Search results for "Grab"

Integrable systems, Frobenius manifolds and cohomological field theories

2022

In this dissertation, we study the underlying geometry of integrable systems, in particular tausymmetric bi-Hamiltonian hierarchies of evolutionary PDEs and differential-difference equations.First, we explore the close connection between the realms of integrable systems and algebraic geometry by giving a new proof of the Witten conjecture, which constructs the string taufunction of the Korteweg-de Vries hierarchy via intersection theory of the moduli spaces of stable curves with marked points. This novel proof is based on the geometry of double ramification cycles, tautological classes whose behavior under pullbacks of the forgetful and gluing maps facilitate the computation of intersection…

La reapertura de la Biblioteca Ambrosiana de Milán

1996

Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems

1999

The tangential Hilbert 16th problem is to place an upper bound for the number of isolated ovals of algebraic level curves { H ( x , y ) = const } \{H(x,y)=\operatorname {const}\} over which the integral of a polynomial 1-form P ( x , y ) d x + Q ( x , y ) d y P(x,y)\,dx+Q(x,y)\,dy (the Abelian integral) may vanish, the answer to be given in terms of the degrees n = deg ⁡ H n=\deg H and d = max ( deg ⁡ P , deg ⁡ Q ) d=\max (\deg P,\deg Q) . We describe an algorithm producing this upper bound in the form of a primitive recursive (in fact, elementary) function of n n and d d for the particular case of hyperelliptic polynomials H ( x , y ) = y 2 + U ( x ) H(x,y)=y^2+U(x) under the additional as…

Generators of Random Processes in Ultrametric Spaces and Their Spectra

2009

The L 2(\( \mathbb{S} \)) space of square integrable functions on an ultrametric space \( \mathbb{S} \) has rather specific structure. As a consequence in a natural way there appear in L 2(\( \mathbb{S} \)) the operators of which unitary counterparts in L 2(ℝn) would be difficult to construct. Such class of self-adjoint operators emerge from theory of random processes on ultrametric spaces. In this paper we collect known material on spectral properties of the generators of random processes on \( \mathbb{S}_B \) an ultrametric space of sequences. (The set of p-adic numbers is a subset of \( \mathbb{S}_B \).) Then we discuss structure of the eigenspaces of the generators.

Some Aspects of Vector-Valued Singular Integrals

2009

Let A, B be Banach spaces and \(1 < p < \infty. \; T\) is said to be a (p, A, B)- CalderoLon–Zygmund type operator if it is of weak type (p, p), and there exist a Banach space E, a bounded bilinear map \(u: E \times A \rightarrow B,\) and a locally integrable function k from \(\mathbb{R}^n \times \mathbb{R}^n \backslash \{(x, x): x \in \mathbb{R}^n\}\) into E such that $$T\;f(x) = \int u(k(x, y), f(y))dy$$ for every A-valued simple function f and \(x \notin \; supp \; f.\)

About Compactness of Faddeev Integral Equations for Three Charged Particles

1999

Momentum space three-body integral equations of the Faddeev type can not be used for Coulomb-like potentials, for energies above the breakup threshold. The reason is the occurrence of singularities in their kernels which destroy the compactness properties known to exist for purely short-range interactions. Using the rigorously equivalent formulation in terms of an effective-two- body theory, we prove that the nondiagonal kernels occurring therein possess on and off the energy shell only integrable singularities, provided all three particles have charges of the same sign (ie., only repulsive Coulomb interactions). In contrast, if some of the charges have opposite signs the nondiagonal kernel…

Thermal solitons in nanotubes

2022

Starting from a recent proposal of a nonlinear Maxwell-Cattaneo equation for the heat transport with relaxational effects at nanoscale, in a special case of thermal-wave propagation we derive a nonlinear Schrodinger equation for the amplitudes of the heatflux perturbation. The complete integrability of the obtained equation is investigated in order to prove the existence of infinite conservation laws, as well as the existence of infinite exact solutions. In this regards, we have considered the simplest nontrivial solutions, namely, the bright and dark (thermal) solitons, which may be interesting for energy transport and for information transmission in phononic circuits. (c) 2022 Elsevier B.…

Spectral approach to the scattering map for the semi-classical defocusing Davey–Stewartson II equation

2019

International audience; The inverse scattering approach for the defocusing Davey–Stewartson II equation is given by a system of D-bar equations. We present a numerical approach to semi-classical D-bar problems for real analytic rapidly decreasing potentials. We treat the D-bar problem as a complex linear second order integral equation which is solved with discrete Fourier transforms complemented by a regularization of the singular parts by explicit analytic computation. The resulting algebraic equation is solved either by fixed point iterations or GMRES. Several examples for small values of the semi-classical parameter in the system are discussed.

Rational solutions to the KPI equation from particular polynomials

2022

Abstract We construct solutions to the Kadomtsev–Petviashvili equation (KPI) from particular polynomials. We obtain rational solutions written as a second spatial derivative of a logarithm of a determinant of order n . We obtain with this method an infinite hierarchy of rational solutions to the KPI equation. We give explicitly the expressions of these solutions for the first five orders.

"Table 2" of "Study of $e^+e^- \rightarrow p\bar{p}$ in the vicinity of $\psi(3770)$"

2014

The two solutions of the dressed cross section and the corresponding phase angles, PHI.