Mollification is a remarkable procedure for approximating real valued functions by smooth C-infinity functions. We describe geometric adaptations of this procedure for approximating certain curves, especially certain planar curves. For C-1 curves with a weak curvature vector we give a mollification procedure leaving curves of constant curvature invariant in analogy to mollification of functions leaving solutions of Laplace's equation (harmonic functions) invariant. Generalization of the basic constructions are also given for curves in higher dimensional Euclidean spaces and curves of lower regularity.
For a regular space X, let CL(X) (1C(X)) be the set of all nonempty closed (compact) subsets of X. Denote the set CL(X) with the Vietoris topology by (CL(X),V), with the locally finite topology by (CL(X), LF) and with the Fell topology by (CL(X), F), respectively. Weak first-countability and first-countability in hyperspaces are studied and the following results are obtained. (1) For a topological space X, (CL(X), LF) is weakly first-countable if and only if it is first-countable. (2) Let X be a topological space. (1C(X), F) is first-countable if and only if X is first-countable, every compact subset of X is (hereditarily) separable and the complement of every nonempty compact subset of X is hemicompact. (3) It is pointed out that for a topological space X, if (CL(X),V) is a Frechet-Urysohn space with an w omega-base, then it is first-countable, which answers a question in literature. A few related questions are posed.
It is known that the functional equation f(n)(z) + g(n)(z) = 1 can be regarded as the Fermat-type equations over function fields, where n is a positive integer. The purpose of the paper is to find out the precise form of the finite order entire solutions of the following Fermat-type difference equations in C-m such as (P-1(z)f(z))(n1) + (P-2(z)f(z + c))(n2) = Q(z) and P-1(2 )(z)(a(1)f (z + c) + a(2)f (z))(2) + P- 2(2) (z)(a(3)f (z + c) + a(4)f (z))(2) = Q(z), where n(1) and n(2) are two positive integers such that n(1) + n(2) > 2, a(j) is an element of C for j = 1, 2, 3, 4 such that a(3)a(2)-a(1)a(4) =/ 0, c is an element of C-m\{0}, P-1(z), P-2(z) and Q(z) are non-zero polynomials in C-m. The results of our paper, significantly generalize and improve the results due to Xu and Cao [34], Haldar [14] and Ahamed and Allu [1]. Also one of the main results of the paper is related to the following Fermat-type difference equations in C-m (P-1(z)f(z))(n1) +(P-2(z)f(z +c))n(2) = 1 for finding finite order meromorphic solutions, where n(1) and n(2) are two positive integers such that n(1) + n(2) > 2, P-1(z) and P-2(z) are two non-zero polynomials in C-m such that P-1(z+c) P-2(z). Moreover plenty of examples are provided to illustrate our findings.
In this paper, we obtain some normality criteria for sequence of meromorphic functions that concerning the sequence of exceptional functions of derivatives, which generalize related results due to Deng-Yang-Fang[4], Chang[1], Yang[18], and Schwick[14].
The power graph of a group G is a graph P(G) with vertex set G and two vertices x and y, x =/ y are adjacent if there exists some integer k such that x = y(k )or y = x(k). We denote the complement of P(G) by P(G) and a proper power graph of G by P & lowast;(G). This paper investigates the Hamiltonian and Eulerian nature of the power graphs of some finite Abelian and non-Abelian groups. We also discuss properties like planarity, outerplanarity, diameter, girth including neighbourhood and separating sets for a power graph, a proper power graph and the complement of a proper power graph of some finite and infinite groups.
