Sequences with idempotent products from finite regular semigroups

We find the smallest integer (n) such that for every regular semigroup S of order n, every sequence of length (n) of elements of S contains a consecutive subsequence whose product is an -element, where = ‘idempotent’, ‘core’ and ‘subgroup and core’. For arbitrary semigroups of order n, we also find (n) where = ‘regular’, ‘group’, ‘core’, ‘regular and core’ and ‘subgroup and core’.     

Some properties of minimal imperfect graphs

The Even Pair Lemma, proved by Meyniel, states that no minimal imperfect graph contains a pair of vertices such that all chordless paths joining them have even lengths. This Lemma has proved to be very useful in the theory of perfect graphs. The Odd Pair Conjecture, with ‘even’ replaced by ‘odd’, is the natural analogue of the Even Pair Lemma. We prove a partial result for this conjecture, namely: no minimal imperfect graph G contains a three-pair, i.e. two nonadjacent vertices u₁, u₂ such that all chordless paths of G joining u₁ to u₂ contain precisely three edges. As a by-product, we obtain short proofs of two previously known theorems: the first one is a well-known theorem of Meyniel (a graph is perfect if each of its odd cycles with at least five vertices contains at least two chords), the second one is a theorem of Olariu (a graph is perfect if it contains no odd antihole, no P₅ and no extended claw as induced subgraphs).     

On the orders of directly indecomposable groups

We investigate the set of those integers n for which directly indecomposable groups of order n exist. For even n such groups are easily constructed. In contrast, we show that the density of the set of odd numbers with this property is zero. For each n we define a graph whose connected components describe uniform direct decompositions of all groups of order n. We prove that for almost all odd numbers (i.e., with the exception of a set of density zero) this graph has a single ‘big’ connected component and all other vertices are isolated. We also give an asymptotic formula for the number of isolated vertices of the graph, i.e., for the number of prime divisors q of n such that every group of order n has a cyclic direct factor of order q.     

Convexities related to path properties on graphs

A feasible family of paths in a connected graph G is a family that contains at least one path between any pair of vertices in G. Any feasible path family defines a convexity on G. Well-known instances are: the geodesics, the induced paths, and all paths. We propose a more general approach for such ‘path properties’. We survey a number of results from this perspective, and present a number of new results. We focus on the behaviour of such convexities on the Cartesian product of graphs and on the classical convexity invariants, such as the Carathéodory, Helly and Radon numbers in relation with graph invariants, such as the clique number and other graph properties.     

A new method of digital simulation for an aircraft gas turbine engine control system

An LD-9 aircraft gas turbine engine with its control system is simulated digitally by a new method, called the ‘method of spare parts’. The computer program of simulation possesses the main capabilities of a real engine altitude test facility and is called a ‘digital engine altitude simulator’. The results of simulation show that the capabilities of this new method are much better than that of the ordinary ‘method of block diagram’. The method can be used for modelling and simulating any type of gas turbine engines or industrial process control systems.     

From greedy to lazy expansions and their driving dynamics

In this paper we study the ergodic properties of non-greedy series expansions to non-integer bases β > 1. It is shown that the so-called ‘lazy’ expansion is isomorphic to the ‘greedy’ expansion. Furthermore, a class of expansions to base β > 1, β , ‘in between’ the lazy and the greedy expansions are introduced and studies. It is shown that these expansions are isomorphic to expansions of the form Tx = βx + (mod 1).     

An introduction to hierarchical displacement elements and the adaptive technique

The ‘adaptive technique’ is the capability for re-analysis with a minimum of additional data preparation and user interaction and is based on a special finite element formulation which is referred to as the ‘hierarchical formulation’. This paper gives an introductory insight into this newly emerging commercial capability and concentrates on displacement-based elements.     

El Naschie’s structures in the electrodynamics of polarizable media

Using the concept of ‘combined field’, an electrodynamics of polarizable media on a fractal space–time is constructed. In this context, using the scale relativity theory, the permanent electric moment, the induced electric moment, the vacuum fluctuations, the paraelectrics, the diaelectrics, the electric Zeeman-type effect, the electric Einstein–de Haas-type effect, the electric Aharonov–Bohm-type effect, the superconductors in the ‘combined field’, the double layers as coherent structures, the magnetic Aharonov–Casher-type effect, are analyzed. Correspondence with the ε^(∞) space–time is accomplished either by admitting an anomal electric Zeeman-type effect, or through a fractal string as in the case of a superconductor in ‘combined field’, or, by phase coherence of the electron–ion pairs from the electric double layers (El Naschie’s coherence). Moreover, the electric double layer or multiple layer may be considered as two-dimensional projections of the same El Naschie’s fractal strings (higher-dimensional strings in ε^(∞) space–time).     

Soliton chaos and lengthscale competition in nonlinear dynamics

Perturbing soliton-bearing completely integrable dynamics can give rise to rich and fascinating behaviour. If the perturbation introduces a lengthscale which is large compared to the spatial extent of the solitons present in the system, the solitons move like particles in an effective potential. Taking into account two-soliton interaction can result in chaotic behaviour called ‘soliton chaos’. In the opposite limit of a small-lengthscale perturbation the solitons acquire a dressing which effectively shields them from the perturbation. If the resulting ‘dressed solitons’ are subject to an additional long-wavelength perturbation they move like renormalised particles. Furthermore they can scatter nearly elastically. If the perturbation contains lengthscales which are comparable to one of the soliton's typical lengthscales then lengthscale competition can occur. Neither the particle approximation nor the dressed-particle approximation for the soliton is valid and complicated spatio-temporal behaviour is observed. We illustrate this scenario by means of the perturbed nonlinear Schrödinger equation. The perturbed sine-Gordon equation and the Ablowitz-Ladik equation are also discussed.     

Topology optimization of periodic Mindlin plates

Periodic structures exhibit unique dynamic characteristics that make them act as tunable mechanical filters for wave propagation. As a result, waves can propagate along the periodic structures only within specific frequency bands called the ‘pass bands’ and wave propagation is completely blocked within other frequency bands called the ‘stop bands’ or ‘band gaps’. The spectral width of these bands can be optimized using topology optimization. In this paper, topology optimization is used to maximize the fundamental natural frequency of Mindlin plates while enforcing periodicity. A finite element model for Mindlin plates is presented and used along with an optimization algorithm that accounts for the periodicity constraint in order to determine the optimal topologies of plates with various periodic configurations. The obtained results demonstrate the effectiveness of the proposed design optimization approach in generating periodic plates with optimal natural frequency and wide stop bands. The presented approach can be invaluable design tool for many structures in order to control the wave propagation in an attempt to stop/confine the propagation of undesirable disturbances.     

Some peculiar boundary phenomena for extremes of rth nearest neighbor links

Let D_n,r denote the largest rth nearest neighbor link for n points drawn independently and uniformly from the unit d-cube C_d. We show that according as r < d or r>d, the limiting behavior of D_n,r, as n → ∞, is determined by the two-dimensional ‘faces’ respectively one-dimensional ‘edges’ of the boundary of C_d. If d = r, a ‘balance’ between faces and edges occurs. In case of a d-dimensional sphere (instead of a cube) the boundary dominates the asymptotic behavior of D_n,r if d 3 or if d = 2, r 3.     

Enhancing the generality of fuzzy relational models for control

A promising area of research in fuzzy control is the model-based fuzzy controller. At the heart of this approach is a fuzzy relational model of the process to be controlled. Since this model is identified directly from process input-output data it is likely that ‘holes’ will be present in the identified relational model. These holes are real problems when the model is incorporated into a model-based controller since the model will be unable to make any predictions whatsoever if the system drifts into an unknown region. The present work deals with the completeness of the fuzzy relational model which forms the core of the controller. This work proposes a scheme of post-processing to ‘fiil in’ the fuzzy relational model once it has been built and thereby improve its applicability for on-line control. A comparative study of the post-processed model and conventional relational model is presented for Box-Jenkins data identification system and a real-time, highly non-linear application of pH control identification.     

Prime power divisors of binomial coefficients

It is known that for sufficiently large n and m and any r the binomial coefficient (ⁿ_m) which is close to the middle coefficient is divisible by p^r where p is a ‘large’ prime. We prove the exact divisibility of (ⁿ_m) by p^r for p> c(n). The lower bound is essentially the best possible. We also prove some other results on divisibility of binomial coefficients.     

What is sustainable development? An attempt to interpret it as a soliton-like phenomenon

An attempt is made to establish a relation between the question of what ‘sustainable development’ means and the non-linear theory of shock waves. Despite the presence of dispersive, i.e. entropy-producing, forces a soliton-like, isentropic, transport of a wilfully desired distribution in a field of traded commodities is possible. Starting with the classical Korteweg de Vries (KdV) equation, two other examples, a sigmoidal and a Gaussian soliton in a diffusional environment, are analyzed in detail as a guide-line of how a ‘sustainable’ transport of an economically defined creation can be carried through time.     

A unified approach for price directive decomposition procedures in integer programming

This paper shows how to generalize the Dantzig-Wolfe decomposition principle to integer programming. It does this in a unified way, regardless of the choice between the two main solution methods: ‘Branch and bound’ or ‘cutting plane’. In both instances the authority at the central level issues price directives in the form of a polyhedral, concave price function, where the purpose is to charge the sublevels for the use of central resources including a penalty for any attempts to violate the integrality of the result. The sublevels then respond with their optimal activities given this price function. Finite convergence of the procedure is established.     

Multiparametric sensitivity analysis of the constraint matrix in linear-plus-linear fractional programming problem

In this paper, we study multiparametric sensitivity analysis for programming problems with linear-plus-linear fractional objective function using the concept of maximum volume in the tolerance region. We construct critical regions for simultaneous and independent perturbations of one row or one column of the constraint matrix in the given problem. Necessary and sufficient conditions are given to classify perturbation parameters as ‘focal’ and ‘nonfocal’. Nonfocal parameters can have unlimited variations, because of their low sensitivity in practice, these parameters can be deleted from the analysis. For focal parameters, a maximum volume tolerance region is characterized. Theoretical results are illustrated with the help of a numerical example.     

An integrative approach to 2D-macromodels of growth, price and inventory dynamics

This paper investigates two variants of a Keynesian model of monetary growth with sluggish price and quantity adjustments. The first model integrates the real growth dynamics of Rose's employment cycle, an inflationary dynamics of the Cagan type and Metzler's inventory dynamics. This model is based on intrinsic nonlinearities solely and it implies for the private sector six laws of motion, two for each of the subdynamics stated above. It is shown that the integrated model does not at all preserve the insights obtained from the three prototype subdynamics. Since this model can give rise to global instabilities even for moderate adjustment speeds of prices and quantities, a variant of this model is then introduced which exhibits one fundamental ‘nonintrinsic’ nonlinearity in its wage adjustment mechanism. This nonlinearity makes the considered 6D-dynamics at the same time extremely ‘viable’ and complex, in particular for a high adjustment speed of nominal wages.     

Some nilpotent and locally nilpotent matrix groups

Say a division ring D is special if for every finite subset X of D there is a homomorphism of the subring of D generated by X into a division ring of finite Schur index a power of its positive characteristic. (D is not assumed to have positive characteristic.) We make a detailed study of nilpotent and locally nilpotent matrix groups over special division rings.

This has been done previously for a number of ‘special’ division rings arising from group algebras and Lie algebras, particularly by A.I. Lichtman. The present paper therefore presents single proofs of all these results. It also covers many division rings not considered before and produces some new results for those that have been considered before.

In view of the definition of ‘special’ it is not surprising that the proofs depend on a detailed analysis of the finite-dimensional case.     

Distributions in fuzzy logic

Probability distributions associated with several ‘ply’-operators are discussed. These exact distributions are compared with relevant Gaussian approximations.     

Betti numbers of perfect homogeneous ideals

In this paper we study the graded minimal free resolution of the ideal, I, of any arithmetically Cohen-Macaulay projective variety. First we determine the range of the shifts (twisting numbers) that can possibly occur in the resolution, in terms of the Hilbert function of I. Then we find conditions under which some of the twisting numbers do not occur. Finally, in some ‘good’ cases, all the Betti numbers are (recursively) computed, in terms of the Hilbert function of I or that of Extⁿ_R(R/I,R), where R is a polynomial ring over a field and n is the height of I in R.