Eigenvalues of the complex Laplacian on compact non-Kähler manifolds

We consider \(\lambda \) is the principle eigenvalue of the complex Laplacian on a compact Hermitian manifold M. We prove that \(\lambda \ge C\) where C depends only on the dimension n, the diameter d, the Ricci curvature of the Levi-Civita connection on M, and a norm, expressed in curvature, that determines how much M fails to be Kähler. We first estimate the principal eigenvalue of a drift Laplacian and then study the structure of Hermitian manifolds using recent results due to Yang and Zheng (on curvature tensors of Hermitian manifolds, 2016. arXiv:1602.01189). We combine these results to obtain the main estimate. We also discuss several special cases in which one can obtain a lower bound solely in terms of the Riemannian geometry.     

Depth and Stanley depth of the edge ideals of square paths and square cycles

In this paper, we compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square path on n vertices. We also compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square cycle on n vertices, when n≡0,3,4( mod 5), and give tight bounds when n≡1,2( mod 5). We also prove a conjecture of Herzog presented in [5 Herzog, J. (2013). A survey on Stanley depth. In: Bigatti, M. A., Gimenez, P., Sáenz-de-Cabezón, E., eds. Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, Vol. 2083. Heidelberg: Springer, pp. 3–45. https://arxiv.org/pdf/1702.00781.pdf.[Crossref] , [Google Scholar]], for the edge ideals of square paths and square cycles.     

A new third order exponentially fitted discretization for the solution of non-linear two point boundary value problems on a graded mesh

This paper puts forward a novel graded mesh implicit scheme resting upon full step discretization of order three for computation of non-linear two point boundary value problems. The suggested method is compact and employs three nodal points for the unknown function $u(x)$ in spatial axis. We have also performed error analysis of the cited method. The given method was tried (implemented) upon multiple problems in Cartesian and Polar coordinates with extremely favorable outcomes. This method, though meant for scalar equations, was further extended to compute the vector equations of two point nonlinear boundary value problems. To check the validity of the proposed scheme, we applied it to multiple problems and obtained supporting numerical computations.     

Inverse problems for quasi-variational inequalities

In this short note, our aim is to investigate the inverse problem of parameter identification in quasi-variational inequalities. We develop an abstract nonsmooth regularization approach that subsumes the total variation regularization and permits the identification of discontinuous parameters. We study the inverse problem in an optimization setting using the output-least squares formulation. We prove the existence of a global minimizer and give convergence results for the considered optimization problem. We also discretize the identification problem for quasi-variational inequalities and provide the convergence analysis for the discrete problem. We give an application to the gradient obstacle problem.     

New concepts in neutrosophic graphs with application

A neutrosophic set is a generalization of an intuitionistic fuzzy set. Neutrosophic models give more flexibility, precisions and compatibility to the system as compared to intuitionistic fuzzy models. In this research study, we apply the concept of neutrosophic sets to graphs and discuss certain concepts of single-valued neutrosophic graphs. We illustrate the concepts by several examples. We investigate some interesting properties. We describe an application of single-valued neutrosophic graph in decision making process. We also present the procedure of our proposed method as an algorithm.     

Novel applications of bipolar fuzzy graphs to decision making problems

Zhang introduced the concept of bipolar fuzzy sets as a generalization of fuzzy sets. Bipolar fuzzy sets have shown advantages in solving decision making problems than fuzzy sets. In this research paper, we study several different types of domination, including equitable domination, k-domination and restrained domination in bipolar fuzzy graphs. We present novel applications of bipolar fuzzy graphs to decision making problems. We also present an algorithm for computing dominating number in our applications.     

A hybrid proximal point algorithm for finding minimizers and fixed points in CAT(0) spaces

We introduce a hybrid proximal point algorithm and establish its strong convergence to a common solution of a proximal point of a lower semi-continuous mapping and a fixed point of a demicontractive mapping in the framework of a CAT(0) space. As applications of our new result, we solve variational inequality problems for these mappings on a Hilbert space. Illustrative example is given to validate theoretical result obtained herein.     

Certain Integral Transforms of Generalized k-Bessel Function

The objective of this note is to provide some(potentially useful) integral transforms(for example, Euler, Laplace, Whittaker etc.) associated with the generalized k-Bessel function defined by Saiful and Nisar [3]. We have also discussed some other transforms as special cases of our main results.     

DICE: exploiting <Emphasis Type="Italic">all</Emphasis> bivariate dependencies in binary and multary search spaces

Although some of the earliest Estimation of Distribution Algorithms (EDAs) utilized bivariate marginal distribution models, up to now, all discrete bivariate EDAs had one serious limitation: they were constrained to exploiting only a limited O(d) subset out of all possible \(O(d^{2})\) bivariate dependencies. As a first we present a family of discrete bivariate EDAs that can learn and exploit all \(O(d^{2})\) dependencies between variables, and yet have the same run-time complexity as their more limited counterparts. This family of algorithms, which we label DICE (DIscrete Correlated Estimation of distribution algorithms), is rigorously based on sound statistical principles, and particularly on a modelling technique from statistical physics: dichotomised multivariate Gaussian distributions. Initially (Lane et al. in European Conference on the Applications of Evolutionary Computation, Springer, 1999), DICE was trialled on a suite of combinatorial optimization problems over binary search spaces. Our proposed dichotomised Gaussian (DG) model in DICE significantly outperformed existing discrete bivariate EDAs; crucially, the performance gap increasingly widened as dimensionality of the problems increased. In this comprehensive treatment, we generalise DICE by successfully extending it to multary search spaces that also allow for categorical variables. Because correlation is not wholly meaningful for categorical variables, interactions between such variables cannot be fully modelled by correlation-based approaches such as in the original formulation of DICE. Therefore, here we extend our original DG model to deal with such situations. We test DICE on a challenging test suite of combinatorial optimization problems, which are defined mostly on multary search spaces. While the two versions of DICE outperform each other on different problem instances, they both outperform all the state-of-the-art bivariate EDAs on almost all of the problem instances. This further illustrates that these innovative DICE methods constitute a significant step change in the domain of discrete bivariate EDAs.     

Existence theorems and Hyers-Ulam stability for a class of Hybrid fractional differential equations with $p$-Laplacian operator

In this paper, we prove necessary conditions for existence and uniqueness of solution (EUS) as well Hyers-Ulam stability for a class of hybrid fractional differential equations (HFDEs) with $p$-Laplacian operator. For these aims, we take help from topological degree theory and Leray Schauder-type fixed point theorem. An example is provided to illustrate the results.