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 A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES

2007 ◽  
Vol 49 (2) ◽  
pp. 367-375 ◽  
Author(s):  
DZMITRY BADZIAHIN ◽  
JASON LEVESLEY
Keyword(s):  
Diophantine Approximation ◽  
Hausdorff Measure ◽  
General Class ◽  
Simultaneous Approximation ◽  
Convergence Result ◽  
Planar Curves ◽  
Planar Curve ◽  
Induced Measure

AbstractLet $\mathbb C$ be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in $\mathbb R^2$ with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.

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.

scholarly journals Nonstandard measure theory–Hausdorff measure

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

The Generalized Baker–Schmidt Problem on Hypersurfaces

2019 ◽  
Author(s):  
Mumtaz Hussain ◽  
Johannes Schleischitz ◽  
David Simmons
Keyword(s):  
Hausdorff Measure ◽  
General Setting ◽  
Dimensional Hausdorff Measure ◽  
Planar Curves ◽  
Open Question

Abstract The generalized Baker–Schmidt problem (1970) concerns the $f$-dimensional Hausdorff measure of the set of $\psi $-approximable points on a nondegenerate manifold. There are two variants of this problem concerning simultaneous and dual approximation. Beresnevich–Dickinson–Velani (in 2006, for the homogeneous setting) and Badziahin–Beresnevich–Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary nondegenerate manifolds. The corresponding convergence counterpart represents a major challenging open question and the progress thus far has only been attained over planar curves. In this paper, we settle this problem for hypersurfaces in a more general setting, that is, for inhomogeneous approximations and with a non-monotonic multivariable approximating function.

scholarly journals XVI. On electrotorsion

1874 ◽  
Vol 164 ◽  
pp. 529-562
Keyword(s):  
Royal Society ◽  
Complete Theory ◽  
Electric Currents ◽  
Iron Wire ◽  
The Subject ◽  
A Current

Wiedemann lias experimentally examined† the influence of magnetism upon the mechanical torsion of iron wire, and has shown that an iron wire hung in the centre of a helix and twisted is more or less untwisted when a current traverses the helix. But as the torsion in his experiments was produced by the combined influence of a voltaic current and previous mechanical twist, and is quite a distinct phenomenon from that produced by the combined influence of electric currents only, which forms the subject of this communication, and as no one appears to have discovered the particular class of phenomena which are treated of in this investigation, I take an opportunity of making known my experiments and the new facts I have found. [Since the publication of the abstract of this paper in the ‘Proceedings of the Royal Society,’ vol. xxii. p. 57, January 8, 1874, Professor Wiedemann (to whom I had sent a copy of that abstract) has kindly written to me as follows:—“ You have found independently some results which I had already published in the year 1862 in Poggen-dorff’s ‘Annalen,’ vol. cxvii. p. 208. A short abstract of these experiments is also given in my ‘Treatise on Galvanism, &c.’ (1st edition, vol. ii. p. 445; 2nd edition, vol. ii. p. 565), where you will find my complete theory of the relations between magnetism and torsion.”]

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].

scholarly journals New approach to some results related to mixed norm sequence spaces

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 83-88 ◽  
Author(s):  
Ivana Djolovic ◽  
Eberhard Malkowsky ◽  
Katarina Petkovic
Keyword(s):  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Diagonal Matrix ◽  
Sequence Spaces ◽  
Mixed Norm ◽  
Hausdorff Measure Of Noncompactness ◽  
Infinite Matrices ◽  
New Approach ◽  
The Subject ◽  
Bk Spaces

In this paper, the mixed norm sequence spaces ?p,q for 1 ? p,q ? ? are the subject of our research; we establish conditions for an operator T? to be compact, where T? is given by a diagonal matrix. This will be achieved by applying the Hausdorff measure of noncompactness and the theory of BK spaces. This problem was treated and solved in [5, 6], but in a different way, without the application of the theory of infinite matrices and BK spaces. Here, we will present a new approach to the problem. Some of our results are known and others are new.

Nonstandard Measure Theory-Hausdorff Measure

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

VII.—On Eisenstein's Continued Fractions

1877 ◽  
Vol 28 (1) ◽  
pp. 135-143 ◽  
Author(s):  
Thomas Muir
Keyword(s):  
Continued Fractions ◽  
General Equation ◽  
Primitive Root ◽  
Complete Theory ◽  
The Subject ◽  
Image Position ◽  
Cubic Forms

In Crelle's Journal for 1844, at the end of a paper on Cubic Forms, Eisenstein gives the following results:—where m is any odd number and ρ a primitive root of the equation zm = 1;whereNo demonstrations are given of these identities; but they are said by the author to be only particular cases of a very general equation, the complete theory connected with which he hoped to give on another occasion.At p. 193 of the same volume of Crelle he returns to the subject, not however for the purpose of giving the promised theory, but to add several other results similar to those before given.

The fractional dimensional theory of continued fractions

1941 ◽  
Vol 37 (3) ◽  
pp. 199-228 ◽  
Author(s):  
I. J. Good
Keyword(s):  
Continued Fractions ◽  
Measure Zero ◽  
Measure Theory ◽  
Linear Measure ◽  
Dimensional Theory ◽  
Partial Quotients ◽  
The Subject ◽  
Almost All ◽  
Fractional Dimensions

The notion of fractional dimensions is one which is now well known. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients. The sets considered are naturally of linear measure zero. Those properties of the partial quotients which hold for almost all continued fractions make up the subject called by Khintchine ‘the measure theory of continued fractions’.

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