The Stanley Conjecture on Intersections of Four Monomial Prime Ideals

We show that the Stanley's Conjecture holds for an intersection of four monomial prime ideals of a polynomial algebra S over a field and for an arbitrary intersection of monomial prime ideals (P _i ) _i∈[s] of S such that each P _i is not contained in the sum of the other (P _j ) _j≠i .     

Oversampling generates super-wavelets

We show that the second oversampling theorem for affine systems generates super-wavelets. These are frames generated by an affine structure on the space .

     

Templated growth of platinum nanowheels using the inhomogeneous reaction environment of bicelles

Novel platinum nanowheels were synthesized by the reduction of aqueous platinum complex with ascorbic acid in the presence of disk-like bicelles. The platinum nanowheels possess thickened centers and flared edges that are connected by dendritic platinum nanosheets. This structural complexity can be attributed to the inhomogeneous micro-environment of the templating bicelles consisting of a central bi-layer region and a high curvature rim. The formation mechanism of the nanowheels was investigated by imaging nanostructures at different stages of the reaction. The templating bicelles were also imaged by TEM with the aid of negative staining. The variation of reaction parameters including platinum concentration, temperature, and total concentration of surfactants (CTAB + FC7) led to other types of platinum nanostructures, such as circular dendritic nanosheets with a tunable diameter and rectangular dendritic nanosheets. Interestingly, under irradiation by a TEM electron beam, the dendritic nanosheet portion of the nanowheels transforms into a metastable holey sheet. In addition, the platinum nanowheels have an electrochemical active surface area comparable to that of ETEK platinum black and thus are expected to have potential applications in catalysis.     

Reduction of bowl-shaped hydrocarbons: dianions and tetraanions of annelated corannulenes

[structure: see text] The reduction of several annelated corannulene derivatives was undertaken using lithium and potassium metals. It was found that annelation affects the annulenic character of corannulene by changing its charge distribution; the dianions of derivatives that are annelated with six-membered rings have less annulenic character and are less paratropic than corannulene dianion. This effect is even more pronounced in corannulenes that are peri-annelated with five-membered rings. The alkali metal used in the reduction process has a great influence on the outcome, especially on the degree of reduction. Most derivatives get reduced to tetraanions only with potassium, and not with lithium, the exception being systems that can stabilize the tetraanion with lithium by special means, such as aggregation or dimerization. One such system is cyclopenta[bc]corannulene (acecorannulylene), which gives a coordinative dimer that consists of two cyclopentacorannulene tetraanions, bound together in a convex-convex fashion by lithium cations. The points of contact in this dimer are two rehybridized carbons from each cyclopentacorannulene unit, which are bridged together by two lithium cations.     

Weak semi-continuity of the duality product in Sobolev spaces

Given a weakly convergent sequence of positive functions in , we prove the equivalence between its convergence in the sense of obstacles and the lower semi-continuity of the term by term duality product associated to (the p-Laplacian of) weakly convergent sequences of p-superharmonic functions of . This result implicitly gives new characterizations for both the convergence in the sense of obstacles of a weakly convergent sequence of positive functions and for the weak l.s.c. of the duality product.