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The EU´s falling share in global manufacturing has fuelled concerns about an overall loss of EU competitiveness, in particular vis-à-vis China. We analyse the empirical evidence underlying these concerns by applying a newly developed decomposition technique to global input-output data spanning the years 2000 to 2014. Our results confirm the diminishing role of the EU in manufacturing value chains, but also show that this is mostly, by nearly 75 per cent, a consequence of the geographical and sectoral reallocation of global demand, reflecting the lower economic growth in the EU relative to the rest of the world. Still, the other almost 25 per cent of the EU’s loss of global share is explained by its lower participation in manufacturing value chains, which confirms a downturn in EU competitiveness. By extending the analysis to individual manufacturing activities we show that this general trend is more pronounced for low-tech (e.g. textiles) than high-tech sectors, with pharmaceuticals emerging as the most resilient EU industry. Policy concerns appear to be most warranted for electronics, a key sector for which the EU´s global share fell even more than for overall manufacturing, without evidence that EU value added from upstream service inputs could significantly mitigate this trend.

Thermal stability of cyclododecylidene-1,1-bishydroperoxide in xylene has been studied and activation parameters of the reaction have been determined. The thermolysis has been accompanied by the reaction with ferrocene and the induced radical chain decomposition of cyclododecylidene-1,1-bishydroperoxide.

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).

The kinetics of thermal decomposition of didecanoyl diperoxyadipate in different organic solvents have been studied. The primary homolytic dissociation of the peroxide bond (O–O) is accompanied by secondary induced chain decomposition processes. The reaction medium affects both the rate of primary homolytic decomposition and secondary decomposition processes. Correlation equations have been proposed for the rate constants of the reactions under study and physicochemical parameters of the solvents.

Thermal decomposition of peroxydecanoic acid has been investigated in various organic solvents. The reaction medium significantly affects the rate of peroxydecanoic acid thermal decomposition. It has been shown that primary homolytic decomposition of peroxy group is followed by induced radical chain decomposition reactions. The solvent nature influences both primary homolytic decomposition and secondary induced reactions. The correlation equations relating kinetic parameters of peroxydecanoic acid thermolysis and basic solvent properties have been derived.

We present linear time in-place algorithms for several fundamental graph problems including the well-known graph search methods (like depth-first search, breadth-first search, maximum cardinality search), connectivity problems (like biconnectivity, 2-edge connectivity), decomposition problem (like chain decomposition) among various others, improving the running time (by polynomial multiplicative factor) of the recent results of Chakraborty et al. [ESA, 2018] who designed \(O(n^3 \lg n)\) time in-place algorithms for some of the above mentioned problems. The running times of all our algorithms are essentially optimal as they run in linear time. One of the main ideas behind obtaining these algorithms is the detection and careful exploitation of sortedness present in the input representation for any graph without loss of generality. This observation alone is powerful enough to design some basic linear time in-place algorithms, but more non-trivial graph problems require extra techniques which, we believe, may find other applications while designing in-place algorithms for different graph problems in future.

Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov--Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The $q,t$-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients. For sequences of length at most 4, we prove that these coefficients enumerate subdiagrams in a certain fixed Young diagram and give an explicit symmetric chain decomposition of the set of such diagrams. This strengthens results of Lee, Li and Loehr for $(4,n)$ rational $q,t$-Catalan numbers., Comment: 33 pages; v2: fixed typos and included referee comments

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.

We consider the problem of minimizing the number of airplanes needed to fly a fixed daily repeating schedule of flights. We use deficit functions (DF) to decompose an aviation schedule of aircraft flights into aircraft chains (routes) called a chain decomposition. Each chain visits periodically a set of airports and is served by several cockpit crews circulating along the airports of this set. The initial step in our approach is to find the minimal number of aircraft needed to carry out the flight schedule. This is achieved by using the fleet size theorem based on a DF representation of an aircraft flight schedule. A DF is a step function associated with an aircraft terminal which changes by + 1 and −1 at flight departure and arrival times, respectively. DF theory was developed in the 1960–70s by Linis and Maksim (1967) and Gertsbakh and Gurevich (1977). Although the initial application of DFs was to the Russian AEROFLOT fleet it has subsequently attracted more attention on bus scheduling than aircraft scheduling. Here we discuss the revival of this method and its crucial use to construct the so-called chain decomposition of the schedule for a single period. We provide a justification for maximizing the number of balanced chains (flight sequences with the same start and end terminals). To do this we propose The Maximal Balanced Chain Problem. These are then converted into a set of infinite periodic flight sequences, each of which can be carried out by a single aircraft. The conversion is carried out by mapping the single period set of chains into an Euler graph. To construct the set of mutiperiod chains that are “balanced” (return to the same terminal at the start) we find all edge disjoint cycle covers of the Euler graph using a modified version of Hierholzer's algorithm. These cycles are converted back into a balanced multiperiod chain solution and modified to conform to any maintenance constraints. To insure maintenance check constraints are satisfied for multiperiod chains, it may be necessary to add deadhead flights. Minimizing the cost of deadhead trips and overnight stays provide the basis for selecting an optimal routing solution. Keywords: Deficit function, Flight schedule, Aircraft fleet, Chain decomposition, Aircraft routing

The Boolean lattice $2^{[n]}$ is the family of all subsets of $[n]=\{1,\dots,n\}$ ordered by inclusion, and a chain is a family of pairwise comparable elements of $2^{[n]}$. Let $s=2^{n}/\binom{n}{\lfloor n/2\rfloor}$, which is the average size of a chain in a minimal chain decomposition of $2^{[n]}$. We prove that $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that all but at most $o(1)$ proportion of the chains have size $s(1+o(1))$. This asymptotically proves a conjecture of F\"uredi from 1985. Our proof is based on probabilistic arguments. To analyze our random partition we develop a weighted variant of the graph container method. Using this result, we also answer a Kalai-type question raised recently by Das, Lamaison and Tran. What is the minimum number of forbidden comparable pairs forcing that the largest subfamily of $2^{[n]}$ not containing any of them has size at most $\binom{n}{\lfloor n/2\rfloor}$? We show that the answer is $(\sqrt{\frac{\pi}{8}}+o(1))2^{n}\sqrt{n}$. Finally, we discuss how these uniform chain decompositions can be used to optimize and simplify various results in extremal set theory., Comment: 22 pages

Thermal decomposition of cellulose can be upgraded by means of an electron-beam irradiation to produce valuable organic products via chain mechanisms. The samples being irradiated decompose effectively at temperatures below the threshold of pyrolysis inception. Cellulose decomposition resembles local “explosion” of the glucopyranose unit when fast elimination of carbon dioxide and water precede formation of residual carbonyl or carboxyl compounds. The dry distillation being performed during an irradiation gives a liquid condensate where furfural and its derivatives are dominant components. Excessively fast heating is adverse, as it results in a decrease of the yield of key organic products because pyrolysis predominates over the radiolytic-controlled decomposition of feedstock. Most likely, conversion of cellulose starts via radiolytic formation of macroradicals do not conform with each other, resulting in instability of the macroradical. As a consequence, glucosidic bond cleavage, elimination of light fragments (water, carbon oxides, formaldehyde, etc.) and formation of furfural take place.

We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is "taut", i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \ldots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions --- the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symmetric chain decompositions \cite{orth}, making progress on a conjecture of Shearer and Kleitman. In general, we show that for a finite graded poset $P$, there exists a canonical bijection between symmetric chain decompositions of $P \times m$ and $P \times n$ for $m, n\ge \text{rk}(P) + 1$, that preserves the existence of taut chains. If $P$ has a unique maximal and minimal element, then we also produce a canonical $(\text{rk}(P) +1)$ to $1$ surjection from symmetric chain decompositions of $P \times (\text{rk}(P) + 1)$ to symmetric chain decompositions of $P \times \text{rk}(P)$ which sends decompositions with taut chains to decompositions with taut chains., Comment: 11 pages, 1 figure, 3 tables, fixed arXiv reference

In 1979, Shearer and Kleitman conjectured that there exist $\lfloor n/2 \rfloor+1$ orthogonal chain decompositions of the hypercube $Q_n$, and constructed two orthogonal chain decompositions. In this paper, we make the first non-trivial progress on this conjecture since by constructing three orthogonal chain decompositions of $Q_n$ for $n$ large enough. To do this, we introduce the notion of "almost orthogonal symmetric chain decompositions". We explicitly describe three such decompositions of $Q_5$ and $Q_7$, and describe conditions which allow us to decompose products of hypercubes into $k$ almost orthogonal symmetric chain decompositions given such decompositions of the original hypercubes., Comment: 17 pages, 4 figures, fixed arXiv reference

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.

There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable M\"obius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests., Comment: 22 pages, 13 figures

Decomposing a graph into a hierarchical structure via k-core analysis is a standard operation in any modern graph-mining toolkit. k-core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere degree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connectedness, and it allows to reveal the structural organization of the graph. Despite the fact that k-core analysis relies on vertex degrees, k-cores do not satisfy a certain, rather natural, density property. Simply put, the most central k-core is not necessarily the densest subgraph. This inconsistency between k-cores and graph density provides the basis of our study. We start by defining what it means for a subgraph to be locally-dense, and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by k-cores, but in this case the components are arranged in order of increasing density. We show that such a locally-dense decomposition for a graph G = (V, E) can be computed in polynomial time. The running time of the exact decomposition algorithm is O(|V|^2|E|) but is significantly faster in practice. In addition, we develop a linear-time algorithm that provides a factor-2 approximation to the optimal locally-dense decomposition. Furthermore, we show that the k-core decomposition is also a factor-2 approximation, however, as demonstrated by our experimental evaluation, in practice k-cores have different structure than locally-dense subgraphs, and as predicted by the theory, k-cores are not always well-aligned with graph density.

We study pattern avoidance by combinatorial objects other than permutations, namely by ordered partitions of an integer and by permutations of a multiset. In the former case we determine the generating function explicitly, for integer compositions of n that avoid a given pattern of length 3 and we show that the answer is the same for all such patterns. We also show that the number of multiset permutations that avoid a given three-letter pattern is the same for all such patterns, thereby extending and refining earlier results of Albert, Aldred et al., and by Atkinson, Walker and Linton. Further, the number of permutations of a multiset S, with a_i copies of i for i = 1, ..., k, that avoid a given permutation pattern in S_3 is a symmetric function of the a_i's, and we will give here a bijective proof of this fact first for the pattern (123), and then for all patterns in S_3 by using a recently discovered bijection of Amy N. Myers., 8 pages, no figure

Consider any sets x ⊆ y ⊆ {1,...,n}. Remove the interval [x, y] = {z ⊆ y|x ⊆ z} from the Boolean lattice of all subsets of {1,...,n}. We show that the resulting poset, ordered by inclusion, has a nested chain decomposition and has the normalized matching property. We also classify the largest antichains in this poset. This generalizes results of Griggs, who resolved these questions in the special case x = θ.

Let 2[n] denote the Boolean lattice of order n, that is, the poset of subsets of {1, …, n} ordered by inclusion. Recall that 2[n] may be partitioned into what we call the canonical symmetric chain decomposition (due to de Bruijn, Tengbergen, and Kruyswijk), or CSCD. Motivated by a question of Füredi, we show that there exists a function d(n)∼12n such that for any n⩾0, 2[n] may be partitioned into (n⌊n/2⌋) chains of size at least d(n). (For comparison, a positive answer to Füredi's question would imply that the same result holds for some d(n)∼π/2n.) More precisely, we first show that for 0⩽j⩽n, the union of the lowest j+1 elements from each of the chains in the CSCD of 2[n] forms a poset Tj(n) with the normalized matching property and log-concave rank numbers. We then use our results on Tj(n) to show that the nodes in the CSCD chains of size less than 2d(n) may be repartitioned into chains of large minimum size, as desired.

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.

A number of experiments have been performed in an effort to better understand the photoinitiated chain decomposition of ClN3. Discharge-flow methods were used to determine the rate of energy exchange between vibrationally excited N2 (a likely chain carrier) and ClN3. The rate constant for energy transfer from N2(v=1) to ClN3 was found to be (2.0 ± 1.0) × 10-13 cm3 s-1 at 300 K. This process is thought to excite the ν2 mode in ClN3 with the release of 281 cm-1 of excess energy. Experiments were also performed in which the decomposition of ClN3 was initiated by photolysis with a pulsed KrF laser at 249 nm, with subsequent observation of the time dependence of the densities of ClN3, electronically excited NCl(a1Δ), and vibrationally excited ClN3(ν2). A kinetic model for the ClN3 decomposition was assembled based on reactions with NCl(a1Δ) and N2(ν).

Loeb et al. (J. Combin. Theory Ser. A. 65 (1994) 151-157) gave an explicit semisymmetric chain decomposition of Pi(n). In this paper we give a second explicit semisymmetric chain decomposition and show that it is different from the one given in Loeb et al. (1995) in the sense that no order automorphism of Pi(n) can carry one to another. (C) 2000 . MSC: primary 05D05; secondary 06A07; 05A18; 05B35.

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)

In the cobalt(II)-catalysed decomposition of t-butyl hydroperoxide at 25°C in a 5 : 1 mixture (v/v) of chlorobenzene and dioxane, cobalt 2-ethylhexanoate is more active as a catalyst than the cobalt(II) complexes of a number of 1,3-dicarbonyl compounds. These complexes become deactivated and, in general, a decrease in the acidity of the parent 1,3-dicarbonyl compound lowers the rate of deactivation. This relationship, as well as the high activity of the 2-ethylhexanoate, is consistent with the radical chain decomposition of the hydroperoxide being initiated by a catalytic cycle which involves bridged dicobalt complexes. This cycle differs from the currently accepted Haber and Weiss cycle in that it does not directly generate an alkoxy radical.

We present new data structures for representing binary relations in an adaptive way, that is, for certain classes of inputs we achieve space below the general information theoretic lower bound, while achieving reasonable space complexities in the worst case. Our approach is derived from a geometric data structure [Arroyuelo et al., TCS 2011]. When used for representing permutations, it converges to a previously known adaptive representation [Barbay and Navarro, STACS 2009]. However, this new way of approaching the problem shows that we can support range searching in the adaptive representation. We extend this approach to representing binary relations, where no other adaptive representations using this chain decomposition have been proposed.

We modify the classical chain decomposition model for vehicle routing in order to formulate a very general class of vehicle routing and scheduling problems with time windows, in which vehicles may pick up loads while on the way to deliver others. We use the model as a framework for a tabu search heuristic that exploits the constraint structure these problems have in common while accommodating complex side constraints and objective functions. We test the heuristic computationally on several random problems and a problem supplied by the U.S. Military Airlift Command.

The chain decomposition of the bis(oxalato)peroxotitanate(IV) complex [Ti(O2)(ox)2]2− (ox = C2O42−) is initiated by the addition of a very small amount of reducing agent, such as Ti(III), Cr(II) or V(II). The chain length increases with increasing the concentrations of added [TiO(ox)2]2− and oxalate ion; the maximum chain length was 600 under the experimental conditions. The decomposition reaction gave 2 mol of CO2 per 1 mol of decomposed [Ti(O2)(ox)2]2−. Ti(III), Cr(II), and V(II) reductions of [Ti(O2)(ox)2]2− produced ·OH, identified by the spin-trapping technique using a 5,5-dimethyl-1-pyrroline-N-oxide radical trap. Oxygen is an inhibitor of the chain reaction in this system. The chain-decomposition mechanism is discussed.

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)

We review the Green/Kleitman/Leeb interpretation of de Bruijn's symmetric chain decomposition of Bn, and explain how it can be used to find a maximal collection of disjoint symmetric chains in the nonsymmetric lattice of partitions of a set.

The title transition has been detected in a mixture of nitrogen trichloride (4%) with helium. The occurrence of a non-linear branching step in the branched-chain reaction shortens the time interval of the development of a chain-thermal flame.

In this paper we give a construction, for any $n$, of an $n$-Venn diagram on the sphere that has antipodal symmetry; that is, the diagram is fixed by the map that takes a point on the sphere to the corresponding antipodal point. Thus, along with certain diagrams due to Anthony Edwards which can be drawn with rotational and reflective symmetry, for any isometry of the sphere that is an involution, there exists an $n$-Venn diagram on the sphere invariant under that involution. Our construction uses a recursively defined chain decomposition of the Boolean lattice.

Let G(V, E) be a digraph (directed graph) with n nodes and e edges. Digraph G* = (V, E*) is the reflexive, transitive closure if (v, u) ∈ E* iff there is a path from v to u in G. Efficient storage of G* is important for supporting reachability queries which are not only common on graph databases, but also serve as fundamental operations used in many graph algorithms. A lot of strategies have been suggested based on the graph labeling, by which each node is assigned with certain labels such that the reachability of any two nodes through a path can be determined by their labels. Among them are interval labelling, chain decomposition, and 2-hop labeling. However, due to the very large size of many real world graphs, the computational cost and size of labels using existing methods would prove too expensive to be practical. In this paper, we propose a new approach to decompose a graph into a series of spanning trees which may share common edges, to transform a reachability query over a graph into a set of queries over trees. We demonstrate both analytically and empirically the efficiency and effectiveness of our method.

A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $P$ is any symmetric chain order, we prove that $P^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $P^n$ by cyclic permutation of the factors., Comment: Final version, 12 pages

Partial ordered sets provide a common frame for many combinatorial configurations. Formally, a partially ordered set (or poset, for short) is a set P together with a binary relation < between its elements which is transitive and antysymmetric: if x

Let $[u,v]$ be a Bruhat interval and $B(u,v)$ be its corresponding Bruhat graph. The combinatorial and topological structure of the longest $u-v$ paths of $B(u,v)$ has been extensively studied and is well-known. Nevertheless, not much is known of the remaining paths. Here we describe combinatorial properties of the shortest $u-v$ paths of $B(u,v)$. We also derive the non-negativity of some coefficients of the complete mcd-index of $[u,v]$., Soit $[u,v]$ un intervalle de Bruhat et $B(u,v)$ le graphe de Bruhat associé. La structure combinatoire et topologique des plus longs chemins de $u$ à $v$ dans $B(u,v)$ est bien comprise, mais on sait peu de chose des autres chemins. Nous décrivons ici les propriétés combinatoires des plus courts de chemins de $u$ à $v$. Nous prouvons aussi que certains coefficients du mcd-indice complet de $[u,v]$ sont positifs.

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

The deactivation Cl23Πou+ + M → Cl21Σg– + M is an important step giving rise to the upper limit of isothermal flame propagation in dilute gaseous mixtures of nitrogen trichloride.

Reduction of the tosylhydrazone of ketone 2 with NaBH 3 CN / ZnCl 2 leads to a 3.5 : 1 mixture of 3 and 4 . It is suggested that this reaction by radical chain decomposition of an intermediate monoalkyl diazene.

3-Methylcyclohexa-1,4-diene-3-carboxylates and 2-methyl-2,5-dihydrofuran-2-carboxylates decompose to generate free alkyl radicals which have been incorporated into chain reactions producing alkyl bromides, alkene adducts and ring-closure products.

On the basis of numerical simulation of the branching chain process of the decomposition of nitrogen trichloride it has been shown that suitable conditions for providing oscillating regimes are the following: (a) inclusion of adsorption-desorption of NCl3 on the reactor walls; (b) inclusion of nonlinear chain branching and breaking reactions.

Monotone chain is an important concept in computational geometry. A general polygon or polygonal chain can be decomposed into monotone chains. Described in this paper are two algorithms to find an optimal direction with respect to which a polygonal chain can be split into the minimal number of monotone chains. The first naive algorithm has O(n 2) time complexity, while the improved algorithm runs in O(n log n) time, where n is the number of vertices of input chain. The optimal direction can improve the performance of the subsequent geometric processing.

Because of the central role Dilworth’s decomposition theorem plays in the structure theory of (partially) ordered sets, it is important to consider it from a computational point of view. The theorem is proved in two steps. The first step is to prove the result for finite ordered sets. Dilworth’s original proof in [1] of the finite result is constructive, but the notion of computational complexity is not addressed. In 1956, Fulkerson [2] gave a network flow proof of Dilworth’s theorem. This proof allows the calculation of a Dilworth decomposition using any of the fast network flow algorithms. In particular, a decomposition can be calculated in O(n 3) time using the network flow algorithms in [3] or [8].

(1994). A Proof of Dilworth's Chain Decomposition Theorem. The American Mathematical Monthly: Vol. 101, No. 4, pp. 352-353.

Reduction of the tosylhydrazone of ketone 2 with NaBH 3 CN / ZnCl 2 leads to a 3.5 : 1 mixture of 3 and 4 . It is suggested that this reaction by radical chain decomposition of an intermediate monoalkyl diazene.

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%.

3-Methylcyclohexa-1,4-diene-3-carboxylates and 2-methyl-2,5-dihydrofuran-2-carboxylates decompose to generate free alkyl radicals which have been incorporated into chain reactions producing alkyl bromides, alkene adducts and ring-closure products.