A visual proof of a result of Knuth on spanning trees of Aztec diamonds in the case of odd order

The even Aztec diamond AD_n is known to have precisely four times more spanning trees than the odd Aztec diamond OD_n—this was conjectured by Stanley and first proved by Knuth. We present a short combinatorial proof of this fact in the case of odd n. Our proof works also for the more general case of odd-by-odd Aztec rectangles.     

On the classification of complex vector bundles of stable rank

One describes, using a detailed analysis of Atiyah-Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constructive terms the Chern classes of a vector bundle. Since deceased.     

The number of spanning trees of plane graphs with reflective symmetry

A plane graph is called symmetric if it is invariant under the reflection across some straight line (called symmetry axis). Let G be a symmetric plane graph. We prove that if there is no edge in G intersected by its symmetry axis then the number of spanning trees of G can be expressed in terms of the product of the number of spanning trees of two smaller graphs, each of which has about half the number of vertices of G.     

Studies on thermal degradation of acrylonitrile–butadiene–styrene copolymer (ABS-Br) containing brominated flame retardant

The thermal degradation of acrylonitrile-butadiene-styrene copolymer (ABS-Br; 10 g) containing brominated flame retardant (Br: 9.59 wt.%) was carried out at 450 °C using a semi batch operation using two different temperature programs. The heating rate was found to affect the quality of the degradation oil and yield of products (liquid, gas and residue). Data on the effect of the temperature program on the accumulation of liquid products was presented. It was found that the majority of the bromine was concentrated in the carbon residue and while majority of the nitrogen accumulates in the liquid products irrespective of degradation mode. The use of a one step constant heating rate process (I) produced a higher liquid yield (39%), than a two step process (29%). Differences were also noted in the Br and N contained in the liquids produced by the two processes.     

The vanishing neighbourhood of non-isolated singularities

We study the vanishing neighbourhood of non-isolated singularities of functions on singular spaces by associating a general linear function. We use the carrousel monodromy in order to show how to get a better control over the attaching of thimbles. For one-dimensional singularities, we prove obstructions to integer (co)homology groups and to the eigenspaces of the monodromy via monodromies of nearby sections. Our standpoint allows one to find, in certain cases, the structure of the Milnor fibre up to the homotopy type.     

Milnor numbers and Euler obstruction*

Using a geometric approach, we determine the relations between the local Euler obstruction Eu_f of a holomorphic function f and several generalizations of the Milnor number for functions on singular spaces. *This work was partially supported by CNRS-CONACYT (12409) Cooperation Program. The first and third named authors partially supported by CONACYT grant G36357-E and DGPA (UNAM) grant IN 101 401.     

The exponential image of simple complex lie groups of exceptional type

Let G be a connected reductive complex Lie group. Let E _G be the image of the exponential map of G and E' _G its complement in G. We give a purely algebraic characterization of the set E _G and also describe an algorithm for finding all conjugacy classes of G in E' _G . We are mainly interested in the case when the Lie algebra of G is simple and exceptional. Full details are provided for groups G of type G ₂, F ₄, and E ₆. If G is of type G ₂ then there are only two such conjugacy classes.This work was supported by NSERC Grant A-5285.     

Matrix Representations for Positive Noncommutative Polynomials

In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails. Specifically, we treat a ``symmetric' polynomial q(x, h) in noncommuting variables, {x<sub>1</sub>, . . . , } and {h<sub>1</sub>, . . . , } for which q(X,H) is positive semidefinite whenever are tuples of selfadjoint matrices with ||X<sub>j</sub>|| ≤ 1 but H<sub>j</sub> unconstrained. The representation we obtain is a Gram representation in the variables h where P<sub>q</sub> is a symmetric matrix whose entries are noncommutative polynomials only in x and V is a ``vector' whose entries are polynomials in both x and h. We show that one can choose P_q such that the matrix P_q(X) is positive semidefinite for all ||X_j|| ≤ 1. The representation covers sum of square results ([Am. Math. (to appear); Linear Algebra Appl. 326 (2001), 193–203; Non commutative Sums of Squares, preprint]) when g_x = 0. Also it allows for arbitrary degree in h, rather than degree two, in the main result of [Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically to appear IOET July 2003] when restricted to x-domains of the type ||X_j|| ≤ 1. Partially supported by NSF, DARPA and Ford Motor Co. Partially supported by NSF grant DMS-0140112 Partially supported by NSF grant DMS-0100367     

Conformational analysis of XA and AX dipeptides in water by electronic circular dichroism and 1H NMR spectroscopy

We measured the temperature-dependent electronic circular dichroism (ECD) spectra of AX, XA, and XG dipeptides in D2O. The spectra of all XA and AX peptides indicate a substantial population of the polyproline II (PPII) conformation, while the ECD spectra of LG, KG, PG, and AG were found to be quantitatively different from the alanine-based dipeptides. Additional UV absorption data indicate that the ECD spectra of the XG peptides stem from electronic coupling between the peptide and the C-terminal group, and that spectral differences reflect different orientations of the latter. We also measured the 1H NMR spectra of the investigated dipeptides to determine the 3JHalphaNH coupling constants for the C-terminal residue. The observed temperature dependence of the ECD spectra and the respective room-temperature 3JHalphaNH coupling constants were analyzed by a two-state model encompassing PPII and a beta-like conformation. The PPII propensity of alanine in the XA series is only slightly modulated by the N-terminal side chain, and is larger than 50%. As compared to AA, XA peptides containing L, P, S, K V, E, T, and I all cause a relative stabilization of the extended beta-strand conformation. The PPII fractions of XA peptides varied between 0.64 for AA and 0.58 for DA, whereas the PPII fractions of AX peptides were much lower. From the investigated AX peptides, only AL and AQ showed the expected PPII propensity. We found that AT, AI, and AV clearly prefer an extended beta-strand conformation. A quantitative comparison of AA, AAA, and AAAA revealed a hierarchy AAAA > AAA approximately AA for the PPII population, in agreement with predictions from MD calculations and results from Raman optical activity studies (McColl et al. J. Am. Chem. Soc. 2004, 126, 5076).