On scalar products and form factors by Separation of Variables: the antiperiodic XXZ model

We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by Separation of Variables (SoV), and the eigenstates can be constructed in terms of Q-functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q-functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues.

Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

Implementation of a Stable Solar-Powered Microgrid Testbed for Remote Applications

An intrinsically stable microgrid, operated by inverter-interfaced distributed energy resources (I-DERs) is introduced in this paper. The microgrid is built upon a systematic design method, which is adapted from the operation of the synchronous machine (SM). The proposed method analogizes the dynamics of the dc-link in I-DERs to the rotor dynamics in synchronous generators (SGs) and utilizes the capacitor as energy storage. Thus, the proposed mechanism relaxes battery usage for frequency control, and by using the capacitive stored energy, provides a high fault ride-through capability, which is suitable for both on-grid and off-grid applications. Based on stability analysis of the SG and the dynamic state matrix eigenvalues for multimachine power system, the dc-link capacitor of I-DERs is characterized in the context of microgrid. The dc-link capacitor stores kinetic energy similar to the rotor of the SG and provides inertia in transients without the need of battery storage. The inverter angle responds to the change of the dc link voltage (energy). The dc-link voltage is then controlled similar to the field control pertaining to the SG. Finally, a governor-like mechanism is applied to maintain dc-link voltage stability. Simulation and experimental results are provided to show the effectiveness of the proposed design mechanisms.

Statistical Research of Intersection of Technology and Entrepreneurship to Modern Practices, Policies and Promises

There are many factors affecting the relationship between entrepreneurship and technology which has not been examined and studied and this has been a daunting problem for researchers in this area. This paper seeks to identify a number of factors that deal with technology and entrepreneurship with a view to understanding the inter-correlation among the identified factors thereby making us to know the intersection between them. It will help provide an overview of the state of the art in terms of technology and offers fresh insights for entrepreneurship policy for technology. This study employed Kendall’s Coefficient of Concordance (KCC) to rank the 32 identified variables and subsequently apply Principal Component Analysis (PCA. KCC was used to rank 32 identified variables in descending order of importance. Furthermore, the PCA was used to analyze a set of questionnaire crafted with the 32 variables and administered to knowledgeable respondents in the area. The outputs gotten from the statistical software include descriptive statistic, correlation matrix, eigenvalues, eigenvector, varimax rotated factor loadings, explained variance and factor plot, among others and thereafter interpretation was given. Result obtained unveiled five principal factors which were labeled creatively. Results obtained by KCC suggested that judges ranking were consistent. Also, PCA was indicating parsimony in data reduction from 32 variables to just five. The most influential variable by its factor loading of 0.954 is innovation. The import of this is that innovation which has the highest factor loading is the nexus between technology and entrepreneurship and should therefore be embraced.

Iterated partial summations applied to finite-support discrete distributions

The problem of iterated partial summations is solved for some discrete distributions defined on finite supports. The power method, usually used as a computational approach to the problem of finding matrix eigenvalues and eigenvectors, is in some cases an effective tool to prove the existence of the limit distribution, which is then expressed as a solution of a system of linear equations. Some examples are presented.