Aequationes mathematicae, ISSN 0001-9054, 8/2018, Volume 92, Issue 4, pp. 709 - 735

We study the Hausdorff and packing measures of typical compact metric spaces belonging to the Gromov–Hausdorff space (of all compact metric spaces) equipped...

The Gromov–Hausdorff metric | Compact metric space | Analysis | Hewitt–Stromberg measure Hausdorff measure | 28A78 | 28A80 | Packing measure | Mathematics | Box dimension | Combinatorics | MATHEMATICS | DIMENSIONS | MATHEMATICS, APPLIED | SETS | The Gromov-Hausdorff metric | Hewitt-Stromberg measure Hausdorff measure

The Gromov–Hausdorff metric | Compact metric space | Analysis | Hewitt–Stromberg measure Hausdorff measure | 28A78 | 28A80 | Packing measure | Mathematics | Box dimension | Combinatorics | MATHEMATICS | DIMENSIONS | MATHEMATICS, APPLIED | SETS | The Gromov-Hausdorff metric | Hewitt-Stromberg measure Hausdorff measure

Journal Article

Real Analysis Exchange, ISSN 0147-1937, 1/2015, Volume 40, Issue 1, pp. 113 - 128

We estimate the -Hausdorff and -packing measures of balanced Cantor sets, and characterize the corresponding dimension partitions. This generalizes results...

Integers | Closed intervals | Open intervals | Cantor set | Symbolism | Mathematical functions | Research Articles | Hausdorff measures | gauge functions | dimension functions | cut-out sets | 28A78 | 28A80 | Cantor sets | packing measures

Integers | Closed intervals | Open intervals | Cantor set | Symbolism | Mathematical functions | Research Articles | Hausdorff measures | gauge functions | dimension functions | cut-out sets | 28A78 | 28A80 | Cantor sets | packing measures

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 06/2019, Volume 474, Issue 1, pp. 143 - 156

We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous. The calculation is based on the local...

Packing measure | Hausdorff measure | Cantor set | MATHEMATICS | MATHEMATICS, APPLIED | DENSITIES | COMPUTABILITY

Packing measure | Hausdorff measure | Cantor set | MATHEMATICS | MATHEMATICS, APPLIED | DENSITIES | COMPUTABILITY

Journal Article

4.
Full Text
Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces

Monatshefte für Mathematik, ISSN 0026-9255, 11/2011, Volume 164, Issue 3, pp. 313 - 323

We give a new estimate for the ratio of s-dimensional Hausdorff measure $${\mathcal{H}^s}$$ and (radius-based) packing measure $${\mathcal{P}^s}$$ of a set in...

Hausdorff measure | Metric space | 28A78 | Mathematics, general | Packing measure | Mathematics | Density | MATHEMATICS | FUNDAMENTAL GEOMETRICAL PROPERTIES | PLANE SETS | POINTS

Hausdorff measure | Metric space | 28A78 | Mathematics, general | Packing measure | Mathematics | Density | MATHEMATICS | FUNDAMENTAL GEOMETRICAL PROPERTIES | PLANE SETS | POINTS

Journal Article

5.
Geometry and dynamics in Gromov hyperbolic metric spaces

: with an emphasis on non-proper settings

2017, Mathematical surveys and monographs, ISBN 9781470434656, Volume 218, xxxv, 281 pages

Geometry, Hyperbolic | Ergodic theory | Measure and integration | Fuchsian groups and their generalizations | Infinite-dimensional Lie groups and their Lie algebras: general properties | Hyperbolic groups and nonpositively curved groups | Special aspects of infinite or finite groups | Semigroups of transformations, etc | Metric spaces | Conformal densities and Hausdorff dimension | Structure and classification of infinite or finite groups | Relations with number theory and harmonic analysis | Classical measure theory | Group theory and generalizations | Other groups of matrices | Complex dynamical systems | Lie groups | Hyperbolic spaces | Semigroups | Groups acting on trees | Topological groups, Lie groups | Dynamical systems and ergodic theory | Hausdorff and packing measures

Book

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 09/2019, Volume 126, pp. 203 - 217

The multifractal formalism for measures holds whenever the existence of corresponding Gibbs-like measures supported on the singularities sets holds. In the...

Multifractal formalism | Mixed cases | Hölderian measures | Hausdorff and packing dimensions | Hausdorff and packing measures | Holderian measures | NUMBERS | PHYSICS, MULTIDISCIPLINARY | SUBSETS | CANTOR SETS | PHYSICS, MATHEMATICAL | SUMS | DIMENSIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | HAUSDORFF MEASURES | DIVERGENCE POINTS | PROJECTIONS

Multifractal formalism | Mixed cases | Hölderian measures | Hausdorff and packing dimensions | Hausdorff and packing measures | Holderian measures | NUMBERS | PHYSICS, MULTIDISCIPLINARY | SUBSETS | CANTOR SETS | PHYSICS, MATHEMATICAL | SUMS | DIMENSIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | HAUSDORFF MEASURES | DIVERGENCE POINTS | PROJECTIONS

Journal Article

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 10/2017, Volume 103, pp. 1 - 11

In this paper, we establish some density results related to the multifractal generalization of the centered Hausdorff and packing measures. We also focus on...

Relative multifractal analysis | Hausdorff and packing dimensions | Hausdorff and packing measures | SPACE | Hausdorffand packing measures | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | Hausdorffand packing dimensions | PHYSICS, MATHEMATICAL | EQUIVALENCE | Specific gravity | Analysis

Relative multifractal analysis | Hausdorff and packing dimensions | Hausdorff and packing measures | SPACE | Hausdorffand packing measures | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | Hausdorffand packing dimensions | PHYSICS, MATHEMATICAL | EQUIVALENCE | Specific gravity | Analysis

Journal Article

Aequationes mathematicae, ISSN 0001-9054, 6/2008, Volume 75, Issue 3, pp. 208 - 225

We analyze the local behaviour of the Hausdorff measure and the packing measure of self-similar sets. In particular, if K is a self-similar set whose Hausdorff...

Hausdorff measure | Analysis | 28A80 | Mathematics | Combinatorics | self-similar measure | packing measure | densities | Self-similar set | Self-similar measure | Packing measure | Densities | MATHEMATICS | MATHEMATICS, APPLIED | Texts | Theorems | Density | Inequalities | Self-similarity

Hausdorff measure | Analysis | 28A80 | Mathematics | Combinatorics | self-similar measure | packing measure | densities | Self-similar set | Self-similar measure | Packing measure | Densities | MATHEMATICS | MATHEMATICS, APPLIED | Texts | Theorems | Density | Inequalities | Self-similarity

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 06/2012, Volume 390, Issue 1, pp. 234 - 243

Let be a dilation-stable Lévy process on . We determine a Hausdorff measure function such that the graph has positive finite -measure. We also investigate the...

Packing measure | Graph | Dilation-stable Lévy processes | Exact Hausdorff measure function | MATHEMATICS | MATHEMATICS, APPLIED | Dilation-stable Levy processes | SAMPLE PATH PROPERTIES

Packing measure | Graph | Dilation-stable Lévy processes | Exact Hausdorff measure function | MATHEMATICS | MATHEMATICS, APPLIED | Dilation-stable Levy processes | SAMPLE PATH PROPERTIES

Journal Article

2013, Volume s 600, 601

Conference Proceeding

Nonlinearity, ISSN 0951-7715, 04/2018, Volume 31, Issue 6, pp. 2571 - 2589

We show that the s-dimensional packing measure P-s(S) of the Sierpinski gasket S, where s = log 3/log 2 is the similarity dimension of S, satisfies 1.6677 <=...

Sierpinski gasket | computability of fractal Measures | packing measure | self-similar sets | algorithm | MATHEMATICS, APPLIED | EQUATIONS | computability of fractal measures | PHYSICS, MATHEMATICAL | CENTERED HAUSDORFF MEASURES | COMPUTABILITY

Sierpinski gasket | computability of fractal Measures | packing measure | self-similar sets | algorithm | MATHEMATICS, APPLIED | EQUATIONS | computability of fractal measures | PHYSICS, MATHEMATICAL | CENTERED HAUSDORFF MEASURES | COMPUTABILITY

Journal Article

Geometriae Dedicata, ISSN 0046-5755, 12/2018, Volume 197, Issue 1, pp. 173 - 192

Let J be the limit set of an iterated function system insatisfying the open set condition. It is well known that the h-dimensional packing measure of J is...

Cantor sets | Packing measure | Fractals | Iterated function systems | Sierpiński triangles | MATHEMATICS | HAUSDORFF | SELF-SIMILAR SETS | Sierpiski triangles | 28A78 | 28A80 | 37C45

Cantor sets | Packing measure | Fractals | Iterated function systems | Sierpiński triangles | MATHEMATICS | HAUSDORFF | SELF-SIMILAR SETS | Sierpiski triangles | 28A78 | 28A80 | 37C45

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2008, Volume 342, Issue 1, pp. 571 - 584

For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the...

Hausdorff dimension | Hausdorff measure | Euler characteristic | Quantization dimension | Packing dimension | Packing measure | Homogeneous Cantor set | Euler exponent | Quantization coefficient | MATHEMATICS, APPLIED | SELF-SIMILAR PROBABILITIES | quantization dimension | DENSITY | MATHEMATICS | DEFINITIONS | FRACTALS | homogeneous Cantor set | DIMENSION | MORAN SETS | ERROR | quantization coefficient

Hausdorff dimension | Hausdorff measure | Euler characteristic | Quantization dimension | Packing dimension | Packing measure | Homogeneous Cantor set | Euler exponent | Quantization coefficient | MATHEMATICS, APPLIED | SELF-SIMILAR PROBABILITIES | quantization dimension | DENSITY | MATHEMATICS | DEFINITIONS | FRACTALS | homogeneous Cantor set | DIMENSION | MORAN SETS | ERROR | quantization coefficient

Journal Article

PROBABILITY THEORY AND RELATED FIELDS, ISSN 0178-8051, 02/2017, Volume 167, Issue 1-2, pp. 201 - 252

We consider super processes whose spatial motion is the d-dimensional Brownian motion and whose branching mechanism is critical or subcritical; such processes...

PATH | Levy snake | LEVY TREES | SUPPORT | BRANCHING-PROCESSES | HAUSDORFF MEASURE | General branching mechanism | Total range | Exact packing measure | STATISTICS & PROBABILITY | Occupation measure | Super Brownian motion | Brownian motion | Occupation | Probability theory | Texts | Constants | Gages | Gauges

PATH | Levy snake | LEVY TREES | SUPPORT | BRANCHING-PROCESSES | HAUSDORFF MEASURE | General branching mechanism | Total range | Exact packing measure | STATISTICS & PROBABILITY | Occupation measure | Super Brownian motion | Brownian motion | Occupation | Probability theory | Texts | Constants | Gages | Gauges

Journal Article

Publicacions Matemàtiques, ISSN 0214-1493, 1/2013, Volume 57, Issue 2, pp. 393 - 420

Packing measures Pg(E) and Hewitt-Stromberg measures ν⁹ (E) and their relatives are investigated. It is shown, for instance, that for any metric spaces X, Y...

Hausdorff measures | Mathematical theorems | Mathematical integrals | Separable spaces | Mathematical functions | Mathematical inequalities | Fractals | Euclidean space | Cartesianism | Lower packing dimension | Packing measure | Lower packing measure | Cartesian product | Packing dimension | MATHEMATICS | cartesian product | HAUSDORFF | SPACES | SETS | lower packing measure | lower packing dimension | packing dimension

Hausdorff measures | Mathematical theorems | Mathematical integrals | Separable spaces | Mathematical functions | Mathematical inequalities | Fractals | Euclidean space | Cartesianism | Lower packing dimension | Packing measure | Lower packing measure | Cartesian product | Packing dimension | MATHEMATICS | cartesian product | HAUSDORFF | SPACES | SETS | lower packing measure | lower packing dimension | packing dimension

Journal Article

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 05/2017, Volume 98, pp. 220 - 232

In this paper we obtain the rates of convergence of the algorithms given in [13] and [14] for an automatic computation of the centered Hausdorff and packing...

Computability of fractal measures | Centered Hausdorff measure | Packing measure | Self-similar sets | Rate of convergence | PHYSICS, MULTIDISCIPLINARY | CANTOR SETS | PHYSICS, MATHEMATICAL | COMPUTABILITY | FRACTALS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DIMENSION | SIERPINSKI GASKET | Analysis | Algorithms | Numerical analysis

Computability of fractal measures | Centered Hausdorff measure | Packing measure | Self-similar sets | Rate of convergence | PHYSICS, MULTIDISCIPLINARY | CANTOR SETS | PHYSICS, MATHEMATICAL | COMPUTABILITY | FRACTALS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | DIMENSION | SIERPINSKI GASKET | Analysis | Algorithms | Numerical analysis

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 2012, Volume 386, Issue 2, pp. 801 - 812

In this paper we consider a class of symmetric Cantor sets in . Under certain separation condition we determine the exact packing measure of such a Cantor set...

Packing measure | Cantor set | Hausdorff measure | Upper and lower density | MATHEMATICS | MATHEMATICS, APPLIED | POINTWISE DENSITIES | HAUSDORFF MEASURES

Packing measure | Cantor set | Hausdorff measure | Upper and lower density | MATHEMATICS | MATHEMATICS, APPLIED | POINTWISE DENSITIES | HAUSDORFF MEASURES

Journal Article

Ergodic Theory and Dynamical Systems, ISSN 0143-3857, 08/2016, Volume 36, Issue 5, pp. 1534 - 1556

We present an algorithm to compute the exact value of the packing measure of self-similar sets satisfying the so called Strong Separation Condition (SSC) and...

MATHEMATICS | MATHEMATICS, APPLIED | CANTOR SETS | DIMENSION | HAUSDORFF MEASURES | Computers | Algorithms | Convergence | Intervals | Separation | Gaskets | Dynamical systems | Information dissemination | Self-similarity | Mathematics - Dynamical Systems

MATHEMATICS | MATHEMATICS, APPLIED | CANTOR SETS | DIMENSION | HAUSDORFF MEASURES | Computers | Algorithms | Convergence | Intervals | Separation | Gaskets | Dynamical systems | Information dissemination | Self-similarity | Mathematics - Dynamical Systems

Journal Article

Indiana University Mathematics Journal, ISSN 0022-2518, 1/2012, Volume 61, Issue 6, pp. 2085 - 2109

In this paper, we get an upper estimate of the Favard length (sometimes called Buffon needle probability) of an arbitrary neighborhood of a big class of...

Integers | Ergodic theory | Cantor set | Algebra | Mathematical theorems | Gaskets | Polynomials | Tessellations | High frequencies | Stochastic geometry | Random sets | Area | Length | Volume | Geometric probability | Fractals | Other geometric measure theory | Hausdorff and packing measures | area | LINEAR INDEPENDENCE | UNITY | length | geometric probability | CANTOR SET | stochastic geometry | fractals | volume | MATHEMATICS | other geometric measure theory | ROOTS | random sets | PROJECTIONS | CONFIGURATIONS

Integers | Ergodic theory | Cantor set | Algebra | Mathematical theorems | Gaskets | Polynomials | Tessellations | High frequencies | Stochastic geometry | Random sets | Area | Length | Volume | Geometric probability | Fractals | Other geometric measure theory | Hausdorff and packing measures | area | LINEAR INDEPENDENCE | UNITY | length | geometric probability | CANTOR SET | stochastic geometry | fractals | volume | MATHEMATICS | other geometric measure theory | ROOTS | random sets | PROJECTIONS | CONFIGURATIONS

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 4/1999, Volume 12, Issue 2, pp. 313 - 346

Burdzy and Khoshnevisan(9) have shown that the Hausdorff dimension of the level sets of an iterated Brownian motion (IBM) is equal to 3/4. In this paper, the...

Hausdorff measure | level set | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | Iterated Brownian motion | packing measure | local time | Packing measure | Level set | Local time | PATH | WIENER-PROCESSES | LAW | MODULUS | SUBORDINATOR | STATISTICS & PROBABILITY | iterated Brownian motion | THEOREMS | LOGARITHM

Hausdorff measure | level set | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | Iterated Brownian motion | packing measure | local time | Packing measure | Level set | Local time | PATH | WIENER-PROCESSES | LAW | MODULUS | SUBORDINATOR | STATISTICS & PROBABILITY | iterated Brownian motion | THEOREMS | LOGARITHM

Journal Article

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