embedding paths and cycles in 3 ary n cubes with faulty nodes and links the k ary n cube denoted by q n k is one of the most important interconnection networks for parallel computing in this paper we consider the problem of embedding cycles and paths into faulty 3 ary n cubes let f be a set of faulty nodes and or edges and n 2 we show that when f 2n 2 there exists a cycle of any length from 3 to v q n 3 f in q n 3 f we also prove that when f 2n 3 there exists a path of any length from 2n 1 to v q n 3 f 1 between two arbitrary nodes in q n 3 f since the k ary n cube is regular of degree 2n the fault tolerant degrees 2n 2 and 2n 3 are optimal

holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space we construct holomorphic families of proper holomorphic embeddings of mathbb c k into mathbb c n 0 textless k textless n 1 so that for any two different parameters in the family no holomorphic automorphism of mathbb c n can map the image of the corresponding two embeddings onto each other as an application to the study of the group of holomorphic automorphisms of mathbb c n we derive the existence of families of holomorphic mathbb c actions on mathbb c n n ge5 so that different actions in the family are not conjugate this result is surprising in view of the long standing holomorphic linearization problem which in particular asked whether there would be more than one conjugacy class of mathbb c actions on mathbb c n with prescribed linear part at a fixed point

path embeddings in faulty 3 ary n cubes the class of k ary n cubes represents the most commonly used interconnection topology for distributed memory parallel systems in this paper we study the problem of embedding paths of various lengths into faulty 3 ary n cubes and prove that a faulty 3 ary n cube with f 2n 3 faulty vertices admits a path of every length from 2n 1 to 3 n f 1 connecting any two distinct healthy vertices this result is optimal with respect to the number of vertex faults tolerated

affine varieties with equivalent cylinders abstract a well known cancellation problem asks when for two algebraic varieties v 1 v 2 c n the isomorphism of the cylinders v 1 c and v 2 c implies the isomorphism of v 1 and v 2 in this paper we address a related problem when the equivalence under an automorphism of c n 1 of two cylinders v 1 c and v 2 c implies the equivalence of their bases v 1 and v 2 under an automorphism of c n we concentrate here on hypersurfaces and show that this problem establishes a strong connection between the cancellation conjecture of zariski and the embedding conjecture of abhyankar and sathaye we settle the problem in the affirmative for a large class of polynomials on the other hand we give examples of equivalent cylinders with inequivalent bases those cylinders however are not hypersurfaces another result of interest is that for an arbitrary field k the equivalence of two polynomials in m variables under an automorphism of k x 1 x n n m implies their equivalence under a tame automorphism of k x 1 x 2 n

hamiltonian connectivity of restricted hypercube like networks under the conditional fault model restricted hypercube like networks rhlns are an important class of interconnection networks for parallel computing systems which include most popular variants of the hypercubes such as crossed cubes mobius cubes twisted cubes and locally twisted cubes this paper deals with the fault tolerant hamiltonian connectivity of rhlns under the conditional fault model let g be an n dimensional rhln and email protected v g e g where n 7 and f 2n 10 we prove that for any two nodes u email protected v g f satisfying a simple necessary condition on neighbors of u and v there exists a hamiltonian or near hamiltonian path between u and v in g f the result extends further the fault tolerant graph embedding capability of rhlns

some new topological properties of the triangular pyramid networks abstract the triangular pyramid proposed by razavi and sarbazi azad the triangular pyramid routing and topological properties information sciences 180 2010 2328 2339 is a hierarchical structure based on triangular meshes in this paper we show that the triangular pyramid shares some nice symmetry properties of the pyramid we also determine the connectivity of the triangular pyramid and prove that it is 1 fault tolerant vertex pancyclic

accurate eigenvalues of certain sign regular matrices we present a new o n 3 algorithm for computing all eigenvalues of certain sign regular matrices to high relative accuracy in floating point arithmetic the accuracy and cost are unaffected by the conventional eigenvalue condition numbers a matrix is called sign regular when the signs of its nonzero minors depend only of the order of the minors the sign regular matrices we consider are the ones which are nonsingular and whose kth order nonzero minors are of sign 1 k k 1 2 for all k this class of matrices can also be characterized as nonsingular totally nonnegative matrices with columns in reverse order we exploit a characterization of these particular sign regular matrices as products of nonnegative bidiagonals and the reverse identity we arrange the computations in such a way that no subtractive cancellation is encountered thus guaranteeing high relative forward accuracy

non linearizable algebraic k actions on affine spaces non linearizable faithful algebraic r r actions of torus on the real affine n space r n are given for each r 1 n 3 when n 4

embedding of cycles in arrangement graphs arrangement graphs have been proposed as an attractive interconnection topology for large multiprocessor systems the authors study these graphs by proving the existence of hamiltonian cycles in any arrangement graph they also prove that an arrangement graph contains cycles of all lengths ranging between 3 and the size of the graph they show that an arrangement graph can be decomposed into node disjoint cycles in many different ways

topological properties of twisted cube twisted cube tq n is derived by changing some connections of hypercube q n according to specific rules recently many topological properties of this variation cube are studied in this paper we prove that its connectivity is n its wide diameter and fault diameter are n 2 2 furthermore we show that tq n is a pancyclic network that is cycles of an arbitrary length at least four

on embedding cycles into faulty twisted cubes the twisted cube tq n is an alternative to the popular hypercube network recently some interesting properties of tq n were investigated in this paper we study the pancycle problem on faulty twisted cubes let f e and f v be the numbers of faulty edges and faulty vertices in tq n respectively we show that with f e f v 3

linear array and ring embeddings in conditional faulty hypercubes the n dimensional hypercube qn is a graph having 2n vertices labeled from 0 to 2n 1 two vertices are connected by an edge if their binary labels differ in exactly one bit position in this paper we consider the faulty hypercube qn with n 3 that each vertex of qn is incident to at least two nonfaulty edges based on this requirement we prove that qn contains a hamiltonian path joining any two different colored vertices even if it has up to 2n 5 edge faults moreover we show that there exists a path of length 2n 2 between any two the same colored vertices in this faulty qn furthermore we also prove that the faulty qn still contains a cycle of every even length from 4 to 2n inclusive

hamiltonian properties of honeycomb meshes meshes are widely used topologies for networks on chip noc honeycomb meshes have better topological properties than meshes in order to communicate efficiently in a linear or cyclic manner it is benefited that there is a hamiltonian path or hamiltonian cycle in noc in this paper we give a necessary and sufficient condition for the existence of hamiltonian path between any pair of vertices in a honeycomb mesh and for the existence of hamiltonian path in a honeycomb mesh with one faulty vertex besides we give a systematic method to construct a hamiltonian path in honeycomb meshes

edge pancyclicity and path embeddability of bijective connection graphs an n dimensional bijective connection graph in brief bc graph is a regular graph with 2 n nodes and n2 n 1 edges the n dimensional hypercube crossed cube mobius cube etc are some examples of the n dimensional bc graphs in this paper we propose a general method to study the edge pancyclicity and path embeddability of the bc graphs first we prove that a path of length l with dist x n x y 2 y in x n where x n n 4 is a n dimensional bc graph satisfying the three specific conditions and v x n is the node set of x n furthermore by this result we can claim that x n is edge pancyclic lastly we show that these results can be applied to not only crossed cubes and mobius cubes but also other bc graphs except crossed cubes and mobius cubes so far the research on edge pancyclicity and path embeddability has been limited in some specific interconnection architectures such as crossed cubes mobius cubes

geodesic pancyclicity of twisted cubes the hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement an n dimensional twisted cube tq n is an important variation of the hypercube q n and preserves many of its desirable properties the problem of embedding linear arrays and cycles into a host graph has attracted substantial attention in recent years the geodesic cycle embedding problem is for any two distinct vertices to find all the possible lengths of cycles that include a shortest path joining them in this paper we prove that tq n is geodesic 2 pancyclic for each odd integer n 3 this result implies that tq n is edge pancyclic for each odd integer n 3 moreover tq nxk 2 is also demonstrated to be geodesic 4 pancyclic

homogeneous locally nilpotent derivations having slices and embeddings of affine spaces abstract let m and n be positive integers such that n m and let b be a polynomial ring in m n 1 variables over a field k of characteristic 0 we give a bijective correspondence between the equivalence classes of embeddings a m a n and the equivalence classes of sequences of mutually commuting locally nilpotent derivations i 1 i m on b in some form which are homogeneous with respect to a z grading on b and have slices the intersection a of the kernels of i for 1 i m inherits the z grading on b we show that a is a polynomial ring with homogeneous coordinates if and only if the corresponding embedding is rectifiable

complete path embeddings in crossed cubes crossed cubes are popular variants of hypercubes in this paper we study path embeddings between any two distinct nodes in crossed cubes we prove two important results in the n dimensional crossed cube a for any two nodes all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1 b for any two integers n 2 and l with 1