This article is an expanded version of my talk at the Gathering for Gardner,
2012.

In this paper I shall try to sketch some typical aspects of Erich Lehmann's
contributions to statistics through his research, his teaching, his service to
the profession and his personality.

In 1693, Gottfried Whilhelm Leibniz published in the Acta Eruditorum a
geometrical proof of the fundamental theorem of the calculus. During his
notorious dispute with Isaac Newton on the development of the calculus, Leibniz
denied any indebtedness to the work of Isaac Barrow. But it is shown here, that
his geometrical proof of this theorem closely resembles Barrow's proof in
Proposition 11, Lecture 10, of his Lectiones Geometricae, published in 1670.

Using an identity arising in the known Banach probability problem on
matchboxes, we prove an unexpected congruence for odd prime $p:$ for $1\leq
k\leq \frac{p-1}{2},\enskip \sum_{i=1}^{p-2k-1}2^{i-1}\binom{k-1+i}{k}\equiv
0\pmod p.$

We find an interesting relationship between the golden ratio, the Moebius
function, the Euler totient function and the natural logarithm -- central
players in the theory of numbers. The theorem and corollaries highlight a
connection between the golden ratio and the factorization of integers that is
not obvious; and display a sort of inverse relationship between the Moebius
function and Euler totient function.

In the famous Three-Door-Game Monte cannot help to win all the time by
signaling location of the prize, using only the freedom he allowed to use. No
matter which strategies played, there is always at least one door where the
prize will not be found. However, already in the game with four doors
cooperative Monte can reveal two useless doors in sequence (leaving two doors
unrevealed), to inform Conie about location of the prize, so enabling her to
beat the only-switching strategies and win all the time.

Incorporating designs into the tiles that form tessellations presents an
interesting challenge for artists. Creating a viable MC Escher like image that
works esthetically as well as functionally requires resolving incongruencies at
a tile's edge while constrained by its shape. Escher was the most well known
practitioner in this style of mathematical visualization, but there are
significant mathematical shapes to which he never applied his artistry.

In this lecture notes we present the equations and the physics involved in
the dynamic of incompressible fluids. We present the mathematical techniques
needed in order to prove the existence and uniqueness result for the case where
we consider Burgers equation. We also explain an useful numerical method when
dealing with this kind of equations. These lecture notes were written for the
2010 JAE-Intro Summer School. This Summer School was organized by ICMAT-CSIC
and takes place in Madrid.

We give an overview about well-known basic properties of two classes of
q-Fibonacci and q-Lucas polynomials and offer a common generalization.

In this article I will address some questions about a mathematical problem
that my friend Patrizio Frederic, a researcher in statistics at the University
of Modena, proposed to me. Given some parallel line segments, is there at least
one straight line that passes through all of them? If there were many lines
that solve the problem, can I choose a "best one" among all of them? I will
fully address the first question. As for the second question, I will illustrate
it with some "experimental" examples and suggest an outline for future
explorations.

Computational science and engineering (CSE) embodies President Obama's
challenge for the future, "ours to win." For decades, CSE has been
misunderstood to require massive computers, whereas breakthroughs in CSE have
historically been the mathematical programs of computing rather than the
machines themselves. Whether scientists and engineers become inventors who make
these breakthroughs depends on circumstances and their educations. The USA
currently has the largest CSE professorate, but the data suggest this
prominence is ephemeral.

SIAM Review is examined for referee delay, citations, and paper length after
the reorganization of the journal in 1999. A single, very-highly cited article
was responsible for all the increase to the impact factor during the past
decade; the reorganization did not improve the journal overall. Some
suggestions are made for additional changes.

The Monty Hall problem is the TV game scenario where you, the contestant, are
presented with three doors, with a car hidden behind one and goats hidden
behind the other two. After you select a door, the host (Monty Hall) opens a
second door to reveal a goat. You are then invited to stay with your original
choice of door, or to switch to the remaining unopened door, and claim whatever
you find behind it. Assuming your objective is to win the car, is your best
strategy to stay or switch, or does it not matter?

This note deals with two topics of linear algebra. We give a simple and short
proof of the multiplicative property of the determinant and provide a
constructive formula for rotations. The derivation of the rotation matrix
relies on simple matrix calculations and thus can be presented in an elementary
linear algebra course. We also classify all invariant subspaces of equiangular
rotations in 4D.

This is a translation of Euler's 1773 "Variae observationes circa angulos in
progressione geometrica progredientes", E561 in the Enestr{\"o}m index.

We propose in this paper a theoretical approach of teachers' professional
development, focusing on teachers' interactions with resources, digital
resources in particular. Documents, entailing resources and schemes of
utilization of these resources, are developed throughout documentational
geneses occurring along teachers' documentation work (selecting resources,
adapting, combining, refining them). The study of teachers' documentation
systems permits to seize the changes brought by digital resources; it also
constitutes a way to embrace teachers' professional change.

The aim of this short note is to realize that the main reason for
non-mechanistic explanation of Newton's gravitational attraction, is explicitly
encapsulated in his famous General Scholium of the second Edition of Principia
Mathematica (1713).

The aim of this paper is twofold. The first is to introduce several
polynomials of one variable as well as two variables defined on the positive
integers with values as congruent numbers. The second is to present a
connection between Pythagorean triples and Pell's equations of the types
$x^2-dy^2=\pm{1}$ which give rise to new congruent numbers n with arbitrarily
many prime factors.

Peano defined 'differentiability' of functions and 'lower tangent cones' in
1887, and 'upper tangent cones' in 1903, but uses the latter concept already in
1887 without giving a formal definition. Both cones were defined for arbitrary
sets, as certain limits of appropriate homothetic relations. Around 1930 Severi
and Guareschi, in a series of mutually fecundating individual papers,
characterized differentiability in terms of 'lower tangent cones' and strict
differentiability in terms of 'lower paratangent cones', a notion introduced,
independently, by Severi and Bouligand in 1928.

The degree of Doctor of Sciences, honoris causa, was conferred on Raoul Bott
by McGill University in 1987. Much of the work to make this happen was done by
Carl Herz. Some of the author's personal recollections of both professors are
included, along with some context for the awarding of this degree and ample
historical tangents. Some cultural aspects occurring in the addresses are
elaborated on, primarily, the Canadian engineer's iron ring. This paper also
reprints both the convocation address of Raoul Bott and the presentation of
Carl Herz on that occasion.

The popular view according to which Category theory provides a support for
Mathematical Structuralism is erroneous. Category-theoretic foundations of
mathematics require a different philosophy of mathematics. While structural
mathematics studies invariant forms (Awodey) categorical mathematics studies
covariant transformations which, generally, don t have any invariants. In this
paper I develop a non-structuralist interpretation of categorical mathematics
and show its consequences for history of mathematics and mathematics education.

In this paper a construction of affine exterior algebra of Grassmann, with a
special attention to the revisitation of this subject operated by Peano and his
School, is examined from a historical viewpoint. Even if the exterior algebra
over a vector space is a well known concept, the construction of an exterior
algebra over an affine space, in which points and vectors coexist, has been
neglected. This paper wants to fill this lack.

Short rigorous solutions to three mathematizations of the famous Monty Hall
problem are given: asking for an unconditional probability, a conditional
probabiliity, or for a game theoretic strategy. It is concluded which
mathematicization ought to be considered as the Only True Solution of the True
Monty Hall Problem: the little known Game Theoretical version.

In this paper a way is suggested for calculating the probability of
consecutive numbers strings within a sequence of n numbers randomly drawn
(without replacement) among the set of the first N consecutive numbers, with N
>> n.

An explicit derivation is carried out for the special case of SuperEnalotto,
nowadays the most famous lottery in Italy, with N=90 and n=6. It turns out
that, on average, one every three drawings presents one or more consecutive
numbers strings inside.

Generalist mathematics journals exhibit bias toward the branches of
mathematics by publishing articles about some subjects in quantities far
disproportionate to the production of papers in those areas within mathematics
as a whole.

We motivate and then prove a generalized pythagorean theorem for
parallelepipeds in Euclidean space.

Georgy Theodosiyovych Voronoi (1868-1908) is famous for his seminal
contributions to number theory,perhaps mostly those involving quadratic forms
and Voronoi tessellations. He was born and grew up in the town of Zhuravka in
the Ukraine, at the time part of the Russian Empire. Having studied at St.
Petersburg University under the supervision of Andrey Markov, in 1894 he became
a professor of pure mathematics at the University of Warsaw.

These notes present an approach to obtaining monoid operations which are
compatible with a given family of mappings in the sense that the mappings
become left translations in the monoid. This can be applied to various
situations such as the addition on the natural numbers and the integers as well
as the concatenation of lists.

It is well known that Al-Khay\^am, for the first time in history, formulated
a complete theory to solve third degree equations using the intersection of
geometric curves and moreover solved the fourteen types of equations using this
method. His solution for these equations was either using the intersection of
two parabolas, a parabola and a circle, a parabola and a hyperbola, a circle
and a hyperbola or two hyperbolas.

The analytical aspects of the "Trait\'e des \'equations" of Sharaf al-D\^in
al-T\^us\^i (2nd half of the XIIth century) have been underlined by R. Rashed
(1974, 1986). In the present paper, we consider again some of those aspects,
when studying the "second part" of the "Trait\'e". We find out that al-T\^us\^i
introduces certain notions, develops reasonings and uses calculating procedures
that may allow his work to be an important reference in the history of the
mathematical analysis.

Democracy functions of wavelet admissible bases are computed for weighted
Orlicz Spaces in terms of its fundamental function. In particular, we prove
that these bases are greedy if and only if the Orlicz space is a Lebesgue
space. Also, sharp embeddings for the approximation spaces are given in terms
of weighted discrete Lorentz spaces. For Lebesgue spaces the approximation
spaces are identified with weighted Besov spaces.

In order to prove irrationality of \sqrt{2} by using only decimal expansions
(and not fractions), we develop in detail a model of real numbers based on
infinite decimals and arithmetic operations with them.

We propose to address the problem of how to know students' knowledge in an
entirely new approach called ?epistemography? which is, roughly, an attempt to
describe the structure of this knowledge. We claim that what is to be known is
made of five tightly interrelated organised systems: the mathematical universe,
the system of semio-linguistic representations, the instruments, the rules of
the mathematical game, and the identifiers.

\'Etienne B\'ezout, member of the Acad\'emie Royale des Sciences, have to
study some works and books sended at the Acad\'emy. In this article, we will
look at this responsibility for Navy, before and after 1764, which is the year
of B\'ezout's nomination at the charge of Examinateur des Gardes du Pavillon et
de la Marine. Each year he must go to Brest, Rochefort and Toulon harbours to
examine the Gardes de la Marine. This give to him titles and qualifications as
expert in sailing. We will see his participation at an Academy polemic :
Blondeau versus Bouguer/Lacaille on a navigation book.

"Mathematicians, like physicists, are pushed by a strong fascination.
Research in mathematics is hard, it is intellectually painful even if it is
rewarding, and you would not do it without some strong urge." [D. Ruelle]. We
shall give some examples from our experience, when we were able to simplify
some serious mathematical models to make them understandable by children,
preserving both aesthetic and intellectual value. The latter is in particularly
measured by whether a given simplification allows setting a sufficient list of
problems feasible for school students.

This paper provides an introduction to trace diagrams at a level suitable for
advanced undergraduates. Trace diagrams are a non-traditional notation for
linear algebra. Vectors are represented by edges in a diagram, and matrices by
markings along the edges of the diagram. The notation is rigorous and permits
proofs more elegant than those written using traditional notation.

In this paper, we study some cards shuffles which are used by magicians. We
focus ourselves on the possibility to hit eventually the initial state after
several shuffles. This is a classical problem arising in discrete dynamical
systems. The computations are performed through an elementary approach, so the
paper is easily accessible.

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In a recent work [JNT \textbf{118}, 192 (2006)], Dancs and He have found new
"Euler-type" formulas for $ \ln{2} $ and $ \zeta{(2 n+1)}$, $ n $ being a
positive integer, each containing a series that apparently can not be evaluated
in closed form, providing some insight into why the odd case is more difficult
than the even, for which the Euler's formula applies, showing that $ \zeta{(2
n)} $ is a rational multiple of $\pi^{2 n}$.

We give two proofs of the identity $$\sqrt{\frac{\cos\frac{2\pi}{5}}
{\cos\frac{\pi}{5}}}+\sqrt{\frac{\cos\frac{\pi}{5}} {\cos\frac{2\pi}{5}}
}=\sqrt{5},$$ using and not using the gold ratio.

A non-traditional proof of the Gregory-Leibniz series, based on the
relationships among the zeta function, Bernoulli coefficients, and the Laurent
expansion of the cotangent is given. New series for calculating pi are
obtained.