We study the de Rham 1-cohomology H 1 DR (M, G) of a smooth manifold M with values in a Lie group G. By definition, this is the quotient of the set of flat connections in the trivial principal bundle M × G by the so-called gauge equivalence. We consider the case when M is a compact Kähler manifold and G is a solvable complex linear algebraic group of a special class which contains the Borel sub...

We discuss an open problem on the discreteness of subgroups of (SL2(R)) (n ≥ 2) generated by n linearly independent upper triangular matrices and n linearly independent lower triangular matrices. According to a conjecture by Margulis, only Hilbert modular groups can arise this way. The purpose of this note is to explain how this open problem is related to another conjecture on the orbit behavio...

Let 1 < p < ∞ and A = (an,k)n,k 1 be a non-negative matrix. Denote by ‖A‖w,p,F , the infimum of those U satisfying the following inequality: ‖Ax‖w,p,F U ‖x‖w,p,I , where x 0 and x ∈ lp(w,I) and also w = (wn)n=1 is a decreasing, non-negative sequence of real numbers. The purpose of this paper is to give a lower bound for ‖A‖w,p,F , where A is a lower triangular matrix. In particular, we apply ou...

*Correspondence: [email protected] Department of Mathematics, Faculty of Science, Cumhuriyet University, Sivas, 58 410, Turkey Abstract The discrete generalized Cesàro matrix At = (ank) is the triangular matrix with nonzero entries ank = tn–k/(n + 1), where t ∈ [0, 1]. In this paper, boundedness, compactness, spectra, the fine spectra and subdivisions of the spectra of discrete gene...

We extend Atanassov’s methods for Halton sequences in two different directions: (1) in the direction of Niederreiter (t,s)−sequences, (2) in the direction of generating matrices for Halton sequences. It is quite remarkable that Atanassov’s method for classical Halton sequences applies almost “word for word” to (t,s)−sequences and gives an upper bound quite comparable to those of Sobol’, Faure a...

Suppose U is an upper-triangular matrix, and D a nonsingular diagonal matrix whose diagonal entries appear in nondescending order of magnitude down the diagonal. It is proved that kD UDk kUk for any matrix norm that is reduced by a pinching. In addition to known examples { weakly unitarily invariant norms { we show that any matrix norm de ned by kAk def = max x 6=0; y 6=0 Re (x Ay) (x) (y) ; wh...

; a ∈ A,m ∈ M, b ∈ B} equipped with the usual 2× 2 matrix-like addition and matrix-like multiplication is an algebra. An algebra T is called a triangular algebra if there exist algebras A and B and nonzero A−B-bimodule M such that T is (algebraically) isomorphic to Tri(A,M,B) under matrixlike addition and matrix-like multiplication; cf. [1]. For example, the algebra Tn of n × n upper triangular...

The hierarchical structure in the quark masses and mixings allows its ten physical parameters to be most conveniently encoded in mass matrices of the upper triangular form. We classify these matrices in the hierarchical, minimal parameter basis where the mismatch between the weak and mass eigenstates involves only small mixing angles. Ten such pairs are obtained for the up and down quarks. This...

We study the freeness problem for matrix semigroups. We show that the freeness problem is decidable for upper-triangular 2 × 2 matrices with rational entries when the products are restricted to certain bounded languages. We also show that this problem becomes undecidable for sufficiently large matrices.

We study the neutrino mixing matrix (the MNS matrix) in the seesaw model. By assuming a large mass hierarchy for the heavy right-handed Majorana mass, we show that, in the diagonal Majorana base, the MNS matrix is determined by a unitary matrix, S, which transforms the neutrino Yukawa matrix, yν , into a triangular form, y△. The mixing matrix of light leptons is VKMS ′∗ , where VKM ≡ VLe VLν an...