相关论文: Integrability of Riccati equation from a group the…
Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.
This is a review of the results related to generalizations of the notion of $\tau$-function and integrable hierarchies and to their interpretation within the group theory framework that admits an immediate quantization procedure. Different…
The Chiellini integrability condition of the first order first kind Abel equation $dy/dx=f(x)y^2+g(x)y^3$ is extended to the case of the general Abel equation of the form $dy/dx=a(x)+b(x)y+f(x)y^{\alpha -1}+g(x)y^{\alpha}$, where $\alpha…
In this note we clarify that the Rcci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.
We discuss the notion of integrability in quantum mechanics. Starting from a review of some definitions commonly used in the literature, we propose a different set of criteria, leading to a classification of models in terms of different…
We extend a positive Ricci curvature gluing theorem of Perelman to a range of positive intermediate curvature conditions, ranging from positive scalar curvature up to (and including) positive sectional curvature. As an application of this,…
We address the problem of finding necessary and sufficient conditions for an arbitrary group, not necessarily finite, to admit a faithful irreducible representation over an arbitrary field.
A renormalization group study of a scalar theory coupled to gravity through a general functional dependence on the Ricci scalar in the action is discussed. A set of non-perturbative flow equations governing the evolution of the new…
We revisit the regularity theory for uniformly elliptic equations.
The main ideas behind nuclear supersymmetry are presented, starting from the basic concepts of symmetry and the methods of group theory in physics. We propose new, more stringent experimental tests that probe the supersymmetry…
Some aspects of phase transitions can be more conveniently studied in the orbit space of the action of the symmetry group. After a brief review of the fundamental ideas of this approach, I shall concentrate on the mathematical aspect and…
The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from…
Motivated by the recent rapid development of complexity theory applied to quantum mechanical processes we present the complete derivation of Nielsen's complexity of unitaries belonging to the representations of oscillator group. Our…
Every abelian (and even every nilpotent) group contains a solution of any finite unimodular system of equations over itself. However, this is not true for infinite systems. We deduced a criterion for a periodic abelian group to contain a…
For initial value problems associated with operator-valued Riccati differential equations posed in the space of Hilbert--Schmidt operators existence of solutions is studied. An existence result known for algebraic Riccati equations is…
In a number of physically important cases, the nonholonomically (nonintegrable) constrained Ricci flows can be modelled by exact solutions of Einstein equations with nonhomogeneous (anisotropic) cosmological constants. We develop two…
A rigorous geometric proof of the Lie's Theorem on nonlinear superposition rules for solutions of non-autonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an…
The theory of superposition rules for solutions of a Lie system of first-order differential equations is extended to deal with analogous systems of second-order and the theory is illustrated with the very rich example of Ermakov-like…
With respect to earlier investigations, the theory of multi-component, concentric, copolar, axisymmetric, rigidly rotating polytropes is improved and extended, including subsystems with nonzero density on the boundary and subsystems with…
We introduce a generalization of Glimm's random choice method, which provides us with an approximation of entropy solutions to quasilinear hyperbolic system of balance laws. The flux-function and the source term of the equations may depend…