Partial projective representations and partial actions

The concept of a partial projective representation of a group is introduced and studied. The interaction with partial actions is explored. It is shown that the factor sets of partial projective representations over a field K are exactly the K-valued twistings of crossed products by partial actions.     

Homomorphisms of matrix semigroups over division rings from dimension two to four

Let D be an arbitrary division ring and M_n(D) the multiplicative semigroup of all n×n matrices over D. We study non-degenerate, injective homomorphisms from M₂(D) to M₄(D). In particular, we present a structural result for the case when D is the ring of quaternions.     

Some problems of conformal mappings of spherical domains

In this study the problem of finding the conformal mapping from a sphere onto a plane with a given scale function independent of longitude is solved for an arbitrary spherical domain. The obtained results are compared with the well-known projections used in cartography and geophysical fluid dynamics. The problem of minimization of the distortion under conformal mappings is solved for domains in the form of the spherical disk. The distortions of some extensively used conformal mappings are compared with the distortions of orthogonal mappings.     

Comparison of different spatial grids for numerical schemes of geophysical fluid dynamics

The generation of computational grids is an important component contributing to the efficiency of numerical schemes of atmosphere/ocean dynamics. In this study the problem of construction of the most uniform grids based on conformal mappings of spherical domains is considered. Stereographic, cylindrical and conic grids for computational rectangles are developed and their uniformity is compared. Numerical experiments with two schemes approximating shallow water equations are performed in order to assess the practical efficiency of the constructed grids and to compare the numerical results with analytical evaluations.     

On grid generation for numerical models of geophysical fluid dynamics

A simple geometric condition that defines the class of classical (stereographic, conic and cylindrical) conformal mappings from a sphere onto a plane is derived. The problem of optimization of computational grid for spherical domains is solved in an entire class of conformal mappings on spherical (geodesic) disk. The characteristics of computational grids of classical mappings are compared for different spherical radii of geodesic disk. For a rectangular computational domain, the optimization problem is solved in the class of classical mappings and respective area of the spherical domain is evaluated.     

Relations between the Clausen and Kazhdan-Lusztig representations of the symmetric group

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x_1,1,…,x_n,n] to give a new construction of the Kazhdan-Lusztig representations of S_n. This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x_1,1,…,x_n,n]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young’s natural representations.     

Bounds on the global dimension of certain piecewise hereditary categories

We give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables.We use this to show that the global dimension of a finite-dimensional, piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of a finite poset or more generally, a sincere algebra. This bound is tight.     

Growth of entire functions of exponential type defined by convolution

Based on the Borel transformation and the Hadamard multiplication theorem on singularities on the convolution of holomorphic functions, results on the growth of entire functions defined by convolution of an entire function of exponential type with a function holomorphic at the origin are obtained. Received: 29 August 2005     

The center of an effect algebra

An effect algebra is a partial algebra modeled on the standard effect algebra of positive self-adjoint operators dominated by the identity on a Hilbert space. Every effect algebra is partially ordered in a natural way, as suggested by the partial order on the standard effect algebra. An effect algebra is said to be distributive if, as a poset, it forms a distributive lattice. We define and study the center of an effect algebra, relate it to cartesian-product factorizations, determine the center of the standard effect algebra, and characterize all finite distributive effect algebras as products of chains and diamonds.     

The weighted complexity and the determinant functions of graphs

The complexity of a graph can be obtained as a derivative of a variation of the zeta function [S. Northshield, A note on the zeta function of a graph, J. Combin. Theory Ser. B 74 (1998) 408-410] or a partial derivative of its generalized characteristic polynomial evaluated at a point [D. Kim, H.K. Kim, J. Lee, Generalized characteristic polynomials of graph bundles, Linear Algebra Appl. 429 (4) (2008) 688-697]. A similar result for the weighted complexity of weighted graphs was found using a determinant function [H. Mizuno, I. Sato, On the weighted complexity of a regular covering of a graph, J. Combin. Theory Ser. B 89 (2003) 17-26]. In this paper, we consider the determinant function of two variables and discover a condition that the weighted complexity of a weighted graph is a partial derivative of the determinant function evaluated at a point. Consequently, we simply obtain the previous results and disclose a new formula for the complexity from a variation of the Bartholdi zeta function. We also consider a new weighted complexity, for which the weights of spanning trees are taken as the sum of weights of edges in the tree, and find a similar formula for this new weighted complexity. As an application, we compute the weighted complexities of the product of the complete graphs.     

Analytic characterizations of Jordan chains by pole cancellation functions of higher order

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of the representing relation of length ?, we show that there exists a pole cancellation function of order at least ?, and, moreover, the construction shows that it is of surprisingly simple form.     

On Klein’s icosahedral solution of the quintic

We present an exposition of the icosahedral solution of the quintic equation first described in Klein’s classic work ‘Lectures on the icosahedron and the solution of equations of the fifth degree’. Although we are heavily influenced by Klein we follow a slightly different approach which enables us to arrive at the solution more directly.     

On the finite generation of the effective monoid of rational surfaces

We give a numerical criterion for ensuring the finite generation of the effective monoid of the surfaces obtained by a blowing-up of the projective plane at the supports of zero dimensional subschemes assuming that these are contained in a degenerate cubic. Furthermore, this criterion also ensures the regularity of any numerically effective divisor on these surfaces. Thus the dimension of any complete linear system is computed. On the other hand, in particular and among these surfaces, we obtain ringed rational surfaces with very large Picard numbers and with only finitely many integral curves of strictly negative self-intersection. These negative integral curves except two (−1)-curves are all contained in the support of an anticanonical divisor. Thus almost all the geometry of such surfaces is concentrated in the anticanonical class.     

Minimal generating set for semi-invariants of quivers of dimension two

A minimal (by inclusion) generating set for the algebra of semi-invariants of a quiver of dimension (2,…,2) is established over an infinite field of arbitrary characteristic. The mentioned generating set consists of the determinants of generic matrices and the traces of tree paths of pairwise different multidegrees, where in the case of characteristic different from two we take only admissible paths. As a consequence, we describe relations modulo decomposable semi-invariants.     

The transvection free groups with polynomial rings of invariants

Let G be a finite linear group containing no transvections. This paper proves that the ring of invariants of G is polynomial if and only if the pointwise stabilizer in G of any subspace is generated by pseudoreflections. Kemper and Malle used the classification of finite irreducible groups generated by pseudoreflections to prove the irreducible case in arbitrary characteristic. We extend their result to the reducible case.     

Completion of operator partial matrices associated with chordal graphs

H. Dym and I. Gohberg established in [6] necessary and sufficient conditions for the existence and uniqueness of an invertible block matrix F=(F_ij)_i,j=1,...,n such that F_ij=R_ij for |i–j|m and F^–1 has a band triangular factorization and so (F^–1)_ij=0 for |i–j|>m. Here R_ij, |i–j|m are given block matrices.The aim of this paper is to generalize these results in two directions. First, we shall allow R_ij to be an (linear bounded) operator acting between the Hilbert spaces H_j and H_i. Secondly, the set of indices of the given R_ij will be more general than banded ones. In fact, we will consider index sets which have an associated graph which is chordal. The case of partial positive operator matrices is also discussed.