CP nonconservation in the leptonic sector

In this paper we use an exact method to impose unitarity on moduli of neutrino PMNS matrix recently determined, and show how one could obtain information on CP nonconservation from a limited experimental information. One suggests a novel type of global fit by expressing all theoretical quantities in terms of convention independent parameters: the Jarlskog invariant $J$ and the moduli $|U_{\alpha i}|$, able to resolve the positivity problem of $|U_{e 3}|$. In this way the fit will directly provide a value for $J$, and if it is different from zero it will prove the existence of CP violation in the available experimental data. If the best fit result, $|U_{e3}|^2<0$, from M. Maltoni {\em et al}, [New J.Phys. {\bf 6} (2004) 122] is confirmed, it will imply a new physics in the leptonic sector

On quaternary complex Hadamard matrices of small orders

One of the main goals of design theory is to classify, characterize and count various combinatorial objects with some prescribed properties. In most cases, however, one quickly encounters a combinatorial explosion and even if the complete enumeration of the objects is possible, there is no apparent way how to study them in details, store them efficiently, or generate a particular one rapidly. In this paper we propose a novel method to deal with these difficulties, and illustrate it by presenting the classification of quaternary complex Hadamard matrices up to order 8. The obtained matrices are members of only a handful of parametric families, and each inequivalent matrix, up to transposition, can be identified through its fingerprint.Comment: 7 page

Exotic complex Hadamard matrices, and their equivalence

In this paper we use a design theoretical approach to construct new, previously unknown complex Hadamard matrices. Our methods generalize and extend the earlier results of de la Harpe--Jones and Munemasa--Watatani and offer a theoretical explanation for the existence of some sporadic examples of complex Hadamard matrices in the existing literature. As it is increasingly difficult to distinguish inequivalent matrices from each other, we propose a new invariant, the fingerprint of complex Hadamard matrices. As a side result, we refute a conjecture of Koukouvinos et al. on (n-8)x(n-8) minors of real Hadamard matrices.Comment: 10 pages. To appear in Cryptography and Communications: Discrete Structures, Boolean Functions and Sequence

Developed Adomian method for quadratic Kaluza-Klein relativity

We develop and modify the Adomian decomposition method (ADecM) to work for a new type of nonlinear matrix differential equations (MDE's) which arise in general relativity (GR) and possibly in other applications. The approach consists in modifying both the ADecM linear operator with highest order derivative and ADecM polynomials. We specialize in the case of a 4$\times$4 nonlinear MDE along with a scalar one describing stationary cylindrically symmetric metrics in quadratic 5-dimensional GR, derive some of their properties using ADecM and construct the \textit{most general unique power series solutions}. However, because of the constraint imposed on the MDE by the scalar one, the series solutions terminate in closed forms exhausting all possible solutions.Comment: 17 pages (minor changes in reference [30]

On two subgroups of U(n), useful for quantum computing

As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit Phi(theta) and the 1-qubit NEGATOR circuit N(theta). Both are roots of the IDENTITY circuit. Indeed: both (NO) and N(0) equal the 2 x 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2(w)); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2(w)). Here, ZU(n) and XU(n) are subgroups of the unitary group U(n): the group XU(n) consists of all n x n unitary matrices with all 2n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU(n) consists of all n x n unitary diagonal matrices with first entry equal to 1. Any U(n) matrix can be decomposed into four parts: U = exp(i alpha) Z(1)XZ(2), where both Z(1) and Z(2) are ZU(n) matrices and X is an XU(n) matrix. We give an algorithm to find the decomposition. For n = 2(w) it leads to a four-block synthesis of an arbitrary quantum computer