Positive definite functions on the Heisenberg group

Work supported in part by the NSF     

Nonvanishing cohomology and classes of Gorenstein rings

We give counterexamples to the following conjecture of Auslander: given a finitely generated module M over an Artin algebra Λ, there exists a positive integer n_M such that for all finitely generated Λ-modules N, if Ext_Λⁱ(M,N)=0 for all i?0, then Ext_Λⁱ(M,N)=0 for all i?n_M. Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.     

Energetic preferences for alpha,beta versus beta,gamma unsaturation

Density functional theory has been applied at the B3LYP/6-311+G(d,p)//B3LYP/6-31G(d) level to examine the energetics of alpha,beta- versus beta,gamma-unsaturation for some common organic functional groups. Specifically, the relative stabilities of allyl-X (H2C=CHCH2X) and 1-propenyl-X (H3CCH=CHX) isomers have been computed for X = methyl, vinyl, phenyl, formyl, acetyl, methoxy, methylthio, methylsulfinyl, methylsulfonyl, sulfamoyl, and methoxysulfonyl, and the results are compared to available experimental data. The intrinsic preference of 3 kcal/mol for the 1-propenyl isomer when X = CH3 is exceeded by 2-4 kcal/mol for first-row conjugating groups, but it is not met for the sulfur-containing groups. In particular, alpha,beta-unsaturation is favored by less than 1 kcal/mol for the sulfone and sulfonamide analogues, while it is preferred by 8 kcal/mol for the vinyl-substituted case. Detailed structural results and torsional energy profiles are also reported.     

Harmonic analysis of fractal measures

We consider affine systems inR ⁿ constructed from a given integral invertible and expansive matrixR, and a finite setB of translates,σ _bx:=R^–1x+b; the corresponding measure μ onR ⁿ is a probability measure fixed by the self-similarity $\mu = \left| B \right|^{ - 1} \sum\nolimits_{b \in B} {\mu o\sigma _b^{ - 1} } $ . There are twoa priori candidates for an associated orthogonal harmonic analysis: (i) the existence of some subset Λ inR ⁿ such that the exponentials {e^iλ·x}Λ form anorthogonal basis forL ²(μ); and (ii) the existence of a certaindual pair of representations of theC ^*-algebraO _N wheren is the cardinality of the setB. (For eachN, theC ^*-algebraO _N is known to be simple; it is also called the Cuntz algebra.) We show that, in the “typical” fractal case, the naive version (i) must be rejected; typically the orthogonal exponentials inL ²(μ) fail to span a dense subspace. Instead we show that theC ^*-algebraic version of an orthogonal harmonic analysis, namely (ii), is a natural substitute. It turns out that this version is still based on exponentialse ^iλ·x, but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of affine systems, based onR andB, such that μ has theC ^*-algebra property (ii). Specifically, we show that μ has an orthogonal harmonic analysis (in the sense (ii)) if the system (R, B) satisfies some specific symmetry conditions (which are geometric in nature). Our conditions for (ii) are stated in terms of two pieces of data: (a) aunitary generalized Hadamard matrix, and (b) a certainsystem of lattices which must exist and, at the same time, be compatible with the Hadamard matrix. A partial converse to this result is also given. Several examples are calculated, and a new maximality condition for exponentials is identified.