The isoperimetric constant of the random graph process

The isoperimetric constant of a graph G on n vertices, i(G), is the minimum of , taken over all nonempty subsets S ⊂ V (G) of size at most n/2, where ∂S denotes the set of edges with precisely one end in S. A random graph process on n vertices, , is a sequence of graphs, where is the edgeless graph on n vertices, and is the result of adding an edge to , uniformly distributed over all the missing edges. The authors show that in almost every graph process equals the minimal degree of as long as the minimal degree is o(log n). Furthermore, it is shown that this result is essentially best possible, by demonstrating that along the period in which the minimum degree is typically Θ(log n), the ratio between the isoperimetric constant and the minimum degree falls from 1 to , its final value. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Potent estrogen agonists based on carborane as a hydrophobic skeletal structure. A new medicinal application of boron clusters

BACKGROUND: Carboranes (dicarba-closo-dodecaboranes) are a class of carbon-containing polyhedral boron-cluster compounds having remarkable thermal stability and exceptional hydrophobicity. Applications of the unique structural and chemical properties offered by icosahedral carboranes in boron neutron capture therapy have received increasing attention over the past 30 years. However, these features of carboranes may allow another application as a hydrophobic pharmacophore in biologically active molecules that interact hydrophobically with receptors. RESULTS: We have designed candidate estrogen-receptor-binding compounds having carborane as a hydrophobic skeletal structure and synthesized them. The most potent compound bearing a carborane cage exhibited activity at least 10-fold greater than that of 17beta-estradiol in the luciferase reporter gene assay. Estrogen receptor-alpha-binding data for the compound were consistent with the results of the luciferase reporter gene assay. The compound also showed potent in vivo effects on the recovery of uterine weight and bone loss in ovariectomized mice. CONCLUSION: Further development of the potent carborane-containing estrogenic agonists described here, having a new skeletal structure and unique characteristics, should yield novel therapeutic agents, especially selective estrogen receptor modulators. Furthermore, the suitability of the spherical carborane cage for binding to the cavity of the estrogen receptor-alpha ligand-binding domain should provide a basis for a similar approach to developing novel ligands for other steroid receptors.     

Stereoselective tandem iridium-catalyzed alkene isomerization-cope rearrangement of ω-diene epoxides: efficient access to acyclic 1,6-dicarbonyl compounds

The Cope rearrangement of 2,3-divinyloxiranes, a rare example of epoxide C–C bond cleavage, results in 4,5-dihydrooxepines which are amenable to hydrolysis, furnishing 1,6-dicarbonyl compounds containing two contiguous stereocenters at the 3- and 4-positions. We employ an Ir-based alkene isomerization catalyst to form the reactive 2,3-divinyloxirane in situ with complete regio- and stereocontrol, which translates into excellent control over the stereochemistry of the resulting oxepines and ultimately to an attractive strategy towards 1,6-dicarbonyl compounds.

Iridium catalyzed alkene isomerization-cope rearrangement of ω-diene epoxide furnishes 3,4-dihydrooxepines. These oxepines are hydrolyzed to diastereomerically pure 1,6-dicarbonyl compound containing two contiguous stereocenters within acyclic system.

1,6-Dicarbonyl compounds are widespread as targets and intermediates in organic synthesis.¹ Due to the “dissonant” polarizing effect induced by the two carbonyl groups,² these motifs are challenging to retrosynthetically disconnect into classical synthons. Unsurprisingly, many approaches toward 1,6-dicarbonyls rely on dimerization of α,β-unsaturated carbonyl compounds (Scheme 1a)³ or oxidative cleavage of substituted cyclohexene derivatives⁴ which significantly limits the range of possible products. Alternative strategies, such as the ring-opening of donor–acceptor cyclopropanes with enolate nucleophiles, efficiently form the 1,6-dicarbonyl skeleton, albeit with limited substrate scope (Scheme 1b).⁵ The Cope rearrangement of 1,5-dienes, featuring oxygen functionality in the 3- and 4-positions,⁶ represents a promising strategy towards 1,6-dicarbonyl compounds but suffers from lack of stereocontrol over the diene substrates, resulting in diastereomeric mixtures of products (Scheme 1c).Open in a separate windowScheme 1Selected approaches towards the formation of 1,6-dicarbonyl compounds and our proposed approach.A conceptually related approach towards the preparation of 1,6-dicarbonyl compounds is through the hydrolysis of 3,4-dihydrooxepines (Scheme 1d), which are in turn generated through the Cope rearrangement of 2,3-divinyloxiranes.⁷ Such a sigmatropic rearrangement is also noteworthy as a rare example where an epoxide C–C bond is selectively cleaved over the usually more reactive C–O bond. This intriguing rearrangement has been studied but its use in synthesis is scarce, presumably due to difficulties in the stereoselective synthesis and handling of the key divinyl epoxides.In line with our interest in the strategic application of alkene isomerization to generate reactive synthetic intermediates in stereodefined form,⁸ we posited to form the reactive 2,3-divinyloxiranes in situ, through alkene isomerization^9,10 of the simpler allyl epoxides, which are accessible in enantiomerically enriched form.¹¹ Such a strategy might greatly facilitate access to these intermediates and therefore uncover a synthetically attractive route toward 1,6-dicarbonyl compounds featuring two contiguous stereocenters.With this idea in mind, we first explored the isomerization and subsequent Cope rearrangement of allyl-vinyl epoxides 1 (Scheme 2). To induce isomerization, we employed a cationic iridium-based catalytic system,¹² which is known to reliably isomerize alkenes with high degrees of regio- and stereocontrol.¹³Open in a separate windowScheme 2Substrate scope for the tandem iridium-catalyzed alkene isomerization-Cope rearrangement of allyl-vinyl epoxides.In line with our expectations, our model substrate 1a (R² = R³ = H, R⁴ = Me, R⁵ = CO₂Et) was smoothly isomerized at 65 °C in the presence of 1.5 mol% of Ir dimer to obtain the corresponding divinyl epoxide with a complete E-selectivity. With suitable conditions for alkene isomerization in hand, we exposed substrate 1a to the Ir-based catalytic system at 120 °C and were equally pleased to observe the 4,5-dihydrooxepine product 2a, resulting from the tandem isomerization-Cope rearrangement as a single diastereoisomer in 81% yield. We proceeded to test the generality of our protocol with respect to different alkene and epoxide substitution patterns. Pleasingly, product 2b was generated with complete stereoselectivity, showcasing the compatibility of the reaction conditions with potentially labile tertiary stereocenters α to the ester group. We then wondered whether the anti-diastereomer could be accessed starting from the corresponding cis allyl-vinyl epoxide. Indeed, in line with the known stereospecific behavior of the Cope rearrangement, we obtained the complementary diastereomer 2c. Turning our attention to more highly substituted epoxides, we were pleased to observe the formation of dihydrooxepines 2d and 2e, which correspond to 1,6-keto-aldehyde and diketone products, respectively. Substrate 1f (R² = R⁴ = R⁵ = H, R³ = Ph), which features an unactivated vinyl group, also underwent the rearrangement, demonstrating that an activated alkenyl group is not required for a successful outcome. Similarly, product 2g featuring two alkyl groups is also generated, with high diastereoselectivity albeit in moderate yield. Products featuring ethyl and methyl ester 2h, 2i could also be obtained in good yields and diastereoselectivity. We next tested substrate 1j (R² = Me, R³ = Ph, R⁴ = CH₂CH₂Ph, R⁵ = H), as a geometric-mixture of the double bond (E : Z = 1.1 : 1) and in accordance with the stereospecificity of the process, the oxepine 2j was obtained as a mixture of two diastereomers with the same ratio. Disappointingly, substrate 1k did not undergo isomerization, presumably due to the Lewis basic nature of the ketone, likely poisoning the Ir-catalyst.During our study, we noticed that allyl-vinyl epoxides bearing electron donating groups on the vinyl moiety tend to decompose during purification by column chromatography on silica gel. This obstacle further motivated us to explore diallyl epoxides 3 as substrates, where the reactive divinyl epoxide would be generated by isomerization of both allyl fragments. Notably, these diallyl epoxides are much more stable compared to their vinyl counterparts and can be readily prepared in two steps from simple alkynes.¹⁴ To our delight, diallyl epoxide 3a (R = CH₂OMe) smoothly underwent the double isomerization-Cope rearrangement cascade at 140 °C, furnishing oxepine 2l with impressive yield and diastereoselectivity (Scheme 3). The use of alkene isomerization to form the reactive divinyl epoxide in situ avoids the isolation of the unstable divinyl epoxide, while controlling the stereochemistry of both double bonds, particularly not trivial to achieve using classical olefination reactions. Products 2m and 2n feature ester and silyl groups, highlighting the functional group tolerance of the catalytic system.Open in a separate windowScheme 3Substrate scope for tandem iridium-catalyzed double alkene isomerization-Cope rearrangement of diallyl epoxides.Our next objective was to hydrolyze the diastereomerically pure oxepines obtained through the rearrangement in a stereoretentive fashion, revealing the acyclic 1,6-dicarbonyl motif. Pleasingly, diversely substituted oxepines 2 underwent smooth hydrolysis either using 5 mol% of Pd(MeCN)₂Cl₂¹⁵ at 50 °C or an acidic aqueous solution to form 1,6-dicarbonyls 4 in diastereomerically pure form (Scheme 4).¹⁶ Dicarbonyl products featuring labile tertiary centers 4a and 4b are formed under these conditions with excellent diastereoselectivities and yields. Without surprise, oxepine 2f (R² = R⁴ = R⁵ = H, R³ = Ph) furnished the keto-substituted product 4c in good yield. The relative stereochemistry of 4b was unambiguously confirmed by single crystal X-ray diffraction analysis of the corresponding carboxylic acid 7 (Scheme 4b).¹⁷ The reaction is scalable to ½ gram of substrate and could be performed in a single-pot operation without isolation of the intermediate oxepine (Scheme 4b). By using this approach, 1h provides 4b in 61% yield as a single diastereomer, underlining the synthetic potential and efficiency of this method.Open in a separate windowScheme 4Hydrolysis of oxepines and one-pot sequence.     

Subdivision Schemes for Positive Definite Matrices

The class of symmetric positive definite matrices is an important class both in theory and application. Although this class is well studied, little is known about how to efficiently interpolate such data within the class. We extend the 4-point interpolatory subdivision scheme, as a method of interpolation, to data consisting of symmetric positive definite matrices. This extension is based on an explicit formula for calculating a binary “geodetic average”. Our method generates a smooth curve of matrices, which retain many important properties of the interpolated matrices. Furthermore, the scheme is robust and easy to implement.     

Theoretical investigation of linalool oxidation

This study concerns the autoxidation of one of the most used fragrances in daily life, linalool (3,7-dimethyl-1,6-octadien-3-ol). It reacts with O2 to form hydroperoxides, which are known to be important contact allergens. Pathways for hydroperoxide formation are investigated by means of quantum mechanical electronic structure calculations. Optimized molecular geometries and harmonic vibrational frequencies are determined using density functional theory (DFT). Insight into how the addition of O2 to linalool occurs is obtained by establishing a theoretical framework and systematically investigating three smaller systems: propene, 2-methyl-2-butene, and 2-methyl-2-pentene. 2-Methyl-2-pentene was chosen as a model system and used to compare with linalool. This theoretical study characterizes the linalool-O2 biradical intermediate state, which constitutes a branching point for the further oxidation reactions pathways. Thus, the observed linalool oxidation product spectrum is discussed in terms of a direct reaction path, the ene-type mechanism, and the radical mechanism. The major hydroperoxide found in experiments is 7-hydroperoxy-3,7-dimethyl-octa-1,5-diene-3-ol, and the calculated results support this finding.     

Shy couplings

A pair (X, Y) of Markov processes on a metric space is called a Markov coupling if X and Y have the same transition probabilities and (X, Y) is a Markov process. We say that a coupling is “shy” if inf _{t ≥ 0} dist(X _t , Y _t ) >  0 with positive probability. We investigate whether shy couplings exist for several classes of Markov processes.     

On the support of harmonic measure for the random walk

We show that harmonic measure for the simple random walk on then ×…×n cube in thed-dimensional lattice is supported on o(n ^d ) vertices. This research was supported in part by NSF grant # DMS-9353149.     

On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002