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.     

Practical, Reliable, Rational Interpolation

An algorithm incorporating features essential for practical,reliable, rational interpolation is explained. This algorithmgenerates a Thiele-Werner continued fraction representationof the interpolant. A backward error analysis is presented forthe algorithm, as well as for its special cases of Newton polynomialinterpolation and Thiele rational interpolation. This is madepossible by introducing into the Newton method, Thiele methodand Werner method a strategy for selecting the interpolationpoints in an optimal order.     

Stability,prediction-correction,and dynamic programming

It is proven here that a bounded perturbation of the discrete dynamic programming functional equation arising from the Bolza problem yields a bounded change in its solution. This stability property encourages the development of approximation techniques for solving such equations. One such technique, involving the backward solution of an approximate functional equation as a prediction step, followed by a forward reconstruction using true equations as a correction step, is then discussed. Bounds for the errors arising from such an approximation procedure are derived. Successive approximations is suggested, in conclusion, as a means for obtaining improved solutions.     

Magnitude,Diversity, Capacities,and Dimensions of Metric Spaces

Magnitude is a numerical invariant of metric spaces introduced by Leinster, motivated by considerations from category theory. This paper extends the original definition for finite spaces to compact spaces, in an equivalent but more natural and direct manner than in previous works by Leinster, Willerton, and the author. The new definition uncovers a previously unknown relationship between magnitude and capacities of sets. Exploiting this relationship, it is shown that for a compact subset of Euclidean space, the magnitude dimension considered by Leinster and Willerton is equal to the Minkowski dimension.     

Fitting Quantiles: Doubling,HR, HQ,and HHH Distributions

Abstract

This article introduces a family of distributional shapes which is flexible in the sense that it contains skewed and symmetric laws as well as heavy-tailed and light-tailed laws. The proposed family is also practically convenient because it is easy to fit to a table of quantiles from any distribution. Inversely, for each of the distributional shapes it is trivial to compute quantiles for any desired probability, and it is possible to compute the corresponding densities.     

Dirichlet's Theorem, Vojta's Inequality, and Vojta's Conjecture

This paper addresses questions involving the sharpness of Vojta's conjecture and Vojta's inequality for algebraic points on curves over number fields. It is shown that one may choose the approximation term mS(D,-) in such a way that Vojta's inequality is sharp in Theorem 2.3. Partial results are obtained for the more difficult problem of showing that Vojta's conjecture is sharp when the approximation term is not included (that is, when D=0). In Theorem 3.7, it is demonstrated that Vojta's conjecture is best possible with D=0 for quadratic points on hyperelliptic curves. It is also shown, in Theorem 4.8, that Vojta's conjecture is sharp with D=0 on a curve C over a number field when an analogous statement holds for the curve obtained by extending the base field of C to a certain function field.     

Economies of scale,competitiveness, and trade patterns within the European community: Clarendon Press,Oxford, 1983, 193 pages, £20.00

We consider two queues in series with input to each queue, which can be controlled by accepting or rejecting arriving customers. The objective is to maximize the discounted or average expected net benefit over a finite or infinite horizon, where net benefit is composed of (random) rewards for entering customers minus holding costs assessed against the customers at each queue. Provided that it costs more to hold a customer at the first queue than at the second, we show that an optimal policy is monotonic in the following senses: Adding a customer to either queue makes it less likely that we will accept a new customer into either queue; moreover moving a customer from the first queue to the second makes it more (less) likely that we will accept a new customer into the first (second) queue. Our model has policy implications for flow control in communication systems, industrial job shops, and traffic-flow systems. We comment on the relation between the control policies implied by our model and those proposed in the communicationa literature.     

Quasi-symmetric designs,codes, quadrics,and hyperplane sections

It is proved that a quasi-symmetric design with theSymmetric Difference Property (SDP) is uniquely embeddable as a derived or a residual design into a symmetric SDP design. Alternatively, any quasi-symmetric SDP design is characterized as the design formed by the minimum weight vectors in a binary code spanned by the simplex code and the incidence vector of a point set in PG(2m-1, 2) that intersects every hyperplane in one of two prescribed numbers of points. Applications of these results for the classification of point sets in PG(2m-1, 2) with the same intersection properties as an elliptic or a hyperbolic quadric, as well as the classification of codes achieving the Grey-Rankin bound are discussed.     

Generic Groups,Collision Resistance,and ECDSA

Proved here is the sufficiency of certain conditions to ensure the Elliptic Curve Digital Signature Algorithm (ECDSA) existentially unforgeable by adaptive chosen-message attacks. The sufficient conditions include (i) a uniformity property and collision-resistance for the underlying hash function, (ii) pseudorandomness in the private key space for the ephemeral private key generator, (iii) generic treatment of the underlying group, and (iv) a further condition on how the ephemeral public keys are mapped into the private key space. For completeness, a brief survey of necessary security conditions is also given. Some of the necessary conditions are weaker than the corresponding sufficient conditions used in the security proofs here, but others are identical. Despite the similarity between DSA and ECDSA, the main result is not appropriate for DSA, because the fourth condition above seems to fail for DSA. (The corresponding necessary condition is plausible for DSA, but is not proved here nor is the security of DSA proved assuming this weaker condition.) Brickell et al. [Vol. 1751 of Lecture Notes in computer Science, pp. 276--292], Jakobsson et al. [Vol. 1976 of Lecture Notes in computer Science, pp. 73--89] and Pointcheval et al. [Vol. 13 of Journal of Cryptology, pp. 361--396] only consider signature schemes that include the ephemeral public key in the hash input, which ECDSA does not do, and moreover, assume a condition on the hash function stronger than the first condition above. This work seems to be the first advance in the provable security of ECDSA.AMS classification: 94A60Supported in part by a National Science and Engineering Research Council of Canada Industrial Research Fellowship.     

AMlet,RAMlet, and GAMlet: Automatic Nonlinear Fitting of Additive Models,Robust and Generalized,With Wavelets

A simple and yet powerful method is presented to estimate nonlinearly and nonparametrically the components of additive models using wavelets. The estimator enjoys the good statistical and computational properties of the Waveshrink scatterplot smoother and it can be efficiently computed using the block coordinate relaxation optimization technique. A rule for the automatic selection of the smoothing parameters, suitable for data mining of large datasets, is derived. The wavelet-based method is then extended to estimate generalized additive models. A primal-dual log-barrier interior point algorithm is proposed to solve the corresponding convex programming problem. Based on an asymptotic analysis, a rule for selecting the smoothing parameters is derived, enabling the estimator to be fully automated in practice. We illustrate the finite sample property with a Gaussian and a Poisson simulation.     

A Two-product,Single-machine,Storage-constrained Production Problem

This paper considers a two-product, single-machine production scheduling problem where there is an added constraint on the amount of finished stock that can be held. The need for a ratio of the cycle times of the two products is a feature both of two-product production scheduling problems contained on one machine and two-product inventory problems with constraints on storage capacity. This means an easy solution algorithm is possible for the problem addressed in the paper which has both types of constraints.     

Distance-regular graphs and (s,c, a,k)-graphs

A diameter-bound theorem for a class of distance-regular graphs which includes all those with even girth is presented. A new class of graphs, called (s, c, a, k)-graphs, is introduced, which are conjectured to contain enough of the local structure of finite distance-regular graphs for them all to be finite. It is proved that they are finite and a bound on the diameter is given in the case a ≤ c.     

Virtual quadruped: Mechanical design,control, simulation,and experimentation

We consider a prototyped walking robot containing a platform and two double-link legs. Thus, it is a five-link mechanism. The front leg models identical motions of the quadruped’s two front legs, and the back leg models identical motions of the quadruped’s two back legs. The legs have passive (uncontrolled) feet that extend in the frontal plane. Because of this the robot is stable in the frontal plane. This robot can be viewed as a “virtual” quadruped. Four DC motors drive the mechanism. Its control system comprises a computer, hardware servo-systems, and power amplifiers. The locomotion of the prototype is planar curvet gait. In the double support our prototype is statically stable and overactuated. In the single support it is an unstable and underactuated system. There is no flight phase. We describe here the scheme of the mechanism, the characteristics of the drives, and the control strategy. The dynamic model of the planar walking is recalled for the double-and single-support phases and for the impact instant. The experiments give results that are close to those of the simulation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 8, pp. 5–28, 2005.     

Convexifactors, Generalized Convexity, and Optimality Conditions

The recently introduced notion of a convexifactor is further studied, and quasiconvex and pseudoconvex functions are characterized in terms of convexifactors. As an application to a chain rule, a necessary optimality condition is deduced for an inequality constrained mathematical programming problem.     

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.     

Communication, complexity, and evolutionary stability

In games with costless preplay communication, some strategies are more complex than others in the sense that they induce a finer partition of the set of states of the world. This paper shows that if the concept of evolutionary stability, which is argued to be a natural solution concept for communication games, is modified to take lexicographic complexity preferences into account, then for a class of games of common interest only communication strategies that induce payoff-dominant Nash outcomes of the underlying game are stable. Received April 1998/Final version September 1998     

Toughness,trees, and walks

A graph is t‐tough if the number of components of G\S is at most |S|/t for every cutset S ⊆ V (G). A k‐walk in a graph is a spanning closed walk using each vertex at most k times. When k = 1, a 1‐walk is a Hamilton cycle, and a longstanding conjecture by Chvátal is that every sufficiently tough graph has a 1‐walk. When k ≥ 3, Jackson and Wormald used a result of Win to show that every sufficiently tough graph has a k‐walk. We fill in the gap between k = 1 and k ≥ 3 by showing that, when k = 2, every sufficiently tough (specifically, 4‐tough) graph has a 2‐walk. To do this we first provide a new proof for and generalize a result by Win on the existence of a k‐tree, a spanning tree with every vertex of degree at most k. We also provide new examples of tough graphs with no k‐walk for k ≥ 2. © 2000 John Wiley & Sons, Inc. J Graph Theory 33:125–137, 2000     

Phase Shift Modulations for Stable,Oscillatory, Traveling,Strongly Nonlinear Waves

Oscillatory dispersive waves propagating in a slowly varying medium are analyzed for Klein-Gordon equations with perturbations. The method of multiple scales is extended to include two fast scales, the usual traveling-wave phase and time, in order to allow initial conditions not usually permitted. An exact wave-action equation is introduced if the traveling wave is stable, involving averages over the periodic wave as well as time. This is equivalent to an extended averaged Lagrangian principle. The equation for the slow modulations of the phase shift of the traveling wave is derived from the higher order terms in the exact action equation and is shown to be the same as in earlier more restrictive studies.     

Nonlinear,finite deformation,finite element analysis

The roles of the consistent Jacobian matrix and the material tangent moduli, which are used in nonlinear incremental finite deformation mechanics problems solved using the finite element method, are emphasized in this paper, and demonstrated using the commercial software ABAQUS standard. In doing so, the necessity for correctly employing user material subroutines to solve nonlinear problems involving large deformation and/or large rotation is clarified. Starting with the rate form of the principle of virtual work, the derivations of the material tangent moduli, the consistent Jacobian matrix, the stress/strain measures, and the objective stress rates are discussed and clarified. The difference between the consistent Jacobian matrix (which, in the ABAQUS UMAT user material subroutine is referred to as DDSDDE) and the material tangent moduli (C^e) needed for the stress update is pointed out and emphasized in this paper. While the former is derived based on the Jaumann rate of the Kirchhoff stress, the latter is derived using the Jaumann rate of the Cauchy stress. Understanding the difference between these two objective stress rates is crucial for correctly implementing a constitutive model, especially a rate form constitutive relation, and for ensuring fast convergence. Specifically, the implementation requires the stresses to be updated correctly. For this, the strains must be computed directly from the deformation gradient and corresponding strain measure (for a total form model). Alternatively, the material tangent moduli derived from the corresponding Jaumann rate of the Cauchy stress of the constitutive relation (for a rate form model) should be used. Given that this requirement is satisfied, the consistent Jacobian matrix only influences the rate of convergence. Its derivation should be based on the Jaumann rate of the Kirchhoff stress to ensure fast convergence; however, the use of a different objective stress rate may also be possible. The error associated with energy conservation and work-conjugacy due to the use of the Jaumann objective stress rate in ABAQUS nonlinear incremental analysis is viewed as a consequence of the implementation of a constitutive model that violates these requirements.