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Diophantine Approximations and Integer Points of Cones

Martin Henk, Robert Weismantel
2002 Combinatorica  
The purpose of this note is to present a relation between directed best approximations of a rational vector and the elements of the minimal Hilbert basis of certain rational pointed cones.  ...  Furthermore, we show that for a special class of these cones the integer Carathéodory property holds true.  ...  We would like to thank the referee for helpful comments and for suggesting simplifications of the proofs.  ... 
doi:10.1007/s004930200019 fatcat:kc33ijcvcvafrmjeamgwm34nw4

Test sets of the knapsack problem and simultaneous diophantine approximation [chapter]

Martin Henk, Robert Weismantel
1997 Lecture Notes in Computer Science  
This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation.  ...  We present best possible inequalities that must be satisfied by all minimal integral solutions of a linear diophantine equation and prove that for the corresponding cone the integer analogue of Caratheodory's  ...  Introduction This paper deals with the study of test sets of the knapsack problem and simultaneous diophantine approximation.  ... 
doi:10.1007/3-540-63397-9_21 fatcat:jx4uafbsdzgjnfdsuvpieuyjye

The subdivision of large simplicial cones in Normaliz [article]

Winfried Bruns, Richard Sieg, Christof Söger
2016 arXiv   pre-print
In the homogeneous case, in which the polyhedron is a cone, the set of generators is the Hilbert basis of the intersection of the cone and the lattice, an affine monoid.  ...  Normaliz is an open-source software for the computation of lattice points in rational polyhedra, or, in a different language, the solutions of linear diophantine systems.  ...  In this case, the approximation method is applied again with a higher level of approximation. Fig. 1 : 1 A cone with the Hilbert basis (circled points) and grading.  ... 
arXiv:1605.07440v1 fatcat:o72ouxkhiffhrno3ojevbeybsu

Page 1633 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
Opgenorth generalizes Koecher’s concept of self-dual cones to a pair of dual cones. This gives a quite general situation, in which perfect points can be defined and determined by Voronoi’s algorithm.  ...  In this paper, the authors find all positive integers d and positive square-free divisors a, b, and c of d such that the Diophantine equation ax? + by? + ez* = dxyz + | has an integer solution.  ... 

Generation of Multiple Dirac Cones in Graphene under Double-Periodic and Quasiperiodic Potentials

Masayuki Tashima, Naomichi Hatano
2013 Journal of the Physical Society of Japan  
We investigate generation of new Dirac cones in graphene under double-periodic and quasiperiodic superlattice potentials.  ...  We first show that double-periodic potentials generate the Dirac cones sporadically, following the Diophantine equation, in spite of the fact that double-periodic potentials are also periodic ones, for  ...  This work is supported by Grant-Aid for scientific Research No. 17340115 from the Ministry of Education, Culture, Sports, Science and Technology.  ... 
doi:10.7566/jpsj.82.113706 fatcat:ssdylnywubdn7epej3ld35wv2m

Page 1891 of Mathematical Reviews Vol. , Issue 94d [page]

1994 Mathematical Reviews  
X C oy of dimension 7 not contained in a hyperplane and with affine cone Y.  ...  The author defines a point x of the limit set L(G) to be t-approximable (t > 1) with respect to a point y € L(G) if x can be approximated by infinitely many g(y) to the order u(g)~*, where u(g) is the  ... 

Page 3913 of Mathematical Reviews Vol. , Issue 81J [page]

1981 Mathematical Reviews  
Theorem 3: A Voronoi polyhedron is a convex closure of the set of all integral points lying in the closure of the cone of positivity Kc E™.” 10F Diophantine approximation 81j: 10038 Lecture Notes in Mathematics  ...  G. 81j: 10039 Achievements and problems of the theory of Diophantine approximations. (Russian) Uspehi Mat. Nauk 35 (1980), no. 4(214), 3-68, 248.  ... 

Quantitative indicators of the solutions of Diophantine equations and systems in the domain of the natural numbers [article]

Victor Volfson
2014 arXiv   pre-print
, non-algebraic Diophantine equations and systems of Diophantine equations in the domain of the natural numbers.  ...  The estimate for the number of positive integer solutions of the second-order Diophantine equations in two, three or more variables is geometrically proved in the paper.  ...  General method of Diophantine approximation is well-known in the study of integer solutions of Diophantine equations [2] .  ... 
arXiv:1411.4845v1 fatcat:r344udw5qfffveet62adgi2bum

Page 6454 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
Clemens Heuberger (A-TGRZ-B; Graz) 2003h:90037 90C10 52C07 Henk, Martin; Weismantel, Robert Diophantine approximations and integer points of cones.  ...  The authors study the minimal integral Hilbert bases of cer- tain rational polyhedral cones associated with the problem of directed simultaneous Diophantine approximation (DSDAP).  ... 

On the Break-Up of Invariant Tori with three Frequencies [chapter]

J. D. Meiss
1999 Hamiltonian Systems with Three or More Degrees of Freedom  
This field has irrational vectors that are most robust in the sense of supremal Diophantine constant. Our renormalization operator has a critical fixed point, but it is not hyperbolic.  ...  We construct an approximate renormalization operator for a two and one half degree of freedom Hamiltonian corresponding to an invariant torus with a frequency in the cubic field Q(τ), where τ 3 +τ 2 -2τ  ...  Acknowledgments This paper is the report of collaborative work with Robert MacKay and Jaroslav Stark.  ... 
doi:10.1007/978-94-011-4673-9_64 fatcat:fascb6k2xfd2dijxhqdujl5acu

On Kleinbock's Diophantine result

NIKOLAY MOSHCHEVITIN
2011 Publicationes mathematicae (Debrecen)  
We give an elementary proof of a metrical Diophantine result by D. Kleinbock related to badly approximable vectors in affine subspaces.  ...  Now we observe that any translation of the 1/2contracted set 1 2 • Ω T + c, c ∈ R d+1 (8) consists of not more than one integer point.  ...  Indeed if two different integer points x, y belong to the same set of the form (8) then 0 = x − y ∈ Ω T . This is not possible.  ... 
doi:10.5486/pmd.2011.5120 fatcat:xd5zvfsmsrfbnenroqr3y5nck4

Page 1221 of Mathematical Reviews Vol. 36, Issue 6 [page]

1968 Mathematical Reviews  
Z. 6348 The distribution of integer points on multidimensional hyperboloids and cones. (Russian) Zap. Nauén. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOM) 1 (1966), 84-113.  ...  More recently Linnik and Mal’cev (following Hecke’s idea of studying the number of primes in a ““Winkelraum” of an ellipse) studied the dis- tribution of integer points on multi-dimensional ellipsoids  ... 

Rational approximation of the maximal commutative subgroups of GL(n,R) [article]

O.Karpenkov, A.Vershik
2009 arXiv   pre-print
In addition we introduce a relation between best approximations and sails of cones and interpret the result for totally real subgroups in geometric terms of sails.  ...  It contains both classical problems of Diophantine and simultaneous approximations as a particular subcases but in general is much wider.  ...  We also suppose that the angle between r 1 and r 2 is non-zero and less than π. Denote the set of all integer points in the closure of the cone except the origin by I r 1 ,r 2 .  ... 
arXiv:0910.3482v1 fatcat:ifayknu3brevnhsod6igecpmrm

Rational approximation of maximal commutative subgroups of $${GL(n,\mathbb{R})}$$

Oleg N. Karpenkov, Anatoly M. Vershik
2010 Journal of Fixed Point Theory and Applications  
In addition we introduce a relation between best approximations and sails of cones and interpret the result for totally real subgroups in geometric terms of sails.  ...  It contains both classical problems of Diophantine and simultaneous approximations as a particular subcases but in general is much wider.  ...  We also suppose that the angle between r 1 and r 2 is non-zero and less than π. Denote the set of all integer points in the closure of the cone except the origin by I r 1 ,r 2 .  ... 
doi:10.1007/s11784-010-0011-2 fatcat:o7334z43dnbjnb5f7li6vkfc7q

The Euclidean algorithm in dimension n

Loïc Pottier
1996 Proceedings of the 1996 international symposium on Symbolic and algebraic computation - ISSAC '96  
As a consequence, this algorithm can be used for example to compute minimal solutions of linear Diophantine systems, the basis of the monoid of integer points of a rational simplicial convex cone (called  ...  Let Az = O, , x > 0 be a linear diophantine system, where A is a matrix with integer coefficients, and @ is an integer vector.  ...  is rational and simplicial. In the basis of integer points of C', elements having 1 as last coordinate project bijectively on integer points of S (by the map (x, y) F+ z). 4.  ... 
doi:10.1145/236869.236894 dblp:conf/issac/Pottier96 fatcat:d4mwu4rmorevrdfcjkod2zojja
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