If φ : L → L′ is a bijection from the set of lines of a linear space (P,L) onto the set of lines of a linear space (P ′,L′) (dim (P,L), dim (P ′,L′) ≥ 3), such that intersecting lines go over to intersecting lines in both directions, then φ is arising from a collineation of (P,L) onto (P ′,L′) or a collineation of (P,L) onto the dual linear space of (P ′,L′). However, the second possibility can...

A method of representing transformations of a finite set pictorially is described. These pictures of a function are used to count certain idempotent transformations, and interesting formulae for the Fibonacci numbers are obtained. No part of any string extends beyond the region ° : : ; z ::; 1, and if two strings intersect then they join together, from the point of intersection, to meet the sam...

Uncontrolled intersections are the intersections where there are no external signs or signals to control the movement of vehicles. In mixed traffic conditions priority rules are often violated by the road users. In All-way-stop-controlled intersections (AWSC), the vehicle should stop themselves before they enter the intersection and should check whether any vehicles are present in the other app...

In this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. Following this, we establish the Minkowski and Brunn-Minkowski inequalities for volumes difference function of the projection and intersection bodies.

As a common generalization of matchings and matroid intersection, W.H. Cunningham and J.F. Geelen introduced the notion of path-matchings, then they introduced the more general notion of even factor in weakly symmetric digraphs. Here we give a min-max formula for the maximum cardinality of an even factor. Our proof is purely combinatorial. We also provide a Gallai-Edmonds-type structure theorem...

Last lecture we covered matroid intersection, and defined matroid union. In this lecture we review the definitions of matroid intersection, and then show that the matroid intersection polytope is TDI. This is Chapter 41 in Schrijver’s book. Next we review matroid union, and show that unlike matroid intersection, the union of two matroids is again a matroid. This material is largely contained in...

the class of ads modules with the sip (briefly‎, ‎$sa$-modules) is studied‎. ‎various conditions for a module to be $sa$-module are given‎. ‎it is proved that for a quasi-continuous module $m$‎, ‎$m$ is a uc-module if and only if $m$ is an $sa$-module‎. ‎also‎, ‎it is proved that the direct sum of two $sa$-modules as $r$-modules is an $sa$-module when $r$ is the sum of the annihilators of these...

in this paper, we introduce a new concept of volumes difference function of the projection and intersection bodies. following this, we establish the minkowski and brunn-minkowski inequalities for volumes difference function of the projection and intersection bodies.

Some of these curves are considered in a rather incidental way in the writings of Poincare.** However, the full concept of pseudo closed trajectories does not seem to have been discussed explicitly heretofore. We assume that all of the variables in the equations (1) are real, that the functions X and Y are continuous in an open connected region R in the xy-plane, that these functions satisfy Li...

For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they ...