Further results on graph equations for line graphs and n-th power graphs

We present solutions of seven graph equations involving the line graph, complement and n-th power operations. One such equation $\text{L(G)}{}^{\mathrm{n}}\text{=}\text{G}\text{?}$ generalizes a result of M. Aigner. In addition, some comments are made about graphs satisfying $\text{G}{}^{\mathrm{n}}\text{=}\text{G}\text{?}$.     

Principle of inclusion-exclusion on semilattices

We introduce an enumeration theorem under lattice action. Let L be a finite semilattice and Ω be a nonempty set. Let $\text{f: L → P(Ω)}$ be a map satisfying f(x ? y) ? f(x) ∩ f(y), where ? and $\text{P}$(Ω) mean “join” and the power set of Ω, respectively. Then

$\text{m}\text{∪}\text{x?L}\text{?(x)}\text{=}\text{Σ}\text{c?C}\text{(?1)}{}^{\mathrm{l(c)}}\text{m}\text{∩}\text{x?c}\text{?(x)}$

, where C is the set of all chains in L and l(c) denotes the length of a chain c. Also the theorem can be dualized. Furthermore, we describe two applications of the theorem to a Boolean lattice of sets and a partition lattice of a set.     

Heat capacities of copper(II) formate tetrahydrate and tetradeuterate: A comparative study of phase transitions in layer hydrate crystals

The heat capacities of copper(II) formate tetrahydrate and tetradeuterate have been measured from 12 to 300 K with an adiabatic calorimeter. They have sigmoidal temperature dependence except near the antiferroelectric-paraelectric transition temperatures, 235.78 ± 0.05 K and 245.64 ± 0.05 K, respectively. The corresponding enthalpy changes are 836.0 ± 1.0 J mol^?1 and 936.9 ± 0.5 J mor^?1. The entropy changes are 3.546 ± 0.005 JK^?1mol^?1 and 3.814 ± 0.002 JK^?1 mol^?1. The heat capacities are larger in the high temperature phase than in the low temperature phase, the difference amounting to 5.74 JK^?1 mol^?1 and 7.15 JK^?1 mol^?1 for the hydrate and the deuterate, respectively. The heat capacity anomaly is compared with those in tin(II) chloride dihydrate and potassium hexacyanoferrate trihydrate and discussed in relation to the structure of the hydrogen bond networks in these substances. The discussion is extended to include possible properties of the hydrogen bond frameworks in ices I_h and II.     

Mechanics of a particle near the center of a rotating ring

The equations of motion of a test particle moving near the center of a massive rotating ring are derived up to the post-post-Newtonian order of approximation, by using the metric tensor for many body system which is Minkowskian at spatial infinity. Logarithmic divergences due to self-interaction of the ring appear in the equations of motion. These divergences can be removed by the procedure which is similar to the renormalization method in particle physics. In the equations of motion there appears a force directing to the rotation axis and depending on the angular velocity of the ring. This force vanishes when the magnitude of the gravitational constant times the mass of the ring divided by the radius of the ring is about one tenth of the square of the velocity of light. Under this condition it is shown that the relative magnitude of the Coriolis force to the centrifugal force in the equations of motion agrees with the expected one from the equations of motion in a rotating reference frame.     

Reaction profiles and substituent effects for the reduction of carbonyl compounds with monomer and dimer simple metal hydride reagents

Reaction profiles and energetics for the reactions of substituted benzaldehydes with a series of different simple metal hydrides (BH(3), BMeH(2), BMe(2)H, AlH(3), and AlMe(2)H) are examined computationally. B3LYP/6-31G optimizations and MP2/6-311G single point energy calculations revealed that the Al reagents are more reactive than B reagents. Replacement of H with Me on BH(3) or AlH(3) makes the reduction transition state (TS) less stable. Accordingly the overall reactivity is in the order AlH(3) > AlMe(2)H > BH(3) > BMe(2)H. The Hammett rho value for substituted benzaldehydes (BAs) is negative for the initial complex formation and positive for the hydride-transfer step. The size and the sign of the apparent rho value depend on the relative stabilities of the separated reactants and the complex. The TS structures vary according to the Hammond postulate for substituted BAs and the variation is reflected in carbonyl-carbon and aldehyde-deuterium isotope effects. Comparison of the reaction profiles of the monomer and dimer reagents reveals that the real reacting species is the monomer in the gas phase for BH(3) but the dimer for BMe(2)H.     

Ruthenium-complex-catalyzed regio- and stereoselective linear codimerization of 2-norbornenes with acrylic compounds

A linear codimerization of 2-norbornenes with acrylic compounds such as acrylates and an acrylamide proceeded efficiently by ruthenium catalyst systems, RuCl3(tpy)/Zn (tpy = 2,2':6',2' '-terpyridine) or [RuCl2(C6H6)]2/Zn in a primary or secondary alcoholic solvent, to afford the corresponding exo-trans-2-norbornylacrylates as major products regio- and stereoselectively along with a small amount of cis isomers. The reaction of 2,5-norbornadiene with methyl acrylate also gave the linear exo-trans codimer, which was effectively catalyzed by the addition of triarylphosphines to the RuCl3(tpy)/Zn catalyst system.     

Feeding attractants for the muricid gastropod Drupella cornus, a coral predator

The muricid gastropods genus Drupella are known to be voracious coral predator. Outbreaks of them have accelerated significant destruction on coral reefs, but its precise mechanism is poorly understood. Here, we describe the identification of montiporic acids C (1) and A (2) isolated from sea water extracts of the coral Montipora sp., which showed potent feeding-attractant activity toward D. cornus.