Reparameterization-Invariant Reduction in the Hamiltonian Description of a Relativistic String

We study the time-reparameterization-invariant dynamics of an open relativistic string using the generalized Dirac–Hamilton theory and resolving the constraints of the first kind. The reparameterization-invariant evolution variable is the time coordinate of the string center of mass. Using a transformation that preserves the diffeomorphism group of the generalized Hamiltonian and the Poincaré covariance of the local constraints, we segregate the center-of-mass coordinates from the local degrees of freedom of the string. We identify the time coordinate of the string center of mass and the proper time measured in the string frame of reference using the Levi-Civita–Shanmugadhasan canonical transformation, which transforms the global constraint (the mass shell) in the new momentum such that the Hamiltonian reduction does not require the corresponding gauge condition. Resolving the local constraints, we obtain an equivalent reduced system whose Hamiltonian describes the evolution w.r.t. the proper time of the string center of mass. The Röhrlich quantum relativistic string theory, which includes the Virasoro operators L_n only with n > 0, is used to quantize this system. In our approach, the standard problems that appear in the traditional quantization scheme, including the space–time dimension D = 26 and the tachyon emergence, arise only in the case of a massless string, M² = 0.     

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.     

Solvability of a Nonlinear Integral Equation in Dynamical String Theory

We investigate an integral equation of the convolution type with a cubic nonlinearity on the entire real line. This equation has a direct application in open-string field theory and in p-adic string theory and describes nonlocal interactions. We prove that there exists a one-parameter family of bounded monotonic solutions and calculate the limits of solutions constructed at infinity.     

On Initial Conditions and Global Existence for Accelerating Cosmologies from String Theory

We construct a solution satisfying initial conditions for accelerating cosmologies from string/M-theory. Gowdy symmetric spacetimes with a positive potential are considered. Also, a global existence theorem for the spacetimes is shown. Communicated by Sergiu Klainerman submitted 18/08/04, accepted 18/01/05     

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.     

Tachyon Effective Actions in String Theory

We discuss the derivation of string effective actions for tachyons and massless modes within the sigma-model approach. We mostly consider the open string case, demonstrating that the renormalized partition function of the boundary sigma model gives the effective action for the massless vector and tachyon in the derivative expansion. We give a manifestly gauge-invariant definition of Z(T,A) in the non-Abelian Neveu–Schwarz–Ramond (NSR) open string theory and verify that its derivative reproduces the tachyon beta function in a particular scheme. We also comment on the derivation of similar actions for tachyons and massless modes in the closed bosonic and the NSR string theories.     

On String Replacement Exponentiation

The string replacement (SR) method was recently proposed as a methodfor exponentiation a ^e in a group G. The canonicalk-SR method operates by replacing a run of i onesin a binary exponent,0k, with i-1 zeroes followedby the single digit b=2 ⁱ -1. After recoding, it was shown in[5] that the expected weight of e tends to n/4 forn-bit exponents. In this paper we show that the canonicalk-SR recoding process can be described as a regular language andthen use generating functions to derive the exact probability distribution ofrecoded exponent weights. We also show that the canonical 2-SR recodingproduces weight distributions very similar to (optimal) signed-digitrecodings, but no group inversions are required.     

The No Ghost Theorem in String Theory

A logically tight proof of the No Ghost Theorem for the standard bosonic open string theory is given, and an extension is established. The latter states that 26 is the critical dimension for all levels N = n > 2 of the number operator N in the sense that for any n > 2, there are no ghosts (negative norm states) on level N = n if and only if the dimension of spacetime is no greater than 26.     

On the Reconstruction of a String from Spectral Data

We consider the problem to reconstruct the mass distribution of a string where the mass is concentrated in a finite number of points, or, equivalently, the problem to reconstruct a simply connected mass spring system with unknown masses and stiffness parameters if the following data are given. Problem 1: The spectra of the string and of a modification of the string, or. Problem 2: The spectra of two different modifications of the string. Here a modification of the string is a string which appears if we link the unknown string with another string of known mass distribution. The paper contains a necessary condition for the existence of a solution of Problem 1, and explicit formulas and an algorithm for the solutions of the Problems 1 and 2 under the condition that there exists a solution. For the case that the mass distribution of the unknown string is not discrete we consider the problem to find discrete approximations of this distribution from the respective spectral data. The methods are based on the spectral theory of generalized second order differential operators as developed by M. G. Krein     

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)     

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.     

On the Description of Partially Saturated Soils by the Theory of Porous Media

In this contribution, a multi‐phase soil model based on the Theory of Porous Media (TPM) is presented. The model is fully coupled in the following constitutive phases: An elasto‐plastic or elasto‐viscoplastic solid skeleton, a materially incompressible pore‐liquid (water) and a materially compressible pore‐gas (air). The interaction of the solid skeleton and the pore‐fluids is specified by a capillary pressure‐saturation relation, whereas the mobilities of the fluid phases in the pore‐space of the solid skeleton are described by the so‐called relative permeabilities. Finally, a gravity governed initial‐boundary‐value problem solved by the FE method is presented.     

Locality in Free String Field Theory - II

We study the covariant free bosonic string field theory and explore its locality (causality) properties. We find string fields which are strictly local and covariant, but act on an unconstrained Hilbert space with an indefinite inner product. From these we construct observable fields which act on the physical Hilbert space with a definite inner product. These are shown to be approximately local.     

On the Well-Posedness of the Kirchhoff String

Let us consider the Cauchy problem for the quasilinear hyperbolic integro-differential equation

where is an open subset of and is a positive function of one real variable which is continuously differentiable. We prove the well-posedness in the Hadamard sense (existence, uniqueness and continuous dependence of the local solution upon the initial data) in Sobolev spaces of low order.

     

On the Description of the Content-Equality by a Relation

In the geometry of polyhedra we understand by an elementary content-functional a real valued, non-negative, finite additive measure on the set of polyhedra which is invariant under isometries. There are close relations between the content-measurement and the relation of equidecomposability. Two polyhedra are called equidecomposable if they are decomposed into pairwise congruent pieces. For an example we consider the set of all polygons in the euclidean plane. It is well known that planar polygons have the same area if and only if they are equidecomposable. In the three-dimensional euclidean space one also can describe the content-equality of polyhedra by a relation. Two polyhedra have the same volume if they are equidecomposable with respect to equiaffine mappings (see [3]). In [4] the concept of an invariant content of polyhedra in a topological Klein space is introduced. Each regular closed quasicompact set ot the space is called polyhedron. Under this supposition two polyhedra have equal contents if they are equivalent by decomposition. The relation “equivalent by decomposition” is closely related to the relation “equidecomposable”.     

Sampling with a String

Kramer's sampling theorem forms a bridge between the Whittaker-Shannon-Kotel'nikov sampling theorem and boundary-value problems. It has been shown that sampling expansions associated with Sturm-Liouville boundary-value problems are Lagrange-type sampling series, i.e., Lagrange series with infinitely many terms converging to entire functions. String theory as developed by Feller, Kac, and Krein, is a generalization of the Sturm-Liouville theory. We investigate sampling series associated with strings and compare them with those associated with Sturm-Liouville problems. We show that unlike sampling series associated with Sturm-Liouville problems, those associated with strings include not only Lagrange-type sampling series, but also Lagrange polynomial interpolation.