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In this paper, we provide some conditions for a K-woven pair of K-g-frames to be preserved under an operator and particularly, we report that applying two different operators to a K-woven pair of K-g-frames can leave them K-woven. Several new methods on the construction of K-woven pair of K-g-frames are also given. We end the paper with a new perturbation result on the weaving of K-g-frames, which shows that, under the perturbation condition involved in one known result on this topic, two K-g-frames can be K-woven in the whole space, not merely in the subspace Range(K).

The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$, where $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$ is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e $$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for each}~ G\in {\it\Omega}^{\alpha}_n. \end{array}$$

In this paper, we develop the elementary theory of inverse semigroups to the cases of type B semigroups. The main aim of this paper is to study proper type B semigroups. We introduce first the concept of a left admissible triple. After obtaining some basic properties of left admissible triple, we give the definition of a Q-semigroup and get a structure theorem of Q-semigroup. In particular, we introduce the notion of an admissible triple and give some characterization of proper type B semigroups. It is proved that an arbitrary Q-semigroup with an admissible triple is an E-unitary type B semigroup.

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 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 Str(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.

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.

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.

[EN] This paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Frobenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Frobenius method, in which the random input coefficients may be expressed via a Karhunen-Loeve expansion., This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the valuable comments raised by the reviewer, which have improved the final version of the paper.

Considering the fact that the development of low carbon economy calls for the low carbon technology sharing between interested enterprises, this paper study a stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises. In the paper, we consider the random interference factors that include the uncertain external environment and the internal understanding limitations of decision maker. In the model, superior enterprises and inferior enterprises are separated entities, and they play Stacklberg master-slave game, Nash non-cooperative game, and cooperative game, respectively. We discuss the feedback equilibrium strategies of superior enterprises and inferior enterprises, and it is found that some random interference factors in sharing system can make the variance of improvement degree of low carbon technology level in the cooperation game higher than the variance in the Stackelberg game, and the result of Stackelberg game is similar to the result of Nash game. Additionally, a government subsidy incentive and a special subsidy that inferior enterprises give to superior enterprises are proposed.

The aim of the paper is to present the binomial transformation formulae of Fibonacci numbers scaled by complex multipliers. Many of these new and nontrivial relations follow from the fundamental properties of the so-called delta-Fibonacci numbers defined by Wituła and Słota. The paper contains some original relations connecting the values of delta-Fibonacci numbers with the respective values of Chebyshev polynomials of the first and second kind.

It is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.

Let SH be the class of functions f = h+g that are harmonic univalent and sense-preserving in the open unit disk U = { z : |z| < 1} for which f (0) = f'(0)-1=0. In this paper, we introduce and study a subclass H( α, β) of the class SH and the subclass NH( α, β) with negative coefficients. We obtain basic results involving sufficient coefficient conditions for a function in the subclass H( α, β) and we show that these conditions are also necessary for negative coefficients, distortion bounds, extreme points, convolution and convex combinations. In this paper an attempt has also been made to discuss some results that uncover some of the connections of hypergeometric functions with a subclass of harmonic univalent functions.

A Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

In this paper, we mainly derive the general solutions of two systems of minus partial ordering equations over von Neumann regular rings. Meanwhile, some special cases are correspondingly presented. As applications, we give some necessary and sufficient conditions for the existence of solutions. It can be seen that some known results can be regarded as the special cases of this paper.

The paper presents a new type of density topology on the real line generated by the pointwise convergence, similarly to the classical density topology which is generated by the convergence in measure. Among other things, this paper demonstrates that the set of pointwise density points of a Lebesgue measurable set does not need to be measurable and the set of pointwise density points of a set having the Baire property does not need to have the Baire property. However, the set of pointwise density points of any Borel set is Lebesgue measurable.

In this paper, the authors study the poly-Bernoulli numbers of the second kind, which are defined by using polyexponential functions introduced by Kims. Also by using unipoly function, we study the unipoly Bernoulli numbers of the second kind, which are attached to an arithmetic function. We derive their explicit expressions and some identities involving poly-Bernoulli numbers of the second kind and unipoly Bernoulli numbers of the second kind.

In this paper, we consider an approximation scheme for fractional evolution equation with variable coefficient. The space derivative is approximated by triangular finite element and the time fractional derivative is evaluated by the L1 approximation. The main aim of this work is to provide convergence and superconvergence analysis and derive a posteriori error estimates. Some numerical examples are presented to demonstrate our theoretical results.

This paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with theϕ-Laplacian equation(ϕ(u′))′+a(t)g(u)=0,(\phi \left(u^{\prime} ))^{\prime} +a\left(t)g\left(u)=0,whereϕis a homeomorphism withϕ(0) = 0,a(t) is a stepwise indefinite weight andg(u) is a continuous function. When dealing with thep-Laplacian differential operatorϕ(s) = ∣s∣p−2swithp > 1, and the nonlinear termg(u) = uγwithγ∈R\gamma \in {\mathbb{R}}, we prove the existence of a unique positive solution whenγ ∈ ]−∞\infty, (1 − 2p)/(p − 1)] ∪ ]p − 1, +∞\infty[.

In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

In this paper, we establish equivalent parameter conditions for the validity of multiple integral half-discrete Hilbert-type inequalities with the nonhomogeneous kernel G ( n λ 1 ∥ x ∥ m , ρ λ 2 ) G\left({n}^{{\lambda }_{1}}\parallel x{\parallel }_{m,\rho }^{{\lambda }_{2}}\hspace{-0.16em}) ( λ 1 λ 2 > 0 {\lambda }_{1}{\lambda }_{2}\gt 0 ) and obtain best constant factors of the inequalities in specific cases. In addition, we also discuss their applications in operator theory.

In this paper, we propose a family of non-stationary combined ternary ( 2 m + 3 ) \left(2m+3) -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach 2 m + 3 2m+3 approximation order. Moreover, we discuss its smoothness and show that it can produce C 2 m + 2 {C}^{2m+2} limit curves. Several numerical examples are given to show the performance of the schemes.

In this paper, we first prove an identity for twice quantum differentiable functions. Then, by utilizing the convexity of ∣ D q 2 b f ∣ | {}^{b}D_{q}^{2}\hspace{0.08em}f| and ∣ D q 2 a f ∣ | {}_{a}D_{q}^{2}\hspace{0.08em}f| , we establish some quantum Ostrowski inequalities for twice quantum differentiable mappings involving q a {q}_{a} and q b {q}^{b} -quantum integrals. The results presented here are the generalization of already published ones.

In this paper, the regularization method of S. A. Lomov is generalized to integro-differential equations with rapidly oscillating coefficients and with a rapidly oscillating right-hand side. The main goal of the work is to reveal the influence of the oscillating components on the structure of the asymptotics of the solution of this problem. The case of coincidence of the frequencies of a rapidly oscillating coefficient and a rapidly oscillating inhomogeneity is considered. In this case, only the identical resonance is observed in the problem. Other cases of the relationship between frequencies can lead to so-called non-identical resonances, the study of which is nontrivial and requires the development of a new approach. It is supposed to study these cases in our further work.

New graph invariant, which is called a mixed metric dimension, has been recently introduced. In this paper, exact results of the mixed metric dimension on two special classes of graphs are found: flower snarks J n {J}_{n} and wheels W n {W}_{n} . It is proved that the mixed metric dimension for J 5 {J}_{5} is equal to 5, while for higher dimensions it is constant and equal to 4. For W n {W}_{n} , the mixed metric dimension is not constant, but it is equal to n n when n ≥ 4 n\ge 4 , while it is equal to 4, for n = 3 n=3 .

In this paper, with the aid of the Faà di Bruno formula and by virtue of properties of the Bell polynomials of the second kind, the authors define a kind of notion of degenerate Narumi numbers and polynomials, establish explicit formulas for degenerate Narumi numbers and polynomials, and derive explicit formulas for the Narumi numbers and polynomials and for (degenerate) Cauchy numbers.

Let T ( X ) T\left(X) be the full transformation semigroup on a set X X . For an equivalence E E on X X , let T E ∗ ( X ) = { α ∈ T ( X ) : ∀ x , y ∈ X , ( x , y ) ∈ E ⇔ ( x α , y α ) ∈ E } . {T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X):\forall x,y\in X,\left(x,y)\in E\iff \left(x\alpha ,y\alpha )\in E\right\}. For each nonempty subset Y Y of X X , we denote the restriction of E E to Y Y by E Y {E}_{Y} . Let T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) be the intersection of the semigroup T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) with the semigroup of all transformations with restricted range Y Y under the condition that ∣ X / E ∣ = ∣ Y / E Y ∣ | X\hspace{-0.1em}\text{/}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| . Equivalently, T E ∗ ( X , Y ) = { α ∈ T E ∗ ( X ) : X α ⊆ Y } {T}_{{E}^{\ast }}\left(X,Y)=\left\{\alpha \in {T}_{{E}^{\ast }}\left(X):X\alpha \subseteq Y\right\} , where ∣ X / E ∣ = ∣ Y / E Y ∣ | X\hspace{-0.1em}\text{/}\hspace{-0.1em}E| =| Y\hspace{-0.16em}\text{/}\hspace{-0.1em}{E}_{Y}| . Then T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) is a subsemigroup of T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) . In this paper, we characterize the natural partial order on T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) . Then we find the elements which are compatible and describe the maximal and minimal elements. We also prove that every element of T E ∗ ( X , Y ) {T}_{{E}^{\ast }}\left(X,Y) lies between maximal and minimal elements. Finally, the existence of an upper cover and a lower cover is investigated.

This paper is devoted to exploring the existence and the forms of entire solutions of several quadratic trinomial differential difference equations with more general forms. Some results about the forms of entire solutions for these equations are some extensions and generalizations of the previous theorems given by Liu, Yang and Cao. We also give a series of examples to explain the existence of the finite order transcendental entire solutions of such equations.

In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity f f satisfying the polynomial growth of arbitrary p − 1 p-1 order. We establish some new estimates, i.e., asymptotic higher-order integrability for the difference of the solutions near the initial time. As an application, we obtain the ( L 2 ( Ω ) , L p ( Ω ) ) \left({L}^{2}\left(\Omega ),{L}^{p}\left(\Omega )) -global attractors immediately; moreover, such an attractor can attract every bounded subset of L 2 ( Ω ) {L}^{2}\left(\Omega ) with the L p + δ {L}^{p+\delta } -norm for any δ ∈ [ 0 , + ∞ ) \delta \in \left[0,+\infty ) .

In this paper, a third-order ordinary differential equation coupled to three-point boundary conditions is considered. The related Green’s function changes its sign on the square of definition. Despite this, we are able to deduce the existence of positive and increasing functions on the whole interval of definition, which are convex in a given subinterval. The nonlinear considered problem consists on the product of a positive real parameter, a nonnegative function that depends on the spatial variable and a time dependent function, with negative sign on the first part of the interval and positive on the second one. The results hold by means of fixed point theorems on suitable cones.

In this paper, we will assume M M to be a compact smooth manifold and f : M → M f:M\to M to be a diffeomorphism. We herein demonstrate that a C 1 {C}^{1} generic diffeomorphism f f is Axiom A and has no cycles if f f is asymptotic measure expansive. Additionally, for a C 1 {C}^{1} generic diffeomorphism f f , if a homoclinic class H ( p , f ) H\left(\hspace{0.08em}p,f) that contains a hyperbolic periodic point p p of f f is asymptotic measure-expansive, then H ( p , f ) H\left(\hspace{0.08em}p,f) is hyperbolic of f f .

In this paper, we study the unicity of entire functions and their derivatives and obtain the following result: letffbe a non-constant entire function, leta1{a}_{1},a2{a}_{2},b1{b}_{1}, andb2{b}_{2}be four small functions offfsuch thata1≢b1{a}_{1}\not\equiv {b}_{1},a2≢b2{a}_{2}\not\equiv {b}_{2}, and none of them is identically equal to∞\infty. Ifffandf(k){f}^{\left(k)}share(a1,a2)\left({a}_{1},{a}_{2})CM and share(b1,b2)\left({b}_{1},{b}_{2})IM, then(a2−b2)f−(a1−b1)f(k)≡a2b1−a1b2\left({a}_{2}-{b}_{2})f-\left({a}_{1}-{b}_{1}){f}^{\left(k)}\equiv {a}_{2}{b}_{1}-{a}_{1}{b}_{2}. This extends the result due to Li and Yang [Value sharing of an entire function and its derivatives, J. Math. Soc. Japan.51(1999), no. 7, 781–799].

In this paper, we disprove a remaining conjecture about Bohemian matrices, in which the numbers of distinct determinants of a normalized Bohemian upper-Hessenberg matrix were conjectured.

Let f k ( z ) = z + ∑ n = 2 k a n z n {f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( z ) / f ′ ( z ) } {\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\} , where f ( z ) f\left(z) belongs to the subclass J p , q m ( μ , α , β ) {{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean ( p , q ) \left(p,q) -differential operator. In addition, the inclusion relations involving N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.

Let X X be a non-empty set and Y Y a non-empty subset of X X . Denote the full transformation semigroup on X X by T ( X ) T\left(X) and write f ( X ) = { f ( x ) ∣ x ∈ X } f\left(X)=\{f\left(x)| x\in X\} for each f ∈ T ( X ) f\in T\left(X) . It is well known that T ( X , Y ) = { f ∈ T ( X ) ∣ f ( X ) ⊆ Y } T\left(X,Y)=\{f\in T\left(X)| f\left(X)\subseteq Y\} is a subsemigroup of T ( X ) T\left(X) and R T ( X , Y ) RT\left(X,Y) , the set of all regular elements of T ( X , Y ) T\left(X,Y) , also forms a subsemigroup of T ( X , Y ) T\left(X,Y) . Green’s ∗ \ast -relations and Green’s ∼ \sim \hspace{0.08em} -relations (with respect to a non-empty subset U U of the set of idempotents) were introduced by Fountain in 1979 and Lawson in 1991, respectively. In this paper, we intend to present certain characterizations of these two sets of Green’s relations of the semigroup T ( X , Y ) T\left(X,Y) . This investigation proves that the semigroup T ( X , Y ) T\left(X,Y) is always a left Ehresmann semigroup, and R T ( X , Y ) RT\left(X,Y) is orthodox (resp. completely regular) if and only if Y Y contains at most two elements.

In this paper, we study the complete consistency for the estimator of nonparametric regression model based on m-END errors and obtain the convergence rates of the complete consistency under more general conditions. Finally, some simulations are illustrated to verify the validity of our results.

In this paper, we investigate the influence of sub-class sizes on a mutually permutable factorized group in which the sub-class sizes of some elements of its factors have certain quantitative properties. Some criteria for a group to be p p -nilpotent or p p -supersolvable are given.

Weierstrass semigroups are well known along the literature. We present a new family of non- Weierstrass semigroups which can be written as an intersection of Weierstrass semigroups. In addition, we provide methods for computing non-Weierstrass semigroups with genus as large as desired., Funding information: Part of this paper was written during a visit of Fernando Torres to the Universidad de Cadiz (Spain) ; his visit was partially supported by Ayudas para Estancias Cortas de Investigadores (EST2018-R0, Programa de Fomento e Impulso de la Investigacion y la Transferencia en la Universidad de Cadiz) . Fernando Torres was partially supported by CNPq/Brazil (Grant 310623/2017-0) . Juan Ignacio Garcia-Garcia, Daniel Marin-Aragon, and Alberto Vigneron-Tenorio were partially supported by Junta de Andalucia research groups FQM-343 and FQM-366, and by the project MTM2017-84890-P (MINECO/FEDER, UE) .

In this paper, we first obtain two new quantum Hermite-Hadamard-type inequalities for newly defined quantum integral. Then we establish several refinements of quantum Hermite-Hadamard inequalities. Natural Science Foundation of ChinaNational Natural Science Foundation of China (NSFC) [11971241] This work was supported by the Natural Science Foundation of China (Grant No. 11971241) . WOS:000684960600001 2-s2.0-85112658736

In this paper, the authors extend determinantal inequalities of the Hua-Marcus-Zhang type for positive definite matrices to the corresponding ones for quaternion matrices.

In this paper, we consider the numerical method for solving the two-dimensional time-fractional convection-diffusion equation with a fractional derivative of order α \alpha ( 1 < α < 2 1\lt \alpha \lt 2 ). By combining the compact difference approach for spatial discretization and the alternating direction implicit (ADI) method in the time stepping, a compact ADI scheme is proposed. The unconditional stability and H 1 {H}^{1} norm convergence of the scheme are proved rigorously. The convergence order is O ( τ 3 − α + h 1 4 + h 2 4 ) O\left({\tau }^{3-\alpha }+{h}_{1}^{4}+{h}_{2}^{4}) , where τ \tau is the temporal grid size and h 1 {h}_{1} , h 2 {h}_{2} are spatial grid sizes in the x x and y y directions, respectively. It is proved that the method can even attain ( 1 + α ) \left(1+\alpha ) order accuracy in temporal for some special cases. Numerical results are presented to demonstrate the effectiveness of theoretical analysis.

In this paper, we consider the regression model with fixed design: Y i = g ( x i ) + ε i {Y}_{i}=g\left({x}_{i})+{\varepsilon }_{i} , 1 ≤ i ≤ n 1\le i\le n , where { x i } \left\{{x}_{i}\right\} are the nonrandom design points, and { ε i } \left\{{\varepsilon }_{i}\right\} is a sequence of martingale, and g g is an unknown function. Nonparametric estimator g n ( x ) {g}_{n}\left(x) of g ( x ) g\left(x) will be introduced and its strong convergence properties are established.

Positive definite polynomials are important in the field of optimization. ℋ-tensors play an important role in identifing the positive definiteness of an even-order homogeneous multivariate form. In this paper, we propose an iterative scheme for identifying ℋ-tensor and prove that the algorithm can terminate within finite iterative steps. Some numerical examples are provided to illustrate the efficiency and validity of methods.

The purpose of this paper is to introduce the class of split regular BiHom-Poisson color algebras, which can be considered as the natural extension of split regular BiHom-Poisson algebras and of split regular Poisson color algebras. Using the property of connections of roots for this kind of algebras, we prove that such a split regular BiHom-Poisson color algebra L L is of the form L = ⊕ [ α ] ∈ Λ / ∼ I [ α ] L={\oplus }_{\left[\alpha ]\in \Lambda \text{/} \sim }{I}_{\left[\alpha ]} with I [ α ] {I}_{\left[\alpha ]} a well described (graded) ideal of L L , satisfying [ I [ α ] , I [ β ] ] + I [ α ] I [ β ] = 0 \left[{I}_{\left[\alpha ]},{I}_{\left[\beta ]}]+{I}_{\left[\alpha ]}{I}_{\left[\beta ]}=0 if [ α ] ≠ [ β ] \left[\alpha ]\ne \left[\beta ] . In particular, a necessary and sufficient condition for the simplicity of this algebra is determined, and it is shown that L L is the direct sum of the family of its simple (graded) ideals.

In this paper, we introduce a new contraction, namely, almost {\mathcal{Z}} contraction with respect to \zeta \in {\mathcal{Z}} , in the setting of complete metric spaces. We proved that such contraction possesses a fixed point and the given theorem covers several existing results in the literature. We consider an example to illustrate our result.

In this paper, we investigate the Cauchy problem for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion. Furthermore, we establish the blow-up result of the positive solutions in finite time under certain conditions on the initial datum. This result complements the early one in the literature, such as [E. Novruzov, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation, J. Math. Phys. 54 (2013), no. 9, 092703, DOI 10.1063/1.4820786] and [Z.Y. Zhang, J.H. Huang, and M.B. Sun, Blow-up phenomena for the weakly dissipative Dullin-Gottwald-Holm equation revisited, J. Math. Phys. 56 (2015), no. 9, 092701, DOI 10.1063/1.4930198].

In this paper, the authors present some inequalities of the generalized trigonometric and hyperbolic functions which occur in the solutions of some linear differential equations and physics. By these results, some well-known classical inequalities for them are improved, such as Wilker inequality, Huygens inequality, Lazarević inequality and Cusa-Huygens inequality.

The latest developments in algebra and graph theory allow us to ask a natural question, what is the application in real world of this graph associated with some mathematical system? Groups can be used to construct new non-associative algebraic structures, loops. Graph theory plays an important role in various fields through edge labeling. In this paper, we shall discuss some applications of bipartite graphs, related with Latin squares of Wilson loops, such as metabolic pathways, chemical reaction networks, routing and wavelength assignment problem, missile guidance, astronomy and x-ray crystallography.

The concept of the domination number plays an important role in both theory and applications of digraphs. Let D = ( V , A ) D=(V,A) be a digraph. A vertex subset T ⊆ V ( D ) T\subseteq V(D) is called a dominating set of D, if there is a vertex t ∈ T t\in T such that t v ∈ A ( D ) tv\in A(D) for every vertex v ∈ V ( D ) \ T v\in V(D)\backslash T . The dominating number of D is the cardinality of a smallest dominating set of D, denoted by γ ( D ) \gamma (D) . In this paper, the domination number of round digraphs is characterized completely.

In 2006, Hubert, Mauduit and Sárközy extended the notion of binary sequences to n-dimensional binary lattices and introduced the measures of pseudorandomness of binary lattices. In 2011, Gyarmati, Mauduit and Sárközy extended the notions of family complexity, collision and avalanche effect from binary sequences to binary lattices. In this paper, we construct pseudorandom binary lattices by using cyclotomic classes in finite fields and study the pseudorandom measure of order k, family complexity, collision and avalanche effect. Results indicate that such binary lattices are “good,” and their families possess a nice structure in terms of family complexity, collision and avalanche effect.

This paper derives elliptic gradient estimates for positive solutions to a nonlinear parabolic equation defined on a complete weighted Riemannian manifold. Applications of these estimates yield Liouville-type theorem, parabolic Harnack inequalities and bounds on weighted heat kernel on the lower boundedness assumption for Bakry-Émery curvature tensor.