Let $G$ be a finite group $G$, $K$ a field of characteristic $p\geq17$ and let $U$ be the group of units in $KG$. We show that if the derived length of $U$ does not exceed $4$, then $G$ must be abelian., Dishari Chaudhuri, Anupam Saikia., and Obsahuje bibliografické odkazy
The paper deals with the problem of finding the field of force that generates a given ($N-1$)-parametric family of orbits for a mechanical system with $N$ degrees of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of celestial mechanics. We state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov's problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function $U$ that is capable of generating a bi-parametric family of orbits for a particle in space. We determine the equations for the sought-for function $U$ and show that on the basis of these equations we can define a system of two linear partial differential equations with respect to $U$ which contains as a particular case the Szebehely equation. We solve completely a special case of the inverse dynamics problem of constructing $U$ that generates a given family of conics known as Bertrand's problem. At the end we establish the relation between Bertrand's problem and the solutions to the Heun differential equation. We illustrate our results by several examples.
The developmental stages and life cycle of the nematode Camallanus anabantis Pcarse, 1933 an intestinal parasite of Anabas testudineus (Bloch) arc described. The copepod Mesocyclops leuckarti (Claus) was used as experimental intermediate host. After being ingested by the copepods the nematode first-stage larvae enter its haemocoel, where they moult twice, 4 d.p.i. and 11 d.p.i., at 21-26°C, respectively to become the infective third-stage larvae. The definitive fish hosts become infected when feeding on copcpods harbouring infective larvae. In the fish host’s intestine the larvae undergo two more moults, the third on day 15 p.i. The fourth moult of “male” larvae occurred on day 68 p.i. and that of “female” larvae on day 86 pi. at water temperatures 24-36°C- A female with eggs and few larvae in the uteri was first observed on day 187 p.i.
The development of Spirocamallanus mysti (Karve, 1952) was studied in the copepod hosts Mesocyclops crassus (Fischer) and M. leuckarti (Claus) and in the fish host Mystus viltatus (Bloch). When eaten by copepods the first-stage larvae burrow through the intestinal wall into the haemocoel and there they moulted twice to become the third, infective stage. The first moulting occurred on day 4 p.i. at 18-2ГС (on day 6 p.i. at 16-20"C) and the second moultingoccurred on day 8 p.i. at 18-19.5"C (on day 11 p.i. at 16-20"C. Further development occurred only after reaching the stomach of the fish definitive host. In the fish stomach two more larval moultings occurred, the third on day 15 p.i. and the fourth (final) on day 37 p.i. in “male” larvae and day 67 p.i. in “female” larvae. The individual developmental stages and the morphological changes occurring during development are described in detail.
On every subspace of $l_{\infty }(\mathbb N)$ which contains an uncountable $\omega $-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin's Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of $l_{\infty }(\mathbb N)$ is infinite. This provides a partial answer to a question asked by Johnson and Odell.
Let G be a finite group. The intersection graph ΔG of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G, and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1, where 1 denotes the trivial subgroup of order 1. A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection graphs of finite non-abelian simple groups have an upper bound 28. In particular, the intersection graph of a finite non-abelian simple group is connected., Xuanlong Ma., and Obsahuje seznam literatury
In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence xn+1 = a0xn + a1xn−1 + . . . + akxn−k ⁄ b0xn + b1xn−1 + . . . + bkxn−k , n = 0, 1, . . . where the parameters ai and bi for i = 0, 1, . . . , k are positive real numbers and the initial conditions x−k, x−k+1, . . . , x0 are arbitrary positive numbers.