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Journal articles on the topic 'Integral equation theories'

Author: Grafiati

Published: 7 December 2024

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1

Song, Junyan. "On Diophantine equation xb=Dy." Theoretical and Natural Science 13, no. 1 (November 30, 2023): 232–36. http://dx.doi.org/10.54254/2753-8818/13/20240852.

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Diophantine equation is an important part of the number theory and it has been widely studied for a long time. There are many studies on solving the integral solutions of Diophantine equations using algebraic methods. This paper uses documentation method, sums up the results of research in different documents of finding the integral solutions of some Diophantine equations in various conditions, especially the utilization of theories on Pell equation, which in the form of xb=Dy. This paper mainly considers the situations when a =2 or a=3, which is a particular type of Diophantine equation. Several of the studies focus on the same equation but using different ways. Also, some subsequent studies on different equations are based on the former theorems provided by other writers and expand these theorems to broader applications. In order to emphasize the theorems obtained by the essays quoted and the methods the authors used, most of the mathematical procedures in their proofs are omitted.

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2

Cann, N. M. "Improvement of integral equation theories for mixtures." Journal of Chemical Physics 110, no. 23 (June 15, 1999): 11466–83. http://dx.doi.org/10.1063/1.479088.

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3

Paci, I., and N. M. Cann. "Integral equation theories for orientionally ordered fluids." Journal of Chemical Physics 119, no. 5 (August 2003): 2638–57. http://dx.doi.org/10.1063/1.1585017.

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4

Pádua, A. A. H., and J. P. M. Trusler. "Application of integral equation theories to the nitrogen molecule." Journal of Chemical Physics 105, no. 14 (October 8, 1996): 5956–67. http://dx.doi.org/10.1063/1.472436.

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5

Liu, Yi, and Toshiko Ichiye. "Integral equation theories for predicting water structure around molecules." Biophysical Chemistry 78, no. 1-2 (April 1999): 97–111. http://dx.doi.org/10.1016/s0301-4622(99)00008-3.

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6

El-Karamany, A. S. "Boundary Integral Equation Formulation in Generalized Linear Thermo-Viscoelasticity With Rheological Volume." Journal of Applied Mechanics 70, no. 5 (September 1, 2003): 661–67. http://dx.doi.org/10.1115/1.1607354.

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A general model of generalized linear thermo-viscoelasticity for isotropic material is established taking into consideration the rheological properties of the volume. The given model is applicable to three generalized theories of thermoelasticity: the generalized theory with one (Lord-Shulman theory) or with two relaxation times (Green-Lindsay theory) and with dual phase-lag (Chandrasekharaiah-Tzou theory) as well as to the dynamic coupled theory. The cases of thermo-viscoelasticity of Kelvin-Voigt model or thermoviscoelasticity ignoring the rheological properties of the volume can be obtained from the given model. The equations of the corresponding thermoelasticity theories result from the given model as special cases. A formulation of the boundary integral equation (BIE) method, fundamental solutions of the corresponding differential equations are obtained and an example illustrating the BIE formulation is given.

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7

Gumede, Sfundo C., Keshlan S. Govinder, and Sunil D. Maharaj. "First Integrals of Shear-Free Fluids and Complexity." Entropy 23, no. 11 (November 19, 2021): 1539. http://dx.doi.org/10.3390/e23111539.

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A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of yxx=f(x)y2, find new solutions, and generate a new first integral. The first integral is subject to an integrability condition which is an integral equation which restricts the function f(x). We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. The solution of the integrability condition is generally given parametrically. A particular form of f(x)∼1x51−1x−15/7 which corresponds to repeated roots of a cubic equation is given explicitly, which is a new result. Our investigation demonstrates that complexity of a self-gravitating shear-free fluid is related to the existence of a first integral, and this may be extendable to general matter distributions.

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8

Tejero, C. F., and E. Lomba. "Density-dependent interactions and thermodynamic consistency in integral equation theories." Molecular Physics 107, no. 4-6 (February 20, 2009): 349–55. http://dx.doi.org/10.1080/00268970902776765.

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9

Pellicane, Giuseppe, Lloyd L. Lee, and Carlo Caccamo. "Integral-equation theories of fluid phase equilibria in simple fluids." Fluid Phase Equilibria 521 (October 2020): 112665. http://dx.doi.org/10.1016/j.fluid.2020.112665.

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10

Huš, Matej, Matja Zalar, and Tomaz Urbic. "Correctness of certain integral equation theories for core-softened fluids." Journal of Chemical Physics 138, no. 22 (June 14, 2013): 224508. http://dx.doi.org/10.1063/1.4809744.

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11

Zhang, Jian, Jinjiao Hou, Jing Niu, Ruifeng Xie, and Xuefei Dai. "A high order approach for nonlinear Volterra-Hammerstein integral equations." AIMS Mathematics 7, no. 1 (2021): 1460–69. http://dx.doi.org/10.3934/math.2022086.

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<abstract><p>Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.</p></abstract>

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12

Malescio, G., and P. V. Giaquinta. "Structural stability of simple fluids and accuracy of integral-equation theories." Physical Review E 62, no. 3 (September 1, 2000): 4439–41. http://dx.doi.org/10.1103/physreve.62.4439.

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13

WIECHEN, Jochen. "The Application of Recent Integral Equation Theories to Molten Salt Mixtures." Zeitschrift für Physikalische Chemie 156, Part_2 (January 1988): 671–75. http://dx.doi.org/10.1524/zpch.1988.156.part_2.671.

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14

Wiechen, J. "On the description of molten sat mixtures by integral equation theories." Journal of Physics C: Solid State Physics 18, no. 24 (August 30, 1985): L717—L724. http://dx.doi.org/10.1088/0022-3719/18/24/002.

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15

Choybekov, S. "Regularization of the Solution of Nonclassical Linear Volterra Equations of the First Kind With Initial Condition." Bulletin of Science and Practice, no. 4 (April 15, 2023): 13–21. http://dx.doi.org/10.33619/2414-2948/89/01.

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Models of many problems of an applied nature are reduced to equations by an integral equation, among which non-classical equations are of special interest and little studied. Integral equations play an important role in the section of the integrodifferential equation. With the help of them, modern sciences and technologies are developing, i.e. they are widely used in the branches of mathematics, are used in physics, in mechanical engineering, in radio engineering, in computer technology, geophysics, control theory, etc. New areas related to the application of integral equations are developing, for example, economic sciences, some sections of biology and management, etc. With the help of modern computer technology, it becomes possible to implement a variety of numerical theories and simulate complex processes. In the same way, many problems are brought to integral equations. In this case, a qualitative study of problem solving comes to the fore. However, equations with two variable limits of integration, which are called non-classical, are poorly understood. This is due to difficulties in constructing a resolvent and in compiling a relation for it, because an analytical representation in general has not yet been obtained, with the exception of some model cases. Therefore, such research decisions are relevant. In this paper, the solution and regularization of the nonlinear nonclassical Volterra integral equation of the first kind is considered. The linear nonclassical Volterra integral equation of the first kind is solved using a derivative and is determined by regularization. The theorem is formulated by the proven fact. An appropriate example will be used, which will fully reveal the solution and evaluation.

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16

Bonotto, Everaldo M., Felipe Federson, and Márcia Federson. "The Schrödinger Equation, Path Integration and Applications." Proceedings of the Singapore National Academy of Science 15, no. 01 (March 2021): 61–75. http://dx.doi.org/10.1142/s259172262140007x.

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The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.

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17

KHORDAD, REZA, MEHRAN MOHEBBI, ABOLLA KESHAVARZI, AHMAD POOSTFORUSH, and FARNAZ GHAJARI HAGHIGHI. "THE STUDY OF GAY–BERNE FLUID: INTEGRAL EQUATIONS METHOD." International Journal of Modern Physics B 23, no. 05 (February 20, 2009): 753–69. http://dx.doi.org/10.1142/s0217979209051991.

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We study a classical fluid of nonspherical molecules. The components of the fluid are the ellipsoidal molecules interacting through the Gay–Berne potential model. A method is described, which allows the Percus–Yevick (PY) and hypernetted-chain (HNC) integral equation theories to be solved numerically for this fluid. Explicit results are given and comparisons are made with recent Monte Carlo (MC) simulations. It is found that, at lower cutoff l max , the HNC and the PY closures give significantly different results. The HNC and PY (approximately) theories, at higher cutoff l max , are superior in predicting the existence of the phase transition in a qualitative agreement with computer simulation.

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18

Kinoshita, M., and F. Lado. "Numerical solution of structure integral equation theories for two-dimensional fluid mixtures." Molecular Physics 83, no. 2 (October 10, 1994): 351–59. http://dx.doi.org/10.1080/00268979400101311.

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19

Lue, Leo, and Daniel Blankschtein. "Application of integral equation theories to predict the structure of diatomic fluids." Journal of Chemical Physics 102, no. 10 (March 8, 1995): 4203–16. http://dx.doi.org/10.1063/1.469468.

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20

Kast, Stefan M., and Daniel Tomazic. "Communication: An exact bound on the bridge function in integral equation theories." Journal of Chemical Physics 137, no. 17 (November 7, 2012): 171102. http://dx.doi.org/10.1063/1.4766465.

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21

Webb, G. M., R. H. Burrows, X. Ao, and G. P. Zank. "Ion acoustic traveling waves." Journal of Plasma Physics 80, no. 2 (January 15, 2014): 147–71. http://dx.doi.org/10.1017/s0022377813001013.

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AbstractModels for traveling waves in multi-fluid plasmas give essential insight into fully nonlinear wave structures in plasmas, not readily available from either numerical simulations or from weakly nonlinear wave theories. We illustrate these ideas using one of the simplest models of an electron–proton multi-fluid plasma for the case where there is no magnetic field or a constant normal magnetic field present. We show that the traveling waves can be reduced to a single first-order differential equation governing the dynamics. We also show that the equations admit a multi-symplectic Hamiltonian formulation in which both the space and time variables can act as the evolution variable. An integral equation useful for calculating adiabatic, electrostatic solitary wave signatures for multi-fluid plasmas with arbitrary mass ratios is presented. The integral equation arises naturally from a fluid dynamics approach for a two fluid plasma, with a given mass ratio of the two species (e.g. the plasma could be an electron–proton or an electron–positron plasma). Besides its intrinsic interest, the integral equation solution provides a useful analytical test for numerical codes that include a proton–electron mass ratio as a fundamental constant, such as for particle in cell (PIC) codes. The integral equation is used to delineate the physical characteristics of ion acoustic traveling waves consisting of hot electron and cold proton fluids.

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22

Strochkov, I. A., and A. A. Khvattcev. "Solution of Linearized Flat Problem of Hydrodynamics (IVF)." Environment. Technology. Resources. Proceedings of the International Scientific and Practical Conference 2 (August 8, 2015): 85. http://dx.doi.org/10.17770/etr2013vol2.864.

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In the present paper a method of the generalized potential to planes is applied for the solution of the linearized according to Oseen of flat problem of hydrodynamics incompressible viscous fluid (IVF). Generalized potential simple layer containing McDonald function serves kernel for generalized potential to planes. For finding of an unknown density of the potential simple layer is received linear integral equation, containing double integral from curvilinear integral along border of the streamlined area. Sharing the pressure is in turn defined by potential simple layer with density of the potential, determined by linear integral equation, hanging from solution specified above integral equation. The offered method of the successive iterations, allowing elaborate the solution of the problem before achievement given to accuracy. As example of exhibit to theories is considered solution of the problem theory of hydrodynamic greasing.

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23

Tamba, Gordien Of’rI’shii. "Sur Une Des Liaisons Des Equations D’euler Lagrange A Celles De Hamilton Par Le Théorème De L. Noether." International Journal of Modern Statistics 4, no. 1 (October 16, 2024): 33–48. http://dx.doi.org/10.47941/ijms.2294.

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In this paper, we have traced some theories in physics which are described by the Lagrangian, by the associated Hamiltonian. Then, we made the connection between the Euler-Lagrange equations to those of Hamilton basing on a result we got : « the functions and are reciprocal diffeomorphisms », L represents the Lagrangian of a phenomen. On his the associated Hamiltonian and respectively the generalized coordinate of phenomenon and the conjugate momentum of the Lagrangian with respect to By proving that , which is a solution of the Euler-Lagrange equation, is also a solution of Hamilton equations therefore a first integral. We have joined Noether’s theorem which states that is a first integral where W is an infinitesimal symmetry of the Lagrangian.

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24

ESPOSITO, GIAMPIERO, DIEGO N. PELLICCIA, and FRANCESCO ZACCARIA. "GRIBOV PROBLEM FOR GAUGE THEORIES: A PEDAGOGICAL INTRODUCTION." International Journal of Geometric Methods in Modern Physics 01, no. 04 (August 2004): 423–41. http://dx.doi.org/10.1142/s0219887804000216.

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The functional-integral quantization of non-Abelian gauge theories is affected by the Gribov problem at non-perturbative level: the requirement of preserving the supplementary conditions under gauge transformations leads to a nonlinear differential equation, and the various solutions of such a nonlinear equation represent different gauge configurations known as Gribov copies. Their occurrence (lack of global cross-sections from the point of view of differential geometry) is called Gribov ambiguity, and is here presented within the framework of a global approach to quantum field theory. We first give a simple (standard) example for the SU(2) group and spherically symmetric potentials, then we discuss this phenomenon in general relativity, and recent developments, including lattice calculations.

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25

DUH, By DER-MING, and DOUGLAS HENDERSON LUIS MIER-Y.-TERAN SOKOLOWSKI. "Application of some second-order integral equation theories to the Lennard-Jones fluid." Molecular Physics 90, no. 4 (March 1997): 563–70. http://dx.doi.org/10.1080/002689797172273.

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26

Ferrari, Franco, and Marcin Pia̧tek. "On a singular Fredholm-type integral equation arising inN=2super-Yang–Mills theories." Physics Letters B 718, no. 3 (January 2013): 1142–47. http://dx.doi.org/10.1016/j.physletb.2012.11.069.

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27

Yu, Hsiang Ai, Martin Karplus, and B. Montgomery Pettitt. "Aqueous solvation of N-methylacetamide conformers: comparison of simulations and integral equation theories." Journal of the American Chemical Society 113, no. 7 (March 1991): 2425–34. http://dx.doi.org/10.1021/ja00007a012.

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28

Lozada‐Cassou, Marcelo, and Enrique Díaz‐Herrera. "Three point extension for hypernetted chain and other integral equation theories: Numerical results." Journal of Chemical Physics 92, no. 2 (January 15, 1990): 1194–210. http://dx.doi.org/10.1063/1.458128.

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29

Ramírez, Andrés A., and Francisco Jurado. "The Regulator Problem to the Convection–Diffusion Equation." Mathematics 11, no. 8 (April 20, 2023): 1944. http://dx.doi.org/10.3390/math11081944.

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In this paper, from linear operator, semigroup and Sturm–Liouville problem theories, an abstract system model for the convection–diffusion (C–D) equation is proposed. The state operator for this abstract system model is here defined as given in the form of the Sturm–Liouville differential operator (SLDO) plus an integral term of the same SLDO. Our aim is to achieve the trajectory tracking task in the presence of external disturbances to the C–D equation invoking the regulator problem theory, where the state from a finite-dimensional exosystem is the state to the feedback law. In this context, the regulator (Francis) equations, established from the abstract system model for the C–D equation, here are solved; i.e., the state feedback regulator problem (SFRP) for the C–D system has a solution. Our proposal is validated via numerical simulation results.

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30

Eggebrecht, J., K. E. Gubbins, and S. M. Thompson. "The liquid–vapor interface of simple polar fluids. I. Integral equation and perturbation theories." Journal of Chemical Physics 86, no. 4 (February 15, 1987): 2286–98. http://dx.doi.org/10.1063/1.452127.

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31

Urbic, T., and M. F. Holovko. "Mercedes–Benz water molecules near hydrophobic wall: Integral equation theories vs Monte Carlo simulations." Journal of Chemical Physics 135, no. 13 (October 7, 2011): 134706. http://dx.doi.org/10.1063/1.3644934.

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32

Fujita, Takatoshi, and Takeshi Yamamoto. "Assessing the accuracy of integral equation theories for nano-sized hydrophobic solutes in water." Journal of Chemical Physics 147, no. 1 (July 7, 2017): 014110. http://dx.doi.org/10.1063/1.4990502.

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33

Perera, A., F. Sokolić, and M. Moreau. "Fluids of linearly fused Lennard‐Jones sites: Comparison between simulations and integral equation theories." Journal of Chemical Physics 97, no. 3 (August 1992): 1969–79. http://dx.doi.org/10.1063/1.463134.

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34

Perera, Aurélien. "Fluids of hard natural and Gaussian ellipsoids: A comparative study by integral equation theories." Journal of Chemical Physics 129, no. 19 (November 21, 2008): 194504. http://dx.doi.org/10.1063/1.3020337.

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35

Cerjan, Charles, Biman Bagchi, and Stuart A. Rice. "A comment on the consistency of truncated nonlinear integral equation based theories of freezing." Journal of Chemical Physics 83, no. 5 (September 1985): 2376–83. http://dx.doi.org/10.1063/1.449281.

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36

Perera, Aurélien, and Bernarda Lovrinčević. "A comparative study of aqueous DMSO mixtures by computer simulations and integral equation theories." Molecular Physics 116, no. 21-22 (June 8, 2018): 3311–22. http://dx.doi.org/10.1080/00268976.2018.1483040.

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37

Zhiming, Ye, and Yeh Kaiyuan. "A Study of Belleville Spring and Diaphragm Spring in Engineering." Journal of Applied Mechanics 57, no. 4 (December 1, 1990): 1026–31. http://dx.doi.org/10.1115/1.2897621.

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This paper deals with the static response of a Belleville spring and a diaphragm spring by using the finite rotation and large deflection theories of a beam and conical shell, and an experimental method as well. The authors propose new mechanical analysis mathematical models. The exact solution of a variable width cantilever beam is obtained. By using the integral equation method and the iterative method to solve the simplified equations and Reissner’s equations of finite rotation and large deflection of a conical shell, this paper has calculated a great number of numerical results. The properties of loads, strains, stresses and displacements, and the distribution rules of strains and stresses of diaphragm springs are investigated in detail by means of the experimental method. The unreasonableness of several assumptions in traditional theories and calculating method is pointed out.

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38

Bassetto, A., and G. Nardelli. "1+1 Dimensional Yang-Mills Theories in Light-Cone Gauge." International Journal of Modern Physics A 12, no. 06 (March 10, 1997): 1075–90. http://dx.doi.org/10.1142/s0217751x97000803.

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In 1+1 dimensions two different formulations exist of SU(N) Yang Mills theories in light-cone gauge; only one of them gives results which comply with the ones obtained in Feynman gauge. Moreover the theory, when considered 1+(D-1) dimensions, looks discontinuous in the limit D = 2. All those features are proven in Wilson loop calculations as well as in the study of the [Formula: see text] bound state integral equation in the large N limit.

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39

Ferrari, Franco, and Marcin Piątek. "On a path integral representation of the Nekrasov instanton partition function and its Nekrasov–Shatashvili limit." Canadian Journal of Physics 92, no. 3 (March 2014): 267–70. http://dx.doi.org/10.1139/cjp-2012-0570.

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In this work we study the Nekrasov–Shatashvili limit of the Nekrasov instanton partition function of Yang–Mills field theories with 𝒩 = 2 supersymmetry and gauge group SU(N). The theories are coupled with fundamental matter. A path integral expression of the full instanton partition function is derived. It is checked that in the Nekrasov–Shatashvili (thermodynamic) limit the action of the field theory obtained in this way reproduces exactly the equation of motion used in the saddle-point calculations.

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40

CHAKRABORTY, SUBENOY. "QUANTUM COSMOLOGY IN ANISOTROPIC COSMOLOGICAL MODELS WITH SCALAR–TENSOR THEORIES." International Journal of Modern Physics D 10, no. 06 (December 2001): 943–56. http://dx.doi.org/10.1142/s0218271801001244.

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This paper deals with quantum cosmological phenomena in anisotropic cosmological models with nonminimally coupled scalor field. With proper transformation of the field variables, the Wheeler–Dewitt (WD) equation looks simple in form and solutions are obtained using separable form of the wave function. Using part integral formulation, the wave function of the Universe has been evaluated by the method of steepest descent. Finally, the causal interpretation has been done using quantum Bohmian trajectories and also we study the classical limit of some particular solutions of these quantum models.

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41

Lue, Leo, and Daniel Blankschtein. "Application of integral equation theories to predict the structure, thermodynamics, and phase behavior of water." Journal of Chemical Physics 102, no. 13 (April 1995): 5427–37. http://dx.doi.org/10.1063/1.469270.

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42

Akiyama, Ryo, Yasuhito Karino, Yasuhiro Hagiwara, and Masahiro Kinoshita. "Remarkable Solvent Effects on Depletion Interaction in Crowding Media: Analyses Using the Integral Equation Theories." Journal of the Physical Society of Japan 75, no. 6 (June 15, 2006): 064804. http://dx.doi.org/10.1143/jpsj.75.064804.

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43

Björling, Mikael, Giuseppe Pellicane, and Carlo Caccamo. "On the application of Flory–Huggins and integral equation theories to asymmetric hard sphere mixtures." Journal of Chemical Physics 111, no. 15 (October 15, 1999): 6884–89. http://dx.doi.org/10.1063/1.479981.

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44

Gazzillo, Domenico. "Fluid–fluid phase separation of nonadditive hard‐sphere mixtures as predicted by integral‐equation theories." Journal of Chemical Physics 95, no. 6 (September 15, 1991): 4565–79. http://dx.doi.org/10.1063/1.461724.

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45

Krienke, H. "Thermodynamical, structural, and dielectric properties of molecular liquids from integral equation theories and from simulations." Pure and Applied Chemistry 76, no. 1 (January 1, 2004): 63–70. http://dx.doi.org/10.1351/pac200476010063.

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A survey is given on our attempts to calculate equilibrium properties of molecular liquids (pure solvents and electrolyte solutions) with the help of spatial pair correlation functions, starting from classical molecular pair interactions. The selection of potential models, especially the influence of molecular polarizability, is discussed as well as the limitations of the different methods of calculation of molecular pair correlation functions (e.g., from molecular and site-site Ornstein-Zernike theories, from MC and from MD simulations). We have performed simulations and integral equation calculations for spatial distribution functions in pure solvents with very low dielectric constants as dioxane and tetrahydrofurane, up to solvents with a very high dielectric constant like n-methylformamide. Ionic solvation is studied in pure solvent systems as well as in solvent mixtures. The general features of ion solvation and association, of the solvent structure around solutes, and their influence on solution properties, are discussed in the framework of the different theoretical approaches.

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46

Kinoshita, M., and D. R. Bérard. "Analysis of the Bulk and Surface-Induced Structure of Electrolyte Solutions Using Integral Equation Theories." Journal of Computational Physics 124, no. 1 (March 1996): 230–41. http://dx.doi.org/10.1006/jcph.1996.0055.

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47

Mannheim, Philip D. "PT symmetry as a necessary and sufficient condition for unitary time evolution." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1989 (April 28, 2013): 20120060. http://dx.doi.org/10.1098/rsta.2012.0060.

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While Hermiticity of a time-independent Hamiltonian leads to unitary time evolution, in and of itself, the requirement of Hermiticity is only sufficient for unitary time evolution. In this paper, we provide conditions that are both necessary and sufficient. We show that symmetry of a time-independent Hamiltonian, or equivalently, reality of the secular equation that determines its eigenvalues, is both necessary and sufficient for unitary time evolution. For any -symmetric Hamiltonian H , there always exists an operator V that relates H to its Hermitian adjoint according to V HV −1 = H † . When the energy spectrum of H is complete, Hilbert space norms 〈 ψ 1 | V | ψ 2 〉 constructed with this V are always preserved in time. With the energy eigenvalues of a real secular equation being either real or appearing in complex conjugate pairs, we thus establish the unitarity of time evolution in both cases. We also establish the unitarity of time evolution for Hamiltonians whose energy spectra are not complete. We show that when the energy eigenvalues of a Hamiltonian are real and complete, the operator V is a positive Hermitian operator, which has an associated square root operator that can be used to bring the Hamiltonian to a Hermitian form. We show that systems with -symmetric Hamiltonians obey causality. We note that Hermitian theories are ordinarily associated with a path integral quantization prescription in which the path integral measure is real, while in contrast, non-Hermitian but -symmetric theories are ordinarily associated with path integrals in which the measure needs to be complex, but in which the Euclidean time continuation of the path integral is nonetheless real. Just as the second-order Klein–Gordon theory is stabilized against transitions to negative frequencies because its Hamiltonian is positive-definite, through symmetry, the fourth-order derivative Pais–Uhlenbeck theory can equally be stabilized.

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48

Xiong, Zonghou. "Symmetry properties of the scattering matrix in 3-D electromagnetic modeling using the integral equation method." GEOPHYSICS 57, no. 9 (September 1992): 1199–202. http://dx.doi.org/10.1190/1.1443334.

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Modeling large three‐dimensional (3-D) earth conductivity structures continues to pose challenges. Although the theories of electromagnetic modeling are well understood, the basic computational problems are practical, involving the quadratically growing requirements on computer storage and cubically growing computation time with the number of cells required to discretize the modeling body.

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49

Bérard, D. R., and G. N. Patey. "The application of integral equation theories to fluids of nonspherical particles near a uniform planar wall." Journal of Chemical Physics 95, no. 7 (October 1991): 5281–88. http://dx.doi.org/10.1063/1.461667.

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50

Guermond, Jean-Luc. "A generalized lifting-line theory for curved and swept wings." Journal of Fluid Mechanics 211 (February 1990): 497–513. http://dx.doi.org/10.1017/s0022112090001665.

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A generalized lifting-line theory is developed in inviscid, incompressible, steady flow for curved, swept wings of large aspect ratio. It is shown in this paper that by using the integral formulation of the problem instead of the partial differential equation formulation, it is possible to circumvent the algebraic complications encountered by the previous approaches using the method of the matched asymptotic expansions. At each approximation order the problem is reduced to inverting a classical Carleman type integral equation. The asymptotic solution in terms of circulation is found up to A−1 and A−1 In (A−1). It is very convenient for illustrating the major three-dimensional effects induced on the flow by curvature and yaw angle. The concept of the finite part integrals, introduced by Hadamard (1932), is shown to be very useful for handling elegantly singularities like 1/x|x| or 1/|x| which occur in the course of our developments. Comparisons of the new, simple approach with lifting-surface theories reveal excellent agreements in terms of circulation. Furthermore, a consistent calculation of the three components of the total force acting on the wing is done in the lifting-line context without re-introducing the inner geometry of the wing.

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