For any two positive integers n and k\geqslant 2, let G(n, k) be a digraph whose set of vertices is {0, 1, ..., n − 1} and such that there is a directed edge from a vertex a to a vertex b if ak ≡ b (mod n). Let n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} be the prime factorization of n. Let P be the set of all primes dividing n and let P_{1},P_{2} \subseteq P be such that P_{1\cup P_{2}}= P and P_{1\cup P_{2}}=\emptyset . A fundamental constituent of G(n, k), denoted by G_{{P_2}}^*(n,k), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \prod\nolimits_{{p_i} \in {P_2}} {{p_i}} and are relatively prime to all primes q\in P_{1}. L. Somer and M. Křižek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic., Amplify Sawkmie, Madan Mohan Singh., and Obsahuje seznam literatury
For every bundle functor we introduce the concept of subordinated functor. Then we describe subordinated functors for fiber product preserving functors defined on the category of fibered manifolds with $m$-dimensional bases and fibered manifold morphisms with local diffeomorphisms as base maps. In this case we also introduce the concept of the underlying functor. We show that there is an affine structure on fiber product preserving functors.
We prove a theorem on the growth of nonconstant solutions of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its linear differential polynomial share a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.
With the aid of the notion of weighted sharing and pseudo sharing of sets we prove three uniqueness results on meromorphic functions sharing three sets, all of which will improve a result of Lin-Yi in Complex Var. Theory Appl. (2003).
The two-point boundary value problem u ′′ + h(x)u p = 0, a < x < b, u(a) = u(b) = 0 is considered, where p > 1, h ∈ C 1 [0, 1] and h(x) > 0 for a ≤ x ≤ b. The existence of positive solutions is well-known. Several sufficient conditions have been obtained for the uniqueness of positive solutions. On the other hand, a non-uniqueness example was given by Moore and Nehari in 1959. In this paper, new uniqueness results are presented.
Bones were obtained from three fish species (brown trout Salmo trutta m. fario, grayling Thymallus thymallus and Carpathian sculpin Cottus poecilopus) for regression analysis. Bones used were chosen based upon frequency of occurrence in spraint samples and diagnostic value. Relationships between the length of diagnostic bones and fish length, fish length and weight, and standard length to total length, were assessed for the three fish species. Polynomial regression was deemed most suitable for the relationship between bone length and fish standard length, multiplicative between fish standard length and fish weight, and linear (brown trout) or polynomial (grayling and Carpathian sculpin) for standard length against total length. All calculated regressions were highly significant and displayed high coefficients of determination, ranging between 93.9 and 99.8 %. The uses of the bones examined, and the equations produced, are discussed in the light of their future use in estimating prey numbers, length and biomass in otter diet analysis.
We consider the initial-boundary value problem for first order differential-functional equations. We present the `vanishing viscosity' method in order to obtain viscosity solutions. Our formulation includes problems with a retarded and deviated argument and differential-integral equations.
We prove that a set of weak solutions of the nonlinear Volterra integral equation has the Kneser property. The main condition in our result is formulated in terms of axiomatic measures of weak noncompactness.
We give sufficient conditions for the existence of at least one integrable solution of equation x(t) = f(t) + ∫ t 0 K(t, s)g(s, x(s)) ds. Our assumptions and proofs are expressed in terms of measures of noncompactness.