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As a type of uncertain differential equations, uncertain spring vibration equation is driven by Liu process. This paper proposes a concept of α -path, and shows that the solution of an uncertain spring vibration equation can be expressed by a family of solutions of second-order ordinary differential equations. This paper also proves that the inverse uncertainty distribution of solution of uncertain spring vibration equation is just the α -path of uncertain spring vibration equation, and a numerical algorithm is designed. Moreover, a formula to calculate the expected value of solution of uncertain spring vibration equation is derived. Finally, several numerical examples are provided to illustrate the efficiency of the numerical method. [ABSTRACT FROM AUTHOR]

The problem of consistency of a solution over time remains an important issue in cooperative dynamic games. Payoffs to players prescribed by an inconsistent solution may not be achievable since such a solution is extremely sensitive to its revision in the course of a game developing along an agreed upon cooperative behavior. The paper proposes a strong time-consistent payment scheme which is stable to a revision of cooperative set solutions, e.g., the core. Using a linear transformation of the solution, it becomes possible to obtain non-negative payments to players. In the paper, we also deal with a limit linear transformation of the solution whose convergence is proved. Developing a non-negative strong time-consistent payment scheme in a closed form, we guarantee that the solution supported by the scheme will not be revised over time. [ABSTRACT FROM AUTHOR]

Convergence rate result for a modified steepest descent method and a modified minimal error method for the solution of nonlinear ill-posed operator equation have been proved with noisy data. To our knowledge, convergence rate result for the steepest descent method and minimal error method with noisy data are not known. We provide a convergence rate results for these methods with noisy data. The result in this paper are obtained under less computational cost when compared to the steepest descent method and minimal error method. We present an academic example which satisfies the assumptions of this paper. [ABSTRACT FROM AUTHOR]

This paper investigates the delay-dependent stability problem of continuous neural networks with a bounded time-varying delay via Lyapunov–Krasovskii functional (LKF) method. This paper focuses on reducing the conservatism of stability criteria by estimating the derivative of the LKF more accurately. Firstly, based on several zero-value equalities, a generalized free-weighting-matrix (GFWM) approach is developed for estimating the single integral term. It is also theoretically proved that the GFWM approach is less conservative than the existing methods commonly used for the same task. Then, the GFWM approach is applied to investigate the stability of delayed neural networks, and several stability criteria are derived. Finally, three numerical examples are given to verify the advantages of the proposed criteria. [ABSTRACT FROM AUTHOR]

The paper proposes a new approach for function approximation by ‘estimating’ the second derivative, using Taylor series, where the samples of the function and its first derivatives at the sample points are known. Without such ‘estimation’, these initial data, can approximate the function traditionally using the first order Taylor approximation, but with more error. If we desire to improve the approximation via second order Taylor series, then we can estimate the ‘pseudo’ second derivatives of the function in three different ways. All these three ways are investigated in this paper. The pseudo second derivatives help in computing many more sample points of the function within each sampling interval. Thus, the approach acts like a mathematical ‘magnifying glass’. Two examples are treated to compare the efficiencies of the methods. Also, a qualitative study for upper bound of error of the approximations is studied in detail. [ABSTRACT FROM AUTHOR]

Let D = ( P , B ) be any nontrivial nonsymmetric 2-( v, k, λ ) design. In this paper we construct a connected, regular and bipartite graph, say IG v , b ( D ) , from D . Some graphic properties of IG v , b ( D ) are investigated. In particular, the results in the paper bridges the equivalency between edge-transitivity of IG v , b ( D ) and flag-transitivity of D . With the equivalency, it is obtained that IG v , b ( D ) is a semisymmetric graph as long as D is a nonsymmetric flag-transitive 2-( v, k, λ ) design. [ABSTRACT FROM AUTHOR]

In this paper we study and obtain the existence of a positive almost periodic solution to the nonlinear differential equation with delays that describes the so-called hematopoiesis model. The main tool utilized to establish our existence result consists of the fixed-point theorem in a cone. Two examples are given at the end of the paper to illustrate our main result. [ABSTRACT FROM AUTHOR]

For uncertain differential equations, we cannot always obtain their analytic solutions. Early researchers have described the Euler method and Runge–Kutta method for solving uncertain differential equations. This paper proposes a new numerical method—Adams method to solve uncertain differential equations. Some numerical experiments are given to illustrate the efficiency of our numerical method. Moreover, this paper also gives two numerical methods for calculating the extreme value and the time integral of solutions of uncertain differential equations. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to investigate some new integral inequalities of Grüss type for k - Riemann–Liouville fractional integrals. From our results, new weighted or classical Grüss type inequalities have been established for some special cases. Moreover, special cases of the integral inequalities in this paper have been obtained by Dahmani and Tabharit, 2010 in [5]. [ABSTRACT FROM AUTHOR]

In the present paper we will give some properties of the subclass of harmonic mappings which is related to m -fold starlike functions in the open unit disc D = { z | | z | < 1 } . Throughout this paper we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that an elegant and complete treatment theory of the harmonic mapping is given in Durens monograph (Duren, 1983). The main aim of us to investigate some properties of the new class of us which represented as in the following form, S ∗ H ( m ) = f = h ( z ) + g ( z ) ‾ | f ∈ SH ( m ) , g ′ ( z ) h ′ ( z ) ≺ b 1 p ( z ) , h ( z ) ∈ S ∗ ( m ) , p ( z ) ∈ P ( m ) , where h ( z ) = z + ∑ n = 1 ∞ a mn + 1 z mn + 1 , g ( z ) = ∑ n = 0 ∞ b mn + 1 z mn + 1 , | b 1 | < 1 . [ABSTRACT FROM AUTHOR]

In this paper, we obtain some gap functions for generalized mixed quasi variational inequality problems in terms of regularized gap function and D -gap function. Further, by using these gap functions we obtain global error bounds for the solution of generalized mixed quasi variational inequality problems in Hilbert spaces. The results obtained in this paper generalize and improve some corresponding known results in literatures. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations. The results of this paper generalize and improve some earlier results in this field. [ABSTRACT FROM AUTHOR]

• A cholera infection model with vaccination strategy is studied. • The basic reproduction number is obtained. • A threshold dynamics is established by using Lyapunov function technique. • Several reasonable optimal control strategies are suggested to the prevention and control of the cholera infection. In this paper, a cholera infection model with vaccination strategy is investigated. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria is established. By means of Lyapunov functions and LaSalle's invariance principle, it is proved that if the basic reproduction number is less than unity, the disease-free equilibrium is globally asymptotically stable. If the basic reproduction number is greater than unity, the endemic equilibrium is globally asymptotically stable. In addition, by using Pontryagin's maximum principle, several reasonable optimal control strategies are suggested to the prevention and control of the cholera infection. Numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

• SIS-type system of equations coupled by impulses at fixed times is analyzed. • Dynamics and structure's interdependence constitute network damping of infection. • Network suppression of pathogen spread indicates units prone to high prevalence. • Based on analytical results a scheme for network R 0 reduction is proposed. • Theoretical results are illustrated by simulations for the patient hospital records. In the paper, we describe the transfer of patients in the healthcare system by SIS-type equations coupled by impulses at fixed times. The first aim for these considerations is to provide a rigorous mathematical analysis of a general theoretical model, which incorporates the structure of patients' transfers. Based on theoretical results, we disprove the intuitions taken from decoupled systems, showing a damping of infection in the case when the transmission rate is locally higher than the recovery rate. Observations on network suppression of pathogen spread allow us to indicate units prone to the high prevalence and propose, possibly low-cost, interventions reducing the infection spread in a whole system. Core results combine the dynamical and structural properties of a system. Finally, we consider a model of the transmission of hospital-acquired multidrug-resistant bacteriae infections, based on real patient hospital records provided by the German insurance company – AOK Lower Saxony, to ilustrate the theoretical considerations. [ABSTRACT FROM AUTHOR]

The paper presents a new semi-analytic numerical method for solving multi-point boundary value problems with linear and nonlinear equations of the second and higher orders. The method is based on the use of the particular solutions of the linearized equation. Presented in the paper examples include the solution of equations with differential operators of the second, third and fourth orders. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates the coexistence and local stability of multiple equilibrium points for competitive neural networks, where the Gaussian-wavelet-type activation functions are employed and the unbounded time-varying delays are considered. Based on geometric formulation, the fixed point theorem, contraction mapping theorem and rigorous mathematical analysis, a series of sufficient conditions are derived to ascertain that the addressed neural networks have exactly 5 n equilibrium points, among which 3 n equilibrium points are locally stable. On this basis, some criteria are also obtained on the multiple exponential stability, multiple power stability and multiple log-stability of Hopfield neural networks with Gaussian-wavelet-type activation functions. The obtained results generalize and improve the existing multistability results of Hopfield neural networks and competitive neural networks without time delays and with Gaussian-wavelet-type activation functions. Moreover, it is highlighted that the competitive neural networks with Gaussian-wavelet-type activation functions can have both more total equilibrium points and more locally stable equilibrium points than the ones with Mexican-hat-type activation function. Finally, two numerical examples with computer simulations are provided to illustrate and validate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce a piecewise reproducing kernel method for impulsive delay differential equations with piecewise constant arguments. The method is an improved reproducing kernel method. Compared with the classical reproducing kernel method, the solutions obtained using the present method can give good approximations for a larger time interval. Some numerical examples are used to show the effectiveness and simplicity of the method. [ABSTRACT FROM AUTHOR]

Abstract A function f : V (G) → { 0 , 1 , ... , k } is called a k -rainbow independent dominating function of G if V i = { x ∈ V (G) : f (x) = i } is independent for 1 ≤ i ≤ k , and for every x ∈ V 0 it follows that N (x) ∩ V i ≠ ∅, for every i ∈ [ k ]. The k -rainbow independent domination number, γ ri k (G), of a graph G is the minimum of w (f) = ∑ i = 1 k | V i | over all such functions. In this paper we show that the problem of determining whether a graph has a k -rainbow independent dominating function of a given weight is NP-complete for bipartite graphs and that there exists a linear-time algorithm to compute γ ri k (G) of trees. In addition, sharp bounds for the k -rainbow independent domination number of the lexicographic product are presented, as well as the exact formula in the case k = 2. [ABSTRACT FROM AUTHOR]

Abstract Let A ∈ C l × l be a diagonalizable matrix with spectrum contained in the set {1, α , 0}. In this paper we derive a general and explicit expression for the solutions X of the Yang–Baxter-like matrix equation A X A = X A X. The idea involves partitioning the matrices to discuss four block-matrix equations and utilizing the eigen-properties of the matrices obtained. When A is an idempotent matrix, the result generates the formula obtained in Mansour et al. (2017). We give examples to illustrate the validity of the results obtained in this note. [ABSTRACT FROM AUTHOR]

Abstract Let G = (V , E) be a simple graph of order n with normalized Laplacian eigenvalues ρ 1 ≥ ρ 2 ≥ ⋯ ≥ ρ n − 1 ≥ ρ n = 0. The normalized Laplacian spread of graph G , denoted by ρ 1 − ρ n − 1 , is the difference between the largest and the second smallest normalized Laplacian eigenvalues of graph G. In this paper, we obtain the first four smallest values on ρ 2 of graphs. Moreover, we give a lower bound on ρ 2 of connected bipartite graph G except the complete bipartite graph and characterize graphs for which the bound is attained. Finally, we present some bounds on the normalized Laplacian spread of graphs and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

Abstract As we know, a ruled surface is the tangent ruled surface of a space curve if and only if its ruled distance density function vanishes identically; a ruled surface is the binormal ruled surface of a space curve if and only if its ruled translation density function vanishes identically. The ruled distance density function and translation density function are differential invariants of ruled surfaces in three dimensional Euclidean space. In this paper we give the necessary and sufficient condition of which a ruled surface is the principal normal ruled surface of a space curve using the theories of ruled invariants of ruled surface in three dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we propose an efficient linearly implicit conservative difference solver for the fractional Klein–Gordon–Zakharov system. First of all, we present a detailed derivation of the energy conservation property of the system in the discrete setting. Then, by using the mathematical induction, it is proved that the proposed scheme is uniquely solvable. Subsequently, by virtue of the discrete energy method and a 'cut-off' function technique, it is shown that the proposed solver possesses the convergence rates of O (Δ t 2 + h 2) in the sense of L ∞- and L 2- norms, respectively, and is unconditionally stable. Finally, numerical results testify the effectiveness of the proposed scheme and exhibit the correctness of theoretical results. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the mathematical analysis of an age-structured multi-group heroin epidemic model, which can be used to describe the spread of heroin habituation and addiction in heterogeneous environment. Under general assumptions on the different level of susceptibility and the relapse to frequent heroin use, we establish sharp criteria for heroin spreading and vanishing. We rigorously investigate the well-posedness of the model, the existence of equilibria, the asymptotic smoothness of solution orbits, and the global stability of equilibria. Specifically, we rigorously show that the drug-free equilibrium is globally asymptotically stable if a threshold value ℜ 0 is less than one, and the unique drug-endemic equilibrium is globally attractive if ℜ 0 is greater than one. In the proofs of global stability of equilibria, we construct suitable Lyapunov functions by using a graph-theoretic method. [ABSTRACT FROM AUTHOR]

Abstract In the paper, the superconvergence of continuous Galerkin finite element methods (CGFEMs) for linear delay differential equations of neutral type is presented. By the orthogonal analysis method and under the suitable condition, it is proven that the finite element solution is superconvergent at the nodal points and Lobatto points. Numerical experiments further confirm the effectiveness and the superconvergence of the CGFEMs. [ABSTRACT FROM AUTHOR]

Abstract An uncertain differential equation with jumps is a type of uncertain differential equation driven by Liu process and uncertain renewal process, which is used to model discontinuous systems. Up to now, the stability in measure and almost sure stability for such an equation have been studied. The above two types of stability cannot be applied to all cases, so this paper aims at presenting a concept of stability in mean for an uncertain differential equation with jumps as a supplement. Most important of all, a stability theorem is given for an uncertain differential equation with jumps being stable in mean. And some examples are proposed to show how to use the theorem to judge whether the uncertain differential equation with jumps is stable in mean. [ABSTRACT FROM AUTHOR]

In the present paper, the ciliary flow of seminal fluid (non-Newtonian fluid) in the vas deferens (ductus efferentes) of the male reproductive tract is examined through a mathematical model. The rheological behavior of seminal fluid is described as a Hershel–Bulkley fluid. The momentum equations governing the flow of Herschel–Bulkley model are simplified by considering long wave-length and low Reynolds number approximations. The analytical solutions are obtained for velocity profile, flow rate, pressure gradient and flow impedance. The variation of plug core radius with respect to the cilia length (ε), the measure of the eccentricity of the elliptical motion (α), the wave number (β) and the time-mean flow rate ( Q av ) is reported for the first time. The time-mean flow rates obtained from the present study are compared with their experimental values of the transport of seminal fluid through the rhesus monkey reproductive track and the ductus efferentes of the male reproductive tract, respectively. It is observed that the theoretical values of time-mean flow rates for seminal fluid (Herschel–Bulkley fluid) have been found to be in good agreement with the experimental results. [ABSTRACT FROM AUTHOR]

A graph G is a ℓ -tree if G = K ℓ + 1 , or G has a vertex v whose neighborhood is a clique of order ℓ, and G − v is a ℓ-tree. A non-increasing sequence π = ( d 1 , … , d n ) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. Yin and Li (2009) proved that if k  ≥ 2, n ≥ 9 2 k 2 + 19 2 k and π = ( d 1 , … , d n ) is a graphic sequence with ∑ i = 1 n d i > ( k − 2 ) n , then π has a realization containing every 1-tree (the usual tree) on k vertices. Moreover, the lower bound ( k − 2 ) n is the best possible. This is a variation of a conjecture due to Erdős and Sós. Recently, Zeng and Yin (2016) investigated an analogue extremal problem for 2-trees and prove that if k  ≥ 3, n ≥ 2 k 2 − k and π = ( d 1 , … , d n ) is a graphic sequence with ∑ i = 1 n d i > 4 k − 5 3 n , then π has a realization containing every 2-tree on k vertices. Moreover, the lower bound 4 k − 5 3 n is almost the best possible. In this paper, we consider the most general case ℓ ≥ 3 and prove that if ℓ ≥ 3, k ≥ ℓ + 1 , n ≥ 2 k 2 − ℓ k + k and π = ( d 1 , … , d n ) is a graphic sequence with ∑ i = 1 n d i > 2 ℓ k − ℓ − 3 ℓ + 1 n , then π has a realization containing every ℓ-tree on k vertices. We also show that the lower bound 2 ℓ k − ℓ − 3 ℓ + 1 n is almost the best possible. [ABSTRACT FROM AUTHOR]

The solution of discrete ill-posed problems has been a subject of research for many years. Among the many methods described in the literature, the Bregman algorithm has attracted a great deal attention and been widely investigated. Recently, a nonstationary preconditioned version of this algorithm, referred to as the nonstationary modified linearized Bregman algorithm, was proposed. The aim of this paper is to discuss numerical aspects of this algorithm and to compare computed results with known theoretical properties. We also discuss the effect of several parameters required by the algorithm on the computed solution. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is using the elementary method and the properties of trigonometric functions to study the computational problem of one kind trigonometric sums, and give some interesting identities involving sin ( x ), cos ( x ), tan ( x ), cot ( x ) , Bernoulli numbers and Dirichlet L -functions. As a result, we obtain a relationship between the summation of cot(x) with character and Dirichlet L -function. [ABSTRACT FROM AUTHOR]

The main goal of this paper is the numerical computation of solutions of the so-called Yang–Baxter-like matrix equation A X A = X A X , where A is a given complex square matrix. Two novel matrix iterations are proposed, both having second-order convergence. A sign modification in one of the iterations gives rise to a third matrix iteration. Strategies for finding starting approximations are discussed as well as a technique for estimating the relative error. One of the methods involves a very small cost per iteration and is shown to be stable. Numerical experiments are carried out to illustrate the effectiveness of the new methods. [ABSTRACT FROM AUTHOR]

The concept of process similarity has attracted the attention of many researchers in the recent literature, since it measures the degree of proximity of processes. Similarities are therefore useful for the efficient management of large repositories, since they allow us to find the appropriate process models among hundreds or thousands of possible candidates. However, time constraints are usually omitted when these similarity measures are defined, so in this paper two similarity measures are defined over a timed extension of Petri nets, the so-called timed-arc Petri nets, in which tokens are assigned an age indicating the time elapsed from creation, and PT-arcs (place to transition arcs) are labeled with time intervals that are used to restrict the age of the tokens that can be used to fire the adjacent transition. [ABSTRACT FROM AUTHOR]

Indubitably, the notion of covering dimension of frames was, extensively, studied. Many searches such as Charalambous, Banashewski and Gilmour (see, for example (Charalambous, 1974; Charalambous, 1974 [11] ; Banaschewski and Gilmour, 1989 [12] ) studied this dimension. Also, in the study [5], the covering dimension of finite lattices has been characterized by using the so called minimal covers. This approach gave the motive to other searches such as Zhang et al., to study properties of this dimension (see Zhang et al. (2017) [9]). In this paper, we study the covering dimension of finite lattices in combination with matrix theory. Essentially, we characterize the minimal covers of finite lattices and the order of those covers using matrices and we compute the covering dimension of the corresponding finite lattices. [ABSTRACT FROM AUTHOR]

In this paper, we present integral representations for the DMP and core-EP inverse, which based on the full-rank decomposition of a given matrix. In particular, integral representations of the core and dual core inverse are given. All of these integral representations do not require any restriction on the spectrum of a certain matrix. [ABSTRACT FROM AUTHOR]

This paper is devoted to the development and analysis of a modified family of Falkner-type methods for solving differential systems of second-order initial-value problems. The approaches of collocation and interpolation are adopted to derive the new methods. These modified methods are implemented in block form to obtain the numerical solutions to the considered problems. The study of the properties of the proposed block Falkner-type methods reveals that they are consistent and zero-stable, and thus, convergent. From the stability analysis, it could be seen that the proposed Falkner methods have non-empty stability regions for k = 2 , 3 , 4 . Some numerical test are presented to illustrate the efficiency of the proposed family. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the convergence and semi-convergence of a class of SSOR-like methods with four real functions for augmented systems. The class takes most existed SSOR-like methods as its special cases. For nonsingular systems, we obtain the minimum of convergence factors of all the SSOR-like methods in the class, and study when it can be reached by the convergence factors of methods in the class. By considering the equivalence of methods, we show that most of the existed SSOR-like methods have the same minimum of convergence factors. [ABSTRACT FROM AUTHOR]

For a connected graph G , the multiplicative eccentricity resistance-distance is defined as ξ R * ( G ) = ∑ { x , y } ⊂ V ( G ) ɛ G ( x ) · ɛ G ( y ) R G ( x , y ) , where ε G ( · ) is the eccentricity of the corresponding vertex and R G ( x, y ) is the effective resistance between vertices x and y in G . A connected graph G is called a cactus if any two of its cycles have at most one common vertex. Let Cat ( n ; t ) be the set of cacti possessing n vertices and t cycles, where 0 ≤ t ≤ n − 1 2 . In this paper, we introduce some edge-grafting transformations which decrease ξ R * ( G ) . As their applications, the extremal graphs with minimum and second minimum ξ R * ( G ) -value in Cat ( n ; t ) are characterized. [ABSTRACT FROM AUTHOR]

The revised Szeged index of a graph is defined as S z * ( G ) = ∑ e = u v ∈ E ( n u ( e ) + n 0 ( e ) 2 ) ( n v ( e ) + n 0 ( e ) 2 ) , where n u ( e ) and n v ( e ) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u , and n 0 ( e ) is the number of vertices equidistant to u and v . In the paper, we identify the lower bound of revised Szeged index among all bicyclic graphs, and also characterize the extremal graphs that attain the lower bound. [ABSTRACT FROM AUTHOR]

This paper investigates the fixed-time consensus tracking problems for second-order multi-agent systems with unknown external disturbances. Firstly, a fixed-time terminal sliding mode (FTTSM) is proposed, which can avoid the singularity problem. Then, two continuous distributed consensus tracking control laws with adaptive updating laws are designed respectively, in which the upper bounds of external disturbances are not required. It is proved that the two control laws can both guarantee the consensus tracking errors converge into the desired regions including the origin in fixed time. A simulation example is given to demonstrate the effectiveness of proposed methods. [ABSTRACT FROM AUTHOR]

NURBS surface is very useful in geometric modeling, animation, image morphing and deformation. Constructing non-self-intersecting (injective) NURBS surfaces is an important process in surface and solid modeling. In this paper, the injective conditions of tensor product NURBS surface are studied, based on the geometric positions of control points, which are equivalent to the surface to be non-self-intersecting for all positive weights. Finally, some representative examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, the error analysis of a two-grid method (TGM) with backward Euler scheme is discussed for semilinear parabolic equation. Contrary to the conventional finite element analysis, the error between exact solution and finite element solution is split into two parts (temporal error and spatial error) by introducing a corresponding time-discrete system. This can lead to the spatial error independent of τ (time step). Secondly, based on the above technique, optimal error estimates in L 2 and H 1 -norms of TGM solution are deduced unconditionally, while previous works always require a certain time step size condition. Finally, a numerical experiment is provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper considers a modular grad-div stabilization method for approximating solutions of the time-dependent Boussinesq model of non-isothermal flows. The proposed method adds a minimally intrusive step to an existing Boussinesq code, with the key idea being that the penalization of the divergence errors, is only in the extra step (i.e. nothing is added to the original equations). The paper provides a full mathematical analysis by proving unconditional stability and optimal convergence of the methods considered. Numerical experiments confirm theoretical findings, and show that the algorithms have a similar positive effect as the usual grad-div stabilization. [ABSTRACT FROM AUTHOR]

The Szeged index of a graph G is defined as S z ( G ) = ∑ e = u v ∈ E n u ( e ) n v ( e ) , where n u ( e ) and n v ( e ) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u . A cactus is a graph in which any two cycles have at most one common vertex. Let C ( n , k ) denote the class of all cacti with order n and k cycles, and C n t denote the class of all cacti with order n and t pendant vertices. In this paper, a lower bound of the Szeged index for cacti of order n with k cycles is determined, and all the graphs that achieve the lower bound are identified. As well, the unique graph in C n t with minimum Szeged index is characterized. [ABSTRACT FROM AUTHOR]

Let d L ( G ) ( e ) be the degree of an edge e in line graph L ( G ) of a graph G . The edge versions of atom-bond connectivity ( ABC e ) and geometric arithmetic ( GA e ) indices of G are defined as ∑ e f ∈ E ( L ( G ) ) ( d L ( G ) ( e ) + d L ( G ) ( f ) − 2 d L ( G ) ( e ) × d L ( G ) ( f ) and G A e ( G ) = ∑ e f ∈ E ( L ( G ) ) 2 d L ( G ) ( e ) d L ( G ) ( f ) d L ( G ) ( e ) + d L ( G ) ( f ) . In this paper, we study the ABC e and GA e indices for joint graphs and certain graph operations. [ABSTRACT FROM AUTHOR]

Entropy-based measures have been used to characterize the structure of complex networks. Various graph entropies based on different graph parameters have been proposed and studied. So, it has been intricate to decide which measure is the right one for solving a particular problem. In this paper, we introduce graph entropy measures based on independent sets and matchings of graphs. The values of entropies of some special graphs are calculated and we draw several conclusions regrading the usability of the measures. Based on our findings, we conclude that it should be demanding studying extremal values of the novel entropies. [ABSTRACT FROM AUTHOR]

The integro-differential population balance equation describing aggregation processes was proposed almost 100 years ago. Aggregation is an important size enlargement process in many industries; the modeling and design of the process can be done using the population balance framework, however it is typically impossible to obtain analytical solutions: in almost every case a numerical solution of the equations must be obtained. In this paper, we present the developed group analysis method for the one-dimensional population balance equation for aggregation in a well-mixed batch system including a crystal growth term. The determining equations are solved, the optimal system, invariant solutions and all the reduced equations are obtained. Furthermore, finding the determining equation by use of the preliminary group classification is also considered. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of nontrivial solutions for an impulsive fractional differential equation with Dirichlet boundary conditions. By using Morse theory coupled with local linking arguments, we obtain some new criteria to guarantee that the impulsive fractional differential equations have at least one nontrivial solution. [ABSTRACT FROM AUTHOR]

A face irregular entire k-labeling of a 2-connected plane graph G is a labeling of vertices, edges and faces of G with labels from the set { 1 , 2 , … , k } in such a way that for any two different faces their weights are distinct. The weight of a face under a k -labeling is the sum of labels carried by that face and all the edges and vertices incident with the face. The minimum k for which a plane graph G has a face irregular entire k -labeling is called the entire face irregularity strength . In this paper, we estimate the bounds of the entire face irregularity strength for disjoint union of multiple copies of a plane graph and prove the sharpness of the lower bound. [ABSTRACT FROM AUTHOR]

In this paper, we define the generalized ( p, q )-Fibonacci polynomials u n, m ( x ) and generalized ( p, q )-Lucas polynomials v n, m ( x ), and further introduce the generalized Humbert polynomials u n , m ( r ) ( x ) as the convolutions of u n, m ( x ). We give many expressions, expansions, recurrence relations and differential recurrence relations of u n , m ( r ) ( x ) , and study the matrices and determinants related to the polynomials u n, m ( x ), v n, m ( x ) and u n , m ( r ) ( x ) . Finally, we present an algebraic interpretation for the generalized Humbert polynomials u n , m ( r ) ( x ) . It can be found that various well-known polynomials are special cases of u n, m ( x ), v n, m ( x ) or u n , m ( r ) ( x ) . Therefore, by introducing the general polynomials u n, m ( x ), v n, m ( x ) and u n , m ( r ) ( x ) , we have a unified approach to dealing with many special polynomials in the literature. [ABSTRACT FROM AUTHOR]

Since the early 2000s, one of the most popular uses of the Internet has been for online social networking. Since 2009, the most popular online social network has been Facebook; roughly 80% of Internet users worldwide have a Facebook account as of the end of 2014. However, in early 2014, Princeton University graduate students Cannarella and Spechler released a manuscript which claimed that Facebook would decline in popularity to essential “death” by the end of 2017. This claim was based on results obtained from an adaptation of the classic SIR epidemic model. In this paper, we will explore a modified version of this model to take into account a changing population size and the possibility of people leaving the social network, but later returning, and compare this model and its results to those of the original model to either confirm or deny the result that Facebook will fade to obscurity by the end of 2017. [ABSTRACT FROM AUTHOR]

Structured total least norm (STLN) and weighted total least squares (WTLS) have been proposed for structured EIV (errors-in-variables) models. STLN is a principle minimizing the L p norm of the perturbation parts of an EIV model, in which p = 1, 2 or ∞. STLN permits affine structure of the matrix A or [ A | y ] such as Toeplitz. STLN has advantages over WTLS on having ∞-norm and robust 1-norm. However, only Hankel or Toeplitz structure was discussed explicitly in STLN, and weight of errors was not discussed. While in some applications, the matrix [ A | y ] has arbitrary linear structure, taking linear regression and coordinate transformation as examples. This paper aims at extending STLN to L-STLN (linear structured total least norm), which can deal with EIV models having linear structures other than Toeplitz or Hankel in [ A | y ]. Additionally, weighted estimation is discussed. A simulated numerical example is computed by STLN and L-STLN under 1-, 2-, and ∞-norm, the results shown that L-STLN can preserve arbitrary linear structure of [ A | y ]. Also, the estimated correction of [ A | y ] by WTLS and L-STLN under 2-norm are compared. The results show that weighted L-STLN under 2-norm is consistent with WTLS. The robustness of L-STLN under 1-norm is demonstrated by simulated outlier. [ABSTRACT FROM AUTHOR]