and sufficient that X K have no connected component with compact closure in X. As is well known, the necessity part of the above theorem follows from the sufficiency part (using the theory of compact Riemann surfaces). In this note, we wish to point out how the famous special polyhedron lemma of Bishop [2], which is an elementary set-topological result, can be used (together with the Riemann-Roch theorem for compact Riemann surfaces) to give a simple proof of the following theorem. Theorem 1.1. Let X be a compact Riemann surface, and K c X a compact subset. Let Q be any subset of X K which contains (precisely) one point qi from each connected component W of X K. Then any holomorphic function on a neighborhood of K can be approximated uniformly on K by meromorphic functions on X whose poles lie in Q. Again it is well known and easy to see that Theorem 1.1 implies the sufficiency part of Behnke-Stein. The main fact needed for this implication is the following: Received by the editors March 24, 1988. 1980 Mathematics Subject Classification (1 985 Revision). Primary 30E 10, 30F10.

Centre and Province in the People's Republic of China: Sichuan and Guizhou, 1955-1965

According to common misconception the Chinese political system is highly centralised. One result of this widely accepted view is that China specialists have often neglected the study of decision-making as a process. Concentrating upon the neighbouring, but contrasting provinces of Sichuan and Guizhou during the decade before the Cultural Revolution, this book examines the interaction between centre and province and, without adopting, a 'centralist' or a 'pluralist' viewpoint, argues that a spatial dimension is of necessity part of the Chinese decision-making process. Particular attention is paid to the variability of this interaction over time.

A note on the frequency-dependent Lyapunov equation

In a recent contribution [l], a necessary and sufficient condition for 2-D stability was presented which was based on the solution of the frequency-dependent Lyapunov equation. In this note, an example is provided which indicates discrepancies in the necessity part of this condition.

Chebyshev centers in normed spaces

Let E be a normed linear space, F a bounded subset, Y a closed subset of E. A nonnegative real number r,,(F) is called the relative Chebyshev radius of F with respect to Y if r,,(F) is the inlimum of all numbers r > 0 for which there exists a y E Y such that F is contained in the closed ball B(v, r) with center y and radius r. Any point y E Y for which F c B(y, r,,(F)) is called a relative Chebyshev center of F with respect to Y. We denote the set of all relative Chebyshev centers of F with respect to Y by zy(F). In this paper we investigate several questions concerning characterization and existence of relative Chebyshev centers, and the continuity of the Chebyshev center map. In Section 1 we give a formula for the relative Chebyshev radius of a bounded set F with respect to Y in terms of the relative radius of F with respect to hyperplanes from the annihilator of Y. For F totally bounded this formula was obtained in [8]. Let F be a bounded set which is contained in the closed ball B(y, r), where r = r(y, F) 3 sup { I] x y )I ; x E F}. In Section 2 we are looking for necessary and sufficient conditions for B(y, r) to be the Chebyshev ball of F. For Hilbert space, a characterization was given in [5]. However, the necessity part of this characterization requires a property valid only for F compact. We give necessary and sufficient conditions for both the compact and noncompact case and deduce several corollaries. In Section 3 we show that every infinite dimensional normed space E has an equivalent norm such that c,(E) does not admit relative centers for all pairs of points in l,(E). In Section 4 we investigate the Lipschitz constants of the Chebyshev center map restricted to certain families of “admissible” pairs of sets, as introduced in [5].

On the oscillation and nonoscillation of second order sublinear equations

An oscillation criterion and a nonoscillation criterion are given for the sublinear equation y + a ( t ) | y | γ sgn ⁡ y = 0 y + a(t)|y{|^\gamma }\operatorname {sgn} y = 0 , 0 &gt; γ &gt; 1 , t ∈ [ 0 , ∞ ) 0 &gt; \gamma &gt; 1,t \in [0,\infty ) , where a ( t ) a(t) is allowed to change sign. When applied to the special case a ( t ) = t λ sin ⁡ t a(t) = {t^\lambda }\sin t , we deduce oscillation for λ &gt; − γ \lambda &gt; - \gamma and nonoscillation for λ &gt; − γ \lambda &gt; - \gamma .

On Teichmüller’s theorem on the quasi-invariance of cross ratios

Teichmuller’s theorem gives necessary and sufficient conditions for mapping one ordered quadruple by aK-quasiconformal map onto a second ordered quadruple. We give a simple non-computational proof of the necessity part. We then characterize such extremal mappings, and obtain as a consequence a new formula for the modular function, with leads to a very simple derivation of the known expression for the Poincare metric on the thrice-punctured sphere.

SETS OF $ H$-FIXED POINTS ARE ABSOLUTE EXTENSORS

The necessity part of a theorem of Jaworowski on the extension of periodic homeomorphisms is strengthened (MR 45 #9319; 47 #9524). Let be the class of all compact Hausdorff spaces (metrizable spaces), and let () be the class of all compact Hausdorff spaces (metrizable spaces) considered with all possible actions of a topological group . Theorems C and M. If a topological space on which a group () acts is an extensor of (), then for every closed subgroup of the set of all "-fixed points" is an extensor of the class . These theorems are also valid for the neighborhood case under the additional condition that for mappings extendible to the dimension , and for equivariant mappings extendible to the dimension . Bibliography: 15 titles.

Corrections to "An Analytical Criterion for the Completeness of Riemannian Manifolds"

The purpose of this present note is to call attention to the that our proof of the necessity part of the theorem contains a serious error, and that a correct proof of this part of the theorem is contained in an article by M. P. Gaffney [3]. (See also [1] and [2] for some earlier results which were tending in this direction.) Our proof of necessity used the. fact that an isometric embedding j: M Rq is closed if the induced metric is complete. This is easily shown to be false; e.g., the embedding of R1 into R2 given by t (et, sin t) is not closed but the induced metric is complete. (On the other hand, the converse statement is true; i.e. a metric induced by a closed embedding is necessarily complete.) A correct proof can be obtained as follows: Let p be a fixed point of M and let f(x) = d(x, p), where d is the distance function corresponding to the given riemannian metric. Then the completeness of M implies that f is proper. (This is merely a restatement of that portion of the Hopf-Rinow theorem which asserts that a riemannian manifold is complete if and only if closed and bounded sets are compact.) The function f is not differentiable, but it is Lipshitz continuous with Lipshitz constant = 1; i.e., /(X) f(y)| < d(x, y). Using this fact, Gaffney [3] shows, by means of various smoothing operations and partition of unity constructions, that for any positive E there exists a smooth

On a Necessary and Sufficient Condition for Admissibility of Estimators When Strictly Convex Loss is Used

1. Introduction. In this paper we consider a necessary and sufficient condition for the admissibility of estimators when strictly convex loss is used. The result is stated as Theorem 1. The sufficiency of the condition is obvious and has served as the basis of admissibility proofs in [1], [6], [11], [3], and [2]. The necessity of such a method of proof is relatively deep. The author claims no practical use of Theorem 1. He has been moved primarily by curiosity about the necessity part of the theorem together with a desire to strengthen the tools of decision theory. The results of this paper depend on Farrell [5] to which one can refer for definitions of some common terms like Bayes if these are not clear.