Invariants, Lie-Bäcklund operators and Bäcklund transformations

This thesis is mainly concerned with the application of groups of transformations to differential equations and in particular with the connection between the group structure of a given equation and the existence of exact solutions and conservation laws. In this respect the Lie-Bäcklund groups of tangent transformations, particular cases of which are the Lie tangent and the Lie point groups, are extensively used.

In Chapter I we first review the classical results of Lie, Bäcklund and Bianchi as well as the more recent ones due mainly to Ovsjannikov. We then concentrate on the Lie-Bäcklund groups (or more precisely on the corresponding Lie-Bäcklund operators), as introduced by Ibragimov and Anderson, and prove some lemmas about them which are useful for the following chapters. Finally we introduce the concept of a conditionally admissible operator (as opposed to an admissible one) and show how this can be used to generate exact solutions.

In Chapter II we establish the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation. It is shown that consideration of only Lie point groups is insufficient. For this purpose a special type of Lie-Bäcklund groups, those equivalent to Lie tangent groups, is used. It is also shown how these generalized groups induce Lie point groups on Hamilton's equations. The generalization of the above results to any first order equation, where the dependent variable does not appear explicitly, is obvious. In the second part of this chapter we investigate admissible operators (or equivalently constants of motion) of the Hamilton-Jacobi equation with polynornial dependence on the momenta. The form of the most general constant of motion linear, quadratic and cubic in the momenta is explicitly found. Emphasis is given to the quadratic case, where the particular case of a fixed (say zero) energy state is also considered; it is shown that in the latter case additional symmetries may appear. Finally, some potentials of physical interest admitting higher symmetries are considered. These include potentials due to two centers and limiting cases thereof. The most general two-center potential admitting a quadratic constant of motion is obtained, as well as the corresponding invariant. Also some new cubic invariants are found.

In Chapter III we first establish the group nature of all separable solutions of any linear, homogeneous equation. We then concentrate on the Schrodinger equation and look for an algorithm which generates a quantum invariant from a classical one. The problem of an isomorphism between functions in classical observables and quantum observables is studied concretely and constructively. For functions at most quadratic in the momenta an isomorphism is possible which agrees with Weyl' s transform and which takes invariants into invariants. It is not possible to extend the isomorphism indefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl' s transform. This is illustrated for the case of cubic invariants. Finally, the case of a specific value of energy is considered; in this case Weyl's transform does not yield an isomorphism even for the quadratic case. However, for this case a correspondence mapping a classical invariant to a quantum orie is explicitly found.

Chapters IV and V are concerned with the general group structure of evolution equations. In Chapter IV we establish a one to one correspondence between admissible Lie-Bäcklund operators of evolution equations (derivable from a variational principle) and conservation laws of these equations. This correspondence takes the form of a simple algorithm.

In Chapter V we first establish the group nature of all Bäcklund transformations (BT) by proving that any solution generated by a BT is invariant under the action of some conditionally admissible operator. We then use an algorithm based on invariance criteria to rederive many known BT and to derive some new ones. Finally, we propose a generalization of BT which, among other advantages, clarifies the connection between the wave-train solution and a BT in the sense that, a BT may be thought of as a variation of parameters of some. special case of the wave-train solution (usually the solitary wave one). Some open problems are indicated.

Most of the material of Chapters II and III is contained in [I], [II], [III] and [IV] and the first part of Chapter V in [V].

Palladium-Catalyzed Decarboxylative and Decarbonylative Transformations in the Synthesis of Fine and Commodity Chemicals

Decarboxylation and decarbonylation are important reactions in synthetic organic chemistry, transforming readily available carboxylic acids and their derivatives into various products through loss of carbon dioxide or carbon monoxide. In the past few decades, palladium-catalyzed decarboxylative and decarbonylative reactions experienced tremendous growth due to the excellent catalytic activity of palladium. Development of new reactions in this category for fine and commodity chemical synthesis continues to draw attention from the chemistry community.

The Stoltz laboratory has established a palladium-catalyzed enantioselective decarboxylative allylic alkylation of β-keto esters for the synthesis of α-quaternary ketones since 2005. Recently, we extended this chemistry to lactams due to the ubiquity and importance of nitrogen-containing heterocycles. A wide variety of α-quaternary and tetrasubstituted α-tertiary lactams were obtained in excellent yields and exceptional enantioselectivities using our palladium-catalyzed decarboxylative allylic alkylation chemistry. Enantioenriched α-quaternary carbonyl compounds are versatile building blocks that can be further elaborated to intercept synthetic intermediates en route to many classical natural products. Thus our chemistry enables catalytic asymmetric formal synthesis of these complex molecules.

In addition to fine chemicals, we became interested in commodity chemical synthesis using renewable feedstocks. In collaboration with the Grubbs group, we developed a palladium-catalyzed decarbonylative dehydration reaction that converts abundant and inexpensive fatty acids into value-added linear alpha olefins. The chemistry proceeds under relatively mild conditions, requires very low catalyst loading, tolerates a variety of functional groups, and is easily performed on a large scale. An additional advantage of this chemistry is that it provides access to expensive odd-numbered alpha olefins.

Finally, combining features of both projects, we applied a small-scale decarbonylative dehydration reaction to the synthesis of α-vinyl carbonyl compounds. Direct α-vinylation is challenging, and asymmetric vinylations are rare. Taking advantage of our decarbonylative dehydration chemistry, we were able to transform enantioenriched δ-oxocarboxylic acids into quaternary α-vinyl carbonyl compounds in good yields with complete retention of stereochemistry. Our explorations culminated in the catalytic enantioselective total synthesis of (–)-aspewentin B, a terpenoid natural product featuring a quaternary α-vinyl ketone. Both decarboxylative and decarbonylative chemistries found application in the late stage of the total synthesis.

Study of polymer thermal and mechanical properties using a fiber Bragg grating thermal sensor

A simple and practical method for the study of polymer thermal and mechanical properties using a fiber Bragg grating (FBG) sensor is presented for the first time, in which the FBG is embedded in a typical epoxy polymer. By measuring the sensitivity change of the FBG sensor, changes of the thermal-mechanical properties of the polymer with temperature and pressure can be measured. The experimental results show that this technique is capable of providing continuous in-line monitoring such properties with high sensitivity during transformation between the glassy state and the rubbery state of a polymer within the temperature and pressure range of 20 to 180 C and 0 to 15 MPa. (c) 2007 Society of Photo-Optical Instrumentation Engineers.

Study of polymer thermal and mechanical properties using a fiber Bragg grating thermal sensor

A simple and practical method for the study of polymer thermal and mechanical properties using a fiber Bragg grating (FBG) sensor is presented for the first time, in which the FBG is embedded in a typical epoxy polymer. By measuring the sensitivity change of the FBG sensor, changes of the thermal-mechanical properties of the polymer with temperature and pressure can be measured. The experimental results show that this technique is capable of providing continuous in-line monitoring such properties with high sensitivity during transformation between the glassy state and the rubbery state of a polymer within the temperature and pressure range of 20 to 180 C and 0 to 15 MPa. (c) 2007 Society of Photo-Optical Instrumentation Engineers.

Thermal and thermal-optical effects in high-power photonic crystal fiber lasers

A series-parallel model is introduced to calculate the effective thermal conductivities of hollow claddings of photonic crystal fibers ( PCFs ). The temperature distribution and thermal-optical properties of PCF lasers are studied by solving the heat transfer equations. The average power scaling of the PCF lasers in respect of the thermal effects is also discussed. (c) 2006 Society of Photo-Optical Instrumentation Engineers.

Thermal and thermal-optical effects in high-power photonic crystal fiber lasers

A series-parallel model is introduced to calculate the effective thermal conductivities of hollow claddings of photonic crystal fibers ( PCFs ). The temperature distribution and thermal-optical properties of PCF lasers are studied by solving the heat transfer equations. The average power scaling of the PCF lasers in respect of the thermal effects is also discussed. (c) 2006 Society of Photo-Optical Instrumentation Engineers.

Novel erbium-doped TeO2-based oxysulfide glasses: thermal and spectroscopic properties

Er3+-doped TeO2-based oxysulfide glasses have been prepared in argon atmosphere in carbon crucibles. The thermal analysis and spectroscopic properties of Er (3+) have been considered in terms of sulfide influence. As a function of composition, we have principally measured optical absorption, spontaneous emission and lifetime measurements. Judd-Ofelt theory was introduced to calculate bandwidth and emission cross-section. The results show the product FVMM x sigma(c) increase from 476.8 8 to 635.04 10(-21) cm(2) nm evidently with the addition of 10 mol% PbS into tellurite glass, which indicates a perfect effect on spectra property of Er3+ ions. (C) 2004 Elsevier B.V. All rights reserved.

Thermal and electric conductivity of near-zero thermal expansion ZrW 2O8//ZrO2 composites

In this work, the microstructure, thermal and electric conductivity properties of near-zero thermal expansion ZrW2O8/ZrO2 and Al2O3 added ZrW2O8/ZrO2 composites were studied. Both the two composites exhibit very low thermal conductivity and the thermal conductivity decreases slightly as the temperature increases. The electric conductivity of the two composites increases with the increasing of the measurement temperature. The Al2O3 added ZrW2O8/ZrO2 composite has higher thermal and electric conductivity than ZrW2O8/ZrO2 composite. The most important factor which causes the difference of the thermal and electric conductivity of the composites is the porosity. (C) 2008 The Ceramic Society of Japan. All rights reserved.