Set Of Functional Differential Equations Research Articles

Towards Stability Analysis of Data Transport Mechanisms: A Fluid Model and Its Applications

The Transmission Control Protocol (TCP) utilizes congestion avoidance and control mechanisms as a preventive measure against congestive collapse and as an adaptive measure in the presence of changing network conditions. The set of available congestion control algorithms is diverse, and while many have been studied from empirical and simulation perspectives, there is a notable lack of analytical work for some variants. To gain more insight into the dynamics of these algorithms, we: (1) propose a general modeling scheme consisting of a set of functional differential equations of retarded type (RFDEs) and of the congestion window as a function of time; (2) apply this scheme to TCP Reno and demonstrate its equivalence to a previous, well known model for TCP Reno; (3) show an application of the new framework to the widely-deployed congestion control algorithm TCP CUBIC, for which analytical models are few and limited; and (4) validate the model using simulations. Our modeling framework yields a fluid model for TCP CUBIC. From a theoretical analysis of this model, we discover that TCP CUBIC is locally uniformly asymptotically stable -- a property of the algorithm previously unknown.

A FUNCTIONAL DIFFERENTIAL EQUATION MODEL FOR BIOLOGICAL CELL SORTING DUE TO DIFFERENTIAL ADHESION

This paper presents a mathematical model to describe the sorting of two different types of cells, arising from differential adhesion mechanisms within biological tissue. The model is based on a continuum approach that takes into account individual cell behavior including aspects of the cell-migration process, dynamics of the adhesions between contacting cells, and finite cell size. Numerical solutions and bifurcation analyses for the case of a collection of two different cell types show a variety of behaviors observed in experiments, including spatially uniform mixing of cells and the formation of two distinct, containing both types of cells or just one. The mathematical model, which is in the form of a set of functional differential equations, represents a novel approach to continuum modeling of cell sorting and migration within biological tissue.

Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces

The two-dimensional analytic optics design method presented in a previous paper [Opt. Express 20, 5576-5585 (2012)] is extended in this work to the three-dimensional case, enabling the coupling of three ray sets with two free-form lens surfaces. Fermat's principle is used to deduce additional sets of functional differential equations which make it possible to calculate the lens surfaces. Ray tracing simulations demonstrate the excellent imaging performance of the resulting free-form lenses described by more than 100 coefficients.

Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles

In this work, a new two-dimensional optics design method is proposed that enables the coupling of three ray sets with two lens surfaces. The method is especially important for optical systems designed for wide field of view and with clearly separated optical surfaces. Fermat's principle is used to deduce a set of functional differential equations fully describing the entire optical system. The presented general analytic solution makes it possible to calculate the lens profiles. Ray tracing results for calculated 15th order Taylor polynomials describing the lens profiles demonstrate excellent imaging performance and the versatility of this new analytic design method.

Approximations for many-body Green's functions: insights from the fundamental equations

Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on many-body perturbation theory. They can be obtained by iterating a set of functional differential equations (DEs) relating the one-particle Green's function (GF) to its functional derivative with respect to an external perturbing potential. In this work, we apply a linear response expansion in order to obtain insights into various approximations for GF calculations. The expansion leads to an effective screening while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations, we discretize them and retain only one point in space, spin and time for all variables. Within this one-point model we obtain an explicit solution for the GF, which allows us to explore the structure of the general family of solutions and to determine the specific solution that corresponds to the physical one. Moreover, we analyze the performances of established approaches like GW over the whole range of interaction strength, and we explore alternative approximations. Finally, we link certain approximations for the exact solution to the corresponding manipulations of the DE which produces them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the DE is known.

Stability of sets of functional differential equations with impulse effect

Abstract In this paper, we investigate stability of sets for a class of impulsive functional differential equations by using piecewise continuous Lyapunov functions with Razumickhin techniques. Some sufficient conditions for stability of sets are established, and some known stability theorems also are generalized.

Complexified path integrals and the phases of quantum field theory

The path integral by which quantum field theories are defined is a particular solution of a set of functional differential equations arising from the Schwinger action principle. In fact these equations have a multitude of additional solutions which are described by integrals over a complexified path. We discuss properties of the additional solutions which, although generally disregarded, may be physical with known examples including spontaneous symmetry breaking and theta vacua. We show that a consideration of the full set of solutions yields a description of phase transitions in quantum field theories which complements the usual description in terms of the accumulation of Lee–Yang zeroes. In particular we argue that non-analyticity due to the accumulation of Lee–Yang zeros is related to Stokes phenomena and the collapse of the solution set in various limits including but not restricted to, the thermodynamic limit. A precise demonstration of this relation is given in terms of a zero dimensional model. Finally, for zero dimensional polynomial actions, we prove that Borel resummation of perturbative expansions, with several choices of singularity avoiding contours in the complex Borel plane, yield inequivalent solutions of the action principle equations.

Functional renormalization group approach – an alternative route to go beyond mode coupling theory

On the coarse-grained level of hydrodynamic fluctuations we briefly present a new approach for analyzing the complex transport behavior in dense compressible fluids. Starting from a generic class of nonlinear Langevin equations we derive a set of functional differential equations for the dimensionless transport coefficients. The functional structure of these flow equations represents the interplay between fast thermal fluctuations and the nonlinear coupling of the slow field variables. We obtain, that these equations do not show a critical physical fixed point. Our results provide an interesting tool to investigate the long-wavelength and low-frequency behavior in dense fluids in terms of a functional renormalization group formalism for not necessarily critical systems.

On the stability of invariant sets of functional differential equations

A system of functional differential equations with delay d z/d t= Z( t, z t ), where z=( x, y), x∈ R n , y∈ R m , and Z is the vector-valued functional, is considered. It is supposed that this system has a positive invariant set x=0. Definitions of its stability, asymptotic stability, and uniform asymptotic stability are given. Theorems on the uniform asymptotic stability are formulated and proved.

A Model of Heterosexual Population Dynamics with Age Structure and Gestation Period

A new set of functional differential equations modelling heterosexual population dynamics is developed and its qualitative study is begun with a proof of existence and uniqueness of solution.

Invariant sets for functional differential equations in Banach spaces and applications

Invariant sets for functional differential equations in Banach spaces and applications

Positively invariant sets for functional differential equations with infinite delay

Positively invariant sets for functional differential equations with infinite delay

Asymptotic self invariant sets and functional differential equations in Banach spaces

The stability properties of asymptotically self-invariant sets of functional differential equations in a Banach space are discussed using a nonlinear variation of constants formula analogous to Alekseev's formula in Euclidean spaces and Lyapunov functions.