On the entropy of a product endomorphism in infinite measure spaces

Summary In this note, a relationship is established between the entropy, defined by Krengel for an endomorphism of a -finite measure space, and the notion of a spreading partition. This relationship is used to answer in the quasi-finite case a question raised by Krengel concerning the entropy of the product endomorphism on the direct product of a finite and -finite measure space.     

A skew-product which is Bernoulli

We show that if is the shift on sequences of {0,1} and is the entropy zero transformation used by Ornstein in constructing a counter-example toPinsker's conjecture, then the skew-product transformationT defined byT(x,y)=(x, ^x0 y) is Bernoulli. ThisT is conditionally mixing with respect to the independent generator for , a partition with full entropy.This research was done while the first author was a visitor at Stanford, supported in part by NSF Grant MP-575-08324.     

Group automorphisms with finite entropy

A compact metrie abelian group X with the normalized Haar measure is a Lebesgue probability space. A group automorphism ofX is an invertible measure preserving transformation of the probability space. This paper is to show that if the entropy of is finite, then there exist totally disconnected subgroupsH andN, a finite-dimensional subgroupS and a subgroupT satisfying the conditions: (i)H, N, S andT are strictly -invariant, (ii)N=HST, (iii)h(| _N )=0, (iv) ifS/N is non-trivial then it is a finite-dimensional solenoidal group with condition (**) (see the definition in §1), (v) ifT/N is non-trivial then it is connected and locally connected, such thatX/N splits into a direct sumX/N=H/NS/NT/N. This result characterizes the structure of finite entropy automorphisms.     

On non-singular transformations of a measure space. I

We consider a Lebesgue measure space (M, , m). By an automorphism of (M, , m) we mean a bi-measurable transformation of (M, , m) that together with its inverse is non-singular with respeot to m. We study an equivalence relation between these automorphisms that we call the weak equivalence. Two automorphisms S and T are weakly equivalent if there is an automorphism U such that for almost all x M U maps the S-orbit of x onto the T-orbit of U x. Ergodicity, the existence of a finite invariant measure, the existenoe of a -finite infinite invariant measure, and the non-existence of such measures are invariants of weak equivalenoe. In this paper and in its sequel we solve the problem of weak equivalenoe for a class of automorphisms that comprises all ergodic automorphisms that admit a -finite invariant measure, and also certain ergodic automorphisms that do not admit such a measure.     

The convolution equation of Choquet and Deny on [IN]-groups

Let be a probability measure on a locally compact groupG. A real Borel functionf onG is called -harmonic if it satisfies the convolution equation *f=f. Given that isnonsingular with its translates, we show that the bounded -harmonic functions are constant on a class of groups including the almost connected [IN]-groups. If is nondegenerate and absolutely continuous, we solve the more general equation *= for positive measure on those groups which are metrizable and separable.Supported by Hong Kong RGC Earmarked Grant and CUHK Direct Grant     

Solenoidal automorphisms with specifications

The specification property for solenoidal automorphisms is discussed; a solenoidal automorphism satisfies specification iff is expansive, and satisfies weak specification but not specification iff is ergodic and central spin. These are problems set up byK. Sigmund for homeomorphisms with specification. The proofs for toral case are given byD. Lind. For solenoidal case, a key ingredient in our proofs is splitting theorems on solenoidal groups with respect to described in § 2. Moreover, the following is proved: (i) If obeys specification then satisfies weak specification and is densely periodic. But the converse is not necessarily true. (ii) Every solenoidal automorphism with specification admits a Markov partition. (iii) Every ergodic solenoidal automorphism without specification does not admit Markov partitions. (iv) There exists an expansive homeomorphism with specification which has not Markov partitions.     

Commuting point transformations

Summary The question considered is the following: If two invertible measure preserving point transformations commute, in what sense is one a function of the other? The main theorem is the following: If two invertible measure preserving transformations commute, and if the first admits of approximation by periodic transformations, then the second transformation is a piecewise power of the first, where we say that is a piecewise power of if there exists a sequence [j(n)] of positive integers such that for each measurable set A the limit of the measure of the symmetric difference of (A) and ^j(n) (A) tends to zero.Research supported in part by NSF grant GP-3752.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

Harmonic functions on nilpotent groups

For a probability measure on a locally compact groupG which is not supported on any proper closed subgroup, an elementF ofL (G) is called -harmonic if F(st)d(t)=F(s), for almost alls inG. Constant functions are -harmonic and it is known that for abelianG all -harmonic functions are constant. For other groups it is known that non constant -harmonic functions exist and the question of whether such functions exist on nilpotent groups is open, though a number of partial results are known. We show that for nilpotent groups of class 2 there are no non constant -harmonic functions. Our methods also enable us to give new proofs of results similar to the known partial results.     

Stationary regenerative phenomena

Summary Corresponding to anyp-functionp, a stationary version of the associated regenerative phenomenon is constructed for which the underlying probability measure may have infinite total mass (though it will always be -finite ifp is standard). As a trivial consequence,p is a positive-definite function. The construction is generalised to quasi-Markov chains.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

A note on the position of the centre of flexure of a beam of isotropic material having a section bounded by a parabola and a straight line

Zusammenfassung Die vorliegende Notiz behandelt das Problem der Auffindung des Biegungszentrums eines aus isotropem Material bestehenden Balkens, der einen durch eine parabolische und eine geradlinige Kante begrenzten Querschnitt besitzt. Die Last ist parallel zur geradlinigen Kante angenommen. Der Verfasser benützt zur Lösung der Aufgabe dasselbe Verfahren, dasSokolnikoff [1] für die Auffindung des Biegungszentrums eines halbkreisförmigen Balkens verwendete.

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Nomenclature x, y, z rectangular co-ordinates - l, m, n direction cosines of outward drawn normal - _x , _y , _z normal components of stress parallel tox, y, z axes - _xy , _yz , _zx shearing stress components in rectangular co-ordinates - u, v, w components of displacement - T (x, y) stress function in rectangular coordinates - Poisson's ratio - shear modulus - _yz , _zx , _xy shearing strain components in rectangular coordinates     

On mixing in infinite measure spaces

Summary Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote -algebra (=tail -field). A transformation T is called mixing if (T ^–n A) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures ₁,₂, in total variation norm.Research of this author is supported by the National Science Foundation (U.S.A.) under grant GP 7693.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

On the existence of regular conditional probabilities

Summary Let (X, A, P) be a measure space with P(X)=1 and a sub--algebra. It is well known (see e.g. Doob, p. 624) that even if A is separable, a regular conditional probability (r.c.p.) on A, given , does not always exist. All theorems assuring the existence of a r.c.p. known hitherto use in addition to the separability of A conditions of an essentially topological nature such as compact approximation (for references see e.g. Pfanzagl-Pierlo (1966), section 7).It is the purpose of this paper to explore the possibility of sufficient conditions of a purely measure-theoretic nature. Without loss of generality we may assume that (X, A, P) is complete and is complete with respect to P ¦A. For many applications it will be sufficient to have the r.c.p. defined on a sub--algebra A ₀ A with A ₀ ~ A(P|A) rather than on the whole -algebra A. It will be shown that a r.c.p. in this more restricted sense always exists if A is the completion of a separable -algebra and if it contains a P-null set of the power of the continuum. The P-null-set -condition cannot be omitted without any replacements. To be more specific: Even a r.c.p. in the restricted sense does not necessarily exist if A is the completion of a separable -algebra (see Example 1). Furthermore, there are instances, in which a r.c.p. in the restricted sense exists but a r.c.p. in the usual sense does not exist (see Example 2).Even if a r.c.p. exists, it is not necessarily proper. The deep study by Blackwell and Ryll-Nardzewski (1963) reveals that this negative statement even holds under rather restrictive conditions (X = complete separable metric space, A = Borel algebra of X). It therefore seems to be of some interest to state that the r.c.p. in the restricted sense can always be chosen such that it is proper with respect to some sub--algebra +B_o +(+B A ₀ with ₀₀(P¦A).     

Riemann-measurable selections

LetX be a Polish space equipped with a -finite regular Borel measure . IfE is a metric space andF a set-valued function:X 2 ^E with complete values, and ifF is lower semicontinuous at almost all points ofX, we prove that there exists a Riemann-measurable selections ofF.     

Noncommutative Valuation Rings of the Quotient Artinian Ring of a Skew Polynomial Ring

Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let Q[X,] be the skew polynomial ring over Q in an indeterminate X, where is an automorphism of Q. Consider the natural map from Q[X,]_XQ[X,] to Q, where Q[X,]_XQ[X,] is the localization of Q[X,] at the maximal ideal XQ[X,] and set , the complete inverse image of R by . It is shown that is a Dubrovin valuation ring of Q(X,) (the quotient ring of Q[X,]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism is classified into five types, in order to study the structure of (the value group of ). It is shown that there is a commutative valuation ring R with automorphism which belongs to each type and which makes Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional. Presented by A. VerschorenMathematics Subject Classifications (2000) 16L99, 16S36, 16W60.     

Subgroups of the full linear group over a semilocal ring

Let be a semilocal ring (a factor ring with respect to the Jacobson-Artin radical) for which the residue field C/m of its center C with respect to each maximal idealmC contains no fewer than seven elements. The structure of subgroups H in the full linear group GL(n, ) containing the group of diagonal matrices is considered. The main theorem: for any subgroup H there is a uniquely determined D-net of ideals such that G()HN(), whereN() is the normalizer of the D-net subgroup . A transparent classification of subgroups GL(n, ) normalizable by diagonal matrices is thus obtained. Further, the factor groupN()/G() is studied.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 32–34, 1978.     

Inegalites de Sobolev pour les sous Laplaciens de certains groupes unimodulaires

Résumé On prouve des inégalités de type Sobolev pour les opérateurs (L-) oùL est un sous-Laplacien sur un groupe de Lie non moyenmable et le bout de son spectre.

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We prove Sobolev inequalities for (L-) whereL is a sub-Laplacian on a non-amenable group and is the bottom of its spectrum.