Word-Representability of Face Subdivisions of Triangular Grid Graphs

A graph \(G=(V,E)\) is word-representable if there exists a word w over the alphabet V such that letters x and y alternate in w if and only if \((x,y)\in E\). A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A face subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph \(K_4\). Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of grid graphs related to convex polyominoes, we characterize word-representable face subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any face subdivision of boundary triangles in the Sierpiński gasket graph is word-representable.     

Structures and some properties of the products of condensation of phthalazone hydrazones with aldehydes and ketones

The products of condensation of 2-H-phthalazone hydrazones with carbonyl compounds in solution and in the crystalline form have phthalazone ylidenehydrazone (unsymmetrical azine) rather than phthalazinylhydrazone I structures, as has previously been assumed. 2-H-Phthalazone arylidenehydrazones exist in the EE form, and their analogs methylated in the 2-N position of the phthalazone ring exist in the ZE' form. Facile ZE' ZZ' isomerization was noted for 2-methyl-4-chlorophthalazone ethylenehydrazone in halogen-containing solvents. Its unmethylated analog — 4-chlorophthalazone ethylenehydrazone (VIIIb) — is a mixture of geometrical isomers (EE'/EZ'=31).Communication LIX from the series Hydrazones. See [1] for communication LVIII.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 5, pp. 690–698, May, 1978.     

The M/G/1 processor-sharing model: transient behavior

This paper deals with the M/G/1 model with processor-sharing service discipline. LetL ^* (t, x) denote the number of jobs present at timet whose attained service time is not greater thanx,x0, andV ₀(t,z) the sojourn time of a tagged job placed in the system at timet and requiringz units of service. Explicit analytical expressions are obtained for the joint distribution ofL ^*(t, ·) andV ⁰(t, ·) under various initial conditions in terms of the Laplace transform with respect tot. It is shown that for initial conditions of special kind (there is one job or none) the results can be expressed in a closed form.     

Theoretical study of H complexes of tetracoordinated phosphorus compounds

Conclusions Energies of formation of complexes of hydrogen fluoride with compounds of tetracoordinated phosphorus, charge transfers in complexing, and other characteristics of these reactions calculated by CNDO/2 methods agree well with the measured values.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 3, pp. 496–502, March, 1979.     

Phthalazone arylhydrazones

The products of the reaction of 1-chloro- and 1, 4-dichlorophthalazine with nitrophenylhydrazines are the E isomers of the nitrophenylhydrazones of the corresponding phthalazone. 2-Methyl-4-chlorophthalazone 4-nitrophenylhydrazone was found to be a mixture of Z and E isomers (32). 1, 4-Dichlorophthalazine reacts with 1-methyl-1-phenyl- and 1, 1-diphenylhydrazines to give monosubstitution produdcts, for which the phthalazone hydrazone (III) or phthalazinylhydrazine (VI) form or a tautomeric mixture of III and VI is realized, depending on the nature of the solvent and other factors. Some oxidation, reduction, and nucleophilic substitution reactions of the compounds obtained are examined.Communication LX from the series Hydrazones. See [1] for communication LIX.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 5, pp. 699–707, May, 1978.     

Geometrical structures of the azines of substituted acetophenones

Conclusions By drawing on measured values of dipole moments and Kerr constants and a study of vibrational spectra, it has been shown that the molecules of the symmetrical azines of the acetophenones exist as EE forms of the s-trans conformer, the angle of rotation of the azomethylidyne fragments around the N-N bond being 130±5° with the phenyl rings displaced out of the C=N bond planes by 35±5°.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 9, pp. 1996–1999, September, 1977.     

Word Problem of the Perkins Semigroup via Directed Acyclic Graphs

For a word w in an alphabet Γ, the alternation word digraph Alt(w), a certain directed acyclic graph associated with w, is presented as a means to analyze the free spectrum of the Perkins monoid \(\mathbf{B_2^1}\). Let \((f_n^{\mathbf{B_2^1}})\) denote the free spectrum of \(\mathbf{B_2^1}\), let a _n be the number of distinct alternation word digraphs on words whose alphabet is contained in {x ₁,..., x _n }, and let p _n denote the number of distinct labeled posets on {1,..., n}. The word problem for the Perkins semigroup \(\mathbf{B_2^1}\) is solved here in terms of alternation word digraphs: Roughly speaking, two words u and v are equivalent over \(\mathbf{B_2^1}\) if and only if certain alternation graphs associated with u and v are equal. This solution provides the main application, the bounds: \(p_n \leq a_n \leq f_n^{\mathbf{B_2^1}} \leq 2^{n}a_{2n}^2\). A result of the second author in a companion paper states that \((\operatorname{log} \; a_n)\in O(n^3)\), from which it follows that \((\operatorname{log} f_n^{\mathbf{B_2^1}})\in O(n^3)\) as well. Alternation word digraphs are of independent interest combinatorially. It is shown here that the computational complexity problem that has as instance {u,v} where u,v are words of finite length, and question “Is Alt(u)?=?Alt(v)?”, is co-NP-complete. Additionally, alternation word digraphs are acyclic, and certain of them are natural extensions of posets; each realizer of a finite poset determines an extension by an alternation word digraph.     

Effect of disorder on the optically amplified photocatalytic efficiency of titania inverse opals

Optically amplified photochemistry with slow photons has been realized in our previous work when a photoactive material such as TiO(2) was molded into a photonic crystal and the corresponding energy of photonic bands overlapped with the electronic excitation. While numerous applications of photonic crystals have been proposed, the real practicality depends on the extent of structural imperfection that can be tolerated before significant deterioration in the optical response deems it unrealistic to use. As a result, it is important to evaluate the amount of structural disorder that can be tolerated in inverse TiO(2) opals if they are to be used as amplified photocatalysts for photolytic degradation of organics in environmental remediation and water purification. We present a systematic study on the effect of disorder with relation to the photocatalytic efficiency of oxidizing methylene blue dye adsorbed on inverse TiO(2) opals by introducing different fractions and sizes of guest spheres into the opal template. Our results show that half of the enhancement originally achieved by the inverse opal made from monodispersed 150-nm spheres is conserved when the domain size of the host spheres remains above a critical threshold. The substitution fraction can be as high as 0.4 when the guest spheres are 1.2 times larger than the host spheres. Such a high tolerance to structural disorder provides strong support for the potential use of inverse TiO(2) opals in environmental cleanup and water treatment applications.     

Facile preparation of highly monodisperse small silica spheres (15 to >200 nm) suitable for colloidal templating and formation of ordered arrays

Highly monodisperse spherical silica nanoparticles with diameters ranging from ca. 15 to 200 nm were prepared using an environmentally friendly water-based synthesis. The size of the spheres can be precisely controlled by using a facile regrowth procedure in the same reaction media. Furthermore, these monodisperse silica spheres can be successfully used as seeds in the well-established St?ber silica preparation. The regrowth approach allows for easy incorporation of functional additives. High monodispersity and charge stabilization renders these nanoparticles highly suitable for close-packed array formation and colloidal templating.     

Thermodynamic Properties of the MZr2(PO4)3 (M=Na,K, Rb or Cs) Compounds

The enthalpies of the solution of MZr₂(PO₄)₃(M=Na, K, Rb or Cs) compounds have been measured by the help of a differential automatic isothermal Calvet calorimeter and the standard enthalpies of formation have been derived. The temperature dependencies of the standard heat capacity of the samples of crystalline NaZr₂(PO₄)₃ and CsZr₂(PO₄)₃ were studied between 7 and 340 K in an automatic adiabatic vacuum calorimeter. The main thermodynamic functions H ⁰(T)–H ⁰(0), S ⁰(T) andG ⁰(T)–H ⁰(0) have been determined. The Gibbs energies of formation of the NaZr₂(PO₄)₃and CsZr₂(PO₄)₃ at 298.15 K were calculated on the basis of these experimental data and the enthalpy of formation data. Qualitative explanations for the results observed were presented. This revised version was published online in July 2006 with corrections to the Cover Date.