Related papers: The birational Lalanne-Kreweras involution

We propose a fibre bundle formulation of the mathematical base of relativistic quantum mechanics. At the present stage the bundle form of the theory is equivalent to its conventional one, but it admits new types of generalizations in…

Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations…

The generalized Lie symmetries of almost regular Lagrangians are studied, and their impact on the evolution of dynamical systems is determined. It is found that if the action has a generalized Lie symmetry, then the Lagrangian is…

A kind of Bargmann symmetry constraints involving Lax pairs and adjoint Lax pairs is proposed for soliton hierarchy. The Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative finite dimensional integrable…

We derive a nonlinear integro-differential transport equation describing collective evolution of weights under gradient descent in large-width neural-network-like models. We characterize stationary points of the evolution and analyze…

This work builds on an existing model of discrete canonical evolution and applies it to the general case of a linear dynamical system, i.e., a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q} $ and linear…

We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly non-reflexive and non-separable spaces. The pivoting point is to establish a novel variational structure, based…

The evolute of a plane curve is the envelope of its normals. Replacing the normals by the lines that make a fixed angle with the curve yields a new curve, called the evolutoid. We prefer the term ``skew evolute", and we study the geometry…

Loewner Theory, based on dynamical viewpoint, proved itself to be a powerful tool in Complex Analysis and its applications. Recently Bracci et al [Bracci et al, to appear in J. Reine Angew. Math. Available on ArXiv 0807.1594; Bracci et al,…

Kaup-Newell soliton hierarchy is derived from a kind of Lax pairs different from the original ones. Binary nonlinearization procedure corresponding to the Bargmann symmetry constraint is carried out for those Lax pairs. The proposed Lax…

The dynamics of certain combinatorial actions and their liftings to actions at the piecewise-linear and birational level have been studied lately with an eye towards questions of periodicity, orbit structure, and invariants. One key…

A field theoretical perturbation theory in inverse powers of coupling constant is developed which is manifestly covariant in every order of the expansion. A dilatation operator serves as an evolution dynamical one in a scale non-invariant…

We develop a variational technique for some wide classes of nonlinear evolutions. The novelty here is that we derive the main information directly from the corresponding Euler-Lagrange equations. In particular, we prove that not only the…

We study piecewise-linear and birational lifts of Sch\"utzenberger promotion, evacuation, and the RSK correspondence defined in terms of toggles. Using this perspective, we prove that certain chain statistics in rectangles shift predictably…

Some mechanical systems with dissipation can be described within the framework of the so-called contact mechanics: a modified form of the Euler-Lagrange equations stemming from Herglotz's variational principle, which admits a geometric…

A Bargmann symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of the Dirac systems. It is shown that the spatial part of the nonlinearized Lax pairs and adjoint Lax pairs is a finite dimensional Liouville integrable…

We introduce an integrable Hamiltonian system which Lax deforms the Dirac operator D=d+d* on a finite simple graph or compact Riemannian manifold. We show that the nonlinear isospectral deformation always leads to an expansion of the…

We study deterministic Loewner evolutions on the complex plane driven by complex-valued functions. This model can be viewed as a generalization of real-driven Loewner evolutions in the upper half-plane, or as the deterministic analogue of…

We present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation, inter-twining lifting, commuting and non-commuting dilations,…

Stochastic Loewner Evolutions (SLE) with a multiple sqrt(kappa)B of Brownian motion B as driving process are random planar curves (if kappa<=4) or growing compact sets generated by a curve (if kappa>4). We consider here more general Levy…