Traditional patterns in Pyrgi of Chios: Mathematics and Community

Ethnomathematical research has revealed interesting artifacts in several cultures all around the world. Although the majority of them come from Africa, some interesting ones exist in Western cultures too. Xysta of Pyrgi are a designing tradition that concerns the construction of mainly geometrical patterns on building fa?ades by scratching plaster. The history and the culture of the community, the way that this tradition is connected with them, as well as the informal mathematical ideas that are incorporated in this tradition are some of the issues that are explored here. H εθνoμαθηματική έρευνα έχει φέρει στην επιφάνεια ενδιαφέρoντα τεχνoυργήματα απó óλo τoν κóσμo Παρóτι η πλειoνóτητά τoυς πρoέρχεται απó την Aφρική ενδιαφέρoντα σχέδια υπάρχoυν και σε δυτικές κoυλτoύρες. Tα ξυστά στo Πυργί της Xίoυ απoτελoύν μια σχεδιαστική παράδoση πoυ αφoρά στην κατασκευή γεωμετρικών, κυρίως, σχεδίων στo σoβά πoυ βρίσκεται στην πρóσoψη των σπιτιών. Σ’ αυτή την εργασία εξερευνάται κυρίως η ιστoρία και η κoυλτoύρα της κoινóτητας τoυ Πυργιoύ, η σύνδεσή τoυς με την παράδoση των ξυστών καθώς και oι άτυπες μαθηματικές ιδέες πoυ είναι ενσωματωμένες σ’ αυτή την παράδoση.     

Self-insurance of investor under repeating catastrophic risks

A decision-making problem of investment into a profitable object in a catastrophic risk area is considered. By a catastrophic risk is meant the probability of severe yet unlikely losses. As a risk hedging mechanism, an insurance fund is considered that is replenished by a part of profit and is used for object renewal. It is shown that methods of insurance mathematics can be used to assess the risk to lose the object. For the plant loss probability as a function of the insurance reserve, integral equations are derived. They can be solved by successive approximations. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 74–83, May–June 2007.     

On Precision in Architecture

Costantino Caciagli discusses the relevance of standards in precision in the surveying of an existing monument and the design process for a new project. In architecture, the term precision, in the sense of “respect for order and exactness”, says everything and nothing.     

Necessary and sufficient conditions of existence and uniqueness of solutions to integral equations of actuarial mathematics

Necessary and sufficient conditions of existence and uniqueness of solutions of integral equations are established for ruin probability as a function of the initial capital of an insurance company. Several sufficient conditions and correctness conditions of the problem of finding ruin probability are also established. A general method of successive approximations for solution of this problem is substantiated. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 157–164, September–October 2006.     

The Complexity of Finding SUBSEQ(<Emphasis Type="Italic">A</Emphasis>)

Higman showed that if A is any language then SUBSEQ(A) is regular. His proof was nonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine M _e , and outputs a DFA for SUBSEQ(L(M _e )), then ∅″≤_T f (f is Σ ₂-hard). We also study the complexity of going from A to SUBSEQ(A) for several representations of A and SUBSEQ(A).     

The Story of Science

The Story of Science is a book series by Joy Hakim of which the first three volumes are now available – Aristotle Leads the Way (2004), Newton at the Center (2005) and Einstein Adds a New Dimension (2007) – and three more are planned. The books represent an on-going project aimed at teaching science in a way that is both specific and interdisciplinary. Kim Williams discussed the project and its goals with author Joy Hakim and art scholar Sabine Russ, who chose the illustrations for the books.     

The method of successive approximation applied to find the probability for an insurance company with random premiums

A risk process that describes the evolution of the capital of an insurance company is analyzed, random premiums and claims being available. Integral equations of nonbankruptcy probability as a function of the initial capital are derived. Necessary and sufficient conditions for the existence and uniqueness of solutions of these integral equations, and convergence conditions for the method of successive approximation for finding their solutions are established __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 112–127, January–February 2006.     

Non-partitionable point sets

Let S be a finite set of n points in d-dimensional space. S is α(n)-partitionable if there exists a set of d mutually intersecting hyperplanes that divide d-space into 2^d open regions, no 2^d ? 1 of which together contain more than α(n) points of S. Willard (1982) has shown that every set in 2-space is $\text{3}\text{4}$n-partitionable. Yao (1983) has shown that every set in 3-space is $\text{23}\text{24}$n-partitionable. It is shown here that there exist sets S of arbitrary cardinality in d-space, d ? 5, for which d² + 1 open regions together contain at least n ? d² points of S, in any partition by d intersecting hyperplanes. Further, at least 2^d ? d² ? 1 open regions contain no points of S. This implies that the powerful balanced quad and oct-trees introduced by the above authors may not be generalized to balanced 2^d-trees in dimension at least 5.     

A tool for model-checking Markov chains

Markov chains are widely used in the context of the performance and reliability modeling of various systems. Model checking of such chains with respect to a given (branching) temporal logic formula has been proposed for both discrete [34, 10] and continuous time settings [7, 12]. In this paper, we describe a prototype model checker for discrete and continuous-time Markov chains, the Erlangen–Twente Markov Chain Checker E⊢MC², where properties are expressed in appropriate extensions of CTL. We illustrate the general benefits of this approach and discuss the structure of the tool. Furthermore, we report on successful applications of the tool to some examples, highlighting lessons learned during the development and application of E⊢MC². Published online: 19 November 2002 Correspondence to: Holger Hermanns     

WORD WAYS,A JOURNAL WORTH GOING YOUR WAY

Abstract

For eight years, the author has taught a course in cryptology for undergraduate non-mathematics majors [15 Charles , F. Rocca , Jr. 2005 . “Cryptology in general education,” Cryptologia , XXIX ( 4 ): 337 – 342.  [Google Scholar]]. However, has his approach to the course been appropriate or has he, along with many others, been doing the history of cryptology an injustice? In this article, the author examines how some textbook authors and mathematicians have approached the connection between math and cryptology. He will look at why people make the connections they do and how they can do better.     

Automatic Construction and Verification of Isotopy Invariants

We extend our previous study of the automatic construction of isomorphic classification theorems for algebraic domains by considering the isotopy equivalence relation. Isotopism is an important generalisation of isomorphism, and is studied by mathematicians in domains such as loop theory. This extension was not straightforward, and we had to solve two major technical problems, namely, generating and verifying isotopy invariants. Concentrating on the domain of loop theory, we have developed three novel techniques for generating isotopic invariants, by using the notion of universal identities and by using constructions based on subblocks. In addition, given the complexity of the theorems that verify that a conjunction of the invariants form an isotopy class, we have developed ways of simplifying the problem of proving these theorems. Our techniques employ an interplay of computer algebra, model generation, theorem proving, and satisfiability-solving methods. To demonstrate the power of the approach, we generate isotopic classification theorems for loops of size 6 and 7, which extend the previously known enumeration results. This work was previously beyond the capabilities of automated reasoning techniques. The author’s work was supported by EPSRC MathFIT grant GR/S31099.     

Evaluation of a web based mathematics teaching material on the subject of functions

The aim of the study is to develop a web-based mathematics teaching (WBMT) material and to evaluate the effectiveness of the WBMT material for 9th grade students learning the concept of mathematical function. Firstly, a WBMT material was designed and piloted. As a result of this pilot study the site was revised as a final form. The study was conducted during the fall term of 2004–2005 academic years and was carried out in two different classes taught by the same teacher. Through mixed methods study the qualitative and quantitative data were collected from the sample, consisting of eighteen teachers and eighty 9th grade students. One of these teachers taught the control and experimental groups at the high school where the main study was conducted. This examination included comparing the results of students experiencing learning with WBMT with those do not, and interpreting the teachers’ responses to the use of WBMT materials. The analysis of the data suggests positive effect of WBMT on student learning of mathematical function and on attitudes towards WBMT. However, the teachers all represented and shared some common ideas that because of the technical problems and readiness of teachers and students there would be some problems in terms of successful implementation of WBMT in schools. Nevertheless, the results provide support for the use of this WBMT material as a complement to traditional classes.     

Mathematics motivation and achievement as predictors of high school students' guessing and help‐seeking with instructional software

The study was conducted to investigate the relation of adolescent students' mathematics motivation and achievement to their appropriate help‐seeking and inappropriate guessing behaviour while using instructional software. High school students (n = 90) completed brief assessments of mathematics motivation and then worked with software for geometry instruction. Students' actions with the software were machine‐classified to identify instances of appropriate help‐seeking and inappropriate guessing. Mathematics teachers provided information about students' achievement (high, average or at risk of failing math class). Results indicated that students with low math self‐concept were most likely to engage in inappropriate guessing behaviour. Students with low math achievement were most likely to engage in appropriate help‐seeking while working with the software.     

From Pentagon to Heptagon: A Discovery on the Generation of the Regular Heptagon from the Equilateral Triangle and Pentagon

Geometer Marcus the Marinite presents a construction for the heptagon that is within an incredibly small percent deviation from the ideal. The relationship between the incircle and excircle of the regular pentagon is the key to this construction, and their ratio is 2 : ϕ. In other words, the golden section plays the critical role in the establishment of this extremely closeto-ideal heptagon construction.     

Real addition and the polynomial hierarchy

The kth alternation level of the theory of real numbers under addition and order is log-complete for the kth level of the polynomial hierarchy.     

Some Kind of Analogtivity: anti-simulation through design

This paper discusses and explores issues surrounding current approaches to the design of technological products and offers two critical design proposals for presentation and debate. Primarily driven by contemporary theoretical writings and thoughts on the subject of ‘technology’ and ‘simulation’, currently being offered by leading thinkers on these subjects and expressed as ‘technological objects’; they are the result of a critical investigation into the emerging design issues surrounding ‘interaction’ and ‘transparency’. By using ‘popular’ language of product design as a vehicle, they exist as ‘cultural offerings’ exploring an alternative future for technological products not necessarily governed by science and economics.     

Nonconstructive Computational Mathematics

We describe a nonconstructive extension to primitive recursive arithmetic, both abstractly and as implemented on the Boyer-Moore prover. Abstractly, this extension is obtained by adding the unbounded µ operator applied to primitive recursive functions; doing so, one can define the Ackermann function and prove the consistency of primitive recursive arithmetic. The implementation does not mention the µ operator explicitly but has the strength to define the µ operator through the built-in functions EVAL$ and V&C$.     

智能算法在数学建模比赛中的应用

数学建模竞赛作为一项成功的数学教学改革实践,有效的促进了学生数学实践能力的提高以及对数学思想的探索,近年来,随着计算机技术及人工智能技术的飞速发展,人工智能算法在数学建模中得到了广泛的应用,智能算法的提出为数学建模的发展起到了至关重要的推动作用.