Polya

AH higher a-olefins, in the presence of Ziegler-Natta catalysts, can easily copolymerise both with other a-olefins and with ethylene (51,59). In these reactions, higher a-olefins are all less reactive than ethylene and propylene (41). Their reactivities in the copolymerisation reactions depend on the sise and the branching degree of their alkyl groups (51) (see Olefin polya rs, linear low density polyethylene). 

Poly(vinyl acetate), the precursor of poly(vinyl alcohol), hydroly2ed to less than 70%, is claimed to be nonbio degradable under conditions similar to those that biodegrade the fuUy hydroly2ed polymer (100) (see Vinylpolya rs poly (vinyl ACETATE polya rs). 

In the eady 1920s, experimentation with urea—formaldehyde resins [9011-05-6] in Germany (4) and Austria (5,6) led to the discovery that these resins might be cast into beautiful clear transparent sheets, and it was proposed that this new synthetic material might serve as an organic glass (5,6). In fact, an experimental product called PoUopas was introduced, but lack of sufficient water resistance prevented commercialization. Melamine—formaldehyde resin [9003-08-1] does have better water resistance but the market for synthetic glass was taken over by new thermoplastic materials such as polystyrene and poly(methyl methacrylate) (see Methacrylic polya rs Styrene plastics). 

Polya.mine-Ba.sed Quaternaries. Another important class of quaternaries are the polyamine-based or polyquats. Generally, polyamine-based quaternaries have been used in the same appHcations as their monomeric counterparts (245). Discussions, including the use of polymeric quaternaries in laundry formulations (246) and in the petroleum industry as damage control agents (247), have been pubHshed. 

Styrene [100-42-5] (phenylethene, viaylben2ene, phenylethylene, styrol, cinnamene), CgH5CH=CH2, is the simplest and by far the most important member of a series of aromatic monomers. Also known commercially as styrene monomer (SM), styrene is produced in large quantities for polymerization. It is a versatile monomer extensively used for the manufacture of plastics, including crystalline polystyrene, mbber-modifted impact polystyrene, expandable polystyrene, acrylonitrile—butadiene—styrene copolymer (ABS), styrene—acrylonitrile resins (SAN), styrene—butadiene latex, styrene—butadiene mbber (qv) (SBR), and unsaturated polyester resins (see Acrylonithile polya rs Styrene plastics). 

Styrene Copolymers. Acrylonitrile, butadiene, a-methylstyrene, acryUc acid, and maleic anhydride have been copolymerized with styrene to yield commercially significant copolymers. Acrylonitrile copolymer with styrene (SAN), the largest-volume styrenic copolymer, is used in appHcations requiring increased strength and chemical resistance over PS. Most of these polymers have been prepared at the cross-over or azeotropic composition, which is ca 24 wt % acrylonitrile (see Acrylonithile polya rs Copolyp rs). 

Some polymers from styrene derivatives seem to meet specific market demands and to have the potential to become commercially significant materials. For example, monomeric chlorostyrene is useful in glass-reinforced polyester recipes because it polymerizes several times as fast as styrene (61). Poly(sodium styrenesulfonate) [9003-59-2] a versatile water-soluble polymer, is used in water-poUution control and as a general flocculant (see Water, INDUSTRIAL WATER TREATMENT FLOCCULATING AGENTs) (63,64). Poly(vinylhenzyl ammonium chloride) [70304-37-9] h.a.s been useful as an electroconductive resin (see Electrically conductive polya rs) (65). 

Ziegler polymerization catalysts may be prepared from Cp—Zr complexes and tri alkyl aluminum. The molecular weight of the polymers can be controlled over a wide range by varying the temperature. The activity of these catalysts is considerably increased by the addition of small amounts of water (263,264) (see Olefin polya rs). 

Polypropylene. There is an added dimension to the catalytic polymerization of propylene, since in addition to the requirement that the catalyst be sufficiently active to allow minute amounts of catalyst to yield large quantities of polymer, it must also give predominantly polypropylene with high tacticity that is, a highly ordered molecular stmcture with high crystallinity. The three stmctures for polypropylene are the isotactic, syndiotactic, and atactic forms (90) (see Olefin polya rs, polypropylene). 

Postpolymerization Reactions. Copolymers can also be formed by postpolymetization reactions on polymers. A well-known example is the partial hydrolysis of polyacrjiamide (PAM) to hydrolyzed polyacrylamide (HPAM). The product becomes a random copolymer of acrylamide and acryUc acid (44) (see Acrylamide polya rs). 

In an earlier paper [Polya, 4], I used "Symmetrieformel" instead of "cycle index". 

Polya 3, 4, 5. The last paper contains a direct proof for the equivalence of the recursion formulas based on combinatorial considerations and on the functional equation (4). 

Biirmann-Lagrange theorem. Polya-Szegd, Problems and Theorems in Analysis, Vol. I, (1972) pp. 145-146. 

THE LEGACY OF POLYA S PAPER FIFTY YEARS OF PdLYA THEORY... 

Polya s paper, translated here for the first time, was a landmark in the history of combinatorial analysis. It presented to mathematicians a unified technique for solving a wide class of combinatorial problems " a technique which is summarized in Polya s main theorem, the "Hauptsatz" of Section 16 of his paper, which will here be referred to as "Polya s Theorem". This theorem can be explained and expounded in many different ways, and at many different levels, ranging from the down-to-earth to highly abstract. It will be convenient for future reference to review the essentials of Polya s Theorem, and to this end I offer the following, rather mundane, way of looking at the type of problem to which the theorem applies and the way that it provides a solution. 

One of the attractions of Polya s paper is the elegant way that generating functions are introduced and used. In order to summarize the information about the figures and their contents, Polya introduced the "figure generating function" (also frequently called the "figure counting series")... 

To make use of the group G we need some way of summarizing those properties of the group that are relevant to the problem. This was provided by Polya in the form of the "cycle index". It is well known that a permutation can be expressed as a product of disjoint... 

We can now display Polya s Theorem in its one-variable form as follows. 

More generally the content of a figure will be a vector of nonnegative integers. Polya frequently used vectors of dimension 3. In that case the generating functions will be functions of three variables, and the statement of Polya s Theorem then gives... 

We note here an important concept introduced by Polya, that of the "corona" or "wreath product" of two permutation groups. If M is an w X M matrix, and G and H are permutation groups of degree m and n, consider all permutations of the mn elements of M obtained as follows ... 

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