The original aim of this paper is to construct a graph associated to a vector space. By inspiration of the classical definition for the Cayley graph related to a group we define Cayley graph of a vector space. The vector space Cayley graph ${rm Cay(mathcal{V},S)}$ is a graph with the vertex set the whole vectors of the vector space $mathcal{V}$ and two vectors $v_1,v_2$ join by an edge whenever...

In this paper, we prove the existence of solutions for a class of nonlinear variational inclusion problems in semi-inner product spaces. We construct an iterative algorithm for approximating the solution for the class of variational inclusions involving A-monotone operators by using the resolvent operator technique.

The strict-contractivity and the convergence of General Linear Methods on the classes of strictly dissipative and dissipative differential systems regarding some inner product are analyzed. New convergence and contractivity results of the methods on semi-infinite intervals are provided for the case of strictly dissipative problems. Some applications of the main results to the class of Runge-Kut...

where 〈·, ·〉 denotes the standard inner product on C. If the Hermitian matrix H is invertible, then the indefinite inner product is nondegenerate. In that case, for every matrix T ∈ C, there is the unique matrix T [∗] satisfying [T x, y] = [x, T y] for all x, y ∈ C, and it is given by T [∗] = HT H . In these spaces, the notion of H-quasihyponormal matrix can be introduced by analogy with the qu...

Many important quantum algebras such as quantum symplectic space, quantum Euclidean space, quantummatrices, q-analogs of the Heisenberg algebra and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras U(L+) of even Lie color algebras are also semi-commutative. In this paper, we generalize work of Montgomery and examine the X-inner automorphisms of such algebras. The t...

Proximal and semi-proximal spaces are defined with the help of a game played on uniform spaces. Proximal spaces are where the entourage-picker has a winning strategy, and semi-proximal spaces are where the opponent, the point-picker, does not have a winning strategy. The class of proximal spaces is closed under Σ-products and closed subspaces. In this paper it is shown that product of two semi-...

is a vector version of p-Laplacian operator. In order to say what we understand by solution for the problem (1.1), (1.2) we remind some basic results concerning the W 1,p T -spaces. Let C T be the space of indefinitely differentiable T -periodic functions from R to R . We denote by 〈·, ·〉 the inner product on R and by ‖ · ‖, the norm generated by this inner product (the same meaning is applied ...

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden’s identity, the expression fo...

If W is an infinite rank 3 Coxeter group, whose Coxeter diagram has no infinite bonds, then the automorphism group of W is generated by the inner automorphisms and any automorphisms induced from automorphisms of the Coxeter diagram. Indeed Aut(W ) is the semi-direct product of Inn(W ) and the group of graph automorphisms.

1 Linear algebra 2 1.1 Inner product, norm, distance, and orthogonality . . . . . . . . . 2 1.2 Angle and inequality . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Vector projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Basics of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . ....