New degenerate Bernoulli, Euler, and Genocchi polynomials

We introduce new generalizations of the Bernoulli, Euler, and Genocchi polynomials and numbers based on the Carlitz-Tsallis degenerate exponential function. Also, we present generalizations of some familiar identities and connection between these types of Bernoulli, Euler, and Genocchi polynomials. Moreover, we establish new analogues of the Euler identity for degenerate Bernoulli polynomials and numbers.

Drawing Graphs

This chapter considers the concept of planar graph and its underlying theory. It begins with a discussion of the Three Houses and Three Utilities Problem and how it can be represented by a graph. It shows that solving the Three Houses and Three Utilities Problem is equivalent to the problem of determining whether the graph that represents it can be drawn in the plane without any of its edges crossing. It then describes the Euler Identity and the Euler Polyhedron Formula, along with the proposition that every planar graph contains a vertex of degree 5 or less. It also examines Kuratowski's Theorem, introduced by the Polish topologist Kazimierz Kuratowski, and the problem of crossing number. Finally, it provides an overview of the Art Gallery Problem, Wagner's Conjecture, and the Brick-Factory Problem.

Convex polytopes and Gaussian covariances

Siegel (1993) presented a covariance identity involving normal variables that seems to flout notions of dependence. We show that it has an explanation from an unexpected quarter: convex geometry and the centroid known as the Steiner point. In the same geometric spirit, we introduce another identity for Gaussian covariances based on an Euler identity for the Steiner point.

The Euler identity

A concise way to establish a well-known identity.