The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities

This paper develops a significant extension of E. Lutwak's dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A new generalization of the well-known Busemann intersection inequality is also proved. Particular attention is given to precise equality conditions, which require results stating that a bounded Borel set, almost all of whose sections of a fixed dimension are essentially convex, is itself essentially convex.     

Stabilization of linear systems by rotation

We introduce the concept of “stabilization by rotation” for deterministic linear systems with negative trace. This concept encompasses the well-known concept of “vibrational stabilization” introduced by Meerkov in the 1970s and is a deterministic version of ‘stabilization by noise’ for stochastic systems as introduced by Arnold and coworkers in the 1980s. It is shown that a linear system with negative trace can be stabilized by adding a skew-symmetric matrix, multiplied by a suitable scalar so-called “gain function” (possibly a constant) which is sufficiently large. To overcome the problem of what is “sufficiently large”, we also present a servo mechanism which tunes the gain function by learning from the trajectory until finally the trajectory tends to zero. This approach allows to show that one of Meerkov's assumptions for vibrational stabilization is superfluous. Moreover, while Meerkov as well as Arnold and coworkers assume that a stabilizing periodic function or the noise has sufficiently large frequency and amplitude, we also provide a servo mechanism to determine this function dynamically in a deterministic setup.     

Linear dependence of stationary distributions in ergodic Markov decision processes

In ergodic MDPs we consider stationary distributions of policies that coincide in all but n states, in which one of two possible actions is chosen. We give conditions and formulas for linear dependence of the stationary distributions of n+2 such policies, and show some results about combinations and mixtures of policies.     

ON DIRECT METHOD OF SOLUTION FOR A CLASS OF SINGULAR INTEGRAL EQUATIONS

In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equations with certain analytic inputs. They obtain both the conditions of solvability and the solutions in closed form. It is noteworthy that the method is different from the classical one that is due to Lu.     

Random orderings of the integers and card shuffling

In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of these orders in terms of a class of generalized riffle shuffles.     

The first <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-cohomology of some groups with one end

Let p be a real number greater than one. In this paper we study the vanishing and nonvanishing of the first L ^p -cohomology space of some groups that have one end. We also make a connection between the first L ^p -cohomolgy space and the Floyd boundary of the Cayley graph of a group. We apply the result about Floyd boundaries to show that there exists a real number p such that the first L ^p -cohomology space of a nonelementary hyperbolic group does not vanish. Received: 4 August 2006 Revised: 2 November 2006     

Solubility of carbon dioxide in aqueous solutions of sodium chloride: Experimental results and correlation

The solubility of carbon dioxide in aqueous solutions of sodium chloride was measured in the temperature range from 40 to 160°C, up to 6 mol-kg salt solutions and total pressures up to 10 MPa. Pitzer's⁽¹⁾ equations as well as the Chen and Evans⁽²⁾ model were used to correlate the new data. Results are reported and compared to literature data and correlations.     

Synthesis of perfluoroheptanal by ozonolysis of perfluoro-1-octene

Perfluoro-1-octene was used as a model for developing a method for the preparative ozonolysis of perfluoro-1-alkenes to the corresponding perfluoro-nor-alkenals. Perfluoroheptanal was synthesized.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 1146–1147, June, 1994.     

Reactions ofN,N-dibromo-tert-butylamine with ketoximes

Reactions of acetone, cyclohexanone, and adamantanone oximes withN,N-dibromo-tert-butylamine result in the correspondingC-bromo-diazene-N-oxides. In the case of benzophenone oxime,O, O'-(diphenylmethylene)-bis-benzophenone oxime has been obtained.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 1, pp. 176–177, January, 1994.     

Sorption characteristics of polyfluorocarboxamides

The dependence of the variations in the differential molar free energy of sorption of polyfluorocarboxamides R_FC(O)NR₂ (R_F = CF₃(CF₂) _i ,i=0 to 5, R=H or Me) on capillary columns on the molecular structure of the amides, the temperature of analysis, and the nature of the stationary phases (OV-101, XE-60, and PEG-40M) were investigated by GLC. The peculiarities of the variation of the sorption characteristics of amides in XE-60 and PEG-40M polar phases, which cause the inversion of the order of retention of the members of pseudohomologous series, were found.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 5, pp. 868–871, May, 1994.We are grateful to R. V. Golovnya and D. N. Grigor'eva (Institute of Food Substances of the RAS, Moscow) for their participation in the discussion of the results obtained.