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The complexity of almost linear diophantine problems

V. Weispfenning
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.  ... 
doi:10.1016/s0747-7171(08)80051-x fatcat:qnchmaoemzdd7gwshilgaiox2u

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

Martin Hromĉík, Radek Frízel, Zdenêk Hurák
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.  ... 
doi:10.3182/20060705-3-fr-2907.00091 fatcat:5uu66lyrjngijkkoadeeztux4m

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]

Anish Ghosh, Alexander Gorodnik, Amos Nevo
2014 arXiv   pre-print
We concentrate specifically on 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.  ... 
arXiv:1401.6581v1 fatcat:gzjtwufewff37iahec2tv622le

Contribution of Jonas Kubilius to the metric theory of Diophantine approximation of dependent variables

Victor V. Beresnevich, Vasily I. Bernik, Friedrich Götze, Elena V. Zasimovich, Nikolai I. Kalosha
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.  ... 
doi:10.33581/2520-6508-2021-3-34-50 fatcat:dckaixcsyvhpth5zuvmdlxzou4

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

Ido Kanter, Evi Kopelowitz, Wolfgang Kinzel
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 .  ... 
doi:10.1103/physrevlett.101.084102 pmid:18764616 fatcat:n6ropiiwpjhfblpnoa25ax4ui4

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

Jeffrey Lagarias
1990 Acta Arithmetica  
Thus, [Belaga 1995] (cf. also [Belaga, Mignotte 1998]) defines a pwpl-function W acting in a two-to-one way on 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.  ... 
doi:10.4064/aa-56-1-33-53 fatcat:jawz2n576nfclbvrx6w567wzpy

On the persistence of pseudo-holomorphic curves on an almost complex torus (with an appendix by J�rgen P�schel)

J�rgen Moser
1995 Inventiones Mathematicae  
This paper originated in a question raised by Gromov, who asked for a 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.  ... 
doi:10.1007/bf01245189 fatcat:5eqrtgtg3jandehnvpubyvgd5u

Phase Precoding for the Compute-and-Forward Protocol [article]

Amin Sakzad, Emanuele Viterbo, Joseph Jean Boutros, Yi Hong
2014 arXiv   pre-print
Using tools from 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.  ... 
arXiv:1404.4157v2 fatcat:fkqwmtb2ybevdlg6smfc23tupe

Smoothness of the uniformization of two-dimensional linear foliation on torus with nonstandard metric [article]

A.A.Glutsuk
2000 arXiv   pre-print
totally geodesic; in other terms, 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).  ... 
arXiv:math/9911154v2 fatcat:bgogljdbt5ekpneexp3kpah6sm

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

Iván Amaya, Luis Gómez, Rodrigo Correa
2014 Dyna  
Thus, 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.  ... 
doi:10.15446/dyna.v81n185.37244 fatcat:t2k64cfhtvdw3ln54e5kpjojky
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