RELATIVISTIC CALCULATION OF WAVELENGTHS AND E1 OSCILLATOR STRENGTHS IN Li-LIKE MULTICHARGED IONS AND GAUGE INVARIANCE PRINCIPLE

The spectral wavelengths and oscillator strengths for 1s22s (2S1/2) → 1s23p (2P1/2) transitions in the Li-like multicharged ions with the nuclear charge Z=28,30 are calculated on the basis of the combined relativistic energy approach and relativistic many-body perturbation theory with the zeroth order optimized Dirac-Kohn-Sham one-particle approximation and gauge invariance principle performance. The comparison of the obtained results with available theoretical and experimental (compilated) data is performed. The important point is linked with an accurate accounting for the complex exchange-correlation (polarization) effect contributions and using the optimized one-quasiparticle representation in the relativistic many-body perturbation theory zeroth order that significantly provides a physically reasonable agreement between theory and precise experiment.

Schrödinger operator with decreasing potential in a cylinder

The Schrödinger operator − Δ + V ( x , y ) -\Delta + V(x,y) is considered in a cylinder R m × U \mathbb {R}^m \times U , where U U is a bounded domain in R d \mathbb {R}^d . The spectrum of such an operator is studied under the assumption that the potential decreases in longitudinal variables, | V ( x , y ) | ≤ C ⟨ x ⟩ − ρ |V(x,y)| \le C \langle x\rangle ^{-\rho } . If ρ > 1 \rho > 1 , then the wave operators exist and are complete; the Birman invariance principle and the limiting absorption principle hold true; the absolute continuous spectrum fills the semiaxis; the singular continuous spectrum is empty; the eigenvalues can accumulate to the thresholds only.

On the weak invariance principle for non-adapted stationary random fields under projective criteria

In this paper, we study the central limit theorem (CLT) and its weak invariance principle (WIP) for sums of stationary random fields non-necessarily adapted, under different normalizations. To do so, we first state sufficient conditions for the validity of a suitable ortho-martingale approximation. Then, with the help of this approximation, we derive projective criteria under which the CLT as well as the WIP hold. These projective criteria are in the spirit of Hannan’s condition and are well adapted to linear random fields with ortho-martingale innovations and which exhibit long memory.

Global Stability for Novel Complicated SIR Epidemic Models with the Nonlinear Recovery Rate and Transfer from Being Infectious to Being Susceptible to Analyze the Transmission of COVID-19

Epidemiological models play pivotal roles in predicting, anticipating, understanding, and controlling present and future epidemics. The dynamics of infectious diseases is complex, and therefore, researchers need to consider more complicated mathematical models. In this paper, we first describe the dynamics of a complex SIR epidemic model with nonstandard nonlinear incidence and recovery rates. In this model, we consider the rate at which individuals lose immunity. Rigorous mathematical results have been established from the point of view of stability and bifurcation. The basic reproduction number ( R 0 ) is determined. We then apply LaSalle’s invariance principle and Lyapunov’s direct method to prove that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 . The model has a unique endemic equilibrium when R 0 > 1 . A nonlinear Lyapunov function is used together with LaSalle’s invariance principle to show that the endemic equilibrium is globally asymptotically stable under some conditions. Further, for the case when R 0 = 1 , we analyze the model and show a backward bifurcation under certain conditions. In the second part of this paper, we analyze a modified SIR model with a vaccination term, which must be a function of time. We show that the modified model agrees well with COVID-19 data in Saudi Arabia. We then investigate different future scenarios. Simulation results suggest that a two-pronged strategy is crucial to control the COVID-19 pandemic in Saudi Arabia.

Higher Regularity, Inverse and Polyadic Semigroups

We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents.

A mathematical analysis of the dynamics of chikungunya virus transmission

In this paper, a deterministic model for the dynamics of chikungunya virus transmission is formulated and analyzed. It is shown that the model has a disease free equilibrium (DFE) and by using the basic reprodution number (R0) local stability of DFE is proved when R0 < 1. Also, the global stability of DFE is investigated by Lyapunov function and LaSalle Invariance Principle. We show that there exists a unique endemic equilibrium (EE) of the model which is locally asymptotically stable whenever R0 > 1 and establish the global stability of the EE when R0 > 1, by using Lyapunov function and LaSalle Invariance Principle for a special case. Numerical simulations and sensitivity analysis show that the destruction of breeding sites and reduction of average life spans of vector would be effective prevention to control the outbreak. Controlling of effective contact rates and reducing transmissions probabilities may reduce the disease prevalence. GANITJ. Bangladesh Math. Soc.41.1 (2021) 41-61