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In January 1972, The Government of Sri Lanka reorganized the formal school system from 8-2-2 to 5-4-2 and introduced a new common curriculum to the junior secondary level in all schools. The National Certificate in General Education Examination (NCGE) was administered in December to the first group of pupils completing the new program. This study deals with certain measures that have been attempted in order to restructure the established examination system and its practices to accommodate the new demands. Specifically, the study deals with four projects: (1) the pilot examination of the NCGE Pattern, including the examination of the pre-vocational study areas, the practical examination in health and physical education, and internal assessment in the social studies program; (2) continuous evaluation of the teaching program at the junior secondary level; (3) conference-marking of examination papers and expediting the distribution of results; and (4) admissions to the university through an examination and district quota system. (Author/MV)

Speech input has proven to be useful for entering text in electronic documents, and seems to be a promising technique also for writing mathematical expressions. Up to now, there is no evidence about the potential advantages of speech input on blind persons who need to edit scientific documents. This paper introduces a technique to enable blind persons to input mathematics by speech. A system prototype has been developed, based on the LAMBDA mathematical editor and the Dragon NaturallySpeakingTM software. An early assessment has been undertaken with a group of blind users. The results of this evaluation will be discussed in this paper and will be exploited to enhance the system prototype. [ABSTRACT FROM AUTHOR]

Lie [104] introduced the geometry of oriented spheres in his dissertation, published as a paper in Mathematische Annalen in 1872. Sphere geometry was also prominent in his study of contact transformations (Lie-Scheffers [105]) and in Volume III of Blaschke's book on differential geometry published in 1929. In recent years, Lie sphere geometry has become a valuable tool in the study of Dupin submanifolds, beginning with Pinkall's [146] dissertation in 1981. In this introduction, we will outline the contents of the book and mention some related results. [ABSTRACT FROM AUTHOR]

Historians distinguish between primary and secondary or even ternary sources. A primary source for, say, a biography would be a birth or death record, personal letters, handwritten drafts of papers by the subject of the biography, or even a published paper by the subject. A secondary source could be a biography written by someone who had examined the primary sources, or a non-photographic copy of a primary source. Ternary sources are things pieced together from secondary sources—encyclopædia or other survey articles, term papers, etc.1 The historian's preference is for primary sources. The further removed from the primary, the less reliable the source: errors are made and propagated in copying; editing and summarising can omit relevant details, and replace facts by interpretations; and speculation becomes established fact even though there is no evidence supporting the "fact".2 [ABSTRACT FROM AUTHOR]

The paper deals with dynamic optimization problems of the Bolza and Mayer types for evolution systems governed by nonconvex Lipschitzian differential inclusions in Banach spaces under endpoint constraints described by finitely many equalities and inequalities with generally nonsmooth functions. We develop a variational analysis of such problems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation. In this way we establish extended results on stability of discrete approximations and derive necessary optimality conditions for nonconvex discrete-time and continuous-time systems in the refined Euler-Lagrange and Weierstrass-Pontryagin forms accompanied by the appropriate transversality inclusions. In contrast to the case of geometric endpoint constraints in infinite dimensions, the necessary optimality conditions obtained in this paper do not impose any nonempty interiority/finite codimension/normal compactness assumptions. [ABSTRACT FROM AUTHOR]

Using vehicle trajectories for a motorway collected from a helicopter, and a newly developed approach to parameter identification, this paper provides new empirical evidence of multi-anticipative car-following by estimating the driver-specific parameters of the different multi-anticipative car-following models. We investigate the nature of the multi-leader stimuli, providing insight into the number of vehicles ahead to which drivers react and the kind of stimuli drivers respond to. The paper also shows that there is large inter-driver variability in multi-leader driving behavior. The implications of our research findings for microscopic modeling are discussed as well. [ABSTRACT FROM AUTHOR]

Insights into pedestrian behavior, and tools to predict this behavior, are essential in the planning and design of public pedestrian facilities, such as transfer stations, shopping malls, and airports. Both macroscopic features of pedestrian flows and microscopic walking behavior underlying these features are important. This paper discusses findings from an experiment with crossing pedestrian flows, and in particular the free speed distribution of the participating pedestrians. Available free speed estimation methods developed for car traffic appear to be not suited for pedestrian traffic. This paper presents a dedicated adaptation of a method used for car traffic, with satisfactory results in pedestrian crossing flows. [ABSTRACT FROM AUTHOR]

The Variance-Gamma (VG) process was introduced by Dilip B. Madan and Eugene Seneta as a model for asset returns in a paper that appeared in 1990, and subsequently used for option pricing in a 1991 paper by Dilip and Frank Milne. This paper serves as a tutorial overview of VG and Monte Carlo, including three methods for sequential simulation of the process, two bridge sampling methods, variance reduction via importance sampling, and estimation of the Greeks. [ABSTRACT FROM AUTHOR]

The article explores how math relates with music, language, art and history. In music, every note has a value and the value comes in beats or counts. There are many words with math meanings, including odd, even, prime, digit, count, pound, negative, product and positive. Palindromes are words that read the same backward and forward, including mom, dad, sees, eve and peep. A scale drawing is done using a pencil and some graph paper. Units of measure in history are discussed.

An interesting surface was discussed by H. F. Minding in the 1830s in papers he published in Crelle's Journal [159] although these papers were then largely forgotten for 30 years; this was a surface of constant negative curvature with the property that geodesics between points are not necessarily unique. Minding's surface is formed by rotating a tractrix about its vertical axis. A tractrix — the name is due to Huyghens — is the curve of the obstinate dog. A point Q on a line ℓ is attached to a point P by a curve of fixed length. P, the dog, is dragged behind Q, the owner, who walks along ℓ, and the path of P is called the tractrix. It is conventional for the walk to start with PQ perpendicular to ℓ. Minding's surface, which is sometimes called the pseudosphere, is formed by rotating the tractrix about ℓ; the point P then generates a circle of singular points (see Figure 11.1). [ABSTRACT FROM AUTHOR]

Focus in the paper is on optimality conditions for bilevel programming problems. We start with a general condition using tangent cones of the feasible set of the bilevel programming problem to derive such conditions for the optimistic bilevel problem. More precise conditions are obtained if the tangent cone possesses an explicit description as it is possible in the case of linear lower level problems. If the optimal solution of the lower level problem is a PC1-function, sufficient conditions for a global optimal solution of the optimistic bilevel problem can be formulated. In the second part of the paper relations of the bilevel programming problem to set-valued optimization problems and to mathematical programs with equilibrium constraints are given which can also be used to formulate optimality conditions for the original problem. Finally, a variational inequality approach is described which works well when the involved functions are monotone. It consists in a variational re-formulation of the optimality conditions and looking for a solution of the thus obtained variational inequality among the points satisfying the initial constraints. A penalty function technique is applied to get a sequence of approximate solutions converging to a solution of the original problem with monotone operators. [ABSTRACT FROM AUTHOR]

In a recent paper, Pan and Başar [19] have studied the H∞ control of large scale Jump Linear systems in which the transitions of the jump Markov chain can be separated into sets having strong and weak interactions. They obtained an approximating reduced-order aggregated problem which is the limit as the rate of transitions of the faster time scale (which is a multiple of some parameter 1/∈) goes to infinity. In this paper we further investigate the solution of that problem as a function of the parameter ϵ. We show that the related optimal feedback policy and the value admit a Taylor series in terms of ϵ, and we compute its coefficients. [ABSTRACT FROM AUTHOR]

This paper exposes in voluntarily simple terms the concept of S-adapted equilibrium introduced to represent and compute economic equilibria on stochastic markets. A model of the European gas market, that has been at the origin of the introduction of the concept, is recalled in this paper and the results obtained in 1987, when the contingent equilibrium has been computed for a time horizon extending until 2020, are compared with the observed trend in these markets over the last two decades. The information structure subsumed by this concept of S-adapted strategies is then analyzed, using different paradigms of dynamic games. The paper terminates with some open and intriguing questions related to the time consistency and subgame perfectness of the dynamic equilibrium thus introduced. [ABSTRACT FROM AUTHOR]

This book is merely an introduction to a vast subject. There is a great deal of additional knowledge concerning these topics, and there are a number of excellent books and expository papers treating much of that material. We briefly describe some of these expositions in order to make it easier for the reader to pursue further study. [ABSTRACT FROM AUTHOR]

Summary: This book is a reprint of papers in the Special Issue published in Education Sciences under the title "Online and Distance Learning during Lockdown Times: COVID-19 Stories". It includes papers covering Higher Education (post-secondary) sector representing international experience of teaching and learning from the start of the first episode of lockdown due to the Covid-19 pandemic.

In the first part of this paper we overview protein structures, their spontaneous formation ("folding") and thermodynamic and kinetic aspects of this phenomenon. It is stressed that universal features of folding are observed near the point of thermodynamic equilibrium between the native and denatured states of the protein. Here the "two-state" ("denatured state" ↔ "native state") transition proceeds without accumulation of metastable intermediates, and only the transition state, i.e., the most unstable state in the folding pathway, is outlined by its essential influence on the folding/unfolding kinetics. In the second part of the paper, a theory of protein folding rates and related phenomena is presented. First, it is shown that the protein size determines the range of protein's folding rates in the vicinity of the point of thermodynamic equilibrium between the native and denatured states of the protein. Then we present methods for calculating folding and unfolding rates of globular proteins from their sizes, stabilities and either 3D structures or amino acid sequences. And, at last, we show that the same theory outlines the location of the protein folding nucleus (i.e., the structured part of transition state) in a reasonable concordance with experimental data. [ABSTRACT FROM AUTHOR]

This set of notes was used to complement my short course on the convergence to equilibrium for the Boltzmann equation, given at Institut Henri Poincaré in November\2-December 2001, as part of the Hydrodynamic limits program organized by Stefano Olla and François Golse. The informal style is in accordance with the fact that this is neither a reference book nor a research paper. The reader can use my review paper, A review of mathematical topics in collisional kinetic theory, as a reference source to dissipate any ambiguity with respect to notation for instance. Apart from minor corrections here and there, the main changes with respect to the original version of the notes were the addition of a final section to present some more recent developments and open directions, and the change of the sign convention for the entropy, to agree with physical tradition. Irene Mazzella is warmly thanked for kindly typesetting a preliminary version of this manuscript. [ABSTRACT FROM AUTHOR]

We describe a discrete model for the dynamics of thin flexible structures, such as hats, leaves, and aluminum cans, which are characterized by a curved undeformed configuration. Previously such thin-shell models required complex continuum mechanics formulations and correspondingly complex algorithms. We show that a simple shell model can be derived geometrically for triangle meshes and implemented quickly by modifying a standard cloth simulator. Our technique convincingly simulates a variety of curved objects with materials ranging from paper to metal, as we demonstrate with several examples including a comparison of a real and simulated falling hat. [ABSTRACT FROM AUTHOR]

Periodic boundary value problems can be reduced to fixed point equations of integral type. After this reduction the Leray-Schauder degree becomes an useful tool for proving the existence of periodic solutions. The purpose of this paper is to discuss the analogous approach for almost periodic problems and to show by means of examples that degree theory is not applicable in this setting. [ABSTRACT FROM AUTHOR]

This paper presents methods using algebraic topology for showing that pseudo-trajectories are close to trajectories of a dynamical system. Our emphasis is the case where the trajectories are unstable in two or more dimensions. We develop the algebraic topology for guaranteeing the existence of such trajectories. [ABSTRACT FROM AUTHOR]

The paper is aimed at providing some results on the almost periodicity of solutions to some general functional or functional differential equations. The term "general" is meant in the sense that the equations involve operators of general form, acting on the space of almost periodic functions. First order and second order equations are dealt with. [ABSTRACT FROM AUTHOR]

The Hochschild and (periodic) cyclic homology of Banach algebras are either trivial or not interesting, see Connes [2], [4], [6]. To correct this deficiency, Connes [3] had produced the entire cyclic cohomology (see also [4], [6], [5]). The entire cyclic cochains are elements of the infinite product (b,B) cohomology bi-complex which satisfy a certain bidegree asymptotic growth condition. The entire cyclic cohomology is a natural target for the asymptotic Chern character of θ-summable Fredholm modules. More recently, Puschnigg [15] introduced the local cyclic cohomology based on precompact subsets of the algebra in an inductive limits system setting. The main purpose of this paper is to create an analogue of the Hochschild and periodic cyclic homology which gives the right result (i.e., the ordinary ℤ2-graded Alexander-Spanier co-homology of the manifold) when applied, at least, onto the algebra of continuous functions on topological manifolds and CW-complexes. This is realized by replacing the Connes periodic bi-complex (b,B), see Connes [2], [4] and Loday [12], by the bi-complex ($$ \tilde b $$, d), where the operator $$ \tilde b $$ is obtained by blending the Hochschild boundary b with the Alexander-Spanier boundary d; the operator $$ \tilde b $$ anti-commutes with the operator d. The homologies of these complexes will be called modified Hochschild, resp. modified periodic cyclic homology. Our construction uses in addition to the algebraic structure solely the locality relationship extracted from the topological structure of the algebra. The modified periodic cyclic homology is invariant under continuous homotopies, while the others are invariant at smooth homotopies (diffeotopies) only. The modified Hochschild and periodic cyclic homology are directly connected to the Alexander-Spanier cohomology. [ABSTRACT FROM AUTHOR]

The current industry standard is to optimize loan portfolios with respect to variance. In this paper we show that optimization with respect to expected shortfall and expected regret is fairly easy to implement. [ABSTRACT FROM AUTHOR]

We consider, in a Hilbert space U, a class of Gaussian processes defined by a linear filter with a cylindrical Wiener process as input process. This noise is used as an additive perturbation to a family of fractional order (in time) partial differential equations. We give conditions such that the stochastic convolution process is well defined, both in finite time horizon and in an infinite interval. An important example of noise that is contained in the paper is the fractional Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper we apply the general framework introduced in [2] to two models with transaction costs: • a dynamic model with an infinite number of assets; • a model with European call options as basic assets. In particular, it is proved that a dynamic model with an infinite number of assets satisfies the No Generalized Arbitrage condition (this notion was introduced in [2]) if and only if there exist an equivalent measure and a martingale with respect to this measure that lies (componentwise) between the discounted ask and bid price processes. Furthermore, the set of fair prices of a contingent claim coincides with the set of expectations of the payoff with respect to these measures. Our approach to arbitrage pricing in models with transaction costs differs from the existing ones. Key words: Delta-martingale, Fair price, Fundamental theorem of asset pricing, General arbitrage pricing model, Generalized arbitrage, Risk-neutral measure, Set of attainable incomes, Transaction costs [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a unified approach to pricing contingent claims through a new concept of generalized arbitrage. First, we prove the fundamental theorem of asset pricing and establish the form of the fair price intervals within the framework of a general arbitrage pricing model. [ABSTRACT FROM AUTHOR]

The paper analyses traffic flow data collected in the Lefortovo tunnel (Moscow) in 2004/05. First, it shows the presence of cooperative traffic dynamics in this tunnel and, second, studies the phase portrait of the vehicle ensemble in the velocity-density plane. In particular, the regions of regular and stochastic dynamics are found and the presence of dynamical traps is demonstrated. [ABSTRACT FROM AUTHOR]

The multi-phase signal control method is one of the important measures to enhance intersection capacities and alleviate urban traffic problems. This paper employs the stopping-line-method to study the capacity model of cross multi-phase signalized intersections and analyses the relevant change of the intersection capacity and cycle length. It studies the intersection capacity in detail under two normal situations, one with one straight lane and one left-turn lane, and the other with two straight lanes and one left-turn lane. Finally, a practical intersection is chosen and its phase design is improved by the method proposed in the paper. [ABSTRACT FROM AUTHOR]

Detection, identification and tracking of multiple moving targets have important applications in transportation and vehicle control areas. In this paper we present our approach to detect, recognize and track the vehicles within the detection region of a moving probe vehicle, based on the data collected by multiple sensors, including LIDAR and GPS. This paper develops a methodology to group the LIDAR measurements into targets, classify the targets as vehicles or fixed objects, and track the vehicular targets within lanes using a Kalman-filter. One important feature of this approach is that we track all of the observations in world coordinates, allowing us to average over many samples and ideally many runs to differentiate between the fixed objects (road boundaries) and moving objects (vehicles). [ABSTRACT FROM AUTHOR]

This paper shows that including the effects of lane-changing activity in kinematic wave theory reveals the physical mechanisms and reproduces the main empirical features that motivated Kerner's three-phase theory. This is shown using a hybrid representation of traffic flow where lane-changing vehicles are treated as discrete particles with realistic accelerations embedded in a continuous multilane kinematic wave stream. We show that this parsimonious four-parameter model reproduces the three phases identified by Kerner, including phase transitions and jam formation. We conclude that synchronized flow and wide-moving jams differ only in their lane-changing spatiotemporal patterns, but obey the same conservation laws and boundary conditions. Freeway segments with one, two and three junctions are analyzed. [ABSTRACT FROM AUTHOR]

This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from the conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence of solutions to the Cauchy problem and we show that the Lipschitz continuous dependence by initial data does not hold. Our method is based on wave front tracking approach, see [4]. [ABSTRACT FROM AUTHOR]

Agent-based and cellular automata models have been widely used in an efficient and effective way for studying granular traffic, but rarely considering the combined effect and interactions of pedestrians and vehicles in urban networks. So from this point of view an attempt has been made to develop a virtual urban environment which considers both vehicular and pedestrian traffic and the interactions arising from their behavior. This paper presents details of the model we have developed. For vehicular traffic a cellular automata model, combining and appropriately modifying (e.g. to account for the pedestrian movement) BML, NaSch and ChSch models, is considered. Pedestrian traffic is simulated using simple behavioral rules combined with an agent-based approach. Different constraints affecting the mobility of the whole system are considered, which can be seen and even changed by the user in the simulated environment. The model belongs to the microscopic category where pedestrians/vehicles behave in their environment by making a sequence of decisions. The interactions among vehicles and pedestrians are also incorporated which signifies various effects, ranging from accident risk of pedestrians to the generation of traffic jams. NetLogo which is a multi-agent based modeling language is used as the programming platform for the simulation. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical approximation of the solutions of a macroscopic model for the description of the flow of pedestrians. Solutions of the associated Riemann problem are known to be possibly nonclassical in the sense that the underlying discontinuities may well violate Oleinik inequalities, which makes their numerical approximation very sensitive. This study proposes to apply the Transport-Equilibrium strategy proposed in [2] for computing nonclassical solutions of scalar conservation laws to this framework. Numerical evidences are proposed. [ABSTRACT FROM AUTHOR]

This paper discusses the validity of first-order traffic flow theory to describe two-dimensional pedestrian flow operations in case of an oversaturated bottleneck upstream of which a large high-density region has formed. Pedestrians passing the same cross-section inside of the congested region appear to encounter different flow conditions. In the lateral center, high densities and low speeds are observed. However, on the boundary of the congested region, pedestrians may walk in nearly free flow conditions. Visualising pedestrian flow data in the flow-density plane results in a large scatter of points having similar flows (bottleneck capacity), but different densities. Observations on congestion of pedestrian traffic over the total width of the cross-section are found to belong to a set of different fundamental diagrams instead of a single one. This has consequences for the estimation of the fundamental diagram describing pedestrian traffic. [ABSTRACT FROM AUTHOR]

In Section 2.7 we encountered a situation in which among all feasible solutions of a linear program, only those with all components integral are of interest in the practical application. A similar situation occurs quite often in attempts to apply linear programming, because objects that can be split into arbitrary fractions are more an exception than the rule.When hiring workers, scheduling buses, or cutting paper rolls one somehow has to deal with the fact that workers, buses, and paper rolls occur only in integral quantities. [ABSTRACT FROM AUTHOR]

The other type of periodic orbits apart from the type encountered in the previous chapter on quadratic systems with a center point is the limit cycle; its introduction into mathematics goes back to Poincaré [P]. The limit cycle problem is considerably more difficult to solve than the center problem. A limited number of papers over the twentieth century was sufficient to clarify the center problem. However, by the end of that century, despite a multitude of papers, the limit cycle problem for quadratic systems is still left with quite a number of unsolved questions, some of them with poor hope for answers on the short run. This is due to the limitations of available methods of investigation and, in some questions, the pure absence of them. For that reason, it is difficult to give a well structured presentation of what is known about limit cycles in quadratic systems, illustrating at the same time that the underlying structure of this non linear phenomenon is not yet well understood. [ABSTRACT FROM AUTHOR]

The goals of the paper are as follows: (i) review some qualitative properties of oil and gas prices in the last 15 years; (ii) propose some mathematical elements towards a definition of mean reversion that would not be reduced to the form of the drift in a stochastic differential equation; (iii) conduct econometric tests in order to conclude whether mean reversion still exists in the energy commodity price behavior. Regarding the third point, a clear "break" in the properties of oil and natural gas prices and volatility can be exhibited in the period 2000-2001. [ABSTRACT FROM AUTHOR]

This paper reviews and extends the mathematical finance literature on bubbles in complete markets. We provide a new characterization theorem for bubbles under the standard no-arbitrage framework, showing that bubbles can be of three types. Type 1 bubbles are uniformly integrable martingales, and these can exist with an infinite lifetime. Type 2 bubbles are nonuniformly integrable martingales, and these can exist for a finite, but unbounded, lifetime. Last, Type 3 bubbles are strict local martingales, and these can exist for a finite lifetime only. When one adds a no-dominance assumption (from Merton [24]), only Type 1 bubbles remain. In addition, under Merton's no-dominance hypothesis, put-call parity holds and there are no bubbles in standard call and put options. Our analysis implies that if one believes asset price bubbles exist and are an important economic phenomena, then asset markets must be incomplete. [ABSTRACT FROM AUTHOR]

Most financial asset-pricing models assume frictionless competitive markets that imply the absence of arbitrage opportunities. Given the absence of ar-bitrage opportunities and complete asset markets, there exists a unique martingale measure that implies martingale pricing formulae and replicating asset portfolios. In incomplete markets, or markets with transaction costs, these results must be modified to admit nonunique measures and the possibility of imperfectly replicating portfolios. Similar di3culties arise in markets with taxation. Some theoretical research has argued that some taxation functions will imply arbitrage opportunities and the nonexistence of a competitive asset economy. In this paper we construct a multiperiod, discrete time/state general equilibrium model of asset markets with transaction costs and taxes. The transaction cost technology and the tax system are quite general, so that we can include most discrete time/state models with transaction costs and taxation. We show that a competitive equilibrium exists. Our results require careful modeling of the government budget constraints to rule out tax arbi-trage possibilities. [ABSTRACT FROM AUTHOR]

The article features fun math activities/lessons that focus on geometry and patterns. One activity involves taking two squares and cutting each one on the diagonal. Another activity includes counting the number of triangles in a picture of a kite. Other math lessons include finding triangles in the home and outside the home, figuring out the doodles, and looking for things that have symmetry.

The purpose of this paper is to describe the main problems concerning the design of assembly lines. It is composed of the following steps: (1) input data preparation (2) elaboration of the logical layout of the line, which consists in the distribution of operations among stations along the line and an assignment of resources to the different stations and (3) finally the mapping phase, allowing to check the results with a commercial simulation package. This work presents a new method to tackle the hybrid assembly lines dealing with multiple objective. The goal is to minimize the total cost of the line by integrating design (congestion, machine cost,...) and operation issues (cycle time, precedence constraints, availability,...). After an overview of the current work in this area, this paper presents, in detail, a very promising approach to solve multiple objective problems: a multiple objective grouping genetic algorithm (a grouping genetic algorithm hybridized with the multi-criteria decision-aid method PROMETHEE II). An approach to deal with user’s preferences in design problems is also introduced. The essential concepts adopted by the method are described. An application of the proposed method to an industrial case study is presented. [ABSTRACT FROM AUTHOR]

Nonlinear optimisation and a pig growth model are combined with the traditional use of linear programming to maximise gross margin per pig place per year for the pig producer. Emphasis in this paper is on description of the problem and analysis of the objective function. [ABSTRACT FROM AUTHOR]

The essential properties of the roots of complex semisimple Lie algebras may be captured in the idea of an abstract ‘root system'. In this chapter, we shall develop the basic theory of root systems. Our eventual aim, achieved in Chapters 13 and 14, will be to use root systems to classify the complex semisimple Lie algebras. Root systems have since been discovered to be important in many other areas of mathematics, so while this is probably your first encounter with root systems, it may well not be your last! In MathSciNet, the main database for research papers in mathematics, there are, at the time of writing, 297 papers whose title contains the words ‘root system', and many thousands more in which root systems are mentioned in the text. [ABSTRACT FROM AUTHOR]

In the original version of Parrondo's paradox, two losing sequences of games of chance are combined to form a winning sequence. The games in the first sequence depend on a single parameter p, while those in the second depend on two parameters p1 and p2. The paradox is said to occur because there exist choices of p, p1 and p2 such that the individual sequences of games are losing but a sequence constructed by choosing randomly between the games at each step is winning. At first sight, such behavior seems surprising. However, we contend in this paper that it should not be seen as surprising. On the contrary, we showthat such behaviour is typical in situations in which we randomly create a sequence from games whose winning regions can be defined on the same parameter space. Before we discuss this issue, we investigate in some detail the issue of when sequences of games, such as those proposed by Parrondo, should be considered to be winning, losing or fair. [ABSTRACT FROM AUTHOR]