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

Analysis of the archetypal functional equation in the non-critical case

We study the archetypal functional equation of the form $y(x)=\iint_{R^2} y(a(x-b))\,\mu(da,db)$ ($x\in R$), where $\mu$ is a probability measure on $R^2$; equivalently, $y(x)=E\{y(\alpha (x-\beta))\}$, where $E$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{R^2}\ln |a| \mu(da,db) =E \{\ln |\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $P(\alpha0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x\rightsquigarrow\alpha (x-\beta)$

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

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

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-

Coordination and resource-related difficulties encountered by Quebec's public health specialists and infectious diseases/medical microbiologists in the management of A (H1N1) - a mixed-method, exploratory survey

<p>Abstract</p> <p>Background</p> <p>In Quebec, the influenza A (H1N1) pandemic was managed using a top-down style that left many involved players with critical views and frustrations. We aimed to describe physicians' perceptions - infectious diseases specialists/medical microbiologists (IDMM) and public health/preventive medicine specialists (PHPMS) - in regards to issues encountered with the pandemics management at the physician level and highlight suggested improvements for future healthcare emergencies.</p> <p>Methods</p> <p>In April 2010, Quebec IDMM and PHPMS physicians were invited to anonymously complete a web-based learning needs assessment. The survey included both open-ended and multiple-choice questions. Descriptive statistics were used to report on the frequency distribution of multiple choice responses whereas thematic content analysis was used to analyse qualitative data generated from the survey and help understand respondents' experience and perceptions with the pandemics.</p> <p>Results</p> <p>Of the 102 respondents, 85.3% reported difficulties or frustrations in their practice during the pandemic. The thematic analysis revealed two core themes describing the problems experienced in the pandemic management: coordination and resource-related difficulties. Coordination issues included communication, clinical practice guidelines, decision-making, roles and responsibilities, epidemiological investigation, and public health expert advisory committees. Resources issues included laboratory resources, patient management, and vaccination process.</p> <p>Conclusion</p> <p>Together, the quantitative and qualitative data suggest a need for improved coordination, a better definition of roles and responsibilities, increased use of information technologies, merged communications, and transparency in the decisional process. Increased flexibility and less contradiction in clinical practice guidelines from different sources and increased laboratory/clinical capacity were felt critical to the proper management of infectious disease emergencies.</p

On Bounded Solutions of the Balanced Generalized Pantograph Equation

The question about the existence and characterization of bounded solutions to linear functional-differential equations with both advanced and delayed arguments was posed in early 1970s by T. Kato in connection with the analysis of the pantograph equation, y ′(x) = ay(qx) + by(x). In the present paper, we answer this question for the balanced generalized pantograph equation of the form −a2y ′′(x)+a1y ′(x)+y(x) = ∫ ∞ 0 y(αx)µ(dα), where a1 ≥ 0, a2 ≥ 0, a2 1 +a22&gt; 0, and µ is a probability measure. Namely, setting K: = ∫ ∞ 0 ln α µ(dα), we prove that if K ≤ 0 then the equation does not have nontrivial (i.e., nonconstant) bounded solutions, while if K&gt; 0 then such a solution exists. The result in the critical case, K = 0, settles a long-standing problem. The proof exploits the link with the theory of Markov processes, in that any solution of the balanced pantograph equation is an L-harmonic function relative to the generator L of a certain diffusion process with “multiplication ” jumps. The paper also includes three “elementary ” proofs for the simple prototype equation y ′(x) + y(x) = 1 1 2 y(qx) + 2 y(x/q), based on perturbation, analytical, and probabilistic techniques, respectively, which may appear useful in other situations as efficient exploratory tools. Key words: Pantograph equation, functional-differential equations, integrodifferential equations, balance condition, bounded solutions, WKB expansion