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.

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.

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

scholarly journals Formulation of scale transformation in a stochastic data assimilation framework

2017 ◽  
Vol 24 (2) ◽  
pp. 279-291 ◽  
Author(s):  
Feng Liu ◽  
Xin Li
Keyword(s):  
Data Assimilation ◽  
Spatial Scale ◽  
Measure Theory ◽  
Stochastic Calculus ◽  
Scale Transformation ◽  
Earth Observations ◽  
Model Unit ◽  
Stochastic Data ◽  
Definition Of ◽  
New Formulation

Abstract. Understanding the errors caused by spatial-scale transformation in Earth observations and simulations requires a rigorous definition of scale. These errors are also an important component of representativeness errors in data assimilation. Several relevant studies have been conducted, but the theory of the scale associated with representativeness errors is still not well developed. We addressed these problems by reformulating the data assimilation framework using measure theory and stochastic calculus. First, measure theory is used to propose that the spatial scale is a Lebesgue measure with respect to the observation footprint or model unit, and the Lebesgue integration by substitution is used to describe the scale transformation. Second, a scale-dependent geophysical variable is defined to consider the heterogeneities and dynamic processes. Finally, the structures of the scale-dependent errors are studied in the Bayesian framework of data assimilation based on stochastic calculus. All the results were presented on the condition that the scale is one-dimensional, and the variations in these errors depend on the differences between scales. This new formulation provides a more general framework to understand the representativeness error in a non-linear and stochastic sense and is a promising way to address the spatial-scale issue.

The measure theory of random fractals

1986 ◽  
Vol 100 (3) ◽  
pp. 383-406 ◽  
Author(s):  
S. James Taylor
Keyword(s):  
Brownian Motion ◽  
Potential Theory ◽  
Measure Theory ◽  
Motion Path ◽  
Natural Question ◽  
Random Fractals ◽  
Limited Space ◽  
Definition Of ◽  
Good Problem

In 1951 A. S. Besicovitch, who was my research supervisor, suggested that I look at the problem of determining the dimension of the range of a Brownian motion path. This problem had been communicated to him by C. Loewner, but it was a natural question which had already attracted the attention of Paul Lévy. It was a good problem to give to an ignorant Ph.D. student because it forced him to learn the potential theory of Frostman [33] and Riesz[75] as well as the Wiener [98] definition of mathematical Brownian motion. In fact the solution of that first problem in [81] used only ideas which were already twenty-five years old, though at the time they seemed both new and original to me. My purpose in this paper is to try to trace the development of these techniques as they have been exploited by many authors and used in diverse situations since 1953. As we do this in the limited space available it will be impossible to even outline all aspects of the development, so I make no apology for giving a biased account concentrating on those areas of most interest to me. At the same time I will make conjectures and suggest some problems which are natural and accessible in the hope of stimulating further research.

scholarly journals The Effect of Angular Momentum and Ostrogradsky-Gauss Theorem in the Equations of Mechanics

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Mathematical Models ◽  
Conservation Laws ◽  
Rarefied Gas ◽  
Classical Mechanics ◽  
Gauss Theorem ◽  
The Media ◽  
Velocity Circulation ◽  
The Times ◽  
The Many ◽  
Definition Of

There are many experimental facts that currently cannot be described theoretically. A possible reason is bad mathematical models and algorithms for calculation, despite the many works in this area of research. The aim of this work is to clarificate the mathematical models of describing for rarefied gas and continuous mechanics and to study the errors that arise when we describe a rarefied gas through distribution function. Writing physical values conservation laws via delta functions, the same classical definition of physical values are obtained as in classical mechanics. Usually the derivation of conservation laws is based using the Ostrogradsky-Gauss theorem for a fixed volume without moving. The theorem is a consequence of the application of the integration in parts at the spatial case. In reality, in mechanics and physics gas and liquid move and not only along a forward path, but also rotate. Discarding the out of integral term means ignoring the velocity circulation over the surface of the selected volume. When taking into account the motion of a gas, this term is difficult to introduce into the differential equation. Therefore, to account for all components of the motion, it is proposed to use an integral formulation. Next question is the role of the discreteness of the description of the medium in the kinetic theory and the interaction of the discreteness and "continuity" of the media. The question of the relationship between the discreteness of a medium and its description with the help of continuum mechanics arises due to the fact that the distances between molecules in a rarefied gas are finite, the times between collisions are finite, but on definition under calculating derivatives on time and space we deal with infinitely small values. We investigate it

scholarly journals On rectifiable measures in Carnot groups: representation

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Gioacchino Antonelli ◽  
Andrea Merlo
Keyword(s):  
Large Class ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Lipschitz Functions ◽  
Carnot Groups ◽  
Geodesic Sphere ◽  
Area Formula ◽  
Definition Of ◽  
Lipschitz Graphs ◽  
Almost All

AbstractThis paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

scholarly journals Riemann Integral: Significance and Limitations

2018 ◽  
Vol 7 ◽  
pp. 31-34
Author(s):  
Laxman Bahadur Kunwar
Keyword(s):  
Closed Interval ◽  
Mathematical Science ◽  
Real Analysis ◽  
Lebesgue Integral ◽  
Riemann Integral ◽  
Definition Of

This article basically defines the Riemann integral starting with the definition of partition of a closed interval. It throws a light on the importance and necessity of the Riemann condition of integralbility of a function and explains how the concept of R- integral only on the basis of upper and lower integral is not always practical. It has been suggested that a better rout is to abandon the Reimann integral for Lebesgue integral in real analysis and other fields of mathematical science.

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