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Documents : Multi angle  Conférences Vidéo | enregistrements trouvés : 565

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Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G, A))$ of the Cayley graph $\Gamma(G, A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding diam$(G) := max_A$ diam$(\Gamma(G, A))$.
It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$. In 2011, Helfgott and Seress gave a quasipolynomial bound, namely, $O\left (e^{(log n)^{4+\epsilon}}\right )$. We will discuss a recent, much simplified version of the proof.
Given a finite group $G$ and a set $A$ of generators, the diameter diam$(\Gamma(G, A))$ of the Cayley graph $\Gamma(G, A)$ is the smallest $\ell$ such that every element of $G$ can be expressed as a word of length at most $\ell$ in $A \cup A^{-1}$. We are concerned with bounding diam$(G) := max_A$ diam$(\Gamma(G, A))$.
It has long been conjectured that the diameter of the symmetric group of degree $n$ is polynomially bounded in $n$. In 2011, ...

20B05 ; 05C25 ; 20B30 ; 20F69 ; 20D60

In this talk, I will introduce the stochastic downscaling method (SDM) that borrows techniques from small scale turbulence (S.B. Pope) for the simulation of wind flows thanks to hybrid methods (deterministic-stochastic). I will present the downscaling method used to refine a wind forecast at a sufficiently small scale, and the way wind turbines are implemented in the model. Comparisons with traditional numerical methods (LES) and validation w.r.t. experimental data will also be provided. In this talk, I will introduce the stochastic downscaling method (SDM) that borrows techniques from small scale turbulence (S.B. Pope) for the simulation of wind flows thanks to hybrid methods (deterministic-stochastic). I will present the downscaling method used to refine a wind forecast at a sufficiently small scale, and the way wind turbines are implemented in the model. Comparisons with traditional numerical methods (LES) and validation ...

60H10 ; 86A10 ; 86-08 ; 76F55 ; 76M35

Multi angle  When he was one hundred years old!
Verdier, Norbert (Auteur de la Conférence) | CIRM (Editeur )

In this talk we will don't speak about Joseph-Louis Lagrange (1736-1813) but about Lagrange's reception at the nineteenth Century. "Who read Lagrange at this Times?", "Why and How?", "What does it mean being a mathematician or doing mathematics at this Century" are some of the questions of our conference. We will give some elements of answers and the case Lagrange will be a pretext in order to explain what are doing historians of mathematics: searching archives and ­ thanks to a methodology ­ trying to understand, read and write the Past.
Lagrange - mathematical press - complete works - bibliographic index of mathematical sciences (1894-1912) - Liouville - Boussinesq - Terquem
In this talk we will don't speak about Joseph-Louis Lagrange (1736-1813) but about Lagrange's reception at the nineteenth Century. "Who read Lagrange at this Times?", "Why and How?", "What does it mean being a mathematician or doing mathematics at this Century" are some of the questions of our conference. We will give some elements of answers and the case Lagrange will be a pretext in order to explain what are doing historians of mathematics: ...

01A50 ; 01A55 ; 01A70 ; 01A74 ; 01A80

Energy investment into maturation encompasses any expenses linked to tissue differentiation, i.e. re-organization of body structure during development. This is different from growth which can be conceptualized as synthesis of more of the same. Energy invested into growth is fixed into the biomass of the organism (with some overheads), but energy invested in maturation is oxidized as metabolic work making it more difficult to quantify in practice. Nonetheless it can be quantified and it can even represent a substantial part of the energy budget of living organisms. In this talk I will give an overview of different studies where investment in maturity was quantified. The focus will be on 4 different types of organisms: cnidarians, ctenophores, teleost fish and frogs. I will further discuss what type of eco-physiological effects might be expected when an organism modifies its investment into these processes. Some intriguing literature studies will be presented which can be re-interpreted in perhaps unexpected ways when investment into maturation is taken into account. This raises the question of just how important and how flexible such costs might actually be. Maturity can be used as a quantifier for internal time. Seven criteria were proposed which should be respected by any such metric: (1) independent of morphology, (2) independent of body size, (3) depend on one a priori homologous event, (4) unaffected by changes in temperature, (5) similar between closely related species, (6) increase with clock time, and (7) physically quantifiable (Reiss 1989). We showed that the maturity concept of Dynamic Energy Budget theory complies with all those criteria and on the basis of this information and the studies presented above I will finish by discussing the potential role of maturity in shaping metabolic flexibility. Energy investment into maturation encompasses any expenses linked to tissue differentiation, i.e. re-organization of body structure during development. This is different from growth which can be conceptualized as synthesis of more of the same. Energy invested into growth is fixed into the biomass of the organism (with some overheads), but energy invested in maturation is oxidized as metabolic work making it more difficult to quantify in ...

92D25 ; 92D40 ; 92C30

We begin by introducing to the diagrammatic Cherednik algebras of Webster. We then summarise some recent results (in joint work with Anton Cox and Liron Speyer) concerning the representation theory of these algebras. In particular we generalise Kleshchev-type decomposition numbers, James-Donkin row and column removal phenomena, and the Kazhdan-Lusztig approach to calculating decomposition numbers.

20G43 ; 20F55 ; 20B30

Multi angle  Waves and microstructures
Weinstein, Michael (Auteur de la Conférence) | CIRM (Editeur )

In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While wavelets are now a well-established tool in numerical signal processing (for instance the JPEG2000 coding standard is based on a wavelet transform) it has been recognized in the past decades that they also possess several shortcomings, in particular with respect to the treatment of multidimensional data where anisotropic structures such as edges in images are typically present. This deficiency of wavelets has given birth to the research area of geometric multiscale analysis where frame constructions which are optimally adapted to anisotropic structures are sought. A milestone in this area has been the construction of curvelet and shearlet frames which are indeed capable of optimally resolving curved singularities in multidimensional data.
In this course we will outline these developments, starting with a short introduction to wavelets and then moving on to more recent constructions of curvelets, shearlets and ridgelets. We will discuss their applicability to diverse problems in signal processing such as compression, denoising, morphological component analysis, or the solution of transport PDEs. Implementation aspects will also be covered. (Slides in attachment).
In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While ...

42C15 ; 42C40

In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While wavelets are now a well-established tool in numerical signal processing (for instance the JPEG2000 coding standard is based on a wavelet transform) it has been recognized in the past decades that they also possess several shortcomings, in particular with respect to the treatment of multidimensional data where anisotropic structures such as edges in images are typically present. This deficiency of wavelets has given birth to the research area of geometric multiscale analysis where frame constructions which are optimally adapted to anisotropic structures are sought. A milestone in this area has been the construction of curvelet and shearlet frames which are indeed capable of optimally resolving curved singularities in multidimensional data.
In this course we will outline these developments, starting with a short introduction to wavelets and then moving on to more recent constructions of curvelets, shearlets and ridgelets. We will discuss their applicability to diverse problems in signal processing such as compression, denoising, morphological component analysis, or the solution of transport PDEs. Implementation aspects will also be covered. (Slides in attachment).
In several applications in signal processing it has proven useful to decompose a given signal in a multiscale dictionary, for instance to achieve compression by coefficient thresholding or to solve inverse problems. The most popular family of such dictionaries are undoubtedly wavelets which have had a tremendous impact in applied mathematics since Daubechies' construction of orthonormal wavelet bases with compact support in the 1980s. While ...

42C15 ; 42C40

We start with a brief historical account of wavelets and of the way they shattered some of the preconceptions of the 20th century theory of statistical signal processing that is founded on the Gaussian hypothesis. The advent of wavelets led to the emergence of the concept of sparsity and resulted in important advances in image processing, compression, and the resolution of ill-posed inverse problems, including compressed sensing. In support of this change in paradigm, we introduce an extended class of stochastic processes specified by a generic (non-Gaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lévy noise. Starting from first principles, we prove that the solutions of such equations are either Gaussian or sparse, at the exclusion of any other behavior. Moreover, we show that these processes admit a representation in a matched wavelet basis that is "sparse" and (approximately) decoupled. The proposed model lends itself well to an analytic treatment. It also has a strong predictive power in that it justifies the type of sparsity-promoting reconstruction methods that are currently being deployed in the field.

Keywords: wavelets - fractals - stochastic processes - sparsity - independent component analysis - differential operators - iterative thresholding - infinitely divisible laws - Lévy processes
We start with a brief historical account of wavelets and of the way they shattered some of the preconceptions of the 20th century theory of statistical signal processing that is founded on the Gaussian hypothesis. The advent of wavelets led to the emergence of the concept of sparsity and resulted in important advances in image processing, compression, and the resolution of ill-posed inverse problems, including compressed sensing. In support of ...

42C40 ; 60G20 ; 60G22 ; 60G18 ; 60H40

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

sparsity - morphological diversity - inpainting - cosmology - weak lensing - cosmic microwave background

65T60 ; 94A12 ; 85A35

Since the last twenty years, Littlewood-Paley analysis and wavelet theory has proved to be a very useful tool for non parametric statistic. This is essentially due to the fact that the regularity spaces (Sobolev and Besov) could be characterized by wavelet coefficients. Then it appeared that that the Euclidian analysis is not always appropriate, and lot of statistical problems have their own geometry. For instance: Wicksell problem and Jacobi Polynomials, Tomography and the harmonic analysis of the ball, the study of the Cosmological Microwave Background and the harmonic analysis of the sphere. In these last years it has been proposed to build a Littlewood-Paley analysis and a wavelet theory associated to the Laplacien of a Riemannian manifold or more generally a positive operator associated to a suitable Dirichlet space with a good behavior of the associated heat kernel. This can help to revisit some classical studies of the regularity of Gaussian field.

Keywords: heat kernel - functional calculus - wavelet - Gaussian process
Since the last twenty years, Littlewood-Paley analysis and wavelet theory has proved to be a very useful tool for non parametric statistic. This is essentially due to the fact that the regularity spaces (Sobolev and Besov) could be characterized by wavelet coefficients. Then it appeared that that the Euclidian analysis is not always appropriate, and lot of statistical problems have their own geometry. For instance: Wicksell problem and Jacobi ...

43A85 ; 60G15 ; 60G17 ; 58C50

Breast cancer is the most common type of cancer among women and despite recent advances in the medical field, there are still some inherent limitations in the currently used screening techniques. The radiological interpretation of X-ray mammograms often leads to over-diagnosis and, as a consequence, to unnecessary traumatic and painful biopsies. First we use the 1D Wavelet Transform Modulus Maxima (WTMM) method to reveal changes in skin temperature dynamics of women breasts with and without malignant tumor. We show that the statistics of temperature temporal fluctuations about the cardiogenic and vasomotor perfusion oscillations do not change across time-scales for cancerous breasts as the signature of homogeneous monofractal fluctuations. This contrasts with the continuous change of temperature fluctuation statistics observed for healthy breasts as the hallmark of complex multifractal scaling. When using the 2D WTMM method to analyze the roughness fluctuations of X-ray mammograms, we reveal some drastic loss of roughness spatial correlations that likely results from some deep architectural change in the microenvironment of a breast tumor. This local breast disorganisation may deeply affect heat transfer and related thermomechanics in the breast tissue and in turn explain the loss of multifractal complexity of temperature temporal fluctuations previously observed in mammary glands with malignant tumor. These promising findings could lead to the future use of combined wavelet-based multifractal processing of dynamic IR thermograms and X-ray mammograms to help identifying women with high risk of breast cancer prior to more traumatic examinations. Besides potential clinical impact, these results shed a new light on physiological changes that may precede anatomical alterations in breast cancer development.

Keywords: breast cancer - X-ray mammography - infrared thermography - multifractal analysis - wavelet transform - wavelet transform modulus maxima method
Breast cancer is the most common type of cancer among women and despite recent advances in the medical field, there are still some inherent limitations in the currently used screening techniques. The radiological interpretation of X-ray mammograms often leads to over-diagnosis and, as a consequence, to unnecessary traumatic and painful biopsies. First we use the 1D Wavelet Transform Modulus Maxima (WTMM) method to reveal changes in skin ...

92-08 ; 92C50 ; 92C55

Coorbit theory was developed in the late eighties as a unifying principle covering (possible non-)orthogonal frame expansions in the wavelet and in the time-frequency context. Very much in the spirit of « coherent frames » or also reproducing kernels for a Moebius invariant Banach space of analytic functions one can describe a family of function spaces associated with a given integrable and irreducible group representation on a Hilbert space by its generalized wavelet transform, and obtain (among others) atomic decomposition results for the resulting spaces. The theory was flexible enough to cover also more recent examples, such as voice transforms related to the Blaschke group or the spaces (and frames) related to the shearlet transform.
As time permits I will talk also on the role of Banach frames and the usefulness of Banach Gelfand triples, especially the one based on the Segal algebra $S_0(G)$, which happens to be a modulation space, in fact the minimal among all time-frequency invariant non-trivial function spaces.

Keywords: wavelet theory - time-frequency analysis - modulation spaces - Banach-Gelfand-triples - Toeplitz operators - atomic decompositions - function spaces - shearlet transform - Blaschke group
Coorbit theory was developed in the late eighties as a unifying principle covering (possible non-)orthogonal frame expansions in the wavelet and in the time-frequency context. Very much in the spirit of « coherent frames » or also reproducing kernels for a Moebius invariant Banach space of analytic functions one can describe a family of function spaces associated with a given integrable and irreducible group representation on a Hilbert space by ...

43-XX ; 46Exx ; 42C40 ; 42C15 ; 42C10

Multi angle  Vertex degrees in planar maps
Drmota, Michael (Auteur de la Conférence) | CIRM (Editeur )

We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. We also discuss some possible extension to maps of higher genus.
This is joint work with Gwendal Collet and Lukas Klausner
We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even inf...

05A19 ; 05A16 ; 05C10 ; 05C30

This talk is about verified numerical algorithms in Isabelle/HOL, with a focus on guaranteed enclosures for solutions of ODEs. The enclosures are represented by zonotopes, arising from the use of affine arithmetic. Enclosures for solutions of ODEs are computed by set-based variants of the well-known Runge-Kutta methods.
All of the algorithms are formally verified with respect to a formalization of ODEs in Isabelle/HOL: The correctness proofs are carried out for abstract algorithms, which are specified in terms of real numbers and sets. These abstract algorithms are automatically refined towards executable specifications based on lists, zonotopes, and software floating point numbers. Optimizations for low-dimensional, nonlinear dynamics allow for an application highlight: the computation of an accurate enclosure for the Lorenz attractor. This contributes to an important proof that originally relied on non-verified numerical computations.
This talk is about verified numerical algorithms in Isabelle/HOL, with a focus on guaranteed enclosures for solutions of ODEs. The enclosures are represented by zonotopes, arising from the use of affine arithmetic. Enclosures for solutions of ODEs are computed by set-based variants of the well-known Runge-Kutta methods.
All of the algorithms are formally verified with respect to a formalization of ODEs in Isabelle/HOL: The correctness proofs are ...

68T15 ; 34-04 ; 34A12 ; 37D45 ; 65G20 ; 65G30 ; 65G50 ; 65L70 ; 68N15 ; 68Q60 ; 68N30 ; 65Y04

Multi angle  Variations on an example of Hirzebruch
Stover, Matthew (Auteur de la Conférence) | CIRM (Editeur )

In '84, Hirzebruch constructed a very explicit noncompact ball quotient manifold in the process of constructing smooth projective surfaces with Chern slope arbitrarily close to 3. I will discuss how this and some closely related ball quotients are useful in answering a variety of other questions. Some of this is joint with Luca Di Cerbo.

14M27 ; 32Q45 ; 57M50

Kardar-Parisi-Zhang fluctuation exponent for the last-passage value of the two-dimensional corner growth model with exponential weights. We sketch the proof of the fluctuation exponent for the stationary corner growth process, and if time permits indicate how the exponent is derived for the percolation process with i.i.d. weights.

60K35 ; 60K37 ; 82C22 ; 82C43 ; 82D60

Busemann functions for the two-dimensional corner growth model with exponential weights. Derivation of the stationary corner growth model and its use for calculating the limit shape and proving existence of Busemann functions.

60K35 ; 60K37 ; 82C22 ; 82C43 ; 82D60

Bayesian posterior distributions can be numerically intractable, even by the means of Markov Chain Monte Carlo methods. Bayesian variational methods can then be used to compute directly (and fast) a deterministic approximation of these posterior distributions. In this course, I describe the principles of the variational methods and their application in Bayesian inference, review main theoretical results and discuss their use on examples.

62F15 ; 62H12 ; 49J40

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