Mathematical Curiosity Research Articles

Understanding information and computation: from Einstein to Web science

Contents: Foreword, Yorick Wilks Foreword. L.J. Rich Preface Introduction Dot-to-dots point the way Hitler, Turing and quantum mechanics A different perspective on numbers, straight lines and other such mathematical curiosities Twists, turns and nature's preference for curves Curves of curves To process or not? Information and computation as a field Why are conic sections important? The gifts that Newton gave, Turing opened and which no chapter one has really appreciated yet Einstein's torch bearers Special relativity General relativity Beyond the fourth dimension Time to reformulate with a little help from information retrieval research Supporting evidence Where does this get us? References Index.

Coagulation Makes Waves

Human life depends on the fact that blood remains fluid while it is in the vessels, but clots promptly when released by injury. Nature accomplishes this balancing act by means of a collection of a dozen or so proteases and cofactors, arranged in a reaction network, as described formally in Panteleev et al. (1xTask-oriented modular decomposition of biological networks: trigger mechanism in blood coagulation. Panteleev, M.A., Balandina, A.N...., and Ataullakhanov, F.I. Biophys. J. 2010; 98: 1751–1761Abstract | Full Text | Full Text PDF | PubMed | Scopus (19)See all References1). The proteases normally circulate in the form of inactive pro-enzymes that can be activated by proteolytic cleavage. Once activated, they cleave and activate other proteases, forming a cascade that powerfully amplifies the initial signal. The cascade begins with so-called “tissue factors,” substances that are exposed to blood when tissues are injured. The end of the cascade is activated factor II (IIa, aka thrombin), which cleaves fibrinogen to form fibrin—the protein polymer that forms the physical body of the clot.Dashkevich et al. (2xThrombin activity propagates in space during blood coagulation as an excitation wave. Dashkevich, N.M., Ovanesov, M.V...., and Ataullakhanov, F.I. Biophys. J. 2012; 103: 2233–2240Abstract | Full Text | Full Text PDF | PubMed | Scopus (24)See all References2) in this issue took note of the fact that this highly nonlinear process contains a positive feedback loop because thrombin can also cleave factor XI, higher up in the cascade (3xFeedback activation of factor XI by thrombin in plasma results in additional formation of thrombin that protects fibrin clots from fibrinolysis. von dem Borne, P.A., Meijers, J.C., and Bouma, B.N. Blood. 1995; 86: 3035–3042PubMedSee all References3). This suggested that the coagulation system might function as an excitable medium, able to propagate autonomous waves of activation in the same way that a nerve fiber conducts an action potential. To test this hypothesis, they devised an apparatus in which a one-dimensional clotting reaction—instigated by tissue factor at x = 0—could be monitored optically.Thrombin itself is invisible, so the investigators devised a peptide that can be cleaved by thrombin, releasing a fluorophore. To work backward from the fluorescence image to the thrombin distribution, they had to reverse-engineer the reaction-diffusion equation describing this process. Despite the well-known numerical difficulties in inverting such equations, they achieved a usable thrombin image, and demonstrated that the clotting process can, in fact, propagate as an autonomous wave front at the constant—if snail-like—velocity of 20–50 μm/min.To further understand the mechanism, they modified their previous mathematical model of the coagulation cascade (1xTask-oriented modular decomposition of biological networks: trigger mechanism in blood coagulation. Panteleev, M.A., Balandina, A.N...., and Ataullakhanov, F.I. Biophys. J. 2010; 98: 1751–1761Abstract | Full Text | Full Text PDF | PubMed | Scopus (19)See all References1) by adding diffusion terms and the presence of the fluorogenic peptide. This gave a system of 24 partial and two ordinary differential equations, which they numerically integrated. The resulting simulation successfully recapitulated the experimental results, and confirmed that the positive feedback loop from thrombin to factor XI was critical for propagation.To test this conclusion, they repeated the experiment using plasma from patients with hemophilia C, a rare genetic bleeding disorder in which factor XI is absent. As predicted, a sustained, autonomous wave did not form. It has been a long-standing puzzle why hemophilia C causes serious bleeding—particularly from large wounds—when factor XI is only a relatively minor component of the clotting cascade, and one that can be bypassed by other pathways. Based on their experimental and theoretical results, the authors hypothesize that the importance of factor XI lies precisely in its ability to propagate the clotting front far from the injured tissue surface, in order to seal large wounds. They also found that propagation depends on the presence of the cofactor factor VIII, shedding further light on the pathophysiology of the more common Hemophilia A in which that factor is deficient.This work is especially notable for the integration (no pun intended) of theoretical, methodological, and experimental studies. Those of us who do modeling are often tempted simply to demonstrate that a certain mechanism is possible and leave it to others to relate the work to the real world. By intertwining experimental, theoretical, and numerical techniques, these investigators were able to go from a mathematical curiosity to an established mechanism and a proposed solution of a riddle of clinical importance. This is an exemplar of how biophysics should work.

SINGULAR SHOCKS: RETROSPECTIVE AND PROSPECTIVE

Singular shocks were first devised over 20 years ago as a tool to resolve some otherwise intractable Riemann problems for hyperbolic conservation laws. Although they appeared at first to be merely a mathematical curiosity, new applications suggest that they may have some greater significance. In this paper, I recount the story of their discovery, which owes much to Michelle Schatzmann, describe some of their old and new appearances, and suggest intriguing possible connections with change of type in conservation law systems.

Drift and the Risk‐Free Rate

It is proven, under a set of assumptions differing from the usual ones in the unboundedness of the time interval, that, in an economy in equilibrium consisting of a risk‐free cash account and an equity whose price process is a geometric Brownian motion on [0, ∞), the drift rate must be close to the risk‐free rate; if the drift rate μ and the risk‐free rate r are constants, then r = μ and the price process is the same under both empirical and risk neutral measures. Contributing in some degree perhaps to interest in this mathematical curiosity is the fact, based on empirical data taken at various times over an assortment of equities and relatively short durations, that no tests of the hypothesis of equality are rejected.

Qualities of examples in learning and teaching

In this paper, we theorise about the different kinds of relationship between examples and the classes of mathematical objects that they exemplify as they arise in mathematical activity and teaching. We ground our theorising in direct experience of creating a polynomial that fits certain constraints to develop our understanding of engagement with examples. We then relate insights about exemplification arising from this experience to a sequence of lessons. Through these cases, we indicate the variety of fluent uses of examples made by mathematicians and experienced teachers. Following Thompson’s concept of “didactic object” (Symbolizing, modeling, and tool use in mathematics education. Kluwer, Dordrecht, The Netherlands, pp 191–212, 2002), we talk about “didacticising” an example and observe that the nature of students’ engagement is important, as well as the teacher’s intentions and actions (Thompson avoids using a verb with the root “didact”. We use the verb “didacticise” but without implying any connection to particular theoretical approaches which use the same verb.). The qualities of examples depend as much on human agency, such as pedagogical intent or mathematical curiosity or what is noticed, as on their mathematical relation to generalities.

Halo clustering with nonlocal non-Gaussianity

We show how the peak-background split can be generalized to predict the effect of non-local primordial non-Gaussianity on the clustering of halos. Our approach is applicable to arbitrary primordial bispectra. We show that the scale-dependence of halo clustering predicted in the peak-background split (PBS) agrees with that of the local-biasing model on large scales. On smaller scales, k >~ 0.01 h/Mpc, the predictions diverge, a consequence of the assumption of separation of scales in the peak-background split. Even on large scales, PBS and local biasing do not generally agree on the amplitude of the effect outside of the high-peak limit. The scale dependence of the biasing - the effect that provides strong constraints to the local-model bispectrum - is far weaker for the equilateral and self-ordering-scalar-field models of non-Gaussianity. The bias scale dependence for the orthogonal and folded models is weaker than in the local model (~ 1/k), but likely still strong enough to be constraining. We show that departures from scale-invariance of the primordial power spectrum may lead to order-unity corrections, relative to predictions made assuming scale-invariance - to the non-Gaussian bias in some of these non-local models for non-Gaussianity. An Appendix shows that a non-local model can produce the local-model bispectrum, a mathematical curiosity we uncovered in the course of this investigation.

Benford's law in the natural sciences

More than 100 years ago it was predicted that the distribution of first digits of real world observations would not be uniform, but instead follow a trend where measurements with lower first digit (1,2,…) occur more frequently than those with higher first digits (…,8,9). This result has long been known but regarded largely as a mathematical curiosity and received little attention in the natural sciences. Here we show that the first digit rule is likely to be a widespread phenomenon and may provide new ways to detect anomalous signals in data. We test 15 sets of modern observations drawn from the fields of physics, astronomy, geophysics, chemistry, engineering and mathematics, and show that Benford's law holds for them all. These include geophysical observables such as the length of time between geomagnetic reversals, depths of earthquakes, models of Earth's gravity, geomagnetic and seismic structure. In addition we find it also holds for other natural science observables such as the rotation frequencies of pulsars; green‐house gas emissions, the masses of exoplanets as well as numbers of infectious diseases reported to the World Health Organization. The wide range of areas where it is manifested opens up new possibilities for exploitation. An illustration is given of how seismic energy from an earthquake can be detected from just the first digit distribution of displacement counts on a seismometer, i.e., without actually looking at the details of a seismogram at all. This led to the first ever detection of an earthquake using first digit information alone.

La comunicación en la didáctica de las Matemáticas

Un profesor de didáctica de las matemáticas tiene que ser un gran comunicador ya que el poco entusiasmo que origina esta materia tiene que contrarrestarse con la fascinación del profesor en transmitir estos conocimientos. Por esto en nuestras clases de didáctica de las matemáticas en los nuevos grados de Magisterio pretendemos que nuestros estudiantes tengan la competencia comunicativa en alto grado. No comunicamos solo de palabra sino también con los gestos, las miradas, todo nuestro cuerpo, yo diría que es imposible no comunicar. Para conseguir esta competencia comunicativa utilizamos distintas actividades a lo largo del año (magia, entretenimientos, juegos, curiosidades matemáticas,…) para que el estudiante vaya familiarizándose con el hecho de que debe conseguir la motivación de sus alumnos hacia las matemáticas.

Swinging in imaginary time

When the small angle approximation is not made, the exact solution of the simple pendulum is a Jacobian elliptic function with one real and one imaginary period. Far from being a physically meaningless mathematical curiosity, the second period represents the imaginary time the pendulum takes to swing upwards and tunnel through the potential barrier in the semi-classical WKB approximation1 in quantum mechanics. The tunneling here provides a simple illustration of similar phenomena in Yang-Mills theories describing weak and strong interactions.

Mathematical amazements and surprises: fascinating figures and noteworthy numbers

What are the odds of finding two people who share the same birth date in a room of thirty-five? Most people would guess they're pretty low. In actuality, the probability is better than 80 percent. This is just one of many entertaining examples of mathematical curiosities presented in this book. If you have been waiting for a book that will evoke the delight and intrigue that mathematics has to offer, this is the book for you. Mathematics educators Alfred S Posamentier and Ingmar Lehmann have created the perfect introduction to the wonders of mathematics for the general reader, requiring only a high school background in the subject. Among the entertaining and useful tricks they teach are shortcuts in arithmetic, such as ways to determine at a glance the exact divisors of any given number. They also demonstrate how the properties of certain numbers can lead to infinite loops. What is particularly exciting is how many correct answers turn out to be counterintuitive. Exploring all these features will instill insights into the nature of numbers, improve your ability to manipulate them, and give you an appreciation for the inherent elegance of mathematics. Posamentier and Lehmann evoke the beauty of mathematics through the visual aspect of geometry. Enjoy with them the riches of the aesthetic and surprising properties to be found in triangles, quadrilaterals, and circles. As you marvel at the many unusual relationships and novelties revealed in this ingenious and delightful presentation, you will be learning more maths than you ever thought possible - and will be relishing every moment of it!

Professor Stewart's Cabinet of Mathematical Curiosities

School maths is not the interesting part. The real fun is elsewhere. Like a magpie, Ian Stewart has collected the most enlightening, entertaining and vexing 'curiosities' of maths over the years...Now, the private collection is displayed in his cabinet. There are some hidden gems of logic, geometry and probability - like how to extract a cherry from a cocktail glass (harder than you think), a pop up dodecahedron, the real reason why you can't divide anything by zero and some tips for making money by proving the obvious. Scattered among these are keys to unlocking the mysteries of Fermat's last theorem, the Poincare Conjecture, chaos theory, and the P/NP problem for which a million dollar prize is on offer. There are beguiling secrets about familiar names like Pythagoras or prime numbers, as well as anecdotes about great mathematicians. Pull out the drawers of the Professor's cabinet and who knows what could happen...

Joint review of Professor Stewart's cabinet of mathematical curiosities by Ian Stewart and Five minute mathematics by Ehrhard Behrends and Aha gotcha!-aha insight! by Martin Gardner and Origami, eleusis, and the soma cube by Martin Gardner and Hexaexagons, probability paradoxes, and the tower of Hanoi by Martin Gardner and Group theory in the bedroom and other mathematical diversions by Brian

Joint review of Professor Stewart's cabinet of mathematical curiosities by Ian Stewart and Five minute mathematics by Ehrhard Behrends and Aha gotcha!-aha insight! by Martin Gardner and Origami, eleusis, and the soma cube by Martin Gardner and Hexaexagons, probability paradoxes, and the tower of Hanoi by Martin Gardner and Group theory in the bedroom and other mathematical diversions by Brian

Early Childhood Corner: Engage Families in Meaningful Mathematics

Young children enter school with natural curiosity and understanding of many mathematical concepts (Clements 2001). Principles and Standards for School Mathematics (NCTM 2000) highlights the necessity for caregivers to support and encourage this natural mathematical curiosity. Together, teachers and families can build on children's conceptual foundations and nurture positive emotional growth in mathematics (Kavkler et al. 2000).

Puzzles by the drawerful (review of Professor Stewart's Cabinet of Mathematical Curiosities by I. Stewart; 2008) [Book review

Open one of the 179 "drawers" in Professor Stewart's cabinet, and you might find just a one-liner (Why did the chicken cross the Mobius band?") or a seven-page essay on Ferman's last theorem. It features a variety of knots, magic hexagons, square wheels, and topology tricks in which you pretend to slice off your finger. Answers are found at the back of the book and many have URLs for further information. The puzzles require at most a bit of algebra and many should be within the grasp of an intelligent 11-year-old.

Teaching Mathematics to Elementary School Students Using a Variety of Tools

This article is based on a first-year, third-grade teacher’s attempt to improve instructional strategies for teaching mathematics in an inner city elementary school. Review of the literature suggested a variety of methods for teaching mathematical concepts to children in grades K-5. The researcher found that carefully-planned units that include group work, intervention groups, parental involvement, entertainment elements, and explicit instruction increase students’ curiosity, enjoyment, and success in learning mathematics.

Generalized Probability Functions

From the integration of nonsymmetrical hyperboles, a one‐parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. Motivated by the mathematical curiosity, we show that these generalized functions are suitable to generalize some probability density functions (pdfs). A very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one‐ and two‐tail stretched exponential functions. We obtain, as particular cases, the generalized error function, the Zipf‐Mandelbrot pdf, the generalized Gaussian and Laplace pdf. Their cumulative functions and moments were also obtained analytically.

Brane-World Models Emerging from Collisions of Plane Waves in 5D

We consider brane-world models embedded in a five-dimensional bulk spacetime with a large extra dimension and a cosmological constant. The cosmology in $5D$ possesses "wave-like" character in the sense that the metric coefficients in the bulk are assumed to have the form of plane waves propagating in the fifth dimension. We model the brane as the "plane" of collision of waves propagating in opposite directions along the extra dimension. This plane is a jump discontinuity which presents the usual ${\bf Z}_2$ symmetry of brane models. The model reproduces the {\em generalized} Friedmann equation for the evolution on the brane, regardless of the specific details in $5D$. Model solutions with spacelike extra coordinate show the usual {\em big-bang} behavior, while those with timelike extra dimension present a {\em big bounce}. This bounce is an genuine effect of a timelike extra dimension. We argue that, based on our current knowledge, models having a large timelike extra dimension cannot be dismissed as mathematical curiosities in non-physical solutions. The size of the extra dimension is small today, but it is {\em increasing} if the universe is expanding with acceleration. Also, the expansion rate of the fifth dimension can be expressed in a simple way through the four-dimensional "deceleration" and Hubble parameters as $- q H$. These predictions could have important observational implications, notably for the time variation of rest mass, electric charge and the gravitational "constant". They hold for the three $(k = 0, + 1, - 1)$ models with arbitrary cosmological constant, and are independent of the signature of the extra dimension.

Fostering Mathematical Curiosity: Highlighting the Mathematics

In “Fostering Mathematical Curiosity” (Knuth 2002), Eric Knuth discusses the idea of problem posing as a means of fostering students' mathematical curiosity. A mathematician colleague, after reading the article, commented that a significant amount of the mathematics in the discussion of the various problems and their solutions had been “left out” —and he was right. The author's primary intent in writing that article, however, was to illustrate “what it might mean to engage students in problem posing and how teachers might begin to create classroom environments that encourage, develop, and foster mathematical curiosity” (Knuth 2002, p. 126), not to discuss in detail the mathematics underlying the solutions to the problems posed.

THE MEMORIA TECHNICA CIPHER

A comprehensive analysis of the ciphers invented by Charles L. Dodgson (Lewis Carroll) and how he used them indicate that his Memoria Technica (1875), a variant of a mnemonic scheme first proposed by Richard Grey in 1730, is properly viewed as Dodgson's fifth cipher system. He used his Memoria Technica cipher as a tool in work that was never published, a projected book whose working title was “Logarithms by Lightning: A Mathematical Curiosity.” The logarithms project, a joint work with his colleague, Robert Edward Baynes, is examined in detail.

Supermassive black holes

OF ALL the legacies of Einstein's general theory of relativity, none is more fascinating than black holes. While we now almost take their existence for granted, for much of the 20th century black holes were viewed as mathematical curiosities with no counterparts in nature. Einstein himself never believed in black holes and wrote two papers in which he argued against their existence. Einstein's resistance to the idea is understandable. Like most physicists of his day, he found it hard to believe that nature could permit the formation of objects as extreme as black holes. Indeed, the gravitational fields of black holes are strong enough to prevent light from escaping, and even distort space and the flow of time around them.