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Löschian numbers are a topic of investigation in scientific fields such as spatial economics, socio-economic geography, geometry, and number theory. This is because these numbers play a significant role in the optimization of spatial interactions, in terms of achieving the best division of space between competing central places. This work presents four new number theoretic theorems and four corollaries concerning the properties of Löschian numbers, which enable a better understanding of the mathematical properties of these numbers. • It has been shown that the two definitions of Löschian numbers given by Konzig and by Dacey are equivalent. • It has been proved that there are essentially only two underlying diophantine equations that generate Löschian numbers. • New theorems and corollaries showing previously unknown properties of Löschian numbers have been formulated and proven. • A formula was established to determine the probability that a randomly chosen natural number is a Löschian number. • The hexagonal networks spatial systems can benefit from the new knowledge concerning Löschian numbers. [ABSTRACT FROM AUTHOR]

• We proposed a unified transform method for general robust property of probabilistic BCNs. • A necessary and sufficient condition is derived for the robust set stabilization with probability one of PBCNs. • All of the time-optimal robust set stabilizers are characterized. This article proposes a unified transform method for general robust property of probabilistic Boolean control networks (PBCNs). Here, the general robust property of PBCNs means any given robust property of any given PBCN with probability one. Specifically, the method is that the investigation on any given robust property of any given PBCN with probability one can be equivalently transformed into the investigation on the given robust property of the corresponding Boolean control network (BCN). Note that the PBCN has time-varying probability distribution and the property can be set according to the needs of specific problems. As an application of the method, we study the robust set stabilization with probability one of PBCNs with positive time-varying probability distribution. We derive the results through a discussion of the robust set stabilization of the corresponding BCNs. Also, we characterize all of the time-optimal robust set stabilizers of the PBCNs. A reduced disturbed PBCN model of the lac operon in the Escherichia coli is provided to show the effectiveness of the achieved results. [ABSTRACT FROM AUTHOR]

In graph theory and its applications, trees, BC-trees, subtrees and BC-subtrees have been extensively studied. We introduce a generalization of the BC-tree, called the multi-granular α -tree, which is a tree (of order at least α + 1) where any two leaves are at a distance that is a multiple of α. We study the number of α -subtrees, through α -subtree generating functions, for generalized Bethe trees, Bethe trees and dendrimers (hyper-branched structures in molecular topology). Our results can also be used to examine the asymptotic behavior of the average order of α -subtrees in dendrimers. [ABSTRACT FROM AUTHOR]

Abstract The Turán number of a graph H , denoted by ex (n, H), is the maximum number of edges in any graph on n vertices containing no H as a subgraph. Let S ℓ denote the star on ℓ + 1 vertices and let k · S ℓ denote disjoint union of k copies of S ℓ. In this paper, for appropriately large n , we determine the Turán numbers for k · S ℓ and F , where F is a forest with components each of order 4, which improve the results of Lidický et al. (2013). [ABSTRACT FROM AUTHOR]

Abstract In a number of applications, the codebook that minimizes the maximal cross-correlation amplitude (I max) is often desirable. There are two major ingredients in this paper. These two contents are to propose two constructions of codebooks. The codebooks generated by these constructions asymptotically meet the Welch bound. What we find is the codebook constructed in Section 2 can also asymptotically meet the Levenshtein bound. The parameters of these codebooks are new. [ABSTRACT FROM AUTHOR]

Abstract Compressed sensing is a relatively new mathematical paradigm that shows a small number of linear measurements are enough to efficiently reconstruct a large dimensional signal under the assumption the signal is sparse. Applications for this technology are ubiquitous, ranging from wireless communications to medical imaging, and there is now a solid foundation of mathematical theory and algorithms to robustly and efficiently reconstruct such signals. However, in many of these applications, the signals of interest do not only have a sparse representation, but have other structure such as lattice-valued coefficients. While there has been a small amount of work in this setting, it is still not very well understood how such extra information can be utilized during sampling and reconstruction. Here, we explore the problem of integer sparse reconstruction, lending insight into when this knowledge can be useful, and what types of sampling designs lead to robust reconstruction guarantees. We use a combination of combinatorial, probabilistic and number-theoretic methods to discuss existence and some constructions of such sensing matrices with concrete examples. We also prove sparse versions of Minkowski’s Convex Body and Linear Forms theorems that exhibit some limitations of this framework. [ABSTRACT FROM AUTHOR]

A linear k -forest of an undirected graph G is a subgraph of G whose components are paths with lengths at most k . The linear k -arboricity of G , denoted by la k ( G ), is the minimum number of linear k -forests needed to partition the edge set E ( G ) of G . In this paper, the exact values of the linear ( n − 1 ) -arboricity of lexicographic product graphs K n ○ K n, n and K n, n ○ K n are obtained. Furthermore, l a k ( K n , n □ K n , n ) are also derived for the Cartesian product graph of two copies of K n, n . These results confirm the conjecture about the upper bound la k ( G ) given in [Discrete Math. 41(1982)219-220] for K n ○ K n, n , K n, n ○ K n and K n , n □ K n , n . [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of evolutionary networked games with different length information via semi-tensor product (STP) method. First, a networked game with different length information is modeled in the form of probabilistic Boolean networks (PBNs) with time delays. Second, based on the utility function of each player, a necessary condition for the existence of a pure Nash equilibrium is obtained. Then a state feedback control is applied to stabilize the considered system. Finally, an example is presented to substantiate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we define a new domination invariant on a graph G , which coincides with the ordinary independent domination number of the generalized prism G □ K k , called the k-rainbow independent domination number and denoted by γ ri k ( G ). Some bounds and exact values concerning this domination concept are determined. As a main result, we prove a Nordhaus–Gaddum-type theorem on the sum for 2-rainbow independent domination number, and show if G is a graph of order n  ≥ 3, then 5 ≤ γ ri 2 ( G ) + γ ri 2 ( G ¯ ) ≤ n + 3 , with both bounds being sharp. [ABSTRACT FROM AUTHOR]

The Sierpinski curve is one of the most known space-filling curves and one with the highest number of applications. We present a recently proposed computational methodology based on the infinite quantity called grossone to investigate the behavior of two different constructions of the Sierpinski curve. We emphasize that, adopting this point of view, we have infinitely many Sierpinski curves depending, contrarily to traditional analysis, on each specific starting configuration. Of particular interest are some power series expansions in the new infinitesimal quantities emerging from the study of the considered curves. [ABSTRACT FROM AUTHOR]

Numerous problems arising in engineering applications can have several objectives to be satisfied. An important class of problems of this kind is lexicographic multi-objective problems where the first objective is incomparably more important than the second one which, in its turn, is incomparably more important than the third one, etc. In this paper, Lexicographic Multi-Objective Linear Programming (LMOLP) problems are considered. To tackle them, traditional approaches either require solution of a series of linear programming problems or apply a scalarization of weighted multiple objectives into a single-objective function. The latter approach requires finding a set of weights that guarantees the equivalence of the original problem and the single-objective one and the search of correct weights can be very time consuming. In this work a new approach for solving LMOLP problems using a recently introduced computational methodology allowing one to work numerically with infinities and infinitesimals is proposed. It is shown that a smart application of infinitesimal weights allows one to construct a single-objective problem avoiding the necessity to determine finite weights. The equivalence between the original multi-objective problem and the new single-objective one is proved. A simplex-based algorithm working with finite and infinitesimal numbers is proposed, implemented, and discussed. Results of some numerical experiments are provided. [ABSTRACT FROM AUTHOR]

We consider the problem of estimating the optimal direction in regression by maximizing the probability that the scalar product between the vector of unknown parameters and the chosen direction is positive. The estimator maximizing this probability is simple in form, and is especially useful for situations where the number of parameters is much larger than the number of observations. We provide examples which show that this estimator is superior to state-of-the-art methods such as the LASSO for estimating the optimal direction. [ABSTRACT FROM AUTHOR]

The study of limit cycles for Liénard system is very important not only in theoretical studies but also in applications. In this paper, we study the number of limit cycles for a class of Liénard systems with restoring forces of seventh degree. Let H ( n, m ) denote the maximum number of limit cycles bifurcated from the generalized Liénard system x ˙ = y , y ˙ = − g ( x ) − f ( x ) y , where f ( x ) and g ( x ) are polynomials in x and deg f = n , det g = m . We greatly improve the existing results of H ( n, m ) for m = 7 , n = 4 and m = 7 , n = 2 n ¯ with 4 ≤ n ¯ ≤ 20 . [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]

The exponential of a matrix plays an important role in the theory of Lie groups. The main purpose of this paper is to examine matrix groups over the split quaternions and the exponential map from their Lie algebras into the groups. Since the set of split quaternions is a noncommutative algebra, the way of computing the exponential of a matrix over the split quaternions is more difficult than calculating the exponential of a matrix over the real or complex numbers. Therefore, we give a method of finding exponential of a split quaternion matrix by its complex adjoint matrix. [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]

In this note we explore the extremal graphs with given matching number with respect to topological indices. We present generalization of previous studies on such graphs and apply our findings to various indices that have not yet been considered for graphs with given matching number. [ABSTRACT FROM AUTHOR]

A new method for the construction of compactly supported Parseval wavelet frames in L 2 ( R d ) with any preassigned approximation order n for arbitrary matrix dilation M is proposed. The number of wavelet functions generating a frame constructed in this way is less or equal to ( d + 1 ) | det M | − d . The method is algorithmic, and the algorithm is simple to use. The number of generating wavelet functions can be reduced to | det M | for a large class of matrices M . [ABSTRACT FROM AUTHOR]

In the canonical framework of evolutionary 2 × 2 games, a binary strategy set comprising cooperation (C) and defection (D) has usually been presumed. Inspired by commonly observed real-world facts, we explore what happens if the resolution of strategy increases. As an extreme limit, the infinite resolution case is both a continuous and a mixed strategy defined by a real number in the range of [0,1]. We find that increasing resolution amplifies cooperation in spatial prisoner's dilemma games as compared with the binary strategy definition; however, this enhancement tendency with increasing resolution is not monotonic in the case of a mixed-strategy setting. [ABSTRACT FROM AUTHOR]

In this paper we investigate a porous thermoelastic system where the heat conduction is given by Cattaneo’s law and where the energy associated with the solution is not necessary positive definite ( μ ξ = b 2 ). We introduce a stability number χ and prove an exponential (resp. polynomial) decay result depending on χ . [ABSTRACT FROM AUTHOR]

A cactus is a connected graph in which any two cycles have at most one common vertex. The distance spectral radius ρ ( G ) of a graph G is the largest eigenvalue of the distance matrix D ( G ). Recently, many researchers proposed the use of ρ ( G ) as a molecular structure descriptor of alkanes. In this paper, we characterize n -vertex cyclic cactus with given matching number m which minimizes the distance spectral radius. The resulting cactus also minimizes the Hosoya index, the Wiener index and the Randić index in the same class of graphs. [ABSTRACT FROM AUTHOR]

Three edges e 1 , e 2 and e 3 in a graph G are consecutive if they form a path (in this order) or a cycle of length 3. The injective edge coloring number χ i ′ ( G ) is the minimum number of colors permitted in a coloring of the edges of G such that if e 1 , e 2 and e 3 are consecutive edges in G , then e 1 and e 3 receive the different colors. Let ω ′ denote the number of edges in a maximum clique of G . A graph G is called an ω ′ edge injective colorable (or perfect EIC-)graph if χ i ′ ( G ) = ω ′ . In this paper, we give a sharp bound of the injective coloring number of a 2-connected graph with some forbidden conditions, and then we also characterize some perfect EIC-graph classes, which extends the results of perfect EIC-graph of Cardoso et al. in [Injective edge chromatic index of a graph, http://arxiv.org/abs/1510.02626 .]. [ABSTRACT FROM AUTHOR]

The Kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph. In this paper, we determine the minimum Kirchhoff index among the unicyclic graphs with fixed number of vertices and matching number, and characterize the extremal graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the composite multiscale cross-sample entropy (CMCSE) which may induce undefined entropies and then further propose the refined composite multiscale cross-sample entropy (RCMCSE) which modifies CMCSE. First, we apply multiscale cross-sample entropy (MCSE), CMCSE and RCMCSE methods to three types of artificial time series in order to test the validity and accuracy of these methods. Results show that RCMCSE reduces not only standard deviation, but also the probability of inducing undefined entropy effectively, which can provide better robustness and more accurate entropies. Then, these three methods are employed to investigate financial time series including US and Chinese stock indices. For the study between stock indices in the same region, some conclusions which are consistent with previous study are drawn by the RCMCSE results. Meanwhile, it can be found that undefined entropies are induced and the numbers of inducing undefined entropy by three methods for investigation between three US stock indices and two Chinese mainland stock indices are given. Compared with MCSE and CMCSE, RCMCSE method is capable of reducing the number of undefined entropy and providing more accurate entropies. Moreover, the differences on inducing undefined entropy between results for US stock indices & two Chinese mainland stock indices and results for US stock indices & HSI demonstrate a much closer relation between US stock markets and HSI than between US stock markets and two Chinese mainland stock markets. Hence, it can be concluded that RCMCSE is more applicable for the study between US and Chinese stock markets. [ABSTRACT FROM AUTHOR]

• The study of anti-Ramsey number a r (G , H) was initiated by Erdőos et al. in 1973 for G ≅ K n , which is close related to Turan number. • Give an algorithm to obtain the upper bound or lower bound of anti-Ramsey numbers. • Determine the exact anti-Ramsey numbers for cycles C 5 and C 6 in the generalized Petersen graphs, respectively. For H ⊆ G , the anti-Ramsey number a r (G , H) is the maximum number of colors in an edge-coloring of G such that each subgraph isomorphic to H has at least two edges in the same color. The study of anti-Ramsey number a r (K n , H) was introduced by Erdős et al. in 1973, and plentiful results were researched for some special graph H. Later, the problem was extended to a r (G , H) when replacing K n by other graph G such as hypergraph, complete split graph, regular bipartite graph, triangulation and so on. In this paper, we consider a generalized Petersen graph P n , k as the host graph and determine the exact anti-Ramsey numbers for cycles C 5 and C 6 in P n , k , respectively. [ABSTRACT FROM AUTHOR]

We consider an earlier much studied quasi-order, defined in terms of matching numbers of graphs, and apply it to graph complements. We establish four transformations on the complements of graphs that increase or decrease the matching numbers accordingly. Several applications of these results are put forward. [ABSTRACT FROM AUTHOR]

Let I ( G ) be a topological index of a graph. If I ( G + e ) < I ( G ) (or I ( G + e ) > I ( G ) , respectively) for each edge e ∉ G , then I ( G ) decreases (or increases, respectively) with addition of edges. In this paper, we determine the extremal values of some topological indices which decrease or increase with addition of edges, and characterize the corresponding extremal graphs in bipartite graphs with a given matching number. [ABSTRACT FROM AUTHOR]

The Wiener polarity index W p ( G ) of a graph G is the number of unordered pairs of vertices { u ,  v } in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in [Y. Zhang, Y. Hu, 2016] [38]. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus–Gaddum-type results for W p ( G ). Our lower bound on W p ( G ) + W p ( G ¯ ) is always better than the previous lower bound given by Zhang and Hu. [ABSTRACT FROM AUTHOR]

In this paper, based on block style spectral decomposition of the block skew circulant with circulant blocks (BSCCB) matrix, the structure perturbation is discussed, which includes the condition number and relative error of the BSCCB linear system. Then the optimal backward perturbation bound of the BSCCB linear system is analyzed. Simultaneously, the algorithm for the optimal backward perturbation bound is presented. At the end of the paper, a numerical example is provided to verify the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper we present a non-stationary binary three-point approximating subdivision scheme which can generate a wide variety of C 3 -continuous limit curves with different initial control parameter α 0 . The continuity of the scheme is proved by using the conclusion in Dyn and Levin, (1995) [5]. The advantage of the scheme is illustrated by a number of examples. [ABSTRACT FROM AUTHOR]

For S ⊆ G , let κ ( S ) denote the maximum number r of edge-disjoint trees T 1 , T 2 , … , T r in G such that V ( T i ) ∩ V ( T j ) = S for any i , j ∈ { 1 , 2 , ⋯ , r } and i ≠ j . For every 2 ≤ k ≤ n , the generalized k - connectivity of G κ k ( G ) is defined as the minimum κ ( S ) over all k -subsets S of vertices, i.e., κ k ( G ) = min { κ ( S ) | S ⊆ V ( G ) a n d | S | = k } . Clearly, κ 2 ( G ) corresponds to the traditional connectivity of G . The generalized k -connectivity can serve for measuring the capability of a network G to connect any k vertices in G . Cayley graphs have been used extensively to design interconnection networks. In this paper, we restrict our attention to two classes of Cayley graphs, the star graphs S n and the bubble-sort graphs B n , and investigate the generalized 3-connectivity of S n and B n . We show that κ 3 ( S n ) = n − 2 and κ 3 ( B n ) = n − 2 . [ABSTRACT FROM AUTHOR]

In terms of the number of participants, almost each collective action has a life cycle where the number grows from zero to its peak where its maximum potential power or influence is acquired, then it decreases to zero eventually. Therefore, we concentrate on modeling, simulating, and predicting the peaks. The model is constructed based on previous models, and the data is collected from simulations. Preliminarily, it suggests that there exists a peak for collective action when its “jointness of supply” is less than one. Under complete homogeneity, the ideal peak is calculated and the ideal peaks function (IPF) is obtained. Then, heterogeneity is introduced into to the model, and the form of real peaks function (RPF) can be obtained based on simulations and statistical methods. For those who intend to organize a collective action and increase the peak of participants should take measures, such as ideology, leadership, and propagation, to enhance homogeneity or try to reduce heterogeneity. [ABSTRACT FROM AUTHOR]

Kim et al. [9] studied some special polynomials and numbers which are closely related to Changhee polynomials and numbers, and Park et al. (2014) [17] studied the twisted Changhee polynomials and numbers. In this paper we consider the twisted Changhee polynomials and numbers. From these polynomials and numbers, we derive some identities. Furthermore, we investigate the higher-order twisted Changhee polynomials and numbers and also discuss some computations of zeros of the twisted Changhee polynomials. [ABSTRACT FROM AUTHOR]

The Wiener polarity index W p ( G ) of a graph G is the number of unordered pairs of vertices { u, v } of G such that the distance between u and v is 3. In this paper, we study the Nordhaus–Gaddum-type inequality for the Wiener polarity index of a graph G of order n . Due to concerns that both W p ( G ) and W p ( G ¯ ) are nontrivial only when d i a m ( G ) = 3 and d i a m ( G ¯ ) = 3 , we firstly consider the crucial case and get that 2 ≤ W p ( G ) + W p ( G ¯ ) ≤ ⌈ n 2 ⌉ ⌊ n 2 ⌋ − n + 2 . Moreover, the bounds are best possible, and the corresponding extremal graphs are also presented. Then we generalize the results to all connected simple graphs. Furthermore, we discuss the Nordhaus–Gaddum-type inequality for trees and unicyclic graphs, respectively. [ABSTRACT FROM AUTHOR]

This paper investigates the Pareto optimization scheduling problem on a single machine with two competing agents A and B in which agent A wants to minimize the number of tardy A -jobs and agent B wants to minimize the maximum cost of B -jobs. In the literature, the constrained optimization problem of minimizing the number of tardy A -jobs under the restriction that the maximum cost of B -jobs is bounded is solved in polynomial time. This implies that the corresponding Pareto optimization scheduling problem can be solved in a weakly polynomial time. In this paper, by presenting a new algorithm for the constrained optimization problem, we provide a strongly polynomial-time algorithm for the corresponding Pareto optimization scheduling problem. Experimentation results show that the proposed algorithm for the considered problem is efficient. [ABSTRACT FROM AUTHOR]

The Wiener index W ( G ) and the edge-Wiener index W e ( G ) of a graph G are defined as the sum of all distances between pairs of vertices in a graph G and the sum of all distances between pairs of edges in G , respectively. The Wiener index, due to its correlation with a large number of physico-chemical properties of organic molecules and its interesting and non-trivial mathematical properties, has been extensively studied in both theoretical and chemical literature. The edge-Wiener index of G is nothing but the Wiener index of the line graph of G . The concept of line graph has been found various applications in chemical research. In this paper, we show that if G is a catacondensed hexagonal system with h hexagons and has t linear segments S 1 , S 2 , … , S t of lengths l ( S i ) = l i ( 1 ≤ i ≤ t ) , then W e ( G ) = 25 16 W ( G ) + 1 16 ( 120 h 2 + 94 h + 29 ) − 1 4 ∑ i = 1 t ( l i − 1 ) 2 . Our main result reduces the problems on the edge-Wiener index to those on the Wiener index in the catacondensed hexagonal systems, which makes the former ones easier. [ABSTRACT FROM AUTHOR]

In this paper, we study the Wiener index (i.e., the total distance or the transmission number) of not necessarily strongly connected digraphs. In order to do so, if there is no directed path from u to v , we follow the convention d ( u , v ) = 0 , which was independently introduced in several studies of directed networks. Under this assumption we naturally generalize the Wiener theorem, as well as a relation between the Wiener index and betweenness centrality to directed graphs. We formulate and study conjectures about orientations of undirected graphs which achieve the extremal values of Wiener index. [ABSTRACT FROM AUTHOR]

We propose and analyze a stage-structured predator–prey model in which the time from birth to maturity is directly related to the number of individuals present. Individuals mature more quickly when there are fewer of them around. The state dependent time delay is taken to be an increasing differentiable bounded function. In this research, we study the dynamics of our model analytically. We present results on positivity and boundedness of all populations. Criteria for the existence of all equilibria and uniqueness of a positive equilibrium are given. In order to observe the effect of state-dependent maturation delay, local stability analysis around all equilibria of the proposed model is discussed due to variation of maturation delay. Also, global stability of trivial and the boundary equilibria is investigated, using Liapunov functional and LaSalle invariant principle. To investigate the effect of state-dependent maturation delay and the harvesting effort of all species we carried out numerical simulations. [ABSTRACT FROM AUTHOR]

The standard goal programming problem allows decision maker to assign an aspiration level to an objective function. In real life decision making problems, the decision maker always seeks for suitable aspiration level i.e. “the more suitable the better”. Therefore, a decision maker is allowed to assign multiple number of aspiration levels to an objective function. The aim of the decision maker is to select an appropriate aspiration level for an objective function that minimizes the deviations between the achievement of goal and the aspiration levels. The traditional goal programming techniques cannot be used for solving such type of multi-choice goal programming problem. This paper presents an equivalent model of the multi-choice goal programming problem by using Vandermonde’s interpolating polynomial, binary variables and least square approximation method. The equivalent model is solved by existing method/software. Two illustrative examples are presented in support of the proposed methodology. [ABSTRACT FROM AUTHOR]

In QSAR/QSPR study, topological indices such as Shultz index, generalized Randic index, Zagreb index, general sum-connectivity index, atom-bond connectivity (ABC) index and geometric-arithmetic (GA) index are utilized to guess the bioactivity of chemical compounds. A topological index in fact relates a chemical structure with a numeric number. Graph theory has established a significant use in this area of research. In this paper we computed ABC 4 and GA 5 indices of the line graph of tadpole, wheel and ladder graphs using the notion of subdivision. [ABSTRACT FROM AUTHOR]

We investigate the graph with minimal Zeroth-order general Randić index in terms of its order n , pendent number N 1 and cyclomatic number γ ≥ 0. Extremal graphs were completely characterized for cases of γ = 0 , 1 , 2 , which can be directly extended for graphs with cyclomatic number γ ≥ 3. [ABSTRACT FROM AUTHOR]

We consider a multi-server queueing system with two types of customers operating in the Markovian random environment. Under the fixed state of the random environment, the arrival flow is described by the marked Markovian arrival process. Type 1 customers have preemptive priority over type 2 customers. Type 1 customer is lost only if, at its arrival moment, all servers are busy by type 1 customers. The service times of both types of customers have an exponential distribution with the rate depending on the type of a customer. Type 2 customer is accepted for service if the number of busy servers at its arrival epoch does not exceed a fixed threshold. Otherwise, the arriving customer makes a randomized choice to leave the system permanently (to balk) or to join so called orbit and try to obtain service later. The inter-retrial times have an exponential distribution. Customers in orbit can be impatient and may leave the system after an exponentially distributed amount of time. Due to preemptive priority of type 1 customers, service of type 2 customer can be interrupted. In this case, the interrupted customer makes a randomized choice to leave the system permanently or go into orbit. When the random environment changes its state, immediately the following parameters of the system change their value: the total number of servers, the number of servers available for type 2 customers, the matrices defining the arrival process, the rates of service of customers, the rate of retrials, the impatience intensity, the probability of balking the system at the arrival moment or the moment of termination of service of type 2 customer. Behavior of the system is described by the level dependent multi-dimensional Markov chain that belongs to the class of asymptotically quasi-Toeplitz Markov chains. This allows us to derive the ergodicity condition for this Markov chain and compute its stationary distribution. The main performance measures of the system are expressed via the stationary state probabilities. Numerical illustrations are presented. [ABSTRACT FROM AUTHOR]

Symbolic methods of umbral nature are exploited to derive series expansion for the products of Bessel functions. It is shown that the product of two cylindrical Bessel functions can be written in terms of Jacobi polynomials. The procedure is extended to products of an arbitrary number of functions and the link with previous researchers is discussed. We show that the technique we propose and the use of the Ramanujan master theorem allow the derivation of integrals of practical interest. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of cubic planar polynomial differential system with non-rational first integral. Using the averaging method at any order, we bound the maximum number of limit cycles which bifurcate from the periodic annulus of the origin when we perturb them inside the class of all polynomial systems of degree n . [ABSTRACT FROM AUTHOR]

The exact asymptotic computational complexity for a problem of indexing cells on a uniform grid intersecting with a union of C ( 1 ) surfaces has been proven. The computational complexity of the recursive partition indexing algorithm, utilized for simulation of septated tissues, is derived and the algorithm is demonstrated as being asymptotically optimal. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend and refine the Ramanujan–Lodge harmonic number expansion into negative powers of a triangular number. We construct a faster asymptotic series and some new sharp inequalities for the harmonic numbers. [ABSTRACT FROM AUTHOR]

It is a classical result that the zeros of the Bell polynomials are real and negative. In this study we deal with the asymptotic growth of the leftmost zeros of the Bell polynomials and generalize the results for the r -Bell polynomials, too. In addition, we offer a heuristic approach for the approximation of the maximizing index of the Stirling numbers of both kind. [ABSTRACT FROM AUTHOR]

The objective of this paper is to construct wave packet frames on local fields of positive characteristic. A necessary and sufficient condition for the wave packet system D p j T u ( n ) a E u ( m ) b ψ j ∈ Z , m , n ∈ N 0 to be a frame for L 2 ( K ) is given by means of the Fourier transform. [ABSTRACT FROM AUTHOR]

k -Tridiagonal matrices have attracted much attention in recent years, which are a generalization of tridiagonal matrices. In this note, a breakdown-free numerical algorithm of O ( n ) is presented for computing the determinants and the permanents of k -tridiagonal matrices. Even though the algorithm is not a symbolic algorithm, it never suffers from breakdown. Furthermore, it produces exact values when all entries of the k -tridiagonal matrices are given in integer. In addition, the algorithm can be simplified for a general symmetric Toeplitz case, and it generates the k th powers of Fibonacci, Pell, and Jacobsthal numbers for a certain symmetric Toeplitz case. [ABSTRACT FROM AUTHOR]

The paper concerns the decompositions of polynomials onto iterated integrals. This is a continuation of our previous paper (Lorenc, submitted for publication), in which the existence of such decomposition for the Faulhaber polynomials is proven. In the current paper we prove the basic theorem (Theorem 2) presenting the necessary and sufficient conditions for the existence of such decomposition. We discuss these conditions in the wider context of theory of the real and complex polynomials. A number of exemplary decompositions onto iterated integrals of the known classical kinds of polynomials are also presented. [ABSTRACT FROM AUTHOR]