The billiard "table" is modeled as an n-dimensional box [0, a(1)] & times; [0, a(2)] & times; & centerdot; & centerdot; & centerdot; & times; [0, a(n)] subset of R-n, with each side having real-valued lengths a(i) that are pairwise commensurable. A ball is launched from the origin in direction d = (1, 1, ... , 1). The ball is reflected if it hits the boundary of the box. It comes to a halt when reaching a corner. We show that the number of intersections of the billiard curve at any given point in the box is either 0 or a power of 2. To prove this, we use algebraic and number theoretic tools to establish a bijection between the set of intersections at a given point of the billiard curve and the set of satisfying assignments of a specific constraint satisfaction problem.
In graph theory, the center function identifies a set of vertices in a connected graph that minimizes the maximum distance from any other vertex. Through an axiomatic characterization of the center function, we identify specific axioms that characterize its behaviour on connected graphs. Universal axioms are those axioms satisfied by the center function on all connected graphs. However, on certain graphs, these universal axioms are insufficient to completely characterize the center function. Non-universal axioms, specific to particular graph classes, were introduced to address this limitation. This work focuses on finding an axiomatic characterization of the center function on graphs with diameter two, employing a combination of both universal and non-universal axioms.
In this paper, motivated by the definition of a non-commutative harmonic oscillator studied by Parmeggiani and Wakayama [Proc. Natl. Acad. Sci. USA 98 (2001), 26-30], we introduce some variants of the Euler operator with the aim of exploring the similarity between recurrence relations and differential operators. By using variants of the Euler operator, we establish a link between Eulerian polynomials of types A and B, and second-order Eulerian polynomials of types A and B as well as Narayana polynomials of types A and B.
Let G ' be a subgraph of a simple finite graph G and U be a subset of a set V. We say that a lambda-fold G '-design (U, C) of order u is embedded into a & micro;- fold G-design (V, B) of order u+w, & micro; >= lambda, if there is an injective function f : C -> B such that C is a subgraph of f (C) for every C is an element of C. The mapping f is called the embedding of (U, C) into (V, B). If w attains the minimum possible value, then f is a minimum embedding. In this paper a complete solution is given to the problem of determining a minimum embedding of a lambda-fold P3-design into a lambda-fold kite system (lambda = & micro;) for any index lambda.
A dominator coloring (DC) of a graph G is a proper coloring in which every vertex dominates all vertices of at least one color class. Let chi(d)(G) denote the minimum number of colors in a DC of G. Arumugam et al. in 2012 proved that chi(d)(G) + 1 = 8. In this paper, by determining the exact values of chi(d)(& micro;(C8)) and chi(d)(& micro;(C9)), we provide counterexamples to the above two results. Motivated by this, we completely characterize when chi(d)(& micro;(C-n)) = chi(d)(C-n) + 2, namely, if and only if n Xi 3 (mod 6) with n not equal 3, or n Xi 5 (mod 6). In addition, we determine the exact values of chi(d)(& micro;(P-n)) for all paths P-n. Our results correct earlier claims and provide a complete dethese graph classes.
