Differential calculi of Poincaré-Birkhoff-Witt type on universal enveloping algebras of Lie algebras g are defined. This definition turns out to be independent of the basis chosen in g. The role of automorphisms of g is explained. It is proved that no differential calculus of PoincaréBirkhoff-Witt type exists on semi-simple Lie algebras. Examples are given, namely gln, Abelian Lie algebras, the...

The polynomials ψk(r,x) were introduced by Ramanujan. Berndt, Evans and Wilson obtained a recurrence relation for ψk(r,x). Shor related to improper edges of rooted tree, leading refinement Cayley's formula. Zeng realized that the Ramanujan coincide with Shor, coincides Wilson. These also arise in work Wang Zhou on orbifold Euler characteristics moduli spaces stable curves. Dumont Ramamonjisoa f...

In this paper we investigate the index theory of transversally elliptic pseudo-differential operators in the framework of noncommutative geometry. We give examples of spectral triples in the sense of Alain Connes and Henri Moscovici in [12] that are transversally elliptic but nonelliptic. We prove that these spectral triples satisfy the conditions which ensure the Connes-Moscovici local index f...

and |g′′(x)| = M. We will show that |f(x)| ≤ g(x) on [0, 1]. The desired conclusion follows. Suppose however that this is false, that there is a number q ∈ [0, 1] for which f(q) > g(q). (The other case, f(q) < −g(q), is similar.) Our strategy will be to show that there are real numbers s and t with s < t such that f ′(s) > g′(s) and f ′(t) < g′(t). See figure. We will then apply the Mean Value ...

the current study presents a new analytical method for buckling analysis of rectangular and annular beams made up of functionally graded materials with constant thickness and poisson’s ratio. the boundary conditions of the beam are assumed to be simply supported and clamped. the stability equations were obtained by using conservation of energy. the critical buckling load and first mode shape we...

the current study presents a new analytical method for buckling analysis of circular plates with constant thickness and poisson’s ratio, made of functionally graded material subjected to radial load. governing equations are based on energy method for thin plates. the boundary conditions of the plate are assumed to be simply supported and clamped. the stability equations were obtained by using c...

In this paper, we use Clifford algebra and the spinor calculus to study the complex structures on Euclidean space R8 and the spheres S4, S6. By the spin representation of G(2, 8) ⊂ Spin(8) we show that the Grassmann manifold G(2, 8) can be looked as the set of complex structures on R8. In this way, we show that G(2, 8) and CP 3 can be looked as twistor spaces of S6 and S4 respectively. Then we ...

Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graph-theoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h : C → G(φ), where G(φ) is a graph associated with φ and C is a coloured...

1. Assume everything you know from the notes on Riemann-Stieltjes integration. (a) Show that if g : [0, 1]→ R is Riemann-integrable on [0, 1] and φ : [a, b]→ R is continuous on an interval [a, b] ⊇ g[[0, 1]], then φ ◦ g is Riemann-integrable on [0, 1]. (b) Assume that f : [0, 1]→ R is a function for which f ′(t) exists at each t ∈ [0, 1] (one-sidedly at endpoints) and that the Riemann integral ...

We describe the local conformal geometry of a Lorentzian spin manifold (M, g) admitting a twistor spinor φ with zero. Moreover, we describe the shape of the zero set of φ. If φ has isolated zeros then the metric g is locally conformally equivalent to a static monopole. In the other case the zero set consists of null geodesic(s) and g is locally conformally equivalent to a Brinkmann metric. Our ...