Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions

Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of p-adic numbers $$\mathbb Q_p$$ Q p . Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in $$\mathbb R$$ R by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in $$\mathbb Q_p$$ Q p . We focus on the dimension two and study the quality of the simultaneous approximation to two p-adic numbers provided by p-adic MCFs, where p is an odd prime. Moreover, given algebraically dependent p-adic numbers, we see when infinitely many simultaneous approximations satisfy the same algebraic relation. This also allows to give a condition that ensures the finiteness of the p-adic Jacobi–Perron algorithm when it processes some kinds of $$\mathbb Q$$ Q -linearly dependent inputs.

Directional nature of Goodman–Kruskal gamma and some consequences: identity of Goodman–Kruskal gamma and Somers delta, and their connection to Jonckheere–Terpstra test statistic

Although usually taken as a symmetric measure, G is shown to be a directional coefficient of association. The direction in G is not related to rows or columns of the cross-table nor the identity of the variables to be a predictor or a criterion variable but, instead, to the number of categories in the scales. Under the conditions where there are no tied pairs in the dataset, G equals Somers’ D so directed that the variable with a wider scale (X) explains the response pattern in the variable with a narrower scale (g), that is, D(g│X). Hence, G = G(g│X) = D(g│X) but G ≠ D(X│g) and G ≠ D(symmetric). If there are tied pairs, the estimates by G = G(g│X) are more liberal in comparison with those by D(g│X). Algebraic relation of G and D with Jonckheere–Terpstra test statistic (JT) is derived. Because of the connection to JT, G = G(g│X) and D = D(g│X) indicate the proportion of logically ordered test-takers in the item after they are ordered by the score. It is strongly recommendable that gamma should not be used as a symmetric measure, and it should be used directionally only when willing to explain the behaviour of a variable with a narrower scale by the variable with a wider scale. This fits well with the measurement modelling settings.

Function Theories in Cayley-Dickson Algebras and Number Theory

In the recent years a lot of effort has been made to extend the theory of hyperholomorphic functions from the setting of associative Clifford algebras to non-associative Cayley-Dickson algebras, starting with the octonions.An important question is whether there appear really essentially different features in the treatment with Cayley-Dickson algebras that cannot be handled in the Clifford analysis setting. Here we give one concrete example: Cayley-Dickson algebras admit the construction of direct analogues of so-called CM-lattices, in particular, lattices that are closed under multiplication.Canonical examples are lattices with components from the algebraic number fields $$\mathbb{Q}{[\sqrt{m1}, \ldots \sqrt{mk}]}$$ Q [ m 1 , … mk ] . Note that the multiplication of two non-integer lattice paravectors does not give anymore a lattice paravector in the Clifford algebra. In this paper we exploit the tools of octonionic function theory to set up an algebraic relation between different octonionic generalized elliptic functions which give rise to octonionic elliptic curves. We present explicit formulas for the trace of the octonionic CM-division values.

Karmarkar scalar condition

In this work we present the Karmarkar condition in terms of the structure scalars obtained from the orthogonal decomposition of the Riemann tensor. This new expression becomes an algebraic relation among the physical variables, and not a differential equation between the metric coefficients. By using the Karmarkar scalar condition we implement a method to obtain all possible embedding class I static spherical solutions, provided the energy density profile is given. We also analyse the dynamic adiabatic case and show the incompatibility of the Karmarkar condition with several commonly assumed simplifications to the study of gravitational collapse. Finally, we consider the dissipative dynamic Karmarkar collapse and find a new solution family.

Ramsey-Turán Numbers for Semi-Algebraic Graphs

A semi-algebraic graph $G = (V,E)$ is a graph where the vertices are points in $\mathbb{R}^d$, and the edge set $E$ is defined by a semi-algebraic relation of constant complexity on $V$. In this note, we establish the following Ramsey-Turán theorem: for every integer $p\geq 3$, every $K_{p}$-free semi-algebraic graph on $n$ vertices with independence number $o(n)$ has at most $\frac{1}{2}\left(1 - \frac{1}{\lceil p/2\rceil-1} + o(1) \right)n^2$ edges. Here, the dependence on the complexity of the semi-algebraic relation is hidden in the $o(1)$ term. Moreover, we show that this bound is tight.

An Algebraic Relation between Consimilarity and Similarity of Quaternion Matrices and Applications

This paper, by means of complex representation of a quaternion matrix, discusses the consimilarity of quaternion matrices, and obtains a relation between consimilarity and similarity of quaternion matrices. It sets up an algebraic bridge between consimilarity and similarity, and turns the theory of consimilarity of quaternion matrices into that of ordinary similarity of complex matrices. This paper also gives algebraic methods for finding coneigenvalues and coneigenvectors of quaternion matrices by means of complex representation of a quaternion matrix.