Signed degree sets in signed graphs

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.     

Additive functions

1<q<2 L:= _n=1 1/q ⁿ=1/q–1. [0,1] _n()=1, A _n:= _i=1 ^n–1 _i(x)/q^i+1/n x _n(x)=0, _n>. , = _{n=1 n}(x)/qⁿ. F: [0,L]R , F(x)= _{n=1 n}(x)a_n, _n=1 ¦a _n¦<. [0,L]. q(1,2), . , q(1, 2), . .     

Continuous additive functions and difference equations of infinite order

Пустьq∈(1, 2) иL=(q?1)^?1. Дляz∈[0,L] обозначимδ(z) функцию, для которойδ(z)=1, еслиz≧1/q иδ(z)=0, еслиz<1/q. Пустьy(z) определяется из урав ненияz= =δ(z)q ^?1+y(z)q ^?1, и регулярное представление \(\mathop \Sigma \limits_{n = 1}^\infty \varepsilon _n \left( x \right)q^{ - n} \) аргументах определя ется из следующих соотношен ий: $$x = x_0 , \varepsilon _n \left( x \right) = \delta \left( {x_n } \right), x_{n + 1} = y\left( {x_n } \right).$$ ФункцияF: [0,L]→C называе тся аддитивной, если о на представляется в вид е $$F\left( x \right) = \mathop \Sigma \limits_{n = 1}^\infty \varepsilon _n \left( x \right)a_n ,$$ где ε ¦a _n ¦<∞. «Бесконеч ное» представление 1=εl _i q ^?1 числа 1 определяется с ледующим образом: еслие _n (1)=1 для б есконечно многихп, т оl _n =ε _n (1) (n=1, 2, ...); если ? максим альный индекс, для которогоε _s (1)=1, то $$l_{ks + 1} = \left\{ \begin{gathered} \varepsilon _i \left( 1 \right) \left( {k = 0, 1, 2, ...; i = 1, ..., s - 1} \right) \hfill \\ 0 \left( {i = 0; k = 1, 2, ...} \right). \hfill \\ \end{gathered} \right.$$ В более ранней работе, опубликованной в это м журнале, авторы доказали, что а ддитивная функция является неп рерывной на отрезке [0,L] тогда и только тогда, когда ра венство $$a_n = \mathop \Sigma \limits_{i = 1}^\infty l_i a_{n + 1} $$ выполняется для всехn∈N. В настоящей работе ра ссматриваются непре рывные функции для которых в ыполняются дополнительные усло вия видаa _n =O(q ^??n ) (0a _n ≧0. Анализируются их свя зи с корнями функцииG(z)=1 +ε l _i z ⁱ . Доказы вается, что непрерывн ая аддитивная функция и ли вляется линейной, или нигде не дифференцир уема на отрезке [0,L].     

Evaluation of the uranium concentration in rocks at the μg/g level by a low-background Geiger-Müller counter

A low-background -counter and a simple direct method, based on total -counting, have been developed in order to estimate the concentration of uranium at the g/g level in ordinary rock samples. The method is a relative one; international standards have been used for calibration. Only 3 g of rock powder is needed. Combining this method with -spectrometry, direct determination of²³⁸U (independently of²²⁶Ra) becomes possible near the above concentration level.     

An alternative method for pucker determination in carbohydrates from residual dipolar couplings: a solution NMR study of the fructofuranosyl ring of sucrose

A simple solution NMR method is presented for pucker determination of five-membered rings using only residual dipolar couplings obtained in a single liquid crystalline medium, DMPC/DHPC bicelles (DMPC = dimyristoylphosphatidylcholine; DHPC = dihexanoylphosphatidylcholine). The method was applied to determine the pucker of the fructofuranosyl ring of sucrose. The results indicate a fructofuranosyl pucker phase in the 20 degrees - 70 degrees range. The pucker phases are in agreement with those from previous NMR and optical spectroscopic studies and, importantly, do not rely on empirically parametrized Karplus curves. Furthermore, the analysis implies more than one stable pucker phase and rapid ring interconversion in this range. The present results suggest that using residual dipolar couplings alone can reveal multiple conformations present in solution. Hence, when a sufficient number of residual dipolar coupling constants is measured, the outcome is a robust, reliable, and independent route for determining carbohydrate and nucleic acid structure by NMR spectroscopy.     

Discriminating the helical forms of peptides by NMR and molecular dynamics simulation

The HNCO NMR pulse sequence was applied to three selectively labeled (15)N and (13)C isotopic homologues of the peptide Ac-WAAAH(AAARA)(3)A-NH(2) to probe directly for hydrogen bonds between residues 8 and 11 (characteristic of a 3(10)-helix), 8 and 12 (alpha-helix), and 8 and 13 (pi-helix). The experiments demonstrate conclusively, and in agreement with circular dichroism studies, that the center of the peptide is alpha-helical; there is no discernible 3(10)- or pi-helix at these specific positions. Molecular dynamics simulations of the preceding peptide and Ac-(AAAAK)(3)A-NH(2) in water using the potential energy parameter set CHARMM22/CMAP correctly yield an alpha-helix, in contrast to simulations with the set CHARMM22, which result in a pi-helix.     

Multiple-steering QM-MM calculation of the free energy profile in chorismate mutase

A novel technique for computing free energy profiles in enzymatic reactions using the multiple steering molecular dynamics approach in the context of an efficient QM-MM density functional scheme is presented. The conversion reaction of chorismate to prephenate catalyzed by the Bacillus subtilis enzyme chorismate mutase has been chosen as an illustrative example.     

Direct assignment of the absolute configuration of a distinct class of deoxyribonucleoside cyclic N-acylphosphoramidites at phosphorus by M-GOESY nuclear magnetic resonance spectroscopy

The determination of the absolute configuration of deoxyribonucleoside cyclic N-acylphosphoramidites at phosphorus toward the synthesis of P-stereodifined phosphorothioated oligodeoxyribonucleotides is easily accomplished with computer-assisted molecular modeling and M-GOESY NMR spectroscopy. Specifically, computer-modeling diasteromeric phosphoramidite 3 has identifed a proximal (2.55 A) through-space interaction between benzylic H-5 and sugar H-2' ', which can predictably be detected by M-GOESY NMR in SP-3 but not in RP-3 because of being too distant (5.85 A). Consistent with computer-assisted modeling predictions, M-GOESY NMR spectra of SP-3 and RP-3 revealed NOE signals generated from nuclei near the selectively excited H-2' ' that are common to both SP-3 and RP-3, namely those of H-2', H-4', H-3', and H-1'. In addition, a diagnostic NOE signal at 5.5 ppm (benzylic H-5) is, as predicted, only detected in SP-3 and thus provides an unequivocal assessment of the configuration of the diastereomer at phosphorus. M-GOESY NMR data also confirm that the condensation of deoxyribonucleoside cyclic N-acylphosphoramidites with base-activated nucleosidic or nucleotidic 5'-hydroxyls proceeds via a single nucleophilic event.