trace like functionals on the double of the taft hopf algebra abstract the cocommutative elements of the dual of the double d h which are the trace like functionals on d h of the taft hopf algebra h over a field k are studied in detail the subalgebra of cocommutative elements of d h is isomorphic to the center of d h trace like functionals on d h determine regular isotopy invariants of oriented knots and links specific calculations made here suggest that further study of these invariants is definitely warranted

tests for two outliers in a linear model the detection of two outliers in a linear model is considered the case of two outliers in the same direction is studied in detail the first bonferroni inequality is used to obtain nominal upper percentage points for a random sample from a normal distribution the proposed statistic reduces to the well known murphy s test statistic similarly for two outliers on either direction our statistic reduces to the studentized range statistic an approximate relationship between the critical points of murphy s test and the studentized range statistic is investigated

connes moscovici characteristic map is a lie algebra morphism let h be a hopf algebra with a modular pair in involution character 1 let a be a module algebra over h equipped with a non degenerated character invariant 1 trace tau we show that connes moscovici characteristic map varphi tau hc character 1 h rightarrow hc lambda a is a morphism of graded lie algebras we also have a morphism phi of batalin vilkovisky algebras from the cotorsion product of h text cotor h bbbk bbbk to the hochschild cohomology of a hh a a let k be both a hopf algebra and a symmetric frobenius algebra suppose that the square of its antipode is an inner automorphism by a group like element then this morphism of batalin vilkovisky algebras phi text cotor k vee mathbb f mathbb f cong text ext k mathbb f mathbb f hookrightarrow hh k k is injective

congruence subgroups and generalized frobenius schur indicators we introduce generalized frobenius schur indicators for pivotal categories in a spherical fusion category mathcal c an equivariant indicator of an object in mathcal c is defined as a functional on the grothendieck algebra of the quantum double z mathcal c via generalized frobenius schur indicators the set of all equivariant indicators admits a natural action of the modular group using the properties of equivariant indicators we prove a congruence subgroup theorem for modular categories as a consequence all modular representations of a modular category have finite images and they satisfy a conjecture of eholzer in addition we obtain two formulae for the generalized indicators one of them a generalization of bantay s second indicator formula for a rational conformal field theory this formula implies a conjecture of pradisi sagnotti stanev as well as a conjecture of borisov halpern schweigert

monte carlo results on several new and existing tests for the error component model abstract the breusch pagan test is a popular test for testing zero variances in a random error component model realizing the one sided nature of this problem honda 1985 derived the one sided lm test for the one way error component model and suggested a handy one sided candidate for the two way error component model in this paper we derive a new one sided test for testing the two way error component model based upon the results of gourieroux holly and monfort 1982 next we derive a lm test for testing zero individual time specific effects assuming that the other time period individual effects are present this is in contrast to the usual lm test which assumes that under the null hypothesis both effects are absent exact power comparisons whenever possible as well as monte carlo experiments are employed to compare the performance of these tests with the corresponding likelihood ratio and anova f tests

on kaplansky s fifth conjecture abstract we prove that the antipode of a semisimple hopf algebra is an involution if the characteristic of the base field is very large

the character table for a hopf algebra arising from the drinfel d double abstract in this paper we give a definition of the character table for a hopf algebra h over a commutative field k of characteristic 0 when h is a finite dimensional semisimple hopf algebra over an algebraically closed field k of characteristic 0 we obtain a formula for the character table which extends the notion of character table for a finite group

invariants of spin three manifolds from chern simons theory and finite dimensional hopf algebras abstract a version of kirby calculus for spin and framed three manifolds is given and is used to construct invariants of spin and framed three manifolds in two situations the first is ribbon categories which possess odd degenerate objects this case includes the quantum group situations corresponding to the half integer level chern simons theories conjectured to give spin tqfts by dijkgraaf and witten 1990 commun math phys 129 393 429 in particular the spin invariants constructed by kirby and melvin 1991 invent math 105 473 545 are shown to be identical to the invariants associated to so 3 second an invariant of spin manifolds analogous to the hennings invariant is constructed beginning with an arbitrary factorizable unimodular quasitriangular hopf algebra in particular a framed manifold invariant is associated to every finite dimensional hopf algebra via its quantum double and is conjectured to be identical to kuperberg s noninvolutory invariant of framed manifolds associated to that hopf algebra

sl 2 z action for ribbon quasi hopf algebras abstract we study the universal hopf algebra l of majid and lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi hopf algebra a we show that l a with coadjoint action and compute the hopf algebra structure morphisms of l in terms of the defining data of a we give explicitly the condition on a which makes rep a factorisable and compute lyubashenko s projective s l 2 z action on the centre of a in this case the point of this exercise is to provide the groundwork for the applications to ribbon categories arising in logarithmic conformal field theories in particular symplectic fermions and w p models and to test a conjectural non semisimple verlinde formula

representations of finite dimensional hopf algebras abstract lethdenote a finite dimensional hopf algebra with antipodesover a field k we give a new proof of the fact due to os thathis a symmetric algebra if and only ifhis unimodular ands2is inner ifhis involutory and not semisimple then the dimensions of all projectiveh modules are shown to be divisible by char k in the case where k is a splitting field forh we give a formula for the rank of the cartan matrix ofh reduced mod char k in terms of an integral forh explicit computations of the cartan matrix the ring structure ofg0 h and the structure of the principal indecomposable modules are carried out for certain specific hopf algebras in particular for the restricted enveloping algebras of completely solvablep lie algebras and ofsl 2 k

hopf algebras of dimension pq ii abstract let h be a hopf algebra of dimension pq over an algebraically closed field of characteristic zero where p q are odd primes with p q 4 p 11 we prove that h is semisimple and thus isomorphic to a group algebra or the dual of a group algebra

the pivotal cover and frobenius schur indicators abstract in this paper we introduce the notion of the pivotal cover c piv of a left rigid monoidal category c to develop a theoretical foundation for the theory of frobenius schur fs indicators in non pivotal settings for an object v c piv the n r th fs indicator n r v is defined by generalizing that of an object of a pivotal monoidal category this notion gives a categorical viewpoint to some recent results on generalizations of fs indicators based on our framework we also study the fs indicators of the adjoint object in a finite tensor category which can be considered as a generalization of the adjoint representation of a hopf algebra the indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor

on the even powers of the antipode of a finite dimensional hopf algebra abstract the trace of powers of the square of the antipode s 2 of a finite dimensional hopf algebra a over a field k is studied it is shown in many cases that the trace function vanishes on s 2 m when s 2 m 1 a finer properties of the antipode are related to this phenomenon

higher genus mapping class group invariants from factorizable hopf algebras abstract lyubashenko s construction associates representations of mapping class groups map g n of riemann surfaces of any genus g with any number n of holes to a factorizable ribbon category we consider this construction as applied to the category of bimodules over a finite dimensional factorizable ribbon hopf algebra h for any such hopf algebra we find an invariant of map g n for all values of g and n more generally we obtain such invariants for any pair h where is a ribbon automorphism of h our results are motivated by the quest to understand higher genus correlation functions of bulk fields in two dimensional conformal field theories with chiral algebras that are not necessarily semisimple so called logarithmic conformal field theories

on the center of the small quantum group using the quantum fourier transform f we describe the block decomposition and multiplicative structure of a subalgebra z f z of the center of the small quantum group uqfin g at a root of unity it contains the known subalgebra z which is isomorphic to the algebra of characters of finite dimensional uqfin g modules we prove that the intersection z f z coincides with the annihilator of the radical of z applying representation theoretical methods we show that z surjects onto the algebra of endomorphisms of certain indecomposable projective modules over uqfin g in particular this leads to the conclusion that the center of uqfin g coincides with z f z in the case g sl2

marginal likelihood score based tests of regression disturbances in the presence of nuisance parameters abstract this paper is concerned with tests of the covariance matrix of the disturbances in the linear regression model that involves nuisance parameters which cannot be eliminated by the usual invariance arguments score based tests namely lagrange multiplier lm and locally most mean powerful lmmp tests are derived from the marginal likelihood applications considered include i testing for random regression coefficients ii testing for second order autoregressive ar 2 disturbances and iii testing for arma 1 1 disturbances each in the presence of ar 1 disturbances an empirical size and power comparison shows that the new tests typically have more accurate asymptotic critical values and slightly more power than their respective conventional counterparts

on the structure of irreducible yetter drinfeld modules over quasi triangular hopf algebras abstract let h r be a finite dimensional semisimple and cosemisimple quasi triangular hopf algebra over a field k in this paper we give the structure of irreducible objects of the yetter drinfeld module category yd h h let h r be the majid s transmuted braided group of h r we show that h r is cosemisimple as a coalgebra let h r d 1 d r be the sum of minimal h adjoint stable subcoalgebras for each i 1 i r we choose a minimal left coideal w i of d i and we can define the r adjoint stable algebra n w i of w i using ostrik s theorem on characterizing module categories over monoidal categories we prove that v h h yd is irreducible if and only if there exists an i 1 i r and an irreducible right n w i module u i such that v u i n w i h w i our structure theorem generalizes the results of dijkgraaf pasquier roche and gould on yetter drinfeld modules over finite group algebras if k is an algebraically closed field of characteristic 0 we stress that the r adjoint stable algebra n w i is an algebra over which the dimension of each irreducible right module divides its dimension

an approach to quasi hopf algebras via frobenius coordinates abstract we study quasi hopf algebras and their subobjects over certain commutative rings from the point of view of frobenius algebras we introduce a type of radford formula involving an anti automorphism and the nakayama automorphism of a frobenius algebra then view several results in quantum algebras from this vantage point in addition separability and strong separability of quasi hopf algebras are studied as frobenius algebras

remarks on quantum integration we give a general integration prescription for finite dimensional braided hopf algebras deriving the n dimensional quantum superplane integral as an example the transformation properties of the integral on the quantum plane are found we also discuss integration on quantum group modules that lack a hopf structure

radford drinfeld and cardy boundary states in the 1 p logarithmic conformal field models we introduce p 1 pseudocharacters in the space of 1 p model vacuum torus amplitudes to complete the distinguished basis in the 2p dimensional fusion algebra to a basis in the whole 3p 1 dimensional space of torus amplitudes and the structure constants in this basis are not necessarily non negative integer numbers we obtain a generalized verlinde formula that gives these structure constants in the context of theories with boundaries we identify the space of vacuum torus amplitudes with the space of ishibashi states then we propose 3p 1 boundary states satisfying the cardy condition

some interrelations between hopf algebras and their duals abstract in this paper we study certain interrelations between subsets of hopf algebras h and their duals which stem from various morphisms t defined on h we define and study t cocommutative elements cocommutative elements generators of h as an h module and invariants under various actions and coactions are all examples an important role is played by nakayama automorphisms these ideas are applied to co quasitriangular hopf algebras in particular when they are factorizable then the drinfeld element gives rise to a nakayama automorphism and to a special element in the dual explicit calculations are carried out for the drinfeld double

invariants of 3 manifolds and projective representations of mapping class groups via quantum groups at roots of unity an example of a finite dimensional factorizable ribbon hopf algebra is given by a quotienth u q g of the quantized universal enveloping algebrau q g at a root of unityq of odd degree the mapping class groupm g 1 of a surface of genusg with one hole projectively acts by automorphisms in theh moduleh g ifh is endowed with the coadjointh module structure there exists a projective representation of the mapping class groupm g n of a surface of genusg withn holes labeled by finite dimensionalh modulesx 1 x n in the vector space hom h x 1 x n h g an invariant of closed oriented 3 manifolds is constructed modifications of these constructions for a class of ribbon hopf algebras satisfying weaker conditions than factorizability including most ofu q g at roots of unityq of even degree are described

kazhdan lusztig dual quantum group for logarithimic extensions of virasoro minimal models we derive and study a quantum group gp q that is kazhdan lusztig dual to the w algebra wp q of the logarithmic p q conformal field theory model the algebra wp q is generated by two currents w z and w z of dimension 2p 1 2q 1 and the energy momentum tensor t z the two currents generate a vertex operator ideal r with the property that the quotient wp q r is the vertex operator algebra of the p q virasoro minimal model the number 2pq of irreducible gp q representations is the same as the number of irreducible wp q representations on which r acts nontrivially we find the center of gp q and show that the modular group representation on it is equivalent to the modular group representation on the wp q characters and pseudocharacters the factorization of the gp q ribbon element leads to a factorization of the modular group representation on the center we also find the gp q grothendieck ring which is presumably the logarithmic fusion of the p q model

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

generalized gravity gradient analysis for 2 d inversion gravity gradient profiles across subsurface structures that are approximately 2 d contain diagnostic information regarding depth size and structure geometry gradient space plots i e plots of horizontal gradient versus vertical gradient present the complete magnitude and phase information in the gradient profiles simultaneously considerable previous work demonstrates the possibility for complete structural interpretation of a truncated plate model from the gradient space plot the qualitative and quantitative diagnostic information contained in gradient space plots is general however examination of the characteristics of gradient space plots reveals that 2 d structures are readily classified as extended or localized for example the truncated plate model is an extended model while the faulted plate model is a localized model comparison of measured or calculated gradient space plots to a model gradient space plot catalog allows a rapid qualitative determination of structure or geometry cor

structure constants related to symmetric hopf algebras abstract let k be an algebraically closed field of characteristic 0 in this paper we continue our study of structure constants for semisimple hopf algebras h whose character algebra is commutative and for non semisimple factorizable ribbon hopf algebras this is done from the point of view of symmetric algebras such as group algebras in particular we consider general fusion rules which are structure constants associated to products of irreducible characters and structure constants associated to generalizations of class sums and conjugacy classes our methods are reminiscent on one hand of the methods employed when analyzing tilting modules for certain quantum groups and on the other hand of splitting modules for certain drinfeld doubles the family of irreducible characters is divided in two according to the vanishing of their quantum dimension the fusion rules on the character algebra are computed with respect to this division

optimal tests when a nuisance parameter is present only under the alternative this paper derives asymptotically optimal tests for testing problems in which a nuisance parameter exists under the alternative hypothesis but not under the null for example the results apply to tests of structural change with unknown changepoint the testing problem considered is nonstandard and the classical asymptotic optimality results for the lagrange multiplier wald and likelihood ratio do not apply a weighted average power criterion is used here to generate optimal tests this criterion is similar to that used by a wald 1943 to obtain the classical asymptotic optimality properties of wald tests in regular testing problems copyright 1994 by the econometric society

indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to sl2 we study the tensor structure of the category of finite dimensional modules of the restricted quantum enveloping algebra associated to sl2 tensor product decomposition rules for all indecomposable modules are explicitly given as a by product it is also shown that the category of finite dimensional modules of the restricted quantum enveloping algebra associated to sl2 is not a braided tensor category

on two invariants of three manifolds from hopf algebras abstract we prove a 20 year old conjecture concerning two quantum invariants of three manifolds that are constructed from finite dimensional hopf algebras namely the kuperberg invariant and the hennings kauffman radford invariant the two invariants can be viewed as a non semisimple generalization of the turaev viro barrett westbury tvbw invariant and the witten reshetikhin turaev wrt invariant respectively by a classical result relating tvbw and wrt it follows that the kuperberg invariant for a semisimple hopf algebra is equal to the hennings kauffman radford invariant for the drinfeld double of the hopf algebra however whether the relation holds for non semisimple hopf algebras has remained open partly because the introduction of framings in this case makes the kuperberg invariant significantly more complicated to handle we give an affirmative answer to this question an important ingredient in the proof involves using a special heegaard diagram in which one family of circles gives the surgery link of the three manifold represented by the heegaard diagram

solvability for semisimple hopf algebras via integrals abstract we use integrals of left coideal subalgebras to develop harmonic analysis for semisimple hopf algebras we show how n the space of functional on n is embedded in h we define a bilinear form on n and show that irreducible n characters are orthogonal with respect to that form we then give an explicit formula for induced characters of n and show how the induced characters are embedded in r h in the second part we give an intrinsic definition for solvable semisimple hopf algebras via left coideal subalgebras and their integrals we show how this definition generalizes solvability for finite groups in particular commutative and nilpotent hopf algebras are solvable we finally prove an analogue of burnside theorem a semisimple quasitriangular hopf algebras of dimension p a q b is solvable

modular group representations and fusion in logarithmic conformal field theories and in the quantum group center the sl 2 representation on the center of the restricted quantum group open image in new window at the primitive 2pth root of unity is shown to be equivalent to the sl 2 representation on the extended characters of the logarithmic 1 p conformal field theory model the multiplicative jordan decomposition of the open image in new window ribbon element determines the decomposition of into a pointwise product of two commuting sl 2 representations one of which restricts to the grothendieck ring this restriction is equivalent to the sl 2 representation on the 1 p characters related to the fusion algebra via a nonsemisimple verlinde formula the grothendieck ring of open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the 1 p logarithmic conformal field theory model as a by product we derive q binomial identities implied by the fusion algebra realized in the center of open image in new window

the asymptotic expansion of certain multi dimensional generalized functions abstract we apply regularization of divergent integrals in the derivation of the asymptotic expansion of certain multi dimensional generalized functions we further present several illustrations to demonstrate that the asymptotic development of generalized functions provides a lucid formulation of many concepts in asymptotic analysis such as the expansion of oscillatory integrals and the expansion of certain series

hilbert transform of gravity gradient profiles special cases of the general gravity gradient tensor in the fourier transform domain the hilbert transform has been extensively used in gravity gradient interpretation since the mid 1970s when stanley 1976 and stanley and green 1976 followed the work on the relationships between magnetic field components nabighian 1972 shuey 1972 by showing that the vertical g z z and horizontal g z x gravity gradients of a 2 d truncated plate are related by the hilbert transform butler 1983 1995 has extended the use of the hilbert transform in interpreting g z z and g z x resulting from additional geometries

small disturbance asymptotics and the durbin watson and related tests in the dynamic regression model abstract until recently it was thought inappropriate to apply the durbin watson dw test to a dynamic linear regression model because of the lack of appropriate critical values recently inder 1986 used a modified small disturbance distribution sdd to find approximate critical values this paper studies the exact sdd of statistics of the same general form as the dw statistic and suggests some changes to inder s result we show how to calculate true small disturbance critical values and bounds for these critical values that take into account the exogenous regressors our results give a justification for the use of the familiar tables of bounds when the dw test is applied to a dynamic regression model

involutory hopf algebras in 1975 kaplansky conjectured that a finite dimensional semisimple hopf algebra is necessarily involutory twelve years later larson and radford proved the conjecture in characterisitic 0 and obtained significant partial results in positive characteristics the goal of this paper is to offer an efficient proof of these results using rather minimal prerequisites no harpoons and gratifyingly few hits let h be a finite dimensional hopf algebra over a field k and let s h h denote its antipode then h is said to be involutory if s2 idh or equivalently if s is an involution on h in k kaplansky conjectured that a semisimple hopf algebra is necessarily involutory and in a fundamental paper lr2 larson and radford proved this assertion at least for fields of characteristic 0 furthermore they showed that if h is semisimple and cosemisimple and if chark p dimk h 2 then h must again be involutory a nice exposition of this material can be found in n while lr2 is reasonably brief it depelids heavily on certain earlier results on characters of hopf algebras as contained in l and lr 1 and the combined proof is substantial the goal of this paper is to present an efficient proof of the main results of lr2 requiring minimal prerequisites as will be apparent our proof follows the same basic outline as that of larson and radford and therefore we offer nothing new in this regard however by computing traces on certain tensor modules we are able to obtain the necessary character formulas quite quickly and this is where the simplifications occur in some sense the hero of this approach is the linear transformation t described below if v is a left h module then h 0 v is well known to be h isomorphic to h 0 vj where vj is equal to v but with the trivial action determined by the counit e this is proved quite simply by considering the map t h v h v given by t h ov hihi h2v for all h e h v e v h indeed one shows that t is invertible with inverse t 1 h x v e hz x s h2 v h received by the editors july 13 1993 and in revised form june 8 1994 originally communicaated to the proceedings of the ams by lance w small 1991 mathematics subject classification primary 1 6w30 research supported in part by nsf grants dms 9224662 and dms 9003374 1995 american mathematical society

optimal weighted average power similar tests for the covariance structure in the linear regression model this paper suggests a procedure for the construction of optimal weighted average power similar tests for the error covariance matrix of a gaussian linear regression model when the alternative model belongs to the exponential family the paper uses a saddlepoint approximation to construct simple test statistics for a large class of problems and overcomes the computational burden of evaluating the complicated integrals arising in the derivation of optimal weighted average power tests extensions to panel data models are considered applications are given to tests for error autocorrelation in the linear regression model and in a panel data framework

kirby elements and quantum invariants we define the notion of a kirby element of a ribbon category mathcal c not necessarily semisimple kirby elements lead to 3 manifold invariants we characterize a class of kirby elements the algebraic kirby elements in terms of the structure maps of a hopf algebra in mathcal c this class is sufficiently large to recover the quantum invariants of 3 manifolds of reshetikhin and turaev of hennings kauffman and radford and of lyubashenko when these are well defined the cases of a semisimple ribbon category and of a category of representations are explored in detail

fast evaluation of the distribution of the durbin watson and other invariant test statistics in time series regression abstract a method is given for evaluating p values in o n operations for a general class of invariant test statistics that can be expressed as the ratio of quadratic forms in time series regression residuals the best known of these is the durbin watson statistic although several others have been discussed in the literature the method is numerically exact in the sense that the user specifies the error tolerance at the outset as with existing exact methods the problem is reexpressed in terms of the distribution function of a single quadratic form in independent normals which is evaluated by numerically inverting its characteristic function existing methods however calculate the characteristic function by reducing the matrix defining the quadratic form to either eigenvalue or tridiagonal form each of which requires o n 3 operations for sample size n whereas the new method uses a modification of the kalman filter to do it in o n operations moreover the new method has minimal storage requiremen

ribbon transformations integrals and triangular decompositions abstract we study the theory of integrals in yetter drinfel d hopf algebras and use the results to determine the integrals of hopf algebras with triangular decomposition

non semisimple hopf algebras of dimension p2 abstract let h be a finite dimensional hopf algebra with antipode s of dimension pq over an algebraically closed field of characteristic 0 where p q are odd primes if h is not semisimple then the order of s 4 is p and tr s 2 p is an integer divisible by p 2 in particular if dim h p 2 we prove that h is isomorphic to a taft algebra this completes the classification for the hopf algebras of dimension p 2

radford s s 4 formula for co frobenius hopf algebras abstract this note extends radford s formula for the fourth power of the antipode of a finite dimensional hopf algebra to co frobenius hopf algebras and studies equivalent conditions to a hopf algebra being involutory for finite dimensional and co frobenius hopf algebras

modular invariant frobenius algebras from ribbon hopf algebra automorphisms abstract for any finite dimensional factorizable ribbon hopf algebra h and any ribbon automorphism of h we establish the existence of the following structure an h bimodule f and a bimodule morphism z from lyubashenko s hopf algebra object k for the bimodule category to f this morphism is invariant under the natural action of the mapping class group of the one punctured torus on the space of bimodule morphisms from k to f we further show that the bimodule f can be endowed with a natural structure of a commutative symmetric frobenius algebra in the monoidal category of h bimodules and that it is a special frobenius algebra iff h is semisimple the bimodules k and f can both be characterized as coends of suitable bifunctors the morphism z is obtained by applying a monodromy operation to the coproduct of f a similar construction for the product of f exists as well our results are motivated by the quest to understand the bulk state space and the bulk partition function in two dimensional conformal field theories with chiral algebras that are not necessarily semisimple

relations between horizontal and vertical gradients of potential fields nabighian 1984 and nelson 1986 have both derived relations between horizontal and vertical gradients of magnetic fields or any other potential field whereby a knowledge of one enables one to calculate the other as pointed out by nelson nabighian s derivation is quite lengthy in the following i offer an alternative derivation in the wavenumber domain which is even shorter than that given by nelson

exact tests in single equation autoregressive distributed lag models abstract for hypotheses on the coefficient values of the lagged dependent variables in the arx class of dynamic regression models test procedures are developed which yield exact inference for given up to an unknown scale factor distribution of the innovation errors they include exact tests on the maximum lag length for structural change and on the presence of seasonal or multiple unit roots i e they cover situations where usually asymptotic and non exact t f aoc adf or hegy tests are employed the various procedures are demonstrated and compared in illustrative empirical models and the approach is critically discussed

the alternative durbin watson test an assessment of durbin and watson s choice of test statistic abstract this paper examines durbin and watson s 1950 choice of test statistic for their test of first 0order autoregressive regression disturbances attention is focused on an alternative statistic d theoretical and empirical power properties of the d test are compared with those of the durbin watson test the former is found to be locally best invariant while the latter is approximately locally best invariant the d test is also found to be more powerful than its counterpart against negative autocorrelation and for small values of the autocorrelation coefficient against positive autocorrelation selected bounds for significance points of d are tabulated

on hopf algebras of dimension 4p abstract in this paper we prove that a non semisimple hopf algebra h of dimension 4 p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided h contains more than two group like elements in particular we prove that non semisimple hopf algebras of dimensions 20 28 and 44 are pointed or their duals are pointed and this completes the classification of hopf algebras in these dimensions

the trace space and kauffman s knot invariants the traces in the construction of kauffman s knot invariants are studied the trace space is determined for a semisimple finite dimensional quantum hopf algebra and the best lower bound of the dimension of the trace space is given for a unimodular finite dimensional quantum hopf algebra

exact inference methods for first order autoregressive distributed lag models methods are proposed to build exact tests and confidence sets in the linear first order autoregressive distributed lag model with i i d disturbances for general linear hypotheses on the regression coefficients inference procedures are obtained which have known level the tests proposed are either similar i e they have constant rejection probability for all data generating processes consistent with the null hypothesis or use bounds which are free of nuisance parameters correspondingly the confidence sets are either similar with known size i e they have constant coverage probability or conservative we also develop exact tests and confidence sets for various nonlinear transformations of model parameters such as long run multipliers and mean lags the practical usefulness of these exact methods which are also asymptotically valid under weak regularity conditions is illustrated by some power comparisons and with applications to a dynamic trend model of money velocity and a model of money demand

characters and a verlinde type formula for symmetric hopf algebras abstract we study certain aspects of finite dimensional non semisimple symmetric hopf algebras h and their duals h we focus on the set i h of characters of projective h modules which is an ideal of the algebra of cocommutative elements of h this ideal corresponds via a symmetrizing form to the projective center higman ideal of h which turns out to be h where is an integral of h and is the left adjoint action of h on itself we describe h via primitive and central primitive idempotents of h we also show that it is stable under the quantum fourier transform our best results are obtained when h is a factorizable ribbon hopf algebra over an algebraically closed field of characteristic 0 in this case h is also the image of i h under a translated drinfel d map we use this fact to prove the existence of a steinberg like character the above ingredients are used to prove a verlinde type formula for h

modified lagrange multiplier tests for problems with one sided alternatives abstract a modified lagrange multiplier test statistic is proposed which takes explicit account of the one sided nature of the alternative in problems where the null hypothesis specifies that the true value of the parameter vector lies on the boundary of the parameter space computation of this statistic requires only the constrained maximum likelihood estimator conditions for the consistency of tests based on this statistic are examined and it is shown that the distribution of the statistic is not affected of nuisance parameters are allowed to lie on the boundary of the parameter space

locally best invariant tests of the error covariance matrix of the linear regression model summary this paper considers a class of hypothesis testing problems concerning the covariance matrix of the disturbances in the classical linear regression model a test that is locally best invariant against one sided alternative hypotheses is constructed and shown to be identical to a one sided version of the lagrange multiplier test