Computing Correlation among the Graphs under Lexicographic Product via Zagreb Indices

A topological index (TI) is a numerical descriptor of a molecule structure or graph that predicts its different physical, biological, and chemical properties in a theoretical way avoiding the difficult and costly procedures of chemical labs. In this paper, for two connected (molecular) graphs G 1 and G 2 , we define the generalized total-sum graph consisting of various (molecular) polygonal chains by the lexicographic product of the graphs T k G 1 and G 2 , where T k G 1 is obtained by applying the generalized total operation T k on G 1 with k ≥ 1 as some integral value. Moreover, we compute the different degree-based TIs such as first Zagreb, second Zagreb, forgotten Zagreb, and hyper-Zagreb. In the end, a comparison among all the aforesaid TIs is also conducted with the help of certain statistical tools for some particular families of generalized total-sum graphs under lexicographic product.

Constraining momentum space correlators using slightly broken higher spin symmetry

In this work, building up on [1] we present momentum space Ward identities related to broken higher spin symmetry as an alternate approach to computing correlators of spinning operators in interacting theories such as the quasi-fermionic and quasi-bosonic theories. The direct Feynman diagram approach to computing correlation functions is intricate and in general has been performed only in specific kinematic regimes. We use higher spin equations to obtain the parity even and parity odd contributions to two-, three- and four-point correlators involving spinning and scalar operators in a general kinematic regime, and match our results with existing results in the literature for cases where they are available.One of the interesting facts about higher spin equations is that one can use them away from the conformal fixed point. We illustrate this by considering mass deformed free boson theory and solving for two-point functions of spinning operators using higher spin equations.

Multiscale structural complexity of natural patterns

Complexity of patterns is key information for human brain to differ objects of about the same size and shape. Like other innate human senses, the complexity perception cannot be easily quantified. We propose a transparent and universal machine method for estimating structural (effective) complexity of two-dimensional and three-dimensional patterns that can be straightforwardly generalized onto other classes of objects. It is based on multistep renormalization of the pattern of interest and computing the overlap between neighboring renormalized layers. This way, we can define a single number characterizing the structural complexity of an object. We apply this definition to quantify complexity of various magnetic patterns and demonstrate that not only does it reflect the intuitive feeling of what is “complex” and what is “simple” but also, can be used to accurately detect different phase transitions and gain information about dynamics of nonequilibrium systems. When employed for that, the proposed scheme is much simpler and numerically cheaper than the standard methods based on computing correlation functions or using machine learning techniques.

Technical note: An improved Grassberger–Procaccia algorithm for analysis of climate system complexity

Understanding the complexity of natural systems, such as climate systems, is critical for various research and application purposes. A range of techniques have been developed to quantify system complexity, among which the Grassberger–Procaccia (G-P) algorithm has been used the most. However, the use of this method is still not adaptive and the choice of scaling regions relies heavily on subjective criteria. To this end, an improved G-P algorithm was proposed, which integrated the normal-based K-means clustering technique and random sample consensus (RANSAC) algorithm for computing correlation dimensions. To test its effectiveness for computing correlation dimensions, the proposed algorithm was compared with traditional methods using the classical Lorenz and Henon chaotic systems. The results revealed that the new method outperformed traditional algorithms in computing correlation dimensions for both chaotic systems, demonstrating the improvement made by the new method. Based on the new algorithm, the complexity of precipitation, and air temperature in the Hai River basin (HRB) in northeastern China was further evaluated. The results showed that there existed considerable regional differences in the complexity of both climatic variables across the HRB. Specifically, precipitation was shown to become progressively more complex from the mountainous area in the northwest to the plain area in the southeast, whereas the complexity of air temperature exhibited an opposite trend, with less complexity in the plain area. Overall, the spatial patterns of the complexity of precipitation and air temperature reflected the influence of the dominant climate system in the region.

Spectral statistics of unitary ensembles

This article focuses on the use of the orthogonal polynomial method for computing correlation functions, cluster functions, gap probability, Janossy density, and spacing distributions for the eigenvalues of matrix ensembles with unitary-invariant probability law. It first considers the classical families of orthogonal polynomials (Hermite, Laguerre, and Jacobi) and some corresponding unitary ensembles before discussing the statistical properties of N-tuples of real numbers. It then reviews the definitions of basic statistical quantities and demonstrates how their distributions can be made explicit in terms of orthogonal polynomials. It also describes the k-point correlation function, Fredholm determinants of finite-rank kernels, and resolvent kernels.

Elucidation of association for yield attributing traits in country bean

Twenty genotypes of country bean were evaluated to explore the association between yield and yield contributing traits in country bean in field condition. Experiment was conducted at field laboratory of department of Genetics and Plant Breeding (GPB), Bangladesh Agricultural University (BAU), Mymensingh. Traits association through computing correlation coefficient and path coefficients were done both at genotypic and phenotypic levels. Among the twelve morphological traits, genotypic and phenotypic correlation studies showed number of raceme per plant and number of flower buds per raceme had significantly positive relationship with seed yield per plant. Genotypic correlation also showed significant positive relation between number of seed per plant and seed yield per plant. Path coefficient analysis revealed that days to 50 per cent flowering, number of raceme per plant, green pod length, 100 dry seed weight, green shelling percentage, dry shelling percentage showed positive effect on seed yield in genotypic level . Also in phenotypic level days to 50% flowering, days to maturity, number of flower buds per plant, green pod length, number of seed per pod, green pod yield per plant, green test weight and dry shelling percentage showed direct effect on seed yield. The association between traits revealed in the present study shall be of great help to choose parents with desirable traits for hybridization, selection method to follow, and selection criteria towards a successful breeding program. Traits with strong association with yield can be improved easily, but with weak relationship, needs more observations to obtain the better segregates.Progressive Agriculture 28 (3): 184-189, 2017

An improved Grassberger-Procaccia algorithm for analysis of climate system complexity

Understanding the complexity of natural systems, such as climate systems, is critical for various research and application purposes. A range of techniques have been developed to quantify system complexity, among which Grassberger-Procaccia (G-P) algorithm has been mostly used. However, the use of this method is still not adaptive and relies heavily on subjective criteria. To this end, an improved G-P algorithm was proposed, which integrated the normal-based K-means clustering technique and Random Sample Consensus algorithm (RANSAC) for computing correlation dimensions. To test its effectiveness for computing correlation dimensions, the proposed algorithm was compared with traditional methods using the classical Lorenz and Henon chaotic systems. The results revealed that the new method outperformed traditional algorithms in computing correlation dimensions for both chaotic systems, demonstrating the improvement made by the new method. Based on the new algorithm, the complexity of precipitation and air temperature in the Haihe River Basin (HRB) in northeast China was further evaluated. The results showed that there existed considerable regional differences in the complexity of both climatic variables across the HRB. Specifically, precipitation was shown to become progressively more complex from the mountainous area in the northwest to the plain area in the southeast; whereas, the complexity of air temperature exhibited an opposite trend with less complexity in the plain area. Overall, the spatial patterns of the complexity of precipitation and air temperature reflected the influence of the dominant climate system in the region.

A Parallel Approach in Computing Correlation Immunity up to Six Variables

We show the use of a reconfigurable computer in computing the correlation immunity of Boolean functions of up to 6 variables. Boolean functions with high correlation immunity are desired in cryptographic systems because they are immune to correlation attacks. The SRC-6 reconfigurable computer was programmed in Verilog to compute the correlation immunity of functions. This computation is performed at a rate that is 190 times faster than a conventional computer. Our analysis of the correlation immunity is across all n-variable Boolean functions, for 2 ≤ n ≤ 6, thus obtaining, for the first time, a complete distribution of such functions. We also compare correlation immunity with two other cryptographic properties, nonlinearity and degree.