5,239 research outputs found
On type sequences and Arf rings
No full textIn this article we explicitly give a description to compute
the type sequence t_1, . . . , t_n of a semigroup S generated by an arithmetic sequence explicitly ; we show that the i-th term t_i is equal to 1 or to the type , depending on its position. Further, for analytically irreducible ring R with the branch sequence R_j ,we give a characterization of the “Arf” property using the type sequence of R and of the rings R_j .
Further, we prove some relations among the integers l*(R) and
l*(R_j ) . These relations allow us to obtaina new charaterization of semigroup rings of minimal multiplicity with l*(R)≤ type (R) in terms of the Arf property, type sequences and relations between
l*(R) and l*(Rj )
On the type sequence of some one dimensional rings
No full textIn this article we describe the holes and their positions of a numerical semigroup and use this description to compute the type sequence of the semigroup generated by an arithmetic explicitly
CM defect and Hilbert function of monomial curves
No full textIn this article we consider a semigroup ring R = K[[Γ ]] of a numerical semigroup Γ
and study the Cohen–Macaulayness of the associated graded ring G(Γ ) := grm(R) :=
⊕n∈N mn/mn+1 and the behaviour of the Hilbert function HR of R.Wedefine a certain (finite)
subset B(Γ ) ⊆ Γ and prove that G(Γ ) is Cohen–Macaulay if and only if B(Γ ) = ∅.
Therefore the subset B(Γ ) is called the Cohen–Macaulay defect of G(Γ ). Further, we prove
that if the degree sequence of elements of the standard basis of Γ is non-decreasing,
then B(Γ ) = ∅ and hence G(Γ ) is Cohen–Macaulay. We consider a class of numerical
semigroups Γ =
Σ3
i=0 Nmi generated by 4 elements m0,m1,m2,m3 such that m1 +m2 =
m0+m3—so called ‘‘balanced semigroups’’. We study the structure of the Cohen–Macaulay
defect B(Γ ) of Γ and particularly we give an estimate on the cardinality |B(Γ , r)| for
every r ∈ N. We use these estimates to prove that the Hilbert function of R is nondecreasing.
Further, we prove that every balanced ‘‘unitary’’ semigroup Γ is ‘‘2-good’’ and
is not ‘‘1-good’’, in particular, in this case, G(Γ ) is not Cohen–Macaulay. We consider a
certain special subclass of balanced semigroups Γ . For this subclass we try to determine
the Cohen–Macaulay defect B(Γ ) using the explicit description of the standard basis of Γ ;
in particular, we prove that these balanced semigroups are 2-good and determine when
exactly G(Γ ) is Cohen–Macaulay
On the Cohen–Macaulayness of some graded rings
No full textLet (R,m) be a 1-dimensional Cohen-Macaulay local ring of multiplicity e and embedding dimension v ≥2 . Let B denote the blowing-up of R along m and let I be the conductor of R in B. Let x be a superficial element in m of degree 1 and I' = (I +xR)/xR . We assume that the length l(I') = 1 . This class of local rings contains the class of 1-dimensional Gorenstein local rings . In section 1, we prove that if the associated graded ring G = gr(R) is Cohen-Macaulay, then I is contained in m^s + xR , where s is the degree of the h-polynomial h(R) of R. In section 2, we give necessary and sufficient conditions for the Cohen-Macaulayness of G.
These conditions are numerical conditions on the h-polynomial h(R) , particularly on its coefficients and the degree in comparison with the difference e − v . In section 3, we give some conditions for the Gorensteinness of G. In section 4, we give a characterisation (see 4.3) of numerical semigroup rings which satisfy the condition l(I') =
On the h–polynomial of certain monomial curves
No full textLet K be a field and let n_1,...,n_e be a sequence of positive integers with gcd(n_1,...,n_e) =1 and n_1 <... < n_e. Let A be the coordinate ring of the associated algebroid monomial curve in the affine algebroid e–space over K . In this article assuming that some e–1 terms of n_1,...,n_e form an arithmetic sequence, we compute ( under some mild additional assumptions , see theorem (2.7) for more precise assumptions ) , the h–polynomial ( and hence the Hilbert function) of A explicitly in terms of the standard basis of the semigroup generated by n_1,...,n_e. Our special assumptions are satisfied in the case e = 3 ; in particular, for the class of algebroid monomial space curves, we can write down the h–polynomial and hence the Hilbert function explicitly
On the length equalities for one–dimensional rings
No full textIn this article we characterize noetherian local one-dimensional analytically irreducible and residually rational domains (R,m) which are non-Gorenstein, the non-negative integer l*(R)=t(R).l(R/C) – l(S/R) is equal to t(R)–1 and l(R/(C+xR))=2, where t(R) is the Cohen–Macaulay type of R , C is the conductor of R in the integral closure S of R in its quotient field Q(R) and xR is a minimal reduction of, by giving some conditions on the numerical semi-group v(R) of R
Distance Agricultural Education: Perspectives in Agricultural Development in India
No full textPCF3 // Working paper presented by S A Nimbalkar, V D Patil, and P O Ingleenter at the Pan-Commonwealth Forum on Open Learning (PCF3) in Dunedin, New Zealand
Polyaniline-polypyrrole nanograined composite via electrostatic adsorption for high performance electrochemical supercapacitors
No full textAbstract not availableDeepak P. Dubal, Sandip V. Patil, G.S. Gund, Chandrakant D. Lokhand
Turmeric, naturally available colorimetric receptor for quantitative detection of fluoride and iron
No full textAbstract not availableMahesh P. Bhat, Madhuprasad, Pravin Patil, S.K. Nataraj, Tariq Altalhi, Ho-Young Jung, Dusan Losic, Mahaveer D. Kurkur
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