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Oscillation criteria for two-dimensional difference systems of first-order linear difference equations are generalized and extended to arbitrary dynamic equations on time scales. This unifies under one theory corresponding results from differential systems, and includes second-order self-adjoint differential, difference, and -difference equations within its scope. Examples are given illustrating a key theorem.
We seek to classify the sets of zero-divisors that form ideals based on their zero-divisor graphs. We offer full classification of these ideals within finite commutative rings with identity. We also provide various results concerning the realizability of a graph as a zero-divisor graph.
Let be a finite abelian group with subgroup and let denote the free abelian monoid with basis . The classical block monoid is the collection of sequences in whose elements sum to zero. The relative block monoid , defined by Halter-Koch, is the collection of all sequences in whose elements sum to an element in . We use a natural transfer homomorphism to enumerate the irreducible elements of given an enumeration of the irreducible elements of .
Let be a polynomial of degree at least two. The associated canonical height is a certain real-valued function on that returns zero precisely at preperiodic rational points of . Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on . A related conjecture claims that at nonpreperiodic rational points, is bounded below by a positive constant (depending only on ) times some kind of height of itself. In this paper, we provide support for these conjectures in the case by computing the set of small height points for several billion cubic polynomials.
The mosaic of the integer is the array of prime numbers resulting from iterating the Fundamental Theorem of Arithmetic on and on any resulting composite exponents. In this paper, we generalize several number theoretic functions to the mosaic of , first based on the primes of the mosaic, second by examining several possible definitions of a divisor in terms of mosaics. Having done so, we examine properties of these functions.
Consider the space of vertical parabolas in the plane interpreted generally to include nonvertical lines. It is proved that an injective map from a closed region bounded by one such parabola into the plane that maps vertical parabolas to other vertical parabolas must be the composition of a Laguerre transformation with a nonisotropic dilation. Here, a Laguerre transformation refers to a linear fractional or antilinear fractional transformation of the underlying dual plane.
Let be the convex kite-shaped quadrilateral with vertices , , , and , where . We wish to dissect into triangles of equal areas. What numbers of triangles are possible? Since is symmetric about the line , admits such a dissection into any even number of triangles. In this article, we prove four results describing that can be dissected into certain odd numbers of triangles.
For a connected graph of order , the detour distance between two vertices and in is the length of a longest path in . A Hamiltonian labeling of is a function such that for every two distinct vertices and of . The value of a Hamiltonian labeling of is the maximum label (functional value) assigned to a vertex of by ; while the Hamiltonian labeling number of is the minimum value of Hamiltonian labelings of . Hamiltonian labeling numbers of some well-known classes of graphs are determined. Sharp upper and lower bounds are established for the Hamiltonian labeling number of a connected graph. The corona of a graph is the graph obtained from by adding exactly one pendant edge at each vertex of . For each integer , let be the set of connected graphs for which there exists a Hamiltonian graph of order such that . It is shown that for each and that both bounds are sharp.
In this work, the concept of a slant helix is extended to Minkowski space-time. In an analogous way, we define type-3 slant helices whose trinormal lines make a constant angle with a fixed direction. Moreover, some characterizations of such curves are presented.