ON THE SPECTRUM OF THE TWO-PARTICLE SCHRÖDINGER OPERATOR WITH POINT POTENTIAL: ONE DIMENTIONAL CASE

In the paper a one-dimensional two-particle quantum system interacted by two identical point interactions is considered. The corresponding Schr\"{o}\-dinger operator (energy operator) $h_\varepsilon$ depending on $\varepsilon,$ is constructed as a self-adjoint extension of the symmetric Laplace operator. The main results of the work are based to the study of the operator $h_\varepsilon.$ First the essential spectrum is described. The existence of unique negative eigenvalue of the Schr\"{o}dinger operator is proved. Further, this eigenvalue and corresponding eigenfunction are found.

Theoretical Designing of Novel Donor Acceptor Type Copolymer Comprising of Thiophene

Using ab initio band structure results of three novel donor acceptor polymers (A)x PCDT, (B)x PMCT and (C)x PFTh as the input, the electronic structures and conduction properties of their periodic and aperiodic copolymer (AmBnCk)x have been investigated. The method involves using negative factor counting method based on Dean’s negative eigenvalue theorem. In this article, the quasi-one-dimensional Type II staggered copolymers comprising of thiophene units on the basis of the band alignments of the constituent homopolymers were studied. The trends in their electronic structures and conduction properties as a function of (i) block sizes (m, n, k) and (ii) arrangement of the blocks (periodic or aperiodic) in the various copolymer chains are discussed. These trends are important guidelines to the experimentalists for designing novel electrically conducting polymers with tailor made conduction properties.

A vortex identification method based on strain and enstrophy production invariants

A new vortex identification method is proposed for extracting vortical structures from homogeneous isotropic turbulence. The method is compared with other identification schemes such as the high rotational method ([Formula: see text]), the vorticity magnitude method ([Formula: see text]), the negative eigenvalue method ([Formula: see text]) and the normalized vorticity method ([Formula: see text]). A new normalization method based on the probability distribution function (PDF) of the identification invariants is also introduced. In addition, a modification for the discriminant criterion known as the [Formula: see text] method is carried out and it is denoted as the modified delta method ([Formula: see text]). The velocity of the isotropic turbulent field is simulated using the lattice Boltzmann method with resolution [Formula: see text]. The new identification method depends on the higher-orders of the invariants of the velocity gradient tensor as well as the strain rate and the enstrophy production terms. The elongated tube-like vortices are extracted successfully using the new method and several features of the vortices are demonstrated and compared with the vortical structures that are extracted using the [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] identification methods. The recommended normalization method enabled the justification of the visualization threshold value to be within the order of unity and the threshold value [Formula: see text] is used in all identification methods. A remarkably similar geometrical worm-like vortices are extracted and a high similarity between the identification methods is observed and statistically studied.

Gradient Descent with Identity Initialization Efficiently Learns Positive-Definite Linear Transformations by Deep Residual Networks

We analyze algorithms for approximating a function [Formula: see text] mapping [Formula: see text] to [Formula: see text] using deep linear neural networks, that is, that learn a function [Formula: see text] parameterized by matrices [Formula: see text] and defined by [Formula: see text]. We focus on algorithms that learn through gradient descent on the population quadratic loss in the case that the distribution over the inputs is isotropic. We provide polynomial bounds on the number of iterations for gradient descent to approximate the least-squares matrix [Formula: see text], in the case where the initial hypothesis [Formula: see text] has excess loss bounded by a small enough constant. We also show that gradient descent fails to converge for [Formula: see text] whose distance from the identity is a larger constant, and we show that some forms of regularization toward the identity in each layer do not help. If [Formula: see text] is symmetric positive definite, we show that an algorithm that initializes [Formula: see text] learns an [Formula: see text]-approximation of [Formula: see text] using a number of updates polynomial in [Formula: see text], the condition number of [Formula: see text], and [Formula: see text]. In contrast, we show that if the least-squares matrix [Formula: see text] is symmetric and has a negative eigenvalue, then all members of a class of algorithms that perform gradient descent with identity initialization, and optionally regularize toward the identity in each layer, fail to converge. We analyze an algorithm for the case that [Formula: see text] satisfies [Formula: see text] for all [Formula: see text] but may not be symmetric. This algorithm uses two regularizers: one that maintains the invariant [Formula: see text] for all [Formula: see text] and the other that “balances” [Formula: see text] so that they have the same singular values.

A NOTE ON VARIANTS OF ZERO FORCING

A small improvement is made to the zero-forcing variants defined by Butler, Grout, and Hall (2015) for matrices with a given number of negative eigenvalues, resulting in a better value for the Barioli-Fallat tree and one negative eigenvalue.

The non-convex geometry of low-rank matrix optimization

This work considers two popular minimization problems: (i) the minimization of a general convex function f(X) with the domain being positive semi-definite matrices, and (ii) the minimization of a general convex function f(X) regularized by the matrix nuclear norm $\|X\|_{*}$ with the domain being general matrices. Despite their optimal statistical performance in the literature, these two optimization problems have a high computational complexity even when solved using tailored fast convex solvers. To develop faster and more scalable algorithms, we follow the proposal of Burer and Monteiro to factor the low-rank variable $X = UU^{\top } $ (for semi-definite matrices) or $X=UV^{\top } $ (for general matrices) and also replace the nuclear norm $\|X\|_{*}$ with $\big(\|U\|_{F}^{2}+\|V\|_{F}^{2}\big)/2$. In spite of the non-convexity of the resulting factored formulations, we prove that each critical point either corresponds to the global optimum of the original convex problems or is a strict saddle where the Hessian matrix has a strictly negative eigenvalue. Such a nice geometric structure of the factored formulations allows many local-search algorithms to find a global optimizer even with random initializations.

Effect of the eigenvalues of the velocity gradient tensor on particle collisions

This study uses the eigenvalues of the local velocity gradient tensor to categorize the local flow structures in incompressible turbulent flows into different types of saddle nodes and vortices and investigates their effect on the local collision kernel of heavy particles. Direct numerical simulation (DNS) results show that most of the collisions occur in converging regions with real and negative eigenvalues. Those regions are associated not only with a stronger preferential clustering of particles, but also with a relatively higher collision kernel. To better understand the DNS results, a conceptual framework is developed to compute the collision kernel of individual flow structures. Converging regions, where two out of three eigenvalues are negative, posses a very high collision kernel, as long as a critical amount of rotation is not exceeded. Diverging regions, where two out of three eigenvalues are positive, have a very low collision kernel, which is governed by the third and negative eigenvalue. This model is not suited for particles with Stokes number $St\gg 1$, where the contribution of particle collisions from caustics is dominant.

Dispersive reservoir influence on the superconducting phase qubit

An analytical description of a superconducting (SC) phase qubit coupled to a torsional resonator, which is damped by a dispersive reservoir, is presented based on the master equation. Therefore, the effect of the qubit phase damping on the dynamical behavior of the entanglement, purity loss and qubit inversion are investigated. It is found that the collapse and revival phenomena of qubit inversion are very sensitive not only to the damping parameter but also to the frequency detuning and the qubit distribution angle of the initial state. It is interesting to note that the purity of the state of the SC-qubit, which is measured by von Neumann entropy, can be completely lost due to the dispersive reservoir parameter. Because of the existence of dispersive reservoir, the von Neumann entropy cannot be a measure for the entanglement in open system. So, the negative eigenvalue of the partially transposed density matrix of qubit-resonator system is used to quantify the entanglement. For certain parameter sets, it is possible to control the degree and the dynamics of entanglement between the qubit and the torsional resonator.