Inhomogeneous Minimum Research Articles

On a question of Uri Shapira and Barak Weiss

Here it is proved that if Q(x1,..., xn) is a positive definite quadratic form which is reduced in the sense of Korkine and Zolotareff and has outer coefficients B1,..., Bn satisfying B1 ≥ 1) Bn ≤ 1 and B1 ⋯ Bn = 1, then its inhomogeneous minimum is at most n/4 for n ≤ 7. This result implies a positive answer to a question of Shapira and Weiss for stable lattices and thereby provides another proof of Minkowski’s Conjecture on the product of n real non-homogeneous linear forms in n variables for n ≤ 7. Our result is an analogue of Woods’ Conjecture which has been proved for n ≤ 9. The analogous problem when B11 is also investigated.

An optimal lower bound for the Frobenius problem

Given N ⩾ 2 positive integers a 1 , a 2 , … , a N with GCD ( a 1 , … , a N ) = 1 , let f N denote the largest natural number which is not a positive integer combination of a 1 , … , a N . This paper gives an optimal lower bound for f N in terms of the absolute inhomogeneous minimum of the standard ( N − 1 ) -simplex.

Inhomogeneous and Euclidean spectra of number fields with unit rank strictly greater than 1

Let K be a number field with unit rank r > 1. In this article we show that the inhomogeneous minimum of K is attained by at least one rational point. In particular, if M ( K ) is the Euclidean minimum of K , we have . This phenomenon has consequences on the decidability of the Euclidean nature of such a field. Moreover, in case K is not a CM-field, we prove that is attained, isolated, and that the inhomogeneous minimum function takes discrete rational values.

Two distinct time scales in the dynamics of a dense hard-sphere liquid.

The dynamic behavior of a dense hard-sphere liquid is studied by numerically integrating a set of Langevin equations that incorporate a free energy functional of the Ramakrishnan-Yussouff form. At relatively low densities, the system remains, during the time scale of our simulation, in the neighborhood of the metastable local minimum of the free energy that represents a uniform liquid. At higher densities, the system is found to fluctuate near the uniform liquid minimum for a characteristic period of time before making a transition to an inhomogeneous minimum of the free energy. The time that the system spends in the vicinity of the liquid minimum before making a transition to another one defines a new time scale of the dynamics. This time scale is found to decrease sharply as the density is increased above a characteristic value. Implications of these observations on the interpretation of experimental and numerical data on the dynamics of supercooled liquids are discussed.

The inhomogeneous minimum of binary quadratic forms

AbstractAn indefinite binary quadratic form ƒ gives rise to a certain function M on the torus. The properties of M, especially those related to its maximum – the so-called inhomogeneous minimum of ƒ – are the subject of numerous papers. Here we continue this study, putting more emphasis on the general behaviour of M.

Inhomogeneous minimum of indefinite quadratic forms in six variables: A conjecture of Watson

The famous conjecture of Watson(11) on the minima of indefinite quadratic forms in n variables has been proved for n ≤ 5, n ≥ 21 and for signatures 0 and ± 1. For the details and history of the conjecture the reader is referred to the author's paper(8). In the succeeding paper (9), we prove Watson's conjecture for signature ± 2 and ± 3 and for all n. Thus only one case for n = 6 (i.e. forms of type (1, 5) or (5, 1)) remains to he proved which we do here; thereby completing the case n = 6. This result is also used in (9) for proving the conjecture for all quadratic forms of signature ± 4. More precisely, here we prove:Theorem 1. Let Q6(x1, …, x6) be a real indefinite quadratic form in six variables of determinant D ( < 0) and of type (5, 1) or (1, 5). Then given any real numbers ci, 1 ≤ i ≤ 6, there exist integers x1,…, x6such that

The inhomogeneous minimum of quadratic forms of signature ± 1

Let Qr be a real indefinite quadratic form in r variables of determinant D ≠ 0 and of type (r1, r2), 0 < r1 < r, r = r1 + r2, S = r1 − r2 being the signature of Qr. It is known (e.g. Blaney (3)) that, given any real numbers c1, c2,…, cr, there exists a constant C depending only on r and s such that the inequalityhas a solution in integers x1, x2, …, xr.

The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms - Corrigenda

On behalf of the authors and the Press, the editor has to notify the following amendments.Volume 48 (1952), 72–86 and 519–20J. W. S. Cassels. ‘The inhomogeneous minimum of binary quadratic, ternary cubic and quaternary quartic forms.’Professor H. W. Lenstra Jr has pointed out that there is a mistake in the proof of Theorem 6, and that the constant 5300 must be replaced by the worse constant 16730. In (iii) of Lemma 16, k should be replaced by k2 and consequently k¼ by k½ in the display at the bottom of page 85. The new estimate results then from putting k = (2·35)2.

A Restricted Inhomogeneous Minimum for Forms

Let us suppose that ƒ(x, y) is an indefinite binary quadratic form that does not represent zero. If P is the real point (x0, y0) then we may define where the infimum is taken over all integral x, y. The inhomogeneous minimum of the form ƒ is defined where the supremum taken over all real points P, need only extend over some complete set of points, incongruent mod 1.

Inhomogeneous minimum of indefinite quaternary quadratic forms

Let Q (x1, …, xn) be a real indefinite quadratic form in n-variables x1,…, xn with signature (r, s),r + s = n and determinant D ≠ 0. Then it is known (see Blaney (2)) that there exists constant Cr, s depending only on r, s such that given any real numbers c1, …,cn we can find integers x1, …, xn satisfying

An inhomogeneous minimum of a class of functions

An inhomogeneous minimum of a class of functions

On the Inhomogeneous Minima of Totally Real Cubic Norm-Forms

On the Inhomogeneous Minima of Totally Real Cubic Norm-Forms

Indefinite Quadratic Forms in Many Variables: The Inhomogeneous Minimum and a Generalization

Indefinite Quadratic Forms in Many Variables: The Inhomogeneous Minimum and a Generalization

The inhomogeneous minima of indefinite quadratic forms

If f(x) is a real indefinite quadratic form in n variables with determinant d ≠ 0, we set for any real α.

The inhomogeneous minimum of an indefinite quadratic form

The inhomogeneous minimum of an indefinite quadratic form

The inhomogeneous minimum of quadratic forms of signature zero

The inhomogeneous minimum of quadratic forms of signature zero

Another transference theorem of the geometry of numbers

In a recent note (1) I sharpened existing theorems, which state, roughly speaking, that unless a symmetric convex body K has an excessively small homogeneous minimum, it must have a reasonably small inhomogeneous minimum; in other words, if the translates of K centred at points of the integer lattice form an efficient packing, then by expanding them we may derive an efficient covering. In this note I will prove the converse, that a bad packing leads to a bad covering; the result is a much less useful one, as the worst case occurs when we foolishly try to pack rather spiky bodies point to point. A result of this kind has previously been proved by Mahler (4), but his result is less precise from our present point of view; a full account is given by Cassels(3).

On the inhomogeneous minimum of the product of n linear forms

On the inhomogeneous minimum of the product of <i>n</i> linear forms

The Covering Of Space By Spheres

Bambah (1) has recently determined the most economical covering of three dimensional space by equal spheres whose centres form a lattice, the density of this covering being1.1.As is well known, this problem may be interpreted in terms of the inhomogeneous minimum of a positive definite quadratic form.

The inhomogeneous minimum of a ternary quadratic form (II)

The inhomogeneous minimum of a ternary quadratic form (II)