an application of the kiefer wolfowitz equivalence theorem to a problem in hadamard transform optics on utilise le theoreme d equivalence de la theorie approchee des plans optimaux pour resoudre des problemes lies a l optique de la transformee de hadamard et les plans de ponderation

an operator valued extension of the super korteweg de vries equations an extension of the super korteweg de vries kdv integrable system in terms of operator valued functions is obtained following the ideas of gardner a general algebraic approach for finding the infinitely many conserved quantities of integrable systems is presented the approach is applied to the above described system and infinitely many conserved quantities are constructed in a particular case they reduce to the corresponding conserved quantities of super kdv

optimal weights for experimental designs on linearly independent support points an explicit formula is derived to compute the a optimal design weights on linearly independent regression vectors for the mean parameters in a linear model with homoscedastic variances the formula emerges as a special case of a general result which holds for a wide class of optimality criteria there are close links to iterative algorithms for computing optimal weights

a new integrable equation valued on a cayley dickson algebra we introduce a new integrable equation valued on a cayley dickson c d algebra in the particular case in which the algebra reduces to a complex one the new interacting term in the equation cancels and the equation becomes the known korteweg de vries kdv equation for each c d algebra the equation has an infinite sequence of local conserved quantities we obtain a backlund transformation in the sense of walhquist estabrook for the equation for any cayley dickson algebra and relate it to a generalized gardner equation from this the infinite sequence of conserved quantities follows directly we give the explicit expression for the first few from the backlund transformation we get the lax pair and the one soliton and two soliton solutions generalizing the known solutions for the quaternion valued kdv equation from the gardner equation we obtain the generalized modified kdv equation which also has an infinite sequence of conserved quantities the new integrable equation is preserved under a subgroup of the automorphisms of the c d algebra in the particular case of the algebra of octonions the equation is invariant under

braided version of shirshov witt theorem the shirshov witt theorem claims that every subalgebra of a free lie algebra is free in character istic zero this theorem can be restated in terms of a free associative algebra every hopf subalgebra of a free algebra kxiwith the coproduct xi xi 1 1 xi is free and it is freely generated by primitive elements our aim is to extend this result to free algebras with a braided coproduct as far as possible by means of p m cohn theory we show that if a subalgebra is a right categorical right coideal then it is free we consider more thoroughly involutive braidings 2 id over a field of zero characteristic in this case every braided hopf subalgebra is generated by primitive elements moreover the space of all primitive elements forms a free lie algebra in the context of this re sult we discuss the situation that arises around the problem of embedding of a lie algebra in its associative universal enveloping algebra 2005 elsevier inc all rights reserved

on twisting of finite dimensional hopf algebras in this paper we study the properties of drinfeld s twisting for finite dimensional hopf algebras we determine how the integral of the dual to a unimodular hopf algebra h changes under twisting of h we show that the classes of cosemisimple unimodular cosemisimple involutive cosemisimple quasitriangular finite dimensional hopf algebras are stable under twisting we also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular hopf algebra by two different twists on two sides such twists are closely related to bi galois extensions and describe the representation theory of its dual next we define the notion of a non degenerate twist for a hopf algebra h and set up a bijection between such twists for h and h this bijection is based on miyashita ulbrich actions of hopf algebras on simple algebras it generalizes to the non commutative case the procedure of inverting a non degenerate skew symmetric bilinear form on a vector space finally we apply these results to classification of twists in group algebras and of cosemisimple triangular finite dimensional hopf algebras in positive characteristic generalizing the previously known classification in characteristic zero

connected braided hopf algebras abstract we prove a braided version of kostant cartier milnor moore theorem the category of connected cocommutative 2 id braided hopf algebras over a field of zero characteristic is equivalent via the enveloping construction to the category of generalized lie algebras this statement includes an embedding of any generalized lie algebra into its universal enveloping algebra the isomorphism h u prim h and primitive generation results the embedding theorem may be derived from the poincare birkhoff witt theorem for quadratic algebras of koszul type remark c we also provide a direct proof that uses neither koszul cohomologies nor the algebraic deformation theory we consider the primitive generation problem for subalgebras biideals and homomorphic images of connected braided hopf algebras in much more general context when the braiding is not necessary involutive and even it is not necessary invertible

on the duality between locally optimal tests and optimal experimental designs abstract locally unbiased tests with maximal power curvature are determined as the solutions of an optimization problem which turns out to be of dual type as compared to the optimal design problem in both cases the proper optimization problem is concerned with matrices only and the transition from the matrix problem to the original variables is a separate second step this approach provides a novel statistical interpretation of the dual problem that arises with the optimal design problem

on the antipode of a quasitriangular hopf algebra let a r be a quasitriangular hopf algebra with antipode s in the category of vector spaces over a field k in this paper we show that s2 is inner which means s is bijective and when a is finite dimensional there is a grouplike element h e a such that s4 a h r for all a e a suppose that a is any hopf algebra with antipode s over the field k proposition 1 of section 1 gives a sufficient condition for 3 to be inner which we apply to quasitriangular hopf algebras in section 2 its statement and proof are based on calculations found in 2 pp 66671 generally s2 need not be inner suppose that a is finite dimensional if s2 were inner then all ideals of a would be invariant under s2 and hence all subcoalgebras of the dual hopf algebra a would be invariant under s where s s is the antipode of a there are examples 5 over algebraically closed lields in which the latter is not the case section 2 begins with a discussion of properties of quasit angular hopf algebras a r over the held k used in this paper write r cr c r ea a and set u cs r r the main result of section 2 is that u is invertible and s2 a uau for all a e a this result was established in z exercise 7 3 61 under the hypothesis that s is bijective if a is any finite dimensional hopf algebra with antipode s over the field k there are distinguished grouplike elements g e a and tl e a which relate left and right integrals and which together describe s4 in section 3 we consider s4 when a r is a unite dimensional quasit angular hopf algebra let h iiu where u c s r r l is defined as above and u s u i by relating h to g and 1 we are able to show that h is a grouplike element and that s4 a hah for all a e a when a is unimodular we prove that h g thus the product vu does not depend on r in the unimodular case we assume that the reader has some familiarity with the elementary aspects of the theory of hopf algebras a recommended reference is 6

further characterizations of design optimality and admissibility for partial parameter estimation in linear regression the paper gives a contribution to the problem of finding optimal linear regression designs when only s out of k regression parameters are to be estimated also a treatment of design admissibility for the parameters of interest is included previous results of kiefer and wolfowitz 1959 karlin and studden 1966 and atwood 1969 are generalized in particular a connection to tchebycheff type approximation of ri valued functions is found which has been known in case s 1 strengthened versions of the results are obtained for invariant designs in situations when invariance properties of the regression setup are available applications are given to multiple quadratic regression and to one dimensional polynomial regression 1 introduction consider a linear regression y x a f x x e with an unknown coefficient vector a e r k and a known r valued function f on the experimental region y the range f y is assumed to be compact as usual observations y1 yn on the dependent variable y taken at points xl xn e e x respectively are assumed to have expectations y x 1 0 and to be pairwise uncorrelated suppose that the experiment aims at estimating ka where k is a given s x k matrix of rank s for example when the first s components of a are to be estimated one has k is 0 where it denotes the unit matrix of order s we will study design optimality and admissibility for estimating ka within the approximate design theory an approximate design is a probability measure on x with finite support the infornation matrix per observation and unit of variance of f is m t ff x f x dd x e f x f x ox f x supp

supersymmetric extension of the korteweg de vries equation it is shown that among a one parameter family of supersymmetric extensions of the korteweg de vries equation there is a special system that has an infinite number of conservation laws which can be formulated in the second hamiltonian structure and which has a nontrivial lax representation its modified version is also discussed

on n 3 super korteweg de vries equation a one parameter family is constructed of n 3 supersymmetric and so 3 symmetric extensions of the korteweg de vries kdv equation as a hamiltonian flow on a n 3 superconformal algebra and it is argued that it is nonintegrable for any choice of the parameter then a modified n 3 super kdv equation is proposed which possesses the higher order conserved quantities and so is a candidate for an integrable system upon reduction to n 2 it yields the recently discussed would be integrable version of the n 2 super kdv equation in the bosonic core it contains a coupled system of the kdv type equation and a three component generalization of the mkdv equation a hamiltonian formulation of the new n 3 super kdv equation as a flow on some contraction of the direct sum of two n 3 superconformal algebras is given

the gardner category and nonlocal conservation laws for n 1 super kdv the nonlocal conserved quantities of the n 1 super kdv are obtained using a gardner map a fermionic substitution semigroup and the resulting gardner category are defined and several propositions concerning their algebraic structure are obtained this algebraic framework makes it possible to define general transformations between different nonlinear susy differential equations a susy ring extension is then introduced to deal with the nonlocal conserved quantities of skdv the algebraic version of the nonlocal conserved quantities is solved in terms of the exponential function applied to the d 1 of the local conserved quantities of skdv finally the same formulas are shown to work for rapidly decreasing superfields