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

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

stochastic hybrid automata with delayed transitions to model biochemical systems with delays to study the effects of a delayed immune response on the growth of an immunogenic neoplasm we introduce stochastic hybrid automata with delayed transitions as a representation of hybrid biochemical systems with delays these transitions abstractly model unknown dynamics for which a constant duration can be estimated i e a delay these automata are inspired by standard stochastic hybrid automata and their semantics is given in terms of piecewise deterministic markov processes the approach is general and can be applied to systems where i components at low concentrations are modeled discretely so to retain their intrinsic stochastic fluctuations ii abundant component e g chemical signals are well approximated by mean field equations so to simulate them efficiently and iii missing components are abstracted with delays via simulations we show in our application that interesting delay induced phenomena arise whose quantification is possible in this new quantitative framework

on bogomolov multiplier and ш rigidity of groups abstract we show the existence of groups with trivial bogomolov multiplier such that the bogomolov multiplier of a central product of these groups is non trivial we then prove that finite quasisimple groups and freest special p groups are rigid in the sense that all their class preserving automorphisms are inner as an application we show the existence of non nilpotent rigid groups and rigid p groups of odd order having non trivial bogomolov multiplier

noncommutative rational functions invariant under the action of a finite solvable group abstract this paper describes the structure of invariant skew fields for linear actions of finite solvable groups on free skew fields in d generators these invariant skew fields are always finitely generated which contrasts with the free algebra case for abelian groups or solvable groups g with a well behaved representation theory it is shown that the invariant skew fields are free on g d 1 1 generators finally positivity certificates for invariant rational functions in terms of sums of squares of invariants are presented

approximate analysis of biological systems by hybrid switching jump diffusion in this paper we consider large state space continuous time markov chains arising in the field of systems biology for a class of such models namely for density dependent families of markov chains that represent the interaction of large groups of identical objects kurtz has proposed two kinds of approximations one is based on ordinary differential equations and provides a deterministic approximation while the other uses a diffusion process with which the resulting approximation is stochastic the computational cost of the deterministic approximation is significantly lower but the diffusion approximation retains stochasticity and is able to reproduce relevant random features like variance bimodality and tail behavior that cannot be captured by a single deterministic quantity in a recent paper for particular stochastic petri net models we proposed a jump diffusion approximation that aims at being applicable beyond the limits of kurtz s diffusion approximation in order to cover the case when the process reaches the boundary with non negligible probability in this paper we generalize the method so that it can be applied to any density dependent markov chains other limitations of the diffusion approximation in its original form are that it can provide inaccurate results when the number of objects in some groups is often or constantly low and that it can be applied only to pure density dependent markov chains in order to overcome these drawbacks in this paper we propose to apply the jump diffusion approximation only to those components of the model that are in density dependent form and are associated with high population levels the remaining components are treated as discrete quantities the resulting process is a hybrid switching jump diffusion i e a diffusion with hybrid state space and jumps where the discrete state changes can be seen as switches that take the diffusion from one condition to another we show that the stochastic differential equations that characterize this process can be derived automatically both from the description of the original markov chains or starting from a higher level description language like stochastic petri nets the proposed approach is illustrated on three models one modeling the so called crazy clock reaction one describing viral infection kinetics and the last considering transcription regulation

a neutral fde is similar to the product of an ode and a shift abstract in finite dimensional control theory a minimal realization of a transfer function is determined only module a similarity transform the infinite dimensional situation is similar there is no real reason to distinguish between two realizations that differ from each other only by a similarity transform the purpose of this work is to show that under natural assumptions the semigroup induced by a neutral functional differential equation with infinite delay is similar to the cross product of a semigroup induced by an ordinary differential equation and a shift semigroup this result is expected to lead to simpler realization results for transfer functions induced by delay equations

rationality of the moduli spaces of plane curves of sufficiently large degree we prove that the moduli space of plane curves of degree d is rational for all sufficiently large d

degree three unramified cohomology groups 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 subfield noether s problem asks whether k g is rational purely transcendental over k the unramified brauer group br nr c g and the unramified cohomology h nr 3 c g q z are obstructions to the rationality of c g see 14 and 5 peyre proves that if p is an odd prime number then there is a group g such that g p 12 br nr c g 0 but h nr 3 c g q z 0 thus c g is not stably c rational 12 using peyre s method we are able to find groups g with g p 9 where p is an odd prime number such that br nr c g 0 h nr 3 c g q z 0

the null space and the range of a convolution operator in a fading memory space we study the convolution equation p x t v x t oo 0 instead we permit and to be of fredholm type i e and v need not vanish on 00 0 we restrict the growth rate of x and at plus and minus infinity and we look at the problem of the existence and uniqueness of solutions of and on the whole real line satisfying conditions like x i ctj i 00 t 00 where c is a constant depending on x and ij is a predefined function some authors use the word admissible when discussing problems of this type in the case when the homoge neous version of has nonzero solutions we decompose the solutions into components with different exponential growth rates and give a priori bounds on the growth rates of the solutions as an application of the basic theory we look at the cauchy problem for a neutral functional differential equation and prove the existence of stable and unstable manifolds

unramified brauer groups of finite and infinite groups the bogomolov multiplier is a group theoretical invariant isomorphic to the unramified brauer group of a given quotient space we derive a homological version of the bogomolov multi plier prove a hopf type formula find a five term exact sequence corresponding to this invariant and describe the role of the bogomolov multiplier in the theory of central extensions a new description of the bogomolov multiplier of a nilpotent group of class two is obtained we define the bogomolov multiplier within k theory and show that proving its triviality is equivalent to solving a long standing problem posed by bass an algorithm for computing the bogomolov multiplier is developed

varietes unirationnelles non rationnelles au dela de l exemple d artin et mumford keywords non rational but unirational variety grothendieck invariants rationality brauer invariants reference cma article 1989 001doi 10 1007 bf01850658 record created on 2008 12 16 modified on 2016 08 08

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

spectral decomposition and invariant manifolds for some functional partial differential equations we study the integrodifferential convolution equationddt x x ax x fon 0 x on 0 as well as a nonlinear perturbation of the corresponding homogeneous equation hereais the generator of an analytic semigroup on a hilbert spaceh and and are operator valued dominated measures with values inl h andl d a h respectively under the assumption that the operator given by the laplace transform of the left hand side of the equation is boundedly invertible on some right half plane and on a line in the left half plane parallel to the imaginary axis we decompose the solutions into components with different exponential growth rates we construct projectors onto the stable and unstable subspaces which are then used for the construction of stable and unstable manifolds for the nonlinear equation which can have a fully nonlinear character the results are applied to two equations of parabolic type moreover the spectrum of the generator of the translation semigroup in various weighted spaces is determined including the stable and unstable subspaces of our problem

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

the groups of order 256 abstract building on earlier work a new method for generating descriptions of p groups is developed the theory and implementation of this method are described and its application in determining the 56 092 groups of order 256 is outlined

on ш rigidity of groups of order p6 abstract let g be a group and out c g be the group of its class preserving outer automorphisms we compute out c g for all the groups g of order p 6 where p is an odd prime as an application we observe that for most of the rigid groups g of order p 6 its bogomolov multiplier b 0 g is zero

unramified cohomology of quadrics i given a quadric x over a field f of characteristic 2 we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 galois cohomology of f to the unramified mod 2 cohomology of f x when dim x 10 and in several smaller dimensional cases applications of these results to real quadrics and to the unramified witt ring are given

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

rationality of three dimensional quotients by monomial actions let g be a finite 2 group and k be a field satisfying that i chark 2 and ii a k for any a k if g acts on the rational function field k x y z by monomial k automorphisms then the fixed field k x y z g is rational purely transcendental over k applications of this theorem will be given

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

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

probabilistic reachability and safety for controlled discrete time stochastic hybrid systems in this work probabilistic reachability over a finite horizon is investigated for a class of discrete time stochastic hybrid systems with control inputs a suitable embedding of the reachability problem in a stochastic control framework reveals that it is amenable to two complementary interpretations leading to dual algorithms for reachability computations in particular the set of initial conditions providing a certain probabilistic guarantee that the system will keep evolving within a desired safe region of the state space is characterized in terms of a value function and maximally safe markov policies are determined via dynamic programming these results are of interest not only for safety analysis and design but also for solving those regulation and stabilization problems that can be reinterpreted as safety problems the temperature regulation problem presented in the paper as a case study is one such case

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

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

unramified brauer groups of finite simple groups of lie type a l we study the subgroup b 0 g of h 2 g inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i consisting of all elements which have trivial restrictions to every abelian subgroup of g the group b 0 g serves as the simplest nontrivial obstruction to stable rationality of algebraic varieties v g where v is a faithful complex linear representation of the group g we prove that b 0 g is trivial for finite simple groups of lie type a l

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

stabilization of hereditary distributed parameter control systems abstract in this paper we study the asymptotic stabilization of semilinear distributed parameter control systems with delay assuming that the semigroup of operators associated with the uncontrolled and undelayed equation is compact we prove that a well known rank condition which implies the feedback stabilization for lumped systems can be extended to this type of systems

computing the nonabelian tensor squares of polycyclic groups abstract in this paper we develop a theory for computing the nonabelian tensor square and related computations for finitely presented groups and specialize it to polycyclic groups this theory provides a framework for making nonabelian tensor square computations for polycyclic groups and is the basis of an algorithm for computing the nonabelian tensor square for any polycyclic group

some well posed functional equations which generate semigroups consider the abstract linear functional equation fe dx t f t t 0 x t t t 0 in a banach space b a theorem is proven which contains the following result as a special case let y r b be a lp space or c0 space on r t8 with a suitable weight function and with values in b let d be a closed unbounded causal linear operator in y r b which commutes with translations suppose that d i has a continuous causal inverse for some complex and that d restricted to those functions in y r b which vanish on r 0 has a continuous causal inverse then fe generates a strongly continuous semigroup of translation type on a banach space which is essentially the cross product of the restriction of the domain of d to r and y r b examples with b cn on how the theory applies to a neutral functional differential equation a difference equation a volterra integrodifferential equation with nonintegrable kernel but integrable resolvent and a fractional order functional differential equation are given also an abstract neutral functional differential equation in a hilbert space is studied and applications to an abstract volterra integrodifferential equation in a banach space are indicated

unramified degree three invariants of reductive groups abstract we prove that if g is a reductive group over an algebraically closed field f then for a prime integer p char f the group of unramified galois cohomology h nr 3 f bg q p z p 2 is trivial for the classifying space bg of g if p is odd or the commutator subgroup of g is simple

universal commutator relations bogomolov multipliers and commuting probability abstract let g be a finite p group we prove that whenever the commuting probability of g is greater than 2 p 2 p 2 p 5 the unramified brauer group of the field of g invariant functions is trivial equivalently all relations between commutators in g are consequences of some universal ones the bound is best possible and gives a global lower bound of 1 4 for all finite groups the result is attained by describing the structure of groups whose bogomolov multipliers are nontrivial and bogomolov multipliers of all of their proper subgroups and quotients are trivial applications include a classification of p groups of minimal order that have nontrivial bogomolov multipliers and are of nilpotency class 2 a nonprobabilistic criterion for the vanishing of the bogomolov multiplier and establishing a sequence of bogomolov s absolute minimal factors which are 2 groups of arbitrarily large nilpotency class thus providing counterexamples to some of bogomolov s claims in relation to this we fill a gap in the proof of triviality of bogomolov multipliers of finite almost simple groups

noether s problem for metacyclic p groups 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 gh for any g h g denote by k g the fixed field k x g g g g noether s problem asks whether k g is rational purely transcendental over k an affirmative answer to noether s problem for metacyclic p groups will be proved provided that k contains enough roots of unity

birational classification of fields of invariants for groups of order 128 let g be a finite group acting on the rational function field c xg g g c xg g g by cc automorphisms h xg xhgh xg xhg for any g h gg h g noether s problem asks whether the invariant field c g k xg g g gc g k xg g g g is rational i e purely transcendental over cc by fischer s theorem c g c g is rational over cc when g is a finite abelian group saltman and bogomolov respectively showed that for any prime p there exist groups g of order p9p9 and of order p6p6 such that c g c g is not rational over cc by showing the non vanishing of the unramified brauer group brnr c g 0brnr c g 0 which is an avatar of the birational invariant h3 x z torsh3 x z tors given by artin and mumford where x is a smooth projective complex variety whose function field is c g c g for p 2p 2 chu hu kang and prokhorov proved that if g is a 2 group of order 32 then c g c g is rational over cc chu hu kang and kunyavskii showed that if g is of order 64 then c g c g is rational over cc except for the groups g belonging to the two isoclinism families 13 13 with brnr c g 0brnr c g 0 and 16 16 with brnr c g c2brnr c g c2 bogomolov and bohning s theorem claims that if g1g1 and g2g2 belong to the same isoclinism family then c g1 c g1 and c g2 c g2 are stably cc isomorphic we investigate the birational classification of c g c g for groups g of order 128 with brnr c g 0brnr c g 0 moravec showed that there exist exactly 220 groups g of order 128 with brnr c g 0brnr c g 0 forming 11 isoclinism families j j we show that if g1g1 and g2g2 belong to 16 16 31 31 37 37 39 39 43 43 58 58 60 60 or 80 80 resp 106 106 or 114 114 then c g1 c g1 and c g2 c g2 are stably cc isomorphic with brnr c gi c2brnr c gi c2 explicit structures of non rational fields c g c g are given for each cases including also the case 30 30 with brnr c g c2 c2brnr c g c2 c2

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

groups of order p 5 and their unramified brauer groups we prove that if p is a prime p 3 and g is a group of order p5 not belonging to the 10 th isoclinism family then the unramified brauer group of g is trivial

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