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## Publications

Publications (345)

This preprint deals with the symmetry of parametrized families of systems and the changes therein as the parameter changes. There are (at least ?) two kinds of symmetry: generic and specific which behave in almost totally opposite ways as the parameter changes: generic symmetry has links with entropy while specific symmetry has to do with symmetry...

There is a relatively well-known description of the algebra of (higher
order) left differential operators on commutative algebras. This note
gives a construction of similar flavor for algebras of differential
operators on not necessarily commutative algebras.

Let A be an associative algebra (or any other kind of algebra for that
matter). A derivation on A is an endomorphism \del of the underlying Abelian
group of A such that \del(ab)=a(\del b)+(\del a)b for all a,b\in A (1.1)
A Hasse-Schmidt derivation is a sequence (d_0=id,d_1,d_2,...,d_n,...) of
endomorphisms of the underlying Abelian group such that...

As already remarked before, the complex representations of the symmetric groups form a natural P SH algebra. Much of this comes for free; the remainder comes from a few basic general theorems from group representation theory. These will be discussed first. 5.1. A little bit of finite group representation theory Let A be an associative algebra with...

The number of published papers on Hopf algebras and their applications in and interrelations with other parts of mathematics and (theoretical) physics is rather large. In October 2004 a biographic search was done and resulted in 10589 hits. It also appeared that Hopf algebras and quantum groups appear in virtually all parts of mathematics. See [316...

There are many results and constructions in mathematics that are * unreasonably nice *. For instance it appears to be difficult
for a set to carry many compatible (algebraic) structures. More precisely, if, say, an algebra carries a compatible *higher*
structure the underlying algebra must be very regular. For instance, if an associative unital alg...

Many things in mathematics seem lamost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the...

This chapter discusses the aspects of Symm, which relate directly to the Witt vector constructions and their properties. The Witt vector construction is a very beautiful one. But it takes one out of the more traditional realms of algebra very quickly. The big and p-adic and truncated Witt vectors carry ring and algebra structures, and hence, natura...

The metaplectic representation describes a class of automorphisms of the Heisenberg group H = H(G), defined for a locally compact abelian group G. For G=ℝd, H is the usual Heisenberg group. For the case when G is the finite cyclic group ℤn, only partial constructions are known. Here we present new results for this case and we obtain an explicit con...

It is proved that MPR is rigid as a Hopf algebra with distinguished basis. I.e. there are no nontrivial automorphisms that preserve the multiplication and comultiplication and take the distinguished basis of all permutations into itself (as a graded set).

A very important Hopf algebra is the graded Hopf algebra Symm of symmetric functions. It can be characterized as the unique graded positive selfdual Hopf algebra with orthonormal graded distinguished basis and just one primitive element from the distinguished basis. This result is due to Andrei Zelevinsky. A noncommutative graded Hopf algebra of th...

As a natural continuation of the first volume of Algebras, Rings and Modules, this book provides both the classical aspects of the theory of groups and their representations as well as a general introduction to the modern theory of representations including the representations of quivers and finite partially ordered sets and their applications to f...

In this paper, we address the problem of automatic keywords assignment to scientific publications. The idea to use textual
traces learned from training data in a supervised manner to identify appropriate keywords is considered. We introduce the
transparent concept of identification cloud as a means to represent the semantics of scientific terms. Th...

Consider stochastic linear dynamical systems, dx=Axdt+Bdw,dy=Cxdt+dv,y(0)=0, x(0) a given initial random variable independent of the standard independent Wiener noise processes w,v. The matrices A,B,C are supposed to be constant. In this paper I consider two problems. For the first one A,B and C are supposed known and the question is how to calcula...

Let ls
n
be the real Lie algebra of all differential operators in n-variables c
x
/ x
, c
R where the sum is over all multi-indices , such that || + || 2. This note describes a certain representation of ls
n
by means of (nonlinear) vectorfields which in a certain sense is all Kalman-Bucy filters for ndimensional linear systems put together. Th...

This note is concerned with the twin questions of how to recognize and define the concept of special structure for linear dynamical systems in a state-space basis independent way and how to exploit the fact that there is special structure to diminish to computational load of various solution procedures.

The first topic of this partial survey paper is that of the growth of adequate lists of key phrase terms for a given field
of science or thesauri for such a field. A very rough ‘taking averages’ deterministic analysis predicts monotonic growth with
saturation effects. A much more sophisticated realistic stochatic model confirms that.
The second, a...

One striking aspect of the class of linear systems is that the controls enter in a way which is independent of the state; that is they are homogeneous, w.r.t. the underlying vectorspace (additive Lie group) structure as far as the controls are concerned, and the autonomous term enjoys reminiscent but not identical G L n\underline{\underline G} \und...

This chapter focuses on David Hilbert's paper, based upon a lecture delivered to the International Congress of Mathematicians in Paris in 1900, Hilbert outlined a range of problems for mathematicians to address in the century about to start. The problems addressed in this paper are (1) Cantor's problem on the cardinal number of the continuum, (2) t...

The Chen-Fox-Lyndon factorization theorem for words over totally ordered sets is a well-known and important theorem; its applications
concern Chen iterated integrals and are used in control and filtering theories. We give a natural generalization for words
over partially ordered sets, a question which came up in the context of generation theorems f...

Firstly this paper summarizes some of the the relations between control theory and coalgebra and Hopf algebra theory, particularly
the fact that the cofree coalgebra over a finite rank free module consists precisely of the realizable power series in noncommuting
variables. The second half of the paper concerns generalizations of the Hopf algebra of...

In late October 2004 I did a search on Hopf algebras in the database ZMATH to find out where they occur and what use is made of them (I.e. what applications there are). The result is a little astonishing, as can be seen from what follows.

In this lecture I discuss some aspects of MKM, Mathematical Knowledge Management, with particuar emphasis on information storage and information retrieval.

Let S be the set of scalings {n
−1:n=1,2,3,...} and let L
z
=z
Z
2, z∈S, be the corresponding set of scaled lattices in R
2. In this paper averaging operators are defined for plaquette functions on L
z
to plaquette functions on L
z′ for all z′, z∈S, z′=dz, d∈{2,3,4,...}, and their coherence is proved. This generalizes the averaging operators introd...

Like its precursor this paper is concerned with the Hopf algebra of noncommutative symmetric functions and its graded dual, the Hopf algebra of quasisymmetric functions. It complements and extends the previous paper but is also selfcontained. Here we concentrate on explicit descriptions (constructions) of a basis of the Lie algebra of primitives of...

Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf al...

In the last decennia two generalizations of the Hopf algebra of symmetric functions have appeared and shown themselves important, the Hopf algebra of noncommutative symmetric functions NSymm and the Hopf algebra of quasisymmetric functions QSymm. It has also become clear that it is important to understand the noncommutative versions of such importa...

Let NSymm be the Hopf algebra of noncommutative symmetric functions over the integers. In this paper a description is given of its Lie algebra of primitives over the integers, Prim(NSymm), in terms of recursion formulas. For each of the primitives of a basis of Prim(NSymm), indexed by Lyndon words, there is a recursively given divided power series...

In (Hazewinkel in Adv. Math. 164:283–300,2001, and CWI preprint,2001) it has been proved that the ring of quasisymmetric functions over the integers is free polynomial. This is a matter that
has been of great interest since 1972; for instance because of the role this statement plays in a classification theory for
noncommutative formal groups that h...

The algebraic structure that we now call a ring originated from several different sources. The concept of a ring — and also that of an ideal — was well known in the nineteenth century and was utilized by R. Dedekind and L. Kronecker in their work on algebraic number
theory, though Kronecker used the word “order” for “ring”. This last term was intro...

For coalgebras over fields, there is a well-known construction which gives the cofree coalgebra over a vector space as a certain completion of the tensor coalgebra. In the case of a one-dimensional vector space this is the coalgebra of recursive sequences. In this paper, it is shown that similar ideas work in the multivariable case over rings (inst...

This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the ques...

Let denote the Leibniz–Hopf algebra, which also turns up as the Solomon descent algebra and the algebra of noncommutative symmetric functions. As an algebra =Z〈Z1, Z2,…〉, the free associative algebra over the integers in countably many indeterminates. The coalgebra structure is given by μ(Zn)=∑ni=0 Zi⊗Zn−i, Z0=1. Let be the graded dual of . This is...

This paper is mainly concerned with the Leibniz-Hopf algebra over the integers and its graded dual, the overlapping shuffle algebra. The Ditters conjecture states that this graded dual is a free commutative polynomial ring over the integers and it specifies a set of conjectured generators. The definition of the overlapping shuffle algebra can be ge...

The paper deals with a mathematical model which describes how the collection of key phrases (and key words) evolves as a field of science develops. The experimental material is based on statistical observations on the sets of key phrases which have been assigned to papers in representative major journals in the field in question. Asymptotic propert...

Let Z denote the Leibniz-Hopf algebra, which also turns up as the Solomon descent algebra, and the algebra of noncommutative symmetric functions. As an algebra Z = Z < Z1, Z2,... >, the free associative algebra over the integers in countably many indeterminates. The co-algebra structure is given by \(\mu({Z_n})=\sum\nolimits_{i = 0}^n{{Z_i}}\otimes...

.. This paper is concerned with information retrieval from large scientific data bases of scientific literature. The central idea is to define metrics on the information space of terms (key phrases) and the information space of documents. This leads naturally to the idea of a weak enriched thesaurus and the semiautomatic generation of such tools. Q...

Let A be a reduced incidence relation between n lines and m points. Suppose that (a) Through each two points there pass l lines (b) Each two lines intersect in points. If l = = 1 assume, moreover, that there are four points no three of which are on one line. Then n m = , l = , and there is a number r such that all lines have r points and through ea...

This paper is supplementary to my paper "Multiparameter Quantum Groups and Multiparameter R-Matrices", [5]. Its main purpose is to point out that among the single block solutions of the Yang-Baxter equation given in [5] there occurs an n+m + 1 parameter quantum deformation of the supergroup GL(mjn) for every n; m 1.

This paper is supplementary to my paper "Multiparameter Quantum Groups and Multiparameter R-Matrices", [5]. Its main purpose is to point out that among the single block solutions of the Yang-Baxter equation given in [5] there occurs an Gamma n+m 2 Delta + 1 parameter quantum deformation of the supergroup GL(mjn) for every n; m 1. AMS Subject Classi...

Let Z denote the free associative algebra ZhZ1 ; Z2 ; : : :i over the integers. This algebra carries a Hopf algebra structure for which the comultiplication is Zn 7! Sigma i+j=nZ iOmega Z j . This the noncommutative Leibniz-Hopf algebra. It carries a natural grading for which gr(Zn) = n. The Ditters-Scholtens theorem says that the graded dual, M, o...

Let A be a bipartite graph between two sets D and T. Then A defines by Hamming distance, metrics on both T and D. The question is studied which pairs of metric spaces can arise this way. If both spaces are trivial the matrix A comes from a Hadamard matrix or is a BIBD. The second question studied is in what ways A can be used to transfer (classific...

## Questions

Question (1)

Hopf algebra module algebras of NSymm are the same thing as an algebra with a Hasse-Schmidt derivation on it. I believe that this can shed light on such things as A_\infty structures.