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.
Study semiclassical measures on complex hyperbolic quotients, identifying measure supports.
problem Understanding Laplacian eigenfunctions on complex hyperbolic quotients.
method Combining fractal uncertainty principle and Ratner theory to analyze measure supports.
result Semiclassical measures support is either cosphere bundle or a compact submanifold.
Formula derived for FUP exponent in quasi-Fuchsian groups.
problem Quantifying the fractal uncertainty principle in higher dimensions.
method Explicit formula derivation for FUP exponent, dependence on porosity parameter quantified.
result Explicit essential spectral gap for quasi-Fuchsian groups in higher dimensions.
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.
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.
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 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.
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.
The study establishes uncertainty principles on harmonic manifolds of rank one.
problem Developing uncertainty principles for harmonic manifolds of rank one.
method Derivation of various uncertainty principles including Heisenberg, Morgen, Schrödinger, and Hömanders principles.
result Generalization of Hausdorff-Young inequality to harmonic manifolds of rank one.
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.
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.
We show that a well known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group.
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.
Mandelbrot unified diverse fields with scaling concept.
problem Understanding Mandelbrot's intellectual approach.
method Tracing Mandelbrot's contributions across math, physics, and economics.
result Scaling concept unified Mandelbrot's diverse work.
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.…
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. 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…
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…
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…
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.
A new method to break down insurance costs into risk and uncertainty.
problem Understanding and quantifying insurance costs in uncertain environments.
method An axiomatic approach to decompose premium principles into risk and deviation measures.
result Maximal risk and minimal deviation measures can be uniquely identified in decompositions.
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.
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 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.
Extends fractional Lp uncertainty principles with extremizers and stability results.
problem Investigating uncertainty principles in fractional Lp settings. method Analyzing the fractional Schrödinger equation to find extremal functions and sharp constants.
result Proves stability of extremizers for fractional uncertainty inequalities.
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.
The paper proves uncertainty principles on Finsler measure spaces.
problem Uncertainty principles on Finsler measure spaces.
method Analyzes Lp-uncertainty principles on Finsler measure spaces with bounded curvatures. result Sharp Lp-uncertainty principles are proven and characterized. 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 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.
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 …
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.
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…
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…
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…
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.
We investigate the local fractal properties of the financial time series based on the evolution of the Warsaw Stock Exchange Index (WIG) connected with the largest developing financial market in Europe. Calculating the local Hurst exponent for the WIG time series we find an interesting dependence between the behavior o…
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…
While developing their software, professional object-oriented (OO) software developers keep in their minds an image of the subtyping relation between types in their software. The goal of this paper is to present an observation about the graph of the subtyping relation in Java, namely the observation that, after the add…