Generalized String Equation Research Articles

Hurwitz numbers and integrable hierarchy of Volterra type

A generating function of the single Hurwitz numbers of the Riemann sphere is a tau function of the lattice KP hierarchy. The associated Lax operator L turns out to be expressed as , where is a difference-differential operator of the form . satisfies a set of Lax equations that form a continuum version of the Bogoyavlensky–Itoh (aka hungry Lotka–Volterra) hierarchies. Emergence of this underlying integrable structure is further explained in the language of generalized string equations for the Lax and Orlov–Schulman operators of the 2D Toda hierarchy. This leads to logarithmic string equations, which are confirmed with the help of a factorization problem of operators.

On the existence of weak solution to the coupled fluid-structure interaction problem for non-Newtonian shear-dependent fluid

We study the existence of weak solution for unsteady fluid-structure interaction problem for shear-thickening flow. The time dependent domain has at one part a flexible elastic wall. The evolution of fluid domain is governed by the generalized string equation with action of the fluid forces. The power-law viscosity model is applied to describe shear-dependent non-Newtonian fluids.

The partition function of the extended r-reduced Kadomtsev–Petviashvili hierarchy

We derive a particular solution of the extended r-reduced KP hierarchy, which is specified by a generalized string equation . The work is a generalization to arbitrary of Buryakʼs recent results of a solution to the extended open KdV hierarchy which corresponds to r = 2.

Kinematic splitting algorithm for fluid–structure interaction in hemodynamics

In this paper we study a kinematic splitting algorithm for fluid–structure interaction problems. This algorithm belongs to the class of loosely-coupled fluid–structure interaction schemes. We will present stability analysis for a coupled problem of non-Newtonian shear-dependent fluids in moving domains with viscoelastic boundaries. Fluid flow is described by the conservation laws with nonlinearities in convective and diffusive terms. For simplicity of presentation the structure is modelled by the generalized string equation, but the results presented in the paper may be generalized to more complex structure models. The arbitrary Lagrangian–Eulerian approach is used in order to take into account moving computational domain. Numerical experiments including numerical error analysis and comparison of hemodynamic parameters for Newtonian and non-Newtonian fluids demonstrate reliability of the proposed scheme.

Generalized string equations for double Hurwitz numbers

The generating function of double Hurwitz numbers is known to become a tau function of the Toda hierarchy. The associated Lax and Orlov–Schulman operators turn out to satisfy a set of generalized string equations. These generalized string equations resemble those of c=1 string theory except that the Orlov–Schulman operators are contained therein in an exponentiated form. These equations are derived from a set of intertwining relations for fermion bilinears in a two-dimensional free fermion system. The intertwiner is constructed from a fermionic counterpart of the cut-and-join operator. A classical limit of these generalized string equations is also obtained. The so-called Lambert curve emerges in a specialization of its solution. This seems to be another way of deriving the spectral curve of the random matrix approach to Hurwitz numbers.

Thermodynamic limit of random partitions and dispersionless Toda hierarchy

We study the thermodynamic limit of random partition models for the instanton sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical observables. The physical observables correspond to external potentials in the statistical model. The partition function is reformulated in terms of the density function of Maya diagrams. The thermodynamic limit is governed by a limit shape of Young diagrams associated with dominant terms in the partition function. The limit shape is characterized by a variational problem, which is further converted to a scalar-valued Riemann–Hilbert problem. This Riemann–Hilbert problem is solved with the aid of a complex curve, which may be thought of as the Seiberg–Witten curve of the deformed U(1) gauge theory. This solution of the Riemann–Hilbert problem is identified with a special solution of the dispersionless Toda hierarchy that satisfies a pair of generalized string equations. The generalized string equations for the 5D gauge theory are shown to be related to hidden symmetries of the statistical model. The prepotential and the Seiberg–Witten differential are also considered.

Non-degenerate solutions of the universal Whitham hierarchy

The notion of non-degenerate solutions for the dispersionless Toda hierarchy is generalized to the universal Whitham hierarchy of genus zero with M + 1 marked points. These solutions are characterized by a Riemann–Hilbert problem (generalized string equations) with respect to two-dimensional canonical transformations and may be thought of as a kind of general solutions of the hierarchy. The Riemann–Hilbert problem contains M arbitrary functions Ha(z0, za), a = 1, …, M, which play the role of generating functions of two-dimensional canonical transformations. The solution of the Riemann–Hilbert problem is described by period maps on the space of (M + 1)-tuples (zα(p): α = 0, 1, …, M) of conformal maps from M disks of the Riemann sphere and their complements to the Riemann sphere. The period maps are defined by an infinite number of contour integrals that generalize the notion of harmonic moments. The F-function (free energy) of these solutions is also shown to have a contour integral representation.

Numerical study of shear-dependent non-Newtonian fluids in compliant vessels

The aim of this contribution is to present recent results on numerical modelling of non-Newtonian flow in compliant stenosed vessels with application in hemodynamics. We consider two models of shear-thinning non-Newtonian fluids and compare them with the Newtonian model. For the structure problem, the generalized string equation for radial symmetric tubes is used and extended to a stenosed vessel. The global iterative approach to approximate the fluid–structure interaction is used. Finally, we present numerical experiments for some non-Newtonian models, comparisons with the Newtonian model and the results for hemodynamic wall parameters such as the wall shear stress and the oscillatory shear index.

Conformal Mappings and Dispersionless Toda Hierarchy II: General String Equations

In this article, we classify the solutions of the dispersionless Toda hierarchy into degenerate and non-degenerate cases. We show that every non-degenerate solution is determined by a function $\mathcal{H}(z_1,z_2)$ of two variables. We interpret these non-degenerate solutions as defining evolutions on the space $\mathfrak{D}$ of pairs of conformal mappings $(g,f)$, where $g$ is a univalent function on the exterior of the unit disc, $f$ is a univalent function on the unit disc, normalized such that $g(\infty)=\infty$, $f(0)=0$ and $f'(0)g'(\infty)=1$. For each solution, we show how to define the natural time variables $t_n, n\in\Z$, as complex coordinates on the space $\mathfrak{D}$. We also find explicit formulas for the tau function of the dispersionless Toda hierarchy in terms of $\mathcal{H}(z_1, z_2)$. Imposing some conditions on the function $\mathcal{H}(z_1, z_2)$, we show that the dispersionless Toda flows can be naturally restricted to the subspace $\Sigma$ of $\mathfrak{D}$ defined by $f(w)=1/\overline{g(1/\bar{w})}$. This recovers the result of Zabrodin.

Numerical modelling of shear‐thinning non‐Newtonian flows in compliant vessels

AbstractThe aim of this paper is to present recent results on numerical modelling of non‐Newtonian flow in two‐dimensional compliant vessels with application in hemodynamics. We consider two models of the shear‐thinning non‐Newtonian fluids and compare them with the Newtonian model. For the structure problem, the generalized string equation for radial symmetric tubes is used and extended to a stenosed vessel. A global iterative approach is used to approximate the fluid–structure interaction. At the end we present numerical experiments for selected non‐Newtonian models, comparisons with the Newtonian model and the hemodynamic wall parameters: the wall shear stress and the oscillatory stress index. Copyright © 2008 John Wiley & Sons, Ltd.

Numerical modelling of shear‐thinning non‐Newtonian flow in compliant vessels

AbstractWe present recent results on numerical modelling of non‐Newtonian flow in two‐dimensional compliant vessels with application in hemodynamics. Two models of the shear‐thinning non‐Newtonian fluids are considered and compared with the Newtonian model. For the structure problem the generalized string equation for radial symmetric tubes is used and extended to a stenosed vessel. A global iterative approach is used to approximate the fluid‐structure interaction. At the end we present numerical experiments for selected non‐Newtonian models, comparisons with the Newtonian model and the hemodynamic wall parameters: the wall shear stress and the oscillatory stress index. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Virtual Moduli Cycles and Gromov-Witten Invariants of Noncompact Symplectic Manifolds

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their variants. The more solutions of WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To realize these purposes we further develop the language introduced by Liu-Tian to describe the virtual moduli cycle (defined by Liu-Tian, Fukaya-Ono, Li-Tian, Ruan and Siebert).

String equation and quantum cohomology for noncompact symplectic manifolds

We use our Gromov–Witten invariant theory in a previous Note for noncompact geometrically bounded symplectic manifolds to get solutions of the generalized string equation and dilation equation and their variants. More solutions of the WDVV equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups. To cite this article: G. Lu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Toda lattice hierarchy and generalized string equations

String equations of thep th generalized Kontsevich model and the compactifiedc=1 string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model atp=−1 does not coincide with thec=1 string theory at selfdual radius. A broader family of solutions of the Toda lattice hierarchy including these models is constructed, and shown to satisfy generalized string equations. The status of a variety ofc≦1 string models is discussed in this new framework.

Stable non-perturbative minimal models coupled to 2D quantum gravity

A generalisation of the non-perturbatively stable solutions of string equations which respect the KdV flows, obtained recently for the (2 m − 1, 2) conformal minimal models coupled to two-dimensional quantum gravity, is presented for the ( p, q) models. These string equations are the most general string equations compatible with the qth generalised KdV flows. They exhibit a close relationship with the bi-hamiltonian structure in these hierarchies. The Ising model is studied as a particular example, for which a real non-singular numerical solution to the string susceptibility is presented.

Multicritical matter from complex matrices

A large N-matrix model of a general complex matrix generates dynamical triangulations in which the triangles can be chequered (i.e. coloured so that neighbours are opposite colours). Gravity and multicritical matter in such triangulations are described by the KdV-type equations found in the Hermitian matrix model but without the doubling of degrees of freedom. However, by tuning further couplings it is possible to obtain a series of more general string equations with redoubled degrees of freedom.

The isomonodromy approach to matric models in 2D quantum gravity

We consider the double-scaling limit in the hermitian matrix model for 2D quantum gravity associated with the measure exp\(\sum\limits_{j = 1}^N {t_{j^{Z^{2j,} } } N \geqq 3} \). We show that after the appropriate modification of the contour of integration the Cross-Migdal-Douglas-Shenker limit to the Painleve I equation (in the generic case of the pure gravity) is valid and calculate the nonperturbative parameters of the corresponding Painleve function. Our approach is based on the WKB-analysis of the L-A pair corresponding to the discrete string equation in the framework of the Inverse Monodromy Method. Here we extend our results, which were obtained before for the particular casesN=2,3. Our analysis complements the isomonodromy approach proposed by G. Moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself.

Harmonic Structure of an Aging Guitar String

By application of various signal processing means, namely FFT, linear estimation, and digital filtering, impulse responses of new and old guitar strings were analyzed to determine the important differences. These measurements were compared to the theoretical results derived from the generalized string equation operator ε∇4+∇2+k2, using appropriate boundary conditions. Results are discussed and synthetically generated samples will be played.