Mostrando recursos 1 - 20 de 20

  1. When a Matrix and Its Inverse Are Nonnegative

    Ding, J.; Rhee, N. H.
    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.

  2. When a Matrix and Its Inverse Are Nonnegative

    Ding, J.; Rhee, N. H.
    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.

  3. Lebesgue's Remarkable Result

    Coppin, Charles A.
    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.

  4. Lebesgue's Remarkable Result

    Coppin, Charles A.
    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.

  5. On Generalized $\omega \beta$-Closed Sets

    Aljarrah, H. H.; Noorani, M. S. M.; Noiri, T.
    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.

  6. On Generalized $\omega \beta$-Closed Sets

    Aljarrah, H. H.; Noorani, M. S. M.; Noiri, T.
    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.

  7. Approximating Continuous Functions with Scattered Translates of the Poisson Kernel

    Ledford, Jeff
    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.

  8. Approximating Continuous Functions with Scattered Translates of the Poisson Kernel

    Ledford, Jeff
    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.

  9. Extending Edwards Likelihood Ratios to Simple One Sided Hypothesis Tests

    Derryberry, Dewayne; Bimali, Milan
    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.

  10. Extending Edwards Likelihood Ratios to Simple One Sided Hypothesis Tests

    Derryberry, Dewayne; Bimali, Milan
    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.

  11. Rates of Uniform Convergence for Riemann Integrals

    Alewine, J. Alan
    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 $\mbox{Var}(g)\approx\mbox{Var}(f)$.

  12. Rates of Uniform Convergence for Riemann Integrals

    Alewine, J. Alan
    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 $\mbox{Var}(g)\approx\mbox{Var}(f)$.

  13. New Forms of Contra-Continuity in Ideal Topology Spaces

    Al-Omeri, Wadei; Noorani, Mohd. Salmi Md.; Al-Omari, A.
    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.

  14. New Forms of Contra-Continuity in Ideal Topology Spaces

    Al-Omeri, Wadei; Noorani, Mohd. Salmi Md.; Al-Omari, A.
    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.

  15. Invariants of Stationary AF-Algebras and Torsion Subgroups of Elliptic Curves with Complex Multiplication

    Nikolaev, Igor
    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.

  16. Invariants of Stationary AF-Algebras and Torsion Subgroups of Elliptic Curves with Complex Multiplication

    Nikolaev, Igor
    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.

  17. Composition Operators on Generalized Weighted Nevanlinna Class

    Al-Rawashdeh, Waleed
    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...

  18. Composition Operators on Generalized Weighted Nevanlinna Class

    Al-Rawashdeh, Waleed
    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...

  19. Polynomials, Binary Trees, and Positive Braids

    Wiley, Chad; Gray, Jeffrey
    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...

  20. Polynomials, Binary Trees, and Positive Braids

    Wiley, Chad; Gray, Jeffrey
    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...

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