Mostrando recursos 1 - 20 de 190

  1. 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.

  2. $P$-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 $X$.

  3. 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.

  4. 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 $.

  5. The primitive ideal space of the partial-isometric crossed product of a system by a single automorphism

    Lewkeeratiyutkul, Wicharn; Zahmatkesh, Saeid
    Let $(A,\alpha )$ be a system consisting of a $C^*$-algebra $A$ and an automorphism $\alpha $ of~$A$. We describe the primitive ideal space of the partial-isometric crossed product $A\times _{\alpha }^{piso }\mathbb{N} $ of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.

  6. A construction of a BSE-algebra of type I which is isomorphic to no $\rm C^*$-algebras

    Inoue, Jyunji; Takahasi, Sin-Ei
    In this note, we construct a BSE-algebra of type I which is isomorphic to no C$^*$-algebras. This affirmatively solves the problem posed by Takahasi and Hatori, ``is there a BSE-algebra of type I isomorphic to no C$^*$-algebras?''

  7. Multigraded Hilbert schemes parametrizing ideals in the Weyl algebra

    Hsiao, Jen-Chieh
    Results of Haiman and Sturmfels \cite {HS04} on multigraded Hilbert schemes are used to establish a quasi-projective scheme which parametrizes all left homogeneous ideals in the Weyl algebra having a fixed Hilbert function with respect to a given grading by an abelian group.

  8. On Ding injective, Ding projective and Ding flat modules and complexes

    Gillespie, James
    We characterize Ding modules and complexes over Ding-Chen rings. We show that, over a Ding-Chen ring $R$, the Ding projective (respectively, Ding injective, respectively, Ding flat) $R$-modules coincide with the Gorenstein projective (respectively, Gorenstein injective, respectively, Gorenstein flat) modules, which, in turn, are noth\-ing more than modules appearing as a cycle of an exact complex of projective (respectively, injective, respectively, flat) modules. We prove a similar characterization for chain complexes of $R$-modules: a complex~$X$ is Ding projective (respectively, Ding injective, respectively, Ding flat) if and only if each component $X_n$ is Ding projective (respectively, Ding injective, respectively, Ding flat). Along...

  9. Mixed chord-integrals of index $i$ and radial Blaschke-Minkowski homomorphisms

    Feng, Yibin; Wu, Shanhe; Wang, Weidong
    In this paper, two Brunn-Minkowski type inequalities for mixed chord-integrals of index~$i$ are established, which are related to the radial Blaschke-Minkowski homomorphisms of star bodies. Moreover, two inequalities similar to Giannopoulos, Hartzoulaki and Paouris's inequality are also considered.

  10. Finding new small degree polynomials with small Mahler measure by genetic algorithms

    Otmani, S. El; Maul, A.; Rhin, G.; Sac-Épée, J.-M.
    In this paper, we propose a new application of genetic-type algorithms to find monic, irreducible, non-cyclotomic integer polynomials with \textit {small degree} and Mahler measure less than $1.3$, which do not appear in Mossinghoff's list of all known polynomials with degree at most 180 and Mahler measure less than 1.3 {Mossinghoff}. The primary focus lies in finding such polynomials of small degree. In particular, the list referred to above is known to be complete through degree 44, and we show that it is not complete from degree 46 on by supplying two new polynomials of small Mahler measure, of degrees...

  11. Rationally connected varieties over the maximally unramified extension of $p$-adic fields

    Duesler, Bradley; Knecht, Amanda
    A result of Graber, Harris and Starr shows that a rationally connected variety defined over the function field of a curve over the complex numbers always has a rational point. Similarly, a separably rationally connected variety over a finite field or the function field of a curve over any algebraically closed field will have a rational point. Here, we show that rationally connected varieties over the maximally unramified extension of the $p$-adics usually, in a precise sense, have rational points. This result is in the spirit of Ax and Kochen's result, which states that the $p$-adics are usually $C_{2}$ fields....

  12. Partial representations and their domains

    Dokuchaev, M.; Lima, H.G.G. de; Pinedo, H.
    We study the structure of the partially or\-dered set of the elementary domains of partial (linear or projective) representations of groups. This provides an important information on the lattice of all domains. Some of these results are obtained through structural facts on the ideals of the semigroup $\mathcal{S} _3(G)$, a quotient of Exel's semigroup $\mathcal{S} (G)$, which plays a crucial role in the theory of partial projective representations. We also fill a gap in the proof of an earlier result on the structure of partial group representations.

  13. Compactness of multiplication, composition and weighted composition operators between some classical sequence spaces: A new approach

    Djolović, Ivana; Malkowsky, Eberhard
    In a previous paper the author studied the compactness of multiplication, composition and weighted composition operators among some sequence spaces. We were motivated by these results and present two different approaches for obtaining some of the results in a previous paper. The first approach is to apply the theory of matrix transformations and the Hausdorff measure of noncompactness, and the second one is to use known results on multiplier spaces and the Hausdorff measure of noncompactness. We also use our techniques and methods from our proofs of the existing results to establish some new results related to the class of...

  14. On a generalization of Wolff's ideal theorem for certain subalgebras of $H^\infty (\mathbb D )$

    Banjade, D.P.; Ephrem, M.; Incognito, A.; Wilkerson, M.
    We settle an open question proposed by Banjade to generalize Wolff's ideal theorem on certain uniformly closed subalgebras of $H^{\infty }(\mathbb{D} )$. Also, we discuss some subalgebras where Wolff's ideal theorem holds without the additional condition $ F(0) \neq 0$.

  15. Mod $p$ equivalence classes of linear recurrence sequences of degree~$2$

    Aoki, Miho; Sakai, Yuho
    Laxton introduced a group structure on the set of equivalence classes of linear recurrence sequences of degree~2. This result yields much information on the divisibilities of such sequences. In this paper, we introduce other equivalence relations for the set of linear recurrence sequences $(G_n)$, which are defined by $G_0, G_1 \in \mathbb{Z} $ and $G_n=TG_{n-1}-NG_{n-2}$ for fixed integers~$T$ and $N=\pm 1$. The relations are given by certain congruences modulo~$p$ for a fixed prime number~$p$, which are different from Laxton's without modulo $p$ equivalence relations. We determine the initial terms $G_0$ and $G_1$ of all of the representatives of the equivalence...

  16. On $n$-trivial extensions of rings

    Anderson, D.D.; Bennis, Driss; Fahid, Brahim; Shaiea, Abdulaziz
    The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research such as cohomology theory, representation theory, category theory and homological algebra. In this paper, we extend this classical ring construction by associating a ring to a ring~$R$ and a family $M=(M_i)_{i=1}^{n}$ of $n$ $R$-modules for a given integer $n\geq 1$. We call this new ring construction an $n$-trivial extension of $R$ by $M$. In particular, the classical trivial extension will merely be the $1$-trivial extension. Thus, we generalize several known results...

  17. Existence of rotating-periodic solutions for nonlinear systems via upper and lower solutions

    Yang, Xue; Zhang, Yu; Li, Yong
    This paper concerns the existence of affine-periodic solutions for nonlinear systems with certain affine-periodic symmetry. The existence result is actually proved based on the existence of upper and lower solutions and the conditions on them. Some applications are also given.

  18. Existence of rotating-periodic solutions for nonlinear systems via upper and lower solutions

    Yang, Xue; Zhang, Yu; Li, Yong
    This paper concerns the existence of affine-periodic solutions for nonlinear systems with certain affine-periodic symmetry. The existence result is actually proved based on the existence of upper and lower solutions and the conditions on them. Some applications are also given.

  19. Existence of rotating-periodic solutions for nonlinear systems via upper and lower solutions

    Yang, Xue; Zhang, Yu; Li, Yong
    This paper concerns the existence of affine-periodic solutions for nonlinear systems with certain affine-periodic symmetry. The existence result is actually proved based on the existence of upper and lower solutions and the conditions on them. Some applications are also given.

  20. Existence of rotating-periodic solutions for nonlinear systems via upper and lower solutions

    Yang, Xue; Zhang, Yu; Li, Yong
    This paper concerns the existence of affine-periodic solutions for nonlinear systems with certain affine-periodic symmetry. The existence result is actually proved based on the existence of upper and lower solutions and the conditions on them. Some applications are also given.

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