baker campbell hausdorff relation for special unitary groups su n multiplication of two elements of the special unitary group su n determines uniquely a third group element a baker campbell hausdorff relation is derived which expresses the group parameters of the product written as an exponential in terms of the parameters of the exponential factors this requires the eigenvalues of three matrices consequently the relation can be stated analytically up to n 4 in principle similarity transformations encoding the time evolution of quantum mechanical observables for example can be obtained by the same means

on global diffeomorphisms of euclidian space as is well known the first attempt at characterizing global diffeomorphisms of ir goes back to the 1906 work of hadamard l since that time progress in the global diffeomorphism problem has been rather slow notable contributions are those of levy 2 who generalized hadamard s theorem to banach spaces and of banach and mazur 3 and caccioppoli 4 who proved the so called properness criterion saying that a proper local homeomorphism of a banach space is a global homeomorphism the standard proofs of the theorems of hadamard and levy and of banach mazur and caccioppoli rely upon ideas of what is now called covering space theory via the important homotopy lifting property of coverings see e g 5 this approach makes it clear that it is only lack of compactness of ir that allows for the existence of noninjective local homeomorphisms when n l 2 the complex exponential when n 2 being the most famous example the theorems mentioned above were found again by various authors sometimes in special cases and sometimes in a different framework but always through some variant of the same argument a synthesis was made by rheinboldt 6 in 1969 part of his results were rediscovered five years later by plastock 7 and given a somewhat simpler exposition for the special case of banach spaces other contributions were made by meisters and olech 8 global homeomorphisms of bounded domains in i meyer 9 complementing the work of levy and radulescu and radulescu lo expanding upon plastock s results many other articles have been devoted to special cases where extra hypotheses are introduced consistently with the problem at hand for instance polynomial mappings only are relevant to the jacobian conjecture of algebraic geometry and hypotheses about the spectrum of the derivative at each point come up in a natural way in global stability questions for differential equations the new results presented in this paper have been obtained through arguments from critical point theory and differential topology here the central notion is that of admissible flow given a mapping f ii ir the main property of what we call an f admissible flow q ir x ii f o i f o is that for every x e ir f o the function t jf q t x l where 1 1 is the euclidean norm is strictly increasing and tends to 0 resp 00 as t tends to co resp 00 the basic feature we prove is that a local diffeomorphism f is a global one if and only if f o zi and there is an f admissible flow this statement makes it clear that 0 plays a special role regarding f admissibility and the condition f o 0 cannot be eliminated as examples will show the introduction of f admissibility followed the author s remark that in hilbert space hadamard s theorem could be derived from the mountain pass theorem of ambrosetti and

polynomial maps with strongly nilpotent jacobian matrix and the jacobian conjecture abstract let h k n k n be a polynomial map it is shown that the jacobian matrix jh is strongly nilpotent if and only if jh is linearly triangularizable if and only if the polynomial map f x h is linearly triangularizable furthermore it is shown that for such maps f sf is linearizable for almost all s k except a finite number of roots of unity

characterizing injectivity of classes of maps via classes of matrices abstract we present a framework for characterizing injectivity of classes of maps on cosets of a linear subspace by injectivity of classes of matrices using our formalism we characterize injectivity of several classes of maps including generalized monomial and monotonic not necessarily continuous maps in fact monotonic maps are special cases of component wise affine maps further we study compositions of maps with a matrix and other composed maps in particular rational functions our framework covers classical injectivity criteria based on mean value theorems for vector valued maps and recent results obtained in the study of chemical reaction networks

d log and formal flow for analytic isomorphisms of n space given a formal map f f1 fn of the form z higher order terms we give tree expansion formulas and as sociated algorithms for the d log of f and the formal flow ft the coefficients which appear in these formulas can be viewed as certain generalizations of the bernoulli numbers and the bernoulli polynomials moreover the coefficient polynomials in the formal flow formula coincide with the strict order polynomials in combina torics for the partially ordered sets induced by trees applications of these formulas to the jacobian conjecture are discussed

quadratic linear keller maps abstract it is proved that a quadratic linear keller map c n c n is linearly triangularizable if its rank is at most 2 or corank at most 1

the jacobian conjecture linear triangularization for homogeneous polynomial maps in dimension three let k be a field of characteristic zero and f k3 k3 a polynomial map of the form f x h where h is homogeneous of degree d 2 we show that the jacobian conjecture is true for such mappings more precisely we show that if jh is nilpotent there exists an invertible linear map t such that t 1ht 0 h2 x1 h3 x1 x2 where the hi are homogeneous of degree d as a consequence of this result we show that all generalized druzkowski mappings f x h x1 l1d xn lnd where li are linear forms for all i and d 2 are linearly triangularizable if jh is nilpotent and rkjh 3

on the existence of global analytic conjugations for polynomial mappings of yagzhev type abstract consider a mappingf c n c nof the form identity plus a term with polynomial components that are homogeneous of the third degree and suppose that the jacobian determinant of this mapping is constant throughout c n polynomial mapping of yagzhev type as a stronger version of the classical jacobian conjecture the question has been posed whether for some values of c 0 there exists a global change of variables conjugation on c nthrough which the mapping fbecomes its linear part at the origin van den essen has recently produced a simple yagzhev mapping for which no suchpolynomialconjugation exists we show here that van den essen s example still admitsglobal analyticconjugations the question on the existence of global conjugations for general yagzhev maps is then still open

on parameter derivatives of the associated legendre function of the first kind with applications to the construction of the associated legendre function of the second kind of integer degree and order abstract a relationship between partial derivatives of the associated legendre function of the first kind with respect to its degree p m z n and to its order p n z m is established for m n n 0 this relationship is used to deduce four new closed form representations of p m z n from those found recently for p n z m by the author r szmytkowski on the derivative of the associated legendre function of the first kind of integer degree with respect to its order with applications to the construction of the associated legendre function of the second kind of integer degree and order j math chem 46 2009 231 several new expressions for the associated legendre function of the second kind of integer degree and order q n m z suitable for numerical purposes are also derived

classical orthogonal polynomials dependence on parameters most of the classical orthogonal polynomials continuous discrete and their q analogues can be considered as functions of several parameters ci a systematic study of the variation innitesimal and nite of these polynomials pn x ci with respect to the parameters ci is proposed a method to get recurrence relations for connection coecients linking r c r i pn x ci t opn x ci is given and in some situations explicit expressions are obtained this allows us to compute new integrals or sums of classical orthogonal polynomials using the digamma function a basic theorem on the zeros of ci pn x ci is also proved c 2000 elsevier science b v all rights reserved msc 33c25 42c05 33b15

quadratic linear keller maps of nilpotency index three let f x h cn cn be a homogeneous polynomial map of degree d 2 and fa y ay d cn cn a power linear map such that f and fa are a generalized gorni zampieri pair we discuss the relation between the nilpotency indices of jh and j ay d and we show that f is linearly triangularizable if and only if fa is linearly triangularizable as a consequence we show that a quadratic linear keller map fa y ay 2 with nilpotency index three i e j ay 2 3 0 is linearly triangularizable

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

foliations and polynomial diffeomorphisms of mathbb r 3 let y f g h mathbb r 3 to mathbb r 3 be a c 2 map and let spec y denote the set of eigenvalues of the derivative dy p when p varies in mathbb r 3 we begin proving that if for some 0 spec y cap epsilon epsilon emptyset then the foliation mathcal f k with k in f g h made up by the level surfaces k constant consists just of planes as a consequence we prove a bijectivity result related to the three dimensional case of jelonek s jacobian conjecture for polynomial maps of mathbb r n

on the derivative of the legendre function of the first kind with respect to its degree we study the derivative of the legendre function of the first kind p z with respect to its degree at first we provide two contour integral representations for p z then we proceed to investigate the case of p z n with n being an integer this case is met in some physical and engineering problems since it holds that we focus on the sub case of n being a non negative integer we show that where rn z is a polynomial in z of degree n we present alternative derivations of several known explicit expressions for rn z and also add some new a generating function for rn z is also constructed properties of the polynomials vn z rn z 1 nrn z 2 and wn 1 z rn z 1 nrn z 2 are also investigated it is found that wn 1 z is the christoffel polynomial well known from the theory of the legendre function of the second kind qn z as examples of applications of the results obtained we present non standard derivations of some representations of qn z sum to closed forms some legendre series evaluate some definite integrals involving legendre polynomials and also derive an explicit representation of the indefinite integral of the legendre polynomial squared

the linear triangularizability of some keller maps abstract in this paper we prove that keller maps are linearly triangularizable under some conditions as a consequence we show that non homogeneous power linear keller maps with jh symmetric are linearly triangularizable

note on the markus yamabe conjecture for gradient dynamical systems abstract let v r n r n be a c 1 vector field which has a singular point o and its linearization is asymptotically stable at every point of r n we say that the vector field v satisfies the markus yamabe conjecture if the critical point o is a global attractor of the dynamical system x v x in this note we prove that if v is a gradient vector field i e v f f c 2 then the basin of attraction of the critical point o is the whole r n thus implying the markus yamabe conjecture for this class of vector fields an analogous result for discrete dynamical systems of the form x m 1 f x m is proved

a sufficient condition in order that the real jacobian conjecture in r 2 holds let f f g r2 r2 be a polynomial map such that det df x y is different from zero for all x y r2 and f 0 0 0 0 we prove that for the injectivity of f it is sufficient to assume that the higher homogeneous terms of the polynomials ffx ggx and ffy ggy do not have real linear factors in common the proofs are based on qualitative theory of dynamical systems

the linear dependence problem for power linear maps abstract let bl l 1 k be m nl complex matrices and let x l c n l l 1 k be complex vector variables we show that the components of the map h b 1 x 1 d 1 b k x k d k are linearly dependent over c if and only if det b 1 b 1 d 1 b k b k d k 0 where means the hadamard product x and x d denote the conjugate transpose and the dth hadamard power of a matrix x respectively connections are established between the homogenous dependence problem hdp n d which arises in the study of the jacobian conjecture and the dependence problem for power linear maps pldp n d an algorithm is given to compute counterexamples to pldp n d from those to hdp n d and counterexamples to pldp n 3 are obtained for all n 67

riemannian geometry of quantum computation abstract a review is given of some recent developments in the differential geometry of quantum computation for which the quantum evolution is described by the special unitary unimodular group s u 2 n using the lie algebra s u 2 n detailed derivations are given of a useful riemannian geometry of s u 2 n including the connection and the geodesic equation for minimal complexity quantum computations

poisson brackets and two generated subalgebras of rings of polynomials let a f x1 x2 xn be a ring of polynomials over a field f on the variables x1 x2 xn it is well known see for example 11 that the study of automorphisms of the algebra a is closely related with the description of its subalgebras by the theorem of p m cohn 4 a subalgebra of the algebra f x is free if and only if it is integrally closed the theorem of a zaks 13 says that the dedekind subalgebras of the algebra a are rings of polynomials in a single variable a nowicki and m nagata 8 proved that the kernel of any nontrivial derivation of the algebra f x y char f 0 is also a ring of polynomials in a single generator an original solution of the occurrence problem for the algebra a using the groebner basis was given by d shannon and m sweedler 9 however the method of the groebner basis does not give any information about the structure of concrete subalgebras recall that the solubility of the occurrence problem for rings of polynomials over fields of characteristic 0 was proved earlier by g noskov 7 the present paper is devoted to the investigation of the structure of twogenerated subalgebras of a in the sequel we always assume that f is an arbitrary field of characteristic 0 let us denote by f the highest homogeneous part of an element f a and by f1 f2 fk the subalgebra of a generated by the elements f1 f2 fk a definition 1 a pair of polynomials f1 f2 a is called reduced if they satisfy the following conditions 1 f1 f2 are algebraically dependent 2 f1 f2 are algebraically independent 3 f1 f2 f2 f1 recall that a pair f1 f2 with condition 3 is usually called reduced condition 1 means that we exclude the trivial case when f1 f2 are algebraically independent we do not consider the case when f1 f2 are algebraically dependent concerning this case recall the well known theorem of s s abhyankar and t t moh 1 which says that if f g f x and f g f x then f or f

on the length of polynomial automorphisms of the affine plane the automorphism group of the affine plane is a mysterious and challenging object although we know that it is an amalgamated product of two well known subgroups many questions are still unsolved moreover the group has the structure of an infinite dimensional algebraic group but the interactions between these two structures are not yet clear in this paper we study the length of an element defined using the amalgamated structure with respect to the algebraic structure if the ground field is of characteristic zero we prove that the length is a lower semicontinuous function on the group

a note on the jacobian conjecture abstract in this note we show that if the druzkowski mappings f x x ax 3 i e f x x 1 a 11 x 1 a 1 n x n 3 x n a n 1 x 1 a nn x n 3 satisfies trj ax 3 0 then rank a 1 2 n where is the number of diagonal elements of a which are equal to zero furthermore we show the jacobian conjecture is true for the druzkowski mappings in dimension 9 in the case i 1 n a ii 0

some remarks on the jacobian conjecture and druzkowski mappings abstract in this paper we first show that the jacobian conjecture is true for non homogeneous power linear mappings under some conditions secondly we prove an equivalent statement about the jacobian conjecture in dimension r 1 and give some partial results for r 2 finally for a homogeneous power linear keller map f x h of degree d 2 we give the inverse polynomial map under the condition that j h 3 0 we shall show that deg f 1 d k if k 2 and j h k 1 0 but also give an example with d 2 and j h 4 0 such that deg f 1 d 3

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

on the algebraic identifiability of finite impulse response channels driven by linearly precoded signals it is common in wireless communications to perform some form of linear precodingoperation on the source signal prior to transmission over a channel although the traditional reason for precoding is to improve the performance of the communication system this paper draws attention to the fact that the receiver can identify the impulse response of the channel without any prior knowledge of the transmitted signal simply by solving a system of polynomial equations since different precoders lead to different systems of equations this paper addresses the fundamental issue of determiningwhich classes of linear precoders lead to a system of equations havinga unique solution in doingso basic properties of polynomial equations which are useful for studyingother identifiability issues commonly encountered in eng ineeringand the applied sciences are presented 2004 elsevier b v all rights reserved

quantifying the complexity of geodesic paths on curved statistical manifolds through information geometric entropies and jacobi fields abstract we characterize the complexity of geodesic paths on a curved statistical manifold m s through the asymptotic computation of the information geometric complexity v m s and the jacobi vector field intensity j m s the manifold m s is a 2 l dimensional gaussian model reproduced by an appropriate embedding in a larger 4 l dimensional gaussian manifold and endowed with a fisher rao information metric g with non trivial off diagonal terms these terms emerge due to the presence of a correlational structure embedding constraints among the statistical variables on the larger manifold and are characterized by macroscopic correlational coefficients r k first we observe a power law decay of the information geometric complexity at a rate determined by the coefficients r k and conclude that the non trivial off diagonal terms lead to the emergence of an asymptotic information geometric compression of the explored macrostates on m s finally we observe that the presence of such embedding constraints leads to an attenuation of the asymptotic exponential divergence of the jacobi vector field intensity

topologies and structures of the cremona groups we study the algebraic structure of the n dimensional cremona group and show that it is not an algebraic group of innite dimension ind group if n 2 we describe the obstruction to this which is of a topological nature by contrast we show the existence of a euclidean topology on the cremona group which extends that of its classical subgroups and makes it a topological group

addendum to on the derivative of the legendre function of the first kind with respect to its degree in our recent paper szmytkowski 2006 j phys a math gen 39 15147 2007 j phys a math theor 40 7819 corrigendum we have investigated the derivative of the legendre function of the first kind p z with respect to its degree in this addendum we derive some further representations of p z n with the results are used to obtain in an elementary way two formulae for the legendre function of the second kind of integer degree qn z

a note on parameter derivatives of the jacobi polynomials on the triangle the aim of the present paper is to obtain derivatives of the jacobi polynomials with two variables on the triangle with respect to their parameters by using the parameter derivatives of the classical jacobi polynomials p n x furthermore we give orthogonality properties of the parametric derivatives on the triangle

the tame and the wild automorphisms of polynomial rings in three variables let c f x1 x2 xn be the polynomial ring in the variables x1 x2 xn over a field f and let autc be the group of automorphisms of c as an algebra over f an automorphism autc is called elementary if it has a form x1 xi 1 xi xi 1 xn 7 x1 xi 1 xi f xi 1 xn where 0 6 f f f x1 xi 1 xi 1 xn the subgroup of autc generated by all the elementary automorphisms is called the tame subgroup and the elements from this subgroup are called tame automorphisms of c non tame automorphisms of the algebra c are called wild it is well known 6 9 10 11 that the automorphisms of polynomial rings and free associative algebras in two variables are tame at present a few new proofs of these results have been found see 5 8 however in the case of three or more variables the similar question was open and known as the generation gap problem 2 3 or tame generators problem 8 the general belief was that the answer is negative and there were several candidate counterexamples see 5 8 12 7 19 the best known of them is the following automorphism aut f x y z constructed by nagata in 1972 see 12 x x x yz z y y 2 x yz x x yz z z z

the lned and lfed conjectures for algebraic algebras abstract let k be a field of characteristic zero and a a k algebra such that all the k subalgebras generated by finitely many elements of a are finite dimensional over k a k e derivation of a is a k linear map of the form i for some k algebra endomorphism of a where i denotes the identity map of a in this paper we first show that for all locally finite k derivations d and locally finite k algebra automorphisms of a the images of d and i do not contain any nonzero idempotent of a we then use this result to show some cases of the lfed and lned conjectures proposed in 21 more precisely we show the lned conjecture for a and the lfed conjecture for all locally finite k derivations of a and all locally finite k e derivations of the form i with being surjective in particular both conjectures are proved for all finite dimensional k algebras furthermore some finite extensions of derivations and automorphism to inner derivations and inner automorphisms respectively have also been established this result is not only crucial in the proofs of the results above but also interesting on its own right

power linear keller maps with ditto triangularizations abstract we show that power linear keller maps f x 1 a 1 x d x 2 a 2 x d x n a n x d are linearly triangularizable if 1 rk a 2 or 2 cork a 2 and d 3 or 3 cork a 3 d 5 and the diagonal of a is nonzero furthermore we show that the triangularizations can be chosen power linear as well

on parameter derivatives of a family of polynomials in two variables the purpose of the present paper is to give the parameter derivative representations of the form p n k x y m 0 n 1 j 0 m d n j m p m j x y j 0 k e n j k p n j x y for a family of orthogonal polynomials of variables x and y with being a parameter and 0 k n n k 0 1 2 first we shall present the representations of the parameter derivatives of the generalized gegenbauer polynomials c n x with the help of the parameter derivatives of the classical jacobi polynomials p n x i e p n x and p n x then by using these derivatives we investigate the parameter derivatives for two variable analogues of the generalized gegenbauer polynomials furthermore we discuss orthogonality properties of the parametric derivatives of these polynomials

on the range of a hadamard power of a positive semidefinite matrix abstract for any n n complex matrix a and any integer k 1 we prove that the span of image of the map x ax k is equal to the range of aa k where x and x k denote the conjugate transpose and the k th hadamard power of a matrix x respectively this settles a conjecture due to gorni and tutaj gasinska related to the study of the jacobian conjecture

incompatibility of diophantine equations arising from the strong factorial conjecture abstract in the present article we establish new evidence in support of the strong factorial conjecture of edo and van den essen 2 by proving it in several special cases for example we show that the conjecture holds for powers of linear forms and sums of prime powers of the variables finally we show one way of constructing new examples of polynomials satisfying the conjecture using known examples

on additive nilpotency of jacobian matrices of polynomial maps abstract let h c n c n be a polynomial map jh the jacobian matrix of h and v h the linear subspace of m n c spanned by j h c n the set of evaluated jacobian matrices of h we describe the dimension of v h through a power linear gorni zampieri mate of h and give a numerical algorithm for computing this dimension and determining whether jh is additive nilpotent i e whether v h is a nilpotent subspace