Mostrando recursos 1 - 20 de 2.368

  1. Stochastic calculus over symmetric Markov processes with time reversal

    Kuwae, K.
    We develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao’s divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on $L^{2}$ -space obtained by lower-order perturbations.

  2. Application and simplified proof of a sharp $L^{2}$ extension theorem

    Ohsawa, Takeo
    As an application of a sharp $L^{2}$ extension theorem for holomorphic functions in Guan and Zhou, a stability theorem for the boundary asymptotics of the Bergman kernel is proved. An alternate proof of the extension theorem is given, too. It is a simplified proof in the sense that it is free from ordinary differential equations.

  3. A global estimate for the Diederich–Fornaess index of weakly pseudoconvex domains

    Adachi, Masanori; Brinkschulte, Judith
    A uniform upper bound for the Diederich–Fornaess index is given for weakly pseudoconvex domains whose Levi form of the boundary vanishes in $\ell$ -directions everywhere.

  4. Remarks on free mutual information and orbital free entropy

    Izumi, Masaki; Ueda, Yoshimichi
    The present notes provide a proof of $i^{*}(\mathbb{C}P+\mathbb{C}(I-P);\mathbb{C}Q+\mathbb{C}(I-Q))=-\chi_{\mathrm{orb}}(P,Q)$ for any pair of projections $P,Q$ with $\tau(P)=\tau(Q)=1/2$ . The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.

  5. Some constructions of modular forms for the Weil representation of $\operatorname{SL}_{2}(\mathbb{Z})$

    Scheithauer, Nils R.
    Modular forms for the Weil representation of $\operatorname{SL}_{2}(\mathbb{Z})$ play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac–Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems $D_{12}^{2}$ , $E_{8}^{3}$ and to the Leech lattice.

  6. $p$ -adic Eisenstein–Kronecker series for CM elliptic curves and the Kronecker limit formulas

    Bannai, Kenichi; Furusho, Hidekazu; Kobayashi, Shinichi
    Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq5$ and with complex multiplication by the full ring of integers $\mathcal{O}_{K}$ of $K$ . In this paper, we construct $p$ -adic analogues of the Eisenstein–Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove $p$ -adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

  7. Instability of periodic traveling waves for the symmetric regularized long wave equation

    Angulo Pava, Jaime; Banquet Brango, Carlos Alberto
    We prove the linear and nonlinear instability of periodic traveling wave solutions for a generalized version of the symmetric regularized long wave (SRLW) equation. Using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. An application of this approach is made to obtain the linear/nonlinear instability of cnoidal wave solutions for the modified SRLW (mSRLW) equation. We also prove the stability of dnoidal wave solutions associated to the equation just mentioned.

  8. Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

    Witte, N. S.
    A $\mathbb{D}$ -semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator $\mathbb{D}$ . It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the $\mathbb{D}$ -semiclassical orthogonal polynomial systems...

  9. Torsion in tensor powers of modules

    Celikbas, Olgur; B. Iyengar, Srikanth; Piepmeyer, Greg; Wiegand, Roger
    Tensor products usually have nonzero torsion. This is a central theme of Auslander’s 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules $N$ with the property that $M\otimes_{R}N$ has nonzero torsion unless $M$ is very special. An important example of such a module $N$ is the Frobenius power ${}^{p^{e}}\!R$ over a complete intersection domain $R$ of characteristic $p\gt 0$ .

  10. On modules of finite projective dimension

    Dutta, S. P.
    We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular, we derive that, in any local ring $R$ of mixed characteristic $p\gt 0$ , where $p$ is a nonzero divisor, if $I$ is an ideal of finite projective dimension over $R$ and $p\inI$ or $p$ is a nonzero divisor on $R/I$ , then every minimal generator of...

  11. A McShane-type identity for closed surfaces

    Huang, Yi
    We prove a McShane-type identity: a series, expressed in terms of geodesic lengths, that sums to $2\pi$ for any closed hyperbolic surface with one distinguished point. To do so, we prove a generalized Birman–Series theorem showing that the set of complete geodesics on a hyperbolic surface with large cone angles is sparse.

  12. Logarithmic abelian varieties, Part IV: Proper models

    Kajiwara, Takeshi; Kato, Kazuya; Nakayama, Chikara
    This is part IV of our series of articles on log abelian varieties. In this part, we study the algebraic theory of proper models of log abelian varieties.

  13. Spherical functors on the Kummer surface

    Krug, Andreas; Meachan, Ciaran
    We find two natural spherical functors associated to the Kummer surface and analyze how their induced twists fit with Bridgeland’s conjecture on the derived autoequivalence group of a complex algebraic K3 surface.

  14. Decay estimates for solutions of nonlocal semilinear equations

    Cappiello, Marco; Gramchev, Todor; Rodino, Luigi
    We investigate the decay for $|x|\rightarrow\infty$ of weak Sobolev-type solutions of semilinear nonlocal equations $Pu=F(u)$ . We consider the case when $P=p(D)$ is an elliptic Fourier multiplier with polyhomogeneous symbol $p(\xi)$ , and we derive algebraic decay estimates in terms of weighted Sobolev norms. Our basic example is the celebrated Benjamin–Ono equation ¶ \begin{equation}(0.1)\quad (|D|+c)u=u^{2},\quad c\gt 0,\end{equation} for internal solitary waves of deep stratified fluids. Their profile presents algebraic decay, in strong contrast with the exponential decay for KdV shallow water waves.

  15. Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

    Ochiai, Tadashi; Shimomoto, Kazuma
    In this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa’s main conjectures, and the deformation theory of Galois representations.

  16. Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

    Holm, Thorsten; Jørgensen, Peter
    The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be $2$ -Calabi–Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a...

  17. Rational points on linear slices of diagonal hypersurfaces

    Brüdern, Jörg; Robert, Olivier
    An asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy–Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent interest.

  18. Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities

    Eyral, Christophe; Ruas, Maria Aparecida Soares
    We show that the possible jump of the order in an $1$ -parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families with isolated singularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman. ¶ As an application, we prove equimultiplicity for new families of nonisolated...

  19. Quantum reconstruction for Fano bundles on projective space

    Strangeway, Andrew
    We present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivization of the bundle from a small number of low-degree Gromov–Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch program.

  20. On certain mean values of the double zeta-function

    Ikeda, Soichi; Matsuoka, Kaneaki; Nagata, Yoshikazu
    In this article we discuss three types of mean values of the Euler double zeta-function. To get the results, we introduce three approximate formulas for this function.

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