On generic complexity of the subset sum problem in monoids and groups of integer matrix of order two

2020 ◽  
Vol 25 (4) ◽  
pp. 10-15
Author(s):  
Alexander Nikolaevich Rybalov
Keyword(s):  
Linear Group ◽  
Modular Group ◽  
Special Linear Group ◽  
Second Order ◽  
Integer Matrix ◽  
Subset Sum ◽  
Algorithmic Problems ◽  
Almost All ◽  
Subset Sum Problems ◽  
Generic Complexity

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

THE GENERAL COMPLEXITY OF THE PROBLEM TO RECOGNIZE HAMILTONIAN PATHS

2021 ◽  
pp. 120-126
Author(s):  
A. N. Rybalov ◽  
Keyword(s):  
Generic Algorithm ◽  
Hamiltonian Paths ◽  
Finite Graphs ◽  
Large Sets ◽  
Cloning Technique ◽  
Algorithmic Problems ◽  
Hard Problems ◽  
Main Component ◽  
Almost All ◽  
Generic Complexity

Generic-case approach to algorithmic problems has been offered by A. Miasnikov, V. Kapovich, P. Schupp, and V. Shpilrain in 2003. This approach studies an algorithm behavior on typical (almost all) inputs and ignores the rest of inputs. In this paper, we study the generic complexity of the problem of recognition of Hamiltonian paths in finite graphs. A path in graph is called Hamiltonian if it passes through all vertices exactly once. We prove that under the conditions P 6= NP and P = BPP for this problem there is no polynomial strongly generic algorithm. A strongly generic algorithm solves a problem not on the whole set of inputs, but on a subset, the sequence of frequencies of which exponentially quickly converges to 1 with increasing size. To prove the theorem, we use the method of generic amplification, which allows to construct generically hard problems from the problems hard in the classical sense. The main component of this method is the cloning technique, which combines the inputs of a problem together into sufficiently large sets of equivalent inputs. Equivalence is understood in the sense that the problem is solved similarly for them.

Representations of the modular group arising from Drinfeld doubles of finite abelian groups

2019 ◽  
pp. 2150011
Author(s):  
Deepak Naidu
Keyword(s):  
Abelian Group ◽  
Linear Group ◽  
Modular Group ◽  
Congruence Subgroup ◽  
Finite Abelian Group ◽  
Special Linear Group ◽  
Principal Congruence ◽  
Finite Abelian Groups ◽  
Ring Of Integers ◽  
Principal Congruence Subgroup

We show that the image of the representation of the modular group [Formula: see text] arising from the representation category [Formula: see text] of the Drinfeld double [Formula: see text] of a finite abelian group [Formula: see text] of exponent [Formula: see text] is isomorphic to the special linear group [Formula: see text], where [Formula: see text] denotes the ring of integers modulo [Formula: see text]. As a consequence, we establish that the kernel of the representation in question is the principal congruence subgroup of level [Formula: see text].

Succinct Certificates for Almost All Subset Sum Problems

1989 ◽  
Vol 18 (3) ◽  
pp. 550-558 ◽  
Author(s):  
Merrick L. Furst ◽  
Ravi Kannan
Keyword(s):  
Subset Sum ◽  
Almost All ◽  
Subset Sum Problems

scholarly journals Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

2021 ◽  
Author(s):  
Adrien Laurent ◽  
Gilles Vilmart
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Linear Group ◽  
Langevin Equation ◽  
Numerical Experiments ◽  
Special Linear Group ◽  
Order Conditions ◽  
High Order ◽  
Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

scholarly journals On Fourier Coefficients of the Symmetric Square L-Function at Piatetski-Shapiro Prime Twins

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1254
Author(s):  
Xue Han ◽  
Xiaofei Yan ◽  
Deyu Zhang
Keyword(s):  
Asymptotic Formula ◽  
Fourier Coefficient ◽  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Mean Value ◽  
Symmetric Square ◽  
The Mean ◽  
Almost All

Let Pc(x)={p≤x|p,[pc]areprimes},c∈R+∖N and λsym2f(n) be the n-th Fourier coefficient associated with the symmetric square L-function L(s,sym2f). For any A>0, we prove that the mean value of λsym2f(n) over Pc(x) is ≪xlog−A−2x for almost all c∈ε,(5+3)/8−ε in the sense of Lebesgue measure. Furthermore, it holds for all c∈(0,1) under the Riemann Hypothesis. Furthermore, we obtain that asymptotic formula for λf2(n) over Pc(x) is ∑p,qprimep≤x,q=[pc]λf2(p)=xclog2x(1+o(1)), for almost all c∈ε,(5+3)/8−ε, where λf(n) is the normalized n-th Fourier coefficient associated with a holomorphic cusp form f for the full modular group.

scholarly journals Integral cohomology and Chern classes of the special linear group over the ring of integers

2001 ◽  
Vol 131 (3) ◽  
pp. 445-457
Author(s):  
DOMINIQUE ARLETTAZ ◽  
CHRISTIAN AUSONI ◽  
MAMORU MIMURA ◽  
NOBUAKI YAGITA
Keyword(s):  
Linear Group ◽  
Upper Bound ◽  
Special Linear Group ◽  
Discrete Groups ◽  
Chern Classes ◽  
Additive Structure ◽  
Special Linear ◽  
Ring Of Integers ◽  
Integral Cohomology ◽  
Complete Calculation

This paper is devoted to the complete calculation of the additive structure of the 2-torsion of the integral cohomology of the infinite special linear group SL(ℤ) over the ring of integers ℤ. This enables us to determine the best upper bound for the order of the Chern classes of all integral and rational representations of discrete groups.

scholarly journals The third homology of the special linear group of a field

2009 ◽  
Vol 213 (9) ◽  
pp. 1665-1680 ◽  
Author(s):  
Kevin Hutchinson ◽  
Liqun Tao
Keyword(s):  
Linear Group ◽  
Special Linear Group ◽  
Special Linear ◽  
The Third
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