A natural problem that arises in the study of derivations on a Banach algebra is to classify the commutators in the algebra. The problem as stated is too broad and we will only consider the algebra of operators acting on a given Banach space X. In particular, we will focus our attention to the spaces $\lambda I and $\linf$. The main results are that the commutators on $\ell_1$ are the operators not of the form $\lambda I + K$ with $\lambda\neq 0$ and $K$ compact and the operators on $\linf$ which are commutators are those not of the form $\lambda I + S$ with $\lambda\neq 0$ and $S$ strictly singular. We generalize Apostol's technique (1972, Rev. Roum. Math. Appl. 17, 1513 - 1534) to obtain these results and use this generalization to obtain partial results about the commutators on spaces $\mathcal{X}$ which can be represented as $\displaystyle \mathcal{X}\simeq \left ( \bigoplus_{i=0}^{\infty} \mathcal{X}\right)_{p}$ for some $1\leq p\leq\infty$ or $p=0$. In particular, it is shown that every non - $E$ operator on $L_1$ is a commutator. A characterization of the commutators on $\ell_{p_1}\oplus\ell_{p_2}\oplus\cdots\oplus\ell_{p_n}$ is also given.

This paper is a study of the properties of commutators, and deals exclusively with finite groups.

University of Macau / Faculty of Science and Technology / Department of Mathematics

<p>We establish several properties of higher commutators, which were<br />introduced by A. Bulatov, in congruence permutable varieties. We use these<br />commutators to prove that the clone of polynomial functions of a finite Mal&rsquo;cev<br />algebra whose congruence lattice is of height at most 2, can be described by a<br />finite set of relations. For a finite nilpotent algebra of finite type that is a product<br />of algebras of prime power order and generates congruence modular variety, we<br />are able to show that the property of affine completeness is decidable. Moreover,<br />polynomial equivalence problem has polynomial complexity in the length of the<br />input polynomials.</p> / <p>Ustanovljavamo osobine viˇsih komutatora, koje je uveo A. Bulatov,<br />u kongruencijki permutabilnim varijetetima. Te komutatore koristimo da bi<br />dokazali da se klon polinomijalnih funkcija konaˇcne Maljcevljeve algebre ˇcija je<br />mreˇza kongruencija visine najviˇse dva moˇze opisati konaˇcnim skupom relacija. Za<br />konaˇcne nilpotentne algebre konaˇcnog tipa koje su proizvod algebri koje imaju red<br />stepena prostog broja i koje generiˇsu kongruencijki modularan varijetet pokazu-jemo da je osobina afine kompletnosti odluˇciva. Takod&macr;e, pokazujemo za istu klasu<br />da problem polinomijalne ekvivalencije ima polinomnu sloˇzenost u zavisnosti od<br />duˇzine unetih polinomijalnih terma.</p>

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 136-138). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 136-138). Also available on the Internet.

The main purpose of this thesis is to use modern goal-oriented adaptive methods of Lie group analysis to construct the optimal sys- tem of Black-Scholes equation. We will show in this thesis how to obtain all invariant solutions by constructing what has now become so popular, optimal system of sub-algebras, the main Lie algebra admit- ted by the Black-Scholes equation. First, we obtain the commutator table of already calculated symmetries of the Black-Scholes equation. We then followed with the calculations of transformation of the gen- erators with the Lie algebra L6 which provides one-parameter group of linear transformations for the operators. Here we make use of the method of Lie equations to solve the partial di®erential equations. Next, we consider the construction of optimal systems of the Black- Scholes equation where the method requires a simpli¯cation of a vector to a general form to each of the transformations of the generators. Further, we construct the invariant solutions for each of the op- timal system. This study is motivated by the analysis of Lie groups which is being taken to another level by ALGA here in Blekinge In- stitute Technology, Sweden. We give a practical and in-depth steps and explanation of how to construct the commutator table, the calcu- lation of the transformation of the generators and the construction of the optimal system as well as their invariant solutions. Keywords: Black-Scholes Equation, commutators, commutator table, Lie equa- tions, invariant solution, optimal system, generators, Airy equation, structure constant, / It was an accolade for us to work with Professor Nail.H. Ibrgimov. +46762600953

Ο σκοπός της εργασίας είναι η παρουσίαση της έννοιας της συμμετρίας ως έναν μετασχηματισμό που απεικονίζει τη λύση μιας Δ.Ε. σε μια άλλη Δ.Ε. διατηρώντας αναλλοίωτη και αμετάβλητη τη μορφή της. Παρουσιάζεται επίσης η μέθοδος της αναλλοίωτης διαφόρισης και ο αλγόριθμος Lie. / -

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2009. / Title from title screen (site viewed June 26, 2009). PDF text: v, 238 p. ; 2 Mb. UMI publication number: AAT 3350445. Includes bibliographical references. Also available in microfilm and microfiche formats.

University of Macau / Faculty of Science and Technology / Department of Mathematics