2010, University lecture series, ISBN 9780821852293, Volume 54, xiii, 143

Book

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 09/2017, Volume 27, Issue 10, p. 1750164

Definitions of Hausdorff–Lebesgue measure and dimension are introduced. Combination of Hausdorff and Lebesgue ideas are used...

Lorenz system | fractal | attractors | Hausdorff-Lebesgue | Hausdorff | dimension | Lyapunov | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MULTIDISCIPLINARY SCIENCES | Mathematics - Dynamical Systems

Lorenz system | fractal | attractors | Hausdorff-Lebesgue | Hausdorff | dimension | Lyapunov | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MULTIDISCIPLINARY SCIENCES | Mathematics - Dynamical Systems

Journal Article

Physics letters. A, ISSN 0375-9601, 2016, Volume 380, Issue 25-26, pp. 2142 - 2149

•Survey on effective analytical approach for Lyapunov dimension estimation, proposed by Leonov, is presented...

Attractors of dynamical systems | Hausdorff dimension | Lyapunov dimension Kaplan–Yorke formula | Leonov method | Invariance with respect to diffeomorphisms | Finite-time Lyapunov exponents | Lyapunov dimension Kaplan-Yorke formula | EXPONENTS | PHYSICS, MULTIDISCIPLINARY | INVARIANT | BOX DIMENSION | TIME | LORENZ | ATTRACTORS | BOUNDS | SETS | TOPOLOGICAL-ENTROPY | Maps | Lyapunov exponents | Upper bounds | Mathematical analysis | Solid state physics | Mathematical models | Dynamical systems | Invariants | Physics - Chaotic Dynamics

Attractors of dynamical systems | Hausdorff dimension | Lyapunov dimension Kaplan–Yorke formula | Leonov method | Invariance with respect to diffeomorphisms | Finite-time Lyapunov exponents | Lyapunov dimension Kaplan-Yorke formula | EXPONENTS | PHYSICS, MULTIDISCIPLINARY | INVARIANT | BOX DIMENSION | TIME | LORENZ | ATTRACTORS | BOUNDS | SETS | TOPOLOGICAL-ENTROPY | Maps | Lyapunov exponents | Upper bounds | Mathematical analysis | Solid state physics | Mathematical models | Dynamical systems | Invariants | Physics - Chaotic Dynamics

Journal Article

Nonlinearity, ISSN 0951-7715, 04/2017, Volume 30, Issue 6, pp. 2268 - 2278

...)-expansion is faithful for the Hausdorff dimension calculations. Applying this result, we give a necessary and sufficient condition for the family of all cylinders of the Q(infinity...

28A80 | faithfulness | Hausdorff dimension Mathematics Subject Classification numbers: 11K55 | Q ∞-expansion | MATHEMATICS, APPLIED | Hausdorff dimension | PROBABILITY-DISTRIBUTIONS | PHYSICS, MATHEMATICAL | Q(infinity)-expansion | Mathematics - Number Theory

28A80 | faithfulness | Hausdorff dimension Mathematics Subject Classification numbers: 11K55 | Q ∞-expansion | MATHEMATICS, APPLIED | Hausdorff dimension | PROBABILITY-DISTRIBUTIONS | PHYSICS, MATHEMATICAL | Q(infinity)-expansion | Mathematics - Number Theory

Journal Article

International Journal of Modern Physics A, ISSN 0217-751X, 11/2017, Volume 32, Issue 31, p. 1750183

We study the Hausdorff dimension of the path of a quantum particle in noncommutative space–time...

Hausdorff dimension | generalized uncertainty relation | kappa-deformed space-time | kappa-deformed space–time | POINCARE | REDUCTION | QUANTUM | ALGEBRA | PHYSICS, NUCLEAR | PLANCK-SCALE | PHYSICS, PARTICLES & FIELDS

Hausdorff dimension | generalized uncertainty relation | kappa-deformed space-time | kappa-deformed space–time | POINCARE | REDUCTION | QUANTUM | ALGEBRA | PHYSICS, NUCLEAR | PLANCK-SCALE | PHYSICS, PARTICLES & FIELDS

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 12/2019, Volume 293, Issue 3, pp. 1015 - 1042

We study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set $$F \subseteq \mathbb {R}$$ F ⊆ R satisfies...

Sumset | Hausdorff dimension | Assouad dimension | Mathematics, general | Primary 28A80 | Mathematics | Box dimension | Distance set | Secondary 11B13 | MATHEMATICS | SELF-SIMILAR SETS | DISTANCE SETS | THEOREMS

Sumset | Hausdorff dimension | Assouad dimension | Mathematics, general | Primary 28A80 | Mathematics | Box dimension | Distance set | Secondary 11B13 | MATHEMATICS | SELF-SIMILAR SETS | DISTANCE SETS | THEOREMS

Journal Article

Communications in nonlinear science & numerical simulation, ISSN 1007-5704, 2016, Volume 41, pp. 84 - 103

.... The exact Lyapunov dimension formula for the Lorenz system for a positive measure set of parameters, including classical values, was analytically obtained first by G.A. Leonov in 2002...

Lyapunov dimension | Lyapunov exponents | Kaplan–Yorke dimension | Lorenz system | Self-excited Lorenz attractor | Kaplan-Yorke dimension | HAUSDORFF DIMENSION | MATHEMATICS, APPLIED | CHAOTIC SYSTEM | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | OSCILLATOR | MULTISTABILITY | SYNCHRONIZATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | HIDDEN ATTRACTORS | CONVECTIVE FLUID MOTION | FEEDBACK | Construction | Computer simulation | Mathematical analysis | Nonlinearity | Mathematical models | Formulas (mathematics)

Lyapunov dimension | Lyapunov exponents | Kaplan–Yorke dimension | Lorenz system | Self-excited Lorenz attractor | Kaplan-Yorke dimension | HAUSDORFF DIMENSION | MATHEMATICS, APPLIED | CHAOTIC SYSTEM | PHYSICS, FLUIDS & PLASMAS | PHYSICS, MATHEMATICAL | OSCILLATOR | MULTISTABILITY | SYNCHRONIZATION | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | HIDDEN ATTRACTORS | CONVECTIVE FLUID MOTION | FEEDBACK | Construction | Computer simulation | Mathematical analysis | Nonlinearity | Mathematical models | Formulas (mathematics)

Journal Article

07/2015, Cambridge studies in advanced mathematics, ISBN 9781107107359, Volume 150, 456

.... This book describes part of that development, concentrating on the relationship between the Fourier transform and Hausdorff dimension...

Mathematical analysis | Fourier analysis | Hausdorff measures

Mathematical analysis | Fourier analysis | Hausdorff measures

eBook

Nonlinear dynamics, ISSN 1573-269X, 2018, Volume 92, Issue 2, pp. 267 - 285

.... The concept of finite-time Lyapunov dimension is developed for numerical study of the dimension of attractors...

Perpetual points | Engineering | Vibration, Dynamical Systems, Control | Hidden attractors | Classical Mechanics | Automotive Engineering | Mechanical Engineering | Adaptive algorithm for the computation of finite-time Lyapunov dimension | Finite-time Lyapunov exponents | HAUSDORFF DIMENSION | FRACTAL BASIN BOUNDARIES | STRANGE ATTRACTORS | ENGINEERING, MECHANICAL | MULTISTABILITY | MECHANICS | DYNAMICAL-SYSTEMS | CONVECTIVE FLUID MOTION | LIMIT-CYCLES | OSCILLATIONS | CHAOTIC ATTRACTORS | TOPOLOGICAL-ENTROPY | Algorithms | Chaos theory | Liapunov exponents | Numerical methods | Adaptive algorithms | Physics - Chaotic Dynamics

Perpetual points | Engineering | Vibration, Dynamical Systems, Control | Hidden attractors | Classical Mechanics | Automotive Engineering | Mechanical Engineering | Adaptive algorithm for the computation of finite-time Lyapunov dimension | Finite-time Lyapunov exponents | HAUSDORFF DIMENSION | FRACTAL BASIN BOUNDARIES | STRANGE ATTRACTORS | ENGINEERING, MECHANICAL | MULTISTABILITY | MECHANICS | DYNAMICAL-SYSTEMS | CONVECTIVE FLUID MOTION | LIMIT-CYCLES | OSCILLATIONS | CHAOTIC ATTRACTORS | TOPOLOGICAL-ENTROPY | Algorithms | Chaos theory | Liapunov exponents | Numerical methods | Adaptive algorithms | Physics - Chaotic Dynamics

Journal Article

Journal of functional analysis, ISSN 0022-1236, 2019, Volume 276, Issue 9, pp. 2731 - 2820

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension n...

Harmonic measure in higher codimension | Dahlberg's theorem | Degenerate elliptic operators | Boundary with co-dimension higher than 1 | HAUSDORFF DIMENSION | HARMONIC MEASURE | MATHEMATICS | DISTORTION | UNIFORM RECTIFIABILITY | HYPERSURFACES | SETS | DIRICHLET PROBLEM | OPERATORS | POISSON KERNELS

Harmonic measure in higher codimension | Dahlberg's theorem | Degenerate elliptic operators | Boundary with co-dimension higher than 1 | HAUSDORFF DIMENSION | HARMONIC MEASURE | MATHEMATICS | DISTORTION | UNIFORM RECTIFIABILITY | HYPERSURFACES | SETS | DIRICHLET PROBLEM | OPERATORS | POISSON KERNELS

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 04/2019, Volume 472, Issue 2, pp. 1820 - 1845

We establish a refinement of Marstrand's projection theorem for Hausdorff dimension functions finer than the usual power functions, including an analogue...

Orthogonal projections | Potential theory | Hausdorff measure/dimension | MATHEMATICS | MATHEMATICS, APPLIED | HAUSDORFF MEASURE

Orthogonal projections | Potential theory | Hausdorff measure/dimension | MATHEMATICS | MATHEMATICS, APPLIED | HAUSDORFF MEASURE

Journal Article

Proceedings of the London Mathematical Society, ISSN 0024-6115, 08/2018, Volume 117, Issue 2, pp. 277 - 302

Given a compact set E⊂Rd−1, d⩾1, write KE:=[0,1]×E⊂Rd. A theorem of Bishop and Tyson states that any set of the form KE is minimal for conformal dimension: If (X,d...

30C65 (primary) | 28A78 (secondary) | MATHEMATICS | HAUSDORFF | SETS

30C65 (primary) | 28A78 (secondary) | MATHEMATICS | HAUSDORFF | SETS

Journal Article

Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, 03/2015, Volume 158, Issue 2, pp. 223 - 238

In this paper we address an interesting question on the computation of the dimension of self-affine sets in Euclidean space...

HAUSDORFF DIMENSION | MATHEMATICS | Geometry | Euclidean space | Algorithms | Theoretical mathematics | Estimating | Mathematical models | Singularities | Two dimensional | Computation

HAUSDORFF DIMENSION | MATHEMATICS | Geometry | Euclidean space | Algorithms | Theoretical mathematics | Estimating | Mathematical models | Singularities | Two dimensional | Computation

Journal Article

The Annals of probability, ISSN 0091-1798, 2008, Volume 36, Issue 4, pp. 1421 - 1452

.... We prove that, with probability one, the Hausdorff dimension of γ is equal to Min(2, 1 + κ/8).

Infinity | Conditional probabilities | Random walk | Geometric planes | Hausdorff dimensions | Mathematical moments | Conformity | Point estimators | Curves | SLE | Hausdorff dimension | EXPONENTS | CONFORMAL-INVARIANCE | ERASED RANDOM-WALKS | PLANE | STATISTICS & PROBABILITY | CRITICAL PERCOLATION | RESTRICTION | Probability | Mathematics | 60D05 | 28A80 | 60G17

Infinity | Conditional probabilities | Random walk | Geometric planes | Hausdorff dimensions | Mathematical moments | Conformity | Point estimators | Curves | SLE | Hausdorff dimension | EXPONENTS | CONFORMAL-INVARIANCE | ERASED RANDOM-WALKS | PLANE | STATISTICS & PROBABILITY | CRITICAL PERCOLATION | RESTRICTION | Probability | Mathematics | 60D05 | 28A80 | 60G17

Journal Article

Journal of the London Mathematical Society, ISSN 0024-6107, 08/2018, Volume 98, Issue 1, pp. 223 - 252

.... In 1988, Falconer proved that, for given matrices, the Hausdorff dimension of the self‐affine set is the affinity dimension for Lebesgue almost every translation vectors...

37C45 (primary) | 28A80 (secondary) | HAUSDORFF DIMENSION | MATHEMATICS | LEDRAPPIER-YOUNG FORMULA | Mathematics - Dynamical Systems

37C45 (primary) | 28A80 (secondary) | HAUSDORFF DIMENSION | MATHEMATICS | LEDRAPPIER-YOUNG FORMULA | Mathematics - Dynamical Systems

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 12/2014, Volume 366, Issue 12, pp. 6687 - 6733

We investigate several aspects of the Assouad dimension and the lower dimension, which together form a natural `dimension pair...

Measurability | Ahlfors regular | Lower dimension | Self-affine carpet | Assouad dimension | Baire hierarchy | measurability | HAUSDORFF DIMENSION | MATHEMATICS | AFFINE FRACTALS | SELF-SIMILAR SETS | SPACES | self-affine carpet | lower dimension | PACKING DIMENSION

Measurability | Ahlfors regular | Lower dimension | Self-affine carpet | Assouad dimension | Baire hierarchy | measurability | HAUSDORFF DIMENSION | MATHEMATICS | AFFINE FRACTALS | SELF-SIMILAR SETS | SPACES | self-affine carpet | lower dimension | PACKING DIMENSION

Journal Article

International Journal of Bifurcation and Chaos, ISSN 0218-1274, 12/2016, Volume 26, Issue 14, p. 1650240

New classes of Lyapunov-type functions are suggested for the estimation of Lyapunov dimension of Lorenz-like systems...

Attractors of dynamical systems | fractal dimension | Hausdorff dimension | Lyapunov dimension | Lorenz and Tigan systems | Lyapunov-Type functions | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Lyapunov-type functions | MULTIDISCIPLINARY SCIENCES | TIME | CHAOTIC ATTRACTORS | ENTROPY

Attractors of dynamical systems | fractal dimension | Hausdorff dimension | Lyapunov dimension | Lorenz and Tigan systems | Lyapunov-Type functions | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | Lyapunov-type functions | MULTIDISCIPLINARY SCIENCES | TIME | CHAOTIC ATTRACTORS | ENTROPY

Journal Article

Advances in Mathematics, ISSN 0001-8708, 01/2017, Volume 305, pp. 165 - 196

.... Completing many former investigations, we give a formula for the Hausdorff dimension D(q) of Uq for each q∈(1,∞). Furthermore, we prove that the dimension function D...

Quasi-greedy expansion | Hausdorff dimension | Non-integer bases | Topological entropy | Greedy expansion | Cantor sets | β-Expansion | Unique expansion | Self-similarity | UNIQUE EXPANSIONS | NONINTEGER BASES | BETA-EXPANSIONS | beta-Expansion | MATHEMATICS | Q-NI | REAL NUMBERS

Quasi-greedy expansion | Hausdorff dimension | Non-integer bases | Topological entropy | Greedy expansion | Cantor sets | β-Expansion | Unique expansion | Self-similarity | UNIQUE EXPANSIONS | NONINTEGER BASES | BETA-EXPANSIONS | beta-Expansion | MATHEMATICS | Q-NI | REAL NUMBERS

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 12/2018, Volume 511, pp. 1 - 17

The box counting dimension dB of a complex network G, and the generalized dimensions {Dq,q∈R...

Multifractals | Hausdorff dimension | Generalized dimensions | Fractal dimensions | Box counting | Complex networks | SELF-SIMILARITY | PHYSICS, MULTIDISCIPLINARY | INFORMATION DIMENSION

Multifractals | Hausdorff dimension | Generalized dimensions | Fractal dimensions | Box counting | Complex networks | SELF-SIMILARITY | PHYSICS, MULTIDISCIPLINARY | INFORMATION DIMENSION

Journal Article