Mostrando recursos 1 - 15 de 15

  1. 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.

  2. 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.

  3. 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...

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. Fatou’s theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open sets

    Park, Hyunchul
    We prove Fatou’s theorem for nonnegative harmonic functions with respect to killed subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential convergence fails on the unit ball.

  9. Strong measure zero sets in Polish groups

    Hrušák, Michael; Zapletal, Jindřich
    In the context of arbitrary Polish groups, we investigate the Galvin–Mycielski–Solovay characterization of strong measure zero sets as those sets for which a meager collection of right translates cannot cover the whole group.

  10. Purely infinite totally disconnected topological graph algebras

    Li, Hui
    We give a sufficient condition on totally disconnected topological graphs such that their associated topological graph algebras are purely infinite.

  11. Weighted local Hardy spaces associated to Schrödinger operators

    Zhu, Hua; Tang, Lin
    In this paper, we characterize the weighted local Hardy spaces $h^{p}_{\rho}(\omega)$ related to the critical radius function $\rho$ and weights $\omega\in A_{\infty}^{\rho,\infty}(\mathbb{R}^{n})$ which locally behave as Muckenhoupt’s weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. Then, we establish the equivalence of the weighted local Hardy space $h^{1}_{\rho}(\omega)$ and the weighted Hardy space $H^{1}_{\mathcal{L}}(\omega)$ associated to Schrödinger operators $\mathcal{L}$ with $\omega\in A_{1}^{\rho,\infty}(\mathbb{R}^{n})$. By the atomic characterization, we also prove the existence of finite atomic decompositions associated with $h^{p}_{\rho}(\omega)$. Furthermore, we establish boundedness in $h^{p}_{\rho}(\omega)$ of quasi-Banach-valued sublinear operators.

  12. On the behavior of singularities at the $F$-pure threshold

    Canton, Eric; Hernández, Daniel J.; Schwede, Karl; Witt, Emily E.
    We provide a family of examples for which the $F$-pure threshold and the log canonical threshold of a polynomial are different, but such that the characteristic $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of Mustaţă–Takagi–Watanabe). We then study the $F$-signature function in the case that either the $F$-pure threshold and log canonical threshold coincide, or that $p$ does not divide the denominator of the $F$-pure threshold. We show that the $F$-signature function behaves similarly in those two cases. Finally, we include an appendix that shows that the test ideal can still behave in...

  13. Constructions of exotic group $C$∗-algebras

    Wiersma, Matthew
    Let $\Gamma$ be a discrete group. When $\Gamma$ is nonamenable, the reduced and full group $C$∗-algebras differ and it is generally believed that there should be many intermediate $C$∗-algebras, however few examples are known. In this paper, we give new constructions and compare existing constructions of intermediate group $C$∗-algebras for both generic and specific groups $\Gamma$.

  14. Local-to-global rigidity of Bruhat–Tits buildings

    De La Salle, Mikael; Tessera, Romain
    A vertex-transitive graph $X$ is called local-to-global rigid if there exists $R$ such that every other graph whose balls of radius $R$ are isometric to the balls of radius $R$ in $X$ is covered by $X$. Let $d\geq4$. We show that the $1$-skeleton of an affine Bruhat–Tits building of type $\widetilde{A}_{d-1}$ is local-to-global rigid if and only if the underlying field has characteristic $0$. For example, the Bruhat–Tits building of $\mathrm{SL}(d,\mathbf{F}_{p}(\!(t)\!))$ is not local-to-global rigid, while the Bruhat–Tits building of $\mathrm{SL}(d,\mathbf{Q}_{p})$ is local-to-global rigid.

  15. Common hypercyclic vectors for certain families of differential operators

    Tsirivas, N.
    Let $(k_{n})$ be a strictly increasing sequence of positive integers. If $\sum_{n=1}^{+\infty}\frac{1}{k_{n}}$ $=+\infty$, we establish the existence of an entire function $f$ such that for every $\lambda\in(0,+\infty)$ the set $\{\lambda^{k_{n}}f^{(k_{n})}(\lambda z):n=1,2,\ldots\}$ is dense in the space of entire functions endowed with the topology of uniform convergence on compact subsets of the complex plane. This provides the best possible strengthened version of a corresponding result due to Costakis and Sambarino (Adv. Math. 182 (2004) 278–306). From this, and using a non-trivial result of Weyl which concerns the uniform distribution modulo $1$ of certain sequences, we also derive an entire function $g$...

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