Let G be a simple connected graph. In this paper, Szeged dimension and PIv dimension of graph G are introduced. It is proved that if G is a graph of Szeged dimension 1 then line graph of G is 2-connected. Trees of Szeged dimension 1 are characterized. The Szeged dimension and PIv dimension of five composite graphs: sum, corona, composition, disjunction and symmetric difference with strongly regular components are computed. Also explicit formulas of Szeged and PIv indices for these composite graphs are obtained.

Modeling the joint distribution of liquidity and return to determine the dependency structure of a 15-stock portfolio using intraday data in 2019 provides a suitable model for the commonalities. Based on the complexities of higher dimensions multivariate modeling (combined distribution of liquidity and return), after univariate modeling of the stocks’,liquidity based on Autoregressive Conditional Poisson model (ACP) and the returns with the Generalized Autoregressive Conditional Heteroscedastic (GARCH), the marginal distributions were incorporated into a Copula Vine to model the dependency. The findings of this study, based on high-dimensional high-frequency data, indicate that there is an extreme nonlinear correlation between the liquidity of stocks and also between liquidity and returns across the portfolio, which is necessary to take into account in risk assessments to prevent inaccurate assessment of risk indicators such as value at risk. In addition, the results have shown that modeling the joint distribution of liquidity and stock returns in high dimensions relying on the Copula Vine model due to the flexibility performs well.

There are remarkable relations between the graded homological dimensions and the ordinary homological dimensions. In this paper, we study the injective dimension of a complex of graded modules and derive its some properties. In particular, we define the  dualizing complex for a graded ring and investigate its consequences.

In this paper, a new homological dimension of modules, co-presented dimension, is defined. We study some basic properties of this homological dimension. Some ring extensions are considered, too. For instance, we prove that if S  R is a finite normalizing extension and SR is a projective module, then for each right S-module MS, the cop-resented dimension of MS does not exceed the copresented dimension of HomR(S, M).

Using a system of axioms among with a modi ed def-inition of boundary on the basis of the intuitionistic fuzzy sets, we formulate an inductive structure for the dimension of fuzzy spaces which has been de ned by Coker. This new de nition of boundary allows to characterize an intuitionistic fuzzy clopen set as a set with zero boundary. Also, some critical properties and applications are established.

1. IntroductionThe study of river flow is one of the most important cases in the designing of a water storage structure and management of extreme events such as floods and droughts.The rate of river flow depends on various parameters and the nonlinear relationship between them has caused river behavior to be dynamic, nonlinear and complex.

In network code setting, a constant dimension subspace code is a set of k-dimensional subspaces of Fn q. If Fn q is a nondegenerate symplectic vector space with bilinear form f, an isotropic subspace U of Fn q is a subspace for which U  U? . We introduce isotropic subspace codes simply as a set of isotropic subspaces and show how the property of being isotropic is used in decoding process, then we demonstrate the structure of an isotropic spread code. In network code setting, a constant dimension subspace code is a set of k-dimensional subspaces of Fn q. If Fn q is a nondegenerate symplectic vector space with bilinear form f, an isotropic subspace U of Fn q is a subspace for which U  U? . We introduce isotropic subspace codes simply as a set of isotropic subspaces and show how the property of being isotropic is used in decoding process, then we demonstrate the structure of an isotropic spread code.

In this paper we investigate the algebraic properties of dimension of fuzzy hypervector spaces. Also, we prove that two isomorphic fuzzy hypervector spaces have the same dimension.