اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the logarithmic probability that a random integral ideal is $mathscr A$-free
83
0
0.0
(
0
)
تأليف
Christian Huck
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This extends a theorem of Davenport and Erdos on sequences of rational integers to sequences of integral ideals in arbitrary number fields $K$. More precisely, we introduce a logarithmic density for sets of integral ideals in $K$ and provide a formula for the logarithmic density of the set of so-called $mathscr A$-free ideals, i.e. integral ideals that are not multiples of any ideal from a fixed set $mathscr A$.
قيم البحث
اقرأ أيضاً
We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this graph invaria
nt depending on the edge density, we establish first that $P(G)$ is bounded below by a positive constant provided that the minimum degree is bounded below by a linear function in the number of vertices. Thereafter, the focus is shifted to the classical ErdH{o}s-Renyi random graph model $G(n,p)$. It is shown that $P(G)$ converges in probability to $e^{-1/(ep_{infty})}$ if $p to p_{infty} > 0$ and to $0$ if $p to 0$.
We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2ntimes 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k}=lim_{nrightarrow
infty}frac {1}{sqrt{2n}} log p_{2n,0}= -frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ where $zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{ngeq 1}$, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k_n}=-frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ provided $lim_{nrightarrow infty} left(n^{-1/2}log(n)right) k_{n}=0$.
The bulk density of a planet, as measured by mass and radius, is a result of planet structure and composition. Relative proportions of iron core, rocky mantle, and gaseous envelopes are degenerate for a given density. This degeneracy is reduced for r
ocky planets without significant gaseous envelopes when the structure is assumed to be a differentiated iron core and rocky mantle, in which the core mass fraction (CMF) is a first-order description of a planets bulk composition. A rocky planets CMF may be derived both from bulk density and by assuming the planet reflects the host stars major rock-building elemental abundances (Fe, Mg, and Si). Contrasting CMF measures, therefore, shed light on the outcome diversity of planet formation from processes including mantle stripping, out-gassing, and/or late-stage volatile delivery. We present a statistically rigorous analysis of the consistency of these two CMF measures accounting for observational uncertainties of planet mass and radius and host-star chemical abundances. We find that these two measures are unlikely to be resolvable as statistically different unless the bulk density CMF is at least 40% greater than or 50% less than the CMF as inferred from the host star. Applied to 11 probable rocky exoplanets, Kepler-107c has a CMF as inferred from bulk density that is significantly greater than the inferred CMF from its host star (2$sigma$) and is therefore likely an iron-enriched super-Mercury. K2-229b, previously described as a super-Mercury, however, does not meet the threshold for a super-Mercury at a 1- or 2- $sigma$ level.
In this short, we study sums of the shape $sum_{nleqslant x}{f([x/n])}/{[x/n]},$ where $f$ is Euler totient function $varphi$, Dedekind function $Psi$, sum-of-divisors function $sigma$ or the alternating sum-of-divisors function $beta.$ We improve pr
evious results when $f=varphi$ and derive new estimates when $f=Psi, f=sigma$ and $f=beta.$
For a partition $lambda$ of $n$, let $I^{rm Sp}_lambda$ be the ideal of $R=K[x_1, ldots, x_n]$ generated by all Specht polynomials of shape $lambda$. We show that if $R/I^{rm Sp}_lambda$ is Cohen--Macaulay then $lambda$ is of the form either $(a, 1,
ldots, 1)$, $(a,b)$, or $(a,a,1)$. We also prove that the converse is true if ${rm char}(K)=0$. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R/I^{rm Sp}_{(n-3,3)}$ is NOT Cohen--Macaulay if and only if ${rm char}(K)=2$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
