Finite Linear Space Research Articles

A geometric characterization of H-sets

The concept of H sets with respect to a finite dimensional linear space of approx imation is extremely important, as these sets unify the theory of Chebyshev approx imaiion by, yielding a characterization of best approximates and conditions for uniqueness, and are also aids in the construction of algorithms for numerica computation. We study here the geometric characterization of H sets in terms of convex polyhedral cones or, in special cases, simplces. In particular we consider the characterization of minimal H sets and use this to find an upper bound for the number of possible minimal H sets with respect to a finite dimensional space.

Finite linear spaces with two consecutive line degrees

Let L be a non-trivial finite linear space in which every line has n or n+1 points. We describe L completely subject to the following restrictions on n and on the number of points p: p≤n2+n−1 and n≥3; n2+n+2≤p≤n2+2n−1 and n≥3; p=n2+2n and n≥4; p=n2+2n+2 and n≥3; p=n2+2n+3 and n≥4.

Characterization of WT-spaces whose derivatives form a WT-space

Let U be a finite dimensional linear space of continuously differentiable functions, containing constants. We prove that U 1, the space of derivatives, is a WT-space iff U has a basis { u o ,…, u n−1 } (that is a complete WT-system with u 0 ≡ 1.

Finite linear spaces and projective planes

In 1948, De Bruijn and Erdös proved that a finite linear space on ν points has at least ν lines, with equality occurring if and only if the space is either a near-pencil (all points but one collinear) or a projective plane. In this paper, we study finite linear spaces which are not near-pencils. We obtain a lower bound for the number of lines (as a function of the number of points) for such linear spaces. A finite linear space which meets this bound can be obtained provided a suitable projective plane exists. We then investigate the converse: can a finite linear space meeting the bound be embedded in a projective plane.

A Common Characterization of Finite Projective Spaces and Affine Planes

Let S be a finite linear space for which there is a non-negative integer s such that for any two disjoint lines L , L ' of S and any point p outside L and L ' there are exactly s lines through p intersecting the two lines L and L '. We prove that one of the following possibilities occurs: (i) S is a generalized projective space, and if the dimension of S is at least 4, then any line of S has exactly two points. (ii) S is an affine plane, an affine plane with one improper point, or a punctured projective plane. (iii) S is the Fano-quasi -plane.

Uniform approximation by generalised polynomials

We here describe methods both numerical and analytic to solve the problem of finding the best uniform approximations to a continuous function by a finite dimensional linear space of functions which does not necessarily satisfy the Haar condition. We show how a knowledge ofH-sets is essential and how the theory is simplified by the use of this concept.

The exceptional case in a theorem of Bose and Shrikhande

AbstractIt is shown that (n being an integer) any non-trivial finite linear space with n2- 1 points, all of degree at most n+1, is embeddable in a finite projective plane of order n. This generalizes a theorem of Bose and Shrikhande and settles the unsolved case n = 6.

Embedding the complement of two lines in a finite projective plane

In this paper we use a result from graph theory on the characterization of the line graphs of the complete bigraphs to show that if n is any integer ≥ 2 then any finite linear space having p = n2 − n or p = n2 − n + 1 points, of which at least n2 − n have degree n + 1, and q ≤ n2 + n − 1 lines is embeddable in an FPP of order n unless n = 4. If n = 4 there is only one possible exception for each of the two values of p, and for p = n2 − n, this exception can be embedded in the FPP of order 5.

Algebraic realization theory of bilinear discrete-time input-output maps

In this paper some new results are presented relating to the abstract realization theory of bilinear input—output maps. Previous results obtained by various authors are also discussed and reconsidered in a unified way.The results obtained are connected with the introduction of the class of realizable series whose structure is strictly related to the possibility of embedding the Nerode state space in a finite dimensional linear space. Updating equations for the state, in the discrete time case, are obtained. The relationships among realizations of bilinear i/o maps and systems described by bilinear equations are also discussed.

Parties équitables d'espaces linéaires

Abstract. — A subset of a linear space L (we don't mean a vector space) is said to be equitable of ordre n (a cardinal which is not necessarily finite) if its intersection with every line of L has cardinality n. Our main results are : a) Every finite linear space L contains at most one non trivial equitable subset. If E is such a subset of L, then the same number v of lines go through all the points of E, the same number w of lines go through all the points of L \ E and v — w divides v — 1 . b) For every integer e > 6, there is a linear space of cardinality e which contains a non trivial equitable subset. c) Let L be an infinite linear space wherein the lines and the set of lines have the same cardinality d. For every cardinal n ⩾ d, there exist in L equitable subsets of order n. d) For every natural number n, there exist infinite linear spaces with the following property : if the point p does not belong to the line D, there are exactly n lines through p which do not cut D.

On power-bounded operators and operators satisfying a resolvent condition

We obtain, by means of a classification of the eigenvalues, local estimates for holomorphic. functions of a class of linear operators on a finite dimensional linear vector space. We apply these methods to find new proofs of some theorems ofKreiss andMorton, and in addition we give a local estimate of the powers of the inverse of any nonsingular operator in this class.

The Structure of Space and the Formalism of Relativistic Quantum Theory. I

The structure of a Euclidean space can be approached, with an unlimited accuracy, by a part of a maximally ordered finite linear space. Accordingly, all the physical theories based on the space-time continuum can also be considered in such a finite space-time. The finiteness of the underlying space makes also some new kinds of theories possible. Among them is a purely group theoretical formalism of relativistic quantum theory, including a free-particle theory as well as a group formalism of interaction of particles. The free-particle formalism of a finite space-time is considered here (Part I). An essential difference in comparison with the formalism of continuous space-time is that there is, as a consequence of the relations of Euclidicity to be imposed on observable 4-vectors, a nontrivial spectrum of momentum, mass, and energy in a finite geometry.

Uniform Approximation of Minimax Point Estimates

bounded risk. When Conditions A and B stated later hold, it is shown that a unique minimax estimate uo(x) exists and if {un(x)} is any uniformly bounded minimax sequence then the risk functions of un(x) converges uniformly to the risk function of uo(x), so that no almost subminimax estimate can exist which, though not a minimax estimate, has for a wide range of values of the parameter 0, a lower value of the risk than that of the minimax estimate. Under some additional conditions, it is shown that an approximation to the minimax estimate uO(x) in the space 5(.. of functions with bounded risk, may be obtained by the minimax estimate UN(X) in the finite dimensional linear space spanned byN basis vectors vi1, * VN of 5(P), so that the maximum risk of UaN(x) converges to that of uo(x). This may help in finding an approximation to a minimax estimate in non-standard problems, where it is difficult to guess a minimax estimate from invariance or other considerations and specially when the problem is a perturbation of a standard problem. 2. Introduction. It has been pointed out by Hodges and Lehmann [3] and Robbins [8], that in certain cases there exist estimates which are not minimax so that their maximum risks may be slightly greater than that of a minimax estimate but their risk functions are considerably less than that of a minimax estimate, for a range of values of the parameter 0. Such estimates have been called E-minimax or subminimax estimates [8], [11]. Since e is an unspecified small quantity above, the situation may be characterised by the existence of a minimax sequence of estimates {un (x)} (which is a minimax solution in the wide sense in Wald's terminology), whose risk functions do not uniformly converge to that of the minimax estimate uo(x). Frank and Kiefer [1], have given examples from the theory of testing of hypotheses, of such almost subminimax solutions. An example is given in the appendix where for any e however small subminimax estimates of this type, exist for a squared error loss function with unbounded range of the parameter. A second difficulty in the choice of a suitable estimate arises when there is a multiplicity of admissible minimax estimates. In the first part of the paper it is shown that Conditions A and B stated later ensures that a unique minimax estimate uo(x) exists and any minimax sequence of estimates { un (x) I converges with respect to a suitable norm topology (Theorem 1) to uo(x) so that the corresponding risk functions also converge uniformly to

On the Cohomology Theory for Associative Algebras

The cohomology theory for associative algebras may be regarded as a superstructure built upon the classical representation theory. Let us consider an algebra 21 over a field 4. Suppose that we are given a finite dimensional linear space 3 over &1 which is simultaneously a representation space and an antirepresentation space for W. Then, to each element a of 21, there correspond two linear transformations u -* u -a and u a -u of A. If, in addition to the usual requirements, these transformations commute with each other, i.e. if a,(ua2) = (a,-u)-a2, for all a1, a2 e 21 and all u e $, we say that $ is an 2-module. The notions of the cohomology theory are based on this set-up, as follows: The set of all m-linear mappings of 21 into $, when regarded as an additive group in the usual way, is called the group of the m-dimensional $-cochains (or (m, $)-cochains) of W. We shall denote it by Cm(2f, p). The group C'(21, p) is identified with the additive group of p. The coboundary operator, 6, as defined by Eilenberg and MacLane, maps each Cm(21, p) linearly into Cm'+(2t, p), as follows: Let f e Cz'(, p), a,, *, am e W; then (6f) {a, , amA} = a, f{aI, * * 7 am?, + Z-m~n (-l~tf{ai, ** *, a~aai +, *, a.+,} + (1)m+lf{ai, am am,+. For any cochain f, we have 6(6f) = 0. Hence the cohomology groups may be defined by the well known procedure of combinatorial topology. An element f E Cm(?2, p) is called an m-dimensional cocycle if cf = 0. An element of the form (5g, where g e Cml(2[, p) is called an m-dimensional coboundary. These form a subgroup B'(2, p) of the group Zm(2, p) of the m-dimensional cocycles. The m-dimensional cohomology group Hm(2, p) is defined as the quotient group Zm(21, p)/Bm(21, p). For m = 1 and m = 2, these groups have been interpreted with reference to classical notions of structure in a previous paper.' Although no such interpretation has as yet been found for the higher dimensional groups it is possible to make a rough forecast of the r6le which the cohomology theory will play in the study of the structure of algebras. There is good evidence that it will provide appropriate methods for investigating a variety of structural characteristics which seem to lie outside the domain of the classical representation theory. The trend which takes its logical departure in Wedderburn's structure theorems has been to intensity and refine the structural analysis only at the level of maximal regularity.2 Thus, the radical of an algebra is split off by Wedder-