Nuclear elements of degree 6 in the free alternative algebra

We construct five new elements of degree 6 in the nucleus of the free alternative algebra. We use the representation theory of the symmetric group to locate the elements. We use the computer algebra system ALBERT and an extension of ALBERT to express the elements in compact form and to show that these new elements are not a consequence of the known clegree-5 elements in the nucleus. We prove that these five new elements and four known elements form a basis for the subspace of nuclear elements of degree 6. Our calculations are done using modular arithmetic to save memory and time. The calculations can be done in characteristic zero or any prime greater than 6, and similar results are expected. We generated the nuclear elements using prime 103. We check our answer using five other primes.

Finite-dimensional non-associative algebras and codimension growth

Let A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded. Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One of the main tools of independent interest is the construction in the free non-associative algebra of multialternating polynomials satisfying special properties. (C) 2010 Elsevier Inc. All rights reserved.

Description of costandard modules for Schur superalgebra S(2 vertical bar 1) in positive characteristic

We describe the characters of simple modules and composition factors of costandard modules for S(2 vertical bar 1) in positive characteristics and verify a conjecture of La Scala-Zubkov regarding polynomial superinvariants for GL(2 vertical bar 1).

The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras

We address two problems with the structure and representation theory of finite W-algebras associated with general linear Lie algebras. Finite W-algebras can be defined using either Kostant`s Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of finite W-algebras. The second main result is a parameterization of finite families of irreducible Gelfand-Tsetlin modules using Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand-Tsetlin modules for finite W-algebras. (C) 2009 Elsevier Inc. All rights reserved.

Nil graded self-similar algebras

In [19], [24] we introduced a family of self-similar nil Lie algebras L over fields of prime characteristic p > 0 whose properties resemble those of Grigorchuk and Gupta-Sidki groups. The Lie algebra L is generated by two derivations v(1) = partial derivative(1) + t(0)(p-1) (partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...))))), v(2) = partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...)))) of the truncated polynomial ring K[t(i), i is an element of N vertical bar t(j)(p) =0, i is an element of N] in countably many variables. The associative algebra A generated by v(1), v(2) is equipped with a natural Z circle plus Z-gradation. In this paper we show that for p, which is not representable as p = m(2) + m + 1, m is an element of Z, the algebra A is graded nil and can be represented as a sum of two locally nilpotent subalgebras. L. Bartholdi [3] andYa. S. Krylyuk [15] proved that for p = m(2) + m + 1 the algebra A is not graded nil. However, we show that the second family of self-similar Lie algebras introduced in [24] and their associative hulls are always Z(p)-graded, graded nil, and are sums of two locally nilpotent subalgebras.

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.

Use of geographical information systems as a tool for predicting fish yield in tropical reservoirs : case study on Sri Lankan reservoirs

Use of geographical information systems (GIS) in inland fisheries has hitherto been essentially restricted to site evaluation for aquaculture development and assessment of limnological changes in time and space in individual water bodies. The present GIS study was conducted on the land-use pattern of the catchments of nine reservoirs in Sri Lanka, for which detailed fishery data, viz. yield, fishing intensity, landing size of major constituent species, together with selected limnological data such as conductivity and chlorophyll-a, were available. Potential statistical relationships (linear, curvilinear, exponential and second-order polynomial) of fish yield (FY, in kg ha−1 yr−1) to different land-use patterns, such as forest cover (FC, in km2) and shrub-land (SL, in km2), either singly, or in combination, and/or the ratio of each land type to reservoir area (RA in km2) and reservoir capacity (RC in km3), were explored. Highly significant relationships were evident between FY to the ratio of SL and/or FC+SL to RA and/or RC. Similarly, the above land-use types to RA and RC ratios were significantly related to limnological features of the reservoirs. The relationships of FY to various parameters obtained in this study were much better correlated than those relationships of FY to limnological and biological parameters used in yield prediction in tropical and temperate lacustrine waters previously.

A model for compressing probabilities in belief networks

Probabilistic reasoning with belief (Bayesian) networks is based on conditional probability matrices. Thus it suffers from NP-hard implementations. In particular, the amount of probabilistic information necessary for the computations is often overwhelming. So, compressing the conditional probability table is one of the most important issues faced by the probabilistic reasoning community. Santos suggested an approach (called linear potential functions) for compressing the information from a combinatorial amount to roughly linear in the number of random variable assignments. However, much of the information in Bayesian networks, in which there are no linear potential functions, would be fitted by polynomial approximating functions rather than by reluctantly linear functions. For this reason, we construct a polynomial method to compress the conditional probability table in this paper. We evaluated the proposed technique, and our experimental results demonstrate that the approach is efficient and promising.

Monotone approximation of aggregation operators using least squares splines

The need for monotone approximation of scattered data often arises in many problems of regression, when the monotonicity is semantically important. One such domain is fuzzy set theory, where membership functions and aggregation operators are order preserving. Least squares polynomial splines provide great flexbility when modeling non-linear functions, but may fail to be monotone. Linear restrictions on spline coefficients provide necessary and sufficient conditions for spline monotonicity. The basis for splines is selected in such a way that these restrictions take an especially simple form. The resulting non-negative least squares problem can be solved by a variety of standard proven techniques. Additional interpolation requirements can also be imposed in the same framework. The method is applied to fuzzy systems, where membership functions and aggregation operators are constructed from empirical data.

Blind source separation based on phase and frequency redundancy of cyclostationary signals

This paper presents a new method for blind source separation by exploiting phase and frequency redundancy of cyclostationary signals in a complementary way. It requires a weaker separation condition than those methods which only exploit the phase diversity or the frequency diversity of the source signals. The separation criterion is to diagonalize a polynomial matrix whose coefficient matrices consist of the correlation and cyclic correlation matrices, at time delay .TAU. = 0, of multiple measurements. An algorithm is proposed to perform the blind source separation. Computer simulation results illustrate the performance of the new algorithm in comparison with the existing ones.

Capitanian (late Guadalupian, permian) global brachiopod palaeobigeography and latitudinal diversity pattern

A global database containing 3365 occurrences, 821 species and 251 genera of the Capitanian (Late Guadalupian, Permian) brachiopod faunas from 24 stations has been analyzed by cluster analysis using the Jaccard and Otsuka coefficients and the probabilistic index of similarity, nonmetric multidimensional scaling and minimum spanning tree. Two supergroups, three groups and six subgroups are revealed and interpreted as representing, respectively, two biotic realms (the Palaeoequatorial and Gondwanan Realms), two regions and six provinces. An additional realm (the Boreal Realm), based on the fauna from Spitsbergen, also appears recognizable although it also shows considerable similarities with southwestern North America and the northern margin of Gondwana as revealed by the statistical analysis. The Palaeoequatorial Realm can be further subdivided into the North America Region and the Asian Tethyan Region. The six biotic provinces are the Cathaysian Province in the Palaeotethys and Mesotethys, the Greenland-Svalbard Province in the Arctic region, the Austrazean Province in eastern Australia and New Zealand, the Grandian Province in western North America and the two transitional zones (the Himalayan Province in the southern temperate zone and the Sino–Mongolian–Japanese Province in the northern temperate zone). Polynomial regression analysis and rarefaction analysis indicate that the generic diversities of brachiopod faunas during the Capitanian peaked in the Palaeoequatorial Cathaysian Province and the two transitional zones (Himalayan Province and Sino–Mongolian–Japanese Province), but fell dramatically in the polar regions. The generic diversity of the Palaeoequatorial Grandian Province is apparently lower than in the two transitional zones of temperate palaeolatitudes, suggesting that the generic diversity of Capitanian brachiopod faunas does not exhibit a strict negative correlation with palaeolatitudes. This in turn would suggest that biogeographical determinants (such as geographical barriers, inhabitable area and ocean currents) other than latitude-related temperature control may also have played an important role in the dispersal of some brachiopods and the characterization of some local provinces and high diversities. The Capitanian global brachiopod palaeobiogeography is generally comparable with those in the Wuchiapingian and Changhsingian, but with some notable differences. These include: (1) that the Grandian Province of the Capitanian in western North America vanished after the end-Guadalupian regression, (2) that the western Tethyan Province of the Lopingian could not be distinguished in the Capitanian, and (3) that the Austrazean Province was larger in area than either in the Wuchiapingian or in the Changhsingian.

Algebraic attacks over GF(q)

Recent algebraic attacks on LFSR-based stream ciphers and S-boxes have generated much interest as they appear to be extremely powerful. Theoretical work has been developed focusing around the Boo- lean function case. In this paper, we generalize this theory to arbitrary finite fields and extend the theory of annihilators and ideals introduced at Eurocrypt 2004 by Meier, Pasalic and Carlet. In particular, we prove that for any function f in the multivariate polynomial ring over GF(q), f has a low degree multiple precisely when two low degree functions appear in the same coset of the annihilator of f ^{q – 1} – 1. In this case, many such low degree multiples exist.

Blind equalization of nonirreducible systems using the CM criterion

We address the blind equalization of finite-impulse-response (FIR) and multiple-input multiple-output (MIMO) channel systems excited by constant modulus (CM) signals. It is known that the algorithms based on the CM criterion can equalize an FIR MIMO system that is irreducible. The irreducible condition is restrictive as it requires all source signals to be received at sensors simultaneously. In this paper, we further show that the CM property of signals can be exploited to construct a zero-forcing equalizer for a system that is nonirreducible. Simulation examples demonstrate the proposed result.

Blind separation of cyclostationary sources based on phase-frequency redundancy

Thin paper presents a new algorithm for blind source separation (BSS) by exploiting phase and frequency redundancy of cyclostationary signals in a complementary way. The separation criterion is to diagonalize a polynomial matrix whose coefficient matrices consist of the correlation and cyclic correlation matrices of multiple measurements. Computer simulation results illustrate, the performance of the new algorithm in comparison with some existing algorithms.

Blind separation of cyclostationary signals from instantaneous mixtures

This paper presents a new approach for blind separation of unknown cyclostationary signals from instantaneous mixtures. The proposed method can perfectly separate the mixed source signals so long as they have either different cyclic frequencies or clock phases. This is a weaker condition than those required by the algorithms. The separation criterion is to diagonalize a polynomial matrix whose coefficient matrices consist of the correlation and cyclic correlation matrices, at time delay τ=0, of multiple measurements.