Some Arithmetic on the Hardy Sums s2（h, κ） and s3（h, κ）

In this work, Rademacher‘s two questions about Dedekind sums are applied to the Hardy sums s2(h,κ) and s3(h,κ). Some relations between these sums are obtained and some inequalities are given.     

The least squares problem of the matrix equation A1X1B1^T ＋ A2X2B2^T = T

The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

The least degree of identities in the subspace M 1 (m,k) (F) of the matrix superalgebra M (m,k)(F)

We determine the least degree of identities in the subspace M ₁ ^(m,k) (F) of the matrix superalgebra M ^(m,k)(F) over a field F for arbitrary m and k. For the subspace M ₁ ^(m,k) (F) (k > 1) we obtain concrete minimal identities and generalize some results by Chang and Domokos.     

A generalization of adjoint crystals for the quantized affine algebras of type An(1) , Cn(1) and Dn+1(2)

We generalize Benkart-Frenkel-Kang-Lee’s adjoint crystals and describe their crystal structure for type A _n ⁽¹⁾, C _n ⁽¹⁾ and D _n+1⁽²⁾.     

A note on the geometry of M 3 and symplectic structures on S 1 × M 3

We investigate the relationship between the geometry of a closed, oriented 3-manifold M and the symplectic structures on S ¹ × M. In most cases the existence of a symplectic structure on S ¹ × M and Thurstonșs geometrization conjecture imply the existence of a geometric structure on M. This observation together with the existence of geometric structures on most 3-manifolds which fiber over the circle suggests a different approach to the problem of finding a fibration of a 3-manifold over the circle in case its product with the circle admits a symplectic structure. This work was supported in part by a GEBIP grant from the Turkish Academy of Sciences and a CAREER grant from the Scientific and Technological Research Council of Turkey.     

The representations of two types of functionals on L∞(Ω, ?) and L∞(Ω, ?, ?)

This article gives the representations of two types of real functionals on L ^∞(Ω, Ƒ) or L ^∞(Ω, Ƒ, ℙ) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respectively.     

Trace maps over M3(K) associated with the substitutions

Let K be a field of characteristic zero and M ₃(K) the ring of matrices 3×3 over K. In this paper, we establish first the certain identities of traces of some algebras of Min3(K), then we define the trace mapping of 3 × 3 matrices associated with a substitution over a two-letter alphabet on some algebraic variety and study their properties.     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.     

Topological singularities in W S,P ( S N , S 1)

We are interested in the location of the singularities of maps u ∈ W ^s,p (S ^N , S ¹) when 1 ≤ sp and 1 < sp < 2. To this end, we consider the distributional Jacobian. We show that the range of this operator on W ^s,p (S ^N , S ¹) is the closure in W ^s−2,p ⋂ W ^−1,sp of the set of N − 2-currents defined as the integration on smooth oriented N − 2-dimensional boundaryless submanifolds.     

A numerical algorithm for solving the matrix equation AX + XTB = C1

An algorithm of the Bartels-Stewart type for solving the matrix equation AX + X ^T B = C is proposed. By applying the QZ algorithm, the original equation is reduced to an equation of the same type having triangular matrix coefficients A and B. The resulting matrix equation is equivalent to a sequence of low-order systems of linear equations for the entries of the desired solution. Through numerical experiments, the situation where the conditions for unique solvability are “nearly” violated is simulated. The loss of the quality of the computed solution in this situation is analyzed.     

A Sufficient Condition for (F1)-Effectiveness of theCT-Method

The study of (F)-effectiveness of theN_p-method was initiated by Hille and Tamarkin. In this direction, Dikshit investigated (F₁)-effectiveness of the product methodC¹N_p-method. Recently Kumar and Prasad established the necessary condition for (F₁)-effectiveness of the product method. In this note, we extend the corresponding sufficient part of Kumar and Prasad to theC¹T-method.     

Sequences of minimal surfaces in S2n+1

For a minimal surface immersed into an odd-dimensional unit sphere S ²ⁿ⁺¹ with the first (n−2) higher-order ellipses of curvature being a circle, we construct a sequence of such surfaces and investigate if some two minimal surfaces in such a sequence can be congruent by an orientation-reversing isometry.     

ltivariate Hausdorff operators on the spaces H1(Rn) and BMO(Rn)

Summary A multivariate Hausdorff operator H = H(, c, A) is defined in terms of a -finite Borel measure on Rⁿ, a Borel measurable function c on Rⁿ, and an n × n matrix A whose entries are Borel measurable functions on rⁿ and such that A is nonsingular -a.e. The operator H*:= H (, c | det A^-1|, A^-1) is the adjoint to H in a well-defined sense. Our goal is to prove sufficient conditions for the boundedness of these operators on the real Hardy space H¹(Rⁿ) and BMO (Rⁿ). Our main tool is proving commuting relations among H, H*, and the Riesz transforms R_j. We also prove commuting relations among H, H*, and the Fourier transform.     

The automorphism group of the Toeplitz algebra onH2(T2)

The automorphism group of the C^* -algebra generated by the Toeplitz operatos with symbols being continuous functions on the bicircle is characterized completely. The investigation is based on the analysis of the behaviour of an automorphism of the Toeplitz algebra on itsC* -ideal chains, and the state of the closed ideals in .