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.
In this pre-print we explore the multi-fractal properties of 1 minute traded volume of the equities which compose the Dow Jones 30. We also evaluate the weights of linear and non-linear dependences in the multi-fractal structure of the observable. Our results show that the multi-fractal nature of traded volume comes es…
Many 0/1 datasets have a very large number of variables; on the other hand, they are sparse and the dependency structure of the variables is simpler than the number of variables would suggest. Defining the effective dimensionality of such a dataset is a nontrivial problem. We consider the problem of defining a robust m…
Smooth fractal trees via analytic generators, preserving combinatorial and geometric properties.
problem Constructing smooth fractal trees from discrete models.
method Using analytic generator fields to integrate smooth vector fields in an internal state space, generating geometric curves as projections of generator trajectories.
result Analytic generators can represent any discrete tree specification and preserve the asymptotic limit geometry.
New framework captures non-autonomous IFS limit set topology.
problem Understanding topological properties of non-autonomous IFS limit sets.
method Homological framework applied to fractal square.
result Provides insights into fractal topology, answering Mandelbrot's percolation problem.
Fractal Lipschitz-Killing curvature measures C^f_k(F,.), k = 0, ..., d, are determined for a large class of self-similar sets F in R^d. They arise as weak limits of the appropriately rescaled classical Lipschitz-Killing curvature measures C_k(F_r,.) from geometric measure theory of parallel sets F_r for small distances…
In the work, a comparative correlation and fractal analysis of time series of Bitcoin crypto currency rate and community activities in social networks associated with Bitcoin was conducted. A significant correlation between the Bitcoin rate and the community activities was detected. Time series fractal analysis indicat…
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.
Fractal Flow enhances normalizing flows with interpretable latent space and hierarchical modeling.
problem High-dimensional density estimation and generative modeling challenges.
method Integrates topic modeling (LDA) and fractal strategy into normalizing flows.
result Achieves latent clustering, controllable generation, and superior estimation accuracy.
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…
The asymptotic behavior of open plane sections of triply periodic surfaces is dictated, for an open dense set of plane directions, by an integer second homology class of the three-torus. The dependence of this homology class on the direction can have a rather rich structure, leading in special cases to a fractal. In th…
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.
Study links fractal structure to generalization in stochastic optimization.
problem Understanding generalization in stochastic optimization algorithms.
method Represented stochastic optimization algorithms as random iterated function systems (IFS) and used dynamical systems theory.
result Proved that generalization error can be bounded based on fractal structure of invariant measure.
In this paper, based on research on rank-one isometries by W.Ballmann and M.Brin and recent research on rank-one isometries of Coxeter groups by P.Caprace and K.Fujiwara, we study a topological fractal structure of boundaries of Coxeter groups. We also show that the limit-point set is dense in a boundary of a Coxeter g…
Gradient descent with large steps leads to chaotic parameter space and unpredictable outcomes.
problem Understanding the behavior of gradient descent with large step sizes in matrix factorization.
method Analyzing the fractal structure of the parameter space and deriving critical step sizes for convergence.
result Gradient descent with large steps exhibits chaotic behavior and sensitivity to initialization, creating a fractal boundary between converging and diverging minimizers.
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.
The paper generalizes Farey tessellation to 3D hyperbolic space.
problem Generalizing Farey tessellation to higher dimensions.
method Introducing conformal bryophylla and classifying them.
result Properties of conformal bryophylla's limiting sets studied.
Study particle dynamics in non-differentiable fractal spaces.
problem Understanding motion in non-smooth, probabilistic geometries.
method Use fiber bundle theory to characterize multivalued geodesic trajectories.
result Developed a hybrid theory combining surface and stochastic process theories.
Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length [Cristea\&Steinsky 2009…
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.
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…
Cohomology fractals illustrate complex 3-manifold properties.
problem Visualizing complex cohomology classes in hyperbolic 3-manifolds.
method Ray-tracing cohomology fractals and proving their distribution.
result Cohomology fractals converge to a distribution on the sphere at infinity.
The paper proves prevalent existence and partially determines moduli space of area-minimizing surfaces with fractal singular sets.
problem Existence and moduli space of area-minimizing surfaces with fractal singular sets.
method Proof of prevalent existence, determination of moduli space, refinement of strata.
result Sharp results on moduli space and refinement of strata, showing fractal singularities do not completely dissolve under generic perturbations.
Labyrinth fractals are dendrites in the unit square. They were introduced and studied in the last decade first in the self-similar case [Cristea & Steinsky (2009,2011)], then in the mixed case [Cristea & Steinsky (2017), Cristea & Leobacher (2017)]. Supermixed fractals constitute a significant generalisation of mixed l…
We present an Expectation-Maximization algorithm for the fractal inverse problem: the problem of fitting a fractal model to data. In our setting the fractals are Iterated Function Systems (IFS), with similitudes as the family of transformations. The data is a point cloud in RH with arbitrary dimension H.…
Given an integer n≥2 and a digit set D⊊0,1,...,n−12, there is a self-similar set F⊂R2 satisfying the set equation: F=(F+D)/n. We call such F a fractal square. By studying a periodic extension H=F+Z2, we classify F into three types accordi…
We investigate the structure of the profit landscape obtained from the most basic, fluctuation based, trading strategy applied for the daily stock price data. The strategy is parameterized by only two variables, p and q. Stocks are sold and bought if the log return is bigger than p and less than -q, respectively. Repet…
Estimates heat kernel gradients on fractal-like cable systems.
problem Bounding gradients of heat kernels on complex fractal structures.
method Pointwise upper estimates for heat kernel gradients.
result Derives Lp-boundedness of quasi-Riesz transforms. 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…
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 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.
Experimental fractal landscape dynamics observed in emulsions.
problem Understanding anomalous motions in soft glassy materials.
method Quantitative analysis of oil droplet trajectories in dense emulsions.
result Experimental fractal geometry matches computational model of soft glassy dynamics.
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.
Constructs area-minimizing submanifolds with fractal singularities.
problem Area-minimizing submanifolds with fractal singular sets.
method Integral currents, mod v currents, stable stationary varifolds.
result Sharp dimensionwise solution to Almgren's conjecture.
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…
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.
Fractal learning rate schedules accelerate vanilla gradient descent.
problem Difficulty in tuning learning rates in iterative optimization.
method Introduce Chebyshev learning rate schedule for gradient descent.
result Locally unstable updates can lead to convergence in deep learning.
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.
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.
Cohomology fractals are visual representations of cohomology classes on hyperbolic 3-manifolds.
problem Visualizing cohomology classes on hyperbolic 3-manifolds.
method Cohomology fractals are images associated to cohomology classes. They are related to limit sets of Kleinian groups but differ in key aspects. An implementation using ideal triangulations and ray-casting is presented.
result Cohomology fractals allow for real-time zooming in any direction at arbitrary depth.
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 …
This study analyzes how the Indian stock market reacts to budget announcements using fractal methods.
problem Understanding the impact of Union Budget announcements on the Indian stock market.
method Utilizes fractal interpolation function and fractal dimensional analysis to study the NIFTY50 index over -15 to +15 days post-budget day.
result The budget announcements significantly affect the Indian stock market, as evidenced by average abnormal return and cumulative abnormal return.
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.
Study generalizes Lebesgue curves to new space-filling and fractal sets.
problem Generating space-filling curves from planar substitutions.
method Generalized Lebesgue's construction to new curves and fractal sets.
result Some substitutions create relatively dense fractal-like sets.
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.
We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves fractal geometry, chaotic dynamics, and an interplay between discrete and continuous…
Paper disproves Milnor's conjecture about manifold fundamental groups.
problem Milnor's conjecture about complete manifolds with nonnegative Ricci curvature.
method New topological construction and analysis of mapping class group.
result Found a counterexample with infinitely generated fundamental group.
FCOC framework improves financial volatility forecasting.
problem Tackles dual challenges of feature fidelity and model responsiveness in financial volatility forecasting.
method Synergizes fractal feature extraction and dynamic chaotic oscillation processing.
result Demonstrates profound and generalizable impact on S\&P 500 and DJI datasets.