Implicit Solitary Waves for One of the Generalized Nonlinear Schrödinger Equations

Application of transformations for dependent and independent variables is used for finding solitary wave solutions of the generalized Schrödinger equations. This new form of equation can be considered as the model for the description of propagation pulse in a nonlinear optics. The method for finding solutions of equation is given in the general case. Solitary waves of equation are obtained as implicit function taking into account the transformation of variables.

Development of a Multivariate Model Focused on the Analysis of Water Availability in Mexico

The present study proposes developing a multivariate model that predicts water availability in Mexico through 26 variables related to aquifers, renewable water, demographic characteristics, rivers and basins, dams, and irrigation factors. The information inherent to them was extracted from the platform of the national water system using records from the 13 administrative hydrological regions between 2010 and 2017. The model is based on the multiple linear regression model and the variable selection method. The results show different versions of the model contrasted concerning the statistical assumptions of the multiple regression. Although the findings presented have implications in the development of strategies focused on a better distribution of the vital liquid, in the face of various projected scenarios based on the variables analyzed, it should be noted that the progressive improvement of the model was carried out through the use of techniques such as the transformation of variables, detection, and elimination of outliers. The final result is water availability in the face of various drought conditions explained by a model with 16 relevant variables. Said prediction model is helpful for the generation of drought mitigation strategies.

Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + d . Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

A Numeric Solution of the Quantum Rigid Rotor

The Variable Fixed Grid Method (VFGM) combines numerical techniques of differentiation and integration to solve the Schrödinger equation in its integral form. The method is extremely simple and allows to overcome difficulties found in conventional resolutions and when coupled equations are produced. However, difficulties arise when curvilinear coordinate systems are used because there are no well-defined boundary conditions for the angular functions. Systematic numerical alternatives were performed using the rigid rotor model. A transformation of variables proved to be efficient for calculating energy and the wave function. The difficulties are located at the poles where the Jacobian of the curvilinear coordinate system is canceled. The method used in this work allows a greater application of VFGM for systems in different curvilinear coordinate systems.

Wall-Crossings for Hassett Descendant Potentials

This paper solves the combinatorics relating the intersection theory of $\psi $-classes of Hassett spaces to that of $\overline{{\mathcal{M}}}_{g,n}$. A generating function for intersection numbers of $\psi $-classes on all Hassett spaces is obtained from the Gromov–Witten potential of a point via a non-invertible transformation of variables. When restricting to diagonal weights, the changes of variables are invertible and explicitly described as polynomial functions. Finally, the comparison of potentials is extended to the level of cycles: the pinwheel cycle potential, a generating function for tautological classes of rational tail type on $\overline{{\mathcal{M}}}_{g,n}$ is the right instrument to describe the pull-back to $\overline{{\mathcal{M}}}_{g,n}$ of all monomials of $\psi $-classes on Hassett spaces.

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

Stability analysis of an interacting holographic dark energy model

This work deals with dynamical system analysis of Holographic Dark Energy (HDE) cosmological model with different infra-red (IR)-cutoff. By suitable transformation of variables, the Einstein field equations are converted to an autonomous system. The critical points are determined and the stability of the equilibrium points are examined by Center Manifold Theory and Lyapunov function method. Possible bifurcation scenarios have also been explained.

Nonlinear Transformations in Organizational Research: Possible Problems and Potential Solutions

We examined the use of nonlinear transformation of variables in a random sample of 323 articles published in six top journals during 2012-2017. Coding categories included the number of transformed variables, the type of transformation, the kinds of variables transformed, reasons provided for transforming variables, how transformed results were reported, and pre- and posttransformation analysis of variables. Common problems include insufficient justification for transforming variables, overreliance on log transformations, failure to report important information on the effects of transformation, and incomplete reporting and discussion of transformed results. Perhaps most importantly, there was frequent misalignment between statements of hypotheses, typically stated in terms of nontransformed variables, and the transformed data used to test them. We discuss the implications of these problems for science and practice, offer recommendations for addressing the issues, and provide illustrations of how to implement the recommendations.