New memory effect discovered in gravitational wave behavior.
problem Understanding gravitational wave behavior in spacetimes with angular momentum.
method Mathematical analysis of Minkowski spacetime and Kerr black holes.
result Angular momentum memory effect observed at future null infinity.
We give useful and simple criteria for determining D_4 singularities of wave fronts. As an application, we investigate behaviors of singular curvatures of cuspidal edges near D_4^+ singularities.
A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing 'controlled Brownian behavior' of financial markets, is formally defined by adaptive nonlinear Schrödinger (NLS) equations, defining the option-pricing wave function in terms of the stock…
New method finds precise late-time behavior of wave equations.
problem Analyzing late-time behavior of wave equations with inverse-square potentials.
method Physical-space-based method for deriving late-time asymptotics.
result Sharp, uniform decay estimates in time for asymptotic late-time tails.
Study focal surfaces of wave fronts with unbounded curvatures.
problem Characterizing singularities of focal surfaces near non-degenerate singular points.
method Characterizations based on types of singularities and geometrical properties of initial fronts.
result Investigation of Gaussian curvature behavior of focal surfaces.
Kundt waves belong to the class of spacetimes which are not distinguished by their scalar curvature invariants. We address the equivalence problem for the metrics in this class via scalar differential invariants with respect to the equivalence pseudo-group of the problem. We compute and finitely represent the algebra o…
The coupled nonlinear volatility and option pricing model presented recently by Ivancevic is investigated, which generates a leverage effect, i.e., stock volatility is (negatively) correlated to stock returns, and can be regarded as a coupled nonlinear wave alternative of the Black-Scholes option pricing model. In this…
Spectral embedding uses eigenfunctions of the discrete Laplacian on a weighted graph to obtain coordinates for an embedding of an abstract data set into Euclidean space. We propose a new pre-processing step of first using the eigenfunctions to simulate a low-frequency wave moving over the data and using both position a…
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.
Let M2 be an oriented 2-manifold and f:M2→R3 a C∞-map. A point p∈M2 is called a singular point if f is not an immersion at p. The map f is called a front (or wave front), if there exists a unit C∞-vector field ν such that the image of each tangent vector df(X) (X∈TM2) is …
Motivated by how transaction amount constrain trading volume and price volatility in stock market, we, in this paper, study the relation between volume and price if amount of transaction is given. We find that accumulative trading volume gradually emerges a kurtosis near the price mean value over a trading price range …
Machine learning predicts dam-break flood wave behavior accurately.
problem Predicting long-term wave behavior in dam-break floods.
method Solved Saint-Venant equations using Lax-Wendroff scheme, trained RC-ESN with flow depth data.
result RC-ESN model predicts 286 time-steps ahead with RMSE < 0.01, outperforming LSTM.
The paper explores conformal groups on plane waves and proves a conjecture in locally homogeneous settings.
problem Proving the Lorentzian conformal Lichnerowicz conjecture in locally homogeneous settings.
method Analyzing conformal groups on plane waves and proving the conjecture in a specific setting.
result The Lorentzian conformal Lichnerowicz conjecture is proven in a locally homogeneous setting.
Model for corporate bond pricing with credit rating migration, solving a double free boundary problem.
problem Corporate bond pricing with credit rating migration risks.
method Established a pricing model as a double free boundary problem, proving existence, uniqueness, and regularity of the solution.
result Two free boundaries are shown to be smooth and converge to a traveling wave solution as time goes to infinity.
Stable blowup profile identified for wave maps in all dimensions.
problem Stability of blowup solutions for wave maps in supercritical energy.
method Novel stability analysis using similarity variables on the whole space.
result Global nonlinear stability of the corotational self-similar blowup profile.
Wave maps can have multiple bubbling solutions at blow-up points.
problem Non-uniqueness of bubbling solutions in wave maps.
method Example construction of multiple bubbling solutions.
result First known example of non-uniqueness of bubbling for dispersive equations.
We consider an evolving plane curve with two endpoints that can move freely on the x-axis with generating constant contact angles. We discuss the asymptotic behavior of global-in-time solutions when the evolution of this plane curve is governed by area-preserving curvature flow equation. The main result shows that an…
A general class of Lorentzian metrics, M0xR2, ds2=<.,.>+2dudv+H(x,u)du2, with (M0,<.,.> any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connected…
New magnetic memory effects found in gravitational waves and memory.
problem Understanding new effects in gravitational waves and memory.
method Analyzing asymptotically-flat spacetimes with slow decay.
result Diverging magnetic memory sourced by curvature and neutrino cloud.
Study traveling waves in hyperbolic space for Fisher-KPP equations.
problem Understanding wave behavior in hyperbolic space for Fisher-KPP equations.
method Analyzes the Cauchy problem in hyperbolic space for heat equation with Fisher-KPP forcing term.
result Proves new results on the dichotomy of solution propagation or vanishing based on diffusion and reaction strength.
In a previous work, the authors gave a definition of `front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory. As an application, we investigate the behavior of principal singular curvatures along A_2-singularities of hypersurfaces w…
Study uses DMD to analyze oceanic features in Strait of Gibraltar.
problem Understanding complex oceanic features in Strait of Gibraltar.
method Dynamic Mode Decomposition (DMD) applied to 3D MIT general circulation model simulations.
result Unveiled new elements and dynamics of the Strait of Gibraltar, including a secondary gyre and wave propagation.
Paper proves trapped surface formation for EMCSF system without symmetry assumptions.
problem Formation of trapped surfaces for the Einstein--Maxwell--charged scalar field system.
method Scale-critical trapped surface formation result established from past null infinity.
result Focusing of gravitational waves, concentration of electromagnetic fields, or condensation of scalar fields can lead to trapped surface formation.
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under C0-limits. method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
Study proves interaction of three impulsive gravitational waves, showing local solution and Lipschitz continuity.
problem Interaction of three impulsive gravitational waves in Einstein vacuum equations.
method Geometric estimates and wave estimates to prove local solution and continuity.
result Local solution to Einstein vacuum equations with three impulsive gravitational waves, Lipschitz continuity.
New Galilean spacetimes found as pp-wave reductions.
problem Understanding isotropic homogeneous Galilean spacetimes.
method Null reductions of pp-wave spacetimes.
result Found novel torsional Galilean spacetimes.
We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate relation between a kind of boundedness of Gaussian curvatures near cuspidal edges an…
Analyzes Gerstner's trochoidal waves and their geometric properties.
problem Understanding the geometry and kinematics of trochoidal waves.
method Derives velocity and arc length conditions for cycloidal, curtate, and prolate trochoids using Galilean transformations.
result Conditions for arc lengths of prolate and curtate trochoids to coincide over a wave cycle.
Wave fronts on certain surfaces become dense.
problem Density of wave fronts on surfaces.
method Proof of density for specific surfaces.
result Wave fronts become dense on flat torus, square billiard, Klein bottle, and cube surface.
Analyzes properties of stiffness tensors for elastic wave imaging.
problem Characterizing stiffness tensor fields for elastic wave imaging.
method Finsler-geometric methods applied to anisotropic stiffness tensor fields.
result Conditions for Finsler-geometric methods to be applicable.
Deep learning model predicts wind-wave relationship.
problem Characterize ocean wave climate for engineering applications.
method Two-stage deep learning model: CNN for spatial features, LSTM for temporal dependencies.
result Predicts spatio-temporal relationship between wind and significant wave height.
Impulsive waves contradict a 1962 conjecture about pp-waves.
problem The failure of the Ehlers--Kundt conjecture in the impulsive case.
method Summarized completeness results for impulsive wave spacetimes.
result Impulsive pp-waves are complete, contradicting the conjecture.
Extends Penrose limit to Finsler spacetimes.
problem Extending Penrose limit to Finsler spacetimes.
method Introducing lightlike coordinates and adapting Lorentzian pp-wave definition.
result New examples of Finsler pp-waves presented.
New findings on plane waves in 3D spacetimes, showing non-unimodular elliptic plane waves are unique.
problem Classifying Lorentz homogeneous spaces of dimension 3, focusing on plane waves.
method Revisiting and relaxing usual completeness assumptions, characterizing homogeneous plane waves.
result Non-unimodular elliptic plane waves are unique and non-extendable, geodesically complete only if symmetric.
Classifies solutions to vacuum weighted Einstein equations on pr-waves.
problem Classifying solutions to vacuum weighted Einstein field equations on pr-waves.
method Classifying solutions using smooth metric measure spacetimes of dimension 4.
result Provides examples of solutions with special geometric properties.
High frequency limit for most of wave phenomena is known as quasiclassical limit or ray optics limit. Propagation of waves in this limit is described in terms of wave fronts and rays. Wave front is a surface of constant phase whose points are moving along rays. As it appears, their motion can be described by Hamilton e…
Study compact plane waves, showing they are essentially standard.
problem Understanding the topology and dynamics of compact plane waves.
method Analyzing quotients of homogeneous plane waves by discrete subgroups.
result Compact quotients of homogeneous plane waves are essentially standard.
We characterize singularities of focal surfaces of wave fronts in terms of differential geometric properties of the initial wave fronts. Moreover, we study relationships between geometric properties of focal surfaces and geometric invariants of the initial wave fronts.
Paper explores non-uniqueness and uniqueness class for wave equations on graphs.
problem Non-uniqueness of solutions to wave equations on infinite graphs.
method Analyticity of solutions in the uniqueness class, extension to a wide class of linear evolution equations.
result Sharp uniqueness class for solutions of wave equations on graphs.
The financial rogue waves are reported analytically in the nonlinear option pricing model due to Ivancevic, which is nonlinear wave alternative of the Black-Scholes model. These solutions may be used to describe the possible physical mechanisms for rogue wave phenomenon in financial markets and related fields.
New metrics link Kähler and pp-wave spacetimes.
problem Connecting Kähler and pp-wave spacetimes.
method Constructing families of complete almost Kähler metrics by deforming pp-waves.
result Established a one-to-one correspondence between almost Kähler metrics and pp-wave spacetimes.
We show that every n-dimensional locally homogeneous pp-wave is a plane wave, provided it is indecomposable and its curvature operator, when acting on 2-forms, has rank greater than one. As a consequence we obtain that indecomposable, Ricci-flat locally homogeneous pp-waves are plane waves. This generalises a classic…
We define cuspidal curvature κc (resp. normalized cuspidal curvature μc) along cuspidal edges (resp. at swallowtail singularity) in Riemannian 3-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show that the product κΠ called the product curva…
Over the last few years, traffic data has been exploding and the transportation discipline has entered the era of big data. It brings out new opportunities for doing data-driven analysis, but it also challenges traditional analytic methods. This paper proposes a new Divide and Combine based approach to do K means clust…
The paper shows how null hypersurfaces behave in Lorentz-Minkowski space.
problem Understanding null hypersurfaces in Lorentz-Minkowski space.
method Analyzing L-complete null hypersurfaces as wave fronts in Euclidean space. result Most null wave fronts can be realized as restrictions of certain L-complete null wave fronts. Stability of a special spacetime solution is proven under certain symmetries.
problem Understanding the long-time behavior of cosmological solutions with symmetries.
method Proves stability of double-cusp spacetime solution under small T2-symmetry-preserving perturbations.
result Double-cusp solution is stable under small T2-symmetry-preserving perturbations.
New metrics constructed dual to specific wave-like geometries.
problem Constructing metrics dual to general plane-fronted wave Lorentzian metrics.
method Explains construction of extremal and non-Kähler almost-Kähler metrics.
result Constructs canonical almost-Kähler metrics dual to general plane-fronted wave Lorentzian metrics.
Study pp-waves with lightlike parallel spinors in vacuum spacetimes.
problem Characterize pp-waves with lightlike parallel spinors in vacuum spacetimes.
method Parametrize pp-wave spacetimes, show correspondence with Riemannian metrics, prove parallel spinor condition.
result A pp-wave spacetime with a lightlike parallel spinor corresponds to a Ricci-flat metric with a parallel spinor.