Related papers: Some formulas for numbers of line segments and lin…

Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$…

We search for triangular numbers that are multiples of other triangular numbers. It is found that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are triangular numbers and…

Let $p(m)$ (respectively, $q(m)$) be the maximum number $k$ such that any tree with $m$ edges can be transformed by contracting edges (respectively, by removing vertices) into a caterpillar with $k$ edges. We derive closed-form expressions…

Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in…

In general graph theory, the only relationship between vertices are expressed via the edges. When the vertices are embedded in an Euclidean space, the geometric relationships between vertices and edges can be interesting objects of study.…

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the…

In this paper we apply a formula of the very-well poised $_{2k+4}\phi_{2k+3}$ to write a $k$-tuple sum of $q$-series as a linear combination of terms wherein each term is a product of expressions of the form $\frac{1}{(qy,…

We give explicit formulas for the number of points on reductions of elliptic curves with complex multiplication by any imaginary quadratic field. We also find models for CM $\mathbf{Q}$-curves in certain cases. This generalizes earlier…

We use new bounds of double exponential sums with ratios of integers from prescribed intervals to get an asymptotic formula for the number of solutions to congruences $$ \sum_{j=1}^n a_j x_jy_j^{-1} \equiv a_0 \pmod p, $$ with variables…

We define a so-called square $k$-zig-zag shape as a part of the regular square grid. Considering the shape as a $k$-zig-zag digraph, we give values of its vertices according to the number of the shortest paths from a base vertex. It…

Given an elliptic curve C, we study here $N_k = #C(F_{q^k})$, the number of points of C over the finite field F_{q^k}. This sequence of numbers, as k runs over positive integers, has numerous remarkable properties of a combinatorial flavor…

Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…

We provide explicit expressions for quadrature rules on the space of $C^1$ quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention…

In this paper, we obtain the counting formulaes of convex pentagons and convex hexagons, respectively, in an $n$-triangular net by solving the corresponding recursive formulaes.

Via multilinear algebra, we formulate a criterion for connectedness in the parametric geometry of numbers in terms of pencils, which are certain algebraic varieties in the space of matrices. As a consequence, we obtain a connectedness…

We use an elementary argument to prove some finite sums involving expressions of the forms $(q)_n$ and $(a;q)_n$ along with inductive formulas for some sequences.

Let S be a set of 2n+1 points in the plane such that no three are collinear and no four are concyclic. A circle will be called point-splitting if it has 3 points of S on its circumference, n-1 points in its interior and n-1 in its exterior.…

We look at the number of solutions of an equation of the form f_1*f_2*...*f_k=a in a finite field, where each f_i is a multilinear polynomial. We use two methods to construct a solution of this problem for the cases a=0, a<>0, and we…

Consider a plane graph G, drawn with straight lines. For every pair a,b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio…

Let A be a quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. Let I_i, i=0,...,m, be two-sided ideals of A, \GL_n(A,I_i) the principal congruence subgroup of level I_i in GL_n(A) and E_n(A,I_i) be the…