On Minkowski Sums of Many Small Sets

It is proved that a weakly closed subset of a Banach space is convex if and only if it can be represented as the sum of sets of arbitrarily small diameter.     

On certain generalizations of the Levi-Civita and Wilson functional equations

We study the functional equation

$$\begin{aligned} \sum _{i=1}^mf_i(b_ix+c_iy)= \sum _{k=1}^nu_k(y)v_k(x) \end{aligned}$$

with \(x,y\in \mathbb {R}^d\) and \(b_i,c_i\in {GL}(d,\mathbb {R})\), both in the classical context of continuous complex-valued functions and in the framework of complex-valued Schwartz distributions, where these equations are properly introduced in two different ways. The solution sets are, typically, exponential polynomials and, in some particular cases, related to the so called characterization problem of the normal distribution in Probability Theory, they reduce to ordinary polynomials.

     

Thermal Degradation of Polymers. IV. Poly-2,2′-(m-phenylene)-5,5′-bibenzimidazole

The thermal decomposition of poly-2,2′-(m-phenylene)-5,5′-bibenzimidazole has been investigated at temperatures to 916°C. Mass-spectrometric thermal analysis (MTA), supported by elemental analysis of residues at various stages of a thermogravimetric analysis, was used to determine product distribution as a function of temperature. Below 550°C, the major product is water. Above 550°C, degradation of the heterocyclic structure to hydrogen cyanide, ammonia, carbon monoxide, and water begins. Hydrogen and methane probably are formed from decomposition and condensation of the carbocyclic structure. Activation energies for the formation of each major product were derived from the MTA data. The average of these is somewhat temperature-dependent but agrees within experimental error with the value of 44 ± 11 kcal obtained from isothermal kinetics. A mechanism involving initial hydrolysis of the polymer to an amine-substituted polyamide is postulated. Subsequent homolytic reactions account for formation of the major products.     

Bismuth valence states and emission centers in Mg-Al-silicate glass

Two Al-Mg-silicate glass sample sets with variable Bi₂O₃ content were prepared. One set was sintered by melting in an iridium crucible at 1850°C in nitrogen, the other—in alumina crucibles at 1550°C in air. Their absorption and emission properties were investigated and compared in the 200–1600 nm spectral range. It was determined that the visible range extinctions values of high-temperature melted samples were almost two orders of magnitude higher than those of the low-temperature melted samples with the same doping level. The concentration dependences of the extinction values at 500, 700 and 800 nm were nonlinear. The investigations allowed us to estimate the absolute concentrations of Bi³⁺ ions and NIR-emitting centers as well as the efficiency of the Bi³⁺ ions conversion into these centers. It was concluded that optical centers emitting at 1100 and 1300 nm contain a pair of bismuth ions. Speculations about some possible structures of Bi dimer NIR-emitting centers were made.     

Not every pseudoalgebra is equivalent to a strict one

We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to be strictifiable. Our counterexample comes from higher category theory: the strict algebras are strict 3-categories, and the pseudoalgebras are a type of semi-strict 3-category lying in between Gray-categories and tricategories. Thus, the result follows from the fact that not every Gray-category is equivalent to a strict 3-category, connecting 2-categorical and higher-categorical coherence theory. In particular, any nontrivially braided monoidal category gives an example of a pseudoalgebra that is not equivalent to a strict one.     

Simulation model for shapiro time delay and light deflection experiments

The time delay experiment proposed by I.I. Shapiro in 1964 and conducted in the seventies was the most precise experiment of general relativity until that time. Further experimentation has improved the accuracy level of both the time delay and the light deflection experiments. A simulation model is proposed that involves only a simple mass and time transformation factor involving velocity of light. The light deflection and the time delay experiments are numerically simulated using this model that does not use the general relativistic equations. The computed values presented in this paper compare well with recent levels of accuracy of their respective experimental results.     

Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action

It is proved that a Lie algebra of compact operators with a non-zero Volterra ideal is reducible (has a nontrivial invariant subspace). A number of other criteria of reducibility for collections of operators is obtained. The results are applied to the structure theory of Lie algebras of compact operators and normed Lie algebras with compact adjoint action.     

Addition to the Solution of the Problem on the Motion of a Free Nonrelativistic Electron in a Magnetic Field

In addition to the solution of the quantum mechanical problem on the motion of a free electron in a magnetic field, given by L. D. Landau in his pioneering work, this quantum mechanical problem is solved in view of the fact that the sum of the components of the free kinetic energy of an electron along two axes is a periodic quantity varying identically within the boundaries of every Landau energy level. This periodicity is a consequence of the quantized motion of an electron in a plane normal to the direction of the external magnetic field.