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We prove the Parseval theorem and the Plancherel Theorem among the analytic Feynman integral and the Fourier Feynman transform and the convolution for the function F : B ν → C in the Class F (B ν) : F ( x → ) = ∫ H exp { i ∑ j = 1 ν (h , x j) ∼ } d μ (h) , μ ∈ M (H) on the Product Abstract Wiener Space (H , B ν , m ν) , where B ν = B × B × ⋯ × B (ν times) and M (H) is the space of a complex valued countably additive measure μ defined on B (H) , the Borel class of a real separable infinite dimensional Hilbert space H. [ABSTRACT FROM AUTHOR]

"Numbers and the World: Essays on Math and Beyond" by David Mumford is a collection of essays that covers a wide range of topics, from mathematics to neuroscience, artificial intelligence, history, and social issues. Mumford, a highly acclaimed mathematician, offers his unique and creative perspectives on these subjects. The book is not a technical exposition of his mathematical work, but rather a collection of his thoughts and musings, originally posted on his blog. Mumford's writing style is clear and accessible, making this book an interesting read for both mathematicians and non-mathematicians alike. [Extracted from the article]

Internal model controllers (IMCs) are popular strategies for controlling stable systems with dead time. The relocated IMC (RIMC) design reportedly achieves the expected performance-robustness tradeoff for stable processes. Hence, an attempt has been made in this work to modify the RIMC (as a bi-loop RIMC proportional-derivative (RIMC-PD)) strategy to make it applicable to a class of unstable and integrating systems involving dead time. The secondary-loop stabilisation is achieved with a PD controller constructed by Routh stability constraints. The primary loop contains a RIMC controller for reference following. Both primary, as well as secondary-loop controller parameters, are optimally tuned in the search space using the equilibrium optimiser subjected to minimal integral square error. The RIMC-PD control strategy delivers reasonable enhancement in performance measures when compared with some of the recently reported strategies. A robust stability investigation is also carried out. Finally, experimental verification of the RIMC-PD strategy is carried out using a magnetic levitation laboratory setup. [ABSTRACT FROM AUTHOR]

The Open Polymer Classical Wigner (OPCW) method [J. Chem. Phys. 152, 094111 (2020)] is applied to a flux-Heaviside trace for calculating reaction rate constants. The obtained expression for the reaction rate constant is tested on a symmetric Eckart potential. The OPCW method is shown to converge toward the exact classical Wigner result. The OPCW method shows some promise for rate constant calculations and suggestions for future tests are made. [ABSTRACT FROM AUTHOR]

Over the last century, theoretical nuclear physicists have been developing and testing approximation methods and numerical techniques to investigate the structure and dynamics of strongly interacting quantum systems. Cells are equipped with a quality control machinery that enables them recognize and eliminate partially or incorrectly folded proteins, through a process called protein degradation. The idea that the folding of biologically relevant proteins could be far more complex than that of mini-proteins had been already suggested, but such folding intermediates were never systematically observed and characterized with atomistic detail. Protein Folding From a chemical point of view, proteins are polypeptide chains made of 20 different types of amino acids. [Extracted from the article]

We report a scaled particle theory-based method for evaluation of second osmotic virial coefficients from molecular simulations of dilute species in solution. In this method, we evaluate the work associated with growing a cavity in solution that is perfectly permeable to the solvent but is completely impermeable to the solutes, thereby establishing an osmotic stress between the cavity interior and exterior. Extrapolating our results to determine the solute concentration in contact with a cavity with an infinite radius, we are able to evaluate the solute osmotic pressure and second osmotic virial coefficient. A finite size correction is introduced to account for the impact of effectively concentrating the solutes in the periphery of the simulation box with increasing cavity size. We demonstrate the utility of the proposed method by evaluating second osmotic virial coefficients for methane in water as a function of temperature. The approach proposed here provides a physically transparent route for calculation of second osmotic virial coefficients by direct interrogation of simulation configurations without having to explicitly evaluate the long-range integral over solute-solute correlations required following McMillan-Mayer theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of the-product which is a very useful tool to obtain theanalytic Fourier–Feynman transform. We then use this concept to obtain various integration formulas involving the-product, the convolution product and the first variation. [ABSTRACT FROM AUTHOR]

It is well–known that Backward Stochastic Differential Equations provide probabilistic formulae for the solution of (systems of) second order elliptic and parabolic equations, thus providing an extension of the Feynman–Kac formula to semilinear PDEs. This method was applied to the class of PDEs with a nonlinear Neumann boundary condition first by Pardoux and Zhang in 1998. However, the proof of continuity of the extended Feynman–Kac formula with respect tox(resp. to (t, x)) is not correct in that paper. Here we consider a more general situation, where both the equation and the boundary condition involve the (possibly multivalued) gradient of a convex function. We prove the required continuity. The result for the class of equations studied Pardoux and Zhang in their paper from 1998, as well as those considered by Maticiuc and Răşcanu in their paper from 2010, are Corollaries of our main results. [ABSTRACT FROM AUTHOR]

In this article we introduce a new concept of an analytic operator-valued Feynman integral for functionals on Wiener space, which we then use to explain various physical phenomena. We then establish the existence of some analytic operator-valued Feynman integrals that prove useful in establishing various applications in quantum mechanics. [ABSTRACT FROM PUBLISHER]

Closed-form expressions are proposed for the Feynman integraloverd=D+mdimensional space with, in the special caseD=1. We show thatcan be expressed in different forms involving real and imaginary parts of the complex variable Gauss hypergeometric function, as well as generalized hypergeometricand, Hornand Appellfunctions. Several interesting relations are derived between the real and imaginary parts ofand the function. [ABSTRACT FROM PUBLISHER]

We investigate the behavior of a first variation of a Fourier transformof a measure of the form :upon the Fourier-Feynman transform and analytic Feynman integral and the convolution, whereand {h1,h2,…,hn} is the orthonormal class of elements inHon the abstract Wiener space (H,B,m). [ABSTRACT FROM PUBLISHER]

In this paper we define a modified analytic function space Fourier-Feynman transform to explain some physical phenomenon. We then establish various relationships involving the modified analytic function space Fourier-Feynman transform with related topics. Finally, we apply the translation theorem to obtain various relationships involving the modified analytic function space Fourier-Feynman transform. Our approach is a new and interesting method to explain some physical phenomenon. [ABSTRACT FROM AUTHOR]

With his death at 90 in March, the interdisciplinary field of theology and science has lost one of its most distinguished pioneers, John C. Polkinghorne. His luminary career in physics began when John entered Trinity College in 1949 where he specialized in quantum mechanics and was lectured by Paul Dirac. With John's death the field of theology and science has lost its three most foundational and visionary scholars: Ian Barbour, Arthur Peacocke, and now John Polkinghorne. [Extracted from the article]

A propagator-description approach of the propagation of EM waves is provided, in complex situations with underlying boundary conditions. To this end, the so-called Schwinger-Feynman causal propagator is explicitly derived, and spelled out, for the first time, for transmission of a photon through, and reflection off, a layer of glass of flat parallel surfaces. In a very general context, the corresponding transition amplitudes of a polarized or unpolarized photon are obtained, and the conservation of probabilities, and the underlying consistency conditions of the results, are established. Inspired by Feynman's intuitive and well-publicized non-technical treatment of such a problem with red and blue light, one, in a quantum view-point, is interested in the probabilities of the events just described in determining the fate of a photon. This analysis provides an alternative way of looking at the problem in question, through propagation of photons, as a physically more appealing approach, in Feynman's spirit, than by matching plane waves at boundaries as we often do. It also shows how a propagator may be derived, in more complex situations with specific boundary conditions, than just simply in infinite extended spaces. Finally, it is hoped that this work will be stimulating and of interest to practitioners with different emphases on EM wave propagation. [ABSTRACT FROM AUTHOR]

In this paper, we define a sequential analytic Feynman integral of functionals on Wiener space. We then give the existence of the sequential analytic Feynman integral of functionals in L 2(C 0[0, T]). The expansion of functionals in L 2(C 0[0, T]) in terms of Fourier–Hermite functionals plays a key role in the study. [ABSTRACT FROM PUBLISHER]

Let C[0, t] denote the function space of all real-valued continuous paths on [0, t]. Define and by X n (x)=(x(t 0), x(t 1), …, x(t n )) and X n+1(x)=(x(t 0), x(t 1), …, x(t n ), x(t n+1)), where 0=t 0

In this paper we study complex scaling in the Feynman–Kac formula, using the fact that for a Dirac delta initial distribution, complex scaling is well defined in white noise analysis. [ABSTRACT FROM AUTHOR]

Closed form expressions are investigated for a family of convergent Mathieu type a-series and its alternating variant, whose terms contain a generalized H-function associated with certain Feynman integrals. The results derived are of general character and provide an elegant generalization for the closed form expressions of these series associated with the H-function by Pogany [Integral expressions of Mathieu-type series whose terms contain Fox's H-function, Appl. Math. Lett. 20 (2007), pp. 764-769], for Fox-Wright Ψ-functions by Pogany and Srivastava [Some Mathieu-type series associated with the Fox-Wright function, Comput. Math. Appl. 57 (1) (2009), pp. 127-140] and for pFq and Meijer's G-function by Pogany and Tomovski [On Mathieu-type series which terms contain generalized hypergeometric function pFq and Meijer's G-function, Math. Comput. Modell. 47 (9-10) (2008), pp. 952-969], and others. [ABSTRACT FROM AUTHOR]

Given n charges interacting with each other according to Feynman's law. Let (rj(t); vj(t)) denote the position and velocity of the charge qj : The list y(t) of all such vectors is called a trajectory. A Lipschitzian trajectory x(t); (t ≤ 0); with continuous derivative, on which the velocities do not exceed some limiting velocity v < c, where c denotes the speed of light, is called an initial trajectory. A locally Lipschitzian trajectory y(t) is called relativistically admissible if the velocities on it stay below the speed of light c. The author constructs operators Φj on the trajectory whose values Φj(y)(t) are linear transformations of R³ into R³: A point t = t1 on a trajectory y is called singular if either some of the charges collide at the time t1 or the determinant is zero for at least one of the transformations Φj(y)(t1): The main result is the following: If x(t) (t ≤ 0) is an initial trajectory with non-singular point t = 0; then there exists a unique relativistically admissible trajectory y(t); defined for t in an interval I ⊂ (0;∞); extending the initial trajectory x(t) and having the following properties. (1) No point t on the trajectory y is singular. (2) The trajectory represents a unique solution of the Newton-Einstein momentum-force system of equations under Lorentz forces induced by electromagnetic field in accord to Feynman's law for moving charges. (3) The trajectory y represents the maximal global solution of the system. [ABSTRACT FROM AUTHOR]

We investigate the behaviour of the first variation under the Fourier-Feynman transform, and we show that the Wiener integral of the first variation behaves successfully under the Lp-analytic Fourier-Feynman transform for 1≤p≤2 for cylinder functions of the form [image omitted] where f: Rn→C∈Lp (Rn), 1≤p≤2 and {h1, h2, ..., hn} is the orthonormal class of elements in H and the Lp-analytic Fourier-Feynman transform of the first variation can be perfectly expressed as the limit of Wiener integrals of the first variation. [ABSTRACT FROM AUTHOR]

The QQT is a quasi-quantum mechanical treatment of the collision between molecules. Instead of a partial wave expansion approach, it uses a kind of Feynman path integral method that exploits the path length differences originating from the different orientations of an anisotropic molecule. As a result, the QQT provides valuable physical insight while requiring very little computational effort. The current paper gives a systematic derivation of the QQT and explains its underlying principles. The expression for the scattering amplitude is shown to be self-consistent, without any normalisation factors, when the rotational energy level spacing is negligible. The constant curvature approximation that is presented makes the QQT conceptually even more simple, and its effect on the calculated differential cross-sections (DCSs) turns out to be small. As examples we present QQT calculations of the DCSs for Ne-CO(1Σ) and He-NO(2Π), at collision energies of, respectively, 511 cm-1 and 514 cm-1. The anisotropy of the hard shell potential energy surface for Ne-CO in terms of the incoming de Broglie wavelength is about twice as large as for He-NO. This leads to state-to-state DCSs that have up to three maxima of comparable amplitude, instead of only one large maximum as is found for He-NO. The QQT results for these two applications are compared with results from close coupling calculations. [ABSTRACT FROM AUTHOR]

We consider linear parabolic stochastic integro-PDEs of Feynman-Kac type associated to Lévy-Itô diffusions. The solution of such equations can be represented as certain Feynman-Kac functionals of the associated diffusion such that taking expectation yields the deterministic Feynamn-Kac formula. We interpret the problem in the framework of white noise analysis and consider differentiation in the sense of stochastic distributions. This concept allows for relaxed assumptions on the equation coefficients, identically to those required in problems of similar deterministic integro-PDEs. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for general second--order uniformly elliptic linear equation in divergence form. We give a stochastic representation of bounded weak solutions of the problem In terms of solutions of associated linear backward stochastic differential equations. Our representation may he considered as an extension of the classical Feynman--Kac formula. [ABSTRACT FROM AUTHOR]

We establish the analytic Wiener and Feynman integral theory for Wiener integrals over paths on C 0 ( B ), the space of abstract Wiener space valued continuous functions on [0, T ], for certain cylinder-type functions of the form: where f ??? L p (R mn ), 1?=? p ?=?8, and 0?=? s 0 ?=? s 1 ?=?&CenterDot;?&CenterDot;?&CenterDot;?=? s n ?=? T is a partition of [0, T ]. In addition, we establish some relationships between analytic Feynman integrals, Fourier–Feynman transforms and Wiener integrals and we prove the change of scale formula for Wiener integrals over paths on C 0 ( B ) in abstract Wiener spaces. [ABSTRACT FROM AUTHOR]

The purpose of this article is to establish some relationships between L 1 -analytic Fourier-Feynman transforms and Wiener integrals for certain bounded functions F in the Fresnel class &Fouriertrf;( B ) of the form: F ( x ) = ∫ H exp{i( h , x ) &nvsim; }dμ( h ), μ &Element; M( H ), on the abstract Wiener space. [ABSTRACT FROM AUTHOR]

A general scheme is presented for using different numbers of 'time slices' for different degrees of freedom in a path integral evaluation of the Boltzmann operator for a large molecular system. This will be particularly useful, for example, in evaluating the 'quantum instanton' rate constant [cf. W.H. Miller, Y. Zhao, M. Ceotto, S. Yang. J. Chem. Phys. , 119 , 1329 (2003)] for H atom transfer reactions, or any applications involving atoms with largely differing masses. [ABSTRACT FROM AUTHOR]

In this paper, we first establish a formula for the conditional generalized Feynman integral of the first variation of a functional F ( x ) with x in a very general function space C a , b [0, T ]. We then use this basic formula to obtain several integration by parts formulas for conditional generalized Feynman integrals and conditional generalized Fourier-Feynman transforms. * E-mail: sejchang@dankook.ac.kr † E-mail: jgchoi@dankook.ac.kr [ABSTRACT FROM AUTHOR]

Ground state and excited state properties of ethylene, C2H4, and several ethylene isomers have been studied by Feynman path integral Monte Carlo (PIMC) simulations. The PIMC treatment of the atomic nuclei has been combined with different electronic Hamiltonians in order to analyse the influence of the nuclear degrees of freedom on electronic quantities. Electronic expectation values at the minimum of the potential energy surface (PES) have been compared with PIMC based ensemble averaged values. Ensemble averaged quantities have been derived by Hamiltonians of the ab initio type and a tight-binding (TB) one-electron model. The combined influence of anharmonicities in the interatomic potential and the quantum fluctuations of the atomic nuclei lead to ensemble averaged bondlengths r[sub g] which are significantly larger than the parameters r[sub e] at the minimum of the PES. The implications of this bond length elongation for the electronic properties of ethylene are discussed. The occupied canonical molecular orbitals (CMOs) of ethylene are destabilized under the influence of the nuclear degrees of freedom while virtual CMOs are stabilized. These shifts of one-electron energies suggest a comparison of electronic excitation energies at the minimum of the PES with PIMC based ensemble averages. The quantum fluctuations of the nuclei cause a strong redistribution in the intensities of electronic transitions. Transitions, which are dipole allowed in the planar D2h geometry of ethylene, lose intensity under the influence of nuclear quantum effects, and vice versa for electronic excitations that are dipole forbidden under D[sub 2h] symmetry. This 'vibrational borrowing' is enhanced with decreasing atomic masses. The Feynman centroid density has been used to calculate the anharmonic vibrational wavenumbers of C[sub 2]H[sub 4] and C[sub 2]D[sub 4]. The results of the present PIMC simulations have been employed to emphasize general problems of electronic structure calculations based on a single nuclear configuration (i.e. the configuration at the minimum of the PES). [ABSTRACT FROM AUTHOR]

Attention is drawn to the similarity between quantum mechanics and finding the simplest explanation for the data from an array of sensors. In particular, comparison of the Durbin-Willshaw approach to solving the travelling salesm an problem with the Feynman path integral for motion of a charged particle in a magnetic field suggests that quantum mechanics might be of practical use both for integrating multiple measurements and for solving nonpolynom ial (N P)-complete problems. [ABSTRACT FROM AUTHOR]

As Royal Society Rutherford Scholar from 1954 to 1958, he spent a year at the Massachusetts Institute of Technology and was appointed Fellow of Balliol College and Lecturer in Theoretical Physics at Oxford in 1958. Graph David M. Brink David Maurice Brink (20 July 1930, Hobart, Tasmania-8 March 2021, Oxford) obtained a B.Sc. from the University of Tasmania in 1951, and a D.Phil. from Oxford in 1955. After Oxford, David moved to Trento, where he helped found the European Centre for Theoretical Studies in Nuclear Physics and Related Areas and acted as vice-director. [Extracted from the article]

The Feynman expressions for the electric and magnetic fields of a charge moving on an arbitrary trajectory are found by two simple methods. The first involves a direct integration of Maxwell's time-dependent equations. The second is based on the formalism of quantum optics which uses the first order equations of motion for creation and annihilation operators. As an illustration of the simplicity of these methods, the fields of an oscillating electric dipole are obtained. [ABSTRACT FROM AUTHOR]