Band 18, Heft 1

We consider the classical functional of the Calculus of Variations of the form I(u)=∫ Ω F(x,u(x),∇ u(x))dx, $$\begin{array}{} \displaystyle I(u)=\int\limits_{{\it\Omega}}F(x, u(x), \nabla u(x))\,dx, \end{array}$$ where Ω is a bounded open subset of ℝ n and F : Ω × ℝ × ℝ n → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω . We formulate some conditions under which a given function in ϕ + W01,p $\begin{array}{} \displaystyle W^{1,p}_0 \end{array}$( Ω ) with I ( u ) < +∞ can be approximated in the W 1, p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω . As a particular case we obtain that the Lavrentiev phenomenon does not occur when F ( x , u , ξ ) = f ( x , u ) + h ( x , ξ ) is convex and x ↦ F ( x , 0, 0) is sufficiently smooth.

Let A and B be nonempty subsets of a normed linear space X . A mapping T : A ∪ B → A ∪ B is said to be a noncyclic relatively nonexpansive mapping if T ( A ) ⊆ A , T ( B ) ⊆ B and ∥ Tx − Ty ∥ ≤ ∥ x − y ∥ for all ( x , y ) ∈ A × B . A best proximity pair for such a mapping T is a point ( p , q ) ∈ A × B such that p = Tp , q = Tq and d ( p , q ) = dist( A , B ). In this work, we introduce a geometric notion of proximal Opiaľs condition on a nonempty, closed and convex pair of subsets of strictly convex Banach spaces. By using this geometric notion, we study the strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We also establish a best proximity pair theorem for noncyclic contraction type mappings in the setting of strictly convex Banach spaces.

The construction of parametric curve and surface plays an important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with a real points set, called generalized toric-Bernstein (GT-Bernstein) basis functions. Then, the generalized toric-Bézier (GT-Bézier) curves and surfaces are constructed based on the GT-Bernstein basis functions, which are the projections of the (irrational) toric varieties in fact and the generalizations of the classical rational Bézier curves/surfaces and toric surface patches. Furthermore, we also study the properties of the presented curves and surfaces, including the limiting properties of weights and knots. Some representative examples verify the properties and results.

Given the adjacency matrix A of a digraph, the eigenvalues of the matrix AA T constitute the so-called non-negative spectrum of this digraph. We investigate the relation between the structure of digraphs and their non-negative spectra and associated eigenvectors. In particular, it turns out that the non-negative spectrum of a digraph can be derived from the traditional (adjacency) spectrum of certain undirected bipartite graphs.

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

In this paper, a discrete orthogonal spline collocation method combining with a second-order Crank-Nicolson weighted and shifted Grünwald integral (WSGI) operator is proposed for solving time-fractional wave equations based on its equivalent partial integro-differential equations. The stability and convergence of the schemes have been strictly proved. Several numerical examples in one variable and in two space variables are given to demonstrate the theoretical analysis.

We obtain some estimates for the operator norms of the dilation operators on Herz-Morrey spaces. These results give us the Hardy’s inequalities and the mapping properties of the integral operators on Herz-Morrey spaces. As applications of this general result, we have the boundedness of the Hadamard fractional integrals on Herz-Morrey spaces. We also obtain the Hilbert inequality on Herz-Morrey spaces.

A pebbling move on a graph G consists of taking two pebbles off one vertex and placing one pebble on an adjacent vertex. The pebbling number of a connected graph G , denoted by f ( G ), is the least n such that any distribution of n pebbles on G allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. In this paper, we determine the 2-pebbling property of squares of paths and Graham’s conjecture on P2n2 $\begin{array}{} P_{2n}^2 \end{array} $.

In this paper, we describe a method to solve the problem of finding periodic solutions for second-order neutral delay-differential equations with piecewise constant arguments of the form x ″( t ) + px ″( t − 1) = qx ([ t ]) + f ( t ), where [⋅] denotes the greatest integer function, p and q are nonzero real or complex constants, and f ( t ) is complex valued periodic function. The method reduces the problem to a system of algebraic equations. We give explicit formula for the solutions of the equation. We also give counter examples to some previous findings concerning uniqueness of solution.

Let W α , ρ = x α (1 – x 2 ) ρ e – Q ( x ) , where α > – 12 $\begin{array}{} \displaystyle \frac12 \end{array}$ and Q is continuous and increasing on [0, 1), with limit ∞ at 1. This paper deals with orthogonal polynomials for the weights Wα ,ρ 2 $\begin{array}{} \displaystyle W^2_{\alpha, \rho} \end{array}$ and gives bounds on orthogonal polynomials, zeros, Christoffel functions and Markov inequalities. In addition, estimates of fundamental polynomials of Lagrange interpolation at the zeros of the orthogonal polynomial and restricted range inequalities are obtained.

In this paper, we present a rough set model based on fuzzy ideals of distributive lattices. In fact, we consider a distributive lattice as a universal set and we apply the concept of a fuzzy ideal for definitions of the lower and upper approximations in a distributive lattice. A novel congruence relation induced by a fuzzy ideal of a distributive lattice is introduced. Moreover, we study the special properties of rough sets which can be constructed by means of the congruence relations determined by fuzzy ideals in distributive lattices. Finally, the properties of the generalized rough sets with respect to fuzzy ideals in distributive lattices are also investigated.

In this article, more general types of fractional proportional integrals and derivatives are proposed. Some properties of these operators are discussed.

An n -sided hyperbolic polygon of type ( ϵ , n ) is a hyperbolic polygon with ordered interior angles π 2 $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ , θ 1 , θ 2 , …, θ n −2 , π 2 $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ , where 0 < ϵ < π 2 $\begin{array}{} \frac{\pi}{2} \end{array} $ and 0 < θ i < π satisfying ∑ i=1n− 2θ i+(π 2+ϵ )+(π 2− ϵ )< (n− 2)π $$\begin{array}{} \displaystyle \sum_{i = 1}^{n-2} \theta_{i}+\Big(\frac{\pi}{2}+\epsilon\Big)+\Big(\frac{\pi}{2}-\epsilon\Big) \lt (n-2)\pi \end{array} $$ and θ i + θ i +1 ≠ π (1 ≤ i ≤ n − 3), θ 1 + ( π 2 $\begin{array}{} \frac{\pi}{2} \end{array} $ + ϵ ) ≠ π , θ n −2 + ( π 2 $\begin{array}{} \frac{\pi}{2} \end{array} $ − ϵ ) ≠ π . In this paper, we present a new characterization of Möbius transformations by using n -sided hyperbolic polygons of type ( ϵ , n ). Our proofs are based on a geometric approach.

In this paper, we introduce a new type of contraction to seek the existence of tripled best proximity point results. Here, using the new contraction and P -property, we generalize and extend results of W. Shatanawi and A. Pitea and prove the existence and uniqueness of some tripled best proximity point results. Examples are also given to support our results.

We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space. As mentioned by Do Carmo, a nonextendible Riemannian manifold can be noncomplete, but in the broader category of metric spaces it becomes extendible. We give a short proof of a characterisation of the Heine-Borel property of the metric completion of a metric space M in terms of the absence of inapproachable finite points in ∗ M .

Let R be a subset of a unital ring Q such that 0 ∈ R . Let us fix an element t ∈ Q . If R is a ( t ; d )-free subset of Q , then T n ( R ) is a ( t ′; d )-free subset of T n ( Q ), where t ′ ∈ T n ( Q ), tll′ $\begin{array}{} t_{ll}' \end{array} $ = t , l = 1, 2, …, n , for any n ∈ N .

Let f ( z ) be an entire function of hyper order strictly less than 1. We prove that if f ( z ) and its n th exact difference Δ c n f ( z ) {\Delta }_{c}^{n}f(z) share 0 CM and 1 IM, then Δ c n f ( z ) ≡ f ( z ) {\Delta }_{c}^{n}f(z)\equiv f(z) . Our result improves the related results of Zhang and Liao [Sci. China A, 2014] and Gao et al. [Anal. Math., 2019] by using a simple method.

In this study, for the first time, we discuss the posteriori error estimates and adaptive algorithm for the non-self-adjoint Steklov eigenvalue problem in inverse scattering. The differential operator corresponding to this problem is non-self-adjoint and the associated weak formulation is not H 1 -elliptic. Based on the study of Armentano et al. [Appl. Numer. Math. 58 (2008), 593–601], we first introduce error indicators for primal eigenfunction, dual eigenfunction, and eigenvalue. Second, we use Gårding’s inequality and duality technique to give the upper and lower bounds for energy norm of error of finite element eigenfunction, which shows that our indicators are reliable and efficient. Finally, we present numerical results to validate our theoretical analysis.

The goal of this article is to prove some coupled common fixed point results by using weakly increasing mappings with two variables. Several examples indicating the usability are provided. Also, we use the results obtained to demonstrate the existence of a common solution to a system of integral equations.

In this paper, we present some fixed point results for multivalued mappings with both closed values and proximinal values on left K -complete quasi metric spaces. We also provide a nontrivial example to illustrate our results.

The eigenvalues and the spectral radius of nonnegative tensors have been extensively studied in recent years. In this paper, we investigate the analytic properties of nonnegative tensors and give some inequalities on the spectral radius.

The self-mappings satisfying implicit relations were introduced in a previous study [Popa, Fixed point theorems for implicit contractive mappings , Stud. Cerc. St. Ser. Mat. Univ. Bacău 7 (1997), 129–133]. In this study, we introduce self-operators satisfying an ordered implicit relation and hence obtain their fixed points in the cone metric space under some additional conditions. We obtain a homotopy result as an application.

A semigroup is called an epigroup if some power of each element lies in a subgroup. Under the universal of epigroups, the aim of the paper is devoted to presenting elements in the groupoid together with the multiplication of Malcev products generated by classes of completely simple semigroups, nil-semigroups and semilattices. The information about the set inclusion relations among them is also provided.

It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K , if K 0 [ S ] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K [ S ] is a right (respectively, left) self-injective imply that S is finite? ( Semigroup Algebras , Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0 [ S ] being right (left) self-injective when K 0 [ S ] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.

In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

In this study, we work with the relative divergence of type s , s ∈ ℝ s,s\in {\mathbb{R}} , which includes the Kullback-Leibler divergence and the Hellinger and χ 2 distances as particular cases. We study the symmetrized divergences in additive and multiplicative forms. Some basic properties such as symmetry, monotonicity and log-convexity are established. An important result from the convexity theory is also proved.

In this study, we develop the concept of multivalued Suzuki-type θ -contractions via a gauge function and established two new related fixed point theorems on metric spaces. We also discuss an example to validate our results.

Using the notion of preconcept, we generalize Pawlak’s approximation operators from a one-dimensional space to a two-dimensional space in a formal context. In a formal context, we present two groups of approximation operators in a two-dimensional space: one is aided by an equivalence relation defined on the attribute set, and another is aided by the lattice theoretical property of the family of preconcepts. In addition, we analyze the properties of those approximation operators. All these results show that we can approximate all the subsets in a formal context assisted by the family of preconcepts using the above groups of approximation operators. Some biological examples show that the two groups of approximation operators provided in this article have potential ability to assist biologists to do the phylogenetic analysis of insects.

In 1985, Howard introduced degenerate Cauchy polynomials together with degenerate Bernoulli polynomials. His degenerate Bernoulli polynomials have been studied by many authors, but his degenerate Cauchy polynomials have been forgotten. In this paper, we introduce some kinds of hypergeometric degenerate Cauchy numbers and polynomials from the different viewpoints. By studying the properties of the first one, we give their expressions and determine the coefficients. Concerning the second one, called H-degenerate Cauchy polynomials, we show several identities and study zeta functions interpolating these polynomials.

In this article, the authors establish the characterizations of a class of anisotropic Herz-type Hardy spaces with two variable exponents associated with a non-isotropic dilation on ℝ n {{\mathbb{R}}}^{n} in terms of molecular decompositions. Using the molecular decompositions, the authors obtain the boundedness of the central δ -Calderón-Zygmund operators on the anisotropic Herz-type Hardy space with two variable exponents.

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

A predator-prey model interaction under fluctuating water level with non-selective harvesting is proposed and studied in this paper. Sufficient conditions for the permanence of two populations and the extinction of predator population are provided. The non-negative equilibrium points are given, and their stability is studied by using the Jacobian matrix. By constructing a suitable Lyapunov function, sufficient conditions that ensure the global stability of the positive equilibrium are obtained. The bionomic equilibrium and the optimal harvesting policy are also presented. Numerical simulations are carried out to show the feasibility of the main results.

The goal of this paper is to examine the structure of split involutive regular BiHom-Lie superalgebras, which can be viewed as the natural generalization of split involutive regular Hom-Lie algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular BiHom-Lie superalgebra L {\mathfrak{L}} is of the form L = U + ∑ α I α {\mathfrak{L}}=U+{\sum }_{\alpha }{I}_{\alpha } with U a subspace of a maximal abelian subalgebra H and any I α , a well-described ideal of L {\mathfrak{L}} , satisfying [ I α , I β ] = 0 if [ α ] ≠ [ β ]. In the case of L {\mathfrak{L}} being of maximal length, the simplicity of L {\mathfrak{L}} is also characterized in terms of connections of roots.

In this article, we study the commutators of Hausdorff operators and establish their boundedness on the weighted Herz spaces in the setting of the Heisenberg group.

The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.

This study pursues an investigation on L -semihypergroups equipped with an L -order. First, the concept of L -ordered L -semihypergroups is introduced by L -posets and L -semihypergroups, and some related results are obtained. Then, prime, weakly prime, and semiprime L -hyperideals of L -ordered L -semihypergroups are studied. Moreover, the relationships among the three types of L -hyperideals are established. Finally, the intra-regular L -ordered L -semihypergroups are characterized in terms of these L -hyperideals. The results of the study show that some well-known results on ordered semihypergroups also hold in the case of L -ordered L -semihypergroups.

Let T > 1 T\gt 1 be an integer, T ≔ [ 1 , T ] Z = { 1 , 2 , … , T } , T ˆ ≔ { 0 , 1 , … , T + 1 } {\mathbb{T}}:= {{[}1,T]}_{{\mathbb{Z}}}=\{1,2,\ldots ,T\},\hspace{.0em}\hat{{\mathbb{T}}}:= \{0,1,\ldots ,T+1\} . In this article, we are concerned with the global structure of the set of sign-changing solutions of the discrete second-order boundary value problem { Δ 2 u ( x − 1 ) + λ h ( x ) f ( u ( x ) ) = 0 , x ∈ T , u ( 0 ) = u ( T + 1 ) = 0 , \left\{\begin{array}{l}{\mathrm{\Delta}}^{2}u(x-1)+\lambda h(x)f(u(x))=0,\hspace{1em}x\in {\mathbb{T}},\\ u(0)=u(T+1)=0,\end{array}\right. where λ > 0 \lambda \gt 0 is a parameter, f ∈ C ( ℝ , ℝ ) f\in C({\mathbb{R}},{\mathbb{R}}) satisfies f ( 0 ) = 0 , s f ( s ) > 0 f(0)=0,\hspace{.1em}sf(s)\gt 0 for all s ≠ 0 s\ne 0 and h : T ˆ → [ 0 , + ∞ ) h:\hat{{\mathbb{T}}}\to {[}0,+\infty ) . By using the directions of a bifurcation, we obtain existence and multiplicity of sign-changing solutions of the above problem for λ \lambda lying in various intervals in ℝ {\mathbb{R}} . Moreover, we point out that these solutions change their sign exactly k − 1 k-1 times, where k ∈ { 1 , 2 , … , T } k\in \{1,2,\ldots ,T\} .

In this work, we show that the existence of fixed points of F- contraction mappings in function weighted metric spaces can be ensured without third condition ( F 3 ) (F3) imposed on Wardowski function F :(0,  ∞ ) → ℜ F\mathrm{:(0,\hspace{0.33em}}\infty )\to \Re . The present article investigates (common) fixed points of rational type F -contractions for single-valued mappings. The article employs Jleli and Samet’s perspective of a new generalization of a metric space, known as a function weighted metric space. The article imposes the contractive condition locally on the closed ball, as well as, globally on the whole space. The study provides two examples in support of the results. The presented theorems reveal some important corollaries. Moreover, the findings further show the usefulness of fixed point theorems in dynamic programming, which is widely used in optimization and computer programming. Thus, the present study extends and generalizes related previous results in the literature in an empirical perspective.

A Cayley graph Γ \Gamma on a group G is called a dual Cayley graph on G if the left regular representation of G is a subgroup of the automorphism group of Γ \Gamma (note that the right regular representation of G is always an automorphism group of Γ \Gamma ). In this article, we study finite dual Cayley graphs regarding identification, construction, transitivity and such graphs with automorphism groups as small as possible. A few problems worth further research are also proposed.

Left and right inverse eigenpairs problem is a special inverse eigenvalue problem. There are many meaningful results about this problem. However, few authors have considered the left and right inverse eigenpairs problem with a submatrix constraint. In this article, we will consider the left and right inverse eigenpairs problem with the leading principal submatrix constraint for the generalized centrosymmetric matrix and its optimal approximation problem. Combining the special properties of left and right eigenpairs and the generalized singular value decomposition, we derive the solvability conditions of the problem and its general solutions. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. We present an algorithm and numerical experiment to give the optimal approximation solution. Our results extend and unify many results for left and right inverse eigenpairs problem and the inverse eigenvalue problem of centrosymmetric matrices with a submatrix constraint.

This article deals with the exact controllability for a class of fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space under the assumption that the semigroup generated by the linear part is noncompact. Our main results are obtained by utilizing stochastic analysis technique, measure of noncompactness and the Mönch fixed point theorem. In the end, an example is presented to illustrate that our theorems guarantee the effectiveness of controllability results in the infinite dimensional spaces.

In this study, Levinson-type inequalities are generalized by using new Green functions and Montgomery identity for the class of k -convex functions ( k ≥ 3). Čebyšev-, Grüss- and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, a new functional is introduced based on f {\mathfrak{f}} divergence and then some estimates for new functional are obtained. Some inequalities for Shannon entropies are obtained too.

In this article, we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse: | | M x − b | | F = min subject to x ∈ ℛ ( M ) , ||Mx-b|{|}_{F}=\hspace{.25em}\min \hspace{1em}\text{subject}\hspace{.25em}\text{to}\hspace{1em}x\in {\mathcal R} (M), where M ∈ ℂ n CM M\in {{\mathbb{C}}}_{n}^{\text{CM}} . We get the unique solution to the problem, provide two Cramer’s rules for the unique solution and establish two new expressions for the core inverse.

In this article, we show that every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of one prime, one prime squares, two prime cubes, and 187 powers of 2.

This article brings together miscellaneous formulas and facts on matrix expressions that are composed by idempotent matrices in one place with cogent introduction and references for further study. The author will present the basic mathematical ideas and methodologies of the matrix analytic theory in a readable, up-to-date, and comprehensive manner, including constructions of various algebraic matrix identities composed by the conventional operations of idempotent matrices, and uses of the block matrix method in the derivation of closed-form formulas for calculating the ranks of matrix expressions that are composed by idempotent matrices. The author also determines the maximum and minimum ranks of some matrix pencils composed by the products of matrices and their generalized inverses and uses the ranks to characterize algebraic performance of the matrix pencils.

In this article, we consider the Laplace-Bessel differential operator Δ B k , n = ∑ i = 1 k ∂ 2 ∂ x i 2 + γ i x i ∂ ∂ x i + ∑ i = k + 1 n ∂ 2 ∂ x i 2 , γ 1 > 0 , … , γ k > 0 . {\Delta }_{{B}_{k,n}}=\mathop{\sum }\limits_{i=1}^{k}\left(\frac{{\partial }^{2}}{\partial {x}_{i}^{2}}+\frac{{\gamma }_{i}}{{x}_{i}}\frac{\partial }{\partial {x}_{i}}\right)+\mathop{\sum }\limits_{i=k+1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},{\gamma }_{1}\gt 0,\ldots ,{\gamma }_{k}\gt 0. Furthermore, we define B -maximal commutators, commutators of B -singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B -maximal commutator M b , γ {M}_{b,\gamma } and the commutator [ b , A γ ] {[}b,{A}_{\gamma }] of the B -singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator [ b , I α , γ ] {[}b,{I}_{\alpha ,\gamma }] of the B -Riesz potential on B -Morrey spaces L p , λ , γ {L}_{p,\lambda ,\gamma } , when b ∈ BMO γ b\in {\text{BMO}}_{\gamma } .

We give the rate of convergence to a Brownian sheet from a family of processes constructed starting from a set of independent standard Poisson processes. These processes have realizations that converge almost surely to the Brownian sheet, uniformly in the unit square.

Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in the Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in L 2 {{\mathbb{L}}}^{2} whose curvature depends on the Lorentzian pseudodistance from the origin, and those ones whose curvature depends on the Lorentzian pseudodistance through the horizontal or vertical geodesic to a fixed lightlike geodesic. Making use of the notions of geometric angular momentum (with respect to the origin) and geometric linear momentum (with respect to the fixed lightlike geodesic), respectively, we get two abstract integrability results to determine such curves through quadratures. In this way, we find out several new families of Lorentzian spiral, special elastic and grim-reaper curves whose intrinsic equations are expressed in terms of elementary functions. In addition, we provide uniqueness results for the generatrix curve of the Enneper surface of second kind and for Lorentzian versions of some well-known curves in the Euclidean setting, like the Bernoulli lemniscate, the cardioid, the sinusoidal spirals and some non-degenerate conics. We are able to get arc-length parametrizations of them and they are depicted graphically.

Change-point detection in categorical time series has recently gained attention as statistical models incorporating change-points are common in practice, especially in the area of biomedicine. In this article, we propose a sequential change-point detection procedure based on the partial likelihood score process for the detection of changes in the coefficients of multinomial logistic regression model. The asymptotic results are presented under both the null of no change and the alternative of changes in coefficients. We carry out a Monte Carlo experiment to evaluate the empirical size of the proposed procedure as well as its average run length. We illustrate the method by using data on a DNA sequence. Monte Carlo experiments and real data analysis demonstrate the effectiveness of the proposed procedure.

Let S = g 1 ⋅ … ⋅ g n S={g}_{1}\cdot \ldots \cdot {g}_{n} be a sequence with elements g i {g}_{i} from an additive finite abelian group G . S is called a tiny zero-sum sequence if S is non-empty, g 1 + … + g n = 0 {g}_{1}+\hspace{0.2em}\ldots \hspace{0.2em}+{g}_{n}=0 and k ( S ) ≔ ∑ i = 1 n 1 ord ( g i ) ≤ 1 k(S):= {\sum }_{i=1}^{n}\frac{1}{\text{ord}({g}_{i})}\le 1 . Let t ( G ) t(G) be the smallest integer t such that every sequence of t elements (repetition allowed) from G contains a tiny zero-sum sequence. In this article, we mainly focus on the explicit value of t ( G ) t(G) and compute this value for a new class of groups, namely ones of the form G = C 3 ⊕ C 3 p G={C}_{3}\oplus {C}_{3p} , where p is a prime number such that p ≥ 5 p\ge 5 .

We extend a boundedness result for Marcinkiewicz integral operator. We find a new space of radial functions for which this class of singular integral operators remains L p {L}^{p} -bounded when its kernel satisfies only the sole integrability condition.

In this article, we obtain the general solution and prove the Hyers-Ulam stability of the following quadratic-multiplicative functional equation: ϕ ( s t − u v ) + ϕ ( s v + t u ) = [ ϕ ( s ) + ϕ ( u ) ] [ ϕ ( t ) + ϕ ( v ) ] \phi (st-uv)+\phi (sv+tu)={[}\phi (s)+\phi (u)]{[}\phi (t)+\phi (v)] by using the direct method and the fixed point method.

In this article, we study some topological properties of the multiplication operator on Orlicz-Cesáro mean sequence spaces equipped with the pre-quasi norm and the pre-quasi operator ideal constructed by this sequence space and s -numbers.

Consider an ordered Banach space and f , g f,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f ( X ) = g ( X ) f(X)=g(X) has a positive solution, whenever f is strictly α \alpha -concave g -monotone or strictly ( − α ) (-\alpha ) -convex g -antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

The aim of this study was to present a simple method for finding the asymptotic relations for products of elements of some positive real sequences. The main reason to carry out this study was the result obtained by Alzer and Sandor concerning an estimation of a sequence of the product of the first k primes.

In this article, we give a sufficient and necessary condition for every Jordan { g , h }-derivation to be a { g , h }-derivation on triangular algebras. As an application, we prove that every Jordan { g , h }-derivation on τ ( N ) \tau ({\mathscr{N}}) is a { g , h }-derivation if and only if dim 0 + ≠ 1 \dim {0}_{+}\ne 1 or dim H − ⊥ ≠ 1 \dim {H}_{-}^{\perp }\ne 1 , where N {\mathscr{N}} is a non-trivial nest on a complex separable Hilbert space H and τ ( N ) \tau ({\mathscr{N}}) is the associated nest algebra.

The first paper in systolic geometry was published by Loewner’s student P. M. Pu over half a century ago. Pu proved an inequality relating the systole and the area of an arbitrary metric in the real projective plane. We prove a stronger version of Pu’s systolic inequality with a remainder term.

For a positive integer n and a prime p , let n p {n}_{p} denote the p -part of n . Let G be a group, cd ( G ) \text{cd}(G) the set of all irreducible character degrees of G G , ρ ( G ) \rho (G) the set of all prime divisors of integers in cd ( G ) \text{cd}(G) , V ( G ) = p e p ( G ) | p ∈ ρ ( G ) V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where p e p ( G ) = max { χ ( 1 ) p | χ ∈ Irr ( G ) } . {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G ≅ L 2 ( p 2 ) G\cong {L}_{2}({p}^{2}) if and only if | G | = | L 2 ( p 2 ) | |G|=|{L}_{2}({p}^{2})| and V ( G ) = V ( L 2 ( p 2 ) ) V(G)=V({L}_{2}({p}^{2})) .

In this article, we discuss the nonlinear boundary value problems involving both left Riemann-Liouville and right Caputo-type fractional derivatives. By using some new techniques and properties of the Mittag-Leffler functions, we introduce a formula of the solutions for the aforementioned problems, which can be regarded as a novelty item. Moreover, we obtain the existence result of solutions for the aforementioned problems and present the Ulam-Hyers stability of the fractional differential equation involving two different fractional derivatives. An example is given to illustrate our theoretical result.

In this article, an errors-in-variables regression model in which the errors are negatively superadditive dependent (NSD) random variables is studied. First, the Marcinkiewicz-type strong law of large numbers for NSD random variables is established. Then, we use the strong law of large numbers to investigate the asymptotic normality of least square (LS) estimators for the unknown parameters. In addition, the mean consistency of LS estimators for the unknown parameters is also obtained. Some results for independent random variables and negatively associated random variables are extended and improved to the case of NSD setting. At last, two simulations are presented to verify the asymptotic normality and mean consistency of LS estimators in the model.

Let A be a square matrix satisfying A 4 = A {A}^{4}=A . We solve the Yang-Baxter-like matrix equation A X A = X A X AXA=XAX to find some solutions, based on analysis of the characteristic polynomial of A and its eigenvalues. We divide the problem into small cases so that we can find the solution easily. Finally, in order to illustrate the results, two numerical examples are presented.

In this article, the general ( p , q )-mixed projection bodies are introduced. Then, some basic properties of the general ( p , q )-mixed projection bodies are discussed, and the extreme values of volumes of the general ( p , q )-mixed projection bodies and their polar bodies are established.

The method of brackets, developed in the context of evaluation of integrals coming from Feynman diagrams, is a procedure to evaluate definite integrals over the half-line. This method consists of a small number of operational rules devoted to convert the integral into a bracket series. A second small set of rules evaluates this bracket series and produces the result as a regular series. The work presented here combines this method with the classical Mellin transform to extend the class of integrands where the method of brackets can be applied. A selected number of examples are used to illustrate this procedure.

The main purpose of this paper is to find some fixed point results with a new approach, particularly in those cases where the existing literature remains silent. More precisely, we introduce partial completeness, f ̄-orbitally completeness, a new type of contractions and many other notions. We also ensure the existence of fixed points for non-contraction maps in the class of incomplete partial b -metric spaces. We have reported some examples in support of our results.

We examine the existence and uniqueness of solutions to two-point boundary value problems involving fourth-order, ordinary differential equations. Such problems have interesting applications to modelling the deflections of beams. We sharpen traditional results by showing that a larger class of problems admit a unique solution. We achieve this by drawing on fixed-point theory in an interesting and alternative way via an application of Rus’s contraction mapping theorem. The idea is to utilize two metrics on a metric space, where one pair is complete. Our theoretical results are applied to the area of elastic beam deflections when the beam is subjected to a loading force and the ends of the beam are either both clamped or one end is clamped and the other end is free. The existence and uniqueness of solutions to the models are guaranteed for certain classes of linear and nonlinear loading forces.

Let G be a finite group. The subgroup intersection graph Γ ( G ) \text{&#x0393;}(G) of G is a graph whose vertices are non-identity elements of G and two distinct vertices x and y are adjacent if and only if | 〈 x 〉 ∩ 〈 y 〉 | > 1 |\langle x\rangle \cap \langle y\rangle |\gt 1 , where 〈 x 〉 \langle x\rangle is the cyclic subgroup of G generated by x . In this paper, we show that two finite abelian groups are isomorphic if and only if their subgroup intersection graphs are isomorphic.

This paper is related to the stochastic smoking model for the purpose of creating the effects of smoking that are not observed in deterministic form. First, formulation of the stochastic model is presented. Then the sufficient conditions for extinction and persistence are determined. Furthermore, the threshold of the proposed stochastic model is discussed, when noises are small or large. Finally, the numerical simulations are shown graphically with the software MATLAB.

In this paper, it is proved that the Ricci operator of an almost Kenmotsu 3- h -manifold M is of transversely Killing-type if and only if M is locally isometric to the hyperbolic 3-space ℍ 3 ( − 1 ) {{\mathbb{H}}}^{3}(-1) or a non-unimodular Lie group endowed with a left invariant non-Kenmotsu almost Kenmotsu structure. This result extends those results obtained by Cho [ Local symmetry on almost Kenmotsu three-manifolds , Hokkaido Math. J. 45 (2016), no. 3, 435–442] and Wang [ Three-dimensional locally symmetric almost Kenmotsu manifolds , Ann. Polon. Math. 116 (2016), no. 1, 79–86; Three-dimensional almost Kenmotsu manifolds with η \eta -parallel Ricci tensor , J. Korean Math. Soc. 54 (2017), no. 3, 793–805].

In 2006, Schuster introduced the radial Blaschke-Minkowski homomorphisms. In this article, associating with the star duality of star bodies and dual quermassintegrals, we establish Brunn-Minkowski inequalities and monotonic inequality for the radial Blaschke-Minkowski homomorphisms. In addition, we consider its Shephard-type problems and give a positive form and a negative answer, respectively.

This paper deals with the existence of mild solutions for a class of non-local stochastic integro-differential equations driven by a fractional Brownian motion with Hurst parameter H ∈ 1 2 , 1 H\in \left(\tfrac{1}{2},1\right) . Discussions are based on resolvent operators in the sense of Grimmer, stochastic analysis theory and fixed-point criteria. As a final point, an example is given to illustrate the effectiveness of the obtained theory.

The aim of this paper is to establish some fixed point results for surrounding quasi-contractions in non-triangular metric spaces. Also, we prove the Banach principle of contraction in non-triangular metric spaces. As applications of our theorems, we deduce certain well-known results in b -metric spaces as corollaries.

This study focuses on finding super edge-magic total (SEMT) labeling and deficiency of imbalanced fork and disjoint union of imbalanced fork with star, bistar and path; in addition, the SEMT strength for Imbalanced Fork is investigated.

The aim of this paper is to investigate a class of nonlinear stochastic reaction-diffusion systems involving fractional Laplacian in a bounded domain. First, the existence and uniqueness of weak solutions are proved by using Galërkin’s method. Second, the existence of optimal controls for the corresponding stochastic optimal control problem is obtained. Finally, several examples are provided to demonstrate the theoretical results.

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

In this paper, firstly, we study the structural form of reflective integral for a given system. Then the sufficient conditions are obtained to ensure there exists the reflective integral with these structured form for such system. Secondly, we discuss the necessary conditions for the equivalence of such systems and a general three-dimensional differential system. And then, we apply the obtained results to the study of the behavior of their periodic solutions when such systems are periodic systems in t .

In this paper, we decorate leaves and edges of planar rooted forests simultaneously and use a part of them to construct free nonunitary Rota-Baxter family algebras. As a corollary, we obtain the construction of free nonunitary Rota-Baxter algebras.

In this paper, by using the ideal theory in residuated lattices, we construct the prime and maximal spectra (Zariski topology), proving that the prime and maximal spectra are compact topological spaces, and in the case of De Morgan residuated lattices they become compact T 0 {T}_{0} topological spaces. At the same time, we define and study the reticulation functor between De Morgan residuated lattices and bounded distributive lattices. Moreover, we study the I -topology ( I comes from ideal) and the stable topology and we define the concept of pure ideal. We conclude that the I -topology is in fact the restriction of Zariski topology to the lattice of ideals, but we use it for simplicity. Finally, based on pure ideals, we define the normal De Morgan residuated lattice ( L is normal iff every proper ideal of L is a pure ideal) and we offer some characterizations.

In this short note, we provide a brief proof for a recent determinantal formula involving a particular family of banded matrices.

In this manuscript, we first consider the diffusive competition and cooperation system subject to Neumann boundary conditions without delay terms and get the conclusion that the unique positive constant equilibrium is locally asymptotically stable. Then, we study the diffusive delayed competition and cooperation system subject to Neumann boundary conditions, and the existence of Hopf bifurcation at the positive equilibrium is obtained by regarding delay term as the parameter. By the theory of center manifold and normal form, an algorithm for determining the direction and stability of Hopf bifurcation is derived. Finally, some numerical simulations and summarizations are carried out for illustrating the theoretical analytic results.

In this article, the error term of the mean value theorem for binary Egyptian fractions is studied. An error term of prime number theorem type is obtained unconditionally. Under Riemann hypothesis, a power saving can be obtained. The mean value in short interval is also considered.

In this paper, we show that if λ 1 , λ 2 , λ 3 {\lambda }_{1},{\lambda }_{2},{\lambda }_{3} are non-zero real numbers, and at least one of the numbers λ 1 , λ 2 , λ 3 {\lambda }_{1},{\lambda }_{2},{\lambda }_{3} is irrational, then the integer parts of λ 1 n 1 2 + λ 2 n 2 3 + λ 3 n 3 4 {\lambda }_{1}{n}_{1}^{2}+{\lambda }_{2}{n}_{2}^{3}+{\lambda }_{3}{n}_{3}^{4} are prime infinitely often for integers n 1 , n 2 , n 3 {n}_{1},{n}_{2},{n}_{3} . This gives an improvement of an earlier result.

This paper focuses on finite-order meromorphic solutions of nonlinear difference equation f n ( z ) + q ( z ) e Q ( z ) Δ c f ( z ) = p ( z ) {f}^{n}(z)+q(z){e}^{Q(z)}{\text{&#x0394;}}_{c}f(z)=p(z) , where p , q , Q p,q,Q are polynomials, n ≥ 2 n\ge 2 is an integer, and Δ c f {\text{&#x0394;}}_{c}f is the forward difference of f . A relationship between the growth and zero distribution of these solutions is obtained. Using this relationship, we obtain the form of these solutions of the aforementioned equation. Some examples are given to illustrate our results.

In this paper, we give a boundedness criterion for the potential operator ℐ α { {\mathcal I} }^{\alpha } in the local generalized Morrey space L M p , φ { t 0 } ( Γ ) L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{&#x0393;}) and the generalized Morrey space M p , φ ( Γ ) {M}_{p,\varphi }(\text{&#x0393;}) defined on Carleson curves Γ \text{&#x0393;} , respectively. For the operator ℐ α { {\mathcal I} }^{\alpha } , we establish necessary and sufficient conditions for the strong and weak Spanne-type boundedness on L M p , φ { t 0 } ( Γ ) L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{&#x0393;}) and the strong and weak Adams-type boundedness on M p , φ ( Γ ) {M}_{p,\varphi }(\text{&#x0393;}) .

In this paper, some new integral curves are defined in three-dimensional Euclidean space by using a new frame of a polynomial spatial curve. The Frenet vectors, curvature and torsion of these curves are obtained by means of new frame and curvatures. We give the characterizations and properties of these integral curves under which conditions they are general helix. Also, the relationships between these curves in terms of being some kinds of associated curves are introduced. Finally, an example is illustrated.

In this article, meromorphic exact solutions for the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff (gCBS) equation are obtained by using the complex method. With the applications of our results, traveling wave exact solutions of the breaking soliton equation are achieved. The dynamic behaviors of exact solutions of the (2 + 1)-dimensional gCBS equation are shown by some graphs. In particular, the graphs of elliptic function solutions are comparatively rare in other literature. The idea of this study can be applied to the complex nonlinear systems of some areas of engineering.

We present a more general homological characterization of the direct summand theorem (DST). Specifically, we state two new conjectures: the socle-parameter conjecture (SPC) in its weak and strong forms. We give a proof for the weak form by showing that it is equivalent to the DST. Furthermore, we prove the SPC in its strong form for the case when the multiplicity of the parameters is smaller than or equal to two. Finally, we present a new proof of the DST in the equicharacteristic case, based on the techniques thus developed.

We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in a standard form). This completes “minimal” arithmetic in free fields since “minimal” constructions for the inverse are already known. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.

A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree n can be restricted with significantly better determinacy than that provided by the Descartes rule of signs. The method also allows the isolation of the zeros of the polynomial quite successfully, and the determined root bounds are significantly narrower than the Cauchy and the Lagrange bounds. The method relies on solving equations of degree smaller than that of the given polynomial. One can determine analytically the exact number of positive and negative zeros of a polynomial of degree up to and including five and also fully isolate the zeros of the polynomial analytically, and with one of the variations of the method, one can analytically approach polynomials of degree up to and including nine by solving equations of degree not more than four. For polynomials of higher degree, either of the two variations of the method should be applied recursively. Numerous examples are given. Presented is the full classification of the roots of the cubic equation, together with their isolation intervals – especially important for various scientific models for which the coefficients of the equation might be functions of the model parameters. An application of the method to a quartic equation with variable coefficients (resulting from the study of wave–current interactions in the physically realistic model of azimuthal two-dimensional non-viscous flows with piecewise constant vorticity in a two-layer fluid with a flat bed and a free surface) has been demonstrated. A step-by-step algorithm for each version of the method has also been presented.

In Applied Mathematics Letters 74 (2017), 147–153, the Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation was investigated when the relevant system has a potential well of finite depth. As a continuous work, we prove in this paper a type of Hyers-Ulam stability of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier of V 0 {V}_{0} in height and 2 c 2c in width, where V 0 {V}_{0} is assumed to be greater than the energy E of the particle under consideration.

In this paper, we prove that the topological spaces of nonzero weighted composition operators acting on some Hilbert spaces of analytic functions on the unit open ball are simply connected.

In this paper, we present the concepts of generalized derivative, directional generalized derivative, subdifferential and conjugate for n -dimensional fuzzy-number-valued functions and discuss the characterizations of generalized derivative and directional generalized derivative by, respectively, using the derivative and directional derivative of crisp functions that are determined by the fuzzy mapping. Furthermore, the relations among generalized derivative, directional generalized derivative, subdifferential and convexity for n -dimensional fuzzy-number-valued functions are investigated. Finally, under two kinds of partial orderings defined on the set of all n -dimensional fuzzy numbers, the duality theorems and saddle point optimality criteria in fuzzy optimization problems with constraints are discussed.

New families of uniformities are introduced on U C ( X , Y ) UC(X,Y) , the class of uniformly continuous mappings between X and Y , where ( X , U ) (X,{\mathcal{U}}) and ( Y , V ) (Y,{\mathcal{V}}) are uniform spaces. Admissibility and splittingness are introduced and investigated for such uniformities. Net theory is developed to provide characterizations of admissibility and splittingness of these spaces. It is shown that the point-entourage uniform space is splitting while the entourage-entourage uniform space is admissible.

In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M = ℳ 0 ( G ; I , Λ ; P ) M={ {\mathcal M} }^{0}(G;I,\text{&#x039B;};P) , we show that the contracted semigroup ring R 0 [ M ] {R}_{0}{[}M] is strongly nil-clean if and only if either | I | = 1 |I|=1 or | Λ | = 1 |\text{&#x039B;}|=1 , and R [ G ] R{[}G] is strongly nil-clean; as a corollary, we characterize the strong nil-cleanness of locally inverse semigroup rings. Moreover, let S = [ Y ; S α , φ α , β ] S={[}Y;{S}_{\alpha },{\varphi }_{\alpha ,\beta }] be a strong semilattice of semigroups, then we prove that R [ S ] R{[}S] is strongly nil-clean if and only if R [ S α ] R{[}{S}_{\alpha }] is strongly nil-clean for each α ∈ Y \alpha \in Y .

In this paper, we develop a technique which enables us to obtain several results from the theory of Γ-semigroups as logical implications of their semigroup theoretical analogues.

In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p -Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p -Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

Let ( Γ , ≤ ) ({\mathrm{\Gamma}},\le ) be a strictly ordered monoid, and let Γ ⁎ = Γ \ { 0 } {{\mathrm{\Gamma}}}^{\ast }\left={\mathrm{\Gamma}}\backslash \{0\} . Let D ⊆ E D\subseteq E be an extension of commutative rings with identity, and let I be a nonzero proper ideal of D . Set D + 〚 E Γ ⁎ , ≤ 〛 ≔ f ∈ 〚 E Γ , ≤ 〛 | f ( 0 ) ∈ D and D + 〚 I Γ ⁎ , ≤ 〛 ≔ f ∈ 〚 D Γ , ≤ 〛 | f ( α ) ∈ I , for all α ∈ Γ ⁎ . \begin{array}{l}D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {E}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(0)\in D\right\}\hspace{.5em}\text{and}\\ \hspace{0.2em}D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] := \left\{f\in [\kern-2pt[ {D}^{{\mathrm{\Gamma}},\le }]\kern-2pt] \hspace{0.15em}|\hspace{0.2em}f(\alpha )\in I,\hspace{.5em}\text{for}\hspace{.25em}\text{all}\hspace{.5em}\alpha \in {{\mathrm{\Gamma}}}^{\ast }\right\}.\end{array} In this paper, we give necessary conditions for the rings D + 〚 E Γ ⁎ , ≤ 〛 D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ( Γ , ≤ ) ({\mathrm{\Gamma}},\le ) is positively ordered, and sufficient conditions for the rings D + 〚 E Γ ⁎ , ≤ 〛 D+[\kern-2pt[ {E}^{{{\mathrm{\Gamma}}}^{\ast },\le }]\kern-2pt] to be Noetherian when ( Γ , ≤ ) ({\mathrm{\Gamma}},\le ) is positively totally ordered. Moreover, we give a necessary and sufficient condition for the ring D + 〚 I Γ ⁎ , ≤ 〛 D+[\kern-2pt[ {I}^{{\Gamma }^{\ast },\le }]\kern-2pt] to be Noetherian when ( Γ , ≤ ) ({\mathrm{\Gamma}},\le ) is positively totally ordered. As corollaries, we give equivalent conditions for the rings D + ( X 1 , … , X n ) E [ X 1 , … , X n ] D+({X}_{1},\ldots ,{X}_{n})E{[}{X}_{1},\ldots ,{X}_{n}] and D + ( X 1 , … , X n ) I [ X 1 , … , X n ] D+({X}_{1},\ldots ,{X}_{n})I{[}{X}_{1},\ldots ,{X}_{n}] to be Noetherian.

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

In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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

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

In this paper, we define the entropy number in probabilistic setting and determine the exact order of entropy number of finite-dimensional space in probabilistic setting. Moreover, we also estimate the sharp order of entropy number of univariate Sobolev space in probabilistic setting by discretization method.

For a partially ordered set ( A , ≤ ) (A,\le ) , let G A {G}_{A} be the simple, undirected graph with vertex set A such that two vertices a ≠ b ∈ A a\ne b\in A are adjacent if either a ≤ b a\le b or b ≤ a b\le a . We call G A {G}_{A} the partial order graph or comparability graph of A . Furthermore, we say that a graph G is a partial order graph if there exists a partially ordered set A such that G = G A G={G}_{A} . For a class C {\mathcal{C}} of simple, undirected graphs and n , m ≥ 1 m\ge 1 , we define the Ramsey number ℛ C ( n , m ) { {\mathcal R} }_{{\mathcal{C}}}(n,m) with respect to C {\mathcal{C}} to be the minimal number of vertices r such that every induced subgraph of an arbitrary graph in C {\mathcal{C}} consisting of r vertices contains either a complete n -clique K n {K}_{n} or an independent set consisting of m vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed.

By using the bifurcation method, we study the existence of an S -shaped connected component in the set of positive solutions for discrete second-order Neumann boundary value problem. By figuring the shape of unbounded connected component of positive solutions, we show that the Neumann boundary value problem has three positive solutions suggesting suitable conditions on the weight function and nonlinearity.

The purpose of this paper is to introduce the logarithmic mean of two convex functionals that extends the logarithmic mean of two positive operators. Some inequalities involving this functional mean are discussed as well. The operator versions of the functional theoretical results obtained here are immediately deduced without referring to the theory of operator means.

In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

For a finitely generated tensor norm α \alpha , we investigate the α \alpha -approximation property ( α \alpha -AP) and the bounded α \alpha -approximation property (bounded α \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ \lambda -bounded α p , q {\alpha }_{p,q} -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ \lambda -bounded g p {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the λ \lambda -bounded g p {g}_{p} -AP, then X has the λ \lambda -bounded w p {w}_{p} -AP.

As an effective and possible method for computing PageRank problem, the inner-outer (IO) iteration has attracted wide interest in the past few years since it was first proposed by Gleich et al. (2010). In this paper, we present a variant of the IO iteration, which is based on multi-step power and multi-step splitting and is denoted by MPMIO. The description and convergence are discussed in detail. Numerical examples are given to illustrate the effectiveness of the proposed method.

A map is called edge-regular if it is edge-transitive but not arc-transitive. In this paper, we show that a complete graph K n {K}_{n} has an orientable edge-regular embedding if and only if n = p d > 3 n={p}^{d}\gt 3 with p an odd prime such that p d ≡ 3 {p}^{d}\equiv 3 ( mod 4 ) (\mathrm{mod}\hspace{.25em}4) . Furthermore, K p d {K}_{{p}^{d}} has p d − 3 4 d ϕ ( p d − 1 2 ) \tfrac{{p}^{d}-3}{4d}\hspace{0.25em}\phi \left(\tfrac{{p}^{d}-1}{2}\right) non-isomorphic orientable edge-regular embeddings.

We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction . We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G . In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

In this paper, we present a new refinement of Jensen’s inequality with applications in information theory. The refinement of Jensen’s inequality is obtained based on the general functional in the work of Popescu et al. As the applications in information theory, we provide new tighter bounds for Shannon’s entropy and some f -divergences.

We characterize skew-symmetric operators on a reproducing kernel Hilbert space in terms of their Berezin symbols. The solution of some operator equations with skew-symmetric operators is studied in terms of Berezin symbols. We also studied essentially unitary operators via Berezin symbols.

We mainly consider the limit behaviors of the Riemann solutions to Chaplygin Euler equations for nonisentropic fluids. The formation of delta shock wave and the appearance of vacuum state are found as parameter ε \varepsilon tends to a certain value. Different from the isentropic fluids, the weight of delta shock wave is determined by variance density ρ \rho and internal energy H . Meanwhile, involving the entropy inequality, the uniqueness of delta shock wave is obtained.

We address the question of identifying non-smooth points in V ℝ ( I ) {{\bf{V}}}_{{\mathbb{R}}}(I) the real part of an affine algebraic variety. Two simple algebraic criteria will be formulated and proven. As an application, we investigate the configuration spaces of the planar four-bar linkage and the delta robot and prove that all singularities are CS-singularities.

In this paper, we introduce the concept of the perfect ϕ \phi -type to describe the growth of the maximal molecule of Laplace-Stieltjes transform by using the more general function than the usual. Based on this concept, we investigate the approximation and growth of analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms convergent in the half plane and obtain some results about the necessary and sufficient conditions on analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms with perfect ϕ \phi -type, which are some generalizations and improvements of the previous results given by Kong [ On generalized orders and types of Laplace-Stieltjes transforms analytic in the right half-plane , Acta Math. Sin. 59A (2016), 91–98], Singhal and Srivastava [ On the approximation of an analytic function represented by Laplace-Stieltjes transformations , Anal. Theory and Appl. 31 (2015), 407–420].

In the article, we present a new ( p , q ) (p,q) -integral identity for the first-order ( p , q ) (p,q) -differentiable functions and establish several new ( p , q ) (p,q) -quantum error estimations for various integral inequalities via ( α , m ) (\alpha ,m) -convexity. We also compare our results with the previously known results and provide two examples to show the superiority of our obtained results.

In this paper, we study boundary value problems of fractional integro-differential equations and inclusions involving Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray-Schauder in the single-valued case, while Martelli’s fixed point theorem, nonlinear alternative for multi-valued maps, and Covitz-Nadler fixed point theorem are used in the inclusion case. Examples illustrating the obtained results are also presented.

We study the boundary layer problem of a Keller-Segel model in a domain of two space dimensions with vanishing chemical diffusion coefficient. By using the method of matched asymptotic expansions of singular perturbation theory, we construct an accurate approximate solution which incorporates the effects of boundary layers and then use the classical energy estimates to prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero.

Let K be an imaginary quadratic field of discriminant d K {d}_{K} with ring of integers O K {{\mathcal{O}}}_{K} , and let τ K {\tau }_{K} be an element of the complex upper half plane so that O K = [ τ K , 1 ] {{\mathcal{O}}}_{K}={[}{\tau }_{K},1] . For a positive integer N , let Q N ( d K ) {{\mathcal{Q}}}_{N}({d}_{K}) be the set of primitive positive definite binary quadratic forms of discriminant d K {d}_{K} with leading coefficients relatively prime to N . Then, with any congruence subgroup Γ {\mathrm{\Gamma}} of SL 2 ( Z ) {\text{SL}}_{2}({\mathbb{Z}}) one can define an equivalence relation ∼ Γ {\sim }_{{\mathrm{\Gamma}}} on Q N ( d K ) {{\mathcal{Q}}}_{N}({d}_{K}) . Let ℱ Γ , ℚ { {\mathcal F} }_{{\mathrm{\Gamma}},{\mathbb{Q}}} denote the field of meromorphic modular functions for Γ {\mathrm{\Gamma}} with rational Fourier coefficients. We show that the set of equivalence classes Q N ( d K ) / ∼ Γ {{\mathcal{Q}}}_{N}({d}_{K})/{\sim }_{{\mathrm{\Gamma}}} can be equipped with a group structure isomorphic to Gal ( K ℱ Γ , ℚ ( τ K ) / K ) \text{Gal}(K{ {\mathcal F} }_{{\mathrm{\Gamma}},{\mathbb{Q}}}({\tau }_{K})/K) for some Γ {\mathrm{\Gamma}} , which generalizes the classical theory of form class groups.

In this paper, we study various strongly convex hyper S -subposets of hyper S -posets in detail. To begin with, we consider the decomposition of hyper S -posets. A unique decomposition theorem for hyper S -posets is given based on strongly convex indecomposable hyper S -subposets. Furthermore, we discuss the properties of minimal and maximal strongly convex hyper S -subposets of hyper S -posets. In the sequel, the concept of hyper C -subposets of a hyper S -poset is introduced, and several related properties are investigated. In particular, we discuss the relationship between greatest strongly convex hyper S -subposets and hyper C -subposets of hyper S -posets. Moreover, we introduce the concept of bases of a hyper S -poset and give out the sufficient and necessary conditions of the existence of the greatest hyper C -subposets of a hyper S -poset by the properties of bases.

Given the abstract evolution equation y ′ ( t ) = A y ( t ) , t ∈ ℝ , y^{\prime} (t)=Ay(t),t\in {\mathbb{R}}, with a scalar type spectral operator A in a complex Banach space, we find conditions on A , formulated exclusively in terms of the location of its spectrum in the complex plane, necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1 \beta \ge 1 , in particular analytic or entire , on ℝ {\mathbb{R}} . We also reveal certain inherent smoothness improvement effects and show that if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded . The important particular case of the equation with a normal operator A in a complex Hilbert space follows immediately.

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

A set W of vertices of a connected graph G strongly resolves two different vertices x ,  y ∉ W if either d G ( x ,  W ) = d G ( x ,  y ) + d G ( y ,  W ) or d G ( y ,  W ) = d G ( y ,  x ) + d G ( x ,  W ), where d G ( x ,  W ) = min{ d ( x , w ):  w ∈ W } and d ( x , w ) represents the length of a shortest x − w path. An ordered vertex partition Π = { U 1 ,  U 2 ,…, U k } of a graph G is a strong resolving partition for G , if every two different vertices of G belonging to the same set of the partition are strongly resolved by some other set of Π . The minimum cardinality of any strong resolving partition for G is the strong partition dimension of G . In this article, we obtain several bounds and closed formulae for the strong partition dimension of some families of graphs and give some realization results relating the strong partition dimension, the strong metric dimension and the order of graphs.

Sullivan stated the conjectures: (1) every oriented graph has a vertex x such that d ++ ( x ) ≥ d − ( x ) and (2) every oriented graph has a vertex x such that d ++ ( x ) + d + ( x ) ≥ 2 d − ( x ). In this paper, we prove that these conjectures hold for local tournaments. In particular, for a local tournament D , there are at least two vertices satisfying (1) and either there exist two vertices satisfying (2) or there exists a vertex v satisfying d ++ ( v ) + d + ( v ) ≥ 2 d − ( v ) + 2 if D has no vertex of in-degree zero.

Given any digraph D , its non-negative spectrum (or N-spectrum, shortly) consists of the eigenvalues of the matrix AA T , where A is the adjacency matrix of D . In this study, we relate the classical spectrum of undirected graphs to the N-spectrum of their oriented counterparts, permitting us to derive spectral bounds. Moreover, we study the spectral effects caused by certain modifications of a given digraph.

Let G be a Hamiltonian graph. A nonempty vertex set X ⊆ V ( G ) X\subseteq V(G) is called a Hamiltonian cycle enforcing set (in short, an H-force set) of G if every X -cycle of G (i.e., a cycle of G containing all vertices of X ) is a Hamiltonian cycle. For the graph G , h ( G ) h(G) (called the H-force number of G ) is the smallest cardinality of an H-force set of G . Ore’s theorem states that an n -vertex graph G is Hamiltonian if d ( u ) + d ( v ) ≥ n d(u)+d(v)\ge n for every pair of nonadjacent vertices u , v u,v of G . In this article, we study the H-force sets of the graphs satisfying the condition of Ore’s theorem, show that the H-force number of these graphs is possibly n , or n − 2 n-2 , or n 2 \frac{n}{2} and give a classification of these graphs due to the H-force number.

Let k k be a positive integer and let G G be a graph with vertex set V ( G ) V(G) . A subset D ⊆ V ( G ) D\subseteq V(G) is a k k -dominating set if every vertex outside D D is adjacent to at least k k vertices in D D . The k k -domination number γ k ( G ) {\gamma }_{k}(G) is the minimum cardinality of a k k -dominating set in G G . For any graph G G , we know that γ k ( G ) ≥ γ ( G ) + k − 2 {\gamma }_{k}(G)\ge \gamma (G)+k-2 where Δ ( G ) ≥ k ≥ 2 \text{&#x0394;}(G)\ge k\ge 2 and this bound is sharp for every k ≥ 2 k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k ≥ 3 k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k = 3 k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

We investigate the abelian sandpile group on modified wheels W ˆ n {\hat{W}}_{n} by using a variant of the dollar game as described in [N. L. Biggs, Chip-Firing and the critical group of a graph , J. Algebr. Comb. 9 (1999), 25–45]. The complete structure of the sandpile group on a class of graphs is given in this paper. In particular, it is shown that the sandpile group on W ˆ n {\hat{W}}_{n} is a direct product of two cyclic subgroups generated by some special configurations. More precisely, the sandpile group on W ˆ n {\hat{W}}_{n} is the direct product of two cyclic subgroups of order a n {a}_{n} and 3 a n 3{a}_{n} for n even and of order a n {a}_{n} and 2 a n 2{a}_{n} for n odd, respectively.

A connected even [ 2 , 2 s ] {[}2,2s] -factor of a graph G is a connected factor with all vertices of degree i ( i = 2 , 4 , … , 2 s ) i(i=2,4,\ldots ,2s) , where s ≥ 1 s\ge 1 is an integer. In this paper, we show that a k + 1 s + 2 \tfrac{k+1}{s+2} -tough k -tree has a connected even [ 2 , 2 s ] {[}2,2s] -factor and thereby generalize the result that a k + 1 3 \tfrac{k+1}{3} -tough k -tree is Hamiltonian in [Hajo Broersma, Liming Xiong, and Kiyoshi Yoshimoto, Toughness and hamiltonicity in k-trees , Discrete Math. 307 (2007), 832–838].

A triangular matroid is a rank-3 matroid whose ground set consists of the points of an n 3 {n}_{3} -configuration and whose bases are the point triples corresponding to non-triangles within the configuration. Raney previously enumerated the n 3 {n}_{3} -configurations which induce triangular matroids for 7 ≤ n ≤ 15 7\le n\le 15 . In this work, the enumeration is extended to configurations having up to 18 points. Several examples of such configurations and their symmetry groups are presented, as well as geometric representations of the triangular matroids induced by these configurations.

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

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

The aim of this study is to investigate the finite approximate controllability of certain Hilfer fractional evolution systems with nonlocal conditions. To achieve this, we first transform the mild solution of the Hilfer fractional evolution system into a fixed point problem for a condensing map. Then, by using the topological degree approach, we present sufficient conditions for the existence and uniqueness of the solution of the Hilfer fractional evolution systems. Using the variational approach, we obtain sufficient conditions for the finite approximate controllability of semilinear controlled systems. Finally, an example is provided to illustrate main results.

The goal of this paper is to discuss the well-posedness and the generalized well-posedness of a new kind of differential quasi-variational-hemivariational inequality (DQHVI) in Hilbert spaces. Employing these concepts, we explore the essential relation between metric characterizations and the well-posedness of DQHVI. Moreover, the compactness of the set of solutions for DQHVI is delivered, when problem DQHVI is well-posed in the generalized sense.

The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.

The primary objective of this research is to establish the generalized fractional integral inequalities of Hermite-Hadamard-type for MT -convex functions and to explore some new Hermite-Hadamard-type inequalities in a form of Riemann-Liouville fractional integrals as well as classical integrals. It is worth mentioning that our work generalizes and extends the results appeared in the literature.

In this article, we study the nonlinear and nonsmooth interval-valued optimization problems in the face of data uncertainty, which are called interval-valued robust optimization problems (IVROPs). We introduce the concept of nondominated solutions for the IVROP. If the interval-valued objective function f and constraint functions g i {g}_{i} are nonsmooth on Banach space E , we establish a nonsmooth and robust Karush-Kuhn-Tucker optimality theorem.

In this paper, an equivalent quasinorm for the Lipschitz space of noncommutative martingales is presented. As an application, we obtain the duality theorem between the noncommutative martingale Hardy space h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) (resp. h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) ) and the Lipschitz space λ β c ( ℳ ) {\lambda }_{\beta }^{c}( {\mathcal M} ) (resp. λ β r ( ℳ ) {\lambda }_{\beta }^{r}( {\mathcal M} ) ) for 0 < p < 1 0\lt p\lt 1 , β = 1 p − 1 \beta =\tfrac{1}{p}-1 . We also prove some equivalent quasinorms for h p c ( ℳ ) {h}_{p}^{c}( {\mathcal M} ) and h p r ( ℳ ) {h}_{p}^{r}( {\mathcal M} ) for p = 1 p=1 or 2 < p < ∞ 2\lt p\lt \infty .

In this paper, we investigate the problem for optimal control of a viscous generalized θ \theta -type dispersive equation (VG θ \theta -type DE) with weak dissipation. First, we prove the existence and uniqueness of weak solution to the equation. Then, we present the optimal control of a VG θ \theta -type DE with weak dissipation under boundary condition and prove the existence of optimal solution to the problem.

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the ( p , q ) (p,q) -Laplace-type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation − Δ p u − Δ q u = f μ ( x , u , ∇ u ) -\hspace{-0.25em}{\text{&#x0394;}}_{p}u-{\text{&#x0394;}}_{q}u={f}_{\mu }(x,u,\nabla u) in a bounded domain Ω ⊂ ℝ N \text{&#x03A9;}\hspace{0.25em}\subset {{\mathbb{R}}}^{N} under certain assumptions on the nonlinearity and with a special attention to the resonance case f μ ( x , u , ∇ u ) = λ 1 ( p ) | u | p − 2 u + μ | u | q − 2 u {f}_{\mu }(x,u,\nabla u)={\lambda }_{1}(p)|u{|}^{p-2}u+\mu |u{|}^{q-2}u , where λ 1 ( p ) {\lambda }_{1}(p) is the first eigenvalue of the p -Laplacian.

We consider a parametric elliptic equation driven by the anisotropic ( p , q ) (p,q) -Laplacian. The reaction is superlinear. We prove a “bifurcation-type” theorem describing the change in the set of positive solutions as the parameter λ \lambda moves in ℝ + = ( 0 , + ∞ ) {{\mathbb{R}}}_{+}=(0,+\infty ) .

This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: − div ∇ v 1 + | ∇ v | 2 = f ( x , v , ∇ v ) in Ω , a 0 v + a 1 ∂ v ∂ ν = 0 on ∂ Ω , \left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{&#x03A9;},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu }=0& \text{on}\hspace{.5em}\partial \text{&#x03A9;},\end{array}\right. with Ω \text{&#x03A9;} an open ball in ℝ N {{\mathbb{R}}}^{N} , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.

The paper deals with the existence of solutions for ( p , Q ) (p,Q) coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set in ℍ n {{\mathbb{H}}}^{n} .

The paper deals with a quasilinear Dirichlet problem involving a competing ( p , q )-Laplacian and a convection term. Due to the lack of ellipticity, monotonicity and variational structure, the known methods to find a weak solution are not applicable. We develop an approximation procedure permitting to establish the existence of solutions in a generalized sense. If in place of competing ( p , q )-Laplacian we consider the usual ( p , q )-Laplacian, our results ensure the existence of weak solutions.

In this paper, we study the Hyers-Ulam-Rassias stability of ( m , n ) (m,n) -Jordan derivations. As applications, we characterize ( m , n ) (m,n) -Jordan derivations on C ⁎ {C}^{\ast } -algebras and some non-self-adjoint operator algebras.

In this paper, we consider the Cauchy problem for a second-order nonlinear equation with mixed fractional derivatives related to the fractional Khokhlov-Zabolotskaya equation. We prove the nonexistence of a classical local in time solution. The obtained instantaneous blow-up result is proved via the nonlinear capacity method.

In this work, we refine the results of Sendov and Shan [ New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures , J. Theor. Probab. 28 (2015), 1689–1725] on subordinators obtained by the class of Bernstein functions stable by the Mellin-Euler differential operator I − x d d x I-x\tfrac{\text{d}}{\text{d}x} by giving a stochastic interpretation, proving monotonicity properties of the related distributions and providing additional extensions.