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

Note on the Braikenridge-Maclaurin theorem

1984 ◽  
Vol 38 (2) ◽  
pp. 235-240 ◽  
Keyword(s):  
Fixed Points ◽  
Conic Section ◽  
Free Vertex ◽  
Straight Line ◽  
Mathematical Problems ◽  
Pass Through ◽  
Straight Lines

In the Mathematical Gazette for 1915, there was a paper (1) of Charles Tweedie, which included an account of the Braikenridge-MacLaurin Theorem: if the sides of a polygon are restricted to pass through fixed points while all the vertices but one lie on fixed straight lines, the free vertex describes a conic section or a straight line. While the account gave a good elucidation of the* mathematical problems surrounding the discovery of the theorem, some MS. letters of Colin MacLaurin indicate that Tweedie’s historical assumption that priority was the reason for the dispute is not quite correct. For Tweedie wrote (2): . . . the name attached to the theorem is justified by the earlier production of Braikenridge, though MacLaurin’s researches seem much more profound. The dispute furnishes an illustration of the sensitiveness of even great mathematicians in their claims to the priority of invention.

Assessment of derailment risk in railway turnouts through quasi-static analysis and dynamic simulation

2021 ◽  
pp. 095440972110240
Author(s):  
Ping Wang ◽  
Jun Lai ◽  
Tao Liao ◽  
Jingmang Xu ◽  
Jian Wang ◽  
...  
Keyword(s):  
Static Analysis ◽  
Dynamic Simulation ◽  
Economic Losses ◽  
Derailment Coefficient ◽  
Straight Line ◽  
Main Track ◽  
Pass Through ◽  
Multi Body ◽  
Yaw Angle ◽  
Curve Radius

Train derailments in railway switches are becoming more and more common, which have caused serious casualties and economic losses. Most previous studies ignored the derailment mechanism when vehicles pass through the turnout. With this consideration, this work aims to research the 3D derailment coefficient limit and passing performance in turnouts through the quasi-static analysis and multi-body dynamic simulation. The proposed derailment criteria have considered the influence of creep force and wheelset yaw angle. Results show that there are two derailing stages in switch panel, which are climbing the switch rail and stock rail, respectively. The 3D derailment coefficient limit at the region of top width 5 mm to 20 mm is much lower than the main track rail, which shows that wheels are more likely to derail in this area. The curve radius before the switch rail is suggested to be set as 350 m. When the curve radius before turnout is 65 m, the length of the straight line between the curve and turnout needs to be larger than 3 m. This work can provide a good understanding of the derailment limit and give guidance to set safety criteria when vehicles pass through the turnout.

scholarly journals A plane quartic curve with twelve undulations

1945 ◽  
Vol 35 ◽  
pp. 10-13 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Homogeneous Coordinates ◽  
Quartic Curve ◽  
Base Points ◽  
Octahedral Group ◽  
Quartic Form ◽  
Image Position ◽  
Quartic Curves ◽  
Set Of Points

The pencil of quartic curveswhere x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.Since (1) may also be writtenthe four linesare, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

Powder Diffraction Data of CdTexSe1-xSolid Solutions

1990 ◽  
Vol 5 (4) ◽  
pp. 206-209 ◽  
Author(s):  
N.A. Razik ◽  
G. Al-Barakati ◽  
S. Al-Heneti
Keyword(s):  
Powder Diffraction ◽  
Diffraction Data ◽  
Hexagonal Wurtzite Structure ◽  
Powder Diffraction Data ◽  
Lattice Constants ◽  
Cubic Form ◽  
Hexagonal Form ◽  
State Diffusion ◽  
Vegard’S Law ◽  
Image Position

AbstractCdTexSe1-xsolid solutions with (x) ranging between zero and one were prepared by solid state diffusion under vacuum and their precise lattice constants and X-ray powder diffraction data were determined. It was found that alloys with 0≤x≤0.4 possess the hexagonal wurtzite structure while those with 0.5≤x≤1.0 have the cubic zincblende structure. The lattice parameters obeyed Vegard's law according to the following formulaeExtrapolated lattice constants were a = 6.066(6) Å for the cubic form of CdSe and a = 4.563(6) Å and c = 7.502 (10) Å for the hexagonal form of CdTe.

Rational normal octavic surfaces with a double line, in space of five dimensions

1933 ◽  
Vol 29 (1) ◽  
pp. 95-102 ◽  
Author(s):  
D. W. Babbage
Keyword(s):  
Rational Surface ◽  
Simple Rule ◽  
Plane Curves ◽  
Double Line ◽  
Five Dimensions ◽  
Multiple Line ◽  
Image Position

The following paper arises from a remark in a recent paper by Professor Baker. In that paper he gives a simple rule, under which a rational surface has a multiple line, expressed in terms of the system of plane curves which represent the prime sections of the surface. The rule is that, if one system of representing curves is given by an equation of the formthe surface being given, in space (x0, x1,…, xr), by the equationsthen the surface contains the linecorresponding to the curve φ = 0; and if the curve φ = 0 has genus q, this line is of multiplicity q + 1.

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.

On Asymptotic Centers and Fixed Points of Nonexpansive Mappings

1980 ◽  
Vol 32 (2) ◽  
pp. 421-430 ◽  
Author(s):  
Teck-Cheong Lim
Keyword(s):  
Banach Space ◽  
Fixed Points ◽  
Nonempty Subset ◽  
Nonexpansive Mappings ◽  
Chebyshev Center ◽  
Bounded Subset ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Asymptotic Centers ◽  
Image Position

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

scholarly journals A proof of Aronhold's theorem upon quartic curves

1940 ◽  
Vol 6 (3) ◽  
pp. 190-191
Author(s):  
H. W. Richmond
Keyword(s):  
Finite Number ◽  
Simple Proof ◽  
Plane Curves ◽  
Cubic Surface ◽  
Quartic Curves ◽  
Order Four ◽  
The Given

It is to be expected that a finite number of plane curves of order four should have seven given lines as bitangents, because the number of conditions imposed is equal to the number of effective free constants in the equation of such a curve, viz., 14. Aronhold made the interesting discovery that one curve could be determined in which no three of the given lines have their six points of contact on a conic. The method, due to Geiser, of obtaining the bitangents as projections of the lines of a cubic surface leads to a simple proof of the existence of this quartic.

On a conjecture of Mordell concerning binary cubic forms

1941 ◽  
Vol 37 (4) ◽  
pp. 325-330 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Strict Inequality ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms

Let f(x, y) be a binary cubic form with real coefficients and determinant D ≠ 0. In a recent paper, Mordell has proved that there exist integral values of x, y, not both zero, for whichThese inequalities are best possible, since they cannot be satisfied with the sign of strict inequality when f(x, y) is equivalent tofor the case D < 0, or tofor the case D > 0.

scholarly journals The thermal decomposition of acetaldehyde and the existence of different activated states

1933 ◽  
Vol 141 (843) ◽  
pp. 41-55 ◽  
Keyword(s):  
High Pressures ◽  
Initial Pressure ◽  
Bimolecular Reaction ◽  
Bimolecular Reactions ◽  
First Order ◽  
Straight Line ◽  
Gas Reactions ◽  
Pass Through ◽  
Low Pressures ◽  
Kinetics Of

Homogeneous thermal gas reactions were at one time tacitly assumed to possess a definite order, unimolecular and bimolecular reactions, for example, being sharply distinguished. The kinetics of the decomposition of acetalde­ hyde, CH 3 CHO = CH 4 + CO, over the pressure range of 100 to 400 mm. were found to satisfy the criterion of a bimolecular reaction, namely, that the reciprocal of the time for half change (1/ t 1/2 ) )plotted against the initial pressure ( p 0 ) gave a straight line inclined to the axes. The line, however, did not pass through the origin, as may be seen in fig. 1 of the present paper. This indicated the presence of some first order reaction, the nature of which was not determined. Subsequently, in accordance with the collision theory of activation and deactivation, it was shown that certain reactions, sometimes called quasiummolecular, change their order from the second at low pressures to the first at high pressures. This apparently was the reverse of the behaviour shown by acetaldehyde.

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