rationality problem of three dimensional monomial group actions abstract let k be a field of characteristic not two and k x y z the rational function field over k with three variables x y z let g be a finite group acting on k x y z by monomial k automorphisms we consider the rationality problem of the fixed field k x y z g under the action of g namely whether k x y z g is rational that is purely transcendental over k or not we may assume that g is a subgroup of gl 3 z and the problem is determined up to conjugacy in gl 3 z there are 73 conjugacy classes of g in gl 3 z by results of endo miyata voskresenski lenstra saltman hajja kang and yamasaki 8 conjugacy classes of 2 groups in gl 3 z have negative answers to the problem under certain monomial actions over some base field k and the necessary and sufficient condition for the rationality of k x y z g over k is given in this paper we show that the fixed field k x y z g under monomial action of g is rational over k except for possibly negative 8 cases of 2 groups and unknown one case of the alternating group of degree four moreover we give explicit transcendental bases of the fixed fields over k for the unknown case we obtain an affirmative solution to the problem under some conditions in particular we show that if k is quadratically closed field then k x y z g is rational over k we also give an application of the result to 4 dimensional linear noether s problem

noether s problem for some semidirect products abstract let k be a field g be a finite group k x g g g be the rational function field with the variables x g where g g the group g acts on k x g g g by k automorphisms where h x g x h g for all h g g let k g be the fixed field defined by k g k x g g g g f k x g g g h f f for all h g noether s problem asks whether the fixed field k g is rational purely transcendental over k let m and n be positive integers and assume that there is an integer t such that t z m z is of order n define a group g m n m n 1 1 t c m c n assume furthermore that i m is an odd integer and ii for any e n the ideal e t m in z e is a principal ideal where e is a primitive e th root of unity theorem if k is a field with m n k then k g m n is rational over k consequently it may be shown that for any positive integer n the set s p p is a prime number such that c g p n is rational over c is of positive dirichlet density in particular s is an infinite set

rationality of finite groups of monomial automorphisms of k x y abstract it has been shown hajja j algebra 85 1983 243 250 that every finite cyclic group of monomial k automorphisms of k x y has a purely transcendental fixed field here the same result is established with the assumption that the group is cyclic dropped

approximation of minimum weight spanners for sparse graphs a t spanner of a graph g is its spanning subgraph s such that the distance between every pair of vertices in s is at most t times their distance in g the sparsestt spanner problem asks to find for a given graph g and an integer t a t spanner of g with the minimum number of edges the problem is known to be np hard for all t 2 and even more it is np hard to approximate it with ratio o logn for every t 2 for t 5 the problem remains np hard for planar graphs and the approximability status of the problem on planar graphs was open we resolve this open issue by showing that the sparsestt spanner problem admits the efficient polynomial time approximation scheme eptas for every t 1 our result holds for a much wider class of graphs namely the class of apex minor free graphs which contains the classes of planar and bounded genus graphs moreover it is possible to extend our results to weighted apex minor free graphs when the maximum edge weight is bounded by some constant

bogomolov multipliers and retract rationality for semidirect products abstract let g be a finite group the bogomolov multiplier b 0 g is constructed as an obstruction to the rationality of c v g where g g l v is a faithful representation over c we prove that for any finite groups g 1 and g 2 b 0 g 1 g 2 b 0 g 1 b 0 g 2 under the restriction map if g n g 0 with gcd n g 0 1 then b 0 g b 0 n g 0 b 0 g 0 under the restriction map for any integer n we show that there are non direct product p groups g 1 and g 2 such that b 0 g 1 and b 0 g 2 contain subgroups isomorphic to z p z n and z p n z respectively on the other hand if k is an infinite field and g n g 0 where n is an abelian normal subgroup of exponent e satisfying that e k we will prove that if k g 0 is retract k rational then k g is also retract k rational provided that certain local conditions are satisfied this result generalizes previous results of saltman and jambor 18

approximation algorithms for np complete problems on planar graphs this paper describes a general technique that can be used to obtain approximation schemes for various np complete problems on planar graphs the strategy depends on decompos ing a planar graph into subgraphs of a form we call k outerplanar for fixed k the problems of interest are solvable optimally in linear time on k outerplanar graphs by dynamic programming for general planar graphs if the problem is a maximization problem such as maximum independent set this technique gives for each k a linear time algorithm that produces a solution whose size is at least k k 1 optimal if the problem is a minimization problem such as minimum vertex cover it gives for each k a linear time algorithm that produces a solution whose size is at most k 1 k optimal taking k c log log nl or k c log nl where n is the number of nodes and c is some constant we get polynomial time approximation algorithms whose solution sizes converge toward optimal as n increases the class of problems for which this approach provides approximation schemes includes maximum independent set maximum tile salvage partition into triangles maximum h matching minimum vertex cover minimum dominat ing set and minimum edge dominating set for these and certain other problems the proof of solvability on k outerplanar graphs also enlarges the class of planar gmphs for which the problems are known to be solvable in polynomial time

retract rational fields abstract let k be an infinite field the notion of retract k rationality was introduced by saltman in the study of noether s problem and other rationality problems we will investigate the retract rationality of a field in this paper theorem 1 let k k l be fields if k is retract k rational and l is retract k rational then l is retract k rational theorem 2 for any finite group g containing an abelian normal subgroup h such that g h is a cyclic group for any complex representation g gl v the fixed field c v g is retract c rational theorem 3 if g is a finite group then all the sylow subgroups of g are cyclic if and only if c m g is retract c rational for all g lattices m for all short exact sequences 0 c m m 0 because the unramified brauer group of a retract c rational field is trivial theorems 2 and 3 generalize previous results of bogomolov and barge respectively see theorems 5 9 and 6 1

the rationality problem for finite subgroups of gl4 q abstract let g be a finite subgroup of gl 4 q the group g induces an action on q x 1 x 2 x 3 x 4 the rational function field of four variables over q theorem the fixed subfield q x 1 x 2 x 3 x 4 g f q x 1 x 2 x 3 x 4 f f for any g is rational i e purely transcendental over q except for two groups which are images of faithful representations of c 8 and c 3 c 8 into gl 4 q both fixed fields for these two exceptional cases are not rational over q there are precisely 227 such groups in gl 4 q up to conjugation the answers to the rationality problem for most of them were proved by kitayama and yamasaki 2009 ky except for four cases we solve these four cases left unsettled by kitayama and yamasaki thus the whole problem is solved completely

pattern formation and bistability in flow between counterrotating cylinders abstract in the taykor coutte experiment on fluid flow counterrotating cylinders there is a bicritical point where the onset of instabilities to taylor vortex flow a steady state bifurcation and spiral vortex flow a hopf bifurcation meet the nonlinear mode interactions near this bicritical point are analyzed exploiting the role of symmetry in the bifurcation theory and with emphasis of the relevance to experiments for a range of raduis ratios 0 43 0 98 the mechanism of the pattern formation is elucidated and several new flow patterns and transitions are predicted including wavy vortices bistability hysteresis and up to 7 quasiperiodic flows

symmetries and dynamics for 2 d navier stokes flow abstract simulations of forced 2 d navier stokes equations are analyzed the forcing is spatially periodic and temporally steady a karhunen loeve analysis is used to identify the structures in phase space that generate the pde behavior their relationship to the invariant subspaces generated by the symmetry group is discussed it is shown that certain modes that are in the stable eigenspace of the kolmogorov flow solution play an essential role for the dynamics of the attractor for the 2 d navier stokes equations below a reynolds number of about 30 in this regime all stable solutions are identified and thier relation to the symmetry structure is elucidated a new type of gluing bifurcation generated by the symmetry is found and analyzed a mechanism for the generation of bursting behavior is suggested

time averages for heteroclinic attractors in neighbourhoods of attracting heteroclinic cycles the time averages fail to converge for almost all initial conditions but spiral closer and closer to the boundary of a polygon this is shown by using a poincare section argument

robust heteroclinic cycles in the one dimensional complex ginzburg landau equation numerical evidence is presented for the existence of stable heteroclinic cycles in large parameter regions of the one dimensional complex ginzburg landau equation cgl on the unit spatially periodic domain these cycles connect difierent spatially and temporally inhomogeneous time periodic solutions as t 1 a careful analysis of the connections is made using a projection onto 5 complex fourier modes it is shown flrst that the time periodic solutions can be treated as relative equilibria after consideration of the symmetries of the cgl second the cycles are shown to be robust since the individual heteroclinic connections exist in invariant

pattern selection in oscillatory rotating convection abstract three dimensional pattern selection in a low prandtl number boussinesq fluid with stress free boundaries where the onset of convection is oscillatory is explored restricting the problem to a square lattice the normal form coefficients are calculated as functions of the square root of the taylor number and the prandtl number there is a large region of the plane where a heteroclinic cycle connecting four travelling roll states is stable as is decreased the cycle undergoes a transverse loss of stability creating quasiperiodic orbits which may themselves become chaotic all these stable dynamics occur at onset although conjectured on the basis of general results from symmetric bifurcation theory and well known for steady convection as the kuppers lortz instability j fluid mech 35 1969 609 cycling behaviour has not previously been demonstrated directly from the hydrodynamic equations in the oscillatory case a second region of the plane contains stable travelling roll solutions we examine their stability to perturbations at varying angles and demonstrate the existence of small angle instabilities of travelling rolls

on noether s problem for central extensions of symmetric and alternating groups for a field k and a finite group g acting regularly on a set of indeterminates x xg g g let k g denote the invariant field k x g we first prove for the alternating group an that if n is odd then q an is rational over q an 1 we then obtain an analogous result where an is replaced by an arbitrary finite central extension of either an or sn valid over q n for suitable n concrete applications of our results yield 1 a new proof of maeda s result on the rationality of q x1 x5 a5 q 2 an affirmative answer to noether s problem over q for both a5 and s5 3 an affirmative answer to noether s problem over c for every finite central extension group of either an or sn with n 5

generic extensions and generic polynomials for linear algebraic groups abstract we construct generic extensions and polynomials for certain linear algebraic groups over finite fields in particular we show the existence of generic polynomials for the cyclic group c 2 m in characteristic p for all odd primes p

heteroclinic cycles in the three dimensional postbifurcation motion of o 2 symmetric fluid conveying tubes abstract the occurrence of heteroclinic cycles is a phenomenon that has been detected in the dynamics of symmetric systems only recently for the physical problem of an o 2 symmetric fluid conveying tube the existence of a robust asymptotically stable heteroclinic cycles is shown

heteroclinic cycles and modulated travelling waves in a system with d 4 symmetry we study a system of two complex ordinary differential equations with d4 symmetry describing the effects of the symmetry breaking o 2 to d4 on an interaction between fourier modes with wavenumbers in the ratio 1 2 this change from continuous to discrete symmetry is relevant to the effect of introducing riblets on a wall to reduce boundary layer drag numerical simulations planar analysis and melnikov s method reveal the persistence under this symmetry breaking of just two of the continuous group orbit of heteroclinic cycles of the o 2 system it is further shown that one of these cycles destabilizes to regular modulated travellin waves while the other loses stability to a new type of solution modulated travelling waves which quasi periodically reverse their direction of propagation a bifurcation diagram for this behavior and the subsequent destabilization of modulated travelling waves to travelling waves is established the method of averaging is used to show that some of the quasiperiodic solutions of the o 2 system persist under the symmetry breaking while others break into sets of periodic orbits

rationality of p group actions abstract let p be a prime number g a p group of order p 4 k a field with char k p if the exponent of g is p e and k contains a primitive p e th root of unity then k v g is rational purely transcendental over k where g gl v is any linear representation of g over k

noether s problem for p groups with a cyclic subgroup of index p2 abstract let k be any field and g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by g x h x g h for any g h g noether s problem asks whether the fixed field k g k x g g g g is rational purely transcendental over k we will prove that if g is a non abelian p group of order p n n 3 containing a cyclic subgroup of index p 2 and k is any field containing a primitive p n 2 th root of unity then k g is rational over k as a corollary if g is a non abelian p group of order p 3 and k is a field containing a primitive p th root of unity then k g is rational

noether s problem on semidirect product groups abstract let k be a field g a finite group let g act on the function field l k x g by x x for any g denote the fixed field of the action by k g l g f g l f g f g g noether s problem asks whether k g is rational purely transcendental over k it is known that if g c m c n is a semidirect product of cyclic groups c m and c n with z n a unique factorization domain and k contains an eth primitive root of unity where e is the exponent of g then k g is rational over k in this paper we give another criteria to determine whether k c m c n is rational over k in particular if p q are prime numbers and there exists x z q such that the norm n q q q x p then c c p c q is rational over c

division algebras over surfaces abstract we consider here a number of topics concerning the theory of division algebras over the function field of a surface one result relates the obstruction for ramification data to be from a division algebra and third etale cohomology another result shows this obstruction is always zero when the surface is spec of a regular local ring with some mild assumptions at the same time we study the brauer group of this function field as it relates to the brauer group of the function field of the henselization finally we prove a result which says that brauer group elements which look like they are of prime index q unequal to any characteristic must have all their ramification split by a cyclic galois extension of the same degree this last result requires a primitive q root of one

a note on monomial automorphisms abstract previous results hajja j algebra73 1981 30 36 on monomial automorphisms are strengthened and the rationality of all monomial automorphisms on k x y is established the relation of certain steps in the proofs to a special case of a problem of zariski on whether quasi rationality implies rationality is also noted

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

collective additive tree spanners of bounded tree breadth graphs with generalizations and consequences in this paper we study collective additive tree spanners for families of graphs enjoying special robertson seymour s tree decompositions and demonstrate interesting consequences of obtained results we say that a graph g admits a system of m collective additive tree r spanners resp multiplicative tree t spanners if there is a system t g of at most m spanning trees of g such that for any two vertices x y of g a spanning tree t t g exists such that d t x y d g x y r resp d t x y t d g x y when m 1 one gets the notion of additive tree r spanner resp multiplicative tree t spanner it is known that if a graph g has a multiplicative tree t spanner then g admits a robertson seymour s tree decomposition with bags of radius at most t 2 in g we use this to demonstrate that there is a polynomial time algorithm that given an n vertex graph g admitting a multiplicative tree t spanner constructs a system of at most log 2n collective additive tree o tlogn spanners of g that is with a slight increase in the number of trees and in the stretch one can turn a multiplicative tree spanner into a small set of collective additive tree spanners we extend this result by showing that if a graph g admits a multiplicative t spanner with tree width k 1 then g admits a robertson seymour s tree decomposition each bag of which can be covered with at most k disks of g of radius at most t 2 each this is used to demonstrate that for every fixed k there is a polynomial time algorithm that given an n vertex graph g admitting a multiplicative t spanner with tree width k 1 constructs a system of at most k 1 log 2n collective additive tree o tlogn spanners of g

pattern selection in three dimensional magnetoconvection abstract weakly nonlinear boussinesq convection in an imposed vertical magnetic field is studied stress free boundary conditions are used at top and bottom of the layer where the magnetic field is assumed to remain vertical both steady and oscillatory convection are considered among all the spatially periodic patterns steady rolls are the preferred pattern when the onset of convection is steady state in the overstable regime a variety of spatially and temporally periodic patterns may be stable depending on the parameters these include in particular a pattern called alternating rolls the location of stable alternating rolls appears to be consistent with recent numerical simulations of compressible magnetoconvection where such patterns have been found in certain cases the conduction solution loses stability directly to spatially periodic patterns with nonperiodic time dependence the analysis of the possible patterns and their stability takes advantage of existing results from equivariant bifurcation theory and uses the critical wavenumber selected by the linear stability calculation as a function of the imposed magnetic field

some actions of symmetrical groups abstract let k be any field v the n dimensional vector space over k on which the symmetric group sn acts by permuting a basis then the fixed fields k skv sn and k k v sn are rational over k where skv and k v are the kth symmetric power and exterior power of v respectively the rationality of some other fixed fields are established also

description of galois unipotent extensions abstract given an arbitrary field f we describe all galois extensions l f whose galois groups are isomorphic to the group of upper triangular unipotent 4 by 4 matrices with entries in the field of two elements

rational invariants for subgroups of s5 and s7 abstract let g be a subgroup of s n the symmetric group of degree n for any field k g acts naturally on the rational function field k x 1 x 2 x n via k automorphisms defined by x i x i for any g any 1 i n theorem if n 5 then the fixed field k x 1 x n g is purely transcendental over k we will show that c x 1 x 7 g is also purely transcendental over c if g is any transitive subgroups of s 7 other than a 7 a similar result is valid for solvable transitive subgroups of s 11

rationality problem of three dimensional purely monomial group actions the last case a k automorphism of the rational function field k x 1 x n is called purely monomial it a sends every variable x i to a monic laurent monomial in the variables x 1 x n let g be a finite subgroup of purely monomial k automorphisms of k x 1 x n the rationality problem of the g action is the problem of whether the g fixed field k x 1 x n g is k rational i e purely transcendental over k or not in 1994 m hajja and m kang gave a positive answer for the rationality problem of the three dimensional purely monomial group actions except one case we show that the remaining case is also affirmative

bursts in oscillatory systems with broken d 4 symmetry a new mechanism responsible for generating regular and irregular bursts of large dynamic range near onset of an oscillatory instability is identified the bursts are present in systems with nearly square symmetry and are the result of heteroclinic cycles involving infinite amplitude states created when the square symmetry is broken all possible cycles of this type are identified and the resulting bursts described global connections involving finite amplitude states are also present the intricate sequence of bifurcations that result is described in several cases 2000 elsevier science b v all rights reserved

rationality problem for some meta abelian groups abstract let k be any field g be a finite group theorem assume that i g contains an abelian normal subgroup h so that g h is cyclic of order n ii z n is a unique factorization domain and iii e k where e is the exponent of g i e e lcm ord g g g if g gl v is any finite dimensional linear representation of g over k then k v g is rational purely transcendental over k

division algebras with psl 2 q galois maximal subfields abstract if g is a finite group and k is a field then g is k admissible if there exists a g galois extension l k such that l is a maximal subfield of a k division algebra we prove that psl 2 7 is k admissible for any number field which either fails to contain 1 or which has two primes lying over the dyadic prime in addition psl 2 11 is shown to be admissible over q or any number field k with at least two extensions of the dyadic prime indeed there exist infinitely many linearly disjoint admissible extensions for these groups

heteroclinic cycles involving periodic solutions in mode interactions with o 2 symmetry in this paper we show that in o 2 symmetric systems structurally stable asymptoticallystable heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions these cycles are found in the third order truncated normal forms of specific codimension two steady state hopf and hopf hopf mode interactions we find these cycles using group theoretic techniques in particular we look for certainpatterns in the lattice of isotropy subgroups once the pattern has been identified the heteroclinic cycle can be constructed by decomposing the vector field on fixed point subspaces into phase amplitude equations it is here that we use the assumption of normal form the final proof of existence and stability relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied

noether s problem for groups of order 243 abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed field noether s problem asks under what situations the fixed field k g will be rational purely transcendental over k according to the data base of gap there are 10 isoclinism families for groups of order 243 it is known that there are precisely 3 groups g of order 243 they consist of the isoclinism family 10 such that the unramified brauer group of c g over c is non trivial thus c g is not rational over c we will prove that if 9 k then k g is rational over k for groups of order 243 other than these 3 groups except possibly for groups belonging to the isoclinism family 7

noether s problem for groups of order 32 abstract let k be any field g be a finite group let g act on the rational function field k x g g g by k automorphisms defined by h x g x h g for any g h g denote by k g k x g g g g the fixed field noether s problem asks under what situations the fixed field k g will be rational purely transcendental over k theorem let g be a finite group of order 32 with exponent e if char k 2 or k is any field containing a primitive eth root of unity then k g is rational over k

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

a trade off between space and efficiency for routing tables two conflicting goals play a crucial role in the design of routing schemes for communication networks a routing scheme should use paths that are as short as possible for routing messages in the network while keeping the routing information stored in the processors local memory as succinct as possible the efficiency of a routing scheme is measured in terms of its stretch factor the maximum ratio between the length of a route computed by the scheme and that of a shortest path connecting the same pair of vertices most previous work has concentrated on finding good routing schemes with a small fixed stretch factor for special classes of network topologies in this paper the problem for general networks is studied and the entire range of possible stretch factors is examined the results exhibit a trade off between the efficiency of a routing scheme and its space requirements almost tight upper and lower bounds for this trade off are presented specifically it is proved that any routing scheme for general n vertex networks that achieves a stretch factor k 1 must use a total of o n 1 1 2 k 4 bits of routing information in the networks this lower bound is complemented by a family k k of hierarchical routing schemes for every k l for unit cost general networks which guarantee a stretch factor of o k require storing a total of o k 3 n 1 1 h log n bits of routing information in the network name the vertices with o log 2 n bit names and use o log n bit headers

frobenius groups and retract rationality abstract let k be any field g be a finite group acting on the rational function field k x g g g by h x g x h g for any h g g define k g k x g g g g noether s problem asks whether k g is rational purely transcendental over k a weaker notion retract rationality introduced by saltman is also very useful for the study of noether s problem we prove that if g is a frobenius group with abelian frobenius kernel then k g is retract k rational for any field k satisfying some mild conditions as an application we show that for any algebraic number field k for any frobenius group g with frobenius complement isomorphic to s l 2 f 5 there is a galois extension field k over k whose galois group is isomorphic to g i e the inverse galois problem is valid for the pair g k the same result is true for any non solvable frobenius group if k 8 is a cyclic extension of k

rationality problem for transitive subgroups of s8 abstract for any field k and any transitive subgroup g of s 8 let g act naturally on k x 1 x 8 by permuting the variables we prove that under some minor conditions k x 1 x 8 g is always k rational except g is a 8 or g is isomorphic to pgl 2 7 we pay special attentions on the characteristic 2 cases

an example of symmetry breaking to heteroclinic cycles is orbitally asymptotically stable suppose now that we consider a small system symmetry breakingperturbation of 1 1 z f z g z 1 2 where is small and g is only 2 equivariant where 2 1 is a lie subgroup when is sufficiently small normal hyperbolicity guarantees that there isa perturbed flow invariant manifold x

lattices over cyclic groups and noether settings abstract for each positive integer m let m be a primitive mth root of 1 given a prime p let a be an integer which is a primitive root of 1 mod p let i p m be the ideal in z m generated by p and m a let g be a cyclic group of order n such that the flasque class of any maximal projective ideal in z g contains a maximal ideal u of index p for some prime p such that n divides p 1 and such that i p n is principal it is not known whether the number of groups satisfying this property is finite or infinite we show that if f is a field of characteristic zero containing all nth roots of 1 then for any z g lattice m the fixed subfield of f m is stably rational over f if f also contains primitive mth roots of 1 for all natural numbers m then for any finite g module t the fixed subfield of the noether setting of the group g t g is stably rational over f

approximating k spanner problems for k 2 given a graph g v e a subgraph g v h h e is a k spanner of g if for any pair of vertices u w v it satisfies dh u w kdg u w the basic k spanner problem is to find a k spanner of a given graph g with the smallest possible number of edges this paper considers approximation algorithms for this and some related problems for k 2 known to be 2log1 n inapproximable the basic k spanner problem over undirected graphs with k 2 has been given a sublinear ratio approximation algorithm with ratio roughly o n2 k 1 but no such algorithms were known for other variants of the problem including the directed and the client server variants as well as for the related k dss problem we present the first approximation algorithms for these problems with sublinear approximation ratio the second contribution of this paper is in characterizing some wide families of graphs on which the problems do admit a logarithmic and a polylogarithmic approximation ratios these families are characterized as containing graphs that have optimal or near optimal spanners with certain desirable properties such as being a tree having low arboricity or having low girth all our results generalize to the directed and the client server variants of the problems as a simple corollary we present an algorithm that given a graph g builds a subgraph with o n edges and stretch bounded by the tree stretch of g namely the minimum maximal stretch of a spanning tree for g the analysis of our algorithms involves the novel notion of edge dominating systems developed in the paper the technique introduced in the paper reduces the studied algorithmic approximability questions on k spanners to purely graph theoretical questions concerning the existence of certain combinatorial objects in families of graphs

heteroclinic cycles in rings of coupled cells abstract symmetry is used to investigate the existence and stability of heteroclinic cycles involving steady state and periodic solutions in coupled cell systems with d n symmetry using the lattice of isotropy subgroups we study the normal form equations restricted to invariant fixed point subspaces and prove that it is possible for the normal form equations to have robust asymptotically stable heteroclinic cycles connecting periodic solutions with steady states and periodic solutions with periodic solutions a center manifold reduction from the ring of cells to the normal form equations is then performed using this reduction we find parameter values of the cell system where asymptotically stable cycles exist simulations of the cycles show trajectories visiting steady states and periodic solutions and reveal interesting spatio temporal patterns in the dynamics of individual cells we discuss how these patterns are forced by normal form symmetries

rationality of some projective linear actions let k be any field which may not be algebraically closed v a finite dimensional vector space over k gl v where the order of can be finite or infinite theorem if dimk v 3 then both k v and k p v are rational purely transcendental over k similar results hold for a cyclic affine action

approximate distance oracles let g v e be an undirected weighted graph with vvv n and vev m let k 1 be an integer we show that g v e can be preprocessed in o kmn1 k expected time constructing a data structure of size o kn1p1 k such that any subsequent distance query can be answered approximately in o k time the approximate distance returned is of stretch at most 2k 1 that is the quotient obtained by dividing the estimated distance by the actual distance lies between 1 and 2k 1 a 1963 girth conjecture of erdos implies that n1p1 k space is needed in the worst case for any real stretch strictly smaller than 2kp1 the space requirement of our algorithm is therefore essentially optimal the most impressive feature of our data structure is its constant query time hence the name oracle previously data structures that used only o n1p1 k space had a query time of n1 k our algorithms are extremely simple and easy to implement efficiently they also provide faster constructions of sparse spanners of weighted graphs and improved tree covers and distance labelings of weighted or unweighted graphs

finite group actions on rational function fields is an invertible n x n matrix with integer entries and where a a e k o if a c 1 vi then cj is called purely monomial it is proved in 6 theorem that if g is any finite group of monomial k automorphisms of k x x2 then its fixed held k x x2 is rational purely transcendental over k this does not generalize to the three variable case since the fixed field of the order 4 cyclic group generated by the monomial q automorphism o defined on q x x2 x3 by is not rational s last paragraph however we prove here that the fixed field of every finite abeliun group of purely monomial k automorphisms of

heteroclinic networks in coupled cell systems we give an intrinsic definition of a heteroclinic network as a flow invariant set that is indecomposable but not recurrent our definition covers many previously discussed examples of heteroclinic behavior in addition it provides a natural framework for discussing cycles between invariant sets more complicated than equilibria or limit cycles we allow for cycles that connect chaotic sets cycling chaos or heteroclinic cycles cycling cycles both phenomena can occur robustly in systems with symmetry

from steady solutions to chaotic flows in a rayleigh benard problem at moderate rayleigh numbers abstract the dynamics of a rayleigh benard convection problem in a cubical cavity at moderate values of the rayleigh number ra 10 5 and a prandtl number of pr 0 71 with extensions to pr 0 75 and 0 80 was investigated the cubical cavity was heated from below and had perfectly conducting sidewalls and uniform temperature distributions on the two horizontal walls a system of ordinary differential equations with a dimension of typically n 11 000 was obtained when the conservation equations were discretized by means of a galerkin method previous knowledge of the bifurcation diagram of steady solutions reported in the literature was used to identify the origin of several branches of periodic orbits that were continued with ra half a dozen of such periodic orbits were found to be stable within narrow ranges of ra at most some 5000 units wide an attracting two torus restricted to a very narrow region of ra was also identified it was found that the instabilization of periodic orbits quite often resulted into the development of complex dynamics such as the creation of homoclinic and heteroclinic orbits instances of both types of global bifurcations were analyzed in some detail one particular instance of chaotic dynamics a strange attractor was also identified chaotic dynamics has been found at pr 0 71 in a flow invariant subspace which can be interpreted as a fixed point subspace in terms of equivariant theory this subspace is not attracting however some regions of attracting chaotic dynamics for moderate rayleigh numbers 9 10 4 r a 10 5 were found at values of pr slightly above 0 71 the role of a particular homoclinic solution found at pr 0 71 in the generation of these chaotic regions was analyzed