On the role of entanglement in Schrödinger’s cat paradox

In this paper we re-investigate the core of Schrödinger’s “cat paradox”. We argue that one has to distinguish clearly between superpositions of macroscopic cat states |?〉 + |?〉 and superpositions of entangled states |?, ↑〉 + |?, ↓〉 which comprise both the state of the cat (?=alive, ?=dead) and the radioactive substance (↑=not decayed, ↓=decayed). It is shown, that in the case of the cat experiment recourse to decoherence or other mechanisms is not necessary in order to explain the absence of macroscopic superpositions. Additionally, we present modified versions of two quantum optical experiments as experimenta crucis. Applied rigorously, quantum mechanical formalism reduces the problem to a mere pseudo-paradox.     

Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

We investigate the small ball problem for d-dimensional fractional Brownian sheets by functional analytic methods. For this reason we show that integration operators of Riemann–Liouville and Weyl type are very close in the sense of their approximation properties, i.e., the Kolmogorov and entropy numbers of their difference tend to zero exponentially. This allows us to carry over properties of the Weyl operator to the Riemann–Liouville one, leading to sharp small ball estimates for some fractional Brownian sheets. In particular, we extend Talagrand's estimate for the 2-dimensional Brownian sheet to the fractional case. When passing from dimension 1 to dimension d2, we use a quite general estimate for the Kolmogorov numbers of the tensor products of linear operators.     

The standard double bubble is the unique stable double bubble in

We prove that the only equilibrium double bubble in which is stable for fixed areas is the standard double bubble. This uniqueness result also holds for small stable double bubbles in surfaces, where it is new even for perimeter-minimizing double bubbles.

     

A Pieri-type formula for isotropic flag manifolds

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from the Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type (respectively, type ) Schubert polynomial by the Schur -polynomial (respectively, the Schur -polynomial ). Geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the infinite Coxeter group , identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show that most of these identities follow from the Pieri-type formula, and our analysis leads to a new partial order on the Coxeter group and formulas for many of these structure constants.

     

Lines tangent to

We show that for there are complex common tangent lines to general spheres in and that there is a choice of spheres with all common tangents real.

     

Dual decompositions of 4-manifolds

This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index . In dimensions results of Smale (trivial ) and Wall (general ) describe analogous decompositions up to diffeomorphism in terms of homotopy type of skeleta or chain complexes. In dimension 4 we show the same data determines decompositions up to 2-deformation of their spines. In higher dimensions spine 2-deformation implies diffeomorphism, but in dimension 4 the fundamental group of the boundary is not determined. Sample results: (1.5) Two 2-complexes are (up to 2-deformation) spines of a dual decomposition of the 4-sphere if and only if they satisfy the conclusions of Alexander-Lefshetz duality ( and ). (3.3) If is 1-connected then there is a ``pseudo' handle decomposition without 1-handles, in the sense that there is a pseudo collar (a relative 2-handlebody with spine that 2-deforms to ) and is obtained from this by attaching handles of index .

     

Construction of the Incipient Infinite Cluster for Spread-out Oriented Percolation Above 4 + 1 Dimensions

 We construct the incipient infinite cluster measure (IIC) for sufficiently spread-out oriented percolation on ℤ ^d × ℤ₊, for d +1 > 4+1. We consider two different constructions. For the first construction, we define ℙ ⁿ (E) by taking the probability of the intersection of an event E with the event that the origin is connected to (x,n)  ℤ ^d × ℤ₊, summing this probability over x  ℤ ^d , and normalising the sum to get a probability measure. We let n → ∞ and prove existence of a limiting measure ℙ_∞, the IIC. For the second construction, we condition the connected cluster of the origin in critical oriented percolation to survive to time n, and let n → ∞. Under the assumption that the critical survival probability is asymptotic to a multiple of n ⁻¹, we prove existence of a limiting measure ℚ_∞, with ℚ_∞ = ℙ_∞. In addition, we study the asymptotic behaviour of the size of the level set of the cluster of the origin, and the dimension of the cluster of the origin, under ℙ_∞. Our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super-Brownian motion, for d+1 > 4+1. Received: 13 December 2001 / Accepted: 11 July 2002 Published online: 29 October 2002 RID="*" ID="*" Present address: Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: rhofstad@win.tue.nl     

Polymer rings and chains consisting of doubly silyl-bridged metallocenes

With the formation of novel organometallic macromolecules in mind, the polycondensation of transition metal ions and bridged cyclopentadienyl ligands was studied. To this end solvated salts MX2 (M = Fe, Ni, and Cr) were treated with a ligand that consisted of two doubly silyl-bridged cyclopentadienyl anions. For M = Fe and diluted solutions a series of rings Oi was obtained that consisted of a minimum of six (O6) and up to 17 (O17) ferrocene moieties in the backbone. They were separated partly by medium pressure liquid chromatography. The macrocycles were established by high-resolution MALDI-TOF mass spectroscopy which also yielded the molecular weight, the polydispersion, and the mean ring size, chi n, of the mixture of reaction products. When the reaction temperature was decreased from 25 degrees C to -20 degrees C, chi n increased from 8.1 to 10.8. Ferrocene-containing chains, lambda j, with 2 < or = j < or = 12 were obtained in addition to rings in the presence of water; the terminal groups were cyclopentadiene moieties. The reaction of two ferrocene-fused cyclopentadienyl anions with [FeCl2(thf)1.5] gave chains consisting of exclusively uneven numbers of ferrocenes. For M = Ni and Cr the formation of doubly silyl-bridged nickelocenes and chromocenes was proven by NMR spectroscopy. MALDI-TOF mass spectroscopy showed nickelocene-containing chains accompanied by some rings. For M = Fe the H,H-DQF COSY spectra established the structure of O7, O8, and O9. The oxidation of the ferrocene-containing ring O7 with I2, NOPF6, and AgPF6 gave ionic species [O7]n+ which suffered from low stability. The ring-closing reaction is discussed, and the relative abundance of the various rings is related to MNDO calculations.