Spec. Matrices 2019; 7:1–19 Research Article Open Access Kazumasa Nomura* and Paul Terwilliger e-Polymers 2019; 19: 385–410 Self-dual Leonard pairs Review Article Open Access https://doi.org/10.1515/spma-2019-0001 Received May 8, 2018; accepted September 22, 2018 Tejas V. Shah and Dilip V. Vasava* Abstract: Let F denote a eld and let V denote a vector space over F with nite positive dimension. Consider A glimpse of biodegradablea pair A, A∗ of diagonalizable polymersF-linear maps and on V ,their each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which biomedical applicationsthere exists an automorphism of the endomorphism algebra of V that swaps A and A∗. Such an automorphism is unique, and called the duality A A∗. In the present paper we give a comprehensive description of this ↔ https://doi.org/10.1515/epoly-2019-0041 duality. In particular,NIPAM we - displayN-isopropylacrylamide an invertible F-linear map T on V such that the map X TXT− is the duality Received December 04, 2018; accepted March 29, 2019. → A A∗. We expressDEAMT -as N, a N-diethyl polynomial acrylamide in A and A∗. We describe how T acts on ags, decompositions, ↔ NVC - N-vinylcaprolactam Abstract: Over the past two decades, biodegradableand 24 bases for V. MVE - Methyl vinyl ether polymers (BPs) have been widely used in biomedical Keywords: LeonardPEO pair,- poly(ethylene tridiagonal matrix, oxide) self-dual applications such as drug carrier, gene delivery, tissue PPO - poly(propylene oxide) engineering, diagnosis, medical devices, and antibac- Classication: 17B37,15A21AAc - Acrylic acid terial/antifouling biomaterials. This can be attributed AAm - Acrylamide to numerous factors such as chemical, mechanical and physiochemical properties of BPs, their improved pro- cessibility, functionality and sensitivity towards Introduction stimuli. The present review intended to highlight main results 1 Introduction of research on advances and improvements in terms of Let F denote aPetroleum-based eld and let V denote polymers a vector (commonly space over knownF with as nite positive dimension. We consider a synthesis, physical properties, stimuli response, and/or pair A, A∗ of diagonalizablePlastics) haveF been-linear used maps since on Va ,long each time of which due to acts its on an eigenbasis for the other one in an applicability of biodegradable plastics (BPs) during last irreducible tridiagonalhandiness, fashion. durability, Such aflexibility pair is called and a Leonardless reactivity pair (see [13, Denition 1.1]). The Leonard pair two decades, and its biomedical applications. Recent A, A∗ is said totowards be self-dual the wheneverwater and there other exists chemicals, an automorphism also these of the endomorphism algebra of V that literature relevant to this study has been cited and their swaps A and A∗products. In this caseare sucheasy anand automorphism cheaper to isprocess. unique, These and called the duality A A∗. developing trends and challenges of BPs have also been ↔ properties allow synthetic plastics to be broadly used in discussed. The literature contains many examples of self-dual Leonard pairs. For instance (i) the Leonard pair associ- ated with an irreduciblemany fields module (1). The for demand the Terwilliger for plastic algebra is boosting of the hypercubeday (see [4, Corollaries 6.8, 8.5]); (ii) a by day, the production of plastic was between 280-290 Keywords: biodegradable polymers; polylacticLeonard acid; drug pair of Krawtchouk type (see [10, Denition 6.1]); (iii) the Leonard pair associated with an irreducible million tons in 2012, exceeded 300 million tons in 2014. delivery; orthopedic application; tissue engineeringmodule for the Terwilliger algebra of a distance-regular graph that has a spin model in the Bose-Mesner alge- It is alleged that only half of this plastic, has been bra (see [1, Theorem], [3, Theorems 4.1, 5.5]); (iv) an appropriately normalized totally bipartite Leonard pair collected for recycling or reuse, while others have been Abbreviations (see [11, Lemmaremained 14.8]); (v) littered the Leonard or thrown pair away consisting in the of oceans any two (2-4). of a modular Leonard triple A, B, C (see [2, PGA - poly(glycolic acid) Denition 1.4]);Most (vi) theof the Leonard petroleum-derived pair consisting polymers of a pair are of resistant opposite generators for the q-tetrahedron alge- PCL - poly(ε-caprolactone) bra, acting on anto evaluationdegradation module and many (see of [5, them Proposition release 9.2]).toxic Thegases, example (i) is a special case of (ii), and the PBS - polybutylene succinate examples (iii), (iv)upon are incineration, special cases they of trigger (v). the pollution and global PBSA - poly(butylene succinate-co-adipate) Let A, A∗ denotewarming. a Leonard Whereas, pair sites on V to. We bury can this determine waste are whether also A, A∗ is self-dual in the following way. d PTMAT - poly (methylene adipate-co- terephthalate)By [13, Lemma 1.3]limited. each Thus, eigenspace upon being of A ,discarded,A∗ has dimension they put a one. serious Let θ denote an ordering of the eigen- { i}i= PVA - polyvinyl alcohol impact on the environment and surroundings (5-9). d values of A. For ≤ i ≤ d let vi denote a θi-eigenvector for A. The ordering θi i= is said to be standard PLA - poly(lactic acid) However, to conquerd the drawbacks of the synthetic { } d whenever A∗ acts on the basis vi i= in an irreducible tridiagonal fashion. If the ordering θi i= is standard PDLA - poly (D- lactic acid) plastics, dresearchers{ } are intensifying their efforts in { } then the ordering θd−i i= is also standard, and no further ordering is standard. Similar comments apply to PHB - poly(hydroxybutyrate) d the{ development} of biodegradable plastics (BPs). d A∗. Let θ denote a standard ordering of the eigenvalues of A. Then A, A∗ is self-dual if and only if θ PBS - polybutylene succinate { i}i= These polymers are vulnerable by variation in pH and { i}i= PBAT - poly (butylene adipate-co-terephthalate)is a standard orderingtemperature, of the eigenvaluesenzymatic and of A ∗non-enzymatic(see [7, Proposition actions 8.7]). of micro-organisms, oxidation, reduction, light which scissions them in smaller portions which are further degraded into simpler and non-harmful products such * Corresponding author: Dilip V. Vasava, Department*Corresponding of Chemistry, Author: Kazumasa Nomura: Tokyo Medical and Dental University, Ichikawa, 272-0827,Japan, as CO2, CH4, water, and hummus. BPs can be used for School of Sciences, Gujarat University, Ahmedabad,E-mail: [email protected] packaging materials, disposable non-woven, hygiene Gujarat- 380009, India, email: [email protected] Terwilliger: Department of Mathematics, University of Wisconsin, Madison, WI53706, USA, E-mail: products, consumer goods, agricultural tools, and much Tejas V. Shah, Department of Chemistry, School of Sciences,[email protected] Gujarat University, Ahmedabad, Gujarat- 380009, India. more (10-13). Open Access. © 2019 Shah and Vasava, published byOpen De Gruyter. Access. ©This2019 work Kazumasa is licensed Nomura under and the Paul Creative Terwilliger, Commons published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Attribution alone 4.0 License. 386 T.V. Shah and D.V. Vasava: A glimpse of biodegradable polymers and their biomedical applications by authors over the globe with their claimed biomedical uses have been communicated in detail. Drug delivery, orthopedics and couple of other applications and trends of degradable polymers are also highlighted. 2 Common polymers used for biomedical applications 2.1 Polyhydroxyalkanoates (PHA) Polyhydroxyalkanoates are the versatile class of biodegrad- able and biocompatible polyesters produced by micro- Figure 1: Classification of biodegradable plastics. organisms as an energy storage material via fermentation process under stressed conditions. A number of bacterial BPs can be classified into three major classes (3,10,14) species from both gram-positive and gram-negative species (Figure 1): are capable of producing PHA ranging from 30-80% of their cellular dry weight. The general formula of PHA is given in (I) Natural polymers: These are the polymers which are Figure 2, and till date more than 150 types of different PHA derived directly from the natural sources monomers have been identified and classified, thus allowing (II) Semi-synthetic polymers: These are the polymers a possibility of preparing an extensive range of polymers and in which raw material is obtained from nature, but copolymers with different properties (34-37). polymerization takes place after some chemical Few common monomers of this family are presented modification. in Table 1. (III) Synthetic polymers: These are purely synthetic Monomer units in PHA possess D (-) configuration and polymers derived from chemical synthesis. the molecular mass of PHA depending upon the source, Owing to their outstanding biocompatibility, low ranges from 50,000 to 1,000,000 Da. In general, PHAs toxicity and chemically tunable properties, biodegradable contain 1 to 14 carbon atoms at C-3 position, and thus polymers have become excellent materials for biomedical can also be classified into three subcategories depending applications and by observing the recent literature, it is upon number of carbons at C-3 position as: not exaggerating to say that, BPs have rife applications in a) short chain