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Register a new userThe present state of the capitulation problem
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Jean-Franc{c}ois Jaulent
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The text is a synthetic presentation of the state of the knowledge about the capitulation for the class-groups of numbers fields, shortly before the demonstration by Suzuki of the main conjecture on this question.
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Building on Boscas method, we extend to tame ray class groups the results on capitulation of ideals of a number field by composition with abelian extensions of a subfield first studied by Gras. More precisely, for every extension of number fields K/k, where at least one infinite place splits completely, and every squarefree divisor m of K, we prove that there exist infinitely many abelian extensions F/k such that the ray class group mod m of K capitulates in KF. As a consequence we generalize to tame ray class groups the results of Kurihara on triviality of class groups for maximal abelian pro-extensions of totally real number fields.
The RAdial Velocity Experiment (RAVE) is an ambitious survey to measure the radial velocities, temperatures, surface gravities, metallicities and abundance ratios for up to a million stars using the 1.2-m UK Schmidt Telescope of the Anglo-Australian Observatory (AAO), over the period 2003 - 2011. The survey represents a big advance in our understanding of our own Milky Way galaxy. The main data product will be a southern hemisphere survey of about a million stars. Their selection is based exclusively on their I--band colour, so avoiding any colour-induced bias. RAVE is expected to be the largest spectroscopic survey of the Solar neighbourhood in the coming decade, but with a significant fraction of giant stars reaching out to 10 kpc from the Sun. RAVE offers the first truly representative inventory of stellar radial velocities for all major components of the Galaxy. Here we present the first scientific results of this survey as well as its second data release which doubles the number of previously released radial velocities. For the first time, the release also provides atmospheric parameters for a large fraction of the second year data, making it an unprecedented tool to study the formation of the Milky Way. Plans for further data releases are outlined.
For any positive integer $n$, define an iterated function $$ f(n)=left{begin{array}{ll} n/2, & mbox{$n$ even,} 3n+1, & mbox{$n$ odd.} end{array} right. $$ Suppose $k$ (if it exists) is the lowest number such that $f^{k}(n)<n$, and there are $O(n)$ multiply by three and add one and $E(n)$ divide by two from $n$ to $f^{k}(n)$, then there must be $$ 2^{E(n)-1}<3^{O(n)}<2^{E(n)}. $$ Our results confirm the conjecture proposed by Terras in 1976.
Let $e(s)$ be the error term of the hyperbolic circle problem, and denote by $e_alpha(s)$ the fractional integral to order $alpha$ of $e(s)$. We prove that for any small $alpha>0$ the asymptotic variance of $e_alpha(s)$ is finite, and given by an explicit expression. Moreover, we prove that $e_alpha(s)$ has a limiting distribution.
The Gauss circle problem concerns the difference $P_2(n)$ between the area of a circle of radius $sqrt{n}$ and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients $P_2(n)^2$, and prove that this series has meromorphic continuation to $mathbb{C}$. Using this series, we prove that the Laplace transform of $P_2(n)^2$ satisfies $int_0^infty P_2(t)^2 e^{-t/X} , dt = C X^{3/2} -X + O(X^{1/2+epsilon})$, which gives a power-savings improvement to a previous result of Ivic [Ivic1996]. Similarly, we study the meromorphic continuation of the Dirichlet series associated to the correlations $r_2(n+h)r_2(n)$, where $h$ is fixed and $r_2(n)$ denotes the number of representations of $n$ as a sum of two squares. We use this Dirichlet series to prove asymptotics for $sum_{n geq 1} r_2(n+h)r_2(n) e^{-n/X}$, and to provide an additional evaluation of the leading coefficient in the asymptotic for $sum_{n leq X} r_2(n+h)r_2(n)$.
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