Tensor Products and Support Varieties for Some Noncocommutative Hopf Algebras

We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations.     

Effect of CO on the Oxidative Addition of Arene C?H Bonds by Cationic Rhodium Complexes

Sequential addition of CO molecules to cationic aryl–hydrido Rh^III complexes of phosphine‐based (PCP) pincer ligands was found to lead first to C? H reductive elimination and then to C? H oxidative addition, thereby demonstrating a dual role of CO. DFT calculations indicate that the oxidative addition reaction is directly promoted by CO, in contrast to the commonly accepted view that CO hinders such reactions. This intriguing effect was traced to repulsive π interactions along the aryl‐Rh‐CO axis, which are augmented by the initially added CO ligand (due to antibonding interactions between occupied Rh d_π orbitals and occupied π orbitals of both CO and the arene moiety), but counteracted by the second CO ligand (due to significant π back‐donation). These repulsive interactions were themselves linked to significant weakening of the π‐acceptor character of CO in the positively charged rhodium complexes, which is concurrent with an enhanced σ‐donating capability. Replacement of the phosphine ligands by an analogous phosphinite‐based (POCOP) pincer ligand led to significant changes in reactivity, whereby addition of CO did not result in C? H reductive elimination, but yielded relatively stable mono‐ and dicarbonyl aryl–hydrido POCOP–Rh^III complexes. DFT calculations showed that the stability of these complexes arises from the higher electrophilicity of the POCOP ligand, relative to PCP, which leads to partial reduction of the excessive π‐electron density along the aryl‐Rh‐CO axis. Finally, comparison between the effects of CO and acetonitrile on C? H oxidative addition revealed that they exhibit similar reactivity, despite their markedly different electronic properties. However, DFT calculations indicate that the two ligands operate by different mechanisms.     

Covering a laminar family by leaf to leaf links

The Tree Augmentation Problem (TAP) is: given a tree T=(V,E) and a set E of edges (called links) on V disjoint to E, find a minimum-size edge-subset F⊆E such that T+F is 2-edge-connected. TAP is equivalent to the problem of finding a minimum-size edge-cover F⊆E of a laminar set-family. We consider the restriction, denoted LL-TAP, of TAP to instances when every link in E connects two leaves of T. The best approximation ratio for TAP is 3/2, obtained by Even et al. (2001, 2009, 2008) [3], [4] and [5], and no better ratio was known for LL-TAP. All the previous approximation algorithms that achieve a ratio better than 2 for TAP, or even for LL-TAP, have been quite involved.For LL-TAP we obtain the following approximation ratios: 17/12 for general trees, 11/8 for trees of height 3, and 4/3 for trees of height 2. We also give a very simple3/2-approximation algorithm (for general trees) and prove that it computes a solution of size at most , where t is the minimum size of an edge-cover of the leaves, and t^∗ is the optimal value of the natural LP-relaxation for the problem of covering the leaf edges only. This provides the first evidence that the integrality gap of a natural LP-relaxation for LL-TAP is less than 2.     

Expanding the Arsenal of PtIV Anticancer Agents: Multi‐action PtIV Anticancer Agents with Bioactive Ligands Possessing a Hydroxy Functional Group

Most multi‐action Pt^IV prodrugs have bioactive ligands containing carboxylates. This is probably due to the ease of carboxylating the OH axial ligands and because following reduction, the active drug is released. A major challenge is to expand the arsenal of bioactive ligands to include those without carboxylates. We describe a general approach for synthesis of Pt^IV prodrugs that release drugs with OH groups. We linked the OH groups of gemcitabine (Gem), paclitaxel (Tax), and estramustine (EM) to the Pt^IV derivative of cisplatin by a carbonate bridge. Following reduction, the axial ligands lost CO₂, rapidly generating the active drugs. In contrast, succinate‐linked drugs did not readily release the free drugs. The carbonate‐bridged ctc‐[Pt(NH₃)₂(PhB)(Gem‐Carb)Cl₂] was significantly more cytotoxic than the succinate‐bridged ctc‐[Pt(NH₃)₂(PhB)(Gem‐Suc)Cl₂], and more potent and less toxic than gemcitabine, cisplatin, and co‐administration of cisplatin and gemcitabine.     

Optical forces arising from phase gradients

We demonstrate both theoretically and experimentally that phase gradients in a light field can be used to create a new category of optical traps complementary to the more familiar intensity-gradient traps known as optical tweezers. We further show that the work done by phase-gradient forces is path dependent and briefly discuss some ramifications of this nonconservativity.     

Single-walled carbon nanotubes embedded in oriented polymeric nanofibers by electrospinning

The electrospinning process was used successfully to embed single-walled carbon nanotubes (SWCNTs) in a poly(ethylene oxide) (PEO) matrix, forming composite nanofibers. Initial dispersion of SWCNTs in water was achieved by the use of an amphiphilic alternating copolymer of styrene and sodium maleate. The resulting dispersions were stable, having a dark, smooth, ink-like appearance. For electrospinning, the dispersions were mixed with PEO solution in an ethanol/water mixture. The distribution and conformation of the nanotubes in the nanofibers were studied by transmission electron microscopy (TEM). Oxygen plasma etching was used to expose the nanotubes within the nanofibers to facilitate direct observation. Nanotube alignment within the nanofibers was shown to depend strongly on the quality of the initial dispersions. Well-dispersed and separated nanotubes were embedded in a straight and aligned form, while entangled nonseparated nanotubes were incorporated as dense aggregates. X-ray diffraction demonstrated a high degree of orientation of the PEO crystals in the electrospun nanofibers with embedded SWCNTs. This result is in pronounced distinction to the detrimental effect of incorporation of multiwalled carbon nanotubes on polymer orientation in electrospun nanofibers, as reported previously.     

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

The majority of first-order methods for large-scale convex–concave saddle point problems and variational inequalities with monotone operators are proximal algorithms. To make such an algorithm practical, the problem’s domain should be proximal-friendly—admit a strongly convex function with easy to minimize linear perturbations. As a by-product, this domain admits a computationally cheap linear minimization oracle (LMO) capable to minimize linear forms. There are, however, important situations where a cheap LMO indeed is available, but the problem domain is not proximal-friendly, which motivates search for algorithms based solely on LMO. For smooth convex minimization, there exists a classical algorithm using LMO—conditional gradient. In contrast, known to us similar techniques for other problems with convex structure (nonsmooth convex minimization, convex–concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper was to develop alternative (and seemingly much simpler) decomposition techniques based on LMO for bilinear saddle point problems and for variational inequalities with affine monotone operators.     

Sums of random symmetric matrices and quadratic optimization under orthogonality constraints

Let B _i be deterministic real symmetric m × m matrices, and ξ _i be independent random scalars with zero mean and “of order of one” (e.g., ). We are interested to know under what conditions “typical norm” of the random matrix is of order of 1. An evident necessary condition is , which, essentially, translates to ; a natural conjecture is that the latter condition is sufficient as well. In the paper, we prove a relaxed version of this conjecture, specifically, that under the above condition the typical norm of S _N is : for all Ω > 0 We outline some applications of this result, primarily in investigating the quality of semidefinite relaxations of a general quadratic optimization problem with orthogonality constraints , where F is quadratic in X = (X ₁,... ,X _k ). We show that when F is convex in every one of X _j , a natural semidefinite relaxation of the problem is tight within a factor slowly growing with the size m of the matrices . Research was partly supported by the Binational Science Foundation grant #2002038.