arXiv subjects

Ikjyot Singh Kohli

Publications and source records attributed to Ikjyot Singh Kohli.

At least 19 recordsLinked to original sources

arXiv2021

The Wavefunction of Continuous-Time Recurrent Neural Networks

In this paper, we explore the possibility of deriving a quantum wavefunction for continuous-time recurrent neural network (CTRNN). We did this by first starting with a two-dimensional dynamical system that describes the classical dynamics of a continuous-time recurrent neural network, and then deriving a Hamiltonian. After this, we quantized this Hamiltonian on a Hilbert space $\mathbb{H} = L^2(\mathbb{R})$ using Weyl quantization. We then solved the Schrodinger equation which gave us the wavefunction in terms of Kummer's confluent hypergeometric function corresponding to the neural network structure. Upon applying spatial boundary conditions at infinity, we were able to derive conditions/restrictions on the weights and hyperparameters of the neural network, which could potentially give insights on the the nature of finding optimal weights of said neural networks.

cs.LG↗

arXiv2020

Einstein's Field Equations as Continuous-Time Recurrent Neural Networks

We examine a subset of spatially homogenous and anisotropic solutions to Einstein's field equations: the Bianchi Type A models, and show that they can be written as a continuous-time recurrent neural network (CTRNN). This reformulation of Einstein's equations allows one to write potentially complicated nonlinear equations as a simpler dynamical system consisting of linear combinations of the neural network weights and logistic sigmoid activation functions. The CTRNN itself is trained by using an explicit Runge-Kutta solver to sample a number of solutions of Einstein's equations for the Bianchi Type A models and then using a nonlinear least-squares approach to find the optimal set of weights, time delay constants, and bias parameters that provide the best fit of the CTRNN equations to the Einstein equations. In terms of numerical examples, we specifically provide solutions to Bianchi Type I and II models. We conclude the paper with some comments on optimal parameter probability distributions and ideas for future work.

gr-qc↗

arXiv2019

On an Optimal Solution to the Film Scheduling and Showtime Staggering Problem

The scheduling of films is a major problem for the movie theatre exhibition business. The problem is two-fold: movie exhibitors ideally would like to schedule films to screens in their various locations to maximize attendance and revenue, but would also like to schedule these films such that neighbouring theatre locations play the same films at different times thus giving guests a multitude of showtime options. We refer to this latter problem as the showtime \emph{staggering} problem. We give an exact formulation of this scheduling problem using binary integer linear optimization, and provide a solved example as well. This work further shows that the optimal scheduling of films cannot be done across all theatre locations at once, but rather, must be done for each cluster of neighbouring locations.

math.OC↗

arXiv2017

Oscillating Towards the de Sitter Universe

In this paper, we consider spatially flat FLRW cosmological models in two contexts in an attempt to derive a set of conditions that characterize when such models exhibit oscillatory behaviour. In the first case, we consider a spatially flat FLRW model with only a minimally-coupled scalar field, described by a nonnegative scalar field potential. In the second case, we extended this model to include barotropic matter. We show that for the case of a positive cosmological constant, spatial flatness, and barotropic matter, oscillatory solutions are certainly possible. However, these solutions do not oscillate indefinitely, they decay over time. Interestingly, it is shown that all such solutions oscillate and decay towards the de Sitter equilibrium point, which represents an inflationary epoch in such models. This implies that for certain families of scalar field potentials, the universe may have undergone several oscillations \emph{before} entering an inflationary epoch.

gr-qc↗

arXiv2017

An Analysis of the Replicator Dynamics for an Asymmetric Hawk-Dove Game

We analyze, using a dynamical systems approach, the replicator dynamics for the asymmetric Hawk-Dove game in which there is a set of four pure strategies with arbitrary payoffs. We give a full account of the equilibrium points and their stability and derive the Nash equilibria. We also give a detailed account of the local bifurcations that the system exhibits based on choices of the typical Hawk-Dove parameters and . We also give details on the connections between the results found in this work and those of the standard two-strategy Hawk-Dove game. We conclude the paper with some examples of numerical simulations that further illustrate some global behaviours of the system.

math.DS↗

arXiv2017

Dynamics of a Vacuum Bianchi Type V Universe with an Arbitrary Cosmological Constant

In this paper, we describe the dynamics of a Bianchi Type V vacuum universe with an arbitrary cosmological constant. We begin by using an orthonormal frame approach to write Einstein's field equations as a coupled system of first-order ordinary differential equations. The equilibrium points of the resulting dynamical system are found to be expanding and contracting de Sitter universe solutions, a Minkowski spacetime solution, and static Anti-de Sitter universe solutions, which are characterized by a negative cosmological constant in addition to constant negative spatial curvature. While the expanding de Sitter universe solution is found to be asymptotically stable for $Λ> 0$, we also found that the AdS solution is stable for $Λ< 0$, as solutions form closed trajectories around it. Further, we show that the different equilibrium points are related to each other by a series of bifurcations of $Λ$.

gr-qc↗

arXiv2017

Dynamics of a Bianchi Type I Model With a Concave Potential

In this paper, we study the dynamics of a Bianchi Type I potential in the presence of a concave potential of the form $V(ϕ) = V_0 \left[1- \left( \fracϕμ\right)^n\right]$, where $V_0$ is a constant, and $μ$ is a mass scale. We show that there are two classes of equilibrium points. The first class corresponds to $ϕ\neq 0$, $μ\neq 0$, $n = 0$, and $\dotϕ = 0$, which describe expanding and contracting de Sitter universes, for which the shear anisotropy is zero. We show that the expanding de Sitter universe is a local sink of the system, and therefore has associated to it a stable manifold. Thus, orbits will approach this point at late times. In other words, such a model is found to inflate and isotropize at late times as long as $n = 0$. The second class of equilibrium points corresponds to an expanding and contracting anisotropic universe. However, these points are found to emerge only when $n > 1$, $ϕ= \dotϕ = 0$, which importantly implies that $V = V_0 < 0$ at this point in order to ensure that the square of the shear scalar, $σ^2$ is real. Therefore, such equilibrium points correspond to the ekpyrotic cosmological models. Further, we show that for $n = 2$, by Lyapunov's stability theorem, the expanding equilibrium point is asymptotically stable, while for $n > 2$, a two-dimensional stable manifold exists corresponding to the fact that for $n > 2$, such an equilibrium point represents a local sink of the system. Finally, we give a general condition for inflation to occur in this model in terms of the deceleration parameter, and show that the expanding ekpyrotic equilibrium point undergoes the phenomenon of anisotropic inflation if $-3Λ/5 < V_0 < 0$, where $Λ> 0$ is the cosmological constant.

gr-qc↗

arXiv2017

Finding Common Characteristics Among NBA Playoff and Championship Teams: A Machine Learning Approach

In this paper, we employ machine learning techniques to analyze seventeen seasons (1999-2000 to 2015-2016) of NBA regular season data from every team to determine the common characteristics among NBA playoff teams. Each team was characterized by 26 predictor variables and one binary response variable taking on a value of "TRUE" if a team had made the playoffs, and value of "FALSE" if a team had missed the playoffs. After fitting an initial classification tree to this problem, this tree was then pruned which decreased the test error rate. Further to this, a random forest of classification trees was grown which provided a very accurate model from which a variable importance plot was generated to determine which predictor variables had the greatest influence on the response variable. The result of this work was the conclusion that the most important factors in characterizing a team's playoff eligibility are a team's opponent number of assists per game, a team's opponent number of made two point shots per game, and a team's number of steals per game. This seems to suggest that defensive factors as opposed to offensive factors are the most important characteristics shared among NBA playoff teams. We then use neural networks to classify championship teams based on regular season data. From this, we show that the most important factor in a team not winning a championship is that team's opponent number of made three-point shots per game. This once again implies that defensive characteristics are of great importance in not only determining a team's playoff eligibility, but certainly, one can conclude that a lack of perimeter defense negatively impacts a team's championship chances in a given season. Further, it is shown that made two-point shots and defensive rebounding are by far the most important factor in a team's chances at winning a championship in a given season.

stat.ML↗

arXiv2017

Stochastic Eternal Inflation in a Bianchi Type I Universe

The phenomenon of stochastic eternal inflation is studied for a chaotic inflation potential in a Bianchi Type I spacetime background. After deriving the appropriate stochastic Klein-Gordon equation, we give details on the conditions for eternal inflation. It is shown that for eternal inflation to occur, the amount of anisotropy must be small. In fact, it is shown that eternal inflation will only take place if the shear anisotropy variables take on values within a small region of the interior of the Kasner circle. We then calculate the probability of eternal inflation occurring based on techniques from stochastic calculus.

gr-qc↗

arXiv2016

Mathematical Issues in Eternal Inflation

In this paper, we consider the problem of existence and uniqueness of solutions to the Einstein field equations for a spatially flat FLRW universe in the context of stochastic eternal inflation where the stochastic mechanism is modelled by adding a stochastic forcing term representing Gaussian white noise to the Klein-Gordon equation. We show that under these considerations, the Klein-Gordon equation actually becomes a stochastic differential equation. Therefore, the existence and uniqueness of solutions to Einstein's equations depend on whether the coefficients of this stochastic differential equation obey Lipschitz continuity conditions. We show that for any choice of $V(ϕ)$, the Einstein field equations are not globally well-posed, hence, any solution found to these equations is not guaranteed to be unique. Instead, the coefficients are at best locally Lipschitz continuous in the physical state space of the dynamical variables, which only exist up to a finite explosion time. We further perform Feller's explosion test for an arbitrary power-law inflaton potential and prove that all solutions to the EFE explode in a finite time with probability one. This implies that the mechanism of stochastic inflation thus considered cannot be described to be eternal, since the very concept of eternal inflation implies that the process continues indefinitely. We therefore argue that stochastic inflation based on a stochastic forcing term would not produce an infinite number of universes in some multiverse ensemble. In general, since the Einstein field equations in both situations are not well-posed, we further conclude that the existence of a multiverse via the stochastic eternal inflation mechanism considered in this paper is still very much an open question that will require much deeper investigation.

math-ph↗

arXiv2016

A Degenerate Bogdanov-Takens Normal Form for FLRW Cosmologies

In this paper, we first show that the Einstein field equations for all perfect-fluid FLRW cosmologies can be written as a planar dynamical system with the equation of state parameter $w$ and cosmological constant $Λ$ as parameters. An important equilibrium point of this dynamical system is the origin which represents Minkowski spacetime. It is shown that the Einstein field equations in a neigbourhood of this point are equivalent to a degenerate Bogdanov-Takens normal form. This normal form admits a set of equilibrium points that describes a set of solutions to the Einstein field equations that have a constant rate of expansion, negative spatial curvature, zero cosmological constant and dust.

gr-qc↗

arXiv2016

On Past Singularities in $k=0$ FLRW Cosmologies

The fundamental singularity theorem of FLRW cosmologies assumes that the matter content in the cosmological model obeys the strong energy condition along with a nonpositive cosmological constant which gives rise to an irrotational geodesic singularity. In this paper, we show that the important case of a spatially flat Friedmann-Lemaître-Robertson-Walker universe with barotropic matter obeying only the \emph{weak} energy condition with a nonnegative cosmological constant also contains a past singularity. We accomplish this using topological methods from dynamical systems theory. The methods employed in this paper are sufficiently general that they could be extended to other models to study the existence of past singularities.

gr-qc↗

arXiv2016

Einstein's Field Equations as a Fold Bifurcation

It is shown that Einstein's field equations for \emph{all} perfect-fluid $k=0$ FLRW cosmologies have the same form as the topological normal form of a fold bifurcation. In particular, we assume that the cosmological constant is a bifurcation parameter, and as such, fold bifurcation behaviour is shown to occur in a neighbourhood of Minkowski spacetime in the phase space. We show that as this cosmological constant parameter is varied, an expanding and contracting de Sitter universe \emph{emerge} via this bifurcation.

physics.gen-ph↗

arXiv2016

The Osgood Criterion and Finite-Time Cosmological Singularities

In this paper, we apply Osgood's criterion from the theory of ordinary differential equations to detect finite-time singularities in a spatially flat FLRW universe in the context of a perfect fluid, a perfect fluid with bulk viscosity, and a Chaplygin and anti-Chaplygin gas. In particular, we applied Osgood's criterion to demonstrate singularity behaviour for Type 0/big crunch singularities as well as Type II/sudden singularities. We show that in each case the choice of initial conditions is important as a certain number of initial conditions leads to finite-time, Type 0 singularities, while other precise choices of initial conditions which depend on the cosmological matter parameters and the cosmological constant can avoid such a finite-time singularity. Osgood's criterion provides a powerful and yet simple way of deducing the existence of these singularities, and also interestingly enough, provides clues of how to eliminate singularities from certain cosmological models.

gr-qc↗

arXiv2016

On Singularities in Cosmic Inflation

In this paper, we examine a flat FLRW spacetime with a scalar field potential and show by applying Osgood's criterion to the Einstein field equations that all such models, irrespective of the particular choice of potential develop finite-time singularities. That is, we show that solutions to the field equations become singular in finite time, which can have important implications for the role of inflation in cosmological models. We further point out that a possible reason for this behaviour is that the solutions to the field equations in such inflationary scenarios do not obey global existence and uniqueness properties, which is a typical characteristic of solutions that diverge in finite time.

gr-qc↗

arXiv2016

Comments On: A Universe From Nothing

We study some claims in Krauss' recent book, \emph{A Universe from Nothing: Why there is something rather than nothing}, that are employed to show that a universe can come from "nothing". In this brief paper, we show that many of the claims are not supported in full by modern general relativity theory or quantum field theory in curved spacetime.

physics.gen-ph↗

arXiv2015

The Dynamics of a Two-Fluid Bianchi Type I Universe

We use a dynamical systems approach based on the method of orthonormal frames to study the dynamics of a two-fluid, non-tilted Bianchi Type I cosmological model. In our model, one of the fluids is a fluid with bulk viscosity, while the other fluid assumes the role of a cosmological constant and represents nonnegative vacuum energy. We begin by completing a detailed fixed-points analysis of the system which gives information about the local sinks, sources and saddles. We then proceeded to analyze the global features of the dynamical system by using topological methods by finding the $α$- and $ω$-limit sets. The fixed points found are a flat FLRW universe, an Einstein-de Sitter universe, a de Sitter universe, a mixed FLRW universe with both vacuum and non-vacuum energy, and a Kasner universe. We then find conditions for which each equilibrium point was a saddle, sink, or source, and attempt to describe the global and past asymptotic behaviour of the model with respect to each fixed point. The flat FLRW universe solution we found with both vacuum and non-vacuum energy is clearly of primary importance with respect to modelling the present-day universe. In particular, we show that this equilibrium point is a local sink and a saddle of the dynamical system, so there are orbits that approach this equilibrium point in the future. Therefore, there exists a time period for which our cosmological model will isotropize and be compatible with present-day observations of a high degree of isotropy of the cosmic microwave background in addition to the existence of both vacuum and non-vacuum energy.

gr-qc↗

arXiv2015

On Optimal Offensive Strategies in Basketball

The purpose of this paper is to determine whether basketball teams who choose to employ an offensive strategy that involves predominantly shooting three point shots is stable and optimal. We employ a game-theoretical approach using techniques from dynamical systems theory to show that taking more three point shots to a point where an offensive strategy is dependent on predominantly shooting threes is not necessarily optimal, and depends on a combination of payoff constraints, where one can establish conditions via the global stability of equilibrium points in addition to Nash equilibria where a predominant two-point offensive strategy would be optimal as well. We perform a detailed fixed-points analysis to establish the local stability of a given offensive strategy. We finally prove the existence of Nash equilibria via global stability techniques via the monotonicity principle. We believe that this work demonstrates that the concept that teams should attempt more three-point shots because a three-point shot is worth more than a two-point shot is therefore, a highly ambiguous statement.

math.DS↗