The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared for its importance with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science.-Clerk Maxwell, 1869. Quaternions came from Hamil...

1.1. The matrix Ξkk′ In the main text, we are given two unit vectors μ1k and μ2k′ in R. We define Ξkk′ = Ξ(μ1k, μ2k′), where Ξ(u, v) ∈ R4×4 is defined by u (q ◦ v) = qΞ(u, v)q, where u = (ui, uj , uk), v = (vi, vj , vk), and q = (qi, qj , qk, qr). By standard quaternion rotation formula, we have u (q ◦ v) = ui uj uk T 1− 2q j − 2q k 2(qiqj − qkqr) 2(qiqk + qjqr) 2(qiqj + qkqr) 1− 2q i − 2...

Let Q be a smooth two-dimensional surface in R, and let a be a point on Q. We consider the Fourier transform of a small portion of the surface near a, localized using a smooth bump function supported near a. After a translation and rotation, without loss of generality we may take a = (0, 0, 0) and assume that (0, 0, 1) is normal to Q at the origin. In this situation, we are looking at the follo...

The Painlev\'e equations possess transcendental solutions $y(t)$ with special initial values that are symmetric under rotation or reflection in the complex $t$-plane. They correspond to monodromy problems explicitly solvable terms of classical functions. In this paper, we show existence such for a $q$-difference equation. We focus on equation known as $q\textrm{P}_{\textrm{IV}}$ $q{\rm P}(A_5^{...

Given a database D of three dimensional (3D) graphs and a query graph Q, the problem of substructure search is deened as nding the graphs in D that contain Q. This is an important search operation in scientiic databases. This paper extends the search operation to nd those graphs D in D that \approximately" contain Q in the presence of rotation , translation, distortion, and node insert/delete i...

Transfer of molecular property tensors (force field, dipole derivatives, polarizabilities, etc.) from smaller fragments to bigger molecules is powerful tool to calculate molecular vibrational spectra. However, we found serious accuracy limits for valinomycin (Phys. Chem. Chem. Phys. 2010, 12, 11021), where the transfer of the Raman optical activity tensors (ROA) had to be avoided. Therefore, in...

In recent years qubit designs such as transmons approached the fidelities of up to 0.999. However, even these devices are still insufficient for realizing quantum error correction requiring better than 0.9999 fidelity. Topologically protected superconducting qubits arguably most prospective building a realistic computer they intrinsically from noise and leakage errors that occur in transmons. W...

We deduce the periodicity 8 for the type of Pin and Spin representations of the orthogonal groups O(n) from simple combinatorial properties of the finite Clifford groups generated by the gamma matrices. We also include the case of arbitrary signature O(p, q). The changes in the type of representation can be seen as a rotation in the complex plane. The essential result is that adding a (+) dimen...

Accurate rotation of the femoral component in total knee arthroplasty is important for normal patella tracking, symmetrical patellofemoral joint contact, neutral varus-valgus positioning in flexion, correct rotational alignment of the tibia in extension and the avoidance of anterior femoral notching . Internal rotation of the femoral component causes a shift into valgus alignment with flexion a...

This paper proposes a novel evolutionary algorithm inspired by quantum computing, called a quantum-inspired evolutionary algorithm (QEA), which is based on the concept and principles of quantum computing, such as a quantum bit and superposition of states. Like other evolutionary algorithms, QEA is also characterized by the representation of the individual, the evaluation function, and the popul...