For more than a century, rigid-body displacements have been viewed as affine transformations described as homogeneous transformation matrices wherein the linear part is a rotation matrix. In group-theoretic terms, this classical description makes rigid-body motions a semi-direct product. The distinction between a rigid-body displacement of Euclidean space and a change in pose from one reference...

In this work we studied the approximation of the fundamental solution of the Laplace operator in two-dimensional space u = log(|x|) by harmonic polynomials. We analyzed the best approximation in the semi-ring with fixed outer radius and inner radius ? tending to zero. We observed exponential convergence in the degree of polynomials used for approximation. However, with inner radius tending to z...

In this paper, we have studied warped products and multiply warped product on quasi-Einstein manifold with semi-symmetric nonmetric connection. Then we have applied our results to generalized Robertson-Walker space times with a semi-symmetric non-metric connection.

Let V ⊗HS W be the completion of V ⊗alg W in the norm defined by this inner product. V ⊗HS W is a Hilbert space; however, as Garrett shows it is not a categorical tensor product, and in fact if V and W are Hilbert spaces there is no Hilbert space that is their categorical tensor product. (We use the subscript HS because soon we will show that V ⊗HS W is isomorphic as a Hilbert space to the Hilb...

A mapping f : M → N between Hilbert C∗-modules approximately preserves the inner product if ‖〈f(x), f(y)〉 − 〈x, y〉‖ ≤ φ(x, y), for an appropriate control function φ(x, y) and all x, y ∈ M. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert C∗modules on more general restricted domains. In particular, we investigate some asympt...

We propose a novel variant of conjugate gradient based on the Reproducing Kernel Hilbert Space (RKHS) inner product. An analysis of the algorithm suggests it enjoys better performance properties than standard iterative methods when applied to learning kernel machines. Experimental results for both classification and regression bear out the theoretical implications. We further address the domina...

We propose a quantization based approach for fast approximate Maximum Inner Product Search (MIPS). Each database vector is quantized in multiple subspaces via a set of codebooks, learned directly by minimizing the inner product quantization error. Then, the inner product of a query to a database vector is approximated as the sum of inner products with the subspace quantizers. Different from rec...