On December 25th, 2021, NASA, in collaboration with the European and Canadian space agencies, achieved a remarkable feat by successfully launching the James Webb Space Telescope (JWST), marking it as the largest and most powerful telescope ever constructed.* The JWST, a monumental collaborative project spanning over two decades, involved scientists and engineers from 300 universities and 14 countries, with a total investment of ten billion dollars.

Pi (π) is a mathematical constant that represents the ratio of the circumference of a circle to its diameter. Although for calculations using the number π usually one only needs a few decimal places, mathematicians have devoted much effort to obtain as many decimal places as they have been able to calculate. For a general description of the methods used to approximate the value of π, see e.g. [1, 2].

In 1742 the Swiss mathematician Gabriel Cramer gave the following seemingly innocuous problem to the young Italian mathematician, Jean de Castillon:

The Cramer-Castillon problem: Given a circle and three points A, B and C, inscribe in the circle a triangle DEF whose sides (extended) pass through each of these points.

A couple of years ago, a non-mathematician friend of mine who takes an interest in mathematics mainly as puzzles mentioned a problem of Euler.

It is well known that the real geometric series ${\sum\limits_{n = 0}^\infty{a{k^n} = a + ak + a{k^2} + }}$ … converges to a definite sum if the common ratio, k, is such that |k| < 1, the sum being ${a}\over{1-k}$. For example, if a = 1 and ${k ={{1}\over{2}}}$ we obtain the series ${{1} + {{1}\over{2}} + ... + {{1}\over{2}^n} + ...}$, whose partial sums are ${{1}, {{3}\over{2}}, {{7}\over{4}}, ..., 2 -{{1}\over{2}^n}}$, …, and these are clearly approaching the value 2 as n becomes larger and larger. As n → ∞, ${2} - {{1}\over{2}^n}{\unicode{x2192}} \,\,2$, in agreement with the formula ${{a}\over{1-k}} {=} {{1}\over{1}-{1\over2}}=2$.

The goal of this paper is to derive a series expansion for standard normal probabilities F(b), where b ≥ 0 and the upper half of the cumulative distribution function (or CDF) of the standard normal is given by(1) See [1, Chapter 4] for information and results about the normal distribution. It is known that this integral is not elementary, and the usual approach is to use the Taylor series expansion for ez in order to evaluate the integral. We know that and therefore that (2).

In the long history of the study of the refraction of light, Willebrord Snell is credited with being the first to show that the ratio of the sines of the angles of incidence and reflection is constant for any given pair of media.

A series of consecutive odd numbers has interesting properties useful for classroom investigation [1]. This Article was inspired by the series of fractions for ${{1}\over{2}}$ using only sums of consecutive odd numbers

$$\frac{1}{3} = \frac{{1 + 3}}{{5 + 7}} = \frac{{1 + 3 + 5}}{{7 + 9 + 11}} = {\text{ }} \ldots {\text{ }} = \frac{{1 + 3 + {\text{ }} \ldots {\text{ }} + (2n - 3) + (2n - 1)}}{{(2n + 1) + (2n + 3) + {\text{ }} \ldots {\text{ }} + (2n + (2n - 1))}}.$$

In [1], the authors presented a class of sequences generated by powers of the kth-order Fibonacci recurrence relation and showed that the sequences converge. Precisely, the following result was shown:

Suppose that k ∈ Z+ and 0 < p < 1. Let x1, x2, …, xk be positive numbers. For n > k, let (1) Then {xn} converges to .

When first learning about geometric series, students often wonder why these series are termed 'geometric'. The geometry of some geometric series is readily apparent for some common series such as

$$\sum\limits_{n = 1}^\infty{\left( {\frac{1}{2}} \right)}^n = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...{\text{ }}.$$

Here we are interested in the familiar identity but only in as far as it provides the solution (1, 2, …, n) of the Diophantine equation (1) where n and the aj are positive integers. In fact, not only is (1, 2, …, n) a solution of (1), it has the remarkable property that, up to a permutation of its entries, it is the only solution of (1) in which the aj are distinct (see Section 4). Thus any other solution will necessarily contain repetitions among the aj and, for this reason, a solution must be defined to be a sequence (which allows repetitions) rather than a set. However, we shall not distinguish between two solutions whose entries are permutations of each other.

This Article is motivated by the observation that the median triangle theorem can be thought of as a dual to a theorem of Pompeiu. It treats questions that arise from the pursual of this duality, and especially of certain imperfections in this duality.

In [1], the following problem appears on page 205.

12. A right-angle tool can be used to draw a line through two points and erect a perpendicular to a given line at a given point. Use this tool to drop a perpendicular from a given point to a given line.

Computing mathematical expectation for an experiment involving a finite number of numerical outcomes is straightforward. Let X denote the random variable having n possible values x1, x2, x3,…, xn. Letting pk denote the probability of xk, the expected value of X is

$$E\,(X)\, = \,\sum\limits_{k\, = \,1}^n {{x_k}{p_k}} ,$$

which can be interpreted as a weighted average of all xk, where the weight of each outcome is represented by its probability. But caution is required when interpreting the sum if there are infinitely many outcomes and the series fails to converge absolutely.

For integers n, , let Sk (n) denote the power sum 1k + 2k +… + nk, with S0 (n) = n. As is well known, Sk (n) can be represented by a polynomial in n of degree k + 1 with constant term equal to zero, i.e. ${S_k}\,(n)\, = \,\mathop \sum \nolimits_{j\, = \,1}^{k\, + \,1} \,{a_{k,j}}{n^j}$, for certain rational coefficients ak,j. In 1973, in the popular science magazine Kvant*, appeared an article by Vladimir Abramovich [1] in which, among other things, the author derived the following minimal recurrence relation which expresses in a very compact way the connection between the power sums Sk (n) and Sk-1 (n):

(1)

where the polynomial $S_{k\, - \,1}^*\,(n)$ is constructed by replacing in Sk-1 (n) the term nj with the expression ${{{n^{j + 1}} - n} \over {j + 1}}$ for each j = 1, 2, … , k. Note that $S_{k\, - \,1}^*\,(n)$ has degree k + 1, as it should be.

In geometry, we sometimes prove that a certain configuration Q has property P. If it turns out that Q is the only configuration that has property P, then we say that P is a characterisation of Q. For example, the centre of a circle is a point in the plane which is equidistant from every point on the circumference of the circle. In fact, the centre is the only point in the plane which is equidistant from every point on the circumference of a given circle. Thus we say that the centre of a circle is characterised by this property.

The Euler numbers are the coefficients En in the expansion

(1)

Since sec x is an even function and tan x an odd one, this decouples into

(2)

and

(3)

where all three series converge if and E2n + 1 are sometimes called the secant numbers (or secant coefficients) and tangent numbers (or tangent coefficients) respectively.

The all-real-roots configurations of the univariant cubic equation allow an easy visualisation of the roots as the orthogonal projection onto the real axis of the ends of three vectors of equal magnitude, radiating symmetrically in the XY plane from the origin of the reduced form [1].

The Routh Array is known to be a very convenient way to determine the number of polynomial roots that lie in the right half of the complex plane using only the polynomial coefficients. Extensive use of this is made particularly in control system stability studies [1]. We now show that the same real-coefficient Routh algorithm can be used to obtain this information for polynomials with complex coefficients, which are prominent in areas of engineering such as electrical inductance theory [2]. The method is easy to apply and is seen to be more straightforward than existing methods for this task [e.g. 3,4]. If required, it also enables the use of appropriate root-finding methods for real-coefficient polynomials to obtain the roots of complex- coefficient polynomials.

We present some less known variations of the the Vecten configuration and give purely geometric proofs for them. It is unlikely that these variations (and even proofs?) are new, probably just well-hidden in the literature. If a reader happens to know references for the variations discussed (or other geometric proofs), please let the authors know. At [1] the reader can find a dynamic webpage on our topic.