182 research outputs found
Bayesian frequentist hybrid inference
Bayesian and frequentist methods differ in many aspects, but share some basic
optimality properties. In practice, there are situations in which one of the
methods is more preferred by some criteria. We consider the case of inference
about a set of multiple parameters, which can be divided into two disjoint
subsets. On one set, a frequentist method may be favored and on the other, the
Bayesian. This motivates a joint estimation procedure in which some of the
parameters are estimated Bayesian, and the rest by the maximum-likelihood
estimator in the same parametric model, and thus keep the strengths of both the
methods and avoid their weaknesses. Such a hybrid procedure gives us more
flexibility in achieving overall inference advantages. We study the consistency
and high-order asymptotic behavior of the proposed estimator, and illustrate
its application. Also, the results imply a new method for constructing
objective prior.Comment: Published in at http://dx.doi.org/10.1214/08-AOS649 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Skorohod and Stratonovich integration in the plane
This article gives an account on various aspects of stochastic calculus in
the plane. Specifically, our aim is 3-fold: (i) Derive a pathwise change of
variable formula for a path indexed by a square, satisfying some H\"older
regularity conditions with a H\"older exponent greater than 1/3. (ii) Get some
Skorohod change of variable formulas for a large class of Gaussian processes
defined on the suqare. (iii) Compare the bidimensional integrals obtained with
those two methods, computing explicit correction terms whenever possible. As a
byproduct, we also give explicit forms of corrections in the respective change
of variable formulas
Multiple binomial sums
International audienceMultiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with algebraic generating function. We study the representation of the generating functions of binomial sums by integrals of rational functions. The outcome is twofold. Firstly, we show that a univariate sequence is a multiple binomial sum if and only if its generating function is the diagonal of a rational function. Secondly, we propose algorithms that decide the equality of multiple binomial sums and that compute recurrence relations for them. In conjunction with geometric simplifications of the integral representations, this approach behaves well in practice. The process avoids the computation of certificates and the problem of the appearance of spurious singularities that afflicts discrete creative telescoping, both in theory and in practice
Evolutionary dynamics of phenotype-structured populations : from individual-level mechanisms to population-level consequences
This research was supported in part by the Australian Research Council (DP140100339) and by the French National Research Agency through the ANR blanche project Kibord [ANR-13-BS01-0004] and the âANR JCâ project Modevol [ANR-13-JS01-0009]. TL was also supported in part by the Hadamard Mathematics Labex, backed by the Fondation MathĂ©matique Jacques Hadamard, through a grant overseen by the French National Research Agency [ANR-11-LABX-0056-LMH]. LD was also supported in part by UniversitĂ© Sorbonne Paris CitĂ© âInvestissements dâAvenirâ[ANR-11-IDEX-0005].Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species that face environmental changes. In particular, empirical work has provided important insights into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a number of verbal explanations have emerged that are suited to logical testing by proof-of-concept mathematical models. Here, we present a stochastic agent-based model and a related deterministic integrodifferential equation model for the evolution of a phenotype-structured population composed of asexually-reproducing and competing organisms which are exposed to novel environmental conditions. This setting has relevance to the study of biological systems where colonising asexual populations must survive and rapidly adapt to hostile environments, like pathogenesis, invasion and tumour metastasis. We explore how evolution might proceed when epigenetic variation in gene expression can change the reproductive capacity of individuals within the population in the new environment. Simulations and analyses of our models clarify the conditions under which certain evolutionary paths are possible, and illustrate that whilst epigenetic mechanisms may facilitate adaptation in asexual species faced with environmental change, they can also lead to a type of âepigenetic loadâ and contribute to extinction. Moreover, our results offer a formal basis for the claim that constant environments favour individuals with low rates of stochastic phenotypic variation. Finally, our model provides a âproof of conceptâ of the verbal hypothesis that phenotypic stability is a key driver in rescuing the adaptive potential of an asexual lineage, and supports the notion that intense selection pressure can, to an extent, offset the deleterious effects of high phenotypic instability and biased epimutations, and steer an asexual population back from the brink of an evolutionary dead end.PostprintPeer reviewe
Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences
Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species that face environmental changes. In particular, empirical work has provided important insights into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a number of verbal explanations have emerged that are suited to logical testing by proof-of-concept mathematical models. Here, we present a stochastic agent-based model and a related deterministic integrodifferential equation model for the evolution of a phenotype-structured population composed of asexually-reproducing and competing organisms which are exposed to novel environmental conditions. This setting has relevance to the study of biological systems where colonising asexual populations must survive and rapidly adapt to hostile environments, like pathogenesis, invasion and tumour metastasis. We explore how evolution might proceed when epigenetic variation in gene expression can change the reproductive capacity of individuals within the population in the new environment. Simulations and analyses of our models clarify the conditions under which certain evolutionary paths are possible and illustrate that while epigenetic mechanisms may facilitate adaptation in asexual species faced with environmental change, they can also lead to a type of âepigenetic loadâ and contribute to extinction. Moreover, our results offer a formal basis for the claim that constant environments favour individuals with low rates of stochastic phenotypic variation. Finally, our model provides a âproof of conceptâ of the verbal hypothesis that phenotypic stability is a key driver in rescuing the adaptive potential of an asexual lineage and supports the notion that intense selection pressure can, to an extent, offset the deleterious effects of high phenotypic instability and biased epimutations, and steer an asexual population back from the brink of an evolutionary dead end
Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line
In this paper, we revisit the infinite iteration scheme of normal form
reductions, introduced by the first and second authors (with Z. Guo), in
constructing solutions to nonlinear dispersive PDEs. Our main goal is to
present a simplified approach to this method. More precisely, we study normal
form reductions in an abstract form and reduce multilinear estimates of
arbitrarily high degrees to successive applications of basic trilinear
estimates. As an application, we prove unconditional well-posedness of
canonical nonlinear dispersive equations on the real line. In particular, we
implement this simplified approach to an infinite iteration of normal form
reductions in the context of the cubic nonlinear Schr\"odinger equation (NLS)
and the modified KdV equation (mKdV) on the real line and prove unconditional
well-posedness in with (i) for the cubic NLS
and (ii) for the mKdV. Our normal form approach also allows us
to construct weak solutions to the cubic NLS in , , and distributional solutions to the mKdV in (with some uniqueness statements).Comment: 60 pages. Typos corrected. To appear in Ann. Fac. Sci. Toulouse Mat
Cognition Without Construction: Kant, Maimon, and the Transcendental Philosophy of Mathematics
In the Critique of Pure Reason, Immanuel Kant takes the ostensive constructions characteristic of Euclidean-style demonstrations to be the paradigm of both mathematical proofs and synthetic a priori cognition in general. However, the development of calculus included a number of techniques for representing infinite series of sums or differences, which could not be represented with the direct geometrical demonstrations of the past. Salomon Maimonâs Essay on Transcendental Philosophy addresses precisely this disparity. Maimon, owing much to G. W. Leibniz, proposes that differentials of sensation achieve what Kantian constructions could not. More importantly, Maimon develops a kind of symbolic cognition that is not delimited by the constraint of the pure forms of intuition. The mind does not construct its objects, but constructs itself through inquiry into the real objects of thought
The calculus according to S. F. Lacroix (1765-1843)
Silvestre François Lacroix (Paris. 1765 - ibid., 1843) was not a prominent mathematical researcher, but he was certainly a most influential mathematical book author. His most famous book is a monumental Traité du calcul différentiel et du calcul intégral (three large volumes, 1797-1800; a second edition appeared in 1810-1819) - an encyclopaedic appraisal of 18th-century calculus. He also published many textbooks, one of which is closely associated to this large Traité: the Traité élémentaire du calcul différentiel et du calcul intégral (first edition in 1802; four more editions in Lacroix's lifetime; four posthumous editions).
Although most historians acknowledge the great influence of Lacroix's large Traité in early 19th-century mathematics it has not been thoroughly studied. This thesis is a contribution for correcting this omission. The focus is on its first edition, but the second edition and the Traité élémentaire, are also addressed.
The thesis starts with a short biography of Lacroix, followed by an overview of the first edition of the large Traité. Next corne five chapters where particular aspects are analyzed in detail: the foundations of the calculus, analytic and differential geometry, approximate integration and conceptions of the integral, types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions), and three aspects related to finite differences and series (the use of subscript indices, types of solutions of finite difference equations, and mixed difference equations); for all these aspects Lacroix's treatment is compared to the 18th-century background, and to his likely sources. Then we examine how the large Traité was adapted to a textbook - the Traité élémentaire, we take a look at the second edition of the large Traité, and conclude the body of the thesis with some final remarks
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