Two photon absorption and coherent control with broadband down-converted light

We experimentally demonstrate two-photon absorption with broadband down-converted light (squeezed vacuum). Although incoherent and exhibiting the statistics of a thermal noise, broadband down-converted light can induce two-photon absorption with the same sharp temporal behavior as femtosecond pulses, while exhibiting the high spectral resolution of the narrow band pump laser. Using pulse-shaping methods, we coherently control two-photon absorption in rubidium, demonstrating spectral and temporal resolutions that are 3-5 orders of magnitude below the actual bandwidth and temporal duration of the light itself. Such properties can be exploited in various applications such as spread-spectrum optical communications, tomography, and nonlinear microscopy.     

Near Optimal Separation Of Tree-Like And General Resolution

We present the best known separation between tree-like and general resolution, improving on the recent exp(n ) separation of [2]. This is done by constructing a natural family of contradictions, of size n, that have O(n)-size resolution refutations, but only exp((n/log n))- size tree-like refutations. This result implies that the most commonly used automated theorem procedures, which produce tree-like resolution refutations, will perform badly on some inputs, while other simple procedures, that produce general resolution refutations, will have polynomial run-time on these very same inputs. We show, furthermore that the gap we present is nearly optimal. Specifically, if S (S _T ) is the minimal size of a (tree-like) refutation, we prove that S _T = exp(O(S log log S/log S)).* This research was supported by Clore Foundation Doctoral Scholarship. Research supported by NSF Award CCR-0098197 and USA–Israel BSF Grant 97-00188. This research was supported by grant number 69/96 of the Israel Science Foundation, founded by the Israel Academy for Sciences and Humanities.     

Catalytic asymmetric hydrosilylation of acetophenone with new chiral thiourea ligands containing the (S)-α-phenylethyl group

New chiral thioureas 1–8 containing 1,2-ethylendiamine or trans-1,2-diaminocyclohexane as the carbon skeleton, and containing an (S)-α-phenylethyl group have been prepared (79–98% yield). Thioureas 1–8 were used as ligands for the zinc-based catalyzed asymmetric hydrosilylation of acetophenone with polymethylhydrosiloxane (PMHS). The best result was achieved with monothiourea 1 (up to 75% ee), in toluene and a catalyst load of 5 mol %.     

Production under periodic demand update prior to a single selling season: A decomposition approach

This paper focuses on dynamic, continuous-time production control problems in the fashion industry. Similar to the classical news-vendor problem, we consider a single product-type and the cumulative demand for items is not known until the end of the production horizon and therefore must be forecasted. Since there are periodic updates before a single selling season, actual demand during a period of time can only be determined by the end of the period. If the overall demand is overestimated, excessive inventory holding and production costs are paid and surpluses are sold at low prices at the end of the production horizon. If it is under-estimated, then sales are lost. The objective is to dynamically determine production orders which minimize overall expected costs. Since the optimal feedback for such a problem is characterized by thresholds evolving with time and system states, there is a significant computational burden in determining them. With the aid of the variational analysis and a decomposition, we derive a closed-form solution for the thresholds. A numerical study carried out to compare the decomposition and straightforward simulation-based solutions indicates the high accuracy of the suggested approach while the computational burden is dramatically reduced.     

On rank vs. communication complexity

This paper concerns the open problem of Lovász and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We first give an example exhibiting the largest gap known. We then prove two related theorems.A preliminary version of this paper appeared in [10].This work was supported by USA-Israel BSF grant 92-00043 and by a Wolfeson research award administered by the Israeli Academy of Sciences.This work was supported by USA-Israel BSF grant 92-00106 and by a Wolfeson research award administered by the Israeli Academy of Sciences.     

Perturbed projections and subgradient projections for the multiple-sets split feasibility problem

We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. By casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case. We further derive and analyze a perturbed projection method for the multiple-sets split feasibility problem and, additionally, furnish alternative proofs to two known results.     

Relations among five radii of circles in a triangle,its sides and other segments

Students use GeoGebra to explore the mathematical relations among different radii of circles in a triangle (circumcircle, incircle, excircles) and the sides and other segments in the triangle. The more formal mathematical development of the relations that follows the explorations is based on known geometrical properties, different formulas relating the radii to the sides and the inequalities between the different averages. The activities described were conducted with pre-service teachers of mathematics, with empirical investigation of the relations using dynamical geometry software, and formal presentation of proofs.     

Dynamic investigation of triangles inscribed in a circle,which tend to an equilateral triangle

We present a geometrical investigation of the process of creating an infinite sequence of triangles inscribed in a circle, whose areas, perimeters and lengths of radii of the inscribed circles tend to a limit in a monotonous manner.

First, using geometrical software, we investigate four theorems that represent interesting geometrical properties, after which we present formal proofs that rest on a combination between different fields of mathematics: trigonometry, algebra and geometry, and the use of the concept of standard deviation that is taken from statistics.