Resolving a problem raised by Norin in 2020, we show that for each k ? N $k \in \mathbb {N}$ , the minimal f ( ) $f(k) with property every graph G $G$ chromatic number at least + 1 $f(k)+1$ contains subgraph H $H$ both connectivity and $k$ satisfies ? 7 \leqslant 7k$ . This result is best-possible up to multiplicative constants, sharpens earlier results of Alon–Kleitman–Thomassen–Saks–Seymour f...

For k > 1, r 6= 0 and large x, let π 2r (x) denote the number of prime pairs (p, p+2r) with p ≤ x. By the Bateman–Horn conjecture the function π 2r(x) should be asymptotic to (2/k)C k 2rli2(x), with certain specific constants C 2r . Heuristic arguments lead to the conjecture that these constants have mean value one, just like the Hardy–Littlewood constants C2r for prime pairs (p, p + 2r). The c...

The acidity constants of some synthesized protonated pyrazolo quinazoline compounds were determined potentiometrically at ionic strength of 0.1 M in DMF: water (60:40 v/v) system at different temperatures (298.15 K to 318.15 K). The pKa values have been found to increase with increasing electron-donating nature of substitutions. Some thermodynamics parameters such as enthalpy (ΔH°), Gibb’s free...

in this paper we introduce two-wavelet constants for square integrable representations of homogeneous spaces. we establishthe orthogonality relations for square integrable representationsof homogeneous spaces which give rise to the existence of aunique self adjoint positive operator on the set of admissiblewavelets. finally, we show that this operator is a constant multiple of identity operator...

Equilibrium constants and rate constants involving Cl"~aq!, Cl, Cl2 "(aq), HO", H2O, and H2O2(aq) determined at 29762 K in the aqueous phase are updated and evaluated. Most of the rate constants and equilibrium constants are obtained by either pulse radiolysis or laser flash photolysis. The recommended values of rate constants and equilibrium constants are achieved by un-weighted averaging of t...

A spectrophotometric method was used to determine the acidity constant of 4'-carboxybenzo-24-crown-8 in 52% (w/w) ethanol-water mixture. The acid dissociation constant, Ka, was found to increase in the presence of Na+, K+, Rb+, Cs+ and Tl+ ions. The formation constants of the resulting 1:1 complexes between the carboxycrown a...

We exhibit classes of polynomials whose sets of kth partial derivatives form Gro bner bases for all k, with respect to all term orders. The classes are deened by syntactic constraints on arithmetical formulas deening the polynomials. Read-once formulas without constants have this property for all k, while those with constants have a weaker \Grr obner-bounding" property introduced here. For k = ...