arXiv subjects

R. A. Sharipov

Publications and source records attributed to R. A. Sharipov.

At least 19 recordsLinked to original sources

arXiv2019

On cylindrical regression in three-dimensional Euclidean space

The three-dimensional cylindrical regression problem is a problem of finding a cylinder best fitting a group of points in three-dimensional Euclidean space. The words best fitting are usually understood in the sense of the minimum root mean square deflection of the given points from a cylinder to be found. In this form the problem has no analytic solution. If one replaces the root mean square averaging by a certain biquadratic averaging, the resulting problem has an almost analytic solution. This solution is reproduced in the present paper in a coordinate-free form.

cs.CG↗

arXiv2019

On linear regression in three-dimensional Euclidean space

The three-dimensional linear regression problem is a problem of finding a spacial straight line best fitting a group of points in three-dimensional Euclidean space. This problem is considered in the present paper and a solution to it is given in a coordinate-free form.

cs.CG↗

arXiv2016

A fast modulo primes algorithm for searching perfect cuboids and its implementation

A perfect cuboid is a rectangular parallelepiped whose all linear extents are given by integer numbers, i. e. its edges, its face diagonals, and its space diagonal are of integer lengths. None of perfect cuboids is known thus far. Their non-existence is also not proved. This is an old unsolved mathematical problem. Three mathematical propositions have been recently associated with the cuboid problem. They are known as three cuboid conjectures. These three conjectures specify three special subcases in the search for perfect cuboids. The case of the second conjecture is associated with solutions of a tenth degree Diophantine equation. In the present paper a fast algorithm for searching solutions of this Diophantine equation using modulo primes seive is suggested and its implementation on 32-bit Windows platform with Intel-compatible processors is presented.

math.NT↗

arXiv2015

A strategy of numeric search for perfect cuboids in the case of the second cuboid conjecture

A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. The problem of finding such cuboids or proving their non-existence is not solved thus far. The second cuboid conjecture specifies a subclass of perfect cuboids described by one Diophantine equation of tenth degree and claims their non-existence within this subclass. Regardless of proving or disproving this conjecture in the present paper the Diophantine equation associated with it is studied and is used in order to build an optimized strategy of computer-assisted search for perfect cuboids within the subclass covered by the second cuboid conjecture.

math.NT↗

arXiv2001

Solitons on a finite-gap background in Bullough-Dodd-Jiber-Shabat model

The determinant formula for N-soliton solutions of the Bullough-Dodd-Jiber-Shabat equation on a finite-gap background is obtained. Nonsingularity conditions for them and their asymptotics are investigated.

math-ph↗

arXiv1999

On a point symmetry analysis for generalized diffusion type equations

Generalized diffusion type equations are considered and point symmetry analysis is applied to them. The equations with extremal order point symmetry algebras are described. Some old geometrical results are rederived in connection with theory of these equation.

math.AP↗

arXiv1999

Magnetization waves in Landau-Lifshitz Model

The solutions of the Landau-Lifshitz equation with finite-gap behavior at infinity are considered. By means of the inverse scattering method the large-time asymptotics is obtained.

solv-int↗

arXiv1999

On some equations that can be brought to the equations of diffusion type

For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.

math.DG↗

arXiv1998

Effective procedure of point-classification for the equation $y'' = P + 3 Q y' + 3 R {y'}^2 + S {y'}^3$

For the equations of the form $y''=P(x,y)+3 Q(x,y) y'+3 R(x,y) {y'}^2 +S(x,y) {y'}^3$ the problem of equivalence in the class of point transformations is considered. Effective procedure for determining the class of point equivalence for the given equation is suggested. This procedure is based on explicit formulas for the invariants.

math.DG↗

arXiv1997

On the geometry of point-expansions for certain class of differential equations of the second order

Second order ordinary differential equations of the form $y'' = P(x,y) + 4 Q(x,y) y' + 6 R(x,y) y'^2 + 4 S(x,y) y'^3 + L(x,y) y'^4$ are considered and their point-expansions are constructed. Geometrical structures connected with these expansions are described.

solv-int↗

arXiv1997

On the point transformations for the equation $y''= P + 3Qy' + 3R{y'}^2 + S{y'}^3$

For the equations $y''=P(x,y) + 3Q(x,y)y' + 3R(x,y){y'}^2 + S(x,y){y'}^3$ the problem of equivalence is considered. Some classical results are resumed in order to prepare the background for the study of special subclass of such equations, which arises in the theory of dynamical systems admitting the normal shift.

solv-int↗

arXiv1997

On the point transformations for the second order differential equations. I

Point transformations for the ordinary differential equations of the form $y''=P(x,y)+3 Q(x,y) y'+3 R(x,y) (y')^2+S(x,y) (y')^3$ are considered. Some classical results are resumed. Solution for the equivalence problem for the equations of general position is described.

solv-int↗

arXiv1995

On the separate algebraicity along the families of algebraic curves

A new generalization of the classical separate algebraicity theorem is suggested and proved.

alg-geom↗

arXiv1994

On $CR$ mappings between algebraic Cauchy--Riemann manifolds and separate algebraicity for holomorphic functions

We prove the algebraicity of smooth $CR$-mappings between algebraic Cauchy--Riemann manifolds. A generalization of separate algebraicity principle is established.

alg-geom↗

arXiv1994

Bäcklund transformation and the construction of the integrable boundary-value problem for the equation $u_{xx}-u_{tt}=e^u-e^{-2u}$

Bäcklund transformation for the Bullough-Dodd-Jiber-Shabat equation $u_{xx}-u_{tt}=e^u-e^{-2u}$ is found. The construction of integrable boundary condition for this equation together with the algebro-geometric solutions satisfying it are suggested.

solv-int↗

arXiv1993

Invariant Integrability Criterion for the Equations of Hydrodynamical Type

Invariant integrability criterion for the equations of hydrodynamical type is found. This criterion is written in the form of vanishing for some tensor which is derived from the velocities matrix of hydrodynamical equations.

solv-int↗

arXiv1993

Complete normality conditions for the dynamical systems on Riemannian manifolds

New additional equations for the Newtonian dynamical systems on Riemannian manifolds are found. They supplement the previously found weak normality conditions up to the complete normality conditions for Newtonian dynamical systems.

astro-ph↗

arXiv1993

Dynamical systems accepting the normal shift on an arbitrary Riemannian manifold

Newtonian dynamical systems which accept the normal shift on an arbitrary Riemannian manifold are considered. For them the determinating equations making the weak normality condition are derived. The expansion for the algebra of tensor fields is constructed.

hep-th↗