Let g be a finite dimensional Lie algebra over a field k of characteristic zero. An element x of g is said to have an abstract Jordan-Chevalley decomposition if there exist unique s, n ∈ g such that x = s + n, [s, n] = 0 and given any finite dimensional representation π : g → gl(V ) the Jordan-Chevalley decomposition of π(x) in gl(V ) is π(x) = π(s) + π(n). In this paper we prove that x ∈ g has...

We review various properties of the exceptional Euclidean Jordan algebra of degree three. Euclidean Jordan algebras of degree three and their corresponding Freudenthal triple systems were recently shown to be intimately related to extremal black holes in N = 2, d = 4 homogeneous supergravities. Using a novel type of eigenvalue problem with eigenmatrix solutions, we elucidate the rich matrix geo...

We complete the analysis of the symmetry algebra L for systems of n second-order linear ODEs with constant real coefficients, by studying the case of coefficient matrices having a non-diagonal Jordan canonical form. We also classify the Levi factor (maximal semisimple subalgebra) of L, showing that it is completely determined by the Jordan form. A universal formula for the dimension of the symm...

The transfer homomorphism from a group G to an abelian subgroup H of finite index in G arises from the natural action of G on the cosets of H. It plays an important role in the study of unsolvable groups and aids in the characterization of groups in which all Sylow p-subgroups are cyclic. In this paper, we will define the transfer homomorphism, and then investigate a few of its properties. Let ...

‎let $m$ be a non-zero finitely generated module over a commutative noetherian local ring $(r,mathfrak{m})$ with $dim_r(m)=t$‎. ‎let $i$ be an ideal of $r$ with $grade(i,m)=c$‎. ‎in this article we will investigate several natural homomorphisms of local cohomology modules‎. ‎the main purpose of this article is to investigate when the natural homomorphisms $gamma‎: ‎tor^{r}_c(k,h^c_i(m))to kotim...

We investigate the convergence of GMRES for an n by n Jordan block J . For each k that divides n we derive the exact form of the kth ideal GMRES polynomial and prove the equality max ‖v‖=1 min p∈πk ‖p(J)v‖ = min p∈πk max ‖v‖=1 ‖p(J)v‖, where πk denotes the set of polynomials of degree at most k and with value one at the origin, and ‖ · ‖ denotes the Euclidean norm. In other words, we show that ...

An n-valued multimap is a continuous multivalued function φ : X ⊸ Y such that φ(x) is an unordered subset of n points of Y for each x ∈ X. If X and Y are finite polyhedra, then φ induces a graded homomorphism of homology with rational coefficients. For φ : X ⊸ X the Lefschetz number L(φ) of φ is defined to be the Lefschetz number of the induced homomorphism. If L(φ) 6= 0, then every n-valued mu...

Let Fn be a free group of rank n and F n the quotient group of Fn by the subgroup [Γn(3),Γn(3)][[Γn(2),Γn(2)],Γn(2)] where Γn(k) denotes the k-th subgroup of the lower central series of the free group Fn. In this paper, we determine the group structure of the graded quotients of the lower central series of the group F n by using a generalized Chen’s integration in free groups. Then we apply it ...

For von Neumann algebras M,N not isomorphic to C C and without type I2 summands, we show that for an order-isomorphism f : AbSub M! AbSub N between the posets of abelian von Neumann subalgebras of M and N , there is a unique Jordan ⇤-isomorphism g : M! N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of ...

This article presents a unified approach to the abstract notions of partial convolution and involution in $L^p$-function spaces over semi-direct product of locally compact groups. Let $H$ and $K$ be locally compact groups and $tau:Hto Aut(K)$ be a continuous homomorphism.  Let $G_tau=Hltimes_tau K$ be the semi-direct product of $H$ and $K$ with respect to $tau$. We define left and right $tau$-c...