On the product of three homogeneous linear forms

1951 ◽  
Vol 47 (2) ◽  
pp. 251-259 ◽  
Author(s):  
J. H. H. Chalk ◽  
C. A. Rogers
Keyword(s):  
Dimensional Space ◽  
General Point ◽  
Linear Forms ◽  
3 Dimensional ◽  
Image Position ◽  
Homogeneous Linear

Let X denote the general point with coordinates (x1, x2, x3) in 3-dimensional space; and let P(X) be the function defined by

On the product of n linear forms

1953 ◽  
Vol 49 (2) ◽  
pp. 190-193 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Complex Conjugate ◽  
Linear Forms ◽  
Image Position ◽  
Complex Coefficients ◽  
Homogeneous Linear

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. For any n numbers α1, …, αn of the same ‘type’ as the forms L1, …, Ln (that is, α1, …, αr real, αr+1, …, αr+s complex, αr+s+j = ᾱr+j), let

On the product of three non-homogeneous linear forms

1947 ◽  
Vol 43 (2) ◽  
pp. 137-152 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Real Numbers ◽  
Linear Forms ◽  
Image Position ◽  
The Right ◽  
Homogeneous Linear

Let ξ, η, ζ be linear forms in u, v, w with real coefficients and determinant Δ ≠ 0. A conjecture of Minkowski, which was subsequently proved by Remak, tells us that for any real numbers a, b, c there exist integral values of u, v, w for whichand the constant ⅛ on the right is best possible.

A general canonical expression

1933 ◽  
Vol 29 (4) ◽  
pp. 465-469 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Linear Space ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
General Point ◽  
Linear Forms ◽  
Powers Of Linear Forms ◽  
Necessary And Sufficient

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

On the Product of Three Homogeneous Linear Forms. IV

1943 ◽  
Vol 39 (1) ◽  
pp. 1-21 ◽  
Author(s):  
H. Davenport
Keyword(s):  
Lower Bound ◽  
Real Number ◽  
Linear Transformation ◽  
Linear Forms ◽  
Special Linear ◽  
Absolute Value ◽  
Cubic Equation ◽  
Image Position ◽  
Homogeneous Linear

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+t2-2t-1 = 0, the special linear forms are equivalent, by an integral unimodular linear transformation, to(in any order), where λ1,λ2,λ3 are real number whose product is In this case, L1L2L3|λ1λ2λ3 is a non-zero integer, and the minimum of its absolute value is 1, giving

scholarly journals HIGH ORDER PARACONTROLLED CALCULUS

2019 ◽  
Vol 7 ◽  
Author(s):  
ISMAËL BAILLEUL ◽  
FRÉDÉRIC BERNICOT
Keyword(s):  
Model Equation ◽  
Dimensional Space ◽  
Parabolic Anderson Model ◽  
General Version ◽  
Kpz Equation ◽  
Elementary Approach ◽  
3 Dimensional ◽  
Regularity Structures ◽  
Dimensional Space Time ◽  
Image Position

We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations $\mathsf{E}$ and  $\mathsf{F}$ . We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701},\end{eqnarray}$$ on a 3-dimensional closed manifold, and the generalized KPZ equation $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701}+g(u)(\unicode[STIX]{x2202}u)^{2},\end{eqnarray}$$ driven by a $(1+1)$ -dimensional space/time white noise.

Note on non-homogeneous linear forms

1953 ◽  
Vol 49 (2) ◽  
pp. 360-362 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Algebraic Number ◽  
Linear Forms ◽  
Image Position ◽  
Homogeneous Linear

Let θ be an algebraic number of degree n, with conjugates θ(1), …, θ(n), where θ(1), …, θ(r) are real and θ(r+j), θ(r+s+j) are complex conjugates for j = 1, …, s. [Here r ≥ 0, s ≥ 0, r + 2s = n.] Let ω1, …, ωn be a basis for the integers of k(θ), and set

Complex homogeneous linear forms

1956 ◽  
Vol 52 (1) ◽  
pp. 35-38 ◽  
Author(s):  
K. Rogers
Keyword(s):  
Complex Space ◽  
Rational Numbers ◽  
Quadratic Equation ◽  
Complex Numbers ◽  
Geometry Of Numbers ◽  
Systematic Method ◽  
Linear Forms ◽  
Rings Of Integers ◽  
Image Position ◽  
Homogeneous Linear

Let Z, Q, C denote respectively the ring of rational integers, the field of rational numbers and the field of complex numbers. Minkowski (4) solved the problem of minimizingfor x, y ∈ Z(i) or Z(ρ), where a, b, c, d ∈ C have fixed determinant Δ ≠ 0. Here ρ = exp 2/3πi, and Z(i) and Z(p) are the rings of integers in Q(i) and Q(ρ) respectively. In fact he found the best possible resultsfor Z(i), andfor Z(ρ), wherewhile Buchner (1) used Minkowski's method to show thatfor Z(i√2). Hlawka(3) has also proved (1·2), and Cassels, Ledermann and Mahler (2) have proved both (1·2) and (1·3). In a paper being prepared jointly by H. P. F. Swinnerton-Dyer and the author, general problems of the geometry of numbers in complex space are discussed and a systematic method given for solving the above problem for all complex quadratic fields Q(ϑ). Here, ϑ is a non-real number satisfying. an irreduc7ible quadratic equation with rational coefficients. The above problem is solved in detail for Q(i√5), for whichand the ‘critical forms’ can be reduced to

Yet another proof of Minkowski's theorem on the product of two inhomogeneous linear forms

1953 ◽  
Vol 49 (2) ◽  
pp. 365-366 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Linear Forms ◽  
Inhomogeneous Linear Forms ◽  
Image Position ◽  
Homogeneous Linear

Theorem 1. Let ξ = αx + βy, η = γx + δy be two homogeneous linear forms in x, y with real coefficients and determinant αδ − βγ = Δ ≠ 0. Then for any real constants p, q there are integers x, y such that

On the product of three non-homogeneous linear forms

1951 ◽  
Vol 47 (2) ◽  
pp. 260-265 ◽  
Author(s):  
L. E. Clarke
Keyword(s):  
Real Numbers ◽  
Linear Forms ◽  
Image Position ◽  
Homogeneous Linear

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 which

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