Feynman Path Integral Using Operator Integration in Banach Space

Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.

CAUCHY TYPE INTEGRALS AND A BOUNDARY VALUE PROBLEM IN A COMPLEX CLIFFORD ANALYSIS

Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.

Design of integral sliding mode guidance law based on disturbance observer

With the increasing precision of guidance,the impact of autopilot dynamic characteristics and target maneuvering abilities on precision guidance is becoming more and more significant.In order to reduce or even eliminate the autopilot dynamic operation and the target maneuvering influence,this paper suggests a guidance system model involving a novel integral sliding mode guidance law(ISMGL).The method utilizes the dynamic characteristics and the impact angle,combined with a sliding mode surface scheme that includes the desired line-ofsight angle,line-of-sight angular rate,and second-order differential of the angular line-of-sight.At the same time,the evaluation scenario considere the target maneuvering in the system as the external disturbance,and the non-homogeneous disturbance observer estimate the target maneuvering as a compensation of the guidance command.The proposed system’s stability is proven based on the Lyapunov stability criterion.The simulations reveale that ISMGL effectively intercepted large maneuvering targets and present a smaller miss-distance compared with traditional linear sliding mode guidance laws and trajectory shaping guidance laws.Furthermore,ISMGL has a more accurate impact angle and fast convergence speed.

Approximate solution of Volterra-Fredholm integral equations using generalized barycentric rational interpolant

It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.

Results Involving Partial Differential Equations and Their Solution by Certain Integral Transform

In this study,we aimto investigate certain triple integral transformand its application to a class of partial differentialequations.We discuss various properties of the new transformincluding inversion, linearity, existence, scaling andshifting, etc. Then,we derive several results enfolding partial derivatives and establish amulti-convolution theorem.Further, we apply the aforementioned transform to some classical functions and many types of partial differentialequations involving heat equations,wave equations, Laplace equations, and Poisson equations aswell.Moreover,wedraw some figures to illustrate 3-D contour plots for exact solutions of some selected examples involving differentvalues in their variables.

Trajectory tracking guidance of interceptor via prescribed performance integral sliding mode with neural network disturbance observer

This paper investigates interception missiles’trajectory tracking guidance problem under wind field and external disturbances in the boost phase.Indeed,the velocity control in such trajectory tracking guidance systems of missiles is challenging.As our contribution,the velocity control channel is designed to deal with the intractable velocity problem and improve tracking accuracy.The global prescribed performance function,which guarantees the tracking error within the set range and the global convergence of the tracking guidance system,is first proposed based on the traditional PPF.Then,a tracking guidance strategy is derived using the integral sliding mode control techniques to make the sliding manifold and tracking errors converge to zero and avoid singularities.Meanwhile,an improved switching control law is introduced into the designed tracking guidance algorithm to deal with the chattering problem.A back propagation neural network(BPNN)extended state observer(BPNNESO)is employed in the inner loop to identify disturbances.The obtained results indicate that the proposed tracking guidance approach achieves the trajectory tracking guidance objective without and with disturbances and outperforms the existing tracking guidance schemes with the lowest tracking errors,convergence times,and overshoots.

The tangential k-Cauchy-Fueter type operator and Penrose type integral formula on the generalized complex Heisenberg group

The tangential k-Cauchy-Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables,respectively.In this paper,we introduce a Lie group that the Heisenberg group can be imbedded into and call it generalized complex Heisenberg.We investigate quaternionic analysis on the generalized complex Heisenberg.We also give the Penrose integral formula for k-CF functions and construct the tangential k-Cauchy-Fueter complex.

Existence of Solutions of a Convolution Integral Equation

In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined on it. An operator P was assigned to the convolution integral operator which was later expressed in terms of the superposition operator and the nonlinear operator. Given a ball Br belonging to the space L it was established that the operator P maps the ball into itself. The Hausdorff measure of noncompactness was then applied by first proving that given a set M∈ B r the set is bounded, closed, convex and nondecreasing. Finally, the Darbo fixed point theorem was applied on the measure obtained from the set E belonging to M. From this application, it was observed that the conditions for the Darbo fixed point theorem was satisfied. This indicated the presence of at least a fixed point for the integral equation which thereby implying the existence of solutions for the integral equation.

Solvability and Construction of a Solution to the Fredholm Integral Equation of the First Kind

The issues of solvability and construction of a solution of the Fredholm integral equation of the first kind are considered. It is done by immersing the original problem into solving an extremal problem in Hilbert space. Necessary and sufficient conditions for the existence of a solution are obtained. A method of constructing a solution of the Fredholm integral equation of the first kind is developed. A constructive theory of solvability and construction of a solution to a boundary value problem of a linear integrodifferential equation with a distributed delay in control, generated by the Fredholm integral equation of the first kind, has been created.

Application of Choquet Integral-Importance-Performance Analysis and TOPSIS Methods in Approaching the Preference Factors of Calligraphy and Seal Engraving Imagery

Classical Chinese characters,presented through calligraphy,seal engraving,or painting,can exhibit different aesthetics and essences of Chinese characters,making them the most important asset of the Chinese people.Calligraphy and seal engraving,as two closely related systems in traditional Chinese art,have developed through the ages.Due to changes in lifestyle and advancements in modern technology,their original functions of daily writing and verification have gradually diminished.Instead,they have increasingly played a significant role in commercial art.This study utilizes the Evaluation Grid Method(EGM)and the Analytic Hierarchy Process(AHP)to research the key preference factors in the application of calligraphy and seal engraving imagery.Different from the traditional 5-point equal interval semantic questionnaire,this study employs a non-equal interval semantic questionnaire with a golden ratio scale,distinguishing the importance ratio of adjacent semantic meanings and highlighting the weighted emphasis on visual aesthetics.Additionally,the study uses Importance-Performance Analysis(IPA)and Technique for Order Preference by Similarity to Ideal Solution(TOPSIS)to obtain the key preference sequence of calligraphy and seal engraving culture.Plus,the Choquet integral comprehensive evaluation is used as a reference for IPA comparison.It is hoped that this study can provide cultural imagery references and research methods,injecting further creativity into industrial design.

Fuzzy Henstock-Kurzweil Triple Integral on a Type 1 Quasi-Fuzzy Parallelepipedal Domain

In this article, we propose by using the Hausdorff distance Simpson’s rule for the triple integral of a fuzzy-valued function and the error bound of this method, one of the variables of which is fuzzy. In addition, thin δ-fine partitions are introduced. The integration domain is a quasi-fuzzy parallelipiped. A numerical example is presented in order to show the application and the significance of the method.

Fuzzy Henstock-Kurzweil Triple Integral

In this article, we use the Hausdorf distance to treat triple Simpson’s rule of the Henstock triple integral of a fuzzy valued function as well as the error bound of the method. We also introduce δ-fine subdivisions for a Henstock triple integral and numerical example is presented in order to show the application and the consequence of the method.

Integral imaging-based tabletop light field 3D display with large viewing angle

Light field 3D display technology is considered a revolutionary technology to address the critical visual fatigue issues in the existing 3D displays.Tabletop light field 3D display provides a brand-new display form that satisfies multi-user shared viewing and collaborative works,and it is poised to become a potential alternative to the traditional wall and portable display forms.However,a large radial viewing angle and correct radial perspective and parallax are still out of reach for most current tabletop light field 3D displays due to the limited amount of spatial information.To address the viewing angle and perspective issues,a novel integral imaging-based tabletop light field 3D display with a simple flat-panel structure is proposed and developed by applying a compound lens array,two spliced 8K liquid crystal display panels,and a light shaping diffuser screen.The compound lens array is designed to be composed of multiple three-piece compound lens units by employing a reverse design scheme,which greatly extends the radial viewing angle in the case of a limited amount of spatial information and balances other important 3D display parameters.The proposed display has a radial viewing angle of 68.7°in a large display size of 43.5 inches,which is larger than the conventional tabletop light field 3D displays.The radial perspective and parallax are correct,and high-resolution 3D images can be reproduced in large radial viewing positions.We envision that this proposed display opens up possibility for redefining the display forms of consumer electronics.

Numerical and experimental study on the particle erosion and gas–particle hydrodynamics in an integral multi-jet swirling spout-fluidized bed

In this paper,using the computational fluid dynamics based on Euler Lagrange and the commercial software Barracuda VR,the gas-particle hydrodynamics and the erosion of particles on the inner wall and internal components of the spouted bed in the integrated multi-jet swirling spout-fluidized bed(IMSSFB)are studied.Erosion experiments have obtained the characterization of particle erosion on internal components and verified the relevant numerical models.The results show that:the particle distribution within the IMSSFB is uneven due to the cyclonic effect of the axial swirl vane(ASV),resulting in particle erosion for the ASV being concentrated on one side;when the gas reaches the top,too high an erosion gas velocity leads to gas backflow.As the filling height increases,there is a tendency for the erosion position of the particles on the ASV to expand upwards.However,the effect of increasing gas velocity on the erosion position is insignificant.

Least square method based on Haar wavelet to solve multi-dimensional stochastic Ito-Volterra integral equations

This paper proposes a method combining blue the Haar wavelet and the least square to solve the multi-dimensional stochastic Ito-Volterra integral equation.This approach is to transform stochastic integral equations into a system of algebraic equations.Meanwhile,the error analysis is proven.Finally,the effectiveness of the approach is verified by two numerical examples.

Application of engineered compressible inclusions to mitigating soilstructure interaction issues in integral bridge abutments

The thermally induced cyclic loading on integral bridge abutments causes soil deformation and lateral stress ratcheting behind the abutment wall due to the expansion and contraction of the bridge deck.The forward and backward movements of the abutment in response to the expansion/contraction of the bridge deck lead to the formation of settlement trough and surface heaving,frequently creating a bump at the bridge approach and increasing the lateral earth pressure behind the abutment.Measures to reduce the bump at the bridge approach,including several treatment methods,such as compaction of selected backfill materials,grout injection,installation of approach slab,and using a layer of compressible inclusion material behind the abutment were proposed.However,these guidelines still lack sufficient design details and there are limited experimental findings to validate design assumptions.In this paper,the use of engineered compressible materials to alleviate the lateral earth pressure ratcheting and settlement at the bridge approach is investigated.The comparative study is presented for the soil-inclusion,material-structure and soil-structure interactions for an integral bridge under three different backfill conditions,i.e.(a)sand,(b)sand and EPS geofoam,and(c)sand and Infinergy®.The study was conducted in a special large-scale test chamber with a semi-scale abutment to gain better insights into the soil-structure interaction(SSI).The kinematics and rearrangement of the soil during the cyclic loading have been investigated to identify the mitigating effects of compressible inclusions.The comparative study indicates that both compressible inclusions perform comparatively well,however,Infinergy®is a better alternative than the medium-density EPS geofoam,as it works more effectively to reduce the backfill settlement and heaving as well as soil ratcheting effects under cyclic translational movement.

Measuring stellar populations,dust attenuation and ionized gas at kpc scales in 10010 nearby galaxies using the integral field spectroscopy from MaNGA

As one of the three major experiments of the fourth-generation Sloan Digital Sky Survey(SDSS-IV),the Mapping Nearby Galaxies at Apatch Point Observatory(MaNGA)survey has obtained high-quality integral field spectroscopy(IFS)with a resolution of 1–2 kpc for104galaxies in the local universe during its six-year operation from July 2014 through August 2020.It is crucial to reliably measure the physical properties of the different components in each spectrum before one can use the data for any scientific study.In the past years we have made lots of efforts to develop a novel technique of full spectral fitting,which estimates a model-independent dust attenuation curve from each spectrum,thus allowing us to break the degeneracy between dust attenuation and stellar population properties when fitting the spectrum with stellar population synthesis models.We have applied our technique to the final data release of Ma NGA,and obtained measurements of stellar population properties and emission line parameters,as well as the kinematics and dust attenuation of both stellar and ionized gas components.In this paper we describe our technique and the content and format of our data products.The whole dataset is publicly available in Science Data Bank with the link https://doi.org/10.57760/sciencedb.j00113.00088.

The Flipping-Free Full-Parallax Tabletop Integral Imaging with Enhanced Viewing Angle Based on Space-Multiplexed Voxel Screen and Compound Lens Array

Tabletop integral imaging display with a more realistic and immersive experience has always been a hot spot in three-dimensional imaging technology,widely used in biomedical imaging and visualization to enhance medical diagnosis.However,the traditional structural characteristics of integral imaging display inevitably introduce the flipping effect outside the effective viewing angle.Here,a full-parallax tabletop integral imaging display without the flipping effect based on space-multiplexed voxel screen and compound lens array is demonstrated,and two holographic functional screens with different parameters are optically designed and fabricated.To eliminate the flipping effect in the reconstruction process,the space-multiplexed voxel screen consisting of a projector array and the holographic functional screen is presented to constrain light beams passing through the corresponding lens.To greatly promote imaging quality within the viewing area,the aspherical structure of the compound lens is optimized to balance the aberrations.It cooperates with the holographic functional screen to modulate the light field spatial distribution.Compared with the simulation results,the distortion rate of the imaging display is reduced to less than 9%from more than 30%.In the experiment,the floating high-quality reconstructed three-dimensional image without the flipping effect can be observed with the correct 3D perception at 96°×96°viewing angle,where 44,100 viewpoints are employed.

q–differ-integral operator on p–valent functions associated with operator on Hilbert space

Making use of multivalent functions with negative coefficients of the type f (z)=z^(p)-~(∑)_(k=p+1)^(∞)a_(k)z^(k),which are analytic in the open unit disk and applying the q-derivative a q–differintegral operator is considered.Furthermore by using the familiar Riesz-Dunford integral,a linear operator on Hilbert space H is introduced.A new subclass of p-valent functions related to an operator on H is defined.Coefficient estimate,distortion bound and extreme points are obtained.The convolution-preserving property is also investigated.

Unique Solution of Integral Equations via Intuitionistic Extended Fuzzy b-Metric-Like Spaces

In this manuscript,our goal is to introduce the notion of intuitionistic extended fuzzy b-metric-like spaces.We establish some fixed point theorems in this setting.Also,we plot some graphs of an example of obtained result for better understanding.We use the concepts of continuous triangular norms and continuous triangular conorms in an intuitionistic fuzzy metric-like space.Triangular norms are used to generalize with the probability distribution of triangle inequality in metric space conditions.Triangular conorms are known as dual operations of triangular norms.The obtained results boost the approaches of existing ones in the literature and are supported by some examples and applications.