Refining Eigenvalue Estimates for a String with a Singular Weight

Differential Equations - We consider the linear differential pencil $$A(\lambda )\equiv L-\lambda V $$ , where $$L\equiv -y^{\prime {}\prime }+q(x)y$$ is the Sturm–Liouville operator with...     

On String Graph Limits and the Structure of a Typical String Graph

We study limits of convergent sequences of string graphs, that is graphs with an intersection representation consisting of curves in the plane. We use these results to study the limiting behavior of a sequence of random string graphs. We also prove similar results for several related graph classes.     

String matching with alphabet sampling

We introduce a novel alphabet sampling technique for speeding up both online and indexed string matching. We choose a subset of the alphabet and extract the corresponding subsequence of the text. Online or indexed searching is then carried out on the extracted subsequence, and candidate matches are verified in the full text. We show that this speeds up online searching, especially for moderate to long patterns, by a factor of up to 5, while using 14% extra space in our experiments. For indexed searching we achieve indexes that are as fast as the classical suffix array, yet occupy less than 50% extra space (instead of the usual 400%). Our experiments show no competitive alternatives exist in a wide space/time range.     

The Periodic Solutions of a Nonhomogeneous String with Dirichlet-Neumann Condition

This paper deals with the existence of periodic solutions of a nonhomogeneous string with Dirichlet-Neumann condition. The authors consider the case that the period is irrational multiple of space length and prove that for some irrational number, zero is not the accumulation point of the spectrum of the associated linear operator. This result can be used to prove the existence of the periodic solution avoid using Nash-Moser iteration.     

Traces of Higher Negative Orders for a String with a Singular Weight

We study the linear operator pencil A(λ) = L?λV, λ ∈ ?, where L is the Sturm–Liouville operator with potential q(x) and V is the operator of multiplication by the weight ρ(x). The potential and the weight are assumed to belong to the space W ₂ ^?1 [0, π]. For the pencil A(λ), we seek formulas for the traces of higher negative orders, i.e., for the sums \(\sum\nolimits_{n = 1}^\infty {\lambda _n^{ - p}} \), p ≥ 2, where λ_n, n ∈ ?, is the sequence of eigenvalues of the pencil numbered in nondescending order of absolute values. Trace formulas in terms of the weight ρ(x) and the integral kernel of the operator L^?1 are obtained, and the relationship between these formulas and the classical results about traces of integral operators is described. The theoretical results are illustrated by examples.     

Perturbed States of a Rotating Relativistic String

We investigate arbitrary small perturbations of the uniform rotation of a rectilinear relativistic string with massive ends. Two classes of such perturbations with different spectra of eigenfrequencies have the form of stationary waves and correspond to string oscillations in the rotation plane and in the orthogonal direction. These states are represented by the Fourier series used to quantize string excitations in the linear neighborhood of the classical rotation. These string oscillations can be used to describe daughter Regge trajectories in the framework of the considered model.     

String cosmology

An overview is given of the formulation of low-energy string cosmologies together with examples of particular solutions, successes and problems of the theory.     

String C-groups with real Schur index 2

We give examples of finite string C-groups (the automorphism groups of abstract regular polytopes) that have irreducible characters of real Schur index 2. This answers a problem of Monson concerning these groups.     

The String Method as a Dynamical System

This paper is devoted to the theoretical analysis of the zero-temperature string method, a scheme for identifying minimum energy paths (MEPs) on a given energy landscape. By definition, MEPs are curves connecting critical points on the energy landscape which are everywhere tangent to the gradient of the potential except possibly at critical points. In practice, MEPs are mountain pass curves that play a special role, e.g., in the context of rare reactive events that occur when one considers a steepest descent dynamics on the potential perturbed by a small random noise. The string method aims to identify MEPs by moving each point of the curve by steepest descent on the energy landscape. Here we address the question of whether such a curve evolution necessarily converges to an MEP. Surprisingly, the answer is no, for an interesting reason: MEPs may not be isolated, in the sense that there may be families of them that can be continuously deformed into one another. This degeneracy is related to the presence of critical points of Morse index 2 or higher along the MEP. In this paper, we elucidate this issue and completely characterize the limit set of a curve evolving by the string method. We establish rigorously that the limit set of such a curve is again a curve when the MEPs are isolated. We also show under the same hypothesis that the string evolution converges to an MEP. However, we identify and classify situations where the limit set is not a curve and may contain higher dimensional parts. We present a collection of examples where the limit set of a path contains a 2D region, a 2D surface, or a region of an arbitrary dimension up to the dimension of the space. In some of our examples the evolving path wanders around without converging to its limit set. In other examples it fills a region, converging to its limit set, which is not an MEP.     

On the Transverse Vibrations of a String Segment Between Two Supports Moving Axially Along a String of Infinite Length

This paper deals with the equation of motion of a string with time-varying boundary conditions. This equation can be transformed to a partial differential equation (PDE) with non-varying boundary conditions. During the solution process, the inverse of the time derivative operator is needed. To ensure the existence of a solution, the time derivative is generalized to a weighted time derivative on a weighted L₂-space. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Modification of Hermite Sampling with a Gaussian Multiplier

The Hermite sampling series is used to approximate bandlimited functions. In this article, we introduce two modifications of Hermite sampling with a Gaussian multiplier to approximate bandlimited and non-bandlimited functions. The convergence rate of those modifications is much higher than the convergence rate of Hermite sampling. Based on complex analysis, we establish some error bounds for approximating different classes of functions by these modifications. Theoretically and numerically, we demonstrate that the approximation by these modifications is highly efficient.     

Sampling with Bessel Functions

The paper deals with sampling of σ-bandlimited functions in R^m with Clifford-valued, where bandlimitedness means that the spectrum is contained in the ball B(0, σ) that is centered at the origin and has radius σ. By comparing with the general setting, what is new in the sampling is using the explicit Bochner-type relations involving spherical harmonics and monogenics in the Clifford algebra setting. Convergence of the sampling formulas in the L² sense and in the uniform and absolute pointwise sense are studied. The study was supported by Research Grant of the University of Macau No. RG059/05- 06S/QT/FST.     

在固定时间抽样的可变抽样区间控制图

本文根据Reynolds在固定时间抽样的可变抽样区间（VSIFT）的x^-控制图^[1]的模型设计中位值x^-和极差R图，规定样本在样等间隔的固定时间点抽取，当过程有变化的迹象时，允许有两个固定时间之间抽取附加样本，本文计算了VSIFTx^~图和R图及联合x^~-R图的发信号前的平均时间，并同固定抽样区间（FSI）的常规x^~和R图作比较，所设计的VSIFTx^~和R图能缩短过程失控时间从而减少不合格品数。     

Unilateral Constrained Bending Vibrations on a Torsional String

Vibrations in long torsional strings result in spatio‐temporal dynamics. In order to actively damp these vibrations the system has to be analysed analytically, numerically and experimentally. Stick‐slip‐effects result in torsional selfexcited vibrations of the string. These vibrations are coupled with bending vibrations which are constrained by the borehole. The straight string was modelled in an experimental setup. The control of the straight string and the unilateral constrained bending vibrations were treated seperately. The dynamics of straight strings were controlled using three different approaches: firstly, a simple PD‐controller with the parameters calculated based on a one‐degree‐of‐freedom oscillator, secondly, the parameters were optimized using a simplex‐method, thirdly, the Karhunen‐Loeve‐transformation was used in order to reduce the dimension of the system. A controller based on the reduced system was implemented and the parameters were optimized with the same simplex algorithm. The unilateral constrained bending motion were examined at a cantilever beam which was assumed to be constrained in one direction in the middle of the beam. First, the beam was modelled analytically as a continuous system. The two states (contact and no contact) were described separately. The transition between these states were modelled with energy assumptions. Second, the beam was modelled as a Finite‐Element‐System. The numerical results of both methods were compared with experimental data.     

String world surfaces in spaces with compact factor-manifolds

The closed string with point-like masses as the string hadron model is considered in the D-dimensional space M = R^1,3 ×T^{D - 4} \mathcal{M} = {R^{1,3}} \times {T^{D - 4}} , which is the direct product of the Minkowski space and the compact manifold T ^D−4 = S ¹ × ⋯ × S ¹ ((D − 4)-dimensional torus). Exact solutions of dynamical equations are obtained; in a particular case of rotational states, they describe a uniform rotation of the system. These rotational states are classified, their physical properties are studied, and Regge trajectories are determined. Central and linear rotational states are tested for stability with respect to small disturbances. It is shown that the central rotational states are not stable if the central mass is less than some threshold value.     

Efficient Algorithms for Approximate String Matching with Swaps

Most research on the edit distance problem and thek-differences problem considered the set of edit operations consisting of changes, insertions, and deletions. In this paper we include theswapoperation that interchanges two adjacent characters into the set of allowable edit operations, and we present anO(t min(m, n))-time algorithm for the extended edit distance problem, wheretis the edit distance between the given strings, and anO(kn)-time algorithm for the extendedk-differences problem. That is, we add swaps into the set of edit operations without increasing the time complexities of previous algorithms that consider only changes, insertions, and deletions for the edit distance andk-differences problems.     

On Starting Control of Vibrations of a String

Within the framework of the theory of games, we consider the problem of starting control of oscillations of points of a string according to a given law. As control parameters for players, the initial position and the starting velocity of the string are taken. We determine the optimal control for players in both discrete case and continuous case.     

On a Coordinate-Independent Description of String Worldsheet Theory

We rewrite the bosonic worldsheet theory in curved background in a language where it describes a single particle moving in an infinite-dimensional curved spacetime. This language is developed at a formal level without regularizing the infinite-dimensional traces. Then, we adopt DeWitt’s (Phys Rev 85:653, 1952) coordinate-independent formulation of quantum mechanics in the present context. This procedure enables us to define coordinate invariant quantum analogue of classical Virasoro generators, which we call DeWitt–Virasoro generators. This framework also enables us to calculate the invariant matrix elements of an arbitrary operator constructed out of the DeWitt–Virasoro generators between two arbitrary scalar states. Using these tools, we further calculate the DeWitt–Virasoro algebra in spin-zero representation. The result is given by the Witt algebra with additional anomalous terms that vanish for Ricci-flat backgrounds. Further analysis need to be performed to precisely relate this with the beta function computation of Friedan and others. Finally, we explain how this analysis improves the understanding of showing conformal invariance for certain pp-wave that has been recently discussed using hamiltonian framework.     

String Jacobi Matrices with Very Fast Decreasing Weights

In this work we investigate a class of string, compact Jacobi matrices with very rapidly decreasing weights. In particular, we have found Hellinger-Weyls function representation as ratio of two entire functions with known Taylor series representation. The asymptotical estimations of zeroes of this entire functions and of eigenvalues of a corresponding operator are found. The application of this theory to Rogers-Ramanujans functions is considered.Submitted: November 18, 2002 Revised: October 30, 2003