Strings of consecutive successes

A sequence of independent Bernoulli trials is conducted. Let Nbe the number of trials required before observing a string of m successes. Using only elementary methods of probability theory, the expected value, variance and probability generating function of N are determined as functions of m and p, the probability of success on any given trial.     

Strings and Pions

We assume that QCD can be effectively described with stringlike variables and that this string is approximately represented by a Nambu–Goto or Polyakov string in some energy region. But to overcome the long-standing problems with the behavior of hadronic string amplitudes at low energies, the string is built over the correct (chirally noninvariant) QCD vacuum using a boundary interaction with background chiral fields associated with pions. Making this interaction compatible with the conformal symmetry of the string and with the unitarity constraint on chiral fields, we reconstruct the equations of motion for the latter and, furthermore, recover the Lagrangian of a nonlinear sigma model of pion interactions. We obtain the chiral structural constants of Gasser and Leutwyler in the next-to-leading order in the derivative expansion. The estimated coefficients fit the phenomenological values well.     

Topological Strings from Quantum Mechanics

We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi–Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov–Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local \({{\mathbb{P}}^2}\), local \({{\mathbb{P}}^1 \times {\mathbb{P}}^1}\) and local \({{\mathbb{F}}_1}\). In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi–Yau manifolds, in which the genus expansion emerges as a ’t Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background-independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry.     

Joint Invariant Signatures

A new, algorithmic theory of moving frames is applied to classify joint invariants and joint differential invariants of transformation groups. Equivalence and symmetry properties of submanifolds are completely determined by their joint signatures, which are parametrized by a suitable collection of joint invariants and/or joint differential invariants. A variety of fundamental geometric examples are developed in detail. Applications to object recognition problems in computer vision and the design of invariant numerical approximations are indicated. August 25, 1999. Final version received: May 3, 2000. Online publication: xxxx.     

Signatures et composantes connexes

Sans résumé     

Strings of special primes in arithmetic progressions

The Green–Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu’s Theorem to certain subsets of the primes such as primes of the form ${\lfloor{\pi n}\rfloor}$ ? π n ? and some of arithmetic density zero such as primes of the form ${\lfloor{n\log\log n}\rfloor}$ ? n log log n ? .     

Rotations of Periodic Strings and Short Superstrings

This paper presents two simple approximation algorithms for the shortest superstring problem with approximation ratios 2 ( ≈ 2.67) and( ≈ 2.596). The framework of our improved algorithms is similar to that of previous algorithms in the sense that they construct a superstring by computing some optimal cycle covers on the distance graph of the given strings and then break and merge the cycles to finally obtain a Hamiltonian path, but we make use of new bounds on the overlap between two strings. We prove that for each periodic semiinfinite string α = a₁a₂··· of periodq, there exists an integerk, such that forany(finite) stringsof periodpwhich isinequivalentto α, the overlap betweensand therotationα[k] = a_ka_{k + 1}··· is at mostp + q. Moreover, ifp ≤ q, then the overlap betweensand α[k] is not larger than (p + q). The bounds are tight. In the previous shortest superstring algorithmsp + qwas used as the standard (tight) bound on overlap between two strings with periodspandq.     

Signatures andS-discrete lattice-ordered groups

The central theme of this article is the approximation of lattice-ordered groups (l-groups) first by Specker groups and, subsequently, by the so-calledS-discretel-groups. The sense of these approximations is made precise via the notion of a signature, defined below, and by that ofa ^*-subgroups. Sample result: ifG is a projectablel-group then it has anl-subgroupH which is Specker and for which the mapPPH defines a boolean isomorphism between the algebras of polars ofG andH.Presented by L. Fuchs.This article was written while this author was a Stouffer Visiting Professor at the University of Kansas. He wishes to thank the members of the Mathematics Department of that institution for their hospitality.     

Matrix Models from Operators and Topological Strings

We propose a new family of matrix models whose 1/N expansion captures the all-genus topological string on toric Calabi–Yau threefolds. These matrix models are constructed from the trace class operators appearing in the quantization of the corresponding mirror curves. The fact that they provide a non-perturbative realization of the (standard) topological string follows from a recent conjecture connecting the spectral properties of these operators, to the enumerative invariants of the underlying Calabi–Yau threefolds. We study in detail the resulting matrix models for some simple geometries, like local \({\mathbb{P}^2}\) and local \({\mathbb{F}_2}\), and we verify that their weak ’t Hooft coupling expansion reproduces the topological string free energies near the conifold singularity. These matrix models are formally similar to those appearing in the Fermi-gas formulation of Chern–Simons matter theories, and their 1/N expansion receives non-perturbative corrections determined by the Nekrasov–Shatashvili limit of the refined topological string.     

Multivariate Signatures of Iterated Torus Links

Functional Analysis and Its Applications - We compute the multivariate signatures of any Seifert link (that is, a union of some fibers in a Seifert homology sphere), in particular, of the union of...     

Hawking Radiation of Relativistic Particles from Black Strings

Theoretical and Mathematical Physics - We study Hawking radiation of relativistic particles from uncharged and charged black strings in 3+1 dimensions in detail. We use the method of quantum...     

Complexity of Substring Search in a Set of Strings

The paper considers the problem of listing all occurrences of an arbitrary pattern in the strings from given set. We obtain a lower bound for time taken by search algorithms. We also obtain the order of memory amount required by algorithms with the best order search time.     

Closure and Convexity Results for Closed Relativistic Strings

We study various properties of closed relativistic strings. In particular, we characterize their closure under uniform convergence, extending a previous result by Y. Brenier on graph-like unbounded strings, and we discuss some related examples. Then we study the collapsing profile of uniformly convex planar strings which start with zero initial velocity, and we obtain a result analogous to the well-known theorem of Gage and Hamilton for the curvature flow of plane curves. We conclude the paper with the discussion of an example of weak Lipschitz evolution starting from the square in the plane.     

Pointwise Stabilization for a Chain of Coupled Vibrating Strings

For the system representing a chain of coupled vibrating strings,we show that the associated semigroup satisfies the assumptionof spectrum-determined growth, and hence obtain conditions forenergy to decay strongly or exponentially. We examine in detailthe three-string case, and our results include those obtainedby others for the two-string case. Permanent address: Beijing Institute of Information and Control,Beijing, China.