The structure of proper and stable bases of rational vector spaces is investigated. We prove that if t(s) is a rational vector space, then among the proper bases of 3(s) there is a subfamily of proper bases which are 1) stable, 2) have no zeros in C\bigcup \{\infty\} and therefore are column (row) reduced at infinity, and 3) their MacMillan degree is minimum among the MacMillan degrees of all other proper bases of 3(s) and it is given by the sum of the MacMillan degrees of their columns taken separately. It is shown that this notion is the counterpart of Forney's concept of a minimal polynomial basis for the family of proper and stable bases of 3(s).

Abstract The algebraic structure of the get of all proper rational vectors contained in a given rational vector space 𝒯(s) is shown to be that of a noothorian ℝpr (s) -motiulo M*. ( Ropf;pr(a) is the ring of proper rational functions.) The proper submodules Mt of M* form an ascending chain of submodules partially ordered by an invariant of Mt defined as the valuation at s= ∞ ofMt , Tho various bases of Mt are examined and classified according to their property of ‘column reduceness at s = ∞’The concept of a prime column reduced at basis of Mt is introduced. It is shown that tho prime bases of Mt can be further classified by their MacMillan degrees and the existence of minimal MacMillan degree bases for Mt is established. A prime and minimal MacMillan degree basis ofMt extends Forney's concept of a minimal polynomial basis of ∞ (s) for the Rpr Mt -module The MacMillan degrees of the columns of such bases form a set of invariants Mt for which are defined as the 𝒯(s) generalized invariant dynamicol indices of Mt and a simple relation is established between (I) the generalized invariant dynamical indices Mt (ii) the orders of zeros at = ∞ s and (iii) the Forney invariant dynamical indices of Finally those results are specialized to the (maximal) noetherian ( Rpr(a)smodule M* it is shown that in this case the ' generalized invariant dynamical indices ' of M*. coincide with the invariant dynamical indices of Forney for 𝒯(s) thus providing an alternative interpretation of the Forney ' invariant dynamical order ' of 𝒯(s) as an absolute minimum of the MacMillan degree of any proper basis for 𝒯(s)

Let L 1 , … , L s {\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s} be s s -distinct lines in A k n + 1 {\mathbf {A}}_k^{n + 1} passing through the origin. Assume s = ( n n + d ) − λ s = (_n^{n + d}) - \lambda where n n , d ⩾ 2 d \geqslant 2 . If L 1 , … , L s {\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s} are in generic s s -position, and λ = 0 \lambda = 0 . 1 , … , n − 1 1, \ldots ,n - 1 , then the Cohen-Macaulay type, t ( L 1 , … , L s ) t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) , of L 1 , … , L s {\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s} is given by the following formula: t ( L 1 , … , L s ) = ( n − 1 n + d − 1 ) − λ t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) = (_{n - 1}^{n + d - 1}) - \lambda . This formula is known to be false for λ = n \lambda = n . In this paper, we show that if L 1 , … , L s {\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s} are in uniform position, and λ = n \lambda = n . then t ( L 1 , … , L s ) = ( n − 1 n + d − 1 ) − n t({\mathfrak {L}_1}, \ldots ,{\mathfrak {L}_s}) = (_{\;n - 1}^{n + d - 1}) - n .

Non-commutative graded algebras and their Hilbert series

Recent efforts in the field of mathematical education have done little to resolve the conflict between ‘traditional’ and ‘modern’ mathematics. In these articles an attempt is made to use the ideas of Linearity as a linkage. Linearity is an important part of mathematics and I try to show how its language may be used at school level to integrate the ideas of algebra and geometry with the modern expression of algebraic and spatial relationships. Part 2 discusses the questions of curve fitting and of constructing the polynomial of lowest degree which most closely fits a set of given values. The polynomial is obtained as a linear expression of a set of difference tables as the basis of a vector space. Part 3 extends the use of difference tables to construct polynomials to give formulae for summation of series, establishes the binomial expansion independently of permutations and combinations, and generalizes the arithmetic progression.

On distribution by rank of bases for vector spaces of matrices over a finite field

Regular products of single-hook characters of the unitary group are defined. Their expansion in terms of irreducible characters is discussed and illustrated, along with the inverse of this expansion for all characters labelled by partitions of n with n=5 and n=9. It is shown that these regular products form an integral basis for the vector space of homogeneous functions.

The stability of bases in finite-dimensional vector spaces

Wolovich generalized the classical resultant matrix to give a test for the relative primeness of two real polynomial matrices T(s), V(s), where G(s) = V(s)T−1 is proper. A new proof is given for this result and it is extended to the case when G(s) = is non-proper. The result is also shown to have connections with the theory of minimal bases for rational vector spaces derived by Forney.

Certain results in the theory of polynomial matrices, free R [s]-modules and minimal bases of rational vector spaces are used in order to investigate the ‘ squaring down ’ and zero placement problem.

Early work combining recursion theory and algebra had (at least) two different sets of motivations. First the precise setting of recursion theory offered a chance to make formal classical concerns as to the effective or algorithmic nature of algebraic constructions. As an added benefit the formalization gives one the opportunity of proving that certain constructions cannot be done effectively even when the original data is presented in a recursive way. One important example of this sort of approach is the work of Frohlich and Shepardson [1955] in field theory. Another motivation for the introduction of recursion theory to algebra is given by Rabin [1960]. One hopes to mathematically enrich algebra by the additional structure provided by the notion of computability much as topological structure enriches group theory. Another example of this sort is provided in Dekker [1969] and [1971] where the added structure is that of recursive equivalence types. (This particular structural view culminates in the monograph of Crossley and Nerode [1974].)More recently there is the work of Metakides and Nerode [1975], [1977] which combines both approaches. Thus, for example, working with vector spaces they show in a very strong way that one cannot always effectively extend a given (even recursive) independent set to a basis for a (recursive) vector space.

This paper presents identities on generating functions for multisectioned partitions of integers by developing in the language of partitions some powerful and essentially combinatorial techniques from the literature of principal differential ideals. D. Mead has stated in Vol. 42 of this journal that one can obtain interesting combinatorial relations by constructing different vector space bases for a subspace of a differential ring and using the fact that the cardinality of all bases is the same. The results of the present paper are of this nature. In particular, we enumerate certain sets of ordered pairs of that have a central role in Mead's paper. Tableaux were used by A. Young and others to study the structure of the symmetric groups Sn. In [3], D. Knuth used an insertion into tableau construction of C. Schensted to give a direct 1-to-l correspondence between generalized permutations and ordered pairs of generalized Young tableaux having the same shape. In [5], Mead independently proved the existence of such a bisection while developing a new vector space basis for the ring of differential polynomials in n independent differential indeterminates. Mead's paper deals with principal differential ideals generated by Wronskians and used determinantal identities going back to Gayley. The ordered pairs of used by Mead appear in a more general setting in the paper [1] by Doubilet, Rota, and Stein.

The development of the flexibility method of analysis of skeletal structures has been hindered by the difficulty of determining a suitable statical basis on which to form the flexibility matrix. A combinatorial approach reduces the difficulty to one of selecting a minimal basis of the cycle vector space. After an introduction to flexibility analysis and a brief review of earlier work using combinatorics, the paper presents a procedure to construct a finite sub-set of the cycle vector space containing the elements of all minimal bases. This makes the generation of the required basis feasible by a finite procedure, such as Welsh’s generalization of the Kruskal algorithm. It is thus possible to have an automatic method for the analysis of skeletal structures which uses an optimal combinatorial approach.

This paper deals with the problem of exact model matching in multivariable systems. The problem is presented using exclusively a transfer matrix for the description of tho system. A solution is obtained by the use of the minimal bases of rational vector spaces.

A minimal basis of a vector space V of n-tuples of rational functions is defined as a polynomial basis such that the sum of the degrees of the basis n-tuples is minimum. Conditions for a matrix G to represent a minimal basis are derived. By imposing additional conditions on G we arrive at a minimal basis for V that is unique. We show how minimal bases can be used to factor a transfer function matrix G in the form $G = ND^{ - 1} $, where N and D are polynomial matrices that display the controllability indices of G and its controller canonical realization. Transfer function matrices G solving equations of the form $PG = Q$ are also obtained by this method; applications to the problem of finding minimal order inverse systems are given. Previous applications to convolutional coding theory are noted. This range of applications suggests that minimal basis ideas will be useful throughout the theory of multivariable linear systems. A restatement of these ideas in the language of valuation theory is given in an Ap...

IN THIS PAPER we consider procedures for going from several individual preferences among several alternatives, called candidates, to something which may be called a collective preference. The individual preferences take the form of (total) orderings of the alternatives, and the collective preference is to take the form of a (total) weak ordering (i.e., ties allowed). We consider certain properties which seem desirable in such and investigate which have these properties. The of view taken here differs from that of other work in this area (e.g., [1, 2, 3, 4]) chiefly in asking that the procedure work for all possible sizes of the voting population, rather than for a fixed population, given in advance. This permits us to require, for example, that if each of two bodies of voters prefers candidate A to candidate B under a given procedure, then the combination of these bodies should prefer A to B under the same procedure. In Section 1 we give the formal definitions of an aggregation procedure and discuss certain desirable features, namely neutrality (treats candidates symmetrically), (the condition mentioned above), monotonicity, and an Archimedean property which says, roughly, that a sufficiently large body with a given distribution of preferences can impose its will on any body of fixed size. In Section 2 we introduce certain procedures: point systems and systems (roughly, allowing infinitesimal points), which are neutral and separable. They are monotonic if and only if the points are arranged in the natural order, and the are, in addition, Archimedean. In Section 3 we prove a converse, namely that any neutral and separable procedure can be realized by a generalized system and, if it is Archimedean, by a system. This part requires some familiarity with the notions of least upper bound of a set of real numbers and bases of vector spaces. In Section 4 (which is largely independent of Section 3), we consider point which use in a succession of eliminations. Such are neither separable nor monotonic but do satisfy some very weak separability and monotonicity conditions. While these probably do not characterize runoff systems, we know of no other satisfying them.

On the ranks of bases of vector spaces of matrices

In this paper one finds a new method to calculate problems concerning affinoid algebras. The method which uses orthonormal bases in normed vector spaces is developed in the first two paragraphs and is applied to affinoid algebras later on. In a simple way there are obtained nearly all results about affinoid algebras which are already known. Further this method gives new information about the functor F which associates to each affinoid space X an affine algebraic variety\(\tilde X\). In detail: F is compatible with extensions of the field k (if affinoid spaces\(X \subset k^{ \circ n} \) are considered, k algebraically closed, n arbitrary) and F is compatible with the cartesian product. These problems are treated in the language of affinoid algebras.

Bases in Vector Spaces and the Axiom of Choice

1. Because of the nonconstructive nature of the axiom of choice there has been much interest in how much of it is needed for various theories. In the case of the theory of vector spaces it appears that one would want to save at least the following two consequences of AC: (1) Every vector space has a basis and (2) Any two bases of a given vector space are equipollent. The question immediately arises: Have we saved the whole axiom of choice; namely is the axiom of choice a logical consequence of (1) and (2) and the other axioms of some appropriate set theory? This question remains open and the author conjectures a negative solution. However, we are able to show that a reasonable strengthening of (1), which is also a consequence of AC, implies AC, namely the universal generalization of Proposition 2 of [1], which we will call the downward basis principle: