On automorphisms of a distance-regular graph with intersection array {204, 175, 48, 1; 1, 12, 175, 204}

The research is focused on the question of proportional development in economic growth modeling. A multilevel dynamic optimization model is developed for the construction of balanced proportions for production factors and investments in a situation of changing prices. At the first level, models with production functions of different types are examined within the classical static optimization approach. It is shown that all these models possess the property of proportionality: in the solution of product maximization and cost minimization problems, production factor levels are directly proportional to each other with coefficients of proportionality depending on prices and elasticities of production functions. At the second level, proportional solutions of the first level are transferred to an economic growth model to solve the problem of dynamic optimization for the investments in production factors. Due to proportionality conditions and the homogeneity condition of degree 1 for the macroeconomic production functions, the original nonlinear dynamics is converted to a linear system of differential equations that describe the dynamics of production factors. In the conversion, all peculiarities of the nonlinear model are hidden in a time-dependent scale factor (total factor productivity) of the linear model, which is determined by proportions between prices and elasticities of the production functions. For a control problem with linear dynamics, analytic formulas are obtained for optimal development trajectories within the Pontryagin maximum principle for statements with finite and infinite horizons. It is shown that solutions of these two problems differ crucially from each other: in finite horizon problems the optimal investment strategy inevitably has the zero regime at the final stage, whereas the infinite horizon problem always has a strictly positive solution. A remarkable result of the proposed model consists in constructive analytical solutions for optimal investments in production factors, which depend on the price dynamics and other economic parameters such as elasticities of production functions, total factor productivity, and depreciation factors. This feature serves as a background for the productive fusion of optimization models for investments in production factors in the framework of a multilevel structure and provides a solid basis for constructing optimal trajectories of economic development.     

Loop invariants,exploration of regularities,and mathematical games

Loop invariants are assertions of regularities that characterize the loop components of algorithms. They are fundamental components of computerprograms verification, but their relevance goes beyond verification—they can be significantly utilized for algorithm design and analysis. Unfortunately, they are only modestly introduced in the teaching of programming and algorithms. One reason for this is an unjustified notion that loop invariants are ‘tied to formality’, hard to illustrate, and difficult to comprehend. In this paper a novel approach is presented for illustrating on a rather intuitive level the significance of loop invariants. The illustration is based on mathematical games, which are attractive examples that require the exploration of regularities via problemsolving heuristics. Throughout the paper students' application of heuristics is described and emphasis is placed on the links between loop invariants, heuristic search activities, recognition of regularities and design and analysis of algorithms.     

Nonsmooth Calculus, Minimality, and Monotonicity of Convexificators

Noncompact convexificators, which provide upper convex and lower concave approximations for a continuous function, are defined. Various calculus rules, including extremality and mean-value properties, are presented. Regularity conditions are given for convexificators to be minimal. A characterization of quasiconvexity of a continuous function is obtained in terms of the quasimonotonicity of convexificators.     

Discontinuous dynamics of a non-linear, self-excited, friction-induced, periodically forced oscillator

The discontinuous dynamics of a non-linear, friction-induced, periodically forced oscillator is studied. The analytical conditions for motion switchability at the velocity boundary in such a nonlinear oscillator are developed to understand the motion switching mechanism. Using such analytical conditions of motion switching, numerical predictions of the switching scenarios varying with excitation frequency and amplitude are carried out, and the parameter maps for specific periodic motions are presented. Chaotic and periodic motions are illustrated through phase planes and switching sections for a better understanding of motion mechanism of the nonlinear friction oscillator. The periodic motions with switching in such a nonlinear, frictional oscillator cannot be obtained from the traditional analysis (e.g., perturbation and harmonic balance method).     

A continuous time,deterministic, nonstationary model of economic ordering

The paper considers the problem of economic ordering for a deterministic, nonstationary environment in continuous time. Previous work on the topic is reviewed. The specification of the cost criterion common in inventory theory is called in question for nonstationary situations as far as interest cost is concerned. It is proposed to account for interest by discounting rather than in a holding cost expression. The main interest of the paper is in three versions of the problem: First an unconstrained version, for which inventory is allowed to become negative (backlogging model), second a model in which inventory is constrained to be nonnegative (non-backlogging model), and third a nonbacklogging model with a storage space constraint. For the first two problems necessary optimality conditions are derived which are based on control theory for continuous time systems with jumps in the state trajectories, especially on Blaquière's impulsive maximum principle. These conditions reduce the problem of finding an optimal ordering plan, i.e. an unknown number of optimal ordering times and for each of them an optimal order size to a one parameter search problem. Due to the possibility of multiple solutions of the optimality conditions for each ordering time, one cannot in general identify a unique candidate ordering plan for each value of the search parameter, but only a tree-structured set of such plans. The optimality conditions for the first two problem versions and for a fourth one with a storage space constraint but without a non-backlogging constraint are eventually combined to yield a solution of the storage space constrained non-backlogging version.     

Modelling,Synthesis, and Simulation of Supervisory Process Control Systems

Modelling, synthesis, and simulation issues of the supervisory systems in process control are investigated in the paper. Petri nets are used as a basic modelling framework for the supervisory part of the system. It is shown how the final verification effort can be minimised by applying formal synthesis methods. A straightforward approach to the industrial implementation of the developed solutions is suggested by means of sequential function chart representation. A batch process cell case study is used to illustrate the described concepts. Corresponding continuous and discrete event models of the process cell units are developed and a co-ordinating supervisor is designed by the method of place invariants. The system is simulated by the continuous simulation tool Matlab-Simulink, which is enhanced for simulation of the sequential control logic represented by sequential function chart.     

Agricultural planning of annual plants under demand,maturation, harvest,and yield risk

In this study we present a planning methodology for a firm whose objective is to match the random supply of annual premium fruits and vegetables from a number of contracted farms and the random demand from the retailers during the planning period. The supply uncertainty is due to the uncertainty of the maturation time, harvest time, and yield. The demand uncertainty is the uncertainty of weekly demand from the retailers. We provide a planning methodology to determine the farm areas and the seeding times for annual plants that survive for only one growing season in such a way that the expected total profit is maximized. Both the single period and the multi period cases are analyzed depending on the type of the plant. The performance of the solution methodology is evaluated by using numerical experiments. These experiments show that the proposed methodology matches random supply and random demand in a very effective way and improves the expected profit substantially compared to the planning approaches where the uncertainties are not taken into consideration.     

Dynamic modelling of dissolved oxygen in the creeks of Sagar island,Hooghly–Matla estuarine system,West Bengal,India

Hooghly–Matla estuarine ecosystem is one of the largest estuarine ecosystems of the world. Sagar island is the largest delta in this estuarine complex. This island is criss-crossed by small and large creeks with mangrove vegetation and all are connected to the principal estuarine water. Decomposition of mangrove litter in soil is major source of inorganic nutrient to phytoplankton of the adjacent creeks. Deforestation of mangrove affects the primary production, which in turn reduces the availability of dissolved oxygen for the organisms residing in the estuary. Considering the importance of dissolved oxygen in various aspects of aquatic life, a dynamic model of dissolved oxygen at Sagar island of Hooghly–Matla estuarine complex with the help of single dimension differential equation is proposed in the present paper. Different physical, chemical and biological factors such as solar irradiance, temperature, salinity of water, particulate organic matter, re-aeration, wind velocity, phytoplankton and zooplankton, which control the fluctuation of dissolved oxygen, are included in the present model. Most of the parameter values are collected directly from the field surveys. The parameter values which are not able to collect from the field, obtained from literatures are calibrated. To make the model realistic it is properly validated with observed data and to know the statistical significance, chi square goodness fit test is performed. Field surveys are performed over two years. During calibration and validation, two sets of data (first year and second year data) are used. Chi-square values are 5.97 and 6.17 for first and second sets of data respectively (p < 0.05). Sensitivity analysis reveals that optimal light intensity is the most sensitive parameter for dissolved oxygen dynamics. Results also show that wind velocity, solar irradiation, salinity of water and temperature are important factors for controlling the dynamics of dissolved oxygen. Macrophytes have very little contribution to oxygen production in the creeks of Sagar island. Model reveals that low dissolved oxygen in the creek water is one of the causes of decline in fish population of the estuary.     

Polynomial solutions of the classical equations of Hermite,Legendre, and Chebyshev

The classical differential equations of Hermite, Legendre, and Chebyshev are well known for their polynomial solutions. These polynomials occur in the solutions to numerous problems in applied mathematics, physics, and engineering. However, since these equations are of second order, they also have second linearly independent solutions that are not polynomials. These solutions usually cannot be expressed in terms of elementary functions alone. In this paper, the classical differential equations of Hermite, Legendre, and Chebyshev are studied when they have a forcing term x ^M on the right-hand side. It will be shown that for each equation, choosing a certain initial condition is a necessary and sufficient condition for ensuring a polynomial solution. Once this initial condition is determined, the exact form of the polynomial solution is presented.     

Backward error,sensitivity, and refinement of computed solutions of algebraic Riccati equations

In this paper, a new backward error criterion, together with a sensitivity measure, is presented for assessing solution accuracy of nonsymmetric and symmetric algebraic Riccati equations (AREs). The usual approach to assessing reliability of computed solutions is to employ standard perturbation and sensitivity results for linear systems and to extend them further to AREs. However, such methods are not altogether appropriate since they do not take account of the underlying structure of these matrix equations. The approach considered here is to first compute the backward error of a computed solution X? that measures the amount by which data must be perturbed so that X? is the exact solution of the perturbed original system. Conventional perturbation theory is used to define structured condition numbers that fully respect the special structure of these matrix equations. The new condition number, together with the backward error of computed solutions, provides accurate estimates for the sensitivity of solutions. Optimal perturbations are then used in an iterative refinement procedure to give further more accurate approximations of actual solutions. The results are derived in their most general setting for nonsymmetric and symmetric AREs. This in turn offers a unifying framework through which it is possible to establish similar results for Sylvester equations, Lyapunov equations, linear systems, and matrix inversions.     

Design of anisotropic structures: Calculation,optimization, and tests

Conclusions A computational procedure based on a limiting state has been worked out which proposes the performance of tests of samples, small-scale models, and full-scale structures and a statistical analysis of the experimental data. Advantages of the method are shown in comparison with the computational method based on fracturing loads which pertain to estimating the accuracy of the computational scheme of structures and determining their safety coefficients. An optimality criterion is formulated within the framework of the method which is related to ensuring a maximum of the limiting (fracturing or critical) load with a constant mass of the structure.Applications of the method are given to estimation of the strength and stability of structures made out of polymeric and metal CM with the use of the Gol'denblat-Kopnov criterion and solutions based on linear shell theory. The conditions for realization of the optimality criterion of structures are determined. Recommendations are made for the rational reinforcement of structures made out of filamentary CM, and a procedure is proposed for the rational design of reinforced structures.As an example of the tests for stability of cylindrical shells made out of aluminum-magnesium alloy under external pressure, the legitimacy of the cause of the disagreement between the calculated and experimental critical loads, which consists of a discrepancy in the nature of wave formation at the instant of stability loss and initial imperfections, and the effectiveness of the computational method based on a limiting state are confirmed. Refined computational formulas for the critical loads of cylindrical shells made out of fiberglass, carbon fiber, and boraluminum under typical kinds of loading are proposed within the framework of the method and on the basis of an analysis of the results of tests, and a comparison is made of the effectiveness of the reinforcement of shells made out of aluminum-magnesium alloy and carbon fiber. The values of the safety coefficients of structures which guarantee their reliability are determined.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 262–271, March–April, 1980.     

Solvability,controllability, and observability of singular systems

The problems of solvability, controllability, and observability for the singular systemKx(t)=Ax(t)+Bu(t) are studied, whereK is a singular, square matrix andu(t) is a complex vector function sufficiently differentiable. The classical theories of matrix pencils are first related to the solvability of singular systems. Then, the concepts of reachability, controllability, and observability of regular systems are extended to singular systems. Finally, the set of reachable states is described. The proposed matrix conditions for testing the controllability and observability of singular systems are simple and always feasible.     

Coherence, local quasiconvexity, and the perimeter of 2-complexes

A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results we prove that one-relator groups with sufficient torsion are coherent and locally quasiconvex and we give an alternative proof of the coherence and local quasiconvexity of certain 3-manifold groups. The main application is to establish the coherence and local quasiconvexity of many small cancellation groups. Received: March 2004 Accepted: August 2004     

Scarcity,competition, and value

We suggest a value for finite coalitional games with transferable utility that are enriched by non-negative weights for the players. In contrast to other weighted values, players stand for types of agents and weights are intended to represent the population sizes of these types. Therefore, weights do not only affect individual payoffs but also the joint payoff. Two principles guide the behavior of this value. Scarcity: the generation of worth is restricted by the scarcest type. Competition: only scarce types are rewarded. We find that the types’ payoffs for this value coincide with the payoffs assigned by the Mertens value to their type populations in an associated infinite game.     

The Stability of Two, Three, and Four Wave Interactions of a Prototype System

The stability of plane wave interactions of coupled nonlinear Schrödinger (CNLS) equations can be analyzed within a bisymplectic framework. This framework is a generalization of the Hamiltonian formulation. The current study considers a family of CNLS equations that are used as a prototype system for studying the combined interaction of unstable and stable component waves in optics. This popular family has a drawback when cast into a bisymplectic framework: the determinant controlling various types of fiber regime is zero. To solve this problem, it is proposed that a limit is taken from a more general CNLS family to the family in question. This method is then bench-marked against known stability results for the simple two plane wave interactions when amplitudes are equal and are found to agree. It is then applied to two wave interactions with unequal amplitudes as well as three and four wave interactions. The latter interactions for this particular system are not spectrally stable. By suggesting a slightly larger family of CNLS equations, it is illustrated that spectral stability can occur. This adapted prototype system may be of use in optics; in particular, to show that long-wave stability is possible given a judicious choice of parameter values.     

Discrete Solutions of Dirichlet Problems, Finite Volumes, and Circle Packings

Convergence results for discrete solutions of Dirichlet problems for Poisson equations are obtained, where discrete solutions are constructed for triangular grids using finite volumes with sides perpendicular to, but not necessarily bisecting, corresponding edges in underlying triangulations. A method, based on properties of circle packings, is described for generating triangular meshes and associated volumes. Also, the approximation of exit probabilities of the Brownian motion by exit probabilities of random walks on circle packings is discussed. Received July 24, 1997, and in revised form November 8, 1998.     

Cyclic reduction,dichotomy, and the estimation of differential equations

In the estimation of ordinary differential equations given observed data it is necessary to parametrize the solutions in a computationally tractable way in order to make comparisons with the given data. Typically, this is done by adjoining initial or boundary conditions, and this requires additional information on the solution structure in order to do this in a manner that leads to stable computation of the comparison solutions. Here, an alternative approach is described in which cyclic reduction is used to reduce the estimation problem to an optimization problem subject to a fixed number of equality constraints. If orthogonal transformations are used in the cyclic reduction process then it appears that stable computations are possible without the need for the structural information needed to devise the stable imbeddings. The aim of this paper is to provide evidence in support of this claim. In particular, it is shown that the cyclic reduction process is linked to a new family of representation of the solutions of the ODE system. Some properties of members of this family are described. These provide insight into the particular advantages of the orthogonal reduction form of cyclic reduction.     

Computational geometric aspects of rhythm,melody, and voice-leading

Many problems concerning the theory and technology of rhythm, melody, and voice-leading are fundamentally geometric in nature. It is therefore not surprising that the field of computational geometry can contribute greatly to these problems. The interaction between computational geometry and music yields new insights into the theories of rhythm, melody, and voice-leading, as well as new problems for research in several areas, ranging from mathematics and computer science to music theory, music perception, and musicology. Recent results on the geometric and computational aspects of rhythm, melody, and voice-leading are reviewed, connections to established areas of computer science, mathematics, statistics, computational biology, and crystallography are pointed out, and new open problems are proposed.     

Local networks for computer communications: A. WEST and P. JANSON (Eds.) Proceedings of the IFIP Working Group 6.4, International Workshop on Local Networks,Zürich,Switzerland, August 27–29, 1980, North-Holland,Amsterdam, 1981, xiii + 470 pages,Dfl.125.00

In this paper we present two lower bounds for the p-median problem, the problem of locating p facilities (medians) on a network. These bounds are based on two separate lagrangean relaxations of a zero-one formulation of the problem with subgradient optimisation being used to maximise these bounds. Penalty tests based on these lower bounds and a heuristically determined upper bound to the problem are developed and shown to result in a large reduction in problem size. The incorporation of the lower bounds and the penalty tests into a tree search procedure is described and computational results are given for problems with an arbitrary number of medians and having up to 200 vertices. A comparison is also made between these algorithms and the dual-based algorithm of Erlenkotter.