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.

In a previous article [1] I showed that the probability of a difference of t units arising when n numbers from a uniform distribution are rounded before, as opposed to after, they are added is(1)Here n is a non-negative integer and t is a positive or negative integer with, the maximum possible error.

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.

This Article is an exposition of two very old theorems. One was first proved by A. A. Markov in 1889 [1], the other by S. N. Bernstein in 1912 [2]. Both Markov (1856–1922) and Bernstein (1880–1968) were authors of numerous pioneering works: there are many other results designated Markov’s theorem or Bernstein’s theorem (or Bernstein’s inequality) in their respective subject areas. “Markov chains” have become a standard concept in probability theory, and “Bernstein polynomials” a standard one in approximation theory.

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.

I show that a neat expression emerges for the number of spheres in each shell of an octahedron composed of close packed spheres, and that from this expression we can compute the value of the packing density for an infinite array of such spheres.

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.

Baillie’s identity is (see [1] or [2], p. 240). Its integral analogue, , is not difficult to prove (see Lemma 1, below). In this Article, we prove a generalisation of the latter result (see Theorem 1). Theorems 2 and 3 are extensions, involving in addition, powers of cosines in the integrand. Theorem 4 answers a question raised after the proof of Theorem 1, and Theorem 5 collects together the preceding results in the form of three identities between trigonometric integrals. Theorem 6 gives a further generalisation.

A Rascal triangle is a Pascal-type numeric triangle developed in 2010 by three middle-school students, Alif Anggoro, Eddy Liu and Angus Tulloch [1]. During their mathematics classes they were challenged to provide the next row for the following sequence:The expected answer was the one that matches Pascal’s triangle, i.e. 1 4 6 4 1, obtained by applying the recurrence rule of binomial coefficients South = East + West. Instead, the young students suggested that the next row should be 1 4 5 4 1. In contrast to Pascal’s triangle rule South = East + West, they produced the new row using the relation they called the diamond formulaBy applying the diamond formula (1), the students produced an original triangular sequence known as the Rascal triangle, see Table 1.

The recent discovery of the aperiodic monotiles known as the hat and the spectre has sparked a surge of interest by the general public in exactly what it means to tile the Euclidean plane in an aperiodic fashion. Explaining a mathematical breakthrough to those without background knowledge can be difficult. However, in this case, communicating the ideas has been facilitated by the highly visual nature of what has been found, much to the delight of the editors of national newspapers and magazines, and also those who love to blog on social media. News of the first tile, the hat, broke in March 2023, with a second tile, the spectre, announced two months later. A festival, hatfest, took place a couple of months after that; the two day event featured lectures for established and aspiring mathematicians, a display of associated art, and an open-to-the-public evening lecture, all at the Mathematical Institute at the University of Oxford. Figure 1 shows the two tiles at the centre of the festivities. To tile the plane aperiodically the hat requires the use of its mirror tile but the spectre, with a minor modification to its edges, is the first true monotile; so adjusted, it, like the hat, can tile the plane aperiodically and not periodically, but without the need for its congruent mirror.

According to Mitchelmore [1], generalisations are the cornerstone of school mathematics, covering various aspects like numerical generalisation in algebra, spatial generalisation in geometry and measurement, as well as logical generalisations in diverse contexts. The process of generalising lies at the heart of mathematical activity, serving as the fundamental method for constructing new knowledge [2, 3]. In this paper we will generalise an interesting geometry problem that appeared in the 1995 edition of the International Mathematical Olympiad (IMO) [4].

Let ABC be a triangle with incentre I, circumcentre O, orthocentre H, centroid G and symmedian point K. In standard notation, the triangle ABC has sides a, b, c, semiperimeter s, circumradius R and inradius r. Euler’s well-known result that the incentre I is always within the orthocentroidal disc DGH, the disc with diameter GH, is probably the first result about the location of the incentre in some disc formed by triangle centres. Investigating the location of the incentre I in other discs, in [1] we proved that the incentre is interior to the Brocard disc DOK, that is, the disc with diameter OK. The disc is named after the French military meteorologist and geometer Henri Brocard (1845-1922), known in triangle geometry for the Brocard points and the Brocard angle (see [2]).

There has long been a recognition of the role history can play in supplementing the mathematics curriculum, perhaps to make the mathematics “easier, clearer, or more interesting” [1], though not all view this positively [2]. Beyond this, history of mathematics is often taught in its own right at university.

Let ABC be a triangle with sides a, b, c, semiperimeter s, circumradius R, inradius r and area Δ. We introduce

$$Q\, = \,{(a - b)^2}\, + \,{(b - c)^2}\, + \,{(c - a)^2}$$

$$M\, = \,{(\left| {a - b} \right|\, + \left| {b - c} \right|\, + \,\left| {c - a} \right|)^2}.$$

At university I was intrigued by a historian friend and his constant quest for gobbets – short, apposite quotations that he avidly collected for possible use in future essays. It strikes me that there might be analogous nuggets of mathematics and I offer the following five short, self-contained items both for amusement and for possible classroom use. Each contains a surprising element and I have deliberately left a few loose ends to encourage further exploration.

In this Article the author hopes to present an unusual view of the Riemann zeta landscape when Re (s) > 0, to show how the prime numbers are discoverable by the altered configuration, and to show a correlation amongst the primes, the regular polygons and a set of complex numbers with real part${1 \over 2}$.

The Chebyshev polynomials: some basic facts

The Chebyshev polynomials Tn, named after the Russian mathematician P. L. Chebyshev (1821-1894), are defined by(1)

$${\text{cos }}nt = {T_n}({\text{cos }}t).$$