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The energy of a graph is calculated by summing the absolute values of the eigenvalues found in its adjacency matrix. In this study, we present examples of line graphs with energy equivalent to the energy of a complete graph, which are called the borderenergetic graphs. As the examples of borderenergetic line graphs are rare, we introduce a new type of graphs called almost borderenergetic graphs whose energy differs from the borderenergetic energy by at most 1. In this paper, we begin by exploring the energy properties of line graphs derived from regular graphs and strongly regular graphs. Specifically, we establish a criterion for a line graph to exhibit borderenergetic characteristics when it consists of p many connected regular graphs and q many complete graphs, even when the original regular graph is disconnected. Additionally, we present a criterion for the line graph of a strongly regular graph to exhibit borderenergetic characteristics. To illustrate these concepts, we offer examples of connected borderenergetic graphs that are not necessarily complete. In the second part, we give the spectrum of the line graph of rK1∇K2, and show that it is an almost borderenergetic graph for r ≥ 5. [ABSTRACT FROM AUTHOR]

This paper investigates the Li–Yorke chaos in linear systems with weak topology on Hilbert spaces. A weak topology induced by bounded linear functionals is first constructed. Under this weak topology, it is shown that the weak Li–Yorke chaos can be equivalently measured by an irregular or a semi-irregular vector, which are utilized to establish criteria for the weak Li–Yorke chaos of diagonalizable operators, Jordan blocks, and upper triangular operators. In particular, for a linear operator that can be decomposed into a direct sum of finite-dimensional Jordan blocks, it is Li–Yorke chaotic in weak topology if its point spectrum contains a pair of real opposite eigenvalues with absolute values not less than 1, or a pair of complex conjugate eigenvalues with moduli not less than 1. Interestingly, as a specific example of upper triangular operator, the existence of Li–Yorke chaos in weak topology can be derived for a class of linear operators expressed as the direct sum of finite-dimensional Jordan blocks and a strongly irreducible operator. [ABSTRACT FROM AUTHOR]

The distance energy of a connected graph G is the sum of absolute values of the eigenvalues of the distance matrix of G. In this paper, we study how the distance energy of the complete split graph G S (m , n) = K m + K ¯ n changes when an edge is deleted from it. The complete split graph G S (m , n) consists of a clique on m vertices and an independent set on n vertices in which each vertex of the clique is adjacent to each vertex of the independent set. We prove that the distance energy of the complete split graph G S (m , n) always increases when an edge is deleted from it. [ABSTRACT FROM AUTHOR]

Variable selection is one of the most important contents of machine learning research and a large number of variable selection methods have been proposed at present. From deep learning perspective, we propose a new variable selection method based on multi-layer artificial neural networks in this paper. The method propagates errors gained form a trained network model through the network from output layer to input layer and the values gained by input layer give information about the relative contribution. The variables with big contribution value are selected, which we call neural inverse propagation (NIP). Specifically, after training a neural network, the gained errors between the actual network outputs and the desired outputs go through the network from top to down and arrive at the input layer. The absolute values gained by input units provide information about the relative importance (contribution) of input units, which is that the larger the value is, the more important the corresponding input variable is. We remove k variables corresponding to the first k minimum absolute value gained by input units to the predictive variables, to perform variable selection. The efficiency of the proposed method is illustrated through its application to classification and regression tasks on a number of synthetic datasets and a real-world image classification dataset. [ABSTRACT FROM AUTHOR]

In this paper, the generalized exponential (GExp) method has been employed to construct novel solitary wave solutions of the nonlinear fractional biological population (FBP) model. This model is used to demonstrate the relation of the population with deaths and births. Many novel traveling wave solutions have been formulated in distinct forms such as exponential, hyperbolic and trigonometric forms. These solutions have been explained in three different axes. The first axis is plotting them in their three optional (real, imaginary and absolute value), the second axis is handling these solutions for constructing the requested conditions for applying trigonometric quintic B-spline (TQBS) scheme. The second one determines the accuracy of the obtained analytical solutions by showing the error's value between the analytical and numerical solutions. At the same time, the third one is comparing our analytical and numerical solutions, which have recently been published that explain the paper's contribution and novelty. [ABSTRACT FROM AUTHOR]

A fractal integral identity with the parameter τ related to twice-differentiable mappings is first proposed in this paper. Based on the identity, the parameterized inequalities over the fractal domains are then derived for the mappings whose second-order derivatives in absolute value at certain powers are generalized n -polynomial convex, which is the main purpose of this investigation. Moreover, a series of fractal findings of some applications, involving the special mean values, the midpoint formulas, the moments of random variable and the wave equations on Cantor sets, are acquired correspondingly. [ABSTRACT FROM AUTHOR]

Applying the local fractional integrals, a generalized identity involving the local second-order differentiable mappings is first developed in this paper. A series of fractal integral inequalities pertaining to Simpson type, for the mappings whose local second-order derivatives are generalized (s , P) -convex in absolute value at some power, are then deduced by the discovered identity. Finally, from an application perspective, a range of fractal outcomes with regard to β -type special means, Simpson numerical integrations, midpoint numerical integrations and wave equations are presented, correspondingly. [ABSTRACT FROM AUTHOR]

If G is a simple graph on n vertices v 1 , v 2 , ... , v n and d i be the degree of i th vertex v i then the average degree matrix of graph G , AD (G) is of order n × n whose (i , j) th entry is d i + d j 2 if the vertices v i and v j are adjacent and zero otherwise. The average degree energy of G , AD (E (G)) is the sum of all absolute value of eigenvalues of average degree matrix of a graph G. In this paper, bounds for average degree energy AD (E (G)) and the relation between average degree energy AD (E (G)) and energy E (G) is discussed. [ABSTRACT FROM AUTHOR]

First, we establish the parametrized integral identity and its improved version via Atangana–Baleanu (AB) fractional integrals. For the focus of this paper, we utilize the resulting identities to derive a series of Simpson-like integral inequalities for mappings whose second-order derivatives belong to the (s , P) -convexity and (s , P) -concavity in absolute value. And a couple of outcomes, concerning the Simpson-like quadrature formulas, the q -digamma functions and the modified Bessel functions, are introduced as applications separately in the end. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the weak-field gravitational lensing properties of a 4D ESTGB black hole, which is surrounded by the plasma medium. The effects of the magnetic charges and the three plasma distribution models in the deflection of light around a 4D ESTGB black hole are investigated in detail. We find that the uniform plasma leads to a larger deflection of light rays in comparison with the singular isothermal sphere (SIS), the non-singular isothermal sphere (NSIS) models. Moreover, the deflection angle increases slightly as the absolute value of the magnetic charge decreases. Finally, we analyze the total magnification of image due to weak gravitational lensing around the black hole. The result shows that the presence of a uniform plasma medium remarkably enhances the total magnification whereas the non-uniform plasma reduces the total magnification. [ABSTRACT FROM AUTHOR]

Structural inherent dynamic characteristics would change when damage occurs, and this fact is the theoretical basis of damage detection method based on vibration features. As one of structural inherent properties, frequency response function (FRF) contains rich information and has more potential in damage detection. On the basis of crack detection researches, a new method based on the change of FRF is proposed in this paper. First, the single-degree-of-freedom (SDOF) system is used to illustrated the change of dynamic features caused by cracks represented by changing stiffness, and it is found that the change of stiffness leads to the obvious nonlinear variation of frequency response function curvature (FRFC) and frequency response function curvature differentiation (FRFCD) near the natural frequency. To describe this phenomenon, the Pearson correlation coefficient is employed to analyze correlation relationship of FRF, FRFC and FRFCD between the undamaged and the damaged system. The results demonstrate that absolute values of these coefficients approach from 1 to 0 with the increase of stiffness change. The rail with cracks is taken as the research object, and relationships between the crack size and the correlation coefficients of FRF, FRFC and FRFCD are investigated through the combination of finite element simulation and frequency response experiment. The correlation coefficients of FRF, FRFC and FRFCD are close to 1 when the structure is intact. When cracks occur, the absolute value of the correlation coefficient will gradually trend from 1 to 0, and the sensitivity of FRFCD correlation coefficient to crack is much higher than that of FRF and FRFC. The selection of calculated frequency bands containing different modes and the damping of the structure would affect the absolute value of FRFCD correlation coefficient, but will not change the characteristics of high sensitivity of FRFCD and the relationship between the coefficient and crack size. Besides, noise signals would enhance the nonlinear relationship between responses of intact structure and the structure with cracks, and the FRFCD correlation coefficient value is influenced accordingly. [ABSTRACT FROM AUTHOR]

Let G be a connected graph of order n and let RD (G) be the reciprocal distance matrix (also called Harary matrix) of the graph G. Let ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n be the eigenvalues of the reciprocal distance matrix RD (G) of the connected graph G called the reciprocal distance eigenvalues of G. The Harary energy HE (G) of a connected graph G is defined as sum of the absolute values of the reciprocal distance eigenvalues of G , that is, HE (G) = ∑ i = 1 n | ρ i |. In this paper, we establish some new lower and upper bounds for HE (G) , in terms of different graph parameters associated with the structure of the graph G. We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of k largest adjacency eigenvalues of a connected graph. [ABSTRACT FROM AUTHOR]

This paper analyzes the properties of the nonequilibrium singular point in one-dimensional elementary catastrophe. For this analysis, the Kosambi–Cartan–Chern (KCC) theory is applied to characterize the dynamical system based on differential geometrical quantities. When both the nonlinear connection and deviation curvature are zero, that is, when the geometric stability of the KCC theory is neutral, two bifurcation curves are obtained: one is the known curve with an equilibrium singular point, and the other is a new curve with a nonequilibrium singular point. The two singular points are distinguished based on the vanishing condition of the Berwald connection. Applied to the ecosystem described by the Hill function, the absolute value of the cuspidal curvature of the nonequilibrium singular point is larger than that of the equilibrium singular point. The ecological interpretation of this result is that the range of bistability of the ecosystem in the nonequilibrium state is greater than that in the equilibrium state. The type of singular points in equilibrium and nonequilibrium bifurcation curves are not necessarily the same. For instance, there is a combination in which even if the former has one cusp, the latter may show various types, depending on the parametric space. These results demonstrate that there are cases where simply shifting the system from the equilibrium to nonequilibrium state expands the range of bistability and changes the type of singularity. Although singularity analysis is often performed near the equilibrium point, nonequilibrium analysis, i.e. analysis based on the KCC theory, provides a useful perspective for analyzing singularity theory according to the bifurcation phenomenon. [ABSTRACT FROM AUTHOR]

In the process of multimodal image fusion, how to improve the visual effect after the image fused, while taking into account the protection of energy and the extraction of details, has attracted more and more attention in recent years. Based on the research of visual saliency and the final action-level measurement of the base layer, a multimodal image fusion method based on a guided filter is proposed in this paper. Firstly, multi-scale decomposition of a guided filter is used to decompose the two source images into a small-scale layer, large-scale layer and base layer. The fusion rule of the maximum absolute value is adopted in the small-scale layer, the weight fusion rule based on regular visual parameters is adopted in the large-scale layer and the fusion rule based on activity-level measurement is adopted in the base layer. Finally, the fused three scales are laminated into the final fused image. The experimental results show that the proposed method can improve the image edge processing and visual effect in multimodal image fusion. [ABSTRACT FROM AUTHOR]

Let G be a graph with n vertices and m edges. The minimum edge dominating energy is equal to the sum of the absolute values of eigenvalues of the minimum edge dominating matrix of graph G. In this paper, we establish some bounds for the minimum edge dominating energy of subgraphs of a graph. [ABSTRACT FROM AUTHOR]