Search results for "symmetrization"
showing 10 items of 37 documents
Hessian equations and symmetrization
2005
In this paper we state same comparisons results for solutions to Hessian type equations in dimension n> 2. These results involve convenient rearrangements of solutions that preserve suitable cross-sectional measures of their level sets.
Comparison results for Hessian equations via symmetrization
2007
where the λ’s are the eigenvalues of the Hessian matrix D2u of u and Sk is the kth elementary symmetric function. For example, for k = 1, S1(Du) = 1u, while, for k = n, Sn(D 2u) = detD2u. Equations involving these operators, and some more general equations of the form F(λ1, . . . , λn) = f in , (1.2) have been widely studied by many authors, who restrict their considerations to convenient cones of solutions with respect to which the operator in (1.2) is elliptic. Following [25] we define the cone 0k of ellipticity for (1.1) to be the connected component containing the positive cone 0 = {λ ∈ R : λi > 0 ∀i = 1, . . . , n} of the set where Sk is positive. Thus 0k is an open, convex, symmetric…
A symmetrization result for Monge–Ampère type equations
2007
In this paper we prove some comparison results for Monge–Ampere type equations in dimension two. We also consider the case of eigenfunctions and we derive a kind of “reverse” inequalities. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Asymmetric Synthesis of Spirocyclic β-Lactams through Copper-Catalyzed Kinugasa/Michael Domino Reactions
2018
The first copper-catalyzed highly chemo-, regio-, diastereo-, and enantioselective Kinugasa/Michael domino reaction for the desymmetrization of prochiral cyclohexadienones is described. In the presence of a chiral copper catalyst, alkyne-tethered cyclohexadienones couple with nitrones to generate the chiral spirocyclic lactams with excellent stereoselectivity (up to 97 % ee, >20:1 dr). The new method provides direct access to versatile highly functionalized spirocyclic β-lactams possessing four contiguous stereocenters, including one quaternary and one tetra-substituted stereocenter.
Recent Advances in Enantioselective Desymmetrizations of Prochiral Oxetanes
2021
Abstract Strain relief of oxetanes offers a plethora of opportunities for the synthesis of chiral alcohols and ethers. In this context, enantioselective desymmetrization has been identified as a powerful tool to construct molecular complexity and this has led to the development of elegant strategies on the basis of transition metal, Lewis acid, and Brønsted acid catalysis. This review highlights recent examples that harness the inherent reactivity of prochiral oxetanes and offers an outlook on the immense possibilities for synthetic application.
Asymmetric Intramolecular Aza-Michael Reaction in Desymmetrization Processes. Total Synthesis of Hippodamine and epi-Hippodamine
2015
The use of chiral sulfinyl amines both as nucleophilic nitrogen sources and chiral inducers has been described for the first time in a desymmetrization-type process involving an intramolecular aza-Michael reaction. The resulting product was employed as an advanced intermediate in the total synthesis of the natural product hippodamine and epi-hippodamine, taking advantage of the special symmetry of these molecules. In addition, this is the first asymmetric total synthesis of epi-hippodamine.
Sharp estimates for eigenfunctions of a Neumann problem
2009
In this paper we provide some bounds for the eigenfunctions of the Laplacian with homogeneous Neumann boundary conditions in a bounded domain Ω of R^n. To this aim we use the so-called symmetrization techniques and the obtained estimates are asymptotically sharp, at least in the bidimensional case, when the isoperimetric constant relative to Ω goes to 0.
Orientation of O(3) and SU(2)⊗CI representations in cubic point groups (Oh,Td) for application to molecular spectroscopy
2003
Abstract We propose a detailed method for the symmetrization of the standard O (3) or SU (2)⊗ C I basis | j τ , m 〉 ( τ = g or u ) into the O h or T d point group. This is realized by means of an orientation matrix called G . The oriented basis obtained in this way allows matrix element calculations for rovibronic spectroscopic problems concerning octahedral or tetrahedral molecules. Particular attention has been put on careful phase choices. A numerical calculation of all the G matrix elements for both integer and half-integer j values up to 399/2 has been performed. Such high angular momentum values are necessary for the case of heavy molecules with high rotational excitation. To calculat…
A scheme for entanglement extraction from a solid
2006
Some thermodynamical properties of solids, such as heat capacity and magnetic susceptibility, have recently been shown to be linked to the amount of entanglement in a solid. However this entanglement may appear a mere mathematical artifact of the typical symmetrization procedure of many-body wave function in solid state physics. Here we show that this entanglement is physical demonstrating the principles of its extraction from a typical solid state system by scattering two particles off the system. Moreover we show how to simulate this process using present-day optical lattices technology. This demonstrates not only that entanglement exists in solids but also that it can be used for quantum…
On the Lindemann criterion for quantum clusters at very low temperature.
2011
The Lindemann criterion to discern the solid-like or liquid-like nature of a quantum cluster at T = 0 is discussed. A critical analysis of current Lindemann parameters is presented and a new parameter is proposed that is appropriate to study quantum clusters made of identical particles. A simple model wave function is introduced to fix the range of variation of these parameters. The model presents two extreme limits that correspond to either a liquid-like or a solid-like system; besides, it fulfills the Bose symmetry and also permits evaluations without symmetrization. Variational and diffusion Monte Carlo calculations are also performed for clusters of spinless bosons interacting through L…