Note on a theorem of Eliahou Hellus, M.; Rechenauer, A.; Waldi, R.
Semigroup forum,
08/2018, Letnik:
97, Številka:
1
Journal Article
Recenzirano
From a result (Wilf’s conjecture and Macaulay’s theorem,
2017
, Theorem 5.11) of Eliahou on the growth of the Hilbert function of a standard graded algebra we derive an inequality related to a ...question in Wilf (Am Math Mon 85,
1978
). This enables us to construct a (to our knowledge new) class of numerical semigroups of embedding dimension 5 and arbitrarily high Cohen-Macaulay-type, for which the question of Wilf (
1978
) has an affirmative answer.
In J. Herzog and E. Kunz (1973) 6 it was shown that for any pair (d,t)∈N×N+ with (d,t)≠(1,1) there exists a local Cohen–Macaulay ring R having deviation d(R)=d and type t(R)=t. By E. Kunz (1974) 7 ...the case d(R)=1,t(R)=1 cannot occur. In this paper certain Cohen–Macaulay rings are studied for which there are close relations between deviation, type and embedding dimension. Similar relations for other classes of local rings have been proved in the recent paper by L. Sharifan (2014) 15. Our relations will be applied to numerical semigroups (or equivalently monomial curves) and lead also to some further cases, generalizing E. Kunz (2016) 8 with ring-theoretic proofs, in which a question of H. Wilf (1978) 16 has a positive answer.
We show that the number of numerical semigroups containing two given coprime numbers
p
and
q
agrees with a quasipolynomial in
q
of degree exactly
p
-
1
and having constant leading coefficient lying ...between
1
(
p
-
1
)
!
·
p
!
and
1
(
p
-
1
)
·
p
!
.
The phenomena of meson-anti-meson flavour oscillation and CP violation are known for several decades as properties in the neutral kaon system. Oscillation in the neutral
B meson system is known since ...1987, and CP violation of
B mesons is still being searched for today. First results on one CP-violating parameter in
B
0 meson decays have become available recently. The present review introduces the phenomenology of both effects, their description within the Standard Model, and the presently available experimental results on
B mesons.
Let
p
,
q
∈
N
be relatively prime integers with
1
<
p
<
q
. In Knebl et al. (J Algebra 348:315–335,
2011
) the numerical semigroups containing
p
and
q
have been illustrated by certain lattice paths ...in the plane. In this paper we derive some further information from this visualization, for example formulas to compute or estimate the number of certain semigroups by counting lattice paths, and formulas for their genus and Frobenius number.
We use 429 fb$^{-1}$ of $e^+e^-$ collision data collected at the $\Upsilon(4S)$ resonance with the BABAR detector to measure the radiative transition rate of $b\rightarrow s\gamma$ with a sum of 38 ...exclusive final states. The inclusive branching fraction with a minimum photon energy of 1.9 GeV is found to be $\mathcal{B}(\bar B \rightarrow X_{s}\gamma)=(3.29\pm 0.19\pm 0.48)\times 10^{-4}$ where the first uncertainty is statistical and the second is systematic. We also measure the first and second moments of the photon energy spectrum and extract the best fit values for the heavy-quark parameters, $m_{b}$ and $\mu_{\pi}^{2}$, in the kinetic and shape function models.
We are interested in the smallest number 𝑠 = 𝑠(𝑛, 𝑞) such that, for any given 𝑛 distinct F𝑞-rational points 𝑃₁, ... ,𝑃𝑛 ∈ P𝑛−1, there exists a hypersurface 𝐻 of degree 𝑠 and defined over ...F𝑞 such that 𝑃₁, ... , 𝑃𝑛−1 ∈ 𝐻, 𝑃𝑛 ∉ 𝐻. Alternately, 𝑠(𝑛, 𝑞) is the maximal Castelnuovo-Mumford regularity of a set of 𝑛 F𝑞-rational points in some projective space. Finally, 𝑠(𝑛, 𝑞) is the index of stability of certain one-dimensional local Cohen-Macaulay rings.
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