Journal articles: 'Free differential calculus'

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Parisi, Francesco, and Jonathan Klick. "The Differential Calculus of Consent." Journal of Public Finance and Public Choice 20, no. 2 (October 1, 2002): 115–24. http://dx.doi.org/10.1332/251569202x15665366114888.

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Abstract Existing treatments of the choice of an optimal voting rule ignore the effects of the rule on political bargaining. Specifically, more stringent majority requirements reduce intra-coalitional free riding in political compromise, leading to greater gains from political trade. Once this benefit of increasing the vote share necessary to enact a proposal is recognized, we suggest that the optimal voting rule in the presence of transactions costs will actually be closer to unanimity than the optimal majority derived by Buchanan - Tullock [1962].

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Shpil'rain, V. �. "Some applications of free differential calculus in Group theory." Mathematical Notes of the Academy of Sciences of the USSR 49, no. 3 (March 1991): 334–35. http://dx.doi.org/10.1007/bf01158317.

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MORENO, GIOVANNI. "ON FAMILIES IN DIFFERENTIAL GEOMETRY." International Journal of Geometric Methods in Modern Physics 10, no. 09 (August 30, 2013): 1350042. http://dx.doi.org/10.1142/s0219887813500424.

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Families of objects appear in several contexts, like algebraic topology, theory of deformations, theoretical physics, etc. An unified coordinate-free algebraic framework for families of geometrical quantities is presented here, which allows one to work without introducing ad hoc spaces, by using the language of differential calculus over commutative algebras. An advantage of such an approach, based on the notion of sliceable structures on cylinders, is that the fundamental theorems of standard calculus are straightforwardly generalized to the context of families. As an example of that, we prove the universal homotopy formula.

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Huang, Zhiyuan, and Shunlong Luo. "Wick Calculus of Generalized Operators and its Applications to Quantum Stochastic Calculus." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 03 (July 1998): 455–66. http://dx.doi.org/10.1142/s0219025798000247.

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A nonlinear and stochastic analysis of free Bose field is established in the framework of white noise calculus. Wick algebra structure is introduced in the space of generalized operators generated by quantum white noise, some fundamental properties of the calculus based on the Wick algebra are investigated. As applications, quantum stochastic integrals and quantum stochastic differential equations are treated from the viewpoint of Wick calculus.

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Karaçuha, Serkan, and Christian Lomp. "Integral calculus on quantum exterior algebras." International Journal of Geometric Methods in Modern Physics 11, no. 04 (April 2014): 1450026. http://dx.doi.org/10.1142/s0219887814500261.

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Hom-connections and associated integral forms have been introduced and studied by Brzeziński as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on a differential calculus (Ω, d) over an algebra A yields the integral complex which for various algebras has been shown to be isomorphic to the non-commutative de Rham complex (in the sense of Brzeziński et al. [Non-commutative integral forms and twisted multi-derivations, J. Noncommut. Geom.4 (2010) 281–312]). In this paper we shed further light on the question when the integral and the de Rham complex are isomorphic for an algebra A with a flat Hom-connection. We specialize our study to the case where an n-dimensional differential calculus can be constructed on a quantum exterior algebra over an A-bimodule. Criteria are given for free bimodules with diagonal or upper-triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum n-space.

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Majid, S. "Free braided differential calculus, braided binomial theorem, and the braided exponential map." Journal of Mathematical Physics 34, no. 10 (October 1993): 4843–56. http://dx.doi.org/10.1063/1.530326.

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MIKHALEV, ALEXANDER A., and ANDREJ A. ZOLOTYKH. "RANK AND PRIMITIVITY OF ELEMENTS OF FREE COLOUR LIE (p-)SUPERALGEBRAS." International Journal of Algebra and Computation 04, no. 04 (December 1994): 617–55. http://dx.doi.org/10.1142/s021819679400018x.

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Using Fox differential calculus we study characteristics of orbits of elements of free colour Lie (p-)superalgebras under action of the automorphism groups of these algebras. In particular, an effective criterion for an element to be primitive and an algorithm for finding the rank of an element are obtained.

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Kenoufi, Abdelouahab. "Linear Algebra and Differential Calculus in Pseudo-Intervals Vector Space." TEMA (São Carlos) 17, no. 3 (December 20, 2016): 283. http://dx.doi.org/10.5540/tema.2016.017.03.0283.

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In this paper one proposes to use a new approach of interval arithmetic, the so-called pseudo- intervals [1, 5, 13]. It uses a construction which is more canonical and based on the semi-group completion into the group, and it allows to build a Banach vector space. This is achieved by embedding the vector space into free algebra of dimensions higher than 4. It permits to perform linear algebra and differential calculus with pseudo-intervals. Some numerical applications for interval matrix eigenmode calculation, inversion and function minimization are exhibited for simple examples.

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Fan, Xiliang, and Yong Ren. "Bismut formulas and applications for stochastic (functional) differential equations driven by fractional Brownian motions." Stochastics and Dynamics 17, no. 04 (May 4, 2017): 1750028. http://dx.doi.org/10.1142/s0219493717500289.

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In this paper by using Malliavin calculus we prove derivative formulas of Bismut type for a class of stochastic (functional) differential equations driven by fractional Brownian motions. As applications, the dimensional-free Harnack type inequalities and the strong Feller property are presented.

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TRAVERSA, FABIO L., and FABRIZIO BONANI. "ASYMPTOTIC STOCHASTIC CHARACTERIZATION OF PHASE AND AMPLITUDE NOISE IN FREE-RUNNING OSCILLATORS." Fluctuation and Noise Letters 10, no. 02 (June 2011): 207–21. http://dx.doi.org/10.1142/s021947751100048x.

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Starting from the definition of the stochastic differential equation for amplitude and phase fluctuations of an oscillator described by an ordinary differential equation, we study the associated Fokker–Planck equation by using tools from stochastic integral calculus, harmonic analysis and Floquet theory. We provide an asymptotic characterization of the relevant correlation functions, showing that within the assumption of a linear perturbative analysis for the amplitude fluctuations phase noise and orbital fluctuations at the same time are asymptotically statistically independent, and therefore the nonlinear perturbative analysis of phase noise recently derived still exactly holds even if orbital noise is taken into account.

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Hartglass, Michael. "Free Product C* -algebras Associated with Graphs, Free Differentials, and Laws of Loops." Canadian Journal of Mathematics 69, no. 3 (June 1, 2017): 548–78. http://dx.doi.org/10.4153/cjm-2016-022-6.

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AbstractWe study a canonical C* -algebra, 𝒮(Г,μ), that arises from a weighted graph (Г,μ), speci fic cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting that ensure simplicity and uniqueness of trace of 𝒮(Г,μ), and study the structure of its positive cone. We then study the *-algebra,𝒜, generated by the generators of 𝒮(Г,μ), and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko as well as Mai, Speicher, and Weber to show that certain “loop” elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements x ∊ Mn(𝒜) have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials inWishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

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Li, Jie, and Na Li. "Study on Free Vibrations of Self-Anchored Suspension Bridges." Advanced Materials Research 243-249 (May 2011): 2014–20. http://dx.doi.org/10.4028/www.scientific.net/amr.243-249.2014.

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Study on characters of suspension bridges free vibration is basic to analyze wind resistance, shock resistance, and other dynamic behaviors. Current studies are usually focused on earth-anchored suspension bridges. Compared with earth-anchored suspension bridges, self-anchored suspension bridges are anchored on the girder end which makes girder under the compression condition, cable longitudinal displacement can not be neglected, in addition, large ratio of rise to span leads to cable large tilt angle. Based on functional analysis method, it uses variation calculus to derive differential equation of two-tower and three-span continuous self-anchored suspension bridges free vibrations, taking into account cable tilt angle and cable longitudinal displacement. According to the simplified differential equation, it draws to a formula that can calculate self-anchored longitudinal and vertical vibration frequencies. Finally, two examples are carried out to check the formula’s reasonableness.

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13

Ayed, Wided. "Module Free White Noise Flows." Open Systems & Information Dynamics 25, no. 04 (December 2018): 1850018. http://dx.doi.org/10.1142/s123016121850018x.

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The main result of this paper is to extend to Hilbert module level the proof of the inclusion of (non-Hamiltonian) stochastic differential equations based on free noise into the class of Hamiltonian equations driven by free white noise. To achieve this goal, free white noise calculus is extended to a trivial Hilbert module. The white noise formulation of the Ito table is radically different from the usual Itô tables, both classical and quantum and, combined with the Accardi–Boukas approach to Ito algebra, allows to drastically simplify calculations. Infinitesimal generators of Hilbert module free flows are characterized in terms of stochastic derivations from an initial algebra into a white noise Itô algebra. We prove that any such derivation is the difference of a ⋆-homomorphism and a trivial embedding.

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Trishin, Vladimir N. "A note on spinor form of Lovelock differential identity." International Journal of Geometric Methods in Modern Physics 16, no. 09 (September 2019): 1950145. http://dx.doi.org/10.1142/s0219887819501457.

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The paper is devoted to 2-spinor calculus methods in general relativity. New spinor form of the Lovelock differential identity is suggested. This identity is second-order identity for the Riemann curvature tensor. We provide an example that our spinorial treatment of Lovelock identity is effective for the description of solutions of Einstein–Maxwell equations. It is shown that the covariant divergence of Lipkin’s zilch tensor for the free Maxwell field vanishes on the solutions of Einstein–Maxwell equations if and only if the energy–momentum tensor of the electromagnetic field is Weyl-compatible.

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FUKUHARA, SHINJI, YUKIO MATSUMOTO, and NORIKO YUI. "NON-COMMUTATIVE POLYNOMIAL RECIPROCITY FORMULAE." International Journal of Mathematics 12, no. 08 (November 2001): 973–86. http://dx.doi.org/10.1142/s0129167x01001088.

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We prove non-commutative reciprocity formulae for certain polynomials using Fox's free differential calculus. The abelianizations of these reciprocity formulae rediscover the polynomial reciprocity formulae of Carlitz and Berndt–Dieter. Further, many other reciprocity formulae related to Dedekind sums are rederived from our polynomial reciprocity formulae; these include, for instance, generalizations of the classical Eisenstein reciprocity formula.

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HAMDI, TAREK. "LIBERATION, FREE MUTUAL INFORMATION AND ORBITAL FREE ENTROPY." Nagoya Mathematical Journal 239 (September 14, 2018): 205–31. http://dx.doi.org/10.1017/nmj.2018.37.

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In this paper, we perform a detailed spectral study of the liberation process associated with two symmetries of arbitrary ranks: $(R,S)\mapsto (R,U_{t}SU_{t}^{\ast })_{t\geqslant 0}$, where $(U_{t})_{t\geqslant 0}$ is a free unitary Brownian motion freely independent from $\{R,S\}$. Our main tool is free stochastic calculus which allows to derive a partial differential equation (PDE) for the Herglotz transform of the unitary process defined by $Y_{t}:=RU_{t}SU_{t}^{\ast }$. It turns out that this is exactly the PDE governing the flow of an analytic function transform of the spectral measure of the operator $X_{t}:=PU_{t}QU_{t}^{\ast }P$ where $P,Q$ are the orthogonal projections associated to $R,S$. Next, we relate the two spectral measures of $RU_{t}SU_{t}^{\ast }$ and of $PU_{t}QU_{t}^{\ast }P$ via their moment sequences and use this relationship to develop a theory of subordination for the boundary values of the Herglotz transform. In particular, we explicitly compute the subordinate function and extend its inverse continuously to the unit circle. As an application, we prove the identity $i^{\ast }(\mathbb{C}P+\mathbb{C}(I-P);\mathbb{C}Q+\mathbb{C}(I-Q))=-\unicode[STIX]{x1D712}_{\text{orb}}(P,Q)$.

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17

Fenn, Roger, and Denis Sjerve. "Massey Products and Lower Central Series of Free Groups." Canadian Journal of Mathematics 39, no. 2 (April 1, 1987): 322–37. http://dx.doi.org/10.4153/cjm-1987-015-5.

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The purpose of this paper is to continue the investigation into the relationships amongst Massey products, lower central series of free groups and the free differential calculus (see [4], [9], [12]). In particular we set forth the notion of a universal Massey product ≪α1, …, αk≫, where the αi are one dimensional cohomology classes. This product is defined with zero indeterminacy, natural and multilinear in its variables.In order to state the results we need some notation. Throughout F will denote the free group on fixed generators x1, …, xn andwill denote the lower central series of F. If I = (i1, …, ik) is a sequence such that 1 ≦ i1, …, ik ≦ n then ∂1 is the iterated Fox derivative and , where is the augmentation. By convention we set ∂1 = identity if I is empty.

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Fang, C. Q., H. Y. Sun, and J. P. Gu. "A Fractional Calculus Approach to the Prediction of Free Recovery Behaviors of Amorphous Shape Memory Polymers." Journal of Mechanics 32, no. 1 (October 27, 2015): 11–17. http://dx.doi.org/10.1017/jmech.2015.82.

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AbstractA fractional model generalized from the Zener model is proposed for the prediction of temperature-dependent free recovery behaviors of amorphous shape memory polymers (SMPs). This model differs from the Zener model in that it involves nonlinear differential equations of fractional, not integer, order. The theoretical solution based on this fractional model is utilized to simulate the isothermal and nonisothermal free recovery of an amorphous SMP compared with the one based on the Zener model. The results show a reasonable improvement in the prediction of the strain recovery response of SMP by the fractional calculus method.

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MAZZA, DAMIANO. "The true concurrency of differential interaction nets." Mathematical Structures in Computer Science 28, no. 7 (November 21, 2016): 1097–125. http://dx.doi.org/10.1017/s0960129516000402.

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We analyse the reduction of differential interaction nets from the point of view of so-called ‘true concurrency,’ that is, employing a non-interleaving model of parallelism. More precisely, we associate with each differential interaction net an event structure describing its reduction. We show how differential interaction nets are only able to generate confusion-free event structures, and we argue that this is a serious limitation in terms of the concurrent behaviours they may express. In fact, confusion is an extremely elementary phenomenon in concurrency (for example, it already appears in CCS with just prefixing and parallel composition) and we show how its presence is preserved by any encoding respecting the degree of distribution and the reduction semantics. We thus infer that no reasonably expressive process calculus may be satisfactorily encoded in differential interaction nets. We conclude with an analysis of one such encoding proposed by Ehrhard and Laurent, and argue that it does not contradict our claims, but rather supports them.

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Khalil, Salma A., Mohammed A. Basheer, and Tarig A. Abdelhaleem. "Types of Derivatives: Concepts and Applications (II)." Journal of Mathematics Research 9, no. 1 (January 23, 2017): 50. http://dx.doi.org/10.5539/jmr.v9n1p50.

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The notion of differential geometry is known to have played a fundamental role in unifying aspects of the physics of particles and fields, and have completely transformed the study of classical mechanics.In this paper we applied the definitions and concepts which we defined and derived in part (I) of our paper: Types of Derivatives: Concepts and Applications to problems arising in Geometry and Fluid Mechanics using exterior calculus. We analyzed this problem, using the geometrical formulation which is global and free of coordinates.

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Saxena, R. K., and S. L. Kalla. "Solution of Volterra-type integro-differential equations with a generalized Lauricella confluent hypergeometric function in the kernels." International Journal of Mathematics and Mathematical Sciences 2005, no. 8 (2005): 1155–70. http://dx.doi.org/10.1155/ijmms.2005.1155.

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The object of this paper is to solve a fractional integro-differential equation involving a generalized Lauricella confluent hypergeometric function in several complex variables and the free term contains a continuous functionf(τ). The method is based on certain properties of fractional calculus and the classical Laplace transform. A Cauchy-type problem involving the Caputo fractional derivatives and a generalized Volterra integral equation are also considered. Several special cases are mentioned. A number of results given recently by various authors follow as particular cases of formulas established here.

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Almeida, Ricardo, Natália Martins, and Cristiana J. Silva. "Global Stability Condition for the Disease-Free Equilibrium Point of Fractional Epidemiological Models." Axioms 10, no. 4 (September 25, 2021): 238. http://dx.doi.org/10.3390/axioms10040238.

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In this paper, we present a new result that allows for studying the global stability of the disease-free equilibrium point when the basic reproduction number is less than 1, in the fractional calculus context. The method only involves basic linear algebra and can be easily applied to study global asymptotic stability. After proving some auxiliary lemmas involving the Mittag–Leffler function, we present the main result of the paper. Under some assumptions, we prove that the disease-free equilibrium point of a fractional differential system is globally asymptotically stable. We then exemplify the procedure with some epidemiological models: a fractional-order SEIR model with classical incidence function, a fractional-order SIRS model with a general incidence function, and a fractional-order model for HIV/AIDS.

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Kohaupt, L. "New upper bounds for free linear and nonlinear vibration systems with applications of the differential calculus of norms." Applied Mathematical Modelling 28, no. 4 (April 2004): 367–88. http://dx.doi.org/10.1016/j.apm.2003.08.004.

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Garcı´a, J., and E. On˜ate. "An Unstructured Finite Element Solver for Ship Hydrodynamics Problems." Journal of Applied Mechanics 70, no. 1 (January 1, 2003): 18–26. http://dx.doi.org/10.1115/1.1530631.

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A stabilized semi-implicit fractional step algorithm based on the finite element method for solving ship wave problems using unstructured meshes is presented. The stabilized governing equations for the viscous incompressible fluid and the free surface are derived at a differential level via a finite calculus procedure. This allows us to obtain a stabilized numerical solution scheme. Some particular aspects of the problem solution, such as the mesh updating procedure and the transom stern treatment, are presented. Examples of the efficiency of the semi-implicit algorithm for the analysis of ship hydrodynamics problems are presented.

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PERRINE, SERGE. "MONODROMY ARISING FROM THE MARKOFF THEORY." International Journal of Modern Physics B 20, no. 11n13 (May 20, 2006): 1819–32. http://dx.doi.org/10.1142/s0217979206034339.

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In a former work, recalling what the Markoff theory is, we summarized some existing links with the group GL(2, ℤ) of 2 × 2 matrices. We also quoted the relation with conformal punctured toruses. The monodromy representation of the Poincaré group of such a torus was considered. Here we explicit the corresponding solution of the associated Riemann-Hilbert problem, and the resulting Fuchs differential equation. We precisely describe how the calculus runs. The main result is the description of a complete family of Fuchs differential equations with, as the monodromy group, the free group with two generators. We also identify a link with some eigenvalues of a Laplacian. The introduction explains the links that we see with information and computation theory (classical or quantum).

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Rahimi, Zaher, Wojciech Sumelka, and Xiao-Jun Yang. "Linear and non-linear free vibration of nano beams based on a new fractional non-local theory." Engineering Computations 34, no. 5 (July 3, 2017): 1754–70. http://dx.doi.org/10.1108/ec-07-2016-0262.

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Purpose Recently, a new formulation has been introduced for non-local mechanics in terms of fractional calculus. Fractional calculus is a branch of mathematical analysis that studies the differential operators of an arbitrary (real or complex) order and is used successfully in various fields such as mathematics, science and engineering. The purpose of this paper is to introduce a new fractional non-local theory which may be applicable in various simple or complex mechanical problems. Design/methodology/approach In this paper (by using fractional calculus), a fractional non-local theory based on the conformable fractional derivative (CFD) definition is presented, which is a generalized form of the Eringen non-local theory (ENT). The theory contains two free parameters: the fractional parameter which controls the stress gradient order in the constitutive relation and could be an integer and a non-integer and the non-local parameter to consider the small-scale effect in the micron and the sub-micron scales. The non-linear governing equation is solved by the Galerkin and the parameter expansion methods. The non-linearity of the governing equation is due to the presence of von-Kármán non-linearity and CFD definition. Findings The theory has been used to study linear and non-linear free vibration of the simply-supported (S-S) and the clamped-free (C-F) nano beams and then the influence of the fractional and the non-local parameters has been shown on the linear and non-linear frequency ratio. Originality/value A new parameter of the theory (the fractional parameter) makes the modeling more fixable – this model can conclude all of integer and non-integer operators and is not limited to special operators such as ENT. In other words, it allows us to use more sophisticated mathematics to model physical phenomena. On the other hand, in the comparison of classic fractional non-local theory, the theory applicable in various simple or complex mechanical problems may be used because of simpler forms of the governing equation owing to the use of CFD definition.

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Vargas, Francisco. "VIRTUAL LABORATORIES AS STRATEGY FOR TEACHING IMPROVEMENT IN MATH SCIENCES AND ENGINEERING IN BOLIVIA." International Journal of Engineering Education 2, no. 1 (June 15, 2020): 52–62. http://dx.doi.org/10.14710/ijee.2.1.52-62.

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The vertiginous technological advancement has made necessary the use of computersoftware that contributes to the improvement of teaching in math sciences and engineering.It is in this context that the last five years the strategy presented in this article has been disseminatedin the main universities of Bolivia, a country where the schools have not yet been ableto offer basic disciplines such as calculus, matrix algebra, physics and/or differential equationsto solve problems considering applicative aspects. To establish this connection, it is necessaryto deduce differential equations associated with practical problems, solve these equationswith different numerical algorithms, and establish the concept of simulation to later introducelanguages like Python/VPython free of license to elaborate Virtual Laboratories that allow obtainingthe solutions in two and three dimensions. The classical problems addressed for thispurpose are the satellite of two degrees of freedom and the inverted pendulum.

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Kohaupt, L. "Two-sided bounds for the asymptotic behaviour of free nonlinear vibration systems with application of the differential calculus of norms." International Journal of Computer Mathematics 87, no. 3 (March 2010): 653–67. http://dx.doi.org/10.1080/00207160802166481.

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AGOSTINI, ALESSANDRA. "COVARIANT FORMULATION OF NOETHER'S THEOREM FOR κ-MINKOWSKI TRANSLATIONS." International Journal of Modern Physics A 24, no. 07 (March 20, 2009): 1333–58. http://dx.doi.org/10.1142/s0217751x09042943.

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The problem of finding a formulation of Noether's theorem in noncommutative geometry is very important in order to obtain conserved currents and charges for particles in noncommutative space–times. In this paper, we formulate Noether's theorem for translations of κ-Minkowski noncommutative space–time on the basis of the five-dimensional κ-Poincaré covariant differential calculus. We focus our analysis on the simple case of free scalar theory. We obtain five conserved Noether currents, which give rise to five energy–momentum charges. By applying our result to plane waves it follows that the energy–momentum charges satisfy a special-relativity dispersion relation with a generalized mass given by the fifth charge. In this paper, we provide also a rigorous derivation of the equation of motion from Hamilton's principle in noncommutative space–time, which is necessary for the Noether analysis.

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Elcrat, Alan, and Bartosz Protas. "A framework for linear stability analysis of finite-area vortices." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2152 (April 8, 2013): 20120709. http://dx.doi.org/10.1098/rspa.2012.0709.

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In this investigation, we revisit the question of linear stability analysis of two-dimensional steady Euler flows characterized by the presence of compact regions with constant vorticity embedded in a potential flow. We give a complete derivation of the linearized perturbation equation which, recognizing that the underlying equilibrium problem is of a free-boundary type, is carried out systematically using methods of shape-differential calculus. Particular attention is given to the proper linearization of contour integrals describing vortex induction. The thus obtained perturbation equation is validated by analytically deducing from it stability analyses of the circular vortex, originally due to Kelvin, and of the elliptic vortex, originally due to Love, as special cases. We also propose and validate a spectrally accurate numerical approach to the solution of the stability problem for vortices of general shape in which all singular integrals are evaluated analytically.

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Reusken, Arnold. "Stream function formulation of surface Stokes equations." IMA Journal of Numerical Analysis 40, no. 1 (October 29, 2018): 109–39. http://dx.doi.org/10.1093/imanum/dry062.

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AbstractIn this paper we present a derivation of the surface Helmholtz decomposition, discuss its relation to the surface Hodge decomposition and derive a well-posed stream function formulation of a class of surface Stokes problems. We consider a $C^2$ connected (not necessarily simply connected) oriented hypersurface $\varGamma \subset \mathbb{R}^3$ without boundary. The surface gradient, divergence, curl and Laplace operators are defined in terms of the standard differential operators of the ambient Euclidean space $\mathbb{R}^3$. These representations are very convenient for the implementation of numerical methods for surface partial differential equations. We introduce surface $\mathbf H({\mathop{\rm div}}_{\varGamma})$ and $\mathbf H({\mathop{\rm curl}}_{\varGamma})$ spaces and derive useful properties of these spaces. A main result of the paper is the derivation of the Helmholtz decomposition, in terms of these surface differential operators, based on elementary differential calculus. As a corollary of this decomposition we obtain that for a simply connected surface to every tangential divergence-free velocity field there corresponds a unique scalar stream function. Using this result the variational form of the surface Stokes equation can be reformulated as a well-posed variational formulation of a fourth-order equation for the stream function. The latter can be rewritten as two coupled second-order equations, which form the basis for a finite element discretization. A particular finite element method is explained and the results of a numerical experiment with this method are presented.

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FIORE, GAETANO. "ON THE HERMITICITY OF q-DIFFERENTIAL OPERATORS AND FORMS ON THE QUANTUM EUCLIDEAN SPACES $\mathbb{R}_q^N$." Reviews in Mathematical Physics 18, no. 01 (February 2006): 79–117. http://dx.doi.org/10.1142/s0129055x06002590.

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We show that the complicated ⋆-structure characterizing for positive q the Uqso(N)-covariant differential calculus on the noncommutative manifold [Formula: see text] boils down to similarity transformations involving the ribbon element of a central extension of Uqso(N) and its formal square root ṽ. Subspaces of the spaces of functions and of p-forms on [Formula: see text] are made into Hilbert spaces by introducing non-conventional "weights" in the integrals defining the corresponding scalar products, namely suitable positive-definite q-pseudodifferential operators ṽ′±1 realizing the action of ṽ±1; this serves to make the partial q-derivatives anti-hermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the "radial coordinate" r. Unless we choose a constant m, then the square-integrables functions/forms must fulfill an additional condition, namely, their analytic continuations to the complex r plane can have poles only on the sites of some special lattice. Among the functions naturally selected by this condition there are q-special functions with "quantized" free parameters.

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Hu, Langhua, Duan Chen, and Guo-Wei Wei. "High-order fractional partial differential equation transform for molecular surface construction." Computational and Mathematical Biophysics 1 (December 20, 2012): 1–25. http://dx.doi.org/10.2478/mlbmb-2012-0001.

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AbstractFractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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34

Barhorst, A. A., and L. J. Everett. "Contact/Impact in Hybrid Parameter Multiple Body Mechanical Systems." Journal of Dynamic Systems, Measurement, and Control 117, no. 4 (December 1, 1995): 559–69. http://dx.doi.org/10.1115/1.2801115.

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The multiple motion regime (free/constrained) dynamics of hybrid parameter multiple body (HPMB) systems is addressed. Impact response has characteristically been analyzed using impulse-momentum techniques. Unfortunately, the classical methods for modeling complex HPMB systems are energy based and have proven ineffectual at modeling impact. The problems are exacerbated by the problematic nature of time varying constraint conditions. This paper outlines the reformulation of a recently developed HPMB system modeling methodology into an impulse-momentum formulation, which systematically handles the constraints and impact. The starting point for this reformulation is a variational calculus based methodology. The variational roots of the methodology allows rigorous equation formulation which includes the complete nonlinear hybrid differential equations and boundary conditions. Because the methodology presented in this paper is formulated in the constraint-free subspace of the configuration space, both holonomic and nonholonomic constraints are automatically satisfied. As a result, the constraint-addition/deletion algorithms are not needed. Generalized forces of constraint can be directly calculated via the methodology, so the condition for switching from one motion regime to another is readily determined. The resulting equations provides a means to determine after impact velocities (and velocity fields for distributed bodies) which provide the after collision initial conditions. Finally the paper demonstrates, via example, how to apply the methodology to contact/impact in robotic manipulators and structural systems.

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Paseka, Alex, and Aerambamoorthy Thavaneswaran. "Bond valuation for generalized Langevin processes with integrated Lévy noise." Journal of Risk Finance 18, no. 5 (November 20, 2017): 541–63. http://dx.doi.org/10.1108/jrf-09-2016-0125.

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Purpose Recently, Stein et al. (2016) studied theoretical properties and parameter estimation of continuous time processes derived as solutions of a generalized Langevin equation (GLE). In this paper, the authors extend the model to a wider class of memory kernels and then propose a bond and bond option valuation model based on the extension of the generalized Langevin process of Stein et al. (2016). Design/methodology/approach Bond and bond option pricing based on the proposed interest rate models presents new difficulties as the standard partial differential equation method of stochastic calculus for bond pricing cannot be used directly. The authors obtain bond and bond option prices by finding the closed form expression of the conditional characteristic function of the integrated short rate process driven by a general Lévy noise. Findings The authors obtain zero-coupon default-free bond and bond option prices for short rate models driven by a variety of Lévy processes, which include Vasicek model and the short rate model obtained by solving a second-order Langevin stochastic differential equation (SDE) as special cases. Originality/value Bond and bond option pricing plays an important role in capital markets and risk management. In this paper, the authors derive closed form expressions for bond and bond option prices for a wider class of interest rate models including second-order SDE models. Closed form expressions may be especially instrumental in facilitating parameter estimation in these models.

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Ugol’nikov, O., B. Demianchuk, N. Kolesnychenko, and O. Malinovsky. "EXACT SOLUTION OF TASKS OF DYNAMIC MODELING OF PROCESSES IN SYSTEMS OF TRANSPORT LOGISTIC." Collection of scientific works of Odesa Military Academy 2, no. 12 (December 27, 2019): 5–13. http://dx.doi.org/10.37129/2313-7509.2019.12.2.5-13.

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The dynamic models of processes in transport logistics systems are considered. In the literature, such complex systems as the military transport logistics system or the combat vehicle support system are often modeled as a set of typical system states. These states are interconnected by a large number of transitions of a given intensity, which are carried out with given probabilities. Graphically, this is represented using the so-called graph of states and transitions, and the probabilities of the system being in a particular state are the subject of research in such a graph. The methods available in the literature for studying the dynamic characteristics of state graphs and transitions are analyzed. A description of the changes in probabilities as a function of time is made using systems of differential equations, usually linear. Based on practical requirements, approximate solutions to such systems are usually sought. One of the approximate methods is the decomposition method, in which, instead of a system of coupled equations, a set of independent equations is considered, the solution of which is not difficult. The results of the solution have an accuracy satisfactory from the point of view of practical use. The assumptions based on which the decomposition method can be used are analyzed. It is shown that the accuracy of the obtained results substantially depends on the given initial conditions and should increase over time, when this dependence weakens. A method is proposed for the exact solution of a system of differential equations, free of any assumptions. The use of operational calculus is substantiated, which reduces the solution of a system of linear differential equations to the solution of a system of linear algebraic equations for unknown images of the sought-for Laplace functions. The method is used to describe the process of technical support for the restoration of the transport flow of military logistics. The boundaries of the possibility of applying the results of a simpler approximate solution are established.

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Strelkovskaya, Irina, Irina Solovskaya, and Anastasiya Makoganiuk. "A Study of the Extremum of the Total Energy of the Selective Signals Constructed by Quadratic Splines." Periodica Polytechnica Electrical Engineering and Computer Science 63, no. 1 (December 20, 2018): 30–36. http://dx.doi.org/10.3311/ppee.12457.

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The development of mobile communication networks in the direction of 5G networks implies the use of the radio interface that is based on new signal-code structures, the choice of which will determine their further development and the ability of operators to provide innovative services. The use of quadratic splines for the synthesis of selective signals with a finite spectrum, free from intersymbol interference, is proposed. An analytical expression for the synthesized signal in the time and frequency domains is obtained. A study was made of the dependence of the total energy of a selective signal on the parameters of a quadratic spline, which is used to interpolate the spectral density in the transition region using the methods of differential calculus of functions of several variables. To study the extremum of the total energy of a selective signal, the parameters of the width of the transition area α was used and the coefficient of rounding the spectrum ρ, the variation of whose spectral density allowed us to establish the limits of the change in the total energy of the signal in question. Conducted studies allow us to synthesize the signal and formulate recommendations on how by changing the parameters of the signal, you can get a signal with the desired properties. This will allow to obtain the optimal waveform in accordance with the selected criteria, providing for the required energy performance of the signal in the radio interface of 5G networks.

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Bonotto, E. M., M. Federson, and P. Muldowney. "The Black–Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral." Proceedings of the Singapore National Academy of Science 15, no. 01 (March 2021): 45–59. http://dx.doi.org/10.1142/s2591722621400068.

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The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka[Formula: see text] formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

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Solomon, Reed. "Ordered Groups: A Case Study in Reverse Mathematics." Bulletin of Symbolic Logic 5, no. 1 (March 1999): 45–58. http://dx.doi.org/10.2307/421140.

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The fundamental question in reverse mathematics is to determine which set existence axioms are required to prove particular theorems of mathematics. In addition to being interesting in their own right, answers to this question have consequences in both effective mathematics and the foundations of mathematics. Before discussing these consequences, we need to be more specific about the motivating question.Reverse mathematics is useful for studying theorems of either countable or essentially countable mathematics. Essentially countable mathematics is a vague term that is best explained by an example. Complete separable metric spaces are essentially countable because, although the spaces may be uncountable, they can be understood in terms of a countable basis. Simpson (1985) gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. Reverse mathematics is less suited to theorems of abstract functional analysis, abstract set theory, universal algebra, or general topology.Section 2 introduces the major subsystems of second order arithmetic used in reverse mathematics: RCA0, WKL0, ACA0, ATR0 and – CA0. Sections 3 through 7 consider various theorems of ordered group theory from the perspective of reverse mathematics. Among the results considered are theorems on ordered quotient groups (including an equivalent of ACA0), groups and semigroup conditions which imply orderability (WKL0), the orderability of free groups (RCA0), Hölder's Theorem (RCA0), Mal'tsev's classification of the order types of countable ordered groups ( – CA0)

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Kosulina, Nataliaya, Stanislav Kosulin, Kostiantyn Korshunov, Tetyana Nosova, and Yana Nosova. "DETERMINATION OF HYDRODYNAMIC PARAMETERS OF THE SEALED PRESSURE EXTRACTOR." Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska 11, no. 2 (June 30, 2021): 44–47. http://dx.doi.org/10.35784/iapgos.2657.

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The subject matter of the article: Sealed extractor with pressure. The goal of the work: Determination of hydrodynamic parameters of the sealed extractor with pressure. The following tasks were solved in the article: Theoretical research on the creation of a sealed extractor with pressure. It is need to development of ways of implementation and practical recommendations for the given technical solutions in the experimental sample. The following methods are used: Mathematical modeling, differential and integral calculus, experimental research methods. The following results were obtained: The processes occurring in sealed extractors are described mathematically. Parameters that affect the performance of aggregates are determined. Conclusions: As a result of the analysis of the technological process and equipment used in the factories for primary processing of wool, shortcomings and problems are identified and means for their elimination are proposed. It is proposed to use small-sized equipment to work on waste-free technology based on a hydrodynamic pressure extractor. Extraction as an efficient mass transfer process for removing organic components from aqueous solutions has the advantages of low operating temperatures and efficiency. The design features of the sealed pressure extractor are as follows: high angular velocities, the moment of inertia of rotating details, powerful pressure, the presence of nodes that provide a supply and discharge of liquids, tightness. The kinematic and geometric parameters of the rotor affect the sealed extractors’ performance). In sealed extractors, the heavy fraction in the field of centrifugal forces will accumulate on a large radius of the inner side of the rotor and for its movement it is necessary to create an excess pressure at the extractor inlet.

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CARTER, J. SCOTT, and MASAHICO SAITO. "SET-THEORETIC YANG–BAXTER SOLUTIONS VIA FOX CALCULUS." Journal of Knot Theory and Its Ramifications 15, no. 08 (October 2006): 949–56. http://dx.doi.org/10.1142/s0218216506004877.

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We construct solutions to the set–theoretic Yang–Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.

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42

CERCHIAI, B. L., J. MADORE, and G. FIORE. "FRAME FORMALISM FOR THE N-DIMENSIONAL QUANTUM EUCLIDEAN SPACES." International Journal of Modern Physics B 14, no. 22n23 (September 20, 2000): 2305–14. http://dx.doi.org/10.1142/s0217979200001849.

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We sketch our application1of a non-commutative version of the Cartan "moving-frame" formalism to the quantum Euclidean space [Formula: see text]the space which is covariant under the action of the quantum group SOq(N). For each of the two covariant differential calculi over [Formula: see text] based on the R-matrix formalism, we summarize our construction of a frame, the dual inner derivations, a metric and two torsion-free almost metric compatible covariant derivatives with a vanishing curvature. To obtain these results we have developed a technique which fully exploits the quantum group covariance of [Formula: see text]. We first find a frame in the larger algebra [Formula: see text]. Then we define homomorphisms from [Formula: see text] to [Formula: see text] which we use to project this frame in [Formula: see text].

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Hassell Sweatman, C. Z. W., G. C. Wake, A. B. Pleasants, C. A. McLean, and A. M. Sheppard. "Linear Models with Response Functions Based on the Laplace Distribution: Statistical Formulae and Their Application to Epigenomics." ISRN Probability and Statistics 2013 (November 2, 2013): 1–22. http://dx.doi.org/10.1155/2013/496180.

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The statistical application considered here arose in epigenomics, linking the DNA methylation proportions measured at specific genomic sites to characteristics such as phenotype or birth order. It was found that the distribution of errors in the proportions of chemical modification (methylation) on DNA, measured at CpG sites, may be successfully modelled by a Laplace distribution which is perturbed by a Hermite polynomial. We use a linear model with such a response function. Hence, the response function is known, or assumed well estimated, but fails to be differentiable in the classical sense due to the modulus function. Our problem was to estimate coefficients for the linear model and the corresponding covariance matrix and to compare models with varying numbers of coefficients. The linear model coefficients may be found using the (derivative-free) simplex method, as in quantile regression. However, this theory does not yield a simple expression for the covariance matrix of the coefficients of the linear model. Assuming response functions which are 𝒞2 except where the modulus function attains zero, we derive simple formulae for the covariance matrix and a log-likelihood ratio statistic, using generalized calculus. These original formulae enable a generalized analysis of variance and further model comparisons.

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Haug, Edward J. "Multibody Dynamics on Differentiable Manifolds." Journal of Computational and Nonlinear Dynamics 16, no. 4 (February 24, 2021). http://dx.doi.org/10.1115/1.4049995.

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Abstract Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.

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Bochniak, Arkadiusz, Andrzej Sitarz, and Pawel Zalecki. "Riemannian Geometry of a Discretized Circle and Torus." Symmetry, Integrability and Geometry: Methods and Applications, December 23, 2020. http://dx.doi.org/10.3842/sigma.2020.143.

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We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and metric compatibility condition in general and show that there are several classes of solutions, out of which only special ones are compatible with a metric that gives a Hilbert C∗-module structure on the space of the one-forms. We compute curvature and scalar curvature for these metrics and find their continuous limits.

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Walker, Shawn W. "Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary." Computational Methods in Applied Mathematics, June 23, 2021. http://dx.doi.org/10.1515/cmam-2020-0123.

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Abstract We establish several useful estimates for a non-conforming 2-norm posed on piecewise linear surface triangulations with boundary, with the main result being a Poincaré inequality. We also obtain equivalence of the non-conforming 2-norm posed on the true surface with the norm posed on a piecewise linear approximation. Moreover, we allow for free boundary conditions. The true surface is assumed to be C 2 , 1 C^{2,1} when free conditions are present; otherwise, C 2 C^{2} is sufficient. The framework uses tools from differential geometry and the closest point map (see [G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, Berlin (1988), 142–155]) for approximating the full surface Hessian operator. We also present a novel way of applying the closest point map when dealing with surfaces with boundary. Connections with surface finite element methods for fourth-order problems are also noted.

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Walker, Shawn W. "The Kirchhoff plate equation on surfaces: the surface Hellan–Herrmann–Johnson method." IMA Journal of Numerical Analysis, August 20, 2021. http://dx.doi.org/10.1093/imanum/drab062.

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Abstract We present a mixed finite element method for approximating a fourth-order elliptic partial differential equation (PDE), the Kirchhoff plate equation, on a surface embedded in ${\mathbb {R}}^{3}$, with or without boundary. Error estimates are given in mesh-dependent norms that account for the surface approximation and the approximation of the surface PDE. The method is built on the classic Hellan–Herrmann–Johnson method (for flat domains), and convergence is established for $C^{k+1}$ surfaces, with degree $k$ (Lagrangian, parametrically curved) approximation of the surface, for any $k \geqslant 1$. Mixed boundary conditions are allowed, including clamped, simply-supported and free conditions; if free conditions are present then the surface must be at least $C^{2,1}$. The framework uses tools from differential geometry and is directly related to the seminal work of Dziuk, G. (1988) Finite elements for the Beltrami operator on arbitrary surfaces. Partial Differential Equations and Calculus of Variations, vol. 1357 (S. Hildebrandt & R. Leis eds). Berlin, Heidelberg: Springer, pp. 142–155. for approximating the Laplace–Beltrami equation. The analysis here is the first to handle the full surface Hessian operator directly. Numerical examples are given on nontrivial surfaces that demonstrate our convergence estimates. In addition, we show how the surface biharmonic equation can be solved with this method.

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Faraji Oskouie, M., R. Ansari, and H. Rouhi. "Investigating vibrations of viscoelastic fluid-conveying carbon nanotubes resting on viscoelastic foundation using a nonlocal fractional Timoshenko beam model." Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, June 18, 2020, 239779142093170. http://dx.doi.org/10.1177/2397791420931701.

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On the basis of fractional viscoelasticity, the size-dependent free-vibration response of viscoelastic carbon nanotubes conveying fluid and resting on viscoelastic foundation is studied in this article. To this end, a nonlocal Timoshenko beam model is developed in the context of fractional calculus. Hamilton’s principle is applied in order to obtain the fractional governing equations including nanoscale effects. The Kelvin–Voigt viscoelastic model is also used for the constitutive equations. The free-vibration problem is solved using two methods. In the first method, which is limited to the simply supported boundary conditions, the Galerkin technique is employed for discretizing the spatial variables and reducing the governing equations to a set of ordinary differential equations on the time domain. Then, the Duffing-type time-dependent equations including fractional derivatives are solved via fractional integrator transfer functions. In the second method, which can be utilized for carbon nanotubes with different types of boundary conditions, the generalized differential quadrature technique is used for discretizing the governing equations on spatial grids, whereas the finite difference technique is used on the time domain. In the results, the influences of nonlocality, geometrical parameters, fractional derivative orders, viscoelastic foundation, and fluid flow velocity on the time responses of carbon nanotubes are analyzed.

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Journal articles on the topic 'Free differential calculus'

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