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Journal articles on the topic 'Infinitesimal quantitie'

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Published: 9 March 2023

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1

Kadianakis, Nikos, and Fotios I. Travlopanos. "Infinitesimally affine deformations of a hypersurface." Mathematics and Mechanics of Solids 23, no. 2 (December 21, 2016): 209–20. http://dx.doi.org/10.1177/1081286516680261.

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Affine deformations serve as basic examples in the continuum mechanics of deformable three-dimensional bodies (usually referred to as homogeneous deformations). They preserve parallelism of straight lines, and are often used as an approximation to general deformations. However, when the deformable body is a membrane, a shell or an interface modeled by a surface, parallelism is defined by the affine connection of this surface. In this work we study the infinitesimally affine time-dependent deformations (motions) of such a continuum, but in a more general context, by considering that it is modeled by a Riemannian hypersurface. First we prove certain equivalent formulas for the variation of the connection of the hypersurface. Some of these formulas are expressed in terms of geometrical quantities, and others in terms of kinematical quantities of the deforming continuum. The latter is achieved by using an adapted version of the polar decomposition theorem, frequently used in continuum mechanics to analyze motion. We also apply our results to special motions like tangential and normal motions. Further, we find necessary and sufficient conditions for this variation to be zero (infinitesimal affine motions), providing insight on the form of these motions and the kind of hypersurfaces that allow such motions. Finally, we give some specific examples of mechanical interest which demonstrate motions that are infinitesimally affine but not infinitesimally isometric.
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Fredericks, E., F. M. Mahomed, and K. Masike. "Lie Infinitesimal Conserved Quantities for Itô Stochastic ODEs." Mathematical and Computational Applications 15, no. 4 (December 1, 2010): 601–12. http://dx.doi.org/10.3390/mca15040601.

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3

Ely, Robert. "Nonstandard Student Conceptions About Infinitesimals." Journal for Research in Mathematics Education 41, no. 2 (March 2010): 117–46. http://dx.doi.org/10.5951/jresematheduc.41.2.0117.

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This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.
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Parameswaran, Revathy. "On Understanding the Notion of Limits and Infinitesimal Quantities." International Journal of Science and Mathematics Education 5, no. 2 (November 25, 2006): 193–216. http://dx.doi.org/10.1007/s10763-006-9050-y.

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5

Shalyt-Margolin, A. E. "Minimal Length and the Existence of Some Infinitesimal Quantities in Quantum Theory and Gravity." Advances in High Energy Physics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/195157.

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It is demonstrated that provided a theory involves a minimal length, this theory must be free from such infinitesimal quantities as infinitely small variations in surface of the holographic screen, its volume, and entropy. The corresponding infinitesimal quantities in this case must be replaced by the “minimal variations possible”—finite quantities dependent on the existent energies. As a result, the initial low-energy theory (quantum theory or general relativity) inevitably must be replaced by a minimal length theory that gives very close results but operates with absolutely other mathematical apparatus.
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Basu, Sanjib, Sreenivasa Rao Jammalamadaka, and Wei Liu. "Stability and infinitesimal robustness of posterior distributions and posterior quantities." Journal of Statistical Planning and Inference 71, no. 1-2 (August 1998): 151–62. http://dx.doi.org/10.1016/s0378-3758(98)00090-1.

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Urabe, M. "Assessment of tableting properties using infinitesimal quantities of powdered medicine." International Journal of Pharmaceutics 263, no. 1-2 (September 16, 2003): 183–87. http://dx.doi.org/10.1016/s0378-5173(03)00340-5.

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8

Cai, J. L., and F. X. Mei. "Conformal Invariance and Conserved Quantity of the Higher-Order Holonomic Systems by Lie Point Transformation." Journal of Mechanics 28, no. 3 (August 9, 2012): 589–96. http://dx.doi.org/10.1017/jmech.2012.67.

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AbstractIn this paper, the conformal invariance and conserved quantities for higher-order holonomic systems are studied. Firstly, by establishing the differential equation of motion for the systems and introducing a one-parameter infinitesimal transformation group together with its infinitesimal generator vector, the determining equation of conformal invariance for the systems are provided, and the conformal factors expression are deduced. Secondly, the relation between conformal invariance and the Lie symmetry by the infinitesimal one-parameter point transformation group for the higher-order holonomic systems are deduced. Thirdly, the conserved quantities of the systems are derived using the structure equation satisfied by the gauge function. Lastly, an example of a higher-order holonomic mechanical system is discussed to illustrate these results.
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9

Paganelli, Simone, and Tony J. G. Apollaro. "Irreversible work versus fidelity susceptibility for infinitesimal quenches." International Journal of Modern Physics B 31, no. 06 (March 5, 2017): 1750065. http://dx.doi.org/10.1142/s0217979217500655.

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We compare the irreversible work produced in an infinitesimal sudden quench of a quantum system at zero temperature with its ground state fidelity susceptibility, giving an explicit relation between the two quantities. We find that the former is proportional to the latter but for an extra term appearing in the irreversible work which includes also contributions from the excited states. We calculate explicitly the two quantities in the case of the quantum Ising chain, showing that at criticality they exhibit different scaling behaviors. The irreversible work, rescaled by square of the quench’s amplitude, exhibits a divergence slower than that of the fidelity susceptibility. As a consequence, the two quantities obey also different finite-size scaling relations.
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10

Urabe, M., S. Ito, S. Itai, and H. Yuasa. "Assessment of tableting properties using infinitesimal quantities of powder medicine II." Journal of Drug Delivery Science and Technology 16, no. 5 (2006): 357–61. http://dx.doi.org/10.1016/s1773-2247(06)50065-6.

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11

Bonsangue, Martin V., and Harris S. Shultz. "In Search of Archimedes: Quadrature of the Parabola." Mathematics Teacher 109, no. 9 (May 2016): 712–16. http://dx.doi.org/10.5951/mathteacher.109.9.0712.

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12

Jiang, Guangxin, and Michael C. Fu. "Technical Note—On Estimating Quantile Sensitivities via Infinitesimal Perturbation Analysis." Operations Research 63, no. 2 (April 2015): 435–41. http://dx.doi.org/10.1287/opre.2015.1356.

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13

Sergeyev, Yaroslav D. "A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities." Informatica 19, no. 4 (January 1, 2008): 567–96. http://dx.doi.org/10.15388/informatica.2008.231.

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14

Jiang, Guangxin, and Michael C. Fu. "Quantile sensitivity estimation for dependent sequences." Journal of Applied Probability 53, no. 3 (September 2016): 715–32. http://dx.doi.org/10.1017/jpr.2016.36.

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AbstractIn this paper we estimate quantile sensitivities for dependent sequences via infinitesimal perturbation analysis, and prove asymptotic unbiasedness, weak consistency, and a central limit theorem for the estimators under some mild conditions. Two common cases, the regenerative setting and ϕ-mixing, are analyzed further, and a new batched estimator is constructed based on regenerative cycles for regenerative processes. Two numerical examples, the G/G/1 queue and the Ornstein–Uhlenbeck process, are given to show the effectiveness of the estimator.
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15

Gong, Sheng-nan, and Jing-li Fu. "Noether’s theorems for the relative motion systems on time scales." Applied Mathematics and Nonlinear Sciences 3, no. 2 (December 1, 2018): 513–26. http://dx.doi.org/10.2478/amns.2018.2.00040.

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AbstractThis paper propose Noether symmetries and the conserved quantities of the relative motion systems on time scales. The Lagrange equations with delta derivatives on time scales are presented for the system. Based upon the invariance of Hamilton action on time scales, under the infinitesimal transformations with respect to the time and generalized coordinates, the Hamilton’s principle, the Noether theorems and conservation quantities are given for the systems on time scales. Lastly, an example is given to show the application the conclusion.
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KERNER, RICHARD, and SALVATORE VITALE. "APPROXIMATE GRAVITATIONAL WAVES VIA DEFORMATIONS OF EMBEDDINGS." International Journal of Modern Physics A 24, no. 08n09 (April 10, 2009): 1465–72. http://dx.doi.org/10.1142/s0217751x09044838.

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Many Einstein spaces can be embedded globally in pseudo-Euclidean spaces of dimension N > 4. The geometrical quantities characterizing the embedded manifold can be expressed by means of derivatives of the embedding functions zA (xµ), A, B, = 1, 2, …N, µ, ν, …= 0, 1, 2, 3. An infinitesimal deformation of embedding can be expanded into a series [Formula: see text], giving rise to a similar expansion of geometrical quantities of the embedded Einstein space, and the Einstein equations in vacuo, too. We show how gravitational wave solutions appear naturally in this context.
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17

Ríos Obregón, Jorge Manuel, Regla María Bernal Gutiérrez, Eberto Pablo Gutiérrez Morales, and Eberto Tuniesky Gutiérrez De León. "Aplicación de una estrategia didáctica para la enseñanza de las integrales indefinidas y definidas en las carreras de Ingeniería." Revista Cognosis. ISSN 2588-0578 5, no. 3 (September 28, 2020): 163. http://dx.doi.org/10.33936/cognosis.v5i3.2956.

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La enseñanza de la matemática requiere ser sustenta en la aplicación de estrategia de enseñanza aprendizaje que contribuyan a una mejor comprensión de sus teorías, por ello este trabajo parte de la necesidad de ejemplificar la aplicación de una estrategia didáctica para la enseñanza del Cálculo Infinitesimal en las carreras de ingeniería. La estrategia se sostiene en un modelo centrado en la sistematización formativa en cálculo infinitesimal es expresión de la relación entre la dimensión analítica infinitesimal del Cálculo para ingenieros y dimensión sistematización contextualizada de recursos del Cálculo Infinitesimal. La estrategia parte del Nivel de Esencialidad dado en objetivos formativos de la Facultad de Ciencias de la Tierra y Agua, de la Universidad Regional Amazónica Ikiam, se valora el Nivel Estratégico con la determinación del estado actual y el estado deseado, así como el entorno donde se desarrolla el proceso de formación en Cálculo Infinitesimal. El Nivel de Concreción, se ejemplifica a través del tema “Cálculo Integral de funciones reales de una variable real”. Este nivel para el Cálculo infinitesimal en las carreras de ingeniería tiene dos subprocesos: subproceso de formación analítica infinitesimal y subproceso de sistematización contextualizada. Ambos subprocesos cuentan con objetivos específicos, orientaciones pedagógicas y acciones; así como ejemplos y ejercicios de integrales indefinidas y definidas orientados a lograr una sistematización teórica y práctica de estos objetos matemáticos que surgen de la necesidad del cálculo de magnitudes físicas. Finalmente, se evalúa la efectividad de la estrategia formativa. PALABRAS CLAVE: Matemática; didáctica; enseñanza aprendizaje; estrategia de enseñanza. APPLICATION OF A DIDACTIC STRATEGY FOR THE TEACHING OF UNDEFINED AND DEFINED INTEGRALS IN ENGINEERING CAREERS ABSTRACT The teaching of mathematics needs to be supported by the application of a teaching-learning strategy that contributes to a better understanding of its theories, therefore this work is based on the need to exemplify the application of a didactic strategy for the teaching of Infinitesimal Calculus in the engineering careers. The strategy is based on a model focused on the formative systematization in infinitesimal calculus, which is an expression of the relationship between the infinitesimal analytical dimension of Calculus for engineers and the contextualized systematization of resources in Infinitesimal Calculus. The strategy starts from the Level of Essentiality given in the training objectives of the Faculty of Earth and Water Sciences, of the Ikiam Regional Amazon University, the Strategic Level is valued with the determination of the current state and the desired state, as well as the environment where the training process in Infinitesimal Calculus is developed. The Level of Specification is exemplified through the topic "Integral Calculus of real functions of a real variable." This level for Infinitesimal Calculus in engineering majors has two threads: infinitesimal analytic training thread and contextualized systematization thread. Both sub-processes have specific objectives, pedagogical orientations and actions; as well as examples and exercises of indefinite and definite integrals aimed at achieving a theoretical and practical systematization of these mathematical objects that arise from the need to calculate physical quantities. Finally, the effectiveness of the training strategy is evaluated. KEYWORDS: Mathematics; didactics; teaching learning; teaching strategy.
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18

Boettcher, Konrad, and Alexander Behr. "Using Virtual Reality for Teaching the Derivation of Conservation Laws in Fluid Mechanics." International Journal of Engineering Pedagogy (iJEP) 11, no. 4 (July 9, 2021): 42. http://dx.doi.org/10.3991/ijep.v11i4.20155.

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In many fields of study, physical sub-areas are treated mathematically in order to teach students the tools for optimization in their professional lives. In the derivation of the fundamental conservation equations theoretical con-structs or infinitesimal elements are used, additionally engaging a Taylor expansion of the variables. For undergraduates, this often means that the understanding of the physical interrelationships is left out in the cold. Practical experiments are not possible for clarification, since important quantities in the mathematical for-mulation can only made visible in experiments with extreme effort or are even in-accessible like theoretical constructs or infinitesimal values. Numerical calcula-tions may be used to show some quantities, but students cannot carry them out for themselves. Therefore, a virtual-reality laboratory for fluid mechanics is creat-ed with the software UNREAL ENGINE 4. This enables the students to learn the derivation of conservation laws and to influence the flow in order to experience and examine the basics of theoretical constructs. The results are evaluated in self-assessments, exercises, tutorials associated to the fluid mechanics course, and the results of an exam. Benefits for the use of virtual reality (VR) in teaching conser-vation laws were ascertained.
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19

Xia, Lili, and Xinsheng Ge. "Lie Symmetry Analysis and Conservation Laws of the Axially Loaded Euler Beam." Mathematics 10, no. 15 (August 3, 2022): 2759. http://dx.doi.org/10.3390/math10152759.

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By applying the Lie symmetry method, group-invariant solutions are constructed for axially loaded Euler beams. The corresponding mathematical models of the beams are formulated. After introducing the infinitesimal transformations, the determining equations of Lie symmetry are proposed via Lie point transformations acting on the original equations. The infinitesimal generators of symmetries of the systems are presented with Maple. The corresponding vector fields are given to span the subalgebra of the systems. Conserved vectors are derived by using two methods, namely, the multipliers method and Noether’s theorem. Noether conserved quantities are obtained using the structure equation, satisfied by the gauge functions. The fluxes of the conservation laws could also be proposed with the multipliers. The relations between them are discussed. Furthermore, the original equations of the systems could be transformed into ODEs and the exact explicit solutions are provided.
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20

Alexander, Amir. "From Indivisibles to Infinitesimals: Studies on Seventeenth-Century Mathematizations of Infinitely Small Quantities. Antoni Malet." Isis 89, no. 1 (March 1998): 131–32. http://dx.doi.org/10.1086/383947.

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21

Zhai, Xiang-Hua, and Yi Zhang. "Lie symmetry analysis on time scales and its application on mechanical systems." Journal of Vibration and Control 25, no. 3 (August 7, 2018): 581–92. http://dx.doi.org/10.1177/1077546318790864.

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The theory of time scales that can unify and extend continuous and discrete analysis has proved to be more accurate in modeling the dynamic process. The Lie symmetry approach, which is an effective way to deal with different kinds of dynamical equations in a variety of areas of applied science, is to be analyzed on time scales. We begin with the Lie group of point infinitesimal transformations on time scales and its corresponding extensions. And the invariance of dynamical equations on time scales under the infinitesimal transformations is discussed. More specifically, the Lie symmetries for dynamical equations of mechanical systems on time scales including Lagrangian systems on time scales, Hamiltonian systems on time scales, and Birkhoffian systems on time scales are investigated as applications. Thus, the corresponding conserved quantities for mechanical systems on time scales are derived by using the Lie symmetries. Examples are given to illustrate the application of the results.
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22

Falcone, Alberto, Alfredo Garro, Marat S. Mukhametzhanov, and Yaroslav D. Sergeyev. "Representation of grossone-based arithmetic in simulink for scientific computing." Soft Computing 24, no. 23 (August 3, 2020): 17525–39. http://dx.doi.org/10.1007/s00500-020-05221-y.

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AbstractNumerical computing is a key part of the traditional computer architecture. Almost all traditional computers implement the IEEE 754-1985 binary floating point standard to represent and work with numbers. The architectural limitations of traditional computers make impossible to work with infinite and infinitesimal quantities numerically. This paper is dedicated to the Infinity Computer, a new kind of a supercomputer that allows one to perform numerical computations with finite, infinite, and infinitesimal numbers. The already available software simulator of the Infinity Computer is used in different research domains for solving important real-world problems, where precision represents a key aspect. However, the software simulator is not suitable for solving problems in control theory and dynamics, where visual programming tools like Simulink are used frequently. In this context, the paper presents an innovative solution that allows one to use the Infinity Computer arithmetic within the Simulink environment. It is shown that the proposed solution is user-friendly, general purpose, and domain independent.
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23

Baalss, Dieter. "The Viscosity Coefficients of Biaxial-Nematic Liquid Crystals. Phenomenology and Affine Transformation Model." Zeitschrift für Naturforschung A 45, no. 1 (January 1, 1990): 7–13. http://dx.doi.org/10.1515/zna-1990-0103.

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AbstractThe viscosity tensor of biaxial-nematic liquid crystals contains 16 independent elements. A complete set of these viscosity coefficients is introduced and related to experimentally accessible quantities. Furthermore, the flow alignment and its stability against an arbitrary infinitesimal disturbance are discussed. By the affine transformation model, formerly established for the uniaxial symmetric case, one can express the viscous anisotropy of perfectly ordered biaxial ellipsoids in terms of the two viscosities of an isotropic reference system and the axes ratios of the nonspherical particles.
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24

Postavaru, Octavian, and Antonela Toma. "Symmetries for Nonconservative Field Theories on Time Scale." Symmetry 13, no. 4 (March 26, 2021): 552. http://dx.doi.org/10.3390/sym13040552.

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Symmetries and their associated conserved quantities are of great importance in the study of dynamic systems. In this paper, we describe nonconservative field theories on time scales—a model that brings together, in a single theory, discrete and continuous cases. After defining Hamilton’s principle for nonconservative field theories on time scales, we obtain the associated Lagrange equations. Next, based on the Hamilton’s action invariance for nonconservative field theories on time scales under the action of some infinitesimal transformations, we establish symmetric and quasi-symmetric Noether transformations, as well as generalized quasi-symmetric Noether transformations. Once the Noether symmetry selection criteria are defined, the conserved quantities for the nonconservative field theories on time scales are identified. We conclude with two examples to illustrate the applicability of the theory.
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25

Sanger, D. J., J. Q. Chen, S. J. Zhang, and D. Howard. "A general method for the stiffness analysis of manipulator mechanisms." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 214, no. 5 (May 1, 2000): 673–85. http://dx.doi.org/10.1243/0954406001523687.

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This paper is concerned with the displacement of the end effector of a manipulator, when subjected to an externally applied force system, which arises because of the flexibility of the actuator drives. General results are developed for the end effector infinitesimal stiffnesses (and compliances) in terms of the actuator infinitesimal stiffnesses (and compliances) for serial and parallel manipulators. It is shown that these quantities are dependent upon the instantaneous force system applied to the end effector, so that the relationship between the applied force system and the resulting displacement is non-linear, even for a given manipulator configuration. These results are of interest for a number of reasons. For example, the accuracy of a manipulator is directly related to its stiffness, and knowledge of the stiffnesses (or compliances) can be used to develop means of simultaneously controlling the force and displacement for a partially constrained end effector. Examples are provided to demonstrate the application of the general results to specific manipulator mechanisms, including both serial and parallel types.
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26

DERELİ, TEKİN, ADNAN TEĞMEN, and TUĞRUL HAKİOĞLU. "CANONICAL TRANSFORMATIONS IN THREE-DIMENSIONAL PHASE-SPACE." International Journal of Modern Physics A 24, no. 25n26 (October 20, 2009): 4769–88. http://dx.doi.org/10.1142/s0217751x09044760.

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Canonical transformation in a three-dimensional phase-space endowed with Nambu bracket is discussed in a general framework. Definition of the canonical transformations is constructed based on canonoid transformations. It is shown that generating functions, transformed Hamilton functions and the transformation itself for given generating functions can be determined by solving Pfaffian differential equations corresponding to that quantities. Types of the generating functions are introduced and all of them are listed. Infinitesimal canonical transformations are also discussed. Finally, we show that the decomposition of canonical transformations is also possible in three-dimensional phase space as in the usual two-dimensional one.
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27

Han, Y. L., X. X. Wang, M. L. Zhang, and L. Q. Jia. "Lie Symmetry and Approximate Hojman Conserved Quantity of Lagrange Equations for a Weakly Nonholonomic System." Journal of Mechanics 30, no. 1 (August 8, 2013): 21–27. http://dx.doi.org/10.1017/jmech.2013.47.

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ABSTRACTThe Lie symmetry and Hojman conserved quantity of Lagrange equations for a weakly nonholonomic system and its first-degree approximate holonomic system are studied. The differential equations of motion for the system are established. Under the special infinitesimal transformations of group in which the time is invariable, the definition of the Lie symmetry for the weakly nonholonomic system and its first-degree approximate holonomic system are given, and the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are obtained. Finally, an example is given to study the exact and approximate Hojman conserved quantity for the system.
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28

Wang, Xing Zhong, Jing Li Fu, and Chao Rong Li. "Noether Symmetry and First Integral of Discrete Nonconservative and Nonholonimic Hamiltoinian System." Applied Mechanics and Materials 117-119 (October 2011): 167–73. http://dx.doi.org/10.4028/www.scientific.net/amm.117-119.167.

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The letter focuses on studying Noether symmetry and conserved quantity of discrete nonconservative and nonholonomic Hamiltonian system. Firstly, the discrete Hamiltonian canonical equations and discrete energy equations of nonconservative and nonholonomic Hamiltonian systems are derived with discrete Hamiltonian action. Secondly, based on the quasi-invariance of discrete Hamiltonian action and equation of lattice under the infinitesimal transformation with respect to time, generalized coordinates and generalized momentums, the discrete analogue of Noether’s identity and determining equation of lattice are obtained for the systems. Thirdly, the discrete analogues of Noether’s theorems and conserved quantities of the systems are presented. Finally, one example is discussed to illustrate the application of the results.
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29

PARK, Y., K. M. SHAW, H. J. CHIEL, and P. J. THOMAS. "The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems." European Journal of Applied Mathematics 29, no. 5 (April 2, 2018): 905–40. http://dx.doi.org/10.1017/s0956792518000128.

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The asymptotic phase θ of an initial point x in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow φt(x) converges as t → ∞. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient ∇x(θ) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, M, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.
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Amodio, Pierluigi, Luigi Brugnano, Felice Iavernaro, and Francesca Mazzia. "On the use of the Infinity Computer architecture to set up a dynamic precision floating-point arithmetic." Soft Computing 24, no. 23 (August 6, 2020): 17589–600. http://dx.doi.org/10.1007/s00500-020-05220-z.

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AbstractWe devise a variable precision floating-point arithmetic by exploiting the framework provided by the Infinity Computer. This is a computational platform implementing the Infinity Arithmetic system, a positional numeral system which can handle both infinite and infinitesimal quantities expressed using the positive and negative finite or infinite powers of the radix $${\textcircled {1}}$$ 1 . The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort. An illustrative example about the solution of a nonlinear equation is also presented.
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31

Abundo, Mario. "First-Passage Problems for Asymmetric Diffusions and Skew-diffusion Processes." Open Systems & Information Dynamics 16, no. 04 (December 2009): 325–50. http://dx.doi.org/10.1142/s1230161209000256.

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For a, b > 0, we consider a temporally homogeneous, one-dimensional diffusion process X(t) defined over I = (-b, a), with infinitesimal parameters depending on the sign of X(t). We suppose that, when X(t) reaches the position 0, it is reflected rightward to δ with probability p > 0 and leftward to -δ with probability 1 - p, where δ > 0. Closed analytical expressions are found for the mean exit time from the interval (-b, a), and for the probability of exit through the right end a, in the limit δ → 0+, generalizing the results of Lefebvre, holding for asymmetric Wiener process. Moreover, in alternative to the heavy analytical calculations, a numerical method is presented to estimate approximately the quantities above. Furthermore, on the analogy of skew Brownian motion, the notion of skew diffusion process is introduced. Some examples and numerical results are also reported.
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32

Wolf, Adam, William R. L. Anderegg, and Stephen W. Pacala. "Optimal stomatal behavior with competition for water and risk of hydraulic impairment." Proceedings of the National Academy of Sciences 113, no. 46 (October 31, 2016): E7222—E7230. http://dx.doi.org/10.1073/pnas.1615144113.

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For over 40 y the dominant theory of stomatal behavior has been that plants should open stomates until the carbon gained by an infinitesimal additional opening balances the additional water lost times a water price that is constant at least over short periods. This theory has persisted because of its remarkable success in explaining strongly supported simple empirical models of stomatal conductance, even though we have also known for over 40 y that the theory is not consistent with competition among plants for water. We develop an alternative theory in which plants maximize carbon gain without pricing water loss and also add two features to both this and the classical theory, which are strongly supported by empirical evidence: (i) water flow through xylem that is progressively impaired as xylem water potential drops and (ii) fitness or carbon costs associated with low water potentials caused by a variety of mechanisms, including xylem damage repair. We show that our alternative carbon-maximization optimization is consistent with plant competition because it yields an evolutionary stable strategy (ESS)—species with the ESS stomatal behavior that will outcompete all others. We further show that, like the classical theory, the alternative theory also explains the functional forms of empirical stomatal models. We derive ways to test between the alternative optimization criteria by introducing a metric—the marginal xylem tension efficiency, which quantifies the amount of photosynthesis a plant will forego from opening stomatal an infinitesimal amount more to avoid a drop in water potential.
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33

YAMAZAKI, AKIRA. "ASSET PRICING WITH NON-GEOMETRIC TYPE OF DIVIDENDS." Annals of Financial Economics 10, no. 02 (December 2015): 1550016. http://dx.doi.org/10.1142/s2010495215500165.

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This paper examines the behavior of the market equilibrium in an endowment economy in continuous time setting, in which a representative investor with exponential utility consumes the dividends generated by multiple risky assets. The dividends of the assets are assumed to be mutually independent and belong to a class of non-negative stochastic processes including square root processes and increasing Lévy processes. We then characterize expected returns and volatilities on the assets as well as interest rates which is defined by risk-free assets. Based on these quantities, we construct the capital asset pricing model in an infinitesimal time period and investigate the impacts of dividend fluctuations and the risk aversion of the investor to the market equilibrium. Numerical examples show that even if the dividends are independent, the asset returns may have strong positive correlations. Moreover, heterogeneous dividend processes generate Jensen’s alpha that can be viewed as the small cap premium.
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34

Hong, Wei, Benrong Mu, and Jun Tao. "Testing the weak cosmic censorship conjecture in torus-like black hole under charged scalar field." International Journal of Modern Physics D 29, no. 12 (July 25, 2020): 2050078. http://dx.doi.org/10.1142/s0218271820500789.

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We investigate weak cosmic censorship conjecture in charged torus-like black hole by the complex scalar field scattering. Using the relation between the conserved quantities of a black hole and the scalar field, we can calculate the change of the energy and charge within the infinitesimal time. The change of the enthalpy is connected to the change of energy, then we use those results to test whether the first law, the second law as well as the weak cosmic censorship conjecture are valid. In the normal phase–space, the first law of thermodynamics and the weak cosmic censorship conjecture are valid, and the second law of thermodynamics is not violated. For the specific black hole under scalar field scattering we consider, in the extended phase–space, the first law of thermodynamics and the weak cosmic censorship conjecture are valid. However, the second law of thermodynamics is violated when the black hole’s initial charge reaches a certain value.
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35

Kreitmeier, F., and P. Lücking. "Demonstration of balance-based space averaging and modelling in view of blading—diffuser interaction." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 217, no. 4 (January 1, 2003): 461–69. http://dx.doi.org/10.1243/095765003322315522.

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Advanced experimental and numerical methods in the field of fluid dynamics and turbomachinery are increasingly successful in describing real flowfields, i.e. fields that are generally three-dimensional and unsteady. For many purposes, e.g. flow characterization, it is necessary to reduce these flowfields step by step to three-, two- or one-dimensional large-scale unsteady flowfields. This procedure permits a lower-level simulation of the flowfields. However, many averaging approaches are arbitrary or succeed in balancing the flowfields in only a few physical aspects. The first author has already shown the steps of a balance-based procedure that avoids this limitation. Small-scale time averaging of (probabilistically) turbulent inhomogeneities by means of irreversible and reversible small-scale time averaging processes on a threefold infinitesimal control volume element has already been demonstrated. The present paper demonstrates the balance-based procedure of space averaging. It is carried out by averaging generally three-dimensional small-scale time-averaged (deterministic) inhomogeneities using irreversible and reversible space averaging processes on onefold infinitesimal and finite control surfaces. The procedure is, similarly to small-scale time averaging, based on conservative and independent non-conservative small-scale time-averaged integral balance equations. The general concept is to represent all the relevant fluxes through the control surface by appropriate average quantities or numbers. The full use of the vector equations for the linear and angular momentum is important. One of the consequences in space averaging is the introduction of a wrench (parallel linear and angular momentum vectors), which is generally used only in mechanics for the reduction of force systems in space. The flowfield inhomogeneity is described on all dimensional levels via the diffusion intensity of the irreversible averaging process, and, only for space averaging, via the distance vector and the parameter of the wrench. A numerical example on different dimensional levels is presented in detail. The procedure also illustrates the basis of a new and more complete two-and one-dimensional large-scale unsteady theory generally in fluid dynamics and especially in turbomachinery.
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36

Huang, Changyu, and Yong-Chang Huang. "Anthropic Principle’s Predicting Symmetric Distribution Matter Strata, Their Physics Laws, and Verifications." Advances in Astronomy 2019 (March 18, 2019): 1–16. http://dx.doi.org/10.1155/2019/2501417.

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This paper shows anthropic principle’s predicting symmetric distribution matter strata, their physics laws, and verifications, concretely deduces characteristic time, energy, and temperature expressions at different scales, discovers four interesting invariant quantities, shows homeomorphic theorem of space map, and naturally presents a supersymmetric scale energy. We further discover that any infinitesimal space has the same proportional structure space; namely, they have renormalization group invariance. Consequently, this paper shows that the region of any nth level Plank-scope is from the nth level Planck scale to the (n+1)​th level Planck scale, where the different matters of the nthlevel Planck scale build up the (n+1)th level Planck scale matter. The branches of physics science for this region include the nth level Planck scale matter dynamics and the nth level Planck scale matter group dynamics. The nth level Planck scale matter group dynamics describe how the nth level Planck scale matter constructs the (n+1)th level Planck scale matter and how the different matters of the nth level Planck scale evolve in the group system. This paper discovers that the different matters below Planck scale can exist with our matter world at the same time and same place and may be some candidates for dark matter; furthermore, this paper shows a relative theorem of matter scale: for the world of any nth level, the matters’ sizes are relative, not absolute. Evidently, the discoveries of both the symmetrical distribution scales and the relations among the corresponding different physics laws from infinitesimal to infinitely large scales give a scientific solid development platform for formation of new scientific branches and deeper development of old scientific branches, because we can precisely construct many kinds of scientific theories relevant to all the corresponding matter strata. All the branch sciences of different matter strata up to now naturally need to be included in the framework of the new scientific system of physics.
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37

Christiansen, Marcus C. "Time-dynamic evaluations under non-monotone information generated by marked point processes." Finance and Stochastics 25, no. 3 (June 14, 2021): 563–96. http://dx.doi.org/10.1007/s00780-021-00456-5.

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AbstractThe information dynamics in finance and insurance applications is usually modelled by a filtration. This paper looks at situations where information restrictions apply so that the information dynamics may become non-monotone. A fundamental tool for calculating and managing risks in finance and insurance are martingale representations. We present a general theory that extends classical martingale representations to non-monotone information generated by marked point processes. The central idea is to focus only on those properties that martingales and compensators show on infinitesimally short intervals. While classical martingale representations describe innovations only, our representations have an additional symmetric counterpart that quantifies the effect of information loss. We exemplify the results with examples from life insurance and credit risk.
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38

ROERO, CLARA SILVIA. "ANTONI MALET, From Indivisibles to Infinitesimals. Studies on Seventeenth-Century Mathematizations of Infinitely Small Quantities, Universitat de Barcelona, Servei de Publicacions Bellaterra, 1996, 163 pp." Nuncius 15, no. 2 (2000): 770–72. http://dx.doi.org/10.1163/182539100x00191.

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39

Akrour, Dalila, Mouloud Tribeche, and Djamel Kalache. "A theoretical and numerical study of thermosolutal convection: stability of a salinity gradient solar pond." Thermal Science 15, no. 1 (2011): 67–80. http://dx.doi.org/10.2298/tsci1101067a.

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A theoretical and numerical study of the effect of thermodiffusion on the stability of a gradient layer is presented. It intends to clarify the mechanisms of fluid dynamics and the processes which occur in a salinity gradient solar pond. A mathematical modelling is developed to describe the thermodiffusion contribution on the solar pond where thermal, radiative, and massive fluxes are coupled in the double diffusion. More realistic boundary conditions for temperature and concentration profiles are used. Our results are compared with those obtained experimentally by authors without extracting the heat flux from the storage zone. We have considered the stability analysis of the equilibrium solution. We assumed that the perturbation of quantities such as velocity, temperature, and concentration are infinitesimal. Linearized equations satisfying appropriate prescribed boundary conditions are then obtained and expanded into polynomials form. The Galerkin method along with a symbolic algebra code (Maple) are used to solve these equations. The effect of the separation coefficient y is analyzed in the positive and negative case. We have also numerically compared the critical Rayleigh numbers for the onset of convection with those obtained by the linear stability analysis for Le = 100, ?a = 0.8, and f = 0.5.
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40

Riaz, Muhammad Bilal, Jan Awrejcewicz, Adil Jhangeer, and Muhammad Junaid-U-Rehman. "A Variety of New Traveling Wave Packets and Conservation Laws to the Nonlinear Low-Pass Electrical Transmission Lines via Lie Analysis." Fractal and Fractional 5, no. 4 (October 18, 2021): 170. http://dx.doi.org/10.3390/fractalfract5040170.

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This research is based on computing the new wave packets and conserved quantities to the nonlinear low-pass electrical transmission lines (NLETLs) via the group-theoretic method. By using the group-theoretic technique, we analyse the NLETLs and compute infinitesimal generators. The resulting equations concede two-dimensional Lie algebra. Then, we have to find the commutation relation of the entire vector field and observe that the obtained generators make an abelian algebra. The optimal system is computed by using the entire vector field and using the concept of abelian algebra. With the help of an optimal system, NLETLs convert into nonlinear ODE. The modified Khater method (MKM) is used to find the wave packets by using the resulting ODEs for a supposed model. To represent the physical importance of the considered model, some 3D, 2D, and density diagrams of acquired results are plotted by using Mathematica under the suitable choice of involving parameter values. Furthermore, all derived results were verified by putting them back into the assumed equation with the aid of Maple software. Further, the conservation laws of NLETLs are computed by the multiplier method.
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41

Solecki, R. "Free-Vibration of an L-Shaped Plate: The General Solution and an Example of a Simply-Supported Plate With a Clamped Cutout." Journal of Vibration and Acoustics 118, no. 1 (January 1, 1996): 107–11. http://dx.doi.org/10.1115/1.2889623.

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This study develops a new accurate method for finding the natural vibration frequencies of plates with cutouts. The method is based on replacing the plate with a cutout by a rectangular plate. This is achieved by filling the cutout with a “dummy” plate made of the same material, and of the same thickness as the original from which it is separated by an infinitesimal gap. Thanks to this device it is possible to apply finite Fourier transformation of discontinuous functions in a rectangular domain. The expression for the deflection now depends on the unknown quantities along the boundary and across the gap. Subsequent application of the available boundary conditions leads to a system of boundary integral equations. An L-shaped plate simply supported along the perimeter, and fixed along the cutout, is analyzed as an example. The frequencies of natural vibration are calculated and compared with the results obtained using the finite element method. The method presented here is also applicable to two- and three-dimensional problems of solids with holes or cavities and to similar thermoelastic problems. Application to plates with curved boundaries is also possible.
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42

Gibbon, J. D. "Regularity and singularity in solutions of the three-dimensional Navier–Stokes equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2121 (March 17, 2010): 2587–604. http://dx.doi.org/10.1098/rspa.2009.0642.

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Higher moments of the vorticity field Ω m ( t ) in the form of L 2 m -norms ( ) are used to explore the regularity problem for solutions of the three-dimensional incompressible Navier–Stokes equations on the domain . It is found that the set of quantities provide a natural scaling in the problem resulting in a bounded set of time averages 〈 D m 〉 T on a finite interval of time [0, T ]. The behaviour of D m +1 / D m is studied on what are called ‘good’ and ‘bad’ intervals of [0, T ], which are interspersed with junction points (neutral) τ i . For large but finite values of m with large initial data ( Ω m (0)≤ ϖ 0 O ( Gr 4 )), it is found that there is an upper bound which is punctured by infinitesimal gaps or windows in the vertical walls between the good/bad intervals through which solutions may escape. While this result is consistent with that of Leray (Leray 1934 Acta Math. 63 , 193–248 ( doi:10.1007/BF02547354 )) and Scheffer (Scheffer 1976 Pacific J. Math. 66 , 535–552),— this estimate for Ω m corresponds to a length scale well below the validity of the Navier–Stokes equations.
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43

Felea, Ioan, Simona Dzitac, Tiberiu Vesselenyi, and Ioan Dzitac. "Decision Support Model for Production Disturbance Estimation." International Journal of Information Technology & Decision Making 13, no. 03 (May 2014): 623–47. http://dx.doi.org/10.1142/s0219622014500576.

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A current modeling framework for disturbance in manufacturing systems (MS) is given by concepts like discrete-event systems, stochastic fluid models and infinitesimal disturbance analysis. The goal of modeling is to achieve control and structural and functional optimization of MS. Objective functions of these optimization models are focused on quantities which reflect the level of reliability, the level of manufactured products, the quality of products or the impact on the environment of MS with disturbances. These models do not allow a dynamic evaluation of consequences of the disturbances which appears in the operation of MS machines and also do not allow an evaluation of the evolution in time of disturbance consequence indicators. Disturbances in technological lines of MS represent local bottlenecks of production with severe economic consequences in what regards production time losses. Good estimation of disturbances dynamics can be very helpful to both technological line designers, who can optimize their projects and production managers who can minimize their losses. Our model allows a dynamic evaluation of consequences of some disturbance of machine operation in MS, using indicators based on time, energy and costs. A MATLAB software package was developed for tests.
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44

Koumoutsakos, P., and D. Shiels. "Simulations of the viscous flow normal to an impulsively started and uniformly accelerated flat plate." Journal of Fluid Mechanics 328 (December 10, 1996): 177–227. http://dx.doi.org/10.1017/s0022112096008695.

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The development of a two-dimensional viscous incompressible flow generated from an infinitesimally thin flat plate, impulsively started or uniformly accelerated normal to the free stream is studied computationally. An adaptive numerical scheme, based on vortex methods, is used to integrate the vorticity–velocity formulation of the Navier–Stokes equations. The results of the computations complement relevant experimental works while providing us with quantities such as the vorticity field and the unsteady forces experienced by the body. For the uniformly accelerated plate the present simulations capture the development of a number of centers of vorticity along the primary separating shear layer. This phenomenon has been observed in experimental works but has not been predicted by inviscid models. The present simulations suggest that this Kelvin–Helmholtz-type instability is driven by the interaction of primary and secondary vorticity near the tips of the plate and depends on the acceleration of the plate.
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45

Hall, Kevin D. "Body fat and fat-free mass inter-relationships: Forbes's theory revisited." British Journal of Nutrition 97, no. 6 (June 2007): 1059–63. http://dx.doi.org/10.1017/s0007114507691946.

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A theoretical equation was developed by Forbes that quantifies the fat-free proportion of a weight change as a function of the initial body fat. However, Forbes's equation was strictly valid only for infinitesimal weight changes. Here, I extended Forbes's equation to account for the magnitude and direction of macroscopic body weight changes. The new equation was also re-expressed in terms of an alternative representation of body composition change defined by an energy partitioning parameter called the P-ratio. The predictions of the resulting equations compared favourably with data from human underfeeding and overfeeding experiments and accounted for previously unexplained trends in the data. The magnitude of the body weight change had a relatively weak effect on the predicted body composition changes and the results were very similar to Forbes's original equation for modest weight changes. However, for large weight changes, such as the massive weight losses found in patients following bariatric surgery, Forbes's original equation consistently underestimated the fat-free mass loss, as expected. The new equation that accounts for the magnitude of the weight loss provides better predictions of body composition changes in such patients.
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46

Nath, G., and Sumeeta Singh. "Similarity solutions for magnetogasdynamic cylindrical shock wave in rotating ideal gas using Lie Group theoretic method: Isothermal flow." International Journal of Geometric Methods in Modern Physics 17, no. 08 (July 2020): 2050123. http://dx.doi.org/10.1142/s0219887820501236.

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The propagation of cylindrical shock wave under the influence of axial magnetic field in rotating medium under isothermal flow condition is investigated. The density, magnetic field and azimuthal and axial components of fluid velocity are assumed to be varying in the undisturbed medium. The arbitrary constants appearing in the expressions for infinitesimals of the Local Lie group of transformations bring about three different cases of solutions, i.e. with power law shock path, exponential law shock path and a particular case of power law shock path. Numerical solutions are obtained in the cases of power law and exponential law shock paths. Distribution of gasdynamical quantities are discussed through figures. The effects of variation in values of Alfven-Mach number [Formula: see text], ambient azimuthal velocity index [Formula: see text] and ambient density index [Formula: see text] are studied on flow variables and on shock strength. The numerical integration is done using software Mathematica. It is obtained that magnetic field has a decaying effect on shock strength. Also, increase in value of ambient density or ambient azimuthal velocity variation index in the case of power law shock path and increase in value of ambient density variation index in case of exponential law shock path have the decaying effect on shock strength.
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47

Lazar, Markus, and Eleni Agiasofitou. "The J-, M- and L-integrals of body charges and body forces: Maxwell meets Eshelby." Journal of Micromechanics and Molecular Physics 03, no. 03n04 (September 2018): 1840012. http://dx.doi.org/10.1142/s242491301840012x.

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In this work, we derive the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals of body charges and point charges in electrostatics, and the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals of body forces and point forces in elasticity and we investigate their physical interpretation. Electrostatics is considered as field theory of an electrostatic scalar potential [Formula: see text] (scalar field theory) and elasticity as field theory of a displacement vector [Formula: see text] (vector field theory). One of the basic quantities appearing in the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals is the electrostatic Maxwell–Minkowski stress tensor in electrostatics and the Eshelby stress tensor in elasticity. Among others, it is shown that the [Formula: see text]-integral of body charges in electrostatics represents the electrostatic part of the Lorentz force, and the [Formula: see text]-integral of body forces in elasticity represents the Cherepanov force. The [Formula: see text]-integral between two-point sources (charges or forces) equals half the electrostatic interaction energy in electrostatics and half the elastic interaction energy in elasticity between these two-point sources. The [Formula: see text]-integral represents the configurational vector moment or torque between two body or point sources (charges or forces). Interesting mathematical and physical features are revealed through the connection of the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals with their corresponding infinitesimal generators in both theories. Several important outcomes arise from the comparison between the examined concepts in electrostatics and elasticity. Differences and similarities, that provide a deeper insight into the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals and the related quantities to them, are pointed out and discussed. The presented results show that the [Formula: see text]-, [Formula: see text]- and [Formula: see text]-integrals are fundamental concepts which can be applied in any field theory.
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48

Bontempo, Rodolfo, and Marcello Manna. "Verification of the Axial Momentum Theory for Propellers with a Uniform Load Distribution." International Journal of Turbomachinery, Propulsion and Power 4, no. 2 (May 14, 2019): 8. http://dx.doi.org/10.3390/ijtpp4020008.

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The paper provides an evaluation of the errors embodied in the Axial Momentum Theory (AMT) as applied to a uniformly loaded actuator disk model without wake rotation. Although this model exhibits some unphysical features, such as the tip singularity and the violation of the angular momentum equation, it is still considered a touchstone in the theoretical aerodynamics of propellers. To simplify the model, a purely mathematical assumption is commonly used in the differential form of the axial momentum equation, i.e., the contribution of the pressure forces on the lateral surface of the infinitesimal streamtubes swallowed by the disk is neglected. In this paper, the errors introduced by this simplifying assumption are evaluated by comparing the results of the AMT with those of a nonlinear method modelling the free wake as the superposition of ring vortices distributed along the wake boundary. Firstly, the validity of this method is verified in terms of global performance coefficients. Then, using a CFD approach, it is also verified in terms of local flow quantities. The comparison between the ring-vortices method and the AMT shows that, for a highly loaded propeller, significant errors exist in the axial velocity at the disk, especially near the tip. Moreover, despite the uniform load, the axial velocity at the disk varies in the radial direction. Instead, the velocity magnitude remains almost uniform only for values of the thrust coefficient lower than 1.
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49

Picerno, Pietro, and Johnny Padulo. "Acute kinematic adaptations to running on a centrifugal track: A pilot study." Proceedings of the Institution of Mechanical Engineers, Part P: Journal of Sports Engineering and Technology 232, no. 3 (January 1, 2018): 275–81. http://dx.doi.org/10.1177/1754337117749515.

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The centrifugal track, a basin-shaped track characterised by a platform with a parabolic section, exploits the centripetal acceleration to increase the bodyweight of the athlete during the foot contact phase of running. Because this overload is produced by an inertial force that is equally distributed to the infinitesimal point masses of the body, no postural changes are expected with respect to level running. This pilot study aimed to compare some selected key kinematic quantities of running on the centrifugal track with respect to level running. A video-based three-dimensional (3D) motion analysis was performed on five sprinters and used to compute spatio-temporal variables, frontal and sagittal trunk kinematics and knee sagittal kinematics at footstrike, midstance and foot-off over two consecutive steps at similar speeds. No significant changes were found in spatio-temporal variables and knee kinematics between the right and the left leg during running on the centrifugal track. Neither step length nor step duration was found statistically different between the two running typologies. Trunk flexion was not altered during the stance phase of running on the centrifugal track. Knee angle at footstrike was found similar to level running. A slightly larger but statistically significant knee flexion at midstance and knee extension at foot-off were found with respect to flat-track running, but these findings appear more beneficial for strength training rather than detrimental for the running technique. The centrifugal track was found to be a viable alternative to the common resisted sprint training techniques as the training effect is produced without localised overloads on the musculoskeletal system and detrimental postural changes.
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50

Huang, Tao, Zhiqiang Fan, Jun Su, and Qi Qiu. "Real-Time Eye Diagram Monitoring for Optical Signals Based on Optical Sampling." Applied Sciences 13, no. 3 (January 19, 2023): 1363. http://dx.doi.org/10.3390/app13031363.

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A real-time eye diagram monitoring method for optical signals is proposed and experimentally demonstrated based on optical sampling. In the system, the optical signals under test are directly sampled by an optical sampling pulse train with a narrow pulse width and a high repetition frequency. The sampling pulse train is achieved in a Mach-Zehnder modulator (MZM), gated on-off by an electrical pulse. The sampled optical signals are then broadened and detected by a photodetector (PD). A low-speed electrical analog-to-digital converter (ADC) will then quantify the detected electrical signals. Combining with an algorithm based on the infinitesimal calculus, the quantified data is then used to achieve the eye diagram, according to which more time-domain parameters, such as period, time jitter, Q value, and bit error rate (BER) for the optical signals under test, are obtained. Thanks to the high repetition rate of the optical sampling pulse train, the eye diagram and the time-domain parameters of the optical signals are observed in real time. Experimental results show that a real time of about 350-μs eye diagram monitoring for a 2.5-Gb/s optical signal with a dynamic range from −10 to −22 dBm is achieved. In addition, time jitters are measured to range from 4.3 to 49.8 ps. Q values are estimated to range from 20.4 to 4.3, corresponding to BERs ranging from 2.3 × 10−92 to 8.5 × 10−6. The results are also verified by a commercial real-time oscilloscope.
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