A general formula for the algebraic degree in semidefinite programming

In this article, we use a natural desingularization of the conormalvariety of (n x n)-symmetric matrices of rank at most r to finda general formula for the algebraic degree in semidefinite programming.     

Computational study on the 1,2-rearrangement in beta-(nitroxy)vinyl and beta-(acetoxy)vinyl radicals

The 1,2-nitroxyl and 1,2-acetoxyl rearrangement in beta-(nitroxy)vinyl and beta-(acetoxy)vinyl radicals 13a and 13b, respectively, has been studied for the gas phase with various ab initio and density functional methods. The energetically most favorable pathway for 13a is calculated to proceed via reversible fragmentation/radical addition through transition state I-19a. In the case of 13b, rearrangement through a five-membered ring transition state III-16b and the fragmentation/radical addition pathway via transition state I-19b are competing processes. Mulliken and natural population analysis reveal a certain degree of charge separation in III-16a/b that may indicate a potential solvent effect on the rearrangement rate. A stepwise group migration through a cyclic radical intermediate V-18a/b or rearrangement through a three-membered ring transition state II-15a/b can be ruled out for both vinyl radicals. A comparison of the results of the calculations with experimental findings provides important insights into the kinetics of "self-terminating radical oxygenations". A significant method dependence on the outcome of the calculations was observed, which revealed the unsuitability of the UHF, MP2, B3LYP, and mPW1PW91 methods for computing these radical rearrangement processes. The results from BHandHLYP/cc-pVDZ calculations showed the best agreement with single-point energy calculations performed at the QCISD and CCSD(T) levels of theory.     

Homogenization of Unidirectional and Arbitrarily Oriented Fiber-Reinforced Materials by the Method of Conditional Moments

In this paper the method of conditional moments is developed for the case of a two–component matrix composite with randomly distributed unidirectional and arbitrarily oriented ellipsoidal inclusions. The algorithm for determination of the effective elastic properties of composites from the given elastic constants of the components and geometrical parameters and orientation of inclusions is discussed. It is assumed that the components of the composite show orthotropic symmetry of thermoelastic properties. As a numerical example arbolite (straw particle inclusions in a cement matrix) is considered. The dependencies of Young's moduli, Poisson's ratios and shear moduli from the concentration of inclusions and for certain orientations of the inclusions are predicted and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nuclear inelastic scattering studies on a dinuclear iron(II) spin crossover complex

The vibrational properties of the (high-spin)-(high-spin) and the (high-spin)- (low-spin) states of the dinuclear Fe(II) spin crossover complex[{Fe(L-N₄Me₂)}₂(BiBzIm)](ClO₄)₂·2EtCN¹ have been studied by means of nuclear inelastic scattering. At a temperature of 80 K typical low spin marker bands are detected in the region around 400 cm^?1, these bands almost completely disappear after increasing temperature to 190 K. Corresponding density functional theory calculations using the functional B3LYP* and the basis set CEP-31G reproduce the experimental data and thus allow a deeper understanding of the vibrational properties of dinuclear Fe(II) spin crossover complexes.     

Experimental evidence of the vibrational coupling of nearest neighbours in 1D spin crossover polymers of rigid bridging ligands

The nuclear inelastic scattering signatures of the low spin centres of the methanosulphonate, tosylate and perchlorate salts of the spin crossover polymer ([Fe(II)(4-amino-1,2,4-triazole)₃]⁺²) _n have been compared for the pure low-spin phase, for the mixed high-spin and low-spin phases, as well as for Zn(II) diluted samples. Within this series a change in the spectral pattern in the 320–500 cm^?1 region is observed involving the decrease of the intensities of a band at ～320 cm^?1 and those over 400 cm^?1 as the molar fraction of the low-spin centres decreases. On the basis of the DFT calculations (B3LYP/CEP-31G) this effect is interpreted in terms of vibrational coupling of the iron centres of the same spin.     

Nonaxisymmetric electroelastic vibrations of a hollow sphere made of functionally gradient piezoelectric material

The nonaxisymmetric problem of natural vibrations of a hollow sphere made of functionally gradient piezoelectric material is solved based on 3D electroelasticity. The properties of the material change continuously along a radial coordinate according to an exponential law. The external surface of the sphere is free of tractions and either insulated or short-circuited by electrodes. After separation of variables and representation of the components of the displacements and of the stress tensor in terms of spherical functions, the initially three-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed by ordinary differential equations. This problem is solved by a stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Moreover, a numerical investigation is performed based on the algorithm used for solving the problem. In particular, we investigate the influence of the geometric and electric parameters on the frequency spectrum at the nonaxisymmetry of natural vibrations of an inhomogeneous piezoceramic thick-walled sphere.     

On the rationality of the moduli space of Lüroth quartics

We prove that the moduli space \mathfrakM_L{\mathfrak{M}_L} of Lüroth quartics in \mathbbP²{\mathbb{P}^2}, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL₃ (\mathbbC){\mathrm{PGL}_3 (\mathbb{C})} is rational, as is the related moduli space of Bateman seven-tuples of points in \mathbbP²{\mathbb{P}^2}.     

Effective thermoelastic properties of discrete-fiber reinforced materials with transversally-isotropic components

In the present paper, we will illustrate the application of the method of conditional moments by constructing the algorithm for determination of the effective elastic properties of composites from the given elastic constants of the components and geometrical parameters of inclusions. A special case of two-component matrix composite with randomly distributed unidirectional spheroidal inclusions is considered. To this end it is assumed that the components of the composite show transversally isotropic symmetry of thermoelastic properties and that the axes of symmetry of the thermoelastic properties of the matrix and inclusions coincide with the coordinate axis x ₃. As a numerical example a composite based on carbon inclusions and epoxide matrix is investigated. The dependencies of Young’s moduli, Poisson’s ratios and shear modulus from the concentration of inclusions and for certain values which characterize the shape of inclusions are analyzed. The results are compared and discussed in context with other theoretical predictions and experimental data.      

The Hilbert series and a-invariant of circle invariants

Let V be a finite-dimensional representation of the complex circle ${\mathbb{C}}^{\times }$ determined by a weight vector $\mathit{a}\in {\mathbb{Z}}^n$. We study the Hilbert series ${\mathrm{Hilb}}_{\mathit{a}}(t)$ of the graded algebra $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ of polynomial ${\mathbb{C}}^{\times }$-invariants in terms of the weight vector a of the ${\mathbb{C}}^{\times }$-action. In particular, we give explicit formulas for ${\mathrm{Hilb}}_{\mathit{a}}(t)$ as well as the first four coefficients of the Laurent expansion of ${\mathrm{Hilb}}_{\mathit{a}}(t)$ at $t=1$. The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomials that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.