scholarly journals INTEGRATION OVER CONNECTIONS IN THE DISCRETIZED GRAVITATIONAL FUNCTIONAL INTEGRALS

2010 ◽  
Vol 25 (05) ◽  
pp. 351-368 ◽  
Author(s):  
V. M. KHATSYMOVSKY
Keyword(s):  
Probability Distribution ◽  
Einstein Gravity ◽  
Generalized Function ◽  
Discrete Form ◽  
Functional Integrals ◽  
Local Maxima ◽  
First Order ◽  
Connection Type ◽  
Probe Functions ◽  
Compact Gauge Group

The result of performing integrations over connection type variables in the path integral for the discrete field theory may be poorly defined in the case of non-compact gauge group with the Haar measure exponentially growing in some directions. This point is studied in the case of the discrete form of the first-order formulation of the Einstein gravity theory. Here the result of interest can be defined as generalized function (of the rest of variables of the type of tetrad or elementary areas), i.e. a functional on a set of probe functions. To define this functional, we calculate its values on the products of components of the area tensors, the so-called moments. The resulting distribution (in fact, probability distribution) has singular (δ-function-like) part with support in the nonphysical region of the complex plane of area tensors and regular part (usual function) which decays exponentially at large areas. As we discuss, this also provides suppression of large edge lengths which is important for internal consistency, if one asks whether gravity on short distances can be discrete. Some other features of the obtained probability distribution including occurrence of the local maxima at a number of the approximately equidistant values of area are also considered.

scholarly journals DEFINING INTEGRALS OVER CONNECTIONS IN THE DISCRETIZED GRAVITATIONAL FUNCTIONAL INTEGRALS

2010 ◽  
Vol 25 (17) ◽  
pp. 1407-1423 ◽  
Author(s):  
V. M. KHATSYMOVSKY
Keyword(s):  
Complex Plane ◽  
Path Integral ◽  
Physical Region ◽  
Relativity Theory ◽  
Generalized Function ◽  
General Relativity Theory ◽  
Functional Integrals ◽  
First Order ◽  
Connection Type ◽  
Area Tensor

Integration over connection type variables in the path integral for the discrete form of the first-order formulation of general relativity theory is studied. The result (a generalized function of the rest of variables of the type of tetrad or elementary areas) can be defined through its moments, i.e. integrals of it with the area tensor monomials. In our previous paper these moments have been defined by deforming integration contours in the complex plane as if we had passed to a Euclidean-like region. In this paper we define and evaluate the moments in the genuine Minkowski region. The distribution of interest resulting from these moments in this non-positively defined region contains the divergences. We prove that the latter contribute only to the singular (δ-function like) part of this distribution with support in the non-physical region of the complex plane of area tensors while in the physical region this distribution (usual function) confirms that defined in our previous paper which decays exponentially at large areas. Besides that, we evaluate the basic integrals over which the integral over connections in the general path integral can be expanded.

A Simple Coding Procedure Enhances a Neuron's Information Capacity

1981 ◽  
Vol 36 (9-10) ◽  
pp. 910-912 ◽  
Author(s):  
Simon Laughlin
Keyword(s):  
Information Theory ◽  
Probability Distribution ◽  
Response Function ◽  
Compound Eye ◽  
Information Capacity ◽  
Cumulative Probability ◽  
Natural Scenes ◽  
First Order ◽  
Contrast Response Function ◽  
Contrast Response

Abstract The contrast-response function of a class of first order intemeurons in the fly's compound eye approximates to the cumulative probability distribution of contrast levels in natural scenes. Elementary information theory shows that this matching enables the neurons to encode contrast fluctuations most efficiently.

Some possible q-exponential type probability distribution in the non-extensive statistical physics

2016 ◽  
Vol 30 (22) ◽  
pp. 1650252 ◽  
Author(s):  
Won Sang Chung
Keyword(s):  
Probability Distribution ◽  
Exponential Type ◽  
Statistical Physics ◽  
Probability Distributions ◽  
Type I ◽  
Quantum Harmonic Oscillator ◽  
Text Type ◽  
First Order ◽  
Gibbs Entropy ◽  
Level Problem

In this paper, we present two exponential type probability distributions which are different from Tsallis’s case which we call Type I: one given by [Formula: see text] (Type IIA) and another given by [Formula: see text] (Type IIIA). Starting with the Boltzman–Gibbs entropy, we obtain the different probability distribution by using the Kolmogorov–Nagumo average for the microstate energies. We present the first-order differential equations related to Types I, II and III. For three types of probability distributions, we discuss the quantum harmonic oscillator, two-level problem and the spin-[Formula: see text] paramagnet.

scholarly journals PROBABILITY DISTRIBUTION FUNCTIONAL FOR EQUAL-TIME CORRELATION FUNCTIONS IN CURVED SPACE

1994 ◽  
Vol 09 (02) ◽  
pp. 221-238 ◽  
Author(s):  
HIROSHI SUZUKI ◽  
MISAO SASAKI ◽  
KAZUHIRO YAMAMOTO ◽  
JUN’ICHI YOKOYAMA
Keyword(s):  
Probability Distribution ◽  
Correlation Functions ◽  
Spatial Configuration ◽  
Scaling Law ◽  
Time Correlation ◽  
Ultraviolet Divergence ◽  
Coupling Constant ◽  
De Sitter ◽  
Curved Space ◽  
First Order

We present a systematic method to calculate the probability distribution functional (PDF) for spatial configuration of an interacting field in curved space-time. As an example, we consider PDF for the minimally coupled massive λΦ4 theory up to the first order of the coupling constant and evaluate it both in Minkowski and de Sitter spacetimes. We observe that PDF has an ultraviolet divergence even after the ultraviolet renormalization. This divergence is unavoidable to reproduce finite expectation values; thus some kind of regularization is necessary to write down PDF. As an application of it, a scaling law among multipoint correlation functions in the de Sitter space is found.

scholarly journals Compilation by stochastic Hamiltonian sparsification

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 235 ◽  
Author(s):  
Yingkai Ouyang ◽  
David R. White ◽  
Earl T. Campbell
Keyword(s):  
Electronic Structure ◽  
Quantum Computing ◽  
Probability Distribution ◽  
Quantum Chemistry ◽  
Error Bounds ◽  
Simulation Methods ◽  
Physical Systems ◽  
First Order ◽  
Convex Optimisation ◽  
Principal Application

Simulation of quantum chemistry is expected to be a principal application of quantum computing. In quantum simulation, a complicated Hamiltonian describing the dynamics of a quantum system is decomposed into its constituent terms, where the effect of each term during time-evolution is individually computed. For many physical systems, the Hamiltonian has a large number of terms, constraining the scalability of established simulation methods. To address this limitation we introduce a new scheme that approximates the actual Hamiltonian with a sparser Hamiltonian containing fewer terms. By stochastically sparsifying weaker Hamiltonian terms, we benefit from a quadratic suppression of errors relative to deterministic approaches. Relying on optimality conditions from convex optimisation theory, we derive an appropriate probability distribution for the weaker Hamiltonian terms, and compare its error bounds with other probability ansatzes for some electronic structure Hamiltonians. Tuning the sparsity of our approximate Hamiltonians allows our scheme to interpolate between two recent random compilers: qDRIFT and randomized first order Trotter. Our scheme is thus an algorithm that combines the strengths of randomised Trotterisation with the efficiency of qDRIFT, and for intermediate gate budgets, outperforms both of these prior methods.

An Efficient Numerical Method for Calculating the Statistical Distribution of Combined First-Order and Wave-Drift Response

1992 ◽  
Vol 114 (3) ◽  
pp. 195-204 ◽  
Author(s):  
A. Naess ◽  
J. M. Johnsen
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Numerical Method ◽  
Probability Distribution Function ◽  
Numerical Procedure ◽  
Offshore Structures ◽  
Numerical Calculations ◽  
Hydrodynamic Loads ◽  
First Order ◽  
Wave Drift

The paper describes an efficient and accurate numerical procedure for calculating the probability distribution function of combined first-order and slowly varying, second-order hydrodynamic loads and response of compliant offshore structures. No approximations are made except those inherent in the numerical calculations. The method does not require extensive computer capacity; in fact it can be implemented on any standard PC. Several example calculations serve to illustrate the method, and its accuracy is demonstrated by comparison with cases where exact analytical results are available. The accuracy of previously proposed approximations are also discussed.

scholarly journals Investigating the holographic complexity in Einsteinian cubic gravity

2019 ◽  
Vol 79 (10) ◽  
Author(s):  
Jie Jiang ◽  
Banglin Deng
Keyword(s):  
Order Approximation ◽  
Mass Parameter ◽  
Small Mass ◽  
Einstein Gravity ◽  
Time Limit ◽  
Late Time ◽  
Full Time ◽  
Time Analysis ◽  
First Order ◽  
First Order Approximation

Abstract In this paper, we investigate the holographic complexity of a small mass AdS black hole in Einsteinian cubic gravity by using the “complexity equals action” (CA) and “complexity equals volume” (CV) conjectures. In the CA context, the late-time growth rate satisfies the Lloyd bound for the $$k=0$$k=0 and $$k=1$$k=1 cases but it violates it for the $$k=-1$$k=-1 case in the first-order approximation of the small mass parameter. However, by a full-time analysis, we find that this late-time limit is approached from above, which implies that this bound in all of these cases will be violated. In the CV context, we considered both the original and the generalized CV conjectures. Differing from the CA conjecture, the late-time rate here is non-vanishing in the zeroth-order approximation, and this shows that the Lloyd bound is exactly violated even in the late-time limit. These results show numerous differences from the neutral case of the Einstein gravity in both the CA and the CV holographic contexts where all of their late-time results saturate the Lloyd bound. These differences illustrate the influence of the higher curvature correction in Einstein gravity.

scholarly journals First-order minisuperspace model for the Faddeev formulation of gravity

2015 ◽  
Vol 30 (32) ◽  
pp. 1550174 ◽  
Author(s):  
V. M. Khatsymovsky
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Field Equations ◽  
Piecewise Constant ◽  
First Order ◽  
Weyl Connection ◽  
Type Form ◽  
Connection Type ◽  
Discrete Theory

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual GR. Earlier we have proposed some minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which [Formula: see text] can be decomposed. Now we study some representation of this (discrete) theory, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR.

Calculating the Distribution of the Serial Correlation Estimator by Saddlepoint Integration

1996 ◽  
Vol 12 (3) ◽  
pp. 458-480 ◽  
Author(s):  
Carl W. Helstrom
Keyword(s):  
Time Series ◽  
Maximum Likelihood ◽  
Probability Distribution ◽  
Finite Number ◽  
Serial Correlation ◽  
Asymptotic Distributions ◽  
Stationary Time Series ◽  
Finite Sample ◽  
Initial Value ◽  
First Order

The efficient method of numerical saddlepoint integration is described and applied to calculating the probability distribution of the maximum likelihood and Yule-Walker estimators of the correlation coefficient a of a first-order autoregressive normal time series with initial value either zero or nonzero when a finite number n of data are at hand. Stationary time series of the same type are also treated. Significance points are computed in a number of examples to show how, as n increases, the finite-sample distributions approach the asymptotic distributions that have appeared in the literature.

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