A strongly aperiodic set of tiles in the hyperbolic plane

We construct the first known example of a strongly aperiodic set of tiles in the hyperbolic plane. Such a set of tiles does admit a tiling, but admits no tiling with an infinite cyclic symmetry. This can also be regarded as a regular production system [5] that does admit bi-infinite orbits, but admits no periodic orbits.     

A Middle School Extension of Pick's Theorem to Areas of Nonsimple Closed Polygonal Regions

The author has extended Pick's theorem for simple closed polygonal regions to unions of simple closed polygonal regions–a topic that is manageable for middle grade students. From sets of data including numbers of boundary points and numbers of interior points, students are guided to discover Pick's theorem. Additionally, with the author's creation of crossing points, Pick's theorem is extended to include areas of other polygonal regions. The article is developed along lines of the 1989 Standards of the NCTM in the use of data tables which lead to the discovery of a formula.     

Personal Crunching

Abstract

Column Editor's Note: When we read about census or survey results, we tend to accept the numbers as facts—for example, that so many people in the United States are Black or White or Hispanic. We may have a fleeting worry about sampling error, or, in the case of racial and ethnic statistics, we may recall that the differential undercount distorts the numbers. But we rarely think about the effects of question wording. Suzann Evinger, a policy analyst in the Statistical Policy Office in the Office of Management and Budget (OMB), reports on research that finds differences in the racial and ethnic composition of the U.S. population as a result of differences in the questions that are asked. This research is part of OMB's review of the federal standards for racial and ethnic classification. Changes that are made in the categories and in how they are ascertained in statistical and administrative data systems could have important public policy implications.     

The nonlocal frequency behavior of nanomechanical mass sensors based on the multi-directional vibrations of a buckled nanoribbon

Considering the potential applications of the buckled structures as nanomechanical mass sensors, this paper presents an analytical method to solve the frequency shifts of the first-order transverse and longitudinal vibration modes when a mass attaches on the surface of a buckled nanoribbon based on the nonlocal elastic theory and the Lagrange's motion equation. The first-order transverse and longitudinal vibration modes of the buckled nanoribbon are introduced. A comparison between the analytic solution and the finite element analysis (FEA) is presented. Then, the effects of the compressive strain, the magnitude and location of attached mass, the nonlocal parameter, and attached mass numbers on the frequency shifts are presented. From example calculations, it is seen that the magnitude of attached mass increases the frequency shifts of the first-order modes, except the first-order transverse vibration mode at the location Z1=0. The frequency shifts for the first-order transverse and longitudinal vibration modes are different, and could be used as important principles in mass sensing. What's more, the compressive strain and the nonlocal parameter play significant roles on the sensing process of the buckled nanoribbon. The results could serve as useful references for the fabrications and applications of buckled structures based nanomechanical mass sensors.     

An extension of Sharpe's single-index model: portfolio selection with expert betas

This paper presents an approach to the portfolio selection problem based on Sharpe's single-index model and on Fuzzy Sets Theory. In this sense, expert estimations about future Betas of each financial asset have been included in the portfolio selection model denoted as ‘Expert Betas’ and modelled as trapezoidal fuzzy numbers. Value, ambiguity and fuzziness are three basic concepts involved in the model which provide enough information about fuzzy numbers representing ‘Expert Betas’ and that are simple to handle. In order to select an optimal portfolio, a Goal Programming model has been proposed including imprecise investor's aspirations concerning asset's proportions of both, high-and low-risk assets. Semantics of these goals are based on the fuzzy membership of a goal satisfaction set. To illustrate the proposed model a real portfolio selection problem is presented.     

Knight's paths towards Catalan numbers

We provide enumerating results for partial knight's paths of a given size. We prove algebraically that zigzag knight's paths of a given size ending on the x-axis are enumerated by the generalized Catalan numbers, and we give a constructive bijection with peakless Motzkin paths of a given length. After enumerating partial knight's paths of a given length, we prove that zigzag knight's paths of a given length ending on the x-axis are counted by the Catalan numbers. Finally, we give a constructive bijection with Dyck paths of a given length.     

Convergence of Rump's method for inverting arbitrarily ill-conditioned matrices

In this paper, the problem of inverting regular matrices with arbitrarily large condition number is treated in double precision defined by IEEE 754 floating point standard. In about 1984, Rump derived a method for inverting arbitrarily ill-conditioned matrices. The method requires the possibility to calculate a dot product in higher precision. Rump's method is of theoretical interest. Rump made it clear that inverting an arbitrarily ill-conditioned matrix in single or double precision does not produce meaningless numbers, but contains a lot of information in it. Rump's method uses such inverses as preconditioners. Numerical experiments exhibit that Rump's method converges rapidly for various matrices with large condition numbers. Why Rump's method is so efficient for inverting arbitrarily ill-conditioned matrices is a little mysterious. Thus, to prove its convergence is an interesting problem in numerical error analysis. In this article, a convergence theorem is presented for a variant of Rump's method.     

An improved two‐stage variance balance approach for constructing partial profile designs for discrete choice experiments

In many discrete choice experiments set up for product innovation, the number of attributes is large, which results in a substantial cognitive burden for the respondents. To reduce the cognitive burden in such cases, Green suggested in the early '70s the use of partial profiles that vary only the levels of a subset of the attributes. In this paper, we present two new methods for constructing Bayesian ‐optimal partial profile designs for estimating main‐effects models. They involve alternative generalizations of Green's approach that makes use of balanced incomplete block designs and take into account the fact that attributes may have differing numbers of levels. We refer to our methods as variance balance I and II because they vary an attribute with a larger number of levels more often than an attribute with fewer levels to stabilize the variances of the individual part‐worth estimates. The two variance balance methods differ in the way attributes with differing numbers of levels are weighted. Both methods provide statistically more efficient partial profile designs for differing numbers of attribute levels than another generalization of Green's approach that does not weight the attributes. This method is called attribute balance. We show results from an actual experiment in software development demonstrating the usefulness of our methods. Copyright © 2014 John Wiley & Sons, Ltd.     

On a new class of hyperbolic functions

This article presents the results of some new research on a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fibonacci and Lucas series. The hyperbolic Fibonacci and Lucas functions, which are the being extension of Binet's formulas for the Fibonacci and Lucas numbers in continuous domain, transform the Fibonacci numbers theory into “continuous” theory because every identity for the hyperbolic Fibonacci and Lucas functions has its discrete analogy in the framework of the Fibonacci and Lucas numbers. Taking into consideration a great role played by the hyperbolic functions in geometry and physics, (“Lobatchevski's hyperbolic geometry”, “Four-dimensional Minkowski's world”, etc.), it is possible to expect that the new theory of the hyperbolic functions will bring to new results and interpretations on mathematics, biology, physics, and cosmology. In particular, the result is vital for understanding the relation between transfinitness i.e. fractal geometry and the hyperbolic symmetrical character of the disintegration of the neural vacuum, as pointed out by El Naschie [Chaos Solitons & Fractals 17 (2003) 631].     

Teaching of real numbers by using the Archimedes–Cantor approach and computer algebra systems

Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of CAS. In the case of real numbers, the Archimedes–Cantor approach satisfies this requirement. The name of Archimedes brings back the exhaustion method. Cantor's name reminds us of the use of Cauchy rational sequences to represent real numbers. The usage of CAS with the Archimedes–Cantor approach enables the discussion of various representations of real numbers such as graphical, decimal, approximate decimal with precision estimates, and representation as points on a straight line. Exercises with numbers such as e, π, the golden ratio ?, and algebraic irrational numbers can help students better understand the real numbers. The Archimedes–Cantor approach also reveals a deep and close relationship between real numbers and continuity, in particular the continuity of functions.     

Are the hyperharmonics integral? A partial answer via the small intervals containing primes

In a recent work, the authors have used Bertrand's postulate to give a partial answer to the conjecture of Mez? which says that the hyperharmonic numbers – iterations of partial sums of harmonic numbers – are not integers. In this Note, using small intervals containing prime numbers, we prove that a great class of hyperharmonic numbers are not integers.     

Hilbert's objectivity

Detlefsen (1986) reads Hilbert's program as a sophisticated defense of instrumentalism, but Feferman (1998) has it that Hilbert's program leaves significant ontological questions unanswered. One such question is of the reference of individual number terms. Hilbert's use of admittedly “meaningless” signs for numbers and formulae appears to impair his ability to establish the reference of mathematical terms and the content of mathematical propositions ( and ). The paper traces the history and context of Hilbert's reasoning about signs, which illuminates Hilbert's account of mathematical objectivity, axiomatics, idealization, and consistency.     

Srinivasa Ramanujan (1887‐1920) A centennial tribute

This centennial tribute commemorates Ramanujan the man and Ramanujan the mathematician. A brief account of his remarkable mathematical contributions is presented to describe the great talent of Srinivasa Ramanujan. Considerable discussion is made of his new and important ideas, results and theorems in various subjects including definite integrals, hypergeometric series, Bernoulli and Euler numbers, the Riemann zeta function, distribution of prime numbers, theory of partitions and statistical mechanics, continued fractions, hypergeometric functions, Rogers‐Ramanujan identities, theory of representation of numbers as sums of squares, Ramanujan's τ‐function, q‐series, elliptic and modular functions.     

Non‐homogeneous infinitely many sites discrete‐time model with exact coalescent

Kingman's coalescent is among the most fertile concepts in mathematical population genetics. However, it only approximates the exact coalescent process associated with the Wright–Fisher model, in which the ancestry of a sample does not have to be a binary tree. The distinction between the approximate and exact coalescent becomes important when population size is small and time has to be measured in discrete units (generations). In the present paper, we explore the exact coalescent, with mutations following the infinitely many sites model. The methods used involve random point processes and generating functionals. This allows obtaining joint distributions of segregating sites in arbitrary intervals or collections of intervals, and generally in arbitrary Borel subsets of two or more chromosomes. Using this framework it is possible to find the moments of the numbers of segregating sites on pairs of chromosomes, as well as the moments of the average of the number of pairwise differences, in the form that is more general than usually. In addition, we demonstrate limit properties of the first two moments under a range of demographic scenarios, including different patterns of population growth. This latter part complements results obtained earlier for Kingman's coalescent. Finally, we discuss various applications, including the analysis of fluctuation experiments, from which mutation rates of biological cells can be inferred. Copyright © 2009 John Wiley & Sons, Ltd.     

Rosenthal’s Inequalities for Asymptotically Almost Negatively Associated Random Variables Under Upper Expectations

In this paper, the authors generalize the concept of asymptotically almost \linebreak negatively associated random variables from the classic probability space to the upper expectation space. Within the framework, the authors prove some different types of Rosenthal''s inequalities for sub-additive expectations. Finally, the authors prove a strong law of large numbers as the application of Rosenthal''s inequalities.     

On Representing Concepts in Finite Models

We present a method of transferring Tarski's technique of classifying finite order concepts by means of truth‐definitions into finite mode theory. The other considered question is the problem of representability relations on words or natural numbers in finite models. We prove that relations representable in finite models are exactly those which are of degree ≤ o′. Finally, we consider theories of sufficiently large finite models. For a given theory T we define sl(T) as the set of all sentences true in almost all finite models for T. For theories of sufficiently large models our version of Tarski's technique becomes practically the same as the classica one. We investigate also degrees of undecidability for theories of sufficiently large finite models. We prove for some special theory ST that its degree is stronger than 0′ but still not more than Σ⁰₂.     

Language Influence on Children's Cognitive Number Representation

This study examined the effect of language on children's cognitive representation of number. The sample for the study consisted of 90 Arabic speaking children with a mean age of 80 months. Children were interviewed individually and asked to represent written two‐digit numbers using base‐10 blocks. A new approach for testing the linguistic relativity hypothesis was used. The results of the study revealed that language played an important role in children's cognitive number representation. On the other hand, the results support the argument that the instructions given with the task alter children's performance on it. This study suggests that language may explain the cross‐cultural differences in mathematics achievement. However, the disadvantage that the language might place on children can be overcome with well‐planned instruction.     

Arithmetical Identities in a 2‐element Model of Tarski's System

All arithmetical identities involving 1, addition, multiplication and exponentiation will be true in a 2‐element model of Tarski's system if a certain sequence of natural numbers is not bounded. That sequence can be bounded only if the set of Fermat's prime numbers is finite.     

Recursive generation of partitionable graphs

Results of Lovász (1972) and Padberg (1974) imply that partitionable graphs contain all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture. A recursive method of generating partitionable graphs was suggested by Chvátal, Graham, Perold, and Whitesides (1979). Results of Seb? (1996) entail that Berge's conjecture holds for all the partitionable graphs obtained by this method. Here we suggest a more general recursion. Computer experiments show that it generates all the partitionable graphs with ω=3,α ≤ 9 (and we conjecture that the same will hold for bigger α, too) and many but not all for (ω,α)=(4,4) and (4,5). Here, α and ω are respectively the clique and stability numbers of a partitionable graph, that is the numbers of vertices in its maximum cliques and stable sets. All the partitionable graphs generated by our method contain a critical ω‐clique, that is an ω‐clique which intersects only 2ω?2 other ω‐cliques. This property might imply that in our class there are no counterexamples to Berge's conjecture (cf. Seb? (1996)), however this question is still open. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 259–285, 2002