Evolution patterns in Collatz problem
The concept of evolution patterns is introduced for Collatz sequences and it is shown that any finite evolution pattern is implemented in some particular Collatz sequence.
math.GM↗
arXiv subjects
Ruslan Sharipov
Publications and source records attributed to Ruslan Sharipov.
Pseudo-Hadamard matrices of the first generation and an algorithm for producing them
Hadamard matrices in $\{0,1\}$ presentation are square $m\times m$ matrices whose entries are zeros and ones and whose rows considered as vectors in $\Bbb R^m$ produce the Gram matrix of a special form with respect to the standard scalar product in $\Bbb R^m$. The concept of Hadamard matrices is extended in the present paper. As a result pseudo-Hadamard matrices of the first generation are defined and investigated. An algorithm for generating these pseudo-Hadamard matrices is designed and is used for testing some conjectures.
cs.DS↗
Hadamard matrices in $\{0,1\}$ presentation and an algorithm for generating them
Hadamard matrices are square $n\times n$ matrices whose entries are ones and minus ones and whose rows are orthogonal to each other with respect to the standard scalar product in $\Bbb R^n$. Each Hadamard matrix can be transformed to a matrix whose entries are zeros and ones. This presentation of Hadamard matrices is investigated in the paper and based on it an algorithm for generating them is designed.
math.CO↗
On a simplified version of Hadamard's maximal determinant problem
Hadamard's maximal determinant problem consists in finding the maximal value of the determinant of a square $n\times n$ matrix whose entries are plus or minus ones. This is a difficult mathematical problem which is not yet solved. In the present paper a simplified version of the problem is considered and studied numerically.
math.NT↗
On simultaneous approximation of several eigenvalues of a semi-definite self-adjoint linear operator in a Hilbert space
A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $λ_{min}=λ_1\leqslant\ldots\leqslantλ_m<σ_{ess}$. The problem of approximation of these eigenvalues by eigenvalues of some linear operator in a finite-dimensional space of the dimension $s$ is considered and solved. The accuracy of the approximation obtained becomes unlimitedly high as $s\to\infty$.
math.SP↗
Umbilical and zero curvature equations in a class of second order ODE's
The class of second order ODE's cubic with respect to the first order derivative is considered. Using geometric structures associated with these equations, the subclasses of umbilical equations, zero mean curvature equations, and zero Gaussian curvature equations are defined. Zero mean curvature equations are studied within the framework of the first case of intermediate degeneration with the stress on their pseudoscalar and scalar invariants.
math.CA↗
Comparison of two classifications of a class of ODE's in the first case of intermediate degeneration
Two classifications of second order ODE's cubic with respect to the first order derivative are compared in the first case of intermediate degeneration. The correspondence of vectorial, pseudovectorial, scalar, and pseudoscalar invariants of the equations in this case is established.
math.CA↗
Comparison of two classifications of a class of ODE's in the case of general position
Two classifications of second order ODE's cubic with respect to the first order derivative are compared in the case of general position, which is common for both classifications. The correspondence of vectorial, pseudovectorial, scalar, and pseudoscalar invariants is established.
math.CA↗
On Walter Wyss's no perfect cuboid paper
The perfect cuboid problem is an old famous unsolved problem in mathematics concerning the existence or non-existence of a rectangular parallelepiped whose edges, face diagonals, and space diagonal are of integer lengths. Recently Walter Wyss has published a paper [arXiv:1506.02215] claiming a solution of this problem. The purpose of this paper is to check out Walter Wyss's result.
math.NT↗
A rough classification of potentially invertible cubic transformations of the real plane
A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called cubic if the degrees of its polynomials are not greater than three. In the present paper a rough classification scheme for cubic transformations of $\Bbb R^2$ is suggested. It is based on quartic forms associated with these transformations.
math.AG↗
On quartic forms associated with cubic transformations of the real plane
A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called cubic if the degrees of its polynomials are not greater than three. It turns out that cubic transformations are associated with some binary and quaternary quartic forms. In the present paper these forms are defined and studied.
math.AG↗
Multiple discriminants and critical values of a multivariate polynomial
A critical value of a function is the value of this function at one of its critical points. Each critical point of a differentiable multivariate function is described by the equations which consist in equating to zero all of its partial derivatives. However, in general case there is no equation for the corresponding critical value. The case of polynomials is different. In the present paper an equation for critical values of a polynomial is derived.
math.AC↗
On positive bivariate quartic forms
A bivariate quartic form is a homogeneous bivariate polynomial of degree four. A criterion of positivity for such a form is known. In the present paper this criterion is reformulated in terms of pseudotensorial invariants of the form.
math.AG↗
On some higher degree sign-definite multivariate polynomials associated with definite quadratic forms
Positive and negative quadratic forms are well known and widely used. They are multivariate homogeneous polynomials of degree two taking positive or negative values respectively for any values of their arguments not all zero. In the present paper a certain higher degree polynomial is associated with each quadratic form such that the form is definite if and only if this polynomial is sign-definite.
math.AG↗
A note on invertible quadratic transformations of the real plane
A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called quadratic if the degrees of its polynomials are not greater than two. In the present paper an exhaustive description of invertible quadratic transformations of the real plane is given. Their application to the perfect cuboid problem is discussed.
math.AG↗
Asymptotic estimates for roots of the cuboid characteristic equation in the nonlinear region
A perfect cuboid is a rectangular parallelepiped. Its edges, its face diagonals, and its space diagonal are of integer lengths. None of such cuboids is known thus far, though the system of Diophantine equations describing them is easily written. The cuboid characteristic equation is a twelfth degree Diophantine equation derived from the initial cuboid equations and equivalent to them. In the case of the second cuboid conjecture it reduces to a tenth degree equation. This equation comprises two parameters. Previously various asymptotics for roots of this equation were studied as its parameters tend to infinity either separately or simultaneously provided some linear combination of them is preserved finite. In the present paper this linear combination is replaced by a certain nonlinear expression.
math.NT↗
Asymptotic estimates for roots of the cuboid characteristic equation in the linear region
A perfect cuboid is a rectangular parallelepiped whose edges, whose face diagonals, and whose space diagonal are of integer lengths. 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. This Diophantine equation is called the cuboid characteristic equation. It has two parameters. The linear region is a domain on the coordinate plane of these two parameters given by certain linear inequalities. In the present paper asymptotic expansions and estimates for roots of the characteristic equation are obtained in the case where both parameters tend to infinity staying within the linear region. Their applications to the cuboid problem are discussed.
math.NT↗
Reverse asymptotic estimates for roots of the cuboid characteristic equation 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 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. This Diophantine equation has two parameters. Previously asymptotic expansions and estimates for roots of this equation were obtained in the case where the first parameter is fixed and the other tends to infinity. In the present paper reverse asymptotic expansions and estimates are derived in the case where the second parameter is fixed and the first one tends to infinity. Their application to the perfect cuboid problem is discussed.
math.NT↗