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Mathematics in Science and Technology   | enregistrements trouvés : 77

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In this talk, I will focus on a Fokker-Planck equation modeling interacting neurons in a network where each neuron is governed by an Integrate and Fire dynamic type. When the network is excitatory, neurons that discharge, instantaneously increased the membrane potential of the neurons of the network with a speed which is proportional to the amplitude of the global activity of the network. The self-excitable nature of these neurons in the case of excitatory networks leads to phenomena of blow-up, once the proportion of neurons that are close to their action potential is too high. In this talk, we are interested in understanding the regimes where solutions globally exist. By new methods of entropy and upper-solution, we give criteria where the phenomena of blow-up can not appear and specify, in some cases, the asymptotic behavior of the solution.

integrate-and-fire - neural networks - Fokker-Planck equation - blow-up
In this talk, I will focus on a Fokker-Planck equation modeling interacting neurons in a network where each neuron is governed by an Integrate and Fire dynamic type. When the network is excitatory, neurons that discharge, instantaneously increased the membrane potential of the neurons of the network with a speed which is proportional to the amplitude of the global activity of the network. The self-excitable nature of these neurons in the case of ...

92B20 ; 82C32 ; 35Q84

In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. These penalty functionals are crucial to force the regularized solution to conform to some notion of simplicity/low complexity. Classical priors of this kind includes sparsity, piecewise regularity and low-rank. These are natural assumptions for many applications, ranging from medical imaging to machine learning.
imaging - image processing - sparsity - convex optimization - inverse problem - super-resolution
In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. These penalty functionals are crucial to force the regularized solution to conform to some notion of simplicity/low complexity. Classical priors of this kind includes sparsity, piecewise regularity and low-rank. These are natural assumptions for many applications, ranging from medical imaging to machine ...

62H35 ; 65D18 ; 94A08 ; 68U10 ; 90C31 ; 80M50 ; 47N10

Mathematical modeling and numerical mathematics of today is very much Lagrangian and modern automated modeling techniques lead to differential-algebraic systems. The optimal control for such systems in general cannot be obtained using the classical Euler-Lagrange approach or the maximum principle, but it is shown how this approach can be extended.
differential-algebraic equations - optimal control - Lagrangian subspace - necessary optimality conditions - Hamiltonian system - symplectic flow
Mathematical modeling and numerical mathematics of today is very much Lagrangian and modern automated modeling techniques lead to differential-algebraic systems. The optimal control for such systems in general cannot be obtained using the classical Euler-Lagrange approach or the maximum principle, but it is shown how this approach can be extended.
differential-algebraic equations - optimal control - Lagrangian subspace - necessary optimality ...

93C05 ; 93C15 ; 49K15 ; 34H05

A popular line of research in evolutionary biology is to use time-calibrated phylogenies in order to infer the underlying diversification process. This involves the use of stochastic models of ultrametric trees, i.e., trees whose tips lie at the same distance from the root. We recast some well-known models of ultrametric trees (infinite regular trees, exchangeable coalescents, coalescent point processes) in the framework of so-called comb metric spaces and give some applications of coalescent point processes to the phylogeny of bird species.

However, these models of diversification assume that species are exchangeable particles, and this always leads to the same (Yule) tree shape in distribution. Here, we propose a non-exchangeable, individual-based, point mutation model of diversification, where interspecific pairwise competition is only felt from the part of individuals belonging to younger species. As the initial (meta)population size grows to infinity, the properly rescaled dynamics of species lineages converge to a one-parameter family of coalescent trees interpolating between the caterpillar tree and the Kingman coalescent.

Keywords: ultrametric tree, inference, phylogenetic tree, phylogeny, birth-death process, population dynamics, evolution
A popular line of research in evolutionary biology is to use time-calibrated phylogenies in order to infer the underlying diversification process. This involves the use of stochastic models of ultrametric trees, i.e., trees whose tips lie at the same distance from the root. We recast some well-known models of ultrametric trees (infinite regular trees, exchangeable coalescents, coalescent point processes) in the framework of so-called comb metric ...

60J80 ; 60J85 ; 92D15 ; 92D25 ; 54E45 ; 54E70

In this international farewell address about 35 years of research on Dynamic energy Budget theory, I review the ontogeny of the theory, and discuss background, motivation, start, milestones and outlook from a personal perspective. The effort is framed as a case study in Theoretical Biology.

Le principe de décision en démocratie consiste à produire, de l'expression des opinions individuelles, un consensus. Il existe de multiples procédures pour passer des unes à l'autre variant suivant les pays, les jurys... Le Président n'est pas élu de la même façon en France, aux USA ou en Irlande. Quelles seraient les procédures qui répondraient à des critères "raisonnables" de qualité ?
Des mathématiciens se sont intéressés à ce type de questions. Du paradoxe de Condorcet au théorème de Black en passant par le théorème de Arrow, leurs réponses sont parfois déconcertantes.
Le principe de décision en démocratie consiste à produire, de l'expression des opinions individuelles, un consensus. Il existe de multiples procédures pour passer des unes à l'autre variant suivant les pays, les jurys... Le Président n'est pas élu de la même façon en France, aux USA ou en Irlande. Quelles seraient les procédures qui répondraient à des critères "raisonnables" de qualité ?
Des mathématiciens se sont intéressés à ce type de ...

91B12 ; 91B08 ; 91B14 ; 91F10

Post-edited  Le problème Graph Motif - Partie 1
Fertin, Guillaume (Auteur de la Conférence) | CIRM (Editeur )

Le problème Graph Motif est défini comme suit : étant donné un graphe sommet colorié G=(V,E) et un multi-ensemble M de couleurs, déterminer s'il existe une occurrence de M dans G, c'est-à-dire un sous ensemble V' de V tel que
(1) le multi-ensemble des couleurs de V' correspond à M,
(2) le sous-graphe G' induit par V' est connexe.
Ce problème a été introduit, il y a un peu plus de 10 ans, dans le but de rechercher des motifs fonctionnels dans des réseaux biologiques, comme par exemple des réseaux d'interaction de protéines ou des réseaux métaboliques. Graph Motif a fait depuis l'objet d'une attention particulière qui se traduit par un nombre relativement élevé de publications, essentiellement orientées autour de sa complexité algorithmique.
Je présenterai un certain nombre de résultats algorithmiques concernant le problème Graph Motif, en particulier des résultats de FPT (Fixed-Parameter Tractability), ainsi que des bornes inférieures de complexité algorithmique.
Ceci m'amènera à détailler diverses techniques de preuves dont certaines sont plutôt originales, et qui seront je l'espère d'intérêt pour le public.
Le problème Graph Motif est défini comme suit : étant donné un graphe sommet colorié G=(V,E) et un multi-ensemble M de couleurs, déterminer s'il existe une occurrence de M dans G, c'est-à-dire un sous ensemble V' de V tel que
(1) le multi-ensemble des couleurs de V' correspond à M,
(2) le sous-graphe G' induit par V' est connexe.
Ce problème a été introduit, il y a un peu plus de 10 ans, dans le but de rechercher des motifs fonctionnels dans des ...

05C15 ; 05C85 ; 05C90 ; 68Q17 ; 68Q25 ; 68R10 ; 92C42 ; 92D20

Motivated by the spectrogram (or short-time Fourier transform) basic principles of linear algebra are explained, preparing for the more general case of Gabor frames in time-frequency analysis. The importance of the singular value decomposition and the four spaces associated with a matrix is pointed out, and based on this the pseudo-inverse (leading later to the dual Gabor frame) and the Loewdin (symmetric) orthogonalization are explained.

15-XX ; 41-XX ; 42-XX ; 46-XX

The aim is to describe the distribution of immune status in an age-structured population on the basis of a within-host sub-model [1] for continuous waning and occasional boosting. Inspired by both Feller's fundamental work [2] and the more recent delay equation formulation of physiologically structured populations [3,4], we derive, for a given force of infection, a linear renewal equation that can be solved by successive approximation, i.e., by generation expansion (with the generation number corresponding to the number of times an individual became infected).
In joint work in progress with Wilfred de Graaf, Peter Teunis and Mirjam Kretzschmar we want to use either the generation expansion or an invariant/stable distribution as the starting point for the efficient computation of coarse statistics.
The aim is to describe the distribution of immune status in an age-structured population on the basis of a within-host sub-model [1] for continuous waning and occasional boosting. Inspired by both Feller's fundamental work [2] and the more recent delay equation formulation of physiologically structured populations [3,4], we derive, for a given force of infection, a linear renewal equation that can be solved by successive approximation, i.e., by ...

92D30 ; 60J75 ; 45D05

Inspired by modeling in neurosciences, we here discuss the well-posedness of a networked integrate-and-fire model describing an infinite population of companies which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the debt of a company increases when some of the others default: precisely, the loss it receives is proportional to the instantaneous proportion of companies that default at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by a, is of great importance as the resulting system is known to blow-up when a takes large values, a blow-up meaning that a macroscopic proportion of companies may default at the same time. In the current talk, we focus on the complementary regime and prove that existence and uniqueness hold in arbitrary time without any blow-up when the excitatory parameter is small enough. Inspired by modeling in neurosciences, we here discuss the well-posedness of a networked integrate-and-fire model describing an infinite population of companies which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the debt of a company increases when some of the others default: precisely, the loss it receives is proportional to the instantaneous proportion ...

35K60 ; 82C31 ; 92B20

Lecture 1. Collective dynamics and self-organization in biological systems : challenges and some examples.

Lecture 2. The Vicsek model as a paradigm for self-organization : from particles to fluid via kinetic descriptions

Lecture 3. Phase transitions in the Vicsek model : mathematical analyses in the kinetic framework.

35L60 ; 82C22 ; 82B26 ; 82C26 ; 92D50

Multi angle  Project evaluation under uncertainty
Zubelli, Jorge P. (Auteur de la Conférence) | CIRM (Editeur )

Industrial strategic decisions have evolved tremendously in the last decades towards a higher degree of quantitative analysis. Such decisions require taking into account a large number of uncertain variables and volatile scenarios, much like financial market investments. Furthermore, they can be evaluated by comparing to portfolios of investments in financial assets such as in stocks, derivatives and commodity futures. This revolution led to the development of a new field of managerial science known as Real Options.
The use of Real Option techniques incorporates also the value of flexibility and gives a broader view of many business decisions that brings in techniques from quantitative finance and risk management. Such techniques are now part of the decision making process of many corporations and require a substantial amount of mathematical background. Yet, there has been substantial debate concerning the use of risk neutral pricing and hedging arguments to the context of project evaluation. We discuss some alternatives to risk neutral pricing that could be suitable to evaluation of projects in a realistic context with special attention to projects dependent on commodities and non-hedgeable uncertainties. More precisely, we make use of a variant of the hedged Monte-Carlo method of Potters, Bouchaud and Sestovic to tackle strategic decisions. Furthermore, we extend this to different investor risk profiles. This is joint work with Edgardo Brigatti, Felipe Macias, and Max O. de Souza.
If time allows we shall also discuss the situation when the historical data for the project evaluation is very limited and we can make use of certain symmetries of the problem to perform (with good estimates) a nonintrusive stratified resampling of the data. This is joint work with E. Gobet and G. Liu.
Industrial strategic decisions have evolved tremendously in the last decades towards a higher degree of quantitative analysis. Such decisions require taking into account a large number of uncertain variables and volatile scenarios, much like financial market investments. Furthermore, they can be evaluated by comparing to portfolios of investments in financial assets such as in stocks, derivatives and commodity futures. This revolution led to the ...

91B26 ; 91B06 ; 91B30 ; 91B24

In this talk, we will briefly look at the history of wavelets, from signal processing algorithms originating in speech and image processing, and harmonic analysis constructions of orthonormal bases. We review the promises, the achievements, and some of the limitations of wavelet applications, with JPEG and JPEG2000 as examples. We then take two key insights from the wavelet and signal processing experience, namely the time-frequency-scale view of the world, and the sparsity property of wavelet expansions, and present two recent results. First, we show new bounds for the time-frequency spread of sequences, and construct maximally compact sequences. Interestingly they differ from sampled Gaussians. Next, we review work on sampling of finite rate of innovation signals, which are sparse continuous-time signals for which sampling theorems are possible. We conclude by arguing that the interface of signal processing and applied harmonic analysis has been both fruitful and fun, and try to identify lessons learned from this experience.

Keywords: wavelets ­ filter banks - subband coding ­ uncertainty principle ­ sampling theory ­ sparse sampling
In this talk, we will briefly look at the history of wavelets, from signal processing algorithms originating in speech and image processing, and harmonic analysis constructions of orthonormal bases. We review the promises, the achievements, and some of the limitations of wavelet applications, with JPEG and JPEG2000 as examples. We then take two key insights from the wavelet and signal processing experience, namely the time-frequency-scale view ...

94A08 ; 94A12 ; 65T60 ; 42C40

Multi angle  Learning on the symmetric group
Vert, Jean-Philippe (Auteur de la Conférence) | CIRM (Editeur )

Many data can be represented as rankings or permutations, raising the question of developing machine learning models on the symmetric group. When the number of items in the permutations gets large, manipulating permutations can quickly become computationally intractable. I will discuss two computationally efficient embeddings of the symmetric groups in Euclidean spaces leading to fast machine learning algorithms, and illustrate their relevance on biological applications and image classification. Many data can be represented as rankings or permutations, raising the question of developing machine learning models on the symmetric group. When the number of items in the permutations gets large, manipulating permutations can quickly become computationally intractable. I will discuss two computationally efficient embeddings of the symmetric groups in Euclidean spaces leading to fast machine learning algorithms, and illustrate their relevance ...

62H30 ; 62P10 ; 68T05

Oxygen is essential for burning food and generate energy, but may become limiting for aquatic organisms that rely on gas exchange under water. This is because breathing under water is challenging: the diffusion of oxygen is orders of magnitude lower in water than in air, while the higher density and viscosity of water greatly enhance the cost of breathing. Given that oxygen may be also be a limiting resource, respiration physiology may be relevant to understand energy budgets in aquatic ectotherms.
Traditionally, respiration physiology has focused on the benefits of extracting sufficient amounts of oxygen and thus prevent asphyxiation. However, breathing oxygen is intrinsically dangerous: while a shortage of oxygen quickly leads to asphyxiation, too much oxygen is toxic. Therefore, the ability to regulate oxygen consumption rates (i.e. respiratory control) is at a premium; good respiratory control will enable ectotherms to balance oxygen toxicity against the risk of asphyxiation across a wide range of temperatures.
In this presentation I will focus on the effects of body size and temperature on this balancing act with regard to oxygen uptake and consumption. Body size is intimately tied to oxygen budgets and hence energy budgets through size related changes in oxygen requirements and respiratory surfaces. Furthermore, a larger body size may represent a respiratory advantage that helps to overcome viscosity. Given that viscous forces are larger in cold water, this respiratory advantage represents a novel explanation for the pattern of larger body sizes in cold water, with polar gigantism as the extreme manifestation.
Temperature is also intimately tied to oxygen budgets and hence energy budgets through thermal controls on metabolism and temperature related changes in the availability of dissolved oxygen (notably diffusivity, viscosity and solubility). Thus, differences in temperatures may act more strongly on ectotherms that rely on aquatic rather than on aerial gas exchange. Comparing four different insect orders, I demonstrate that thermal tolerance is indeed affected more by the prevalent oxygen conditions in species with poor respiration control. In conclusion, the ability to regulate gas exchange (i.e. respiratory control) is thus a key attribute of species that helps to explain thermal responses from an oxygen perspective.
Oxygen is essential for burning food and generate energy, but may become limiting for aquatic organisms that rely on gas exchange under water. This is because breathing under water is challenging: the diffusion of oxygen is orders of magnitude lower in water than in air, while the higher density and viscosity of water greatly enhance the cost of breathing. Given that oxygen may be also be a limiting resource, respiration physiology may be ...

92D25 ; 92D50 ; 92C15 ; 92C30

Recent population genomics studies focus prevalently on the aspects of demography and adaptation, whereas age structure (for example, in plants via the maintenance of seed banks) has attracted less attention. Germ banking, that is, seed or egg dormancy, is a prevalent and important life-history trait in plants and invertebrates, which buffers against environmental variability and modulates species extinction in fragmented habitats. I will here summarize our recent findings investigating the intertwined effect of germ banking, time-varying population size and selection on genetic polymorphism in the wild tomato species. First, we examine the effect of seed banking on within species variability and local adaptation in the wild tomato Solanum chilense. Population genetic analyses and statistical inference of past demography was conducted on pooled-sequencing from 30 genes from an exhaustive sampling of 23 populations over Chile and Peru. We reveal a north-south colonization associated with relaxed purifying selection in the south as shown by a decrease of genetic variation and an increasing proportion of nonsynonymous polymorphism from north to south and population substructure with at least four genetic groups. We also uncover 1) a decreasing proportion of adaptive amino acid substitutions from north to south suggesting that adaptation is favoured in large populations, while 2) signatures of local adaptation predominantly occur in the smaller populations from the marginal ranges in the south. These results combined with additional germination data suggest that colonization of new habitats was accompanied by local adaptation for shorter seed banks in the marginal populations, shaping in return the available nucleotide diversity and effectiveness of purifying and positive selection. Second, we use ABC and polymorphism data to estimate population divergence times between two wild tomato species in presence of seed banks. We show that unknown seed banking also impedes our knowledge of the speciation process. Joint work with Katharina B. Böndel, Wolfgang Stephan. Recent population genomics studies focus prevalently on the aspects of demography and adaptation, whereas age structure (for example, in plants via the maintenance of seed banks) has attracted less attention. Germ banking, that is, seed or egg dormancy, is a prevalent and important life-history trait in plants and invertebrates, which buffers against environmental variability and modulates species extinction in fragmented habitats. I will here ...

92D40 ; 92C80 ; 92D15

The flexibility of the Bayesian approach to uncertainty, and its notable practical successes, have made it an increasingly popular tool for uncertainty quantification. The scope of application has widened from the finite sample spaces considered by Bayes and Laplace to very high-dimensional systems, or even infinite-dimensional ones such as PDEs. It is natural to ask about the accuracy of Bayesian procedures from several perspectives: e.g., the frequentist questions of well-specification and consistency, or the numerical analysis questions of stability and well-posedness with respect to perturbations of the prior, the likelihood, or the data. This talk will outline positive and negative results (both classical ones from the literature and new ones due to the authors) on the accuracy of Bayesian inference. There will be a particular emphasis on the consequences for high- and infinite-dimensional complex systems. In particular, for such systems, subtle details of geometry and topology play a critical role in determining the accuracy or instability of Bayesian procedures. Joint with with Houman Owhadi and Clint Scovel (Caltech). The flexibility of the Bayesian approach to uncertainty, and its notable practical successes, have made it an increasingly popular tool for uncertainty quantification. The scope of application has widened from the finite sample spaces considered by Bayes and Laplace to very high-dimensional systems, or even infinite-dimensional ones such as PDEs. It is natural to ask about the accuracy of Bayesian procedures from several perspectives: e.g., the ...

62F15 ; 62G35

Microparasites (virus, bactéries, protozoaires ...) et macroparasites (métazoaires : helminthes, arthropodes...) sont omniprésents dans les écosystèmes terrestres et marins. Le nombre total d'espèces parasites sur la planète est supérieur à celui des espèces libres qu'ils colonisent, temporairement ou non, au point que ces organismes interfèrent à toutes les échelles d'organisation du vivant. Les pathologies qu'ils peuvent parfois engendrer sont dépendantes de conditions particulières, soit liées à leur propre virulence, soit à un ensemble de facteurs environnementaux. Dans ce contexte, les modèles mathématiques constituent des outils précieux en épidémiologie, permettant de mieux comprendre les modalités de leur propagation dans les populations d'hôtes. Aborder les stratégies démographiques des micro ou des macroparasites implique des approches mathématiques différentes. Le développement de ces modèles ouvre des perspectives intéressantes pour décrire, analyser et même prévoir les comportements démographiques de ces systèmes couplés. En milieu marin, les macroparasites peuvent aussi poser des problèmes de santé à leurs hôtes quand les équilibres de différentes natures sont déplacés, avec ou sans l'intervention de l'homme (espace protégé, pêche, aquaculture...). En prenant l'exemple de parasites de Poissons téléostéens, l'accent sera mis sur la complexité des processus biologiques en cause, et son intégration dans des modèles mathématiques.

Microparasites (virus, bactéries, protozoaires ...) et macroparasites (métazoaires : helminthes, arthropodes...) sont omniprésents dans les écosystèmes terrestres et marins. Le nombre total d'espèces parasites sur la planète est supérieur à celui des espèces libres qu'ils colonisent, temporairement ou non, au point que ces organismes interfèrent à toutes les échelles d'organisation du vivant. Les pathologies qu'ils peuvent parfois engendrer sont ...

00A06 ; 00A08 ; 92-XX

Dans une première partie, je présenterai différentes problématiques liées à des statistiques d'occurrences de mots dans des génomes et décortiquerai plus en détail la question de savoir comment détecter si un mot a une fréquence d'apparition significativement anormale dans une séquence. Dans une deuxième partie, je présenterai différentes extensions pour tenir compte du fait qu'un motif d'ADN fonctionnel n'est pas toujours un « mot », mais qu'il peut avoir une structure plus complexe qui nécessite le développement de nouvelles méthodes statistiques. Dans une première partie, je présenterai différentes problématiques liées à des statistiques d'occurrences de mots dans des génomes et décortiquerai plus en détail la question de savoir comment détecter si un mot a une fréquence d'apparition significativement anormale dans une séquence. Dans une deuxième partie, je présenterai différentes extensions pour tenir compte du fait qu'un motif d'ADN fonctionnel n'est pas toujours un « mot », mais qu'il ...

92C40 ; 62P10 ; 60J20 ; 92C42

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