Elimination in Operator Algebras

A large class of special functions are solutions of systems of linear difference and differential equations with polynomial coefficients. For a given function, these equations considered as operator polynomials generate a left ideal in a noncommutative algebra called Ore algebra. This ideal with finitely many conditions characterizes the function uniquely so that Gröbner basis techniques can be applied. Many problems related to special functions which can be described by such ideals can be solved by performing elimination of appropriate noncommutative variables in these ideals. In this work, we mainly achieve the following: 1. We give an overview of the theoretical algebraic background as well as the algorithmic aspects of different methods using noncommutative Gröbner elimination techniques in Ore algebras in order to solve problems related to special functions. 2. We describe in detail algorithms which are based on Gröbner elimination techniques and perform the creative telescoping method for sums and integrals of special functions. 3. We investigate and compare these algorithms by illustrative examples which are performed by the computer algebra system Maple. This investigation has the objective to test how far noncommutative Gröbner elimination techniques may be efficiently applied to perform creative telescoping.

A Basis for Z-Graded Identities of Matrices over Infinite Fields

2000 Mathematics Subject Classification: 16R10, 16R20, 16R50

Efficient selective identity-based encryption without random oracles

We construct two efficient Identity-Based Encryption (IBE) systems that admit selective-identity security reductions without random oracles in groups equipped with a bilinear map. Selective-identity secure IBE is a slightly weaker security model than the standard security model for IBE. In this model the adversary must commit ahead of time to the identity that it intends to attack, whereas in an adaptive-identity attack the adversary is allowed to choose this identity adaptively. Our first system—BB1—is based on the well studied decisional bilinear Diffie–Hellman assumption, and extends naturally to systems with hierarchical identities, or HIBE. Our second system—BB2—is based on a stronger assumption which we call the Bilinear Diffie–Hellman Inversion assumption and provides another approach to building IBE systems. Our first system, BB1, is very versatile and well suited for practical applications: the basic hierarchical construction can be efficiently secured against chosen-ciphertext attacks, and further extended to support efficient non-interactive threshold decryption, among others, all without using random oracles. Both systems, BB1 and BB2, can be modified generically to provide “full” IBE security (i.e., against adaptive-identity attacks), either using random oracles, or in the standard model at the expense of a non-polynomial but easy-to-compensate security reduction.

Robust network reconstruction in polynomial time

This paper presents an efficient algorithm for robust network reconstruction of Linear Time-Invariant (LTI) systems in the presence of noise, estimation errors and unmodelled nonlinearities. The method here builds on previous work [1] on robust reconstruction to provide a practical implementation with polynomial computational complexity. Following the same experimental protocol, the algorithm obtains a set of structurally-related candidate solutions spanning every level of sparsity. We prove the existence of a magnitude bound on the noise, which if satisfied, guarantees that one of these structures is the correct solution. A problem-specific model-selection procedure then selects a single solution from this set and provides a measure of confidence in that solution. Extensive simulations quantify the expected performance for different levels of noise and show that significantly more noise can be tolerated in comparison to the original method. © 2012 IEEE.

Two thymosin-repeated molecules with structural and functional diversity coexist in Chinese mitten crab Eriocheir sinensis

Recently, beta-thymosin-like proteins with multiple thymosin domains (defined as thymosin-repeated proteins) have been identified from invertebrate. In the present study, the cDNAs of two thymosin-repeated proteins (designated EsTRP1 and EsTRP2) were cloned from Chinese mitten crab by expressed sequence tags (EST) techniques. BLAST analysis presented three and two thymosin domains in EsTRP1 and EsTRP2, respectively, with the identities amongst the five domains varying from 47% to 100%. Both EsTRP1 and EsTRP2 shared high similarities with previously identified vertebrate beta-thynnosins and invertebrate thymosin-repeated proteins (TRPs) with the identities ranging from 43% to 78%, indicating that EsTRPs were new members of the beta-thymosin family. Real-time RT-PCR assay was adopted to determine the tissue distribution of EsTRPs and their temporal expression profile in hemocytes after pathogen stimulation and injury challenge. The expression of EsTRP1 transcript was predominantly detectable in the tissues of hemocytes, hepatopancreas and gonad with the highest expression in hemocytes, while the highest expression level of EsTRP2 was found in heart. EsTRP1 mRNA expression in hemocytes significantly increased at 3 and 48 h after Listonella anguillarum challenge, but there was no significant variation in EsTRP2 temporal expression profile. The injury challenge reduced the mRNA expression of EsTRPs, with the down-regulation of EsTRP2 expression occurred earlier than that of EsTRP1. The cDNA fragments encoding their mature peptides of EsTRP1 and EsTRP2 were recombined and expressed in Escherichia coli. The activities of recombinant proteins (rEsTRP1 and rEsTRP2) were examined by MTT (3-(4,5-dimethylthiazol-2-yl)-2, 5-diphenyltetrazoliumbromide) and lysoplate assay. rEsTRP2 could significantly accelerate the growth of human hepatocellular carcinoma cell line, but there was no significant effect of rEsTRP1 on the tumor cell proliferation. Both rEsTRP1 and rEsTRP2 did not possess the ability of killing Micrococcus luteus and L. anguillarum. The differences in the tissue distribution of mRNA transcripts, the response to pathogen stimulation and injury challenge, and the effect of recombinant proteins on human cell proliferation, indicated that there were functional diversity between the two structurally different molecules, EsTRP1 and EsTRP2. (C) 2009 Elsevier Ltd. All rights reserved.

On the weak amenability of β(X)

We investigate the weak amenability of the Banach algebra ß(X) of all bounded linear operators on a Banach space X. Sufficient conditions are given for weak amenability of this and other Banach operator algebras with bounded one-sided approximate identities.

Semiring induced valuation algebras: Exact and approximate local computation algorithms

Local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure, called valuation algebra.There are many models of this algebraic structure ranging from probability theory to numerical analysis, relational databases and various classical and non-classical logics. It turns out that many interesting models of valuation algebras may be derived from semiring valued mappings. In this paper we study how valuation algebras are induced by semirings and how the structure of the valuation algebra is related to the algebraic structure of the semiring. In particular, c-semirings with idempotent multiplication induce idempotent valuation algebras and therefore permit particularly efficient architectures for local computation. Also important are semirings whose multiplicative semigroup is embedded in a union of groups. They induce valuation algebras with a partially defined division. For these valuation algebras, the well-known architectures for Bayesian networks apply. We also extend the general computational framework to allow derivation of bounds and approximations, for when exact computation is not feasible.

Generating finite integral relation algebras

Relation algebras and categories of relations in particular have proven to be extremely useful as a fundamental tool in mathematics and computer science. Since relation algebras are Boolean algebras with some well-behaved operations, every such algebra provides an atom structure, i.e., a relational structure on its set of atoms. In the case of complete and atomic structure (e.g. finite algebras), the original algebra can be recovered from its atom structure by using the complex algebra construction. This gives a representation of relation algebras as the complex algebra of a certain relational structure. This property is of particular interest because storing the atom structure requires less space than the entire algebra. In this thesis I want to introduce and implement three structures representing atom structures of integral heterogeneous relation algebras, i.e., categorical versions of relation algebras. The first structure will simply embed a homogeneous atom structure of a relation algebra into the heterogeneous context. The second structure is obtained by splitting all symmetric idempotent relations. This new algebra is in almost all cases an heterogeneous structure having more objects than the original one. Finally, I will define two different union operations to combine two algebras into a single one.

Representation type of Jordan algebras

The problem of classification of Jordan bit-nodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0. (c) 2010 Elsevier Inc. All rights reserved.

ASSOCIATIVE IDENTITIES OF OCTONIONS

We work to find a basis of identities for an octonion algebra modulo an associator ideal of a free alternative algebra, or, in other words, a basis for an associative replica of an ideal of identities of an octonion algebra.

Nonhomogeneous subalgebras of Lie and special Jordan superalgebras

We consider polynomial identities satisfied by nonhomogeneous subalgebras of Lie and special Jordan superalgebras: we ignore the grading and regard the superalgebra as an ordinary algebra. The Lie case has been studied by Volichenko and Baranov: they found identities in degrees 3, 4 and 5 which imply all the identities in degrees <= 6. We simplify their identities in degree 5, and show that there are no new identities in degree 7. The Jordan case has not previously been studied: we find identities in degrees 3, 4, 5 and 6 which imply all the identities in degrees <= 6, and demonstrate the existence of further new identities in degree 7. our proofs depend on computer algebra: we use the representation theory of the symmetric group, the Hermite normal form of an integer matrix, the LLL algorithm for lattice basis reduction, and the Chinese remainder theorem. (C) 2009 Elsevier Inc. All rights reserved.

Exemplification and deconstructive writing: dancing with Massumi, Derrida, and others

I am experimenting with ‘exemplification’ (Massumi, 2002) and ‘deconstrucitve writing’ (Maclure, 2010) by applying them to selected data from research for my doctoral thesis. Central to both is the power of examples to expose detail and divergence that lead to new and different possibilities and connections. I have been inspired by Derrida’s (Maclure, 2003; Spitzer, 2011) notion of the fabric of a text with ‘tears’, ‘cuts’, ‘knots or aporias’. Therefore, I have created a fabric of text with tears and aporias: one that encourages readers to think and read doubly (Derrida, cited in Spitzer, 2011). I write in an ‘inattentive’ way (Massumi, 2002): juxtaposing examples (narratives) with the identities of the narrators, my responses to their narratives, and theoretical and other musings to play with the ambiguity of reality and identity. The interplay of inattention, exemplification, and deconstructive writing allows for interpretation, reinterpretation, affective responses, and new and different impressions and possibilities. My aim in writing this paper is threefold: (1) to explore the application of a Derridean perspective to some of my research, (2) to practise exemplification and deconstructive writing, and (3) to write in an inattentive and creative way.

Finiteness of hermitian levels of some algebras

We characterize the hermitian levels of quaternion and octonion algebras and of an 8-dimensional algebra D over the ground field F, constructed using a weak version of the Cayley-Dickson double process. It is shown that all values of the hermitian levels of quaternion algebras with the hat-involution also occur as hermitian levels of D. We give some limits to the levels of the algebra D over some ground field. © Soc. Paran. de Mat.

Resonant properties, of modified triangular plasmonic nanoparticles with higher field concentration

In this paper, we present an analysis of the resonant response of modified triangular metallic nanoparticles with polynomial sides. The particles are illuminated by an incident plane wave and the method of moments is used to solve numerically the electromagnetic scattering problem. We investigate spectral response and near field distribution in function of the length and polynomial order of the nanoparticles. Our results show that in the analyzed wavelength range (0.5-1.8) µm these particles possess smaller number of resonances and their resonant wavelengths, near field enhancement and field confinement are higher than those of the conventional triangular particle with linear sides.

Lattices from maximal orders into quaternion algebras

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)