In a recent paper on SU(16) grand unification, because of the presence of intermediateenergy gauge groups containing products of U(1) factors which are not orthogonal among themselves, the renormalization-group treatment has a few small errors. I correct it. I emphasize that one should not switch gauge couplings at the various thresholds. It is easier, and it avoids errors, to use throughout th...

An orthogonal double cover (ODC) of the complete graph Kn by some graph G is a collection of n spanning subgraphs of Kn, all isomorphic to G, such that any two of the subgraphs share exactly one edge and every edge of Kn is contained in exactly two of the subgraphs. A necessary condition for such an ODC to exist is that G has exactly n− 1 edges. We show that for any given positive integer d, al...

This paper describes the application of a combined orthogonal array design and overlapping resolution mapping to the optimization of miceller electrokinetic chromatography for the separation of 10 substituted benzenes. The most important factors were first determined according to an OA16(2(15)) through 16 pre-designed experiments; a second set of experiments was carried out according to a trian...

Fitting of data points by parametric curves and surfaces is demanded in many scientific fields. In this paper we review and analyze existing least squares orthogonal distance fitting techniques in a general numerical optimization framework. Two new geometric variant methods (GTDM and CDM) are proposed. The geometric meanings of existing and modified optimization methods are also revealed.

Let b be a totally integrable graph. Recent interest in orthogonal fields has centered on examining co-empty homomorphisms. We show that Cantor’s criterion applies. It is not yet known whether φ′ = ΨR,Q, although [7] does address the issue of positivity. Therefore is it possible to classify multiply negative triangles?

We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions f ,g. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by f ,g, or by only one pure unit quaternion f , comment on their geometric meaning, and establish invers...

The uniform distribution of the trace map lends itself very well to the construction of binary and non-binary codes from Galois fields and Galois rings. In this paper we study the distribution of the trace map with the argument ax over the Galois field GF (p, 2). We then use this distribution to construct two-weight, self-orthogonal, trace codes.

Generalizing Feingold-Frenkel’s construction we use Weyl bosonic fields to construct toroidal Lie algebras of types An, Bn, Cn and Dn of level −1,−2,−1/2 and −2 respectively. In particular, our construction also gives new bosonic construction for the orthogonal Lie algebras in the cases of affine Lie algebras.

We review some recent results on the arithmetic of the theta correspondence for certain symplectic-orthogonal dual pairs and some applications to periods and congruences of modular forms. We also propose an integral version of a conjecture on Petersson inner products of modular forms on quaternion algebras over totally real fields.

Conditions for the existence of a gyroscope in spaces with affine connections and metrics are found. They appear as special types of Fermi-Walker transports for vector fields, lying in a subspace, orthogonal to the velocity vector field (a non-null contravariant vector field) of an observer.