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 On the representations of numbers by binary cubic forms

1985 ◽  
Vol 27 ◽  
pp. 95-98 ◽  
Author(s):  
C. Hooley
Keyword(s):  
Positive Integer ◽  
Recent Work ◽  
Present Note ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms ◽  
Theory Of Numbers

There are not a few situations in the theory of numbers where it is desirable to have as sharp an estimate as possible for the number r(n) of representations of a positive integer n by an irreducible binary cubic formA variety of approaches are available for this problem but, as they stand, they are all defective in that they introduce unwanted factors into the estimate. For instance, an estimate involving the discriminant of f(x, y) is obtained if we adopt the Lagrange procedure [5] of using congruences of the type f(σ, 1)≡0, mod n, to reduce the problem to one where n=1. Alternatively, following Oppenheim (vid. [2]), Greaves [3], and others, we may appeal to the theory of factorization of ideals, which leads to unwanted logarithmic factors owing to the involvement of algebraic units. Having had need, however, in some recent work on quartic forms [4] for an estimate without such extraneous imperfections, we intend in the present note to prove thatuniformly with respect to the coefficients of f(x, y), where ds(n) denotes the number of ways of expressing n as a product of s factors.

Cubic forms representing arithmetic progressions

1959 ◽  
Vol 55 (3) ◽  
pp. 270-273 ◽  
Author(s):  
G. L. Watson
Keyword(s):  
Arithmetic Progressions ◽  
Cubic Form ◽  
Image Position ◽  
Cubic Forms

The following result has recently been proved by Lewis (3), Davenport (2) and Birch (1).There exists an integer n0 such that every cubic form with rational coefficients and at least n0 integral variables represents zero non-trivially.The arguments of Lewis and Birch are simple, and yield also various generalizations of this result. Davenport's proof is complicated, but it shows that the minimal n0 satisfies

Non-homogeneous binary cubic forms

1951 ◽  
Vol 47 (3) ◽  
pp. 457-460 ◽  
Author(s):  
R. P. Bambah
Keyword(s):  
Lower Bound ◽  
Finite Volume ◽  
Upper Bound ◽  
The Other ◽  
Real Numbers ◽  
Cubic Form ◽  
Homogeneous Form ◽  
Other Hand ◽  
Image Position ◽  
Cubic Forms

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the regionhas a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that

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.

scholarly journals Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities

2005 ◽  
Vol 135 (3) ◽  
pp. 643-662 ◽  
Author(s):  
Hichem Hajaiej
Keyword(s):  
Strict Inequality ◽  
Schwarz Symmetrization ◽  
Image Position

Extended Hardy-Littlewood inequalities are where {ui}1≤i≤m are non-negative functions and denote their Schwarz symmetrization.In this paper, we determine appropriate conditions under which equality in (*) occurs if and only if {ui}1≤i≤m are Schwarz symmetric.

Cubic forms in 29 variables

1962 ◽  
Vol 266 (1326) ◽  
pp. 287-298 ◽  
Keyword(s):  
Cubic Form ◽  
Cubic Forms

It is proved that if C (x 1 ..., x n) is any cubic form in n variables, with integral coefficients, then the equation C ( x 1 x n ) = 0 has a solution in integers x 1 ..., x n not all 0, provided n is at least 29. This is an improvement on a previous result (Davenport 1959).

Cubic forms in sixteen variables

1963 ◽  
Vol 272 (1350) ◽  
pp. 285-303 ◽  
Keyword(s):  
Cubic Form ◽  
Cubic Forms

It is proved that if C ( x 1 , ..., x n ) is any cubic form in n variables, with integral coefficients, then the equation C ( x 1 , ..., x n ) = 0 has a solution in integers x 1 , ..., x n not all 0, provided n is at least 16. This is an improvement upon earlier results (Davenport 1959, 1962).

Cubic forms in thirty-two variables

1959 ◽  
Vol 251 (993) ◽  
pp. 193-232 ◽  
Keyword(s):  
Exponential Sum ◽  
Definite Integral ◽  
Cubic Form ◽  
New Form ◽  
Cubic Forms ◽  
Hardy Littlewood Method

It is proved that if C(xu...,*„) is any cubic form in n variables, with integral coefficients, then the equation C{xu ...,*„) = 0 has a solution in integers xXi...,xn, not all 0, provided n is at least 32. The proof is based on the Hardy-Littlewood method, involving the dissection into parts of a definite integral, but new principles are needed for estimating an exponential sum containing a general cubic form. The estimates obtained here are conditional on the form not splitting in a particular manner; when it does so split, the same treatment is applied to the new form, and ultimately the proof is made to depend on known results.

scholarly journals Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities

2005 ◽  
Vol 135 (3) ◽  
pp. 643-662 ◽  
Author(s):  
Hichem Hajaiej
Keyword(s):  
Strict Inequality ◽  
Schwarz Symmetrization ◽  
Image Position

Extended Hardy-Littlewood inequalities are where {ui}1≤i≤m are non-negative functions and denote their Schwarz symmetrization.In this paper, we determine appropriate conditions under which equality in (*) occurs if and only if {ui}1≤i≤m are Schwarz symmetric.

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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