An Enantiospecific Formal Total Synthesis of the 5-8-5 Tricyclic Diterpene ent-Fusicoauritone

An enantiospecific formal total synthesis of the 5-8-5 tricyclic diterpene fusicoauritone has been accomplished, starting from 5-isopropyl-2-methylcyclopent-1-enemethanol [available in three steps from (R)-dihydrolimonene] employing two ring-closing-metathesis reactions for the construction of the eight- and five-membered rings.

Some questions on integral geometry on noncompact symmetric spaces of higher rank

Let be a noncompact symmetric space of higher rank. We consider two types of averages of functions: one, over level sets of the heat kernel on and the other, over geodesic spheres. We prove injectivity results for functions in which extend the results in Pati and Sitaram (Sankya Ser A 62:419-424, 2000).

REPRODUCING KERNEL FOR A CLASS OF WEIGHTED BERGMAN SPACES ON THE SYMMETRIZED POLYDISC

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions includes the Szego and the Bergman kernel on the symmetrized polydisc.

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order H-m (H-n), m is an element of N-n, under the heat kernel transform on H-n, using direct sum and direct integral of Bergmann spaces and certain unitary representations of H-n which can be realized on the Hilbert space of Hilbert-Schmidt operators on L-2 (R-n). We also show that the image of Sobolev space of negative order H-s (H-n), s(> 0) is an element of R is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on H-n under the heat kernel transform. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Lipschitz Correspondence between Metric Measure Spaces and Random Distance Matrices

Given a metric space with a Borel probability measure, for each integer N, we obtain a probability distribution on N x N distance matrices by considering the distances between pairs of points in a sample consisting of N points chosen independently from the metric space with respect to the given measure. We show that this gives an asymptotically bi-Lipschitz relation between metric measure spaces and the corresponding distance matrices. This is an effective version of a result of Vershik that metric measure spaces are determined by associated distributions on infinite random matrices.

RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES

In the study of holomorphic maps, the term ``rigidity'' refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one holomorphic map f :Y -> X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products X = X-1 x X-2 and Y = Y-1 x Y-2 of compact connected complex manifolds. When X-1 is a Riemann surface of genus >= 2, we show that any non-constant holomorphic map F:Y -> X is of a special form.

End point estimates for Radon transform of radial functions on non-Euclidean spaces

We prove end point estimate for Radon transform of radial functions on affine Grasamannian and real hyperbolic space. We also discuss analogs of these results on the sphere.

Formal Synthesis of Actin Binding Macrolide Rhizopodin

Formal synthesis, of an actin binding macrolide rhizopodin was achieved in 19 longest linear steps. The key features of the synthesis include a stereoselective Mukaiyama aldol reaction, dual role of a Nagao auxiliary (first, as a chiral auxiliary of choice for installing hydroxy centers and, later, as an acylating agent to form an amide bond with an amino alcohol), late stage oxazole formation, and Stille coupling reactions.

Unexpected mechanism for formal 2+2] cycloadditions of metallacyclocumulenes

The formation of radialene complex 6M proceeds through a three-membered metallacyclopropene complex 7M, contrary to the prevailing notion of simple dimerization of metallacyclocumulene 1M. The 1M-7M equilibrium, which is predominantly governed by the size-dependent ligand binding of the metal atoms, plays a decisive role in the chemistry of Cp2M-ligand complexes. This size dependency is further fine-tuned by the substituents on the substrates and helps in exploiting these classes of metallacycles to generate new chemistry.

Enantiospecific Formal Total Synthesis of Iriomoteolide 3a

A formal total synthesis of the marine macrolide iriomoteolide3a is described. Salient features of the synthesis include the elaboration of a -keto phosphonate derived from D-(-)-tartaric acid and the extension of a chiral butyrolactone derived from L-glutamic acid. Ring-closing metathesis is employed to construct the macrolactone core of the natural product.

Using formal reasoning on a model of tasks for FreeRTOS

FreeRTOS is an open-source real-time microkernel that has a wide community of users. We present the formal specification of the behaviour of the task part of FreeRTOS that deals with the creation, management, and scheduling of tasks using priority-based preemption. Our model is written in the Z notation, and we verify its consistency using the Z/Eves theorem prover. This includes a precise statement of the preconditions for all API commands. This task model forms the basis for three dimensions of further work: (a) the modelling of the rest of the behaviour of queues, time, mutex, and interrupts in FreeRTOS; (b) refinement of the models to code to produce a verified implementation; and (c) extension of the behaviour of FreeRTOS to multi-core architectures. We propose all three dimensions as benchmark challenge problems for Hoare's Verified Software Initiative.

Organocatalytic Enantioselective Formal C(sp(2))-H Alkylation

An organocatalytic enantioselective formal C(sp(2))-H alkylation is reported. This alkylative desymmetrization of prochiral 2,2-disubstituted cyclopentene-1,3-dione is catalyzed by a bifunctional tertiary aminourea derivative, utilizes air-stable and inexpensive nitroalkanes as the alkylating agents, and delivers synthetically versatile five-membered carbocycles containing an all-carbon quaternary stereogenic center remote from the reaction site in excellent enantioselectivity.

A versatile, RCM based approach to eudesmane and dihydroagarofuran sesquiterpenoids from (-)-carvone: a formal synthesis of (-)-isocelorbicol

An enantiospecific and diversity oriented approach to a range of functionalized eudesmane, nor-, iso-, and dihydroagarofuran frameworks from (-)-carvone is delineated. The cornerstone of this approach is the installation of the quaternary carbon center through reductive opening of the carvone epoxide and setting-up of RCM reaction to generate the bicyclic eudesmane framework. Various options like carbocation mediated oxycyclization and intramolecular hydroxy directed epoxide opening have been explored for the construction of the bridged tetrahydrofuran moiety. Among the several eudesmane and dihydroagarofurans accessed during the present study, one has been previously elaborated to iso-celorbicol, thus constituting its formal synthesis. (C) 2015 Elsevier Ltd. All rights reserved.

Formal synthesis of degraded sterol (+)-aplykurodinone-1

The formal synthesis of aplykurodinone-1 is accomplished starting from a suitably functionalized bicyclic lactone having the requisite cis-fused ring junction with a quaternary chiral center that was assembled following a Cp2TiCl-mediated radical cyclization protocol. Our synthetic route further elaborates implementation of Grubbs ring closing metathesis (RCM), Eschenmoser-Claisen rearrangement and iodo-lactonization reactions for the synthesis of the final tricyclic precursor of the target molecule. (C) 2015 Elsevier Ltd. All rights reserved.

Exact Phase Retrieval in Principal Shift-Invariant Spaces

We address the problem of phase retrieval from Fourier transform magnitude spectrum for continuous-time signals that lie in a shift-invariant space spanned by integer shifts of a generator kernel. The phase retrieval problem for such signals is formulated as one of reconstructing the combining coefficients in the shift-invariant basis expansion. We develop sufficient conditions on the coefficients and the bases to guarantee exact phase retrieval, by which we mean reconstruction up to a global phase factor. We present a new class of discrete-domain signals that are not necessarily minimum-phase, but allow for exact phase retrieval from their Fourier magnitude spectra. We also establish Hilbert transform relations between log-magnitude and phase spectra for this class of discrete signals. It turns out that the corresponding continuous-domain counterparts need not satisfy a Hilbert transform relation; notwithstanding, the continuous-domain signals can be reconstructed from their Fourier magnitude spectra. We validate the reconstruction guarantees through simulations for some important classes of signals such as bandlimited signals and piecewise-smooth signals. We also present an application of the proposed phase retrieval technique for artifact-free signal reconstruction in frequency-domain optical-coherence tomography (FDOCT).