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Search: All articles in the CMB digital archive with keyword Logarithm

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1. CMB Online first

Cui, Xiaohui; Wang, Chunjie; Zhu, Kehe
Area Integral Means of Analytic Functions in the Unit Disk
For an analytic function $f$ on the unit disk $\mathbb D$ we show that the $L^2$ integral mean of $f$ on $c\lt |z|\lt r$ with respect to the weighted area measure $(1-|z|^2)^\alpha\,dA(z)$ is a logarithmically convex function of $r$ on $(c,1)$, where $-3\le\alpha\le0$ and $c\in[0,1)$. Moreover, the range $[-3,0]$ for $\alpha$ is best possible. When $c=0$, our arguments here also simplify the proof for several results we obtained in earlier papers.

Keywords:logarithmic convexity, area integral mean, Bergman space, Hardy space
Categories:30H10, 30H20

2. CMB 2017 (vol 60 pp. 490)

Fiori, Andrew
A Riemann-Hurwitz Theorem for the Algebraic Euler Characteristic
We prove an analogue of the Riemann-Hurwitz theorem for computing Euler characteristics of pullbacks of coherent sheaves through finite maps of smooth projective varieties in arbitrary dimensions, subject only to the condition that the irreducible components of the branch and ramification locus have simple normal crossings.

Keywords:Riemann-Hurwitz, logarithmic-Chern class, Euler characteristic
Categories:14F05, 14C17

3. CMB Online first

Miranda-Neto, Cleto Brasileiro
A module-theoretic characterization of algebraic hypersurfaces
In this note we prove the following surprising characterization: if $X\subset {\mathbb A}^n$ is an (embedded, non-empty, proper) algebraic variety defined over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ of logarithmic vector fields of $X$ is a reflexive ${\mathcal O}_{{\mathbb A}^n}$-module. As a consequence of this result, we derive that if $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ is a free ${\mathcal O}_{{\mathbb A}^n}$-module, which is shown to be equivalent to the freeness of the $t$th exterior power of $T_{{\mathcal O}_{{\mathbb A}^n}/k}(X)$ for some (in fact, any) $t\leq n$, then necessarily $X$ is a Saito free divisor.

Keywords:hypersurface, logarithmic vector field, logarithmic derivation, free divisor
Categories:14J70, 13N15, 32S22, 13C05, 13C10, 14N20, , , , , 14C20, 32M25

4. CMB 2016 (vol 60 pp. 184)

Pathak, Siddhi
On a Conjecture of Livingston
In an attempt to resolve a folklore conjecture of Erdös regarding the non-vanishing at $s=1$ of the $L$-series attached to a periodic arithmetical function with period $q$ and values in $\{ -1, 1\} $, Livingston conjectured the $\bar{\mathbb{Q}}$ - linear independence of logarithms of certain algebraic numbers. In this paper, we disprove Livingston's conjecture for composite $q \geq 4$, highlighting that a new approach is required to settle Erdös's conjecture. We also prove that the conjecture is true for prime $q \geq 3$, and indicate that more ingredients will be needed to settle Erdös's conjecture for prime $q$.

Keywords:non-vanishing of L-series, linear independence of logarithms of algebraic numbers
Categories:11J86, 11J72

5. CMB 2005 (vol 48 pp. 473)

Zeron, E. S.
Logarithms and the Topology of the Complement of a Hypersurface
This paper is devoted to analysing the relation between the logarithm of a non-constant holomorphic polynomial $Q(z)$ and the topology of the complement of the hypersurface defined by $Q(z)=0$.

Keywords:Logarithm, homology groups and periods
Categories:32Q55, 14F45

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