Python app for drawing Bode diagram asymptotes of transfer function for minimum and non-minimal phase systems

The purpose of this article is to introduce an application to draw the asymptotes of Bode diagram module and phase from each constituent elementary factors of any transfer function for minimum and non-minimal phase systems without transport delay. The Bode diagram is the most used tool in the frequency response method. Python was used to program the application to perform the operations as well as the Qt5 Design for the simple graphical interface for the application and all this in the Linux operating system. The application purpose is to assist students in learning the concept and drawing of Bode diagram. For students the non-minimum phase system Bode diagram is more difficult to draw than a minimum phase system due to the presence of zeros and/or poles on right half of s-plane. The phase asymptotes of a quadratic factor was closest to the real phase curve around the corresponding undamped natural frequency and this can be observed in the example showed in this article. This example must be used as a help and not a simply to solve a problem.

A one parameter family of Calabi-Yau manifolds with attractor points of rank two

In the process of studying the ζ-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the ζ-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in ℚ, so independent of any particular prime. Such factorisations are expected to be modular with each quadratic factor associated to a modular form. If the parameter is defined over ℚ this modularity is assured by the proof of the Serre Conjecture. We identify three values of the parameter that give rise to persistent factorisations, one of which is defined over ℚ, and identify, for all three cases, the associated modular groups. We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. The values of the periods and their covariant derivatives, at the attractor points, are identified in terms of critical values of the L-functions of the modular groups. Thus the critical L-values enter into the calculation of physical quantities such as the area of the black hole in the 4D spacetime. In our search for additional rank two attractor points, we perform a statistical analysis of the numerator of the ζ-function and are led to conjecture that the coefficients in this polynomial are distributed according to the statistics of random USp(4) matrices.

Tight Convergence Rate of Gradient Descent for Eigenvalue Computation

Riemannian gradient descent (RGD) is a simple, popular and efficient algorithm for leading eigenvector computation [AMS08]. However, the existing analysis of RGD for eigenproblem is still not tight, which is O(log(n/epsilon)/Delta^2) due to [Xu et al., 2018]. In this paper, we show that RGD in fact converges at rate O(log(n/epsilon)/Delta), and give instances to shows the tightness of our result. This improves the best prior analysis by a quadratic factor. Besides, we also give tight convergence analysis of a deterministic variant of Oja's rule due to [Oja, 1982]. We show that it also enjoys fast convergence rate of O(log(n/epsilon)/Delta). Previous papers only gave asymptotic characterizations [Oja, 1982; Oja, 1989; Yi et al., 2005]. Our tools for proving convergence results include an innovative reduction and chaining technique, and a noisy fixed point iteration argument. Besides, we also give empirical justifications of our convergence rates over synthetic and real data.

A quadratic approach to allometry yields promising results for the study of growth

Julian Huxley (1924) came to the conclusion that intra-specific growth usually follows a sequence of power curves. So Huxley claimed that during growth sudden changes in the growth rate can occur. The restudy of his material, however, reveals that his observations closely follow single quadratic curves. As a result the intra-specific allometry studied by Huxley is comparable to ontogenetic allometry. The quadratic factor of the quadratic equations obtained, represents the growth rate; it shows the constant increase (positive factor) or decrease (minus factor) of one of the measurements for a constant increase in the other measurement with which it is compared. The quadratic factor explains the entire growth process and is the same for the smaller (younger) and larger (older) specimens. It could probably permit the prediction of the shape of larger and/or smaller animals not yet found, or give a clue to some evolutionary changes. By using the quadratic parabola there is no need to postulate “sudden changes in the growth curve” and so it appears that Huxley’s power curve can be abandoned.

Insert a Root to Extract a Root of Quintic Quickly

The usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up, leading to the solutions. In this paper, a relation among the coefficients of a principal quintic equation is established by effecting a change of variable and inserting a root to the quintic equation, and then equating odd-powers of the resulting sextic equation to zero. This leads to an even-powered sextic equation, or equivalently a cubic equation; thus one needs to solve the cubic equation.We break from this tradition, rather factor the even-powered sextic equation in a novel fashion, such that the inserted root is identified quickly along with one root of the quintic equation in a quadratic factor of the form, u2− g2 = (u + g)(u − g). Thus there is no need to solve any cubic equation. As an extra benefit, this root is a function of only one coefficient of the given quintic equation.

A design of experiment aided sensitivity analysis and parameterization for hydrological modeling

To provide a better understanding of the water balance in the Deer River watershed of the Hudson Bay lowlands, the Semi-distributed Land Use-based Runoff Process hydrological model was applied to simulate the runoff over a 20 year period. The purpose of this study is to develop an approach to examine the sensitivity of the ten parameters and their interactions via statistical design of experiment methodology. Using the proposed approach, the contribution of each parameter and how they interact with one another were evaluated. The results indicated that the interaction between “retention constant for fast storage” and “precipitation factor” had the greatest positive impact on the Nash–Sutcliffe efficiency (NSE) and the quadratic factor term of “precipitation factor” had the greatest negative effect on the NSE. The proposed approach provided an effective tool for evaluating the contribution of the input parameters and could also be applied for calibration of other hydrological models.

NORDSIECK METHODS WITH INHERENT QUADRATIC STABILITY

We derive suffcient conditions which guarantee that the stability polynomial of Nordsieck method for ordinary differential equations has only two nonzero roots. Examples of such methods up to order four are presented which are A-and L-stable. These examples were obtained by computer search using the Schurcriterion applied to the quadratic factor of the resulting stability polynomials.