Mostrando recursos 1 - 20 de 30

  1. Announcements


  2. Soccer Balls, Golf Balls, and the Euler Identity

    Lesniak, Linda; White, Arthur T.
    We show, with simple combinatorics, that if the dimples on a golf ball are all 5-sided and 6-sided polygons, with three dimples at each “vertex”, then no matter how many dimples there are and no matter the sizes and distribution of the dimples, there will always be exactly twelve 5-sided dimples. Of course, the same is true of a soccer ball and its faces.

  3. On a Problem of Harary

    Kainen, Paul C.
    An exercise in Harary [1, p. 100] states that the product of the vertex independence number and the vertex covering number is an upper bound on the number of edges in a bipartite graph. In this note, we extend the bound to triangle-free graphs, and show that equality holds if and only if the graph is complete bipartite.

  4. On $(\in_\alpha,\in_\alpha\vee q_\beta)$-fuzzy Soft $BCI$-algebras

    Jana, Chiranjibe; Pal, Madhumangal
    Molodtsov initiated soft set theory which has provided a general mathematical framework for handling uncertainties that occur in various real life problems. The aim of this paper is to provide fuzzy soft algebraic tools in considering many problems that contain uncertainties. In this article, the notion of $(\in_\alpha,\in_\alpha\vee q_\beta)$-fuzzy soft $BCI$-subalgebra of $BCI$-algebra is introduced. Some operational properties on $(\in_\alpha,\in_\alpha\vee q_\beta)$-fuzzy soft $BCI$-subalgebras are discussed as well as lattice structures of this kind of fuzzy soft set on $BCI$-subalgebras are derived.

  5. Weakly JU Rings

    Danchev, Peter V.
    We define and completely explore the so-called WJU rings. This class properly encompasses the class of JU rings, introduced and studied by the present author in detail in Toyama Math. J. (2016).

  6. Sieving for the Primes to Prove Their Infinitude

    Eba, Hunde
    We prove the infinitude of prime numbers by the principle of contradiction, that is different from Euclid's proof in a way that it uses an explicit property of prime numbers. A sieve method that applies the inclusion-exclusion principle is used to give the property of the prime numbers in terms of the prime counting function.

  7. Geometry of Polynomials with Three Roots

    Frayer, Christopher
    Given a complex-valued polynomial of the form $p(z) = (z-1)^k(z-r_1)^m(z-r_2)^n$ with $|r_1|=|r_2|=1$; $k,m,n \in \mathbb{N}$ and $m \neq n$, where are the critical points? The Gauss-Lucas Theorem guarantees that the critical points of such a polynomial will lie within the unit disk. This paper further explores the location and structure of these critical points. Surprisingly, the unit disk contains two ‘desert’ regions in which critical points cannot occur, and each $c$ inside the unit disk and outside of the desert regions is the critical point of exactly two such polynomials.

  8. $(\in,\in\vee q)$-bipolar Fuzzy $BCK/BCI$-algebras

    Jana, Chiranjibe; Pal, Madhumangal; Saied, Arsham Borumand
    In this paper, the concept of quasi-coincidence of a bipolar fuzzy point within a bipolar fuzzy set is introduced. The notion of $(\in,\in\vee q)$-bipolar fuzzy subalgebras and ideals of $BCK/BCI$-algebras are introduced and their related properties are investigated by some examples. We study bipolar fuzzy $BCK/BCI$-subalgebras and bipolar fuzzy $BCK/BCI$-ideals by their level subalgebras and level ideals. We also provide the relationship between $(\in,\in\vee q)$-bipolar fuzzy $BCK/BCI$-subalgebras and bipolar fuzzy $BCK/BCI$-subalgebras, and $(\in,\in\vee q)$-bipolar fuzzy $BCK/BCI$-ideals and bipolar fuzzy $BCK/BCI$-ideals by counter examples.

  9. Cubic Implicative Ideals of $BCK$-algebras

    Senapati, Tapan; Shum, K. P.
    In this paper, we apply the concept of cubic sets to implicative ideals of $BCK$-algebras, and then characterize their basic properties. We discuss relations among cubic implicative ideals, cubic subalgebras and cubic ideals of $BCK$-algebras. We provide a condition for a cubic ideal to be a cubic implicative ideal. We define inverse images of cubic implicative ideals and establish how the inverse images of a cubic implicative ideal become a cubic implicative ideal. Finally we introduce products of cubic $BCK$-algebras.

  10. An Elementary Approach to the Diophantine Equation $ax^m + by^n = z^r$ Using Center of Mass

    Rahimi, Amir M.
    This paper takes an interesting approach to conceptualize some power sum inequalities and uses them to develop limits on possible solutions to some Diophantine equations. In this work, we introduce how to apply center of mass of a $k$-mass-system to discuss a class of Diophantine equations (with fixed positive coefficients) and a class of equations related to Fermat's Last Theorem. By a constructive method, we find a lower bound for all positive integers that are not the solution for these type of equations. Also, we find an upper bound for any possible integral solution for these type of equations. We...

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. 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.

  18. 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.

  19. 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.

  20. 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.

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