Geometric tools study two- and threepeakons in Camassa-Holm equation.
problem Dynamics of peakon solutions in the Camassa-Holm equation.
method Geometric tools applied to prove asymptotic behavior and compute curvature.
result New proofs and curvature computations for two- and threepeakons.
New framework explains neural network behavior through geometric postulates.
problem Understanding neural network mechanisms and making them more transparent.
method Introducing the Pursuit of Subspaces (PoS) hypothesis as an axiomatic framework.
result Unified geometric perspective on neural network representation, computation, and generalization.
Study on Einstein solitons with bounds and asymptotic behavior.
problem Understanding the properties of Einstein solitons.
method Computed lower bounds for scalar curvature, established asymptotic behavior, proved finiteness of fundamental group and weighted volume.
result Established finiteness of fundamental group and weighted volume for gradient shrinking Einstein solitons.
We study differential geometric properties of cuspidal edges with boundary. There are several differential geometric invariants which are related with the behavior of the boundary in addition to usual differential geometric invariants of cuspidal edges. We study the relation of these invariants with several other invar…
Study the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
problem Understanding the long-time behavior of Hermitian-Yang-Mills flow on non-Kähler manifolds.
method Monotonicity of eigenvalues of mean curvature, convergence to geometric invariants.
result Eigenvalues of mean curvature converge to geometric invariants in the Gauduchon case.
Bistable structures associated with non-linear deformation behavior, exemplified by the Venus flytrap and slap bracelet, can switch between different functional shapes upon actuation. Despite numerous efforts in modeling such large deformation behavior of shells, the roles of mechanical and nonlinear geometric effects …
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
The paper studies heat kernel behavior on symmetric spaces.
problem Large-time behavior of heat operator traces on symmetric spaces.
method Uses representation theory and Carmona's proof of Vogan's lambda map.
result Provides an asymptotic formula for heat kernel behavior.
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
Anisotropic curvature flow of networks shows unique solutions and behavior under finite time.
problem Existence and behavior of networks under anisotropic curvature flow.
method Existence, uniqueness, and regularity of maximal geometric solutions proven.
result Existence of maximal geometric solutions and behavior under finite time.
Study on harmonic functions in spaces with collapsing behaviors.
problem Harmonic functions on spaces with inhomogeneous collapsing behaviors at infinity.
method Analysis of complete and incomplete spaces with nonnegative Ricci curvature.
result Any nonconstant harmonic function yields a definite exponential growth rate.
Geometric optics describes wave behavior near convex obstacles.
problem Wave behavior near convex obstacles.
method Geometric optics in L2 and H1 spaces. result Oscillations transport along grazing rays to any order.
Recently, the behavior of different epidemic models and their relation both to different types of geometries and to some biological models has been revisited . Path equations representing the behavior of epidemic models and their corresponding deviation vectors are examined. A comparison between paths and their deviati…
A new oscillator measures trending behavior of financial instruments.
problem Detecting underlying deterministic components in financial market prices.
method Financial market geometry and tube oscillator derived from past history.
result Simple trading strategy based on tube oscillator leads to consistent positive returns.
The paper studies geometric flows of spacelike curves in Lorentz-Minkowski plane and proves their long-term behavior.
problem Investigating geometric flows of spacelike curves in Lorentz-Minkowski plane.
method Examining the evolution of spacelike curves along prescribed geometric flows, including curve shortening and mean curvature flows.
result The geometric flows of spacelike curves in Lorentz-Minkowski plane exist for all time and converge to specific curves as time tends to infinity.
Study bi-harmonic flow with forcing term on smooth curves.
problem Analyzing the evolution of smooth, closed planar curves under bi-harmonic flow with a forcing term.
method Reformulated geometric flow using support function, scalar PDE characterization, Monge Ampére structure analysis.
result Convexity is preserved and steady-state solutions converge over long times under specific conditions.
Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.
problem Investigate differential geometric properties of lightcone framed surfaces.
method Introduced modified frame to study the properties of lightcone framed surfaces.
result Showed behavior of Gaussian and mean curvatures at lightlike and singular points.
The paper examines short-term volatilities in equity indexes using a ranking procedure.
problem Understanding short-term behaviors of implied volatility in equity markets.
method Using a ranking procedure to model equity index dynamics, the paper investigates the short-term volatilities of derivatives written on indexes.
result The models reconcile the long memory of volatilities and power law of ATM skews in equity markets.
The paper examines torsions in Minkowskian product of Finsler metrics.
problem Investigating Cartan torsion and mean Cartan torsion in Minkowskian product of Finsler metrics.
method Deriving explicit formulas for Cartan torsion and mean Cartan torsion, analyzing their geometric behavior, and providing conditions for bounded norm.
result Both Cartan torsion and mean Cartan torsion decompose additively if and only if the Minkowskian product is Euclidean, and a necessary and sufficient condition for the norm of the mean Cartan torsion to remain bounded in the Euclidean case is provided.
New measure shows how links can be untangled as twists increase.
problem Understanding how links can be simplified through repeated twists.
method Introduced the stable unknotting number to analyze links in a twist family.
result The stable unknotting number depends only on the winding number of the link, not the wrapping number.
In this paper we study asymptotic behavior of n-superharmonic functions at isolated singularity using the Wolff potential and n-capacity estimates in nonlinear potential theory. Our results are inspired by and extend those of Arsove-Huber and Taliaferro in 2 dimensions. To study n-superharmonic functions we use a…
Study ruled surfaces with finite multiplicity, focusing on their curves and singularities.
problem Understanding ruled surfaces with finite multiplicity.
method Analyzing striction curves and singularities of ruled surfaces.
result Geometric meanings of invariants related to ruled surfaces.
Study irregular behavior of ball averages for non-amenable group actions on foliations.
problem Exploring irregular behavior of ball averages for non-amenable group actions.
method Introducing a new mechanism based on group structure to analyze irregular behavior.
result First examples of codimension one foliations with non-existent length averages.
We characterize sequences of Kleinian surface groups with convergent subsequences in terms of the asymptotic behavior of the ending invariants of the associated hyperbolic 3-manifolds. Asymptotic behavior of end invariants in a convergent sequence predicts the parabolic locus of the algebraic limit as well as how the a…
Constructs Cartan geometries from automorphism behaviors.
problem Determining Cartan geometries from automorphism local behavior.
method Introduces a construction for Cartan geometries capturing automorphism local behavior.
result The sprawl uniquely characterizes Cartan geometries with equivalent local behavior.
In this paper we study the global behavior of the Ricci flow equation for two classes of homogeneous manifolds with two isotropy summands. Using methods of the qualitative theory of differential equations, we present the global phase portrait of such systems and derive some geometrical consequences on the structure of …
In this study, we present and analyze a framework for geometric and topological estimation for mapping of unknown environments. We consider agents mimicking motion behaviors of cyborg insects, known as biobots, and exploit coordinate-free local interactions among them to infer geometric and topological information abou…
The paper studies dynamical systems with evolving geometric structure using numerical methods.
problem Qualitative behavior of ODEs with varying geometric structure.
method Fourth-order Runge-Kutta scheme for numerical analysis.
result Qualitative transitions in system dynamics as rotation parameter varies.
Although consistency is a minimum requirement of any estimator, little is known about consistency of the mean partition approach in consensus clustering. This contribution studies the asymptotic behavior of mean partitions. We show that under normal assumptions, the mean partition approach is consistent and asymptotic …
New integrators preserve geometric structure in Hamiltonian systems.
problem Preserving geometric structure in Hamiltonian systems on Jacobi manifolds.
method Combining Poissonization and symplectic bi-realizations to construct structure-preserving integrators.
result Explicit construction and application of Jacobi Hamiltonian integrators.
Study finds many nonplanar minimal spheres in elongated ellipsoids.
problem Existence of nonplanar minimal spheres in elongated ellipsoids.
method Global bifurcation techniques to establish existence and quantify number.
result Arbitrarily many nonplanar minimal spheres exist in elongated ellipsoids.
Study explains Zipf's law using geometric mechanisms from a finite alphabet.
problem Explains Zipf's law in language without relying on linguistic elements.
method Uses the Full Combinatorial Word Model (FCWM) to generate geometric distributions of word lengths.
result Supports predictions of power-law rank-frequency curves, matching various languages.
We study how resetting affects geometric Brownian motion, showing it becomes stationary but remains non-ergodic.
problem Effects of stochastic resetting on geometric Brownian motion.
method Analysis of geometric Brownian motion under stochastic resetting.
result Resetting makes geometric Brownian motion stationary but non-ergodic.
We recall fundamental aspects of the pluriclosed flow equation and survey various existence and convergence results, and the various analytic techniques used to establish them. Building on this, we formulate a precise conjectural description of the long time behavior of the flow on complex surfaces. This suggests an at…
The aim of this work is to study how the asymptotic boundary of a minimal hypersurface in H^nxR determines the behavior of the hypersurface at finite points, in several geometric situations.
We describe some relations between the long-time asymptotic behavior of the vacuum Einstein evolution equations and the geometrization of 3-manifolds. These relations are expressed in terms of evolution of CMC hypersurfaces in the vacuum space-time.Some results are also obtained on the singularity avoidance of CMC foli…
We provide two examples of spectral analysis techniques of Schroedinger operators applied to geometric Laplacians. In particular we show how to adapt the method of analytic dilation to Laplacians on complete manifolds with corners of codimension 2 finding the absence of singular continuous spectrum for these operators,…
In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some …
We construct CAT(0) groups containing subgroups whose Dehn functions are given by xs, for a dense set of numbers s∈[2,∞). This significantly expands the known geometric behavior of subgroups of CAT(0) groups.
Many studies in Economics and other disciplines have been reporting distributions following power-law behavior (i.e distributions of incomes (Pareto's law), city sizes (Zipf's law), frequencies of words in long sequences of text etc.)[1, 6, 7]. This widespread observed regularity has been explained in many ways: genera…
Study geometric properties of SGL submanifolds in a specific manifold.
problem Analyzing geometric characteristics of SGL submanifolds.
method Examines integrability conditions and parallelism properties of distributions.
result Provides insights into geometric behavior of SGL submanifolds.
Study of Type IIA flow on symplectic Lie algebras for geometric structures.
problem Detecting geometric structures like Lagrangian torus fibrations and harmonic almost complex structures.
method Investigate F-harmonic forms and the long-time behavior of the Type IIA flow. result The Type IIA flow helps in detecting desired geometric structures.
The study finds many tight contact structures on hyperbolic 3-spheres.
problem Finding tight contact structures on hyperbolic 3-spheres.
method Constructing hyperbolic homology 3-spheres and analyzing their tight contact structures.
result Produces hyperbolic homology 3-spheres with multiple distinct tight contact structures.
This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general, Ricci Flow is a nonlinear PDE whose solutions are rather difficult to calculate; ho…
This paper develops a general method for constructing Poisson integrators.
problem Lack of a general theory for Poisson integrators due to geometric challenges.
method Adapting structural results about symplectic realizations to create geometric approximations.
result Developed a general approach for constructing geometric integrators on Poisson manifolds.
A new spinorial heat flow framework studies geometric degeneration on 3-manifolds.
problem Analyzing geometric degeneration on 3-manifolds via spinor dynamics.
method Introducing a spinorial heat flow governed by the squared Dirac operator, where the metric is induced conformally by the spinor amplitude.
result Degeneration of the induced metric corresponds to nodal behavior of the spinor field.
The paper examines torsional rigidity bounds under geometric flows.
problem Torsional rigidity behavior under geometric flows.
method Bounds on torsional rigidity derived under Ricci Flow and Inverse Mean Curvature Flow.
result Inequalities of comparison with the flat disk for torsional rigidity.
Modeling price dynamics in AMMs with fees using geometric Brownian motion.
problem Understanding price dynamics in AMMs with transaction fees.
method Geometric Brownian motion, local times, excursion theory.
result Derivation of time-changed representation and limiting behavior of AMM prices.