Band 17, Heft 1

It is shown that, if all weak solutions of the evolution equation y′(t)=Ay(t),t≥0, $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array} $$ with a scalar type spectral operator A in a complex Banach space are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded.

By σ ∈ S km we denote a permutation of the cycle-type k m and also the induced automorphism permuting subscripts of free generators in the free group F km . It is known that the centralizer of the permutation σ in S km is isomorphic to a wreath product Z k ≀ S m and is generated by its two subgroups: the first one is isomorphic to Zkm $\begin{array}{} \displaystyle Z_k^m \end{array}$, the direct product of m cyclic groups of order k , and the second one is S m . We show that the centralizer of the automorphism σ ∈ Aut( F km ) is generated by its subgroups isomorphic to Zkm $\begin{array}{} \displaystyle Z_k^m \end{array}$ and Aut( F m ).

There are three types of results in this paper. The first, extending a representation theorem on a conformal mapping that omits two values of equal modulus. This was due to Brickman and Wilken. They constructed a representation as a convex combination with two terms. Our representation constructs convex combinations with unlimited number of terms. In the limit one can think of it as an integration over a probability space with the uniform distribution. The second result determines the sign of ℜ L ( z 0 ( f ( z )) 2 ) up to a remainder term which is expressed using a certain integral that involves the Löwner chain induced by f ( z ), for a support point f ( z ) which maximizes ℜ L . Here L is a continuous linear functional on H ( U ), the topological vector space of the holomorphic functions in the unit disk U = { z ∈ ℂ | | z | < 1}. Such a support point is known to be a slit mapping and f ( z 0 ) is the tip of the slit ℂ − f ( U ). The third demonstrates some properties of support points of the subspace S n of S . S n contains all the polynomials in S of degree n or less. For instance such a support point p ( z ) has a zero of its derivative p ′( z ) on ∂U .

Given positive integers n , n ′ and k , we investigate the Möbius-Bernoulli numbers M k ( n ), double Möbius-Bernoulli numbers M k (n, n ′), and Möbius-Bernoulli polynomials M k ( n )( x ). We find new identities involving double Möbius-Bernoulli, Barnes-Bernoulli numbers and Dedekind sums. In part of this paper, the Möbius-Bernoulli polynomials M k ( n )( x ), can be interpreted as critical values of the following Dirichlet type L -function LHM(s;n,x):=∑d|n∑m=0∞μ(d)(md+x)s(for Re(s)>1), $$\begin{array}{} \displaystyle L_{HM}(s;n,x):=\sum_{d|n} \sum_{m= 0}^\infty \frac{\mu(d)}{(md+x)^s} \, \, \text{(for Re} (s) \gt 1), \end{array} $$ which has analytic continuation to the whole s -complex plane, where μ is the Möbius function.

As the real common generalisations of both orthodox transversals and adequate transversals in the abundant case, the concept of refined generalised quasi - adequate transversal , for short, RGQA transversal was introduced by Kong and Wang. In this paper, an interesting characterization for a generalised quasi-adequate transversal to be refined is acquired. It is shown that the product of every two quasi-ideal RGQA transversals of the abundant semigroup S satisfying the regularity condition is a quasi-ideal RGQA transversal of S and that all quasi-ideal RGQA transversals of S compose a rectangular band. The related results concerning adequate transversals are generalised and enriched.

This paper provides some new characterizations of the solution sets for non-differentiable generalized convex fuzzy optimization problem. Firstly, we introduce some new generalized convex fuzzy functions and discuss the relationships among them. Secondly, some properties of these new generalized convex fuzzy functions are given. Finally, as applications, some characterizations of the solution sets for non-differentiable generalized convex fuzzy optimization problem are obtained.

The concept of generalized digroup was proposed by Salazar-Díaz, Velásquez and Wills-Toro in their paper “Generalized digroups” as a non trivial extension of groups. In this way, many concepts and results given in the category of groups can be extended in a natural form to the category of generalized digroups. The aim of this paper is to present the construction of the free generalized digroup and study its properties. Although this construction is vastly different from the one given for the case of groups, we will use this concept, the classical construction for groups and the semidirect product to construct the tensor generalized digroup as well as the semidirect product of generalized digroups. Additionally, we give a new structural result for generalized digroups using compatible actions of groups and an equivariant map from a group set to the group corresponding to notions of associative dialgebras and augmented racks.

In this article, we consider the long-time behavior of solutions for the wave equation with nonlinear damping and linear memory. Within the theory of process on time-dependent spaces, we verify the process is asymptotically compact by using the contractive functions method, and then obtain the existence of the time-dependent attractor in H01(Ω )× L2(Ω )× Lμ 2(R+;H01(Ω )) $\begin{array}{} H^{1}_0({\it\Omega})\times L^{2}({\it\Omega})\times L^{2}_{\mu}(\mathbb{R}^{+};H^{1}_0({\it\Omega})) \end{array}$.

In this paper, a new improved smoothing Newton algorithm for the nonlinear complementarity problem was proposed. This method has two-fold advantages. First, compared with the classical smoothing Newton method, our proposed method needn’t nonsingular of the smoothing approximation function; second, the method also inherits the advantage of the classical smoothing Newton method, it only needs to solve one linear system of equations at each iteration. Without the need of strict complementarity conditions and the assumption of P 0 property, we get the global and local quadratic convergence properties of the proposed method. Numerical experiments show that the efficiency of the proposed method.

A discrete nonlinear almost periodic multispecies competitive system with delays and feedback controls is proposed and investigated. We obtain sufficient conditions to ensure the permanence of the system. Also, we establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. In additional, an example together with its numerical simulation are presented to illustrate the feasibility of the main result.

Let K be an imaginary quadratic field, and let 𝔣 be a nontrivial integral ideal of K . Hasse and Ramachandra asked whether the ray class field of K modulo 𝔣 can be generated by a single value of the Weber function. We completely resolve this question when 𝔣 = ( N ) for any positive integer N excluding 2, 3, 4 and 6.

In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

In this paper we derive the formula of ( P + Q ) D under the conditions Q ( P + Q ) P ( P + Q ) = 0, P ( P + Q ) P ( P + Q ) = 0 and QPQ 2 = 0. Then, a corollary is given which satisfies the conditions ( P + Q ) P ( P + Q ) = 0 and QPQ 2 = 0. Meanwhile, we show that the additive formula provided by Bu et al. (J. Appl. Math. Comput. 38 (2012) 631-640) is not valid for all matrices which satisfies the conditions ( P + Q ) P ( P + Q ) = 0 and QPQ 2 = 0. Also, the representation can be simplified from Vi š nji ć (Filomat 30 (2016) 125-130) which satisfies given conditions. Furthermore, we apply our result to establish a new representation for the Drazin inverse of a complex block matrix having generalized Schur complement equal to zero under some conditions. Finally, a numerical example is given to illustrate our result.

The aim of the paper is to prove the completeness theorem for probability models with finitely many valued measure.

This paper is devoted to the existence of a periodic solution for ϕ -Laplacian neutral differential equation as follows (ϕ (x(t)− cx(t− τ ))′)′=f(t,x(t),x′(t)). $$\begin{array}{} (\phi(x(t)-cx(t-\tau))')'=f(t,x(t),x'(t)). \end{array}$$ By applications of an extension of Mawhin’s continuous theorem due to Ge and Ren, we obtain that given equation has at least one periodic solution. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from the corresponding ones of the known literature.

Let M be a closed smooth Riemannian manifold and let f : M → M be a diffeomorphism. We show that if f has the C 1 robustly asymptotic orbital shadowing property then it is an Anosov diffeomorphism. Moreover, for a C 1 generic diffeomorphism f , if f has the asymptotic orbital shadowing property then it is a transitive Anosov diffeomorphism. In particular, we apply our results to volume-preserving diffeomorphisms.

We study a continued fraction X ( τ ) of order six by using the modular function theory. We first prove the modularity of X ( τ ), and then we obtain the modular equation of X ( τ ) of level n for any positive integer n ; this includes the result of Vasuki et al. for n = 2, 3, 5, 7 and 11. As examples, we present the explicit modular equation of level p for all primes p less than 19. We also prove that the ray class field modulo 6 over an imaginary quadratic field K can be obtained by the value X 2 ( τ ). Furthermore, we show that the value 1/ X ( τ ) is an algebraic integer, and we present an explicit procedure for evaluating the values of X ( τ ) for infinitely many τ ’s in K .

The solutions to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas are constructed for all kinds of situations by using the method of phase plane analysis. The asymptotic limits of solutions to the Riemann problem for the relativistic extended Chaplygin Euler system are investigated in detail when the pressure given by the equation of state of extended Chaplygin gas becomes that of the pressureless gas. During the process of vanishing pressure, the phenomenon of concentration can be identified and analyzed when the two-shock Riemann solution tends to a delta shock wave solution as well as the phenomenon of cavitation also being captured and observed when the two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution with a vacuum state between them.

The nonlinear stability and the existence of periodic orbits of the equilibrium states of the Clebsch’s system are discussed.. Numerical integration using the Lie-Trotter integrator and the analytic approximate solutions using Multistage Optimal Homotopy Asymptotic Method are presented, too.

The degree-based topological indices are numerical graph invariants which are used to correlate the physical and chemical properties of a molecule with its structure. Para-line graphs are used to represent the structures of molecules in another way and these representations are important in structural chemistry. In this article, we study certain well-known degree-based topological indices for the para-line graphs of V -Phenylenic 2 D lattice, V -Phenylenic nanotube and nanotorus by using the symmetries of their molecular graphs.

In order to begin an approach to the structure of arbitrary Lie color triple systems, (with no restrictions neither on the dimension nor on the base field), we introduce the class of split Lie color triple systems as the natural generalization of split Lie triple systems. By developing techniques of connections of roots for this kind of triple systems, we show that any of such Lie color triple systems T with a symmetric root system is of the form T = U + ∑ [ α ]∈ Λ 1 /∼ I [ α ] with U a subspace of T 0 and any I [ α ] a well described (graded) ideal of T , satisfying { I [ α ] , T , I [ β ] } = 0 if [ α ] ≠ [ β ]. Under certain conditions, in the case of T being of maximal length, the simplicity of the triple system is characterized.

Based on the relationship between probability operators and curve/surface modeling, a new kind of surface modeling method is introduced in this paper. According to a kind of bivariate Meyer-König-Zeller operator, we study the corresponding basis functions called triangular Meyer-König-Zeller basis functions which are defined over a triangular domain. The main properties of the basis functions are studied, which guarantee that the basis functions are suitable for surface modeling. Then, the corresponding triangular surface patch called a triangular Meyer-König-Zeller surface patch is constructed. We prove that the new surface patch has the important properties of surface modeling, such as affine invariance, convex hull property and so on. Finally, based on given control vertices, whose number is finite, a truncated triangular Meyer-König-Zeller surface and a redistributed triangular Meyer-König-Zeller surface are constructed and studied.

The Conway groups are the three sporadic simple groups Co 1 , Co 2 and Co 3 . There are total of 22 maximal subgroups of Co 1 and generators of 6 maximal subgroups are provided in web Atlas of finite simple groups. The aim of this paper is to give generators of remaining 16 maximal subgroups.

In this work, we will introduce two novel positivity preserving operator splitting nonstandard finite difference (NSFD) schemes for the numerical solution of SEIR reaction diffusion epidemic model. In epidemic model of infection diseases, positivity is an important property of the continuous system because negative value of a subpopulation is meaningless. The proposed operator splitting NSFD schemes are dynamically consistent with the solution of the continuous model. First scheme is conditionally stable while second operator splitting scheme is unconditionally stable. The stability of the diffusive SEIR model is also verified numerically with the help of Routh-Hurwitz stability condition. Bifurcation value of transmission coefficient is also carried out with and without diffusion. The proposed operator splitting NSFD schemes are compared with the well-known operator splitting finite difference (FD) schemes.

In this paper we use the notion of derived sets to study convex spaces. By axiomatizing the derived sets on convex spaces, we define c-derived operators and restricted c-derived operators. Results show that convex structures can be characterized in terms of c-derived operators. Furthermore, the link between c-derived operators and Shi’s m-derived operators is studied. Specifically, it is proved that a c-derived operator is an m-derived operator if and only if it satisfies the Exchange Law. At last, we show an application of c-derived operators to anti-matroids.

In this study we examine some combinatorial properties of the Arf semigroup. In previous work, the author and Karakaş, Gümüşbaş defined an Arf partition of a positive integer n . Here, we continue this work and give new results on Arf partitions. In particular, we analyze the relation among an Arf partition, its Young dual diagram, and the corresponding rational Young diagram. Additionally, this study contains some results that present the relations between partitions and Arf semigroup polynomials.

Singular overpartition functions were defined by Andrews. Let C k , i ( n ) denote the number of ( k , i )-singular overpartitions of n , which counts the number of overpartitions of n in which no part is divisible by k and only parts ± i (mod k ) may be overlined. A number of congruences modulo 3, 9 and congruences modulo powers of 2 for C k , i ( n ) were discovered by Ahmed and Baruah, Andrews, Chen, Hirschhorn and Sellers, Naika and Gireesh, Shen and Yao for some pairs ( k , i ). In this paper, we prove some congruences modulo powers of 2 for C 48, 6 ( n ) and C 48, 18 ( n ).

On estimations of the lower and upper bounds for the spectral and nuclear norm of a tensor, Li established neat bounds for the two norms based on regular tensor partitions, and proposed a conjecture for the same bounds to be hold based on general tensor partitions [Z. Li, Bounds on the spectral norm and the nuclear norm of a tensor based on tensor partition, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1440-1452]. Later, Chen and Li provided a solution to the conjecture [Chen B., Li Z., On the tensor spectral p -norm and its dual norm via partitions]. In this short paper, we present a concise and different proof for the validity of the conjecture, which also offers a new and simpler proof to the bounds of the spectral and nuclear norms established by Li for regular tensor partitions. Two numerical examples are provided to illustrate tightness of these bounds.

In this paper, we establish the axiomatic conditions of hull operators and introduce the category of interval spaces. We also investigate their relations with convex spaces from a categorical sense. It is shown that the category CS of convex spaces is isomorphic to the category HS of hull spaces, and they are all topological over Set . Also, it is proved that there is an adjunction between the category IS of interval spaces and the category CS of convex spaces. In particular, the category CS(2) of arity 2 convex spaces can be embedded in IS as a reflective subcategory.

Overf the last few years, by utilizing Mawhin’s continuation theorem of coincidence degree theory and Lyapunov functional, many scholars have been concerned with the global asymptotical stability of positive periodic solutions for the non-linear ecosystems. In the real world, almost periodicity is usually more realistic and more general than periodicity, but there are scarcely any papers concerning the issue of the global asymptotical stability of positive almost periodic solutions of non-linear ecosystems. In this paper we consider a kind of delayed two-species competitive model with stage structure. By means of Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions are obtained for the existence of at least one positive almost periodic solutions for the above model with nonnegative coefficients. Furthermore, the global asymptotical stability of positive almost periodic solution of the model is also studied. The work of this paper extends and improves some results in recent years. An example and simulations are employed to illustrate the main results of this paper.

Recently, Dai proved new infinite families of congruences modulo 2 for broken 11-diamond partition functions by using Hecke operators. In this note, we establish new parity results for broken 11-diamond partition functions. In particular, we generalize the congruences due to Dai by utilizing an identity due to Newman. Furthermore, we prove some strange congruences modulo 2 for broken 11-diamond partition functions. For example, we prove that if p is a prime with p ≠ 23 and Δ 11 (2 p + 1) ≡ 1 (mod 2), then for any k ≥ 0, Δ 11 (2 p 3 k +3 + 1) ≡ 1 (mod 2), where Δ 11 ( n ) is the number of broken 11-diamond partitions of n .

The main purpose of this paper is to use analytic methods and properties of quartic Gauss sums to study a special fourth power mean of a two-term exponential sums mod p , with p an odd prime, and prove interesting new identities. As an application of our results, we also obtain a sharp asymptotic formula for the fourth power mean.

We introduce the notion of K-ideals associated with Kuratowski partitions. Using new operations on complete ideals we show connections between K-ideals and precipitous ideals and prove that every complete ideal can be represented by some K-ideal.

Soft set theory and rough set theory are two new tools to discuss uncertainty. Graph theory is a nice way to depict certain information. Particularly soft graphs serve the purpose beautifully. In order to discuss uncertainty in soft graphs, some new types of graphs called soft covering based rough graphs are introduced. Several basic properties of these newly defined graphs are explored. Applications of soft covering based rough graphs in decision making can be very fruitful. In this regard an algorithm has been proposed.

Let { X n , k , 1 ≤ k ≤ m n , n ≥ 1} be an array of independent random variables from the Pareto distribution. Let X n ( k ) be the k th largest order statistic from the n th row of the array and set R n , i n , j n = X n ( j n ) / X n ( i n ) where j n < i n . The aim of this paper is to study the complete convergence of the ratios { R n , i n , j n , n ≥ 1}.

In the present paper, the sufficient and necessary conditions of the complete convergence and complete moment convergence for ρ͠ -mixing random variables are established, which extend some well-known results.

In this article the properties of attractors of dynamical systems in locally compact metric space are discussed. Existing conditions of attractors and related results are obtained by the near isolating block which we present.

In this paper we study the asymptotic behavior for a class of stochastic retarded strongly damped wave equation with additive noise on a bounded smooth domain in ℝ d . We get the existence of the random attractor for the random dynamical systems associated with the equation.

In this paper, we introduce a new generalization of λ -Bernstein operators based on q -integers, we obtain the moments and central moments of these operators, establish a statistical approximation theorem and give an example to show the convergence of these operators to f ( x ). It can be seen that in some cases the absolute error bounds are smaller than the case of classical q -Bernstein operators to f ( x ).

In this paper the authors prove the existence as well as approximations of the solutions for the Bagley-Torvik equation admitting only the existence of a lower (coupled lower and upper) solution. Our results rely on an appropriate fixed point theorem in partially ordered normed linear spaces. Illustrative examples are included to demonstrate the validity and applicability of our technique.

A Cayley graph Γ is said to be arc - transitive if its full automorphism group Aut Γ is transitive on the arc set of Γ . In this paper we give a characterization of pentavalent arc-transitive Cayley graphs on a class of Frobenius groups with soluble vertex stabilizer.

The main purpose of this paper is using the analytic method, the properties of trigonometric sums and Gauss sums to study the computational problem of one kind hybrid power mean involving two different trigonometric sums, and give an interesting computational formula for it.

A graph ℘ is said to be edge - magic total (EMT if there is a bijection Υ : V ( ℘ ) ∪ E ( ℘ ) → {1, 2, …, | V ( ℘ ) ∪ E ( ℘ )|} s.t ., Υ ( υ ) + Υ ( υν ) + Υ ( ν ) is a constant for every edge υν ∈ E ( ℘ ). An EMT graph ℘ will be called strong edge - magic total (SEMT) if Υ ( V ( ℘ )) = {1, 2, …, | V ( ℘ )|}. The SEMT strength , sm ( ℘ ), of a graph ℘ is the minimum of all magic constants a ( Υ ), where the minimum runs over all the SEMT valuations of ℘ , this minimum is defined only if the graph has at least one such SEMT valuation. Furthermore, the SEMT deficiency of a graph ℘ , μ s ( ℘ ), is either the minimum non-negative integer n such that ℘ ∪ nK 1 is SEMT or +∞ if there will be no such integer n . In this paper, we will present the strong edge-magicness and deficiency of disjoint union of 2-sided generalized comb with bistar, path and caterpillar, moreover we will evaluate the SEMT strength for 2-sided generalized comb.

Suppose that λ 1 , λ 2 , λ 3 , λ 4 , λ 5 are nonzero real numbers, not all of the same sign, λ 1 / λ 2 is irrational, λ 2 / λ 4 and λ 3 / λ 5 are rational. Let η real, and ε > 0. Then there are infinitely many solutions in primes p j to the inequality |λ 1p1+λ 2p22+λ 3p33+λ 4p44+λ 5p55+η |< (maxpjj)− 1/32+ε $\begin{array}{} \displaystyle |\lambda_1p_1+\lambda_2p_2^2+\lambda_3p_3^3+\lambda_4p_4^4+\lambda_5p_5^5+\eta| \lt (\max{p_j^j})^{-1/32+\varepsilon} \end{array}$. This improves an earlier result under extra conditions of λ j .

We prove a general version of Bezout’s form of the Nullstellensatz for arbitrary fields. The corresponding sufficient and necessary condition only involves the local existence of multi-valued roots for each of the polynomials belonging to the ideal in consideration. Finally, this version implies the standard Nullstellensatz when the coefficient field is algebraically closed.

This paper presents a new representation of α -openness, α -continuity, α -irresoluteness, and α -compactness based on L -fuzzy α -open operators introduced by Nannan and Ruiying [1] and implication operation. The proposed representation extends the properties of α -openness, α -continuity, α -irresoluteness, and α -compactness to the setting of L -fuzzy pretopological spaces based on graded concepts. Moreover, we introduce and establish the relationships among the new concepts.

In this paper, we study the approximation of π through the semiperimeter or area of a random n -sided polygon inscribed in a unit circle in ℝ 2 . We show that, with probability 1, the approximation error goes to 0 as n → ∞, and is roughly sextupled when compared with the classical Archimedean approach of using a regular n -sided polygon. By combining both the semiperimeter and area of these random inscribed polygons, we also construct extrapolation improvements that can significantly speed up the convergence of these approximations.

Given a distribution of pebbles on the vertices of a connected graph G , a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The optimal pebbling number of G , denoted by π opt ( G ), is the smallest number m such that for some distribution of m pebbles on G , one pebble can be moved to any vertex of G by a sequence of pebbling moves. Let P k be the path on k vertices. Snevily defined the n – k spindle graph as follows: take n copies of P k and two extra vertices x and y , and then join the left endpoint (respectively, the right endpoint) of each P k to x (respectively, y ), the resulting graph is denoted by S ( n , k ), and called the n – k spindle graph . In this paper, we determine the optimal pebbling number for spindle graphs.

The notion of MBJ-neutrosophic ideal is introduced, and its properties are investigated. Conditions for an MBJ-neutrosophic set to be an MBJ-neutrosophic ideal are provided. In a BCK/BCI -algebra, a condition for an MBJ-neutrosophic set to be an MBJ-neutrosophic ideal is given. In a BCK -algebra, a condition for an MBJ-neutrosophic subalgebra to be an MBJ-neutrosophic ideal is given. In a BCI -algebra, conditions for an MBJ-neutrosophic ideal to be an MBJ-neutrosophic subalgebra are considered. In an ( S )- BCK -algebra, we show that every MBJ-neutrosophic ideal is an MBJ-neutrosophic ∘-subalgebra, and a characterization of an MBJ-neutrosophic ideal is established.

Let G be a finite group and H ≤ G . The authors study the structure of finite groups G having a subgroup H which is maximal in 〈 H , H g 〉 for some g ∈ G . Some results on the structure of 〈 H , H g 〉 and G are set up. Especially, a generalization of Baer′s theorem is established.

A multi-objective linear programming problem (ITF-MOLP) is presented in this paper, in which coefficients of both the objective functions and constraints are interval-typed triangular fuzzy numbers. An algorithm of the ITF-MOLP is provided by introducing the cut set of interval-typed triangular fuzzy numbers and the dominance possibility criterion. In particular, for a given level, the ITF-MOLP is converted to the maximization of the sum of membership degrees of each objective in ITF-MOLP, whose membership degrees are established based on the deviation from optimal solutions of individual objectives, and the constraints are transformed to normal inequalities by utilizing the dominance possibility criterion when compared with two interval-typed triangular fuzzy numbers. Then the equivalent linear programming model is obtained which could be solved by Matlab toolbox. Finally several examples are provided to illuminate the proposed method by comparing with the existing methods and sensitive analysis demonstrates the stability of the optimal solution.

The existing results on the variational inequality problems for fuzzy mappings and their applications were based on Zadeh’s decomposition theorem and were formally characterized by the precise sets which are the fuzzy mappings’ cut sets directly. That is, the existence of the fuzzy variational inequality problems in essence has not been solved. In this paper, the fuzzy variational-like inequality problems is incorporated into the framework of n -dimensional fuzzy number space by means of the new ordering of two n -dimensional fuzzy-number-valued functions we proposed [Fuzzy Sets and Systems 295 (2016) 19-36]. As a theoretical basis, the existence and the basic properties of the fuzzy variational inequality problems are discussed. Furthermore, the relationship between the variational-like inequality problems and the fuzzy optimization problems is discussed. Finally, we investigate the optimality conditions for the fuzzy multiobjective optimization problems.

We revisit a prey-predator model with stage structure for predator, which was proposed by Tapan Kumar Kar. By using the differential inequality theory and the comparison theorem of the differential equation, we show that the prey free equilibrium is globally asymptotically stable under some suitable assumption. Our study shows that although the predator species has other food resource, if the amount of the predator species is too large, it could also do irreversible harm to the prey species, and this could finally lead to the extinction of the prey species. Our result supplement and complement some known results.

In the linear response eigenvalue problem arising from computational quantum chemistry and physics, one needs to compute a few of smallest positive eigenvalues together with the corresponding eigenvectors. For such a task, most of efficient algorithms are based on an important notion that is the so-called pair of deflating subspaces. If a pair of deflating subspaces is at hand, the computed approximated eigenvalues are partial eigenvalues of the linear response eigenvalue problem. In the case the pair of deflating subspaces is not available, only approximate one, in a recent paper [SIAM J. Matrix Anal. Appl., 35(2), pp.765-782, 2014], Zhang, Xue and Li obtained the relationships between the accuracy in eigenvalue approximations and the distances from the exact deflating subspaces to their approximate ones. In this paper, we establish majorization type results for these relationships. From our majorization results, various bounds are readily available to estimate how accurate the approximate eigenvalues based on information on the approximate accuracy of a pair of approximate deflating subspaces. These results will provide theoretical foundations for assessing the relative performance of certain iterative methods in the linear response eigenvalue problem.

Let G be a connected graph and u and v two vertices of G . The hyper-Wiener index of graph G is WW(G)=12∑u,v∈V(G)(dG(u,v)+dG2(u,v)) $\begin{array}{} WW(G)=\frac{1}{2}\sum\limits_{u,v\in V(G)}(d_{G}(u,v)+d^{2}_{G}(u,v)) \end{array}$, where d G ( u , v ) is the distance between u and v . In this paper, we first give the recurrence formulae for computing the hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. We then obtain the sharp upper and lower bounds for the hyper-Wiener index among polyphenyl chains and polyphenyl spiders, respectively. Moreover, the corresponding extremal graphs are determined.

This work is mainly concerned with the exponential stability of time-changed stochastic functional differential equations with Markovian switching. By expanding the time-changed Itô formula and the Razumikhin theorem, we obtain the exponential stability results for the time-changed stochastic functional differential equations with Markovian switching. What’s more, we get many useful stability results by applying our new results to several important types of functional differential equations. Finally, an example is given to demonstrate the effectiveness of the main results.

In this paper, we investigate the relationships between fixed points of meromorphic functions, and their higher order differences and shifts, and generalize the case of fixed points into the more general case for first order difference and shift. Concretely, some estimates on the order and the exponents of convergence of special points of meromorphic functions and their differences and shifts are obtained.

In the present paper, we introduce a new form of generalized Rayleigh distribution called the Alpha Power generalized Rayleigh (APGR) distribution by following the idea of extension of the distribution families with the Alpha Power transformation. The introduced distribution has the more general form than both the Rayleigh and generalized Rayleigh distributions and provides a better fit than the Rayleigh and generalized Rayleigh distributions for more various forms of the data sets. In the paper, we also obtain explicit forms of some important statistical characteristics of the APGR distribution such as hazard function, survival function, mode, moments, characteristic function, Shannon and Rényi entropies, stress-strength probability, Lorenz and Bonferroni curves and order statistics. The statistical inference problem for the APGR distribution is investigated by using the maximum likelihood and least-square methods. The estimation performances of the obtained estimators are compared based on the bias and mean square error criteria by a conducted Monte-Carlo simulation on small, moderate and large sample sizes. Finally, a real data analysis is given to show how the proposed model works in practice.

This paper considers the nonfragile observer-based guaranteed cost finite-time control of discrete-time positive impulsive switched systems(DPISS). Firstly, the positive observer and nonfragile positive observer are designed to estimate the actual state of the underlying systems, respectively. Secondly, by using the average dwell time(ADT) approach and multiple linear co-positive Lyapunov function (MLCLF), two guaranteed cost finite-time controller are designed and sufficient conditions are obtained to guarantee the corresponding closed-loop systems are guaranteed cost finite-time stability(GCFTS). Such conditions can be solved by linear programming. Finally, a numerical example is provided to show the effectiveness of the proposed method.

In this paper, we investigate a partially single-index varying-coefficient model, and suggest two empirical log-likelihood ratio statistics for the unknown parameters in the model. The first statistic is asymptotically distributed as a weighted sum of independent chi-square variables under some mild conditions. It is proved that another statistic, with adjustment factor, is asymptotically standard chi-square under some suitable conditions. These useful statistics could be used to construct the confidence regions of the parameters. A simulation study indicates that, with the increase of sample size, the coverage probability of the confidence region constructed by us gradually approaches the theoretical value.

In this paper, we study the algebraic loop structures on the set of Lie algebra comultiplications. More specifically, we investigate the fundamental concepts of algebraic loop structures and the set of Lie algebra comultiplications which have inversive, power-associative and Moufang properties depending on the Lie algebra comultiplications up to all the possible quadratic and cubic Lie algebra comultiplications. We also apply those notions to the rational cohomology of Hopf spaces.

In the present paper, the authors investigate the two weight, weak-( p , q ) type norm inequalities for a class of sublinear operators 𝓣 γ and their commutators [ b , 𝓣 γ ] on weighted Morrey and Amalgam spaces. What should be stressed is that we introduce the new BMO type space and our results generalize known results before.

In this paper, we consider a nonautonomous two-species impulsive competitive system with infinite delays. By the impulsive comparison theorem and some mathematical analysis, we investigate the permanence, extinction and global attractivity of the system, as well as the influence of impulse perturbation on the dynamic behaviors of this system. For the logistic type impulsive equation with infinite delay, our results improve those of Xuxin Yang, Weibing Wang and Jianhua Shen [Permanence of a logistic type impulsive equation with infinite delay, Applied Mathematics Letters, 24(2011), 420-427]. For the corresponding nonautonomous two-species impulsive competitive system without delays, we discuss its permanence, extinction and global attractivity, which weaken and complement the results of Zhijun Liu and Qinglong Wang [An almost periodic competitive system subject to impulsive perturbations, Applied Mathematics and Computation, 231(2014), 377-385].

The 2-closure G (2) of a permutation group G on a finite set Ω is the largest subgroup of Sym( Ω ) which has the same orbits as G in the induced action on Ω × Ω . In this paper, the 2-closures of certain primitive permutation groups of holomorph simple and holomorph compound types are determined.

In the paper, the authors present some monotonicity properties and some sharp inequalities for the generalized Grötzsch ring function and related elementary functions. Consequently, the authors obtain new bounds for solutions of the Ramanujan generalized modular equation.

This paper establishes the boundedness of the variation operators associated with Riesz transforms and commutators generated by the Riesz transforms and BMO-type functions in the Schrödinger setting on the weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.

Based on the “three bottom line” and stakeholder theory, the paper considers the relationship and cooperation strategy between the government and the supplier and manufacturer of the green supply chain. By constructing the dynamic differential game model, the paper discusses the differences in the optimal effort level, green degree of product, reputation and the optimal benefit under the three situations of noncooperation, government promotion and collaborative cooperation. The results show that the optimal effort level, green degree of product, reputation and the optimal benefit in collaborative cooperation are obviously higher than the situations of non-cooperation and government promotion, and the cooperation of the three parties can promote the development of green supply chain. Government promotion is better than noncooperation. The government plays an active role in improving the optimal benefit and reputation of green supply chain. Finally, the reliability of the proposed proposition is verified by an example analysis, which provides an important reference for improving the efficiency of green supply chain.

The extinction property of a two species competitive stage-structured phytoplankton system with harvesting is studied in this paper. Several sets of sufficient conditions which ensure that one of the components will be driven to extinction are established. Our results supplement and complement the results of Li and Chen [Extinction in periodic competitive stage-structured Lotka-Volterra model with the effects of toxic substances, J. Comput. Appl. Math., 2009, 231(1), 143-153] and Liu, Chen, Luo et al. [Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl., 2002, 274(2), 667-684].

Let ( M , g ) be a non-Kenmotsu ( κ , μ )′-almost Kenmotsu manifold of dimension 2 n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍ n +1 (−4)×ℝ n or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of ( κ , μ )′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

Given a simple graph G , its Laplacian-energy-like invariant LEL ( G ) and incidence energy IE ( G ) are the sum of square root of its all Laplacian eigenvalues and signless Laplacian eigenvalues, respectively. This paper obtains some improved bounds on LEL and IE of the 𝓡-graph and 𝓠-graph for a regular graph. Theoretical analysis indicates that these results improve some known results. In addition, some new lower bounds on LEL and IE of the line graph of a semiregular graph are also given.

In this paper, we study the discrete-time super-replication problem of contingent claims with respect to an acceptable terminal discounted cash flow. Based on the concept of Immediate Profit, i.e., a negative price which super-replicates the zero contingent claim, we establish a weak version of the fundamental theorem of asset pricing. Moreover, time consistency is discussed and we obtain a representation formula for the minimal super-hedging prices of bounded contingent claims.

For a finite noncyclic group G , let Cyc( G ) be the set of elements a of G such that 〈 a , b 〉 is cyclic for each b of G . The noncyclic graph of G is a graph with the vertex set G ∖ Cyc( G ), having an edge between two distinct vertices x and y if 〈 x , y 〉 is not cyclic. In this paper, we classify all finite noncyclic groups whose noncyclic graphs are K 1, n -free, where K 1, n is a star and 3 ≤ n ≤ 6.

In this paper, by means of the implication operator → on a completely distributive lattice M , we define the approximate degrees of M -fuzzifying convex structures, M -fuzzifying closure systems and M -fuzzifying Alexandrov topologies to interpret the approximate degrees to which a mapping is an M -fuzzifying convex structure, an M -fuzzifying closure system and an M -fuzzifying Alexandrov topology from a logical aspect. Moreover, we represent some properties of M -fuzzifying convex structures as well as its relations with M -fuzzifying closure systems and M -fuzzifying Alexandrov topologies by inequalities.

In this paper, we show the existence of an S -shaped connected component in the set of radial positive solutions of boundary value problem − div(ϕN(∇y))=λa(|x|)f(y)inA,∂y∂ν=0 on Γ1,y=0 onΓ2, $$\begin{array}{} \displaystyle \left\{\,\begin{array}{} -\text{ div}\big(\phi_N(\nabla y)\big)=\lambda a(|x|)f(y)\, \, \, \, \, \text{in}\, \, \mathcal{A},\\\frac{\partial y}{\partial \nu}=0\, \, \, \,\, \text{ on }\, \, {\it\Gamma}_1,\qquad y=0\, \, \, \, \text{ on}\, \, {\it\Gamma}_2,\\ \end{array} \right. \end{array} $$ where R 2 ∈ (0, ∞) and R 1 ∈ (0, R 2 ) is a given constant, 𝓐 = { x ∈ ℝ N : R 1 < ∣ x ∣ < R 2 }, Γ 1 = { x ∈ ℝ N : ∣ x ∣ = R 1 }, Γ 2 = { x ∈ ℝ N : ∣ x ∣ = R 2 }, ϕN(s)=s1−|s|2, $\begin{array}{} \phi_N(s)=\frac{s}{\sqrt{1-|s|^2}}, \end{array} $ s ∈ ℝ N , λ is a positive parameter, a ∈ C [ R 1 , R 2 ], f ∈ C [0, ∞), ∂y∂ν $\begin{array}{} \frac{\partial y}{\partial \nu} \end{array} $ denotes the outward normal derivative of y and ∣⋅∣ denotes the Euclidean norm in ℝ N . The proof of main result is based upon bifurcation techniques.

In this paper, we shall study the Diophantine equation u n = R ( m ) P ( m ) Q ( m ) , where u n is a Lucas sequence and R , P and Q are polynomials (under weak assumptions).

In this paper we will show a new way to represent functions as infinite series, finding some conditions under which a function is expandable with this method, and showing how it allows us to find the values of many interesting series. At the end, we will prove one of the main results of the paper, a Representation Theorem.

In this paper, we consider a class of delay coupled Lotka-Volterra ring systems. Based on the symmetric bifurcation theory of delay differential equations and representation theory of standard dihedral groups, properties of phase locked periodic solutions are given. Moreover, the direction and the stability of the Hopf bifurcation periodic orbits are obtained by using normal form and center manifold theory. Finally, the research results are verified by numerical simulation.

Linear regression models are foundation of current statistical theory and have been a prominent object of study in statistical data analysis and inference. A special class of linear regression models is called the seemingly unrelated regression models (SURMs) which allow correlated observations between different regression equations. In this article, we present a general approach to SURMs under some general assumptions, including establishing closed-form expressions of the best linear unbiased predictors (BLUPs) and the best linear unbiased estimators (BLUEs) of all unknown parameters in the models, establishing necessary and sufficient conditions for a family of equalities of the predictors and estimators under the single models and the combined model to hold. Some fundamental and valuable properties of the BLUPs and BLUEs under the SURM are also presented.

In this paper, we show that for generic C 1 , if a flow X t has the shadowing property on a bi-Lyapunov stable homoclinic class, then it does not contain any singularity and it is hyperbolic.

The main goal of this paper is to introduce the notion of stabilizers in EQ-algebras and develop stabilizer theory in EQ-algebras. In the paper, we introduce (fuzzy) left and right stabilizers and investigate some related properties of them. Then, we discuss the relations among (fuzzy) stabilizers, (fuzzy) prefilters (filters) and (fuzzy) co-annihilators. Also, we obtain that the set of all prefilters in a good EQ-algebra forms a relative pseudo-complemented lattice, where St r ( F , G ) is the relative pseudo-complemented of F with respect to G . These results will provide a solid algebraic foundation for the consequence connectives in higher fuzzy logic.

The purpose of this manuscript is to study some properties on meromorphic solutions for several types of q -difference equations. Some exponents of convergence of zeros, poles and fixed points related to meromorphic solutions for some q -difference equations are obtained. Our theorems are some extension and improvements to those results given by Qi, Peng, Chen, and Zhang.

For an integer p ≥ 1, let Γ p be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm N Γ p (.) and the property ∑k=1∞|λk(A)|p≤apNΓpp(A)(A∈Γp), $$\begin{array}{} \displaystyle \sum_{k=1}^{\infty} |\lambda_k(A)|^p\le a_p N_{{\it\Gamma}_p}^p(A) \;\;(A\in {\it\Gamma}_p), \end{array}$$ where λ k ( A ) ( k = 1, 2, …) are the eigenvalues of A and a p is a constant independent of A . Let A , Ã ∈ Γ p and Δp(A,A~):=NΓp(A−A~)expapbpp1+12(NΓp(A+A~)+NΓp(A−A~))p, $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A):= N_{{\it\Gamma}_p}(A-\tilde A) \;\exp\;\left[a_p b_p^p \;\left(1+\frac{1}2 (N_{{\it\Gamma}_p}(A+\tilde A) + N_{{\it\Gamma}_p}(A-\tilde A))\right)^p\right], \end{array}$$ where b p is the quasi-triangle constant in Γ p . It is proved the following result: let I be the unit operator, I – A p be boundedly invertible and Δp(A,A~)expapNΓpp(A)ψp(A)<1, $$\begin{array}{} \displaystyle {\it\Delta}_p(A, \tilde A)\exp\;\left[\frac{a_pN^p_{{\it\Gamma}_p}(A) } {\psi_p(A)}\right] \lt 1, \end{array}$$ where ψ p ( A ) = inf k =1,2,… |1 – λkp $\begin{array}{} \displaystyle \lambda_k^{p} \end{array}$( A )|. Then I – Ã p is also boundedly invertible. Applications of that result to the spectrum perturbations of absolutely p -summing and absolutely ( p , 2) summing operators are also discussed. As examples we consider the Hille-Tamarkin integral operators and matrices.

The aim of this paper is to construct and study the properties of a certain graph associated with a non-central hypergroup, i.e. a hypergroup having non-commutative the associated fundamental group. The method of the construction of the graph is similar to that one proposed by Paul Erdős, when he defined a graph associated with a non-commutative group. We establish necessary and /or sufficient conditions for the associated graph to be Hamiltonian or planar.

In this paper, Lie symmetry analysis is performed for the equation derived from $(2+1)$-dimensional higher order Broer-Kaup equation. Meanwhile, the optimal system and similarity reductions based on the Lie group method are obtained. Furthermore, the conservation law is studied via the Ibragimov’s method.

In this paper, by using critical point theory, we obtain some sufficient conditions on the existence of infinitely many positive solutions of the discrete Dirichlet problem involving the mean curvature operator. We show that the suitable oscillating behavior of the nonlinear term near at the origin and at infinity will lead to the existence of a sequence of pairwise distinct nontrivial positive solutions. We also give two examples to illustrate our main results.

In this paper we introduce dislocated and dislocated quasi version of a cone b-metric space over a Banach algebra as well as weak semi α -admissible and α -identical pair of mappings and prove common fixed point theorems for a pair of α -identical and weak α -admissible mappings in the aforesaid spaces. Our results are supported with suitable examples and an application to a system of m-tupled functional equations.

Given the abstract evolution equation y′(t)=Ay(t),t≥ 0, $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire , on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

In this paper, by using the idea of truncated counting functions, we study the uniqueness of transcendental meromorphic functions and L-functions whose certain nonlinear differential polynomials share one finite nonzero value. The result in this paper extends the theorem given by Liu, Li and Yi.

In this paper, we break the index set N into disjoint subsets S and its complement, and propose two S -type exclusion sets that all the eigenvalues do not belong to them. Furthermore, we establish new S -type eigenvalue inclusion sets, which can reduce computations and obtain more accurate numerical results. At the same time, we give two criteria for identifying nonsingular tensors. Finally, new S -type eigenvalue inclusion sets are shown to be sharper than existing results via two examples.

Let ( Ω , Σ , μ ) be a complete σ -finite measure space, φ a Young function and X and Y be Banach spaces. Let L φ ( X ) denote the corresponding Orlicz-Bochner space and Tφ ∧ $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$ denote the finest Lebesgue topology on L φ ( X ). We examine different classes of ( Tφ ∧ $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$, ∥ ⋅ ∥ Y )-continuous linear operators T : L φ ( X ) → Y : weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. The relationships among these classes of operators are established.

The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term − △ u+V(x)u− l(x)ϕ |u|3u=η K(x)f(u),  in  R3,− △ ϕ =l(x)|u|5,                                   in  R3, $$\begin{array}{} \displaystyle \begin{cases} -\triangle u+V(x)u-l(x)\phi|u|^{3}u=\eta K(x)f(u),~~\mbox{in}~~\mathbb{R}^{3}, \notag\\ -\triangle\phi=l(x)|u|^{5},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mbox{in}~~\mathbb{R}^{3}, \end{cases} \end{array}$$(1.1) where the potential V ( x ) and K ( x ) are positive continuous functions that vanish at infinity, and l ( x ) is bounded, nonnegative continuous function. Under some simple assumptions on V , K , l and f , we prove that the problem (1.1) has a non-trivial solution.

A new class of generalized convex set-valued maps termed relatively solid generalized cone-subconvexlike maps is introduced in real linear spaces not equipped with any topology. This class is a generalization of generalized cone-subconvexlike maps and relatively solid cone-subconvexlike maps. Necessary and sufficient conditions for Benson proper efficiency of set-valued optimization problem are established by means of scalarization, Lagrange multipliers, saddle points and duality. The results generalize and improve some corresponding ones in the literature. Some examples are afforded to illustrate our results.

We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound.

A Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species is proposed and studied. For non-delay case, such topics as the persistent of the system, the local stability property of the equilibria, the global stability of the positive equilibrium are investigated. For the system with infinite delay, by using the iterative method, a set of sufficient conditions which ensure the global attractivity of the positive equilibrium is obtained. By introducing the density dependent birth rate, the dynamic behaviors of the system becomes complicated. The system maybe collapse in the sense that both the species will be driven to extinction, or the two species could be coexist in a stable state. Numeric simulations are carried out to show the feasibility of the main results.

The coexistence of species sustains the ecological balance in nature. This paper focuses on sufficient conditions for the coexistence of a three-species stochastic competitive model, where the model has non-linear diffusion parts. Three values λ 3 z , λ 3 x and λ 3 y are introduced and calculated from the coefficients, which can be considered as threshold values. Moreover, convergence in distribution of the positive solution of the model is also addressed. A few numerical simulations are carried out to illustrate the theoretical results.

The aim of this paper is the analysis, from algebraic point of view and singularities studies, of the 5-parametric family of differential equations yy′=(α xm+k− 1+β xm− k− 1)y+γ x2m− 2k− 1,y′=dydx $$\begin{array}{} \displaystyle yy'=(\alpha x^{m+k-1}+\beta x^{m-k-1})y+\gamma x^{2m-2k-1}, \quad y'=\frac{dy}{dx} \end{array}$$ where a , b , c ∈ ℂ, m , k ∈ ℤ and α =a(2m+k)β =b(2m− k),γ =− (a2mx4k+cx2k+b2m). $$\begin{array}{} \displaystyle \alpha=a(2m+k) \quad \beta=b(2m-k), \quad \gamma=-(a^2mx^{4k}+cx^{2k}+b^2m). \end{array}$$ This family is very important because include Van Der Pol equation. Moreover, this family seems to appear as exercise in the celebrated book of Polyanin and Zaitsev. Unfortunately, the exercise presented a typo which does not allow to solve correctly it. We present the corrected exercise, which corresponds to the title of this paper. We solve the exercise and afterwards we make algebraic and of singularities studies to this family of differential equations.

The main aim of this paper is to use the analytic methods and the properties of the classical Gauss sums to research the computational problem of one kind hybrid power mean containing the character sums of polynomials and two-term exponential sums modulo p , an odd prime, and acquire several accurate asymptotic formulas for them.

Let ψ :( M , g ) → ( N , h ) be a map between Riemannian manifolds ( M , g ) and ( N , h ). We introduce the notion of the F -bienergy functional EF,2(ψ )=∫ MF|τ (ψ )|22dVg, $$\begin{array}{} \displaystyle E_{F,2}(\psi)=\int\limits_{M}F\left(\frac{|\tau(\psi)|^{2}}{2}\right)\text{d}V_{g}, \end{array}$$ where F : [0, ∞) → [0, ∞) be C 3 function such that F ′ > 0 on (0, ∞), τ ( ψ ) is the tension field of ψ . Critical points of τ F ,2 are called F -biharmonic maps. In this paper, we prove a nonexistence result for F -biharmonic maps from a complete non-compact Riemannian manifold of dimension m = dimM ≥ 3 with infinite volume that admit an Euclidean type Sobolev inequality into general Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the L p -norm ( p > 1) of the tension field is bounded and the m -energy of the maps is sufficiently small, then every F -biharmonic map must be harmonic. We also get a Liouville-type result under proper integral conditions which generalize the result of [Branding V., Luo Y., A nonexistence theorem for proper biharmonic maps into general Riemannian manifolds, 2018, arXiv: 1806.11441v2].

The main result of this paper is the embedding Bβ s,r(Ω )↪ Bβ +(n− 1)(1s− 1s1)s1,r1(Ω ), $$\begin{array}{} \displaystyle \mathcal{B}^{s,r}_\beta({\it\Omega})\hookrightarrow \mathcal{B}^{s_1,r_1}_{\beta+(n-1)\big(\frac 1s-\frac 1{s_1}\big)}({\it\Omega}), \end{array}$$ 0 < r ≤ r 1 ≤ ∞, 0 < s ≤ s 1 ≤ ∞, β > –1, of harmonic functions mixed norm spaces on a smoothly bounded domain Ω ⊂ ℝ n . We also extend a result on boundedness, in mixed norm, of a maximal function-type operator from the case of the unit disc and the unit ball to general domains in ℝ n .

We study the asymptotic behavior of solutions to the non-autonomous stochastic plate equation driven by additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence and upper semicontinuity of random attractors.

Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

In this paper we define and investigate nearly Hurewicz spaces and their star version. It is shown that a nearly Hurewicz space fits between Hurewicz and almost Hurewicz spaces. As a counter example it is shown that a particular point topology which fails to be Lindelöf, Menger, Hurewicz is a nearly Hurewicz space.

Suppose that Γ is a graph of order n and A ( Γ ) = [ a i , j ] is its adjacency matrix such that a i , j is equal to 1 if v i is adjacent to v j and a i , j is zero otherwise, where 1 ≤ i , j ≤ n . In a family of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix is minimum in the set of the least eigenvalues of all the graphs. Petrovi ć et al. [On the least eigenvalue of cacti, Linear Algebra Appl., 2011, 435, 2357-2364] characterized a minimizing graph in the family of all cacti such that the complement of this minimizing graph is disconnected. In this paper, we characterize the minimizing graphs G ∈ Ω nc $\begin{array}{} {\it\Omega}^c_n \end{array}$, i.e. λ min(G)≤ λ min(Cc) $$\begin{array}{} \displaystyle \lambda_{min}(G)\leq\lambda_{min}(C^c) \end{array}$$ for each C c ∈ Ω nc $\begin{array}{} {\it\Omega}^c_n \end{array}$, where Ω nc $\begin{array}{} {\it\Omega}^c_n \end{array}$ is a collection of connected graphs such that the complement of each graph of order n is a cactus with the condition that either its each block is only an edge or it has at least one block which is an edge and at least one block which is a cycle.

The presented paper is strictly connected, among others, with the paper On the sum of some alternating series , Comp. Math. Appl. (2011), written by Wituła and Słota. A problem concerning the form of determinants formulated in the cited paper is solved here. Next, the obtained result is adapted to solve some system of linear equations and the description of the sum of alternating series.

The aim of this paper is to study the translational hull of a strongly right type B semigroup. Our main result is to prove that the translational hull of a strongly right type B semigroup is itself a strongly right type B semigroup. As an application, we give a proof of a problem posted by Petrich on translational hulls of inverse semigroups in Petrich (Inverse Semigroups, Wiley, New York, 1984) to the cases of some strongly right type B semigroups.

The purpose of this paper is to find common fixed point results for two families of multivalued mappings fulfilling generalized rational type A –dominated contractive conditions on a closed ball in complete dislocated b -metric spaces. Some new fixed point results with graphic contractions on a closed ball for two families of multi-graph dominated mappings on dislocated b -metric space have been established. An application to the unique common solution of two families of nonlinear integral equations is presented to show the novelty of our results.

This paper is concerned with establishing L p estimates for a class of maximal operators associated to surfaces of revolution with kernels in L q ( S n −1 × S m −1 ), q > 1. These estimates are used in extrapolation to obtain the L p boundedness of the maximal operators and the related singular integral operators when their kernels are in the L ( logL ) κ ( S n −1 × S m −1 ) or in the block space Bq0,κ − 1 $\begin{array}{} B^{0,\kappa-1}_ q \end{array}$( S n −1 × S m −1 ). Our results substantially improve and extend some known results.

Given a simple graph with the vertex set X , we discuss a closure operator on X induced by a set of paths with identical lengths in the graph. We introduce a certain set of paths of the same length in the 2-adjacency graph on the digital line ℤ and consider the closure operators on ℤ m ( m a positive integer) that are induced by a special product of m copies of the introduced set of paths. We focus on the case m = 3 and show that the closure operator considered provides the digital space ℤ 3 with a connectedness that may be used for defining digital surfaces satisfying a Jordan surface theorem.

Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P -expansion between these modules.

In this paper, we concern with a 2 n th-order discrete system. Using the critical point theory, we establish various sets of sufficient conditions for the existence of periodic solutions with prescribed minimal period. To the best of our knowledge, this is the first time to discuss the periodic solutions with prescribed minimal period for a 2 n th-order discrete system.

This paper is devoted to the study of injectivity for ordered universal algebras. We first characterize injectives in the category OALΣ⩽ $\begin{array}{} \displaystyle {\mathsf{OAL}_{{\it\Sigma}}^{\leqslant}} \end{array}$ of ordered Σ -algebras with lax morphisms as sup- Σ -algebras. Second, we show that every ordered Σ -algebra has an σ ⩽ -injective hull, and give its concrete form.

We first introduce the concept of the random uniform exponential attractor for a jointly continuous non-autonomous random dynamical system (NRDS) and give a theorem on the existence of the random uniform exponential attractor for a jointly continuous NRDS. Then we study the existence of the random uniform exponential attractor for reaction-diffusion equation with quasi-periodic external force and multiplicative noise in ℝ 3 .

In this paper we construct a class of virus dynamics models with impairment of B-cell functions. Two forms of the incidence rate have been considered, saturated and general. The well-posedness of the models is justified. The models admit two equilibria which are determined by the basic reproduction number R 0 . The global stability of each equilibrium is proven by utilizing Lyapunov function and LaSalle’s invariance principle. The theoretical results are illustrated by numerical simulations.

In this paper, we investigate the monotonicity of the functions x↦ 1x1− a+23ax2+a2arctan⁡ x,x↦ 1x4π 2+4π 2x2+aarctan⁡ x $$\begin{array}{} \begin{split}{} \displaystyle x &\mapsto &\frac{1}{x}\left( 1-a+\sqrt{\frac{2}{3}ax^{2}+a^{2}}\right) \arctan x, \\ x &\mapsto &\frac{1}{x}\left( \frac{4}{\pi ^{2}}+\sqrt{\frac{4}{\pi ^{2}}% x^{2}+a}\right) \arctan x \end{split} \end{array}$$ on (0, ∞) for a > 0, which not only gives relative errors of known bounds with quadratic for arctan x , but also yields some new accurate bounds. Moreover, the known bounds are extended and a more accurate estimate for arctan x is presented.

The set of subsums of the series ∑n=1∞ $\begin{array}{} \sum_{n=1}^{\infty} \end{array}$ x n is known to be one of three types: a finite union of intervals, homeomorphic to the Cantor set, or of the type known as a Cantorval. Bartoszewicz, Filipczak and Szymonik have described a family of series which contained all known examples of subsum sets which are Cantorvals. We construct another family of series which produces new examples of subsum sets which are Cantorvals.

Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] studied the fluid asymptotics of the joint queue length process for an overloaded cyclic polling system with multigated service discipline by exploiting the connection with multi-type branching processes. In contrast to the heavy traffic behaviors, the cycle time of the overloaded polling system increases by a deterministic times over times under passage to the fluid dynamics and the fluid limit preserves some randomness. The present paper aims to extend the overloaded asymptotics in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] to the corresponding polling system with general branching-type service disciplines and customer re-routing policy. A unifying overloaded asymptotic property is derived. Due to the exhaustiveness, the property is a natural extension of the classical polling model with multigated service discipline in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] and provides new exact results that have not been observed before for rerouting policy. Additionally, a stochastic simulation is undertaken for the validation of the fluid limit and the optimization of the gating indexes to minimize the total population is considered as an example to demonstrate the usefulness of the random fluid limit.

In this paper, we consider an almost periodic commensal symbiosis model with nonlinear harvesting on time scales. We establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Our results show that the continuous system and discrete system can be unify well. Examples and their numerical simulations are carried out to illustrate the feasibility of our main results.

This paper aims to present a new pathwise approximation method, which gives approximate solutions of order 32 $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals I α by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals I α . Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

In this paper, we apply the concept of fuzzy sets to Novikov algebras, and introduce the concepts of L -fuzzy ideals and L -fuzzy subalgebras. We get a sufficient and neccessary condition such that an L -fuzzy subspace is an L -fuzzy ideal. Moreover, we show that the quotient algebra A / μ of the L -fuzzy ideal μ is isomorphic to the algebra A / A μ of the non-fuzzy ideal A μ . Finally, we discuss the algebraic properties of surjective homomorphic image and preimage of an L -fuzzy ideal.

In this paper, axiomatic definitions of both L -convex bases and L -convex subbases are introduced and their relations with L -convex spaces are studied. Based on this, the notion of L -topological-convex space is introduced as a triple ( X , 𝓣, 𝓒), where X is a nonempty set, 𝓒 is an L -convex structure on X and 𝓣 is an L -cotopology on X compatible with 𝓒. It can be characterized by many means.

In this paper, we derive and analyze a new one-parameter family of modified Cauchy method free from second derivative for obtaining simple roots of nonlinear equations by using Pad é approximant. The convergence analysis of the family is also considered, and the methods have convergence order three. Based on the family of third-order method, in order to increase the order of the convergence, a new optimal fourth-order family of modified Cauchy methods is obtained by using weight function. We also perform some numerical tests and the comparison with existing optimal fourth-order methods to show the high computational efficiency of the proposed scheme, which confirm our theoretical results. The basins of attraction of this optimal fourth-order family and existing fourth-order methods are presented and compared to illustrate some elements of the proposed family have equal or better stable behavior in many aspects. Furthermore, from the fractal graphics, with the increase of the value m of the series in iterative methods, the chaotic behaviors of the methods become more and more complex, which also reflected in some existing fourth-order methods.

A new error bound for the linear complementarity problem (LCP) of Σ -SDD matrices is given, which depends only on the entries of the involved matrices. Numerical examples are given to show that the new bound is better than that provided by García-Esnaola and Peña [Linear Algebra Appl., 2013, 438, 1339–1446] in some cases. Based on the obtained results, we also give an error bound for the LCP of SB -matrices. It is proved that the new bound is sharper than that provided by Dai et al. [Numer. Algor., 2012, 61, 121–139] under certain assumptions.

In this paper we define and consider some familiar subsets of analytic functions associated with sine functions in the region of unit disk on the complex plane. For these classes our aim is to find the Hankel determinant of order three.

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

In this paper we consider a class of nD Navier-Stokes equations of Kirchhoff type and prove the global existence of solutions by using a new approach introduced in [Jday R., Zennir Kh., Georgiev S.G., Existence and smoothness for new class of n -dimentional Navier-Stokes equations, Rocky Mountain J. Math., 2019, 49(5), 1595–1615].

This paper investigates the abstract-valued Orlicz space of range-varying type. We firstly give the notions and examples of partially continuous modular net and regular Banach space net of type (II), then deal with the definitions, constructions, and geometrical properties of the range-varying Orlicz spaces, including representation of the dual L+φ $\begin{array}{} L_{+}^{\varphi} \end{array}$( I , X θ (⋅) ) * , and reflexivity of L φ ( I , X θ (⋅) ), under some reasonable conditions. As an application, we finally make another approach to the real interpolation spaces constructed by a generalized Φ -function.

Let E be a generalized Cesáro sequence space defined by weighted means and by using s -numbers of operators from a Banach space X into a Banach space Y . We give the sufficient (not necessary) conditions on E such that the components SE(X,Y):={T∈ L(X,Y):((sn(T))n=0∞ ∈ E}, $$\begin{array}{} \displaystyle S_{E}(X, Y):=\Big\{T\in L(X, Y):((s_{n}(T))_{n=0}^{\infty}\in E\Big\}, \end{array}$$ of the class S E form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal S E , the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small and the pre-quasi Banach operator ideal constructed by s -numbers is simple Banach space. Finally the pre-quasi operator ideal formed by the sequence of s -numbers and this sequence space is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to this sequence space.

In this paper, we introduce a new way-below relation in T 0 topological spaces based on cuts and give the concepts of SI 2 -continuous spaces and weakly irreducible topologies. It is proved that a space is SI 2 -continuous if and only if its weakly irreducible topology is completely distributive under inclusion order. Finally, we introduce the concept of 𝓓-convergence and show that a space is SI 2 -continuous if and only if its 𝓓-convergence with respect to the topology τ SI 2 ( X ) is topological. In general, a space is SI -continuous if and only if its 𝓓-convergence with respect to the topology τ SI ( X ) is topological.

The purpose of this paper is to study behavior of a rational type contraction introduced in [A fixed point theorem for contractions of rational type in partially ordered metric spaces, Ann. Univ. Ferrara, 2013, 59, 251–258] in context of ordered dualistic partial metric spaces and to investigate sufficient conditions for the existence of a fixed point in this space. These results extend various comparable results, existing in the literature. We give examples to explain our findings. We apply our result to prove the existence of the solution of functional equation.

Here, a coupled system of nonlinear weighted Cauchy-type problem of a diffre-integral equations of fractional order will be considered. We study the existence of at least one integrable solution of this system by using Schauder fixed point Theorem. The continuous dependence of the uniqueness of the solution is proved.

In this paper, we introduce a rough quotient. Also, we present conditions ensuring that G / H are partitions of G . The rough projection map is also presented. We discuss first, second and third rough isomorphism theorems and other related results. At the end, an orbit and a stabilizer in topological rough groups are considered.

In this paper, the concepts of weak quasi-hypercontinuous posets and weak generalized finitely regular relations are introduced. The main results are: (1) when a binary relation ρ : X ⇀ Y satisfies a certain condition, ρ is weak generalized finitely regular if and only if ( φ ρ ( X , Y ), ⊆) is a weak quasi-hypercontinuous poset if and only if the interval topology on ( φ ρ ( X , Y ), ⊆) is split T 2 ; (2) the relation ≰ on a poset P is weak generalized finitely regular if and only if P is a weak quasi-hypercontinuous poset if and only if the interval topology on P is split T 2 .

In this paper we investigate theoretically and numerically the new preconditioned method to accelerate over-relaxation (AOR) and succesive over-relaxation (SOR) schemes, which are used to the large sparse linear systems. The iterative method that is usually measured by the convergence rate is an important method for solving large linear equations, so we focus on the convergence rate of the different preconditioned iterative methods. Our results indicate that the proposed new method is highly effective to improve the convergence rate and it is the best one in three preconditioned methods that are revealed in the comparison theorems and numerical experiment.

In this work, we propose to approximate the Gaussian integral, the error function and the cumulative distribution function by using the power series extender method (PSEM). The approximations proposed in this paper present a high accuracy for the complete domain [–∞,∞]. Furthermore, the approximations are handy and easy computable, avoiding the application of special numerical algorithms. In order to show its high accuracy, three case studies are presented with applications to science and engineering.

The general position number gp( G ) of a connected graph G is the cardinality of a largest set S of vertices such that no three pairwise distinct vertices from S lie on a common geodesic. It is proved that gp( G ) ≥ ω ( G SR ), where G SR is the strong resolving graph of G , and ω ( G SR ) is its clique number. That the bound is sharp is demonstrated with numerous constructions including for instance direct products of complete graphs and different families of strong products, of generalized lexicographic products, and of rooted product graphs. For the strong product it is proved that gp( G ⊠ H ) ≥ gp( G )gp( H ), and asked whether the equality holds for arbitrary connected graphs G and H . It is proved that the answer is in particular positive for strong products with a complete factor, for strong products of complete bipartite graphs, and for certain strong cylinders.

The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G . If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G . Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P 3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with K k for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.

The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs Ω nc=Ω 1,nc∪ Ω 2,nc, $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where Ω 1,nc $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and Ω 2,nc $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

A graph is said to be NSSD (=non-singular with a singular deck) if it has no eigenvalue equal to zero, whereas all its vertex-deleted subgraphs have eigenvalues equal to zero. NSSD graphs are of importance in the theory of conductance of organic compounds. In this paper, a novel method is described for constructing NSSD molecular graphs from the commuting graphs of the H v -group. An algorithm is presented to construct the NSSD graphs from these commuting graphs.