Involutions of \operatorname{{SL}} (nk), (n>2)

A first characterization of the isomorphism classes of -involutions for any reductive algebraic group defined over a perfect field was given in [7] using three invariants. In this paper we give a simple characterization of the isomorphism classes of involutions of with any field of characteristic not equal to . We classify the isomorphism classes of involutions for algebraically closed, the real numbers, the -adic numbers and finite fields. We also determine in which cases the corresponding fixed point group is -anisotropic. In those cases the corresponding symmetric -variety consists of semisimple elements.Aloysius G. Helminck was partially supported by N.S.F. Grant DMS-9977392.     

Graph von Neumann Algebras

In this paper, we will define a graph von Neumann algebra over a fixed von Neumann algebra M, where G is a countable directed graph, by a crossed product algebra = M × _α , where is the graph groupoid of G and α is the graph-representation. After defining a certain conditional expectation from onto its M-diagonal subalgebra we can see that this crossed product algebra is *-isomorphic to an amalgamated free product where = vN(M × _α where is the subset of consisting of all reduced words in {e, e ^–1} and M × _α is a W ^*-subalgebra of as a new graph von Neumann algebra induced by a graph G _e . Also, we will show that, as a Banach space, a graph von Neumann algebra is isomorphic to a Banach space ⊕ where is a certain subset of the set E(G)* of all words in the edge set E(G) of G. The author really appreciates to Prof F. Radulescu and Prof P. Jorgensen for the valuable discussion and kind advice. Also, he appreciates all supports from St. Ambrose Univ.. In particular, he thanks to Prof T. Anderson and Prof V. Vega for the useful conversations and suggestions.     

Nonlinear Parabolic Equations with Regularized Derivatives in Colombeau Algebra

We consider several types of nonlinear parabolic equations with singular like potential and initial data. To prove the existence-uniqueness theorems we employ regularized derivatives. As a framework we use Colombeau space and Colombeau vector space      

SU(d)-Biinvariant Random Walks on SL\,{\left( {d,\,\mathbb{C}} \right)} and their Euclidean Counterparts

We establish a deformation isomorphism between the algebras of -biinvariant compactly supported measures on and -conjugation invariant measures on the Euclidean space of all Hermitian -matrices with trace . This isomorphism concisely explains a close connection between the spectral problem for sums of Hermititan matrices on one hand and the singular spectral problem for products of matrices from on the other, which has recently been observed by Klyachko [13]. From this deformation we further obtain an explicit, probability preserving and isometric isomorphism between the Banach algebra of bounded -biinvariant measures on and a certain (non-invariant) subalgebra of the bounded signed measures on . We demonstrate how this probability preserving isomorphism leads to limit theorems for the singular spectrum of -biinvariant random walks on in a simple way. Our construction relies on deformations of hypergroup convolutions and will be carried out in the general setting of complex semisimple Lie groups.Margit Rösler was partially supported by the Netherlands Organisation for Scientific Research (NWO), project nr. B 61-544.     

The Pluri-Fine Topology is Locally Connected

We prove that the pluri-fine topology on any open set in is locally connected. This answers a question by Fuglede.     

Quadratic-Residue Codes and Cyclotomic Fields

We present an exposition of quadratic-residue codes through their embedding in codes over the quadratic subfield of the th cyclotomic field, the algebraic number field of th roots of unity. This representation allows the development of effective syndrome decoding algorithms that can fully exploit the code’s error-correcting capability. This is accomplished via Galois automorphisms of cyclotomic fields. For each fixed , the general results hold for all pairs , with a finite number of exceptions that depends on . A complete discussion of the set of quadratic-residue codes of length and dimension illustrates these results. This set includes the Golay code, the only perfect binary three-error-correcting code.A preliminary version of this paper was presented at the International Conference on Statistics, Combinatorics and Related Areas, October 3–5, 2003, University of Southern Maine, Portland, ME, USA.     

Beurling Zeta Functions, Generalised Primes, and Fractal Membranes

We study generalised prime systems , with tending to infinity) and the associated Beurling zeta function . Under appropriate assumptions, we establish various analytic properties of , including its analytic continuation, and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of . Further we study ‘well-behaved’ g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on . Some of the above results are relevant to the second author’s theory of ‘fractal membranes’, whose spectral partition functions are given by Beurling-type zeta functions, as well as to joint work of that author and R. Nest on zeta functions attached to quasicrystals.The work of M. L. Lapidus was partially supported by the U. S. National Science Foundation under grant DMS-0070497.     

Characterization of the Newtonian Free Particle System in m\geqslant 2 Dependent Variables

We treat the problem of linearizability of a system of second order ordinary differential equations. The criterion we provide has applications to nonlinear Newtonian mechanics, especially in three-dimensional space. Let or , let , let , let and let

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

Small weight codewords in LDPC codes defined by (dual) classical generalized quadrangles

We find lower bounds on the minimum distance and characterize codewords of small weight in low-density parity check (LDPC) codes defined by (dual) classical generalized quadrangles. We analyze the geometry of the non-singular parabolic quadric in PG(4,q) to find information about the LDPC codes defined by Q (4,q), and . For , and , we are able to describe small weight codewords geometrically. For , q odd, and for , we improve the best known lower bounds on the minimum distance, again only using geometric arguments. Similar results are also presented for the LDPC codes LU(3,q) given in [Kim, (2004) IEEE Trans. Inform. Theory, Vol. 50: 2378–2388]     

The Frobenius Formalism in Galois Quantum Systems

Quantum systems in which the position and momentum take values in the ring and which are described with -dimensional Hilbert space, are considered. When is the power of a prime, the position and momentum take values in the Galois field , the position-momentum phase space is a finite geometry and the corresponding ‘Galois quantum systems’ have stronger properties. The study of these systems uses ideas from the subject of field extension in the context of quantum mechanics. The Frobenius automorphism in Galois fields leads to Frobenius subspaces and Frobenius transformations in Galois quantum systems. Links between the Frobenius formalism and Riemann surfaces, are discussed.     

Weak Quantum Enveloping Algebras of Borcherds Superalgebras

In present paper we define a new kind of weak quantized enveloping algebra of Borcherds superalgebras . It is a noncommutative and noncocommutative weak graded Hopf algebra. Using localizing with some Ore set, we obtain a different kind of quantized enveloping algebras of Borcherds superalgebras . It has a homomorphic image which is isomorphic to the usual quantum enveloping algebra of . Moreover, is isomorphic to a direct sum of and an other algebra as algebras. The author is sponsored by ZJNSF No. Y607136.     

Noncommutative Integrable Systems of Hydrodynamic Type

On noncommutative spaces, integrable hierarchies of hydrodynamic type systems (1-order quasilinear PDE’s) do not, in general, exist. Nevertheless, an infinite-component hydrodynamic chain defined below is shown to be integrable. Its modified version is also constructed and it exhibits a new purely noncommutative phenomenon: the number of modified variables is either or .     

On the Complexity of Parallel Algorithms for Computing Inverses in GF(2^m) with m Prime

Inversion over a finite field is usually an expensive operation which is a crucial issue in many applications, such as cryptography and error-control codes. In this paper, three different strategies for computing the inverse over binary finite fields , called Eulerian, Gaussian, and Euclidean, respectively, are discussed and compared in terms of time and space complexity. In particular, some new upper and lower bounds to the respective complexities are evaluated.     

A quadrilateral nonconforming finite element for linear elasticity problem

In this paper, a four-parameter quadrilateral nonconforming finite element with DSP (double set parameters) is presented. Then we discuss the quadrilateral nonconforming finite element approximation to the linear elastic equations with pure displacement boundary. The optimal convergence rate of the method is established in the broken energy and -norms, and in particular, the convergence is uniform with respect to the Lamé parameter . Also the performance of the scheme does not deteriorate as the material becomes nearly incompressible. Lastly, a numerical test is carried out, which coincides with our theoretical analysis. The research is supported by NSF of China (No. 10471133) and the project of Creative Engineering of Henan Province of China.     

Coherence for Product Monoids and their Actions

Let and be two monoids (algebras) in a monoidal category . Further let be a distributive law in the sense of [J. Beck, Lect. Notes Math., 80:119–140, 1969]; naturally yields a monoid . Consider a word in the symbols , , and . The first coherence theorem proved in this paper asserts that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , and . Assume now that an object is endowed with both an -object structure , and an -object structure . Further assume that these two structures are compatible, in the sense that they naturally yield an -object . Let be a word in , , , and , which contains a single instance of , in the rightmost position. The second coherence theorem states that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , , , and .     

A non-MRA C^r Frame Wavelet with Rapid Decay

A generalized filter construction is used to build an example of a non-MRA normalized tight frame wavelet for dilation by 2 in . This example has the same multiplicity function as the Journé wavelet, yet has a Fourier transform and can be made to be for any fixed postive integer . L. Baggett and P. Jorgensen were supported by a US–NSF Focused Research Group (FRG) grant.     

Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit

is the category of archimedean -groups with distinguished weak order unit, with -group homomorphisms which preserve unit. This category includes all rings of continuous functions and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in . There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each – object, we define a topology and a convergence . These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject of is dense iff is epically embedded. The topology is , but only sometimes Hausdorff or an -group topology. The convergence is a Hausdorff -group convergence, but only sometimes topological. The associations of to , and to , are functorial. Dedicated to Bernhard Banaschewski for his 80th birthday.     

Absolute Lax 2-categories

We have introduced, in a previous paper, the fundamental lax 2-category of a ‘directed space’ . Here we show that, when has a -topology, this structure can be embedded into a larger one, with the same objects (the points of ), the same arrows (the directed paths) and the same cells (based on directed homotopies of paths), but a larger system of comparison cells. The new comparison cells are absolute, in the sense that they only depend on the arrows themselves rather than on their syntactic expression, as in the usual settings of lax or weak structures. It follows that, in the original structure, all the diagrams of comparison cells commute, even if not constructed in a natural way and even if the composed cells need not stay within the old system.Work supported by MIUR Research Projects.     

Special Quasigraded Lie Algebras and Integrable Hamiltonian Systems

We construct a family of special quasigraded Lie algebras of functions of one complex variables with values in finite-dimensional Lie algebra , labeled by the special 2-cocycles F on . The main property of the constructed Lie algebras is that they admit Kostant-Adler-Symes scheme. Using them we obtain new integrable finite-dimensional Hamiltonian systems and new hierarchies of soliton equations.