Combinatorial structure of exceptional sets in resolutions of singularities

The dual complex can be associated to any resolution of singularities whose exceptional set is a divisor with simple normal crossings. It generalizes to higher dimensions the notion of the dual graph of a resolution of surface singularity. The homotopy type of the dual complex does not depend on the choice of a resolution and thus can be considered as an invariant of singularity. In this preprint we show that the dual complex is homotopy trivial for resolutions of 3-dimensional terminal singularities and for resolutions of Brieskorn singularities. We also review our earlier results on resolutions of rational and hypersurface singularities.Comment: 18 pages; to appear as a preprint of the Max-Planck-Institut, Bon

The Law of Conservation of Energy in Chemical Reactions

Earlier it has been supposed that the law of conservation of energy in chemical reactions has the following form: DU=DQ-PDV+SUM(muiDN) In the present paper it has been proved by means of the theory of ordinary differential equations that in the biggest part of the chemical reactions it must have the following form: DU=DQ+PDV+SUM(muiDN) The result obtained allows to explain a paradox in chemical thermodynamics: the heat of chemical processes measured by calorimetry and by the Vant-Hoff equation differs very much from each other. The result is confirmed by many experiments.Comment: e-mail me [email protected], 10 page

Full perturbation solution for the flow in a rotated torus

We present a perturbation solution for a pressure-driven fluid flow in a rotating toroidal channel. The analysis shows the difference between the solutions of full and simplified equations studied earlier. The result is found to be reliable for {\it low} Reynolds number ($\R$) as was the case for a previously studied solution for high $\R$. The convergence conditions are defined for the whole range of governing parameters. The viscous flow exhibits some interesting features in flow pattern and hydrodynamic characteristics.Comment: 4 pages, 6 figure

The 1st Law of Thermodynamics in Chemical Reactions

In the previous papers of the author it has been shown that the 1st law of thermodynamics in chemical reactions is the following one: dU=dQ+PdV+SUM In the present paper this theory was developed and it has been shown that the 1st law of thermodynamics in chemical reactions has the following form: dC=-dU+dA and -dU=dQ where dC is the change in the chemical energy, dU is the change in the internal energy. Internal energy is the energy of thermal motion of molecules.Comment: e-mail me [email protected], 6 page

Dependence of the Energy of Molecules on Interatomic Distance at Large Distances

Earlier it has been supposed that energy of molecules depends on interatomic distance according to the curve 1, Fig. 1. However, dissociation of molecules (for example, Te2=2Te) often is a chemical reaction. According to chemical kinetics, chemical reactions overcome a potential barrier. This barrier is absent at the curve 1. It is a very strong argument against the curve 1. It is shown that the molecule energy dependence on interatomic distance can behave at large distances not so but like the curve 2, Fig. 1. Earlier it has been supposed that quantum chemical methods give a wrong result at big distances if the wave function does not turn to zero. In this paper, it is been shown that it must not turn to zero. The wave function can be a piecewise function.Comment: e-mail me [email protected], 6 pages; accepted at 32nd EGAS Conference, Vilnius, 200

The Heats of Dilution. Calorimetry and Van't-Hoff

Earlier it has been found that there is a big difference between heats of dilution measured by calorimetry and by the Van't-Hoff equation. In the present paper a reason for that is proposed. Experimental data for dilution of benzene and n-hexane in water were used.Comment: e-mail me [email protected], 6 page

Smooth 3-dimensional canonical thresholds

If $X$ is an algebraic variety with at worst canonical singularities and $S$ is a \Q-Cartier hypersurface in $X$, the canonical threshold of the pair $(X,S)$ is the supremum of $c\in\R$ such that the pair $(X,cS)$ is canonical. We show that the set of all possible canonical thresholds of the pairs $(X,S)$, where $X$ is a germ of smooth 3-dimensional variety, satisfies the ascending chain condition. We also deduce a formula for the canonical threshold of (\C^3,S), where S is a Brieskorn singularity.Comment: Dedicated to the memory of my advisor Vasilii Alekseevich Iskovskikh. 14 pages; v3: minor correction

Thermodynamics of Substances with Negative Thermal Expansion Coefficient

The 1st law of thermodynamics for heat exchange is dQ=dU+PdV. According to K. Martinas etc., J. Non-Equil. Thermod. 23 (4), 351-375 (1988), for substances with negative thermal expansion coefficient, P in this law is negative. In the present paper it has been shown that P for such substances is positive but the sign before P must be minus not plus: dQ=dU-PdV.Comment: There was a misprint in Eq. (11), e-mail me [email protected], 7 page

Non-rational divisors over non-Gorenstein terminal singularities

Let $(X,o)$ be a germ of a 3-dimensional terminal singularity of index $m\geq 2$. If $(X,o)$ has type cAx/4, cD/3-3, cD/2-2, or cE/2, then assume that the standard equation of $X$ in $\mathbb{C}^4/\mathbb{Z}_m$ is non-degenerate with respect to its Newton diagram. Let $\pi\colon Y\to X$ be a resolution. We show that there are not more than 2 non-rational divisors $E_i$, $i=1,2$, on $Y$ such that $\pi(E_i)=o$ and discrepancy $a(E_i,X)\leq 1$. When such divisors exist, we describe them as exceptional divisors of certain blowups of $X$ and study their birational type.Comment: 17 pages, LaTeX2

Analysis of spatial correlations in a model 2D liquid through eigenvalues and eigenvectors of atomic level stress matrices

Considerations of local atomic level stresses associated with each atom represent a particular approach to address structures of disordered materials at the atomic level. We studied structural correlations in a two-dimensional model liquid using molecular dynamics simulations in the following way. We diagonalized the atomic level stress tensors of every atom and investigated correlations between the eigenvalues and orientations of the eigenvectors of different atoms as a function of distance between them. It is demonstrated that the suggested approach can be used to characterize structural correlations in disordered materials. In particular, we found that changes in the stress correlation functions on decrease of temperature are the most pronounced for the pairs of atoms with separation distance that corresponds to the first minimum in the pair density function. We also show that the angular dependencies of the stress correlation functions previously reported in [Phys. Rev. E v.91, 032301 (2015)] related not to the alleged anisotropies of the Eshelby's stress fields, but to the rotational properties of the stress tensors.Comment: 14 pages, 9 figure