All-optical measurement-based quantum-information processing in quantum dots

Parity measurements on qubits can generate the entanglement resource necessary for scalable quantum computation. Here we describe a method for fast optical parity measurements on electron spin qubits within coupled quantum dots. The measurement scheme, which can be realized with existing technology, consists of the optical excitation of excitonic states followed by monitored relaxation. Conditional on the observation of a photon, the system is projected into the odd/even-parity subspaces. Our model incorporates all the primary sources of error, including detector inefficiency, effects of spatial separation and nonresonance of the dots, and also unwanted excitations. Through an analytical treatment we establish that the scheme is robust to such effects. Two applications are presented: a realization of a controlled-NOT gate, and a technique for growing large scale graph states.     

Common Fixed Point and Invariant Approximation Results for Gregus Type I-Contractions

A fixed point theorem of Ciric, Diviccaro et al., Fisher and Sessa, Gregus, Jungck, and Mukherjee and Verma is generalized to weakly compatible maps. As applications, common fixed point and approximation results for Gregus type I-contractions are obtained. Our results unify and generalize various known results to the more general classes of noncommuting mappings.     

2-Selmer groups and the Birch-Swinnerton-Dyer Conjecture for the congruent number curves

We take an approach toward counting the number of integers n for which the curve En: y²=x³−n²x has 2-Selmer groups of a given size. This question was also discussed in a pair of papers by Roger Heath-Brown. In contrast to earlier work, our analysis focuses on restricting the number of prime factors of n. Additionally, we discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the “independence” of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.     

Linear relations among Poincaré series via harmonic weak Maass forms

We discuss the problem of the vanishing of Poincaré series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan??s mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincaré series.     

Cobalt‐Catalyzed Aminocarbonylation of Alkyl Tosylates: Stereospecific Synthesis of Amides

Metal‐catalyzed aminocarbonylation is a standard approach for installing amide functionality in chemical synthesis. Despite broad application of this transformation using aryl or vinyl electrophiles, there are few examples involving unactivated aliphatic substrates. Furthermore, there are no stereocontrolled aminocarbonylations of alkyl electrophiles known. Herein, we report a stereospecific aminocarbonylation of unactivated alkyl tosylates for the synthesis of enantioenriched amides. This cobalt‐catalyzed transformation uses a remarkably broad range of amines and proceeds with excellent stereospecificity and chemoselectivity.     

Toward optimized high-relaxivity MRI agents: the effect of ligand basicity on the thermodynamic stability of hexadentate hydroxypyridonate/catecholate gadolinium(III) complexes

The thermodynamic stabilities of the Gd(III) complexes of five hexadentate ligands, which incorporate the 2,3-dihydroxyterephthalamide and 2,3-hydroxypyridonate chelating moieties, have been determined by potentiometric and spectrophotometric titration. The ligands were chosen to span a range of basicities while maintaining a similar tripodal structural motif, facilitating a study of the effect of ligand basicity on the thermodynamic stability of the Gd(III) complexes. The relative stability of the five complexes is found to be highly pH dependent, with the most acidic ligands forming the most stable complexes at low pH and more basic ligands forming more stable complexes at high pH. The most stable Gd(III) complex at a physiological pH of 7.4 is formed with a ligand of intermediate basicity and is of stability comparable to that of Gd(III) complexes that feature eight-coordinate amino-carboxylate ligands and are currently used as magnetic resonance imaging contrast agents in diagnostic medicine. A single-crystal X-ray structure of the intermediate compound 3-hydroxy-6-methyl-2-oxo-1,2-dihydropyridine-4-carboxylic acid ethyl ester is described: This compound crystallizes in the triclinic space group P1 with a = 7.4801(3) A, b = 8.0671(3) A, c = 8.3457(4) A, alpha = 72.242(2) degrees, beta = 80.693(2) degrees, gamma = 69.943(3) degrees, V = 449.60(3) A(3), Z = 2, and R = 0.042.