A Nonlocal Approach to the Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

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Missing the point in noncommutative geometry

Synthese ◽

10.1007/s11229-020-02998-1 ◽

2021 ◽

Author(s):

Nick Huggett ◽

Fedele Lizzi ◽

Tushar Menon

Keyword(s):

Scalar Field ◽

Noncommutative Geometry ◽

Smooth Manifold ◽

Operational Definition ◽

Spectral Triple ◽

Fundamental Quantity

AbstractNoncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

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Wormhole Solutions in Symmetric Teleparallel Gravity with Noncommutative Geometry

Symmetry ◽

10.3390/sym13071260 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1260

Author(s):

Zinnat Hassan ◽

Ghulam Mustafa ◽

Pradyumn Kumar Sahoo

Keyword(s):

Linear Model ◽

Noncommutative Geometry ◽

Teleparallel Gravity ◽

Exponential Model ◽

Model Parameter ◽

Exponential Models ◽

Feasible Solutions ◽

The Stability ◽

Tov Equation

This article describes the study of wormhole solutions in f(Q) gravity with noncommutative geometry. Here, we considered two different f(Q) models—a linear model f(Q)=αQ and an exponential model f(Q)=Q−α1−e−Q, where Q is the non-metricity and α is the model parameter. In addition, we discussed the existence of wormhole solutions with the help of the Gaussian and Lorentzian distributions of these linear and exponential models. We investigated the feasible solutions and graphically analyzed the different properties of these models by taking appropriate values for the parameter. Moreover, we used the Tolman–Oppenheimer–Volkov (TOV) equation to check the stability of the wormhole solutions that we obtained. Hence, we found that the wormhole solutions obtained with our models are physically capable and stable.

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Embedding non-linear filtering in autonomous underwater vehicle dynamics via the Kolmogorov backward equation

Transactions of the Institute of Measurement and Control ◽

10.1177/01423312211019662 ◽

2021 ◽

pp. 014233122110196

Author(s):

Ravish H. Hirpara ◽

Shambhu N. Sharma

Keyword(s):

State Vector ◽

Autonomous Underwater Vehicle ◽

Underwater Vehicle ◽

Linear Filtering ◽

Homogeneous Markov Process ◽

Non Linear ◽

Backward Equation ◽

Noise Correction ◽

Ito Process ◽

Correction Terms

This paper revisits the state vector of an autonomous underwater vehicle (AUV) dynamics coupled with the underwater Markovian stochasticity in the ‘non-linear filtering’ context. The underwater stochasticity is attributed to atmospheric turbulence, planetary interactions, sea surface conditions and astronomical phenomena. In this paper, we adopt the Itô process, a homogeneous Markov process, to describe the AUV state vector evolution equation. This paper accounts for the process noise as well as observation noise correction terms by considering the underwater filtering model. The non-linear filtering of the paper is achieved using the Kolmogorov backward equation and the evolution of the conditional characteristic function. The non-linear filtering equation is the cornerstone formalism of stochastic optimal control systems. Most notably, this paper introduces the non-linear filtering theory into an underwater vehicle stochastic system by constructing a lemma and a theorem for the underwater vehicle stochastic differential equation that were not available in the literature.

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