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In this paper, we look at three common theorems in number theory: the Chinese remainder theorem, the multiplicative property of the Euler totient function, and a decomposition property of reduced residue systems. We use a grid of squares to give simple transparent visual proofs.
A “dead end” in the Cayley graph of a finitely generated group is an element beyond which no geodesic ray issuing from the identity can be extended. We study the so-called “strong dead-end depth” of group elements and its relationship with the set of infinite quasigeodesic rays issuing from the identity. We show that the ratio of strong depth to word length is bounded above by in every finitely generated group and that for any element in a finitely generated group , there is an infinite -quasigeodesic ray issuing from the identity and passing through . Applying the Švarc–Milnor lemma to a finitely generated group acting geometrically on a geodesically connected metric space, we obtain the result that for any two points in such a space, there is an infinite quasigeodesic ray starting at one and passing through the other with quasigeodesic constants independent of the points selected.
Generalized factorization theory for integral domains was initiated by D. D. Anderson and A. Frazier in 2011 and has received considerable attention in recent years. There has been significant progress made in studying the relation for the integers in previous undergraduate and graduate research projects. In 2013, the second author extended the general theory of factorization to commutative rings with zero-divisors. In this paper, we consider the same relation over the modular integers, . We are particularly interested in which choices of yield a ring which satisfies the various -atomicity properties. In certain circumstances, we are able to say more about these -finite factorization properties of .
We study the cocircular relative equilibria (planar central configurations) in the four-vortex problem using methods suggested by the study of cocircular central configurations in the Newtonian four-body problem in recent work of Cors and Roberts. Using mutual distance coordinates, we show that the set of four-vortex relative equilibria is a two-dimensional surface with boundary curves representing kite configurations, isosceles trapezoids, and degenerate configurations with one zero vorticity. We also show that there is a constraint on the signs of the vorticities in these configurations; either three or four of the vorticities must have the same sign, in contrast to the noncocircular cases studied by Hampton, Roberts, and Santoprete.
We say that a point in a specific rectangular array of lattice points is weakly visible from a lattice point not in the array if no point in the array other than lies on the line connecting the external point to . A necessary and sufficient condition for determining if a point in the array is weakly viewable by the external point, as well as the number of points that are weakly visible from the external point, is determined.
We study the vertex-connectivity and edge-connectivity of the zero-divisor graph associated to a finite commutative ring . We show that the edge-connectivity of always coincides with the minimum degree, and that vertex-connectivity also equals the minimum degree when is nonlocal. When is local, we provide conditions for the equality of all three parameters to hold, give examples showing that the vertex-connectivity can be much smaller than minimum degree, and prove a general lower bound on the vertex-connectivity.
An -endomorphism on a free semigroup is an endomorphism that sends every generator to a word of length . Two -endomorphisms are combinatorially equivalent if they are conjugate under an automorphism of the semigroup. In this paper, we specialize an argument of N. G. de Bruijn to produce a formula for the number of combinatorial equivalence classes of -endomorphisms on a rank- semigroup. From this formula, we derive several little-known integer sequences.
We study some combinatorial objects related to the flag manifold of Lie type . Using the moment graph of , we calculate all the curve neighborhoods for Schubert classes. We use this calculation to investigate the ordinary and quantum cohomology rings of . As an application, we obtain positive Schubert polynomials for the cohomology ring of and we find quantum Schubert polynomials which represent Schubert classes in the quantum cohomology ring of .
In 1908, Schur raised the question of the irreducibility over of polynomials of the form , where the are distinct integers and . Since then, many authors have addressed variations and generalizations of this question. In this article, we investigate the irreducibility of and , where the integers are consecutive terms of an arithmetic progression and is a nonzero integer.
In this paper, we apply the calculus of variations to solve the elastica problem. We examine a more general elastica problem in which the material under consideration need not be uniformly rigid. Using, the Euler–Lagrange equations, we derive a system of nonlinear differential equations whose solutions are given by these generalized elastica curves. We consider certain simplifying cases in which we can solve the system of differential equations. Finally, we use novel numerical techniques to approach solutions to the problem in full generality.
Let be a commutative ring. When is a subgroup of an ideal of ? We investigate this problem for the rings and . In the cases of and , our results give, for any given subgroup of these rings, a computable criterion for the problem under consideration. We also compute the probability that a randomly chosen subgroup from is an ideal.
We study the -pseudospectra of square matrices . We give a complete characterization of the -pseudospectra of matrices and describe the asymptotic behavior (as ) of for every square matrix . We also present explicit upper and lower bounds for the -pseudospectra of bidiagonal matrices, as well as for finite-rank operators.