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Journal articles on the topic 'Calculus of variations'

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Published: 4 June 2021
Last updated: 29 July 2024

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1

De Lellis, Camillo, Gerhard Huisken, and Robert Jerrard. "Calculus of Variations." Oberwolfach Reports 9, no. 3 (2012): 2205–68. http://dx.doi.org/10.4171/owr/2012/36.

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2

Brendle, Simon, Camillo De Lellis, and Robert Jerrard. "Calculus of Variations." Oberwolfach Reports 11, no. 3 (2014): 1801–60. http://dx.doi.org/10.4171/owr/2014/33.

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3

Brendle, Simon, Alessio Figalli, Robert Jerrard, and Neshan Wickramasekera. "Calculus of Variations." Oberwolfach Reports 13, no. 3 (2016): 1943–2008. http://dx.doi.org/10.4171/owr/2016/34.

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4

Figalli, Alessio, Robert Kohn, Tatiana Toro, and Neshan Wickramasekera. "Calculus of Variations." Oberwolfach Reports 15, no. 3 (August 26, 2019): 2077–156. http://dx.doi.org/10.4171/owr/2018/35.

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5

Figalli, Alessio, Robert V. Kohn, Tatiana Toro, and Neshan Wickramasekera. "Calculus of Variations." Oberwolfach Reports 17, no. 2 (July 1, 2021): 1139–96. http://dx.doi.org/10.4171/owr/2020/22.

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6

Bronsard, Lia, László Székelyhidi Jr., Yoshihiro Tonegawa, and Tatiana Toro. "Calculus of Variations." Oberwolfach Reports 19, no. 3 (June 13, 2023): 2129–93. http://dx.doi.org/10.4171/owr/2022/37.

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7

Dods, Victor. "Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus." Mathematics 10, no. 18 (September 6, 2022): 3231. http://dx.doi.org/10.3390/math10183231.

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In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived.
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8

Chambers, Ll G., N. I. Akhiezer, Michael E. Alferieff, and B. Dacorogna. "The Calculus of Variations." Mathematical Gazette 74, no. 468 (June 1990): 191. http://dx.doi.org/10.2307/3619399.

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9

Gondran, M., and R. Hoblos. "Complex Calculus of Variations." IFAC Proceedings Volumes 34, no. 13 (August 2001): 727–30. http://dx.doi.org/10.1016/s1474-6670(17)39079-1.

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10

Neves, V. "Nonstandard Calculus of Variations." Journal of Mathematical Sciences 120, no. 1 (March 2004): 940–54. http://dx.doi.org/10.1023/b:joth.0000013557.48018.2a.

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11

Barron, E. N., and W. Liu. "Calculus of variations inL ∞." Applied Mathematics & Optimization 35, no. 3 (May 1997): 237–63. http://dx.doi.org/10.1007/bf02683330.

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12

Chrastina, Jan. "Examples from the calculus of variations. I. Nondegenerate problems." Mathematica Bohemica 125, no. 1 (2000): 55–76. http://dx.doi.org/10.21136/mb.2000.126263.

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13

Chrastina, Jan. "Examples from the calculus of variations. IV. Concluding review." Mathematica Bohemica 126, no. 4 (2001): 691–710. http://dx.doi.org/10.21136/mb.2001.134113.

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14

Massa, Enrico, Danilo Bruno, Gianvittorio Luria, and Enrico Pagani. "Geometric constrained variational calculus. II: The second variation (Part I)." International Journal of Geometric Methods in Modern Physics 13, no. 01 (January 2016): 1550132. http://dx.doi.org/10.1142/s0219887815501327.

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Within the geometrical framework developed in [Geometric constrained variational calculus. I: Piecewise smooth extremals, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550061], the problem of minimality for constrained calculus of variations is analyzed among the class of differentiable curves. A fully covariant representation of the second variation of the action functional, based on a suitable gauge transformation of the Lagrangian, is explicitly worked out. Both necessary and sufficient conditions for minimality are proved, and reinterpreted in terms of Jacobi fields.
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15

Barron, E. N., and W. Liu. "Calculus of Variations in L ∞." Applied Mathematics and Optimization 35, no. 3 (May 1, 1997): 237–63. http://dx.doi.org/10.1007/s002459900047.

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16

Hildebrandt, Stefan. "The calculus of variations today." Mathematical Intelligencer 11, no. 4 (September 1989): 50–60. http://dx.doi.org/10.1007/bf03025887.

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17

Saunders, David J. "Jets and the variational calculus." Communications in Mathematics 29, no. 1 (April 30, 2021): 91–114. http://dx.doi.org/10.2478/cm-2021-0004.

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Abstract We review the approach to the calculus of variations using Ehresmann’s theory of jets. We describe different types of jet manifold, different types of variational problem and different cohomological structures associated with such problems.
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18

Torres, Delfim F. M. "On a Non-Newtonian Calculus of Variations." Axioms 10, no. 3 (July 29, 2021): 171. http://dx.doi.org/10.3390/axioms10030171.

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The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.
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19

Zine, Houssine, and Delfim F. M. Torres. "A Stochastic Fractional Calculus with Applications to Variational Principles." Fractal and Fractional 4, no. 3 (August 1, 2020): 38. http://dx.doi.org/10.3390/fractalfract4030038.

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We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.
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20

Odzijewicz, Tatiana, Agnieszka B. Malinowska, and Delfim F. M. Torres. "Generalized fractional calculus with applications to the calculus of variations." Computers & Mathematics with Applications 64, no. 10 (November 2012): 3351–66. http://dx.doi.org/10.1016/j.camwa.2012.01.073.

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21

Chrastina, Jan. "Solution of the inverse problem of the calculus of variations." Mathematica Bohemica 119, no. 2 (1994): 157–201. http://dx.doi.org/10.21136/mb.1994.126079.

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22

Chrastina, Jan. "Examples from the calculus of variations. II. A degenerate problem." Mathematica Bohemica 125, no. 2 (2000): 187–97. http://dx.doi.org/10.21136/mb.2000.125951.

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23

Bandyopadhyay, Saugata, Bernard Dacorogna, and Swarnendu Sil. "Calculus of variations with differential forms." Journal of the European Mathematical Society 17, no. 4 (2015): 1009–39. http://dx.doi.org/10.4171/jems/525.

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24

Nurbekyan, L. "Calculus of variations in Hilbert spaces." Journal of Contemporary Mathematical Analysis 47, no. 3 (May 2012): 148–60. http://dx.doi.org/10.3103/s1068362312030053.

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25

Chorlton, Frank, and U. Brechtken-Manderscheid. "Introduction to the Calculus of Variations." Mathematical Gazette 78, no. 481 (March 1994): 96. http://dx.doi.org/10.2307/3619470.

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26

Dickey, Leonid A. "Do Dogs Know Calculus of Variations?" College Mathematics Journal 37, no. 1 (January 1, 2006): 20. http://dx.doi.org/10.2307/27646267.

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27

Dickey, Leonid A. "Do Dogs Know Calculus of Variations?" College Mathematics Journal 37, no. 1 (January 2006): 20–23. http://dx.doi.org/10.1080/07468342.2006.11922162.

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28

Carriero, Michele, Antonio Leaci, and Franco Tomarelli. "Calculus of variations and image segmentation." Journal of Physiology-Paris 97, no. 2-3 (March 2003): 343–53. http://dx.doi.org/10.1016/j.jphysparis.2003.09.008.

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29

Tan, H. H., and R. B. Potts. "A Discrete calculus of variations algorithm." Bulletin of the Australian Mathematical Society 38, no. 3 (December 1988): 365–71. http://dx.doi.org/10.1017/s0004972700027726.

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30

Maalaoui, Ali. "Bubbling phenomena in calculus of variations." Arabian Journal of Mathematics 6, no. 3 (December 26, 2016): 213–37. http://dx.doi.org/10.1007/s40065-016-0157-x.

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31

Ramos, J. l. "Introduction to the calculus of variations." Applied Mathematical Modelling 17, no. 1 (January 1993): 53. http://dx.doi.org/10.1016/0307-904x(93)90134-3.

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32

Serfaty, Sylvia. "Lagrange and the calculus of variations." Lettera Matematica 2, no. 1-2 (May 3, 2014): 39–46. http://dx.doi.org/10.1007/s40329-014-0049-x.

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33

Bailey, C. D. "Dynamics and the calculus of variations." Computer Methods in Applied Mechanics and Engineering 60, no. 3 (February 1987): 275–87. http://dx.doi.org/10.1016/0045-7825(87)90135-6.

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34

Gift, S. J. G. "Contributions to the calculus of variations." Journal of Optimization Theory and Applications 52, no. 1 (January 1987): 25–51. http://dx.doi.org/10.1007/bf00938463.

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35

Bögelein, Verena, Frank Duzaar, and Paolo Marcellini. "Existence of evolutionary variational solutions via the calculus of variations." Journal of Differential Equations 256, no. 12 (June 2014): 3912–42. http://dx.doi.org/10.1016/j.jde.2014.03.005.

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36

Chrastina, Jan. "Examples from the calculus of variations. III. Legendre and Jacobi conditions." Mathematica Bohemica 126, no. 1 (2001): 93–111. http://dx.doi.org/10.21136/mb.2001.133926.

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37

Matějík, M. "Calculus of variations and its application to division of forest land." Journal of Forest Science 50, No. 9 (January 11, 2012): 439–46. http://dx.doi.org/10.17221/4639-jfs.

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The paper deals with an application of the least squares method (LSM) for the purposes of division and evaluation of land. This method can be used in all cases with redundant number of measurements, in this case of segments of plots. From the mathematical aspect, the minimisation condition of the LSM is a standardised condition &sum; pvv = min., minimising the Euclidean norm ||v||<sub>E</sub> of an n-dimensional vector of residues of plot segments at simultaneous satisfaction of the given conditions. The traditional procedure of calculus of variations with the use of Lagrangian function is shown. If some additional conditions are included in the calculation, on the basis of the criteria presented in this article it is possible to evaluate the degree of deformation of the selected solution in relation to the measured quantities. The application of the method of adjustment of condition measurements may help solve the problems of parcel division on the basis of intersection of the parcel layers according to the real-estate cadastre and according to previous land records, valuation, typological, price and other map sources.
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38

Anza Hafsa, Omar, and Jean-Philippe Mandallena. "Γ-convergence of nonconvex integrals in Cheeger--Sobolev spaces and homogenization." Advances in Calculus of Variations 10, no. 4 (October 1, 2017): 381–405. http://dx.doi.org/10.1515/acv-2015-0053.

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39

Cellina, Arrigo. "Some Problems in the Calculus of Variations." Annales Mathematicae Silesianae 31, no. 1 (September 26, 2017): 5–55. http://dx.doi.org/10.1515/amsil-2017-0005.

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40

Shah, Yogendra Prasad. "A Study of Geodesic Equation From Variational Principle." Cognition 6, no. 1 (April 8, 2024): 147–51. http://dx.doi.org/10.3126/cognition.v6i1.64458.

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This paper employs variational principle in studying geodesic. In mathematical studies, a variational principle enables a problem to be solved employing calculus of variations that concerns seeking functions that increase the values of quantities that rely on those functions. For instance, the problem of ascertaining the shape of a hanging chain suspended at both ends can be solved using variational calculus. Hence, the variational principle is a function that lessens the gravitational potential energy of the chain. Geodesicis a procedure used in mathematics, specifically in Riemannian geometry that results in obtaining geodesics. Actually, these represent the paths of particles with no proper acceleration, their motion pleasing the geodesic equations. As the particles are subject to no appropriate acceleration, the geodesics generally signify the straightest path between two points in a bent spacetime. The article has investigated into the relation between the variation principle and geodesic equations that are particularly used in general relativity in physics as well.
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41

Kupershmidt, B. A. "An algebraic model of graded calculus of variations." Mathematical Proceedings of the Cambridge Philosophical Society 101, no. 1 (January 1987): 151–66. http://dx.doi.org/10.1017/s0305004100066494.

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The modern theory of integrable systems rests on two fundamental pillars: the classification of Lax [13] and zero-curvature equations [14, 1, 2]; and algebraic models of the classical calculus of variations [9,5] specialized to the residue calculus in modules of differential forms over rings of matrix pseudo-differential operators [9, 6]. Both these aspects of the theory are by now very well understood for integrable systems in one space dimension.
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42

Aksenenko, I. A. "SOLVING THE PROBLEM OF LONGITUDINAL BENDING OF A ROD BY METHODS OF DISCRETE CALCULUS OF VARIATIONS." Applied Mathematics and Control Sciences, no. 3 (December 15, 2022): 26–42. http://dx.doi.org/10.15593/2499-9873/2022.03.02.

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The question of the longitudinal bending of a rod consisting of rigid links connected by hinges is being studied. It is shown that, as in the classical version of a solid rod, the problem can be posed as a variational minimum energy problem, but given by a functional defined on a class of functions with a discrete domain of definition. The main provisions of the classical calculus of variations are transferred to functionals of this type: the formula of variation is found, a generalization of the main lemma of the calculus of variations is proved, an analogue of the Euler equation, which is a difference equation, is obtained. Applying the results obtained and the known properties of classical difference equations, we succeeded in solving an analogue of the Euler problem for two types of a hinge rod: for a rod consisting of links of the same length, and for an arbitrary choice of lengths of links. In both cases we find the critical Euler force, as well as the equation and the form of the deflection curve.
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43

Ioffe, Alexander D., and Alexander J. Zaslavski. "Variational Principles and Well-Posedness in Optimization and Calculus of Variations." SIAM Journal on Control and Optimization 38, no. 2 (January 2000): 566–81. http://dx.doi.org/10.1137/s0363012998335632.

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44

Kohan, MahdiNakhaie, and Hamid Behnam. "Denoising medical images using calculus of variations." Journal of Medical Signals & Sensors 1, no. 3 (2011): 5. http://dx.doi.org/10.4103/2228-7477.95413.

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45

Giannetti, Flavia. "Elliptic complexes in the calculus of variations." Dissertationes Mathematicae 418 (2003): 1–63. http://dx.doi.org/10.4064/dm418-0-1.

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46

Bandyopadhyay, Saugata, and Swarnendu Sil. "Exterior convexity and classical calculus of variations." ESAIM: Control, Optimisation and Calculus of Variations 22, no. 2 (March 4, 2016): 338–54. http://dx.doi.org/10.1051/cocv/2015007.

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47

Donaldson, Simon. "Karen Uhlenbeck and the Calculus of Variations." Notices of the American Mathematical Society 66, no. 03 (March 1, 2019): 1. http://dx.doi.org/10.1090/noti1806.

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48

Sil, Swarnendu. "Calculus of variations: A differential form approach." Advances in Calculus of Variations 12, no. 1 (January 1, 2019): 57–84. http://dx.doi.org/10.1515/acv-2016-0058.

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AbstractWe study integrals of the form {\int_{\Omega}f(d\omega_{1},\dots,d\omega_{m})}, where {m\geq 1} is a given integer, {1\leq k_{i}\leq n} are integers, {\omega_{i}} is a {(k_{i}-1)}-form for all {1\leq i\leq m} and {f:\prod_{i=1}^{m}\Lambda^{k_{i}}(\mathbb{R}^{n})\rightarrow\mathbb{R}} is a continuous function. We introduce the appropriate notions of convexity, namely vectorial ext. one convexity, vectorial ext. quasiconvexity and vectorial ext. polyconvexity. We prove weak lower semicontinuity theorems and weak continuity theorems and conclude with applications to minimization problems. These results generalize the corresponding results in both classical vectorial calculus of variations and the calculus of variations for a single differential form.
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49

Vyridís, Panayotis. "Bifurcation in calculus of variations with constraints." Acta Universitaria 23 (December 1, 2013): 27–31. http://dx.doi.org/10.15174/au.2013.583.

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We describe a variational problem on a domain of a plane under a constraint of geometrical character. We provide sufficient and necessary conditions for the existence of bifurcation points. The problem in 2 coordinate form, corresponds to a quasilinear elliptic boundary value problem. The problem provides a physical model for several applications referring to continuum media and membranes.
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50

Clarke, F. H., Yu S. Ledyaev, and J. B. Hiriart-Urruty. "Global optimality in the calculus of variations." Nonlinear Analysis: Theory, Methods & Applications 28, no. 7 (April 1997): 1187–92. http://dx.doi.org/10.1016/s0362-546x(97)82868-4.

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