Counting solutions to binomial complete intersections

We study the problem of counting the total number of affine solutions of a system of n binomials in n   variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is #P$\#P$-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors.     

Proton reduction and dihydrogen oxidation on models of the [2Fe]H cluster of [Fe] hydrogenases. A density functional theory investigation

Density functional theory was used to compare reaction pathways for H2 formation and H+ reduction catalyzed by models of the binuclear cluster found in the active site of [Fe] hydrogenases. Terminal H+ binding to an Fe(I)-Fe(I) form, followed by monoelectron reduction and protonation of the di(thiomethyl)amine ligand, can conveniently lead to H2 formation and release, suggesting that this mechanism could be operative within the enzyme active site. However, a pathway that implies the initial formation of Fe(II)-Fe(II) mu-H species and release of H2 from an Fe(II)-Fe(I) form is characterized by only slightly less favored energy profiles. In both cases, H2 formation becomes less favored when taking into account the competition between CN and amine groups for H+ binding, an observation that can be relevant for the design of novel synthetic catalysts. H2 cleavage can take place on Fe(II)-Fe(II) redox species, in agreement with previous proposals [Fan, H.-J.; Hall, M. B. J. Am. Chem. Soc. 2001, 123, 3828] and, in complexes characterized by terminal CO groups, does not need the involvement of an external base. The step in H2 oxidation characterized by larger energy barriers corresponds to the second H+ extraction from the cluster, both considering Fe(II)-Fe(II) and Fe(II)-Fe(III) species. A comparison of the different reaction pathways reveals that H2 formation could involve only Fe(I)-Fe(I), Fe(II)-Fe(I), and Fe(II)-Fe(II) species, whereas Fe(III)-Fe(II) species might be relevant in H2 cleavage.     

Legendre spectral method for the fractional Bratu problem

In this paper, the Legendre spectral collocation method (LSCM) is applied for the solution of the fractional Bratu's equation. It shows the high accuracy and low computational cost of the LSCM compared with some other numerical methods. The fractional Bratu differential equation is transformed into a nonlinear system of algebraic equations for the unknown Legendre coefficients and solved with some spectral collocation methods. Some illustrative examples are also given to show the validity and applicability of this method, and the obtained results are compared with the existing studies to highlight its high efficiency and neglectable error.     

Plane Waves in Cubically Nonlinear Elastic Media

The theory of plane waves in nonlinear materials described by the Murnaghan potential is proposed. The theory takes into account both the classical quadratic nonlinearity and the cubic nonlinearity of the basic wave equations. Some new opportunities for the wave interaction analysis are commented on: in addition to the second harmonics, a longitudinal plane wave generates the third one, a transverse plane wave generates the third harmonics, and horizontally and vertically polarized transverse plane waves jointly generate new waves     

Well adjusted: Using expediting and cancelation to manage store replenishment inventory for a seasonal good

We consider an inventory problem that can be translated into a two-period newsvendor setting where the day prior to sales, the newsvendor places an initial preliminary order—a semi-binding forecast—with the publisher. At the beginning of the actual day of sales, the newsvendor has a better forecast for the day’s demand: based on knowing the actual content of the paper, he knows whether it will be a high-demand day due to breaking news or a low-demand day due to slow news. He then can revise the preliminary order quantity by expediting additional papers or canceling all or part of the order, but each of these activities has an associated cost.     

Self-broadening of the ammonia inversion spectrum

We show that self-broadening of the ammonia inversion lines with J = K can be explained by taking account of the rotational resonance effect.     

σ−π Contributions in metal-co bond: A theoretical study of RhCO and PdCO

The electronic structure of linear RhC0 and PdCO molecules has been studied by means of effective core potential calculations including configuration interaction. The study of the ⁴Δ and ²Σ⁺ states of RhCO and the ³Σ⁺ and ¹Σ⁺ states of PdCO has shown that their low-spin states are bound and their high-spin states are repulsive. The binding energies of RhCO and PdCO are 4.0 and 7.2 kcal/mol, for the equilibrium bond lengths of 3.87 and 4.03 au, respectively. The interaction between the diffuse 5s metal orbital and the carbon lone pair produces a repulsion in the α-space, as depicted by the negative region along the metal-carbon axis in the electron density difference contour maps. The bonding in low-spin states of RhCO and PdCO is due to the metal-CO π “backdonation”. The present investigation on RhCO and PdCO, preliminary results on ScCO, as well as published results on FeCO, NiCO, CuCO and PtCO allows us to make a general conjecture that a low occupancy of the outer s atomic orbital of the transition metal atom in the complex favours the bonding interaction with the CO moiety. The analogy with properties of LiCO and NaCO systems is pointed out.     

Conformational behaviour of the antineoplastic peptide dolastatin-10 and of two mutated derivatives

Summary The three-dimensional structure of dolastatin-10, an extremely potent cytostatic and antineoplastic peptide extracted from the mollusc Dolabella auricularia, has not yet been fully characterized in an experimental way. By means of a systematic conformational search of the natural peptide and of two mutated analogs, carried out both in vacuo and in aqueous solution, the present work allows to obtain insights into the conformational preferences of this remarkable compound. In addition, the ability to form intra- and intermolecular H-bonds as a function both of the sequence and of the conformation is discussed. The search for the best molecular conformations has been carried out using a molecular mechanics approach, based on the CVFF potential. Dolastatin-10 contains some unusual amino acids for which no experimental structural data are available. In order to check the reliability of the CVFF potential in predicting structures of such nonconventional amino acids, geometry optimizations have been carried out using the ab initio Hartree-Fock procedure. The CVFF parameterization is found to be adequate also for nonconventional amino acids.     

Dissecting the intimate mechanism of cyanation of {2Fe3S} complexes related to the active site of all-iron hydrogenases by DFT analysis of energetics, transition states, intermediates and products in the carbonyl substitution pathway

A bridging carbonyl intermediate with key structural elements of the diiron sub-site of all-iron hydrogenase has been experimentally observed in the CN/CO substitution pathway of the {2Fe3S} carbonyl precursor, [Fe(2)(CO)(5){MeSCH(2)C(Me)(CH(2)S)(2)}]. Herein we have used density functional theory (DFT) to dissect the overall substitution pathway in terms of the energetics and the structures of transition states, intermediates and products. We show that the formation of bridging CO transitions states is explicitly involved in the intimate mechanism of dicyanation. The enhanced rate of monocyanation of {2Fe3S} over the {2Fe2S} species [Fe(2)(CO)(6){CH(2)(CH(2)S)(2)}] is found to rest with the ability of the thioether ligand to both stabilise a mu-CO transition state and act as a good leaving group. In contrast, the second cyanation step of the {2Fe3S} species is kinetically slower than for the {2Fe2S} monocyanide because the Fe2 atom is deactivated by coordination of the electron-donating thioether group. In addition, hindered rotation and the reaction coordinate of the approaching CN(-) group, are other factors which explain reactivity differences in {2Fe2S} and {2Fe3S} systems. The intermediate species formed in the second cyanation step of {2Fe3S} species is a mu-CO species, confirming the structural assignment made on the basis of FT-IR data (S. J. George, Z. Cui, M. Razavet, C. J. Pickett, Chem. Eur. J. 2002, 8, 4037-4046). In support of this we find that computed and experimental IR frequencies of structurally characterised {2Fe3S} species and those of the bridging carbonyl intermediate are in excellent agreement. In a wider context, the study may provide some insight into the reactivity of dinuclear systems in which neighbouring group on-off coordination plays a role in substitution pathways.