Complex, Lorentzian, and Euclidean simplicial quantum gravity: numerical methods and physical prospects

Evaluating gravitational path integrals in the Lorentzian has been a long-standing challenge due to the numerical sign problem. We show that this challenge can be overcome in simplicial quantum gravity. By deforming the integration contour into the complex, the sign fluctuations can be suppressed, for instance using the holomorphic gradient flow algorithm. Working through simple models, we show that this algorithm enables efficient Monte Carlo simulations for Lorentzian simplicial quantum gravity. In order to allow complex deformations of the integration contour, we provide a manifestly holomorphic formula for Lorentzian simplicial gravity. This leads to a complex version of simplicial gravity that generalizes the Euclidean and Lorentzian cases. Outside the context of numerical computation, complex simplicial gravity is also relevant to studies of singularity resolving processes with complex semi-classical solutions. Along the way, we prove a complex version of the Gauss-Bonnet theorem, which may be of independent interest.

A speed preserving Hilbert gradient flow for generalized integral Menger curvature

We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.

Spiral waves within a bistability parameter region of an excitable medium

Spiral waves are a well-known and intensively studied dynamic phenomenon in excitable media of various types. Most studies have considered an excitable medium with a single stable resting state. However, spiral waves can be maintained in an excitable medium with bistability. Our calculations, performed using the widely used Barkley model, clearly show that spiral waves in the bistability region exhibit unique properties. For example, a spiral wave can either rotate around a core that is in an unexcited state, or the tip of the spiral wave describes a circular trajectory located inside an excited region. The boundaries of the parameter regions with positive and "negative" cores have been defined numerically and analytically evaluated. It is also shown that the creation of a positive or "negative" core may depend on the initial conditions, which leads to hysteresis of spiral waves. In addition, the influence of gradient flow on the dynamics of the spiral wave, which is related to the tension of the scroll wave filaments in a three-dimensional medium, is studied.

Cryptocurrency Price Estimation Using Hyperparameterized Oscillatory Activation Functions in LSTM Networks

Activation functions are critical components of neural networks, helping the model learn highly-intricate dependencies, trends, and patterns. Non-linear activation functions allow the model to behave as a functional approximator, learning complex decision boundaries and multi-dimensional patterns in the data. Activation functions can be combined with one another to learn better representations with the objective of improving gradient flow, performance metrics reducing training time and computational cost. Recent work on oscillatory activation functions\cite{noel2021growing}\cite{noel2021biologically} showcased their ability to perform competitively on image classification tasks using a compact architecture. Our work proposes the utilization of these oscillatory activation functions for predicting the volume-weighted average of Bitcoin on the G-Research Cryptocurrency Dataset. We utilize a popular LSTM architecture for this task achieving competitive results when compared to popular activation functions formally used.

Global existence and stability for the modified Mullins–Sekerka and surface diffusion flow

<abstract><p>In this survey we present the state of the art about the asymptotic behavior and stability of the <italic>modified Mullins</italic>–<italic>Sekerka flow</italic> and the <italic>surface diffusion flow</italic> of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the <italic>strict stability</italic> property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under $ W^{2, p} $–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently "close" to a smooth <italic>strictly stable critical</italic> set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.</p></abstract>

The static force from generalized Wilson loops using gradient flow

We explore a novel approach to compute the force between a static quark-antiquark pair with the gradient flow algorithm on the lattice. The approach is based on inserting a chromoelectric field in a Wilson loop. The renormalization issues, associated with the finite size of the chromoelectric field on the lattice, can be solved with the use of gradient flow. We compare numerical results for the flowed static potential to our previous measurement of the same observable without a gradient flow.

VALIDATION OF STABILITY INDICATING RP-HPLC METHOD FOR THE SIMULTANEOUS ESTIMATION OF TELMISARTAN AND HYDROCHLOROTHIAZIDE CONTENT IN BULK AND PHARMACEUTICAL DOSAGE FORM

A simple, accurate, precise and rapid stability indicating reverse phase High performance chromatography method was used for estimation of Telmisartan and Hydrochlorothiazide in bulk and fixed-dose combination solid oral dosage form. The proposed analytical method has been validated for specificity, Linearity, Accuracy, Precision and Robustness. The chromatography was achieved in a GL science, Inertsil C8 (Length 125x Diameter 4.0mm Particle size 5µm) column with gradient flow. The optimal chromatographic condition consisted of mobile phase pH 3.0 at a flow rate of 1.2mL/min, with a column temperature of 40°C, run time 14 minutes and detector wavelength of 270nm.

Urban Spatial Structures From Human Flow By Hodge-Kodaira Decomposition

Human flow in cities indicates social activity and can reveal urban spatial structures based on human behaviours for relevant applications. Scalar potential is a mathematical concept, and if successfully introduced, it can provide an intuitive perspective of human flow. However, the definition of such a potential to the origin-destination flow matrix and determination of its plausibility remain unsolved. Here, we apply Hodge-Kodaira decomposition, in which a matrix is uniquely decomposed into a potential-driven (gradient) flow and a curl flow. We depict the potential landscapes in cities due to commuting flow and reveal how the landscapes have been changed or unchanged by years or transport methods. We then determine how well the commuting flow is described by the potential, by evaluating the percentage of the gradient component for metropolitan areas in the USA and show that the gradient component is almost 100% in several areas; in other areas, however, the curl component is dominant, indicating the importance of circular flow along triangles of places. The potential landscape provides an easy-to-use visualisation tool to show the attractive places of human flow and will aid in various applications in commerce, urban design, and epidemic spreading.

Gradient flow and holography from a local Wilsonian cutoff

We consider the vacuum partition function of a 4d scalar QFT in a curved background as function of bare marginal and relevant couplings. A local UV cutoff $$\Lambda (x)$$ Λ ( x ) transforming under Weyl rescalings allows to construct Weyl invariant kinetic terms including Wilsonian cutoff functions. The local cutoff can be absorbed completely by a rescaling of the metric and the bare couplings. The vacuum partition function satisfies consistency conditions which follow from the Abelian nature of local redefinitions of the cutoff, and which differ from Weyl rescalings. These imply a gradient flow for beta functions describing the cutoff dependence of rescaled bare couplings. The consistency conditions allow to satisfy all but one Hamiltonian constraints required for a holographic description of the flow of bare couplings with the cutoff.