Semi-Conjugate Direction Methods for Real Positive Definite Systems

In this preliminary work, left and right conjugate direction vectors are defined for nonsymmetric, nonsingular matrices A and some properties of these vectors are studied. A left conjugate direction (LCD) method for solving nonsymmetric systems of linear equations is proposed. The method has no breakdown for real positive definite systems. The method reduces to the usual conjugate gradient method when A is symmetric positive definite. A finite termination property of the semi-conjugate direction method is shown, providing a new simple proof of the finite termination property of conjugate gradient methods. The new method is well defined for all nonsingular M-matrices. Some techniques for overcoming breakdown are suggested for general nonsymmetric A. The connection between the semi-conjugate direction method and LU decomposition is established. The semi-conjugate direction method is successfully applied to solve some sample linear systems arising from linear partial differential equations, with attractive convergence rates. Some numerical experiments show the benefits of this method in comparison to well-known methods. This revised version was published online in July 2006 with corrections to the Cover Date.     

On spaces of modular forms spanned by eta-quotients

An eta-quotient of level N   is a modular form of the shape f(z)=_∏δ|Nη(δz)^_rδ$f(z)={\prod }_{\delta |N}\eta {(\delta z)}^{r_{\delta }}$. We study the problem of determining levels N   for which the graded ring of holomorphic modular forms for _Γ0(N)${\Gamma }_0(N)$ is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N  . In addition, we prove that if f(z)$f(z)$ is a holomorphic modular form that is non-vanishing on the upper half plane and has integer Fourier coefficients at infinity, then f(z)$f(z)$ is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of _J0(2^k)(Q)$J_0(2^k)(\mathbb{Q})$.     

Toward a Symphony of Reactivity: Cascades Involving Catalysis and Sigmatropic Rearrangements

Catalysis and synthesis are intimately linked in modern organic chemistry. The synthesis of complex molecules is an ever evolving area of science. In many regards, the inherent beauty associated with a synthetic sequence can be linked to a certain combination of the creativity with which a sequence is designed and the overall efficiency with which the ultimate process is performed. In synthesis, as in other endeavors, beauty is very much in the eyes of the beholder. It is with this in mind that we will attempt to review an area of synthesis that has fascinated us and that we find extraordinarily beautiful, namely the combination of catalysis and sigmatropic rearrangements in consecutive and cascade sequences.

1 Sometimes the assessment of beauty is nearly unanimous. The first four notes of Beethoven’s 5th Symphony (Symphony No.5 in C minor, Op.67) represent perhaps the most well‐known and popular motif in classical music. The orchestral score is shown in the background of the cover graphic. Accessed March 20, 2013 from http://imslp.org/wiki/Symphony_No.5,_Op.67_%28Beethoven,_Ludwig_van%29 .

     

Nonnegative matrix factorization for spectral data analysis

Data analysis is pervasive throughout business, engineering and science. Very often the data to be analyzed is nonnegative, and it is often preferable to take this constraint into account in the analysis process. Here we are concerned with the application of analyzing data obtained using astronomical spectrometers, which provide spectral data, which is inherently nonnegative. The identification and classification of space objects that cannot be imaged in the normal way with telescopes is an important but difficult problem for tracking thousands of objects, including satellites, rocket bodies, debris, and asteroids, in orbit around the earth. In this paper, we develop an effective nonnegative matrix factorization algorithm with novel smoothness constraints for unmixing spectral reflectance data for space object identification and classification purposes. Promising numerical results are presented using laboratory and simulated datasets.     

Explicit bounds for second-order difference equations and a solution to a question of Stevi?

This note gives explicit, applicable bounds for solutions of a wide class of second-order difference equations with nonconstant coefficients. Among the applications is an affirmative answer to a recent question of Stevi?.     

An Old Argument Against Co-location

I defend an old argument against co-location—the view that human animals are distinct from, but co-located with human persons. The argument is drawn from St. Thomas Aquinas. In order to respond to the argument, co-locationists have to endorse at least one of a trio of claims, none of which is obviously correct. Further, two of the options do not seem to be the sort of positions that should be flowing out of the acceptance of a general metaphysical position. I conclude that co-locationism is more costly than generally thought.     

Hemoglobin as a nitrite anhydrase: modeling methemoglobin-mediated N2O3 formation

Nitrite has recently been recognized as a storage form of NO in blood and as playing a key role in hypoxic vasodilation. The nitrite ion is readily reduced to NO by hemoglobin in red blood cells, which, as it happens, also presents a conundrum. Given NO’s enormous affinity for ferrous heme, a key question concerns how it escapes capture by hemoglobin as it diffuses out of the red cells and to the endothelium, where vasodilation takes place. Dinitrogen trioxide (N₂O₃) has been proposed as a vehicle that transports NO to the endothelium, where it dissociates to NO and NO₂. Although N₂O₃ formation might be readily explained by the reaction Hb‐Fe³⁺+NO₂^?+NO?Hb‐Fe²⁺+N₂O₃, the exact manner in which methemoglobin (Hb‐Fe³⁺), nitrite and NO interact with one another is unclear. Both an “Hb‐Fe³⁺‐NO₂^?+NO” pathway and an “Hb‐Fe³⁺‐NO+NO₂^?” pathway have been proposed. Neither pathway has been established experimentally. Nor has there been any attempt until now to theoretically model N₂O₃ formation, the so‐called nitrite anhydrase reaction. Both pathways have been examined here in a detailed density functional theory (DFT, B3LYP/TZP) study and both have been found to be feasible based on energetics criteria. Modeling the “Hb‐Fe³⁺‐NO₂^?+NO” pathway proved complex. Not only are multiple linkage‐isomeric (N‐ and O‐coordinated) structures conceivable for methemoglobin–nitrite, multiple isomeric forms are also possible for N₂O₃ (the lowest‐energy state has an N? N‐bonded nitronitrosyl structure, O₂N? NO). We considered multiple spin states of methemoglobin–nitrite as well as ferromagnetic and antiferromagnetic coupling of the Fe³⁺ and NO spins. Together, the isomerism and spin variables result in a diabolically complex combinatorial space of reaction pathways. Fortunately, transition states could be successfully calculated for the vast majority of these reaction channels, both M_S=0 and M_S=1. For a six‐coordinate Fe³⁺‐O‐nitrito starting geometry, which is plausible for methemoglobin–nitrite, we found that N₂O₃ formation entails barriers of about 17–20 kcal mol^?1, which is reasonable for a physiologically relevant reaction. For the “Hb‐Fe³⁺‐NO+NO₂^?” pathway, which was also found to be energetically reasonable, our calculations indicate a two‐step mechanism. The first step involves transfer of an electron from NO₂^? to the Fe³⁺–heme–NO center ({FeNO}⁶) , resulting in formation of nitrogen dioxide and an Fe²⁺–heme–NO center ({FeNO}⁷). Subsequent formation of N₂O₃ entails a barrier of only 8.1 kcal mol^?1. From an energetics point of view, the nitrite anhydrase reaction thus is a reasonable proposition. Although it is tempting to interpret our results as favoring the “{FeNO}⁶+NO₂^?” pathway over the “Fe³⁺‐nitrite+NO” pathway, both pathways should be considered energetically reasonable for a biological reaction and it seems inadvisable to favor a unique reaction channel based solely on quantum chemical modeling.     

Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm

We introduce a generalization of the Robinson–Schensted–Knuth insertion algorithm for semi-standard augmented fillings whose basement is an arbitrary permutation σ∈S _n . If σ is the identity, then our insertion algorithm reduces to the insertion algorithm introduced by the second author (Sémin. Lothar. Comb. 57:B57e, 2006) for semi-standard augmented fillings and if σ is the reverse of the identity, then our insertion algorithm reduces to the original Robinson–Schensted–Knuth row insertion algorithm. We use our generalized insertion algorithm to obtain new decompositions of the Schur functions into nonsymmetric elements called generalized Demazure atoms (which become Demazure atoms when σ is the identity). Other applications include Pieri rules for multiplying a generalized Demazure atom by a complete homogeneous symmetric function or an elementary symmetric function, a generalization of Knuth’s correspondence between matrices of non-negative integers and pairs of tableaux, and a version of evacuation for composition tableaux whose basement is an arbitrary permutation σ.     

Enhanced catalytic performance of Ce-MCM-41-supported Rh for CO oxidation

Research on Chemical Intermediates - 1wt%Rh/Ce-MCM-41 catalysts were prepared by the incipient wet impregnation method using mesoporous Ce-MCM-41 materials as supports where the Si/Ce molar ratios...