scholarly journals An alternative approach to the ergodic theory of measured foliations on surfaces

1981 ◽  
Vol 1 (4) ◽  
pp. 461-488 ◽  
Author(s):  
Mary Rees
Keyword(s):  
Projective Space ◽  
Lebesgue Measure ◽  
Ergodic Theory ◽  
Diffeomorphism Group ◽  
Alternative Approach ◽  
Interval Exchanges ◽  
Almost All ◽  
Uniquely Ergodic

AbstractWe consider measured foliations on surfaces, and interval exchanges. We give alternative proofs of the following theorems first proved by Masur and (independently) Veech. The action of the diffeomorphism group of the surface on the projective space of measured foliations (with respect to a natural ‘Lebesgue’ measure) is ergodic. Almost all measured foliations are uniquely ergodic. Almost all interval exchanges (again, with respect to a natural ‘Lebesgue’ measure) are uniquely ergodic.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

Identification of ’Effective’ Linear Joints Using Coupling and Joint Identification Techniques

1998 ◽  
Vol 120 (2) ◽  
pp. 331-338 ◽  
Author(s):  
Y. Ren ◽  
C. F. Beards
Keyword(s):  
Dynamic Properties ◽  
Real Life ◽  
Model Parameters ◽  
Coupling Technique ◽  
Alternative Approach ◽  
Identification Technique ◽  
Joint Identification ◽  
Stiff Joint ◽  
Almost All ◽  
Identification Techniques

Almost all real-life structures are assembled from components connected by various types of joints. Unlike many other parts, the dynamic properties of a joint are difficult to model analytically. An alternative approach for establishing a theoretical model of a joint is to extract the model parameters from experimental data using joint identification techniques. The accuracy of the identification is significantly affected by the properties of the joints themselves. If a joint is stiff, its properties are often difficult to identify accurately. This is because the responses at both ends of the joint are linearly-dependent. To make things worse, the existence of a stiff joint can also affect the accuracy of identification of other effective joints (the term “effective joints” in this paper refers to those joints which otherwise can be identified accurately). This problem is tackled by coupling these stiff joints using a generalized coupling technique, and then the properties of the remaining joints are identified using a joint identification technique. The accuracy of the joint identification can usually be improved by using this approach. Both numerically simulated and experimental results are presented.

The trajectory of the turning point is dense for almost all tent maps

1996 ◽  
Vol 16 (6) ◽  
pp. 1173-1183 ◽  
Author(s):  
Karen Brucks ◽  
Michal Misiurewicz
Keyword(s):  
Lebesgue Measure ◽  
Turning Point ◽  
Tent Map ◽  
Tent Maps ◽  
Almost All

AbstractWe prove that for almost every (with respect to the Lebesgue measure) a ∈ [√2, 2], the forward trajectory of the turning point of the tent map fa with slope a is dense in the interval of transitivity of fa.

An Alternative Approach for Evaluating the Binormal ROC Curve

2013 ◽  
Vol 4 (1) ◽  
pp. 1-17 ◽  
Author(s):  
R. Amala ◽  
R. Vishnu Vardhan
Keyword(s):  
Roc Curve ◽  
Error Function ◽  
False Positive Rate ◽  
True Positive Rate ◽  
Roc Curve Analysis ◽  
New Approach ◽  
Alternative Approach ◽  
Positive Rate ◽  
Almost All ◽  
Two Populations

In recent years the ROC curve analysis has got its attention in almost all diversified fields. Basing on the data pattern and its distribution various forms of ROC models have been derived. In this paper, the authors have assumed that the data of two populations (healthy and diseased) follows normal distribution, it is one of the most commonly used forms under parametric approach. The present paper focuses on providing an alternative approach for the tradeoff plot of ROC curve and the computation of AUC using a special function of sigmoid shape called Error function. It is assumed that the test scores of particular biomarker are normally distributed. The entire work has been carried out for providing a new approach for the construction of Binormal ROC curve, which makes use of Error function which can be called as ErROC curve. The summary measure AUC of the resulting ErROC curve has been estimated and defined as ErAUC. The authors have also focused on deriving the expression for obtaining the optimal cut-off point. The new ErROC curve model will provide the true positive rate value at each and every point of false positive rate unlike conventional Binormal ROC model.

A Family of Generalized Riesz Products

1996 ◽  
Vol 48 (2) ◽  
pp. 302-315 ◽  
Author(s):  
A. H. Dooley ◽  
S. J. Eigen
Keyword(s):  
Lebesgue Measure ◽  
Spectral Measure ◽  
Probability Space ◽  
Rank One ◽  
Almost All ◽  
Riesz Products

AbstractGeneralized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure.

scholarly journals Canonical and log canonical thresholds of multiple projective spaces

2019 ◽  
Author(s):  
Aleksandr V. Pukhlikov
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Regularity Conditions ◽  
Large Classes ◽  
Log Canonical Threshold ◽  
Birational Rigidity ◽  
Log Canonical ◽  
Finite Set ◽  
Dimensional Projective Space ◽  
Almost All

AbstractWe show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where $$d\geqslant 4$$d⩾4, is equal to 1 for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano–Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

Decidability of the “almost all” theory of degrees

1972 ◽  
Vol 37 (3) ◽  
pp. 501-506 ◽  
Author(s):  
John Stillwell
Keyword(s):  
Lebesgue Measure ◽  
Measure Theory ◽  
Random Sequence ◽  
Recursion Theory ◽  
Sharp Contrast ◽  
Standard Theory ◽  
Simple Manner ◽  
Fubini’S Theorem ◽  
Almost All ◽  
Random Pair

Ever since Spector's brilliant application of measure theory to recursion theory in 1958 [6] it has been realized that measure theory promotes sweeping simplifications in the theory of degrees. Results previously thought to be pathological were shown by Spector, and later Sacks [4], [5], to hold for almost all degrees (“almost all” in the sense of Lebesgue measure), often with much simpler proofs. Good examples of this phenomenon are Spector's demonstration that almost all pairs of sets are of incomparable degree (as an immediate consequence of Fubini's theorem) and Sacks' exquisitely simple deduction from this result that almost every degree is the join of two incomparable degrees (for if a random sequence is decomposed into its even and odd parts, the result is a random pair).The present paper attempts to vindicate the feeling that almost all degrees behave in a simple manner by showing that if the quantifier in the theory of degrees with ′(jump), ∪ (join) and ∩ (meet) is taken to be (almost ∀a) instead of (∀a) then the theory is decidable. We are able to use ∩ because it will be shown that if t1, t2 are any terms built from degree variables a1, …, am with ′ and ∪ then t1 ∩ t2 exists for almost all a1, …, am. Thus the “almost all” theory presents a sharp contrast to the standard theory, where ∩ is not always defined (Kleene-Post [1]) and which is known to be undecidable (Lachlan [2]).

scholarly journals Simplicial systems for interval exchange maps and measured foliations

1985 ◽  
Vol 5 (2) ◽  
pp. 257-271 ◽  
Author(s):  
S. P. Kerckhoff
Keyword(s):  
General Theorem ◽  
Interval Exchange ◽  
Alternative Proof ◽  
Simple Assumption ◽  
Refinement Process ◽  
Almost All ◽  
Interval Exchange Maps ◽  
Uniquely Ergodic

AbstractThe spaces of interval exchange maps and measured foliations are considered and an alternative proof that almost all interval exchange maps and measured foliations are uniquely ergodic is given. These spaces are endowed with a refinement process, called a simplicial system, which is studied abstractly and is shown to be normal under a simple assumption. The results follow and thus are a corollary of a more general theorem in a broader setting.

Some theorems in the metric theory of diophantine approximation

1986 ◽  
Vol 99 (3) ◽  
pp. 385-394 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Lebesgue Measure ◽  
Diophantine Approximation ◽  
Additional Constraint ◽  
Number Of Solutions ◽  
Initial Question ◽  
Image Position ◽  
Almost All

An excellent introduction to the metric theory of diophantine approximation is provided by [19], where, in chapter 1·7, the reader may find a discussion of the first two problems considered in this paper. Our initial question concerns the number of solutions of the inequalityfor almost all α(in the sense of Lebesgue measure on ℝ). Here ∥ ∥ denotes distance to a nearest integer, {βr}, {ar} are given sequences of reals and distinct integers respectively, and f is a function taking values in [0, ½] and with Σf(r) divergent (for convenience we write ℱ for the set of all such functions). It is reasonable to expect that, for almost all α and with some additional constraint on f, the number of solutions of (1) is asymptotically equal toas k tends to infinity.

scholarly journals PENINGKATAN KETERSEDIAAN NUTRISI MIKRO PADA TANAMAN: UPAYA MENGURANGI MALNUTRISI PADA MANUSIA

2018 ◽  
Vol 31 (2) ◽  
pp. 118
Author(s):  
Amalia Tetrani Sakya
Keyword(s):  
Food System ◽  
Vitamin A Deficiency ◽  
Well Being ◽  
Population Pressure ◽  
Nutritional Deficiencies ◽  
Nutrient Deficiencies ◽  
Food Ingredient ◽  
Micronutrient Intake ◽  
Alternative Approach ◽  
Almost All

<div class="WordSection1"><p><em>Malnutrition is still one of the big problems the majority of devel<strong>op</strong>ing countries including Indonesia. Malnutrition is the result of insufficient intake of available nutrients in the human diet. The availability of nutrients is mainly determined by the output of food produced from agricultural systems. Plants provide almost all the necessary vitamins and minerals, but due to low mineral content in staple crops, resulting in the intake becomes less and lead to malnutrition or lack of nutrients. Unfortunately, as a result of population pressure, a lot of the current global food system does not provide enough micronutrients to ensure adequate micronutrient intake for everyone. This has resulted in an increase in the prevalence of micro-nutrient deficiencies (for example, iron deficiency, vitamin A deficiency, and iodine), which now afflicts many poor women resources, infants and children in developing countries. To get a balanced nutrition and adequate then improve the quality of agriculture as a food ingredient indispensable. Various attempts to overcome nutritional deficiencies especially regarding micro nutrient deficiencies, such as supplementation, food fortification and diversification of the food has a lot to do, but did not provide maximum results. Another alternative approach to address the problem of shortage of micronutrients is biofortification, genetic biofortification or agronomic biofortification. This approach emerged due to health and human well-being depends entirely on the plants, either directly or indirectly.</em></p></div>

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