scholarly journals Construction of a Hermitian lattice without a basis of minimal vectors

2012 ◽  
Vol 88 (5) ◽  
pp. 75-77
Author(s):  
Poo-Sung Park
Keyword(s):  
Minimal Vectors

On Eutactic Forms

1977 ◽  
Vol 29 (5) ◽  
pp. 1040-1054 ◽  
Author(s):  
Avner Ash
Keyword(s):  
Quadratic Form ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Column Vector ◽  
Positive Definite Quadratic Form ◽  
Minimal Vectors

Let (aij) = A be a positive definite n × n symmetric matrix with real entries. To it corresponds a positive definite quadratic form ƒ on Rn: ƒ(x) = txAx = ∑ aijXiXj for x any column vector in Rn. The set of values ƒ(y) for y in Zn — {0} has a minimum m (A) > 0 and the number of “minimal vectors“ y1, … , yr in Zn for which ƒ(yi) = m (A) is finite. By definition, ƒ and A are called eutactic if and only if there are positive numbers s1 ,… , sr such that

scholarly journals BASES OF MINIMAL VECTORS IN LATTICES, III

2012 ◽  
Vol 08 (02) ◽  
pp. 551-567 ◽  
Author(s):  
JACQUES MARTINET ◽  
ACHILL SCHÜRMANN
Keyword(s):  
Euclidean Lattices ◽  
Minimal Vectors

We prove that all Euclidean lattices of dimension n ≤ 9 which are generated by their minimal vectors, also possess a basis of minimal vectors. By providing a new counterexample, we show that this is not the case for all dimensions n ≥ 10.

scholarly journals Efficient production of superior dumbbell-shaped DNA minimal vectors for small hairpin RNA expression

2015 ◽  
Vol 43 (18) ◽  
pp. e120-e120 ◽  
Author(s):  
Han Yu ◽  
Xiaoou Jiang ◽  
Kar Tong Tan ◽  
Liting Hang ◽  
Volker Patzel
Keyword(s):  
Efficient Production ◽  
Rna Expression ◽  
Hairpin Rna ◽  
Small Hairpin Rna ◽  
Minimal Vectors

Bases of minimal vectors in lattices, II

2007 ◽  
Vol 89 (6) ◽  
pp. 541-551
Author(s):  
Jacques Martinet
Keyword(s):  
Minimal Vectors

On the unimodularity of minimal vectors of Humbert forms

2004 ◽  
Vol 83 (6) ◽  
pp. 528-535
Author(s):  
R. Baeza ◽  
M. I. Icaza
Keyword(s):  
Minimal Vectors

scholarly journals Minimal Vectors in the Second Exterior Power of a Lattice

1997 ◽  
Vol 194 (2) ◽  
pp. 467-476 ◽  
Author(s):  
Renaud Coulangeon
Keyword(s):  
Exterior Power ◽  
Minimal Vectors

The Construction of Perfect and Extreme Forms

1966 ◽  
Vol 18 ◽  
pp. 147-158 ◽  
Author(s):  
P. R. Scott
Keyword(s):  
Quadratic Form ◽  
Finite Number ◽  
Positive Quadratic Form ◽  
Image Position ◽  
Minimal Vectors

Letbe a positive quadratic form of determinantD,and letMbe the minimum off(x) for integral x ≠ 0. Thenf(x) assumes the valueMfor a finite number of integral vectors x =±mk(k= 1 , … ,s)called theminimal vectors.

scholarly journals ON WELL-ROUNDED IDEAL LATTICES

2012 ◽  
Vol 08 (01) ◽  
pp. 189-206 ◽  
Author(s):  
LENNY FUKSHANSKY ◽  
KATHLEEN PETERSEN
Keyword(s):  
Full Rank ◽  
Number Fields ◽  
Cyclotomic Fields ◽  
Quadratic Number ◽  
Ideal Lattices ◽  
Rings Of Integers ◽  
Euclidean Lattices ◽  
Whole Space ◽  
Imaginary Quadratic Number ◽  
Minimal Vectors

We investigate a connection between two important classes of Euclidean lattices: well-rounded and ideal lattices. A lattice of full rank in a Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. We consider lattices coming from full rings of integers in number fields, proving that only cyclotomic fields give rise to well-rounded lattices. We further study the well-rounded lattices coming from ideals in quadratic rings of integers, showing that there exist infinitely many real and imaginary quadratic number fields containing ideals which give rise to well-rounded lattices in the plane.

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