W. Charles Holland, 75<Superscript>th</Superscript> birthday

This issue of Mathematica Slovaca is in honour of W. Charles Holland’s 75th birthday. We present here a brief account of some of his research (to date) and a couple of brief personal sketches of the man.     

Estimate of error of an approximated solution by the method of moments of an operator equation

For an equationAu = f whereA is a closed densely defined operator in a Hilbert spaceH, f εH, we estimate the deviation of its approximated solution obtained by the moment method from the exact solution. All presented theorems are of direct and inverse character. The paper refers to direct methods of mathematical physics, the development of which was promoted by Yu. D. Sokolov, the well-known Ukrainian mathematician and mechanic, a great humanitarian and righteous man. We dedicate this paper to his blessed memory.     

Die Habilitation von John von Neumann an der Friedrich-Wilhelms-Universität in Berlin: Urteile über einen ungarisch-jüdischen Mathematiker in Deutschland im Jahr 1927

The mathematician John von Neumann was born in Hungary but principally received his scientific education and socialization in the German science system. He received his Habilitation from the Friedrich-Wilhelms–Universität in Berlin in 1927, where he lectured as a Privatdozent until his emigration to the USA. This article aims at making a contribution to this early part of Neumann’s scientific biography by analyzing in detail the procedure that led to his Habilitation as well as the beginnings of Neumann’s research on functional analysis. An analysis of the relevant sources shows that in Berlin in the year 1927 Neumann was not yet regarded as the outstanding mathematical genius of the 20th century. Furthermore it will be seen that Neumann had great difficulties in developing the fundamental concepts for his path breaking work in spectral theory and only managed to do so with the support of the Berlin mathematician Erhard Schmidt.     

A dialogue on the use of arithmetic in geometry: Van Ceulen’s and Snellius’s Fundamenta Arithmetica et Geometrica

Snellius’s Fundamenta Arithmetica et Geometrica (1615) is much more than a Latin translation of Ludolph van Ceulen’s Arithmetische en Geometrische Fondamenten. Willebrord Snellius both adapted and commented on the Dutch original in his Fundamenta, and thus his Latin version can be read as a dialogue between representatives of two different approaches to mathematics in the early modern period: Snellius’s humanist approach and Van Ceulen’s practitioner’s approach. This article considers the relationship between the Dutch and Latin versions of the text and, in particular, puts some of their statements on the use of numbers in geometry under the microscope. In addition, Snellius’s use of the Fundamenta as an instrument to further his career is explained.     

On a generalization of Kaplansky's game

In this paper, games of the following general kind are studied: Two players move alternately by selecting unselected integer coordinate points in the plane. On each move, the first player selects exactly r points and the second player selects exactly one point. The first player wins if he can select p points on a line having none of his opponent's points before his opponent selects q points on a line having none of his own. If this latter eventuality occurs first, the second player wins. It is shown that if p ? c(r)q, then the second player can always win.     

Documents on the mathematical education of Edmund Külp (1800–1862), the mathematics teacher of Georg Cantor

Despite some studies by the historian Wilhelm Lorey, Edmund Külp is rather unknown today. His role in the development of mathematics and mathematics teaching in the nineteenth century, however, deserves closer attention. Having been the director of the höhere Gewerbe—und Realschule in Darmstadt, he can be counted among the founders of the Technische Hochschule Darmstadt. Moreover, he had been, still at the Realschule, the mathematics teacher of Georg Cantor. Recently detected documents concerning Külp’s mathematical formation in Brussels by A. Quetelet permit revealing insights into the evolution of Külp’s mathematical ideas and of his views on the context of mathematics in Germany. The contribution presents extracts from these documents (in French) and analyses them. Furthermore, the paper discusses possible influences exerted by Külp on Cantor.     

Asymmetric evaluation of fitness enhances spatial reciprocity in social dilemmas

Evolutionary game theory has been a powerful tool to understand the ubiquity of cooperation in many real-world systems which ranges from biological to economical and social sciences. In this paper, we propose a novel game model, which considers different fitness evaluation means for focal agent and his randomly chosen neighbor, to characterize the asymmetry of information during the game playing. When playing the game, the focal agent obtains its own fitness which includes its own payoff and the average payoff of his neighbors. However, due to the limited information for the chosen neighbor, his fitness can not be evaluated as the focal agent but just the traditional way, namely his fitness equals to his payoff. Large-scale numerical simulations indicate that this kind of asymmetric fitness computation can have substantial effects on the cooperative behaviors on the regular lattice. Interestingly, the larger the asymmetric factor α, the higher the cooperation level ρ_C. Meanwhile, introducing the asymmetric fitness evaluation can induce the cooperative clusters to become larger and larger as α increases. The phase space between b and α can be enlarged, and thereby the ranges for cooperators to survive in the sea of defectors can be widely extended. Our findings can greatly be conducive to interpret the emergence of cooperation within the population.     

The Chinese concept of Cavalieri's principle and its applications

Bonaventura Cavalieri (1598–1647) was noted for his method of indivisibles which led to the principle which bears his name. In the third century, while attempting to derive the volume of a sphere, Liu Hui applied a similar principle to determine the ratio of the volumes of a sphere and a solid circumscribing the sphere. This solid is formed by the intersection of two perpendicular cylinders circumscribing the sphere and is called mou he fang gai. Liu Hui left unresolved the problem of finding the volume of the mou he fang gai. In the fifth century Zu Geng, also applying Cavalieri's principle, solved this problem and thus derived the volume of a sphere. The influence of Zu Geng's method on later mathematicians is discussed in the latter part of the article.     

Improved and self-tuned occurrence heuristics

In this note we present three efficient variations of the occurrence heuristic, adopted by many exact string matching algorithms and first introduced in the well-known Boyer–Moore algorithm. Our first heuristic, called improved-occurrence heuristic, is a simple improvement of the rule introduced by Sunday in his Quick-Search algorithm. Our second heuristic, called worst-occurrence heuristic, achieves its speed-up by selecting the relative position which yields the largest average advancement. Finally, our third heuristic, called jumping-occurrence heuristic, uses two characters for computing the next shift. Setting the distance between these two characters optimally allows one to maximize the average advancement. The worst-occurrence and jumping-occurrence heuristics tune their parameters according to the distribution of the characters in the text. Experimental results show that the newly proposed heuristics achieve very good results on average, especially in the case of small alphabets.     

Exceptions and counterexamples: Understanding Abel's comment on Cauchy&#x27;s Theorem

It may seem odd that Abel, a protagonist of Cauchy's new rigor, spoke of “exceptions” when he criticized Cauchy's theorem on the continuity of sums of continuous functions. However, when interpreted contextually, exceptions appear as both valid and viable entities in the early 19th century. First, Abel's use of the term “exception” and the role of the exception in his binomial paper is documented and analyzed. Second, it is suggested how Abel may have acquainted himself with the exception and his use of it in a process denoted critical revision is discussed. Finally, an interpretation of Abel's exception is given that identifies it as a representative example of a more general transition in the understanding of mathematical objects that took place during the period. With this interpretation, exceptions find their place in a fundamental transition during the early 19th century from a formal approach to analysis toward a more conceptual one.     

Smale’s Fundamental Theorem of Algebra Reconsidered

In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s upper bound estimate was infinite average cost. Ours is polynomial in the Bézout number and the dimension of the input. Hence it is polynomial for any range of dimensions where the Bézout number is polynomial in the input size. In particular it is not just for the case that Smale considered but for a range of dimensions as considered by Bürgisser–Cucker, where the max of the degrees is greater than or equal to n ^1+? for some fixed ?. It is possible that Smale’s algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem.     

Selling to the “Newsvendor” with a forecast update: Analysis of a dual purchase contract

We consider a supply chain in which a manufacturer sells to a procure-to-stock retailer facing a newsvendor problem with a forecast update. Under a wholesale price contract, the retailer waits as long as she can and optimally places her order after observing the forecast update. We show that the retailer’s wait-and-decide strategy, induced by the wholesale price contract, hinders the manufacturer’s ability to (1) set the wholesale price and maximize his profit, (2) hedge against excess inventory risk, and (3) reduce his profit uncertainty. To mitigate the adverse effect of wholesale price contract, we propose the dual purchase contract, through which the manufacturer provides a discount for orders placed before the forecast update. We characterize how and when a dual purchase contract creates strict Pareto improvement over a wholesale price contract. To do so, we establish the retailer’s optimal ordering policy and the manufacturer’s optimal pricing and production policies. We show how the dual purchase contract reduces profit variability and how it can be used as a risk hedging tool for a risk averse manufacturer. Through a numerical study, we provide additional managerial insights and show, for example, that market uncertainty is a key factor that defines when the dual purchase contract provides strict Pareto improvement over the wholesale price contract.     

Richard von Mises and the “problem of two races”: A statistical satire in 1934

The article “Problem of two races,” which appeared in 1934 in French and Russian in the Matematichesky Sbornik (Moscow), authored by the noted applied mathematician and statistician Richard von Mises (1883–1953), is interpreted as having a twofold agenda. On its surface it is merely a detailed mathematical analysis of the statistical problem of comparing the distributions of a quantitative characteristic in two different classes. But on closer examination it serves two auxiliary purposes. On the one hand, von Mises, who as a Jew fled Nazi Germany in 1933, satirically attacked Nazi racial doctrines by resorting to statistical parody. On the other hand, von Mises employed his theoretical results to further his own distinctive program of objective Bayesian statistics.     

Cárnico-ICSPEA2—A metaheuristic co-evolutionary navigator for a complex co-evolutionary farming system

This paper introduces Cárnico-ICSPEA2, a metaheuristic co-evolutionary navigator designed by its end-user as an aid for the analysis and multi-objective optimisation of a beef cattle enterprise running on temperate pastures and fodder crops in Chalco, Mexico State, in the central plateau of Mexico. By combining simulation routines and a multi-objective evolutionary algorithm with a deterministic and stochastic framework, the software imitates the evolutionary behaviour of the system of interest, helping the farm manager to ‘navigate’ through his system’s dynamic phase space. The ultimate goal was to enhance the manager’s decision-making process and co-evolutionary skills, through an increased understanding of his system and the discovery of new, improved heuristics. This paper describes the numerical simulation and optimisation resulting from the application of Cárnico-ICSPEA2 to solve a specific multi-objective optimisation problem, along with implications for the management of the system of interest.     

L-stable Simpson’s 3/8 rule and Burgers’ equation

In this paper we develop an unconditionally stable third order time integration formula for the diffusion equation with Neumann boundary condition. We use a suitable arithmetic average approximation and explicit backward Euler formula and then develop a third order L-stable Simpson’s 3/8 type formula. We also observe that the arithmetic average approximation is not unique. Then L-stable Simpson’s 3/8 type formula and Hopf-Cole transformation is used to solve Burger’s equation with Dirichlet boundary condition. It is also observed that this numerical method deals efficiently in case of inconsistencies in initial and boundary conditions.     

On the linear search problem

A man in an automobile searches for another man who is located at some point of a certain road. He starts at a given point and knows in advance the probability that the second man is at any given point of the road. Since the man being sought might be in either direction from the starting point, the searcher will, in general, have to turn around many times before finding his target. How does he search so as to minimize the expected distance travelled? When can this minimum expectation actually be achieved? This paper answers the second of these questions.     

Cayley's anticipation of a generalised Cayley-Hamilton theorem

The principal result of Cayley's famouus memoir on matrices of 1858 is his contribution to what is now known as ‘the Cayley-Hamilton theorem’. We discuss this theorem and show that prior to its publication Cayley was aware of a more general theorem, a result that he left unpublished. This theorem is associated with the binary algebraic form det (μP ? λQ) analogous to the standard characteristic polynomial det (A ? λI).     

Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy

In 2007, Huang proposed the optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy, in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides its customer a permissible delay period N (with N < M). In this paper, we extend his EPQ model to complement the shortcoming of his model. In addition, we relax the dispensable assumptions of N < M and others. We then establish an appropriate EPQ model to the problem, and develop the proper theoretical results to obtain the optimal solution. Finally, a numerical example is used to illustrate the proposed model and its optimal solution.     

Note on Halin's theorem on minimally connected graphs

In this note it is shown that any finite directed graph of strong connectivity n contains either a vertex with indegree n, a vertex with outdegree n, or an edge whose removal does not decrease the connectivity. This is a directed graph counterpart of Halin's theorem on undirected graphs. It is pointed out that only a few preparations and modifications are necessary to make his proof valid for directed graphs.     

Analysis of Local Search Landscapes for <Emphasis Type="Italic">k</Emphasis>-SAT Instances

Stochastic local search is a successful technique in diverse areas of combinatorial optimisation and is predominantly applied to hard problems. When dealing with individual instances of hard problems, gathering information about specific properties of instances in a pre-processing phase is helpful for an appropriate parameter adjustment of local search-based procedures. In the present paper, we address parameter estimations in the context of landscapes induced by k-SAT instances: at first, we utilise a sampling method devised by Garnier and Kallel in 2002 for approximations of the number of local maxima in landscapes generated by individual k-SAT instances and a simple neighbourhood relation. The objective function is given by the number of satisfied clauses. The procedure provides good approximations of the actual number of local maxima, with a deviation typically around 10%. Secondly, we provide a method for obtaining upper bounds for the average number of local maxima in k-SAT instances. The method allows us to obtain the upper bound \(2^{n-O(\sqrt{n/k})}\) for the average number of local maxima, if m is in the region of 2 ^k · n/k.