Additive complexity in directed computations

AbstractA straight-line additive computation which computes a set A of linear forms can be presented as a product of elementary matrices (one instruction of such a computation corresponds to a multiplication by an elementary matrix). For the general complexity measure no methods for obtaining nonlinear lower bounds for concrete natural sets of linear forms are known at the moment (under the general complexity measure of A we mean the minimal number of multipliers in products computing A). In the paper three complexity measures (triangular, directed and a modification of the latter—reduced directed complexity) close in spirit each to others are defined and investigated. For these measures some nonlinear lower bounds are obtained. Moreover, the problem of the exact explicit calculation of the directed complexity is solved for which a suitable algebraic apparatus (the generalized Bruhat decomposition) is developed. Apparatus is exposed in the appendix to the paper

Hexacarbonyl­technetium(I) perchlorate

The title compound, [Tc(CO)6]ClO4, was synthesized by the reaction of [TcCl(CO)5] with AgClO4, followed by acidification with HClO4 under a CO atmosphere. The [Tc(CO)6]+ cation has close to idealized octa­hedral geometry, with the bond angles between cis-CO groups close to 90° and the Tc—C bond lengths in the range 2.025 (3)–2.029 (3)Å. The perchlorate anion is disordered over two crystallographically equivalent half-occupied positions. The Tc atom in the [Tc(CO)6]+ cation is located on an inversion centre

Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients

We discuss two conjectures. (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k) \forall i,j,k \in {1,...,n} (x_i \cdot x_j=x_k \Rightarrow y_i \cdot y_j=y_k) (II) Let G be an additive subgroup of C. Then for each x_1,...,x_n \in G there exist y_1,...,y_n \in G \cap Q such that \forall i \in {1,...,n} |y_i| \leq 2^{n-1} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n} (x_i+x_j=x_k \Rightarrow y_i+y_j=y_k)Comment: LaTeX2e, 28 pages, a shortened and revised version will appear in Mathematical Logic Quarterly 56 (2010), no.2, under the title ``Two conjectures on the arithmetic in R and C'

Reinvestigation of tris­odium dihydroxido­tetra­oxidoneptunate(VII) dihydrate

The title compound, Na3[NpO4(OH)2]·2H2O, contains distorted tetra­gonal–bipyramidal centrosymmetric [NpO4(OH)2]3− complex anions. The Np—O distances are 1.8975 (7) and 1.8891 (7) Å in the NpO4 group and 2.3451 (7) Å to the OH group. Both Na atoms (one in a general position, the second in a special position on an inversion centre) have a distorted octahedral oxygen environment

Computing the Rank Profile Matrix

The row (resp. column) rank profile of a matrix describes the staircase shape of its row (resp. column) echelon form. In an ISSAC'13 paper, we proposed a recursive Gaussian elimination that can compute simultaneously the row and column rank profiles of a matrix as well as those of all of its leading sub-matrices, in the same time as state of the art Gaussian elimination algorithms. Here we first study the conditions making a Gaus-sian elimination algorithm reveal this information. Therefore, we propose the definition of a new matrix invariant, the rank profile matrix, summarizing all information on the row and column rank profiles of all the leading sub-matrices. We also explore the conditions for a Gaussian elimination algorithm to compute all or part of this invariant, through the corresponding PLUQ decomposition. As a consequence, we show that the classical iterative CUP decomposition algorithm can actually be adapted to compute the rank profile matrix. Used, in a Crout variant, as a base-case to our ISSAC'13 implementation, it delivers a significant improvement in efficiency. Second, the row (resp. column) echelon form of a matrix are usually computed via different dedicated triangular decompositions. We show here that, from some PLUQ decompositions, it is possible to recover the row and column echelon forms of a matrix and of any of its leading sub-matrices thanks to an elementary post-processing algorithm

Конституційне судочинство як спосіб конституційного контролю : історична ретроспектива європейської моделі конституційної юстиції в Україні.

Григор'єв, В.А. Конституційне судочинство як спосіб конституційного контролю : історична ретроспектива європейської моделі конституційної юстиції в Україні. / В.А. Григор'єв // Актуальні проблеми політики : зб. наук. пр. / редкол. : С. В. Ківалов (голов. ред.), Л. І. Кормич (заст. голов. ред.), Ю. П. Аленін [та ін.] ; МОНмолодьспорт України, НУ ОЮА. – Одеса : Фенікс, 2011. – Вип. 43. С. 218 - 226.The given article is devoted to the constitutional legal proceedings as a way of the constitutional control realization. Comparing the constitutional legal proceedings to the constitutional control, the author comes to a conclusion that the concept the constitutional legal proceedings is wider, than the constitutional control as powers of the constitutional courts concern not only the questions of conformity check of the Constitution, but also questions of constitutional procedure observance the of investigation and a legal investigation on dis- placement of the President from a post as impeachment, giving an offi cial interpretation of the Constitution laws and others

Totalitarianism and geography: L.S. Berg and the defence of an academic discipline in the age of Stalin

In considering the complex relationship between science and politics, the article focuses upon the career of the eminent Russian scholar, Lev Semenovich Berg (1876–1950), one of the leading geographers of the Stalin period. Already before the Russian Revolution, Berg had developed a naturalistic notion of landscape geography which later appeared to contradict some aspects of Marxist–Leninist ideology. Based partly upon Berg's personal archive, the article discusses the effects of the 1917 revolution, the radical changes which Stalin's cultural revolution (from the late 1920s) brought upon Soviet science, and the attacks made upon Berg and his concept of landscape geography thereafter. The ways in which Berg managed to defend his notion of geography (sometimes in surprisingly bold ways) are considered. It is argued that geography's position under Stalin was different from that of certain other disciplines in that its ideological disputes may have been regarded as of little significance by the party leaders, certainly by comparison with its practical importance, thus providing a degree of ‘freedom’ for some geographers at least analogous to that which has been described by Weiner (1999. A little corner of freedom: Russian nature protection from Stalin to Gorbachev. Berkeley: University of California Press) for conservationists. It is concluded that Berg and others successfully upheld a concept of scientific integrity and limited autonomy even under Stalinism, and that, in an era of ‘Big Science’, no modernizing state could or can afford to emasculate these things entirely

REGULAR QUATERNIONIC FUNCTIONS AND THEIR APPLICATIONS

The theory of regular quaternionic functions of a reduced quaternionic variable is a 3-dimensional generalization of complex analysis. The Moisil-Theodorescu system (MTS) is a regularity condition for such functions depending on the radius vector r = ix+jy+kz seen as a reduced quaternionic variable. The analogues of the main theorems of complex analysis for the MTS in quaternion forms are established: Cauchy, Cauchy integral formula, Taylor and Laurent series, approximation theorems and Cauchy type integral properties. The analogues of positive powers (inner spherical monogenics) are investigated: the set of recurrence formulas between the inner spherical monogenics and the explicit formulas are established. Some applications of the regular function in the elasticity theory and hydrodynamics are given