We introduce a new measure of dimension for binary datasets.
problem Defining the effective dimensionality of sparse binary datasets.
method Adapted fractal dimension concept for binary data and introduced normalized fractal dimension.
result Normalized fractal dimension measures the degree of dependency structure of binary datasets.
A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system(RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal su…
Study fractal dimension for motion without crossing a subset.
problem Fractal dimension of a subset X in R^n for motion without crossing.
method Analyzes fractal dimension of subset X in R^n.
result Determines conditions for motion without crossing a subset.
This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with segments of equal size, the composite dimension can be expressed as a function …
Paper develops a fractal dimension-based generalization measure.
problem Developing a robust generalization measure for machine learning models.
method Analyzes decision boundaries using fractal dimension concept.
result Developed a generalization measure based on fractal dimension.
New framework links fractal complexity to separation dimension.
problem Quantifying the complexity of fractal partitions.
method Introducing Separation Dimension ($\sepdim$) and Geometrically Regular Partitions (GRPs).
result Sharp upper bound for chromatic number of fractal partitions.
Estimates box dimension of fractal interpolation surfaces using oscillation vectors.
problem Estimating the complexity of fractal interpolation surfaces.
method Defined vertical scaling matrices and used them to relate oscillation vectors of different levels.
result Obtained the box dimension of generalized affine fractal interpolation surfaces.
This paper evaluates fractal dimension and persistent homology for neural network generalization.
problem Bounding and predicting the generalization gap of neural networks.
method Empirical evaluation of fractal dimension and persistent homology as generalization measures.
result Fractal dimension and persistent homology fail to predict generalization of models trained from poor initializations.
Algorithm reconstructs fractals from point clouds with high precision.
problem Fitting fractal models to data.
method Expectation-Maximization algorithm for IFS models.
result Algorithm reconstructs fractals with high precision.
Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.
problem Rigidity of geodesic planes in Hitchin manifolds.
method Constructing a specific surface group and analyzing its action on the Hitchin manifold.
result Existence of floating geodesic planes in Hitchin manifolds with fractal closures.
Study permeable sets and their dimensions, with applications to fractals.
problem Understanding permeability and dimensions of sets.
method Investigate permeable sets and their properties, establish theorems on permeability and dimension relations.
result Most subsets of \(\mathbb{R}^d\) with dimension less than \(d-1\) are permeable.
The paper generalizes fractals to higher dimensions using affine transformations.
problem Generalizing fractals to higher dimensions.
method Using affine transformations and characterizations of affinely-equivalent Sierpinski carpet.
result Menger sponge and Sierpinski simplex in 4-dimensional space can be drawn out clearly.
Unified bounds linking compressibility, fractal dimensions, and mutual information.
problem Understanding generalization in stochastic learning algorithms.
method Rate-distortion theory applied to machine learning generalization.
result Unified bounds linking compressibility, fractal dimensions, and mutual information.
In this paper two zero-dimensional compact sets with equal topological and fractal dimensions but embedded in Euclidean space by different ways are under study. Diffraction of plane electromagnetic wave propagated and reflected by fractal surfaces is considered for each of these compact sets placed in vacuum. It is obt…
We give a brief overview of the theory of complex dimensions of real (archimedean) fractal strings via an illustrative example, the ordinary Cantor string, and a detailed survey of the theory of p-adic (nonarchimedean) fractal strings and their complex dimensions. Moreover, we present an explicit volume formula for the…
This paper introduces Hausdorff measure and its applications in fractal geometry.
problem Defining and applying Hausdorff measure to fractal geometry.
method Definition of Hausdorff outer measure, Caratheodory's criterion, construction of Hausdorff measure, and introduction of Hausdorff dimension.
result Demonstrates the Hausdorff dimension of the Cantor ternary set.
Study on topological order on fractal geometries, proving no-go theorem and fault-tolerant gates.
problem Investigating topological order on fractal geometries embedded in n dimensions.
method Using quantum error-correcting codes and systolic geometry to diagnose topological order.
result Proves no-go theorem for topological order on 2D fractals, survival on higher dimensions, and construction of fault-tolerant gates.
This work proves generalization bounds for neural networks without Lipschitz assumptions.
problem Proving generalization guarantees for neural networks without Lipschitz continuity.
method Introduces a data-dependent fractal dimension and uses it to prove generalization bounds.
result Generalization bounds are proven without requiring Lipschitz continuity.
Infinite fractal tree solves shortest connection problem.
problem Finding the shortest connection for a fractal set.
method Constructing an infinite planar self-similar binary tree.
result The tree is the unique solution to the Steiner problem.
Study fractal and regular geometry in deep neural networks.
problem Investigate geometric properties of neural networks.
method Analyze boundary volumes of excursion sets for different activations.
result Hausdorff dimension increases with depth for non-regular activations.
Algorithm identifies fractal system's scaling exponents in high dimensions.
problem Statistical identification of Hurst distribution in high-dimensional fractal systems.
method Wavelet random matrices, modified spectral clustering, model selection.
result Algorithm consistently estimates Hurst distribution in moderately high dimensions.
New manifolds show Riesz transform unbounded for p > 2.
problem Understanding Riesz transform behavior on manifolds.
method Constructing Riemannian manifolds with specific properties.
result Riesz transform unbounded on Lp(M) for all p>2. New fractal spaces not quasisymmetric to Loewner spaces discovered.
problem Finding new fractal spaces not quasisymmetric to Loewner spaces.
method Introduced iterated graph systems (IGS) to create new fractal spaces.
result Disproved Kleiner's conjecture about self-similar fractals.
Reservoir computer dimensions estimated using three methods.
problem Estimating the dimension of reservoir computer signals.
method Used three dimension estimation methods: false nearest neighbor, covariance, and Kaplan-Yorke.
result Signals in reservoir system exist on a low dimensional surface.
Analytic patch trees reveal new geometric structures and dimension fields.
problem Understanding the geometric and analytical properties of surface patch trees.
method Developed analytic surface patch trees and introduced interface curves to transmit state.
result Surface patch trees have natural foliations with one-dimensional curve trees and dimension fields.
New framework constructs holographic tensor networks using hyperbolic buildings.
problem Building holographic tensor networks for non-integer dimensions and fractal spaces.
method Introducing a unifying framework based on hyperbolic buildings and dualities.
result Constructs a family of bulk regions satisfying complementary recovery and Ryu-Takayanagi formula.
This paper presents an analysis of the study variables such as gdp, employment levels, the level of R & D and technology that will serve as the basis for stochastic modeling of production possibilities frontier in the goodness of fractal dimensions Ex Ante and Ex Post a priori to determine the levels of causality immed…
We utilize long-term memory, fractal dimension and approximate entropy as input variables for the Efficiency Index [Kristoufek & Vosvrda (2013), Physica A 392]. This way, we are able to comment on stock market efficiency after controlling for different types of inefficiencies. Applying the methodology on 38 stock marke…
We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on the dimensions and intersections of boundary sets. As an application, we descri…
The present paper shows that for a given integer k greater than 2 it is possible to construct an at least k-differentiable Riemannian metric on the sphere of a certain dimension such that the cut locus of a point of it becomes a fractal. Moreover, we show that this construction can be extended to the case of Finsler sp…
New integration theory on topological spaces, including fractals.
problem Developing a universal integration theory for arbitrary topological spaces.
method Introducing a new integration framework using unital magma valued functions and measures.
result Integration, differentiation, and orientation defined for arbitrary topological spaces.
This talk reviews some mathematical and physical ideas related to the notion of dimension. After a brief historical introduction, various modern constructions from fractal geometry, noncommutative geometry, and theoretical physics are invoked and compared.
Paper introduces quadrilateral labyrinth fractals and their properties.
problem None explicitly stated; focus on fractal construction and properties.
method Construction of quadrilateral labyrinth fractals and study of their topological properties.
result Properties of quadrilateral labyrinth fractals are studied.
New bounds on nearly maximally predictive features help improve data prediction.
problem Inferring maximally predictive features from data sets is often uncountably infinite.
method Derived upper-bounds on the number and coding cost of nearly maximally predictive features.
result Mixed-state predictive features offer a substantial improvement over finite-order Markov models.
We introduce a new measure for the capital market efficiency. The measure takes into consideration the correlation structure of the returns (long-term and short-term memory) and local herding behavior (fractal dimension). The efficiency measure is taken as a distance from an ideal efficient market situation. Methodolog…
The paper studies dimensions of attractors for modified Leray-alpha equation on various surfaces.
problem Investigate attractor dimensions of the modified Leray-alpha equation.
method Existence and uniqueness of weak solutions, global attractor existence, estimates for vorticity scalar equations, Kolmogorov flows.
result Established upper and lower bounds for Hausdorff and fractal dimensions of global attractors on S2 and T2. Supermixed labyrinth fractals extend mixed fractals by using multiple patterns.
problem Extending labyrinth fractals to use multiple patterns.
method Iterative construction using multiple labyrinth patterns.
result Sufficient condition for infinite length of arcs in supermixed labyrinth fractals.
Study on heat content for domains with fractal boundaries.
problem Analyzing short-time asymptotics of heat content for domains with fractal boundaries.
method Developing mathematical analysis on de Gennes' hypothesis and exploring fractal curvatures.
result Fractal curvatures and their scaling exponents may emerge in the short-time heat content asymptotics of domains with fractal boundaries.
New method estimates deep neural network's intrinsic dimension for better generalization.
problem Estimating intrinsic dimension of deep neural networks for generalization.
method Topological data analysis (TDA) and persistent homology (PHD).
result Efficient algorithm to estimate PHD in deep neural networks.
Globalization is one of the central concepts of our age. The common perception of the process is that, due to declining communication and transport costs, distance becomes less and less important. However, the distance coefficient in the gravity model of trade, which grows in time, indicates that the role of distance i…
Lazy, perfectly informed investors trade infrequently due to costs.
problem The paradox of an omniscient yet lazy investor trading infrequently.
method Formalized the paradox using geometric and fractional Brownian motion models, derived closed-form profit functions, and proved existence and uniqueness of the optimal trading frequency.
result The optimal trading frequency can be interpreted through the fractal dimension of the price path.
New theory explains how chaotic training improves neural network generalization.
problem Understanding how chaotic training improves neural network generalization.
method Representing stochastic optimizers as random dynamical systems and introducing a new dimension concept.
result Generalization in chaotic training depends on the complete Hessian spectrum and partial determinants.
FGN models networks with fractal structures using Gaussian Multiplicative Chaos.
problem Modeling networks with fractal structures.
method FGN model based on Gaussian Multiplicative Chaos.
result FGNs reveal distinct scaling patterns in edge and clique counts.
Knots can be embedded into fractals like the Menger Sponge and Sierpinski Tetrahedron.
problem Embedding knots into fractals to compare complexity.
method Proved all knots can be embedded into Menger Sponge, and Pretzel knots into Sierpinski Tetrahedron. Compared iterations needed for given knots.
result Comparison of fractal complexity through knot embedding.
Mixed labyrinth fractals can have finite or infinite arc lengths.
problem Characterizing the length of arcs in mixed labyrinth fractals.
method Analyzing sequences of labyrinth patterns to determine arc lengths.
result Arc lengths can be finite or infinite depending on pattern choice.
Study on hyperbolic manifolds finds measures of Laplace eigenfunctions restricted to cosphere bundles.
problem Analyzing semiclassical measures on hyperbolic manifolds.
method Adapting Dyatlov and Jin's argument to higher dimensions and using Ratner theory.
result Semiclassical measures' support contains the cosphere bundle of a compact totally geodesic submanifold.
eDCF estimates intrinsic dimension using local connectivity.
problem Challenges in estimating intrinsic dimension due to scale dependence.
method eDCF: a novel, scalable, and parallelizable method based on Connectivity Factor (CF).
result eDCF consistently matches leading estimators with comparable MAE and higher exact intrinsic dimension match rates.
Paper proves existence of area-minimizing submanifolds on almost any manifold with fractal singular sets.
problem Existence of area-minimizing submanifolds with fractal singular sets.
method Constructing and proving existence on almost any smooth manifold.
result Existence of area-minimizing submanifolds with fractal singular sets on almost any smooth manifold.