Some remarks on a paper by L. Carlitz

We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials.     

Some remarks on a paper by Liu and van Rooij

Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability.     

Some remarks on a paper by samimi on nonuniqueness criteria for ordinary differential equations

In [4] M. SAMIMI gives nonuniqueness theorems for initial value problems x′= f(t,x), x(0) =x₀. where f is not defined at t = 0. But the result for the scalar case only holds with the additional assumption that one solution exists. The applicability of the revised theorem is shown. The result for the n-dimensional case is generalized.     

Some remarks on the Euler ring U(G)

Let G be a compact Lie group and let U(G) denote the Euler ring of G constructed by tom Dieck [5, 6]. The main result of the paper (Theorem 4.1) describes the homomorphism induced by the inclusion . Dedicated to Professor Vladimir Arnold     

Some remarks on the (f, g)-linear connections

In this note we consider structures defined on a differentiable manifoldM by a tensor fieldf of type (1,1) satisfying the conditionf ⁴+f ²=0 and by a Riemannian structureg such thatg(f(X),Y)=−g(Xf(Y)) for all vector fieldsX,YεT ¹ ₀(M). Then we determine linear connections compatible with those structures. Facultatea de Matematica, Str. Academiei 14, 70109 Bucaresti, Romania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 3, pp. 383–387, July–September, 1997.     

Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476]. We shall prove that there are no normal cones with normal constant M<1 and for each k>1 there are cones with normal constant M>k. Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], we obtain generalizations of the results.     

Classification theorems for central simple algebras with involution (with an appendix by R. Parimala)

The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian forms. Analogues of the classical invariants of quadratic forms (discriminant, Clifford algebra, signature) have been defined for arbitrary central simple algebras with involution. In this paper it is shown that over certain fields these invariants are sufficient to classify involutions up to conjugation. For algebras of low degree a classification is obtained over an arbitrary field. Received: 29 April 1999     

Comment to “interactive fuzzy goal programming approach for bilevel programming problem” by S.R. Arora and R. Gupta

The fuzzy bilevel programming problem is solved in the paper [1] using an approach of fuzzy goal programming. Using a simple example we show that this will not lead to a satisfactory solution. Especially the fuzzy constraint that the upper level variable is near some desired one has no influence on the computed solution in the example. This constraint is used to model the hierarchy in the approach in [1]. At the end of the paper we suggest one straightforward possibility for modeling the fuzzy bilevel programming problem and converting it into a crisp substitute.     

On the paper “Symmetry analysis of wave equation on sphere” by H. Azad and M.T. Mustafa

Using the scalar curvature of the product manifold S²×R and the complete group classification of nonlinear Poisson equation on (pseudo) Riemannian manifolds, we extend the previous results on symmetry analysis of homogeneous wave equation obtained by H. Azad and M.T. Mustafa [H. Azad, M.T. Mustafa, Symmetry analysis of wave equation on sphere, J. Math. Anal. Appl. 333 (2007) 1180-1188] to nonlinear Klein-Gordon equations on the two-dimensional sphere.     

A note on the paper “Fractional programming with convex quadratic forms and functions” by H.P. Benson

In this technical note, we give a short proof based on some standard results in convex analysis of some important characterization results listed in Theorems 3 and 4 of Benson [Benson, H.P., 2006. Fractional programming with convex quadratic forms and functions. European Journal of Operational Research]. Actually our result is slightly more general since we do not specify the nonempty convex set X. For clarity we use the same notation for the different equivalent optimization problems as done in Benson (2006).