arXiv subjects
Publications and source records attributed to V. N. Dumachev.
The paper proposes a polynomial formula for solution quadratic congruences in $\mathbb{Z}_p$. This formula gives the correct answer for quadratic residue and zeroes for quadratic nonresidue. The general form of the formula for $p=3 \;\rm{mod}\,4$, $p=5 \;\rm{mod}\,8$ and for $p=9 \;\rm{mod}\,16$ are suggested.
We give a generalization of the Nambu mechanics based on vector Hamiltonians theory. It is shown that any divergence-free phase flow in $\mathbb{R}^n$ can be represented as a generalized Nambu mechanics with $n-1$ integral invariants. For the case when the phase flow in $\mathbb{R}^n$ has $n-3$ or less first integrals, we introduce the Cartan concept of mechanics. As an example we give the fifth integral invariant of Euler top.
We present a generalization of the Nambu mechanics on the base of Liouville's theorem. We prove that the Poisson structure of an n-dimensional multisymplectic phase space is induced by (n-1)-Hamiltonian k-vector field seach of which requires introduction of k-Hamiltonians.
In work the internal structure of de Rham cohomology is considered. As examples the phase flows in $\mathbb {R}^3$ admitting the Nambu Poisson structure are studied.
On the basis of Liouville theorem the generalization of the Nambu mechanics is considered. For three-dimensional phase space the concept of vector hamiltonian and vector lagrangian is entered.
On the basis of Liouville theorem the generalization of the Nambu mechanics is considered. Is shown, that Poisson manifolds of n-dimensional multi-symplectic phase space have inducting by (n-1) Hamiltonian k-vector fields, each of which requires of (k)-hamiltonians.
The task of reception of a copy of an arbitrary quantum state with use of a minimum quantity of quantum operations is considered.
The influence of twin boundaries on the heat capacity and diffuse scattering in a crystal is described within the framework of a macroscopic dynamic theory.
A system of unitary transformations providing two optimal copies of an arbitrary input cubit is obtained. An algorithm based on classical Boolean algebra and allowing one to find any unitary transformation realized by the quantum CNOT operators is proposed.
In frameworks of the macroscopic dynamic theory of coherent interfaces in crystals the acoustic radiation of coherent boundary at passage along it of localized wave packages is considered. It is shown that the coherent twin boundary of acoustic waves does not radiate.
The influence of coherent interface on dissipations of mechanical energy of driven dislocations near to a point of martensite type phase transition is considered. The expressions for dynamic braking of dislocations, owing to losses of energy on excitation of deformations-type phase transition are received.