Short Proof Of Theorem Research Articles

A short proof of Tomita's theorem

Tomita-Takesaki theory associates a positive operator called the “modular operator” with a von Neumann algebra and a cyclic-separating vector. Tomita's theorem says that the unitary flow generated by the modular operator leaves the algebra invariant. I give a new, short proof of this theorem which only uses the analytic structure of unitary flows, and which avoids operator-valued Fourier transforms (as in van Daele's proof) and operator-valued Mellin transforms (as in Zsidó's and Woronowicz's proofs). The proof is similar to one given by Bratteli and Robinson in the special case that the modular operator is bounded.

Short proofs of theorems of Malyutin and Margulis

The Ghys-Margulis alternative asserts that a subgroup $G$ of homeomorphisms of the circle which does not contain a free subgroup on two generators must admit an invariant probability measure. Malyutin's theorem classifies minimal actions of $G$. We present a short proof of Malyutin's theorem and then deduce Margulis' theorem which confirms the G-M alternative. The basic ideas are borrowed from the original work of Malyutin but the use of the apparatus of the enveloping semigroup enables us to shorten the proof considerably.

Conics and convexity

In [1], W. D. Munn proved the following result.Theorem 1: Infinitely many ellipses pass through the four vertices of a given convex quadrilateral.Much of the geometry that I studied as an undergraduate in the 1950s concerned complex projective space, in which convexity plays no part. So I found Theorem 1 especially piquant and sought to understand it better. This article is the result. After examining the convexity of quadrilaterals in general, especially those inscribed in conics, I consider the following problem. Let P be a variable point in the plane, distinct from the vertices of a given convex quadrilateral ABCD. It is well known that there is a unique conic, S (P), through the five points A, B, C, D and P. How does the nature of this conic depend on the position of P? As a spin-off, we get a very short proof of Theorem 1. Finally I look at what happens when the quadrilateral ABCD is not convex. In this case, S (P) is always a hyperbola, but the distribution of A, B, C and D on its branches is still of interest.

A SURVEY OF RAMANUJAN EXPANSIONS

This paper summarizes the development of Ramanujan expansions of arithmetic functions since Ramanujan's paper in 1918, following Carmichael's mean-value-based concept from 1932 up to 1994. A new technique, based on the concept of related arithmetic functions, is introduced that leads to considerable extensions of preceding results on Ramanujan expansions. In particular, very short proofs of theorems for additive and multiplicative functions going far beyond previous borders are presented, and Ramanujan expansions that formerly have been considered mysterious are explained.

GREEN'S RELATIONS AND MINIMAL QUASI-IDEALS IN RINGS

ABSTRACT In this paper we study the Green's relations , , , , as well as the relations concerning principal quasi-ideals in rings, whose definitions mimic definitions of Green's relations and relation in semigroups. We show that, differently from semigroups in rings we have in general , and we provide a sufficient, but not necessary, condition for , to hold in a ring. We prove some results about Green's relation in rings, which are similar to the analogous result in semigroups. The main result is an analogous of Green's Theorem for semigroups. By means of Green's Theorem for rings we give a short proof of Theorem about minimal quasi-ideal in rings. We provide various sufficient, but not necessary conditions on a minimal quasi-ideal of a ring for the set to be a -class. Finally we prove that, if for some elements of a ring we have and the principal quasi-ideal is minimal and has the intersection property, then the principal quasi-ideal is minimal and has the intersection property, too.

Meagre functions and asymptotic behaviour of dynamical systems

A measurable function x : J ⊂R → X (X a metric space) is said to be C-meagre if C ⊂X is non-empty and, for every closed set K ⊂X with K ∩ C = ∅; x−1(K) has 5nite Lebesgue measure. This concept of meagreness is shown to provide a unifying framework which facilitates a variety of characterizations, extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems. By way of motivation, consider the initial-value problem ẋ = f(x); x(0) = ∈ RN , with f : RN → RN locally Lipschitz, and denote its unique solution by x : t → (t; ) de5ned on its maximal interval of existence I = [0; ). Let g : RN → R be continuous. Assume that x is a bounded solution of the initial-value problem (in which case, I = [0;∞)), then g ◦ x is uniformly continuous. If, in addition, g ◦ x ∈ L1, then we may conclude (by an elementary observation frequently referred to as Barb> alat’s lemma [5] (see, also, [18, Section 21, Lemma 1])) that g(x(t)) → 0 as t → 0. Since, by boundedness, x(t) approaches, as t → ∞, its non-empty, compact, invariant !-limit set B, we also conclude that B⊂ g−1(0). It follows that x(t) tends, as t → ∞, to the largest invariant subset of g−1(0). This is a short proof of Theorem 1:2 of [6] (the proof in [6] is based on arguments involving properties of the Dow generated by the diEerential

Short proofs of theorems of Lekkerkerker and Ballieu

For any irrational number ξ let A(ξ) denote the set of all accumulation points of {z : z = q(qξ − p), p ∈ ℤ, q ∈ ℤ − {0}, p and q relatively prime}. In this paper the following theorem of Lekkerkerker is proved in a short and elementary way: The set A(ξ) is discrete and does not contain zero if and only if ξ is a quadratic irrational. The procedure used for this proof simultaneously takes care of a theorem of Ballieu.

Note on "Positive Cesaro Means of Numerical Series"

We give a short proof of a result due to R. Askey. In this note we will give a short proof of Theorem 1 in the preceding paper by R. Askey. The proof utilizes a trick applied earlier by Askey and Pollard LI]. We restate Askey's result for convenience. Theorem. If y > a. > 1, and the (C, y) means of E a n are nonnegative then the (C, a) means of Ia rn are nonnegative for 0 Proof. We need to show that A n(r) > 0 for 0 (1w)-aIa rnwn= EA(r)wn. We may write (1 w)a1 a r-wn ) -var)1( + I(Y(rw) E a nrnwn The hypothesis gives that (I rw) 1E a rnwn has nonnegative power n series coefficients for r > 0. Now we need only show that h(w; r) = (1 w)-a-I(, rw)P+l has nonnegative power series coefficients for r in the given interval. Taking logs we get log h(w; r) = L(a + 1) (y + I)r n] _n and log h(w; r) then has nonnegative coefficients for 0 The same must be true of h(w; r) and so An(r) > 0 in this interval as claimed. BIBLIOGRAPHY 1. R. Askey and H. Pollard, Some absolutely monotonic and completely mono- tonic functions, SIAM. J. Math. Anal. 5 (1974). DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CINCINNATI, CINCINNATI, OHIO 45221 Received by the editors February 14, 1974. AMS (MOS) subject classifications (1970). Primary 40G05.

A short proof of the levy continuity theorem in Hilbert space

A short proof of the Levy continuity theorem in Hilbert space. In the theory of the normal distribution on a real Hilbert spaceH, certain functionsφ have been shown by L. Gross to give rise to random variablesφ∼ in a natural way; in particular, this is the case for functions which are “uniformly τ-continuous near zero”. Among such functions are the characteristic functionsφ of probability distributionsm onH, given byφ(y)=∫e i(y,x)dm(x). The following analogue of the Levy continuity theorem has been proved by Gross: Letφ j be the characteristic function of the probability measurem j onH, Then necessary and sufficient that ∫f dm j → ∫f dm for some probability measurem and all bounded continuousf, is that there exists a functionφ, uniformly τ-continuous near zero, withφ j∼ →φ∼ in probability.φ turns out, of course, to be the characteristic function ofm. In the present paper we give a short proof of this theorem.