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On the Product of Three Homogeneous Linear Forms. IV

Mathematical Proceedings of the Cambridge Philosophical Society, 1938
Let L1, L2, L3 be three homogeneous linear forms in u, v, w with real coefficients and determinant 1. Let M denote the lower bound offor integral values of u, v, w, not all zero. I proved a few years ago (1) thatmore precisely, thatexcept when L1, L2, L3 are of a special type, in which case If we denote by θ, ø, ψ the roots of the cubic equation t3+t2-
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On the product of n linear forms

Mathematical Proceedings of the Cambridge Philosophical Society, 1953
Let L1, …, Ln be n homogeneous linear forms in n variables u1, …, un, with non-zero determinant Δ. Suppose that L1, …, Lr have real coefficients, that Lr+1, …, Lr+s have complex coefficients, and that the form Lr+s+j is the complex conjugate of the form Lr+j for j = 1, …, s, where r + 2s = n. Letfor integral u1, …, un, not all zero.
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On the Product of Two Linear Homogeneous Forms

Journal of the London Mathematical Society, 1938
Verf. führt einen kurzen, arithmetischen Beweis für den bekannten Satz: Sind \(L\) und \(M\) zwei reelle, homogene lineare Formen in \(x, y\) mit der Determinante 1, so gibt es ganze \(x, y \ne 0, 0\), so daß \(\vert LM \vert \le \frac1{\sqrt{5}}\). Beim Beweis wird der Minkowskische Linearformensatz and die Ungleichung \(\vert w^2 + 3w + 1\vert\le 1\),
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The product of linear nonhomogeneous forms

Mathematical Notes of the Academy of Sciences of the USSR, 1974
We show that for an arbitrary unimodular lattice Λ of dimension n and an arbitrary point C =(c1, c2...cn) ɛ Rn a point Y = (y1, y2,..., yn) e Λ can be found and also a number h, satisfying the condition 1 ⩽h ⩽ 2−n/2 θ−1 + 1 (0 < θ ⩽ 2−n/2), such that the inequality $$\prod\nolimits_{i = 1}^n {\left| {Y_i + hc_i } \right|}< \theta $$ will be ...
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The Product of Two Non-Homogeneous Linear Forms

Journal of the London Mathematical Society, 1948
Einfacher zahlengeometrischer Beweis des bekannten Satzes: Es gibt ganze Zahlen \(x, y\), so daß \[ \vert(ax + by + p)(cx + dy + q)\vert \le \tfrac14 \vert ad - bc\vert \quad (a, b, c, d, p, q \text{ reell}). \]
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On the product of three homogeneous linear forms

Mathematical Proceedings of the Cambridge Philosophical Society, 1951
Let X denote the general point with coordinates (x1, x2, x3) in 3-dimensional space; and let P(X) be the function defined ...
Chalk, John H. H., Rogers, C. A.
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On the Product of Two Non-Homogeneous Linear Forms

Journal of the London Mathematical Society, 1941
Yet another proof of Minkowski's theorem on the product of two inhomogeneous linear forms. From the text: Though a number of proofs of this result are known (see references on p. 19 in [\textit{J. F. Koksma}, Diophantische Approximationen. Berlin: Springer (1936; Zbl 0012.39602)], the proof given here be of interest.
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Multilinear forms which are products of linear forms

2014
The conditions under which, multilinear forms (the symmetric case and the non symmetric case),can be written as a product of linear forms, are considered. Also we generalize a result due to S.Kurepa for 2n-functionals in a group G.
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On the product of three non-homogeneous linear forms

Mathematical Proceedings of the Cambridge Philosophical Society, 1951
Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. Then there exists a number ℳ such that, corresponding to any real numbers a, b, c, there exist rational integers u, v, w for ...
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On the Product of Four Non-Homogeneous Linear Forms

The Annals of Mathematics, 1948
where u1, U2 U3 I U4 are four integral unimodular forms in x1, X2 I X3 X4X the Pi, P2, P3, P4 are four real numbers whose product is A, and ni , n2 , n3 , n4 are integers. The proof will be carried out in two stages, the results of which are stated separately in the following two theorems A and B. THEOREM A.
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