toeplitz and hankel operators on bergman spaces in this paper we consider toeplitz and hankel operators on the bergman spaces of the unit ball and the polydisk in c n whose symbols are bounded measurable functions we giv e necessary and sufficient conditions on the symbols for these operators to be compact we study the fredholm theory of topelitz operators for which the corresponding hankel operator is compact for these toeplitz operators the essential spectrum is computed and shown to be connected we also consider symbols that extend to continuous functions on the maximal ideal space of h for these symbols we describe when the toeplitz or hankel operators are compact

products of bergman space toeplitz operators on the polydisk motivated by recent works of ahern and breve rm c u breve rm c kovic on the disk we study the generalized zero product problem for toeplitz operators acting on the bergman space of the polydisk first we extend the results to the polydisk next we study the generalized compact product problem our results are new even on the disk as a consequence on higher dimensional polydisks we show that the generalized zero and compact product properties are the same for toeplitz operators in a certain case

carleson measures and multipliers for dirichlet spaces abstract we characterize carleson measures on the dirichlet spaces our result leads to necessary and sufficient conditions for multipliers of the dirichlet spaces

on the connectedness of solution sets in linear complementarity problems abstract we investigate conditions on a square matrix m for which every lcp m q with q arbitrary has a connected solution set we show that a matrix with this property is necessarily fully semimonotone using degree theory we show that the solution set of lcp m q corresponding to a p 0 matrix is connected if there is a bounded connected component in the solution set

finite rank sums of products of toeplitz and hankel operators abstract on the dirichlet space of the unit disk we consider operators which are finite sums of toeplitz products hankel products or products of toeplitz and hankel operators we then give characterizations of when such operators have finite rank on the dirichlet space

dines fourier motzkin quantifier elimination and an application of corresponding transfer principles over ordered fields a constructive procedure using dines fourier motzkin elimination is given for eliminating quantifiers in a linear first order formula over ordered fields an ensuing transfer principle is illustrated by showing that a locally one to one affine map is globally one to one and onto all over ordered fields

the predual and second predual of wa we identify the predual and second predual of the space w when 0 w bmoa our results are similar to fefferman s and sarasons duality theorems the study of boundedness and compactness of hankel operators on dirichiet type spaces is the key to our approach a new result for the schatten p class of this operator is also indicated

an invariant volume mean value property if is harmonic and integrable over the open unit disc u then so is for every moebius transformation of u and therefore 1 u d a 0 for every conversely does this mean value property imply that is harmonic a more general question with the unit ball bn of c for arbitrary n 1 in place of the disc is investigated in the present paper the answer is found to be affirmative if n 11 negative if n 12

asymptotic behavior of bergman kernels with logarithmic weight abstract we describe singularities of weighted bergman kernels on the unit disc with respect to radial logarithmic weights

toeplitz and hankel operators with l 1 symbols on dirichlet space abstract we show that toeplitz or small hankel operators with symbols in l 1 is a generalization of the case with the harmonic symbol in c d d where d is the dirichlet space on d we also give a decomposition theorem for the sobolev space of first order on d using this result some characterizations for algebraic properties of toeplitz or small hankel operators with symbols in l 1 are given

a lil type result for the product limit estimator let x1 x2 xn be i i d r v s with p x u f u and y1 y2 yn be i i d p y u g u where both f and g are unknown continuous survival functions for i 1 2 n set i 1 if xi yi and 0 if xi yi and zi min itxi yi one way to estimate f from the observations zi i i l n is by means of the product limit p l estimator fn kaplan meier 1958 6

mellin transform monomial symbols and commuting toeplitz operators abstract in this paper we give necessary and sufficient conditions for a symbol that produces a toeplitz operator on the bergman space that commutes with another such operator whose symbol is a monomial the result is stated in terms of the mellin transform of the symbol as a corollary we show that a toeplitz operator with a radial symbol may only commute with another such operator with a radial symbol

upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices abstract upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained we provide a unified approach that yields or improves upon most of the bounds that have been obtained so far

some asymptotic properties of kernel estimators of a density function in case of censored data on considere l adaptation de l estimateur du noyau a des donnees censurees en utilisant l estimateur de kaplan meier

invariant polynomial curves of piecewise linear maps consider a system which evolves in n space from x to f x as time moves from t to t 1 assuming f is piecewise linear our interest is in establishing conditions for existence of and an algori

finite rank toeplitz products with harmonic symbols abstract on the bergman space of the unit ball of c n we study the finite rank problem for toeplitz products with harmonic symbols we first solve the problem with two factors in case symbols have local continuous extension property up to the boundary also in case symbols have additional lipschitz continuity up to some part of the boundary we solve the problem for multiple products with number of factors depending on the dimension n analogous theorems on the polydisk are also obtained

dual truncated toeplitz operators abstract let u be a nonconstant inner function in this paper we study the dual truncated toeplitz operators on the orthogonal complement of the model space k u 2 this is a new class of toeplitz operator we show the product of two dual truncated toeplitz operators d f d g to be zero if and only if either f or g is zero we give a necessary and sufficient condition for the product of two dual truncated toeplitz operators to be a finite rank operator furthermore a necessary and sufficient condition is found for the product of two dual truncated toeplitz operators to be a dual truncated toeplitz operator the last two results are different from the classical toeplitz operator theory

commutativity of toeplitz operators on the harmonic dirichlet space in this paper we completely characterize semi commutativity of toeplitz operators with harmonic symbols on harmonic dirichlet space

homeomorphism conditions for coherently oriented piecewise affine mappings this article is mainly concerned with the homeomorphism problem for piecewise affine mappings pa maps i e mappings which coincide with an affine mapping on each polyhedron of some finite polyhedral subdivision of rn in the first part we prove that a pa map can be defined without referring to a subdivision of rn as a continuous mapping which coincides at every point x rn with at least one function from a finite collection of affine functions the second part studies the recession function of a pa map it is shown that the recession function is piecewise linear and that a coherently oriented pa map is a homeomorphism if and only if its recession function is a homeomorphism in the last part we prove that a coherently oriented pa map is a homeomorphism if it admits a corresponding polyhedral subdivision of rn such that for some number k 2 n every face of codimension k is contained in at most 2k polyhedra provided the subdivision contains at least one face of codimension k

toeplitz products with pluriharmonic symbols on the hardy space over the ball on the hardy space over the unit ball in cn we consider operators which have the form of a finite sum of products of several toeplitz operators we study characterizing problems of when such an operator is compact or of finite rank some of our results show higher dimensional phenomena

hankel operators and toeplitz operators on the bergman space abstract we consider in this paper the question of when the semi commutator t fg t f t g on the bergman space with bounded harmonic symbols is compact several conditions equivalent to compactness of t fg t f t g are given as a consequence we prove a conjecture of axler that for bounded analytic functions f and g on the unit disk t fg t g t f is compact iff either f or g is constant on each gleason part p m except d

algebraic properties of toeplitz operators on generalized fock spaces on cd abstract we study two problems involving algebraic properties of toeplitz operators on generalized fock spaces on c d with weights of the form z 2 s e z 2 m m 1 s 0 we determine the commutant of a given toeplitz operator with a radial symbol which satisfies certain growth conditions we also discuss the equation t f t g 0 when f or g is radial

a new proof of robinson s homeomorphism theorem for pl normal maps abstract we study a certain class of piecewise linear functions from r n to r n namely robinson s normal maps induced by linear mappings and polyhedral convex sets called pl normal maps robinson s homeomorphism theorem characterizes the pl normal maps which are homeomorphisms as those which have nonzero determinants of the same sign on all pieces of linearity this paper presents a new shorter proof of the result pl normal systems include many optimization and equilibrium problems they arise from variational inequalities or equivalently generalized equations specified by linear maps and polyhedral convex sets unique continuous solvability of these systems which is important in theory and computation is captured by the homeomorphism result

a theorem of brown halmos type for bergman space toeplitz operators abstract we study the analogues of the brown halmos theorem for toeplitz operators on the bergman space we show that for f and g harmonic t f t g t h only in the trivial case provided that h is of class c 2 with the invariant laplacian bounded here the trivial cases are f or g holomorphic from this we conclude that the zero product problem for harmonic symbols has only the trivial solution finally we provide examples that show that the brown halmos theorem fails for general symbols even for symbols continuous up to the boundary

a reproducing kernel thesis for operators on bergman type function spaces abstract in this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on bergman type spaces in particular the results in this paper apply to the weighted bergman space on the unit ball the unit polydisc and more generally to weighted fock spaces

relationships of properties of piecewise affine maps over ordered fields abstract with an emphasis on general ordered fields we survey relationships of properties of piecewise linear maps on vector spaces the properties of interest are local and global variations of univalence openness onto condition nonexpansiveness and orientation and eigenvalue conditions on derivatives

minimum convex cost dynamic network flows this paper presents and solves in polynomial time the minimum convex cost dynamic network flow problem an infinite horizon integer programming problem which involves network flows evolving over time the model is a finite network in which each arc has an associated transit time for flow to pass through it an integral amount of flow is to be sent through arcs of the network in each period over an infinite horizon so as to satisfy conservation of flow from some fixed period on furthermore the net amount of flow in transit is assumed fixed over the infinite horizon the objective is to minimize the average convex cost per period of sending flow this problem is a generalization of the minimum convex cost network flow problem the maximum throughput dynamic network flow problem and the minimum cost to time ratio circuit problem furthermore the model has applications in periodic production and transshipment airplane scheduling cyclic capacity scheduling and cyclic staffing to solve the dynamic network flow problem one first obtains an optimal continuous valued flow which repeats every period the fractional variables of this solution may be rounded in such a way as to obtain an optimal integral flow which repeats every q periods where q is the least common denominator of the fractional parts of the continuous flow furthermore this integral flow is also an optimal continuous flow

a discrete time average cost flexible manufacturing and operator scheduling model solved by deconvexification over time a flexible manufacturing and operator scheduling problem is introduced and solved the principal concern is to schedule operators over time to various activities of a manufacturing system for the purpose of optimizing some steady state criteria in mathematical terms the problem is modeled as a deterministic infinite horizon discrete dynamic program our solution procedure is to convexify the problem to obtain a linear program and then to deconvexify the solution of the linear program over time to arrive at an optimal solution apparent loss in objective value due to the deconvexifications is circumvented with buffer inventories the procedure is reduced to solving a sequence of linear programs and the complexity is stated in these terms

a note on the variable kernel estimator of the hazard function from randomly censored data in a recent paper tanner and wong 1983b a family of data based nonparametric hazard estimators was introduced several of these estimators were studied in an extensive simulation experiment the estimator which allows for variable bandwidth was found to have a superior performance in this note sufficient conditions for the variable kernel estimator to be strongly consistent are presented

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

asymptotically optimal bandwidth selection for kernel density estimators from randomly right censored samples abstract this paper makes two important contributions to the theory of bandwidth selection for kernel density estimators under right censorship first an asymptotic representation of the integrated squared error into easily understood variance and squared bias components is given second it is shown that if the bandwidth is chosen by the data based method of least squares cross validation then it is asymptotically optimal in a compelling sense a by product of the first part is an interesting comparison of the two most popular kernel estimators keywords nonparametric density estimation smoothing parameter

toeplitz operators on the dirichlet space we study algebraic properties of toeplitz operators acting on the dirichlet space we first characterize two harmonic symbols of commuting toeplitz operators also we give characterizations of the harmonic symbol for which the corresponding toeplitz operator is self adjoint or an isometry

commuting toeplitz operators with pluriharmonic symbols on the fock space abstract in the setting of the bergman space over the disk or the ball it has been known that two toeplitz operators with bounded pluriharmonic symbols can semi commute only in the trivial cases in this paper we study the analogues on the fock space over the multi dimensional complex space as is the case in various other settings we are naturally led to the problem of characterizing a certain type of fixed points of the berezin transform for such fixed points we obtain a complete characterization by means of eigenfunctions of the laplacian we also obtain other characterizations in particular it turns out that there are many nontrivial cases on the fock space for semi commuting toeplitz operators with pluriharmonic symbols all in all our results reveal that the situation on the fock space appears to be much more complicated than that on the classical bergman space setting which partly is caused by the unboundedness of the operator symbols some of our results are restricted to the one variable case and the corresponding several variable case is left open

a coburn type theorem for toeplitz operators on the dirichlet space abstract a celebrated theorem of coburn asserts that on the setting of the hardy space if a toeplitz operator is nonzero then either it is one to one or its adjoint operator is one to one in this paper we show that an analogous result holds for toeplitz operators acting on the dirichlet space

on branching numbers of normal manifolds kuhn and lowen l and robinson 2 have provided deep and elegant results that characterize the homeomorphic members of certain classes of piecewise linear pl mappings the purpose of this paper is to show that the class of mappings examined by robinson is contained in the class of mappings examined by kuhn and lowen let 312 be a finite piecewise linear pl manifold of r and p r r be pl with respect to 3n i e p is affine on each set in 9x the branching number of 9x l is the maximum number of sets in 317 that can contain any common face of codimension 2 in l theorem 5 31 it is shown that if 312 has a branching number less than or equal to 4 then p is a homeomorphism if and only if it is coherently oriented i e the determinants of p on each set in 312 have the same nonzero sign let c be a nonempty polyhedral convex set in real n dimensional space ir and a belong to the space of real n x n matrices f robinson 2 defines a finite pl manifold 32 of k 2 proposition 2 41 called the normal manifold of c and the normal map a i t r induced by a c which is pl with respect to e 2 proposition 2 51 the main result 2 theorem 4 31 shows that ac is homeomorphic if and only if it is coherently oriented see ralph 3 for an alternative proof and robinson 4 for a much simpler proof in the special case that a is symmetric it is pointed out in 2 section l that if xc could be shown to have a branching number less than or equal to 4 then 2 theorem 4 31 would follow from l theorem 5 31 indeed showing that normal manifolds have this branching number property is precisely our goal throughout the paper c and a will be as above let q be the euclidean projection onto c that is rcc maps each point of ir to its nearest point in c with respect to the euclidean norm the identity in ir x is i

kernels and spectrum of toeplitz operators on the dirichlet space abstract in this paper we study the spectrum of toeplitz operators on the dirichlet space d with bounded conjugate analytic symbols and give a characterization of the kernels of toeplitz operators with harmonic symbols in p m d using this characterization we show that the spectrum of a toeplitz operator with symbol the sum of an analytic polynomial with degree at most two and the conjugate of linear polynomial is connected but there are toeplitz operators with general harmonic polynomial symbols having disconnected spectrum

commuting toeplitz operators on the polydisk we obtain characterizations of essentially commuting toeplitz operators with pluriharmonic symbols on the bergman space of the polydisk we show that commuting and essential commuting properties are the same for dimensions bigger than 2 while they are not for dimensions less than or equal to 2 also the corresponding results for semi commutators are obtained

a theorem of brown halmos type on the bergman space modulo finite rank operators abstract in this paper we study the product problem of toeplitz operators on the bergman space of the unit disk we characterize when the product of two toeplitz operators t f t g is a finite rank perturbation of another toeplitz operator t h with f g bounded harmonic and h in c 2 class with invariant laplacian in l 1 as a consequence we show that there is no nontrivial rank one perturbation however in the case rank m 2 we construct an example that shows there are bounded harmonic functions f g and h such that t f t g t h has rank exactly m

zero products of toeplitz operators we prove that the product of finitely many toeplitz operators oil the hardy space is zero if and only if at least one of the operators is zero we use some new vector valued techniques that not only lead to a vector valued version of this result but also appear to be needed in the scalar case

characterizations of bergman space toeplitz operators with harmonic symbols it is well known that toeplitz operators on the hardy space of the unit disc are characterized by the equality where s1 is the hardy shift operator in this paper we give a generalized equality of this type which characterizes toeplitz operators with harmonic symbols in a class of standard weighted bergman spaces of the unit disc containing the hardy space and the unweighted bergman space the operators satisfying this equality are also naturally described using a slightly extended form of the sz nagy foias functional calculus for contractions this leads us to consider toeplitz operators as integrals of naturally associated positive operator measures in order to take properties of balayage into account

piecewise affine bijections of rn and the equation sx tx y we prove two criteria for bijectivity of piecewise affine maps f of rn it is assumed that det f has the same nonzero sign everywhere the first criterion is based solely on the branching properties of the partition of rn into domains of linearity of f the second one concerns the map itself and depends on the images under f of certain two dimensional angles this generalizes and improves results of r schramm the methods are topological our results imply k g murty s solution of the linear complementarity problem and some other partly known facts which we show are equivalent to it for example the map x sx tx is bijective iff all principal minors of s 1t are strictly positive

zero products of toeplitz operators with harmonic symbols on the bergman space of the unit ball in cn we solve the zero product problem for two toeplitz operators with harmonic symbols that have continuous extensions to some part of the boundary in the case where symbols have lipschitz continuous extensions to the boundary we solve the zero product problem for multiple products with the number of factors depending on the dimension n of the underlying space the number of factors is n 3 we also prove a local version of this result but with loss of a factor

the finite rank perturbations of the product of hankel and toeplitz operators in this paper we completely characterize when the product of a hankel operator and a toeplitz operator on the hardy space is a finite rank perturbation of a hankel operator and when the commutator of a hankel operator and a toeplitz operators has finite rank

algebraic properties of toeplitz operators on the dirichlet space we study some algebraic properties of toeplitz operators on the dirichlet space we first characterize semi commuting toeplitz operators with harmonic symbols next we study the product problem of when product of two toeplitz operators is another toeplitz operator as an application we show that the zero product of two toeplitz operators with harmonic symbol has only a trivial solution also the corresponding compact product problem is studied

a continuous time average cost flexible manufacturing and operator scheduling model solved by deconvexification over time abstract a continuous time flexible manufacturing and operator scheduling problem is introduced and solved the principle concern is with scheduling operators over time to various activities of a manufacturing system with the purpose of optimizing some steady state criterion in mathematical terms the problem is modeled as a deterministic infinite horizon continuous time discrete dynamic program our solution procedure is to convexify the problem to obtain a linear program and then to deconvexify the solution of the linear program over time to arrive at an optimal solution which is periodic and piecewise constant apparent loss in object value due to the deconvexifications is circumvented with buffer inventories the procedure can be reduced to solving a sequence of linear programs and the complexity can be stated in these terms

complex symmetric toeplitz operators on the dirichlet space abstract we study when a toeplitz operator t on the dirichlet space of the unit disk is complex symmetric with respect to a class of conjugations and find surprisingly that the case of complex symmetries of toeplitz operators according to these conjugations is very few we also show that if t is complex symmetric then the curve t t must be nowhere winding furthermore the spectrum and invertibility of complex symmetric toeplitz operators are described

operators on the orthogonal complement of the dirichlet space ii in this paper we first prove that a dual hankel operator r on the orthogonal complement of the dirichlet space is compact for w 1 d and then that a semicommutator of two toeplitz operators on the dirichlet space or two dual toeplitz operators on the orthogonal complement of the dirichlet space in sobolev space is compact we also prove that a dual hankel operator r with w 1 d is of finite rank if and only if b is orthogonal to the dirichlet space for some finite blaschke product b and give a sufficient and necessary condition for the semicommutator of two dual toeplitz operators to be of finite rank

density and hazard rate estimation for censored data via strong representation of the kaplan meier estimator we study the estimation of a density and a hazard rate function based on censored data by the kernel smoothing method our technique is facilitated by a recent result of lo and singh 1986 which establishes a strong uniform approximation of the kaplan meier estimator by an average of independent random variables note that the approximation is carried out on the original probability space which should be distinguished from the hungarian embedding approach pointwise strong consistency and a law of iterated logarithm are derived as well as the mean squared error expression and asymptotic normality which is obtain using a more traditional method as compared with the hajek projection employed by tanner and wong 1983

on optimal packing of randomly arriving objects objects of finitely many types arrive at a facility according to independent stationary poisson processes the objects are to be placed in boxes as they arrive there are finitely many types of boxes and each box type has its own packing configuration there is an unlimited supply of boxes of each type and once an object is placed in a box it cannot be moved to another box in the future when a box is completely filled it produces a reward which depends on its configuration the objective is to select a rule for placing the arriving objects in boxes so as to maximize expected reward we show how to construct optimal and e optimal policies under which the expected number of partially packed boxes does not explode

sensitivity analysis of composite piecewise smooth equations this paper is a contribution to the sensitivity analysis of piecewise smooth equations a piecewise smooth function is a lipschitzian homeomorphism near a given point if and only if it is coherently oriented and has an invertible b derivative at this point we emphasise the role of functions of the typef g h whereg is piecewise smooth andh is smooth and present verifiable conditions which ensure that the functionf g tilde h is a lipschitzian homeomorphism near a given point for every tilde h sufficiently close toh with respect to thec1 topology

finite sums of toeplitz products on the dirichlet space on the dirichlet space of the unit disk we consider a class of operators which contain finite sums of products of two toeplitz operators with harmonic symbols we give characterizations of when an operator in that class is zero or compact also we solve the zero product problem for products of finitely many toeplitz operators with harmonic symbols

hankel operators on the dirichlet space abstract we study small hankel operators on the dirichlet space d with symbols in a class of function space and show that such small hankel operators are closely related to the corresponding hankel operators on the bergman space l a 2 and the hardy space h 2

density estimation with bivariate censored data abstract in this article we construct a kernel estimate of the probability density function from bivariate data that have been randomly censored we study the large sample properties of the proposed estimator using a strong approximation result we establish consistency and asymptotic normality and give a convenient representation of the kernel density estimator simulation studies show that the proposed procedure gives a good estimate of the true density function even when the sample size is moderate we discuss various issues about implementation of the estimator including bandwidth selection and boundary effects the procedure can be generalized to higher dimensional variables in a straightforward manner

kernels and ranks of hankel operators on the dirichlet spaces abstract we consider hankel operators and little hankel operators on the dirichlet space and pluriharmonic dirichlet space of the unit ball we first describe kernels of hankel operators with pluriharmonic symbol next we prove that there are no non trivial finite rank hankel operators with pluriharmonic symbol finally we characterize finite rank little hankel operators with pluriharmonic symbol our results show that there are several differences between the results on the dirichlet space and pluriharmonic dirichlet space

on the range of the berezin transform abstract we study the range of the berezin transform b more precisely we characterize all triples f g u where f and g are non constant holomorphic functions on the unit disc d in the complex plane and u is integrable on d such that f g bu it turns out that there are very few such triples this problem arose in the study of bergman space toeplitz operators and its solution has application to the theory of such operators

on the lipschitzian properties of polyhedral multifunctions in this paper we show that for a polyhedral multifunctionf rn rm with convex range the inverse functionf 1 is locally lower lipschitzian at every point of the range off equivalently lipschitzian on the range off if and only if the functionf is open as a consequence we show that for a piecewise affine functionf rn rn f is surjective andf 1 is lipschitzian if and only iff is coherently oriented an application via robinson s normal map formulation leads to the following result in the context of affine variational inequalities the solution mapping as a function of the data vector is nonempty valued and lipschitzian on the entire space if and only if the solution mapping is single valued this extends a recent result of murthy parthasarathy and sabatini proved in the setting of linear complementarity problems

tests for location with k samples and censored data summary tests which are specific for location are developed in the k sample situation with right censored data the tests are based on general estimating functions and do not require full parametric assumptions for large samples but in general they are not based only on ranks one class of tests allows shapes to differ in the k populations and includes as a special case the median test of brookmeyer crowley 1982 if the distributions are assumed to differ at most in location modified tests are suggested in which survival function estimates are obtained by pooling the data from all samples prentice s 1978 median test with minor modifications is included in this class

berezin versus mellin we study zero products of two bergman space toeplitz operators where one symbol is harmonic our results point in the direction of the zero product problem having only a trivial solution the techniques we use are based on a formula that connects the berezin and mellin transform

the essential norm of a sum of toeplitz products on the dirichlet space abstract for an operator which is a finite sum of products of several toeplitz operators on the dirichlet space we describe the essential norm in terms of the boundary value of the corresponding sum of products of symbols

a proof of the branching number bound for normal manifolds abstract we prove that every face of dimension n 2 of the normal manifold induced by a polyhedron in r n is contained in precisely four distinct n dimensional polyhedra of the normal manifold this result implies that the branching number of the normal manifold is 4 and refines ralph s result which only gave 4 as an upper bound the proof is based on the well known fact that every ridge of a polyhedron is contained in precisely two facets

commuting toeplitz operators on the dirichlet space in this paper we study the commutativity of toeplitz operators with continuous symbols on the dirichlet space first under a mild condition concerning absolute continuity we characterize semi commuting toeplitz operators this is a generalization of the case of harmonic symbols also if one of the symbol is radial or analytic we get another characterization which is different from the case on the bergman space

the pluriharmonic toeplitz operators on the polydisk in this paper we completely characterize the compact semi commutator of two toeplitz operators with bounded pluriharmonic symbols on the bergman space of the polydisk several necessary and sufficient conditions are obtained for the commutator of two toeplitz operators with bounded pluriharmonic symbols on the polydisk to be compact

products of hankel operators on the fock space abstract for entire functions f and g we study the problem of when the hankel product h f h g is bounded or compact on the fock space f 2 this is a companion to sarason s toeplitz product problem which was completely solved for the fock space by cho park zhu in 2014 surprisingly our results here are different from the natural conjectures motivated by the toeplitz product problem

a theory on extending algorithms for parametric problems given certain algorithms for certain problems these algorithms can be incorporated in a lift solve and lower sequence to form algorithms for solving parametric versions of the problems in the first phase of the sequence the parametric problem is lifted to an extended ordered field where it becomes a single nonparametric problem second the original algorithm is run in the extended ordered field to solve the single problem finally this solution to the single problem is lowered to the original ordered field to obtain a parametric solution to the original parametric problem