Cryptography, the mathematics of protecting secret or sensitive information, is a continuously evolving area of interest. A concrete example of this is the fact that worldwide spending on information security and risk management technology and services has been estimated to reach over $150 billion in 2021. Modern cryptography is actually not a single separate domain of mathematics but an advanced encryption scheme can be based on applications of results in number theory, such as the Euler–Fermat Theorem, or involve the discrete logarithm problem using either a primitive root of a large prime or an element of an elliptic curve over a prime field. In the background, probability theory, statistics, studies on computational models and finite geometries, etc. play a major role. Recent research has considered even DNA-based molecular cryptography systems.

Given a polynomial equation x3 + ax2 + bx + c = 0 of degree 3 with real coefficients, we may translate the variable by replacing x with x − to make the quadratic term vanish. We then obtain a simpler equation x3 + px + q = 0 where Therefore, in order to solve a polynomial equation of degree 3, it is sufficient to solve equations of the form x3 + px + q = 0.

Imagine that you go for a walk with your friend. You walk along a closed loop around a university campus, finishing at the point where you started. You and your friend keep the same distance throughout your walk. You walk on the ‘outside’.

Many readers are familiar with the celebrated Finsler-Hadwiger and Weitzenböck’s inequality in triangle geometry. On the other hand, Wolstenholme’s inequality is not widely known, but is equally important and in an indirect way can be used to derive these two. We will introduce it and then show its strength.

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph G is a pair G = (V, E), where V and E are the vertex set and the edge set of G, respectively. The order and size of G is the number of vertices and edges of G, respectively. The degree or valency of a vertex u in a graph G (loopless), denoted by deg (u), is the number of edges meeting at u. If, for every vertex ν in G, deg (ν) = k, we say that G is a k-regular graph. The cycle of order n is denoted by Cn and is a connected 2-regular graph. The path graph of order n is denoted by Pn and obtain by deleting an edge of Cn. A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected undirected graph without cycle. A leaf (or pendant vertex) of a tree is a vertex of the tree of degree 1. An edge of a graph is said to be pendant if one of its vertices is a pendant vertex. A complete bipartite graph is a graph G with and such that every vertex of the set (part) X is connected to every vertex of the set (part) Y. If , then this graph is denoted by Km,n. The complete bipartite graph K1,n is called the star graph which has n + 1 vertices. The distance between two vertices u and ν of G, denoted by d (u, ν), is defined as the minimum number of edges of the walks between them. The complement of graph G is denoted by and is a graph with the same vertices such that two distinct vertices of are adjacent if, and only if, they are not adjacent in G. For more information on graphs, refer to [1].

Several years ago, Professor Martin Josefsson told the first-named author, in a private communication, that his ‘characterisations of’ and ‘properties of’ series of papers that comprised tangential, extangential, bicentric, orthodiagonal, equidiagonal, and bisect-diagonal quadrilaterals, and rhombi and trapezoids, did not include parallelograms because the many characterisations of these figures are well known and are easily accessed via the net, and that nothing interesting can be added. In this Article, we present some characterisations of parallelograms that are very likely unknown and that will hopefully appeal to the readers of the Gazette. Other interesting characterisations that we have obtained and that could not find their place here are intended to form the material of another Article.

The following theorems appear in [1, pp. 62-63]: Theorem 1 If similar triangles A1B0C0, A0B1C0, A0B0C1 are erected externally on the sides of ΔA0B0C0, then the circumcircles of these three triangles have a common point, F.

The celebrated theorem of Weierstrass, dating from 1885, states that continuous functions can be uniformly approximated by polynomials on any bounded, closed interval. But just how well can we approximate by polynomials of a certain degree? Let us introduce some notation to facilitate the discussion. For an interval I (which will usually be ), denote by C (I) the space of continuous functions on I, and write for (the notation is often used). Uniform convergence of fn to f (on I) equates to the statement that . Denote by the space of polynomials of degree not more than n. This is a linear subspace of C (I) of dimension n + 1. We write

In teaching mathematics to first-year undergraduates, and thus in the appropriate calculus textbooks, the task of calculating an integral that satisfies a specific first-order or second-order recurrence relation often appears. These relations are obtained mainly by applying the method of integration by parts. Calculating such integrals is usually tedious, especially for an integer n > 2, time-consuming, and presents the possibility of making a large number of errors when computing involves multiple iterative steps. In [1], it is shown that in two cases (Theorems 2.1. and 2.3), the process of calculating integrals satisfying first-order recurrence relations can be performed quickly using easily memorised closed-form formulas for corresponding primitive functions. The question can rightly be asked whether there is a faster way to calculate other integrals of this type. In this paper, our goal is to give an affirmative answer to such a question, though without convering all situations. Since each recurrence relation is equivalent to a difference equation of the same order, the calculation of integrals mentioned above can be reduced to solving the corresponding difference equations. Since every first-order or second-order linear difference equation is solvable, it follows that for every integral which can be reduced to a first- order or second-order recurrence formula, it is possible to find corresponding primitive functions directly. Sometimes such a procedure is much faster than iterative solving of the integral. Closed-form formulas for the integrals discussed in the following sections are not unknown (see [2]). However, here our goal is to present the idea of computing indefinite integrals using difference equations. We will discuss it in more detail in Section 2. In Section 3, we discuss the application of the results obtained to calculate several improper integrals and the application of some of them in different sciences. An exciting example of such an application is the integral , which in the case n = 1 is used in the kinetic theory of gases, particularly in the Maxwell-Boltzmann distribution of gas molecules by energies (see Remark 4). Also, we compare the formulas obtained by the method of difference equations with the formulas obtained using Wolfram Alpha software (see Remark 5).

When children are first taught about multiplication of the natural numbers, it is usually presented as repeated addition. Later, learning about raising to an exponent is presented as repeated multiplication. Then the following analogy is obvious: addition is to multiplication as multiplication is to raising to an exponent. An interesting question is to ask what happens if we go in the other direction. That is, multiplication is to addition as addition is to what? In this paper, we will answer this question, and show that there are several possible operations that could be used to answer the question. Some of them will be presented in connection with algebraic semirings. A semiring is a set, together with two operations, semiring addition and semiring multiplication, satisfying certain properties. We will define several semirings, such that the semiring multiplication is ordinary addition. In each case, the semiring addition will then be an answer to the question in the title of this paper.

In order to state our primary results, we must first establish some notation. Let S-1 (x) = sin x and C-1 (x) = cos x, then for each non-negative integer n, let

these are the remainders of the Maclaurin series for sine and cosine, respectively. Note that for each and for each . It is known that

See [1] for several different proofs of the well-known fact that

the values of αn and βn for then follow rather easily using induction and integration by parts. (Details are provided in the Appendix.)

In 1879, the Oxford mathematician Charles Dodgson (1832-1898) published Euclid and his modern rivals. This book compared the work of contemporary geometers to that of the Greek master from two millennia earlier. Dodgson, best known by his pen name of ‘Lewis Carroll’, much preferred Euclid. For him, the ancients outshone the moderns.