We present an introduction to the mathematics of quantum physics and quantum computation which put emphasis on the basic mathematical aspects of definition and operations on qubits. We start by a comprehensive introduction of a qubit as a unit element of \( \mathbb{C}^2 \), and its representations on spheres in \( \mathbb{R}^3 \). This introduction leads to the interpretation of Pauli operators as basic rotations in \( \mathbb{R}^3 \). Then we study unitary operators. Their link to rotations in \( \mathbb{R}^3 \) is established using the density operator associated to a qubit. We complete this paper by some decomposition, or splitting, problems of unitary operators on \( \mathbb{C}^2 \) based on decomposition results of rotations in \( \mathbb{R}^3 \). These decomposition results are useful for the construction of quantum gates.
In this research article, the authors introduce the refinements of some special inequalities, like Lah-Ribarič type, Giaccardi, and Petrović’s inequalities. Also, the authors define Fejér, Giaccardi, and Petrović’s types of inequalities for different classes of convex functions.
In this study, an approximate solution of the Sitnikov problem was investigated using fourth-order Runge – Kutta method. We confirmed the periodicity and the symmetric nature of the orbits. The various values of eccentricities were obtained which showed that at eccentricity e = 0, the orbit moves in a circular shape and otherwise when e < 0. Also at every values of e, we found the numerical results which we demonstrated by simulations using MATCAD which showed that the range for the search of eccentricities can be narrowed down at different values of e, different sinusoidal frequencies were obtained.
Predation models have come close to modelling dynamic and complex economic factors despite its simplistic criticism. Based on Goodwin’s predator-prey framework, this study modelled the dynamics between employment rate and wage share of Ghana’s national output. Empirical data simulations revealed clear cyclical patterns in employment rates and wage shares, reflecting the dynamics in Goodwin’s class struggle theory. The employment rate and wage share exhibited a symbiotic relationship, where changes in one variable significantly influenced the other. The analysis further revealed that although both employment rate and wage share periodically declined, these variables were never annihilated indicative that the economy was resilient. Sensitivity analysis also demonstrated the robustness of the model, showing consistent patterns despite variations in initial conditions. After subjecting the model to stability test, the study showed that despite the economic fluctuations during the study period, the economy was generally stable mathematically, with a projected economic growth assured.
It is well known that positive Green’s operators are not necessarily positivity preserving. This result is important, because many physical problems require positivity in their solutions in order to make sense. In this paper we investigate the matter of just how far from being positivity preserving a positive Green’s operator can be. In particular, we will see that there exists positive Green’s operators that takes some positive functions to functions with negative mean values. We will also identify a broad class of Green’s operators that are not necessarily positivity preserving but have properties related to positivity preservation that one expects from positivity preserving Green’s operators. Finally, we will compare the results contained in this paper with those that already exist in the literature on the subject.
Using the \(q\)-Jackson integral and some elements of the \(q\)-harmonic analysis associated with the generalized q-Bessel operator for fixed \(0<q<1\), we introduce the generalized q-Bessel multiplier operators and we give some new results related to these operators as Plancherel’s, Calderón’s reproducing formulas and Heisenberg’s, Donoho-Stark’s uncertainty principles. Next, using the theory of reproducing kernels we give best estimates and an integral representation of the extremal functions related to these operators on weighted Sobolev spaces.
