scholarly journals Nonstandard measure theory–Hausdorff measure

1977 ◽  
Vol 65 (2) ◽  
pp. 326-326
Author(s):  
Frank Wattenberg
Keyword(s):  
Hausdorff Measure ◽  
Measure Theory

scholarly journals Hausdorff theory of dual approximation on planar curves

2018 ◽  
Vol 2018 (740) ◽  
pp. 63-76 ◽  
Author(s):  
Jing-Jing Huang
Keyword(s):  
Hausdorff Measure ◽  
General Class ◽  
Measure Theory ◽  
Complete Theory ◽  
Planar Curves ◽  
The Subject ◽  
Hausdorff Theory ◽  
Challenging Question

AbstractTen years ago, Beresnevich–Dickinson–Velani [Mem. Amer. Math. Soc. 179 (2006), no. 846] initiated a project that develops the general Hausdorff measure theory of dual approximation on non-degenerate manifolds. In particular, they established the divergence part of the theory based on their general ubiquity framework. However, the convergence counterpart of the project remains wide open and represents a major challenging question in the subject. Until recently, it was not even known for any single non-degenerate manifold. In this paper, we settle this problem for all curves in{\mathbb{R}^{2}}, which represents the first complete theory of its kind for a general class of manifolds.

FRACTAL STOKES’ THEOREM BASED ON INTEGRALS ON FRACTAL MANIFOLDS

Fractals ◽  
2020 ◽  
Vol 28 (01) ◽  
pp. 2050010
Author(s):  
JUNRU WU ◽  
CHENGYUAN WANG
Keyword(s):  
Hausdorff Measure ◽  
Measure Theory ◽  
Lower Dimension ◽  
Riemann Integral ◽  
Fractal Sets ◽  
Gauss Theorem ◽  
Variable Substitution ◽  
Definition Of ◽  
Substitution Theorem

In this paper, with the Hausdorff measure, the Hausdorff integral on fractal sets with one or lower dimension is firstly introduced via measure theory. Then the definition of the integral on fractal sets in [Formula: see text] is given. With the variable substitution theorem in the Riemann integral generalized to the integral on fractal sets, the integral on fractal manifolds is defined. As a result, with the generalization of Gauss’ theorem, Stokes’ theorem is generalized to the integral on fractal manifolds in [Formula: see text].

Measure and dimension functions: measurability and densities

1997 ◽  
Vol 121 (1) ◽  
pp. 81-100 ◽  
Author(s):  
PERTTI MATTILA ◽  
R. DANIEL MAULDIN
Keyword(s):  
Hausdorff Measure ◽  
Measure Theory ◽  
Packing Dimension ◽  
Dimension Function ◽  
Packing Measure ◽  
Compact Sets ◽  
Analytic Sets ◽  
Dimension Functions ◽  
Definition Of ◽  
Borel Measurable

During the past several years, new types of geometric measure and dimension have been introduced; the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdorff measure in that it is defined in terms of packings rather than coverings. However, in contrast to Hausdorff measure, the usual definition of packing measure requires two limiting procedures, first the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whether there is some simpler method for defining packing measure and dimension. In this paper, we find a basic limitation on this possibility. We do this by determining the descriptive set-theoretic complexity of the packing functions. Whereas the Hausdorff dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand, we show that the packing dimension functions are measurable with respect to the σ-algebra generated by the analytic sets. Thus, the usual sorts of measurability properties used in connection with Hausdorff measure, for example measures of sections and projections, remain true for packing measure.

Matrix Completions, Moments, and Sums of Hermitian Squares

2011 ◽  
Author(s):  
Mihály Bakonyi ◽  
Hugo J. Woerdeman
Keyword(s):  
Linear Algebra ◽  
Undergraduate Students ◽  
Complex Analysis ◽  
Measure Theory ◽  
Complex Function ◽  
Complex Function Theory ◽  
Extensive Discussion ◽  
The Subject ◽  
Processing Control ◽  
Developing Area

Intensive research in matrix completions, moments, and sums of Hermitian squares has yielded a multitude of results in recent decades. This book provides a comprehensive account of this quickly developing area of mathematics and applications and gives complete proofs of many recently solved problems. With MATLAB codes and more than two hundred exercises, the book is ideal for a special topics course for graduate or advanced undergraduate students in mathematics or engineering, and will also be a valuable resource for researchers. Often driven by questions from signal processing, control theory, and quantum information, the subject of this book has inspired mathematicians from many subdisciplines, including linear algebra, operator theory, measure theory, and complex function theory. In turn, the applications are being pursued by researchers in areas such as electrical engineering, computer science, and physics. The book is self-contained, has many examples, and for the most part requires only a basic background in undergraduate mathematics, primarily linear algebra and some complex analysis. The book also includes an extensive discussion of the literature, with close to six hundred references from books and journals from a wide variety of disciplines.

Measure Theory and Fair Arbiters.

1987 ◽  
Author(s):  
David M. McKeown ◽  
Jr
Keyword(s):  
Measure Theory

DERIVATION BASES AND THE HAUSDORFF MEASURE

1986 ◽  
Vol 12 (1) ◽  
pp. 118
Author(s):  
Meinershagen
Keyword(s):  
Hausdorff Measure

Derivation Bases and the Hausdorff Measure

1987 ◽  
Vol 13 (1) ◽  
pp. 223
Author(s):  
Meinershagen
Keyword(s):  
Hausdorff Measure

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

Sign in / Sign up

Export Citation Format

Share Document