Narrow Results

We present the bosonic operator Hamiltonian models of two Josephson junctions which are connected in series and in parallel, respectively. By introducing the entangled state representations (|η〉 representation and Cooper pairs representation) and projecting the operator Hamiltonian onto the |η〉 representation we obtain the correct Josephson current and phase equations consistent with the classical Hamiltonian cases.

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e. looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.

Improving the ability of nonlinear energy sink (NES) to suppress large vibrations under strong excitation has been widely concerned. Moreover, distributing NES to suppress multimodal resonances is also an issue that deserves more attention. This paper investigates the distribution strategy of the series NES in suppressing the vibration of a two-degree-of-freedom (2-DOF) system. Dynamic models are developed for two strategies: centralized and distributed series NES. The vibration reduction efficiency of these strategies is compared, and the approximate results obtained from the analytical analysis were verified using numerical methods. The research results indicate that distributing the series NES among different vibration sources can lead to improved vibration reduction performance. Moreover, the distributed vibration reduction strategy exhibits greater adaptability to different modes of primary system. Furthermore, while keeping the total additional weight of the NES unchanged, the effects of the series NES and 1DOF NES on vibration control were compared. The results demonstrate that, under large amplitude excitation, the series NES has a better vibration reduction effect than the 1DOF NES when using distributed NES to control the vibration of a 2-DOF system. Additionally, by studying the energy transfer between NES oscillators in series, the series NES with high stiffness asymmetry exhibits superior vibration reduction. The findings of this study contribute to the advancement of research on the NES control of strong excitation vibration and multi-mode structure vibration.

For n ≥ 1, let p_n be the nth prime number and for n ≥ 1. Using several results of Erdős, we study the sequence (q_n)_{n ≥ 1} and we prove similar results as for the sequence (d_n)_{n ≥ 1}, d_n = p_n+1 - p_n. We also consider the sequence for n ≥ 1 and denote by G_n and A_n its geometrical and arithmetical averages. We prove that for n ≥ 4.

The existing rate of convergence of series requires a known limit of that series. Direct computation of Brun’s constant as the sum of twin prime reciprocals is challenging as the convergence is agonizingly slow. Here, I introduce the relative increase of the partial sum of a series as a new rate of convergence regardless of the limit of the series. With this concept, I extract a slowly convergent part from the original sum for Brun’s constant so that the remaining one converges faster. Computed data demonstrate the usefulness of the concept. As a result, I obtain an improved lower bound for Brun’s constant using less computing resource and establish an approximate mapping that gives a new estimate for the constant at 1.89558413.