Size direction games over the real line. I

Two related infinite two person games of perfect information are defined and studied.     

Size direction games over the real line. II

Variants of two basic infinite games of perfect information are studied. A notion of continuous strategy for the playerS (Size) is shown to be related to a notion of convergence norm for sequences of reals. With each such norm, a variant of each of the basic games is associated in which the size player has to see that each play obeys the norm. Restriction to choose only rational numbers is also imposed onS. Some games are completely solved, and in this caseS has a winning strategy iff his set includes a perfect subset, andD has a winning strategy iffS's set is at most denumerable. Some other games, in whichS has to choose only rationals and obey a norm, induce a hierarchy structure on the class of nowhere dense perfect sets, that is embedded cofinally in the lattice of infinite sequences of integers modulo finite differences.     

On line bundles over real algebraic curves

Let X be a geometrically irreducible smooth projective curve defined over the real numbers. Let nX be the number of connected components of the locus of real points of X. Let x₁,…,x? be real points from ? distinct components, with ?<nX. We prove that the divisor x₁+?+x? is rigid. We also give a very simple proof of the Harnack's inequality.     

Asymptotically optimal quadrature rules for uniform splines over the real line

Numerical Algorithms - We provide explicit asymptotically optimal quadrature rules for uniform Ck-splines, k =?0,1, over the real line. The nodes of these quadrature rules are given in terms...     

Colouring the real line

The problem of colouring the real line so that the distance between like coloured numbers does not lie in some specified set D, called the distance set, is discussed. In particular, the minimum number of colours needed for various distance sets are determined.     

Geometry of real grassmann manifolds. Part III

A canonical decomposition for an element of the tangent bundle of the Grassmann manifold G _p,n ⁺ in its Plücker model is constructed. This decomposition is used for introducing the notion of stationary angles between oriented planes. The relation with the stationary angles in the nonoriented case is established. An explicit formula for the exponential mapping is given, which allows calculating the diameter and the injectivity radius of the manifold G _p,n ⁺ . The problem of uniqueness of the canonical decomposition is reduced to a similar problem on the decomposition of bivectors that realizes their mass (the latter has earlier been solved by the author). The techniques developed are used for describing the structure of the closure of an arbitrary geodesic in the manifolds G _p,n ⁺ and G_p,n. (For the manifolds G_p,n, this result was announced by Wong without proof.) Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 108–129. Translated by N. Yu. Netsvetaev.     

Shift-invariant spaces on the real line

We investigate the structure of shift-invariant spaces generated by a finite number of compactly supported functions in . Based on a study of linear independence of the shifts of the generators, we characterize such shift-invariant spaces in terms of the semi-convolutions of the generators with sequences on . Moreover, we show that such a shift-invariant space provides -approximation order if and only if it contains all polynomials of degree less than .

     

Fuzzy P-measure on the real line

The fuzzy measurable space on the real line, β_ϱ say, is defined by means of fuzzy relation ‘less or equal’ [5]. The main properties of β_ϱ are given. These results are used for investigation of fuzzy P-measure on β_ϱ. The principal emphasis is laid on the connections between a fuzzy P-measure and its cumulative distribution function. A theorem extending a fuzzy P-measure to β_ϱ is shown.     

Fourier interpolation on the real line

Publications mathématiques de l'IHÉS - In this paper we construct an explicit interpolation formula for Schwartz functions on the real line. The formula expresses the value of a...     

Group topologies on the real line

It is shown that there exist 2^2c non-isomorphic group topologies on the real line, among them 2^2c are non-locally compact, ?₀ are compact (connected), ?₀ are locally compact and compactly generated. We also give some characterizations of the usual topology among group topologies on the real line.     

Isoperimetric problem on the real line

The isoperimetric problem on the real line for distributions with continuous positive densities is considered. Necessary and sufficient conditions under which the intervals (−∞, a) are extremal are suggested. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 5–9. Translated by V. Sudakov.     

Rational wavelets on the real line

In this note a representation of the discrete Green's function of a compact discretization of a two point boundary value problem of order n 2 is given which among other things allows a direct proff of the convergence (and divergence) properties.     

On partitions of the real line

Answering a question of Sierpinski, we prove that the real line is not necessarily the disjoint union of {btℵ} ₁ non-emptyG _σ sets.     

Asymmetric marking games on line graphs

This paper investigates the asymmetric marking games on line graphs. Suppose G is a graph with maximum degree Δ and G has an orientation with maximum outdegree k, we show that the (a,1)-game coloring number of the line graph of G is at most . When a=1, this improves some known results of the game coloring number of the line graphs.     

Shannon switching games without terminals III

Shannon switching game has been analyzed by Lehman [3] in matroid theory. Kano [2] has studied games with many terminals where Short intends to connect at least two of them. This result is included in [6] and [7], where the games are described on graphs with group action. These results will be generalized in this paper which deals with games on a finite set with two matroid structures. Our new theory can be applied to the game where Short's purpose is to connect all the given pairs of terminals and to the game where Short's purpose is to connect at least one of given two pairs of terminals. On the other hand, the game is not reversible (see [9]) if Short's purpose is to connect at least one of given three pairs of terminals.     

Ordering fuzzy sets over the real line: An approach based on decision making under uncertainty

The ordering of fuzzy sets over the real line is approached from the point of view of ordering intervals rather than ordering numbers. First, the maximax and maximin criteria which are commonly used for ordering intervals are expressed in terms of characteristic functions. These criteria and the Hurwicz criterion for decision making under complete ignorance are then reformulated in a manner that allows for their generalization to the fuzzy case. Transitivity is established for these ordering rules. A criterion based on the principle of diminishing marginal utility is also presented.     

A surprising covering of the real line

We construct an increasing sequence of Borel subsets of , such that their union is , but cannot be covered with countably many translations of one set. The proof uses a random method.