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

associativity of quantum multiplication we proved the associativity of the multiplication of quantum cohomology for a monotone compact symplectic manifold v for which c 1 a 1 for any effective class the same proof also works for any positive compact symplectic manifold with c 1 a 1

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

asymptotic behaviors of governing equation of gauged sigma model for heisenberg ferromagnet abstract in this note we study weak solutions of equation 0 1 u 4 e u 1 e u 4 i 1 n p i 4 j 1 m q j in r 2 where p i i 1 n resp q j j 1 m are dirac masses concentrated at the points p i i 1 n resp q j j 1 m and n m 1 eq 0 1 represents a governing equation of gauged sigma model for heisenberg ferromagnet we show that it has a sequence of solutions u having behaviors as ln x o 1 at infinity with a free parameter 2 2 n m and our concern in this paper is to study the asymptotic behavior of b as approaching the extremal values 2 and 2 n m

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

on anisotropic singularities for semi linear elliptic equations in r2 abstract in this note we study weak solutions of 1 u g u 8 0 x 2 in r 2 g u x 0 as x where g s 2 e s 1 e s 1 is an odd increasing and bounded function and 0 x 2 is defined in the sense of distribution 0 x 2 0 x 2 for c c 1 r 2 we obtain that problem 1 admits a unique x 2 odd weak solution which has anisotropic singularity 4 x 2 x 2 near the origin

on the conformal gaussian curvature equation inr2 abstract in this paper we consider the equation u k x e 2u 0 in r 2 0 1 wherek x k x inr2andk x does not decay at x large from the geometric viewpoint it is quite interesting to ask what the best possible range is of total curvature of all solutions of 0 1 in this paper we study this problem for radial solutions we also construct some particularkto demonstrate the rich phenomenon in particular we show by examples that whenk x is negative for x large eq 0 1 possess a branch of solutions which satisfies some monotonicity property

qualitative properties of solutions to some nonlinear elliptic equations in r 2 o introduction elliptic equations in this paper we investigate properties of the solutions to the au r x e u x x r 2 for functions r x which are positive near infinity equations of this kind arise from a variety of situations such as from prescribing gaussian curvature in geometry i 2 and from combustion theory in physics 3 recently a series ofworks have been sought to understand the existence and the qualitative properties of the solutions of ni i 5 and ni cheng 4 1 considered the case where r x is nonpositive mcowen 6 1 and aviles 7 investigated the situation where r x 0 in some order as ixl in our previous paper 1 we consider a special case where r is a constant we proved that the solutions are radially symmetric and hence classified all the solutions in this paper we consider more general functions r x first we obtain the asymptotic behavior of the solution near infinity consequently we prove that all the solutions satisfy an identity which is somewhat of a generalization of the well known kazdan warner condition finally using the asymptotic behavior together with the further development of the method employed in our previous paper i 1 we show that all the solutions are radially symmetric provided r is radially symmetric and nonincreasing this part can be viewed as the completion of 1 throughout this paper we assume that the function r x is positive near infinity in 1 we study the asymptotic behavior of the solution u x of let fl 1 2n r2 r x e dx under some appropriate conditions we show that the solutions approach

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

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

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

classification of solutions of elliptic equations arising from a gravitational o 3 gauge field model abstract in this paper we study an elliptic equation arising from the self dual maxwell gauged o 3 sigma model coupled with gravity when the parameter equals 1 and there is only one singular source we consider radially symmetric solutions there appear three important constants a positive parameter a representing a scaled gravitational constant a nonnegative integer n 1 representing the total string number and a nonnegative integer n 2 representing the total anti string number the values of the products a n 1 a n 2 0 play a crucial role in classifying radial solutions by using the decay rates of solutions at infinity we provide a complete classification of solutions for all possible values of a n 1 and a n 2 this improves previously known results

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

conformal metrics with prescribed gaussian curvature on s2 we consider on s 2 the problem of which functions k can be the scalar curvature of a metric conformal to the standard metric on s 2 we assume that k is a function of one variable and we obtain a necessary and sufficient condition for the problem to be solvable we also obtain several new sufficient conditions on k which are easy to check in order that the problem be solvable

evaluating solutions on an elliptic problem in a gravitational gauge field theory abstract an elliptic equation arising from the study of static solutions with prescribed zeros and poles of the einstein equations coupled with the classical sigma model and an abelian gauge field is considered we classify the solutions and establish the uniqueness of radially symmetric solutions we also complete a classification of symmetric solutions of an elliptic equation on the sphere

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

existence of topological multi string solutions in abelian gauge field theories in this paper we consider a general form of self dual equations arising from abelian gauge field theories coupled with the einstein equations by applying the super subsolution method we prove that topological multi string solutions exist for any coupling constant which improves previously known results we provide two examples for application the self dual einstein maxwell higgs model and the gravitational maxwell gauged o 3 sigma model

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

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

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

the solution structure of the o 3 sigma model in a maxwell chern simons theory in this paper a system of semilinear elliptic equations arising from a relativistic self dual maxwell chern simons o 3 sigma model is considered we reveal the uniqueness aspect of the topological solutions for the model the uniqueness result is associated with a clear solution structure of the equations of the radially symmetric case we locate each solution set denoted by a planar diagram