61 research outputs found
A cell growth model revisited
In this paper a stochastic model for the simultaneous growth and division
of a cell-population cohort structured by size is formulated. This probabilistic approach
gives straightforward proof of the existence of the steady-size distribution and a simple
derivation of the functional-differential equation for it. The latter one is the celebrated
pantograph equation (of advanced type). This firmly establishes the existence of the
steady-size distribution and gives a form for it in terms of a sequence of probability distribution functions. Also it shows that the pantograph equation is a key equation for other situations where there is a distinct stochastic framework
The Pantograph Equation in the Complex Plane
AbstractThe subject matter of this paper focuses on two functional differential equations with complex lag functions. We address ourselves to the existence and uniqueness of solutions and to their asymptotic behaviour
Functional-Differential Equations with Compressed Arguments and Polynomial Coefficients: Asymptotics of the Solutions
AbstractFunctional-differential equations with linearly compressed arguments and polynomial coefficients are considered. We prove, under some mild restrictions on the coefficients, that each solution y(t) of such an equation, satisfying estimate |y(t)| ≤ C exp{γ In2 |t|} (t → ∞), where 0 < γ < γ̃, is polynomial
On bounded continuous solutions of the archetypal equation with rescaling
The `archetypal' equation with rescaling is given by
(), where is a probability measure; equivalently,
, with random and
denoting expectation. Examples include: (i) functional equation
; (ii) functional-differential
(`pantograph') equation (,
). Interpreting solutions as harmonic functions of the
associated Markov chain , we obtain Liouville-type results asserting
that any bounded continuous solution is constant. In particular, in the
`critical' case such a theorem holds subject to
uniform continuity of ; the latter is guaranteed under mild regularity
assumptions on , satisfied e.g.\ for the pantograph equation (ii). For
equation (i) with (, ), the
result can be proved without the uniform continuity assumption. The proofs
utilize the iterated equation (with a
suitable stopping time ) due to Doob's optional stopping theorem applied
to the martingale .Comment: Substantially revised. The title is modifie
Analysis of the archetypal functional equation in the non-critical case
We study the archetypal functional equation of the form (), where is a probability measure on ; equivalently, , where is expectation with respect to the distribution of random coefficients . Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value ; namely, under mild technical conditions no such solutions exist whenever (and ) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with . Further results are obtained in the supercritical case , including existence, uniqueness and a maximum principle. The case with is drastically different from that with ; in particular, we prove that a bounded solution possessing limits at must be constant. The proofs employ martingale techniques applied to the martingale , where is an associated Markov chain with jumps of the form
Generalized Refinement Equations and Subdivision Processes
AbstractThe concept of subdivision schemes is generalized to schemes with a continuous mask, generating compactly supported solutions of corresponding functional equations in integral form. A necessary and a sufficient condition for uniform convergence of these schemes are derived. The equivalence of weak convergence of subdivision schemes with the existence of weak compactly supported solutions to the corresponding functional equations is shown for both the discrete and integral cases. For certain non-negative masks stronger results are derived by probabilistic methods. The solution of integral functional equations whose continuous masks solve discrete functional equations, are shown to be limits of discrete nonstationary schemes with masks of increasing support. Interesting functions created by these schemes are C∞ functions of compact support including the up-function of Rvachev
Spectral analysis on infinite Sierpinski fractafolds
A fractafold, a space that is locally modeled on a specified fractal, is the
fractal equivalent of a manifold. For compact fractafolds based on the
Sierpinski gasket, it was shown by the first author how to compute the discrete
spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian.
A similar problem was solved by the second author for the case of infinite
blowups of a Sierpinski gasket, where spectrum is pure point of infinite
multiplicity. Both works used the method of spectral decimations to obtain
explicit description of the eigenvalues and eigenfunctions. In this paper we
combine the ideas from these earlier works to obtain a description of the
spectral resolution of the Laplacian for noncompact fractafolds. Our main
abstract results enable us to obtain a completely explicit description of the
spectral resolution of the fractafold Laplacian. For some specific examples we
turn the spectral resolution into a "Plancherel formula". We also present such
a formula for the graph Laplacian on the 3-regular tree, which appears to be a
new result of independent interest. In the end we discuss periodic fractafolds
and fractal fields
Laplace Operators on Fractals and Related Functional Equations
We give an overview over the application of functional equations, namely the
classical Poincar\'e and renewal equations, to the study of the spectrum of
Laplace operators on self-similar fractals. We compare the techniques used to
those used in the euclidean situation. Furthermore, we use the obtained
information on the spectral zeta function to define the Casimir energy of
fractals. We give numerical values for this energy for the Sierpi\'nski gasket
Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals
We investigate the existence of the meromorphic extension of the spectral
zeta function of the Laplacian on self-similar fractals using the classical
results of Kigami and Lapidus (based on the renewal theory) and new results of
Hambly and Kajino based on the heat kernel estimates and other probabilistic
techniques. We also formulate conjectures which hold true in the examples that
have been analyzed in the existing literature
The monomer-dimer problem and moment Lyapunov exponents of homogeneous Gaussian random fields
We consider an "elastic" version of the statistical mechanical monomer-dimer
problem on the n-dimensional integer lattice. Our setting includes the
classical "rigid" formulation as a special case and extends it by allowing each
dimer to consist of particles at arbitrarily distant sites of the lattice, with
the energy of interaction between the particles in a dimer depending on their
relative position. We reduce the free energy of the elastic dimer-monomer (EDM)
system per lattice site in the thermodynamic limit to the moment Lyapunov
exponent (MLE) of a homogeneous Gaussian random field (GRF) whose mean value
and covariance function are the Boltzmann factors associated with the monomer
energy and dimer potential. In particular, the classical monomer-dimer problem
becomes related to the MLE of a moving average GRF. We outline an approach to
recursive computation of the partition function for "Manhattan" EDM systems
where the dimer potential is a weighted l1-distance and the auxiliary GRF is a
Markov random field of Pickard type which behaves in space like autoregressive
processes do in time. For one-dimensional Manhattan EDM systems, we compute the
MLE of the resulting Gaussian Markov chain as the largest eigenvalue of a
compact transfer operator on a Hilbert space which is related to the
annihilation and creation operators of the quantum harmonic oscillator and also
recast it as the eigenvalue problem for a pantograph functional-differential
equation.Comment: 24 pages, 4 figures, submitted on 14 October 2011 to a special issue
of DCDS-
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