A generalization of the little Picard theorem

We show the little Picard theorem for a map of a quasicompact manifold to a relatively compact domain of Pn whose degeneracy locus of the Kobayashi pseudodistance is contained in some algebraic hypersurface.     

A generalization of the little Picard theorem

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 46, pp. 90–100, 1986.     

A remark on Fermat's last theorem

It is proved that the equation x^p + y^p + Z^p=Q, (xyz, p)=1 has no solutions in rational integers x, y, z for all odd prime numbers p for which q=pk + 1 is a prime number, k 82, k 0 (mod 3).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 67, pp. 223–224, 1977.     

A generalization of Plantholt's theorem

Let K denote the complete graph K_2n+1 with each edge replicated r times and let χ′(G) denote the chromatic index of a multigraph G. A multigraph G is critical if χ′(G) > χ′(G/e) for each edge e of G. Let S be a set of sn – 1 edges of K. We show that, for 0 < s ≦ r, G/S is critical and that χ′ (G/(S ∪{e})) = 2rn + r – s for all e ∈ E(G/S). Plantholt [M. Plantholt, The chromatic index of graphs with a spanning star. J. Graph Theory 5 (1981) 5–13] proved this result in the case when r = 1.     

A generalization of Jung's theorem

Jung's theorem establishes a relation between circumradius and diameter of a convex body. Half of the diameter can be interpreted as the maximum of circumradii of all 1-dimensional sections or 1-dimensional orthogonal projections of a convex body. This point of view leads to two series of j-dimensional circumradii, defined via sections or projections. In this paper we study some relations between these circumradii and by this we find a natural generalization of Jung's theorem.I would like to thank Prof. Dr J. M. Wills, who called my attention to these generalized circumradii.     

A generalization of Barbashin-Krasovski theorem

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a function V(t,x), a(‖x‖)?V(t,x)?b(‖x‖) for some a,b∈K, such that for some c∈K. In this paper we prove that if f(t,x) is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , t∈[−T,T], where , uniformly for (t,x)∈[−T,T]×BH.     

Fermat's theorem as a corollary to the multinomial theorem

The multinomial theorem and a basic property of multinomial coefficients yield a way of counting the number of chains of coloured beads of a fixed length where each colour can be used repeatedly. This coupled with a simple counting argument forms a concise proof of Fermat's theorem.     

A generalization of the remainder theorem and factor theorem

We propose a generalization of the classical Remainder Theorem for polynomials over commutative coefficient rings that allows calculating the remainder without using the long division method. As a consequence we obtain an extension of the classical Factor Theorem that provides a general divisibility criterion for polynomials. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school students.     

A generalization of the Looman-Menchoff theorem

In this paper we give a generalization of the classical Looman-Menchoff theorem:If f is a complex-valued continuous function of a complex variable in a domain G, f has partial derivatives f _x and f _y everywhere in G and the Cauchy Riemann equations f _x +if _y = 0are satisfied almost everywhere, then f is holomorphic in G. From our generalization of this theorem, we deduce a theroem of Sindalovskii [9] as a corollary and also answer some of the questions raised in [9]. We note in this context that, as far as we know, Sindalovskii’s result is the best published to date in this area.     

A generalization of the Riesz-Fischer theorem

An analog is established, in a certain sense, of the Riesz-Fischer theorem for the space L^P, p1, and a corollary derived.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 365–372, October, 1972.