E. G. Straus

A Simple Non-Euclidean Geometry and Its Physical Basis

1. What is geometry?.- 2. What is mechanics?.- I. Distance and Angle Triangles and Quadrilaterals.- 3. Distance between points and angle between lines.- 4. The triangle.- 5. Principle of duality coparallelograms and cotrapezoids.- 6. Proof s of the principle of duality.- II. Circles and Cycles.- 7. Definition of a cycle radius and curvature.- 8. Cyclic rotation diameters of a cycle.- 9. The circumcycle and incycle of a triangle.- 10. Power of a point with respect to a circle or cycle inversion.- Conclusion.- 11. Einsteins principle of relativity and Lorentz transformations.- 12. Minkowskian geometry.- 13. Galilean geometry as a limiting case of Euclidean and Minkowskian geometry.- Supplement A. Nine plane geometries.- Supplement B. Axiomatic characterization of the nine plane geometries.- Supplement C. Analytic models of the nine plane geometries.- Answers and Hints to Problems and Exercises.- Index of Names.- Index of Subjects.

Dissection Graphs of Planar Point Sets

This chapter discusses the dissection graphs of planar point sets. It discusses the general properties of the graphs G k . The graph G k can be constructed as follows. Let l be any oriented line containing no points of S and having k + 1 points of S on its positive side. l is to be translated to its left until it meets a point p 1 of S . This line will be called l (0). Then, l (0) is to be rotated counterclockwise by θ about p 1 into line l (θ) until it meets a second point p 2 of S at l (θ 1 ) = 1 1 . l (0) is then rotated counterclockwise about p 2 until l (θ) meets a point p 3 of S at l (θ 2 ) = l 2 , etc. This gives a sequence of points p 1 , p 2 , …, p n of S with p N +1 = p 1 , p N +2 = p 2 and a sequence of directed lines l 1 , l 2 , l N , l N +1 with l N +2 = l 1 .

Elementary inclusion relations for generalized numerical ranges

Abstract Let γ 1 ,…,γ n be complex constants. The set W (γ 1 ,…,γ n ) ( A ) = {Σγ j ( Ax j , x j )},where ( x 1 ,…, x n ) vary over all orthonormal systems in C n , is called a generalized numerical range of a given n × n matrix A . In this paper we study inclusion relations of the form W (γ 1 ,…,γ n ) ⊂λ W (γ′ 1 ,…,γ′ n ) which hold uniformly for all n -square matrices A In particular we concentrate on the case where the coefficients are real. Such inclusion relations yield simple inequalities among generalized numerical radii. Finally, a further generalization of the above numerical range is discussed.

Euclidean Ramsey Theorems. I

Abstract The general Ramsey problem can be described as follows: Let A and B be two sets, and R a subset of A × B . For a ϵ A denote by R ( a ) the set { b ϵ B | ( a , b ) ϵ R }. R is called r -Ramsey if for any r -part partition of B there is some a ϵ A with R ( a ) in one part. We investigate questions of whether or not certain R are r -Ramsey where B is a Euclidean space and R is defined geometrically.

Norm properties of C-numerical radii

Abstract Given n × n complex matrices A , C , the C -numerical radius of A is the nonnegative quantity r c (A)≡ma{|tr(CU ∗ AU)|:U unitary} . For C = diag (1,0,…,0) it reduces to the classical numerical radius r(A)= max{|x ∗ Ax|:x ∗ x=1} . We show that r c is a generalized matrix norm if and only if C is nonscalar and tr C ≠0. Next, we consider an arbitrary generalized matrix norm and characterize all constants v ⩾0 for which vN is multiplicative. A technique to obtain such v is then applied to C -numerical radii with Hermitian C . In particular we find that vr is a matrix norm if and only if v ⩾4.

Sorting X + Y

This paper consists of the application of well- known results to a new problem. The problem, posed by E. Berlekamp, was to save computation time in sorting sets of numbers of the form X q- Y. The main appeal of the new results, aside from their practical uses, is the diversity of the old ideas behind them. It is also interesting to see how the modeling of data and computing affect the results: when the model is of sorting n-tupules of real numbers with binary compari- sons, the best result we can do is nalogan comparisons, which is shown to be sharp under certain circumstances; when the model is of sorting n-tuples of integers in the range 0 to n -- 1, on a random-access stored program computer, the time complexity is shown to be on the order of n(logan).

On a problem of D. R. Hughes

F. I. Fleischer, Sur les espaces normes non-archimtdiens, Neder. Akad. Wetensch. vol. 16 (1954) pp. 165-168. G. K. A. H. Gravett, Valued linear spaces, Quart. J. Math. (Oxford) vol. 24. (1955) pp. 309-315. I. A. W. Ingleton, The Hahn-Banach theorem for non-archimedian valued fields, Proc. Cambridge Philos. Soc. vol. 48 (1952) pp. 41-45. S. O. F. G. Schilling, The theory of valuations, Amer. Math. Soc, New York, 1950.

Operator norms, multiplicativity factors, and C-numerical radii

Abstract Let V be a normed vector space over C , let B ( V ) denote the algebra of linear bounded operators on V , and let N be an arbitrary seminorm or norm on B ( V ). In this paper we discuss multiplicativity factors for N , i.e., constants μ>0 for which N μ ≡ μN is submultiplicative. We find that, while in the finite dimensional case nontrivial indefinite seminorms have no multiplicativity factors and norms do have multiplicativity factors, in the infinite dimensional case N may or may not have such factors. Our results are then applied in order to compute multiplicativity factors for certain generalizations of the classical numerical radius, called C -numerical radii. This is done with the help of a combinatorial inequality which seems to be of independent interest.

Multiplicativity of lp norms for matrices

Abstract The l p norm and the l p operator norm of an m × n complex matrix A = ( α ij ) are given by |A| p = ∑ i, j |α ij | p 1 p and ‖A‖ p = max {|Ax| p :xe C n , |x| p =1} , respectively. The main purpose of this paper is to investigate the multipicativity of the l p norms and their relation to the l p operator norms.

Extremals of functions on graphs with applications to graphs and hypergraphs

The method used in an article by T. S. Matzkin and E. G. Straus [Canad. J. Math. 17 (1965), 533–540] is generalized by attaching nonnegative weights to t-tuples of vertices in a hypergraph subject to a suitable normalization condition. The edges of the hypergraph are given weights which are functions of the weights of its t-tuples and the graph is given the sum of the weights of its edges. The extremal values and the extremal points of these functions are determined. The results can be applied to various extremal problems on graphs and hypergraphs which are analogous to P. Turans Theorem [Colloq. Math. 3 (1954), 19–30: (Hungarian) Mat. Fiz. Lapok 48 (1941), 436–452].