Search Results (4)

The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph $\mathcal{D}$ having at least one doubly adjacent vertex in terms of indegree is proposed. Particularly, it is exhibited that $\rho (\mathcal{D})\ge {\alpha }^j=\sqrt{{\displaystyle \frac{{\sum }_{p=1}^m{({\chi }_{j+1}^{(p)})}^2}{{\sum }_{p=1}^m{({\chi }_j^{(p)})}^2}}}$, such that equality is attained iff $\mathcal{D}=\overleftrightarrow{\mathfrak{G}}+$ {$\mathcal{DE}\notin$ Cycle}, where each component of associated graph is a k-regular or $(k_1,k_2)$ semiregular bipartite. By utilizing the sequence of lower bounds of the spectral radius of $\mathcal{D}$, the sequence of upper bounds of energy of $\mathcal{D}$, where the sequence decreases when $\sqrt{{\displaystyle \frac{\mathbf{e}}{\mathfrak{U}}}}\le {\alpha }^j$ and increases when $\sqrt{{\displaystyle \frac{\mathbf{e}}{\mathfrak{U}}}}>{\alpha }^j$, are also proposed. All of the obtained inequalities are elaborated using examples. We also discuss the monotonicity of these sequences. Full article

Understanding how systems relax to equilibrium is a core theme of statistical physics, especially in economics, where systems are known to be subject to extrinsic noise not included in simple agent-based models. In models of binary choice—ones not much more complicated than Kirman’s model of ant recruitment—such relaxation dynamics become difficult to determine analytically and require solving a three-term recurrence relation in the eigendecomposition of the stochastic process. In this paper, we derive a concise closed-form solution to this linear three-term recurrence relation. Its solution has traditionally relied on cumbersome continued fractions, and we instead employ a linear algebraic approach that leverages the properties of lower-triangular and tridiagonal matrices to express the terms in the recurrence relation using a finite set of orthogonal polynomials. We pay special attention to the power series coefficients of Heun functions, which are also important in fields such as quantum mechanics and general relativity, as well as the binary choice models studied here. We then apply the solution to find equations describing the relaxation to steady-state behavior in social choice models through eigendecomposition. This application showcases the potential of our solution as an off-the-shelf solution to the recurrence that has not previously been reported, allowing for the easy identification of the eigenspectra of one-dimensional, one-step, continuous-time Markov processes. Full article

We employ the Tight Binding Fishbone-Wire Model to study the electronic structure and coherent transfer of a hole (the absence of an electron created by oxidation) in all possible ideal B-DNA dimers as well as in homopolymers (one base pair repeated along the whole sequence with purine on purine). The sites considered are the base pairs and the deoxyriboses, with no backbone disorder. For the time-independent problem, we calculate the eigenspectra and the density of states. For the time-dependent problem after oxidation (i.e., the creation of a hole either at a base pair or at a deoxyribose), we calculate the mean-over-time probabilities to find the hole at each site and establish the frequency content of coherent carrier transfer by computing the Weighted Mean Frequency at each site and the Total Weighted Mean Frequency of a dimer or polymer. We also evaluate the main oscillation frequencies of the dipole moment along the macromolecule axis and the relevant amplitudes. Finally, we focus on the mean transfer rates from an initial site to all others. We study the dependence of these quantities on the number of monomers that are used to construct the polymer. Since the value of the interaction integral between base pairs and deoxyriboses is not well-established, we treat it as a variable and examine its influence on the calculated quantities. Full article

Large scale coherent structures in the atmospheric boundary layer (ABL) are known to contribute to the power generation in wind farms. In order to understand the dynamics of large scale structures, we perform proper orthogonal decomposition (POD) analysis of a finite sized wind turbine array canopy in the current paper. The POD analysis sheds light on the dynamics of large scale coherent modes as well as on the scaling of the eigenspectra in the heterogeneous wind farm. We also propose adapting a novel Fourier-POD (FPOD) modal decomposition which performs POD analysis of spanwise Fourier-transformed velocity. The FPOD methodology helps us in decoupling the length scales in the spanwise and streamwise direction when studying the 3D energetic coherent modes. Additionally, the FPOD eigenspectra also provide deeper insights for understanding the scaling trends of the three-dimensional POD eigenspectra and its convergence, which is inherently tied to turbulent dynamics. Understanding the behaviour of large scale structures in wind farm flows would not only help better assess reduced order models (ROM) for forecasting the flow and power generation but would also play a vital role in improving the decision making abilities in wind farm optimization algorithms in future. Additionally, this study also provides guidance for better understanding of the POD analysis in the turbulence and wind farm community. Full article