In this paper we describe the irreducible decomposition of the Magnus representation of the Torelli group.

We study the concepts of continuous shadowing and continuous inverse shadowing with respect to various classes of admissible pseudo orbits, and characterize hyperbolicity and structural stability using the notion of continuous inverse shadowing.

The authors provide precise lower bounds for the completely bounded norm of the operator Ta,b: ß(H) → ß(H) defined by Ta,b (x) = axb + bxa and the injective norm of the corresponding tensor. Further, they compute the norm of the operator x ↦ a*xb + b*xa acting on the space of all conjugate-linear operators on H.

Let T be an eventually expansive transformation on the unit interval satisfying the Markov condition. The T is an ergodic transformation on (X, ß, μ) where X = [0, 1), ß is the Borel σ-algebra on the unit interval and μ is the T invariant absolutely continuous measure. Let G be a finite subgroup of the circle group or the whole circle group and φ: X → G be a measurable function with finite discontinuity points. We investigate ergodicity of skew product transformations Tφ on X × G by showing the solvability of the coboundary equation φ(x) g (Tx) = λg (x), |λ| = 1. Its relation with the uniform distribution mod M is also shown.

We obtain a basic B,-Y. Chen's inequality for a C-totally real submanifold in a (κ,μ)-contact space form involving intrinsic invariants, namely the scalar curvature and the sectional curvatures of the submanifold on left hand side and the main extrinsic invariant, namely the squared mean curvature on the right hand side. Inequalities between the squared mean curvature and Ricci curvature and between the squared mean curvature and κ-Ricci curvature are also obtained. These results are applied to get corresponding results for C-totally real submanifolds in a Sasakian space form.

In this paper, we introduce a new class of set-valued mappings in a non-convex setting called D-KKM mappings and prove a general D-KKM theorem. This extends and improves the KKM theorem for several families of set-valued mappings, such as (X, Y), C(X, Y), C (X, Y), C (X, Y) and C (X, Y). In the sequel, we apply our theorem to get some existence results for maximal elements, generalised variational inequalities, and price equilibria.

Let Y = {z ∈ C: z3 ∈ [0, 1]} (equipped with subspace topology of the complex space C) and let f: Y → Y be a continuous map. We show that if f is pointwise chain recurrent (that is, every point of Y is chain recurrent under f), then either f12 is the identity map or f12 is turbulent. This result is a generalisation to Y of a result of Block and Coven for pointwise chain recurrent maps of the interval.

Let C be a cancellative cone and consider a subcone C0 of C. We study the natural problem of obtaining conditions on a non negative homogeneous function φ: C → R+ so that for each linear functional f defined in C0 which is bounded by φ, there exists a linear extension to C. In order to do this we assume several geometric conditions for cones related to the existence of special algebraic basis of the linear span of these cones.

In this paper we define Weyl semiparallel ( · h = 0) and Weyl 2-semiparallel ( · h = 0) submanifolds. We consider n-dimensional normally flat submanifolds satisfying these curvature conditions in (n + d)-dimensional Euclidean space n+d. We also consider normally flat submanifolds in (n + d)-dimensional Euclidean space n+d satisfying the condition  · h = LQ (g, h) and  · h = LQ(g, h).

Let N be a fixed positive integer. In this paper, we apply a newly developed method of Eie and Lai to the evaluation of certain infinite trigonometric series. Indeed, we are able to express the sums of these series in terms of Bernoulli polynomials. That is, we obtain several new Bernoulli identities. A similar method leads to Ramanujan's identities.

Let c ≥ 0, d ≥ 2 be integers and be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that is locally nilpotent? We prove that if c ≤ 2d + 2d−1 − 3 then the variety is locally nilpotent and we reduce the question of Gupta about the periodic groups in to the prime power exponent groups in this variety.

Without the need of a constraint qualification, we establish the optimality necessary and sufficient conditions for generalised minimax programming. Using these optimality conditions, we construct a parametric dual model and a parameter-free mixed dual model. Several duality theorems are established.

We recall the definition and properties of an algebra cone C of a complex unital Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A, and A is then called an ordered Banach algebra. The Banach algebra ℒ(E) of all bounded linear operators on a complex Banach lattice E is an example of an ordered Banach algebra, and an interesting aspect of research in ordered Banach algebras is that of investigating in an ordered Banach algebra-context certain problems that originated in ℒ(E). In this paper we investigate the problems of providing conditions under which (1) a positive element a with spectrum consisting of 1 only will necessarily be greater than or equal to 1, and (2) f (a) will be positive if a is positive, where f (a) is the element defined by the holomorphic functional calculus.

In this paper, we discover a sufficient and necessary condition under which two geodesic segments joining the base point and another point in an infinite-dimensional Teichmüller space are the same.

Let G be a finite p-solvable group for a fixed prime p. We determine the structure of G when the set of p-regular conjugacy class sizes of G is {1, m} for an arbitrary integer m > 1.

In this note, we prove that if a torsion group G has an Abelian subgroup B such that G/B is Abelian and R is a G-adapted ring with the property that R (G/B) has only trivial units then G has the normaliser property in RG.