35,739 research outputs found
Extended Real-Valued Double Sequence and Its Convergence
AbstractIn this article we introduce the convergence of extended realvalued double sequences [16], [17]. It is similar to our previous articles [15], [10]. In addition, we also prove Fatou’s lemma and the monotone convergence theorem for double sequences.This work was supported by JSPS KAKENHI 2350002Gifu National College of Technology, Gifu, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Józef Białas. Infimum and supremum of the set of real numbers. Measure theory. Formalized Mathematics, 2(1):163-171, 1991.Józef Białas. Series of positive real numbers. Measure theory. Formalized Mathematics, 2(1):173-183, 1991.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Noboru Endou. Double series and sums. Formalized Mathematics, 22(1):57-68, 2014. doi:10.2478/forma-2014-0006. [Crossref]Noboru Endou and Yasunari Shidama. Integral of measurable function. Formalized Mathematics, 14(2):53-70, 2006. doi:10.2478/v10037-006-0008-x. [Crossref]Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Basic properties of extended real numbers. Formalized Mathematics, 9(3):491-494, 2001.Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definitions and basic properties of measurable functions. Formalized Mathematics, 9(3):495-500, 2001.Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1. [Crossref]Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. Double sequences and limits. Formalized Mathematics, 21(3):163-170, 2013. doi:10.2478/forma-2013-0018. [Crossref]Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, 2 edition, 1999.D.J.H. Garling. A Course in Mathematical Analysis: Volume 1, Foundations and Elementary Real Analysis, volume 1. Cambridge University Press, 2013.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1 (3):471-475, 1990.Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki, Noboru Endou, Yasunari Shidama, and Hiroyuki Okazaki. Inferior limit, superior limit and convergence of sequences of extended real numbers. Formalized Mathematics, 15(4):231-236, 2007. doi:10.2478/v10037-007-0026-3. [Crossref
Linear Processes for Functional Data
Linear processes on functional spaces were born about fifteen years ago. And
this original topic went through the same fast development as the other areas
of functional data modeling such as PCA or regression. They aim at generalizing
to random curves the classical ARMA models widely known in time series
analysis. They offer a wide spectrum of models suited to the statistical
inference on continuous time stochastic processes within the paradigm of
functional data. Essentially designed to improve the quality and the range of
prediction, they give birth to challenging theoretical and applied problems. We
propose here a state of the art which emphasizes recent advances and we present
some promising perspectives based on our experience in this area
Stable convergence of generalized stochastic integrals and the principle of conditioning: L^2 theory
Consider generalized adapted stochastic integrals with respect to
independently scattered random measures with second moments. We use a
decoupling technique, known as the "principle of conditioning", to study their
stable convergence towards mixtures of infinitely divisible distributions. Our
results apply, in particular, to multiple integrals with respect to
independently scattered and square integrable random measures, as well as to
Skorohod integrals on abstract Wiener spaces. As a specific application, we
establish a Central Limit Theorem for sequences of double integrals with
respect to a general Poisson measure, thus extending the results contained in
Nualart and Peccati (2005) and Peccati and Tudor (2004) to a non-Gaussian
context.Comment: 44 page
Contributions of Issai Schur to Analysis
The name Schur is associated with many terms and concepts that are widely
used in a number of diverse fields of mathematics and engineering. This survey
article focuses on Schur's work in analysis. Here too, Schur's name is
commonplace: The Schur test and Schur-Hadamard multipliers (in the study of
estimates for Hermitian forms), Schur convexity, Schur complements, Schur's
results in summation theory for sequences (in particular, the fundamental
Kojima-Schur theorem), the Schur-Cohn test, the Schur algorithm, Schur
parameters and the Schur interpolation problem for functions that are
holomorphic and bounded by one in the unit disk. In this survey, we discuss all
of the above mentioned topics and then some, as well as some of the
generalizations that they inspired. There are nine sections of text, each of
which is devoted to a separate theme based on Schur's work. Each of these
sections has an independent bibliography. There is very little overlap. A tenth
section presents a list of the papers of Schur that focus on topics that are
commonly considered to be analysis. We begin with a review of Schur's less
familiar papers on the theory of commuting differential operators
Numerical approach to -problems with the second order elliptic operators
For a second order differential operator A(\msx) =-\nabla a(\msx)\nabla +
b'(\msx)\nabla+ \nabla \big(\msb''(\msx) \cdot\big) on a bounded domain
with the Dirichlet boundary conditions on there exists the inverse
in . If is a Radon
(probability) measure on Borel algebra of subsets of , then . We construct the numerical
approximations to in two steps. In the first one we
construct grid-solutions and in the second step we embed
grid-solutions into the linear space of hat functions . The strong convergence to the original solutions is
established in and the weak convergence in .Comment: 33 page
Set-valued differentiation as an operator
We introduce real vector spaces composed of set-valued maps on an open set.
They are also complete metric spaces, lattices, commutative rings. The set of
differentiable functions is a dense subset of these spaces and the classical
gradient may be extended in these spaces as a closed operator. If a function f
belongs to the domain of such extension, then f is locally lipschitzian and the
values of extended gradient coincide with the values of Clarke's gradient.
However, unlike Clarke's gradient, our generalized gradient is a linear
operator
General solution of overdamped Josephson junction equation in the case of phase-lock
The first order nonlinear ODE d phi(t)/d t + sin phi(t)=B+A cos(omega t),
(A,B,omega are real constants) is investigated. Its general solution is derived
in the case of the choice of parameters ensuring the phase-lock mode. It is
represented in terms of Floquet solution of double confluent Heun equation.Comment: 28 page
Double dimers, conformal loop ensembles and isomonodromic deformations
The double-dimer model consists in superimposing two independent, identically
distributed perfect matchings on a planar graph, which produces an ensemble of
non-intersecting loops. Kenyon established conformal invariance in the small
mesh limit by considering topological observables of the model parameterized by
\SL_2(\C) representations of the fundamental group of the punctured domain.
The scaling limit is conjectured to be \CLE_4, the Conformal Loop Ensemble at
.
In support of this conjecture, we prove that a large subclass of these
topological correlators converge to their putative \CLE_4 limit. Both the
small mesh limit of the double-dimer correlators and the corresponding \CLE_4
correlators are identified in terms of the -functions introduced by
Jimbo, Miwa and Ueno in the context of isomonodromic deformations.Comment: 40 page
An SLE loop measure
There is an essentially unique way to associate to any Riemann surface a
measure on its simple loops, such that the collection of measures satisfy a
strong conformal invariance property. Wendelin Werner constructed these random
simple loops as boundaries of Brownian loops, and so they correspond in the zoo
of statistical mechanics models to central charge , or Schramm-Loewner
Evolution (SLE) parameter . The goal of this paper is to construct
a family of measures on simple loops on Riemann surfaces that satisfies a
conformal covariance property, and that would correspond to SLE parameter
(central charge ). On planar annuli, this loop measure was
already built by Adrien Kassel and Rick Kenyon. We will give an alternative
construction of this loop measure on planar annuli, investigate its conformal
covariance, and finally extend this measure to general Riemann surfaces. This
gives an example of a Malliavin-Kontsevich-Suhov loop measure in non-zero
central charge.Comment: 28 pages, 8 figures, To appear in Annales de l'Institut Henri
Poincar\'e (B
Structural Compactness and Stability of Pseudo-Monotone Flows
Fitzpatrick's variational representation of maximal monotone operators is
here extended to a class of pseudo-monotone operators in Banach spaces. On this
basis, the initial-value problem associated with the first-order flow of such
an operator is here reformulated as a minimization principle, extending a
method that was pioneered by Brezis, Ekeland and Nayroles for gradient flows.
This formulation is used to prove that the problem is stable w.r.t.\ arbitrary
perturbations not only of data but also of operators. This is achieved by using
the notion of evolutionary -convergence w.r.t.\ a nonlinear topology of
weak type.
These results are applied to the Cauchy problem for quasilinear parabolic
PDEs. This provides the structural compactness and stability of the model of
several physical phenomena: nonlinear diffusion, incompressible viscous flow,
phase transitions, and so on.Comment: arXiv admin note: text overlap with arXiv:1509.0381
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