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In this paper we elaborate on the gauge invariant frame-like Lagrangian description for the wide class of the so-called infinite (or continuous) spin representations of Poincar\'e group. We use our previous results on the gauge invariant…
We consider a linear $2\times2$ matrix ODE with two coalescing regular singularities. This coalescence is restricted with an isomonodromy condition with respect to the distance between the merging singularities in a way consistent with the…
Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a…
Simple derivation is presented of the four families of infinitely many shape invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. Darboux-Crum transformations are applied to connect the well-known shape…
We show that the radial harmonic oscillator problem in the position-dependent mass background of the type $m(\alpha;r) = (1+\alpha r^2)^{-2}$, $\alpha>0$, can be solved by using a point canonical transformation mapping the corresponding…
The authors previously described an algebraic analogue of the JSJ-decomposition of a 3-manifold. This analogue is defined for any finitely presented, one-ended group. We study this analogue in the special case of Poincar\'e duality pairs.
We establish a necessary and sufficient condition for all zeros of a self-reciprocal polynomial to lie on the unit circle. Moreover, we relate the necessary and sufficient condition with a canonical system of linear differential equations…
As a sequel to Kawakami-Nakamura-Sakai (arXiv:1209.3836), this series of papers constructs the complete degeneration scheme of four-dimensional Painlev\'e-type equations which includes the Painlev\'e-type equations associated with linear…
We are investigating the dynamics of a new Poincar\'e gauge theory of gravity model, which has cross coupling between the spin-0$^+$ and spin-0$^-$ modes. To this end we here consider a very appropriate situation---homogeneous-isotropic…
We found Fuchs--Garnier pairs in 3X3 matrices for the first and second Painleve' equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve' equation we use the generalized Laplace transform…
We introduce the notion of a monodromy for gauge fields with vanishing curvature on the noncommutative torus. Similar to the ordinary gauge theory, traces of the monodromies define noncommutative Wilson lines. Our main result is that these…
We consider a wide class of nonlinear canonical quantum systems described by a one-particle Schroedinger equation containing a complex nonlinearity. We introduce a nonlinear unitary transformation which permits us to linearize the…
We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases: (i) F is an algebraically closed field of characteristic…
The main goal of this paper is to study compactifications of polynomial slow-fast systems. More precisely, the aim is to give conditions in order to guarantee normal hyperbolicity at infinity of the Poincar\'e-Lyapunov sphere for slow-fast…
For Hamiltonian systems with degeneracy of any higher order, we study the persistence of resonant invariant tori, which as some lower-dimensional invariant tori might be elliptic, hyperbolic or of mixed types. Hence we prove a quasiperiodic…
In order to derive a large set of Hamiltonian dynamical systems, but with only first order Lagrangian, we resort to the formulation in terms of Lagrange-Souriau 2-form formalism. A wide class of systems derived in different phenomenological…
We present a variational approach to a general Lienard-type equation in order to linearize it and, as an example, the Van der Pol oscillator is discussed. The new equation which is almost linear is factorized. The point symmetries of the…
Reduction is a process that uses symmetry to lower the order of a Hamiltonian system. The new variables in the reduced picture are often not canonical: there are no clear variables representing positions and momenta, and the Poisson bracket…
Geometry of sparse systems of polynomial equations (i.e. the ones with prescribed monomials and generic coefficients) is well studied in terms of their Newton polytopes. The results of this study are colloquially known as the…
We study the global behavior of the trajectories of the polynomial system $\dot x = x - x^2 y+p x y^2+ y^3, \ \dot y=y+p y^3 , \ \ p\in \mathbb{R}$. Our study is related to the paper {\it Alarcon B., Castro S.B.S.D., Labouriau I.S.} Glodal…