‎Let $R$ be a commutative ring with non-zero identity. ‎We describe all $C_3$‎- ‎and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. ‎Also, ‎we shall describe all complete, ‎regular and $n$-claw-free intersection graphs. ‎Finally, ‎we shall prove that almost all Artin rings $R$ have Hamiltonian intersection graphs. ...

An algebraic procedure of getting of canonical variables in a rigid body dynamics is presented. The method is based on using a structure of an algebra of Lie—Poisson brackets with which a Hamiltonian dynamics is set. In a particular case of a problem of a top in a homogeneous gravitation field the method leads to well–known Andoyer—Deprit variables. Earlier, the method of getting of them was ba...

One of the main highlights of the previous semester was an interplay between the following objects: the nilpotent cone in g, the cotangent bundle T ∗(G/B), the universal enveloping algebra U(g) (or more precisely, its central reduction Uλ(g)) and the sheaf D G/B of λ-twisted differential operators on G/B. In our present story, Symn(C) is an analog of the nilpotent cone, Hilbn(C) is an analog of...

We find an infinite dimensional free algebra which lives at large N in any SU(N)-invariant action or Hamiltonian theory of bosonic matrices. The natural basis of this algebra is a free-algebraic generalization of Chebyshev polynomials and the dual basis is closely related to the planar connected parts. This leads to a number of free-algebraic forms of the master field including an algebraic der...

In this paper, we treat moduli spaces of parabolic connections. We take an affine open covering the spaces, and construct a Hamiltonian structure algebraic vector field determined by isomonodromic deformation for each subset.

Via a special dimensional reduction, that is, Fourier transforming over one of the coordinates of Casimir operator of su(2) Lie algebra and 4-oscillator Hamiltonian, we have obtained 2 and 3 dimensional Hamiltonian with shape invariance symmetry. Using this symmetry we have obtained their eigenspectrum. In the mean time we show equivalence of shape invariance symmetry and Lie algebraic symmetry...

The system of a closed vortex filament is an integrable Hamiltonian one, namely, a Hamiltonian system with an infinite sequense of constants of motion in involution. An algebraic framework is given for the aim of describing differential geometry of this system. A geometrical structure related to the integrability of this system is revealed. It is not a bi-Hamiltonian structure but similar one. ...

We consider the Hamiltonian cycle problem (HCP) embedded in a controlled Markov decision process. In this setting, HCP reduces to an optimization problem on a set of Markov chains corresponding to a given graph. We prove that Hamiltonian cycles are minimizers for the trace of the fundamental matrix on a set of all stochastic transition matrices. In case of doubly stochastic matrices with symmet...

A new backward stable, structure preserving method of complexity O(n) is presented for computing the stable invariant subspace of a real Hamiltonian matrix and the stabilizing solution of the continuous-time algebraic Riccati equation. The new method is based on the relationship between the invariant subspaces of the Hamiltonian matrix H and the extended matrix [ 0 H H 0 ] and makes use of the ...

A quantum Hamiltonian which evolves the gravitational field according to time as measured by constant surfaces of a scalar field is defined through a regularization procedure based on the loop representation, and is shown to be finite and diffeomorphism invariant. The problem of constructing this Hamiltonian is reduced to a combinatorial and algebraic problem which involves the rearrangements o...