Let T be a Banach space operator. In this paper we characterize a-Browder’s theorem for T by the localized single valued extension property. Also, we characterize a-Weyl’s theorem under the condition E a (T) = π a (T), where E a (T) is the set of all eigenvalues of T which are isolated in the approximate point spectrum and π a (T) is the set of all left poles of T. Some applications are also given.
We consider the Cahn-Hilliard equation in H 1 (ℝ N ) with two types of critically growing nonlinearities: nonlinearities satisfying a certain limit condition as |u| → ∞ and logistic type nonlinearities. In both situations we prove the H 2 (ℝ N )-bound on the solutions and show that the individual solutions are suitably attracted by the set of equilibria. This complements the results in the literature; see J.W. Cholewa, A. Rodriguez-Bernal (2012).
In the paper we discuss the following type congruences: $$ \biggl ({np^k\atop mp^k}\biggr ) \equiv \left (m \atop n\right ) \pmod {p^r}, $$ where $p$ is a prime, $n$, $m$, $k$ and $r$ are various positive integers with $n\ge m\ge 1$, $k\ge 1$ and $r\ge 1$. Given positive integers $k$ and $r$, denote by $W(k,r)$ the set of all primes $p$ such that the above congruence holds for every pair of integers $n\ge m\ge 1$. Using Ljunggren's and Jacobsthal's type congruences, we establish several characterizations of sets $W(k,r)$ and inclusion relations between them for various values $k$ and $r$. In particular, we prove that $W(k+i,r)=W(k-1,r)$ for all $k\ge 2$, $i\ge 0$ and $3\le r\le 3k$, and $W(k,r)=W(1,r)$ for all $3\le r\le 6$ and $k\ge 2$. We also noticed that some of these properties may be used for computational purposes related to congruences given above.
In this paper we characterize those bounded linear transformations $Tf$ carrying $L^{1}( \mathbb {R}^{1}) $ into the space of bounded continuous functions on $\mathbb {R}^{1}$, for which the convolution identity $T(f\ast g) =Tf\cdot Tg $ holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.
The cubical dimension of a graph $G$ is the smallest dimension of a hypercube into which $G$ is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with $2^n$ vertices, $n\geq 1$, is $n$. The 2-rooted complete binary tree of depth $n$ is obtained from two copies of the complete binary tree of depth $n$ by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice a 2-rooted complete binary tree and prove that every such balanced tree satisfies the conjecture of Havel.
The paper deals with the different ways in which 20th century Hindi writers introduced the theme of religion into their work. A selection of authors has been made in order to highlight some important issues connected with religion. As may be expected, basically two points of view are to be found, depending on the ideological stand of the writer – either politically committed or committed to man. Thus, whilst religion is deemed by one author to be a factor which divides communities, it is considered by another to be an important tool for exploring the human soul.
Let $a$, $b$, $c$, $r$ be positive integers such that $a^{2}+b^{2}=c^{r}$, $\min (a,b,c,r)>1$, $\gcd (a,b)=1, a$ is even and $r$ is odd. In this paper we prove that if $b\equiv 3\hspace{4.44443pt}(\@mod \; 4)$ and either $b$ or $c$ is an odd prime power, then the equation $x^{2}+b^{y}=c^{z}$ has only the positive integer solution $(x,y,z)=(a,2,r)$ with $\min (y,z)>1$.
If G is a simple graph of size n without isolated vertices and G is its complement, we show that the domination numbers of G and G satisfy γ(G) + γ(G) ≤ { n − δ + 2 if γ(G) > 3, δ + 3 if γ(G) > 3, where δ is the minimum degree of vertices in G.
The paper is devoted to the question whether some kind of additional information makes it possible to determine the fundamental matrix of variational equations in ℝ3 . An application concerning computation of a derivative of a scalar Poincaré mapping is given.