String Equation Research Articles

진동 측정법을 이용한 PC교량 외부텐던의 장력 추정

Park, Sung Woo Jung, Ha Tae Jung, Soo HyungAbstractIn this study, vibrational tendon tension measurement methods are applied to estimate tension of external tendons used in segmental post-tensioned bridges. The acceleration of various length type of tendons is measured and natural frequencies are obtained using FFT (Fast Fourier Transform). The obtained natural frequencies are within 1% error regardless of sensor direction and location. On the basis of natural frequency of tendon, estimation of the tendon tension is performed by using many types ofsolutions such as string theory equation, multi mode estimation, practical formula estimation and stiff string with clamped-clamped boundary conditions. The results are compared with each other and have shown that the flexural stiffness is not negligible in tendons of this type causing the vibration mode to be inharmonically related. The results have shown that the method using stiffstring equation with clamped-clamped boundary conditions is more accurate than the other methods. Application example of in-service bridges has shown that force distribution effects from friction at deviation blocks can be effectively detected. Keywords : Tendon tension, Vibrational measurement method, Natural frequency, Flexural stiffness, String equation

Self-dual string and higher instanton solutions

We present and discuss explicit solutions to the non-abelian self-dual string equation as well as to the non-abelian self-duality equation in six dimensions. These solutions are generalizations of the 't Hooft-Polyakov monopole and the BPST instanton to higher gauge theory. We expect that these solutions are relevant to the effective description of M2- and M5-branes.

Non-Abelian Tensor Multiplet Equations from Twistor Space

We establish a Penrose–Ward transform yielding a bijection between holomorphic principal 2-bundles over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. Extending the twistor space to supertwistor space, we derive sets of manifestly \({\mathcal{N} = (1, 0)}\) and \({\mathcal{N} = (2, 0)}\) supersymmetric non-Abelian constraint equations containing the tensor multiplet. We also demonstrate how this construction leads to constraint equations for non-Abelian supersymmetric self-dual strings.

On variable length induced vibrations of a vertical string

Abstract The purpose of this paper is to study the free lateral responses of vertically translating media with variable length, velocity and tension, subject to general initial conditions. The translating media are modeled as taut strings with fixed boundaries. The problem can be used as a simple model to describe the lateral vibrations of an elevator cable, for which the length changes linearly in time, or for which the length changes harmonically about a constant mean length. In this paper an initial-boundary value problem for a linear, axially moving string equation is formulated. In the given model a rigid body is attached to the lower end of the string, and the suspension of this rigid body against the guide rails is assumed to be rigid. For linearly length variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to car mass, and for the harmonically length variations small oscillation amplitudes are assumed and it is also assumed that the string mass is small compared to the total mass of the string and the car. A multiple-timescales perturbation method is used to construct formal asymptotic approximations of the solutions to show the complicated dynamical behavior of the string. For the linearly varying length analytic approximations of the exact solution are compared with numerical solution. For the harmonically varying length it will be shown that Galerkin׳s truncation method cannot be applied in all cases to obtain approximations valid on long timescales.

The (n,1)-Reduced DKP Hierarchy, the String Equation and W Constraints

The total descendent potential of a simple singularity satisfies the Kac-Wakimoto principal hierarchy. Bakalov and Milanov showed recently that it is also a highest weight vector for the corresponding W-algebra. This was used by Liu, Yang and Zhang to prove its uniqueness. We construct this principal hierarchy of type D in a different way, viz. as a reduction of some DKP hierarchy. This gives a Lax type and a Grassmannian formulation of this hierarchy. We show in particular that the string equation induces a large part of the W constraints of Bakalov and Milanov. These constraints are not only given on the tau function, but also in terms of the Lax and Orlov-Schulman operators.

On the solution of some nonlinear inverse problems in geophysics

We show that a canonical three-dimensional domain can be mapped onto unknown source of masses with the help of dispersionless two-dimensional (2D) Toda integrable hierarchies. The maps are determined by a particular approximate solution to a set of 2D Toda hierarchies singled out by the condition known in the 2D case as ‘string equations’. The same hierarchy locally solves 2D inverse potential problem, i.e. reconstruction of the 2D domain out of a set of its harmonic moments. We solve the non-linear inverse problem of geophysics by setting a conditional variational problem to determine the coefficients of the Laurent expansion of the function realizing the conformal map.

Controllability Problem for the String Equation with External Load with the use of Atomic Functions

The necessary and sufficient conditions for the approximate controllability and null-controllability by atomic functions are obtained for the string equation controlled by an external load on a rectangular domain. The function of external load is presented as a scalar product of atomic functions and controls. The controls that solve these problems are found explicitly.

Erratum to “Weierstrass’ Elliptic Function Solution to the Autonomous Limit of the String Equation of Type (2,5)” [Advances in Pure Mathematics 4 (2014), 494-497

In this note, we analyze a few major claims about . As a consequence, we rewrite a major theorem, nullify its proof and one remark of importance, and offer a valid proof for it. The most important gift of this paper is probably the reasoning involved in all: We observe that a constant, namely t, has been changed into a variable, and we then tell why such a move could not have been made, we observe the discrepancy between the claimed domain and the actual domain of a supposed function that is created and we then explain why such a function could not, or should not, have been created, along with others.

Weierstrass’ Elliptic Function Solution to the Autonomous Limit of the String Equation of Type (2,5)*

In this article, we study the string equation of type (2,5), which is derived from 2D gravity theory or the string theory. We consider the equation as a 4th order analogue of the first Painlevé equation, take the autonomous limit, and solve it concretely by use of the Weierstrass’ elliptic function.

Global and Blow-Up Solutions for Nonlinear Hyperbolic Equations with Initial-Boundary Conditions

We consider an initial-boundary value problem to a nonlinear string equations with linear damping term. It is proved that under suitable conditions the solution is global in time and the solution with a negative initial energy blows up in finite time.

Minimal Liouville gravity correlation numbers from Douglas string equation

We continue the study of $(q,p)$ Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of \cite{Moore:1991ir}, \cite{Belavin:2008kv}, where Lee-Yang series $(2,2s+1)$ was studied, to $(3,3s+p_0)$ Minimal Liouville Gravity, where $p_0=1,2$. We demonstrate that there exist such coordinates $\tau_{m,n}$ on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates $\tau_{m,n}$ are related in a non-linear fashion to the natural coupling constants $\lambda_{m,n}$ of the perturbations of Minimal Lioville Gravity by the physical operators $O_{m,n}$. We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature \cite{Goulian:1990qr}, \cite{Zamolodchikov:2005sj}, \cite{Belavin:2006ex}.

Influence Analysis of the Vibration Frequency to the Dynamic Response of Deepwater Drilling Casing String

The dynamic response of the casing string under the mud line is concerned by engineer when deepwater drilling operating. Considering with the nonlinear characteristics of seabed soil, a dynamic differential equation of the casing string below mud line was established. Then the Newmark method was adapted to solving this equation by time step. Results illustrate that, the deformation and internal force of the casing string in low frequency region have little difference with the static situation. The amplitude of the string displacement increases gradually at the natural frequency. And the hysteretic phenomenon more and more obviously at this point also. In general, there are almost no influences to the dynamic lateral bearing of the casing string when not considering the wellhead vertical force. Use a static method to analyses the wellhead stability and the string bearing capacity can meet the project requirement. The dynamic analysis of casing string coupled with the marine riser is very necessary to future work.

A Deformation of the Method of Characteristics and the Cauchy Problem for Hamiltonian PDEs in the Small Dispersion Limit

We introduce a deformation of the method of characteristics valid for Hamiltonian perturbations of a scalar conservation law in the small dispersion limit. Our method of analysis is based on the “variational string equation”, a functional-differential relation originally introduced by Dubrovin in a particular case, of which we lay the mathematical foundation. Starting from first principles, we construct the string equation explicitly up to the fourth order in perturbation theory, and we show that the solution to the Cauchy problem of the Hamiltonian partial differential equation (PDE) satisfies the appropriate string equation in the small dispersion limit. We apply our construction to explicitly compute the first two perturbative corrections of the solution to the general Hamiltonian PDE. In the Korteweg–de Vries (KdV) case, we prove the existence of a quasitriviality transformation at any order and for arbitrary initial data.

On the problem of precise control of the system obeying the delay string equation

Consideration was given to the problem of precise control of a system obeying the equation of string with integral "memory." This system was proved to be reducible to the quiescent state in a finite time with the use of a distributed action bound in magnitude. The time to stop oscillations was also estimated.

Fluid mechanics of strings

We consider conserved currents in an interacting network of one-dimensional objects (or strings). Singular currents localized on a single string are considered in general, and a formal procedure for coarse-graining over many strings is developed. This procedure is applied to strings described by the Nambu-Goto action such as cosmic strings. In addition to conserved currents corresponding to the energy-momentum tensor, we consider an antisymmetric tensor of conserved currents related to the string tangent vector. Under the assumption of local equilibrium we derive a complete set of hydrodynamic equations for strings.

The string equation and the τ-function Witt constraints for the discrete Kadomtsev-Petviashvili hierarchy

Using the tool of pseudo-difference operators, the string equation for the discrete Kadomtsev-Petviashvili (KP) hierarchy is introduced and a general expression for the string equation is formulated. Furthermore, the operators of the constraints that the string equation imposes on the τ-function of the p-reduced dKP hierarchy are presented. The algebra, which part of the constraints spans, and eigenvalues of part of the constraints corresponding to the τ-function are calculated. It is also shown that this algebra is exactly a Witt algebra not only for p = 2, but also for a general \documentclass[12pt]{minimal}\begin{document}$p\in \mathbb {N}$\end{document}p∈N, and that the τ-function of the p-reduced dKP hierarchy constrained by the string equation is a vacuum vector of a Witt algebra.

Another representation of the β form of the inhomogeneous Picard-Fuchs equation

In this letter we give another representation of the β form in the inhomogeneous Picard-Fuchs equation for open topological string for some one-parameter Calabi-Yau hypersurfaces in weighted projective spaces. Furthermore, the corresponding domain wall tensions calculated by using these β forms are consistent with the results that appear in literature. The β form is essential for the calculation of the D-brane domain wall tension, and a convenient choice of β forms should simplify the calculation. The freedom of the choice of β forms shows some symmetries in Calabi-Yau space.

Exact Controllability of the Distributed System, Governed by String Equation with Memory

We will consider the exact controllability of the distributed system, governed by string equation with memory. It will be proved that this mechanical system can be driven to an equilibrium point in a finite time, the absolute value of the distributed control function being bounded. In this case, the memory kernel is a linear combination of exponentials.

Kinematic splitting algorithm for fluid–structure interaction in hemodynamics

Abstract 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.

Laplacian growth in a channel and Hurwitz numbers

We study the integrable structure of the 2D Laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in a transverse direction. Similarly to the Laplacian growth in radial geometry, this problem can be embedded into the 2D Toda lattice hierarchy in the zero dispersion limit. However, the relevant solution to the hierarchy is different. We characterize this solution by the string equations and construct the corresponding dispersionless tau-function. This tau-function is shown to coincide with the genus-zero part of the generating function for double Hurwitz numbers.