Space-time Point Research Articles

Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

We introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The algorithm is obtained through a delicate combination of the Feynman–Kac and the Bismut–Elworthy–Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman–Kac formula, a new class of seminorms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearities that the computational complexity of the proposed algorithm is bounded by O(d,{varepsilon }^{-(4+delta )}) for any delta in (0,infty ) under suitable assumptions, where din {{mathbb {N}}} is the dimensionality of the problem and {varepsilon }in (0,infty ) is the prescribed accuracy. Moreover, the introduced class of numerical algorithms is also powerful for proving high-dimensional approximation capacities for deep neural networks.

From dual-unitary to quantum Bernoulli circuits: Role of the entangling power in constructing a quantum ergodic hierarchy

Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are "as random as a coin-toss". Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power $e_p(U)$ of the basic two-particle unitary building block, $U$, of the circuit, that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally we show, both analytically and numerically, how local-averaging over random realizations of the single-particle unitaries, $u_i$ and $v_i$ such that the building block is $U^\prime = (u_1 \otimes u_2 ) U (v_1 \otimes v_2 )$ leads to an identification of the average mixing rate as being determined predominantly by the entangling power $e_p(U)$. Finally we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits.

Asymptotically flat black hole solutions in quadratic gravity

Black holes constitute some of the most fascinating objects in our universe. According to Einstein’s theory of general relativity, they are also deceivingly simple: Schwarzschild black holes are completely determined by their mass. Moreover, the singularity theorems by Penrose and Hawking indicate that they host a curvature singularity within their event horizon. The presence of the latter invites the question whether these dead-end points of spacetime can be made regular by considering (quantum) corrections to the classical field equations. In this light, we use the Frobenius method to investigate the phase space of asymptotically flat, static, and spherically symmetric black hole solutions in quadratic gravity. We argue that the only asymptotically flat black hole solution visible in this approach is the Schwarzschild solution.

An entropy current and the second law in higher derivative theories of gravity

We construct a proof of the second law of thermodynamics in an arbitrary diffeomorphism invariant theory of gravity working within the approximation of linearized dynamical fluctuations around stationary black holes. We achieve this by establishing the existence of an entropy current defined on the horizon of the dynamically perturbed black hole in such theories. By construction, this entropy current has non-negative divergence, suggestive of a mechanism for the dynamical black hole to approach a final equilibrium configuration via entropy production as well as the spatial flow of it on the null horizon. This enables us to argue for the second law in its strongest possible form, which has a manifest locality at each space-time point. We explicitly check that the form of the entropy current that we construct in this paper exactly matches with previously reported expressions computed considering specific four derivative theories of higher curvature gravity. Using the same set up we also provide an alternative proof of the physical process version of the first law applicable to arbitrary higher derivative theories of gravity.

Solving the Kolmogorov PDE by Means of Deep Learning

Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly all approximation methods for Kolmogorov PDEs in the literature suffer under the curse of dimensionality or only provide approximations of the solution of the PDE at a single fixed space-time point. In this paper we derive and propose a numerical approximation method which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region \([a,b]^d\) without suffering from the curse of dimensionality. Numerical results on examples including the heat equation, the Black–Scholes model, the stochastic Lorenz equation, and the Heston model suggest that the proposed approximation algorithm is quite effective in high dimensions in terms of both accuracy and speed.

Tractor atom interferometry

We propose a tractor atom interferometer (TAI) based on three-dimensional (3D) confinement and transport of split atomic wave-function components in potential wells that follow programmed paths. The paths are programmed to split and recombine atomic wave functions at well-defined space-time points, guaranteeing closure of the interferometer. Uninterrupted 3D confinement of the interfering wave-function components in the tractor wells eliminates coherence loss due to wave-packet dispersion. Using Crank-Nicolson simulation of the time-dependent Schr\"odinger equation, we compute the quantum evolution of scalar and spinor wave functions in several TAI sample scenarios. The interferometric phases extracted from the wave functions allow us to quantify gravimeter sensitivity for the TAI scenarios studied. We show that spinor TAI supports matter-wave beam splitters that are more robust against nonadiabatic effects than their scalar-TAI counterparts. We confirm the validity of semiclassical path-integral phases taken along the programmed paths of the TAI. Aspects for future experimental realizations of TAI are discussed.

ALIENS AND UFO’S

Are aliens transdimensional entities?, By this I have meant to say those beings rather living beings that are existent on different dimensions greater than our 4 that is our normal space-time dimensions. We have enough evidence from the Lorentz generators of the string theory that the maximum space-time dimensions can take the value upto 10, if time is considered as a singular dimension. However, in certain theories like the M-theory where there exists the 11th dimensions as SUGRA or supersymmetric gravity (graviton with gravitino) which on F-theory the dimensional regularization has been taken to 12 by splitting time into 2 dimensions with a higher degrees of freedom. It is very much possible and probable that in a higher dimensions d≫4 time can take the form of 2D where the local nature of 1D time would behave as a singular loop of various multiplicities as a non-local 2D element. In our 4 dimensions also, if we are sustained to believe time being a non-local element with the Einstein’s principle of “spontaneously happening past, present and future then we ought to believe that 1D local time itself acts as a 2D non-local time. Considering the dimensional range of spatial and temporal factor ~ 3+1 it can be said that there are higher ~ 6 additional spatial dimensions which are connected with the lower ~ 3+1 co-dimensions in the form of various inter-dimensional tunnels that specifies the initial and final positions from a lower dimensions to a higher dimensions. Without considering, in this paper, the respective size of the dimensions, if all those space-time dimensions ~ 6+1 are there, then it’s probable that there exists some particular creatures over those various dimensions. However, just as 4D consists of all the 3, 2, 1D’s, similarly the higher order dimensions like 10D would also contains 9, 8, 7, 6, 5…. 1D’s, therefore it can be said that 10D is very unstable and chaotic because of the intersections of various lower co-dimensions that exists in reality. There must be an inter-dimensional membrane that protects one dimensions from the other and those inter-dimensional tunnels, that exists in between (connecting) those dimensions must have an unstable mouth wandering unpredictably from one hyper-surface to another where if anyone gets caught in any of those tunnels mouth’ devoid of any singularities and horizons, then they ought to travel to the other dimensions that will always be >4 as permitted by the laws of physics standing on the 4D universe. Now, analyzing the vehicles that they used to transport from one point in spacetime to other, A detailed analysis on the engineering and phenomenology with respect to mechanisms of the unidentified aerial objects has been carried out extensively on the paper depicting why they are more advanced and on what mechanisms are they capable of the interstellar & intergalactic travel by virtue of electrohydrodynamics and semi-quantum kinetics. KEYWORDS – Hyper-surfaces; Hyper-membranes; Inter-dimensional tunnel, Cauchy horizon, Singularity, Cross-sections, Monodromy, Time slices, Co-dimensions; String theory; Average null energy conditions; f(R,T) gravity.

Gravitational waves with colliding or noncolliding wave fronts

The known exact solutions of Einstein's vacuum field equations modeling the gravitational fields of pure gravitational radiation involve wave fronts which are either planar or roughly spherical. We describe a scheme designed to check explicitly whether or not the wave fronts collide. From the spacetime point of view the scheme determines whether or not the null hypersurface histories of the wave fronts intersect and, in particular, allows easy identification of the cases in which the null hypersurfaces do not intersect.

Singularity Universe or Parallel Universes? – MSP Theory (Multiple Singular and Parallel Theory)

This paper analyzes the issue of the singularity universe and parallel universes. Multiple universes exist, but not inside one to another, as presented in the work “Is It Plausible to Have Universes Inside Universes - The Interuniversal Numbers”. The interuniversal numbers show that every universe has a singularity structure, but parallel cases. The parallel cases are deactivated forms. This means that making time travel causes changes to all the space-time patterns. Then, it is not plausible to go to another point of space-time and this to happen to multiple universes. The parallel universes are different cases of the same universe and then we live a different case of reality. The reason for this is the singularity. Without singularity, it is not plausible to have multiple universes. But, parallel universes are inactive cases of the current universe, and the only way to break them is by following the way described in the paper “Time travel - According to Mathematical Axiomatic Physics”.

Banach manifold structure and infinite-dimensional analysis for causal fermion systems

A mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating derivatives of functions on Banach spaces which are differentiable only in certain directions. A chain rule is proven for Hölder continuous functions which are differentiable on expedient subspaces. These results are made applicable to causal fermion systems by proving that the causal Lagrangian is Hölder continuous. Moreover, Hölder continuity is analyzed for the integrated causal Lagrangian.

Gauge-invariant renormalization scheme in QCD: Application to fermion bilinears and the energy-momentum tensor

We consider a gauge-invariant, mass-independent prescription for renormalizing composite operators, regularized on the lattice, in the spirit of the coordinate space (X-space) renormalization scheme. The prescription involves only Green's functions of products of gauge-invariant operators, situated at distinct space-time points, in a way as to avoid potential contact singularities. Such Green's functions can be computed nonperturbatively in numerical simulations, with no need to fix a gauge: thus, renormalization to this "intermediate" scheme can be carried out in a completely nonperturbative manner. Expressing renormalized operators in the $\overline{\rm MS}$ scheme requires the calculation of corresponding conversion factors. The latter can only be computed in perturbation theory, by the very nature of the $\overline{\rm MS}$; however, the computations are greatly simplified by virtue of the following attributes: i) In the absense of operator mixing, they involve only massless, two-point functions; such quantities are calculable to very high perturbative order. ii) They are gauge invariant; thus, they may be computed in a convenient gauge. iii) Where operator mixing may occur, only gauge-invariant operators will appear in the mixing pattern: Unlike other schemes, involving mixing with gauge-variant operators (which may contain ghost fields), the mixing matrices in the present scheme are greatly reduced. Still, computation of some three-point functions may not be altogether avoidable. We exemplify the procedure by computing, to lowest order, the conversion factors for fermion bilinear operators of the form $\bar\psi\Gamma\psi$ in QCD. We also employ the gauge-invariant scheme in the study of mixing between gluon and quark energy-momentum tensor operators: We compute to one loop the conversion factors relating the nonperturbative mixing matrix to the $\overline{\rm MS}$ scheme.

Modeling wildfires via marked spatio-temporal Poisson processes

From a statistical viewpoint, characteristics such as ignition time, location and duration are relevant components for wildfire modeling. The observed ignition sites and starting times constitute a space-time point pattern, and a natural framework to model this type of data is via point processes. In this work, we propose a marked Poisson process to model fire patterns in space-time, considering durations as marks. The collected data correspond to fires observed in the Valencian Community, Spain, between 2010 and 2015. The methodology relies on writing the intensity function of such a process, jointly for starting times, locations and durations, as a weighted Dirichlet process mixture model. A particular choice of the kernel that determines such mixture was made, compatible with data features. We conducted posterior inference on some characteristics of interest for understanding wildfire behavior, showing high flexibility to emulate data patterns.

Relationship between degree of polarization and two-time degree of coherence of electromagnetic fields

For stationary light fields, the degree of polarization (DOP) of light relates to the maximum correlation between two arbitrarily decomposed orthogonal electric field components at a space–time point. We generalize this understanding theoretically by showing a physically insightful connection between DOP and the two-time degree of coherence of electromagnetic fields. Further, by using a variable path Mach–Zehnder interferometer, we experimentally demonstrate that the DOP of light can be directly determined by measuring the correlation between orthogonal electric field components at two-time points, showing an excellent match between the experiment and the theory. This study also leads to methods for constructing a tunable DOP source.

Chance Debugged

A ‘Big Bad Bug’ threatens Lewis’s Humean metaphysics of chance (Lewis 1986a, p. XIV); his Principal Principle provides an intuitive link between chance and credence. Yet on the one hand, certain future developments are incompatible with the true theory of chance, but on the other hand, such future developments have a positive chance to occur. The combination of these two claims with the Principal Principle leads to inconsistent credences. I present a Humean solution to the Bug: chances are relative to a limited perspective. The perspective comprises facts available as evidence to an ideal cognizer at a point in space-time. As a consequence, the same future event can have different chances of occurring provided the perspective is different. I show how this dissolves the Bug.

Artificial Intelligence Models for Crime Prediction in Urban Spaces

This work presents research based on evidence with neural networks for the development of predictive crime models, finding the data sets used are focused on historical crime data, crime classification, types of theft at different scales of space and time, counting crime and conflict points in urban areas. Among some results, 81% precision is observed in the prediction of the Neural Network algorithm and ranges in the prediction of crime occurrence at a space-time point between 75% and 90% using LSTM (Long-ShortSpace-Time). It is also observed in this review, that in the field of justice, systems based on intelligent technologies have been incorporated, to carry out activities such as legal advice, prediction and decisionmaking, national and international cooperation in the fight against crime, police and intelligence services, control systems with facial recognition, search and processing of legal information, predictive surveillance, the definition of criminal models under the criteria of criminal records, history of incidents in different regions of the city, location of the police force, established businesses, etc., that is, they make predictions in the urban context of public security and justice. Finally, the ethical considerations and principles related to predictive developments based on artificial intelligence are presented, which seek to guarantee aspects such as privacy, privacy and the impartiality of the algorithms, as well as avoid the processing of data under biases or distinctions. Therefore, it is concluded that the scenario for the development, research, and operation of predictive crime solutions with neural networks and artificial intelligence in urban contexts, is viable and necessary in Mexico, representing an innovative and effective alternative that contributes to the attention of insecurity, since according to the indices of intentional homicides, the crime rates of organized crime and violence with firearms, according to statistics from INEGI, the Global Peace Index and the Government of Mexico, remain in increase.

Gauge-invariant renormalization of the gluino-glue operator

We study the Gluino-Glue operator in the context of Supersymmetric N=1 Yang-Mills (SYM) theory. This composite operator is gauge invariant, and it is directly connected to light bound states of the theory; its renormalization is very important as a necessary step for the study of low-lying bound states via numerical simulations. We make use of a Gauge-Invariant Renormalization Scheme (GIRS). This requires the calculation of the Green's function of a product of two Gluino-Glue operators, situated at distinct space-time points. Within this scheme, the mixing with non-gauge invariant operators which have the same quantum numbers is inconsequential. We compute the one-loop conversion factor relating the GIRS scheme to MS‾. This conversion factor can be used in order to convert to MS‾ Green's functions which are obtained via lattice simulations and are renormalized nonperturbatively in GIRS.

Backgrounds from tensor models: A proposal

Although tensor models are serious candidates for a theory of quantum gravity, a connection with classical spacetimes has been elusive thus far. I aim to fill this gap by proposing a neat connection between tensor theory and Euclidean gravity at the classical level. The main departure from the usual approach is the use of Schur invariants (instead of monomial invariants) as manifold partners. Classical spacetime features can be identified naturally on the tensor side in this new setup. A notion of locality is shown to emerge through Ward identities, where proximity between spacetime points translates into vicinity between Young diagram corners.

Complete complementarity relations in curved spacetimes

We extend complete complementarity relations to curved spacetimes by considering a succession of infinitesimal local Lorentz transformations, which implies that complementarity remains valid as the quanton travels through its world line and the complementarity aspects in different points of spacetime are connected. This result allows the study of these different complementary aspects of a quantum system as it travels through spacetime. In particular, we study the behavior of these different complementary properties of massive spin-$1/2$ particles in the Schwarzschild spacetime. For geodetic circular orbits, we find that the spin state of one particle oscillates between a separable and an entangled state. For non-geodetic circular orbits, we notice that the frequency of these oscillations gets bigger as the orbit gets nearer to the Schwarzschild radius $r_s$.

KPZ equation correlations in time

We consider the narrow wedge solution to the Kardar–Parisi–Zhang stochastic PDE under the characteristic 3:2:1 scaling of time, space and fluctuations. We study the correlation of fluctuations at two different times. We show that, when the times are close to each other, the correlation approaches one at a power-law rate with exponent 2/3, while, when the two times are remote from each other, the correlation tends to zero at a power-law rate with exponent −1/3. We also prove exponential-type tail bounds for differences of the solution at two space-time points. Three main tools are pivotal to proving these results: (1) a representation for the two-time distribution in terms of two independent narrow wedge solutions, (2) the Brownian Gibbs property of the KPZ line ensemble and (3) recently proved one-point tail bounds on the narrow wedge solution.

Video Frame Interpolation via Generalized Deformable Convolution

Video frame interpolation aims at synthesizing intermediate frames from nearby source frames while maintaining spatial and temporal consistencies. The existing deep-learning-based video frame interpolation methods can be roughly divided into two categories: flow-based methods and kernel-based methods. The performance of flow-based methods is often jeopardized by the inaccuracy of flow map estimation due to oversimplified motion models, while that of kernel-based methods tends to be constrained by the rigidity of kernel shape. To address these performance-limiting issues, a novel mechanism named generalized deformable convolution is proposed, which can effectively learn motion information in a data-driven manner and freely select sampling points in space-time. We further develop a new video frame interpolation method based on this mechanism. Our extensive experiments demonstrate that the new method performs favorably against the state-of-the-art, especially when dealing with complex motions. Code is available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/zhshi0816/GDConvNet</uri> .