Methods Of Mathematical Physics Research Articles

Searching for a response: the intriguing mystery of Feynman’s theoretical reference amplifier

We analyze Feynman's work on the response of an amplifier performed at Los Alamos and described in a technical report of 1946, as well as lectured on at the Cornell University in 1946-47 during his course on Mathematical Methods. The motivation for such a work was Feynman's involvement in the Manhattan Project, for which the necessity emerged of feeding the output pulses of counters into amplifiers or several other circuits, with the risk of introducing distortion at each step. In order to deal with such a problem, Feynman designed a theoretical "reference amplifier", thus enabling a characterization of the distortion by means of a benchmark relationship between phase and amplification for each frequency, and providing a standard tool for comparing the operation of real devices. A general theory was elaborated, from which he was able to deduce the basic features of an amplifier just from its response to a pulse or to a sine wave of definite frequency. Moreover, in order to apply such a theory to practical problems, a couple of remarkable examples were worked out, both for high-frequency cutoff amplifiers and for low-frequency ones. A special consideration deserves a mysteriously exceptional amplifier with best stability behavior introduced by Feynman, for which different physical interpretations are here envisaged. Feynman's earlier work then later flowed in the Hughes lectures on Mathematical Methods in Physics and Engineering of 1970-71, where he also remarked on causality properties of an amplifier, that is on certain relations between frequency and phase shift that a real amplifier has to satisfy in order not to allow output signals to appear before input ones. Quite interestingly, dispersion relations to be satisfied by the response function were introduced.

Enhancing teaching and learning of fluid mechanics with interactive computational modelling

Contemporary science, technology, engineering and mathematics (STEM) increasingly require advanced knowledge about mathematical physics models and methods of scientific computation. In STEM educational environments this entails the necessity to implement pedagogical curricula and methodologies that epistemologically balance the inclusion of interactive engagement sequences of computational modelling activities, and offer students opportunities to develop meaningful knowledge of physics, mathematics and scientific computation, as well as of the specific STEM concepts and processes. Our approach in this context is based on interactive engagement activities built around computational modelling experiments implemented in the Modellus environment that span the range of different kinds of modelling, from exploratory to expressive modelling. In this paper we describe research concerning a sequence of activities about hydrostatic pressure forces and torques, a theme of an introductory fluid mechanics course for undergraduate university engineering students having only elementary knowledge of secondary education physics and mathematics and no significant prior knowledge about scientific computation. We analyse student’s perceptions about the activities and the effects generated on the learning process. Using a Likert scale questionnaire, we show that students reacted positively to the Modellus based activities, considering them helpful in the learning process of the mathematics and physics of fluid mechanics and for their overall professional training. The results also show that students considered Modellus a useful software for computational activities that help the learning of mathematical physics models, sufficiently easy to learn and user-friendly. Based on the analysis of student’s work content, we show that students were able to construct and explore the proposed mathematical physics models and simulations, and establish meaningful and operationally reified relations with the relevant hydrostatic phenomena. We also show that the computational modelling activities were effective in resolving several difficulties persisting after theoretical lectures and problem-solving paper and pen activities.

A Proof of First Digit Law from Laplace Transform**Supported by the National Natural Science Foundation of China under Grant No 11475006.

The first digit law, also known as Benford’s law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form log(1 + 1/d), where d = 1, 2,…, 9. Such a law has been elusive for over 100 years because it has been obscure whether this law is due to the logical consequence of the number system or some mysterious mechanism of nature. We provide a simple and elegant proof of this law from the application of the Laplace transform, which is an important tool of mathematical methods in physics. It is revealed that the first digit law originates from the basic property of the number system, thus it should be attributed as a basic mathematical knowledge for wide applications.

Predictive control modeling of ADS’s MEBT using BPNN to reduce the impact of noise on the control system

In the beam debugging of China ADS Injector II, the mathematical matrix operation, software simulation or physical derivation methods are mainly used. These methods do not take into account noise conditions, so it is difficult to accurately control the deflection position of the beam. For this reason, this paper takes MEBT as an example to establish a function model of quadrupole magnet control parameter and beam position, and based on the BPNN to fit the function model to reduce the influence of noise on the control process, and then realizes control prediction model in beam debugging. The experiments results show that the prediction accuracy of the quadrupole magnet control voltages Q2 and Q3 are 0.9072 and 0.9352, respectively, and the MSE is 0.0064. This paper is the first attempt of deep neural network in China ADS injector II, which provides a new method for beam debugging of the system.

Production forecast of a multistage fractured horizontal well by an analytical method in shale gas reservoir

Industry benefits cannot be obtained from shale gas reservoir without stimulations, due to the ultra-low porosity and permeability of shale. A series of integrated technical measures have been developed for developing shale gas economically, such as horizontal wells and multistage hydraulic fracturing. Combining the above works, we can achieve higher productivity by enlarging stimulated reservoir volume (SRV) and linking fracture network in shale gas reservoirs. In this paper, a novel analytical mathematical model for production forecast of multistage horizontal well was developed based on the seepage theory of fractured well in the dual-medium gas reservoirs. In this model, multi-scale migration mechanism and the complicated morphology of hydraulic fractures in fractured shale gas reservoir were considered. It has been closely solved by the method of well test analysis and mathematical physics. To validate the accuracy of the model in this paper, a well from Changning–Weiyuan shale gas reservoir in China is taken as a real-case application. The calculation results of the model and the actual production data of the well are in good accordance. Meanwhile, the impacts of sensitive factors including desorption, Knudsen diffusion, slip flow, stress sensitivity of micro-fractures and high-velocity non-Darcy flow within hydraulic fractures on cumulative production were analyzed. At last, fracture morphology has been optimized through the model.

Quantum Calcium-Ion Interactions with EEG

Background: Previous papers have developed a statistical mechanics of neocortical interactions (SMNI) fit to short-term memory and EEG data. Adaptive Simulated Annealing (ASA) has been developed to perform fits to such nonlinear stochastic systems. An N-dimensional path-integral algorithm for quantum systems, qPATHINT, has been developed from classical PATHINT. Both fold short-time propagators (distributions or wave functions) over long times. Previous papers applied qPATHINT to two systems, in neocortical interactions and financial options. Objective: In this paper the quantum path-integral for Calcium ions is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Using fits of this SMNI model to EEG data, including these effects, will help determine if this is a reasonable approach. Method: Methods of mathematical-physics for optimization and for path integrals in classical and quantum spaces are used for this project. Studies using supercomputer resources tested various dimensions for their scaling limits. In this paper the quantum path-integral is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Results: The mathematical-physics and computer parts of the study are successful, in that there is modest improvement of cost/objective functions used to fit EEG data using these models. Conclusions: This project points to directions for more detailed calculations using more EEG data and qPATHINT at each time slice to propagate quantum calcium waves, synchronized with PATHINT propagation of classical SMNI.

DETERMINATION OF EQUIVALENT QUANTITIES OF ELECTROMAGNETIC FIELD IN MASSIVE FERROUS DETAILS OF ELECTRIC EQUIPMENT UNDER STRONGLY VARYING LOADS

Purpose. To derive the relationships for calculating determination of equivalent quantities for electromagnetic field in massive ferrous bodies at strongly varying loads.Methodology. We have applied methodological fundamentals of skin-effect theory for massive ferrous bodies combined with methods of mathematical physics as well as physical and computer simulation.Findings. We have found the mathematical expressions for the equivalence ratios allowing calculate basic electromagnetic field quantities, such as magnetic flux and losses, in massive ferrous bodies under strongly varying loads. Their adequacy is confirmed by research on physical and computer models, as well as existing computational procedures of the additional losses in the massive units of energy-intensive electrical equipment for special purposes.Originality. For the first time, we have established mathematical dependencies between skin losses and magnetic flux in massive ferrous body and total and individual harmonic distortion of periodically varying magnetic and electric fields. Found dependences are the basis for the development of refined computational procedures for determining the parameters of the electromagnetic field in massive ferromagnetic elements of electrical equipment under strongly-varying loads.Practical value. The use of derived expressions for the equivalence ratios enable successfully solve the problems aimed to optimize constructive components when designing electrical equipment for supplying consumers with strongly-varying behavior of the load.

Modeling of multiphase flow at solid particles hydropelling

Purpose. Development of methods for calculating the parameters of the solid particles lifting from the bottom of reservoirs based on the three-phase (gas-liquid-solid particles) flow simulation in an air-lift pipe. Methodology. Mathematical modeling of multiphase flows based on hydraulic ratios. In doing so, heights of up to 10 m (short airlift) are considered with corresponding physical assumptions. Findings. A methodological support has been developed for calculating rational parameters of modes of transporting solid particles with a short airlift. The solution of the problem of the solid particles hydraulic lift by a short airlift is reduced to solving a system of nonlinear algebraic equations. Parametric calculations were carried out and the dependences of the gas flow rate required to ensure a given airlift performance were determined, and the influence of the geometric characteristics of the pipelines on the flow of the solid phase was estimated. Originality. The proposed method allows one to determine the parameters of the solid particles hydraulic lift in a short airlift by solving a system of nonlinear algebraic equations, without using the apparatus of mathematical physics methods. Practical value. It consists in the development of a method for calculating and determining rational parameters of pulp transportation processes in short airlifts. These methods make it possible to determine the dependence of the air flow on the solid particles flow and substantiate the corresponding geometric and flow parameters of the hydraulic lift system. Key words: short airlift, multiphase flow, solid particles hydraulic lift, technological calculatio.

Quantum Calcium-Ion Interactions with EEG

Background: Previous papers have developed a statistical mechanics of neocortical interactions (SMNI) fit to short-term memory and EEG data. Adaptive Simulated Annealing (ASA) has been developed to perform fits to such nonlinear stochastic systems. An N-dimensional path-integral algorithm for quantum systems, qPATHINT, has been developed from classical PATHINT. Both fold short-time propagators (distributions or wave functions) over long times. Previous papers applied qPATHINT to two systems, in neocortical interactions and financial options. Objective: In this paper the quantum path-integral for Calcium ions is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Using fits of this SMNI model to EEG data, including these effects, will help determine if this is a reasonable approach. Method: Methods of mathematical-physics for optimization and for path integrals in classical and quantum spaces are used for this project. Studies using supercomputer resources tested various dimensions for their scaling limits. In this paper the quantum path-integral is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Results: The mathematical-physics and computer parts of the study are successful, in that there is modest improvement of cost/objective functions used to fit EEG data using these models. Conclusions: This project points to directions for more detailed calculations using more EEG data and qPATHINT at each time slice to propagate quantum calcium waves, synchronized with PATHINT propagation of classical SMNI.

Quantum Calcium-Ion Interactions with EEG

Background: Previous papers have developed a statistical mechanics of neocortical interactions (SMNI) fit to short-term memory and EEG data. Adaptive Simulated Annealing (ASA) has been developed to perform fits to such nonlinear stochastic systems. An N-dimensional path-integral algorithm for quantum systems, qPATHINT, has been developed from classical PATHINT. Both fold short-time propagators (distributions or wave functions) over long times. Previous papers applied qPATHINT to two systems, in neocortical interactions and financial options. Objective: In this paper the quantum path-integral for Calcium ions is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Using fits of this SMNI model to EEG data, including these effects, will help determine if this is a reasonable approach. Method: Methods of mathematical-physics for optimization and for path integrals in classical and quantum spaces are used for this project. Studies using supercomputer resources tested various dimensions for their scaling limits. In this paper the quantum path-integral is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Results: The mathematical-physics and computer parts of the study are successful, in that there is modest improvement of cost/objective functions used to fit EEG data using these models. Conclusions: This project points to directions for more detailed calculations using more EEG data and qPATHINT at each time slice to propagate quantum calcium waves, synchronized with PATHINT propagation of classical SMNI.

Mathematical Physics:With Applications, Problems & Solutions

This is a very unusual book, 852 pages long with 32 chapters. The author, V. Balakrishnan is an eminent theoretical physicist who has inspired a generation of students at IIT Madras over more than three decades. Bala, as popularly known amongst friends, is well known for his breadth of knowledge in various areas of theoretical physics. There are very few theoretical physicists who have written important papers in widely different areas like high energy physics, condensed matter physics, equilibrium and non-equilibrium statistical mechanics and nonlinear dynamics as Bala has done. So when I heard of a book on mathematical physics by Bala, I was curious about its contents. After carefully going through it, I can only compare it with some of the classics of mathematical physics like Methods of Theoretical Physics , vols I and II by Morse and Feshback, and Methods of Mathematical Physics , vols I and II by Courant and Hilbert. And I am happy to note that it is of similar level, but very different from the other two books. It is not only different in its content (which is to be expected since many new topics have become important in the last 50 years or so), but more important in its emphasis. In particular, this book assigns a prominent role to the applications of the relevant mathematics to different areas of physics ranging from fluid dynamics, electromagnetic theory, quantum mechanics, special theory of relativity, quantum optics, random processes, linear response theory, and so on. The emphasis in the book is not on formal proofs but rather on motivating and elaborating the results and even more important, discussing the relevance of the results in different areas of physics. Bala knows very well that one can never learn mathematical physics (actually even theoretical physics) without solving problems. With this in mind, he has given about 370 problems, many of them with several parts and sub-parts. He has made the problems contiguous with the text and has provided solutions to them either in the outline or in detail at the end of each chapter. This I consider as an important aspect of the book. I believe that over the years this book will occupy a place similar to the other classics in the field, like the two books mentioned above.

Mathematical Methods in Engineering and Physics

Mathematical Methods in Engineering and Physics

A Block Diagram of Electromagnetoelastic Actuator Nanodisplacement for Communications Systems

The parametric block diagram of the electromagnetoelastic actuator nanodisplacement or the piezoactuator is determined in contrast the electrical equivalent circuit types Cady or Mason for the calculation of the piezoelectric transmitter and receiver, the vibration piezomotor with the mechanical parameters in form the velosity and the pressure. The method of mathematical physics is used. The parametric block diagram of electromagnetoelastic actuator is obtained with the mechanical parameters the displacement and the force. The transfer functions of the electroelastic actuator are determined. The the generalized parametric block diagram, the generalized matrix equation for the electromagnetoelastic actuator nanodisplacement are obtained. The deformations of the electroelastic actuator for the nanotechnology are described by the matrix equation. Block diagram and structural-parametric model of electromagnetoelastic actuator nanodisplacement for nanodisplacement of the communications systems are obtained, its transfer functions are bult. Effects of geometric and physical parameters of electromagnetoelastic actuators and external load on its dynamic characteristics are determined. For calculations the communications systems with the piezoactuator for nanodisplacement the parametric block diagram and the transfer functions of the piezoactuator are obtained.

Analyzing linear spatial features in ecology.

The spatial analysis of dimensionless points (e.g., tree locations on a plot map) is common in ecology, for instance using point-process statistics to detect and compare patterns. However, the treatment of one-dimensional linear features (fiber processes) is rarely attempted. Here we appropriate the methods of vector sums and dot products, used regularly in fields like astrophysics, to analyze a data set of mapped linear features (logs) measured in 12×1-ha forest plots. For this demonstrative case study, we ask two deceptively simple questions: do trees tend to fall downhill, and if so, does slope gradient matter? Despite noisy data and many potential confounders, we show clearly that topography (slope direction and steepness) of forest plots does matter to treefall. More generally, these results underscore the value of mathematical methods of physics to problems in the spatial analysis of linear features, and the opportunities that interdisciplinary collaboration provides. This work provides scope for a variety of future ecological analyzes of fiber processes in space.

Electroelastic Actuator Nano- and Microdisplacement for Precision Mechanics

In the present work the structural-parametric model of the piezoactuator is determined in contrast electrical equivalent circuit types Cady or Mason for the calculation of the piezoelectric transmitter and receiver, the vibration piezoactuator and the vibration piezomotor with the mechanical parameters in form the velosity and the pressure. The aim of this work is to obtain the structural-parametric model of the electroelastic actuator with the mechanical parameters the displacement and the force. The method of mathematical physics is used. Structural scheme of electroelastic actuator for nanotechnology is obtained. The transfer functions of the actuators are determined. For calculations control systems for nanotechnology with piezoactuator the structural scheme and the transfer functions of piezoactuator are obtained. The generalized structural-parametric model, the generalized structural scheme, the generalized matrix equation for the electroelastic actuator nano- and microdisplacement are obtained in the matrix form. The deformations of the electroelastic actuator for the precision mechanics are described by the matrix equation.

Parametric structural schematic diagram of electroelastic actuator in nanotechnology.

In this work the parametric structural schematic diagram of the electroelastic actuator or the piezoactuator is determined in contrast the electrical equivalent circuit types Cady or Mason for the calculation of the piezoelectric transmitter and receiver, the vibration piezomotor with the mechanical parameters in form the velocity and the pressure. The method of mathematical physics is used. The parametric structural schematic diagram of electroelastic actuator is obtained with the mechanical parameters the displacement and the force. The transfer functions of the electroelastic actuator are determined. The generalized parametric structural schematic diagram, the generalized matrix equation for the electroelastic actuator nano displacement are obtained. The deformations of the electroelastic actuator for the nanotechnology are described by the matrix equation.

Quantum Variables in Finance and Neuroscience II

Background: A path-integral algorithm, PATHINT used previously for several systems, has been generalized from 1 dimension to N dimensions, and from classical to quantum systems into qPATHINT. Previous publications applied qPATHINT to two systems developed by the author, in neocortical interactions and financial options. Also, previous publications using classical PATHINT have developed a statistical mechanics of neocortical interactions (SMNI) that has been fitted to EEG data under attentional experimental paradigms. Classical PATHINT also has been published demonstrating development of Eurodollar options in industrial applications. Objective: A study is required to see if the qPATHINT algorithm can scale sufficiently to further develop real-world calculations in these two systems, requiring interactions between classical and quantum scales. A new algorithm is needed to develop interactions between classical and quantum scales. Method: Both systems are developed using mathematical-physics methods of path integrals in quantum spaces. Supercomputer pilot studies using XSEDE.org resources tested various dimensions for their scaling limits. For the financial options study, all traded Greeks are calculated for Eurodollar options in quantum-money spaces. For the neuroscience study, tripartite neuron-astrocyte-neuron Ca-ion waves are propagated for 100's of msec. For the neuroscience system, the quantum path-integral is used to derive a closed-form analytic solution at arbitrary time, in the absence of shocks, that is used to calculate classical-physics interactions among scales. Results: The mathematical-physics and computer parts of the study are successful for both systems. A 3-dimensional path-integral propagation of qPATHINT for both systems is within normal computational bounds on supercomputers. The neuroscience quantum path-integral also has a closed solution at arbitrary time that tests the multiple-scale model including the quantum scale. Conclusion: Each of the two systems considered contribute insight into applications of qPATHINT to the other system, leading to new algorithms presenting time-dependent propagation of interacting quantum and classical scales. This can be achieved by propagating qPATHINT and PATHINT in synchronous time for the interacting systems, which is a future set of studies.

Serial, Anthropomorphic, Spatial, Mechatronic Systems Can Be Studied More Simply in a Plan

The mobile, mechatronic, robotic, serial, spatial, anthropomorphic type systems, which are currently the most used in the machine building industry, can be studied much more simply in a plan instead of the usual spatial study. This not only simplifies the understanding of these systems (including from a didactic point of view) but also facilitates computational methods, moving from matrix analytical methods to more simple classical methods. Usage is done by conversion, so nothing is lost from the essence of physic-mathematical phenomena. The idea of moving from spatial to planar study has been centered over time on all major mechanisms when it was possible, precisely for ease of calculation and working methods, but also for a better understanding of physical phenomena. The vast majority of classical mechanisms can be treated as they move in most cases into a master plan. This is the case with known mechanical transmissions, classic motors of all types, working mechanisms, engine or lucrative cars, etc. In anthropomorphic robots, the method is no longer used because they work clearly in a well-defined space. We used the idea to divide this space into a main work plan that can rotate around a main axis so that the study of all movements is done in the workplan and then the rotations of these parameters corresponding to the rotation of the work plane around the main axis of rotation. The physical-mathematical methods are greatly simplified in this way, from matrix difficult calculations to simple, classical analytical calculation methods.

Investigation on Releasing of a Stuck Drill String by Means of a Mechanical Jar

Purpose. In this article the most important part is dedicated to the research of elimination of accident that is caused by drill string sticking during the process. That is why it is necessary to develop a mathematical model of the mechanic system: travelling system + drill string + mechanical jar + rock, to develop a computer model for numerical calculation of dynamic characteristics of firing gear. The aim is to use the results of the research and to work out recommendations for expediency of jar application. Methods. For description of the drill string we are using synthesis of the wave theory and theory of the local distortions. For mathematical modeling of firing device we are offering the use of the combined method that combines static solutions of the theory of elasticity for contact zone of drill string and method of a plain wave of Saint-Venant. We solved systems of differential equations using the methods of mathematical physics. An algorithm of the numerical decision which mounted in the computing environment were used at simulation of the longitudinal impact to the stuck drill pipe. In this article we designed a wave chart of the equation system of the drill pipe and conducted step-by-step calculation of a collision momentum. We also designed a computer program for numerical modeling of the drill pipe mechanism with firing gear. We also designed a method of calculation of main dynamic characteristics of firing device that will help analyze and prove the performance of the mechanical jar. A wave diagram was built that shows the impact forces and speeds on the boundary surfaces of the sections of the drill string. There were calculated main dynamic characteristic of mechanical the jar. Originality. Authors also developed a dynamic mathematical model that combined elastic vibrations of continual system of loose part drill pipe, impact mechanisms and discrete movements of a given drill pipe. The process of a mechanical jar work that considers elastic deformations of contact areas was not published by researchers. Practical implications. Received results could be used for further research and perfection of existing engineering methods of modeling and calculations of drill pipes at the stage of their designing and their construction.

Euclidian and fractal geometry applied to the diagnosis of lesions from cervical cells

A diagnostic methodology for preneoplastic and neoplastic lesions of cervical cells from simultaneous fractals and euclidean measures was developed. In this work the objective was to confirm the diagnostic concordance of the methodology in normal cells and in different stages of progression to disease. For this, 60 photographs of cervical squamous epithelium cells were taken: 10 normal, 10 ASCUS, 20 with low grade – squamous intraepithelial lesion (L-SIL) and 20 high-grade lesion (H-SIL). Measures of fractal dimension and space occupancy of the surface and the edge of the nucleus and cytoplasm were performed in Box Counting Fractal space, establishing their physical-mathematical diagnosis. Measurements of the nucleus surface for normality were between 305 and 651, for ASCUS between 1293 and 4588, for L-SIL, between 986 and 4873 and for H-SIL between 567 and 2311. The subtraction of Cytoplasm-Nucleus border was found between 238 and 477 for normality; between 185 and 417 for ASCUS; between 131 and 342 for L-SIL and between 43 and 117 for H-SIL, finding sensitivity and specificity of 100%, likelihood ratio of 0 and kappa coefficient of 1. Conclusions: the diagnostic concordance at clinical level of the physical-mathematical method was confirmed, objective and reproducible quantifying the degree of injury of cervical cells and establishing an objective diagnosis for ASCUS cells from simultaneous fractals and euclidean measures, which improves qualitative classification methods.