On solvability of linear differential equations in ${\Bbb R}^{\Bbb N}$

We construct $x^0$ in ${\Bbb R}^{\Bbb N}$ and a row-finite matrix $T=\{T_{i,j}(t)\}_{i,j\in\N}$ of polynomials of one real variable $t$ such that the Cauchy problem $\dot x(t)=T_tx(t)$, $x(0)=x^0$ in the Fr\'echet space $\R^\N$ has no solutions. We also construct a row-finite matrix $A=\{A_{i,j}(t)\}_{i,j\in\N}$ of $C^\infty(\R)$ functions such that the Cauchy problem $\dot x(t)=A_tx(t)$, $x(0)=x^0$ in ${\Bbb R}^{\Bbb N}$ has no solutions for any $x^0\in{\Bbb R}^{\Bbb N}\setminus\{0\}$. We provide some sufficient condition of solvability and of unique solvability for linear ordinary differential equations $\dot x(t)=T_tx(t)$ with matrix elements $T_{i,j}(t)$ analytically dependent on $t$.

Sequentially continuous non-linear fundamental systems of solutions of affine equations in locally convex spaces

According to the Mickael's selection theorem any surjective continuous linear operator from one Fr\'echet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if $E$ is a Fr\'echet space and $T:E\to E$ is a continuous linear operator such that the Cauchy problem $\dot x=Tx$, $x(0)=x_0$ is solvable in $[0,1]$ for any $x_0\in E$, then for any $f\in C([0,1],E)$, there exists a continuos map $S:[0,1]\times E\to E$, $(t,x)\mapsto S_tx$ such that for any $x_0\in E$, the function $x(t)=S_tx_0$ is a solution of the Cauchy problem $\dot x(t)=Tx(t)+f(t)$, $x(0)=x_0$ (they call $S$ a fundamental system of solutions of the equation $\dot x=Tx+f$). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Fr\'echet spaces and strong duals of Fr\'echet--Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.

Some results on solvability of ordinary linear differential equations in locally convex spaces

Let $\Gamma$ be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem $\dot x = Ax$, $x(0) = x_0$ with respect to functions $x: R\to E$. It is proved that if $E\in \Gamma$, then $E\times R^A$ is-an-element-of $\Gamma$ for an arbitrary set $A$. It is also proved that a topological product of infinitely many infinite-dimensional Frechet spaces, each not isomorphic to $\omega$, does not belong to $\Gamma$.

Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations

We propose a new approach for modeling nonlinear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of nonlinear SDEs that are reducible to Ornstein--Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a nonlinear transformation function. The main advantage of this approach is that these SDEs can account for nonlinear features, observed in short-term interest rate series, while at the same time leading to exact discretization and closed-form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted as OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed-form likelihood functions. The transition density, the conditional distribution function, and the steady-state density function are derived in closed form as well as the conditional and unconditional moments for both processes. In order to obtain a more flexible functional form over time, we allow the transformation function to be time varying. Results from our study of U.S. and UK short-term interest rates suggest that the new models outperform existing parametric models with closed-form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. To examine the joint behavior of interest rate series, we propose flexible nonlinear multivariate models by joining univariate nonlinear processes via appropriate copulas. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying symmetrized Joe--Clayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. (JEL: C13, C32, G12) Copyright The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org, Oxford University Press.

A compendium of modelling techniques

This Integration Insight provides a brief overview of the most popular modelling techniques used to analyse complex real-world problems, as well as some less popular but highly relevant techniques. The modelling methods are divided into three categories, with each encompassing a number of methods, as follows: 1) Qualitative Aggregate Models (Soft Systems Methodology, Concept Maps and Mind Mapping, Scenario Planning, Causal (Loop) Diagrams), 2) Quantitative Aggregate Models (Function fitting and Regression, Bayesian Nets, System of differential equations / Dynamical systems, System Dynamics, Evolutionary Algorithms) and 3) Individual Oriented Models (Cellular Automata, Microsimulation, Agent Based Models, Discrete Event Simulation, Social Network
Analysis). Each technique is broadly described with example uses, key attributes and reference material.

Application of Volterra Equations to Solve Unit Commitment Problem of Optimised Energy Storage and Generation

Development of reliable methods for optimised energy storage and generation is one of the most imminent challenges in modern power systems. In this paper an adaptive approach to load leveling problem using novel dynamic models based on the Volterra integral equations of the first kind with piecewise continuous kernels. These integral equations efficiently solve such inverse problem taking into account both the time dependent efficiencies and the availability of generation/storage of each energy storage technology. In this analysis a direct numerical method is employed to find the least-cost dispatch of available storages. The proposed collocation type numerical method has second order accuracy and enjoys self-regularization properties, which is associated with confidence levels of system demand. This adaptive approach is suitable for energy storage optimisation in real time. The efficiency of the proposed methodology is demonstrated on the Single Electricity Market of Republic of Ireland and Northern Ireland.

Correspondence between continuous-variable and discrete quantum systems of arbitrary dimensions

We establish a mapping between a continuous-variable (CV) quantum system and a discrete quantum system of arbitrary dimension. This opens up the general possibility to perform any quantum information task with a CV system as if it were a discrete system. The Einstein-Podolsky-Rosen state is mapped onto the maximally entangled state in any finite-dimensional Hilbert space and thus can be considered as a universal resource of entanglement. An explicit example of the map and a proposal for its experimental realization are discussed.

Ovariole number - a predictor of differential reproductive success among worker subfamilies in queenless honeybee (Apis mellifera L.) colonies

A honeybee queen normally mates with 10-20 drones, and reproductive conflicts may arise among a colony's different worker patrilines, especially after a colony has lost its single queen and the workers commence egg laying. In this study, we employed microsatellite markers to study aspects of worker reproductive competition in two queenless Africanized honeybee colonies. First, we determined whether there was a bias among worker patrilines in their maternity of drones and, second, we asked whether this bias could be attributed to differences in the degree of ovary activation of workers. Third, we relate these behavioral and physiological factors to ontogenetic differences between workers with respect to ovariole number. Workers from each of three (colony A) and one (colony B) patrilineal genotypes represented less than 6% of the worker population, yet each produced at least 13% of the drones in a colony, and collectively they produced 73% of the drones. Workers representing these genotypes also had more developed follicles and a greater number of ovarioles per ovary. Across all workers, ovariole development and number were closely correlated. This suggests a strong effect of worker genotype on the development of the ovary already in the postembryonic stages and sets a precedent to adult fertility, so that

Male mating behaviour and mating systems of bees: an overview

Considerable interspecific diversity exists among bees in the rendezvous sites where males search for females and in the behaviours employed by males in their efforts to secure matings. I present an evolutionary framework in which to interpret this variation, and highlight the importance for the framework of (i) the distribution of receptive ( typically immediate post-emergence) females, which ordinarily translates into the distribution of nests, and (ii) the density of competing males. Other than the highly polyandrous honey bees ( Apis), most female bees are thought to be monandrous, though genetic data with which to support this view are generally lacking. Given the opportunity, male bees are typically polygamous. I highlight intraspecific diversity in rendezvous site, male behaviour and mating system, which is in part predicted from the conceptual framework. Finally, I suggest that inbreeding may be far more widespread among bees than has hitherto been considered the case.

Regulation and consequences of differential gene expression in diabetic kidney disease

DIN (diabetic nephropathy) is the leading cause of end-stage renal disease worldwide and develops in 25-40% of patients with Type 1 or Type 2 diabetes mellitus. Elevated blood glucose over long periods together with glomerular hypertension leads to progressive glomerulosclerosis and tubulointerstitial fibrosis in susceptible individuals. Central to the pathology of DIN are cytokines and growth factors such as TGF-beta (transforming growth factor beta) superfamily members, including BMPs (bone morphogenetic protein) and TGF-beta 1, which play key roles in fibrogenic responses of the kidney, including podocyte loss, mesangial cell hypertrophy, matrix accumulation and tubulointerstitial fibrosis. Many of these responses can be mimicked in in vitro models of cells cultured in high glucose. We have applied differential gene expression technologies to identify novel genes expressed in in vitro and in vivo models of DN and, importantly, in human renal tissue. By mining these datasets and probing the regulation of expression and actions of specific molecules, we have identified novel roles for molecules such as Gremlin, IHG-1 (induced in high glucose-1) and CTGF (connective tissue growth factor) in DIN and potential regulators of their bioactions.

A statistical framework for the design of microarray experiments and effective detection of differential gene expression

Motivation: Microarray experiments generate a high data volume. However, often due to financial or experimental considerations, e.g. lack of sample, there is little or no replication of the experiments or hybridizations. These factors combined with the intrinsic variability associated with the measurement of gene expression can result in an unsatisfactory detection rate of differential gene expression (DGE). Our motivation was to provide an easy to use measure of the success rate of DGE detection that could find routine use in the design of microarray experiments or in post-experiment assessment.