Abstract:
Sodium thiopental, used in a narcotic dose, makes it possible to identify the nervous processes that underlie consciousness and establish the causes of its disorder. When studying the cortical EEG activity, the impulses of individual nerve cells and the electromyographic activity of the muscles of the forelimb, it was found that thiopental blocks a number of neuronal reactions requiring energy support: tonic activating reactions to acetylcholine, applied to neurons, cease; the rate of spontaneous neuronal activity drops; the stage of non-specific activation in response to electrocutaneous stimulation disappears. So, thiopental blocks consciousness by significant limitation of the brain energy metabolism. This results in a loss of the adaptive function of the central nervous system. At the same time, glutamatergic excitation, the formation of which does not depend on energy support, is resistant to the action of thiopental. The blocking of the brain’s energy supply caused by thiopental, in accordance with its depth, develops in two stages—hypoxic and narcotic. The hypoxic stage is accompanied by hyperactivity in the nervous system, which is manifested by epileptiform discharges on the EEG and powerful unmotivated movement; the narcotic stage is associated with blockade of motor activity and flattening of EEG oscillations. The post-narcotic state associated with the consequence of the hypoxic effect of thiopental leads to the loss of ionic homeostasis and is accompanied by a steady drop in the amplitude of cortical neuron spikes.

Abstract:
We discuss brane wormhole solution when classical brane action contains 4d curvature. The equations of motion for the cases with R=0 and $R\ne 0$ are obtained. Their numerical solutions corresponding to wormhole are found for specific boundary conditions.

Abstract:
An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $\alpha, \beta\in \Bbb F$ such that $\alpha\neq 0$ and $[[a,b],c]=\alpha [a,[b,c]]+\beta[[a,c],b]$. Examples of Lie type algebras are associative algebras, Lie algebras, Leibniz algebras, etc. It is proved that if a Lie type algebra $L$ admits an automorphism of finite order $n$ with finite-dimensional fixed-point subalgebra of dimension $m$, then $L$ has a soluble ideal of finite codimension bounded in terms of $n$ and $m$ and of derived length bounded in terms of $n$.

Abstract:
We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations. In this paper we consider a number of exact solutions obtained by the method of separation of variables. Some applications to Cosmology and BH entropy are briefly mentioned.

Abstract:
Internal waves in the atmosphere and ocean are generated frequently from the interaction of mean flow with bottom obstacles such as mountains and submarine ridges. Analysis of these environmental phenomena involves theoretical models of non-homogeneous fluid affected by the gravity. In this paper, a semi-analytical model of stratified flow over the mountain range is considered under the assumption of small amplitude of the topography. Attention is focused on stationary wave patterns forced above the rough terrain. Adapted to account for such terrain, model equations involves exact topographic condition settled on the uneven ground surface. Wave solutions corresponding to sinusoidal topography with a finite number of peaks are calculated and examined.

Abstract:
For the description of the Universe expansion, compatible with observational data, a model of modified gravity - Lovelock gravity with dilaton - is investigated. D-dimensional space with 3- and (D-4)-dimensional maximally symmetric subspaces is considered. Space without matter and space with perfect fluid are under test. In various forms of the theory under way (third order without dilaton and second order - Einstein-Gauss-Bonnet gravity - with dilaton and without it) stationary, power-law, exponential and exponent-of-exponent form cosmological solutions are obtained. Last two forms include solutions which are clear to describe accelerating expansion of 3-dimensional subspace. Also there is a set of solutions describing cosmological expansion which does not tend to isotropization in the presence of matter.

Abstract:
We study cosmological models derived from higher-order Gauss-Bonnet gravity $F(R,G)$ by using the Lagrange multiplier approach without assuming the presence of additional fields with the exception of standard perfect fluid matter. The presence of Lagrange multipliers reduces the number of allowed solutions. We need to introduce compatibility conditions of the FRW equations, which impose strict restrictions on the metric or require the introduction of additional exotic matter. Several classes of $F(R,G)$ models are generated and discussed.

Abstract:
Suppose that a finite group $G$ admits a Frobenius group of automorphisms FH of coprime order with cyclic kernel F and complement H such that the fixed point subgroup $C_G(H)$ of the complement is nilpotent of class $c$. It is proved that $G$ has a nilpotent characteristic subgroup of index bounded in terms of $c$, $|C_G(F)|$, and $|FH|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. This generalizes the previous theorem of the authors and P. Shumyatsky, where for the case of $C_G(F)=1$ the whole group was proved to be nilpotent of $(c,|H|)$-bounded class. Examples show that the condition of $F$ being cyclic is essential. B. Hartley's theorem based on the classification provides reduction to soluble groups. Then representation theory arguments are used to bound the index of the Fitting subgroup. Lie ring methods are used for nilpotent groups. A similar theorem on Lie rings with a metacyclic Frobenius group of automorphisms $FH$ is also proved.

Abstract:
Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ such that the characteristic of the ground field does not divide $|H|$. It is proved that if the subalgebra $C_L(F)$ of fixed points of the kernel has finite dimension $m$ and the subalgebra $C_L(H)$ of fixed points of the complement is nilpotent of class $c$, then $L$ has a nilpotent subalgebra of finite codimension bounded in terms of $m$, $c$, $|H|$, and $|F|$ whose nilpotency class is bounded in terms of only $|H|$ and $c$. Examples show that the condition of the kernel $F$ being cyclic is essential.

Abstract:
Suppose that a finite $p$-group $P$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ that is a cyclic $p$-group and with complement $H$. It is proved that if the fixed-point subgroup $C_P(H)$ of the complement is nilpotent of class $c$, then $P$ has a characteristic subgroup of index bounded in terms of $c$, $|C_P(F)|$, and $|F|$ whose nilpotency class is bounded in terms of $c$ and $|H|$ only. Examples show that the condition of $F$ being cyclic is essential. The proof is based on a Lie ring method and a theorem of the authors and P. Shumyatsky about Lie rings with a metacyclic Frobenius group of automorphisms $FH$. It is also proved that $P$ has a characteristic subgroup of $(|C_P(F)|, |F|)$-bounded index whose order and rank are bounded in terms of $|H|$ and the order and rank of $C_P(H)$, respectively, and whose exponent is bounded in terms of the exponent of $C_P(H)$.