Related papers: Linear independence of trigonometric numbers
We examine implications of angles having their own dimension, in the same sense as do lengths, masses, {\it etc.} The conventional practice in scientific applications involving trigonometric or exponential functions of angles is to assume…
In this paper it was shown that all prime numbers lie on 96 half-lines. At the same time, it was shown that if a given number does not lie on any of the above half-lines, then it is a composite number. A corresponding linear mathematical…
A triangle $T^{(r)}$ in an $r$-uniform hypergraph is a set of $r+1$ edges such that $r$ of them share a common $(r-1)$-set of vertices and the last edge contains the remaining vertex from each of the first $r$ edges. Our main result is that…
Pairwise comparison matrices often exhibit inconsistency, therefore many indices have been suggested to measure their deviation from a consistent matrix. A set of axioms has been proposed recently that is required to be satisfied by any…
Using the sieve for Frobenius, we show that, in a certain sense, the roots of the L-functions of "most" algebraic curves over finite fields do not satisfy any non-trivial (linear or multiplicative) rational dependency relations. This can be…
We develop a new notion of independence suggested by Scanlon (th-independence). We prove that in a large class of theories (which includes all simple theories) this notion has many of the properties needed for an adequate geometric…
Free independence is an important tool for studying the structure of operator algebras. It is natural to ask from the model-theoretic standpoint whether free independence is captured well in first-order model theory via the notion of a…
It is proved that the rectilinear crossing number of every graph with bounded tree-width and bounded degree is linear in the number of vertices. **** This paper has been withdrawn by the author. **** The results have been superseeded by the…
Let $G$ be a simple graph and let $L(G)$ denote the \emph{line graph} of $G$. A \emph{$p$-independent} set in $G$ is a set of vertices $S \subseteq V(G)$ such that the subgraph induced by $S$ has maximum degree at most $p$. The…
Let $G=\{e^{tA}:t\in\mathbb{R}\}$ be a closed one-parameter subgroup of the general linear group of matrices of order $n$ acting on $\mathbb{R}^{n}$ by matrix-vector multiplications. We assume that all eigenvalues of $A$ are rationally…
We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain…
An $r$-graph is a triangle if there exists a positive integer $i \le \lceil r/2 \rceil$ such that it is isomorphic to the following $r$-graph with three edges: \begin{align*} \left\{\{1, \ldots, r\},~\{1, \ldots, i, r+1, \ldots,…
Graphical models are a key class of probabilistic models for studying the conditional independence structure of a set of random variables. Circular variables are special variables, characterized by periodicity, arising in several contexts…
We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to…
We classify all sets of nonzero vectors in $\mathbb{R}^3$ such that the angle formed by each pair is a rational multiple of $\pi$. The special case of four-element subsets lets us classify all tetrahedra whose dihedral angles are multiples…
An independent set in a simple graph $G$ is a set of pairwise non-adjacent vertices in $G$. The independence polynomial of $G$, denoted by $I_G$ is defined as $1 + a_1 z + a_2 z^2+\cdots+a_d z^{d}$, where $a_i$ denotes the number of…
The notion of lacunary infinite numerical sequence is introduced. It is shown that for an arbitrary linear difference operator L with coefficients belonging to the set R of infinite numerical sequences, a criterion (i.e., a necessary and…
A sequence of rational points on an algebraic planar curve is said to form an $r$-geometric progression sequence if either the abscissae or the ordinates of these points form a geometric progression sequence with ratio $r$. In this work, we…
From around 2010 onward, Elsner et al.,developed and applied a method in which the algebraic independence of n quantities x_1,...,x_n over a field is transferred to further n quantities y_1,...,y_n by means of a system of polynomials in 2n…
For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…