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.

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.

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.

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.

The resolution of monoidal Cremona transformations of three-dimensional space

1938 ◽  
Vol 34 (1) ◽  
pp. 22-26
Author(s):  
D. W. Babbage
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Suitable Choice ◽  
Cremona Transformation ◽  
Cremona Transformations ◽  
Image Position ◽  
Three Dimensional Space

A Cremona transformation Tn, n′ between two three-dimensional spaces is said to be monoidal if the surfaces of order n in one space which form the homaloidal system corresponding to the planes of the second space have a fixed (n − 1)-ple point O. If the surfaces of order n′ forming the homaloidal system in the second space have a fixed (n′ − 1)-ple point O′, the transformation is said to be bimonoidal. A particularly simple bimonoidal transformation is that which transforms lines through O into lines through O′, and planes through O into planes through O′. Such a transformation we shall call an M-transformation. Its equations can, by suitable choice of coordinates, be expressed in the formwhere φn−1(x, y, z, w) = 0, φn(x, y, z, w) = 0 are monoids with vertex (0, 0, 0, 1).

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.

A generalisation of the arbelos theorem of Archimedes

2011 ◽  
Vol 95 (533) ◽  
pp. 197-205
Author(s):  
Shailesh A Shirali
Keyword(s):  
Straight Line ◽  
Archimedean Property ◽  
Remarkable Result ◽  
Image Position

Over two thousand years ago, Archimedes discovered a remarkable result concerning two circles drawn with reference to a configuration of three circles and a straight line. Figure 1 displays this result.In the figure, A, B, C are three collinear points, with B between A and C; circles ω1, ω2, ω3 are drawn on AB, BC, AC as diameters, respectively; a line l is drawn through B, perpendicular to AC; a circle ω4 is inscribed in the region bounded by {ωl, ω3, l}; and a circle ω5 is inscribed in the region bounded by {ω2, ω3, l}. The Archimedean property is that ω4 and ω5 have equal radii.

On certain nets of plane curves

1924 ◽  
Vol 22 (1) ◽  
pp. 1-10 ◽  
Author(s):  
F. P. White
Keyword(s):  
Fixed Points ◽  
Plane Curves ◽  
Cubic Form ◽  
Straight Line ◽  
Pass Through ◽  
Image Position ◽  
Quartic Curves

1. The plane quartic curves which pass through twelve fixed points g, of which no three lie on a straight line, no six on a conic and no ten on a cubic, form a net of quartics represented by the equation

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