In this note, we revisit the classical question of which integers can be represented as norms of algebraic integers in a given Euclidean quadratic field. We present a proof based on basic properties of finite groups.
H & aring;kon Kolderup is associate professor at the Western Norway University of Applied Sciences. He obtained his Ph.D. from the University of Oslo, where he specialized in motivic homotopy theory. He has broad interests within mathematics, ranging from number theory, homotopy theory, and algebraic geometry to mathematics education and the interaction between mathematics and other disciplines.
In 1974, Stoka solved Buffon's needle problem in ℝ^d, d ≥ 2, i.e. he found a closed form solution for the probability that a line segment ("needle") with length ℓ intersects a grid of parallel hyperplanes with mutual distance a≥ℓ. For the Laplace needle problem in ℝ^d, where there are d families of parallel hyperplanes with distances a_1,…,a_d fulfilling min(a_1,…,a_d)≥ℓ, and normal vectors in the direction of the coordinate axes x_1,…,x_d, he was only able to give a closed solution for the case that the needle intersects hyperplanes of all families simultaneously. In the present paper, we calculate the probabilities p_d(i) of exactly i, 0≤ i≤ d, intersection points between the needle and the hyperrectangular grid formed by the d families, and conclude the expected value and the variance for the number of intersection points. Furthermore, we present a simulation program and some numerical results.
A mathematical donut is a rectangle of integral side length with a smaller rectangle (called the hole of the donut), also of integral side length, strictly inside it and with sides of the rectangles parallel to each other, where the area of the larger rectangle is twice that of the smaller. Necessary and sufficient conditions are determined for when the hole of the donut can be rotated 90^∘ and a donut still exists, and a complete classification of all square or square-holed donuts is given, with the square donut classification being intimately related to Pythagorean triples.
In this note, we explore the connections between the confluent Vandermonde matrix over an arbitrary field and several mathematical topics, including interpolation polynomials, Hasse derivatives, LU factorization, companion matrices and their Jordan forms, and the partial fraction decomposition. Using a unified approach based on polynomial evaluations and derivative computations at selected points, we provide accessible proofs that not only clarify key results but also offer insights for both experienced researchers and those new to the subject.
It is well known that the power sums 1(k-1)+2(k-1)+& ctdot;+n(k-1) are polynomials p(k)(n) of degree k in n. We provide an elementary proof of the fact that the polynomial p(k)(x) is symmetric with respect to the line x=-1/2 or the point (-1/2,0), depending on whether k is even or odd, respectively.
Suppose 2n people participate in a sports tournament that consists of multiple rounds. In each round, two teams of n people are formed to play against each other. We require that every two players play at least once in opposing teams and once in the same team. For this variant of the round-robin scheduling problem, an explicit formula for the minimum number of rounds needed to satisfy both conditions has recently been published. In this short note, an alternative and short proof of this is given.
The sides of a hexagon whose diagonals intersect at a point define two conic sections: one through the intersection points of sides that are neither adjacent nor opposite, and a second that all sides touch. If the corners of the hexagon also lie on a conic section, then you have an ensemble of three conic sections whose centers - provided they are not parabolas - lie on a common straight line. If only one conic section is a parabola, then the line connecting the centers of the other two is a minor axis of the parabola. This note provides a proof of this statement.
Liliana Gheorghe, originally from Bucharest, Romania, completed her mathematical degrees there, including a PhD in harmonic analysis at the Institute of Mathematics of the Romanian Academy under the guidance of Prof. Nicolae Popa. Thanks to a stimulating environment and gifted teachers, she developed an early yet lasting interest in classical geometry during her junior high school years. Since the late 1990s, she has made Recife, Brazil, her home and has been a professor at UFPE. Ronaldo Garcia is a professor at the Federal University of Goi & aacute;s (UFG), Brazil. He holds a BSc in electrical engineering and received his doctorate in mathematics from IMPA, Rio de Janeiro, Brazil, under Jorge Sotomayor (in memoriam). His research interests include metric and differential geometry, differential equations, and dynamical systems, in particular the qualitative theory of principal lines and others geometric singular foliations of surfaces and hypersurfaces of Euclidean spaces.
