Narrow Results

A k-container of a graph G is a set of k internally disjoint paths between two distinct vertices. A k-container of G is a k*-container if it contains all vertices of G. A graph G is k*-connected if there exists a k*-container between any two distinct vertices, and a bipartite graph G is k*-laceable if there exists a k*-container between any two vertices from different partite sets of G. A k-connected graph (respectively, bipartite graph) G is super spanning connected (respectively, laceable) if G is r*-connected (r*-laceable) for any r with 1 ≤ r ≤ k. This paper shows that the two-dimensional torus Torus(m, n), m, n ≥ 3, is super spanning connected if at least one of m and n is odd and super spanning laceable if both m and n are even. Furthermore, the super spanning connectivity and spanning laceability of tori with faulty elements have been discussed.

In this paper, we perform our algorithm developed in [Yu & Lee, 2001] to present the entire branches of quasiperiodic solutions starting from the bifurcation points in the branches of periodic solutions in an interval of parameters for the 2-mode damped, driven sine-Gordon ODE.

In this paper, we report the dynamical behaviors of a four-dimensional autonomous system, that is, the modified canonical Chua's circuit. An interesting transition of three-tori–period-3 doubling–chaos is observed when the circuit parameters are varied in the range of our choice. Furthermore, the detailed numerical studies of the system behavior with supporting PSPICE simulations and hardware experiments are also presented here.

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type $D_n$, with specific emphasis on the case of number fields and $p$-adic fields. This includes the forms associated to quadratic spaces, all of their inner and outer forms as well as the Spin groups, their simply connected covers. In particular, in this work, we handle all (simply connected) outer forms of $D_4$.

A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla–Selberg series formula, for which an alternative proof is thereby given. In addition, a new proof of the functional determinant on the torus results, which does not use the Kronecker first limit formula nor the functional equation of the non-holomorphic Eisenstein series. As a bonus, several identities involving the Dedekind eta function are obtained as well.

Let G be the product of an abelian variety and a torus defined over a number field K. The aim of this paper is detecting the dependence among some given rational points of G by studying their reductions modulo all primes of K. We show that if some simple conditions on the order of the reductions of the points are satisfied then there must be a dependency relation over the ring of K-endomorphisms of G. We generalize Larsen's result on the support problem to several points on products of abelian varieties and tori.

We explore algebraic subgroups of the Cremona group over an algebraically closed field of characteristic zero. First, we consider some class of algebraic subgroups of that we call flattenable. It contains all tori. Linearizability of the natural rational actions of flattenable subgroups on Aⁿ is intimately related to rationality of the invariant fields and, for tori, is equivalent to it. We prove stable linearizability of these actions and show the existence of nonlinearizable actions among them. This is applied to exploring maximal tori in and to proving the existence of nonlinearizable, but stably linearizable elements of infinite order in for n ≽ 5. Then we consider some subgroups of that we call the rational de Jonquières subgroups. We prove that every affine algebraic subgroup of is solvable and the group of its connected components is Abelian. We also prove that every reductive algebraic subgroup of is diagonalizable. Further, we prove that the natural rational action on Aⁿ of any unipotent algebraic subgroup of admits a rational cross-section which is an affine subspace of Aⁿ. We show that in this statement “unipotent” cannot be replaced by “connected solvable”. This is applied to proving a conjecture of A. Joseph on the existence of “rational slices” for the coadjoint representations of finite-dimensional algebraic Lie algebras 𝔤 under the assumption that the Levi decomposition of 𝔤 is a direct product. We then consider some overgroup of and prove that every torus in is linearizable. Finally, we prove the existence of an element of order 2 such that g ∉ G for every connected affine algebraic subgroup G of ; in particular, g is not stably linearizable.