We characterize the analogues of Householder reflectors in matrix groups associated with scalar products. Examples of such groups include the symplectic and pseudounitary groups. We also precisely delimit the mapping capabilities of these Householder analogues: given a matrix group G and vectors x, y, we give necessary and sufficient conditions for the existence of a Householder-like analogue G...

Compound identification is often achieved by matching the experimental mass spectra to the mass spectra stored in a reference library based on mass spectral similarity. Because the number of compounds in the reference library is much larger than the range of mass-to-charge ratio (m/z) values so that the data become high dimensional data suffering from singularity. For this reason, penalized lin...

8 Vectors and Quaternions 40 8.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 40 8.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 41 8.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 42 8.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 43...

In mining and integrating data from multiple sources, there are many privacy and security issues. In several different contexts, the security of the full privacy-preserving data mining protocol depends on the security of the underlying private scalar product protocol. We show that two of the private scalar product protocols, one of which was proposed in a leading data mining conference, are ins...

8 Vectors and Quaternions 145 8.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 145 8.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 146 8.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 147 8.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . ....

Numerical methods for stochastic ordinary differential equations typically estimate moments of the solution from sampled paths. Instead, in this paper we directly target the deterministic equation satisfied by the first and second moments. For the canonical examples with additive noise (Ornstein–Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic...

9 Vectors and Quaternions 51 9.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 9.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 51 9.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 52 9.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 53 9.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 54...

4 Vectors and Quaternions 47 4.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Displacement Vectors . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 The Parallelogram Law of Vector Addition . . . . . . . . . . . 49 4.4 The Length of Vectors . . . . . . . . . . . . . . . . . . . . . . 50 4.5 Scalar Multiples of Vectors . . . . . . . . . . . . . . . . . . . . 51...

A generalization of Kamke's uniqueness theorem in ordinary differential equations is obtained for the limit Cauchy problem, viz x'{t) = f(t, x(t)), x{t) -> x0 as 1J10, where / and x take values in an arbitrary normed linear space X and the initial point {t0, x0) is permitted to be on the boundary of the domain of/. Kamke's hypothesis that \\f(t,x)-f{t,y)\\ < <(>(\t-to\, ||x-,y||) is replaced by...