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Автор: Grafiati

Опубліковано: 10 березня 2023

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Статті в журналах з теми "Monotonicity formulae"

Engelfriet, J. "Monotonicity and Persistence in Preferential Logics." Journal of Artificial Intelligence Research 8 (January 1, 1998): 1–21. http://dx.doi.org/10.1613/jair.461.

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An important characteristic of many logics for Artificial Intelligence is their nonmonotonicity. This means that adding a formula to the premises can invalidate some of the consequences. There may, however, exist formulae that can always be safely added to the premises without destroying any of the consequences: we say they respect monotonicity. Also, there may be formulae that, when they are a consequence, can not be invalidated when adding any formula to the premises: we call them conservative. We study these two classes of formulae for preferential logics, and show that they are closely linked to the formulae whose truth-value is preserved along the (preferential) ordering. We will consider some preferential logics for illustration, and prove syntactic characterization results for them. The results in this paper may improve the efficiency of theorem provers for preferential logics.

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Djaa, Nour Elhouda, Ahmed Mohamed Cherif, and Kaddour Zegga. "Monotonicity formulae and F-stress energy." Bulletin of the Transilvania University of Brasov Series III Mathematics and Computer Science 1(63), no. 1 (October 7, 2021): 81–96. http://dx.doi.org/10.31926/but.mif.2021.1.63.1.7.

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The goal of this work is the application of the f-stress energy of differ-ential forms to study the generalized monotonicity formulae and generalizedvanishing theorems. We obtain some generalized monotonicity formulas forp-formsω∈Ap(ξ), which satisfy the generalizedf-conservation laws, withf∈C∞(M×R) satisfying some conditions

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Evans, Lawrence C. "Monotonicity formulae for variational problems." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 2005 (December 28, 2013): 20120339. http://dx.doi.org/10.1098/rsta.2012.0339.

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Li, Jintang. "Monotonicity formulae and vanishing theorems." Pacific Journal of Mathematics 281, no. 1 (February 9, 2016): 125–36. http://dx.doi.org/10.2140/pjm.2016.281.125.

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Fazly, Mostafa, and Juncheng Wei. "On stable solutions of the fractional Hénon–Lane–Emden equation." Communications in Contemporary Mathematics 18, no. 05 (July 18, 2016): 1650005. http://dx.doi.org/10.1142/s021919971650005x.

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Анотація:

We derive monotonicity formulae for solutions of the fractional Hénon–Lane–Emden equation [Formula: see text] when [Formula: see text], [Formula: see text] and [Formula: see text]. Then, we apply these formulae to classify stable solutions of the above equation.

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Fazly, Mostafa, and Henrik Shahgholian. "Monotonicity formulas for coupled elliptic gradient systems with applications." Advances in Nonlinear Analysis 9, no. 1 (June 6, 2019): 479–95. http://dx.doi.org/10.1515/anona-2020-0010.

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Abstract Consider the following coupled elliptic system of equations $$\begin{array}{} \displaystyle (-{\it\Delta})^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} ~~ \mathbb{R}^n , \end{array}$$ where 0 < s ≤ 2, p > 1, m ≥ 1, u = $\begin{array}{} \displaystyle (u_i)_{i=1}^m \end{array}$ and ui : ℝn → ℝ. The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is when m = 1 and s = 1, Gidas and Spruck in [26] and later Caffarelli, Gidas and Spruck in [6] provided the classification of solutions for Sobolev sub-critical and critical exponents. More recently, for the case of local system of equations that is when m ≥ 1 and s = 1 a similar classification result is given by Druet, Hebey and Vétois in [17] and references therein. In this paper, we derive monotonicity formulae for entire solutions of the above local, when s = 1, 2, and nonlocal, when 0 < s < 1 and 1 < s < 2, system. These monotonicity formulae are of great interests due to the fact that a counterpart of the celebrated monotonicity formula of Alt-Caffarelli-Friedman [1] seems to be challenging to derive for system of equations. Then, we apply these formulae to give a classification of finite Morse index solutions. In the end, we provide an open problem in regards to monotonicity formulae for Lane-Emden systems.

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Wang, Jiaxiang, and Bin Zhou. "Monotonicity formulae for the complex Hessian equations." Methods and Applications of Analysis 28, no. 1 (2021): 77–84. http://dx.doi.org/10.4310/maa.2021.v28.n1.a6.

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Nikolov, Geno. "On the monotonicity of sequences of quadrature formulae." Numerische Mathematik 62, no. 1 (December 1992): 557–65. http://dx.doi.org/10.1007/bf01396243.

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Fried, Eliot, and Luca Lussardi. "Monotonicity formulae for smooth extremizers of integral functionals." Rendiconti Lincei - Matematica e Applicazioni 30, no. 2 (June 21, 2019): 365–77. http://dx.doi.org/10.4171/rlm/851.

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Dong, Y., and H. Lin. "MONOTONICITY FORMULAE, VANISHING THEOREMS AND SOME GEOMETRIC APPLICATIONS." Quarterly Journal of Mathematics 65, no. 2 (March 28, 2013): 365–97. http://dx.doi.org/10.1093/qmath/hat009.

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Більше джерел

Дисертації з теми "Monotonicity formulae"

Edquist, Anders. "Monotonicity formulas and applications in free boundary problems." Doctoral thesis, KTH, Matematik (Avd.), 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-12405.

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This thesis consists of three papers devoted to the study of monotonicity formulas and their applications in elliptic and parabolic free boundary problems. The first paper concerns an inhomogeneous parabolic problem. We obtain global and local almost monotonicity formulas and apply one of them to show a regularity result of a problem that arises in connection with continuation of heat potentials.In the second paper, we consider an elliptic two-phase problem with coefficients bellow the Lipschitz threshold. Optimal $C^{1,1}$ regularity of the solution and a regularity result of the free boundary are established.The third and last paper deals with a parabolic free boundary problem with Hölder continuous coefficients. Optimal $C^{1,1}\cap C^{0,1}$ regularity of the solution is proven.
QC20100621

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OGNIBENE, ROBERTO. "Monotonicity formulas and blow-up methods for the study of spectral stability and fractional obstacle problems." Doctoral thesis, Università degli studi di Pavia, 2021. http://hdl.handle.net/11571/1431735.

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The present dissertation is essentially divided into two parts. In the first part, we investigate questions of spectral stability for eigenvalue problems driven by the Laplace operator, under certain specific kinds of singular perturbations. More precisely, we start by considering the spectrum of the Laplacian on a fixed, bounded domain with prescribed homogeneous boundary conditions (of pure Dirichlet or Neumann type); then, we introduce a singular perturbation of the problem, which gives rise to a perturbed sequence of eigenvalues. Our goal is to understand the asymptotic behavior of the perturbed spectrum as long as the perturbation tends to disappear. In particular, we consider two different types of singular perturbation. On one hand, in the case of homogeneous Dirichlet boundary conditions, we consider a perturbation of the domain, which consists in attaching a thin cylindrical tube to the fixed limit domain and let its section shrink to a point. In this framework, we combine energy estimates coming from a tailor-made Almgren type monotonicity formula with the Courant-Fischer min-max characterization and then we perform a careful blow-up analysis for scaled eigenfunctions; with these ingredients, we identify the sharp rate of convergence of a perturbed eigenvalue in the case in which it is approaching a simple eigenvalue of the limit problem. On the other hand, we deal with a perturbation of the boundary conditions. More specifically, we start with the homogeneous Neumann eigenvalue problem for the Laplacian and we perturb it by prescribing zero Dirichlet boundary conditions on a small subset of the boundary. In this context, we describe the sharp asymptotic behavior of a perturbed eigenvalue when it is converging to a simple eigenvalue of the limit Neumann problem. In particular, the first term in the asymptotic expansion turns out to depend on the Sobolev capacity of the subset where the perturbed eigenfunction is vanishing. We also provide a more ‘explicit’ expression for the eigenvalue variation in the particular case of Dirichlet boundary conditions imposed on a subset which is scaling to a point. In the second part of this thesis, we deal with two problems, both governed by the fractional Laplace operator, i.e. the power of order between 0 and 1 of the classical (negative) Laplacian. First, we address the question of positivity of a nonlocal Schrödinger operator, driven by the fractional Laplacian and with singular multipolar Hardy-type potentials. Namely, we provide necessary and sufficient conditions on the coefficients of the potential for the existence of a configuration of poles that ensures the positivity of the corresponding Schrödinger operator. This result is based, in turn, on a criterion in the spirit of the Agmon-Allegretto-Piepenbrink principle and on a tool fitting in the theory of localization of binding. The second topic we investigate in this part concerns geometric properties of the free boundary of solutions of a two-phase penalized obstacle-type problem for the fractional Laplacian. In view of the Caffarelli-Silvestre extension, we can interpret it as a thin obstacle-type problem driven by a second-order differential operator living in one dimension more and with a Muckenhoupt weight, that can be either singular or degenerate on the thin space. Working in this framework, by means of Almgren and Monneau type monotonicity formulas and blow-up analysis, we first prove a classification of the possible vanishing orders on the thin space and, as a consequence, the boundary strong unique continuation principle. We finally establish a stratification result for the nodal set (which coincides with the free boundary) on the thin space and we provide sharp estimates on the Hausdorff dimension of its regular and singular part.

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Lan, Yang. "Dynamique asymptotique pour des équations de KdV généralisées L2 critiques et surcritiques." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS123/document.

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Dans cette thèse, nous étudions la dynamique à temps long des solutions des équations de KdV généralisées (gKdV) critiques et surcritiques pour la masse.La première partie de cette thèse est consacrée à la construction d’une dynamique explosive auto-similaire stable pour des équations de gKdV légèrement L2 surcritique dans l’espace d’énergie H1. La preuve repose sur le profil auto-similaire construit par H. Koch. Nous donner une escription précise de la formation des singularité près du temps d’explosion.La deuxième partie est consacrée à la construction de solutions explosive aux équations de gKdV légèrement L2 surcritiques avec plusieurs points d’explosion. L’idée clé est d’envisager des solutions qui se comportent comme une somme de bulles découplée, chaque bulle se comportent comme un solution auto-similaire explosent en un seul point. Nous utilisons les argument topologique classique pour s’assurer que chaque bulle explose en même temps. Ici, nous avons besoin de données initiales plus grande régularité pour contrôler la solution entre les différents points d’explosion.Enfin, dans la troisième partie, nous considérons les équations de gKdV L2 critiques avec une perturbation saturée. Dans ce cas, toute solution avec des données initiales dans H1 est toujours globale en le temps et bornée dans H1. Nous donner une classification explicite de la dynamique près du solitons. Sous certaines hypothèses de décroissance, il n’y a que trois possibilités : (i) la solution converge asymptotiquement vers une onde solitaire ; (ii) la solution reste dans un petit voisinage de la famille modulée de l’état fondamental, en s’étalant par de temps infiniment grande (Blow down) ; (iii) la solution quitte tout petit voisinage de la famille modulée de solitons
In this thesis, we deal with the long time dynamics for solutions of the L2 critical and supercritical generalized KdV equations.The first part of this work is devoted to construct a stable self-similar blow up dynamics for slightly L2 supercritical gKdV equations in the energy space H1. The proof relies on the self-similar profile constructed by H. Koch. We will also give a specific description of the formation of singularity near the blow up time.The second part is devoted to construct blow up solutions to the slightly L2 supercritical gKdV equations with multiple blow up points. The key idea is to consider solutions which behaves like a decoupled sum of bubbles. And each bubble behaves like a self-similar blow up solutions with a single blow up point. Then we can use a classic topological argument to ensure that each bubble blows up at the same time. Here, we require a higher regularity of the initial data to control the solution between the different blow up points.Finally, in the third part, we consider the L2 critical gKdV equations with a saturated perturbation. In this case, any solution with initial data in H1 is always global in time and bounded in H1. We will give a explicit classification of the flow near the ground states. Under some suitable decay assumptions, there are only three possibilities: (i) the solution converges asymptotically to a solitary wave; (ii) the solution is always in some small neighborhood of the modulated family of the ground state, but blows down at infinite time; (iii) the solution leaves any small neighborhood of the modulated family of the ground state

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Eriksson, Jonatan. "On the pricing equations of some path-dependent options." Doctoral thesis, Uppsala : Department of Mathematics, Univ. [distributör], 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-6329.

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SOAVE, NICOLA. "Variational and geometric methods for nonlinear differential equations." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2014. http://hdl.handle.net/10281/49889.

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This thesis is devoted to the study of several problems arising in the field of nonlinear analysis. The work is divided in two parts: the first one concerns existence of oscillating solutions, in a suitable sense, for some nonlinear ODEs and PDEs, while the second one regards the study of qualitative properties, such as monotonicity and symmetry, for solutions to some elliptic problems in unbounded domains. Although the topics faced in this work can appear far away one from the other, the techniques employed in different chapters share several common features. In the firts part, the variational structure of the considered problems plays an essential role, and in particular we obtain existence of oscillating solutions by means of non-standard versions of the Nehari's method and of the Seifert's broken geodesics argument. In the second part, classical tools of geometric analysis, such as the moving planes method and the application of Liouville-type theorems, are used to prove 1-dimensional symmetry of solutions in different situations.

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"Monotonicity formulae in geometric variational problems." 2002. http://library.cuhk.edu.hk/record=b5896025.

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Анотація:

Ip Tsz Ho.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2002.
Includes bibliographical references (leaves 86-89).
Chapter 0 --- Introduction --- p.5
Chapter 1 --- Preliminary --- p.11
Chapter 1.1 --- Background in analysis --- p.11
Chapter 1.1.1 --- Holder Continuity --- p.11
Chapter 1.1.2 --- Hausdorff Measure --- p.12
Chapter 1.1.3 --- Weak Derivatives --- p.13
Chapter 1.2 --- Basic Facts of Harmonic Functions --- p.14
Chapter 1.2.1 --- Harmonic Approximation --- p.14
Chapter 1.2.2 --- Elliptic Regularity --- p.15
Chapter 1.3 --- Background in geometry --- p.16
Chapter 1.3.1 --- Notations and Symbols --- p.16
Chapter 1.3.2 --- Nearest Point Projection --- p.16
Chapter 2 --- Monotonicity formula and Regularity of Harmonic maps --- p.17
Chapter 2.1 --- Energy Minimizing Maps --- p.17
Chapter 2.2 --- Variational Equations --- p.18
Chapter 2.3 --- Monotonicity Formula --- p.21
Chapter 2.4 --- A Technical Lemma --- p.22
Chapter 2.5 --- Luckhau's Lemma --- p.28
Chapter 2.6 --- Reverse Poincare Inequality --- p.40
Chapter 2.7 --- ε-Regularity of Energy Minimizing Maps --- p.45
Chapter 3 --- Monotonicity Formulae and Vanishing Theorems --- p.52
Chapter 3.1 --- Stress energy tensor and basic formulae for harmonic p´ؤforms --- p.52
Chapter 3.2 --- Monotonicity formula --- p.59
Chapter 3.2.1 --- Monotonicity Formula for Harmonic Maps --- p.64
Chapter 3.2.2 --- Bochner-Weitzenbock Formula --- p.65
Chapter 3.3 --- Conservation Law and Vanishing Theorem --- p.68
Chapter 4 --- On conformally compact Einstein Manifolds --- p.71
Chapter 4.1 --- Energy Decay of Harmonic Maps with Finite Total Energy --- p.73
Chapter 4.2 --- Vanishing Theorem of Harmonic Maps --- p.81
Bibliography --- p.86

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GERACI, FRANCESCO. "The Classical Obstacle Problem for nonlinear variational energies and related problems." Doctoral thesis, 2017. http://hdl.handle.net/2158/1079281.

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In this thesis we investigate the classical obstacal problem for nonlinear variational energies and related problems. We prove quasi-monotonicity formulae for classical obstacle-type problems with quadratic energies with coefficients in fractional Sobolev spaces, and a linear term with a type-Dini continuity property. These formulae are used to obtain the regularity of free boundary points following the approaches by Caffarelli, Monneau and Weiss. We develop the complete free boundary analysis for solutions to classical obstacle problems related to nondegenerate nonlinear variational energies. The key tools are optimal C 1,1 regularity, which we review more generally for solutions to variational inequalities driven by nonlinear coercive smooth vector fields, and the results in Focardi et al. (2015) concerning the obstacle problem for quadratic energies with Lipschitz coefficients. Furthermore, we highlight similar conclusions for locally coercive vector fields having in mind applications to the area functional, or more generally to area-type functionals, as well. We prove also an epiperimetric inequality for the fractional obstacle problem thus extending the pioneering results by Weiss (1999) on the classical obstacle problem and the results of Focardi and Spadaro (2016) in the thin obstacle problem. We deduce the regularity of a suitable subset of the free boundary as a consequence of a decay estimate of a boundary adjusted energy “à la Weiss”, the non degeneracy of the solution and the uniqueness of the limits of suitable rescaled funciotns.

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Частини книг з теми "Monotonicity formulae"

Bellettini, Giovanni. "Huisken’s monotonicity formula." In Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations, 59–68. Pisa: Scuola Normale Superiore, 2013. http://dx.doi.org/10.1007/978-88-7642-429-8_4.

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Ritoré, Manuel, and Carlo Sinestrari. "Huisken’s monotonicity formula." In Mean Curvature Flow and Isoperimetric Inequalities, 28–32. Basel: Birkhäuser Basel, 2010. http://dx.doi.org/10.1007/978-3-0346-0213-6_9.

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Caffarelli, Luis, and Sandro Salsa. "Monotonicity formulas and applications." In Graduate Studies in Mathematics, 211–34. Providence, Rhode Island: American Mathematical Society, 2005. http://dx.doi.org/10.1090/gsm/068/12.

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Ecker, Klaus. "Integral Estimates and Monotonicity Formulas." In Regularity Theory for Mean Curvature Flow, 47–79. Boston, MA: Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-0-8176-8210-1_4.

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Velichkov, Bozhidar. "The Weiss Monotonicity Formula and Its Consequences." In Lecture Notes of the Unione Matematica Italiana, 125–47. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-13238-4_9.

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Blanchette, Jasmin Christian, and Alexander Krauss. "Monotonicity Inference for Higher-Order Formulas." In Automated Reasoning, 91–106. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-14203-1_8.

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Mantegazza, Carlo. "Monotonicity Formula and Type I Singularities." In Lecture Notes on Mean Curvature Flow, 49–84. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0145-4_3.

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Shahgholian, Henrik. "The Impact of Monotonicity Formulas in Regularity of Free Boundaries." In European Congress of Mathematics, 313–18. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8266-8_27.

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Borghini, Stefano, and Lorenzo Mazzieri. "Monotonicity Formulas for Static Metrics with Non-zero Cosmological Constant." In Contemporary Research in Elliptic PDEs and Related Topics, 129–202. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-18921-1_3.

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Focardi, Matteo, Francesco Geraci, and Emanuele Spadaro. "Quasi-Monotonicity Formulas for Classical Obstacle Problems with Sobolev Coefficients and Applications." In Advances in Mechanics and Mathematics, 185–203. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-90051-9_7.

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Тези доповідей конференцій з теми "Monotonicity formulae"

Azarm, S., and Wei-Chu Li. "Optimal Design Using a Two-Level Monotonicity-Based Decomposition Method." In ASME 1987 Design Technology Conferences. American Society of Mechanical Engineers, 1987. http://dx.doi.org/10.1115/detc1987-0006.

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Abstract In this paper, a two-level decomposition method for optimal design is described. Using this method, an optimal design problem is decomposed into several subproblems in the first-level and a coordinating problem in the second-level. In the first-level, the subproblems are analyzed using the global monotonicity concepts, then in the second-level the analyses of the subproblems are coordinated to obtain the optimal solution. Two engineering design examples, namely a gear reducer (formulated and solved in the literature) and a flywheel (formulated and solved here), illustrate applications of the developed method.

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Wilde, Douglass J. "Monotonicity Analysis of Taguchi’s Robust Circuit Design Problem." In ASME 1990 Design Technical Conferences. American Society of Mechanical Engineers, 1990. http://dx.doi.org/10.1115/detc1990-0052.

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Abstract A recent article showed how to formulate Taguchi’s robust circuit design problem rigorously as an optimization problem. A necessary condition for optimality was found to be that the control range be centered about the target value. This generates a constraint on the two design variables which cannot be solved for either variable. The present article shows that by approximating this unsolvable constraint with a simpler constraint that is solvable, one variable can be eliminated and the problem reduced to an unconstrained one in a single variable. Since this reduced objective turns out to be monotonic in the remaining design variable, its optimum value must be at the limit of its range. The corresponding optimum value of the other variable is then determined exactly from the true, not approximate, constraint. Since no model construction, experimentation, statistical analysis or numerical iteration is needed, this procedure is recommended whenever the input-output relation is known to be a monotonic algebraic function.

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Fujita, Kikuo, Naoki Ono, Yui Mitsuhashi, and Yutaka Nomaguchi. "Lineup Design Method for Intermediate Product Family by Monotonicity-Guided Optimization of Nested Mini-Max Problem." In ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/detc2020-22211.

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Abstract As a product has become complicated, and its requirements have been diversified, the simultaneous and integrative design of a series of products has become so important. However, the contents of design activities have become complex across product variety and supply chain. This paper views such a situation through a chain of parts, intermediate products, and final products, and focuses on the lineup design problem of an intermediate product. The lineup design means here to secure the worst-case performance of intermediate products across a range of specifications. The problem is formulated as a mathematical problem to maximize the worst-case efficiency, i.e., the minimum efficiency of products across the range by arranging commonalization strategy, segmentation of the range, and original design variables. This paper proposes a two-phase approach, designing a framework, and optimizing contents under the framework. The latter phase is formulated as a nested mini-max optimization problem. An effective and efficient optimization scheme is configured with employing monotonicity analysis. Finally, an application to universal motors is demonstrated for ascertaining the validity and promises of the proposed design method.

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Cha, J. C., and R. W. Wayne. "Optimization With Discrete Variables via Recursive Quadratic Programming: Part II — Algorithm and Results." In ASME 1987 Design Technology Conferences. American Society of Mechanical Engineers, 1987. http://dx.doi.org/10.1115/detc1987-0003.

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Abstract A discrete recursive quadratic programming algorithm is developed for mixed discrete constrained nonlinear progrmming (MDCNP) problems. The symmetric rank one (SR1) Hessian update formula is used to generate second order information. Also, strategies, such as the watchdog technique (WT), the monotonicity analysis technique (MA), the contour analysis technique (CA) and the restoration strategy of feasibility have been considered. Heuristic aspects of handling discrete variables are treated via the concepts and convergence discussions of Part I. This paper summarizes the details of the algorithm and its implementation. Test results for 25 different problems are presented to allow evaluation of this approach and provide a basis for performance comparison. The results show that the suggested method is a promising one, efficient and robust for the MDCNP problem.

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Zhao, Wenzhong, and Shapour Azarm. "A Cross-Sectional Shape Multiplier Method for Two-Level Optimum Design of Frames." In ASME 1990 Design Technical Conferences. American Society of Mechanical Engineers, 1990. http://dx.doi.org/10.1115/detc1990-0068.

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Анотація:

Abstract In this paper, a new method for optimum design of frame structures is presented. The method is based on a hierarchical decomposition of the structure into two-levels, namely, the bottom- and the top-level. The bottom-level consists of several subproblems each dealing with the cross-sectional sizing of a given frame-element. The top-level consists of one subproblem which is formulated for configuration design of the frame structure. Since there may be a large number of frame elements, a new shape multiplier method has been developed to simplify the formulation of the bottom-level subproblems. Furthermore, a two-level solution procedure has been developed which first solves the bottom-level subproblems based on their monotonicity analysis. It then solves the top-level subproblem as it coordinates, based on a linear approximation, the solutions to the bottom-level subproblems. Three examples with increasing degree of difficulty are presented to demonstrate the effectiveness of the method.

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