Related papers: Non-Linear Algebra and Bogolubov's Recursion
The non-relativistic current algebra approach is analyzed subject to its application to studying the distribution functions of many-particle systems at the temperature equilibrium and their stability properties. We show that the classical…
We apply the generalized Abel-Plana formula for the investigation of one-loop quantum effects on manifolds with boundaries. This allows to extract from the vacuum expectation values of local physical observables the parts corresponding to…
The textbook Newton's iteration is practically inapplicable on solutions of nonlinear systems with singular Jacobians. By a simple modification, a novel extension of Newton's iteration regains its local quadratic convergence toward…
We analyse nonperturbatively signal transmission patterns in Green's functions of interacting quantum fields. Quantum field theory is re-formulated in terms of the nonlinear quantum-statistical response of the field. This formulation…
Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of B\'ezout. This was…
Approximately 10 years ago, the method of renormalization-group symmetries entered the field of boundary value problems of classical mathematical physics, stemming from the concepts of functional self-similarity and of the Bogoliubov…
In the number-conserving Bogoliubov theory of BEC the Bogoliubov transformation between quasiparticles and particles is nonlinear. We invert this nonlinear transformation and give general expression for eigenstates of the Bogoliubov…
Solutions of nonlinear acoustic equations describing propagation of strong sound pulses with account of curvature of wave fronts in multi-dimensional geometry are obtained from simple physical considerations. The form of these solutions…
The development and applications of the method of quasiaverages developed by N. N. Bogoliubov to quantum statistical physics and to quantum solid state theory and, in particular, to quantum theory of magnetism, were analyzed. The problem of…
We start with a simple introduction into the renormalization group (RG) in quantum field theory and give an overview of the renormalization group method. The third section is devoted to essential topics of the renorm-group use in the QFT.…
This paper introduces the concept of renormalized solution for a general class of non-coercive nonlinear parabolic problems, including both singularities and unbounded lower order terms. We prove existence and uniqueness of renormalized…
Generalizations of the three main equations of quantum physics, namely, the Schr\"odinger, Klein-Gordon, and Dirac equations, are proposed. Nonlinear terms, characterized by exponents depending on an index $q$, are considered in such a way…
A practical approach is presented which allows the use of a non-invariant regularization scheme for the computation of quantum corrections in perturbative quantum field theory. The theoretical control of algebraic renormalization over…
We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the non-associative algebra generated by the convolution product of the…
The work presents two examples of simple mathematical formulas which are natural nonlinear modifications (one being a generalization) of Gielis' formula. These formulas involve a comparable number of parameters and provide non-Platonic…
Following the previously developed approach to the calculation of quantum corrections to the effective potential in arbitrary scalar field theories in the leading logarithmic approximation, we extended it to the next-to-leading order. Based…
An explicit perturbative solution to all orders is given for a general class of nonlinear differential equations. This solution is written as a sum indexed by rooted trees and uses the Green function of a linearization of the equations. The…
Nonlinear Hamiltonian systems describing the abstract Vlasov and Hartree equations are considered in the framework of algebraic Poissonian theory. The concept of uniformization is introduced; it generalizes the method of second quantization…
Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical…
We generalise the two-sided Bogoliubov inequality for classical particles from [L. Delle Site et al., J.Stat.Mech.Th.Exp. 083201 (2017)] to systems of quantum particles. As in the classical set-up, the inequality leads to upper and lower…