Preserving Transformation Research Articles

Hopf's ratio ergodic theorem by inducing

We present a very quick and easy proof of the classical StepanovHopf ratio ergodic theorem, deriving it from Birkho¤’s ergodic theorem by a simple inducing argument. During the last few years, there has been some interest in short and easy proofs of (pointwise) ergodic theorems, naturally focussing on the most fundamental one, i.e. on Birkho¤’s result for probability preserving transformations, see e.g. [KW], [Ke], [P], and [Sh]. In [KK] a similar proof of an important extension was given, which came shortly after the discovery of the rst ergodic theorems ([N] and [B], see [Z] for historical comments): the Stepanov-Hopf ratio ergodic theorem ([St], [H]) which is the proper version of the pointwise ergodic theorem for in nite measure preserving transformations (there is no way to get a.e. convergence for ergodic sums normalized by a sequence of constants, cf. [A], §2.4). The aim of the present note is to point out that this result can also be derived as a direct consequence of Birkho¤’s theorem, via a (very) simple inducing argument (which doesn’t seem to be available or hinted at in the literature I know). We are going to prove Theorem 1 (Hopf’s Ratio Ergodic Theorem). Let T be a measure preserving transformation on the nite measure space (X;A; ). Let f; g 2 L1( ) with g 0 and R X g d > 0. Then there exists a measurable function Q(f; g) : X ! R such that Sn(f) Sn(g) = Pn 1 k=0 f T k Pn 1 k=0 g T k ! Q(f; g) a.e. on fsupn Sn(g) > 0g as n!1. On the conservative part the limit function Q(f; g) is measurable w.r.t. the algebra I A of T -invariant sets and satis es R I Q(f; g) g d = R I f d for all I 2 I. In particular, if g > 0 a.e., then Q(f; g) = E g f g kI , where d g := g d , and if T is ergodic, then Q(f; g) = R X f d = R X g d a.e. 2000 Mathematics Subject Classi cation. Primary 28D05, 37A30.

System Modeling and Transformational Design Refinement in ForSyDe

The scope of the formal system design (ForSyDe) methodology is high-level modeling and refinement of systems-on-a-chip and embedded systems. Starting with a formal specification model, that captures the functionality of the system at a high abstraction level, it provides formal design-transformation methods for a transparent refinement process of the system model into an implementation model that is optimized for synthesis. The main contribution of this paper is the ForSyDe modeling technique and the formal treatment of transformational design refinement. We introduce process constructors, that cleanly separate the computation part of a process from the synchronization and communication part. We develop the characteristic function for each process type and use it to define semantic preserving and design decision transformations. In a study of a digital equalizer example, we illustrate the modeling and refinement process and focus in particular on refinement of the clock domain, communication refinement, and resource sharing.

Steering Resources to Safe-Harbor Repositories: The Need for Reliable, Accurate and Affordable Ingest Services

With the emergence of centralized, large-scale digital archives, geography becomes a key factor in the preservation of cultural heritage materials. Objects “inside” archives will be actively monitored and managed, whereas those “outside” will be at greater risk of loss and obsolescence. Developing ingest systems and services to process, package and transport objects into managed safe-harbor repositories is an immediate need. Standards, frameworks, and business models for digital archiving must also evolve in due time to support these services.Ingest solutions must address a range of challenges: legal, technical, and financial. Software development, however, is a logical starting point, since tools that automate pre-archiving tasks meet technical requirements for viability and economic ones for affordability. The key tasks to automate are production of preservation metadata, transformation and validation of formats, and creation of repository-compliant transfer packages.The Harvard University Library (HUL) Office for Information Systems (OIS) has developed two applications to promote use of the HUL Digital Repository Service. JHOVE, developed with JSTOR, is a format-validation program; Dmart is a batch deposit tool for audio preservation packages. In Harvard's experience, the target user for such applications has typically been a professional depositing agent with technical expertise, who consults as needed with curatorial experts.With greater understanding of ingest requirements, and the profiles of persons or agencies likely to perform these services, it is hoped that industry will develop and support tools for the digital archiving market.

BabyJ: From Object Based to Class Based Programming via Types

Object oriented programming can be classified into the object based, and the class based paradigm. Object based languages typically are weakly typed and interpreted, allow member addition and removal, and thus they support flexibility and prototyping. Class based languages are usually strongly typed and compiled, require a rigid class structure and class membership, and thus they support more robust, type safe programs.The two paradigms therefore address the needs of different stages in the programming lifecycle: object based programming better fits the earlier, exploratory phases, whereas class based better fits the latter, consolidation and maintenance phases. Because the transition from one paradigm to the other is not straightforward, programs tend to be developed in one of the two paradigms, thus foregoing the advantages of the other.We suggest that this need not be so, and that the benefits of the two paradigms can be combined: The earlier, exploratory, programming phases should take place in an object based setting. Then, the program should be incrementally annotated with types. Once fully typed, the program can be mapped onto an equivalent class based program.We demonstrate these ideas through the introduction of BabyJ, a formalization of Javascript. We define BabyJT, a typed extension of BabyJ. A permissive type in BabyJ allows the typing process to be incremental. We then define a meaning preserving transformation of BabyJT programs to Java programs.

A method for calculating subdifferential of convex vector functions

Generalized convex vector functions are characterized by using order preserving transformations and some calculus rules for subdifferential of convex vector functions are obtained.

Orthogonality Preserving Transformations on Indefinite Inner Product Spaces: Generalization of Uhlhorn's Version of Wigner's Theorem

We present an analogue of Uhlhorn's version of Wigner's theorem on symmetry transformations for the case of indefinite inner product spaces. This significantly generalizes a result of Van den Broek. The proof is based on our main theorem, which describes the form of all bijective transformations on the set of all rank-one idempotents of a Banach space which preserve zero products in both directions.

On the relative variational principle for fibre expanding maps

Relative transfer operators are used in Denker and Gordin's 1999 paper to construct Gibbs families of conditional measures on fibres of a uniformly-expanding and exact-fibred system. Here we investigate the relationship between these operators and the relative variational principle (extending results for non-fibred expanding systems).It turns out that the maximal value for the free energy in the relative variational problem can be represented in terms of the transfer operator. However, for a general potential, the possibility to reduce the construction of an equilibrium measure to the search for an appropriate family of conditional measures on the fibres, critically depends on the invertibility of the base transformation.A certain class of potentials (called basic) which allow the last-mentioned reduction is introduced and the properties of the corresponding equilibrium measures are studied. Any measure of this kind gives rise to a regular factor; under a natural assumption the latter property is shown to be equivalent to the validity of the relative version of Rokhlin's formula for the entropy of a measure preserving transformation. Several examples are presented, among them families of polynomial skew products in \mathbb{C}^n restricted to their Julia sets.

Some open problems concerning relatively free and free profinite semigroups

Besides constituting an important and often crucial tool in studying semigroup varieties and pseudovarieties, relatively free and free profinite semigroups have proved to be of interest from many other natural standpoints. We present three groups of new problems concerning certain relatively free and free profinite semigroups and give background and motivation for them. The problems deal with the first-order theories of free profinite semigroups, the structure of free Burnside semigroups with commuting idempotents and of free Burnside inverse semigroups, and the representation of finitely generated free bands by order preserving transformations of a finite chain.

Finitary reconstruction of a measure preserving transformation

This paper considers thefinitary reconstruction of an ergodic measure preserving transformationT of a complete separable metric spaceX from a single trajectoryx, Tx, …, or more generally, from a suitable reconstruction sequence x=x1,x2, … withxi∈X. Ann-sample reconstruction is a functionTn: Xn+1 →X; the map\(\hat T_n \)(·;x1, …,xn)is treated as an estimate ofT(·) based on then initial elements of x. Given a reference probability measureμ0 and constantM>1, functionsT1,T2, … are defined, and it is shown that for everyμ with 1/M≤dμ/dμ0≤M, everyμ-preserving transformationT, and every reconstruction sequence x forT, the estimates\(\hat T_n \)(·;x1, …,xnconverge toT in the weak topology.

Strongly Mixing Sequences of Measure Preserving Transformations

We call a sequence (Tn) of measure preserving transformations strongly mixing if \(P(T_n^{ - 1} A \cap B)\) tends to P(A)P(B) for arbitrary measurable A, B. We investigate whether one can pass to a suitable subsequence \(T_{n_k } \) such that \(\frac{1}{K}\sum\limits_{k = 1}^k {f(T_{n_k } )} \to \smallint fdP\) almost surely for all (or many) integrable f.

Multiple recurrence of Markov shifts and other infinite measure preserving transformations

We discuss the concept of multiple recurrence, considering an ergodic version of a conjecture of Erdős. This conjecture applies to infinite measure preserving transformations. We prove a result stronger than the ergodic conjecture for the class of Markov shifts and show by example that our stronger result is not true for all measure preserving transformations.

Supersymmetric quantum-well shape optimization for intersubband bound–continuum second harmonic generation

A method is described for the optimized design of quantum-well structures, with respect to maximizing the second-order susceptibilities relevant for second harmonic generation. The possibility is explored of obtaining resonantly enhanced nonlinear optical susceptibilities in quantum wells with two bound states and a continuum resonance state as the dominant third state. The method relies on applying the isospectral (energy structure preserving) transformations to an initial Hamiltonian in order to generate a parameter-controlled family of Hamiltonians. By changing the values of control parameters one changes the potential shape and thus the values of matrix elements relevant to susceptibility to be maximized. The method was used for the design of AlxGa1−xAs -based QWs. The results indicate the possibility of employing continuum states in resonant second harmonic generation at higher photon energies,ℏω= 200–300 meV.

REVISION OF NONFUNCTIONING KOCK POUCH EFFERENT LIMB: CONTINENT, TISSUE PRESERVING TECHNIQUE

Despite extensive surgical experience with the intussuscepted efferent nipple of the Kock pouch, complications are not unusual. Although most repairs are relatively simple, the use of intestinal segments is necessary for reconstruction of the complete efferent limb in cases of severe stenosis, pre-stenotic diverticular enlargement or partial necrosis. We describe the tissue preserving transformation of an inadequate efferent Kock pouch outlet into a flap-valve continence mechanism. In 5 women a new efferent limb for the Kock pouch was created by transverse retubularization of the short intact ileal segment of the original limb. Continence was preserved through the construction of a Mitrofanoff-like flap valve, created by embedding the new ileal tube in an extramural trough. At followup ranging from 6 to 28 months all patients were continent and experienced no problems with catheterization. This technique of transverse retubularization of the inadequate efferent ileal limb and creation of a flap valve has obvious advantages. No new small bowel segments are required, thereby simplifying and shortening the procedure. The newly created ileal tube is wide (16 to 18Fr) and easy to catheterize. The mucosal folds are longitudinal and do not impede catheterization. The remaining thin layer of mesentery is set in the center of the tube and permits construction of a straight extramural tunnel.

Entropy of Endomorphisms and Relative Entropy in Finite von Neumann Algebras

We show the analogue for the entropy of automorphisms of finite von Neumann algebras of the classical formula H(T)=H(∨∞i=0T−iP∣∨∞i=1T−iP), where T is a measure preserving transformation of a probability space, and P is a generator.

A variant of the Hausdorff theorem for multiindex matrices I

An infinite matrix is regular if it is a limit preserving transformation in the domain of convergent sequences. A classical Hausdorff theorem states that some matrices composed of diagonal matrices and matrices, whose elements are defined as alternating Newton symbols, are regular iff the diagonal elements of the above diagonal matrices are consecutive moments of a Stieltjes integral. In this paper an analogous theorem for multiindex infinite matrices is proved.

Quantum-well shape optimization for intersubband-related electro-optic modulation properties

A method is described for the optimized design of quantum-well structures, in respect to maximizing the second-order susceptibilities relevant for electro-optic applications. It relies on applying the isospectral (energy structure preserving) transformations to an initial Hamiltonian, in order to generate parameter(s) controlled family of Hamiltonians that (i) are isospectral to the initial one and, (ii) have the potential variation proportional to the effective mass variation, being thus realizable by graded ternary alloys like ${\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}.$ By changing the values of control parameters one changes the potential shape and thus the values of matrix elements relevant for the susceptibility to be maximized. The use of the method is demonstrated by designing optimal quantum wells for the Stark effect or quantum interference derived electro-optical effects.

Adjacency Preserving Transformations of Grassmann Spaces

Adjacency Preserving Transformations of Grassmann Spaces

Multiple Recurrence Theorem for Measure Preserving Actions of a Nilpotent Group

The multidimensional ergodic Szemeredi theorem of Furstenberg and Katznelson, which deals with commuting transformations, is extended to the case where the transformations generate a nilpotent group: ¶Theorem. Let \((X, \frak B, \mu\)) be a measure space with \( \mu (X) \le \infty \) and let \(T_1, \dots, T_k\) be measure preserving transformations of X generating a nilpotent group. Then for any \(A \in \frak B\) with \(\mu (A) \ge 0,\)¶¶\( \liminf \limits_{N \rightarrow \infty} {1\over N} \sum \limits^{N-1} \limits_{n=0}\mu (T_1^{-n} A \cap \cdots \cap T_k^{-n} A) \ge 0 \).¶¶ Our main result also generalizes the polynomial Szemeredi Theorem in [BL1]. In the course of the proof we describe a relatively simple form to which any unitary action of a finitely generated nilpotent group on a Hilbert space and any measure preserving action of a finitely generated nilpotent group on a probability space can be reduced.

Testing for Pairwise Serial Independence Via the Empirical Distribution Function

Summary Built on Skaug and Tjøstheim's approach, this paper proposes a new test for serial independence by comparing the pairwise empirical distribution functions of a time series with the products of its marginals for various lags, where the number of lags increases with the sample size and different lags are assigned different weights. Typically, the more recent information receives a larger weight. The test has some appealing attributes. It is consistent against all pairwise dependences and is powerful against alternatives whose dependence decays to zero as the lag increases. Although the test statistic is a weighted sum of degenerate Cramér–von Mises statistics, it has a null asymptotic N(0, 1) distribution. The test statistic and its limit distribution are invariant to any order preserving transformation. The test applies to time series whose distributions can be discrete or continuous, with possibly infinite moments. Finally, the test statistic only involves ranking the observations and is computationally simple. It has the advantage of avoiding smoothed nonparametric estimation. A simulation experiment is conducted to study the finite sample performance of the proposed test in comparison with some related tests.

Semigroups of order preserving mappings on a finite chain: A new class of divisors

In this paper we aim to prove that every semigroup of the pseudovariety generated by all semigroups of partial, injective and order preserving transformations on a finite chain belongs to the pseudovariety generated by all semigroups of order preserving mappings on a finite chain.