Patlak-Keller-Segel model

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Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model

Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model

1.1 The PKS system and its critical mass The Patlak-Keller-Segel system [39, 27] is one of the simplest models of chemotaxis, describing the evolution of the population density of a cell colony which is diffusing across a two dimensional surface. In addition to the diffusion, as the cells move across the surface, they continually emit a chemical attractant, which itself diffuses across the surface. The cells tend to move towards higher concentrations of the attractant, and this induces a drift term tending to concentrate the population, and countering the spreading effects of the diffusion. A model organism for this type of behavior is the Dictyostelium Discoideum which segregates cyclic adenosine monophosphate, another
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Infinite time blow-up in the Keller-Segel system: existence and stability

Infinite time blow-up in the Keller-Segel system: existence and stability

Patlak-Keller-Segel model in a disk. SIAM J. Math. Anal. 40 (2008/09), no. 5, 1852-1881. [18] E. F. Keller; L. A. Segel, Initiation of slide mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399-415. [19] Merle, F.; Raphael, P. Sharp upper bound on the blow-up rate for the critical nonlin- earSchrdinger equation. Geom. Funct. Anal.13(2003), no. 3, 591-642.

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Existence and stability of infinite time blow-up in the Keller-Segel system

Existence and stability of infinite time blow-up in the Keller-Segel system

There is a huge literature on chemotaxis in biology and in mathematics. The Patlak-Keller-Segel model [ 43 , 35 ] is used in mathematical biology to describe the motion of mono-cellular organisms, like Dictyostelium Discoideum, which move ran- domly but experience a drift in presence of a chemo-attractant. Under certain cir- cumstances, these cells are able to emit the chemo-attractant themselves. Through the chemical signal, they coordinate their motion and eventually aggregate. Such a self-organization scenario is at the basis of many models of chemotaxis and is con- sidered as a fundamental mechanism in biology. Of course, the aggregation induced by the drift competes with the noise associated with the random motion so that aggregation occurs only if the chemical signal is strong enough. A classical survey of the mathematical problems in chemotaxis models can be found in [ 31 , 32 ]. After a proper adimensionalization, it turns out that all coefficients in the Patlak-Keller- Segel model studied in this paper can be taken equal to 1 and that the only free parameter left is the total mass. For further considerations on chemotaxis, we shall refer to [ 30 ] for biological models and to [ 11 ] for physics backgrounds.
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Existence and stability of infinite time blow-up in the Keller-Segel system

Existence and stability of infinite time blow-up in the Keller-Segel system

There is a huge literature on chemotaxis in biology and in mathematics. The Patlak-Keller-Segel model [ 43 , 35 ] is used in mathematical biology to describe the motion of mono-cellular organisms, like Dictyostelium Discoideum, which move ran- domly but experience a drift in presence of a chemo-attractant. Under certain cir- cumstances, these cells are able to emit the chemo-attractant themselves. Through the chemical signal, they coordinate their motion and eventually aggregate. Such a self-organization scenario is at the basis of many models of chemotaxis and is con- sidered as a fundamental mechanism in biology. Of course, the aggregation induced by the drift competes with the noise associated with the random motion so that aggregation occurs only if the chemical signal is strong enough. A classical survey of the mathematical problems in chemotaxis models can be found in [ 31 , 32 ]. After a proper adimensionalization, it turns out that all coefficients in the Patlak-Keller- Segel model studied in this paper can be taken equal to 1 and that the only free parameter left is the total mass. For further considerations on chemotaxis, we shall refer to [ 30 ] for biological models and to [ 11 ] for physics backgrounds.
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A finite volume scheme for a Keller-Segel model with additional cross-diffusion

A finite volume scheme for a Keller-Segel model with additional cross-diffusion

WITH ADDITIONAL CROSS-DIFFUSION MARIANNE BESSEMOULIN-CHATARD AND ANSGAR J ¨ UNGEL Abstract. A finite volume scheme for the (Patlak-) Keller-Segel model in two space di- mensions with an additional cross-diffusion term in the elliptic equation for the chemical signal is analyzed. The main feature of the model is that there exists a new entropy func- tional yielding gradient estimates for the cell density and chemical concentration. The main features of the numerical scheme are positivity preservation, mass conservation, en- tropy stability, and—under additional assumptions—entropy dissipation. The existence of a discrete solution and its numerical convergence to the continuous solution is proved. Furthermore, temporal decay rates for convergence of the discrete solution to the homo- geneous steady state is shown using a new discrete logarithmic Sobolev inequality. Nu- merical examples point out that the solutions exhibit intermediate states and that there exist nonhomogeneous stationary solutions with a finite cell density peak at the domain boundary.
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Approches numeriques par des volumes finis du modele de keller-segel

Approches numeriques par des volumes finis du modele de keller-segel

Theoretical and mathematical modelling of chemotaxis dates to the pio- neering works of Patlak in the 1950s [ 72 ] and Keller and Segel in the 1970s [ 49 ]. The review article by Horstmann [ 46 ] provides a detailed introduction into the KellerSegel mathematics model for chemotaxis. In its original form this model consists of four coupled reaction-advection-diffusion equations, which can be reduced under quasi-steady-state assumptions to a model of two unknown functions u and c. This model is the basis of our study in this research work. The general form of this model is given by the following set of partial differential equations:
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The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\RR^d$, $d \ge 3$

The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $\RR^d$, $d \ge 3$

describe the aggregation of cells by chemotaxis: the diffusion of the amoebae in a Petri dish is counterbalanced by the attraction toward higher concentrations of chemo-attractant that they themselves emit. This model has been widely studied mathematically in the last two decades with a main focus on the so-called parabolic- elliptic Keller-Segel system (also referred to as the Patlak-Keller-Segel system or the Smoluchowski-Poisson equation in astrophysics) which corresponds to the choice τ = 0, see [5, 14, 15, 22] for a review. In particular, a striking feature of the Patlak- Keller-Segel system is that, given a non-negative and integrable initial condition ρ 0 ,
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Keller-Lieb-Thirring inequalities for Schrödinger operators on cylinders

Keller-Lieb-Thirring inequalities for Schrödinger operators on cylinders

R´ esum´ e. Cette note est consacr´ee ` a des estimations spectrales de Keller-Lieb-Thirring pour des op´erateurs de Schr¨ odinger sur des cylindres infinis : la valeur absolue de l’´etat fondamental est born´ee par une fonction d’une norme du potentiel. Il est montr´e que les potentiels optimaux de petite norme ne d´ependent que d’une seule variable : il s’agit d’un r´esultat de sym´etrie. La preuve provient d’un argument de perturbation qui repose sur des r´esultats de rigidit´e r´ecents pour des ´equations elliptiques non-lin´eaires sur des cylindres. A l’inverse, les potentiels optimaux de grande norme qui ne d´ependent que d’une seule variable sont instables : cela fournit un r´esultat de brisure de sym´ etrie. La valeur optimale qui s´epare les deux r´egimes est ´etablie dans le cas du produit d’une sph`ere et d’une droite.
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Towards Scalable Model Views on Heterogeneous Model Resources

Towards Scalable Model Views on Heterogeneous Model Resources

We significantly improved EMF Views for large model resources by following these ideas. As introduced in Section 3.3, our key tenet was: delay actual hits to the resource as much as possible. Another important improvement of EMF Views concerned the view loading process. As said previously, weaving models can be large depending on the number (and contents) of virtual references. Previously, EMF Views eagerly populated these virtual references when loading the view. Each virtual reference thus delayed loading the view further: for larger weaving models, this meant several seconds or even minutes. Here again, the optimization lies in lazi- ness: delaying work that can be done later. In this case, we have to populate virtual references only when they are first accessed. If some virtual associations are never looked up then we never have to load them from the weaving model, thus avoiding the loading cost. Making this change to EMF Views enabled loading views with large weaving models with no overhead in terms of time.
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Efficient Model Partitioning for Distributed Model Transformations

Efficient Model Partitioning for Distributed Model Transformations

Keywords Model Transformation, ATL, Data Distribution, Static Analysis, MapReduce 1. Introduction The Model-Driven Engineering (MDE) paradigm has been successfully embraced in several domains, for manufactur- ing maintainable software while decreasing cost and effort. For instance, recent works have shown its benefits in applica- tions for the construction industry [26] (for communication of building information and interoperation with different tools and actors), modernization of legacy systems [3] (for aiding the migration of large legacy codebases into novel versions meeting particular requirements), learning and big data ana- lytics [9] (for reducing the expertise necessary to implement probabilistic models for machine learning, and speed up de- velopment).
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On Model Subtyping

On Model Subtyping

Abstract. Various approaches have recently been proposed to ease the manipu- lation of models for specific purposes (e.g., automatic model adaptation or reuse of model transformations). Such approaches raise the need for a unified theory that would ease their combination, but would also outline the scope of what can be expected in terms of engineering to put model manipulation into action. In this work, we address this problem from the model substitutability point of view, through model typing. We introduce four mechanisms to achieve model substi- tutability, each formally defined by a subtyping relation. We then discuss how to declare and check these subtyping relations. This work provides a formal refer- ence specification establishing a family of model-oriented type systems. These type systems enable many facilities that are well known at the programming lan- guage level. Such facilities range from abstraction, reuse and safety to impact analyses and auto-completion. Key words: SLE, Modeling Languages, Model Typing, Model Substitutability
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Distributed Model-to-Model Transformation with ATL on MapReduce

Distributed Model-to-Model Transformation with ATL on MapReduce

Shared-memory parallelization is a closely related prob- lem to distribution. For model transformation, Tisi et al. [30] present a systematic two-steps approach to parallelize ATL transformations. The authors provided a multi-threaded im- plementation of the ATL engine, where every rule is executed in a separate thread for both steps. The parallel ATL compiler and virtual machine have been adapted to enable a parallel execution and reduce synchronization overhead. A similar approach for parallel graph transformations in multicore sys- tems [14] introduces a two-phase algorithm (matching and modifier) similar to ours. Bergmann et al. propose an ap- proach to parallelize graph transformations based on incre- mental pattern matching [3]. This approach uses a message passing mechanism to notify of model changes. The incre- mental pattern matcher is split into different containers, each one is responsible for a set of patterns. The lack of distributed memory concerns make these solutions difficult to adapt to the distributed computing scenario. Moreover in these cases the authors investigate task distribution, while we focus on data distribution, especially for handling VLMs.
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Analysis of singularities in elliptic equations : the Ginzburg-Landau model of superconductivity, the Lin-Ni-Takagi problem, the Keller-Segel model of chemotaxis, and conformal geometry

Analysis of singularities in elliptic equations : the Ginzburg-Landau model of superconductivity, the Lin-Ni-Takagi problem, the Keller-Segel model of chemotaxis, and conformal geometry

quantity is the gauge-invariant version of the Jacobian determinant of u, and is the ana- logue of the vorticity of a fluid. To analyze the vortices, people have been developing tools, in particular the ball construction method and Jacobian estimates. The first one was introduced independently by Jerrard [Jer99] and Sandier [San98]. It allows one to obtain universal lower bounds for two-dimensional Ginzburg-Landau energies in terms of the topology of the vortices. These lower bounds capture the known fact that vortices of degree d cost at least an order π|d| log 1 ε of energy. The second tool, that has been widely used in the analysis of the Ginzburg-Landau model in any dimension after the work by Jerrard and Soner [JS02], is the Jacobian (or vorticity) estimate. This estimate allows one to relate the vorticity µ(u, A) with Dirac masses (supported on co-dimension 2 objects), which in 2D are naturally derived from the ball construction method.
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Keller-Lieb-Thirring inequalities for Schrödinger operators on cylinders

Keller-Lieb-Thirring inequalities for Schrödinger operators on cylinders

R´ esum´ e. Cette note est consacr´ee ` a des estimations spectrales de Keller-Lieb-Thirring pour des op´erateurs de Schr¨ odinger sur des cylindres infinis : la valeur absolue de l’´etat fondamental est born´ee par une fonction d’une norme du potentiel. Il est montr´e que les potentiels optimaux de petite norme ne d´ependent que d’une seule variable : il s’agit d’un r´esultat de sym´etrie. La preuve provient d’un argument de perturbation qui repose sur des r´esultats de rigidit´e r´ecents pour des ´equations elliptiques non-lin´eaires sur des cylindres. A l’inverse, les potentiels optimaux de grande norme qui ne d´ependent que d’une seule variable sont instables : cela fournit un r´esultat de brisure de sym´ etrie. La valeur optimale qui s´epare les deux r´egimes est ´etablie dans le cas du produit d’une sph`ere et d’une droite.
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A complete resolution of the Keller maximum clique problem

A complete resolution of the Keller maximum clique problem

Jeff Lagarias heard about the Keller conjecture early in 1991 from Victor Klee, who gave a lecture on interesting geometry problems at a meeting in Ober- wolfach, Germany. Jeff told Peter Shor about it and in 1992, they found a counterexample in 12 dimensions which they were promptly able to reduce to ten di- mensions [13]. In 2000, John Mackey [14] found an eight-dimensional counterexample leaving the conjec- ture open only for seven dimensions. The computation described in this paper proves that the Keller graph G 7 has no clique of order 128, which implies that there is no counterexample to Keller’s conjecture in seven di- mensions with cubes whose coordinates are integers or half-integers. This is very strong evidence that Keller’s conjecture is true in seven dimensions, as all the other counterexamples that have been found have been tilings with integer and half-integer coordinates.
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L'insoumise : Helen Keller, ou, Le trajet de soi vers l'autre : essai sur l'appropriation du langage et analyse heuristique d'une pratique interdisciplinaire

L'insoumise : Helen Keller, ou, Le trajet de soi vers l'autre : essai sur l'appropriation du langage et analyse heuristique d'une pratique interdisciplinaire

L'interdisciplinarité étant reliée historiquement au courant de la performance , la perspective heuristique me semble constituer un écrin pertinent pour étudier cette [r]

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Automatic Model Generation Strategies for Model Transformation Testing

Automatic Model Generation Strategies for Model Transformation Testing

These model fragments are transformed to Alloy predicates by Cartier . For instance, model fragment mfAllRanges7 is transformed to the predicate : pred mfAllRanges7(){some c:Class|#c.attrs=1} As mentioned in our previous paper [8] if a test set contains models where all model fragments are contained in at least one model then we say that the input domain is com- pletely covered. However, these model fragments are generated considering only the concepts and relationships in the Ecore model and they do not take into account the constraints on the Ecore model. Therefore, not all model fragments are consistent with the input meta-model because the generated models that contain these model fragments do not satisfy the constraints on the meta-model. Cartier invokes the Alloy Analyzer [15] to automatically check if a model containing a model fragment and satisfying the input domain can be synthesized for a general scope of number of objects. This allows us to detect inconsistent model fragments. For example, the following predicate, mfAll- Ranges7a, is the Alloy representation of a model fragment specifying that some Class
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Model Transformation Reuse: A Graph-based Model Typing Approach

Model Transformation Reuse: A Graph-based Model Typing Approach

4.6 Prototype Summary and Discussion In this chapter, we have described a prototype of a transformation migration toolkit, named MetaModMap, which provides supports for transformation reuse through adaptation with user-defined directives. The toolkit has been developed to work with model-to-model transformations written in the MOMENT2-GT language. The toolkit was implemented as Eclipsed-based plug-ins that includes 1) a tex- tual editor used to define semantic correspondences between metamodels written in a DSL language, i.e. MetaModMap, 2) a Higher-Order-Transformation interpreter used for automatically migrating an existing transformation definition according to a valid mapping definition defined in the provided editor. For the MetaModMap editor, we have presented how to develop a modern editor that allows users to work with a language at the abstract syntax representation level by using the Xtext lan- guage development framework. Such an editor can provide on-the-fly features such as scoping and validation which are related to the integration of our graph-based typing approach presented in Chapter 3 into practice. For the interpreter of the language, we have shown how an on-the-fly transformation adaptation engine can be integrated into the editor by means of the code generator API provided by Xtext. By such integrations, end-users are provided with an easy to use transformation migration toolkit within Eclipse. Thereby, they can use our toolkit for improving transforma- tion re-usability in incorporating others EMF facilities for creating, managing models, metamodels, and specially with MOMENT2-GT framework for defining and running transformations as a complete MDE ecology.
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A Model-Driven Engineering Framework for Constrained Model Search

A Model-Driven Engineering Framework for Constrained Model Search

metamodels and low-level languages. As an open-source tool, it becomes a good alternative to conguration tools. 4.3.2 Limitations and future work The rst limitation of this implementation concerns the Alloy metamodel. The metamodel is adapted to work directly with the TCS parser, and as such has some syntactical constructs that would deserve a semantical analysis to completely check the validity of a textual in- put during model injections. For the same parsing reasons, some of the Alloy metamodel concepts are purely syntactical and may be confusing when developping transformations from/to Alloy. We plan to realize a more tted metamodel that leaves aside the syntactical constructs that were necessary for the TCS parser, and to develop a two-direction (ATL) transformation between the two metamodels. By transparently chaining the TCS process and the transformation, the syntactical metamodel would not be visible anymore for exter- nal users. This notion of syntactical and semantical analysis is well-known in the eld of compilation, and we believe that, from a general point of view, it also applies to metamodels and their textual versions.
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Multiphase model for transformation induced plasticity Extended Leblond's model

Multiphase model for transformation induced plasticity Extended Leblond's model

S k Average stress deviator in the k-th phase 1. Introduction When phase transformations occur under applied mechanical loads that can be much lower than the yield stress, significant plastic strains are observed. This phenomenon called transfor- mation induced plasticity or sometimes super-plasticity received particular attention since the eighties because of numerous applications in mechanical engineering. For instance welding or sheet metal manufacturing (especially the run out process that consists in cooling down a sheet under tension) cannot be approached without considering transformation plasticity. Two di fferent mechanisms have been proposed to explain transformation plasticity by Greenwood and Johnson (1965) and Magee and Paxton (1966). One of the most fruitful model has been proposed by Leblond et al. (1986a) and relies on an homogenization procedure without adding a priori contributions in the plastic strain tensor that would be proportional to phase propor- tion rate. This classic contribution results directly from the homogenization procedure itself and both Greenwood and Johnson and Magee mechanisms are thus identified in the obtained contributions. In order to give a more specific and usable form to the homogenized model, a morphological assumption which consists in an idealization of the microstructure has been pro- posed and solved analytically by Leblond et al. (1986b, 1989) without considering hardening e ffects and by Leblond (1989) if hardening is taken into account. These four papers form the base of the Leblond’s transformation induced plasticity model. Because of its solid theoretical basis and simple explicit formulas for the overall plastic strain increment, Leblond’s model has been intensively used for various engineering applications and has been included (see Bergheau and Leblond (1991)) in a commercial Finite Element software SYSWELD ® (2012) dedicated to welding applications. Among many contributions, one can mention for instance quite re- cent applications of the Leblond’s model to welding proposed by Bate et al. (2009); Xu et al. (2012) and sheet metal manufacturing by Lee et al. (2009). Moreover, Kim et al. (2005) devel- oped a numerical implementation of Leblond’s model within the framework of multiplicative elastic-plasticity considering welding applications.
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