Measure Zero Research Articles

Theory of $(a,b)$-continued fraction transformations and applications

We study a two-parameter family of one-dimensional maps and the related $(a,b)$-continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) the associated natural extension mapshave attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional zero measure that we completely describe; (ii) for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations.We also give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.

Ergodic billiards that are not quantum unique ergodic

Partially rectangular domains are compact two-dimensional Riemannian manifolds $X$, either closed or with boundary, that contain a flat rectangle or cylinder. In this paper we are interested in partially rectangular domains with ergodic billiard flow; examples are the Bunimovich stadium, the Sinai billiard or Donnelly surfaces. We consider a one-parameter family $X_t$ of such domains parametrized by the aspect ratio $t$ of their rectangular part. There is convincing theoretical and numerical evidence that the Laplacian on $X_t$ with Dirichlet or Neumann boundary conditions is not quantum unique ergodic (QUE). We prove that this is true for all $t \in [1,2]$ excluding, possibly, a set of Lebesgue measure zero. This yields the first examples of ergodic billiard systems proven to be non-QUE.

Uniqueness for the signature of a path of bounded variation and the reduced path group

We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length. We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are definite iterated integrals of the path. We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature. In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for reparameterisation in the general setting. It is suggestive to think of this result as a non-commutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients. As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra.

Stable Sets in Multi-Good Pillage Games are Small

It is known that, in one – good pillage games, stable sets are finite. For m goods, it has been conjectured that the stable sets have measure zero. We introduce a class of sets, termed pseudo-indiff erence sets, which includes level sets of utility functions, quasi-indiff erence classes associated with a preference relation not given by a utility function, production possibility frontiers, and Pareto efficient sets. We establish the truth of the conjecture by proving that pseudo-indifference sets in R^p have p-dimensional measure zero. This implies that stable sets in n-agent, m-good pillage games have m(n-1) – dimensional measure zero.We then prove that each pseudo-indi fference set in R^p has Hausdor ff dimension at most p-1, a much stronger result than measure zero. Finally, we establish a stronger version of the conjecture: stable sets in n-agent, m-good pillage games have dimension at most m(n-1)-1.

Most games violate the common priors doctrine

The type of a player in a game describes the beliefs of that player about the types of others. We show that the subset of vectors of such player‐type beliefs which obey the consistency condition sometimes called the Harsanyi doctrine is of Lebesgue measure zero. Furthermore, as the number of players becomes large the ratio of the dimension Harsanyi‐consistent beliefs to the set of all beliefs tends to zero.

The BV-capacity in metric spaces

We study basic properties of the BV-capacity and Sobolev capacity of order one in a complete metric space equipped with a doubling measure and supporting a weak Poincare inequality. In particular, we show that the BV-capacity is a Choquet capacity and the Sobolev 1-capacity is not. However, these quantities are equivalent by two sided estimates and they have the same null sets as the Hausdorff measure of codimension one. The theory of functions of bounded variation plays an essential role in our arguments. The main tool is a modified version of the boxing inequality.

Misiurewicz parameters in the exponential family

A complex exponential map is said to be Misiurewicz if the forward trajectory of the asymptotic value 0 lies in the Julia set and is bounded. We prove that the set of Misiurewicz parameters in the exponential family \({\lambda\exp(z),\lambda\in\mathbb{C}{\setminus}\{0\}}\), has the Lebesgue measure zero.

Computer validation and statistical correlations of a modern decompression diving algorithm

A diving algorithm is a safe combination of model and data to efficiently stage diver ascents following arbitrary underwater exposures. To that end, we detail a modern one, the LANL reduced gradient bubble model (RGBM), dynamical principles, and correlations with the LANL Data Bank data. Table, profile, and meter fit and risk parameters are obtained in statistical likelihood analysis from decompression exposure data. The RGBM algorithm enjoys extensive and utilitarian application in mixed gas diving, both in recreational and technical sectors, and forms the bases for released tables, software, and decompression meters used by scientific, commercial, and research divers. The LANL Data Bank is described, and the methods used to deduce risk are detailed. Risk functions for dissolved gas and bubbles are summarized. Parameters that can be used to estimate profile risk are tallied. To fit data, a modified Levenberg–Marquardt routine is employed. The LANL Data Bank presently contains 2879 profiles with 20 cases of DCS across nitrox, trimix, and heliox deep and decompression diving. This work establishes needed correlation between global mixed gas diving, specific bubble model, and deep stop data. Our objective is operational diving, not clinical science. The fit of bubble model to deep stop data is chi squared significant to 93%, using the logarithmic likelihood ratio of null set (actual set) to fit set. The RGBM algorithm is thus validated within the LANL Data Bank. Extensive and safe utilization of the model reported in field user statistics for tables, meters, and software also suggests real world validation, that is, one without noted nor reported DCS spikes in the field.

Rademacher Theorem for Fréchet Spaces

Let X be a separable Frechet space. In this paper we define a class A of null sets in X that is properly contained in the class of Aronszajn null sets, and we prove that a Lipschitz map from an open subset of X into a Gelfand-Frechet space is Gateaux differentiable outside a set belonging to A. This is an extension to Frechet spaces of a result (see [PZ]) due to D. Preiss and L. Zajicek.

Non existence of principal values of signed Riesz transforms of non integer dimension

In this paper we prove that, given s ≥ 0, and a Borel non zero measure µ in R m , if for µ-almost every x ∈ R m the limit lim

Universally measurable sets in generic extensions

A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $\sigma$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality $\aleph_{1}$, and thus that there exist at least $2^{\aleph_{1}}$ such sets. Laver showed in the 1970's that consistently there are just continuum many universally null sets of reals. The question of whether there exist more than continuum many universally measurable sets of reals was asked by Mauldin in 1978. We show that consistently there exist only continuum many universally measurable sets. This result also follows from work of Ciesielski and Pawlikowski on the iterated Sacks model. In the models we consider (forcing extensions by suitably-sized random algebras) every set of reals is universally measurable if and only if it and its complement are unions of ground model continuum many Borel sets.

P-Harmonic measure is not additive on null sets

When 1 < p < ∞ and p = 2 the p-harmonic measure on the boundary of the half plane R+ is not additive on null sets. In fact, there are finitely many sets E1, E2,...,Eκ in R, of p-harmonic measure zero, such that E1 ∪ E2 ∪ ... ∪ Eκ =R. Mathematics Subject Classification (2000): 31A15 (primary); 35J70, 60G46 (secondary).

Identifiability of parameters in latent structure models with many observed variables

While hidden class models of various types arise in many statistical applications, it is often difficult to establish the identifiability of their parameters. Focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones, we demonstrate a general approach for establishing identifiability utilizing algebraic arguments. A theorem of J. Kruskal for a simple latent-class model with finite state space lies at the core of our results, though we apply it to a diverse set of models. These include mixtures of both finite and nonparametric product distributions, hidden Markov models and random graph mixture models, and lead to a number of new results and improvements to old ones. In the parametric setting, this approach indicates that for such models, the classical definition of identifiability is typically too strong. Instead generic identifiability holds, which implies that the set of nonidentifiable parameters has measure zero, so that parameter inference is still meaningful. In particular, this sheds light on the properties of finite mixtures of Bernoulli products, which have been used for decades despite being known to have nonidentifiable parameters. In the nonparametric setting, we again obtain identifiability only when certain restrictions are placed on the distributions that are mixed, but we explicitly describe the conditions.

Remarks on small sets on the real line Remarks on small sets on the real line

Abstract We consider two kinds of small subsets of the real line: the sets of strong measure zero and the microscopic sets. There are investigated the properties of these sets. The example of a microscopic set, which is not a set of strong measure zero, is given.

Orlicz-Sobolev capacities and their null sets

We define a class of Young functions and prove that certain Orlicz-Sobolev capacities have the same null sets in a fixed Euclidean ball, when a Young function belongs to this class. The main idea of the proof is to show that for these Young functions there is a quasi-norm which is equivalent to the Luxemburg norm.

Periodic and Bloch Solutions to a Magnetic Nonlinear Schrödinger Equation

Abstract We study the equation (℘A) (-i▽ + A)2u + V u = |u|p-2u, where A ∈C1,α(ℝN,ℝN) and V ∈C0,α(ℝN) are 2π-periodic in each variable, V &gt; 0; and p ∈(2; 2*) with 2* := ∞ if N = 2 and We address two questions: first, the gauge-dependence problem for 2π-periodic solutions u : ℝN → C and second, the multiplicity of Bloch solutions. Unlike the nonperiodic case where problem (℘A) is basically independent of A (it is gauge invariant), in the periodic case this is far from being true. Under some assumptions on A we show that, if there exists a one-to-one correspondence between the 2π-periodic solutions of (℘A) and those of (℘A + z) preserving their absolute value, then z lies in a subset of measure zero of ℝN. We use this fact to show the existence of an uncountable set of Bloch solutions with real quasimomentum.

An in vivo comparison of working length determination of two frequency‐based electronic apex locators

To compare in vivo the accuracy of two electronic apex locators (EALs) by means of digital radiographic imaging system. Electronic working lengths of 831 canals were determined with the DentaPort ZX and Raypex 5 apex locators and confirmed radiographically. The radiographic images acquired with the aid of a digital radiographic imaging system (VisualiX eHD; Gendex Dental Systems, Des Plaines, IL, USA) were blindly analysed by two independent evaluators. The distance between the file tip and the radiographic apex was measured using dedicated software (VixWin Pro, Gendex Dental Systems, Des Plaines, IL, USA) and the mean distance achieved between different tooth type and EALs were compared statistically. Statistical analyses were performed using the t-test for independent samples and one-way anova with the null hypothesis set as 5%. Positive or negative values were recorded when the file tip was detected beyond or short of the radiographic apex, respectively. The mean distance between file tip and radiographic apex were -1.08 +/- 0.73 and -1.0 +/- 0.67 mm considering DentaPort ZX and Raypex 5 groups, respectively, with no significant differences (P > 0.05). No statistically significant differences were found amongst the same tooth type when comparing both groups (P > 0.05) or amongst different teeth type in the same group (P > 0.05). Within the limitations of this in vivo study, the DentaPort ZX and Raypex 5 were similar in terms of accuracy.

Cores of many-player games; nonemptiness and equal treatment

This paper provides sufficient conditions to ensure nonemptiness of approximate cores of many-player games and symmetry of approximate core payoffs (the equal treatment property). The conditions are: (a) essential superadditivity—an option open to a group of players is to partition into smaller groups and realize the worths of these groups and (b) small group effectiveness (SGE)—almost all gains to collective activities can be realized by cooperation only within members of some partition of players into relatively small groups. Another condition, small group negligibility (SGN), is introduced and shown to be equivalent to SGE. SGN dictates that small groups of players cannot have significant effects on average (i.e., per capita) payoffs of large populations; thus, SGN is a analogue, for games with a finite player set, of the condition built into models with a continuum of player that sets of measure zero can be ignored. SGE implies per capita boundedness (PCB), that the supremum of average or per capita payoffs is uniformly bounded above. Further characterization of SGE in terms of its relationship to PCB is provided. It is known that if SGE does not hold, then approximate cores of many-player games may be empty. Examples are developed to show that if SGE does not hold and if there are players of “scarce types” (in other works, players with scarce attributes) then even if there is only a finite number of types of players and approximate cores are non-empty, symmetry may be lost; moreover, even players of abundant types may be treated asymmetrically by the core.

Besov capacity redux

Relations between the Besov capacities and other set functions, for example, the Hausdorff capacities are considered. A unique approach suggested here is based on the ideas of the previous works of the author, the results of Netrusov, and the classical characterization of Frostman for Hausdorff capacity. In particular, the Besov capacity C(·; B α p, q ), α > 0, 0 < p, q ⩽ ∞, is reconsidered in light of Netrusov’s recent contribution to the subject–identifying the nature of the null sets when either p or q is less than or equal to 1. We also give a Frostman type argument to replace one of Netrusov’s arguments (for 0 < q ⩽ 1) and present a Frostman type characterization of these set functions: a condition in terms of Borel measures applied to Euclidean balls. Bibliography: 7 titles. Illustrations: 1 figure.

TG‐43 U1 based dosimetric characterization of model 67‐6520 Cs‐137 brachytherapy source

Brachytherapy treatment has been a cornerstone for management of various cancer sites, particularly for the treatment of gynecological malignancies. In low dose rate brachytherapy treatments, 137Cs sources have been used for several decades. A new 137Cs source design has been introduced (model 67-6520, source B3-561) by Isotope Products Laboratories (IPL) for clinical application. The goal of the present work is to implement the TG-43 U1 protocol in the characterization of the aforementioned 137Cs source. The dosimetric characteristics of the IPL 137Cs source are measured using LiF thermoluminescent dosimeters in a Solid Water phantom material and calculated using Monte Carlo simulations with the GEANT4 code in Solid Water and liquid water. The dose rate constant, radial dose function, and two-dimensional anisotropy function of this source model were obtained following the TG-43 U1 recommendations. In addition, the primary and scatter dose separation (PSS) formalism that could be used in convolution/superposition methods to calculate dose distributions around brachytherapy sources in heterogeneous media was studied. The measured and calculated dose rate constants of the IPL 137Cs source in Solid Water were found to be 0.930 (+/-7.3%) and 0.928 (+/-2.6%) cGy h(-1) U(-1), respectively. The agreement between these two methods was within our experimental uncertainties. The Monte Carlo calculated value in liquid water of the dose rate constant was null set=0.948 (+/-2.6%) cGy h(-1) U(-1). Similarly, the agreement between measured and calculated radial dose functions and the anisotropy functions was found to be within +/-5%. In addition, the tabulated data that are required to characterize the source using the PSS formalism were derived. In this article the complete dosimetry of the newly designed 137Cs IPL source following the AAPM TG-43 U1 dosimetric protocol and the PSS formalism is provided.