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Idealized polymethines are defined as conjugated chain molecules with complete bond equilization and charge alternation. A constant π-bond order along the chain with N atoms has been derived for three types of polymethines which differ by the nature of the heteroatomic end groups. Corresponding to the number of π-electrons they are termined (N + 1)π-, Nπ- or (N − 1)π-polymethines. This classification embraces closed shell as well as open shell systems. The relations for the energy eigenvalues, coefficients and derived quantities have been obtained in closed form. The results for the different classes of polymethines are strongly related. Their molecular properties should be similar.

Poly-γ-benzyl-glutamate with strict alternation of l and d residues is shown to exist in α and possibly ω helical conformations, and in a new helical structure specific to poly-d-l-peptides, the πDL helix. Infrared, X-ray and electron diffraction data on these structures are given. The transconformation mechanism from αDL to πDL, helices, which implies the formation of a new set of hydrogen bonds, is discussed. The multiplicity of conformations observed with this polymer (which can also exist in an unusual sheet structure) is discussed from the point of view of the sequence-structure relation.

The condensation of 1,4-dilithium-1,4-dihydronaphthalene with phenyltrichlorosilane has been investigated. It was found that during separation of the polyfunctional silyldihydronaphthalene derivative, partial hydrolysis occurs with formation of 1-bis-[1,4-bis-(diphenyldichlorosilyl) dihydronapthalene] cyclodioxide. The polyfunctional cyclic systems containing bis-silyldihydronaphthalene units were subjected to hydrolytic polycondensation and the products were polymerized in the presence of metal hydroxides. This yielded soluble, cyclo-linear polymers with regularly alternating siloxane and silyldihydronaphthalene units.

An extensive study has been made of the trialuminides of the rare-earth metals. In addition to showing the existence of several Ni3Sn-type trialuminides hitherto unknown (those of Ce, La, Pr and Nd) the detailed structures are given of a number of rhombohedral trialuminides (those of Y and Tb which are isotypic with BaPb3, and those of HoAl3 and DyAl3 which both have the HoAl3 type of structure). These structures, together with the Ni3Ti type, encountered with the ternary trialuminides (Gdx, Er1−x)Al3 and (Yx, Er1−x)Al3, and also the PuAl3 type form a family of transition structures between the Ni3Sn and the Cu3Au types. As the rare-earth/Al radius ratio decreases the trialuminides adopt structures passing from predominantly hexagonal to predominantly cubic, subject to the two rules that the layer repeat period of the lattice tends to be a minimum and that the alternation of hexagonal and cubic stacking is a maximum.

For suitable Cr:Fe and Ti:Zr ratios in the system Cr2O3Fe2O3TiO2ZrO2 the composition range MO1.667 to MO2 contains a quasicontinuous series of compounds whose structures are ordered intergrowths of the V3O5 and α-PbO2 structural types. The stoichiometries of the intergrowth phases are described by the general formula ( Cr, Fe ) 2 p ( Ti, Zr ) p +2 qO 5 p +4 q  pM 3 O 5 · qM 2 O 4 , where p and q can take any integral values. The compounds are monoclinic, with unit cells defined by α-PbO2 unit cell vectors such that in all cases a ⋍ 7.0 A, b ⋍ 5.0 A and β = 138°, and c is defined by the periodicity of the M3O5 and M2O4 slabs. X-Ray and electron diffraction results are consistent with a structural model in which there is a maximum alternation of the M3O5 and M2O4 slabs. An alternative description in terms of crystallographic shear (CS) is also possible, whereby each of the compounds may be considered to derive from rutile by CS on planes (hkl) where ( hkl ) r = p (121) r + q (011) r .

This note describes an efficient class of procedures for finding a solution to the functional equations \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$v^{*}_{i}=\max_{k} \biggl[q^{k}_{i}-g^{*}T^{k}_{i}+ \sum^{j=n}_{j=1}P^{k}_{ij}v_{j}^{*}\biggr],\quad 1\leq i \leq N,$$ \end{document} of undiscounted Markov renewal programming. First, for the special case of a single possible policy, the problem is proved equivalent to solving two related ordinary Markov chain problems, which leads to an algorithm for the general problem whose exact form depends on the specification of a decision rule for alternation of two types of iterations. At one extreme, the technique is exactly "policy iteration," with iterative techniques replacing solution of N equations for each improved policy; at the other extreme, the algorithm becomes essentially "value iteration," generalizing the method of successive approximations proposed by D. J. White for Markovian decision processes. The latter version of the technique is related to another generalization being currently proposed by Paul J. Schweitzer; the methods being proposed here, however, do not deteriorate when the minimum transition time between states becomes very small.

Twelve absorption bands of C2D2 have been observed in the photographic infrared. Eleven of these bands show simple P and R branches with intensity alternation, i.e., they are Σu+—Σg+ bands while one band, without intensity alternation, is a difference band of type Π—Π. From the positions of the bands some of the vibrational constants ωi0 and xik are determined. In addition, from the rotational fine structure, the rotational constants B[v] and D[v] are determined. For B[0] the value 0.84787 cm−1 is obtained. The αi values are α1=+0.00578, α2=+0.00302, α3=+0.00455, α4=−0.00120, α5=−0.0026 cm−1. From the αi and B[0] one obtains Be=0.85075 cm−1. Combining Be(C2D2) with Be(C2H2) yields, for the internuclear distances in the acetylene molecule, re(C≡C)=1.2010×10−8 and re(C–H)=1.0637×10−8 cm in very satisfactory agreement with the values previously obtained from Be(C2HD) and Be(C2H2). Evidence for l‐type doubling in C2D2 is given and approximate values for the l‐type doubling constants q4 and q5 are obtained.

The enthalpies and temperatures of fusion and of transition for twenty arsinic acids, R 2 As(O)OH, have been measured by DTA, where R = n -alkyl, CH 3 through n -C 20 H 41 inclusive. A crystalline transition occurs for acids n -C 6 H 13 through n -C 20 H 41 , but not for CH 3 through n -C 5 H 11 . The enthalpies of transition vary from 3.9 kcal/mole (C 6 ) to 12.2 kcal/mole (C 17 ), and show an alternation with change from an even acid to an odd acid. The enthalpies of melting show a reverse alternation, so that the total enthalpy of fusion plus transition does not show a regular pattern of alternation. The X-ray powder patterns for the twenty acids have also been obtained, as well as single crystal data for the C 4 and C 8 acids. These data indicate that all of the compounds crystallize in the triclinic system and that the [010] spacing of the cell chosen increases regularly with chain length.

IN CHAPTER 3 of his book Magic Squares and Cubes W. S. Andrews states in concise form what is known about the particular magic square invented by Benjamin Franklin. Included in this chapter is "An Analysis of the Franklin Square" by Paul Carus. Each of these authors gives a most penetrating treat ment of the subject, and an ingenious method of construction is described. Of this method, however, Carus admits: "We do not know the method employed by Franklin; we possess only the result, but it is not probable that he derived his square according to the scheme employed here" (P112). Without recapitulating the discussion in Andrews's book, to which the reader is referred, it may be observed that the chief peculiarity of Franklin's squares?8 X 8 and 16 X 16?is their excess of rows add ing up to the required sum. The ordinary magic square has horizontal rows, vertical rows, and 2 oblique rows?that is, 2(n + 1), making 18 such rows for an 8X8 square and 34 for a 16 X 16 square. In the Franklin squares there are no oblique rows, but in place of these there are 2n horizontal "bent" rows and 2n vertical bent rows?that is, 6n rows in all, making 48 such rows for an 8 X 8 square and 96 for a 16 X 16 square. (It is understood, of course, that the usual assumption holds true? that the left and right edges and the upper and lower edges in each case touch and are continuous.) Construction of the Franklin Square The construction of Franklin's two squares and the unlimited extension of the Franklin series (as well as an equivalent number of similar squares) is simplicity itself, not involving anything comparable to Carus's "simple alternation" and "quaternate transposition" but consisting of little more than an elementary exercise in counting and copying, with the applica tion of two or three simple rules. If we are looking for some method so simple that Franklin could amuse himself with it while performing his duties as clerk in the Pennsylvania Assembly, this method recommends itself as a likely candidate. The following steps will illustrate the formation of the 8X8 square: 1. Draw an 8 X 8 square. 2. Counting from the left edge, draw a heavy broken line from top to bottom between the second and third vertical rows (that is, n/4 squares from the edge). 3. Enter the numbers from 1 through 8 along this broken line in the following zigzag manner: o) Enter a quarter (1, 2) at the bottom, a quarter (3, 4) at the top, proceed ing upward with the odd numbers at the left of the broken line, the even numbers at the right, as seen in (a) of figure 1. b) Enter the remaining half (5-8) proceeding downward in the remain ing squares, odd numbers at the right of the broken line, even num bers at the left, also as seen in (a) of figure 1.

Disorder diffuse scattering has been observed for N -oxyphenazine which indicates the existence of domains of short-range order in (100) planes, of average size 13 A (along b) by 50 A (along c). Along a the alternation of centrosymmetrically arranged molecules is random. Mixed crystals with phenazine show diffuse scattering which indicates that up to about 60 mole % of phenazine can occur in the structure without disrupting these short-range-order domains. Mixed crystals of low N -oxyphenazine content have these molecules arranged randomly in the phenazine matrix. Between these two phases there appears to be a miscibility gap. It is probable that two components, one of which is ordered and the other showing only short-range order cannot form a continuous mixed-crystal series.

By the methods of photographic photometry, the alternating intensities in emission of the $\ensuremath{\Delta}v=0$ sequence of the Fulcher bands of deuterium have been investigated in order to determine the nuclear spin. The alternation of intensities was determined for the $P$, $Q$ and $R$ branches of each of the 5 bands studied. The average value of $\frac{{g}_{s}}{{g}_{a}}$ obtained from 4 plates is 1.97\ifmmode\pm\else\textpm\fi{}0.03. This value agrees best with the theoretical ratio of 2 and a consequent nuclear spin of 1. Since the even rotational levels are more intense than the odd levels, the nucleus obeys Bose-Einstein statistics.

The infinite polyacene (C 4 H 2 ) ∞ has been studied by methods similar to those previously applied to the infinite polyene. Whereas in the latter the most stable configuration is one in which the C—C bonds are alternately long and short, it is found that in the infinite polyacene the external C—C bonds all have the same length in the most stable configuration, being slightly shorter than the cross bonds. The reasons for this are examined.

There is a definite alternation of intensity in the successive lines of a given branch of this spectrum. This is particularly evident in the Q branches, which are the strongest, the lines usually denoted by Q (1), Q (3) and Q (5) being much more intense than the intervening lines Q (2) and Q (4). For some time I have had evidence which, though in some respects incomplete, pointed to the con­clusion that in any band the mean value of the intensities of any two consecutive odd-numbered lines such as Q (1) and Q (3) was nearly equal to three times the intensity of the intervening line such as Q (2). An examination of the much more comprehensive measurements which have recently been obtained by McLennan, Grayson-Smith and Collins seems to remove all doubt from this matter. In the second row of Table I, I tabulate a the sums of their measured inten­sities of the lines Q (1) and Q (3) for the successive bands denoted by 0 x 0 , 1 x 1 , 2 x 2 , 3 x 3 and 4 x 4 in Structure, Part V. These are followed in the third row by b , twice the intensity as measured of the corresponding Q (2) line. The fourth row contains the resulting values of b ÷ a . All the foregoing are the data got at Boom Temperature (R. T.). These are followed in the same order by similar data got with the thin-walled discharge tube immersed in liquid air (L. A.). Finally, the eighth row contains the average of the two values of b ÷ a got at the two temperatures.

The apparent dipole moments calculated from solutions of mercuric chloride and dimethylmercury have been found to alternate between high and low values over the temperature range extending from 8.5 to 50 °C. The significance of this alternation is discussed.

ESR spectra of the 2,6-dimethyl-4-methoxyphenoxy radical show line-width alternation at temperatures from 0 to −80°C. The effect is ascribed to the rotation of the methoxy group. The line-width analysis is given in terms of the relaxation matrix theory. From the spectra a value of 7.8 kcal/mole is estimated for the experimental energy of activation. This value is compared with dielectric relaxation data.

The e.s.r. spectrum of the durosemiquinone cation exhibits a marked line width alternation effect which is dependent on the temperature and the solvent. The effect is ascribed to cis-trans isomerism, the lifetime of each isomer being comparable with the inverse frequency separation between the proton hyperfine components.

The two most intense bands of the 370 nm electronic band system of tropolone have been rotationally analysed. They are separated by 18·93 cm-1 and it has been shown that the high wavenumber band is the 0--0- (H1 1) transition in what is almost certainly the internal hydrogen-bonding vibration v H : the low wavenumber band is the 0+-0+ (00 0) transition. A rotational contour analysis of both bands shows that there is an intensity alternation in K a″ such that the ratio K a″ even : odd is 10 : 6 in the 0+-0+ band and 6 : 10 in the 0--0- band. The intensity alternation, the nearly equal intensities of the 0+-0+ and 0--0- bands, the separation of these two bands and the anharmonic behaviour of v H show that the separation of the 0+ and 0- levels is small in the ground electronic state (probably less than 50 cm-1) and is 18·93 cm-1 larger in the excited electronic state. The 0+-0+ and 0--0- bands are both type B showing that the electronic transition is A 1 B 2-X 1 A 1 and therefore π*-π rather than π*-n. The ...

Electron spin resonance spectra of ion-pairs of 2,3- and 2,5-dimethyl-p-benzosemiquinone are reported. Numerical relationships between the various splitting constants suggest the equality of QC–Mc and QC–H. Unexpected sharp lines in the spectra recorded under conditions of line-width alternation are interpreted in terms of these numerical relationships.

This is a study of the structural arrangement in a polymer which is on a size level larger than the crystallite but smaller than the spherulite. It is based on the data from x-ray diagrams at small angles, combined with crystallite dimensions measured from wide-angle x-ray patterns. The earlier interpretations of polymer texture by Hess and Kiessig which describe alternating regions of order and disorder appear to be most probable. However, those earlier interpretations are extended by showing that oriented crystalline polymers can be thought of as having aligned elementary fibrils which consist along their length of a regular alternation of bulky and condensed regions. The width of these elementary fibrils is proportional to the width of the crystallites. The amount of regularity of crystallite positions along an elementary fibril and the length of the repeat period are both variable; they depend on the manner of original nucleation and the thermal treatment after nucleation. The length of the long period is a function of the length of the crystallites. These elementary fibrils can pack closely to form a higher state of ordered texture. The amount of close packing is variable; it is optimal after a relaxation treatment. Elongated voids exist between the elementary fibrils. The difference between the effects of original nucleation and thermal after-treatment is emphasized. Elementary fibrils are proposed as a basic unit of aggregation in polymer melts, solutions, and fibers.

ESR spectra of radical anions from 1,2,3,4-tetrachloro-5,6-diphenylcalicene (PhC) and 1,2,3,4-tetrachloro-5,6-di-n-propylcalicene (PrC) were studied. An unpaired electron spin is localized on 5- and 6-carbon atoms and its substituents for both. PrC shows a typical line width alternation, which was explained by a hindered internal rotation (the three-membered and five-membered rings are not coplanar) of two n-propyl groups with a barrier of ∼1.5 kcal/mole.

The ESR of the 4-phenyl-1,3-dioxane anion shows a line-width alternation effect which can be attributed to the restricted rotation between the 1,3-dioxane and benzene rings. From the temperature dependence of the line-width alternation effect, the activation energy was determined to be 180±110 cal·mol−1. The preferential conformation between the two rings was estimated from the hyperfine splitting constant of the proton at the 4 position in the 1,3-dioxane ring, to be that where the two rings have some angle close to 90° rather than the structure where they are parallel. It was also found that an odd electron, which mainly occupies a vacant π orbital in the benzene ring, delocalizes appreciably into a part of the 1,3-dioxane ring.

A very weak but sharp absorption system of s -tetrazine vapor has been found to extend from 8000 to 6000 A. It is characterized by a single progression of bands whose relative intensities require an appreciable change of geometric structure, confined mainly to the NN bonds and NCN angles. A 134-m absorbing path reveals rotational structure in the OO band at 13 608 cm −1 . Analysis of the structure is consistent with one of the two geometry changes allowed by the vibrational analysis, namely, Δ r( NN ) = −0.06 A and Δ NCN = +3.4° . Intensity alternation in the rotational structure shows that the excited state is triplet. Since the five observed vibrational frequencies of the new state have nearly the same respective values as those of the 1 B 3 u state at 18 128 cm −1 , it is assumed that the orbital configuration of the excited state also is the same, that is, the excited state is 3 B 3 u .

Extended Huckel calculations have been carried out on the monoprotonated pyrazine radical. The sites for ion-pairing are unequivocally located in the molecular plane on a twofold symmetry axis in the region of the nitrogen lone-pair electrons. The most probable path of migration is above the molecular plane but does not lie in a pyrazine symmetry plane. The molecular processes involved in the phenomenon of linewidth variation in electron spin resonance spectra are discussed in the light of these results.

X-ray analyses of chain-shaped and of coupled polymethines show an enlargement of the bond angles of the positively charged methine atoms and a contraction of the bond angles of the negatively charged methine atoms. The bond angle alteration can be explained by deviations from the trigonal hybridisation of the methine atoms. There exists a simple linear function between the square of the hybridisation parameter λ and the total charge densities calculated by means of the CNDO/2 method: λ2 ≅ 2 – Q.

The alternation of intensities in absorption in the band spectrum of sodium was investigated for the purpose of determining the spin of the sodium nucleus. Measurements were confined to the (0,2) band in the $^{1}\ensuremath{\Pi}\ensuremath{\leftarrow}^{1}\ensuremath{\Sigma}$ transition, which alone is reasonably free from overlapping bands. Relative absorption coefficients for the $Q$ branch lines, with $J=48$ to $J=73$ inclusive, were calculated. These yielded an average value for the alternation of intensities of 1.71, the average being obtained from 5 different plates. These results agree best with the theoretical alternation ratio of 1.67 corresponding to a nuclear spin of $\frac{3}{2}$.

Vibration spectrum. The three normal vibrations of ${\mathrm{N}}_{2}$O are all active. The fundamental bands at 17.0\ensuremath{\mu}, 7.78\ensuremath{\mu} and 4.50\ensuremath{\mu}, and the three harmonics have all been observed; also six combination bands at wave-lengths greater than 2\ensuremath{\mu}. The 17.0\ensuremath{\mu} fundamental and two combination bands involving this fundamental have strong zero branches, while the harmonic and two combination bands involving the harmonic transition have no zero branches. All other bands are doublets.Rotational analysis. Three bands at 17.0\ensuremath{\mu}, 7.78\ensuremath{\mu} and 8.60\ensuremath{\mu} have been resolved under high dispersion. All show rotation lines with continuously varying intensities (no alternation) and almost uniform spacing, the same for each band, viz. 0.84 ${\mathrm{cm}}^{\ensuremath{-}1}$. The corresponding value of the moment of inertia, 59.4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}40}$, is practically equivalent to that computed from the doublet separations of the various bands.Form of the molecule. The molecule is apparently linear but not symmetrical. The observations are not consistent with a triangular configuration, nor with the symmetrical linear configuration N---O---N, but they may be correlated precisely by means of the model $\mathrm{N}\ensuremath{\equiv}\mathrm{N}=\mathrm{O}$.

A theoretical study of the observed visible and ultra-violet spectra of C 18 H 18 , C 24 H 24 and C 30 H 30 indicates strongly that in these molecules the C—C bonds alternate in length. It is shown, furthermore, that the diamagnetic anisotropies and proton chemical shifts should indicate very sensitively the extent of bond alternation; as yet no measurements on these properties are available.

Electron spin resonance spectra of paradinitrobenzene anions in methanoldioxane mixtures exhibit a very marked line-width alternation. It is shown that such alternation, stemming from a relatively small difference between the isotropic hyperfine coupling constants of the two nitrogen atoms can account for the apparently anomalous spectra in solvents of low dielectric constant.

The electron spin resonance (e.s.r.) spectrum of 4,4′-dimethoxybiphenyl in aluminium chloride–nitromethane is remarkably temperature dependent. The changes are explained by postulating the rapid interconversion of conformational isomers, leading to line-width alternation.

The concentration of the original monomers in the reaction medium influences the alteranation effect during the copolymerization of vinyl ethers with acrylates. A concentration of the active monomer may be chosen such that the polymer of methyl methacrylate will be formed at the beginning.

Publisher Summary Cyanine dyes and related compounds cover the whole visible spectrum and have high tinctorial power but owing to their poor stability to light they are not important as textile dyes. Some of these dyes show high light stability on acrylonitrile homo-and co-polymers, and have attained technological importance. For convenience, cyanine dyes and related compounds are studied under the groups: Cyanines; Azacyanines; Polynuclear cyanines; Dye bases; Merocyanines; Linear tri- and poly-nuclear cyanines and merocyanines; Oxonols; nonheterocyclic cyanines; Styryl dyes and N-hemicyanines. The vast majority of cyanine dyes are mono-acid salts, in which two heterocyclic nuclei containing nitrogen are linked through an odd numbered methine chain in such a way that resonance occurs through the conjugated system between tertiary and quaternary nitrogen atoms. Azacyanines are dyes, in which one or more of the methine groups in the conjugated chain have been replaced by nitrogen. When cyanine dyes are heated with a high-boiling amine, such as N,N-diethylaniline, the more basic heterocyclic ring becomes dequaternized to produce the dye base. Merocyanines may be defined as nonionic dyes, in which a double bond or an even-numbered methine chain unites a dihydrogenated nitrogen-containing heterocycle with the residue of a compound having an active methylene group, usually a cyclic ketomethylene compound.

Yet, this condition of the cosmic system was still different from the world as it is today, since all of this diakosmesis was still unfit to bring about an alternation of day and night. For still the entire perichoresis was everywhere taking place around the one axis chosen in the beginning when Nous started cosmogony from that tiny district. Consequently, the circles described around that axis by the sun and the moon could only lie in the plane of the earth disk itself, and the orbits of the other stars were either in this very plane or parallel with it.