Combinatorics of Bousquet-M\'elou-Schaeffer numbers in the light of topological recursion

In this paper we prove, in a purely combinatorial way, a structural quasi-polynomiality property for the Bousquet-M\'elou-Schaeffer numbers. Conjecturally, this property should follow from the Chekhov-Eynard-Orantin topological recursion for these numbers (or, to be more precise, the Bouchard-Eynard version of the topological recursion for higher order critical points), which we derive in this paper from the recent result of Alexandrov-Chapuy-Eynard-Harnad. To this end, the missing ingredient is a generalization to the case of higher order critical points on the underlying spectral curve of the existing correspondence between the topological recursion and Givental's theory for cohomological field theories.Comment: 33 page

Exact and optimal quadratization of nonlinear finite-dimensional non-autonomous dynamical systems

Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms and software capabilities for quadratization of non-autonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semi-discretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both non-autonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages

Explicit closed algebraic formulas for Orlov-Scherbin nn-point functions

We derive a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions). Notably, we use the change of variables suggested by the associated spectral curve, and our formula turns out to be a polynomial expression in a certain small set of formal functions defined on the spectral curve.Comment: 35 pages; statements regarding the $\hbar$-deformations were corrected; further minor change

Generalised ordinary vs fully simple duality for nn-point functions and a proof of the Borot--Garcia-Failde conjecture

We study a duality for the $n$-point functions in VEV formalism that we call the ordinary vs fully simple duality. It provides an ultimate generalisation and a proper context for the duality between maps and fully simple maps observed by Borot and Garcia-Failde. Our approach allows to transfer the algebraicity properties between the systems of $n$-point functions related by this duality, and gives direct tools for the analysis of singularities. As an application, we give a proof of a recent conjecture of Borot and Garcia-Failde on topological recursion for fully simple maps.Comment: 25 pages; minor correction

KP integrability through the x−yx-y swap relation

We discuss a universal relation that we call the $x-y$ swap relation, which plays a prominent role in the theory of topological recursion, Hurwitz theory, and free probability theory. We describe in a very precise and detailed way the interaction of the $x-y$ swap relation and KP integrability. As an application, we prove a recent conjecture that relates some particular instances of topological recursion to the Mironov-Morozov-Semenoff matrix integrals.Comment: 27 pages; minor correction

On certain approaches to the control methods development for the precipitation formation processes in convective clouds

The article aims at searching for the optimal way of emission of ice nucleating agent in convective cloud in order to prevent ‎formation of harmful hail by analyzing simulations of this process within a numerical model ‎of cloud‎. The state of the physics of clouds and active influences on them is discussed. It is noted that at the present time studies of the regularities of the formation and development of clouds as a whole begin taking into account their systemic properties. The main directions of research at the next stage of its development are discussed. The features of the existing methods of active action on convective clouds are noted, the main tasks encountered in the development of methods for controlling sedimentation in convective clouds by introducing reagents are formulated. It is noted that research on the development of methods for active influence on clouds should be conducted on the basis of new and more effective approaches, which should be based on the extensive use of mathematical modeling. Some approaches to solving this problem are discussed. According to the authors, the most promising of them are approaches based on the theory of optimal control and bifurcation theory. Some results of numerical modeling of the active effect on convective clouds are given

Triplet pairing due to spin-orbit-assisted electron-phonon coupling

We propose a microscopic mechanism for triplet pairing due to spin-orbit-assisted electron interaction with optical phonons in a crystal with a complex unit cell. Using two examples of electrons with symmetric Fermi surfaces in crystals with either a cubic or a layered square lattice, we show that spin-orbit-assisted electron-phonon coupling can, indeed, generate triplet pairing and that, in each case, it predetermines the tensor structure of a p-wave order parameter