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- Abramov, Viktor, et al.
(författare)
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3-Hom-Lie Algebras Based on σ-Derivation and Involution
- 2020
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Ingår i: Advances in Applied Clifford Algebras. - : Springer. - 0188-7009 .- 1661-4909. ; 30:3
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Tidskriftsartikel (refereegranskat)abstract
- We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to two Hom-Lie algebras constructed on an associative, commutative algebra using σ-derivation and involution, and we obtain two 3-Hom-Lie algebras.
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| 4. |
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Algebra, Geometry, and Mathematical Physics 2010
- 2012
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Ingår i: Journal of Physics: Conference Series. - : IOP Publishing. - 1742-6596 .- 1742-6588.
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Samlingsverk (redaktörskap) (övrigt vetenskapligt/konstnärligt)abstract
- This proceedings volume presents results obtained by the participants of the 6th Baltic–Nordic workshop 'Algebra, Geometry, and Mathematical Physics (AGMP-6)' held at the Sven Lovén Centre for Marine Sciences in Tjärnö, Sweden on October 25–30, 2010. The Baltic–Nordic Network AGMP 'Algebra, Geometry, and Mathematical Physics' http://www.agmp.eu was created in 2005 on the initiative of two Estonian universities and two Swedish universities: Tallinn University of Technology represented by Eugen Paal (coordinator of the network), Tartu University represented by Viktor Abramov, Lund University represented by Sergei Silvestrov, and Chalmers University of Technology and the University of Gothenburg represented by Alexander Stolin. The goal was to promote international and interdisciplinary cooperation between scientists and research groups in the countries of the Baltic–Nordic region in mathematics and mathematical physics, with special emphasis on the important role played by algebra and geometry in modern physics, engineering and technologies. The main activities of the AGMP network consist of a series of regular annual international workshops, conferences and research schools. The AGMP network also constitutes an important educational forum for scientific exchange and dissimilation of research results for PhD students and Postdocs. The network has expanded since its creation, and nowadays its activities extend beyond countries in the Baltic–Nordic region to universities in other European countries and participants from elsewhere in the world. As one of the important research-dissimilation outcomes of its activities, the network has a tradition of producing high-quality research proceedings volumes after network events, publishing them with various international publishers.
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| 6. |
- Kamiya, Noriaki, et al.
(författare)
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A review on $\delta$-structurable algebras
- 2011
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Ingår i: Banach Center Publications. - : Institute of Mathematics, Polish Academy of Sciences. - 1730-6299 .- 0137-6934. ; 93, s. 59-67
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Konferensbidrag (refereegranskat)abstract
- In this paper we give a review on $\delta$-structurable algebras. A connection between Malcev algebras and a generalization of $\delta$-structurable algebras is also given.
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| 8. |
- Musonda, John
(författare)
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Orthogonal Polynomials, Operators and Commutation Relations
- 2017
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Orthogonal polynomials, operators and commutation relations appear in many areas of mathematics, physics and engineering where they play a vital role. For instance, orthogonal functions in general are central to the development of Fourier series and wavelets which are essential to signal processing. In particular, as demonstrated in this thesis, orthogonal polynomials can be used to establish the L2-boundedness of singular integral operators which is a fundamental problem in harmonic analysis and a subject of extensive investigations. The Lp-convergence of Fourier series is closely related to the Lp-boundedness of singular integral operators. Many important relations in physical sciences are represented by operators satisfying various commutation relations. Such commutation relations play key roles in such areas as quantum mechanics, wavelet analysis, representation theory, spectral theory, and many others.This thesis consists of three main parts. The first part presents a new system of orthogonal polynomials, and establishes its relation to the previously studied systems in the class of Meixner–Pollaczek polynomials. Boundedness properties of two singular integral operators of convolution type are investigated in the Hilbert spaces related to the relevant orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on L2-spaces, and estimates of the norms are obtained.The second part extends the investigation of the boundedness properties of the two singular integral operators to Lp-spaces on the real line, both in the weighted and unweighted spaces. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for L2 and weak boundedness for L1, and then using interpolation to obtain boundedness for the intermediate spaces. To obtain boundedness for the remaining spaces, duality is used in the translation invariant case, while the weighted case is partly based on the methods developed by M. Riesz in his paper of 1928 for the conjugate function operator.The third and final part derives simple and explicit formulas for reordering elements in an algebra with three generators and Lie type relations. Centralizers and centers are computed as an example of an application of the formulas.
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| 9. |
- Musonda, John, 1981-
(författare)
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Reordering in Noncommutative Algebras, Orthogonal Polynomials and Operators
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The main object studied in this thesis is the multi-parametric family of unital associative complex algebras generated by the element $Q$ and the finite or infinite set $\{S_j\}_{j\in J}$ of elements satisfying the commutation relations $S_jQ=\sigma_j(Q)S_j$, where $\sigma_j$ is a polynomial for all $j\in J$. A concrete representation is given by the operators $Q_x(f)(x)=xf(x)$ and $\alpha_{\sigma_j}(f)(x)=f(\sigma_j(x))$ acting on polynomials or other suitable functions. The main goal is to reorder arbitrary elements in this family and some of its generalizations, and to study properties of operators in some representing operator algebras, including their connections to orthogonal polynomials. For $J=\{1\}$ and $\sigma(x)=x+1$, the above commutation relations reduce to the famous classical Heisenberg--Lie commutation relation $SQ-QS=S$. Reordering an element in $S$ and $Q$ means to bring it, using the commutation relation, into a form where all elements $Q$ stand either to the left or to the right. For example, $SQ^2=Q^2S+2QS+S$. In general, one can use the commutation relation $SQ-QS=S$ successively and transform for any positive integer $n$ the element $SQ^n$ into a form where all elements $Q$ stand to the left. The coefficients which appear upon reordering in this case are the binomial coefficients. General reordering formulas for arbitrary elements in noncommutative algebras defined by commutation relations are important in many research directions, open problems and applications of the algebras and their operator representations. In investigation of the structure, representation theory and applications of noncommutative algebras, an important role is played by the explicit description of suitable normal forms for noncommutative expressions or functions of generators. Further investigation of the operator representations of the commutation relations by difference type operators on Hilbert function spaces leads to interesting connections to functional analysis and orthogonal polynomials. This thesis consists of two main parts. The first part is devoted to the multi-parametric family of algebras introduced above. General reordering formulas for arbitrary elements in this family are derived, generalizing some well-known results. As an example of an application of the formulas, centralizers and centers are computed. Some operator representations of the above algebras are also described, including considering them in the context of twisted derivations. The second part of this thesis is devoted to a special representation of these algebras by difference operators associated with action by shifts on the complex plane. It is shown that there are three systems of orthogonal polynomials of the class of Meixner--Pollaczek polynomials that are connected by these operators. Boundedness properties of two singular integral operators of convolution type connected to these difference operators are investigated in the Hilbert spaces related to these systems of orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on the $L^2$-spaces and estimates of the norms are obtained. This investigation is also extended to $L^p$-spaces on the real line where it is proved again that the two operators are bounded.
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| 10. |
- Nohrouzian, Hossein, 1980-
(författare)
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A Cubature Method for Solving Stochastic Equations : A Modern Monte-Carlo Approach with Applications to Financial Market
- 2022
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Before the financial crisis started in 2007, there were no significant spreads between the forward rate curves constructed either using the market quotes of overnight indexed swaps or those of forward rate agreements. After the crisis, we observe such spreads in the form of forward spread curves.In a popular approach pioneered by Heath, Jarrow, and Morton, the above curves satisfy a system of infinite-dimensional stochastic integral equations. In fact, the solution is a random field, or a random function of two real variables. By fixing the value of the second variable, one obtains a finite set of random forward spread curves, one for each maturity. Varying the above value generates the curves “in motion”.A standard approach to solve such a system is to replace it by a “discrete” version in the following order: first introduce discrete space, then discrete time, and finally, a discrete set of solutions. A modern approach starts by introducing a discrete space of solutions called a “cubature formulae on Wiener space”. An advantage of the modern approach is that the obtained system of equations becomes deterministic rather than stochastic and may be easily solved by standard finite-difference or finite-element methods.The thesis contains the followings new important results. The market model under consideration is large, that is, it includes infinitely many financial instruments. We reviewed existing approaches for finding conditions of no arbitrage on such a market with only one forward spread curve. First, we extended one of the approaches to the case of multiple curves and proved sufficient conditions for absence of arbitrage on such a large market. Second, we found conditions under which the solution to our system of equations is unique and non-negative. Third, using the theory of free Lie algebra, we found new cubature formulae on Wiener space and extensively tested them using the celebrated Black–Scholes equation as an input. Forth, using the results of cubature formula of degree 5, we evaluated the forward and short rates in the Heath–Jarrow–Morton and Hull–White (one-factor) models. Finally, using the same results, we constructed a new trinomial tree model for Black–Scholes–Merton and Black models.In future research, we plan to apply the obtained formulae to solve some systems of infinite-dimensional stochastic equations describing mathematical models of spread curves. Further, we plan to use the obtained formulae to deal with backward stochastic differential equations.
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