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The complexity of almost linear diophantine problems

1990
*
Journal of symbolic computation
*

*The*results form a common extension

*of*known

*complexity*bounds for PrA and for

*the*existential

*almost*

*linear*

*problems*studied by Gurari & lbarra. ... We show that

*the*extension ALA (

*almost*

*linear*arithmetic)

*of*PrA obtained in this way, has essentially

*the*same upper and lower

*complexity*bounds as

*the*original theory. ...

*The*resulting theory ALA (

*almost*

*linear*arithmetic) is an extension

*of*PrA and hence satisfies

*the*same lower

*complexity*bounds. ...

##
###
Page 2309 of Mathematical Reviews Vol. , Issue 87e
[page]

1987
*
Mathematical Reviews
*

Gol- ubeva (

*The*approximation*of*zero by polynomials over imaginary quadratic field for*almost*all*complex*numbers), V. A. ... Kotov (Some classes*of**Diophantine*equations*of**the*form f(z, y) = 0), N. M. Korobov (Some*problems**of**the*theory*of**Diophantine*ap- proximations), B. G. ...##
###
Page 11 of Mathematical Reviews Vol. , Issue 92c
[page]

1992
*
Mathematical Reviews
*

*The*current state

*of*

*the*use

*of*

*the*lower bound on

*linear*forms in logarithms

*of*algebraic numbers to solve

*Diophantine*equations is surveyed. ... If Z is a

*Diophantine*subset

*of*a ring R, then

*the*negative solution

*of*Hilbert’s tenth

*problem*in Z lifts to a negative solution

*of*

*the*corresponding

*problem*for R. J. Denef [Trans. Amer. Math. ...

##
###
BANDED MATRIX SOLVERS AND POLYNOMIAL DIOPHANTINE EQUATIONS

2006
*
IFAC Proceedings Volumes
*

*The*algorithms and programs developed are based on

*the*observation that a set

*of*constant

*linear*equations resulting from

*the*polynomial

*problem*features a special structure. ... Numerical procedures and codes for

*linear*

*Diophantine*polynomial equations are proposed in this paper based on

*the*banded matrix algorithms and solvers. Both

*the*scalar and matrix cases are covered. ... It provides routines for solving systems

*of*

*linear*equations, least-squares solutions

*of*

*linear*systems

*of*equations, eigenvalue

*problems*, and singular value

*problems*. ...

##
###
Page 1204 of Mathematical Reviews Vol. 44, Issue 6
[page]

1972
*
Mathematical Reviews
*

In Section 6 he gives some basic metric results on

*Diophantine*approximation including*the*theorems*of*Dirichlet, Hinéin and others on*the*kind*of*approximation possible for all,*almost*all or*almost*no ...*The*limitations on*the*possible ex- tensions*of*Schanuel’s conjecture and other open*problems*are discussed. ...##
###
Diophantine approximation exponents on homogeneous varieties
[article]

2014
*
arXiv
*
pre-print

We concentrate specifically on

arXiv:1401.6581v1
fatcat:gzjtwufewff37iahec2tv622le
*the**problem**of*estimating exponents*of**Diophantine*approximation by arithmetic lattices acting on algebraic varieties. ... These include*the**linear*and affine actions on affine spaces, and*the*action on*the*variety*of*matrices*of*fixed determinant. In some cases, these exponents are shown to be best possible. ...*The*first author acknowledges support*of**the*Royal Society.*The*second author acknowledges support*of*EPSRC, ERC and RCUK.*The*third author acknowledges support*of*ISF. ...##
###
Contribution of Jonas Kubilius to the metric theory of Diophantine approximation of dependent variables

2021
*
Journal of the Belarusian State University. Mathematics and Informatics
*

*The*article is devoted to

*the*latest results in metric theory

*of*

*Diophantine*approximation. One

*of*

*the*first major result in area

*of*number theory was a theorem by academician Jonas Kubilius. ... Over

*the*last 70 years,

*the*area

*of*

*Diophantine*approximation yielded a number

*of*significant results by great mathematicians, including Fields prize winners Alan Baker and Grigori Margulis. ... Groshev for system

*of*

*linear*forms [8]. One important generalisation

*of*

*the*above setting considered by A. ...

##
###
Public Channel Cryptography: Chaos Synchronization and Hilbert's Tenth Problem

2008
*
Physical Review Letters
*

*The*task

*of*a passive attacker is mapped onto Hilbert's tenth

*problem*, solving a set

*of*nonlinear

*Diophantine*equations, which was proven to be in

*the*class

*of*NP-Complete

*problems*. ... This bridge between two different disciplines, synchronization in nonlinear dynamical processes and

*the*realm

*of*

*the*NPC

*problems*, opens a horizon for a new type

*of*secure public-channel protocols. ...

*The*number

*of*solutions is at least one, but can be unbounded, hence,

*the*

*complexity*

*of*

*the*attacker is at least NPC, where

*the*

*complexity*

*of*

*the*

*problem*increases with N . ...

##
###
The set of rational cycles for the 3x+1 problem

1990
*
Acta Arithmetica
*

Thus, [Belaga 1995] (cf. also [Belaga, Mignotte 1998]) defines a pwpl-function W acting in a two-to-one way on

doi:10.4064/aa-56-1-33-53
fatcat:jawz2n576nfclbvrx6w567wzpy
*the*set (6)*of*positive integers not divisible by 6. ... actual number*of*parameters (a j , b j ) does not need to be as big as 2s. ... than, say,*of**diophantine**problems*. ...##
###
On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by J�rgen P�schel)

1995
*
Inventiones Mathematicae
*

This paper originated in a question raised by Gromov, who asked for a

doi:10.1007/bf01245189
fatcat:5eqrtgtg3jandehnvpubyvgd5u
*complex*analogue*of**the*result [19] on*the*persistence*of*minimal surfaces*of*codimension 1. ... During*the*development*of*this work*the*author had*the*benefit*of*informative discussions with D. Burns, C. Fefferman, R. Narasimhan, S. Webster and R. Ye. ... This would provide*the*answer to*the*open*problem*2*of**the*introduction, where we admit*almost**complex*structures, however. ...##
###
Phase Precoding for the Compute-and-Forward Protocol
[article]

2014
*
arXiv
*
pre-print

Using tools from

arXiv:1404.4157v2
fatcat:fkqwmtb2ybevdlg6smfc23tupe
*Diophantine*approximation and algebraic geometry, we prove*the*existence*of*a phase precoder that approaches*the*maximum DoF when*the*number*of*transmitters tends to infinity. ... We further give a new low-*complexity*method for finding network equations. We finally show that*the*proposed precoding scheme increases*the*degrees-*of*-freedom (DoF)*of*CoF protocol. ...*The**Diophantine*approximation theory that works with*almost*all numbers is called metric number theory. ...##
###
Smoothness of the uniformization of two-dimensional linear foliation on torus with nonstandard metric
[article]

2000
*
arXiv
*
pre-print

totally geodesic; in other terms,

arXiv:math/9911154v2
fatcat:bgogljdbt5ekpneexp3kpah6sm
*the*correspondent triple consisting*of**the*torus,*the*foliation and*the**complex*structure on*the*leaves is isomorphic to a triple correspondent to a*linear*foliation ... This*Diophantine*condition is exact. We state and prove*the*analytic versions*of*these statements. ...*The*paper was written while I was staying at*the*Independent University*of*Moscow and Institut des HautesÉtudes Scientifiques (IHÉS, Bures-sur-Yvette, France). ...##
###
Page 457 of Mathematical Reviews Vol. 49, Issue 2
[page]

1975
*
Mathematical Reviews
*

y)=m, where f is a binary form, (3) effective bounds on

*the**Diophantine*approximations*of*algebraic numbers, (4)*the*resolution*of**the*class number 2*problem*, (5) a result on*the*numbers represented by ... In it*the*author gives more recent developments concern- ing (1) lower bounds for*linear*forms in logarithms*of*algebraic numbers, (2) effective bounds on*the*solutions to*the**Diophantine*equation f(x, ...##
###
Page 4445 of Mathematical Reviews Vol. , Issue 86j
[page]

1986
*
Mathematical Reviews
*

*The*author studies a factorization

*problem*for

*linear*recurrent sequences and gives some applications. David Lee Hilliker (Costa Mesa, Calif.) Horak, P. [Hordk, Peter] (CS-PRKN); 86j:11018 Skula, L. ...

*The*author considers

*complex*k-vectors u,, defined recursively by (*) Un+1 = Aun, where A is a k x k matrix with

*complex*entries.

*The*author’s purpose is not very clear. ...

##
###
Discrete Particle Swarm Optimization in the numerical solution of a system of linear Diophantine equations

2014
*
Dyna
*

Thus,

doi:10.15446/dyna.v81n185.37244
fatcat:t2k64cfhtvdw3ln54e5kpjojky
*the**problem**of*determining whether a*linear**Diophantine*equation has a solution or not, is reduced to showing if*the*greatest common divisor*of**the*coefficients divide or not. ... This article proposes to solve, in case*the*solution exists in*the*given search domain, a*linear*system*of**Diophantine*equations. ...
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