scholarly journals On the meaning of an equation in dual coordinates

1927 ◽  
Vol 1 (1) ◽  
pp. 39-40
Author(s):  
H. W. Richmond
Keyword(s):  
Homogeneous Equation ◽  
Special Kind ◽  
Dual Numbers ◽  
Edinburgh Math ◽  
Straight Line ◽  
Image Position

If L, M, N denote Prof. Study's Dual Coordinates of a straight line (see Proc. Edinburgh Math. Soc., 44 (1926), 90–97), any (homogeneous) equation F (L, M, N) = 0 must define a certain system of lines. By the nature of dual numbers we must havewhere U and V are functions of l, m, n, λ, μ, ν, the ordinary (Pluckerian) coordinates. Since F = 0 implies U = 0 and V = 0 the system of lines is a congruence. But it is a congruence of a very special kind, whose nature will now be considered.

XVIII.—The Invariant Theory of Forms in Six Variables relating to the Line Complex

1927 ◽  
Vol 46 ◽  
pp. 210-222 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Invariant Theory ◽  
Line Geometry ◽  
Five Dimensions ◽  
Ordinary Space ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Algebraic Theories ◽  
Straight Line ◽  
Image Position ◽  
Theory Of Forms

It is well known that the Plücker coordinates of a straight line in ordinary space satisfy a quadratic identitywhich may also be considered as the equation of a point-quadric in five dimensions, if the six coordinates Pij are treated as six homogeneous coordinates of a point. Projective properties of line geometry may therefore be treated as projective properties of point geometry in five dimensions. This suggests that certain algebraic theories of quaternary forms (corresponding to the geometry of ordinary space) can best be treated as algebraic theories of senary forms: that is, forms in six homogeneous variables.

Trace inequalities for mixtures of Markov chains

1977 ◽  
Vol 9 (04) ◽  
pp. 747-764
Author(s):  
Burton Singer ◽  
Seymour Spilerman
Keyword(s):  
Markov Chains ◽  
Special Kind ◽  
Probability Measures ◽  
Stochastic Matrices ◽  
Transition Matrices ◽  
Time Intervals ◽  
Panel Studies ◽  
Image Position ◽  
Trace Inequalities

In a wide variety of multi-wave panel studies in economics and sociology, comparisons between the observed transition matrices and predictions of them based on time-homogeneous Markov chains have revealed a special kind of discrepancy: the trace of the observed matrices tends to be larger than the trace of the predicted matrices. A common explanation for this discrepancy has been via mixtures of Markov chains. Specializing to mixtures of Markov semi-groups of the form we exhibit classes of stochastic matrices M, probability measures µ and time intervals Δ such that for k = 2, 3 and 4. These examples contradict the substantial literature which suggests — implicitly — that the above inequality should be reversed for general mixtures of Markov semi-groups.

scholarly journals Star multigraphs with three vertices of maximum degree

1986 ◽  
Vol 100 (2) ◽  
pp. 303-317 ◽  
Author(s):  
A. G. Chetwynd ◽  
A. J. W. Hilton
Keyword(s):  
Chromatic Number ◽  
Maximum Degree ◽  
Special Kind ◽  
Chromatic Index ◽  
Simple Graphs ◽  
Multiple Edges ◽  
Image Position ◽  
Edge Colouring

The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with each non-simple edge. An edge-colouring of a multigraph G is a map ø: E(G)→, where is a set of colours and E(G) is the set of edges of G, such that no two edges receiving the same colour have a vertex in common. The chromatic index, or edge-chromatic numberχ′(G) of G is the least value of || for which an edge-colouring of G exists. Generalizing a well-known theorem of Vizing [14], we showed in [6] that, for a star-multigraph G,where Δ(G) denotes the maximum degree (that is, the maximum number of edges incident with a vertex) of G. Star-multigraphs for which χ′(G) = Δ(G) are said to be Class 1, and otherwise they are Class 2.

scholarly journals On the stationary eccentricity of a system of four-tangent conics

1909 ◽  
Vol 28 ◽  
pp. 2-5
Author(s):  
F. E. Edwards
Keyword(s):  
Variable Parameter ◽  
Convex Quadrilateral ◽  
Straight Line ◽  
Image Position

Let the convex quadrilateral formed by the four given tangents be ABA′B′, and O the intersection of the diagonals. Let OA and OB be taken as axes of x and y. Denote OA, OA′, OB and OB′ by a, a′, b and b′, a and b being positive, and a′ and b′ negative. The tangential equation of the system is thenwhere k is a variable parameter; for the equation is satisfied when the straight line lx + my + 1 = 0 passes through any two adjacent angular points of the quadrilateral.

scholarly journals XIII.—The General Form of the Involutive 1-1 Quadric Transformation in a Plane

1905 ◽  
Vol 40 (2) ◽  
pp. 253-262
Author(s):  
Charles Tweedie
Keyword(s):  
Special Property ◽  
Cremona Transformation ◽  
Straight Line ◽  
Image Position

§ 1. In a communication read before the Society, 3rd December 1900, Dr Muir discusses the generalisation, for more than two pairs of variables, of the proposition that: IfthenIf we interpret (x, y) and (ξ, η) iis points in a plane, it is manifest that the transformation thereby obtained is a Cremona transformation. It has the special property of being reciprocal or involutive in character; i.e., if the point P is transformed into Q, then the repetition of the same transformation on Q transforms Q into P. Symbolically, if the transformation is denoted by T. T(P) = Q, and T(Q) = T2(P) = P; so that T2 = 1, and T = T−1. Moreover, if the locus of P (x, y) is a straight line, the locus of Q (ξ, η) is in general a conic.

The Problems of Random Flight and Conduction of Heat

1924 ◽  
Vol 22 (2) ◽  
pp. 167-168
Author(s):  
W. Burnside
Keyword(s):  
Constant Velocity ◽  
Time Interval ◽  
Straight Line ◽  
Random Flight ◽  
Image Position

In a paper on random flight Lord Rayleigh proved the following result: A number is formed by adding together n numbers each of which is equally likely to have any value from − a to + a. Then, if f (n, s) ds is the probability that the number so formed lies between s and s + ds, and if n is sufficiently great,This result may be stated as follows: A point moves discontinuously in a straight line. For a time τ it has a constant velocity. During the next time-interval τ it again has a constant velocity, and so on. Then if each of these velocities is equally likely to have any value from − v to + v, the probability that in the time nτ, the point moves a distance lying between s and s + ds is f (n, s) ds, with vτ written for a.

A Geometrical treatment of the Correspondence between Lines in Threefold Space and Points of a Quadric in Fivefold space

1925 ◽  
Vol 22 (5) ◽  
pp. 694-699 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Quadratic Relation ◽  
Four Dimensions ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Straight Line ◽  
Set Up ◽  
Image Position ◽  
Three Dimensional Space

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relationwhich immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

XXVII.—Bothrodon pridii, an Extinct Serpent of Gigantic Dimensions

1927 ◽  
Vol 46 ◽  
pp. 314-315
Author(s):  
J. Graham Kerr
Keyword(s):  
Straight Line ◽  
Text Figure ◽  
The One ◽  
Image Position ◽  
Time Existence

The object of this note is to place on record the existence of a specimen which demonstrates the one-time existence of poisonous serpents of hitherto quite unheard-of dimensions.The specimen, shown in the text-figure, is a poison-fang remarkable for (1) its enormous size, and (2) its curvature. As regards size, the fang measures nearly 65 mm. along the outside of the curve, and 46 mm. in astraight line along the chord of the curve. It measures close to but clear of the swollen base 11·4 mm. in diameter and 35 mm. in circumference.

scholarly journals Numerical ranges of powers of hermitian elements

1986 ◽  
Vol 28 (1) ◽  
pp. 37-45 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Previous Knowledge ◽  
Line Segments ◽  
Straight Line ◽  
Image Position ◽  
Unital Banach Algebra

An element k of a unital Banach algebra A is said to be Hermitian if its numerical rangeis contained in ℝ; equivalently, ∥eitk∥ = 1(t ∈ ℝ)—see Bonsall and Duncan [3] and [4]. Here we find the largest possible extent of V(kn), n ∈ ℕ, given V(k) ⊆ [−1, 1], and so ∥k∥ ≤ 1: previous knowledge is in Bollobás [2] and Crabb, Duncan and McGregor [7]. The largest possible sets all occur in a single example. Surprisingly, they all have straight line segments in their boundaries. The example is in [2] and [7], but here we give A. Browder's construction from [5], partly published in [6]. We are grateful to him for a copy of [5], and for discussions which led to the present work. We are also grateful to J. Duncan for useful discussions.

scholarly journals A link between the shape of the austenite–martensite interface and the behaviour of the surface energy

2004 ◽  
Vol 134 (6) ◽  
pp. 1099-1113
Author(s):  
A. Elfanni ◽  
M. Fuchs
Keyword(s):  
Surface Energy ◽  
Sobolev Class ◽  
Minimizing Sequences ◽  
Elastic Energy Density ◽  
Almost Everywhere ◽  
Straight Line ◽  
Existence Of Minimizers ◽  
Bounded Lipschitz Domain ◽  
Vertical Tangent ◽  
Image Position

Let Ω ⊂ R2 denote a bounded Lipschitz domain and consider some portion Γ0 of ∂Ω representing the austenite–twinned-martensite interface which is not assumed to be a straight segment. We prove that for an elastic energy density ϖ: R2 → [0 ∞) such that ϖ(0, ±1) = 0. Here, W(Ω) consists of all functions u from the Sobolev class W1, ∞(Ω) such that |uy| = 1 almost everywhere on Ω together with u = 0 on Γ0. We will first show that, for Γ0 having a vertical tangent, one cannot always expect a finite surface energy, i.e. in the above problem, the condition in general cannot be included. This generalizes a result of [12] where Γ0is a vertical straight line. Property (*) is established by constructing some minimizing sequences vanishing on the whole boundary ∂Ω, that is, one can even take Γ0 = ∂Ω. We also show that the existence or non-existence of minimizers depends on the shape of the austenite–twinned-martensite interface Γ0.

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