Relevant bibliographies by topics / Calculus of variations. Multiple integrals

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Calculus of variations. Multiple integrals'

Author: Grafiati
Published: 4 June 2021
Last updated: 8 February 2022

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Journal articles on the topic "Calculus of variations. Multiple integrals"

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Bousquet, Pierre. "The Euler Equation in the Multiple Integrals Calculus of Variations." SIAM Journal on Control and Optimization 51, no. 2 (January 2013): 1047–62. http://dx.doi.org/10.1137/120882561.

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Arcoya, David, and Lucio Boccardo. "Critical points for multiple integrals of the calculus of variations." Archive for Rational Mechanics and Analysis 134, no. 3 (1996): 249–74. http://dx.doi.org/10.1007/bf00379536.

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Ball, J. M., and K. W. Zhang. "Lower semicontinuity of multiple integrals and the Biting Lemma." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 114, no. 3-4 (1990): 367–79. http://dx.doi.org/10.1017/s0308210500024483.

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SynopsisWeak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.
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Rund, Hanno. "LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS." Quaestiones Mathematicae 12, no. 2 (January 1989): 205–29. http://dx.doi.org/10.1080/16073606.1989.9632177.

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Rund, Hanno. "LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS." Quaestiones Mathematicae 12, no. 3 (January 1989): 315–39. http://dx.doi.org/10.1080/16073606.1989.9632186.

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Almeida, Ricardo, Agnieszka B. Malinowska, and Delfim F. M. Torres. "A fractional calculus of variations for multiple integrals with application to vibrating string." Journal of Mathematical Physics 51, no. 3 (2010): 033503. http://dx.doi.org/10.1063/1.3319559.

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Cesari, L., P. Brandi, and A. Salvadori. "Existence theorems for multiple integrals of the calculus of variations for discontinuous solutions." Annali di Matematica Pura ed Applicata 152, no. 1 (December 1988): 95–121. http://dx.doi.org/10.1007/bf01766143.

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Fonseca, Irene, and Giovanni Leoni. "On lower semicontinuity and relaxation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 3 (June 2001): 519–65. http://dx.doi.org/10.1017/s0308210500000998.

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Lower semicontinuity and relaxation results in BV are obtained for multiple integrals where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.
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Fonseca, Irene, and Giovanni Leoni. "On lower semicontinuity and relaxation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 3 (June 2001): 519–65. http://dx.doi.org/10.1017/s0308210501000245.

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Lower semicontinuity and relaxation results in BV are obtained for multiple integrals where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.
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Taheri, Ali. "Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 1 (February 2001): 155–84. http://dx.doi.org/10.1017/s0308210500000822.

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Let Ω ⊂ Rn be a bounded domain and let f : Ω × RN × RN×n → R. Consider the functional over the class of Sobolev functions W1,q(Ω;RN) (1 ≤ q ≤ ∞) for which the integral on the right is well defined. In this paper we establish sufficient conditions on a given function u0 and f to ensure that u0 provides an Lr local minimizer for I where 1 ≤ r ≤ ∞. The case r = ∞ is somewhat known and there is a considerable literature on the subject treating the case min(n, N) = 1, mostly based on the field theory of the calculus of variations. The main contribution here is to present a set of sufficient conditions for the case 1 ≤ r < ∞. Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of ‘directional convergence’.
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Dissertations / Theses on the topic "Calculus of variations. Multiple integrals"

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Zhang, Chengdian. "Calculus of variations with multiple integration." Bonn : [s.n.], 1989. http://catalog.hathitrust.org/api/volumes/oclc/20436929.html.

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Shahrokhi-Dehkordi, Mohammad Sadegh. "Topological methods for strong local minimizers and extremals of multiple integrals in the calculus of variations." Thesis, University of Sussex, 2011. http://sro.sussex.ac.uk/id/eprint/6913/.

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Let Ω ⊂ Rn be a bounded Lipschitz domain and consider the energy functional F[u, Ω] := ∫ Ω F(∇u(x)) dx, over the space Ap(Ω) := {u ∈ W 1,p(Ω, Rn): u|∂Ω = x, det ∇u> 0 a.e. in Ω}, where the integrand F : Mn×n → R is quasiconvex, sufficiently regular and satisfies a p-coercivity and p-growth for some exponent p ∈ [1, ∞[. A motivation for the study of above energy functional comes from nonlinear elasticity where F represents the elastic energy of a homogeneous hyperelastic material and Ap(Ω) represents the space of orientation preserving deformations of Ω fixing the boundary pointwise. The aim of this thesis is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of F and the relation it bares to the domain topology. Our work, building upon previous works of others, explicitly and quantitatively confirms the significant role of domain topology, and provides explicit and new examples as well as methods for constructing such maps. Our approach for constructing strong local minimizers is topological in nature and is based on defining suitable homotopy classes in Ap(Ω) (for p ≥ n), whereby minimizing F on each class results in, modulo technicalities, a strong local minimizer. Here we work on a prototypical example of a topologically non-trivial domain, namely, a generalised annulus, Ω= {x ∈ Rn : a< |x|
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Soneji, Parth. "Lower semicontinuity and relaxation in BV of integrals with superlinear growth." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:c7174516-588e-46ae-93dc-56d4a95f1e6f.

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Coine, Clément. "Continuous linear and bilinear Schur multipliers and applications to perturbation theory." Thesis, Bourgogne Franche-Comté, 2017. http://www.theses.fr/2017UBFCD074/document.

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Dans le premier chapitre, nous commençons par définir certains produits tensoriels et identifions leur dual. Nous donnons ensuite quelques propriétés des classes de Schatten. La fin du chapitre est dédiée à l’étude des espaces de Bochner à valeurs dans l'espace des opérateurs factorisables par un espace de Hilbert. Le deuxième chapitre est consacré aux multiplicateurs de Schur linéaires. Nous caractérisons les multiplicateurs bornés sur B(Lp, Lq) lorsque p est inférieur à q puis appliquons ce résultat pour obtenir de nouvelles relations d'inclusion entre espaces de multiplicateurs. Dans le troisième chapitre, nous caractérisons, au moyen de multiplicateurs de Schur linéaires, les multiplicateurs de Schur bilinéaires continus à valeurs dans l'espace des opérateurs à trace. Dans le quatrième chapitre, nous donnons divers résultats concernant les opérateurs intégraux multiples. En particulier, nous caractérisons les opérateurs intégraux triples à valeurs dans l'espace des opérateurs à trace puis nous donnons une condition nécessaire et suffisante pour qu'un opérateur intégral triple définisse une application complètement bornée sur le produit de Haagerup de l'espace des opérateurs compacts. Enfin, le cinquième chapitre est dédié à la résolution des problèmes de Peller. Nous commençons par étudier le lien entre opérateurs intégraux multiples et théorie de la perturbation pour le calcul fonctionnel des opérateurs autoadjoints pour finir par la construction de contre-exemples à ces problèmes
In the first chapter, we define some tensor products and we identify their dual space. Then, we give some properties of Schatten classes. The end of the chapter is dedicated to the study of Bochner spaces valued in the space of operators that can be factorized by a Hilbert space.The second chapter is dedicated to linear Schur multipliers. We characterize bounded multipliers on B(Lp, Lq) when p is less than q and then apply this result to obtain new inclusion relationships among spaces of multipliers.In the third chapter, we characterize, by means of linear Schur multipliers, continuous bilinear Schur multipliers valued in the space of trace class operators. In the fourth chapter, we give several results concerning multiple operator integrals. In particular, we characterize triple operator integrals mapping valued in trace class operators and then we give a necessary and sufficient condition for a triple operator integral to define a completely bounded map on the Haagerup tensor product of compact operators. Finally, the fifth chapter is dedicated to the resolution of Peller's problems. We first study the connection between multiple operator integrals and perturbation theory for functional calculus of selfadjoint operators and we finish with the construction of counter-examples for those problems
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Chá, Sílvia Alexandra Carrapato. "Problemas convexos e não-convexos do cálculo das variações." Doctoral thesis, Universidade de Évora, 2014. http://hdl.handle.net/10174/17929.

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Nas aplicações do Cálculo das Variações, Controlo Óptimo & Inclusões Diferenciais, muitos problemas importantes da vida real são vectoriais não-convexos e sujeitos a restrições pontuais. O teorema clássico da convexidade de Liapunov é uma ferramenta crucial para resolver problemas vectoriais não-convexos envolvendo integrais simples. No entanto, a possibilidade da extensão deste teorema para lidar com restrições pontuais manteve-se um problema em aberto durante duas décadas, no caso mais realista usando controlos vectoriais variáveis. Nesta tese apresentamos condições necessárias e condições suficientes para a resolução deste problema; CONVEX AND NONCONVEX PROBLEMS OF THE CALCULUS OF VARIATIONS Abstract In applications of the Calculus of Variations, Optimal Control & Differential Inclusions, very important real-life problems are nonconvex vectorial and subject to pointwise constraints. The classical Liapunov convexity theorem is a crucial tool allowing researchers to solve nonconvex vectorial problems involving single integrals. However, the possibility of extending such theorem so as to deal with pointwise constraints has remained an open problem for two decades, in the more realistic case using variable vectorial controls. In this thesis we present necessary conditions and sufficient conditions for solvability of such problem.
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Moyo, Sibusiso. "Noether's theorem and first integrals of ordinary differential equations." Thesis, 1997. http://hdl.handle.net/10413/5061.

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The Lie theory of extended groups is a practical tool in the analysis of differential equations, particularly in the construction of solutions. A formalism of the Lie theory is given and contrasted with Noether's theorem which plays a prominent role in the analysis of differential equations derivable from a Lagrangian. The relationship between the Lie and Noether approach to differential equations is investigated. The standard separation of Lie point symmetries into Noetherian and nonNoetherian symmetries is shown to be irrelevant within the context of nonlocality. This also emphasises the role played by nonlocal symmetries in such an approach.
Thesis (M.Sc.)-University of Natal, Durban, 1997.
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(6368468), Daesung Kim. "Stability for functional and geometric inequalities and a stochastic representation of fractional integrals and nonlocal operators." Thesis, 2019.

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The dissertation consists of two research topics.

The first research direction is to study stability of functional and geometric inequalities. Stability problem is to estimate the deficit of a functional or geometric inequality in terms of the distance from the class of optimizers or a functional that identifies the optimizers. In particular, we investigate the logarithmic Sobolev inequality, the Beckner-Hirschman inequality (the entropic uncertainty principle), and isoperimetric type inequalities for the expected lifetime of Brownian motion.

The second topic of the thesis is a stochastic representation of fractional integrals and nonlocal operators. We extend the Hardy-Littlewood-Sobolev inequality to symmetric Markov semigroups. To this end, we construct a stochastic representation of the fractional integral using the background radiation process. The inequality follows from a new inequality for the fractional Littlewood-Paley square function. We also prove the Hardy-Stein identity for non-symmetric pure jump Levy processes and the L^p boundedness of a certain class of Fourier multiplier operators arising from non-symmetric pure jump Levy processes. The proof is based on Ito's formula for general jump processes and the symmetrization of Levy processes.
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Pass, Brendan. "Structural Results on Optimal Transportation Plans." Thesis, 2011. http://hdl.handle.net/1807/31893.

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In this thesis we prove several results on the structure of solutions to optimal transportation problems. The second chapter represents joint work with Robert McCann and Micah Warren; the main result is that, under a non-degeneracy condition on the cost function, the optimal is concentrated on a $n$-dimensional Lipschitz submanifold of the product space. As a consequence, we provide a simple, new proof that the optimal map satisfies a Jacobian equation almost everywhere. In the third chapter, we prove an analogous result for the multi-marginal optimal transportation problem; in this context, the dimension of the support of the solution depends on the signatures of a $2^{m-1}$ vertex convex polytope of semi-Riemannian metrics on the product space, induce by the cost function. In the fourth chapter, we identify sufficient conditions under which the solution to the multi-marginal problem is concentrated on the graph of a function over one of the marginals. In the fifth chapter, we investigate the regularity of the optimal map when the dimensions of the two spaces fail to coincide. We prove that a regularity theory can be developed only for very special cost functions, in which case a quotient construction can be used to reduce the problem to an optimal transport problem between spaces of equal dimension. The final chapter applies the results of chapter 5 to the principal-agent problem in mathematical economics when the space of types and the space of available goods differ. When the dimension of the space of types exceeds the dimension of the space of goods, we show if the problem can be formulated as a maximization over a convex set, a quotient procedure can reduce the problem to one where the two dimensions coincide. Analogous conditions are investigated when the dimension of the space of goods exceeds that of the space of types.
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Books on the topic "Calculus of variations. Multiple integrals"

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service), SpringerLink (Online, ed. Multiple Integrals in the Calculus of Variations. Berlin, Heidelberg: Springer-Verlag Berlin Heidelberg, 2008.

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Giaquinta, Mariano. Cartesian Currents in the Calculus of Variations II: Variational Integrals. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998.

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Spandagos, Vaggelēs. Oloklērōtikos logismos: Theōria-methodologia, 1600 lymenes askēseis. Athēna: Aithra, 1988.

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1947-, Guzman Alberto. Derivatives and integrals of multivariable functions. Boston, MA: Birkhauser, 2003.

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Muldowney, P. A modern theory of random variation: With applications in stochastic calculus, financial mathematics, and Feynman integration. Hoboken, N.J: Wiley, 2012.

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Muldowney, P. A modern theory of random variation: With applications in stochastic calculus, financial mathematics, and Feynman integration. Hoboken, N.J: Wiley, 2012.

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Zhukova, Galina, and Margarita Rushaylo. The mathematical analysis. Volume 2. ru: INFRA-M Academic Publishing LLC., 2020. http://dx.doi.org/10.12737/1072172.

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The aim of the tutorial is to help students to master the basic concepts and methods of the study of calculus. In volume 2 we study analytic geometry in space; differential calculus of functions of several variables; local, conditional, global extrema of functions of several variables; multiple, curvilinear and surface integrals; elements of field theory; numerical, power series, Taylor series and Maclaurin, and Fourier series; applications to the analysis and solution of applied problems. Great attention is paid to comparison of these methods, the proper choice of study design tasks, analyze complex situations that arise in the study of these branches of mathematical analysis. For self-training and quality control knowledge given test questions. For teachers, students and postgraduate students studying mathematical analysis.
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Zhukova, Galina, and Margarita Rushaylo. Mathematical analysis in examples and tasks. Part 2. ru: INFRA-M Academic Publishing LLC., 2020. http://dx.doi.org/10.12737/1072162.

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The purpose of the textbook is to help students to master basic concepts and research methods used in mathematical analysis. In part 2 of the proposed cycle of workshops on the following topics: analytic geometry in space; differential calculus of functions of several variables; local, conditional, global extrema of functions of several variables; multiple, curvilinear and surface integrals; elements of field theory; numerical, power series, Fourier series; applications to the analysis and solution of applied problems. These topics are studied in universities, usually in the second semester in the discipline "Mathematical analysis" or the course "Higher mathematics", "Mathematics". For the development of each topic the necessary theoretical and background material, reviewed a large number of examples with detailed analysis and solutions, the options for independent work. For self-training and quality control of the acquired knowledge in each section designed exercises and tasks with answers and guidance. It is recommended that teachers, students and graduate students studying advanced mathematics.
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Giaquinta, Mariano. Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. (AM-105), Volume 105. Princeton University Press, 2016.

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Dedecker, Paul. Foundations of the Multiple Integrals Calculus of Variations: The Hamilton-Jacobi-E. Cartan Approach (Hadronic Press Monographs in Mathematics, No 2). Hadronic Press, 1985.

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Book chapters on the topic "Calculus of variations. Multiple integrals"

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Brechtken-Manderscheid, U. "Variational problems with multiple integrals." In Introduction to the Calculus of Variations, 145–66. Boston, MA: Springer US, 1991. http://dx.doi.org/10.1007/978-1-4899-3172-6_10.

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Boccardo, Lucio, and Benedetta Pellacci. "Bounded Positive Critical Points of Some Multiple Integrals of the Calculus of Variations." In Nonlinear Equations: Methods, Models and Applications, 33–51. Basel: Birkhäuser Basel, 2003. http://dx.doi.org/10.1007/978-3-0348-8087-9_3.

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Morrey, Charles B. "Multiple Integral Peoblems in the Calculus of Variations and Related Topics." In Il principio di minimo e sue applicazioni alle equazioni funzionali, 93–153. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-10926-3_3.

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Bressoud, David M. "Line Integrals, Multiple Integrals." In Second Year Calculus, 111–38. New York, NY: Springer New York, 1991. http://dx.doi.org/10.1007/978-1-4612-0959-1_5.

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Gilbert, Robert P., Michael Shoushani, and Yvonne Ou. "Multiple Integrals." In Multivariable Calculus with Mathematica, 179–242. Boca Raton : Chapman & Hall/CRC Press, 2020.: Chapman and Hall/CRC, 2020. http://dx.doi.org/10.1201/9781315161471-5.

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Lipsman, Ronald L., and Jonathan M. Rosenberg. "Multiple Integrals." In Multivariable Calculus with MATLAB®, 147–83. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-65070-8_8.

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Courant, Richard, and Fritz John. "Multiple Integrals." In Introduction to Calculus and Analysis, 367–542. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4613-8958-3_4.

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Coombes, Kevin R., Ronald L. Lipsman, and Jonathan M. Rosenberg. "Multiple Integrals." In Multivariable Calculus and Mattiematica®, 153–83. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-1698-8_8.

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Pao, Karen, and Frederick Soon. "Multiple Integrals." In Student’s Guide to Basic Multivariable Calculus, 89–116. New York, NY: Springer New York, 1993. http://dx.doi.org/10.1007/978-1-4757-4300-5_5.

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Courant, Richard, and Fritz John. "Multiple Integrals." In Introduction to Calculus and Analysis Volume II/1, 367–542. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-642-57149-7_4.

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Conference papers on the topic "Calculus of variations. Multiple integrals"

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Barhorst, Alan A. "Closed Form Modeling of Continuous Parameter Robotic Systems-Contact/Impact and Wave Propagation." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0142.

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Abstract The high fidelity modeling of robotic systems is addressed. The operation of a robot usually consists of a free-flight regime, a contact/impact regime, and a constrained motion regime. Presented in this paper is a demonstration of a systematic method for modeling the complete motion of a manipulator. The model is complete in the sense that all motion regimes are seamlessly handled and the inherently distributed nature of elasticity and mass are rigorously modeled. Since the model properly integrates the distributed parameters with the discrete parameters (boundary conditions too), the effects of disturbance waves propagating in the robotic chain of bodies can be studied. The methodology demonstrated in this work is a recent development. The methodology is based in variational principles but its operational aspects hide the variational calculus. The method seamlessly handles holonomic and nonholonomic motion constraints. The method also allows the determination of post impact velocities and pointwise velocity fields for hybrid parameter multiple body (HPMB) systems. Exact relationships used to determine when the separation of colliding bodies has occurred are also readily generated. For analysts familiar with Kane’s form of the Gibbs-Appell equations, the method will be affable. The system modeled in this paper will allow all the curious HPMB system dynamics to be studied. That is: a two flexible link planar manipulator undergoing all three motion regimes mentioned above. The flexible beams are taken as Euler-Bernoulli beams with large deflections. The modeling technique will be demonstrated by deriving the complete model of this system. Some of the advantages of the technique demonstrated herein include: a) closed form hybrid parameter models, b) automatic boundary conditions, c) holonomic and nonholonomic motion, d) wave propagation, and e) rapid regeneration of system models.
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Tangpong, X. W., and Om P. Agrawal. "Fractional Optimal Control of Distributed Systems." In ASME 2007 International Mechanical Engineering Congress and Exposition. ASMEDC, 2007. http://dx.doi.org/10.1115/imece2007-43046.

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This paper presents a formulation and a numerical scheme for Fractional Optimal Control (FOC) for a class of distributed systems. The fractional derivative is defined in the Caputo sense. The performance index of a Fractional Optimal Control Problem (FOCP) is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). The scheme presented rely on reducing the equations for distributed system into a set of equations that have no space parameter. Several strategies are pointed out for this task, and one of them is discussed in detail. This involves discretizing the space domain into several segments, and writing the spatial derivatives in terms of variables at space node points. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the problem. The numerical technique presented in [1] for scalar case is extended for the vector case. In this technique, the FOC equations are reduced to Volterra type integral equations. The time domain is also descretized into several segments. For the linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for various order of fractional derivatives and various order of space and time discretizations. Numerical results show that for the problem considered, only a few space grid points are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other distributed systems.
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Agrawal, Om P. "Fractional Optimal Control of a Distributed System Using Eigenfunctions." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-35921.

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This paper presents a formulation and a numerical scheme for Fractional Optimal Control (FOC) for a class of distributed systems. The fractional derivative is defined in the Caputo sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equations (PFDEs). Eigenfunctions are used to eliminate the space parameter, and to define the problem in terms of a set of state and control variables. This leads to a multi FOCP in which each FOCP could be solved independently. Several other strategies are pointed out to reduce the problem to a finite dimensional space, some of which may not provide a decoupled set of equations. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the problem. The numerical technique presented in [1] is used to obtain the state and the control variables. In this technique, the FOC equations are reduced to Volterra type integral equations. The time domain is descretized into several segments and a time marching scheme is used to obtain the response at discrete time points. For a linear case, the numerical technique results into a set of algebraic equations which can be solved using a direct or an iterative scheme. The problem is solved for different number of eigenfunctions and time discretizations. Numerical results show that only a few eigenfunctions are sufficient to obtain good results, and the solutions converge as the size of the time step is reduced. The formulation presented is simple and can be extended to FOC of other distributed systems.
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Finch, William W., and Allen C. Ward. "Quantified Relations: A Class of Predicate Logic Design Constraints Among Sets of Manufacturing, Operating, and Other Variations." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/dtm-1550.

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Abstract This paper addresses a class of engineering design problems in which multiple sources of variations affect a product’s design, manufacture, and performance. Examples of these sources include uncertainty in nominal dimensions, variations in manufacture, changing environmental or operating conditions, and operator adjustments. Quantified relations (QR’s) are defined as a class of predicate logic expressions representing constraints between sets of design variations. Within QR’s, each variable’s quantifier and the order of quantification express a physical system’s causal relationships. This paper also presents an algorithm which propagates intervals through QR’s involving continuous, monotonic equations. Causal relationships between variables in engineering systems are discussed, and a tabular representation for them is presented. This work aims to broaden the application of automated constraint satisfaction algorithms, shortening design cycles for this class of problem by reducing modeling, and possibly computing effort. It seems to subsume Ward’s prior work on the Label Interval Calculus, extending the approach to a wider range of engineering design problems.
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You might also be interested in the extended bibliographies on the topic 'Calculus of variations. Multiple integrals' for particular source types:
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