Gauge invariant grid discretization of Schr\"odinger equation

Using the Wilson formulation of lattice gauge theories, a gauge invariant grid discretization of a one-particle Hamiltonian in the presence of an external electromagnetic field is proposed. This Hamiltonian is compared both with that obtained by a straightforward discretization of the continuous Hamiltonian by means of balanced difference methods, and with a tight-binding Hamiltonian. The proposed Hamiltonian and the balanced difference one are used to compute the energy spectrum of a charged particle in a two-dimensional parabolic potential in the presence of a perpendicular, constant magnetic field. With this example we point out how a "naive" discretization gives rise to an explicit breaking of the gauge invariance and to large errors in the computed eigenvalues and corresponding probability densities; in particular, the error on the eigenfunctions may lead to very poor estimates of the mean values of some relevant physical quantities on the corresponding states. On the contrary, the proposed discretized Hamiltonian allows a reliable computation of both the energy spectrum and the probability densities.Comment: 7 pages, 4 figures, discussion about tight-binding Hamiltonians adde

Black rings in more than five dimensions

We construct balanced black ring solutions in $d\geq 6$ spacetime dimensions, by solving the Einstein field equations numerically with suitable boundary conditions. The black ring solutions have a regular event horizon with $S^1\times S^{d-3}$ topology, and they approach the Minkowski background asymptotically. We analyze their global and horizon properties. The Smarr formula is well satisfied.Comment: 8 pages, 3 figure

Structural Equation Modeling and simultaneous clustering through the Partial Least Squares algorithm

The identification of different homogeneous groups of observations and their appropriate analysis in PLS-SEM has become a critical issue in many appli- cation fields. Usually, both SEM and PLS-SEM assume the homogeneity of all units on which the model is estimated, and approaches of segmentation present in literature, consist in estimating separate models for each segments of statistical units, which have been obtained either by assigning the units to segments a priori defined. However, these approaches are not fully accept- able because no causal structure among the variables is postulated. In other words, a modeling approach should be used, where the obtained clusters are homogeneous with respect to the structural causal relationships. In this paper, a new methodology for simultaneous non-hierarchical clus- tering and PLS-SEM is proposed. This methodology is motivated by the fact that the sequential approach of applying first SEM or PLS-SEM and second the clustering algorithm such as K-means on the latent scores of the SEM/PLS-SEM may fail to find the correct clustering structure existing in the data. A simulation study and an application on real data are included to evaluate the performance of the proposed methodology

Explicit examples of DIM constraints for network matrix models

Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.Comment: 46 page