Mostrando recursos 1 - 20 de 163

  1. On rational triangles via algebraic curves

    Sadek, Mohammad; Shahata, Farida
    A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between each family and the set of rational points of an algebraic curve. These algebraic curves are: a curve of genus 0, an elliptic curve and a genus~3 curve. We study the set of rational points on each of these curves and explicitly describe some of its rational points.

  2. Spectral inclusion for unbounded diagonally dominant $n\times n$ operator matrices

    Rasulov, Tulkin H.; Tretter, Christiane
    In this paper, we establish an analytic enclosure for the spectrum of unbounded linear operators~$\mathcal{A} $ admitting an $n \times n$ matrix representation in a Hilbert space $\mathcal{H} =\mathcal{H} _1\oplus \cdots \oplus \mathcal{H} _n$. For diagonally dominant operator matrices of order 0, we show that this new enclosing set, the block numerical range $W^n(\mathcal{A} )$, contains the eigenvalues of $\mathcal{A} $ and that the approximate point spectrum of $\mathcal{A} $ is contained in its closure $\overline {W^n(\mathcal{A} )}$. Since the block numerical range turns out to be a subset of the usual numerical range, $W^n(\mathcal{A} )\subset W(\mathcal{A} )$, it may...

  3. An identity for cocycles on coset spaces of locally compact groups

    Dharmadasa, H. Kumudini; Moran, William
    We prove here an identity for cocycles associated with homogeneous spaces in the context of locally compact groups. Mackey introduced cocycles ($\lambda $-functions) in his work on representation theory of such groups. For a given locally compact group $G$ and a closed subgroup $H$ of $G$, with right coset space $G/H$, a cocycle $\lambda $ is a real-valued Borel function on $G/H \times G$ satisfying the cocycle identity \[ \lambda (x, st)=\lambda (x.s,t)\lambda (x,s), \] \[\mbox {almost everywhere } x\in G/H,\ s,t\in G, \] where the ``almost everywhere" is with respect to a measure whose null sets pull back to...

  4. $\alpha $-positive/$\alpha $-negative definite functions on groups

    Heo, Jaeseong
    In this paper, we introduce the notions of an $\alpha $-positive/$\alpha $-negative definite function on a (discrete) group. We first construct the Naimark-GNS type representation associated to an $\alpha $-positive definite function and prove the Schoenberg type theorem for a matricially bounded $\alpha $-negative definite function. Using a $J$-representation on a Krein space $(\mathcal{K} ,J)$ associated to a nonnegative normalized $\alpha $-negative definite function, we also construct a $J$-cocycle associated to a $J$-representation. Using a $J$-cocycle, we show that there exist two sequences of $\alpha $-positive definite functions and proper $(\alpha ,J)$-actions on a Krein space $(\mathcal{K} ,J)$ corresponding to...

  5. Recurrence relation for computing a bipartition function

    Gireesh, D.S.; Naika, M.S. Mahadeva
    Recently, Merca found the recurrence relation for computing the partition function $p(n)$ which requires only the values of $p(k)$ for $k\leq n/2$. In this article, we find the recurrence relation to compute the bipartition function $p_{-2}(n)$ which requires only the values of $p_{-2}(k)$ for $k\leq n/2$. In addition, we also find recurrences for $p(n)$ and $q(n)$ (number of partitions of $n$ into distinct parts), relations connecting $p(n)$ and $q_0(n)$ (number of partitions of $n$ into distinct odd parts).

  6. Generalized palindromic continued fractions

    Freeman, David M.
    In this paper, we introduce a generalization of palindromic continued fractions as studied by Adamczewski and Bugeaud. We refer to these generalized palindromes as $m$-palindromes, where $m$ ranges over the positive integers. We provide a simple transcendency criterion for $m$-palindromes, extending and slightly refining an analogous result of Adamczewski and Bugeaud. We also provide methods for constructing examples of $m$-palindromes. Such examples allow us to illustrate our transcendency criterion and to explore the relationship between $m$-palindromes and stammering continued fractions, another concept introduced by Adamczewski and Bugeaud.

  7. Construction of globalizations for partial actions on rings, algebras, C$^*$-algebras and Hilbert bimodules

    Ferraro, Damián
    We give a necessary condition for a partial action on a ring to have globalization. We also show that every partial action on a C$^*$-algebra satisfying this condition admits a globalization and, finally, we use the linking algebra of a Hilbert module to translate our condition to the realm of partial actions on Hilbert modules.

  8. On quasi-normality of function rings

    Dube, Themba
    An $f$-ring is called quasi-normal if the sum of any two different minimal prime $\ell $-ideals is either a maximal $\ell $-ideal or the entire $f$-ring. Recall that the \textit {zero-component} of a prime ideal $P$ of a commutative ring $A$ is the ideal \[ O_P=\{a\in A\mid ab=0 \mbox { for some } b\in A\setminus P\}. \] \vspace {0.5pt} ¶ \noindent Let $C(X)$ be the $f$-ring of continuous real-valued functions on a Tychonoff space $X$. Larson proved that $C(\beta X)$ is quasi-normal precisely when $C(X)$ is quasi-normal and the zero-component of every hyper-real ideal of $C(X)$ is prime. We show that...

  9. On topological spaces that have a bounded complete DCPO model

    Dongsheng, Zhao; Xiaoyong, Xi
    A dcpo model of a topological space $X$ is a dcpo (directed complete poset) $P$ such that $X$ is homeomorphic to the maximal point space of $P$ with the subspace topology of the Scott space of $P$. It has been previously proved by Xi and Zhao that every $T_1$ space has a dcpo model. It is, however, still unknown whether every $T_1$ space has a bounded complete dcpo model (a poset is bounded complete if each of its upper bounded subsets has a supremum). In this paper, we first show that the set of natural numbers equipped with the co-finite...

  10. Reciprocal relations for trigonometric sums

    Chu, Wenchang
    By means of the partial fraction decomposition method, a general reciprocal theorem on trigonometric sums is established. Several trigonometric reciprocities and summation formulae are derived as consequences.

  11. The real-rootedness of generalized Narayana polynomials related to the Boros-Moll polynomials

    Chen, Herman Z.Q.; Yang, Arthur L.B.; Zhang, Philip B.
    In this paper, we prove the real-rootedness of a family of generalized Narayana polynomials which arose in the study of the infinite log-concavity of the Boros-Moll polynomials. We establish certain recurrence relations for these Narayana polynomials, from which we derive the real-rootedness. In order to prove the real-rootedness, we use a sufficient condition due to Liu and Wang to determine whether two polynomials have interlaced zeros. The recurrence relations are verified with the help of the $Mathematica$ package $HolonomicFunctions$.

  12. Constrained shape preserving rational cubic fractal interpolation functions

    Chand, A.K.B.; Tyada, K.R.
    In this paper, we discuss the construction of $\mathcal {C}^1$-rational cubic fractal interpolation function (RCFIF) and its application in preserving the constrained nature of a given data set. The $\mathcal {C}^1$-RCFIF is the fractal design of the traditional rational cubic interpolant of the form ${p_i(\theta )}/{q_i(\theta )}$, where $p_i(\theta )$ and $q_i(\theta )$ are cubic and quadratic polynomials with three tension parameters. We present the error estimate of the approximation of RCFIF with the original function in $\mathcal {C}^k[x_1,x_n]$, $k=1,3$. When the data set is constrained between two piecewise straight lines, we derive the sufficient conditions on the IFS parameters...

  13. Root power sums and Chebyshev polynomials

    Capparelli, Stefano
    We determine sequences of polynomials with rational coefficients that have certain postulated values for their root power sums. These, in turn, determine four families of orthogonal polynomials that can be expressed in terms of Chebyshev polynomials by a change of variables. We examine properties of these sequences of polynomials, two of which have already been studied in a previous paper.

  14. Fixed points of augmented generalized happy functions

    Swart, Breeanne Baker; Beck, Kristen A.; Crook, Susan; Eubanks-Turner, Christina; Grundman, Helen G.; Mei, May; Zack, Laurie
    An augmented generalized happy function $S_{[c,b]} $ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. In this paper, we study various pro\-perties of the fixed points of $S_{[c,b]} $; count the number of fixed points of $S_{[c,b]} $ for $b \geq 2$ and $0\lt c\lt 3b-3$; and prove that, for each $b \geq 2$, there exist arbitrarily many consecutive values of~$c$ for which $S_[{c,b]} $ has no fixed point.

  15. Multivariable isometries related to certain convex domains

    Athavale, Ameer
    Several interesting results exist in the literature on subnormal operator tuples having their spectral properties tied to the geometry of strictly pseudoconvex domains or to that of bounded symmetric domains in $\mathbb{C} ^n$. We introduce a class $\Omega ^{(n)}$ of convex domains in $\mathbb{C} ^n$ which, for $n \geq 2$, is distinct from the class of strictly pseudoconvex domains and the class of bounded symmetric domains and which lends itself to the application of theories related to the abstract inner function problem and the $\overline \partial $-Neumann problem, allowing us to make a number of interesting observations about certain subnormal...

  16. On a sine polynomial of Turán

    Alzer, Horst; Kwong, Man Kam
    In 1935, Tur\'an proved that \[ S_{n,a}(x)= \sum _{j=1}^n{n+a-j\choose n-j} \sin (jx)>0, \] \[n,a\in \mathbf {N},\quad 0\lt x\lt \pi .\] We present various related inequalities. Among others, we show that the refinements $$ S_{2n-1,a}(x)\geq \sin (x) \quad \mbox {and} \quad {S_{2n,a}(x)\geq 2\sin (x)(1+\cos (x))} $$ are valid for all integers $n\geq 1$ and real numbers $a\geq 1$ and $x\in (0,\pi )$. Moreover, we apply our theorems on sine sums to obtain inequalities for Chebyshev polynomials of the second kind.

  17. The probability that the number of points on a complete intersection is squarefree

    Schmidt, Eric
    We consider the asymptotic probability that integers chosen according to a binomial distribution will have certain properties: (i) that such an integer is not divisible by the $k$th power of a prime, (ii) that any $k$ of $s$ chosen integers are relatively prime and (iii) that a chosen integer is prime. We also prove an analog of the Dirichlet divisor problem for the binomial distribution. We show how these results yield corresponding facts concerning the number of points on a smooth complete intersection over a finite field.

  18. $P$-spaces and intermediate rings of continuous functions

    Murray, Will; Sack, Joshua; Watson, Saleem
    A completely regular topological space $X$ is called a $P$-space if every zero-set in $X$ is open. An intermediate ring is a ring $A(X)$ of real-valued continuous functions on $X$ containing all the bounded continuous functions. In this paper, we find new characterizations of $P$-spaces $X$ in terms of properties of correspondences between ideals in $A(X)$ and $z$-filters on $X$. We also show that some characterizations of $P$-spaces that are described in terms of properties of $C(X)$ actually characterize $C(X)$ among intermediate rings on $X$.

  19. The first mixed problem for the nonstationary Lamé system

    Makhmudov, O.I.; Tarkhanov, N.
    We find an adequate interpretation of the stationary Lam\'{e} operator within the framework of elliptic complexes and study the first mixed problem for the nonstationary Lam\'{e} system.

  20. Some finite generalizations of Gauss's square exponent identity

    Liu, Ji-Cai
    We obtain three finite generalizations of Gauss's square exponent identity. For example, we prove that, for any non-negative integer $m$, \[ \sum _{k=-m}^{m}(-1)^k \left [{3m-k+1}\atop {m+k}\right ] (-q;q)_{m-k}q^{k^2}=1, \] where \[ \left [{\vphantom {3}n}\atop {\vphantom {k}m}\right ] =\prod _{k=1}^m\frac {1-q^{n-k+1}}{1-q^k} \quad \mbox {and}\quad (a;q)_n=\prod _{k=0}^{n-1}(1-aq^k). \] These identities reduce to Gauss's famous identity $$\sum _{k=-\infty }^{\infty }(-1)^kq^{k^2}=\frac {(q;q)_{\infty }}{(-q;q)_{\infty }}$$ by letting $m\to \infty $.

Aviso de cookies: Usamos cookies propias y de terceros para mejorar nuestros servicios, para análisis estadístico y para mostrarle publicidad. Si continua navegando consideramos que acepta su uso en los términos establecidos en la Política de cookies.