A generic method for measuring the potential number of structure‐preserving transformations

In many fields of psychology, it may be interesting to measure the potential number of structure‐preserving transformations that exist between succeeding structures. The aim of this article is to present a methodology for measuring the potential number of structure‐preserving transformations between succeeding structures and to illustrate the applicability of the methodology through a case study. The article concludes by discussing the lessons and implications of the proposed methodology for microgenetic research. © Wiley Periodicals, Inc. Complexity, 2012     

Direct sums and refinement

In this paper we present a division theorem for the pseudovariety of semigroups OP generated by all semigroups of orientation preserving full transformations on a finite chain, which is achieved by using a natural representation of a certain monoid of transformations of a set X as a monoid of transformations of a subset of X.     

Certain Characterizations of Normal Distribution via Transformations

It is well known that i.i.d. (independent and identically distributed) normal random variables are transformed into i.i.d. normal random variables by any orthogonal transformation. Less well known are nonlinear transformations with the above-mentioned property. In this work we present nonlinear transformations preserving normality, which are more general than the existing ones in the literature.     

Nonanticipative transformations of the two-parameter Wiener process and a Girsanov theorem

Nonanticipative linear transformations of the two-parameter Wiener process W are studied. It is shown that they induce measures equivalent to two-parameter Wiener measure and the corresponding Radon-Nikodym derivatives are calculated. A two-parameter extension of Girsanov's theorem is established for a class of nonanticipative, possibly nonlinear transformations of W.     

Mixing Limit Theorems for Ergodic Transformations

We show that distributional and weak functional limit theorems for ergodic processes often hold for arbitrary absolutely continuous initial distributions. This principle is illustrated in the setup of ergodic sums, renewal-theoretic variables, and hitting times for ergodic measure preserving transformations.     

Ergodic flows of positive entropy can be time changed to becomeK-flows

It is shown that every Kakutani equivalence class of ergodic measure preserving transformations of positive entropy containsK-automorphisms. Also, every ergodic flow of positive entropy can be time changed to become aK-flow and every ergodic automorphism of positive entropy is a cross-section of someK-flow.     

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.     

Measure space automorphisms,the normalizers of their full groups,and approximate finiteness

We deal with the normalizer $\text{N}$[T] of the full group [T] of a nonsingular transformation T of a Lebesgue measure space in the group of all nonsingular transformations. We solve the conjugacy problem in $\text{N}$[T]/[T] for a measure preserving and ergodic T. Our results show that a locally finite extension of a solvable group is approximately finite.     

Pointwise convergence of ergodic averages along cubes

Let (X, B, μ, T) be a measure preserving system. We prove the pointwise convergence of ergodic averages along cubes of 2 ^k − 1 bounded and measurable functions for all k. We show that this result can be derived from estimates about bounded sequences of real numbers and apply these estimates to establish the pointwise convergence of some weighted ergodic averages and ergodic averages along cubes for not necessarily commuting measure preserving transformations.     

Lorentz–Minkowski Distances in Hilbert Spaces

For arbitrary pre-Hilbert spaces X of dimension at least 2, we define the notion of Lorentz–Minkowski distance. We determine all Lorentz transformations, i.e. all distance preserving mappings of X into itself, introduce relativistic addition, and characterize the 2-point-invariants of the group of bijective Lorentz tranformations and, especially, the distance in question.     

Normal ergodic actions

The von Neumann-Halmos theory of ergodic transformations with discrete spectrum makes use of the duality theory of locally compact abelian groups to characterize those transformations preserving a probability measure, which are defined by a rotation on a compact abelian group. We use the recently developed duality between general locally compact groups and Hopf-von Neumann algebras to characterize those actions of a locally compact group, preserving a σ-finite measure, which are defined by a dense embedding in another group. They are characterized by the property of normality, previously introduced by the author, and motivated by Mackey's theory of virtual groups. The discrete spectrum theory is readily seen to come out as the special case in which the invariant measure is finite.     

On the ergodic theory of non-integrable functions and infinite measure spaces

We prove a Chow-Robbins type result for an ergodic, non-negative SSSP, and a similar result for transformations preserving infinite measure, which implies that for these transformations, no “absolute” version of Hopf's theorem can hold.     

High school students’ understandings of geometric transformations in the context of a technological environment

This study investigated the nature of students’ understandings of geometric transformations, which included translations, reflections, rotations, and dilations, in the context of the technological tool, The Geometer’s Sketchpad. The researcher implemented a seven-week instructional unit on geometric transformations within an Honors Geometry class. Students’ conceptions of transformations as functions were analyzed using the APOS theory and were informed by an analysis of students’ interpretations and uses of representations of geometrical objects using the constructs of drawing and figure. The analysis suggests students’ understandings of key concepts including domain, variables and parameters, and relationships and properties of transformations were critical for supporting the development of deeper understandings of transformations as functions.     

Distributional limit theorems in infinite ergodic theory

We present a unified approach to the Darling-Kac theorem and the arcsine laws for occupation times and waiting times for ergodic transformations preserving an infinite measure. Our method is based on control of the transfer operator up to the first entrance to a suitable reference set rather than on the full asymptotics of the operator. We illustrate our abstract results by showing that they easily apply to a significant class of infinite measure preserving interval maps. We also show that some of the tools introduced here are useful in the setup of pointwise dual ergodic transformations.     

On the asymptotics of a 1-parameter family of infinite measure preserving transformations

We estimate various aspects of the growth rates of ergodic sums for some infinite measure preserving transformations which are not rationally ergodic.Dedicated to R. Mañé     

Understanding rigid geometric transformations: Jeff's learning path for translation

This article describes the development of knowledge and understanding of translations of Jeff, a prospective elementary teacher, during a teaching experiment that also included other rigid transformations. His initial conceptions of translations and other rigid transformations were characterized as undefined motions of a single object. He conceived of transformations as movement and showed no indication about what defines a transformation. The results of the study indicate that the development of his thinking about translations and other rigid transformations followed an order of (1) transformations as undefined motions of a single object, (2) transformations as defined motions of a single object, and (3) transformations as defined motions of all points on the plane. The case of Jeff is part of a bigger study that included four prospective teachers and analyzed their development in understanding of rigid transformations. The other participants also showed a similar evolution.     

Bernoulli shifts and local density property

We consider the property LocDen for the squaring mapping into the space of all measure preserving transformations and into the space of mixing transformations. It is proved that Bernoulli shifts with infinite entropy do not possess this property.     

Le Groupe des Transformations de [0, 1] Qui Preservent la Mesure de Lebesgue Est un Groupe Simple

We prove that the group of measure preserving transformations of [0, 1] is a simple group, i.e. has no non-trivial normal subgroup.      

One-cohomology and the uniqueness of the group measure space decomposition of a II1 factor

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II₁ factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γ the associated II₁ factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan–Peterson, but we replace the usage of Peterson’s unbounded derivations by Thomas Sinclair’s dilation into a malleable deformation by a one-parameter group of automorphisms.     

Bi-orthogonal systems on the unit circle, regular semi-classical weights and integrable systems — II

We derive the Christoffel–Geronimus–Uvarov transformations of a system of bi-orthogonal polynomials and associated functions on the unit circle, that is to say the modification of the system corresponding to a rational modification of the weight function. In the specialisation of the weight function to the regular semi-classical case with an arbitrary number of regular singularities {z₁,…,z_M} the bi-orthogonal system is known to be monodromy preserving with respect to deformations of the singular points. If the zeros and poles of the Christoffel–Geronimus–Uvarov factors coincide with the singularities then we have the Schlesinger transformations of this isomonodromic system. Compatibility of the Schlesinger transformations with the other structures of the system — the recurrence relations, the spectral derivatives and deformation derivatives is explicitly deduced. Various forms of Hirota–Miwa equations are derived for the τ-functions or equivalently Toeplitz determinants of the system.