Mostrando recursos 1 - 20 de 28

  1. Some combinatorial number theory problems over finite valuation rings

    Pham, Thang; Vinh, Le Anh
    Let $\mathcal{R}$ be a finite valuation ring of order $q^{r}$. In this paper, we generalize and improve several well-known results, which were studied over finite fields $\mathbb{F}_{q}$ and finite cyclic rings $\mathbb{Z}/p^{r}\mathbb{Z}$, in the setting of finite valuation rings.

  2. Sum of Toeplitz products on the Hardy space over the polydisk

    Yu, Tao
    In this paper, we obtain several sufficient and necessary conditions for a finite sum of Toeplitz products with form $\sum_{m=1}^{M}T_{f_{m}}T_{g_{m}}$ on the Hardy space over the polydisk to be zero. The methods used in this note are Berezin transform and the essential fiber dimension.

  3. The expected number of complex zeros of complex random polynomials

    Ferrier, Katrina; Jackson, Micah; Ledoan, Andrew; Patel, Dhir; Tran, Huong
    By using the technique introduced in 1995 by Shepp and Vanderbei, we derive an exact formula for the expected number of complex zeros of a complex random polynomial due to Kac. The explicit evaluation of the average intensity function is obtained in closed form in the case of standard normal coefficients. In addition, we provide the limiting expressions for the intensity function and the expected number of zeros in open circular disks in the complex plane.

  4. On representations of error terms related to the derivatives for some Dirichlet series

    Furuya, Jun; Minamide, T. Makoto; Tanigawa, Yoshio
    In previous papers, we examined several properties of an error term in a certain divisor problem related to the derivatives of the Riemann zeta-function. In this paper, we obtain representations of error terms related to the derivatives of some Dirichlet series, which can be regarded as generalized versions of a Dirichlet divisor problem and a Gauss circle problem. We also give the upper bounds of the error terms in terms of exponent pairs.

  5. Evaluation of Tornheim’s type of double series

    Kadota, Shin-ya; Okamoto, Takuya; Tasaka, Koji
    We examine values of certain Tornheim’s type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra $G_{2}$, proposed by Komori, Matsumoto and Tsumura, is given.

  6. Maximal torus theory for compact quantum groups

    Banica, Teodor; Patri, Issan
    Associated to any compact quantum group $G\subset U_{N}^{+}$ is a canonical family of group dual subgroups $\widehat{\Gamma }_{Q}\subset G$, parametrized by unitaries $Q\in U_{N}$, playing the role of “maximal tori” for $G$. We present here a series of conjectures, relating the various algebraic and analytic properties of $G$ to those of the family $\{\widehat{\Gamma }_{Q}|Q\in U_{N}\}$.

  7. Structure of porous sets in Carnot groups

    Pinamonti, Andrea; Speight, Gareth
    We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $\sigma $-porous with respect to the Carnot–Carathéodory (CC) distance. In the first Heisenberg group, we observe that there exist sets which are porous with respect to the CC distance but not the Euclidean distance and vice-versa. In Carnot groups, we then construct a Lipschitz function which is Pansu differentiable at no point of a given $\sigma $-porous set and show preimages of open sets under the horizontal gradient are far from being porous.

  8. On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds

    Bourguin, Solesne; Durastanti, Claudio
    In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point...

  9. Bounds on the norm of the backward shift and related operators in Hardy and Bergman spaces

    Ferguson, Timothy
    We study bounds for the backward shift operator $f\mapsto(f(z)-f(0))/z$ and the related operator $f\mapsto f-f(0)$ on Hardy and Bergman spaces of analytic and harmonic functions. If $u$ is a real valued harmonic function, we also find a sharp bound on $M_{1}(r,u-u(0))$ in terms of $\|u\|_{h^{1}}$, where $M_{1}$ is the integral mean with $p=1$.

  10. Bi-parameter Littlewood–Paley operators with upper doubling measures

    Cao, Mingming; Xue, Qingying
    Let $\mu=\mu_{n_{1}}\times\mu_{n_{2}}$, where $\mu_{n_{1}}$ and $\mu_{n_{2}}$ are upper doubling measures on $\mathbb{R}^{n_{1}}$ and $\mathbb{R}^{n_{2}}$, respectively. Let the pseudo-accretive function $b=b_{1}\otimes b_{2}$ satisfy a bi-parameter Carleson condition. In this paper, we established the $L^{2}(\mu)$ boundedness of non-homogeneous Littlewood–Paley $g_{\lambda}^{*}$-function with non-convolution type kernels on product spaces. This was mainly done by means of dyadic analysis and non-homogenous methods. The result is new even in the setting of Lebesgue measures.

  11. Almost conformally flat hypersurfaces

    Onti, Christos-Raent; Vlachos, Theodoros
    We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact $n$-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a consequence, we determine the homology of almost conformally flat hypersurfaces. Furthermore, we provide a necessary condition for a compact Riemannian manifold to admit an isometric minimal immersion as a hypersurface in the round sphere and extend a result due to Shiohama and Xu (J. Geom. Anal. 7 (1997) 377–386) for compact hypersurfaces in any space form.

  12. The Hörmander multiplier theorem, I: The linear case revisited

    Grafakos, Loukas; He, Danqing; Honzik, Petr; Van Nguyen, Hanh
    We discuss $L^{p}(\mathbb{R}^{n})$ boundedness for Fourier multiplier operators that satisfy the hypotheses of the Hörmander multiplier theorem in terms of an optimal condition that relates the distance $\vert \frac{1}{p}-\frac{1}{2}\vert $ to the smoothness $s$ of the associated multiplier measured in some Sobolev norm. We provide new counterexamples to justify the optimality of the condition $\vert \frac{1}{p}-\frac{1}{2}\vert <\frac{s}{n}$ and we discuss the endpoint case $\vert \frac{1}{p}-\frac{1}{2}\vert =\frac{s}{n}$.

  13. On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets

    Bachir, Mohammed
    We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi $-extreme points of a $\Phi $-convex compact metrizable space are replaced by the $\Phi $-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.

  14. On the classification of rational sphere maps

    D’Angelo, John P.
    We prove a new classification result for (CR) rational maps from the unit sphere in some $\mathbb{C}^{n}$ to the unit sphere in $\mathbb{C}^{N}$. To do so, we work at the level of Hermitian forms, and we introduce ancestors and descendants.

  15. Newton’s lemma for differential equations

    Aroca, Fuensanta; Ilardi, Giovanna
    The Newton method for plane algebraic curves is based on the following remark: the first term of a series, root of a polynomial with coefficients in the ring of series in one variable, is a solution of an initial equation that can be determined by the Newton polygon. ¶ Given a monomial ordering in the ring of polynomials in several variables, we describe the systems of initial equations that satisfy the first terms of the solutions of a system of partial differential equations. As a consequence, we extend Mora and Robbiano’s Groebner fan to differential ideals.

  16. Asymptotic stabilization of Betti diagrams of generic initial systems

    Mayes-Tang, Sarah
    Several authors investigating the asymptotic behaviour of the Betti diagrams of the graded system $\{I^{k}\}$ independently showed that the shape of the nonzero entries in the diagrams stabilizes when $I$ is a homogeneous ideal with generators of the same degree. In this paper, we study the Betti diagrams of graded systems of ideals built by taking the initial ideals or generic initial ideals of powers, and discuss the stabilization of additional collections of Betti diagrams. Our main result shows that when $I$ has generators of the same degree, the entries in the Betti diagrams of the reverse lexicographic generic initial...

  17. Koszul factorization and the Cohen–Gabber theorem

    Skalit, C.
    We present a sharpened version of the Cohen–Gabber theorem for equicharacteristic, complete local domains $(A,\mathfrak{m},k)$ with algebraically closed residue field and dimension $d>0$. Namely, we show that for any prime number $p$, $\operatorname{Spec}A$ admits a dominant, finite map to $\operatorname{Spec}k[[X_{1},\ldots,X_{d}]]$ with generic degree relatively prime to $p$. Our result follows from Gabber’s original theorem, elementary Hilbert–Samuel multiplicity theory, and a “factorization” of the map induced on the Grothendieck group $\mathbf{G}_{0}(A)$ by the Koszul complex.

  18. On the injective dimension of $\mathscr{F}$-finite modules and holonomic $\mathscr{D}$-modules

    Dorreh, Mehdi
    Let $R$ be a regular local ring containing a field $k$ of characteristic $p$ and $M$ be an $\mathscr{F}$-finite module. In this paper, we study the injective dimension of $M$. We prove that $\operatorname{dim}_{R}(M)-1\leq\operatorname{inj.dim}_{R}(M)$. If $R=k[[x_{1},\ldots,x_{n}]]$ where $k$ is a field of characteristic $0$ we prove the analogous result for a class of holonomic $\mathscr{D}$-modules which contains local cohomology modules.

  19. A note on nonexistence of multiple black holes in static vacuum Einstein space–times

    Baltazar, H.; Leandro, B.
    The purpose of this note is to study the static vacuum Einstein space–time with half harmonic Weyl tensor, that is, $\delta W^{+}=0$. We prove that there are no multiple black holes on a four-dimensional static vacuum Einstein space–time with half harmonic Weyl tensor.

  20. Non-compact subsets of the Zariski space of an integral domain

    Spirito, Dario
    Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\operatorname{Zar}(D)$ be the Zariski space of the valuation overrings of $D$. Starting from a result in the theory of semistar operations, we prove a criterion under which the set $\operatorname{Zar}(D)\setminus\{V\}$ is not compact. We then use it to prove that, in many cases, $\operatorname{Zar}(D)$ is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.

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