Dual pairs and Kostant-Sekiguchi correspondence. II. Classification of nilpotent elements

We classify the homogeneous nilpotent orbits in certain Lie color algebras and specialize the results to the setting of a real reductive dual pair. For any member of a dual pair, we prove the bijectivity of the two Kostant-Sekiguchi maps by straightforward argument. For a dual pair we determine the correspondence of the real orbits, the correspondence of the complex orbits and explain how these two relations behave under the Kostant-Sekiguchi maps. In particular we prove that for a dual pair in the stable range there is a Kostant-Sekiguchi map such that the conjecture formulated in [6] holds. We also show that the conjecture cannot be true in general.     

Total linearization of probability density evolution equations

Total linearization of the Boltzmann and Vlasov equations is used to present a technique applicable to equations which have polynomial-type nonlinearities and describe the evolution of probability density functions or distributions. It is an extension of an earlier result, where a method yielding a linear equation with a nonlinear constraint was presented.     

Turán numbers for odd wheels

The Turán number $\text{ex}(n,G)$ is the maximum number of edges in any $n$-vertex graph that does not contain a subgraph isomorphic to $G$. A wheel$W_n$ is a graph on $n$ vertices obtained from a $C_{n?1}$ by adding one vertex $w$ and making $w$ adjacent to all vertices of the $C_{n?1}$. We obtain two exact values for small wheels:

$\text{ex}(n,W_5)=?\frac{n^2}{4}+\frac{n}{2}?,$

$\text{ex}(n,W_7)=?\frac{n^2}{4}+\frac{n}{2}+1?.$

Given that $\text{ex}(n,W_6)$ is already known, this paper completes the spectrum for all wheels up to 7 vertices. In addition, we present the construction which gives us the lower bound $\text{ex}(n,W_{2k+1})>?\frac{n^2}{4}?+?\frac{n}{2}?$ in general case.     

Background removal procedure for rapid scan EPR

In rapid scan EPR the changing magnetic field creates a background signal with components at the scan frequency and its harmonics. The amplitude of the background signal increases with scan width and is more significant for weak EPR signals such as are obtained in the presence of magnetic field gradients. A procedure for distinguishing this background from the EPR signal is proposed, mathematically described, and tested for various experimental conditions.     

On some three color Ramsey numbers for paths and cycles

For given graphs G₁,G₂,…,G_k, k≥2, the multicolor Ramsey number, denoted by R(G₁,G₂,…,G_k), is the smallest integer n such that if we arbitrarily color the edges of a complete graph on n vertices with k colors, there is always a monochromatic copy of G_i colored with i, for some 1≤i≤k. Let P_k (resp. C_k) be the path (resp. cycle) on k vertices. In the paper we consider the value for numbers of type R(P_i,P_k,C_m) for odd m, k≥m≥3 and when i is odd, and when i is even. In addition, we provide the exact values for Ramsey numbers R(P₃,P_k,C₄) for all integers k≥3.     

Time and a physical Hamiltonian for quantum gravity

We present a nonperturbative quantization of general relativity coupled to dust and other matter fields. The dust provides a natural time variable, leading to a physical Hamiltonian with spatial diffeomorphism symmetry. The surprising feature is that the Hamiltonian is not a square root. This property, together with the kinematical structure of loop quantum gravity, provides a complete theory of quantum gravity, and puts applications to cosmology, quantum gravitational collapse, and Hawking radiation within technical reach.     

Positive Effect of Cannabis sativa L. Herb Extracts on Skin Cells and Assessment of Cannabinoid-Based Hydrogels Properties

The skin is an organ that is constantly exposed to many external factors that can affect its structure and function. Due to the presence of different cannabinoid receptors on many types of skin cells, cannabinoids can interact directly with them. Therefore, as part of this work, the impact of two types of Cannabis sativa L. herb extracts on keratinocytes and fibroblasts was assessed. The content of biologically active compounds such as phenols, flavonoids, chlorophylls and cannabinoids was evaluated. The antioxidant capacity of prepared extracts using the DPPH radical, H₂DCFDA probe and measurement of superoxide dismutase activity was also assessed. The cytotoxicity of hemp extracts was determined using the Alamar Blue, Neutral Red and LDH assays. The ability of the extracts to inhibit the activity of matrix metalloproteinases, collagenase and elastase, was assessed. Preparations of model hydrogels were also prepared and their effect on transepidermal water loss and skin hydration was measured. The obtained results indicate that hemp extracts can be a valuable source of biologically active substances that reduce oxidative stress, inhibit skin aging processes and positively affect the viability of skin cells. The analysis also showed that hydrogels based on cannabis extracts have a positive effect on skin hydration.