Euler's pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler's pentagonal number theorem., Ji-Cai Liu., and Seznam literatury
In this paper, we introduce the concept of a logarithmic convex structure. Let X be a set and D : X × X → [1, ∞) a function satisfying the following conditions: (i) For all x, y ∈ X, D(x, y) ≥ 1 and D(x, y) = 1 if and only if x = y. (ii) For all x, y ∈ X, D(x, y) = D(y, x). (iii) For all x, y, z ∈ X, D(x, y) ≤ D(x, z)D(z, y). (iv) For all x, y, z ∈ X, z ≠ x, y and λ ∈ (0, 1), D(z, W(x, y, λ)) ≤ D λ (x, z)D 1−λ (y, z), D(x, y) = D(x, W(x, y, λ))D(y, W(x, y, λ)), where W : X ×X ×[0, 1] → X is a continuous mapping. We name this the logarithmic convex structure. In this work we prove some fixed point theorems in the logarithmic convex structure.
A generalization is obtained for some of the fixed point theorems of Khan, Swaleh and Sessa, Pathak and Rekha Sharma, and Sastry and Babu for a self-map on a metric space, which involve the idea of alteration of distances between points.
Using generalized absolute continuity, we characterize additive interval functions which are indefinite Henstock-Kurzweil integrals in the Euclidean space.
Let ∑ ∞ n=1 an be a convergent series of positive real numbers. L. Olivier proved that if the sequence (an) is non-increasing, then lim n→∞ nan = 0. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having lim n→∞ nan = 0; Olivier’s theorem is a consequence of our Theorem 2.1. (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the I-convergence, that is a convergence according to an ideal I of subsets of ℕ. Again, Olivier’s theorem is a consequence of our Theorem 3.1, when one takes as I the ideal of all finite subsets of ℕ.
n this paper we investigate two new classes of torsion-free Abelian groups which arise in a natural way from the notion of a torsion-free Crawley group. A group $G$ is said to be an Erdős group if for any pair of isomorphic pure subgroups $H,K$ with $G/H \cong G/K$, there is an automorphism of $G$ mapping $H$ onto $K$; it is said to be a weak Crawley group if for any pair $H, K$ of isomorphic dense maximal pure subgroups, there is an automorphism mapping $H$ onto $K$. We show that these classes are extensive and pay attention to the relationship of the Baer-Specker group to these classes. In particular, we show that the class of Crawley groups is strictly contained in the class of weak Crawley groups and that the class of Erdős groups is strictly contained in the class of weak Crawley groups.
Let $W_{n}=K_{1}\vee C_{n-1}$ be the wheel graph on $n$ vertices, and let $S(n,c,k)$ be the graph on $n$ vertices obtained by attaching $n-2c-2k-1$ pendant edges together with $k$ hanging paths of length two at vertex $v_{0}$, where $v_{0}$ is the unique common vertex of $c$ triangles. In this paper we show that $S(n,c,k)$ ($c\geq 1$, $k\geq 1$) and $W_{n}$ are determined by their signless Laplacian spectra, respectively. Moreover, we also prove that $S(n,c,k)$ and its complement graph are determined by their Laplacian spectra, respectively, for $c\geq 0$ and $k\geq 1$.
The basic fuzzy logic BL is extended by two unary connectives L, U
(lower, upper) whose standard semantics is, given a continuous t-norm, the function assigning to each x € [0,1] the biggest idempotent < x (least idempotent > x). An axiom system is presented and shown complete with respect to the corresponding class of algebras. But the set of tautologies for a fixed continuous t-norm may have an arbitrarily high degree of insolvability.