On the total mean curvature of a nonrigid surface

Using the Green’s theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field, which immediately yields the following well-known theorem: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.     

The complexity of recursive constraint satisfaction problems

We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint satisfaction problems where the problem of finding a solution to the infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through finitely branching recursive trees or recursive binary trees. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to give equivalent results about the complexity of the problem of finding a solution to a recursive constraint satisfaction problem.     

A two-parameter generalization of the complete elliptic integral of second kind

A two-parameter generalization of the complete elliptic integral of second kind is expressed in terms of the Appell function F ₄. This function is further reduced to a quite simple bilinear form in the complete elliptic integrals K and E. Some physical applications are briefly mentioned.     

On H. Weyl and J. Steiner Polynomials

The paper deals with root location problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set . A polynomial of this class describes the volume of the set V + tB ⁿ as a function of t, where t is a positive number and B ⁿ denotes the unit ball in . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold , where is isometrically embedded with positive codimension in . A Weyl polynomial describes the volume of a tubular neighborhood of its associated as a function of the tube’s radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of into for m > n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial’s roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold.

Erasmus Darwin, the nephew of the great scientist Charles Darwin, believed that sometimes one should perform the most unusual experiments. They usually yield no results but when they do . . . . So once he played trumpet in front of tulips for the whole day. The experiment yielded no results.

Submitted: March 5, 2007., Revised: February 1, 2008., Accepted: February 2, 2008.     

Modeling of ligation-induced helix/loop displacements in myoglobin: toward an understanding of hemoglobin allostery

Combining quantum and molecular mechanics (QM/MM) methods and protein structure prediction algorithms, helix and loop movements are computed along the pathway of CO dissociation from myoglobin (Mb). The results are compared with high-resolution crystallographic data using sequence-displacement graphs. These graphs provide an unbiased method for evaluating main-chain segmental motions; they resolve an apparent disagreement between two sets of high-resolution crystal structures for MbCO and deoxyMb. The QM/MM modeling of the CO deligation reproduces the experimentally observed spin states and photodissociated crystal structure. The principal effect of CO dissociation is shown to be a concerted rotation of the E and F helices, which hold the heme like a clamshell. The rotation is a response to deligation forces, which impel the F helix away from the heme because of the Fe spin conversion, and which allow the E helix to collapse toward the heme as nonbonded contacts on the distal side are relieved. Additional helix and loop displacements stem from these primary events. In particular, the CD loop is found to be repositioned as a result of steric interactions with the water molecule that becomes H-bonded to the distal histidine in deoxyMb. A similar EF rotation and CD loop displacement are proposed to be the first steps along the allosteric pathway from the R to the T state in hemoglobin.     

Synthesis of silica-gold nanocomposites and their porous nanoparticles by an in-situ approach

We demonstrate the one-step synthesis of a silica-gold nanocomposite by simultaneous hydrolysis and reduction of gold chloride. The aminophenyl group was used as a reducing agent, and the trimethoxy silane group acts a precursor for the formation of silica. The porous gold nanoparticles were formed by etching out the silica-gold nanocomposite by hydrofluoric acid. The electron diffraction of porous gold nanoparticles showed that the particle are polycrystalline with FCC structure. The silica-gold nanocomposite exhibited nonlinear current-voltage behavior, and the porous gold nanoparticles displayed linear current-voltage behavior.     

Influence of Hartree-Fock exchange on the calculated Mössbauer isomer shifts and quadrupole splittings in ferrocene derivatives using density functional theory

Influence of molecular geometry, type of exchange-correlation functional, and contraction scheme of basis set applied at the iron nuclei have been tested in the calculation of 57Fe M?ssbauer isomer shifts and quadrupole splittings for a wide range of ligand types, as well as oxidation and spin states, in inorganic and organometallic systems. It has been found that uncontraction of the s-part of Wachter's full-electron basis set at the iron nuclei does not appreciably improve the calculated isomer shifts. The observed correlations for all tested sets of geometries are close to each other and predominantly depend on the employed exchange-correlation functional with B3LYP functional being slightly better as compared to BPW91. Both hybrid (B3LYP) and pure (BPW91) exchange-correlation functionals are suitable for the calculation of isomer shifts in organometallic compounds. Surprisingly, it has been found that the hybrid B3LYP exchange-correlation functional completely fails in accurate prediction of quadrupole splittings in ferrocenes, while performance of the pure BPW91 functional for the same systems was excellent. This observation has been explained on the basis of relationship between the amount of Hartree-Fock exchange involved in the applied exchange-correlation functional and the calculated HOMO-LUMO energy gap in ferrocenes. On the basis of this explanation, use of only pure exchange-correlation functionals has been suggested for accurate prediction of M?ssbauer spectra parameters in ferrocenes.     

2-iodylphenol ethers: preparation, X-ray crystal structure, and reactivity of new hypervalent iodine(V) oxidizing reagents

2-Iodylphenol ethers were prepared by the dimethyldioxirane oxidation of the corresponding 2-iodophenol ethers and isolated as chemically stable, microcrystalline products. Single-crystal X-ray diffraction analysis of 1-iodyl-2-isopropoxybenzene 8c and 1-iodyl-2-butoxybenzene 8d revealed pseudopolymeric arrangements in the solid state formed by intermolecular interactions between IO2 groups of different molecules. 2-Iodylphenol ethers can selectively oxidize sulfides to sulfoxides and alcohols to the respective aldehydes or ketones.     

Separation performance of single-stranded DNA electrophoresis in photopolymerized cross-linked polyacrylamide gels

Considerable effort has been directed toward optimizing performance and maximizing throughput in ssDNA electrophoresis because it is a critical analytical step in a variety of genomic assays. Ultimately, it would be desirable to quantitatively determine the achievable level of separation resolution directly from measurements of fundamental physical properties associated with the gel matrix rather than by the trial and error process often employed. Unfortunately, this predictive capability is currently lacking, due in large part to the need for a more detailed understanding of the fundamental parameters governing separation performance (mobility, diffusion, and dispersion). We seek to address this issue by systematically characterizing electrophoretic mobility, diffusion, and dispersion behavior of ssDNA fragments in the 70-1,000 base range in a photopolymerized cross-linked polyacrylamide matrix using a slab gel DNA sequencer. Data are collected for gel concentrations of 6, 9, and 12%T at electric fields ranging from 15 to 40 V/cm, and resolution predictions are compared with corresponding experimentally measured values. The data exhibit a transition from behavior consistent with the Ogston model for small fragments to behavior in agreement with the biased reptation model at larger fragment sizes. Mobility data are also used to estimate the mean gel pore size and compare the predictions of several models.