1 Let K be a field, V a finite-dimensional vector space over K, and T ∈ EndK(V ). For k ∈ K and S ∈ EndK(V ), define kS : V → V, v → k S(v). (a) Let Φ : K[x] → EndK(V ) be defined by Φ(k0 + k1x + k2x + . . . + knx) = k0T 0 + k1T 1 + k2T 2 + . . . + knT, where T 0 = idV . This is well-defined because a polynomial is completely determined by its coefficients. We will show that this is a homomorph...

4.3. (i) Since HomA(A (I),M) ∼= M (I) for a set I and ⊕ is exact, it is clear that HomA(A (I),−) is exact, i.e. A(I) is projective. (ii) “Only if” part: Let A(P ) = ⊕ p∈P Ap be the free A-module indexed by P and ψ : A (P ) → p be the A-homomorphism such that ψ(1p) = p. Then ψ is surjective. If P is projective, then ψ has a section, i.e. an A-homomorphism φ : P → A(P ) such that ψ ◦ φ = idP . Th...

Let R be any commutative ring. Then by GLN (R) we mean the group of all N × N matrices M with entries in R, and which are invertible: det(M) ∈ R×. The determininant map gives a homomorphism of groups which is easily seen to be surjective: defining the kernel to be SLN (R), we get an extension of groups The center Z of GLN (R) consists of the scalar matrices R× · IN , so as a group isomorphic to...

Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(\Omega_{\mathbb{T}}(M\times \mathbb{T}))$ $M$, is mapped to Bismut-Chern under Chen integral map. It also shown that assignment $g\mapsto \mathrm{Ch}^-(g)$ induces wel...

A homomorphism J: 7Tk_1(SOw) -> nm+k_1(S ) from the homotopy groups of rotation groups to the homotopy groups of spheres has been defined by H. Hopf and G. W. Whitehead. This homomorphism plays an important role in the study of differentiable manifolds. We will study its relation to one particular problem: the question of possible Pontrjagin numbers of an 'almost parallelizable' manifold. Defin...

in this paper, we show that every surjective $n$-homomorphism ($n$-anti-homomorphism) from a banach algebra $a$ into a semisimple banach algebra $b$ is continuous.

In this paper we prove that every n-Jordan homomorphis varphi:mathcal {A} longrightarrowmathcal {B} from unital Banach algebras mathcal {A} into varphi -commutative Banach algebra mathcal {B} satisfiying the condition varphi (x^2)=0 Longrightarrow varphi (x)=0, xin mathcal {A}, is an n-homomorphism. In this paper we prove that every n-Jordan homomorphism varphi:mathcal {A} longrightarrowmathcal...

Several existence theorems of positive solutions are established for nonlinear m-point boundary value problem for the following dynamic equations on time scales φu Δ ∇ atft, ut 0, t ∈ 0, T, φu Δ 0 m−2 i1 a i φu Δ ξ i , uT m−2 i1 b i uξ i , where φ : R→R is an increasing homeomorphism and homomorphism and φ0 0. As an application, an example to demonstrate our results is given.

In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let π be a group and let M → N be a homomorphism between projective Z[π]-modules such that Zp⊗Z[π]M → Zp⊗Z[π]N is injective; for which other right Z[π]-modules V is the induced map V ⊗Z[π] M → V ⊗Z[π] N also injective? Our main theorem gives a new criterion which combin...

The algebra Ψ(M) of order zero pseudodifferential operators on a compact manifold M defines a well-known C∗-extension of the algebra C(S∗M) of continuous functions on the cospherical bundle S∗M ⊂ T ∗M by the algebra K of compact operators. In his proof of the index theorem, Higson defined and used an asymptotic homomorphism T from C0(T ∗M) to K, which plays the role of a deformation for the com...