Academic literature on the topic 'Periodic Unfolding Method'

At a saddle-center bifurcation for Hamiltonian systems, the homoclinic orbit is cusp shaped at the nonlinear nonhyperbolic saddle point. Near but before the bifurcation, orbits are periodic corresponding to the unfolding of the homoclinic orbit, while after the bifurcation a double homoclinic orbit is formed with a local and global branch. The saddle-center bifurcation is dynamically unfolded due to a slowly varying potential. Near the unfolding of the homoclinic orbit, the period and action are analyzed. Asymptotic expansions for the action, period and dissipation are obtained in an overlap region near the homoclinic orbit of the saddle-center bifurcation. In addition, the unfoldings of the action and dissipation functions associated with zero energy orbits (periodic and homoclinic) near the saddle-center bifurcation are determined using the method of matched asymptotic expansions for integrals.

This work creates a version of the periodic unfolding method suitable for domains with very small inclusions in \(\mathbb{R}^N\) for \(N\geq 3\). In the first part, we explore the properties of the associated operators. The second part involves the application of the method in obtaining the asymptotic behavior of a stationary heat dissipation problem depending on the parameter \( \gamma < 0\). In particular, we consider the cases when \(\gamma \in (-1,0)\), \( \gamma < -1\) and \(\gamma = -1\). We also include here the corresponding corrector results for the solution of the problem, to complete the homogenization process. For more information see https://ejde.math.txstate.edu/Volumes/2023/85/abstr.html

Cette thèse se focalise sur l'analyse mathématique, l'analyse asymptotique et la simulation numérique de certains problèmes d'équations aux dérivées partielles issus de la modélisation en micromagnétisme. Une large partie de cette thèse est consacrée à des travaux en homogénéisation, s'attachant à établir rigoureusement des modèles dits effectifs permettant de substituer un problème posé dans un milieu homogène à un problème posé dans un milieu très fortement hétérogène. On s'attache au cas particulier où le passage d'une hétérogénéité à l'autre induit un défaut de transmission, matérialisé par un saut de la valeur de l'inconnue du problème. Ce saut est supposé proportionnel au flux traversant l'interface entre les deux hétérogénéités. Il est de plus pondéré par un coefficient dépendant de la taille des hétérogénéités. Dans tous les travaux présentés dans ce mémoire, l'influence de ce coefficient, donc du ratio entre les défauts de transmission et la taille des hétérogénéités est systématiquement étudié. La méthode privilégiée pour l’homogénéisation est celle de l'éclatement périodique initiée par D. Cioranescu, A. Damlamain et G. Griso. Le premier cas d'étude est original dans le contexte du micromagnétisme. En effet, il s'agit d'un modèle de transmission de l'information dans un système dense de capteurs sans fil. Kalantari et Shayman ont proposé des méthodes de routage gouvernées par des modèles inspirés de modèles électrostatiques. En premier lieu, on montre que l'optimisation du transport de l'information dans le réseau, dans le cas où de plus on a des défauts de transmission identifiés, conduit à considérer une forme de modèle du micromagnétisme avec saut de l'inconnue. Par la suite, on s'est intéressé à l'adaptation du modèle introduit lorsque les défauts de transmission sont distribués périodiquement dans une partie du réseau et lorsque leur nombre croit asymptotiquement. Une procédure d'homogénéisation pour définir rigoureusement le modèle homogénéisé correspondant, est développée. Les résultats de convergence obtenus justifient l'efficacité du modèle dans la détermination du routage optimal dans le réseau, en prenant en considération sa vulnérabilité et la magnitude des défauts ou des attaques. On a donc mis en place des simulations numériques pour comparer le routage optimal dans les cas avec et sans défauts. Il s'est avéré que si les trajectoires sont calculées sans tenir compte des défauts de transmission, elles peuvent être très éloignées des véritables trajectoires optimales. Comme les défauts de transmission ne sont évidemment pas toujours prévisibles, on propose une nouvelle méthode de calcul des trajectoires, beaucoup plus robuste en cas de défauts de transmission, en superposant à l'approche précédente le problème de navigation de Zermelo. Le modèle est codé en revenant à l'équation Eikonale pour éprouver numériquement sa robustesse. Le deuxième cas d'étude est l’homogénéisation de l'équation de Landau--Lifshitz--Gilbert dans un domaine e-périodique composé de deux constituants, séparés par une interface non idéale au travers de laquelle on prescrit la continuité de la dérivée conormale et un saut de la solution, saut qui est proportionnel à la dérivée conormale. Les résultats sont encore obtenus par la méthode d'éclatement périodique. Mais pour gérer les non-linéarités on doit introduire des opérateurs d'extension appropriés afin d'identifier le problème limite lorsque e tend vers zéro. Finalement, le dernier chapitre de la thèse est consacré à l'analyse mathématique d'un modèle fractionnaire décrivant la transition de phase dans les matériaux ferromagnétiques en tenant compte de l'évolution tridimensionnelle des propriétés thermodynamiques et électromagnétiques du matériel. En se basant sur la méthode de Faedo--Galerkin, on démontre l'existence et l'unicité globale de la solution faible du problème
This thesis focuses on the mathematical and asymptotical analysis, together with numerical simulations, of some PDEs problems arising from the modeling of micromagnetism. A large part of this thesis is devoted to homogenization results, for rigorously obtaining so-called effective models allowing to substitute a problem posed in a homogeneous medium for a problem posed in a very strongly heterogeneous medium. We focus on the particular case where the passage from one heterogeneity to the other induces a transmission defect, materialized by a jump in the value of the unknown of the problem. This jump is assumed to be proportional to the flux crossing the interface between the two heterogeneities. It is further weighted by a coefficient depending on the size of the heterogeneities. In all the works presented in this thesis, the influence of this coefficient, thus of the ratio between the transmission defects and the size of the heterogeneities is systematically studied. The preferred method for homogenization is that of periodic unfolding initiated by D. Ciuranescu, A. Damlamain and G. Griso. The first case study is original in the context of micromagnetism. Indeed, it is a model of information transmission in a dense wireless sensor system. Kalantari and Shayman have proposed routing methods governed by models inspired by electrostatic models. First, it is shown that the optimization of the information transport in the network, in the case where moreover we have identified transmission faults, leads to consider a form of micromagnetism model with jump of the unknown as described above. Subsequently, we are interested in the adaptation of the introduced model when the transmission defects are periodically distributed in a part of the network and when their number grows asymptotically. A homogenization procedure is developed to rigorously define the corresponding homogenized model. The convergence results obtained justify the efficiency of the model in determining the optimal routing in the network, taking into consideration its vulnerability and the magnitude of defects or attacks. Therefore, numerical simulations were set up to compare the optimal routing in the cases with and without defects. It turned out that if the trajectories are calculated without considering the transmission faults, they can be far away from the true optimal trajectories. Since transmission defects are obviously not always predictable, a new method for computing trajectories, much more robust in the case of transmission faults, is proposed by superimposing the Zermelo navigation problem on the previous approach. The model is coded by returning to the Eikonal equation to numerically test its robustness. The second case of study is the homogenization of the Landau-Lifshitz-Gilbert equation in a e-periodic domain composed of two constituents, separated by a nonideal interface through which continuity of the conormal derivative and a jump in the solution, a jump that is proportional to the conormal derivative, are prescribed. The results are once again obtained by the periodic unfolding method. But to handle the nonlinearities, appropriate extension operators are introduced to identify the limit problem when e vanishes. Finally, the last chapter of the thesis is devoted to the mathematical analysis of a fractional model describing the phase transition in ferromagnetic materials when taking into account the three-dimensional evolution of the thermodynamic and electromagnetic properties of the material. Based on the Faedo-Galerkin method, the existence and global uniqueness of the weak solution of the problem is demonstrated

This book traces the development of a critical practice within African American literature, art, and activism that identifies and critiques the widespread appropriation of classical tradition to the projects of exceptionalist historiography and cultural white supremacy in the United States. This appropriative method has typically figured the United States as the inheritor of the best traditions of classical antiquity and thus as the standard bearer for the idea of civilization. Where dominant narratives—articulated through political speeches and editorials, poetry and the visual arts, and the monumental architecture of Washington, DC—envision the political project of the United States as modeled on ancient Rome yet destined to surpass it in the unfolding of an exceptional history, the writers, artists, and activists this book considers have connected modern America to the ancient world through the institution of slavery and the geopolitics of empire. The book tracks this critique over more than two centuries, from Phillis Wheatley’s poetry in the era of Revolution, through the antislavery writings of David Walker, William Wells Brown, and the black newspapers of the antebellum period, to the works of Charles Chesnutt, Toni Morrison, and other twentieth-century writers, before concluding with the monumental sculpture of the contemporary artist Kara Walker.

This chapter draws upon statistical reports generated from all levels of the statistical system—internal work bulletins, and materials from conferences at the local, provincial, and national levels—to uncover the messiness of actual statistical work and its relationship to planning. It captures not only the centralizing impetus of the expansion but also the varieties of challenges that were encountered in putting into practice the methods that were at the heart of socialist statistics: the periodic reporting system, and the various forms of typical sampling. Statistical work was carried out within the larger unfolding context of increasing complexity, changing (economic) focus, and struggles over administrative devolution. As work became more complicated and demanding, engagement with and discussions about statistical activities grew more sophisticated—methods were expanded, analyses were undertaken, and the results of such analyses were circulated among leaders and bureaucrats. However, a key problem remained—the system that had been set up incentivized the overproduction of reports, and the state had little or no capacity to handle the resultant excess reports.

Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix Φ(t,α) associated with the linear part of the equation can be expressed in terms of the periodic Lyapunov-Floquét transformation matrix Q(t,α) and a time-invariant matrix R(α). Computation of Q(t,α) and R(α) in a symbolic form as a function of system parameters α is of paramount importance in stability, bifurcation analysis, and control system design. In the past, a methodology has been presented for computing Φ(t,α) in a symbolic form, however Q(t,α) and R(α) have never been calculated in a symbolic form. Since Q(t,α) and R(α) were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In this work a technique for symbolic computation of Q(t,α), and R(α) matrices is presented. First, Φ(t,α) is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then R(α) is computed using the Gaussian quadrature integral formula. Finally Q(t,α) is computed using the matrix exponential summation method. Using Mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

Two-dimensional (2D) nanofabrication processes such as lithography are the primary tools for building functional nanostructures. The third spatial dimension enables completely new devices to be realized, such as photonic crystals with arbitrary defect structures and materials with negative index of refraction [1]. Presently, available methods for three-dimensional (3D) nanopatterning tend to be either cost inefficient or limited to periodic structures. The Nanostructured Origami method fabricates 3D devices by first patterning nanostructures (electronic, optical, mechanical, etc) onto a 2D substrate and subsequently folding segments along predefined creases until the final design is obtained [2]. This approach allows almost arbitrary 3D nanostructured systems to be fabricated using exclusively 2D nanopatterning tools. In this paper, we present two approaches to the kinematic and dynamic modeling of folding origami structures. The first approach deals with the kinematics of unfolding single-vertex origami. This work is based on research conducted in the origami mathematics community, which is making rapid progress in understanding the geometry of origami and folding in general [3]. First, a unit positive “charge” is assigned to the creases of the structure in its folded state. Thus, each configuration of the structure as it unfolds can be assigned a value of electrostatic (Coulomb) energy. Because of repulsion between the positive charges, the structure will unfold if allowed to decrease its energy. If the energy minimization can be carried out all the way to the completely unfolded state, we are simultaneously guaranteed of the absence of collisions for the determined path. The second method deals with dynamic modeling of folding multi-segment (accordion style) origamis. The actuation method for folding the segments uses a thin, stressed metal layer that is deposited as a hinge on a relatively stress free structural layer. Through the use of robotics routines, the hinges are modeled as revolute joints, and the system dynamics are calculated.

The goal of the article is to describe the transformation of retail and to reveal how representatives of retail turn to the idea of authenticity. The study focuses on 3 areas in the city of Novosibirsk: the city center, Akademgorodok, and Zatulinskiy residential area. This research is conducted by a mixed-method involving quantitative and qualitative approaches. The quantitative approach is an analysis of interactive map data (2GIS): the information about all businesses in Novosibirsk’s districts in 2007 and 2023 years. The qualitative approach is visual analysis of the commercial fabric of the districts using the authenticity concept. The main findings of the research. During the period from 2007 to 2023, there was a significant increase in the number of retails in the selected areas. Moreover, in the case of the city center and Akademgorodok, the proportion of the HORECA category, primarily catering establishments, has increased. The retail in Zatulinskiy residential area is mainly represented by chain establishments, among which various grocery stores and supermarkets stand out, and the HORECA sphere is not expressed vividly. At the design level, establishments can refer to the authenticity of the place in various ways: by indicating official, expert evaluation of the age of the building, through names that refer to the roots of the place, or through images that would be associated with cultural symbols of the place. However, it is also typical for situations when symbols of places located in other countries are commodified, or when the only concern in the design is the visibility of the establishment. The results of this study may be of interest to business representatives who apply local symbols in their design.

Within an organization, there will always be more project proposals than the resources available. Having this in mind, selecting projects must be made through a system of priorities as those projects which contribute to reaching the organization's aims should be selected. Planning a project is complete after having scheduled all activities, taking into account the entire resource basis available in the organization. We should also keep in mind the fact that in order to successfully run a project one should focus on the project finalizing effort in accordance with the deadline. Thus, the planning process should comprise the fact that the organization resources are to be employed neither excessively nor deficiently. All the above lead to a rather complex planning process, should we keep in mind the scheduling or the organization could face great unbalance in carrying out the project or situations could appear in which the resources necessary to the project could be more numerous than the ones available in the organization. Unfolding a project under an ideal execution program cannot be always a fact, but using rules and best working tools definitely lead to a rational and efficient scheduling of resources. Employing an adequate method could support starting a project in a short period of time only if scarce resources are available and can be legally allotted to the respective project. While allotting resources, most of the times observation can be made that available resources do not allow unfolding activities quickly enough so that the project could reach the final deadline. Projects are limited in using resources by the level and the bulk of available resources and the maximum deadline set in finalizing the respective project.

A point cloud of reference points forms the programmable basis of a new method of urban and architectural modeling. Points in space from the smallest identifiable units that are informed to communicate with each other to form complex data structures. The data are visualized as spatial voxels [3d pixels] as to represent spaces and volumes that maintain their mutual relationships under varying circumstances. The subsequent steps in the development from point cloud to the multimodal urban strategy are driven by variable local and global parameters. Step by step new and more detailed actors are introduced in the serious design game. Values feeding the voxel units may be fixed, variables based on experience, or randomly generated. The target value may be fixed or kept open. Using lines or curves and groups of points from the original large along the X, Y and Z-axes organized crystalline set of points are selected to form the shape of actual working space. The concept of radical multimodality at the level of the smallest grain requires that at each stage in the design game individual units are addressed as to adopt a unique function during a unique amount of time. Each unit may be a home, a workplace, a workshop, a shop, a lounge area, a school, a garden or just an empty voxel anytime and anywhere in the selected working space. The concept of multimodality [MANIC, K Oosterhuis, 2018] is taken to its extreme as to stimulate the development of diversity over time and in its spatial arrangement. The programmable framework for urban multimodality acknowledges the rise and shine of the new international citizen, who travels the world, lives nowhere and everywhere, inhabits places and spaces for ultrashort, shorter or longer periods of time, lives her/his life as a new nomad [New Babylon, Constant Nieuwenhuys, 1958]. The new nomad lives on her/his own or in groups of like-minded people, effectuated by setting preferences and choices being made via the ubiquitous multimodality app, which organizes the unfolding of her / his life. In the serious design game nomadic life is facilitated by real time activation of a complex set of programmable monads. Playing and further developing the design journey was executed in 4 workshop sessions with different professional stakeholders, architects, engineers, entrepreneurs and project developers.