A review of statistical models for global optimization

A review of statistical models for global optimization is presented. Rationality of the search for a global minimum is formulated axiomatically and the features of the corresponding algorithm are derived from the axioms. Furthermore the results of some applications of the proposed algorithm are presented and the perspectives of the approach are discussed.     

Discrete limit theorems for general Dirichlet series. III

Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given. Partially supported by Lithuanian Foundation of Studies and Science     

Surface integral and Gauss-Ostrogradskij theorem from the viewpoint of applications

Making use of a surface integral defined without use of the partition of unity, trace theorems and the Gauss-Ostrogradskij theorem are proved in the case of three-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces H ^1,p () (1 p < ). The paper is a generalization of the previous author's paper which is devoted to the line integral.     

Plane Sections of Convex Bodies of Maximal Volume

Let K = {K ₀ ,... ,K _k } be a family of convex bodies in R ⁿ , 1≤ k≤ n-1 . We prove, generalizing results from [9], [10], [13], and [14], that there always exists an affine k -dimensional plane A _k (subset, dbl equals) R ⁿ , called a common maximal k-transversal of K , such that, for each i∈ {0,... ,k} and each x∈ R ⁿ , where V _k is the k -dimensional Lebesgue measure in A _k and A _k +x . Given a family K = {K _i } _i=0 ^l of convex bodies in R ⁿ , l < k , the set C _k ( K ) of all common maximal k -transversals of K is not only nonempty but has to be ``large' both from the measure theoretic and the topological point of view. It is shown that C <sub>k</sub> ( K ) cannot be included in a ν -dimensional C <sup>1</sup> submanifold (or more generally in an ( H <sup>ν</sup> , ν) -rectifiable, H <sup>ν</sup> -measurable subset) of the affine Grassmannian AGr <sub>n,k</sub> of all affine k -dimensional planes of R <sup>n</sup> , of O(n+1) -invariant ν -dimensional (Hausdorff) measure less than some positive constant c <sub>n,k,l</sub> , where ν = (k-l)(n-k) . As usual, the ``affine' Grassmannian AGr _n,k is viewed as a subspace of the Grassmannian Gr _n+1,k+1 of all linear (k+1) -dimensional subspaces of R ⁿ⁺¹ . On the topological side we show that there exists a nonzero cohomology class θ∈ H ^n-k (G _n+1,k+1 ;Z ₂ ) such that the class θ ^l+1 is concentrated in an arbitrarily small neighborhood of C _k ( K ) . As an immediate consequence we deduce that the Lyusternik—Shnirel'man category of the space C _k ( K ) relative to Gr _n+1,k+1 is ≥ k-l . Finally, we show that there exists a link between these two results by showing that a cohomologically ``big' subspace of Gr _n+1,k+1 has to be large also in a measure theoretic sense. Received May 22, 1998, and in revised form March 27, 2000. Online publication September 22, 2000.     

Transient time in homogeneous nucleation

The problem of the transient time in nucleation is studied. The size-dependent transient time at a constant temperature is defined and the approximate analytical solution is found and compared with the exact numerical solution for the model Li₂O.2 SiO₂ melt. It is shown that the analytical solution for the size-dependent transient time is in agreement with the numerical result.     

Discrete Universality in the Selberg Class

The Selberg class S consists of functions L(s) that are defined by Dirichlet series and satisfy four axioms (Ramanujan conjecture, analytic continuation, functional equation, and Euler product). It has been known that functions in S that satisfy the mean value condition on primes are universal in the sense of Voronin, i.e., every function in a sufficiently wide class of analytic functions can be approximated by the shifts L(s + iτ ), τ ∈ R. In this paper we show that every function in the same class of analytic functions can be approximated by the discrete shifts L(s + ikh), k = 0, 1,..., where h > 0 is an arbitrary fixed number.     

Insight into the chiral induction in supramolecular stacks through preferential chiral solvation

Preferred handedness in the supramolecular chirality of self-assembled achiral oligo(p-phenylenevinylene) (OPV) derivatives is induced by chiral solvents and spectroscopic probing provides insight into the mechanistic aspects of this chiral induction through chiral solvation.     

Reactions of Doubly Ionized Benzene with Nitrogen and Water: A Nitrogen‐Mediated Entry into Superacid Chemistry

Even in the highly diluted gas phase, rather than electron transfer the benzene dication C₆H₆²⁺ undergoes association with dinitrogen to form a transient C₆H₆N₂²⁺ dication which is best described as a ring‐protonated phenyl diazonium ion. Isotopic labeling studies, photoionization experiments using synchrotron radiation, and quantum chemical computations fully support the formation of protonated diazonium, which is in turn a prototype species of superacidic chemistry in solution. Additionally, reactions of C₆H₆²⁺ with background water involve the transient formation of diprotonated phenol and, among other things, afford a long‐lived C₆H₆OH₂²⁺ dication, which is attributed to the hydration product of Hogeveen’s elusive pyramidal structure of C₆H₆²⁺, as the global minimum of doubly ionized benzene. Nitrogen is essential for the formation of the C₆H₆OH₂²⁺ dication in that it mediates the formation of the water adduct, while the bimolecular encounter of the C₆H₆²⁺ dication with water only leads to (dissociative) electron transfer.