On the one-commodity pickup-and-delivery traveling salesman problem with stochastic demands

This paper studies how to set the vehicle capacity for traveling Salesman Problems where some of the customer demands are stochastic. The analyses are done for the one-commodity pickup-and-delivery TSP, as this problem also includes the setting of the initial load. The paper first considers feasibility issues. This includes finding the smallest vehicle capacity and some initial load such that a given tour is feasible for all scenarios. Different variants are considered as a function of the time when information becomes available. The paper then analyzes the case where some penalties are paid for routing a tour unable to handle customer demands. Various types of penalties are considered. The paper studies properties of the minimal expected penalty of a given tour, which are then used to provide approaches to find near-optimal tours. Computational results are presented.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Invariants of finite aspherical CW-complexes

Let X be a finite aspherical CW-complex whose fundamental group π ₁(X) possesses a subnormal series with a non-trivial elementary amenable group G ₀. We investigate the L ²-invariants of the universal covering of such a CW-complex X. The main result is the proof of the vanishing of the L ²-torsion under the condition that π ₁(X) has semi-integral determinant. We further show that the Novikov–Shubin invariants are positive.     

Irreducible algebraic integers in short intervals

We study distribution of irreducible algebraic integers in short intervals and prove that if the class number of an algebraic number field K exceeds 2, every interval of the form (x, x + x ^θ) with a fixed θ > 1/2 contains absolute value of the norm of an irreducible algebraic integer from K provided x ≥ x ₀. The constant x ₀ effectively depends on K and θ. The author was supported in part by the grant no. N N201 1482 33 from the Polish Ministry of Science and Higher Education.     

Non-annulation effective et positivité locale des fibrés en droites amples adjoints

We show how to use effective non-vanishing to prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. We prove the effective non-vanishing conjecture of Ionescu–Kawamata in dimension 3 in the case of line bundles of “high” volume.     

Dimensional asymptotics of determinants on <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>, and proof of Bär–Schopka’s conjecture

We study the dimensional asymptotics of the effective actions, or functional determinants, for the Dirac operator D and Laplacians \({\Delta +\beta R}\) on round S ⁿ . For Laplacians the behavior depends on “the coupling strength” β, and one cannot in general expect a finite limit of \({\zeta'(0)}\) , and for the ordinary Laplacian, \({\beta=0}\) , we prove it to be \({+\infty}\) , for odd dimensions. For the Dirac operator, Bär and Schopka conjectured a limit of unity for the determinant (Bär and Schopka, Geometric Analysis and Nonlinear PDEs, pp. 39–67, 2003), i.e.

$\lim_{n\to\infty} {\rm{det}}(D, S^n_{\rm{can}})=1.$

We prove their conjecture rigorously, giving asymptotics, as well as a pattern of inequalities satisfied by the determinants. The limiting value of unity is a virtue of having “enough scalar curvature” and no kernel. Thus, for the important (conformally covariant) Yamabe operator, \({\beta=(n-2)/(4(n-1))}\) , the determinant tends to unity. For the ordinary Laplacian it is natural to rescale spheres to unit volume, since

$\lim_{k\to\infty} {\rm{det}}(\Delta,S^{2k+1}_{\rm rescaled})=\frac{1}{2\pi e}$

.

     

A characterization of domains in <Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript> with noncompact automorphism group

Let D be a bounded domain in C ² with a non-compact group of holomorphic automorphisms. Model domains for D are obtained under the hypotheses that at least one orbit accumulates at a boundary point near which the boundary is smooth, real analytic and of finite type. The author was supported by DST (India) Grant No.: SR/S4/MS-283/05 and in part by a grant from UGC under DSA-SAP, Phase IV.     

Motives of some Fano varieties

We study the Fano varieties of projective k-planes lying in hypersurfaces and investigate the associated motives. The first author is partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The second author is partially supported by TüBİTAK-BDP funds and Bilkent University research development funds.     

A splitting theorem for holomorphic Banach bundles

This paper is motivated by Grothendieck’s splitting theorem. In the 1960s, Gohberg generalized this to a class of Banach bundles. We consider a compact complex manifold X and a holomorphic Banach bundle E → X that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that E splits into the sum of a finite rank bundle and a trivial bundle, provided .     

Affine restriction for radial surfaces

Suppose dμ is affine surface measure on a convex radial surface Γ(x) = (x, γ(|x|)), a ≤ |x| < b, in . Under appropriate smoothness and growth conditions on γ, we prove and Fourier restriction estimates for Γ.     

Lower bounds on the canonical height associated to the morphism $${\phi(z)= z^d+c}$$

A lower bound is obtained of the canonical height associated to the morphism evaluated at wandering points α. The lower bound is of the form Ch(c), for some constant C depending on the the number of primes of bad reduction for , and the degree of the number field . The author was supported by a PDF grant from NSERC.