Projective maps of rank ≥2 are strongly projective

We investigate the structure of projective maps between manifolds with linear connections, showing in particular that a projective map on a connected manifold that attains a rank ≥2 at some point is strongly projective     

Dynamics of commuting rational maps on Berkovich projective space over ℂ p

We study the dynamics of commuting rational maps with coefficients in Cp. By lifting the dynamics from P1（Cp） to Berkovich projective space P1 Berk, we prove that two nonlinear commuting maps have the same Berkovich Julia set and the same canonical measure. As a consequence, two nonlinear commuting maps with coefficient in Cp have the same classical Julia set. We also prove that they have the same pre-periodic Berkovich Fatou components.     

Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane

Let \( E \subset \mathbb{C} \) be a compact set, \( g:\mathbb{C} \to \mathbb{C} \) be a K-quasiconformal map, and let 0 < t < 2. Let \( {\mathcal{H}^t} \) denote t-dimensional Hausdorff measure. Then

$ {\mathcal{H}^t}(E) = 0\quad \Rightarrow \quad {\mathcal{H}^{t'}}\left( {gE} \right) = 0,\quad t' = \frac{{2Kt}}{{2 + \left( {K - 1} \right)t}}. $

This is a refinement of a set of inequalities on the distortion of Hausdorff dimensions by quasiconformal maps proved by K. Astala in [2] and answers in the positive a conjecture of K. Astala in op. cit.     

The Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors

With a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincaré series of this filtration turns out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincaré series of the corresponding multi-index filtration similar to the one associated with plane germs. Here we show that the Poincaré series of a set of divisorial valuations on the ring of germs of functions of two variables defines “the topology of the set of the divisors” in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which shows that the Alexander polynomial is equivalent to the embedded topology.     

Simultaneous siting and sizing of distribution centers on a plane

The benefits of simultaneous consideration of siting and sizing of distribution centers have been well acknowledged in supply chain design. Most formulations assume that the potential DC sites are known and the decision on location is to select sites from the finite potential DC sites. However, the quality of this discrete version problem depends on the selection of potential DC sites. In this paper we present a planar version of the problem, which assumes that there is no a priori knowledge of DC sites and DCs can be located anywhere in the plane. The goal of the problem is to simultaneously find locations and sizing of DC sites. The solution of the planar problem provides a lower bound for the discrete problem. The objective of the problem is to minimize the total of inbound and outbound transportation costs and distribution center construction costs—which include its fixed charge cost and concave sizing cost. The problem is initially formulated as a nonlinear programming model. We then reformulate it as a set covering problem after establishing certain key properties. A greedy drop heuristic and a column generation heuristic are developed to solve the problem. Computational experiments are provided.     

Quantum cohomology of the Hilbert scheme of points in the plane

We determine the ring structure of the equivariant quantum cohomology of the Hilbert scheme of points of ℂ². The operator of quantum multiplication by the divisor class is a nonstationary deformation of the quantum Calogero-Sutherland many-body system. A relationship between the quantum cohomology of the Hilbert scheme and the Gromov-Witten/Donaldson-Thomas correspondence for local curves is proven.     

The evaluation of Tornheim double sums. Part 2

We provide an explicit formula for the Tornheim double series T(a,0,c) in terms of an integral involving the Hurwitz zeta function. For integer values of the parameters, a=m, c=n, we show that in the most interesting case of even weight N:=m+n the Tornheim sum T(m,0,n) can be expressed in terms of zeta values and the family of integrals

The braid groups of the projective plane and the Fadell–Neuwirth short exact sequence

We study the pure braid groups of the real projective plane , and in particular the possible splitting of the Fadell–Neuwirth short exact sequence , where n ≥ 2 and m ≥ 1, and p _* is the homomorphism which corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration of configuration spaces. Van Buskirk proved (1966, Trans. Am. Math. Soc., 122:81–97) that p and p _* admit a section if n = 2 and m = 1. Our main result in this paper is to prove that there is no section if n ≥ 3. As a corollary, it follows that n = 2 and m = 1 are the only values for which a section exists. As part of the proof, we derive a presentation of : this appears to be the first time that such a presentation has been given in the literature.      

Fredholm properties of Riemannian exponential maps on diffeomorphism groups

We prove that exponential maps of right-invariant Sobolev H ^r metrics on a variety of diffeomorphism groups of compact manifolds are nonlinear Fredholm maps of index zero as long as r is sufficiently large. This generalizes the result of Ebin et al. (Geom. Funct. Anal. 16, 2006) for the L ² metric on the group of volume-preserving diffeomorphisms important in hydrodynamics. In particular, our results apply to many other equations of interest in mathematical physics. We also prove an infinite-dimensional Morse Index Theorem, settling a question raised by Arnold and Khesin (Topological methods in hydrodynamics. Springer, New York, 1998) on stable perturbations of flows in hydrodynamics. Finally, we include some applications to the global geometry of diffeomorphism groups.     

New identities involving sums of the tails related to real quadratic fields

In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of $\mathbb {Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ . Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.     

On the representation of large integers as sums of four almost equal squares of primes

In this paper, we are able to establish two localized results on the conjecture that each large integer congruent to 4 modulo 24 can be written as the sum of four squares of primes. The proof is based on the new estimates for exponential sums over primes in short intervals and a technique to get the asymptotic formula on the enlarged major arcs in the circle method. This work is supported by the National Natural Science Foundation of China (Grant Nos. 10701048 and 10771127).     

Rationality of the moduli spaces of plane curves of sufficiently large degree

We prove that the moduli space of plane curves of degree d is rational for all sufficiently large d.     

Another proof of Grothendieck’s theorem on the splitting of vector bundles on the projective line

This note contains another proof of Grothendieck‘s theorem on the splitting of vector bundles on the projective line over a field k. Actually the proof is formulated entirely in the classical terms of a lattice \(\Lambda \cong k[T]^d\), discretely embedded into the vector space \(V \cong K_\infty ^d\), where \(K_\infty \cong k((1/T))\) is the completion of the field of rational functions k(T) at the place \(\infty \) with the usual valuation.     

Quantization of Pseudo-differential Operators on the Torus

Pseudo-differential and Fourier series operators on the torus \mathbbTⁿ=(\BbbR/2p\BbbZ)ⁿ{{\mathbb{T}}^{n}}=(\Bbb{R}/2\pi\Bbb{Z})^{n} are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on L ² under certain conditions on their phases and amplitudes.     

Vector Interpretation of the Matrix Orthogonality on the Real Line

In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients that do not obey to any type of symmetry. In this sense the vectorial reinterpretation allows us to study a non-symmetric case of the matrix orthogonality. We also prove that our systems of polynomials are indeed orthonormal with respect to a complex measure of orthogonality. Approximation problems of Hermite-Padé type are also discussed. Finally, a Markov’s type theorem is presented.     

Efficient simulation of tail probabilities of sums of correlated lognormals

We consider the problem of efficient estimation of tail probabilities of sums of correlated lognormals via simulation. This problem is motivated by the tail analysis of portfolios of assets driven by correlated Black-Scholes models. We propose two estimators that can be rigorously shown to be efficient as the tail probability of interest decreases to zero. The first estimator, based on importance sampling, involves a scaling of the whole covariance matrix and can be shown to be asymptotically optimal. A further study, based on the Cross-Entropy algorithm, is also performed in order to adaptively optimize the scaling parameter of the covariance. The second estimator decomposes the probability of interest in two contributions and takes advantage of the fact that large deviations for a sum of correlated lognormals are (asymptotically) caused by the largest increment. Importance sampling is then applied to each of these contributions to obtain a combined estimator with asymptotically vanishing relative error.     

On the number of points of bounded height¶on arithmetic projective spaces

Let K be a number field and its ring of integers. Let be a Hermitian vector bundle over . In the first part of this paper we estimate the number of points of bounded height in (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙ ^N _K blown up at a linear subspace of codimension two. Received: 20 February 1998 / Revised version: 9 November 1998     

Optimisation of maintenance scheduling strategies on the grid

The emerging paradigm of Grid Computing provides a powerful platform for the optimisation of complex computer models, such as those used to simulate real-world logistics and supply chain operations. This paper introduces a Grid-based optimisation framework that provides a powerful tool for the optimisation of such computationally intensive objective functions. This framework is then used in the optimisation of maintenance scheduling strategies for fleets of aero-engines, a computationally intensive problem with a high-degree of stochastic noise, achieving substantial improvements in the execution time of the algorithm.     

The spectrum of the twisted Dirac operator on K?hler submanifolds of the complex projective space

We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on K?hler submanifolds in K?hler manifolds carrying K?hlerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding ${{\mathbb C}P^d\rightarrow {\mathbb C}P^n}$ in order to test the sharpness of the upper bounds.