Tensor product in symmetric function spaces

A concept of multiplicator of symmetric function space concerning to projective tensor product is introduced and studied. This allows to obtain some concrete results. In particular, the well-known theorem of R. O'Neil about the boundedness of tensor product in the Lorentz spaces L_{p,q} is discussed.Comment: 17 page

On uniqueness of distribution of a random variable whose independent copies span a subspace in L_p

Let 1\leq p<2 and let L_p=L_p[0,1] be the classical L_p-space of all (classes of) p-integrable functions on [0,1]. It is known that a sequence of independent copies of a mean zero random variable f from L_p spans in L_p a subspace isomorphic to some Orlicz sequence space l_M. We present precise connections between M and f and establish conditions under which the distribution of a random variable f whose independent copies span l_M in L_p is essentially unique.Comment: 14 pages, submitte

Best constants in Rosenthal-type inequalities and the Kruglov operator

Let $X$ be a symmetric Banach function space on $[0,1]$ with the Kruglov property, and let $\mathbf{f}=\{f_k\}_{{k=1}}^n$, $n\ge1$ be an arbitrary sequence of independent random variables in $X$. This paper presents sharp estimates in the deterministic characterization of the quantities $$\Biggl\|\sum_{{k=1}}^nf_k\Biggr\|_X,\Biggl\|\Biggl(\sum_{{k=1}}^n|f_k|^p\Biggr)^{1/p}\Biggr\|_X,\qquad 1\leq p<\infty,$$ in terms of the sum of disjoint copies of individual terms of $\mathbf{f}$. Our method is novel and based on the important recent advances in the study of the Kruglov property through an operator approach made earlier by the authors. In particular, we discover that the sharp constants in the characterization above are equivalent to the norm of the Kruglov operator in $X$.Comment: Published in at http://dx.doi.org/10.1214/10-AOP529 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On unconditionality of fractional Rademacher chaos in symmetric spaces

We study density estimates of an index set $\mathcal{A}$, under which unconditionality (or even a weaker property of the random unconditional divergence) of the corresponding Rademacher fractional chaos $\{r_{j_1}(t)\cdot r_{j_2}(t)\cdot\dots\cdot r_{j_d}(t)\}_{(j_1,j_2,\dots,j_d)\in \mathcal{A}}$ in a symmetric space $X$ implies its equivalence in $X$ to the canonical basis in $\ell_2$. In the special case of Orlicz spaces $L_M$, unconditionality of this system is also equivalent to the fact that a certain exponential Orlicz space embeds into $L_M$.Comment: to appear in Izv. RA

Position of Persia on the problem of «Russian transit» in the XVI–XVII centuries: a plot about Shirvan cities

The article attempts to consider the position of the Persian Safavid state on the question of using the territory of the Muscovy empire for the development of the European-Asian relations in the period of the XVI–XVII centuries with the use of the method of diachronic analysis and historical periodization. The aforementioned problem is studied as a separate aspect of scientific topics related to the place of the then Russia in the process of the European expansion into the countries of the East. The object of the analysis is the practical activities of the Persian administrations, which had both a direct and indirect relationship to the conventional topic of «Russian transit». Nowadays the experience of the comprehensive and conceptual study of this issue is absent not only in the domestic, but also in the foreign historical science. The specific subject of the study is related to the repeated promises of official Iran to transfer control over the largest centers of the historical region of Shirvan to the Russian tsars. The source base for this work was an array of diplomatic documents on the history of Russian-Persian interstate relations of the XVI–XVII centuries. These materials, in particular, reflected Moscow’s reaction to the Safavid proposals for territorial concessions. The scientific works of Russian-speaking and foreign specialists, including the largest representatives of classical and modern historical Iranian studies, were also used. The analysis made it possible to fit the diplomatic issue about the fate of the Shirvan cities into the context of the general problem of «Russian transit». It is hypothesized that the Persian government deliberately initiated this discussion taking into account the economic potential of the Volga-Caspian trade route. It is concluded that if the appropriate agreements between the two states were reached, the Russian empire could in the future produce a certain revolution in the intercontinental trade, especially in the silk market

Electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with nuclei

We review and summarize recent theoretical and experimental work on electron spin dynamics in quantum dots and related nanostructures due to hyperfine interaction with surrounding nuclear spins. This topic is of particular interest with respect to several proposals for quantum information processing in solid state systems. Specifically, we investigate the hyperfine interaction of an electron spin confined in a quantum dot in an s-type conduction band with the nuclear spins in the dot. This interaction is proportional to the square modulus of the electron wave function at the location of each nucleus leading to an inhomogeneous coupling, i.e. nuclei in different locations are coupled with different strength. In the case of an initially fully polarized nuclear spin system an exact analytical solution for the spin dynamics can be found. For not completely polarized nuclei, approximation-free results can only be obtained numerically in sufficiently small systems. We compare these exact results with findings from several approximation strategies.Comment: 26 pages, 9 figures. Topical Review to appear in J. Phys.: Condens. Matte