Band 20, Heft 1

In this paper, we prove a theorem that confirms, under a supplementary condition, a conjecture concerning random permutations of sequences of partitions of the unit interval.

In this paper, we establish sufficient and necessary conditions for the existence of abelian subgroups of maximal order of a finite group G G , by means of its commuting graph. The order of these subgroups attains the bound c = ∣ [ x 1 ] ∣ + ⋯ + ∣ [ x m ] ∣ c=| \left[{x}_{1}]| \hspace{-0.25em}+\cdots +\hspace{-0.25em}\hspace{0.33em}| \left[{x}_{m}]| , where [ x i ] \left[{x}_{i}] denotes the conjugacy class of x i {x}_{i} in G G and m m is the smallest integer j j such that ∣ [ x 1 ] ∣ + ⋯ + ∣ [ x j ] ∣ ≥ ∣ C G ( x j ) ∣ | \left[{x}_{1}]| \hspace{-0.25em}+\cdots +| \left[{x}_{j}]| \ge | {C}_{G}\left({x}_{j})| , where C G ( x j ) {C}_{G}\left({x}_{j}) is the centralizer of x j {x}_{j} in G G .

The asset-liability management problem with cash flow under an uncertain exit time has been investigated in this article, which is based on the fundamental framework of the mean-variance model in the multi-period version. The liability and random cash flow will affect asset optimization, while the investor may be forced to withdraw from investments with a random probability at each period in our model. The closed-form expressions for the mean-variance optimal portfolio selection and its corresponding efficient frontier are obtained by employing the mean-field formulation and dynamic programming approach. Moreover, some numerical examples are provided to illustrate the validity and accuracy of the theoretical results.

In this paper, the stochastic asymptotic behavior of the nonautonomous stochastic higher-order Kirchhoff equation with variable coefficients is studied. By using the Galerkin method, the solution of this kind of equation is obtained, and stochastic dynamical system under this kind of equation is obtained; by using the uniform estimation, the existence of the family of D k {{\mathcal{D}}}_{k} -absorbing sets of the stochastic dynamical system Φ k {\Phi }_{k} is obtained, and the asymptotic compactness of Φ k {\Phi }_{k} is proved by the decomposition method. Finally, the D k {{\mathcal{D}}}_{k} -stochastic attractor family of the stochastic dynamical system Φ k {\Phi }_{k} in V m + k ( Ω ) × V k ( Ω ) {V}_{m+k}\left(\Omega )\times {V}_{k}\left(\Omega ) is obtained.

In this article, we introduce and study the intersection graph of graded submodules of a graded module. Let M M be a left G G -graded R R -module. We define the intersection graph of G G -graded R R -submodules of M M , denoted by Γ ( G , R , M ) \Gamma \left(G,R,M) , to be a simple undirected graph whose set of vertices consists of all nontrivial G G -graded R R -submodules of M M , where two vertices are adjacent if their intersection is nonzero. We study properties of these graphs, such as connectivity, diameter, and girth. We also investigate the intersection graph of graded submodules for certain types of gradings such as faithful and strong gradings.

Motivated by the celebrated work of Lutwak, Yang and Zhang [1] on ( p , q ) \left(p,q) -mixed volumes and that of Feng and He [2] on ( p , q ) \left(p,q) -mixed geominimal surface areas, we in the present paper establish and confirm the affine isoperimetric and Brunn-Minkowski-type inequalities with respect to ( p , q ) \left(p,q) -mixed geominimal surface areas.

This work is concerned with dynamical behavior of a second-order fuzzy discrete population model: x n = A x n − 1 1 + x n − 1 + B x n − 2 , n = 1 , 2 , … , {x}_{n}=\frac{A{x}_{n-1}}{1+{x}_{n-1}+B{x}_{n-2}},\hspace{1em}n=1,2,\ldots , where A , B A,B are positive fuzzy numbers. x n {x}_{n} is a positive fuzzy number and represents the population size at the observation instant n . According to a generalization of division ( g g -division) of fuzzy number, we study the dynamical behaviors including boundedness, global asymptotical stability, and persistence of positive fuzzy solution. Finally, two examples are given to demonstrate the effectiveness of the results obtained.

The purpose of this paper is to prove the following result. Let R R be prime ring of characteristic different from two and three, and let F : R → R F:R\to R be an additive mapping satisfying the relation F ( x 3 ) = F ( x 2 ) x − x F ( x ) x + x F ( x 2 ) F\left({x}^{3})=F\left({x}^{2})x-xF\left(x)x+xF\left({x}^{2}) for all x ∈ R x\in R . In this case, F F is of the form 4 F ( x ) = D ( x ) + q x + x q 4F\left(x)=D\left(x)+qx+xq for all x ∈ R x\in R , where D : R → R D:R\to R is a derivation, and q q is some fixed element from the symmetric Martindale ring of quotients of R R .

Abardia and Bernig proposed the notions of complex projection body and complex mixed projection body. In this paper, we introduce the concepts of the general complex L p {L}_{p} projection body and complex L p {L}_{p} mixed projection body. Furthermore, we establish the Brunn-Minkowski-type inequalities for the general complex L p {L}_{p} projection bodies and the Aleksandrov-Fenchel-type inequalities for the general complex L p {L}_{p} mixed projection bodies.

In this paper, we first investigate the total proper connection number of a graph G G according to some constraints of G ¯ \overline{G} . Next, we investigate the total proper connection numbers of graph G G with large clique number ω ( G ) = n − s \omega \left(G)=n-s for 1 ≤ s ≤ 3 1\le s\le 3 . Finally, we determine the total proper k k -connection numbers of circular ladders, Möbius ladders and all small cubic graphs of order 8 or less.

In this paper, our aims are to study the stability with general decay rate of hybrid stochastic fractional differential equations driven by Lévy noise with impulsive effects. Using Lyapunov function, nonnegative semi-martingale convergence theorem, we obtain the almost sure stability with general decay rate, including the exponential stability and the polynomial stability. Moreover, we give an upper bound of each coefficient at any mode according to the theory of M-matrix. Finally, one example is given to show the effectiveness of the obtained theory.

In this paper, we introduce a new metric space called the mixed-norm Lebesgue space, which allows its norm decay to zero with different rates as ∣ x ∣ → ∞ | x| \to \infty in different spatial directions. Then we study the well posedness for the system of magnetohydrodynamic equations in 3D mixed-norm Lebesgue spaces. By using some fundamental analysis theories in mixed-norm Lebesgue space such as Young’s inequality, time decaying of solutions for heat equations, and the boundedness of the Helmholtz-Leray projection, we prove local well posedness and global well posedness of the solutions.

In this paper, we study the problem of finding a common solution of the pseudomonotone variational inequality problem and fixed point problem for demicontractive mappings. We introduce a new inertial iterative scheme that combines Tseng’s extragradient method with the viscosity method together with the adaptive step size technique for finding a common solution of the investigated problem. We prove a strong convergence result for our proposed algorithm under mild conditions and without prior knowledge of the Lipschitz constant of the pseudomonotone operator in Hilbert spaces. Finally, we present some numerical experiments to show the efficiency of our method in comparison with some of the existing methods in the literature.

The generic uniqueness of saddle point for two-person zero-sum differential games, within the class of open-loop, against the perturbation of the right-hand side function of the control system is investigated. By employing set-valued mapping theory, it is proved that the majority of the two-person zero-sum differential games have unique saddle point in the sense of Baire’s category.

In this paper, we define the concepts of strongly regular relation, finitely strongly regular relation, and generalized finitely strongly regular relation, and get the relational representations of strongly algebraic, hyperalgebraic, and quasi-hyperalgebraic lattices. The main results are as follows: (1) a binary relation ρ : X ⇀ Y \rho :X\hspace{0.33em}\rightharpoonup \hspace{0.33em}Y is strongly regular if and only if the complete lattice ( Φ ρ ( X ) , ⊆ ) \left({\Phi }_{\rho }\left(X),\subseteq ) is a strongly algebraic lattice; (2) a binary relation ρ : X ⇀ Y \rho :X\hspace{0.33em}\rightharpoonup \hspace{0.33em}Y is finitely strongly regular if and only if ( Φ ρ ( X ) , ⊆ ) \left({\Phi }_{\rho }\left(X),\subseteq ) is a hyperalgebraic lattice if and only if the finite extension of ρ \rho is strongly regular; and (3) a binary relation ρ : X ⇀ Y \rho :X\hspace{0.33em}\rightharpoonup \hspace{0.33em}Y is generalized finitely strongly regular if and only if ( Φ ρ ( X ) , ⊆ ) \left({\Phi }_{\rho }\left(X),\subseteq ) is a quasi-hyperalgebraic lattice if and only if ( Φ ρ ( X ) , ⊆ ) \left({\Phi }_{\rho }\left(X),\subseteq ) equipped with the interval topology is a Priestley space.

Extending our previous work we construct weakly holomorphic Hecke eigenforms whose period polynomials correspond to elements in a basis consisting of odd and even Hecke eigenpolynomials induced by only cusp forms. As an application of our results, we give an explicit construction of the holomorphic parts of harmonic weak Maass forms that are good for Hecke eigenforms. Moreover, we give an explicit construction of the Hecke-equivariant map between the space of weakly holomorphic cusp forms and two copies of the spaces of cusp forms, and show that the map is compatible with the corresponding map on the spaces of period polynomials.

In this paper, we introduce the concepts of G G -Lipschitz tracking property and G G -asymptotic tracking property in metric G G -space and obtain the equivalent conditions of G G -asymptotic tracking property in metric G G -space. In addition, we prove that the self-map f f has the G G -Lipschitz tracking property if and only if the shift map σ \sigma has the G ¯ \overline{G} -Lipschitz tracking property in the inverse limit space under the topological group action. These results generalize the corresponding results in [Proc. Amer. Math. Soc. 115 (1992), 573–580].

The aim of this paper is to find arithmetic convolution sums of some restricted divisor functions. When divisors of a certain natural number satisfy a suitable condition for modulo 12, those restricted divisor functions are expressed by the coefficients of certain eta quotients. The coefficients of eta quotients are expressed by the sine function and cosine function, and this fact is used to derive formulas for the convolution sums of restricted divisor functions and of the number of divisors. In the sine function used to find the coefficients of eta quotients, the result is obtained by utilizing a feature with symmetry between the divisor and the corresponding divisor. Let N , r N,r be positive integers and d d be a positive divisor of N N . Let e r ( N ; 12 ) {e}_{r}\left(N;\hspace{0.33em}12) denote the difference between the number of 2 N d − d \frac{2N}{d}-d congruent to r r modulo 12 and the number of those congruent to − r -r modulo 12. The main results of this article are to find the arithmetic convolution identities for ∑ a 1 + ⋯ + a j = N ( ∏ i = 1 j e ˆ ( a i ) ) {\sum }_{{a}_{1}+\cdots +{a}_{j}=N}({\prod }_{i=1}^{j}\hat{e}\left({a}_{i})) with e ˆ ( a i ) = e 1 ( a i ; 12 ) + 2 e 3 ( a i ; 12 ) + e 5 ( a i ; 12 ) \hat{e}\left({a}_{i})={e}_{1}\left({a}_{i};\hspace{0.33em}12)+2{e}_{3}\left({a}_{i};\hspace{0.33em}12)+{e}_{5}\left({a}_{i};\hspace{0.33em}12) and j = 1 , 2 , 3 , 4 j=1,2,3,4 . All results are obtained using elementary number theory and modular form theory.

Difference equations are often used to create discrete mathematical models. In this paper, we mainly study the dynamical behaviors of positive solutions of a nonlinear fuzzy difference equation: x n + 1 = x n A + B x n − k ( n = 0 , 1 , 2 , … ) , {x}_{n+1}=\frac{{x}_{n}}{A+B{x}_{n-k}}\hspace{0.33em}\left(n=0,1,2,\ldots ), where parameters A , B A,B and initial value x − k , x − k + 1 , … , x − 1 , x 0 {x}_{-k},{x}_{-k+1},\ldots ,{x}_{-1},{x}_{0} , k ∈ { 0 , 1 , … } k\in \{0,1,\ldots \} are positive fuzzy numbers. We investigate the existence, boundedness, convergence, and asymptotic stability of the positive solutions of the fuzzy difference equation. At last, we give numerical examples to intuitively reflect the global behavior. The conclusion of the global stability of this paper can be applied directly to production practice.

In this paper, we study the structures of the invariant subspaces under the action of orthogonal group O 2 ν ( F q , S ) {O}_{2\nu }\left({F}_{q},S) . In particular, we give a detailed description of 2-codimensional invariant subspaces. Moreover, we show that the height of transfer ideal Im ( Tr O 2 ν ( F q , S ) ) {\rm{Im}}({{\rm{Tr}}}^{{O}_{2\nu }\left({F}_{q},S)}) is 2 and give a primary decomposition for the radical ideal of this transfer ideal.

The main theorem of this article is to evaluate and express the multinomial convolution sum of the divisor function σ r ♯ ( n ; N / 4 , N ) {\sigma }_{r}^{\sharp }\left(n;\hspace{0.33em}N\hspace{-0.08em}\text{/}\hspace{-0.08em}4,N) in as a simple form as possible, where N / 4 N\hspace{-0.08em}\text{/}\hspace{-0.08em}4 is an arbitrary odd positive integer. This generalizes previous result in combination with Cho and Kim, which is about the case N = 4 N=4 . While obtaining our main theorem, we derive some generalizations of other identities to the case that we are dealing with.

In this paper, we establish several kinds of integral inequalities in two independent variables, which improve well-known versions of Gronwall-Bellman inequalities and extend them to fractional integral form. By using these inequalities, we can provide explicit bounds on unknown functions. The integral inequalities play an important role in the qualitative theory of differential and integral equations and partial differential equations.

In this paper, we study the unicity of meromorphic functions concerning differences and small functions and mainly prove two results: 1. Let f f be a transcendental entire function of finite order with a Borel exceptional entire small function a ( z ) a\left(z) , and let η \eta be a constant such that Δ η 2 f ≢ 0 {\Delta }_{\eta }^{2}\hspace{0.25em}f\not\equiv 0 . If Δ η 2 f {\Delta }_{\eta }^{2}\hspace{0.25em}f and Δ η f {\Delta }_{\eta }\hspace{0.25em}f share Δ η a {\Delta }_{\eta }a CM, then a ( z ) a\left(z) is a constant a a and f ( z ) = a + B e A z f\left(z)=a+B{e}^{Az} , where A , B A,B are two nonzero constants; 2. Let f f be a transcendental meromorphic function with ρ 2 ( f ) < 1 {\rho }_{2}(f)\lt 1 , let a 1 {a}_{1} , a 2 {a}_{2} be two distinct small functions of f f , let L ( z , f ) L\left(z,f) be a linear difference polynomial, and let a 1 ≢ L ( z , a 2 ) {a}_{1}\not\equiv L\left(z,{a}_{2}) . If δ ( a 2 , f ) > 0 \delta \left({a}_{2},f)\gt 0 , and f f and L ( z , f ) L\left(z,f) share a 1 {a}_{1} and ∞ \infty CM, then L ( z , f ) − a 1 f − a 1 = c , \frac{L\left(z,f)-{a}_{1}}{f-{a}_{1}}=c, for some constant c ≠ 0 c\ne 0 . The results improve some results following C. X. Chen and R. R. Zhang [ Uniqueness theorems related difference operators of entire functions , Chinese Ann. Math. Ser. A 42 (2021), no. 1, 11–22] and R. R. Zhang, C. X. Chen, and Z. B. Huang [ Uniqueness on linear difference polynomials of meromorphic functions , AIMS Math. 6 (2021), no. 4, 3874–3888].

Let S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}) , where a 1 , … , a m {a}_{1},\ldots ,{a}_{m} and b 1 , … , b n {b}_{1},\ldots ,{b}_{n} are two nonincreasing sequences of nonnegative integers. The pair S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}) is said to be a bigraphic pair if there is a simple bipartite graph G = ( X ∪ Y , E ) G=\left(X\cup Y,E) such that a 1 , … , a m {a}_{1},\ldots ,{a}_{m} and b 1 , … , b n {b}_{1},\ldots ,{b}_{n} are the degrees of the vertices in X X and Y Y , respectively. In this case, G G is referred to as a realization of S S . Given a bigraphic pair S S , and a complete bipartite graph K s , t {K}_{s,t} , we say that S S is a potentially K s , t {K}_{s,t} - bigraphic pair if some realization of S S contains K s , t {K}_{s,t} as a subgraph (with s s vertices in the part of size m m and t t in the part of size n n ). Ferrara et al. ( Potentially H-bigraphic sequences , Discuss. Math. Graph Theory 29 (2009), 583–596) defined σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) to be the minimum integer k k such that every bigraphic pair S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};{b}_{1},\ldots ,{b}_{n}) with σ ( S ) = a 1 + ⋯ + a m ≥ k \sigma \left(S)={a}_{1}+\cdots +{a}_{m}\ge k is a potentially K s , t {K}_{s,t} -bigraphic pair. This problem can be viewed as a “potential” degree sequence relaxation of the (forcible) Turán problem. Ferrara et al. determined σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) for n ≥ m ≥ 9 s 4 t 4 n\ge m\ge 9{s}^{4}{t}^{4} . In this paper, we further determine σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) for n ≥ m ≥ s n\ge m\ge s and n + m ≥ 2 t 2 + t + s n+m\ge 2{t}^{2}+t+s . As two corollaries, if n ≥ m ≥ t 2 + t + s 2 n\ge m\ge {t}^{2}+\frac{t+s}{2} or if n ≥ m ≥ s n\ge m\ge s and n ≥ 2 t 2 + t n\ge 2{t}^{2}+t , the values σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) are determined completely. These results give a solution to a problem due to Ferrara et al. and a solution to a problem due to Yin and Wang.

In this paper, we consider the following fractional p p -equation with a gradient term: ( − Δ ) p s u ( x ) = f ( x , u ( x ) , ∇ u ( x ) ) . {\left(-\Delta )}_{p}^{s}u\left(x)=f\left(x,u\left(x),\nabla u\left(x)). We first prove the uniqueness and monotonicity of positive solutions in a bounded domain. Then by estimating the singular integrals which define the fractional p p -laplacian along a sequence of approximate maximum points, we obtain monotonicity of positive solutions in the whole space via the sliding method. In order to solve the difficulties caused by the gradient term, we introduce some new techniques which may also be applied to investigate the qualitative properties of solutions for many problems with gradient terms. Our results are extensions of Berestycki and Nirenberg [ Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations , J. Geom. Phys. 5 (1988), 237–275] and Wu and Chen [ The sliding methods for the fractional p-Laplacian , Adv. Math. 361 (2020), 106933].

The aim of this paper is to present a new stable method for smoothing and differentiating noisy data defined on a bounded domain Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} with N ≥ 1 N\ge 1 . The proposed method stems from the smoothing properties of the classical diffusion equation; the smoothed data are obtained by solving a diffusion equation with the noisy data imposed as the initial condition. We analyze the stability and convergence of the proposed method and we give optimal convergence rates. One of the main advantages of this method lies in multivariable problems, where some of the other approaches are not easily generalized. Moreover, this approach does not require strong smoothness assumptions on the underlying data, which makes it appealing for detecting data corners or edges. Numerical examples demonstrate the feasibility and robustness of the method even with the presence of a large amounts of noise.

A positive definite homogeneous multivariate form plays an important role in the field of optimization, and positive definiteness of the form can be identified by a special structured tensor. In this paper, based on the equivalence between the form and the corresponding tensor, and the links of the positive definiteness of a tensor with ℋ-tensor, we propose an ℋ-tensor-based criterion for identifying the positive definiteness of multivariate homogeneous forms. Some numerical examples are provided to illustrate the efficiency and validity of our results.

Let A {\mathcal{A}} be a ∗ \ast -algebra, ℳ {\mathcal{ {\mathcal M} }} be a ∗ \ast - A {\mathcal{A}} -bimodule, and δ \delta be a linear mapping from A {\mathcal{A}} into ℳ {\mathcal{ {\mathcal M} }} . δ \delta is called a ∗ \ast - derivation if δ ( A B ) = A δ ( B ) + δ ( A ) B \delta \left(AB)=A\delta \left(B)+\delta \left(A)B and δ ( A ∗ ) = δ ( A ) ∗ \delta \left({A}^{\ast })=\delta {\left(A)}^{\ast } for each A , B A,B in A {\mathcal{A}} . Let G G be an element in A {\mathcal{A}} , δ \delta is called a ∗ \ast - antiderivable mapping at G G if A B ∗ = G ⇒ δ ( G ) = B ∗ δ ( A ) + δ ( B ) ∗ A A{B}^{\ast }=G\Rightarrow \delta \left(G)={B}^{\ast }\delta \left(A)+\delta {\left(B)}^{\ast }A for each A , B A,B in A {\mathcal{A}} . We prove that if A {\mathcal{A}} is a C ∗ {C}^{\ast } -algebra, ℳ {\mathcal{ {\mathcal M} }} is a Banach ∗ \ast - A {\mathcal{A}} -bimodule and G G in A {\mathcal{A}} is a separating point of ℳ {\mathcal{ {\mathcal M} }} with A G = G A AG=GA for every A A in A {\mathcal{A}} , then every ∗ \ast -antiderivable mapping at G G from A {\mathcal{A}} into ℳ {\mathcal{ {\mathcal M} }} is a ∗ \ast -derivation. We also prove that if A {\mathcal{A}} is a zero product determined Banach ∗ \ast -algebra with a bounded approximate identity, ℳ {\mathcal{ {\mathcal M} }} is an essential Banach ∗ \ast - A {\mathcal{A}} -bimodule and δ \delta is a continuous ∗ \ast -antiderivable mapping at the point zero from A {\mathcal{A}} into ℳ {\mathcal{ {\mathcal M} }} , then there exists a ∗ \ast -Jordan derivation Δ \Delta from A {\mathcal{A}} into ℳ ♯ ♯ {{\mathcal{ {\mathcal M} }}}^{\sharp \sharp } and an element ξ \xi in ℳ ♯ ♯ {{\mathcal{ {\mathcal M} }}}^{\sharp \sharp } such that δ ( A ) = Δ ( A ) + A ξ \delta \left(A)=\Delta \left(A)+A\xi for every A A in A {\mathcal{A}} . Finally, we show that if A {\mathcal{A}} is a von Neumann algebra and δ \delta is a ∗ \ast -antiderivable mapping (not necessary continuous) at the point zero from A {\mathcal{A}} into itself, then there exists a ∗ \ast -derivation Δ \Delta from A {\mathcal{A}} into itself such that δ ( A ) = Δ ( A ) + A δ ( I ) \delta \left(A)=\Delta \left(A)+A\delta \left(I) for every A A in A {\mathcal{A}} .

In this paper, we study the initial boundary problem of fifth-order Korteweg-de Vries equation with nonlinear boundary values. First, we establish a so-called sharp boundary trace regularity associated with the linearized fifth-order Korteweg-de Vries equation. Then, aided by the sharp boundary trace regularity, we verify that initial-boundary value problem of fifth-order Korteweg-de Vries equation with nonlinear boundary conditions is locally well-posed when initial and boundary values are properly chosen.

In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. Furthermore, by introducing the Bernoulli number and the rational fraction expansion of tangent function, some special and interesting Hilbert-type inequalities and their equivalent hardy-type inequalities are presented at the end of the paper.

The aim of this paper is to study the structure of arbitrary split twisted inner derivation triple systems. We obtain a sufficient condition for the decomposition of arbitrary twisted inner derivation triple system T {\mathscr{T}} which is of the form T = U + ∑ [ θ ] ∈ Λ T / ∼ I [ θ ] {\mathscr{T}}=U+{\sum }_{\left[\theta ]\in {\Lambda }^{{\mathscr{T}}}\text{/} \sim }{I}_{\left[\theta ]} with U U a subspace of T 0 {{\mathscr{T}}}_{0} and any I [ θ ] {I}_{\left[\theta ]} a well-described ideal of T {\mathscr{T}} , satisfying { I [ θ ] , T , I [ η ] } \left\{{I}_{\left[\theta ]},{\mathscr{T}},{I}_{\left[\eta ]}\right\} = { I [ θ ] , I [ η ] , T } \left\{{I}_{\left[\theta ]},{I}_{\left[\eta ]},{\mathscr{T}}\right\} = { T , I [ θ ] , I [ η ] } \left\{{\mathscr{T}},{I}_{\left[\theta ]},{I}_{\left[\eta ]}\right\} = { I [ θ ] , T , I [ η ] } ′ \left\{{I}_{\left[\theta ]},{\mathscr{T}},{I}_{\left[\eta ]}\right\}^{\prime} = { I [ θ ] , I [ η ] , T } ′ \left\{{I}_{\left[\theta ]},{I}_{\left[\eta ]},{\mathscr{T}}\right\}^{\prime} = { T , I [ θ ] , I [ η ] } ′ = 0 \left\{{\mathscr{T}},{I}_{\left[\theta ]},{I}_{\left[\eta ]}\right\}^{\prime} =0 if [ θ ] ≠ [ η ] \left[\theta ]\ne \left[\eta ] . In particular, a necessary and sufficient condition for the simplicity of the triple system is given.

We prove that if an η \eta -Einstein para-Kenmotsu manifold admits a conformal η \eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η \eta -Ricci soliton is Einstein if its potential vector field V V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η \eta -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal η \eta -Ricci soliton and satisfy our results. We also have studied conformal η \eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.

In this paper, a discrete Leslie-Gower predator-prey system with Michaelis-Menten type harvesting is studied. Conditions on the existence and stability of fixed points are obtained. It is shown that the system can undergo fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations are presented to illustrate the main theoretical results. Compared to the continuous analog, the discrete system here possesses much richer dynamical behaviors including orbits of period-16, 21, 35, 49, 54, invariant cycles, cascades of period-doubling bifurcation in orbits of period-2, 4, 8, and chaotic sets.

In this article, we consider fully discrete characteristic mixed finite elements for convection-diffusion optimal control problems. We use the characteristic line method to treat the hyperbolic part of the state equation as a directional derivative. The state and the co-state are discretized by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. Using some proper duality problems, we derive a posteriori error estimates for the scalar functions. Such estimates are not available in the literature. A numerical example is presented to validate the theoretical results.

We propose and analyze a Lotka-Volterra commensal model with an additive Allee effect in this article. First, we study the existence and local stability of possible equilibria. Second, the conditions for the existence of saddle-node bifurcations and transcritical bifurcations are derived by using Sotomayor’s theorem. Third, we give sufficient conditions for the global stability of the boundary equilibrium and positive equilibrium. Finally, we use numerical simulations to verify the above theoretical results. This study shows that for the weak Allee effect case, the additive Allee effect has a negative effect on the final density of both species, with increasing Allee effect, the densities of both species are decreasing. For the strong Allee effect case, the additive Allee effect is one of the most important factors that leads to the extinction of the second species. The additive Allee effect leads to the complex dynamic behaviors of the system.

In this article, an effective finite element method based on dimension reduction scheme is proposed for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.

We discuss certain closure operators that generalize the Alexandroff topologies. Such a closure operator is defined for every ordinal α > 0 \alpha \gt 0 in such a way that the closure of a set A A is given by closures of certain α \alpha -indexed sequences formed by points of A A . It is shown that connectivity with respect to such a closure operator can be viewed as a special type of path connectivity. This makes it possible to apply the operators in solving problems based on employing a convenient connectivity such as problems of digital image processing. One such application is presented providing a digital analogue of the Jordan curve theorem.

This paper is concerned with the existence and uniqueness theorems for stochastic functional differential equations with Markovian switching and jump, where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Lyapunov-type functions.

In this study, we establish some new Hermite-Hadamard type inequalities for s -convex functions in the second sense using the post-quantum calculus. Moreover, we prove a new ( p , q ) \left(p,q) -integral identity to prove some new Ostrowski type inequalities for ( p , q ) \left(p,q) -differentiable functions. We also show that the newly discovered results are generalizations of comparable results in the literature. Finally, we give application to special means of real numbers using the newly proved inequalities.

In the article, we establish some monotonicity and convexity (concavity) properties for certain combinations of polynomials and the Ramanujan R -function by use of the monotone form of L’Hôpital’s rule and present serval new asymptotically sharp bounds for the Ramanujan R -function that improve some previously known results.

In this paper, we use the shooting method to study the solvability of the boundary value problem of differential equations with sign-changing weight function: u ″ ( t ) + ( λ a + ( t ) − μ a − ( t ) ) g ( u ) = 0 , 0 < t < T , u ′ ( 0 ) = 0 , u ′ ( T ) = 0 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}^{^{\prime\prime} }\left(t)+\left(\lambda {a}^{+}\left(t)-\mu {a}^{-}\left(t))g\left(u)=0,\hspace{1.0em}0\lt t\lt T,\\ u^{\prime} \left(0)=0,\hspace{1.0em}u^{\prime} \left(T)=0,\end{array}\right. where a ∈ L [ 0 , T ] a\in L\left[0,T] is sign-changing and the nonlinearity g : [ 0 , ∞ ) → R g:{[}0,\infty )\to {\mathbb{R}} is continuous such that g ( 0 ) = g ( 1 ) = g ( 2 ) = 0 g\left(0)=g\left(1)=g\left(2)=0 , g ( s ) > 0 g\left(s)\gt 0 for s ∈ ( 0 , 1 ) s\in \left(0,1) , g ( s ) < 0 g\left(s)\lt 0 for s ∈ ( 1 , 2 ) s\in \left(1,2) .

In this article, we study two classes of Kirchhoff-type equations as follows: − a + b ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u + V ( x ) u = ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + f ( u ) , in R 3 , u ∈ H 1 ( R 3 ) , \left\{\begin{array}{l}-\left(a+b\underset{{{\mathbb{R}}}^{3}}{\overset{}{\int }}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+V\left(x)u=({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u+f\left(u),\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{3}),\end{array}\right.\hspace{1.85em} and − a + b ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u + V ( x ) u = ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + m ∣ u ∣ l − 2 u , in R 3 , u ∈ H 1 ( R 3 ) , \left\{\begin{array}{l}-\left(a+b\underset{{{\mathbb{R}}}^{3}}{\overset{}{\int }}| \nabla u{| }^{2}{\rm{d}}x\right)\Delta u+V\left(x)u=({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u+m| u{| }^{l-2}u,\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{3}),\end{array}\right. where a > 0 a\gt 0 , b ≥ 0 b\ge 0 , α ∈ ( 0 , 3 ) \alpha \in \left(0,3) , ( 3 + α ) / 3 < p < ( 3 + α ) \left(3+\alpha )\hspace{0.1em}\text{/}\hspace{0.1em}3\lt p\lt \left(3+\alpha ) , l ≥ 6 l\ge 6 , m > 0 m\gt 0 , V : R 3 → R V:{{\mathbb{R}}}^{3}\to {\mathbb{R}} is a potential function and I α {I}_{\alpha } is a Riesz potential whose order is α ∈ ( 0 , 3 ) \alpha \in \left(0,3) . Under some assumptions on V ( x ) V\left(x) and f ( u ) f\left(u) , we can prove that the equations have ground state solutions by variational methods.

The Picard iteration method is used to study the existence and uniqueness of solutions for the stochastic Volterra-Levin equation with variable delays. Several sufficient conditions are specified to ensure that the equation has a unique solution. First, the stochastic Volterra-Levin equation is transformed into an integral equation. Then, to obtain the solution of the integral equation, the successive approximation sequences are constructed, and the existence and uniqueness of solutions for the stochastic Volterra-Levin equation are derived by the convergence of the sequences. Finally, two examples are given to demonstrate the validity of the theoretical results.

In this article, we are concerned with the periodic solutions of first-order difference equation Δ u ( t − 1 ) = f ( t , u ( t ) ) − s , t ∈ Z , ( P ) \Delta u\left(t-1)=f\left(t,u\left(t))-s,\hspace{1em}t\in {\mathbb{Z}},\hspace{1.0em}\hspace{1.0em}\left(P) where s ∈ R s\in {\mathbb{R}} , f : Z × R → R f:{\mathbb{Z}}\times {\mathbb{R}}\to {\mathbb{R}} is continuous with respect to u ∈ R u\in {\mathbb{R}} , f ( t , u ) = f ( t + T , u ) f\left(t,u)=f\left(t+T,u) , T > 1 T\gt 1 is an integer, Δ u ( t − 1 ) = u ( t ) − u ( t − 1 ) \Delta u\left(t-1)=u\left(t)-u\left(t-1) . We prove a result of Ambrosetti-Prodi-type for ( P ) \left(P) by using the method of lower and upper solutions and topological degree. We relax the coercivity assumption on f f in Bereanu and Mawhin [1] and obtain Ambrosetti-Prodi-type results.

To cope with the development of digital economy, it is particularly important to discuss the digital transformation strategy of government-enterprise cooperation. Taking government and enterprise as game subjects, this paper constructs a game model of government subsidy behavior and enterprise digital transformation effort in the process of digital transformation based on differential game theory and studies different game strategies in noncooperative game, Stackelberg game, and cooperative game. The results show that (1) as an effective regulation mechanism, the government subsidy coefficient can significantly improve the optimal effort of the digital transformation of enterprises; (2) the optimal strategy, the optimal benefit, and the total benefit of the digital ecosystem formed by the government and enterprises in the cooperative game are better than those in the non-cooperative game. Pareto optimality is achieved; (3) the results of theoretical derivation are verified by numerical simulation. This study provides a theoretical basis for the digital transformation cooperation between government and enterprises.

This article is devoted to study the existence conditions of solutions to several complex differential-difference equations. We obtain some Malmquist theorems related to complex differential-difference equations with a more general form than the previous equations given by Zhang, Huang, and others. Moreover, some examples are provided to demonstrate why some restrictive conditions in some of our theorems cannot be removed.

In this article, we will discuss disjoint diskcyclicity for finitely many operators acting on a separable, infinite dimensional Fréchet space X X . More precisely, we provide disjoint disk blow-up/collapse property and disjoint diskcyclicity criterion. In addition, we characterize the disjoint diskcyclicity for weighted shifts both in the bilateral and unilateral cases.

A scheme for the analytical stochastization of ordinary differential equations (ODEs) is presented in this article. Using Itô calculus, an ODE is transformed into a stochastic differential equation (SDE) in such a way that the analytical solutions of the obtained equation can be constructed. Furthermore, the constructed stochastic trajectories remain bounded in the same interval as the deterministic solutions. The proposed approach is in a stark contrast to methods based on the randomization of solution trajectories and is not focused on the analysis of martingales. This article extends the theory of Itô calculus by directly implementing it into analytical schemes for the solution of differential equations based on the generalized operator of differentiation. The efficacy of the presented analytical stochastization techniques is demonstrated by deriving stochastic soliton solutions to the Riccati differential equation. The presented semi-analytical stochastization scheme is relevant for the investigation of the global dynamics of different biological and biomedical processes where the variation interval of the stochastic solution is predetermined by the rationale of the model.

In this manuscript, some remarks on the papers [H. A. Hammad, P. Agarwal, S. Momani, and F. Alsharari, Solving a fractional-order differential equation using rational symmetric contraction mappings , Fractal Fract. 5 (2021), 159] and [A. Hussain, F. Jarad, and E. Karapinar, A study of symmetric contractions with an application to generalized fractional differential equations , Adv. Differ. Equ. 2021 (2021), 300] are given. In the light of remarks, we introduce a new property that makes it convenient to investigate the existence of fixed points of the interpolative contractions in the orthogonal metric spaces. We derive several new results based on known contractions from the main theorems. As an application, we resolve a Urysohn integral equation.

In this article, we mainly consider a repairable degradation system consisting of a single component, a repairman, and repair equipment. Suppose that the system cannot be repaired immediately after failure and cannot be repaired “as good as new.” Herein, the repair equipment may fail during repair and the system will replace a new one after failures. In particular, the repair time follows the general distribution. Under the above assumptions, a partial differential equation model is established through the geometric process and supplementary variable technique. By Laplace transform, we obtain the availability of the system, and from the expression one can see that the availability of the system will tend to zero after running for a long time. Therefore, we further study a replacement policy N N based on the failed times of the system. We give the explicit expression of the system average cost C ( N ) C\left(N) and obtain the optimal replacement policy N ∗ {N}^{\ast } by minimizing the average cost rate C ( N ∗ ) C\left({N}^{\ast }) . That is, the system will be replaced when the failure number of the system reaches N ∗ {N}^{\ast } . Furthermore, the extended degenerate system is proposed by assuming that the system is not always successive degenerative after repair, and then the optimal replacement policy is studied. Finally, the numerical analysis is given to illustrate the theoretical results.

This paper intend to study the following critical fractional Schrödinger-Kirchhoff-Poisson equations with electromagnetic fields in R 3 {{\mathbb{R}}}^{3} : ε 2 s M ( [ u ] s , A 2 ) ( − Δ ) A s u + V ( x ) u + ( ∣ x ∣ 2 t − 3 ∗ ∣ u ∣ 2 ) u = f ( x , ∣ u ∣ 2 ) u + ∣ u ∣ 2 s ∗ − 2 u , x ∈ R 3 . {\varepsilon }^{2s}{\mathfrak{M}}\left({\left[u]}_{s,A}^{2}){\left(-\Delta )}_{A}^{s}u+V\left(x)u+\left(| x\hspace{-0.25em}{| }^{2t-3}\ast | u\hspace{-0.25em}{| }^{2})u=f\left(x,| u\hspace{-0.25em}{| }^{2})u+| u\hspace{-0.25em}{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{3}. Under suitable assumptions, together with the concentration compactness principle and variational method, we prove that the existence and multiplicity of semiclassical solutions for above problem as ε → 0 \varepsilon \to 0 .

The exact traveling wave solutions of generalized Davey-Stewartson equations with arbitrary power nonlinearities are studied using the dynamical system and the first integral methods. Taking different parameter conditions, we obtain periodic wave solutions, exact solitary wave solutions, kink wave solutions, and anti-kink wave solutions.

We consider a jump Markov process X = ( X t ) t ≥ 0 X={\left({X}_{t})}_{t\ge 0} , with values in a state space ( E , ℰ ) \left(E,{\mathcal{ {\mathcal E} }}) . We suppose that the corresponding infinitesimal generator π θ ( x , d y ) , x ∈ E {\pi }_{\theta }\left(x,{\rm{d}}y),x\in E , hence the law P x , θ {{\mathbb{P}}}_{x,\theta } of X X , depends on a parameter θ ∈ Θ \theta \in \Theta . We prove that several models (filtered or not) associated with X X are linked, by their regularity according to a certain scheme. In particular, we show that the regularity of the model ( π θ ( x , d y ) ) θ ∈ Θ {\left({\pi }_{\theta }\left(x,{\rm{d}}y))}_{\theta \in \Theta } is equivalent to the local regularity of ( P x , θ ) θ ∈ Θ {\left({{\mathbb{P}}}_{x,\theta })}_{\theta \in \Theta } .

While it is known that one can consider the existence of solutions to boundary-value problems for fractional differential equations with derivative terms, the situations for the multiplicity of weak solutions for the p-Laplacian fractional differential equations with derivative terms are less considered. In this article, we propose a new class of p-Laplacian fractional differential equations with the Caputo derivatives. The multiplicity of weak solutions is proved by the variational method and critical point theorem. At the end of the article, two examples are given to illustrate the validity and practicality of our main results.

By applying the combination of discrete variational method and approximation, developed in a previous study [J. Kuang, W. Chen, and Z. Guo, Periodic solutions with prescribed minimal period for second-order even Hamiltonian systems , Commun. Pure Appl. Anal. 21 (2022), no. 1, 47–59], we overcome some difficulties in the absence of Ambrosetti-Rabinowitz condition and obtain new sufficient conditions for the existence of periodic solutions with prescribed minimal period for second-order Hamiltonian systems with asymptotically linear nonlinearities.

In this article, we analyze the convergence rate of the modified Levenberg-Marquardt (MLM) method under the Hölderian local error bound condition and the Hölderian continuity of the Jacobian, which are more general than the local error bound condition and the Lipschitz continuity of the Jacobian. Under special circumstances, the convergence rate of the MLM method coincides with the results presented by Fan. A globally convergent MLM algorithm by the trust region technique will also be given.

Let q = p m q={p}^{m} , p p be an odd prime, and R k = F q [ u 1 , u 2 , … , u k ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ {R}_{k}={{\mathbb{F}}}_{q}\left[{u}_{1},{u}_{2},\ldots ,{u}_{k}]\hspace{-0.08em}\text{/}\hspace{-0.08em}\langle {u}_{i}^{3}={u}_{i},{u}_{i}{u}_{j}={u}_{j}{u}_{i}\rangle , where k ≥ 1 k\ge 1 and 1 ≤ i , j ≤ k 1\le i,j\le k . In this article, we define a Gray map from R k n {R}_{k}^{n} to F q 3 k n {{\mathbb{F}}}_{q}^{{3}^{k}n} . We study constacyclic codes over R k {R}_{k} and construct non-binary quantum codes over F q {{\mathbb{F}}}_{q} .

The general position problem is to find the cardinality of the largest vertex subset S S such that no triple of vertices of S S lies on a common geodesic. For a connected graph G G , the cardinality of S S is denoted by gp ( G ) {\rm{gp}}\left(G) and called the gp {\rm{gp}} -number (or general position number) of G G . In the paper, we obtain an upper bound and a lower bound regarding the gp {\rm{gp}} -number in all cacti with k k cycles and t t pendant edges. Furthermore, the exact value of the gp {\rm{gp}} -number on wheel graphs is determined.

In this article, we consider a two-dimensional inverse heat conduction problem that determines the surface temperature distribution from measured data at the fixed location. This problem is severely ill-posed, i.e., the solution does not depend continuously on the data. A quasi-boundary value regularization method in conjunction with the a posteriori parameter choice strategy is proposed to solve the problem. A Hölder-type error estimate between the approximate solution and its exact solution is also given. The error estimate shows that the regularized solution is dependent continuously on the data.

In this article, we deal with orbital stability and Zhukovskiǐ quasi-stability of periodic or recurrent orbits in an impulsive dynamical system defined in the n -dimensional Euclidean space R n {{\mathbb{R}}}^{n} . We show that for a periodic orbit of an impulsive system, its asymptotically orbital stability is equivalent to the asymptotically Zhukovskiǐ quasi-stability, and for a recurrent orbit, the orbital stability is equivalent to the Zhukovskiǐ quasi-stability.

In this paper, the authors present some monotonicity properties for certain functions involving the complete p -elliptic integrals of the first and second kinds, by showing the monotonicity and concavity-convexity properties of certain combinations defined in terms of K p {{\mathscr{K}}}_{p} , E p {{\mathscr{E}}}_{p} and the inverse hyperbolic tangent arth p {{\rm{arth}}}_{p} , which is of importance in the computation of the generalized pi and in the elementary proof of Ramanujan’s cubic transformation. By these results, several well-known results for the classical complete elliptic integrals including its bounds and logarithmic inequalities are remarkably improved.

In this article, the authors consider the commutators of strongly singular Calderón-Zygmund operator with Lipschitz functions. A sufficient condition is given for the boundedness of the commutators from Lebesgue spaces L p ( R n ) {L}^{p}\left({{\mathbb{R}}}^{n}) to certain Campanato spaces C p , β ( R n ) {C}^{p,\beta }\left({{\mathbb{R}}}^{n}) .

Generalized Munn rings exist extensively in the theory of rings. The aim of this note is to answer when a generalized Munn ring is primitive (semiprimitive, semiprime and prime, respectively). Sufficient and necessary conditions are obtained for a generalized Munn ring with a regular sandwich matrix to be primitive (semiprimitive, semiprime and prime, respectively). Also, we obtain sufficient and necessary conditions for a Munn ring over principal ideal domains to be prime (semiprime, respectively). Our results can be regarded as the generalizations of the famous result in the theory of rings that for a ring R R , R R is primitive (semiprimitive and semiprime, respectively) if and only if so is M n ( R ) {M}_{n}\left(R) . As applications of our results, we consider the primeness and the primitivity of generalized matrix rings and generalized path algebras. In particular, it is proved that a path algebra is a semiprime if and only if it is semiprimitive.

In graph G G , a vertex dominates itself and its neighbors. A subset S ⊆ V ( G ) S\subseteq V\left(G) is said to be a double-dominating set of G G if S S dominates every vertex of G G at least twice. The double domination number γ × 2 ( G ) {\gamma }_{\times 2}\left(G) is the minimum cardinality of a double dominating set of G G . We show that if G G is a maximal outerplanar graph on n ≥ 3 n\ge 3 vertices, then γ × 2 ( G ) ≤ 2 n 3 {\gamma }_{\times 2}\left(G)\le &#x230A;\frac{2n}{3}&#x230B; . Further, if n ≥ 4 n\ge 4 , then γ × 2 ( G ) ≤ min n + t 2 , n − t {\gamma }_{\times 2}\left(G)\le \min \left\{&#x230A;\frac{n+t}{2}&#x230B;,n-t\right\} , where t t is the number of vertices of degree 2 in G G . These bounds are shown to be tight. In addition, we also study the case that G G is a striped maximal outerplanar graph.

In this article, based on the path homology theory of digraphs, which has been initiated and studied by Grigor’yan, Lin, Muranov, and Yau, we prove the existence and uniqueness of solutions to the problem ∥ w ∥ = min u ∈ Ω 2 ( G ) , u ≠ 0 1 2 ∥ ∂ u − w ∥ 2 2 + ∣ u ∣ 1 \parallel w\parallel =\mathop{\min }\limits_{u\in {\Omega }_{2}\left(G),u\ne 0}\left\{\phantom{\rule[-1.25em]{}{0ex}},\frac{1}{2}{\parallel \partial u-w\parallel }_{2}^{2}+{| u| }_{1}\right\} for w ∈ H 1 ( G ) w\in {H}_{1}\left(G) and any digraph G G generated by squares and triangles belonging to the same cluster.

The control systems described by the Urysohn-type integral equations and integral constraints on the control functions are considered. The functions from the closed ball of the space L p {L}_{p} , p > 1 , p\gt 1, with radius r r , are chosen as admissible control functions. The trajectory of the system is defined as a p -integrable function, satisfying the system’s equation almost everywhere. The boundedness and path-connectedness of the set of p -integrable trajectories are discussed. It is illustrated that the set of trajectories, in general, is not a closed subset of the space L p {L}_{p} . The robustness of a trajectory with respect to the fast consumption of the remaining control resource is established, and it is proved that every trajectory of the system can be approximated by the trajectory obtained by the full consumption of the control resource.

In this article, we present a new robust estimation procedure based on the exponential squared loss function for varying coefficient partially functional linear regression models, where the slope function and nonparametric coefficients are approximated by functional principal component basis functions and B splines, respectively. Under some mild conditions, the convergence rates of the resulted estimators are obtained. Simulation studies indicate that our proposed method can achieve robustness against outliers or heavy-tail error distributions and perform no worse than the popular least-squares estimation method for the normal error case. Finally, a real data example is used to illustrate the application of the proposed method.

In this article, we are concerned with the equations of Krylov type on compact Hermitian manifolds, which are in the form of the linear combinations of the elementary symmetric functions of a Hermitian matrix. Under the assumption of the 𝒞-subsolution, we obtain a priori estimates in Γ k − 1 {\Gamma }_{k-1} cone. By using the method of continuity, we prove an existence theorem, which generalizes the relevant results. As an application, we give an alternative way to solve the deformed Hermitian Yang-Mills equation on compact Kähler threefold.

Let K K be an imaginary quadratic field different from Q ( − 1 ) {\mathbb{Q}}\left(\sqrt{-1}) and Q ( − 3 ) {\mathbb{Q}}\left(\sqrt{-3}) . For a nontrivial integral ideal m {\mathfrak{m}} of K K , let K m {K}_{{\mathfrak{m}}} be the ray class field modulo m {\mathfrak{m}} . By using some inequalities on special values of modular functions, we show that a single x x -coordinate of a certain elliptic curve generates K m {K}_{{\mathfrak{m}}} over K K .

All the (2, 1)-congruences on a left restriction semigroup become a complete sublattice of its lattice of congruences. The aim of this article is to study certain fundamental properties of this complete sublattice. We introduce k k -, K K -, t t -, and T T -operators on this sublattice and obtain some properties. As applications, the remarkable (2, 1)-congruences are characterized. These results extend the corresponding results on inverse semigroups and ample semigroups.

We study a linear boundary value problem for systems of essentially loaded differential equations with an integro-multipoint condition. We make use of the numerical implementation of the Dzhumabaev parametrization method to obtain the desired result, which is well supported by two numerical examples.

This article studies the regression learning problem from given sample data by using stochastic approximation (SA) type algorithm, namely, the accelerated SA. We focus on problems without strong convexity, for which all well-known algorithms achieve a convergence rate for function values of O ( 1 / n ) O\left(1\hspace{0.1em}\text{/}\hspace{0.1em}n) . We consider and analyze accelerated SA algorithm that achieves a rate of O ( 1 / n ) O\left(1\hspace{0.1em}\text{/}\hspace{0.1em}n) for classical least-square regression and logistic regression problems, respectively. Comparing with the well-known results, we only need fewer conditions to obtain the tight convergence rate for least-square regression and logistic regression problems.

A matrix A A is said to have the displacement structure if the rank of the Sylvester displacement A U − V A AU-VA or the Stein displacement A − V A U A-VAU is much smaller than the rank of A A . In this article, we study the displacement structure of the DMP inverse A d , † {A}^{d,\dagger } . Estimations for the Sylvester displacement rank of DMP inverse are presented under some restrictions. The generalized displacement is also discussed. The general results are applied to the core inverse.

In this article, we study the eigenvalue dependence of Sturm-Liouville problems on time scales with spectral parameter in the boundary conditions. We obtain that the eigenvalues not only continuously but also smoothly depend on the parameters of the problem. Moreover, the differential expressions of the eigenvalues with respect to the data are given.

In this article, we consider a discrete nonlinear third-order boundary value problem Δ 3 u ( k − 1 ) = λ a ( k ) f ( k , u ( k ) ) , k ∈ [ 1 , N − 2 ] Z , Δ 2 u ( η ) = α Δ u ( N − 1 ) , Δ u ( 0 ) = − β u ( 0 ) , u ( N ) = 0 , \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\Delta }^{3}u\left(k-1)=\lambda a\left(k)f\left(k,u\left(k)),\hspace{1em}k\in {\left[1,N-2]}_{{\mathbb{Z}}},\hspace{1.0em}\\ {\Delta }^{2}u\left(\eta )=\alpha \Delta u\left(N-1),\Delta u\left(0)=-\beta u\left(0),\hspace{1em}u\left(N)=0,\hspace{1.0em}\end{array}\right. where N > 4 N\gt 4 is an integer, λ > 0 \lambda \gt 0 is a parameter. f : [ 1 , N − 2 ] Z × [ 0 , + ∞ ) → [ 0 , + ∞ ) f:{\left[1,N-2]}_{{\mathbb{Z}}}\times \left[0,+\infty )\to \left[0,+\infty ) is continuous, a : [ 1 , N − 2 ] Z → ( 0 , + ∞ ) a:{\left[1,N-2]}_{{\mathbb{Z}}}\to \left(0,+\infty ) , α ∈ 0 , 1 N − 1 \alpha \in \left[0,\frac{1}{N-1}\right) , β ∈ 0 , 2 ( 1 − α ( N − 1 ) ) N ( 2 − α ( N − 1 ) ) \beta \in \left[0,\frac{2\left(1-\alpha \left(N-1))}{N\left(2-\alpha \left(N-1))}\right) , and η ∈ N − 2 2 + 1 , N − 2 Z \eta \in {\left[\left[\frac{N-2}{2}\right]+1,N-2\right]}_{{\mathbb{Z}}} . With the sign-changing Green’s function, we obtain not only the existence of positive solutions but also the multiplicity of positive solutions to this problem.

In this article, we present some new fixed point existence results for nonexpansive-type mappings in geodesic spaces. We also give a number of illustrative examples to settle our claims. We study the asymptotic behavior of Picard iterates generated by these class of mappings under different conditions. Finally, we approximate the solutions of the constrained minimization problem in the setting of Cartan, Alexandrov, and Toponogov (CAT(0)) spaces.

The connectivity is an important measurement for the fault-tolerance of a network. The generalized connectivity is a natural generalization of the classical connectivity. An S S -tree of a connected graph G G is a tree T = ( V ′ , E ′ ) T=\left(V^{\prime} ,E^{\prime} ) that contains all the vertices in S S subject to S ⊆ V ( G ) S\subseteq V\left(G) . Two S S -trees T T and T ′ T^{\prime} are internally disjoint if and only if E ( T ) ∩ E ( T ′ ) = ∅ E\left(T)\cap E\left(T^{\prime} )=\varnothing and V ( T ) ∩ V ( T ′ ) = S V\left(T)\cap V\left(T^{\prime} )=S . Denote by κ ( S ) \kappa \left(S) the maximum number of internally disjoint S S -trees in graph G G . The generalized k k -connectivity is defined as κ k ( G ) = min { κ ( S ) ∣ S ⊆ V ( G ) and ∣ S ∣ = k } {\kappa }_{k}\left(G)=\min \left\{\kappa \left(S)| S\subseteq V\left(G)\hspace{0.33em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{0.33em}| S| \hspace{0.33em}=\hspace{0.33em}k\right\} . Clearly, κ 2 ( G ) = κ ( G ) {\kappa }_{2}\left(G)=\kappa \left(G) . In this article, we show that κ 4 ( H S n ) = n − 1 {\kappa }_{4}\left(H{S}_{n})=n-1 , where H S n H{S}_{n} is the hierarchical star network.

In this article, we further study the filter theory of semihoops. Moreover, we use the prime (maximal) filters to construct the prime (maximal) spectrum on semihoops, and prove that the prime spectrum is a compact T 0 {T}_{0} topological space and that the maximal spectrum is a compact T 2 {T}_{2} topological space. As an application, in order to study the relationship between the spectrum of semihoops and the spectrum of lattices, we introduce the reticulations of semihoops and obtain some related results.

In this article, a generalization of the well-known Stein-Weiss inequality for the fractional integral operator on functions with different integrability properties in the radial and the angular direction in local Morrey spaces is established. We find that some conditions can be relaxed for the Stein-Weiss inequality for local mixed radial-angular Morrey spaces.

In this article, we design a family of scale-free networks and study its random target access time and weighted spanning trees through the eigenvalues of transition weight matrix. First, we build a type of fractal network with a weight factor r r and a parameter m m . Then, we obtain all the eigenvalues of its transition weight matrix by revealing the recursive relationship between eigenvalues in every two consecutive time steps and obtain the multiplicities corresponding to these eigenvalues. Furthermore, we provide a closed-form expression of the random target access time for the network studied. The obtained results show that the random target access is not affected by the weight; it is only affected by parameters m m and t t . Finally, we also enumerate the weighted spanning trees of the studied networks through the obtained eigenvalues.

In this article, we consider the identification of an unknown steady source in a class of fractional diffusion equations. A modified Tikhonov regularization method based on Hermite expansion is presented to deal with the ill-posedness of the problem. By using the properties of Hermitian functions, we construct a modified penalty term for the Tikhonov functional. It can be proved that the method can adaptively achieve the order optimal results when we choose the regularization parameter by the discrepancy principle. Some examples are also provided to verify the effectiveness of the method.

Let k k be a nonnegative integer. Let K K be a number field and O K {{\mathcal{O}}}_{K} be the ring of integers of K K . Let ℓ ≥ 5 \ell \ge 5 be a prime and v v be a prime ideal of O K {{\mathcal{O}}}_{K} over ℓ \ell . Let f f be a modular form of weight k + 1 2 k+\frac{1}{2} on Γ 0 {\Gamma }_{0} (4) such that its Fourier coefficients are in O K {{\mathcal{O}}}_{K} . In this article, we study sufficient conditions that if f f has the form f ( z ) ≡ ∑ n = 1 ∞ ∑ i = 1 t a f ( s i n 2 ) q s i n 2 ( mod v ) f\left(z)\equiv \mathop{\sum }\limits_{n=1}^{\infty }\mathop{\sum }\limits_{i=1}^{t}{a}_{f}\left({s}_{i}{n}^{2}){q}^{{s}_{i}{n}^{2}}\hspace{0.5em}\left({\rm{mod}}\hspace{0.33em}v) with square-free integers s i {s}_{i} , then f f is congruent to a linear combination of iterated derivatives of a single theta function modulo v v .

We obtain a class of commutators of bilinear pseudo-differential operators on products of Hardy spaces by applying the accurate estimates of the Hörmander class. And we also prove another version of these types of commutators on Herz-type spaces.

The main purpose of this article is to develop a theory that extends the domain of any local isometry to the whole space containing the domain, where a local isometry is an isometry between two proper subsets. In fact, the main purpose of this article consists of the following three detailed objectives: The first objective is to extend the bounded domain of any local isometry to the first-order generalized linear span. The second one is to extend the bounded domain of any local isometry to the second-order generalized linear span. The third objective of this article is to extend the bounded domain of any local isometry to the whole Hilbert space.

This study elucidates the sufficient conditions for the first-order nonlinear differential equations with periodic coefficients and time-varying delays to have positive periodic solutions. Our results are proved using the Krasnosel’skii fixed point theorem. In this article, we have identified two sets Δ \Delta and ∇ \nabla and proved that at least one positive periodic solution exists in the interval between the point belonging to Δ \Delta and the point belonging to ∇ \nabla . We propose simple conditions that guarantee the existence of sets Δ \Delta and ∇ \nabla . In addition, we obtain the necessary conditions for the existence of positive periodic solutions of the first-order nonlinear differential equations when the periodic coefficients satisfy certain conditions. Finally, examples and numerical simulations are used to illustrate the validity of our results.

Let A {\mathcal{A}} be a unital primitive C ∗ \ast -algebra. This article studies the properties of the B-Fredholm elements, the B-Weyl elements and the B-Browder elements in A {\mathcal{A}} . Particularly, this article describes the B-Fredholm element as the sum of a Fredholm element and a nilpotent element. In addition, the socle of A {\mathcal{A}} is characterized by the B-Fredholm elements.

In this work, the solvability for an inverse problem of a nonlinear parabolic equation with nonlocal integral overdetermination supplementary condition is examined. The proof of the existence and uniqueness of the solution of the inverse nonlinear parabolic problem upon the data is established by using the fixed-point technique. In addition, the inverse problem is investigated by using the cubic B-spline collocation technique together with the Tikhonov regularization. The resulting nonlinear system of parabolic equation is approximated using the MATLAB subroutine lsqnonlin . The obtained results demonstrate the accuracy and efficiency of the technique, and the stability of the approximate solutions even in the existence of noisy data. The stability analysis is also conducted for the discretized system of the direct problem.

Discovering maximal frequent itemsets is an important issue and key technique in many data mining problems such as association rule mining. In the literature, generating maximal frequent itemsets proves either to be NP-hard or to have O ( l 3 4 l ( m + n ) ) O\left({l}^{3}{4}^{l}\left(m+n)) complexity in the worst case from the perspective of generating maximal complete bipartite graphs of a bipartite graph, where m m , n n are the item number and the transaction number, respectively, and l l denotes the maximum of ∣ C ∣ ∣ Ψ ( C ) ∣ / ( ∣ C ∣ + ∣ Ψ ( C ) ∣ − 1 ) | C| | \Psi \left(C)| \hspace{0.1em}\text{/}\hspace{0.1em}\left(| C| +| \Psi \left(C)| -1) , with the maximum taken over all maximal frequent itemsets C C . In this article, we put forward a method for discovering maximal frequent itemsets, whose complexity is O ( 3 m n 2 β + 4 β n ) O\left(3mn{2}^{\beta }+{4}^{\beta }n) , lower than the known complexity both in the worst case, from the perspective of semigroup algebra, where β \beta is the number of items whose support is more than the minimum support threshold. Experiments also show that an algorithm based on the algebraic method performs better than the other three well-known algorithms. Meanwhile, we explore some algebraic properties with respect to items and transactions, prove that the maximal frequent itemsets are exactly the simplified generators of frequent itemsets, give a necessary and sufficient condition for a maximal i + 1 i+1 -frequent itemset being a subset of a closed i i -frequent itemset, and provide a recurrence formula of maximal frequent itemsets.

The normalizer problem of integral group rings has been studied extensively in recent years due to its connection with the longstanding isomorphism problem of integral group rings. Class-preserving Coleman automorphisms of finite groups occur naturally in the study of the normalizer problem. Let G G be a finite group with a nilpotent subgroup N N . Suppose that G / N G\hspace{-0.0em}\text{/}\hspace{-0.0em}N acts faithfully on the center of each Sylow subgroup of N N . Then it is proved that every class-preserving Coleman automorphism of G G is an inner automorphism. In addition, if G G is the product of a cyclic normal subgroup and an abelian subgroup, then it is also proved that every class-preserving Coleman automorphism of G G is an inner automorphism. Other similar results are also obtained in this article. As direct consequence, the normalizer problem has a positive answer for such groups.

This article is concerned with the nonlinear stability of traveling waves of a delayed susceptible-infective-removed (SIR) epidemic model with nonlocal dispersal, which can be seen as a continuity work of Li et al. [ Traveling waves for a nonlocal dispersal SIR model with delay and external supplies , Appl. Math. Comput. 247 (2014), 723–740]. We prove that the traveling wave solution is exponentially stable when the initial perturbation around the traveling wave is relatively small in a weighted norm. The time decay rate is also obtained by weighted-energy estimates.

We are concerned with Dirichlet problems of impulsive differential equations − u ″ ( x ) − λ u ( x ) + g ( x , u ( x ) ) + ∑ j = 1 p I j ( u ( x ) ) δ ( x − y j ) = f ( x ) for a.e. x ∈ ( 0 , π ) , u ( 0 ) = u ( π ) = 0 , \left\{\begin{array}{l}-{u}^{^{\prime\prime} }\left(x)-\lambda u\left(x)+g\left(x,u\left(x))+\mathop{\displaystyle \sum }\limits_{j=1}^{p}{I}_{j}\left(u\left(x))\delta \left(x-{y}_{j})=f\left(x)\hspace{1em}\hspace{0.1em}\text{for a.e.}\hspace{0.1em}\hspace{0.33em}x\in \left(0,\pi ),\\ u\left(0)=u\left(\pi )=0,\\ \end{array}\right. where λ \lambda is a parameter and runs near 1, f ∈ L 2 ( 0 , π ) f\in {L}^{2}\left(0,\pi ) , I j ∈ C ( R , R ) {I}_{j}\in C\left({\mathbb{R}},{\mathbb{R}}) , j = 1 , 2 , … , p j=1,2,\ldots ,p , p ∈ N p\in {\mathbb{N}} , the nonlinearity g : [ 0 , π ] × R → R g:\left[0,\pi ]\times {\mathbb{R}}\to {\mathbb{R}} satisfies the Carathéodory condition, δ = δ ( x ) \delta =\delta \left(x) denote the Dirac delta impulses concentrated at 0, which are applied at given points 0 < y 1 < y 2 < ⋯ < y p < π 0\lt {y}_{1}\lt {y}_{2}\hspace{0.33em}\lt \cdots \lt {y}_{p}\lt \pi . We show the existence and multiplicity of solutions to the aforementioned problem for λ \lambda in a neighborhood of 1 by using degree theory and bifurcation theory.

The conjugacy relation plays an important role in group theory and the conjugacy relation of groups has been generalized to semigroups in various methods by several authors. If a a and b b are elements of a semigroup S S , then a a is called primarily conjugate to b b if a = u v a=uv and b = v u b=vu for some u , v ∈ S 1 u,v\in {S}^{1} . In general, primary conjugacy is reflexive and symmetric, but not transitive. Finding the classes of semigroups in which the primary conjugacy is transitive is an open problem raised by Araújo et al. in the literature. In this article, among other things we prove that the primary conjugacy is transitive in regular ω \omega -semigroups.

We consider a discrete-time Markov controlled process endowed with the expected total discounted reward. We assume that the distribution of the underlying random vectors is unknown and that it is approximated by an appropriate known distribution. We found upper bounds of a decrease in reward when the policy, optimal for the approximating process, is applied to control the original process.

In this article, we introduce two new classes of modules over a ϕ \phi -ring that generalize the classes of coherent modules and Noetherian modules. We next study the possible transfer of the properties of being nonnil-Noetherian rings, ϕ \phi -coherent rings, and nonnil-coherent rings in the amalgamated algebra along an ideal.

In this article, we present a new concept named the N -tuples weighted noncommutative Orlicz space ⊕ j = 1 n L p , λ ( Φ j ) ( ℳ ˜ , τ ) {\oplus }_{j=1}^{n}{L}_{p,\lambda }^{\left({\Phi }_{j})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ) , where L ( Φ j ) ( ℳ ˜ , τ ) {L}^{\left({\Phi }_{j})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ) is the noncommutative Orlicz space. Based on the maximum principle, the Riesz-Thorin interpolation theorem of ⊕ j = 1 n L p , λ ( Φ j ) ( ℳ ˜ , τ ) {\oplus }_{j=1}^{n}{L}_{p,\lambda }^{\left({\Phi }_{j})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ) is given. As applications, we obtain the Clarkson inequality and some other geometrical properties which include the uniform convexity and uniform smoothness of noncommutative Orlicz spaces L ( Φ s ) ( ℳ ˜ , τ ) , 0 < s ≤ 1 {L}^{\left({\Phi }_{s})}\left(\widetilde{{\mathcal{ {\mathcal M} }}},\tau ),0\lt s\le 1 .

We mainly study the dimension-free L p {L}^{p} -inequality of the truncated maximal operator M n a f ( x ) = sup t > 0 1 ∣ B a 1 ∣ ∫ B a 1 f ( x − t y ) d y , {M}_{n}^{a}f\left(x)=\mathop{\sup }\limits_{t\gt 0}\frac{1}{| {B}_{a}^{1}| }\left|\mathop{\int }\limits_{{B}_{a}^{1}}f\left(x-ty){\rm{d}}y\hspace{-0.25em}\right|, where B a 1 = { x : a ≤ ∣ x ∣ ≤ 1 } {B}_{a}^{1}=\left\{x:a\le | x| \le 1\right\} . When 0 ≤ a < 1 0\le a\lt 1 , we prove that ‖ M n a ‖ L p ( R n ) ≤ C ( p ) ‖ f ‖ L p ( R n ) \Vert {M}_{n}^{a}{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}\le C\left(p)\Vert f{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})} for p > n / ( n − 1 ) p\gt n\hspace{0.1em}\text{/}\hspace{0.1em}\left(n-1) . When a = 1 a=1 , we prove that ‖ M n 1 ‖ L p ( R n ) ≤ C ( p ) ‖ f ‖ L p ( R n ) \Vert {M}_{n}^{1}{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})}\le C\left(p)\Vert f{\Vert }_{{L}^{p}\left({{\mathbb{R}}}^{n})} for p ≥ 2 p\ge 2 .

In the implementation of management error proofing, enterprises need to carry out risk evaluation and ranking of management activities error, to determine error proofing improvement order. We note that management function failure is different from industrial failure, and the management activity error as its cause is also different from operational or device error, so the method generally used in industry is not suitable for the risk assessment of management activity error. To address this issue, this study integrates the FMEA (failure modes and effect analysis) with the ISO9001 standardized management system framework describing multilevel and multi-attribute management activities, and the human errors that may lead to functional failure of quality management activities are identified at each level and each category. The three factors (severity (S), occurrence (O), and detection (D)) of errors in each management activity are evaluated and integrated into the risk priority number for each quality management activity error. The two major defects of the traditional FMEA in the practical application process are as follows: (1) the evaluation of error attributes of various management activities is basically qualitative language description, which cannot be quantified; (2) the other one disadvantage is that it ignores the fact that three factors have the different weights in the system rather than equality, and not considering the relative importance of between them. In this article, the fuzzy set theory and the grey relational decision theory are, respectively, applied to improve the two defects and to improve the accuracy of the decision. Finally, a numerical example of a coal mine is given to verify the effectiveness of the proposed method. This study provides a basis for managers and practitioners to quantitatively evaluate and prevent errors in management activities.

Almost s s -mappings and almost compact mappings have been introduced and studied. In this article, we continue to research some questions related to the almost s -images (resp., almost compact images) of metric spaces. The following results are obtained. (1) A space X X is a quotient and almost compact image of a metric space if and only if X X is a sequential space having a c s ∗ c{s}^{\ast } -network which is point-regular at nonisolated points, which gives an affirmative answer to Question 4.9 in the article “S. Lin, X. W. Ling, and Y. Ge, Point-regular covers and sequence-covering compact mappings , Topology Appl. 271 (2020), 106987.” (2) There exists a bi-quotient and almost compact image of a metric space satisfying no base, which is point-countable at nonisolated points, which gives negative answers to Question 3.1 in the article “X. W. Ling and S. Lin, On open almost s-images of metric spaces , Adv. Math. (China) 48 (2019), no. 4, 489–496” and Question 3.7 in the article “X. W. Ling, S. Lin, and W. He, Point-countable covers and sequence-covering s-mappings at subsets , Topology Appl. 290 (2021), 107572.” (3) Some characterizations of countably bi-quotient and almost s s -images (resp., pseudo-open and almost compact images) of metric spaces.

In this article, we propose a Gompertz-two-parameter-Lindley distribution by mixing the frailty parameter of the Gompertz distribution with a two-parameter Lindley distribution. The structural properties of the model, such as shape properties, cumulative distribution, quantile functions, moment, moment generating function, failure rate function, mean residual function, and stochastic orders, were derived. Moreover, the unknown parameters are estimated by the profile log likelihood algorithm, and their performance is examined by simulation studies. Finally, a real data example is used to demonstrate the application of the proposed model.

In this article, we focus on the existence of positive solutions and establish a corresponding iterative scheme for a nonlinear fourth-order equation with indefinite weight and Neumann boundary conditions y ( 4 ) ( x ) + ( k 1 + k 2 ) y ″ ( x ) + k 1 k 2 y ( x ) = λ h ( x ) f ( y ( x ) ) , x ∈ [ 0 , 1 ] , y ′ ( 0 ) = y ′ ( 1 ) = y ‴ ( 0 ) = y ‴ ( 1 ) = 0 , \left\{\begin{array}{l}{y}^{\left(4)}\left(x)+\left({k}_{1}+{k}_{2}){y}^{^{\prime\prime} }\left(x)+{k}_{1}{k}_{2}y\left(x)=\lambda h\left(x)f(y\left(x)),\hspace{1em}x\in \left[0,1],\\ y^{\prime} \left(0)=y^{\prime} \left(1)={y}^{\prime\prime\prime }\left(0)={y}^{\prime\prime\prime }\left(1)=0,\\ \end{array}\right. where k 1 {k}_{1} and k 2 {k}_{2} are constants, λ > 0 \lambda \gt 0 is a parameter, h ( x ) ∈ L 1 ( 0 , 1 ) h\left(x)\in {L}^{1}\left(0,1) may change sign, and f ∈ C ( [ 0 , 1 ] × R + , R ) f\in C\left(\left[0,1]\times {{\mathbb{R}}}^{+},{\mathbb{R}}) , R + ≔ [ 0 , ∞ ) {{\mathbb{R}}}^{+}:= \left[0,\infty ) . We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we establish some new results of the existence of positive solutions to this problem if the nonlinearity f f is monotone on R + {{\mathbb{R}}}^{+} . The technique for dealing with this article relies on a monotone iteration technique and Schauder’s fixed point theorem. Finally, an example is presented to illustrate the application of our main results.

Most previous studies concerning delta waves have been focused on the overcompressible ones. To study the non-overcompressible delta waves, this article is concerned with the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas. The Riemann problem is completely solved. The solutions exhibit four kinds of wave patterns: the first contains three contact discontinuities; the second includes a single overcompressible delta wave, and the third and fourth involve a contact discontinuity and a non-overcompressible delta wave.

In this article, we use the Nehari manifold method to study a class of fourth-order even difference systems. First, we show that there exist multiple periodic solutions to the non-autonomous system. Second, we obtain sufficient conditions to guarantee the existence of solutions with prescribed minimal periods to the autonomous system. Our results generalize the results in reference.

Let f n {f}_{n} be the n n th Fibonacci number with f 1 = f 2 = 1 {f}_{1}={f}_{2}=1 . Recently, the exact values of ∑ k = n ∞ 1 f k s − 1 &#x230A;{\left({\sum }_{k=n}^{\infty }\frac{1}{{f}_{k}^{s}}\right)}^{-1}&#x230B; have been obtained only for s = 1 , 2 s=1,2 , where ⌊ x ⌋ \lfloor x\rfloor is the floor function. It has been an open problem for s ≥ 3 s\ge 3 . In this article, we consider the case of s = 4 s=4 and show that ∑ k = n ∞ 1 f k 4 − 1 = f n 4 − f n − 1 4 + 2 ( − 1 ) n 5 f 2 n − 1 − n + 2 5 , &#x230A;{\left(\mathop{\sum }\limits_{k=n}^{\infty }\frac{1}{{f}_{k}^{4}}\right)}^{-1}&#x230B;={f}_{n}^{4}-{f}_{n-1}^{4}+\frac{2{\left(-1)}^{n}}{5}{f}_{2n-1}-\left\{\frac{n+2}{5}\right\}, where { x } ≔ x − ⌊ x ⌋ \left\{x\right\}:= x-\lfloor x\rfloor .

This article is devoted to studying the averaging principle for two-time-scale stochastic differential equations with correlated noise. By the technique of multiscale expansion of the solution to the backward Kolmogorov equation and consequent elimination of variables, we obtain the Kolmogorov equation corresponding to the reduced simplified system. The approximation of the slow component of the original system to the solution of the corresponding averaged equation is in the weak sense. An example is also provided to illustrate our result.

This article investigates the spatial behavior of the solutions of the Brinkman equations in a semi-infinite cylinder. We no longer require the solutions to satisfy any a priori assumptions at infinity. Using the energy estimation method and the differential inequality technology, the differential inequality about the solutions is derived. By solving this differential inequality, it is proved that the solutions grow polynomially or decay exponentially with spatial variables. In the case of decay, the structural stability of Brinkman fluid is also proved.

The degenerate Stirling numbers of the second kind and of the first kind, which are, respectively, degenerate versions of the Stirling numbers of the second kind and of the first kind, appear frequently when we study various degenerate versions of some special numbers and polynomials. The aim of this article is to consider the r -truncated degenerate Stirling numbers of the second kind, which reduce to the degenerate Stirling numbers of the second for r = 1, and to investigate their explicit expressions, some properties and related identities, in connection with several other degenerate special numbers and polynomials.

A graph is said to be symmetric if its automorphism group is transitive on its arcs. This article is one of a series of articles devoted to characterizing prime-valent arc-transitive graphs of square-free order or twice square-free order. In this article, we determine all 7-valent symmetric graphs of order 2 pq and 11-valent symmetric graphs of order 4 pq .

In this article, some new sufficient conditions of p -nilpotency of finite groups are obtained by using c -normality and Φ-supplementary of the maximal or the 2-maximal subgroups of the Sylow p -subgroups.

This article is devoted to the study of the Bowen topological entropy for nonautonomous dynamical systems, which is an extension of the classical definition of Bowen topological entropy. We show that the Bowen topological entropy can be determined by the local entropies of measures for nonautonomous dynamical systems, which extends Ma and Wen’s result.

Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 and ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E 6 {E}_{6} -Higgs bundles over X X . We consider the automorphisms σ + {\sigma }_{+} of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ + ( E , φ ) = ( E ∗ , − φ t ) {\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t}) , induced by the action of the outer involution of E 6 {E}_{6} in ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , and σ − {\sigma }_{-} defined by σ − ( E , φ ) = ( E ∗ , φ t ) {\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t}) , which results from the combination of σ + {\sigma }_{+} with the involution of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ + {\sigma }_{+} and σ − {\sigma }_{-} , as F 4 {F}_{4} -Higgs bundles, F 4 {F}_{4} -Higgs pairs associated with the fundamental irreducible representation of F 4 {F}_{4} , and PSp ( 8 , C ) {\rm{PSp}}\left(8,{\mathbb{C}}) -Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp ( 8 , C ) {\rm{Sp}}\left(8,{\mathbb{C}}) . Finally, we describe the reduced notions of semistability and polystability for these objects.

In this paper, we first investigate the relationships between the McCoy module and related modules based on their relationships in rings. After that, we improve some properties of McCoy modules and introduce ZPZC modules which extend the notion of McCoy modules. We observe the structure of ZPZC modules providing a number of examples of problems that arise naturally in the process. Finally, answers to some open questions related to the ZPZC condition are provided.

In a real Banach space, let the VI indicate a variational inclusion for two accretive operators and let the CFPP denote a common fixed point problem of countably many nonexpansive mappings. In this article, we introduce a generalized extragradient implicit method for solving a general system of variational inequalities (GSVI) with the VI and CFPP constraints. Strong convergence of the suggested method to a solution of the GSVI with the VI and CFPP constraints under some suitable assumptions is established.

This work is to analyze a nonlocal dispersal model governed by a Volterra type integral and two space integrals. A weighted integral is included, and an existence result of solutions for this model is proved. We assume a suitably Hartman-type sign condition and use a sufficiently regular measure of noncompactness. Finally, the degree theory referring to condensing operators is applied.

A finite group G G is called an MSN-group if all maximal subgroups of the Sylow subgroups of G G are subnormal in G G . In this article, we investigate the structure of finite groups G G such that G G is a non-MSN-group of even order in which every maximal subgroup of even order is an MSN-group. In addition, we determine the minimal simple groups all of whose second maximal subgroups are MSN-groups.

In this article, we study the elliptic system: − Δ u + μ 1 u = ∣ x ∣ α u 3 + λ v , x ∈ Ω − Δ v + μ 2 v = ∣ x ∣ α v 3 + λ u , x ∈ Ω u , v > 0 , x ∈ Ω , u = v = 0 , x ∈ ∂ Ω , \left\{\begin{array}{ll}-\Delta u+{\mu }_{1}u=| x\hspace{-0.25em}{| }^{\alpha }{u}^{3}+\lambda v,& x\in \Omega \\ -\Delta v+{\mu }_{2}v=| x\hspace{-0.25em}{| }^{\alpha }{v}^{3}+\lambda u,& x\in \Omega \\ u,v\gt 0,x\in \Omega ,u=v=0,x\in \partial \Omega ,\end{array}\right. where Ω ⊂ R 3 \Omega \subset {{\mathbb{R}}}^{3} is the unit ball. By the variational method, we prove that if α \alpha is sufficiently small, the ground state solutions of the system are radial symmetric, and if α > 0 \alpha \gt 0 is sufficiently large, the ground state solutions are nonradial; however, the solutions are Schwarz symmetry.

It is well known that a sufficient and necessary condition for a continuous function g g to be almost periodic on time scale T {\mathbb{T}} is the existence of an almost periodic function f f on R {\mathbb{R}} such that f f is an extension of g g . The purpose of this article is to extend these results to S p {S}^{p} -almost periodic functions. We prove that the necessity is true, that is, an S p {S}^{p} -almost periodic function on T {\mathbb{T}} can be extended to an S p {S}^{p} -almost periodic function on R {\mathbb{R}} . However, a counterexample is given to show that the sufficiency is not true in general. By introducing a concept of minor translation set and characterizing the almost periodicity on T {\mathbb{T}} in terms of this new concept, we obtain a condition to ensure the sufficiency. Moreover, we show the necessity of this condition by a counterexample.

In this study, inspired by the definition and a previous study [F. Eryılmaz, SS-lifting modules and rings , Miskolc Math. Notes 22 (2021), no. 2, 655–662], left Harada rings are adapted to ss-Harada rings, and the important properties of these rings are provided. The characterization of a left ss-Harada ring R R with R R left perfect and Rad ( R ) {\rm{Rad}}\left(R) included in Soc ( R R ) {\rm{Soc}}\left({}_{R}R) was found with the help of strongly local R R -modules.

A subgroup functor τ \tau is said Φ \Phi -regular if for all primitive groups G G , whenever H ∈ τ ( G ) H\in \tau \left(G) is a p p -subgroup and N N is a minimal normal subgroup of G G , then ∣ G : N G ( H ∩ N ) ∣ = p d | G:{N}_{G}\left(H\cap N)| ={p}^{d} for some integer d d . In this article, we investigate groups in which some primary subgroups are τ \tau -subgroups for a Φ \Phi -regular subgroup functor τ \tau , and we obtain new criteria for the supersolubility or p p -nilpotency of a group.

In this article, new iterative algorithms for solving the discrete Riccati and Lyapunov equations are derived in the case where the transition matrix is diagonalizable with real eigenvalues. It is shown that the proposed iterative algorithms are faster than the classical ones, even for a small number of iterations.

Let f f be a C 2 {C}^{2} -diffeomorphism with Axiom A and no cycle condition on a two-dimensional smooth manifold. In this article, we prove that if f f is C 2 {C}^{2} -robustly weak measure expansive, then it is Q 2 {Q}^{2} -Anosov. Moreover, we expand the results of the C 2 {C}^{2} -diffeomorphism case into the C 2 {C}^{2} -vector field on a three-dimensional smooth manifold. Let X X be a C 2 {C}^{2} -vector field with Axiom A and no cycle condition. We prove that if X X is C 2 {C}^{2} -robustly weak measure expansive, then it is Q 2 {Q}^{2} -Anosov.

The main aim of this article is to study admissible congruences on a type B semigroup. First, we give characterizations of the minimum admissible congruence whose trace is a normal congruence on a type B semigroup. After obtaining some properties of admissible congruences with the same trace on a type B semigroup, we introduce the notion of a normal subsemigroup and give characterizations of the minimum and maximum admissible congruences whose kernels are normal subsemigroups. Finally, the concept of a congruence pair of a type B semigroup is given, and two congruences associated with the congruence pair are obtained.

In this article, we introduce the notions of generalized fractional integrals for the interval-valued functions (IVFs) of two variables. We establish Hermite-Hadamard (H-H) type inequalities and some related inequalities for co-ordinated convex IVFs by using the newly defined integrals. The fundamental benefit of these inequalities is that these can be turned into classical H-H inequalities and Riemann-Liouville fractional H-H inequalities, and new k k -Riemann-Liouville fractional H-H inequalities can be obtained for co-ordinated convex IVFs without having to prove each one separately.

This article is concerned with the inverse eigenvalue problem for rank one perturbations of the Sturm-Liouville operator. I obtain the relationship between the spectra of the Sturm-Liouville operator and its rank one perturbations, and from the spectra I reconstruct the perturbed operators, provided that the potential is known a priori .

Because the communication link connection in the Internet of Vehicles is transient and intermittent, the established connection may be disconnected at any time, which cannot meet the requirements of the data transmission of the Internet of Vehicles. Therefore, based on the traditional data transmission mechanism, the fuzzy comprehensive evaluation method is used to achieve the optimal design of the data transmission mechanism of the Internet of Vehicles. First, according to the operation mechanism of the Internet of Vehicles build the Internet of Vehicles model. Then in this model, the fuzzy comprehensive evaluation is used to analyze the data transmission characteristics of the Internet of Vehicles, and the characteristics evaluation results of data transmission are obtained. Finally, set the data transmission protocol of the Internet of Vehicles, select the appropriate data transmission path, allocate the data transmission load, and realize the data transmission of the Internet of Vehicles. The experimental results show that compared with the traditional data transmission mechanism of the Internet of Vehicles, the speed of the designed transmission mechanism is increased by 3.58 MB/s, and the packet loss rate is reduced by 41%.

The notion of dual uniformity is introduced on U C ( Y , Z ) UC(Y,Z) , the uniform space of uniformly continuous mappings between Y Y and Z Z , where ( Y , V ) (Y,{\mathcal{V}}) and ( Z , U ) (Z,{\mathcal{U}}) are two uniform spaces. It is shown that a function space uniformity on U C ( Y , Z ) UC(Y,Z) is admissible (resp. splitting) if and only if its dual uniformity on U Z ( Y ) = { f 2 − 1 ( U ) ∣ f ∈ U C ( Y , Z ) , U ∈ U } {{\mathcal{U}}}_{Z}(Y)=\{{f}_{2}^{-1}(U)\hspace{0.33em}| \hspace{0.33em}f\in UC(Y,Z),U\in {\mathcal{U}}\} is admissible (resp. splitting). It is also shown that a uniformity on U Z ( Y ) {{\mathcal{U}}}_{Z}(Y) is admissible (resp. splitting) if and only if its dual uniformity on U C ( Y , Z ) UC(Y,Z) is admissible (resp. splitting). Using duality theorems, it is also proved that the greatest splitting uniformity and the greatest splitting family open uniformity exist on U Z ( Y ) {{\mathcal{U}}}_{Z}(Y) and U C ( Y , Z ) UC(Y,Z) , respectively, and these two uniformities are mutually dual splitting uniformities.

First, this work provides an overview of some of the Hahn-Banach type theorems. Of note, some of these extension results for linear operators found recent applications to isotonicity of convex operators on a convex cone. Next, the work investigates applications of the Krein-Milman theorem to representation theory and elements of Choquet theory. A sandwich theorem of intercalating an affine function h h between f f and g , g, where f f\hspace{.25em} and – g \mbox{--}g are convex, f ≤ g f\le g on a finite-simplicial set, is recalled. Its recent topological version is also noted. Here, the novelty is that a finite-simplicial set may be unbounded in any locally convex topology on the domain space. Third, the paper summarizes and comments on recently published applications of a Hahn-Banach extension result for positive linear operators, combined with polynomial approximation on unbounded subsets, to the Markov moment problem. Some applications to concrete spaces are detailed as well. Finally, this work provides a characterization of a finite-dimensional convex bounded subset in terms of the property that any convex function defined on that subset is bounded below. This last property remains valid for a large class of convex operators.

Following the results of our recently published book [F. Lev, Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory. With Applications to Gravity and Particle Theory , Springer, 2020, ISBN 978-3-030-61101-9], we discuss different aspects of classical and finite mathematics and explain why finite mathematics based on a finite ring of characteristic p p is more general (fundamental) than classical mathematics: the former does not have foundational problems, and the latter is a special degenerate case of the former in the formal limit p → ∞ p\to \infty . In particular, quantum theory based on a finite ring of characteristic p p is more general than standard quantum theory because the latter is a special degenerate case of the former in the formal limit p → ∞ p\to \infty .

One of the most common challenges in multivariate statistical analysis is estimating the mean parameters. A well-known approach of estimating the mean parameters is the maximum likelihood estimator (MLE). However, the MLE becomes inefficient in the case of having large-dimensional parameter space. A popular estimator that tackles this issue is the James-Stein estimator. Therefore, we aim to use the shrinkage method based on the balanced loss function to construct estimators for the mean parameters of the multivariate normal (MVN) distribution that dominates both the MLE and James-Stein estimators. Two classes of shrinkage estimators have been established that generalized the James-Stein estimator. We study their domination and minimaxity properties to the MLE and their performances to the James-Stein estimators. The efficiency of the proposed estimators is explored through simulation studies.

In this paper, we consider the problem of representing any polynomial in terms of the degenerate Daehee polynomials and more generally of the higher-order degenerate Daehee polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

Davie’s exact coupling technique for stochastic differential equations may be used to enhance the convergence of the multilevel Monte Carlo (MC) methodology. Giles developed the multilevel MC technique, which is based on executing the MC method several times with various time increments. It cuts computing costs significantly by executing most simulations at a low cost. The essential concept behind the multilevel MC approach with the exact coupling is discussed in this article. Numerical implementation reveals significant computational savings, which supports the analysis.

In this article, by using critical point theory, we prove the existence of multiple T T -periodic solutions for difference equations with the mean curvature operator: − Δ ( ϕ c ( Δ u ( t − 1 ) ) ) + q ( t ) u ( t ) = λ f ( t , u ( t ) ) , t ∈ Z , -\Delta ({\phi }_{c}\left(\Delta u\left(t-1)))+q\left(t)u\left(t)=\lambda f\left(t,u\left(t)),\hspace{1em}t\in {\mathbb{Z}}, where Z {\mathbb{Z}} is the set of integers. As a T T -periodic problem, it does not require the nonlinear term is unbounded or bounded, and thus, our results are supplements to some well-known periodic problems. Finally, we give one example to illustrate our main results.

The aim of this article is to study the results of the fixed-point in coupled and tripled measure of noncompactness (MNC). We will use the technique of MNC for coupled and tripled MNC. Also, we tend to prove some results of coupled and tripled MNC for the family of JS-contractive-type mappings. Moreover, an application with an example is provided to illustrate the results.

In this paper, we study the existence of solutions for the neutral evolution equations with nonlocal conditions and delay in α \alpha -norm, which are more general than in many previous publications. We assume that the linear part generates an analytic semigroup and transforms them into suitable integral equations. By using the Kuratowski measure of noncompactness and fixed-point theory, some existence theorems are established without the assumption of compactness on the associated semigroup. Particularly, our results cover the cases where the nonlinear term F F takes values in different spaces such as X α {X}_{\alpha } . An example of neutral partial differential system is also given to illustrate the feasibility of our abstract results.

In this paper, we prove the global existence of incompressible Navier-Stokes equations with damping α ( e β ∣ u ∣ 2 − 1 ) u \alpha \left({e}^{\beta | u{| }^{2}}-1)u , where we use the Friedrich method and some new tools. The delicate problem in the construction of a global solution is the passage to the limit in exponential nonlinear term. To solve this problem, we use a polynomial approximation of the damping part and a new type of interpolation between L ∞ ( R + , L 2 ( R 3 ) ) {L}^{\infty }\left({{\mathbb{R}}}^{+},{L}^{2}\left({{\mathbb{R}}}^{3})) and the space of functions f f such that ( e β ∣ f ∣ 2 − 1 ) ∣ f ∣ 2 ∈ L 1 ( R + × R 3 ) \left({e}^{\beta | f{| }^{2}}-1)| f{| }^{2}\in {L}^{1}\left({{\mathbb{R}}}^{+}\times {{\mathbb{R}}}^{3}) . Fourier analysis and standard techniques are used.

The purpose of this paper is to investigate the ground state solutions for the following nonlinear Schrödinger equations involving the fractional p-Laplacian ( − Δ ) p s u ( x ) + λ V ( x ) u ( x ) p − 1 = u ( x ) q − 1 , u ( x ) ≥ 0 , x ∈ R N , {\left(-\Delta )}_{p}^{s}u\left(x)+\lambda V\left(x)u{\left(x)}^{p-1}=u{\left(x)}^{q-1},\hspace{1em}u\left(x)\ge 0,\hspace{0.33em}x\in {{\mathbb{R}}}^{N}, where λ > 0 \lambda \gt 0 is a parameter, 1 < p < q < N p N − s p 1\lt p\lt q\lt \frac{Np}{N-sp} , N ≥ 2 N\ge 2 , and V ( x ) V\left(x) is a real continuous function on R N {{\mathbb{R}}}^{N} . For λ \lambda large enough, the existence of ground state solutions are obtained, and they localize near the potential well int ( V − 1 ( 0 ) ) \left({V}^{-1}\left(0)) .

In this article, we study a class of Kirchhoff-type equation driven by the variable s ( x , ⋅)-order fractional p 1 ( x , ⋅) & p 2 ( x , ⋅)-Laplacian. With the help of three different critical point theories, we obtain the existence and multiplicity of solutions in an appropriate space of functions. The main difficulties and innovations are the Kirchhoff functions with double Laplace operators in the whole space ℝ N . Moreover, the approach is variational, but we do not impose any Ambrosetti-Rabinowitz condition for the nonlinear term.

Inequalities play an important role in pure and applied mathematics. In particular, Jensen’s inequality, one of the most famous inequalities, plays the main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. In this work, we prove some new Jensen-type inequalities for m -convex functions and apply them to generalized Riemann-Liouville-type integral operators. Furthermore, as a remarkable consequence, some new inequalities for convex functions are obtained.

In this article, we take into account the stochastic Kuramoto-Sivashinsky equation forced by multiplicative noise in the Itô sense. To obtain the exact stochastic solutions of the stochastic Kuramoto-Sivashinsky equation, we apply the G ′ G \frac{{G}^{^{\prime} }}{G} -expansion method. Furthermore, we extend some previous results where this equation has not been previously studied in the presence of multiplicative noise. Also, we show the influence of multiplicative noise on the analytical solutions of the stochastic Kuramoto-Sivashinsky equation.

In this paper, we introduce and study Golden-like statistical manifolds. We obtain some basic inequalities for curvature invariants of statistical submanifolds in Golden-like statistical manifolds. Also, in support of our definition, we provide a couple of examples.

This article is devoted to study the existence of global solutions and finite time blow-up of local solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term. By applying the potential well and energy estimation method, in low initial energy and critical initial energy, we derive some sufficient conditions which are global existence and explosion of the solutions for this type of Klein-Gordon equation.

In this article, we study Hopf bifurcation and Turing instability of a diffusive predator-prey model with hunting cooperation. For the local model, we analyze the stability of the equilibrium and derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, first it is shown that Turing instability occurs, then the direction and stability of the Hopf bifurcation is reached. Our results show that hunting cooperation plays a crucial role in the dynamics of the model, that is, it can be beneficial to the predator population and induce the rise of Turing instability. Finally, numerical simulations are performed to visualize the complex dynamic behavior.

The aim of this article is to design a new iteration process for solving certain fixed-point problems. In particular, we prove weak and strong convergence theorems for generalized nonexpansive mappings in the framework of uniformly convex Banach spaces. In addition, we discuss the stability of the solution under mild conditions. Further, we provide some numerical examples to indicate that the proposed method works properly.