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In 1970s, Griggs conjectured that every normalized matching rank-unimodal poset has a nested chain decomposition. This conjecture is proved to be true only for some posets of small ranks (Wang Discrete Math. 145(3), 493–497, 2005; Hsu et al. Discrete Math. 309(3), 521–531, 2009; Escamilla et al. Order 28, 357–373, 2011). In this paper, we provide some sufficient conditions on the rank numbers of posets of rank 3 to satisfy the Griggs’s conjecture by refining the proofs in the two papers (Hsu et al. Discrete Math. 309(3), 521–531, 2009; Escamilla et al. Order 28, 357–373, 2011).

We extend the study on shadowable points recently introduced by Morales in relation to chaotic or non-chaotic properties. Firstly, some sufficient conditions for a quantitative shadowable point to be approximated by an entropy point are given. As a corollary, we get different three chaotic conditions from which a shadowable point becomes an entropy point. Secondly, we provide a dichotomy on the interior of the set of shadowable chain recurrent points by two canonical chaotic and non-chaotic dynamics, the full shift and odometers.

In 1975, J. Griggs conjectured that a normalized matching rank-unimodal poset possesses a nested chain decomposition. This elegant conjecture remains open even for posets of rank 3. Recently, Hsu, Logan, and Shahriari have made progress by developing techniques that produce nested chain decompositions for posets with certain rank numbers. As a demonstration of their methods, they prove that the conjecture is true for all rank 3 posets of width at most 7. In this paper, we present new general techniques for creating nested chain decompositions, and, as a corollary, we demonstrate the validity of the conjecture for all rank 3 posets of width at most 11.

Di(tert-butyl) trioxide in a solution of CFCl3 (Freon-11) at –23 °C exists in equilibrium with the tert-butoxyl and tert-butylperoxyl radicals virtually without irreversible decomposition. The above radicals decompose ozone to dioxygen with a high effective rate constant, which is proprotional to the square root of the ButOOOBut concentration. The kinetic scheme describing the found relationships was proposed.

An edge decomposition of a poset P is a collection of chains such that every pair of elements of which one covers the other belongs to exactly one chain. We consider this and the related notion of the line poset L(P) which consists of pairs of adjacent elements of P so that (x⋖y)

The symmetric chain decomposition of the lattice of divisors,D N, is applied to prove results about the strict unimodality of the Whitney numbers ofD N, about minimum interval covers for the union of consecutive rank-sets ofD N, and about the distribution of sums of vectors in which each vector can be included several times (an extension of the famous Littlewood-Offord problem)

The free radical initiated chain decomposition of 1,4-cis-polyisoprene was carried out in 13C-enriched tetrachloromethane.

The quantum yield for the photodissociation of phenyl azide in the concentration range from 10−4 to 10−1M is 0.4–0.5, while the yield of azobenzene does not exceed 50%.

1. n-Butylacetamide (HPNBAA) and ethylacetamide (HPEAA) hydroperoxides in n-butylacetamide medium (NBAA) thermally decompose into free radicals at similar rates according to the equation vi=kui1[ROOH]/ [NBAA] + ki2[ROOH]2. Rapid formation of radicals was found in the o-dichlorobenzene-ROOH-NBAA system. 2. Overall decomposition of HPNBAA and HPEAA in an inert atmosphere takes place primarily as radical-induced decomposition of ROOH, and the rate when [NBAA] ≥ 3.4 mole/liter obeys the equation vd=kd· [ROOH]. A diagram of the chain process which explains this dependence and the composition of the products of oxidation was proposed, and the rate constant of the reaction of the alkyl radicals in NBAA with ROOH was estimated. Of the inhibitors of free-radical processes studied,α-naphthol was not effective in inhibition of chain decomposition of hydroperoxides above 387°K.

The laser heating procedure has been used to study chain reactions under strictly homogeneous conditions. The rate constants for the initiation and overall chain decomposition of methanol have been determined.

We establish some inequalities connecting natural parameters of a partial order P. For example, if every interval [a,b] contains at most λ maximal chains, if some antichain has cardinality v, and if there are χ1 chains whose union is cofinal and coinitial in P, then the chain decomposition number for P is ⩽χ1λv (Theorem 2.2), and the inequality is sharp in a certain sense (Section 3).

1. Study of kinetics and reaction products has shown that ozone undergoes chain decomposition in the CH3CHO-O3-O2 system. A scheme, based on the CH3\(\dot O\)2 + O3 → CH3\(\dot O\) + 2O2, CH3\(\dot O\) + O3 → CH3\(\dot O\)2 + O2 reactions, is proposed. 2. Rate constants for initiation and the primary reaction of CH3CHO with O3, valid at 27°C, have been determined; the rate constant for the reaction of ozone with the methylperoxide radical has been determined from the measured efficiency of initiation.

The hydroperoxides resulting from the oxidation of the dicaprylate of diethylene glycol and the tetravalerate of pentaerythrite can undergo either radical or chain decomposition. Rate constants have been determined for the reactions of alkyl radicals with hydroperoxides.

1. The process of chain-thermal flame propagation with a simplified chain decomposition scheme was investigated and a principle for determining the flame velocity from the greatest possible rate of heat evolution was proposed. 2. Approximate formulas were deduced to calculate flame velocity and these gave values which practically coincided with the results of numerical integration of the initial system of equations. The formulas obtained were used for numerical calculations on examples of flame decomposition of hydrazine.

The thermal gross decomposition of dimethylacetamide hydroperoxide (ROOH) in dimethylacetamide (RH) as the medium proceeds in harmony with the equation: Wp=kind [ROOH]0m/[RH]0n, where m=1.5−2; n=2−0.5. Alkyl radicals with a chain length of 195−3 units are mainly responsible for the induced decomposition. We determined the ratio between the molecular and radical directions of the decomposition and kind in the range 80–132°.