Develops local theory for singular spacetimes becoming asymptotically self-similar.
problem Construction of singular spacetimes in all dimensions.
method Local theory and construction of exact self-similar solutions.
result Construction of exact self-similar solutions corresponding to formal asymptotic expansions.
Paper defines new sets and calculates their Hausdorff dimensions.
problem Calculating Hausdorff dimensions of new self-similar sets.
method Introduced asymptotic self-similar sets and new geometric constructions.
result Determined Hausdorff dimensions of new sets.
New clustering method for financial data with known cluster number.
problem Clustering financial data with known number of clusters.
method Introduced a covariance-based dissimilarity measure for multifractional Brownian motions.
result Asymptotically consistent clustering algorithms for multifractional Brownian motions.
We present new examples of complete embedded self-similar surfaces under mean curvature by gluing a sphere and a plane. These surfaces have finite genus and are the first examples of self-shrinkers in R3 that are not rotationally symmetric. The strategy for the construction is to start with a family of initi…
We consider the class of self-similar Gaussian stochastic volatility models, and compute the small-time (near-maturity) asymptotics for the corresponding asset price density, the call and put pricing functions, and the implied volatilities. Unlike the well-known model-free behavior for extreme-strike asymptotics, small…
Self-similar solutions to geometric flows are stable under small perturbations.
problem Stability of self-similar solutions in geometric flows.
method Global analytic solutions, compactness arguments, spatial equi-decay properties, and estimates of linearized operator.
result Perturbed solutions are asymptotically self-similar as time tends to infinity.
Stable shrinkers found for heat flow of harmonic maps.
problem Stability analysis of self-similar blowup in parabolic evolution equations.
method Systematic, robust, and constructive approach avoiding delicate techniques.
result Nonlinear asymptotic stability of a self-similar shrinker proved.
We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the to…
The study proves uniqueness and symmetry of self-similar solutions in warped product spaces.
problem Uniqueness and symmetry of self-similar solutions in warped product spaces.
method Analysis of curvature flows with homogeneous speed functions in warped product spaces.
result Compact star-shaped self-similar solutions in warped product spaces are slices.
The self-similar solutions to the mean curvature flows have been defined and studied on the Euclidean space. In this paper we initiate a general treatment of the self-similar solutions to the mean curvature flows on Riemannian cone manifolds. As a typical result we extend the well-known result of Huisken about the asym…
The paper studies a flow of Legendre curves, generalizing the inverse curvature flow of regular curves.
problem Analyzing the inverse curvature flow of Legendre curves.
method Investigates the unique existence, monotonicity, and asymptotic behavior of the flow.
result The flow asymptotically converges to a self-similar solution, categorized by initial curve.
The paper constructs and analyzes self-similar blowup solutions for a wave map equation.
problem Existence and stability of self-similar blowup solutions for a wave map equation.
method Construction of self-similar solutions, detailed nonlinear stability analysis, spectral analysis of linearized operators.
result Sharp semigroup bounds and nonlinear stability of all discretely self-similar profiles in all dimensions.
The study examines the long-term behavior of a flow on Lie groups.
problem Understanding the long-time behavior of the pluriclosed flow on Lie groups.
method Analysis of left-invariant Hermitian structures on Lie groups, proving convergence and existence of solutions.
result Solutions on certain Lie groups converge to self-similar solutions, some of which are shrinking solitons.
New self-similarity for Einstein vacuum equations identified.
problem Understanding spacetime behavior near singularities.
method Systematic geometric characterization and formal expansions.
result Twisted self-similar solutions cover all asymptotic behaviors.
Let C⊂Rn+1 be a regular cone with vertex at the origin. In this paper, we show the uniqueness for smooth properly embedded self-shrinking ends in Rn+1 that are asymptotic to C. As an application, we prove that not every regular cone with vertex at the origin has a smooth complete pro…
Asymptotically consistent clustering algorithms for ergodic stochastic processes are developed.
problem Clustering stochastic processes with consistency guarantees.
method Review and development of clustering algorithms for ergodic stochastic processes.
result Asymptotically consistent clustering algorithms can be obtained for ergodic stochastic processes.
Study provides LDP for non self-similar stochastic volatility models.
problem Analyzing non self-similar stochastic volatility models.
method Short-time large deviation principle (LDP) for models with Volterra process.
result Derives consequences for option prices, implied volatility surfaces, and skew.
Global stock markets exhibit exponential growth and Gaussian fluctuations with self-similar monthly patterns.
problem Understanding regularities in stock market fluctuations across different countries.
method Analysis of daily and monthly stock indices from six countries.
result Monthly stock growth is statistically self-similar to daily growth and follows a Wiener process.
Study Ricci flows on manifolds, proving they behave like self-similar solutions and confirming a conjecture.
problem Understanding the behavior of Ricci flows on higher-dimensional manifolds.
method Analyzing n-dimensional Ricci flows with non-negative Ricci curvature, starting at metric cones. result Ricci flows behave like self-similar solutions up to an exponential error in time.
This work addresses the {\em singularity formation} of complete non-compact solutions to the conformally flat Yamabe flow whose conformal factors have {\em cylindrical behavior at infinity}. Their singularity profiles happen to be {\em Yamabe solitons}, which are {\em self-similar solutions} to the fast diffusion equat…
In [LW], we construct examples of two-dimensional Hamiltonian stationary self-shrinkers and self-expanders for Lagrangian mean curvature flows, which are asymptotic to the union of two Schoen-Wolfson cones. These self-shrinkers and self-expanders can be glued together to yield solutions of the Brakke flow - a weak form…
In this paper, we consider the heat flow for Yang-Mills connections on R5×SO(5). In the SO(5)−equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove …
We prove that the only self-similar surfaces of Euclidean 3-space which are foliated by circles are the self-similar surfaces of revolution discovered by S. Angenent and that the only ruled, self-similar surfaces are the cylinders over planar self-similar curves.
The paper introduces a new method to detect rough volatility and market states using fractional derivatives.
problem Testing self-similarity in fractional processes from a single observed trajectory is difficult under long-range dependence.
method The paper introduces a regime-adaptive KS/GL--KS framework based on the discrete Grünwald--Letnikov (GL) fractional derivative.
result The method detects rough volatility and persistent, anti-persistent, or efficient market states in financial applications.
New method for constructing space-filling curves for self-similar sets.
problem Constructing space-filling curves for self-similar sets.
method Skeleton concept and neighbor graph analysis.
result Connected self-similar sets satisfying the finite type condition always possess skeletons.
In this paper we construct an end of a self-similar shrinking solution of the mean curvature flow asymptotic to an isoparametric cone C and lying outside of C. We call a cone C in Rn+1 an isoparametric cone if C is the cone over a compact embedded isoparametric hypersurface Γ⊂Sn. The theory of isoparamet…
Lipschitz equivalence of self-similar sets is an important area in the study of fractal geometry. It is known that two dust-like self-similar sets with the same contraction ratios are always Lipschitz equivalent. However, when self-similar sets have touching structures the problem of Lipschitz equivalence becomes much …
Two self-similar solutions found for time-like hypersurfaces in Minkowski spacetime.
problem Finding self-similar solutions for time-like extremal hypersurfaces in Minkowski spacetime.
method Explicit construction of two self-similar solutions.
result An untable eigenvalue found in the linearized equation around the solutions.
The paper lists all self-similar solutions for a flow in 2D space.
problem Finding solutions to the inverse mean curvature flow in 2D.
method Obtained a complete list of self-similar solutions.
result Completely enumerated all self-similar solutions for the flow.
Classifies self-similar curve shortening flows in hyperbolic 2-space.
problem Classifying self-similar curve shortening flows in hyperbolic 2-space.
method Analyzes and classifies solutions in hyperbolic 2-space.
result Completes the classification of self-similar curve shortening flows in constant curvature model spaces in 2-dimensions.
Study properties of self-similar continua with finite intersection property.
problem Characterize self-similar continua with finite intersection property.
method Prove intersection graph criterion, finite order theorem, and parameter matching theorem.
result All Jordan arcs starting from a intersection point in such continuum on a plane should have the same slope parameter at that point.
We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…
The article contains a construction of a self-similar dendryte which cannot be the attractor of any self-similar zipper.
Stability of singularity formation in Yang-Mills fields in higher dimensions.
problem Stability of self-similar blowup profiles for Yang-Mills equations in (1+d)-dimensions. method Analysis of explicitly known equivariant self-similar blowup solution and small equivariant perturbations.
result Global-in-space asymptotic stability of the self-similar blowup solution for Yang-Mills equations in (1+d)-dimensions for d≥5. The paper analyzes self-similar solutions for mean curvature flow in 3D.
problem Analyzing self-similar solutions for mean curvature flow in R3. method Analysis of self-similar solutions for surfaces of revolution, ruled surfaces, and cylindrical surfaces under homothetic helicoidal motions.
result Characterization and explicit families of exact solutions for cylindrical surfaces.
Study proves existence and uniqueness of ancient flows from cones.
problem Existence and uniqueness of ancient rescaled mean curvature flows.
method Proved existence and uniqueness using strong uniqueness theorem.
result Proved existence and uniqueness of ancient flows from cones.
The paper examines self-similar solutions in warped products.
problem Analyzing self-similar solutions in warped products.
method Investigates solutions satisfying F−F=gˉ(λ(r)∂r,ν), focusing on slices and uniqueness in specific spaces. result Slices are the only closed strictly convex self-similar solutions in the hemisphere for certain curvature functions.
This paper proves uniqueness of Kähler-Ricci flow on non-compact manifolds.
problem Uniqueness of asymptotically conical Kähler-Ricci flow on non-compact manifolds.
method Analysis of complete gradient expanding Kähler-Ricci solitons and their tangent cones.
result A complete solution to the Kähler-Ricci flow emerging from the soliton's tangent cone at infinity coincides with the forward self-similar Kähler-Ricci flow associated with the soliton.
Stability of specific solitons proven in higher dimensions.
problem Stability of homothetically shrinking Yang-Mills solitons in higher dimensions.
method Heat flow for Yang-Mills connections, small equivariant perturbations, general framework for spectral problems.
result Nonlinear asymptotic stability of the Weinkove solution in higher dimensions.
Paper proves rigidity for self-similar solutions in 3D flows.
problem Proving rigidity for self-similar solutions in curvature flows.
method Proves rigidity results for self-similar solutions of fully non-linear parabolic flows in R^3.
result Self-similar solutions are round spheres with genus zero.
Wave maps with noise can lead to self-similar blowup from arbitrary initial data.
problem Analyzing self-similar blowup in wave maps with additive noise.
method Stochastic perturbation of wave maps in supercritical dimensions.
result Self-similar blowup with positive probability for arbitrary corotational initial data.
Study finds solutions for degenerate affine curve shortening flow.
problem Analyzing degenerate affine curve shortening flow.
method Solved equations for affine self-similar solutions.
result New special solutions discovered for affine curve shortening flow.
Study on self-similar surfaces and their mapping class groups generated by involutions.
problem When do big mapping class groups of self-similar surfaces generated by involutions?
method Investigation of self-similar surfaces with self-similar ends, focusing on infinite and one maximal ends.
result For self-similar surfaces with infinite maximal ends, their mapping class groups are generated by involutions and are uniformly perfect.
Study finds solutions to flows by negative curvature powers.
problem Curvature flows with negative powers.
method Closed self-similar solutions in warped product manifolds, proving non-strict convexity.
result Proves self-similar solutions are slices of warped product manifolds.
The study examines two types of fractals and their topological properties.
problem Analyzing the topological properties of self-similar fractals with a shifting parameter.
method Detailed discussion and proof for disk-likeness and connectivity.
result Conditions for Tε to be quasi-periodic and connected. We give a classification of all self-similar solutions to the curve shortening flow in the plane.
Introduces CSST and characterizes its topology.
problem Characterize the topology of the continuum random tree.
method Introduce continuum self-similar tree (CSST) and apply it.
result Characterizes the topology of CSST and other trees.
In [9] Kaimanovich introduced the concept of augmented tree on the symbolic space of a self-similar set. It is hyperbolic in the sense of Gromov, and it was shown in [13] that under the open set condition, a self-similar set can be identified with the hyperbolic boundary of the tree. In the paper, we investigate in det…