Local Spacetime Research Articles

Extending Getis–Ord Statistics to Account for Local Space–Time Autocorrelation in Spatial Panel Data

Space and time are both crucial characteristic dimensions of geographic events and phenomena. Although exploratory spatial data analysis (ESDA) can be used to visualize and summarize complex spatial patterns, it has limitations in capturing the temporal dynamics of geographic features. Efforts have been made to incorporate the time dimension into ESDA techniques to detect space–time clustering or trends. Localized space–time statistics that could help in exploratory space–time data analysis (ESTDA), however, are still lacking. Focusing on spatial panel data, our work extended Getis–Ord and statistics using a space–time contemporaneous weight matrix and a space–time lagged weight matrix to account for local space–time autocorrelation. Two applications in this article show that the newly developed method can be used to summarize space–time patterns from spatial panel data, identify changes of landscape more consistently, and lend the results readily to visualization.

Quasi-stationary solutions in gravity theories with modified dispersion relations and Finsler–Lagrange–Hamilton Geometry

Modified gravity theories, MGTs, with modified (nonlinear) dispersion relations, MDRs, encode via indicator functionals possible modifications and effects of quantum gravity; in string/brane, noncommutative and/or nonassociative gravity theories etc. MDRs can be with global and/or local Lorentz invariance violations, LIVs, determined by quantum fluctuations, random, kinetic, statistical and/or thermodynamical processes etc. Such MGTs with MDRs and corresponding models of locally anisotropic spacetime and curved phase spaces can be geometrized in an axiomatic form for theories constructed on (co) tangent bundles with base spacetime Lorentz manifolds. In certain canonical nonholonomic variables, the geometric/physical objects are defined equivalently as in generalized Einstein-Finsler and/or Lagrange-Hamilton spaces. In such Finsler like MGTs, the coefficients of metrics and connections depend both on local Lorentz spacetime coordinates and, additionally, on (co) fiber velocity and/or momentum type variables. The main goal of this work is to elaborate on a nonholonomic diadic "shell by shell" formulation of MGTs with MDRs, with a conventional (2+2)+(2+2) splitting of total phase space dimensions, when the (dual) modified Einstein-Hamilton equations can be decoupled in general forms. We show how this geometric formalism allows us to construct various classes of exact and parametric solutions determined by generating and integration functions and effective sources depending, in principle, on all phase space coordinates. This work provides a self-consistent geometric and analytic method for constructing in our further partner papers different types of black hole solutions for theories with MDRs and LIVs and elaborating various applications in modern cosmology and astrophysics.

An Efficient ADER Discontinuous Galerkin Scheme for Directly Solving Hamilton-Jacobi Equation

This paper proposes an efficient ADER (Arbitrary DERivatives in space and time) discontinuous Galerkin (DG) scheme to directly solve the Hamilton-Jacobi equation. Unlike multi-stage Runge-Kutta methods used in the Runge-Kutta DG (RKDG) schemes, the ADER scheme is one-stage in time discretization, which is desirable in many applications. The ADER scheme used here relies on a local continuous spacetime Galerkin predictor instead of the usual Cauchy-Kovalewski procedure to achieve high order accuracy both in space and time. In such predictor step, a local Cauchy problem in each cell is solved based on a weak formulation of the original equations in spacetime. The resulting spacetime representation of the numerical solution provides the temporal accuracy that matches the spatial accuracy of the underlying DG solution. The scheme is formulated in the modal space and the volume integral and the numerical fluxes at the cell interfaces can be explicitly written. The explicit formulas of the scheme at third order is provided on two-dimensional structured meshes. The computational complexity of the ADER-DG scheme is compared to that of the RKDG scheme. Numerical experiments are also provided to demonstrate the accuracy and efficiency of our scheme.

Accelerating lattice quantum Monte Carlo simulations using artificial neural networks: Application to the Holstein model

Monte Carlo (MC) simulations are essential computational approaches with widespread use throughout all areas of science. We present a method for accelerating lattice MC simulations using fully connected and convolutional artificial neural networks that are trained to perform local and global moves in configuration space, respectively. Both networks take local spacetime MC configurations as input features and can, therefore, be trained using samples generated by conventional MC runs on smaller lattices before being utilized for simulations on larger systems. This new approach is benchmarked for the case of determinant quantum Monte Carlo (DQMC) studies of the two-dimensional Holstein model. We find that both artificial neural networks are capable of learning an unspecified effective model that accurately reproduces the MC configuration weights of the original Hamiltonian and achieve an order of magnitude speedup over the conventional DQMC algorithm. Our approach is broadly applicable to many classical and quantum lattice MC algorithms.

Local Space–Time Dirichlet–Neumann Nonreflecting Boundary Values: Half-Space Spatial Orthogonal Mode Projections

This paper follows the time-honored analyses to provide a formal bridge connecting the PML and the well-understood (classical) Dirichlet and Neumann boundary value problems. The connection offers many theoretical and practical advantages not immediately obvious from any PML reconstructions. For example, the analysis shows that the reflection-free of PML boundaries is composed of the superposition of one Dirichlet and one Neumann solution acting out of phase constituents. In fact, it uncovers that the existing PML solutions are readily obtained from the projection of the well-understood 1-D Robin impedance-matched wave equations. The projection hence not only explains how the PML split fields fundamentally come about but also that the PML implicitly uses only an orthogonal projection. Naturally, we further extend the construction to include nonorthogonal decomposition. Local-to-global coordinate transformation is introduced to give a general nonreflecting boundary value solution valid for planar orientations not aligned with the Cartesian coordinates. Nonorthogonal generalizations also make it geometrically explicit as to what causes an incorrect PML problem that can arise from mediums lacking orthogonal symmetry invariant terminations.

Partial Regularity for Type Two Doubly Nonlinear Parabolic Systems

We study weak solutions ${\bf v}:U\times (0,T)\rightarrow \mathbb{R}^m$ of the nonlinear parabolic system $$ D\psi({\bf v}_t)=\text{div}DF(D{\bf v}), $$ where $\psi$ and $F$ are convex functions. This is a prototype for more general doubly nonlinear evolutions which arise in the study of structural properties of materials. Under the assumption that the second derivatives of $F$ are H\"older continuous, we show that $D^2{\bf v}$ and ${\bf v}_t$ are locally H\"older continuous except for possibly on a lower dimensional subset of $U\times (0,T)$. Our approach relies on two integral identities, decay of the local space-time energy of solutions, and fractional time derivative estimates for $D^2{\bf v}$ and ${\bf v}_t$.

Kerr-de Sitter spacetime, Penrose process, and the generalized area theorem

We investigate various aspects of energy extraction via the Penrose process in the Kerr-de Sitter spacetime. We show that the increase in the value of a positive cosmological constant, $\mathrm{\ensuremath{\Lambda}}$, always reduces the efficiency of this process. The Kerr-de Sitter spacetime has two ergospheres associated with the black hole and the cosmological event horizons. We prove by analyzing turning points of the trajectory that the Penrose process in the cosmological ergoregion is never possible. We next show that in this process both the black hole and cosmological event horizons' areas increase, and the latter becomes possible when the particle coming from the black hole ergoregion escapes through the cosmological event horizon. We identify a new, local mass function instead of the mass parameter, to prove this generalized area theorem. This mass function takes care of the local spacetime energy due to the cosmological constant as well, including that which arises due to the frame-dragging effect due to spacetime rotation. While the current observed value of $\mathrm{\ensuremath{\Lambda}}$ is quite small, its effect in this process could be considerable in the early Universe scenario where its value is much larger, where the two horizons could have comparable sizes. In particular, the various results we obtain here are also evaluated in a triply degenerate limit of the Kerr-de Sitter spacetime we find, in which radial values of the inner, the black hole and the cosmological event horizons are nearly coincident.

A family of metric gravities

.The goal of this paper is to take a completely fresh approach to metric gravity, in which the metric principle is strictly adhered to but its properties in local space-time are derived from conservation principles, not inferred from a global field equation. The global field strength variation then gains some flexibility, but only in the regime of very strong fields (2nd-order terms) whose measurement is now being contemplated. So doing provides a family of similar gravities, differing only in strong fields, which could be developed into meaningful verification targets for strong fields after the manner in which far-field variations were used in the 20th century. General Relativity (GR) is shown to be a member of the family and this is demonstrated by deriving the Schwarzschild metric exactly from a suitable field strength assumption. The method of doing so is interesting in itself because it involves only one differential equation rather than the usual four. Exact static symmetric field solutions are also given for one pedagogical alternative based on potential, and one theoretical alternative based on inertia, and the prospects of experimentally differentiating these are analyzed. Whether the method overturns the conventional wisdom that GR is the only metric theory of gravity and that alternatives must introduce additional interactions and fields is somewhat semantical, depending on whether one views the field strength assumption as a field and whether the assumption that produces GR is considered unique in some way. It is of course possible to have other fields, and the local space-time principle can be applied to field gravities which usually are weak-field approximations having only time dilation, giving them the spatial factor and promoting them to full metric theories. Though usually pedagogical, some of them are interesting from a quantum gravity perspective. Cases are noted where mass measurement errors, or distributions of dark matter, can cause one theory to mimic another implying that such estimates or distributions should be first obtained from weak-field measurements before being used to discriminate verification candidates. By this method theorists gain insight into the local constraints on space-time, and GR verification gains strong-field comparative objectives.

Space–Time Signal Analysis and the 3D Shearlet Transform

In this work, we address the problem of analyzing video sequences by representing meaningful local space–time neighborhoods. We propose a mathematical model to describe relevant points as local singularities of a 3D signal, and we show that these local patterns can be nicely highlighted by the 3D shearlet transform, which is at the root of our work. Based on this mathematical framework, we derive an algorithm to represent space–time points which is very effective in analyzing video sequences. In particular, we show how points of the same nature have a very similar representation, allowing us to compute different space–time primitives for a video sequence in an unsupervised way.

Massive graviton geons

We find vacuum solutions such that massive gravitons are confined in a local spacetime region by their gravitational energy in asymptotically flat spacetimes in the context of the bigravity theory. We call such self-gravitating objects massive graviton geons. The basic equations can be reduced to the Schr\"odinger-Poisson equations with the tensor "wavefunction" in the Newtonian limit. We obtain a non-spherically symmetric solution with $j=2,\ell=0$ as well as a spherically symmetric solution with $j=0,\ell=2$ in this system where $j$ is the total angular momentum quantum number and $\ell$ is the orbital angular momentum quantum number, respectively. The energy eigenvalue of the Schr\"odinger equation in the non-spherical solution is smaller than that in the spherical solution. We then study the perturbative stability of the spherical solution and find that there is an unstable mode in the quadrupole mode perturbations which may be interpreted as the transition mode to the non-spherical solution. The results suggest that the non-spherically symmetric solution is the ground state of the massive graviton geon. The massive graviton geons may decay in time due to emissions of gravitational waves but this timescale can be quite long when the massive gravitons are non-relativistic and then the geons can be long-lived. We also argue possible prospects of the massive graviton geons: applications to the ultralight dark matter scenario, nonlinear (in)stability of the Minkowski spacetime, and a quantum transition of the spacetime.

Geometrothermodynamics for black holes and de Sitter space

A general method to extract thermodynamic quantities from solutions of the Einstein equation is developed. In 1994, Wald established that the entropy of a black hole could be identified as a Noether charge associated with a Killing vector of a global space-time (pseudo-Riemann) manifold. We reconstruct Wald’s method using geometrical language, e.g., via differential forms defined on the local space-time (Minkowski) manifold. Concurrently, the abstract thermodynamics are also reconstructed using geometrical terminology, which is parallel to general relativity. The correspondence between the thermodynamics and general relativity can be seen clearly by comparing the two expressions. This comparison requires a modification of Wald’s method. The new method is applied to Schwarzschild, Kerr, and Kerr–Newman black holes and de Sitter space. The results are consistent with previous results obtained using various independent methods. This strongly supports the validity of the area theorem for black holes.

Insights from polymer crystallization: Chirality, recognition and competition

Crystallization of flexible polymer chains reveals distinct characters compared to small molecules, which provides a platform to study molecular self-assembly and morphogenesis. In this review, some examples, e.g., twisting chirality of polymer lamellar crystals, recognition of different chain units and competitive nucleation of different polymorphs and different lamellar thicknesses are briefly discussed. It is shown that the polymer crystallization process far from equilibrium is in practically minimization of the system free energy in local space and finite time, leading to formation of twisted crystals, metastable polymorphism and lamellar crystals with finite thickness. Though each molecule is blind to others, the peculiar ordered configurations with stronger long-range interactions are chosen from the enormous random trials. At the end, we list some remaining questions and outlook the perspectives.

Gravitational quanta and unification

A major issue that has challenged physics is the goal of bringing gravity into a complete unification of the interactions and the quantization of gravity. We build upon the appreciation that electromagnetic waves are also gravitational waves because they transport local spacetime curvature. Logical steps lead us to appreciate the totality of the traditional quantized particles in nature as gravitational quanta. Thus, our present particle physics model is seen to be unified with the gravitational interaction. Whether there remains a scope for the existence of a spin-2 graviton in vacuum is brought into question by our invariant energy construct centered upon the Ricci tensor. We suggest that this construct holds the key to ultra-strong gravity’s essential role in quantization.

Unification mechanism for gauge and spacetime symmetries

A group theoretical mechanism for unification of local gauge and spacetime symmetries is introduced. No-go theorems prohibiting such unification are circumvented by slightly relaxing the usual requirement on the gauge group: only the so called Levi factor of the gauge group needs to be compact semisimple, not the entire gauge group. This allows a non-conventional supersymmetry-like extension of the gauge group, glueing together the gauge and spacetime symmetries, but not needing any new exotic gauge particles. It is shown that this new relaxed requirement on the gauge group is nothing but the minimal condition for energy positivity. The mechanism is demonstrated to be mathematically possible and physically plausible on a based gauge theory setting. The unified group, being an extension of the group of spacetime symmetries, is shown to be different than that of the conventional supersymmetry group, thus overcoming the McGlinn and Coleman–Mandula no-go theorems in a non-supersymmetric way.

Recognizing Human Actions by Local Space Time and LS-TSVM over CUDA

Local space-time features can be used to make the events adapted to the velocity of moving patterns, size of the object and the frequency in captured video. This paper purposed the new implementation approach of Human Action Reorganization (HAR) using Compute Unified Device Architecture (CUDA). Initially, local space-time features extracted from the customized dataset of videos. The video features are extracted by utilizing the Histogram of Optical Flow (HOF) and Harris detector algorithm descriptor. A new extended version of SVM classifier which is four time faster and has better precision than classical SVM known as the Least Square Twin SVM (LS-TSVM); a binary classifier which use two non-parallel hyperplanes, is applied on extracted video features. Paper evaluates the LS-TSVM performance on the customized data and experimental result showed the significant improvements.

High Order Direct Arbitrary-Lagrangian–Eulerian (ALE) Finite Volume Schemes for Hyperbolic Systems on Unstructured Meshes

In this work we develop a new class of high order accurate Arbitrary-Lagrangian–Eulerian (ALE) one-step finite volume schemes for the solution of nonlinear systems of conservative and non-conservative hyperbolic partial differential equations. The numerical algorithm is designed for two and three space dimensions, considering moving unstructured triangular and tetrahedral meshes, respectively. As usual for finite volume schemes, data are represented within each control volume by piecewise constant values that evolve in time, hence implying the use of some strategies to improve the order of accuracy of the algorithm. In our approach high order of accuracy in space is obtained by adopting a WENO reconstruction technique, which produces piecewise polynomials of higher degree starting from the known cell averages. Such spatial high order accurate reconstruction is then employed to achieve high order of accuracy also in time using an element-local space–time finite element predictor, which performs a one-step time discretization. Specifically, we adopt a discontinuous Galerkin predictor which can handle stiff source terms that might produce jumps in the local space–time solution. Since we are dealing with moving meshes the elements deform while the solution is evolving in time, hence making the use of a reference system very convenient. Therefore, within the space–time predictor, the physical element is mapped onto a reference element using a high order isoparametric approach, where the space–time basis and test functions are given by the Lagrange interpolation polynomials passing through a predefined set of space–time nodes. The computational mesh continuously changes its configuration in time, following as closely as possible the flow motion. The entire mesh motion procedure is composed by three main steps, namely the Lagrangian step, the rezoning step and the relaxation step. In order to obtain a continuous mesh configuration at any time level, the mesh motion is evaluated by assigning each node of the computational mesh with a unique velocity vector at each timestep. The nodal solver algorithm preforms the Lagrangian stage, while we rely on a rezoning algorithm to improve the mesh quality when the flow motion becomes very complex, hence producing highly deformed computational elements. A so-called relaxation algorithm is finally employed to partially recover the optimal Lagrangian accuracy where the computational elements are not distorted too much. We underline that our scheme is supposed to be an ALE algorithm, where the local mesh velocity can be chosen independently from the local fluid velocity. Once the vertex velocity and thus the new node location has been determined, the old element configuration at time $$t^n$$ is connected with the new one at time $$t^{n+1}$$ with straight edges to represent the local mesh motion, in order to maintain algorithmic simplicity. The final ALE finite volume scheme is based directly on a space–time conservation formulation of the governing system of hyperbolic balance laws. The nonlinear system is reformulated more compactly using a space–time divergence operator and is then integrated on a moving space–time control volume. We adopt a linear parametrization of the space–time element boundaries and Gaussian quadrature rules of suitable order of accuracy to compute the integrals. We apply the new high order direct ALE finite volume schemes to several hyperbolic systems, namely the multidimensional Euler equations of compressible gas dynamics, the ideal classical magneto-hydrodynamics equations and the non-conservative seven-equation Baer–Nunziato model of compressible multi-phase flows with stiff relaxation source terms. Numerical convergence studies as well as several classical test problems will be shown to assess the accuracy and the robustness of our schemes. Finally we briefly present some variants of the algorithm that aim at improving the overall computational efficiency.

A Novel Solver for the Generalized Riemann Problem Based on a Simplified LeFloch–Raviart Expansion and a Local Space–Time Discontinuous Galerkin Formulation

In a wide class of high order shock-capturing methods for hyperbolic conservation laws, the solution of the conservation law is represented at each time-step by a piecewise smooth function (say, a polynomial reconstructed from cell-averages or an approximation in a finite element space). To maintain a sharp resolution of shock waves, jumps at the cell boundaries are allowed. The resulting initial value problem with piecewise smooth but discontinuous initial data is called the generalized Riemann problem. We present a new solver for the generalized Riemann problem based on a simplified version of a local asymptotic series expansion constructed by LeFloch and Raviart (Ann Inst H Poincare Anal Non Lineaire 5:179---207, 1988). Contrary to the original approach, in our new solver no higher order flux derivatives and other nonlinear terms need to be computed. Moreover, we introduce a new variant of the local space---time DG method of Dumbser et al. (J Comput Phys 227:3971---4001, 2008), that allows us to use a direct solution strategy for the generalized Riemann problem without relying on a Cauchy---Kovalevskaya procedure for the flux computation.

Velocity of Light in Anisotropic Spacetime

The task of the present study is to describe local anisotropic spacetime and to discuss the possibility of its experimental detection. Anisotropic spacetime is treated as the flat isotropic Minkowski space with anisotropic perturbations. A determination of the components of the metric tensor is bound up with measurements of the velocity of light in different directions.

Enhanced bag of words using multilevel k-means for human activity recognition

This paper aims to enhance the bag of features in order to improve the accuracy of human activity recognition. In this paper, human activity recognition process consists of four stages: local space time features detection, feature description, bag of features representation, and SVMs classification. The k-means step in the bag of features is enhanced by applying three levels of clustering: clustering per video, clustering per action class, and clustering for the final code book. The experimental results show that the proposed method of enhancement reduces the time and memory requirements, and enables the use of all training data in the k-means clustering algorithm. The evaluation of accuracy of action classification on two popular datasets (KTH and Weizmann) has been performed. In addition, the proposed method improves the human activity recognition accuracy by 5.57% on the KTH dataset using the same detector, descriptor, and classifier.

Nonlinear cosmological spherical collapse of quintessence

We present a study of the fully relativistic spherical collapse in presence of quintessence using on Numerical Relativity, following the method proposed by the authors in a previous article [arXiv:1409.3476]. We ascertain the validity of the method by studying the evolution of a spherically symmetric quintessence inhomogeneity on a de Sitter background and we find that it has an impact on the local expansion around the centre of coordinates. We then proceed to compare the results of our method to those of the more largely adopted top-hat model. We find that quintessence inhomogeneities do build up under the effect that matter inhomogeneities have on the local space-time yet remain very small due to the presence of momentum transfer from the over-dense to the background regions. We expect that these might have an even more important role in modified theories of gravitation.