scholarly journals Reduction of the Zakai equation by invariance group techniques

1998 ◽  
Vol 73 (1) ◽  
pp. 119-130 ◽  
Author(s):  
Michel Cohen de Lara
Keyword(s):  
Invariance Group ◽  
Zakai Equation

scholarly journals Unbiased estimation of the solution to Zakai’s equation

2020 ◽  
Vol 26 (2) ◽  
pp. 113-129
Author(s):  
Hamza M. Ruzayqat ◽  
Ajay Jasra
Keyword(s):  
Continuous Time ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Linear Filtering ◽  
Multilevel Monte Carlo ◽  
Zakai Equation ◽  
Unbiased Estimators ◽  
First Order ◽  
Multilevel Monte Carlo Method ◽  
Filtering Problem

AbstractIn the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.

scholarly journals Dynamical Symmetries of the H Atom, One of the Most Important Tools of Modern Physics: SO(4) to SO(4,2), Background, Theory, and Use in Calculating Radiative Shifts

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1323 ◽  
Author(s):  
G. Jordan Maclay
Keyword(s):  
Hydrogen Atom ◽  
Quantum Electrodynamics ◽  
Particle Physics ◽  
Bound States ◽  
Integrated Treatment ◽  
Invariance Group ◽  
Elementary Particle Physics ◽  
Background Theory ◽  
Modern Physics ◽  
History Of

Understanding the hydrogen atom has been at the heart of modern physics. Exploring the symmetry of the most fundamental two body system has led to advances in atomic physics, quantum mechanics, quantum electrodynamics, and elementary particle physics. In this pedagogic review, we present an integrated treatment of the symmetries of the Schrodinger hydrogen atom, including the classical atom, the SO(4) degeneracy group, the non-invariance group or spectrum generating group SO(4,1), and the expanded group SO(4,2). After giving a brief history of these discoveries, most of which took place from 1935–1975, we focus on the physics of the hydrogen atom, providing a background discussion of the symmetries, providing explicit expressions for all of the manifestly Hermitian generators in terms of position and momenta operators in a Cartesian space, explaining the action of the generators on the basis states, and giving a unified treatment of the bound and continuum states in terms of eigenfunctions that have the same quantum numbers as the ordinary bound states. We present some new results from SO(4,2) group theory that are useful in a practical application, the computation of the first order Lamb shift in the hydrogen atom. By using SO(4,2) methods, we are able to obtain a generating function for the radiative shift for all levels. Students, non-experts, and the new generation of scientists may find the clearer, integrated presentation of the symmetries of the hydrogen atom helpful and illuminating. Experts will find new perspectives, even some surprises.

Quantum fluctuations of the metric, three-geometries and conformal embeddings in four-manifolds

2021 ◽  
Author(s):  
Simon Davis
Keyword(s):  
Path Integrals ◽  
Invariance Group ◽  
Two Dimensional ◽  
Four Dimensions ◽  
Physical Conditions ◽  
Exponential Expansion ◽  
Cotangent Bundles ◽  
Four Manifolds ◽  
Conformal Embeddings ◽  
Natural Relation

In this paper, connections between the path integrals for four-dimensional quantum gravity and string theory are emphasized. It is shown that there is a natural relation between these two path integrals based on the theorems on embeddings of two-dimensional surfaces in four dimensions and four-dimensional manifolds in ten dimensions. The isometry groups of the three-geometries that are spatial hypersurfaces confomally embedded in the four-manifolds are required to be subgroups of [Formula: see text], which is the invariance group of the Pfaffian differential system satisfied by one form in the cotangent bundles on the four-manifolds. Based on this and other physical conditions, the three-geometries are restricted to be [Formula: see text], [Formula: see text] and [Formula: see text] with a boundary, which may be included in the quantum gravitational path integral over four-manifolds which are closed at initial times followed by an exponential expansion compatible with supersymmetry.

scholarly journals ON THE TWO-POINT CORRELATION FUNCTION IN DYNAMICAL SCALING AND SCHRÖDINGER INVARIANCE

1992 ◽  
Vol 03 (05) ◽  
pp. 1011-1017 ◽  
Author(s):  
MALTE HENKEL
Keyword(s):  
Correlation Function ◽  
Invariance Group ◽  
Exactly Solvable Models ◽  
Solvable Models ◽  
Dynamical Exponent ◽  
Dynamical Scaling ◽  
Exactly Solvable ◽  
Local Space ◽  
Point Correlation ◽  
Point Correlation Function

The extension of dynamical scaling to local, space-time dependent rescaling factors is investigated. For a dynamical exponent z=2, the corresponding invariance group is the Schrödinger group. Schrödinger invariance is shown to determine completely the two-point correlation function. The result is checked in two exactly solvable models.

On Certain Extensions of Function Rings

1959 ◽  
Vol 11 ◽  
pp. 87-96
Author(s):  
Bernhard Banaschewski
Keyword(s):  
Topological Space ◽  
Galois Extension ◽  
Present Note ◽  
Banach Algebras ◽  
Faithful Representation ◽  
Normal Extension ◽  
Invariance Group ◽  
Group Of Automorphisms ◽  
Normed Ring ◽  
Function Ring

The present note is concerned with the existence and properties of certain types of extensions of Banach algebras which allow a faithful representation as the normed ring C(E) of all bounded continuous real functions on some topological space E. These Banach algebras can be characterized intrinsically in various ways (1); they will be called function rings here. A function ring E will be called a normal extension of a function ring G if E is directly indecomposable, contains C as a Banach subalgebra and possesses a group G of automorphisms for which C is the ring of invariants, that is, the set of all elements fixed under G. G will then be called a group of automorphisms of E over C. If E is a normal extension of C with precisely one group of automorphisms over C, which is then the invariance group of C in E, then E will be called a Galois extension of C. Such an extension will be called finite if its group is finite.

Fractional generalizations of filtering problems and their associated fractional Zakai equations

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Sabir Umarov ◽  
Frederick Daum ◽  
Kenric Nelson
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Lévy Process ◽  
Nonlinear Filtering ◽  
Levy Process ◽  
Zakai Equation ◽  
Filtering Problem

AbstractIn this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

A Zakai equation derivation of the extended Kalman filter

Automatica ◽  
2010 ◽  
Vol 46 (3) ◽  
pp. 620-624 ◽  
Author(s):  
Robert J. Elliott ◽  
Simon Haykin
Keyword(s):  
Kalman Filter ◽  
Extended Kalman Filter ◽  
Zakai Equation
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