pinecone/core-2020-05-10-deduplication huggingface dataset:
Complete systems of recursive integrals and Taylor series for solutions
  of Sturm-Liouville equations

Consider an arbitrary complex-valued, twice continuously differentiable,
nonvanishing function $\phi$ defined on a finite segment $[a,b]\subset
\mathbb{R}$. Let us introduce an infinite system of functions constructed in
the following way. Each subsequent function is a primitive of the preceding one
multiplied or divided by $\phi$ alternately. The obtained system of functions
is a generalization of the system of powers ${(x-x_{0}%)^{k}}_{k=0}^{\infty}$.
We study its completeness as well as the completeness of its subsets in
different functional spaces. This system of recursive integrals results to be
closely related to so-called $L$-bases arising in the theory of transmutation
operators for linear ordinary differential equations. Besides the results on
the completeness of the system of recursive integrals we show a deep analogy
between the expansions in terms of the recursive integrals and Taylor
expansions. We prove a generalization of the Taylor theorem with the Lagrange
form of the remainder term and find an explicit formula for transforming a
generalized Taylor expansion of a function in terms of the recursive integrals
into a usual Taylor expansion. As a direct corollary of the formula we obtain
the following new result concerning solutions of the Sturm-Liouville equation.
Given a regular nonvanishing complex valued solution $y_{0}$ of the equation
$y^{\prime\prime}+q(x)y=0$, $x\in(a,b)$, assume that it is $n$ times
differentiable at a point $x_{0}% \in\lbrack a,b]$. We present explicit
formulas for calculating the first $n$ derivatives at $x_{0}$ for any solution
of the equation $u^{\prime\prime}+q(x)u=\lambda u$. That is, an explicit map
transforming the Taylor expansion of $y_{0}$ into the Taylor expansion of $u$
is constructed.Comment: 17 pages, 0 figur
Clifford geometric parameterization of inequivalent vacua

We propose a geometric method to parameterize inequivalent vacua by dynamical
data. Introducing quantum Clifford algebras with arbitrary bilinear forms we
distinguish isomorphic algebras --as Clifford algebras-- by different
filtrations resp. induced gradings. The idea of a vacuum is introduced as the
unique algebraic projection on the base field embedded in the Clifford algebra,
which is however equivalent to the term vacuum in axiomatic quantum field
theory and the GNS construction in C^*-algebras. This approach is shown to be
equivalent to the usual picture which fixes one product but employs a variety
of GNS states. The most striking novelty of the geometric approach is the fact
that dynamical data fix uniquely the vacuum and that positivity is not
required. The usual concept of a statistical quantum state can be generalized
to geometric meaningful but non-statistical, non-definite, situations.
Furthermore, an algebraization of states takes place. An application to physics
is provided by an U(2)-symmetry producing a gap-equation which governs a phase
transition. The parameterization of all vacua is explicitly calculated from
propagator matrix elements. A discussion of the relation to BCS theory and
Bogoliubov-Valatin transformations is given.Comment: Major update, new chapters, 30 pages one Fig. (prev. 15p, no Fig.
A nonlinear elliptic problem with terms concentrating in the boundary

In this paper we investigate the behavior of a family of steady state
solutions of a nonlinear reaction diffusion equation when some reaction and
potential terms are concentrated in a $\epsilon$-neighborhood of a portion
$\Gamma$ of the boundary. We assume that this $\epsilon$-neighborhood shrinks
to $\Gamma$ as the small parameter $\epsilon$ goes to zero. Also, we suppose
the upper boundary of this $\epsilon$-strip presents a highly oscillatory
behavior. Our main goal here is to show that this family of solutions converges
to the solutions of a limit problem, a nonlinear elliptic equation that
captures the oscillatory behavior. Indeed, the reaction term and concentrating
potential are transformed into a flux condition and a potential on $\Gamma$,
which depends on the oscillating neighborhood
Kinematic quantities of finite elastic and plastic deformation

Kinematic quantities for finite elastic and plastic deformations are defined
via an approach that does not rely on auxiliary elements like reference frame
and reference configuration, and that gives account of the inertial-noninertial
aspects explicitly. These features are achieved by working on Galilean
spacetime directly. The quantity expressing elastic deformations is introduced
according to its expected role: to measure how different the current metric is
from the relaxed/stressless metric. Further, the plastic kinematic quantity is
the change rate of the stressless metric. The properties of both are analyzed,
and their relationship to frequently used elastic and plastic kinematic
quantities is discussed. One important result is that no objective elastic or
plastic quantities can be defined from deformation gradient.Comment: v5: minor changes, one section moved to an Appendix, 26 pages, 2
  figure
MIMO capacity for deterministic channel models: sublinear growth

This is the second paper of the authors in a series concerned with the
development of a deterministic model for the transfer matrix of a MIMO system.
Starting from the Maxwell equations, we have described in \cite{BCFM} the
generic structure of such a deterministic transfer matrix. In the current paper
we apply the results of \cite{BCFM} in order to study the (Shannon-Foschini)
capacity behavior of a MIMO system as a function of the deterministic spread
function of the environment, and the number of transmitting and receiving
antennas. The antennas are assumed to fill in a given, fixed volume. Under some
generic assumptions, we prove that the capacity grows much more slowly than
linearly with the number of antennas. These results reinforce previous
heuristic results obtained from statistical models of the transfer matrix,
which also predict a sublinear behavior.Comment: 12 pages, to appear in Math. Meth. Appl. Sc
On the connections between symmetries and conservation rules of
  dynamical systems

The strict connection between Lie point-symmetries of a dynamical system and
its constants of motion is discussed and emphasized, through old and new
results. It is shown in particular how the knowledge of a symmetry of a
dynamical system can allow to obtain conserved quantities which are invariant
under the symmetry. In the case of Hamiltonian dynamical systems it is shown
that, if the system admits a symmetry of "weaker" type (specifically, a
\lambda\ or a \Lambda-symmetry), then the generating function of the symmetry
is not a conserved quantity, but the deviation from the exact conservation is
"controlled" in a well defined way. Several examples illustrate the various
aspects.Comment: 13 pages, no fi
Algebraic foundations of split hypercomplex nonlinear adaptive filtering

A split hypercomplex learning algorithm for the training of nonlinear finite
impulse response adaptive filters for the processing of hypercomplex signals of
any dimension is proposed. The derivation strictly takes into account the laws
of hypercomplex algebra and hypercomplex calculus, some of which have been
neglected in existing learning approaches (e.g. for quaternions). Already in
the case of quaternions we can predict improvements in performance of
hypercomplex processes. The convergence of the proposed algorithms is
rigorously analyzed.
  Keywords: Quaternionic adaptive filtering, Hypercomplex adaptive filtering,
Nonlinear adaptive filtering, Hypercomplex Multilayer Perceptron, Clifford
geometric algebraComment: 14 pages, 1 figur
Clifford algebras, Fourier transforms and quantum mechanics

In this review, an overview is given of several recent generalizations of the
Fourier transform, related to either the Lie algebra sl_2 or the Lie
superalgebra osp(1|2). In the former case, one obtains scalar generalizations
of the Fourier transform, including the fractional Fourier transform, the Dunkl
transform, the radially deformed Fourier transform and the super Fourier
transform. In the latter case, one has to use the framework of Clifford
analysis and arrives at the Clifford-Fourier transform and the radially
deformed hypercomplex Fourier transform. A detailed exposition of all these
transforms is given, with emphasis on aspects such as eigenfunctions and
spectrum of the transform, characterization of the integral kernel and
connection with various special functions.Comment: Review paper, 39 pages, to appear in Math. Methods. Appl. Sc
On the well-posedness of a mathematical model describing water-mud
  interaction

In this paper we consider a mathematical model describing the two-phase
interaction between water and mud in a water canal when the width of the canal
is small compared to its depth. The mud is treated as a non-Netwonian fluid and
the interface between the mud and fluid is allowed to move under the influence
of gravity and surface tension. We reduce the mathematical formulation, for
small boundary and initial data, to a fully nonlocal and nonlinear problem and
prove its local well-posedness by using abstract parabolic theory.Comment: 16 page
A singularly perturbed nonlinear traction problem in a periodically
  perforated domain. A functional analytic approach

We consider a periodically perforated domain obtained by making in R^n a
periodic set of holes, each of them of size proportional to \epsilon. Then we
introduce a nonlinear boundary value problem for the Lam\'e equations in such a
periodically perforated domain. The unknown of the problem is a vector valued
function u which represents the displacement attained in the equilibrium
configuration by the points of a periodic linearly elastic matrix with a hole
of size \epsilon contained in each periodic cell. We assume that the traction
exerted by the matrix on the boundary of each hole depends (nonlinearly) on the
displacement attained by the points of the boundary of the hole. Then our aim
is to describe what happens to the displacement vector function u when \epsilon
tends to 0. Under suitable assumptions we prove the existence of a family of
solutions {u(\epsilon,.) : \epsilon \in ]0,\epsilon'[} with a prescribed
limiting behaviour when \epsilon approaches 0. Moreover, the family
{u(\epsilon,.) : \epsilon \in ]0,\epsilon'[} is in a sense locally unique and
can be continued real analytically for negative values of \epsilon
Existence and nonexistence results for a singular boundary value problem
  arising in the theory of epitaxial growth

The existence of stationary radial solutions to a partial differential
equation arising in the theory of epitaxial growth is studied. Our results
depend on the size of a parameter that plays the role of the velocity at which
mass is introduced into the system. For small values of this parameter we prove
existence of solutions to this boundary value problem. For large values of the
same parameter we prove nonexistence of solutions. We also provide rigorous
bounds for the values of this parameter which separate existence from
nonexistence. The proofs come as a combination of several differential
inequalities and the method of upper and lower functions.Comment: Mathematical Methods in the Applied Sciences. Published onlin
Relativistic Finsler geometry

We briefly review some basic concepts of parallel displacement in Finsler
geometry. In general relativity, the parallel translation of objects along the
congruence of the fundamental observer corresponds to the evolution in time. By
dropping the quadratic restriction on the measurement of an infinitesimal
distance, the geometry is generalized to a Finsler structure. Apart from
curvature a new property of the manifold complicates the geometrodynamics, the
color. The color brings forth an intrinsic local anisotropy and many quantities
depend on position and to a "supporting" direction. We discuss this direction
dependence and some physical interpretations. Also, we highlight that in
Finsler geometry the parallel displacement isn't necessarily always along the
"supporting" direction. The latter is a fundamental congruence of the manifold
and induces a natural 1+3 decomposition. Its internal deformation is given
through the evolution of the irreducible components of vorticity, shear and
expansion.Comment: references added, Contribution to the proceedings of "Modern
  Mathematical Methods in Science and Technology 2012 (M3ST2012)", Kalamata,
  Greece, August, 201
Monogenic pseudo-complex power functions and their applications

The use of a non-commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D-elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan-Sofo type binomial identity is also proved
Monogenic pseudo-complex power functions and their applications

The use of a non-commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D-elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan–Sofo type binomial identity is also proved.Fundação para a Ciência e a Tecnologia (FCT
Generalized notions of symmetry of ODE's and reduction procedures

This paper describes the notion of \sigma -symmetry, which extends the one of
\lambda-symmetry, and its application to reduction procedures of systems of
ordinary differential equations and of dynamical systems as well. We also
consider orbital symmetries, which give rise to a different form of reduction
of dynamical systems. Finally, we discuss how dynamical systems can be
transformed into higher-order ordinary differential equations, and how these
symmetry properties of the dynamical systems can be transferred into reduction
properties of the corresponding ordinary differential equations. Many examples
illustrate the various situations.Comment: 14 pages, no fig., Math. Meth. Appl. Sci. (ICNAAM Proc.), to appea
Infinite energy solutions for the Cahn-Hilliard equation in cylindrical domains

We give a detailed study of the infinite-energy solutions of the Cahn-Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well-posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties
Corner asymptotics of the magnetic potential in the eddy-current model

International audienceIn this paper, we describe the scalar magnetic potential in the vicinity of a corner of a conducting body embedded in a dielectric medium in a bidimensional setting. We make explicit the corner asymptotic expansion for this potential as the distance to the corner goes to zero. This expansion involves singular functions and singular coefficients. We introduce a method for the calculation of the singular functions near the corner and we provide two methods to compute the singular coefficients: the method of moments and the method of quasi-dual singular functions. Estimates for the convergence of both approximate methods are proven. We eventually illustrate the theoretical results with finite element computations. The specific non-standard feature of this problem lies in the structure of its singular functions: They have the form of series whose first terms are harmonic polynomials and further terms are genuine non-smooth functions generated by the piecewise constant zeroth order term of the operator
Invasion waves in the presence of a mutualist

This paper studies invasion waves in the diffusive
Competitor-Competitor-Mutualist model generalizing the 2-species Lotka-Volterra
model studied by Weinberger et al. The mutualist may benefit the invading or
the resident species producing two different types of invasions. Sufficient
conditions for linear determinacy are derived in both cases, and when they
hold, explicit formulas for linear spreading speeds of the invasions are
obtained by linearizing the model. While in the first case the linear speed is
increased by the mutualist, it is unaffected in the second case. Mathematical
methods are based on converting the model into a cooperative reaction-diffusion
system.Comment: 21 pages, 4 figure
Decay of a model system of radiating gas

This paper is concerned with optimal time-decay estimates of solutions of the
Cauchy problem to a model system of the radiating gas in $\mathbb{R}^n$.
Compared to Liu and Kawashima (2011) \cite{Liu1} and Wang and Wang (2009)
\cite{Wang}, without smallness assumption of initial perturbation in
$L^1$-norm, we study large time behavior of small amplitude classical solutions
to the Cauchy problem. The optimal $H^N$-norm time-decay rates of the solutions
in $\mathbb{R}^n$ with $1\leq n\leq4$ are obtained by applying the Fourier
splitting method introduced in Schonbek (1980) \cite{Schonbek1} with a slight
modification and an energy method. Furthermore, basing on a refined pure energy
method introduced in Guo and Wang \cite{Guo} (2011), we give optimal
$L^p$-$L^2(\mathbb{R}^3)$ decay estimates of the derivatives of solutions when
initial perturbation is bounded in $L^p$-norm with some $p\in(1,2]$.Comment: 15 page
Conservative-dissipative approximation schemes for a generalized Kramers equation

We propose three new discrete variational schemes that capture the conservative-dissipative structure of a generalized Kramers equation. The first two schemes are single-step minimization schemes, whereas the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin-Wentzell theory of large deviations for the underlying stochastic system. We prove that all three schemes converge to the solution of the generalized Kramers equation
An Analytical and Numerical Study of Liquid Dynamics in a 1D Capillary
  under Entrapped Gas Action

Capillary dynamics has been and is yet an important field of research,
because of its very relevant role played as the core mechanism at the base of
many applications. In this context, we are particularly interested in the
liquid penetration inspection technique. Due to the obviously needed level of
reliability involved with such a non-destructive test, this paper is devoted to
study how the presence of an entrapped gas in a close-end capillary may affect
the inspection outcome. This study is carried out through a 1D ordinary
differential model that despite its simplicity is able to point out quite well
the capillary dynamics under the effect of an entrapped gas. The paper is
divided into two main parts; the first starts from an introductory historical
review of capillary flows modeling, goes on presenting the 1D second order
ordinary differential model, taking into account the presence of an entrapped
gas and therefore ends by showing some numerical simulation results. The second
part is devoted to the analytical study of the model by separating the initial
transitory behavior from the stationary one. Besides, these solutions are
compared with the numerical ones and finally an expression is deduced for the
threshold radius switching from a fully damped transitory to an oscillatory
one.Comment: 23 pages; 9 figures, 1 tabl
Multiplicity of solutions for fractional Hamiltonian systems with
  Liouville-Weyl fractional derivative

In this paper, we investigate the existence of infinitely many solutions for
the following fractional Hamiltonian systems: \begin{eqnarray}\label{eq00}
_{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = & \nabla
W(t,u(t))\\ u\in H^{\alpha}(\mathbb{R}, \mathbb{R}^{N}).\nonumber
\end{eqnarray} where $\alpha \in (1/2, 1)$, $t\in \mathbb{R}$, $u\in
\mathbb{R}^{n}$, $L\in C(\mathbb{R}, \mathbb{R}^{n^2})$ is a symmetric and
positive definite matrix for all $t\in \mathbb{R}$, $W\in
C^{1}(\mathbb{R}\times \mathbb{R}^{n}, \mathbb{R})$, and $\nabla W$ is the
gradient of $W$ at $u$. The novelty of this paper is that, assuming there
exists $l\in C(\mathbb{R}, \mathbb{R})$ such that $(L(t)u,u)\geq l(t)|u|^{2}$
for all $t\in \mathbb{R}$, $u\in \mathbb{R}^{n}$ and the following conditions
on $l$: $\inf_{t\in \mathbb{R}}l(t) >0$ and there exists $r_{0}>0$ such that,
for any $M>0$
  $$
  m(\{t\in (y-r_{0}, y+r_{0})/\;\;l(t)\leq M\}) \to 0\;\;\mbox{as}\;\;|y|\to
\infty.
  $$
  are satisfied and $W$ is of subquadratic growth as $|u| \to +\infty$, we show
that (\ref{eq00}) possesses infinitely many solutions via the genus properties
in the critical theory. Recent results in [Z. Zhang and R. Yuan, Solutions for
subquadratic fractional Hamiltonian systems without coercive conditions, Math.
Methods Appl. Sci., DOI: 10.1002/mma.3031] are significantly improved
Gaps in the spectrum of the Neumann Laplacian generated by a system of
  periodically distributed trap

The article deals with a convergence of the spectrum of the Neumann Laplacian
in a periodic unbounded domain $\Omega^\varepsilon$ depending on a small
parameter $\varepsilon>0$. The domain has the form
$\Omega^\varepsilon=\mathbb{R}^n\setminus S^\varepsilon$, where $S^\varepsilon$
is an $\varepsilon\mathbb{Z}^n$-periodic family of trap-like screens. We prove
that for an arbitrarily large $L$ the spectrum has just one gap in $[0,L]$ when
$\varepsilon$ small enough, moreover when $\varepsilon\to 0$ this gap converges
to some interval whose edges can be controlled by a suitable choice of geometry
of the screens. An application to the theory of 2D-photonic crystals is
discussed.Comment: 18 pages, 2 figure
An analytic approach to a class of fractional differential-difference equations of rational type via symbolic computation

Fractional derivatives are powerful tools in solving the problems of science and engineering. In this paper, an analytical algorithm for solving fractional differential-difference equations in the sense of Jumarie's modified Riemann-Liouville derivative has been described and demonstrated. The algorithm has been tested against time-fractional differentialdifference equations of rational type via symbolic computation. Three examples are given to elucidate the solution procedure. Our analyses lead to closed form exact solutions in terms of hyperbolic, trigonometric, and rational functions, which might be subject to some adequate physical interpretations in the future. Copyright © 2013 JohnWiley & Sons, Ltd