on how we solve problems in applied physics. Recently published extended or gamma sine function sin∗(α, γ)

and cosine function cos∗(α, γ) — along with their upgraded identity angle sum and subtraction rules sin∗(A ± B, γ) and

cos∗(A ± B, γ) — have enabled a new approach on how to tackle practical problems using mathematics (a published example

is the energy-coupled mass-spring oscilatory system). The usefullness of a theory is measured by both the insight

it generates, and the solution it produces, when applied to physical problems with pertinent applications. Its acceptance

amongst peers depends on the availability of such examples, as way-showers of how the theory can be applied in practice,

and how useful results can be derived by employing it in similar or related examples/problems. This article has the

purpose of providing this bridge between the above theories and its application in some common scientific fields. Several

exercises are solved employing these new formulae, and new potential applications are identified that cover various topics

in physics such as civil engineering (i.e., measuring distances in bridges), aerospace and aeronautics (i.e., turbine velocity

triangles and optimum orbital deployment for a satellite constellation) and telecommunications (i.e., antenna array

beamforming and steering, as well as new modulations based on quadrature phase-shift keying). These problems (and

solutions) are designed to indicate the usefullness of these new expanded functions, and can become practical classroom

exercises applicable to both academic and professional environments.]]>

Model (SM) as for example, looking for the dark photon with Feynman diagrams, in the process γγ −→ e+e , is

still inconclusive (Xu, I. et al., 2022). However, empirical-like methods can give the proof about the existence of dark

matter, see for instance (Clowe, D. et al., 2006). Hence it is reasonable trying to understand as to why ordinary and novel

(dm) particles differ so much from each other. This we wish to do with solutions of the bicubic equation for particle

limiting velocities ( ˇ Soln, J., 2014-2022). Once we have the solutions for novel and ordinary particle limiting velocities

from ( ˇ Soln, J. 2021.1.2, 2022), we first establish, with the help of evolutionary congruent parameters, ordinary z1 and

novel z2, satisfying z1 ⪯ 1and z2 ⪰ 1,the smooth matching point of equal values for ordinary and novel particles at

z1 = z2 = 1. At this point the limiting velocities and other physical quantities of ordinary and novel particles have

equal values, which can be also characterized by z1× z2 = 1; this, consistent with Discriminants of ordinary and novel

limiting velocity solutions, is extended everywhere, so that z2 = 1/ z1. the novel particle limiting velocity solutions

reveal congruent angle α, contained now in z1 and z2, and as such can also serve as another evolutionary parameter. The

smooth matching point is now α = π/2. If physically equivalent ordinary and novel particles move away from this point

to α ̸= π/2, they will physically be different from each other. In other words, the novel particle is in z2 ⪰ 1 territory, and

the ordinary particle is in z1 ⪯ 1 territory and direct interactions are likely impossible. With this formalism, we investigate

physical differences between ordinary and novel particles, when moving away from α = π/2. In tis article, we largely

are dealing with high energy leptons together with relevant photons with congruent parameter ranges of 0 ≺ α ⪯ π/2,

0 ≺ z1 ⪯ 1,∞≻ z2 ⪰ 1. In fact due to a large interest in photons, here, within this formalism, we evaluate very precisely

limiting velocities for the ordinary and novel photons. From these evaluations, we deduce numerically that congruent angles

of novel and ordinary photons are related through the quantum jump α(γN) = 2α(γ), which is verified also for other

particles. Hence, the general quantum jump between congruent angles of limiting velocities associated with ordinary and

novel particles is α(xN) = α(x), where x = γ, e, ν, etc. The congruent angle quantum jump connects every ordinary

particle, such as electron e, or neutrino ν,respectively, to novel electron eN and novel neutrino νN. This, definitively is a

rather simple way to identify novel particles. All that one needs is to find them.]]>

in multi-electron Rydberg states, such as the Rydberg states of group II elements in the periodic table. It often leads to

autoionization due to the fact that a discrete state couples with an autoionizing state. Unlike the typical asymmetric line

shape, here we report a symmetric unusual Fano line shape that has been predicted. In addition, this profile is caused

by molecular states or polymer states instead of multi-electron atomic states. Moreover, it is shown that the fano profile

caused by the multi-electron polymer states does not necessarily lead to autoionization. We propose that the Fano profile

studied is caused by the interference between discrete levels and the multipole-multipole coupled continuum, or energy

band.]]>