let $r=k[x_1,x_2,cdots, x_n]$ be a polynomial ring over a field $k$. we prove that for any positive integers $m, n$, $text{reg}(i^mj^nk)leq mtext{reg}(i)+ntext{reg}(j)+text{reg}(k)$ if $i, j, ksubseteq r$ are three monomial complete intersections ($i$, $j$, $k$ are not necessarily proper ideals of the polynomial ring $r$), and $i, j$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l...

in this study, b12n12 nano ring has been selected because it consist of four 6-side rings and polar bonds b-n which in comparison with non-polar bonds c-c, is more suitable for the study of the absorption of other compounds. so reactivity and stability of benzene alone and in the presence b12n12 nano ring field checked. to determine the non-bonded interaction energies between benzene and b12n12...

Let $R=k[x_1,x_2,cdots, x_N]$ be a polynomial ring over a field $k$. We prove that for any positive integers $m, n$, $text{reg}(I^mJ^nK)leq mtext{reg}(I)+ntext{reg}(J)+text{reg}(K)$ if $I, J, Ksubseteq R$ are three monomial complete intersections ($I$, $J$, $K$ are not necessarily proper ideals of the polynomial ring $R$), and $I, J$ are of the form $(x_{i_1}^{a_1}, x_{i_2}^{a_2}, cdots, x_{i_l...

Two potassium-dialkyl-TMP-zincate bases [(pmdeta)K(mu-Et)(mu-tmp)Zn(Et)] (1) (PMDETA = N,N,N',N'',N''-pentamethyldiethylenetriamine, TMP = 2,2,6,6-tetramethylpiperidide), and [(pmdeta)K(mu-nBu)(mu-tmp)Zn(nBu)] (2), have been synthesized by a simple co-complexation procedure. Treatment of 1 with a series of substituted 4-R-pyridines (R = Me(2)N, H, Et, iPr, tBu, and Ph) gave 2-zincated products ...

Coarctate reactions are defined as reactions that include atoms at which two bonds are made and two bonds are broken simultaneously. In the pursuit of the discovery of new coarctate reactions we investigate the fragmentation reactions of cyclic ketals. Three ketals with different ring sizes derived from indan-2-one were decomposed by photolysis and pyrolysis. Particularly clean is the photolysi...

As a function of the magnetic flux threading the object, the Little-Parks oscillation in the critical temperature of a large-radius, thin-walled superconducting ring or hollow cylinder has a period given by h /2e, due to the binding of electrons into Cooper pairs. On the other hand, the single-electron Aharonov-Bohm oscillation in the resistance or persistent current for a clean i.e., ballistic...

The synthesis of cis-N-[a-carbobenzyloxy-b-(p-hydroxyphenyl)ethyl]-3-phthalimido-4-styryl-2-azetidinone is described. We have found that the electron-rich Schiff bases can afford the cis-b-lactam ring even in the presence of the free hydroxyl functions. The mechanisms of cis and trans-b-lactam ring formation are discuss...