In knot theory, a common task is to take a given knot diagram and generate from it a polynomial. One method for accomplishing this is to employ a skein relation to convert the knot into a type of labeled binary tree and from this tree derive a two-variable polynomial. The purpose of this paper is to determine, in a simplified setting, which polynomials can be generated from labeled binary trees. We give necessary and sufficient conditions for a polynomial to be constructible in this fashion and we will provide a method for reconstructing the generating tree from such a polynomial. We conclude with an application of this theorem to a class of knots and links given by closed positive braids.

Let $\varphi$ be an analytic self-map of open unit disk $\mathbb{D}$. The operator given by $(C_{\varphi}f)(z)=f(\varphi(z))$, for $z \in \mathbb{D}$ and $f$ analytic on $\mathbb{D}$ is called a composition operator. Let $\omega$ be a weight function such that $\omega\in L^{1}(\mathbb{D}, dA)$. The space we consider is a generalized weighted Nevanlinna class $\mathcal{N}_{\omega}$, which consists of all analytic functions $f$ on $\mathbb{D}$ such that $\displaystyle \|f\|_{\omega}=\int_{\mathbb{D}}\log^{+}(|f(z)|) \omega(z) dA(z)$ is finite; that is, $\mathcal{N}_{\omega}$ is the space of all analytic functions belong to $L_{\log^+}(\mathbb{D}, \omega dA)$. In this paper we investigate, in terms of function-theoretic, composition operators on the space $\mathcal{N}_{\omega}$. We give sufficient conditions for the boundedness and compactness of these composition operators.

Let $G_A$ be an $AF$-algebra given by a periodic Bratteli diagram with the incidence matrix $A\in GL(n, {\Bbb Z})$. For a given polynomial $p(x)\in {\Bbb Z}[x]$ we assign to $G_A$ a finite abelian group $Ab_{p(x)}(G_A)={\Bbb Z}^n/p(A){\Bbb Z}^n$. It is shown that if $p(0)=\pm 1$ and ${\Bbb Z}[x]/\langle p(x)\rangle$ is a principal ideal domain, then $Ab_{p(x)}(G_A)$ is an invariant of the strong stable isomorphism class of $G_A$. For $n=2$ and $p(x)=x-1$ we conjecture a formula linking values of the invariant and torsion subgroup of elliptic curves with complex multiplication.

In this paper, we apply the notion of $e$-$\I$-open sets \cite{Wadei6} in ideal topological spaces to present and study new classes of functions called contra $e$-$\I$-continuous functions, almost-$e$-$\I$-continuous, almost contra-$e$-$\I$-continuous, and almost weakly-$e$-$\I$-continuous along with their several properties, characterizations and mutual relationships. Relationships between their new classes and other classes of functions are established and some characterizations of their new classes of functions are studied. Further, we introduce new types of graphs, called $e$-$\I$-closed, contra-$e$-$\I$-closed, and strongly contra-$e$-$\I$-closed graphs via $e$-$\I$-open sets. Several characterizations and properties of such notions are investigated.

A function $f \colon [0,1]\to {\mathbb R}$ is Riemann integrable if and only if its Riemann sums $f(T)$ and $f(T')$ get closer to each other as $\delta\to 0$, uniformly over all $\delta$-fine tagged divisions $T$ and $T'$. We show that $\delta^{-1}|f(T)-f(T')|\asymp \mbox{Var}(f)$. We also give an example of a function $f\notin \mbox{BV}$ with $|f(T)-f(T')|= {\mathcal O}(\delta|\ln \delta|)$. As a lemma, we show that any $f\in\mbox{BV}$ can be approximated uniformly by a step function $g$ with $\mbox{Var}(g)\approx\mbox{Var}(f)$.

With regard to the one sided hypothesis test, we propose a likelihood ratio that might be viewed as a Bayes/Non-Bayes compromise in the spirit of I. J. Good (1983). The influence of A. W. F. Edwards (1972) will also be apparent. Although we will develop some general ideas, most of our effort will focus on tests of a single unknown mean and the specific case of a sample from a normal population with unknown mean and known variance.

This article shows that continuous functions on compact intervals may be approximated uniformly with scattered translates of the Poisson kernel $(\alpha^2+x^2)^{-1}$, where $\alpha>0$ is a fixed real parameter.

The aim of this paper is to introduce and study the class of $g\omega \beta$-closed sets. This class of sets is finer than $g$-closed sets and $\omega \beta- $closed sets. We study the fundamental properties of this class of sets. Further, we introduce and study $g\omega \beta$-open sets, $g\omega \beta$-neighborhoodsets, $g\omega \beta$-continuous functions, $g\omega \beta$-irresolute functions and $g\omega \beta$-closed functions.

We present a proof based on a 1905 paper by Henri Lebesgue that any continuous function defined on an interval has an antiderivative {\em without first proving the existence of the definite integral of the function}. We also demonstrate how the definite integral is a byproduct of this proof. Instead of merely presenting an efficient proof using modern techniques, we have chosen to present a more instructive proof actually following the steps of Lebesgue in the spirit of Otto Toeplitz's~\cite{Toeplitz} genetic approach.

In this article we prove that $A$ and $A^{-1}$ are stochastic if and only of $A$ is a permutation matrix. Then we extend this result to show that $A$ and $A^{-1}$ are nonnegative if and only if it is a product of a diagonal matrix with all positive diagonal entries and a permutation matrix.