The Quasi-Isomorphism Theorem

2019 ◽  
pp. 185-194
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Affine Transformation ◽  
Equivalence Theorem ◽  
Isomorphism Theorem ◽  
Orbit Equivalence ◽  
Geometric Properties ◽  
Projection Theorem ◽  
Canonical Bijection ◽  
Arithmetic Graph

This chapter proves the Quasi-Isomorphism Theorem modulo two technical lemmas, which will be dealt with in the next two chapters. Section 18.2 introduces the affine transformation TA from the Quasi-Isomorphism Theorem. Section 18.3 defines the graph grid GA = TA(Z2) and states the Grid Geometry Lemma, a result about the basic geometric properties of GA. Section 18.4 introduces the set Z* that appears in the Renormalization Theorem and states the main result about it, the Intertwining Lemma. Section 18.5 explains how the Orbit Equivalence Theorem sets up a canonical bijection between the nontrivial orbits of the plaid PET and the orbits of the graph PET. Section 18.6 reinterprets the orbit correspondence in terms of the plaid polygons and the arithmetic graph polygons. Everything is then put together to complete the proof of the Quasi-Isomorphism Theorem. Section 18.7 deduces the Projection Theorem (Theorem 0.2) from the Quasi-Isomorphism Theorem.

The Orbit Equivalence Theorem

2019 ◽  
pp. 173-184
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Convex Polytopes ◽  
Equivalence Theorem ◽  
Computational Techniques ◽  
Isomorphism Theorem ◽  
Orbit Equivalence ◽  
Large Subset ◽  
Computer Calculations

This chapter begins Part 4 of the monograph. The goal of this part is to prove the Orbit Equivalence Theorem and the Quasi-Isomorphism Theorem. Theorem 17.1 (Orbit Equivalence) states that there is a dynamically large subset Z ⊂ X and a map Ω‎: Z → Y. Section 17.2 defines Z. Section 17.3 defines Ω‎. Section 17.4 characterizes the image Ω‎(Z). Section 17.5 defines a partition of Z into small convex polytopes which have the property that all the maps in Equations 17.1 and 1 are entirely defined and projective on each polytope. This allows us to verify the properties in the Orbit Equivalence Theorem just by checking what the two relevant maps do to the vertices of the new partition. Section 17.6 puts everything together and prove the Orbit Equivalence Theorem modulo some integer computer calculations. Section 17.7 discusses the computational techniques used to carry out the calculations from Section 17.6. Section 17.8 explains the calculations.

RECOGNITION OF AFFINE TRANSFORMED PLANAR CURVES BY EXTREMAL GEOMETRIC PROPERTIES

1993 ◽  
Vol 03 (02) ◽  
pp. 183-202 ◽  
Author(s):  
CRAIG GOTSMAN ◽  
MICHAEL WERMAN
Keyword(s):  
Digital Image ◽  
Affine Transformation ◽  
Time Complexity ◽  
Affine Invariant ◽  
Geometric Properties ◽  
Planar Curves ◽  
Geometric Methods ◽  
Planar Curve ◽  
Image Pixels

An algorithm for the recognition of a digital image of a planar curve which has undergone an affine transformation is presented. The algorithm is based on affine-invariant extremal geometric properties of curves, utilizes existing computational-geometric methods, and is relatively insensitive to noise. Its time complexity is linear in the number of image pixels on the curve. Extensions of our algorithm to deal with some cases of occlusion of the image curve and recognition under perspective transformations are also described. These algorithms are almost linear in the number of image pixels on the curve.

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

Some Properties of Para-Sasakian Manifolds

2017 ◽  
Vol 5 (2) ◽  
pp. 73-78
Author(s):  
Jay Prakash Singh ◽  
Keyword(s):  
Vector Field ◽  
Killing Vector ◽  
Sasakian Manifold ◽  
Subject Classification ◽  
Killing Vector Field ◽  
Sasakian Manifolds ◽  
Geometric Properties ◽  
Paper Author

In this paper author present an investigation of some differential geometric properties of Para-Sasakian manifolds. Condition for a vector field to be Killing vector field in Para-Sasakian manifold is obtained. Mathematics Subject Classification (2010). 53B20, 53C15.

Effect of 12 months of creatine supplementation and whole-body resistance training on measures of bone, muscle and strength in older males

2020 ◽  
pp. 026010602097524
Author(s):  
Darren G Candow ◽  
Philip D Chilibeck ◽  
Julianne Gordon ◽  
Emelie Vogt ◽  
Tim Landeryou ◽  
...  
Keyword(s):  
Bone Mineral Density ◽  
Resistance Training ◽  
Bone Mineral ◽  
Creatine Supplementation ◽  
Whole Body ◽  
Mineral Density ◽  
Geometric Properties ◽  
Section Modulus ◽  
Body Resistance ◽  
Older Males

Background: The combination of creatine supplementation and resistance training (10–12 weeks) has been shown to increase bone mineral content and reduce a urinary indicator of bone resorption in older males compared with placebo. However, the longer-term effects (12 months) of creatine and resistance training on bone mineral density and bone geometric properties in older males is unknown. Aim: To assess the effects of 12 months of creatine supplementation and supervised, whole-body resistance training on bone mineral density, bone geometric properties, muscle accretion, and strength in older males. Methods: Participants were randomized to supplement with creatine ( n = 18, 49–69 years, 0.1 g·kg-1·d-1) or placebo ( n = 20, 49–67 years, 0.1 g·kg-1·d-1) during 12 months of supervised, whole-body resistance training. Results: After 12 months of training, both groups experienced similar changes in bone mineral density and geometry, bone speed of sound, lean tissue and fat mass, muscle thickness, and muscle strength. There was a trend ( p = 0.061) for creatine to increase the section modulus of the narrow part of the femoral neck, an indicator of bone bending strength, compared with placebo. Adverse events did not differ between creatine and placebo. Conclusions: Twelve months of creatine supplementation and supervised, whole-body resistance training had no greater effect on measures of bone, muscle, or strength in older males compared with placebo.

Equivalence theorem of light waves on scattering from anisotropic media

Optik ◽  
2020 ◽  
Vol 206 ◽  
pp. 164300
Author(s):  
Xiaoning Pan ◽  
Ke Cheng ◽  
Xiaoling Ji ◽  
Tao Wang
Keyword(s):  
Anisotropic Media ◽  
Equivalence Theorem ◽  
Light Waves
Sign in / Sign up

Export Citation Format

Share Document