Relevant bibliographies by topics / Isoperimetric problems

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

Academic literature on the topic 'Isoperimetric problems'

Author: Grafiati
Published: 28 June 2021
Last updated: 8 February 2022

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Journal articles on the topic "Isoperimetric problems"

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Petty, C. M. "AFFINE ISOPERIMETRIC PROBLEMS." Annals of the New York Academy of Sciences 440, no. 1 Discrete Geom (May 1985): 113–27. http://dx.doi.org/10.1111/j.1749-6632.1985.tb14545.x.

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Apostol, Tom M., and Mamikon A. Mnatsakanian. "Isoperimetric and Isoparametric Problems." American Mathematical Monthly 111, no. 2 (February 2004): 118. http://dx.doi.org/10.2307/4145213.

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Apostol, Tom M., and Mamikon A. Mnatsakanian. "Isoperimetric and Isoparametric Problems." American Mathematical Monthly 111, no. 2 (February 2004): 118–36. http://dx.doi.org/10.1080/00029890.2004.11920056.

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Tóth, L. Fejes. "Isoperimetric problems for tilings." Mathematika 32, no. 1 (June 1985): 10–15. http://dx.doi.org/10.1112/s0025579300010792.

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BOLLOBÁS, BÉLA, and IMRE LEADER. "Isoperimetric Problems for r-sets." Combinatorics, Probability and Computing 13, no. 2 (March 2004): 277–79. http://dx.doi.org/10.1017/s0963548304006078.

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Tóth, L. Fejes. "Isoperimetric problems for tilings, corrigendum." Mathematika 33, no. 2 (December 1986): 189–91. http://dx.doi.org/10.1112/s0025579300011177.

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Siegel, Jerrold, and Frank Williams. "Uniform bounds for isoperimetric problems." Proceedings of the American Mathematical Society 107, no. 2 (February 1, 1989): 459. http://dx.doi.org/10.1090/s0002-9939-1989-0984815-2.

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Ritoré, Manuel, and Antonio Ros. "Some updates on isoperimetric problems." Mathematical Intelligencer 24, no. 3 (June 2002): 9–14. http://dx.doi.org/10.1007/bf03024725.

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Clarenz, Ulrich, and Heiko von der Mosel. "Isoperimetric inequalities for parametric variational problems." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 19, no. 5 (2002): 617–29. http://dx.doi.org/10.1016/s0294-1449(02)00096-3.

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Demyanov, V. F., and G. Sh Tamasyan. "Exact penalty functions in isoperimetric problems." Optimization 60, no. 1-2 (January 2011): 153–77. http://dx.doi.org/10.1080/02331934.2010.534166.

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Dissertations / Theses on the topic "Isoperimetric problems"

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SILVA, MARCELO CHAVES. "ISOPERIMETRIC PROBLEMS IN THE MINKOWSKI PLANE." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2015. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=25618@1.

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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
O objetivo principal deste trabalho é resolver o problema isoperimétrico no plano de Minkowski, isto é, determinar dentre todas as curvas convexas, fechadas, simples e suaves de perímetro fixo de um plano munido com uma norma qualquer, qual é aquela que delimita a maior área. Mostraremos que a solução para este problema não é necessariamente o círculo como no caso euclideano e sim uma curva conhecida como isoperimetrix. Para isto, vamos demonstrar a desigualdade de Minkowski a partir do conceito de área mista. Em seguida, vamos determinar se há outros casos (além do caso euclideano) em que o círculo coincide com o isoperimetrix. Também iremos mostrar que o perímetro da bola nestes planos pode assumir qualquer valor real entre seis e oito, sendo seis quando a bola for um hexágono regular afim e oito quando for um paralelogramo.
The main objective of this work is to solve the isoperimetric problem in the Minkowski plane, i. e., determine among all smooth simple closed convex curves of a normed plane with fixed perimeter, what is that which defines the largest area. We will show that the solution to this problem is not necessarily the circle as in the Euclidean case, but a curve known as isoperimetrix. For this, we will demonstrate the Minkowski inequality from the concept of mixed area. Then, we determine if there are other cases (apart from the Euclidean case) in which the circle coincides with the isoperimetrix. We will also show that the ball perimeter in a normed plane can take any real value between six and eight. It is six when the ball is an affine regular hexagon and eight when it is a parallelogram.
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Serra, Montolí Joaquim. "Elliptic and parabolic PDEs : regularity for nonlocal diffusion equations and two isoperimetric problems." Doctoral thesis, Universitat Politècnica de Catalunya, 2014. http://hdl.handle.net/10803/279290.

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The thesis is divided into two parts. The first part is mainly concerned with regularity issues for integro-differential (or nonlocal) elliptic and parabolic equations. In the same way that densities of particles with Brownian motion solve second order elliptic or parabolic equations, densities of particles with Lévy diffusion satisfy these more general nonlocal equations. In this context, fully nonlinear nonlocal equations arise in Stochastic control problems or differential games. The typical example of elliptic nonlocal operator is the fractional Laplacian, which is the only translation, rotation and scaling invariant nonlocal elliptic operator. There many classical regularity results for the fractional Laplacian ---whose ``inverse'' is the Riesz potential. For instance, the explicit Poisson kernel for a ball is an ``old'' result, as well as the linear solvability theory in L^p spaces. However, very little was known on boundary regularity for these problems. A main topic of this thesis is the study of this boundary regularity, which is qualitatively very different from that for second order equations. We stablish a new boundary regularity theory for fully nonlinear (and linear) elliptic integro-differential equations. Our proofs require a combination of original techniques and appropriate versions of classical ones for second order equations (such as Krylov's method). We also obtain new interior regularity results for fully nonlinear parabolic nonlocal equation with rough kernels. To do it, we develop a blow up and compactness method for viscosity solutions to fully nonlinear equations that allows us to prove regularity from Liouville type theorems.This method is a main contribution of the thesis. The new boundary regularity results mentioned above are crucially used in the proof of another main result of the thesis: the Pohozaev identity for the fractional Laplacian. This identity is has a flavor of integration by parts formula for the fractional Laplacian, with the important novely there appears a local boundary term (this was unusual with nolocal equations). In the second part of the thesis we give two instances of interaction between isoperimetry and Partial Differential Equations. In the first one we use the Alexandrov-Bakelman-Pucci method for elliptic PDE to obtain new sharp isoperimetric inequalities in cones with densities by generalizing a proof of the classical isoperimetric inequality due to Cabré. Our new results contain as particular cases the classical Wulff inequality and the isoperimetric inequality in cones of Lions and Pacella. In the second instance we use the isoperimetric inequality and the classical Pohozaev identity to establish a radial symmetry result for second order reaction-diffusion equations. The novelty here is to include discontinuous nonlinearities. For this, we extend a two-dimensional argument of P.-L. Lions from 1981 to obtain now results in higher dimensions
La tesi està dividida en dues parts. La primera part es centra principalment en questions de regularitat per equacions integro - iferencials (o no locals) el·líptiques i parbòliques. De la mateixa manera que les densitats de partícules amb un moviment Brownià resolen equacions el·líptiques o parbòliques de segon ordre, les densitats de partícules amb una difusió de tipus Lévy resolen aquestes equacions no locals més generals. En aquest context, les equacions completament no lineals sorgeixen de problemes de control estocàstic o "differential games''. L'exemple típic d'operador el·liptic no local és el laplacià fraccionari, el qual és l'únic d'aquests operadors que és invariant per translacions, rotacions, i reescalament. Hi ha molts resultats clàssics de regularitat per el laplacià fraccionari --- "l'invers'' del qual és el potencial de Riesz. Per exemple, el nucli de Poisson (explícit) per la bola és un resultat "vell'', així com la teoria de resolubilitat en espais L^p. No obstant això, se sabia ben poc sobre la regularitat a la vora per a aquests problemes. Un tema principal d'aquesta tesi és l'estudi d'aquesta regularitat a la vora, que és qualitativament molt diferent de la de les equacions de segon ordre . A la tesi s'estableix una nova teoria regularitat a la vora per completament no lineals ( i lineals ) equacions integro - diferencials el·líptiques . Les nostres demostracions requeixen una combinació de tècniques originals i versions apropiades de les clàssiques equacions de segon ordre ( com ara el mètode de Krylov ). També obtenim nous resultats de regularitat interior per equacions parabòliques no locals completament no lineals i amb "rough kernels''. A tal efecte, desenvolupem un mètode de blow-up i compacitat per a equacions completament no lineals que en permet provar regularitat a partir de teoremes de tipus Liouville. Aquest mètode és una contribució principal de la tesi. Els nous resultats de regularitat a la vora esmentats anteriorment són essencials en la prova d'un altre resultat principal de la tesi: la identitat Pohozaev per al Laplacià fraccionari. Aquesta identitat recorda a una fórmula d'integració per parts, però amb el Laplacià fraccionari. La novetat important és que apareix un terme de vora locals (això era inusual amb equacions no locals) . A la segona part de la tesi que donem dos exemples d'interacció entre isoperimetria i Equacions en Derivades Parcials. En el primer, s'utilitza el mètode d'Alexandrov- Bakelman-Pucci per a EDP el·líptiques a fi d'obtenir noves desigualtats isoperimètriques en cons convexos amb densitats, generalitzant una prova de la desigualtat isoperimètric clàssica de X. Cabré. Els nostres nous resultats contenen com a casos particularsla desigualtat clàssica de Wulff i la desigualtat isoperimètrica en cons de Lions-Pacella. En el segon exemple s'utilitza la desigualtat isoperimètrica i la identitat Pohozaev clàssica per establir un resultat de simetria radial per equacions de reacció-difusió de segon ordre. La novetat en aquest cas és que s'inclouen no-linealitats discontínues. Per a provar aquest resultat, estenem un argument en dues dimensions de P.-L. Lions de 1981 i podem obtenir ara resultass en dimensions superiors.
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Costa, Marcos Antônio da. "Máximos e Mínimos: uma abordagem para o ensino médio." Universidade Federal de Goiás, 2013. http://repositorio.bc.ufg.br/tede/handle/tde/2947.

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Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2014-08-28T15:33:45Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertacao Marcos Antonio da Costa.pdf: 1237372 bytes, checksum: e4392b806f26f0293114e12b4ed829c9 (MD5)
Made available in DSpace on 2014-08-28T15:33:45Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Dissertacao Marcos Antonio da Costa.pdf: 1237372 bytes, checksum: e4392b806f26f0293114e12b4ed829c9 (MD5) Previous issue date: 2013-04-12
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
We deal with extremum values problems. Our focus is the high school students. We present simple ideas and techniques on solving classical optimization problems. Among other problems we cite the classical isoperimetric ploblem and the Heron0s problem. We are based on the book Stories About Maxima and Minima by Tikhomirov which lead with these classical problems using only elementary mathematical subjects.
Estudamos problemas envolvendo valores extremos, com foco nos estudantes do Ensino Médio. Apresentamos de forma simples e resumida, algumas ideias e teorias para a solução de tais problemas. Dentre os quais citamos o Problema de Dido e o de Heron. O principal referencial teórico para confecção deste trabalho foi o livro de Tikhomirov intitulado Stories About Maxima and Minima. Baseados em tal livro, aplicamos métodos e teorias elementares para solucionarmos problemas clássicos de máximos e mínimos.
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Do, Minh Nhat Vo. "The Constrained Isoperimetric Problem." BYU ScholarsArchive, 2011. https://scholarsarchive.byu.edu/etd/2700.

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Let X be a space and let S ⊂ X with a measure of set size |S| and boundary size |∂S|. Fix a set C ⊂ X called the constraining set. The constrained isoperimetric problem asks when we can find a subset S of C that maximizes the Følner ratio FR(S) = |S|/|∂S|. We consider different measures for subsets of R^2,R^3,Z^2,Z^3 and describe the properties that must be satisfied for sets S that maximize the Folner ratio. We give explicit examples.
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Lomas, Fernando Herrero. "Problemas isoperimétricos: uma abordagem no ensino médio." Universidade de São Paulo, 2016. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-24112016-210117/.

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Nesta dissertação foram discutidas abordagens do problema isoperimétrico que podem ser aplicadas no ensino médio e para alunos de Licenciatura plena em Matemática. Foi realizada inicialmente uma abordagem histórica e posteriormente a discussão de casos particulares e gerais de desigualdade isoperimétrica tanto no plano como no espaço. A abordagem principal deste texto é no plano, no qual foram analisadas as áreas dos triângulos, quadriláteros e polígonos regulares dado um perímetro fixo.
In this dissertation isoperimetric problem approaches were discussed that can be applied in high school and full degree students in mathematics. It was initially performed a historical approach and then the discussion of individual and general cases of isoperimetric inequality both in the plane and in space . The main approach of this text is in the plan, in which the areas of the triangles were analyzed , quadrilaterals and regular polygons given a fixed perimeter.
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Corucci, Mariachiara. "La soluzione di Hurwitz del problema isoperimetrico." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/19220/.

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In questo elaborato viene trattata la soluzione analitica del problema isoperimetrico nel piano proposta da Adolf Hurwitz. Utilizzando i risultati della teoria delle serie di Fourier e l’applicazione del Teorema di Gauss Green per il calcolo dell’area, dimostriamo che “fra tutte le curve semplici, chiuse, regolari e rettificabili di lunghezza fissata, la circonferenza è la sola a racchiudere la regione di maggior area”.
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Lee, Sunmi. "The edge-isoperimetric problem for the square tessellation of plane." CSUSB ScholarWorks, 2000. https://scholarworks.lib.csusb.edu/etd-project/1622.

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The solution for the edge-isoperimetric problem (EIP) of the square tessellation of plane is investigated and solved. Summaries of the stabilization theory and previous research dealing with the EIP are stated. These techniques enable us to solve the EIP of the cubical tessellation.
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Louis, Anand. "The complexity of expansion problems." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/52294.

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Graph-partitioning problems are a central topic of research in the study of algorithms and complexity theory. They are of interest to theoreticians with connections to error correcting codes, sampling algorithms, metric embeddings, among others, and to practitioners, as algorithms for graph partitioning can be used as fundamental building blocks in many applications. One of the central problems studied in this field is the sparsest cut problem, where we want to compute the cut which has the least ratio of number of edges cut to size of smaller side of the cut. This ratio is known as the expansion of the cut. In spite of over 3 decades of intensive research, the approximability of this parameter remains an open question. The study of this optimization problem has lead to powerful techniques for both upper bounds and lower bounds for various other problems, and interesting conjectures such as the SSE conjecture. Cheeger's Inequality, a central inequality in Spectral Graph Theory, establishes a bound on expansion via the spectrum of the graph. This inequality and its many (minor) variants have played a major role in the design of algorithms as well as in understanding the limits of computation. In this thesis we study three notions of expansion, namely edge expansion in graphs, vertex expansion in graphs and hypergraph expansion. We define suitable notions of spectra w.r.t. these notions of expansion. We show how the notion Cheeger's Inequality goes across these three problems. We study higher order variants of these notions of expansion (i.e. notions of expansion corresponding to partitioning the graph/hypergraph into more than two pieces, etc.) and relate them to higher eigenvalues of graphs/hypergraphs. We also study approximation algorithms for these problems. Unlike the case of graph eigenvalues, the eigenvalues corresponding to vertex expansion and hypergraph expansion are intractable. We give optimal approximation algorithms and computational lower bounds for computing them.
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John, Daniel. "Symmetrization procedures for the isoperimetric problem in symmetric spaces of noncompact type." [S.l.] : [s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=974086401.

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Dos, Santos Viana C. "Index one minimal surfaces and the isoperimetric problem in spherical space forms." Thesis, University College London (University of London), 2018. http://discovery.ucl.ac.uk/10061744/.

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The research carried out in this thesis concerns two important class of stationary surfaces in Differential Geometry, namely isoperimetric surfaces and index one minimal surfaces. The former are solutions of the so called isoperimetric problem, which is to determine the regions of least perimeter among regions of same volume in a given manifold. The latter are critical points of the area functional with Morse index one, i.e., minimal surfaces which admits only one direction where the surface can be deformed so to decrease its area. These are usually constructed via mountain pass arguments. This work focus on the study of these objects when the ambient space is a 3-dimensional spherical space forms, i.e., space form with positive curvature. Our main results classify, at the level of topology, such stationary surfaces in the spherical space forms with large fundamental group. Our first result proves that the solutions of the isoperimetric problem in spherical space forms with large fundamental group are either spheres or tori. It was previously known that solutions with genus zero and one are respectively totally umbilical and flat. Combining our result and this geometric description, we derive that the solutions of the isoperimetric problem are either geodesic spheres or quotients of Clifford tori. Our second result proves that orientable minimal surfaces with index one in the aforementioned spherical space forms have genus at most two. This is a sharp estimate as one can use the continuous one-parameter min-max theory to construct in every 3-dimensional spherical space form an index one minimal surface with genus equal the Heegaard genus of such space which is known to be at most two. Our result confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.
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Books on the topic "Isoperimetric problems"

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B, Zegarlinski, ed. Entropy bounds and isoperimetry. Providence, RI: American Mathematical Society, 2005.

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Ecole d'été de probabilités de Saint-Flour (24th 1994). Lectures on probability theory and statistics: Ecole d'été de probabilités de Saint-Flour XXIV, 1994. Edited by Dobrushin R. L. 1929-, Groeneboom P, Ledoux Michel 1958-, and Bernard P. 1944-. Berlin: Springer, 1996.

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Tyson, Jeremy T., ed. An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Basel: Birkhäuser Basel, 2007. http://dx.doi.org/10.1007/978-3-7643-8133-2.

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Harper, L. H. Global Methods for Combinatorial Isoperimetric Problems (Cambridge Studies in Advanced Mathematics). Cambridge University Press, 2004.

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1928-, Bakelʹman I. I͡A︡, ed. Geometric analysis and nonlinear partial differential equations. New York: M. Dekker, 1993.

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Coopersmith, Jennifer. Antecedents. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.003.0002.

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Early ideas about optimization principles were brought in by an eclectic group of extraordinary thinkers: the Ancients (Hero, and Princess Dido), Fermat with his Principle of Least Time, the Bernoullis, Leibniz, Maupertuis, Euler, and d’Alembert. Also, Stevin was the first to invoke the impossibility of perpetual motion in a proof, and Huygens was the first to put Galilean Relativity to a quantitative test. The Swiss family of mathematical geniuses, the Bernoullis, tackled isoperimetric problems, such as the brachystochrone, and Johann Bernoulli discovered the Principle of Virtual Velocities. The flavour of the eighteenth century is shown in the evocative tale of the König affair, and the correspondence between Daniel Bernoulli and Euler. It is shown how symmetry arguments, leading ultimately to an energy-analysis, were competing with Newton’s force-analysis. The Principle of Least Action and Variational Mechanics, proper, were developed by Lagrange, Hamilton, and Jacobi.
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Ledoux, Michel, R. Dobrushin, and P. Groeneboom. Lectures on Probability Theory and Statistics: Ecole D'Ete De Probabilities De St. Flour Xxiv - 1994 (Lecture Notes in Mathematics). Springer, 1997.

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An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem (Progress in Mathematics). Birkhäuser Basel, 2007.

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Book chapters on the topic "Isoperimetric problems"

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Buldygin, V. V., and A. B. Kharazishvili. "Two classical isoperimetric problems." In Geometric Aspects of Probability Theory and Mathematical Statistics, 49–56. Dordrecht: Springer Netherlands, 2000. http://dx.doi.org/10.1007/978-94-017-1687-1_4.

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Kobelev, V. V. "Isoperimetric Inequalities in Stability Problems." In Optimization of Large Structural Systems, 1155–64. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-010-9577-8_60.

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Gulliver, Robert. "Isoperimetric problems having continua of solutions." In Lecture Notes in Mathematics, 256–63. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/bfb0082870.

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Filliman, P. "Symmetric solutions to isoperimetric problems for polytopes." In DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 289–300. Providence, Rhode Island: American Mathematical Society, 1991. http://dx.doi.org/10.1090/dimacs/004/21.

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Hildebrandt, S. "On Two Isoperimetric Problems with Free Boundary Conditions." In Variational Methods for Free Surface Interfaces, 43–51. New York, NY: Springer New York, 1987. http://dx.doi.org/10.1007/978-1-4612-4656-5_5.

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Morini, Massimiliano. "Local and global minimality results for an isoperimetric problem with long-range interactions." In Free Discontinuity Problems, 153–224. Pisa: Scuola Normale Superiore, 2016. http://dx.doi.org/10.1007/978-88-7642-593-6_3.

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Bezrukov, Sergei L., and Robert Elsässer. "Edge-Isoperimetric Problems for Cartesian Powers of Regular Graphs." In Graph-Theoretic Concepts in Computer Science, 9–20. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/3-540-45477-2_3.

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Pacella, Filomena. "Some Relative Isoperimetric Inequalities and Applications to Nonlinear Problems." In Variational Methods, 219–35. Boston, MA: Birkhäuser Boston, 1990. http://dx.doi.org/10.1007/978-1-4757-1080-9_15.

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Neumayer, Robin. "On Minimizers and Critical Points for Anisotropic Isoperimetric Problems." In 2018 MATRIX Annals, 293–302. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-38230-8_20.

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Guba, V. S. "Polynomial Isoperimetric Inequalities for Richard Thompson’s Groups F, T, and V." In Algorithmic Problems in Groups and Semigroups, 91–120. Boston, MA: Birkhäuser Boston, 2000. http://dx.doi.org/10.1007/978-1-4612-1388-8_5.

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Conference papers on the topic "Isoperimetric problems"

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Bognár, Gabriella. "Isoperimetric inequalities for some nonlinear eigenvalue problems." In The 7'th Colloquium on the Qualitative Theory of Differential Equations. Szeged: Bolyai Institute, SZTE, 2003. http://dx.doi.org/10.14232/ejqtde.2003.6.4.

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Glizer, Valery Y., Vladimir Turetsky, and Emil Bashkansky. "Application of Pontryagin’s Maximum Principle to Statistical Process Control Optimization." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70104.

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The optimization of a statistical process control with a variable sampling interval is studied. A control performance index is the expected loss, caused by delay in detecting process change. It is to be minimized by a proper choice of a sampling interval. The mathematical model of this problem is a nonstandard variational calculus problem with two types of constraints, an isoperimetric constraint and two geometric constraints. The integrands in the cost functional and the isoperimetric constraint are independent of the derivative of the minimizing function. Therefore, the classical Euler-Lagrange equation approach is not applicable when analyzing this extremal problem. The optimization problem depends on the signal-to-noise ratio parameter. The original problem is transformed to an equivalent optimal control problem. Based on the value of the parameter, the latter is decomposed into two simpler problems, solved by application of Pontryagin’s Maximum Principle. The theoretical results are evaluated by numerical simulations.
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3

Pandey, Rajesh K., and Om P. Agrawal. "Comparison of Four Numerical Schemes for Isoperimetric Constraint Fractional Variational Problems With A-Operator." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46570.

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This paper presents a comparative study of four numerical schemes for a class of Isoperimetric Constraint Fractional Variational Problems (ICFVPs) defined in terms of an A-operator introduced recently. The A-operator is defined in a more general way which in special cases reduces to Riemann-Liouville, Caputo, Riesz-Riemann-Liouville and Riesz-Caputo, and several other fractional derivatives defined in the literature. Four different schemes, namely linear, quadratic, quadratic-linear and Bernsteins polynomials approximations, are used to obtain approximate solutions of an ICFVP. All four schemes work well, and when the number of terms approximating the solution are increased, the desired solution is achieved. Results for a modified power kernel in A-operator for different fractional orders are presented to demonstrate the effectiveness of the proposed schemes. The accuracy of the numerical schemes with respect to parameters such as fractional order α and step size h are analyzed and illustrated in detail through various figures and tables. Finally, comparative performances of the schemes are discussed.
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4

Pandey, Rajesh K., and Om P. Agrawal. "Numerical Scheme for Generalized Isoparametric Constraint Variational Problems With A-Operator." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-12388.

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This paper presents a numerical scheme for a class of Isoperimetric Constraint Variational Problems (ICVPs) defined in terms of an A-operator introduced recently. In this scheme, Bernstein’s polynomials are used to approximate the desired function and to reduce the problem from a functional space to an eigenvalue problem in a finite dimensional space. Properties of the eigenvalues and eigenvectors of this problem are used to obtain approximate solutions to the problem. Results for two examples are presented to demonstrate the effectiveness of the proposed scheme. In special cases the A-operator reduce to Riemann-Liouville, Caputo, Riesz-Riemann-Liouville and Riesz-Caputo, and several other fractional derivatives defined in the literature. Thus, the approach presented here provides a general scheme for ICVPs defined using different types of fractional derivatives. Although, only Bernstein’s polynomials are used here to approximate the solutions, many other approximation schemes are possible. Effectiveness of these approximation schemes will be presented in the future.
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5

Zhang, Ruochun, and Xiaoping Qian. "Triangulation Based Isogeometric Analysis of the Cahn-Hilliard Phase-Field Model." In ASME 2018 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/detc2018-85576.

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This paper presents the triangulation based isogeometric analysis of the Cahn–Hilliard phase-field model. We validate our method by convergence analysis, show detailed system evolution from a randomly perturbed initial condition and then discuss related isoperimetric problems. Lastly an example highlighting its efficacy in complex geometry is provided. Triangulation based isogeometric analysis shows time step stability and complex geometry adaptability in our experiments.
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6

Rodrigues, Helder C., and Paulo A. Fernandes. "Generalized Topology Optimization of Linear Elastic Structures Subjected to Thermal Loads." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0370.

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Abstract This paper presents the development of a computational model for the generalized topology optimization problem, using a material distribution approach, of 2-D linear elastic solids subjected to thermal loads, with compliance objective function and an isoperimetric constraint on volume. The model relies on homogenization asymptotic methods to characterize the influence of the material periodic microstructure and a finite element displacement formulation is used to approximate the homogenized equilibrium equations obtained. The computational model developed is tested in several examples considering different finite element approximations and the influence of the design variables (material density and orientation) is analyzed.
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7

Date, Hisashi, and Yoshihiro Takita. "Control of 3D Snake-Like Locomotive Mechanism Based on Continuum Modeling." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-85130.

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An effective control method that achieves movement over a small ridge as an example of three-dimensional (3D) snake-like creeping locomotion is presented. The creeping robot is modeled as a continuum with zero thickness capable of generating bending moment at arbitrary points. Under a simplified contact condition, the optimal bending moment distribution in terms of a quadratic cost function of input can be obtained as a function of curvature by solving an isoperimetric problem. The solution is well suited to an articulated body consisting of finite number of links. The model is demonstrated through simulations and experiments using a prototype robot to be effective for traversing smooth 3D terrain.
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