Some quartic curves with no points in any cubic field

2018 ◽  
Vol 116 (4) ◽  
pp. 1028-1028
Author(s):  
Andrew Bremner
Keyword(s):  
Cubic Field ◽  
Quartic Curves

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.

The determination of units in totally real cubic fields

1960 ◽  
Vol 56 (4) ◽  
pp. 318-321 ◽  
Author(s):  
H. J. Godwin
Keyword(s):  
Trial And Error ◽  
Integral Lattice ◽  
Cubic Field ◽  
Cubic Fields ◽  
Trial And Error Method ◽  
Totally Real ◽  
Rational Integral ◽  
Fundamental Units ◽  
Error Method

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

On the occurrence of a metastable tetragonal t′-phase in a ZrO2−13.6 mole % MgO ceramic and its microscopic thermal evolution

1993 ◽  
Vol 8 (3) ◽  
pp. 605-610 ◽  
Author(s):  
M.C. Caracoche ◽  
P.C. Rivas ◽  
A.F. Pasquevich ◽  
A.R. López García ◽  
E. Aglietti ◽  
...  
Keyword(s):  
Thermal Behavior ◽  
Room Temperature ◽  
Oxygen Vacancies ◽  
Single Phase ◽  
Thermal Evolution ◽  
Hyperfine Interactions ◽  
Correlation Technique ◽  
Cubic Field ◽  
Time Differential ◽  
T Phase

The time-differential perturbed angular correlation technique has been used to investigate the thermal behavior of a ZrO2−13.6 mole % MgO ceramic between room temperature and 1423 K. Two different quadrupole hyperfine interactions corresponding to a tetragonal structure have been found to result on cooling the ceramic from the single-phase cubic field. One of them agrees with that depicting the pure t-ZrO2 tetragonal phase and the other one has been interpreted as describing a high-MgO-content nontransformable t'–ZrO2 phase. As temperature increases, the latter gives rise to a similar but fluctuating interaction related to the oxygen vacancies mobility and which shows a thermal behavior analogous to that already reported for the stabilized cubic ZrO2. Above 1100 K these dynamic t'-sites transform into pure tetragonal ones which behave ordinarily, suffering the t → m phase transition when cooling to room temperature. Differences found between TDPAC results and information drawn from other techniques are discussed.

On Five Lines, in Space of Four Dimensions, which lie upon a, Quadric Threefold and the Normal Quartic Curves of which they are Chords

1924 ◽  
Vol 22 (2) ◽  
pp. 189-199
Author(s):  
F. Bath
Keyword(s):  
Three Dimensions ◽  
Apparent Contradiction ◽  
Four Dimensions ◽  
Quartic Curve ◽  
Lines In Space ◽  
Quadric Threefold ◽  
Quartic Curves

The connexion between the conditions for five lines of S4(i) to lie upon a quadric threefold,and (ii) to be chords of a normal quartic curve,leads to an apparent contradiction. This difficulty is explained in the first paragraph below and, subsequently, two investigations are given of which the first uses, mainly, properties of space of three dimensions.

Transformations depending on sets of associated points

1952 ◽  
Vol 48 (3) ◽  
pp. 383-391
Author(s):  
T. G. Room
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Three Dimensional ◽  
Cubic Curves ◽  
Birational Transformations ◽  
Quadric Surfaces ◽  
Base Points ◽  
Dimensional Projective Space ◽  
Quartic Curves

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

scholarly journals Remarks on principal factors in a relative cubic field

1971 ◽  
Vol 3 (2) ◽  
pp. 226-239 ◽  
Author(s):  
Pierre Barrucand ◽  
Harvey Cohn
Keyword(s):  
Cubic Field ◽  
Principal Factors

Ground-State Splitting ford5S6Ions in a Cubic Field

1960 ◽  
Vol 5 (4) ◽  
pp. 145-146 ◽  
Author(s):  
M. J. D. Powell ◽  
J. R. Gabriel ◽  
D. F. Johnston
Keyword(s):  
Ground State ◽  
Cubic Field ◽  
State Splitting

Zeeman Effect of the Purely Cubic Field Fluorescence Line of MgO:Cr3+Crystals

1960 ◽  
Vol 120 (6) ◽  
pp. 2045-2053 ◽  
Author(s):  
S. Sugano ◽  
A. L. Schawlow ◽  
F. Varsanyi
Keyword(s):  
Zeeman Effect ◽  
Cubic Field ◽  
Fluorescence Line
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