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For any discrete valuation ring R, any R-linear mapping u from an R-module E into an R-module F and any \(y_0\in F\), a necessary and sufficient condition for the solvability of the equation \(u(x)=y_0\) is established, and an application of this result is presented.

In order to study the deformations of foliations of codimension 1 of a smooth manifold L, de Bartolomeis and Iordan defined the DGLA $$ \mathcal {Z}^{*}\left( L\right) $$ , where $$\mathcal {Z}^{*}\left( L\right) $$ is a subset of differential forms on L. In another paper, de Bartolomeis and Iordan studied the deformations of foliations of a smooth manifold L by defining the canonical solutions of Maurer–Cartan equation in the DGLA of graded derivations $$\mathcal {D}^{*}\left( L\right) $$ . Let L be a Levi flat hypersurface in a complex manifold. Then the deformation theories in $$\mathcal {Z}^{*}\left( L\right) $$ and $$\mathcal {D }^{*}\left( L\right) $$ lead to the moduli space for the Levi flat deformations of L. In this paper we discuss the relationship between the infinitesimal deformations of L defined by the solutions of Maurer–Cartan equation in $$\mathcal {Z}^{*}\left( L\right) $$ and the infinitesimal deformations of L obtained by means of the canonical solutions of Maurer–Cartan equation in the DGLA of graded derivations $$\mathcal {D}^{*}\left( L\right) $$ .

We study etale extensions of rings that have FIP., Comment: The paper entitled FIP and FCP products of ring morphisms (arXiv: 1312.1250 [math.AC]) is now split into three papers. The present paper contains the last section of the original paper and many other results on etale FIP extensions

In this paper, we study some uniqueness problems of meromorphic functions when certain nonlinear differential polynomials generated by them share a nonconstant polynomial. The results of the paper improve the concerning results due to Xu et al. (Mat Vesnik 64:1–16, 2012).

In 1886 Giuseppe Peano presents the first proof of the existence of a solution of an initial value problem \(y^\prime =f(x,y)\), \(y(a)=b\), under the assumption of the continuity of the function f. The present paper gives a detailed description of Peano’s original statements and proofs, filling gaps, clarifying obscure points and avoiding ambiguous use of mathematical symbols. Peano’s 1886 work is compared with later papers of Peano himself as well as of Mie (Math Ann 43:553–568, 1893), Osgood (Monatsh Math Phys 9:331–345, 1898) and Perron (Math. Ann. 76:471–484, 1915).

International audience; In this paper, we prove some new results on operational second order differential equations of elliptic type with general Robin boundary conditions in a non-commutative framework. The study is developed in Hölder spaces under some natural assumptions generalizing those in [4]. We give necessary and sufficient conditions on the data to obtain a unique strict solution satisfying the maximal regularity property, see Theorems 1 and 2. This work completes the one given in [4] and [12].

In this paper we prove a strengthening of the generic vanishing result in characteristic $p>0$ given in [HP16]. As a consequence of this result, we show that irreducible $\Theta$ divisors are strongly F-regular and we prove a related result for pluri-theta divisors., Comment: Comments are more than welcome

In this paper, we propose a new inertial extrapolation method for solving a certain class of generalized split feasibility problems in two real Hilbert spaces. We prove that the proposed method converges strongly to a minimum norm solution of the problem when the underlying operator is pseudomonotone and uniformly continuous which are much more weaker assumptions than the inverse strongly monotonicity assumptions used in the literature. Moreover, our method uses stepsizes that are generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the operator norm or the coefficient of a underlying operator. Furthermore, some examples and numerical experiments to show the efficiency and implementation of our method (in comparison with other methods in the literature) were also discussed in the framework of infinite dimensional Hilbert spaces.

We consider a system of pseudomonotone equilibrium problem and split common fixed point problem in the framework of real Hilbert spaces. We propose a modified extragradient method with line searching technique for approximating a common element in the sets of solutions of the two nonlinear problems. The convergence result is proved without prior knowledge of the Lipschitz-like constants of the equilibrium bifunctions and the norm of the bounded linear operator of the split common fixed point problem. We further provide some application and numerical example to show the importance of the obtained results in the paper.

Hypercyclic operators are operators with dense orbits. A contraction cannot be hypercyclic since its orbits are bounded sets. Nevertheless, by multiplying a contraction with a scalar of absolute value larger than 1, the resulting scaled contraction can occasionally be a hypercyclic operator. In this paper, we investigate which Hilbert space contractions have that property and which don’t. We introduce the set $$\Lambda (T)$$ of all scalars which produce a hypercyclic operator, by scaling the operator T, and determine $$\Lambda (T)$$ in various cases. New properties of hyperciclic operators are discovered in this process. For instance, it is proved that any connected component of the essential spectrum of a hypercyclic operator must meet the unit circle.

In this paper it will be considered a non-linear system consisting of two wave equations, non-homogeneous and coupled under the effect of non-linear source and damping terms. In addition, one of them will also act as a memory term. The structure of the dynamic system associated with the solutions of this system will allow the use of the quasi-stability theory in order to obtain the existence of global and exponential attractors.

In this paper we prove that if $$X\subset {\mathbb P}^r$$ X ⊂ P r is a 2-smooth, irreducible, projective non-degenerate variety of dimension n such that $${\text {Sec}}_2(X)={\mathbb P}^r$$ Sec 2 ( X ) = P r , if $$n > \frac{4}{7}(r-2)$$ n > 4 7 ( r - 2 ) and if $$X'$$ X ′ is the projection of X to $${\mathbb P}^{r-1}$$ P r - 1 from a general point, then the set of length 3 subschemes of $$X'$$ X ′ which lie on a line form an irreducible variety.

In this paper we study Vainikko integral operators which are similar to so-called cordial integral operators and contain the classical Hardy operator, the Schur operator, and the Hilbert transform as special cases. For such operators we obtain norm estimates and equalities, mainly in BV type spaces in the sense of Jordan, Wiener, Riesz, and Waterman. Several examples are also discussed.

In this paper we present three recently obtained important generalizations of the Choquet integral. First generalization is based on different collection of finite systems. Second generalization of the Choquet integral uses sublinear means. Third generalization is based on two fuzzy measures, one of which is pseudo-additive.

We take a magical tour in algebraic logic, which is the natural interface between universal algebra and mathematical logic, starting from classical results on neat embeddings due to Andreḱa, Henkin, Nemeti, Monk and Tarski, all the way to recent novel results in algebraic logic using so-called rainbow constructions. Highlighting the connections with graph theory, model theory, and finite combinatorics, this article aspires to present topics of broad interest in a way that is hopefully accessible to a large audience. Other topics dealt with include the interaction of algebraic and modal logic, the so-called (central still active) finitizability problem, Godels’s incompleteness Theorem in guarded fragments, counting the number of subvarieties of $$\textsf {RCA}_{\omega }$$ which is reminiscent of Shelah’s classification theory and the interaction of algebraic logic and descriptive set theory as means to approach Vaught’s conjecture in model theory. The paper has a survey character but it contains new results and new approaches to old ones (such as the interaction of algebraic logic and descriptive set theory).

In this survey paper we discuss a series of recent results concerning nonnegative solutions to nonlinear diffusion equations of porous-medium type on Cartan–Hadamard manifolds, a special class of negatively-curved Riemannian manifolds that generalize the Euclidean space. We focus on sharp barrier estimates, asymptotic convergence and smoothing effects, describing quantitatively how the curvature behavior at infinity affects the way solutions depart from having a Euclidean-like structure.

In this paper, the existence results of the solutions and stability are focused for impulsive implicit sequential fractional differential equation with multi-points boundary conditions. In view of the definitions of Caputo fractional order, the existence, uniqueness, and at least one solution of the aforesaid equation are presented. Beside this, Ulam’s type stabilities are discussed. To support our main results, we present some examples.

In this paper we study a class of first-order Ventcel’ boundary value problems for second-order, elliptic Waldenfels integro-differential operators. More precisely, by using real analysis techniques such as Strichartz norms and the complex interpolation method we prove existence and uniqueness theorems in the framework of Sobolev and Besov spaces of $$L^{p}$$ type which extend earlier theorems due to Bony–Courrege–Priouret and Runst–Youssfi to the general degenerate case. Our proof is based on various maximum principles for second-order, elliptic Waldenfels operators with discontinuous coefficients in the framework of $$L^{p}$$ Sobolev spaces.

In this paper, the two-variable one-parameter generalized Hermite-based hybrid polynomials are introduced by means of generating function, series definition and determinant definition. The recurrence relations, shift operators, differential, integro-differential and partial differential equations for these polynomials are established via factorization method. The two-variable one-parameter generalized Hermite-based Bernoulli, Euler and Genocchi polynomials are studied as the particular members and some examples are considered in terms of these polynomials to give the applications of main results. The graphical representation and interpretation is also shown for these polynomials.

Denote by $$\mathfrak {M}$$ the class of all multioperator T-groups satisfying the minimality and maximality conditions for T-subgroups, where T is a set of algebraic group operations. In each T-group G, we select a system of T-subgroups $$\tau (G)$$ and say that $$\tau $$ is a T-subgroup functor if it is satisfying the following two conditions: (1) $$G \in \tau (G)$$ for every T-group G; and (2) for every epimorphism $$\varrho : A \rightarrow B$$ and any $$H \in \tau (A)$$ and $$K \in \tau (B)$$, we have $$ H^\varrho \in \tau (B) \, \text{ and } \, K^{\varrho ^{-1}} \in \tau (A). $$ Let n be a positive integer, and $$\mathfrak {F}$$, $$\mathfrak {H}$$ be a $$\tau $$-closed n-multiply $$\Omega _1$$-foliated $$\mathfrak {M}$$-formations with direction $$\varphi $$ such that $$\varphi _0 \leqslant \varphi $$, and $$\mathfrak {H} \subseteq \mathfrak {F}$$. Denote by $$\mathfrak {F} /^\tau _{\Omega _1 F_n^\varphi } \mathfrak {H}$$ the lattice of all $$\tau $$-closed n-multiply $$\Omega _1$$-foliated $$\mathfrak {M}$$-formations (with direction $$\varphi $$ such that $$\varphi _0 \leqslant \varphi $$) such that $$\mathfrak {H} \subseteq \mathfrak {X} \subseteq \mathfrak {F}$$. If $$\mathfrak {X} \subset \mathfrak {F}$$ and the lattice $$\mathfrak {F} /^\tau _{\Omega _1 F_n^\varphi } \mathfrak {X}$$ consists of only two elements then $$\mathfrak {X}$$ is called a maximal$$\tau $$-closed n-multiply $$\Omega _1$$-foliated $$\mathfrak {M}$$-formation (with direction $$\varphi $$ such that $$\varphi _0 \leqslant \varphi $$) of $$\mathfrak {F}$$. In the present paper, the properties of the intersection of these formations are studied.

In this paper, we prove a result about the vanishing of the moments for a class of Abel equation and we give a partial positive answer to a conjecture proposed by Lijun and Yun (J Math Anal Appl 261:100–112, 2001).

For surfaces of revolution we prove the existence of infinitely many rotationally symmetric solutions to a wide class of semilinear Laplace–Beltrami equations. Our results extend those in Castro and Fischer (Can Math Bull 58(4):723–729, 2015) where for the same equations the existence of infinitely many even (symmetric about the equator) rotationally symmetric solutions on spheres was established. Unlike the results in that paper, where shooting from a singularity to an ordinary point was used, here we obtain regular solutions shooting from a singular point to another singular point. Shooting from a singularity to an ordinary point has been extensively used in establishing the existence of radial solutions to semilinear equations in balls, annulii, or $$\mathbb {R}^N$$ .

Let $$(X, *, 0)$$ be a d-algebra and $$\alpha , \beta $$ are endomorphisms on X. Motivated by some results on derivations, $$(\alpha ,\beta )$$ -derivation in rings, and the generalizations of BCK and BCI-algebras, in this paper, we introduce the notion of $$(\alpha ,\beta )$$ -derivations on d-algebras, construct several examples and investigate some simple and important results.

The main purpose of this paper is to investigate the character amenability of semigroup algebras. In this regard, the new concept character amenability modulo an ideal of Banach algebras are introduced. For a large class of semigroups such as E-inversive E-semigroup and eventually inverse semigroups, it is shown that the semigroup S is amenable if and only if the semigroup algebra $$l^1(S)$$ is character amenable modulo an ideal. Some characterizations of character amenability modulo an ideal of Banach algebras are studied and interesting examples are presented.

This paper aims at giving a unified framework for a number of well-known extremal Kahler metrics with a big group of symmetries, with a special emphasis on the case of scalar-flat Kahler metrics, all stemming from the general construction of $$\mathrm{U} (m)$$ -invariant extremal Kahler metrics on the space $$\mathbb {C}^m {\setminus } \{0\}$$ , based on the momentum profile introduced by Hwang and Singer (Trans Am Math Soc 354:2285–2325, 2002).

In this paper, we investigate the problem of describing the form of Lie generalized derivations on trivial extension algebras. We show, under some conditions, that every Lie generalized derivation on a trivial extension algebra is an sum of a generalized derivation and a center valued map which vanishes on all commutators. As an application we characterize Lie generalized derivation on a triangular algebra.

We consider a functional of pullbacks of metrics on the space of maps between Riemannian manifolds. Stationary maps for this functional are called symphonic maps (Kawai and Nakauchi in Nonlinear Anal 74:2284–2295, 2011; Differ Geom Appl 44:161–177, 2016; Misawa and Nakauchi in Nonlinear Anal 75:5971–5974, 2012; Calc Var Partial Differ Equ 55:1–20, 2016; Adv Differ Equ, 2018; Nakauchi and Takakuwa in Nonlinear Anal 108:87–98, 2014; Nakauchi and Takenaka in Ricerche di Matematica 60:219–235, 2011). In this paper we introduce a stress energy tensor for symphonic maps and give some results.

Building on the recent paper (Blanc et al. preprint, arXiv:1607.03012 ), we present an $$\ell $$ -adic trace formula for smooth and proper dg-categories over a base $$\mathbb {E}_\infty $$ -algebra B. We also give a variant when B is just an $$\mathbb {E}_2$$ -algebra. As an application of this trace formula, we propose a strategy of proof of Bloch’s conductor conjecture. This is a research announcement and detailed proofs will appear elsewhere.

In this paper we study two semilinear Dirichlet problems; the linear parts (in some sense, in duality) are a problem with singular convection term and a problem with singular drift. The nonlinear lower order terms have a regularizing effect: the solutions of the corresponding linear problems are less regular.

Let p be an odd prime. In this paper we determine the group of length-preserving automorphisms for the ordinary Steenrod algebra A(p) and for $${\mathcal {B}}(p)$$ , the algebra of cohomology operations for the cohomology of cocommutative $$\mathbb {F}_p$$ -Hopf algebras. Contrarily to the $$p=2$$ case, no length-preserving strict monomorphism turns out to exist.

Let G be a group with identity e and let R be a commutative G-graded ring. In this paper, we will investigate commutative graded rings which satisfy the condition \((*)\). We say that a graded ring R satisfy the condition \((*)\) if P is a graded prime ideal of R and if \(\{I_{\alpha }\}_{\alpha \in \Delta }\) is a family of graded ideals of R, then P contains \(\cap _{\alpha \in \Delta }I_{\alpha }\) only if P contains some \(I_{\alpha }\).

In this paper, a generalization of max modules is introduced by applying the concept of corational submodules.

In this paper, identities related to \((\sigma , \tau )\)-Lie derivations and \((\sigma , \tau \))-derivations are considered. Let \(m,n \ge 1\) be integers and \(\mathcal {R}\) be an M!-torsion free unital ring, where \(M = \max \{m, n\}\). Suppose that \(\sigma , \tau : \mathcal {R} \rightarrow \mathcal {R}\) are two endomorphisms such that \(\sigma (\mathbf e )\),\(\tau (\mathbf e ) \in Z(\mathcal {R})\), where \(\mathbf e \) denotes the unit element of \(\mathcal {R}\). If \(D:\mathcal {R} \rightarrow \mathcal {R}\) is an additive map satisfying \(D[x^n, y^m] = [D(x^n), \sigma (y^m)] + [\tau (x^n), D(y^m)]\) for all \(x, y \in \mathcal {R}\), then D is a \((\sigma , \tau \))-Lie derivation. Moreover, we offer a characterization of (\(\sigma , \tau \))-derivations from a \(C^{*}\)-algebra \(\mathcal {A}\) into a Banach \(\mathcal {A}\)-bimodule \(\mathcal {M}\) which reads as follows. Let \(\mathcal {A}\) be a unital \(C^{*}\)-algebra, \(\mathcal {M}\) be a unital Banach \(\mathcal {A}\)-bimodule, and let \(\sigma , \tau :\mathcal {A} \rightarrow \mathcal {A}\) be continuous endomorphisms such that \(\sigma (\mathbf e ) = \mathbf e = \tau (\mathbf e )\), where \(\mathbf e \) denotes the unit element of \(\mathcal {A}\). Suppose that \(n > 1\) is an integer and \(d:\mathcal {A} \rightarrow \mathcal {M}\) is a linear map satisfying \(d(a^{n}) = \sum _{j = 1}^{n}\tau (a)^{n - j}d(a) \sigma (a)^{j - 1}\) for all \(a \in \mathcal {A}\). Then, d is a continuous \((\sigma , \tau \))-derivation.

In this paper, we establish a criterion of parabolicity for complete two-sided hypersurfaces immersed in a Riemannian warped product of the type $$I\times _fM^n$$ , where $$M^{n}$$ is a connected n-dimensional oriented Riemannian manifold and $$f:I\rightarrow \mathbb {R}$$ is a positive smooth function. As applications, we obtain several uniqueness results concerning these hypersurfaces with constant mean curvature, under standard constraints on the Ricci curvature of $$M^n$$ and on the warping function f. Moreover, considering the higher order mean curvatures, we also obtain estimates for the index of relative nullity.

Let X be a Stein manifold of dimension $$n \ge 3$$ . Let $$\Omega _{1}$$ be a weakly q-convex and $$\Omega _{2}$$ be a weakly $$(n-q-1)$$ -convex in X with smooth boundaries such that $$\overline{\Omega }_{2}\Subset \Omega _{1}\Subset X$$ . Assume that $$\Omega =\Omega _{1}\backslash \overline{\Omega }_{2}$$ . In this paper, we establish sufficient conditions for the closed range of $$\overline{\partial }$$ on $$\Omega $$ . Moreover, we study the global boundary regularity of the $$\overline{\partial }$$ -problem on $$\Omega $$ .

The paper discusses uniqueness of solutions to stationary elliptic problems of the type $$\begin{aligned} A(u)+H(u)=f\in {\mathcal {D}}'(\Omega ), \end{aligned}$$ where $$\Omega \ \in R^{N},\ $$ $$u\in W^{1,p}(\Omega )\ (1\le p\le +\infty ),\ A(u)\ $$ is an elliptic operator, $$H(u)\ $$ is an Hamiltonian that grows with $$\left| {\nabla u}\right| ^{p}$$ and f is given. Methods introduced in Artola (Boll UMI 6(5-B):51–71, 1986), (Proceedings of the International Conference on Generalized Functions, (ICGF 2000). Cambridge Scientific Publishers, Cambridge, 51–92, 2004), (Ricerche di Matematica XLIV, fasc. 2:400–420, 1995) for quasilinear parabolic or elliptic equations, together with properties for some continuity moduli, are used to improve some results from Barles and Murat (Arch Ration Mech Anal 133(1):77–101, 1995) for bounded solutions and from Barles and Porretta (Ann Scuola Norm Sup Pisa Cl Sci 5(1):107–136, 2006), Lions (J Anal Math 45: 234–254, 1985) for unbounded solutions, when 1 $$\le p\le 2.$$ Unilateral problems are considered and the case where f depends on the solution u is also discussed.

In the first part of this paper, we give a sufficient condition for a particular case of the symmetric moment problem to be determinate using standards methods of q-Bessel Fourier analysis. This condition it cannot be deduced from any other classical criterion of determinacy. In the second part, we study the q-Strum–Liouville equation in the non-real case and we elaborate an analogue of the well known theorem due to Hermann Weyl concerning the Strum–Liouville equation. This emphasizes the connection between the moment problem associated to a particular class of orthonormal polynomials $$(P_n)$$ and the uniqueness of solution which belong to the $$L^2$$ space. The third part is devoted to the study of the q-Strum–Liouville equation in the real case and the behavior of solutions at infinity, which give more information about this type of orthonormal polynomials.

In this paper, we introduce and study the notions of Gorenstein n-FP-injective and Gorenstein n-flat complexes, which are special cases of Gorenstein FP-injective and Gorenstein flat complexes respectively. We prove that over a left n-coherent ring R the class of all Gorenstein n-FP-injective (resp., Gorenstein n-flat) complexes is injectively (resp., projectively) resolving and we discuss the relationship between Gorenstein n-FP-injective and Gorenstein n-flat complexes.

In this paper, we generalize classical Hecke algebras \(\mathcal {H}(G_{p})\) over the generalized linear groups \(G_{p} = GL_{2}(\mathbb {Q}_{p})\) induced by the p-adic number fields \(\mathbb {Q}_{p}\), for primes p. For a given group \(G_{p},\) construct a suitable semigroup \(W^{*}\)-dynamical system \((M, \sigma (G_{p}), \pi ),\) where M is a fixed von Neumann algebra, and \( \pi \) is a semigroup-action of the \(\sigma \)-algebra \(\sigma (G_{p})\) of \( G_{p}\) acting on M. By constructing the corresponding crossed product \( W^{*}\)-algebra \(M \times _{\pi } \sigma (G_{p})\) generated by \((M, \sigma (G_{p}), \pi ),\) we study free probability on the \(W^{*}\) -subalgebra \(\mathcal {H}_{M}(G_{p})\) of \(M \times _{\pi } \sigma (G_{p})\) . One can understand our von Neumann algebra \(\mathcal {H}_{M}(G_{p})\) as a generalized \(*\)-algebra over both M and a Hecke algebra \(\mathcal {H} (G_{p}),\) for a prime p.

Let (G, X) be a flow such that X is a locally finite graph and G is a finitely generated group. In this paper, it is shown that the following properties are equivalent: We show that every pointwise recurrent flow of a locally finite graph is equicontinuous. We also give some qualitative properties of an equicontinuous flow.

The self-idealization of a commutative ring R is isomorphic to the ring \(R[x]/(x^2)\) or, equivalently, the ring of upper-triangular Toeplitz matrices \({{\mathrm{\mathcal {T}}}}(R)=\left\{ \left( \begin{matrix} a &{} b \\ 0 &{} a \end{matrix}\right) :a,b\in R\right\} \). Recently, Chang and Smertnig characterized the sets of lengths of factorizations in \({{\mathrm{\mathcal {T}}}}(D)\) where D is a principal ideal domain. In this work, in addition to correcting an error in their paper, we extend the study to \({{\mathrm{\mathcal {T}}}}(R)\) when R is either a principal ideal ring or a unique factorization ring.

In this paper we study stability of multivariate wave packet frames. Necessary and sufficient conditions for a certain system to be multivariate wave packet frames are obtained.

In the present paper we consider the following question: does there exist a Neron model for families of Jacobians of curves with sections? By applying a construction of the author of universal compactified Jacobians over the moduli stack of reduced curves with markings and a result by J. Kass, we give a positive answer to the question holding for curves with planar singularities.

In this paper we exhibit new classes of Banach spaces for which strong notions of optimization can be lifted from quotient spaces. Motivated by a well known result of Cheney and Wulbert on lifting of proximinality from a quotient space to a subspace, for closed subspaces, \(Z \subset Y \subset X\), we consider stronger forms of optimization, that Z has in X and the quotient space Y / Z has in X / Z should lead to the conclusion Y has the same property in X. The versions we consider have been studied under various names in the literature as L-proximinal subspaces or subspaces that have the strong-\(1\frac{1}{2}\)-ball property. We give an example where the strong-\(1\frac{1}{2}\)-ball property fails to lift to the quotient. We show that if every M-ideal in Y is a M-summand, for a finite codimensional subspace \(Z \subset Y\), that is a M-ideal in X with the strong-\(1\frac{1}{2}\)-ball property in X and if Y / Z has the \(1\frac{1}{2}\)-ball property in X / Z, then Y has the strong-\(1\frac{1}{2}\)-ball property in X.

The purpose of this paper is to obtain sufficient conditions for the existence and uniqueness of points of coincidence and common fixed points for mappings defined on cone modular spaces endowed with a graph. Our results will improve and supplement several recent results in the literature.

In this paper, we describe a family of meromorphic functions in $$\mathbf {C}$$ from analyzing some properties of these L-functions in the extended Selberg class and show two uniqueness results of such a function, in terms of shared values with a general meromorphic function in $$\mathbf {C}$$ . In particular, we show the condition “ $$\displaystyle {1\mathsf{CM}+3\mathsf{IM}}$$ value-sharing” suffices.

In this paper, we apply (p, q)-calculus to construct generalized bivariate Bleimann–Butzer–Hahn operators based on (p, q)-integers and obtain Korovkin type approximation theorem. Furthermore, we compute the rate of convergence for these operators by using the modulus of continuity and Lipschitz type maximal function.

We consider in the paper the situation when an entire function shares a polynomial with its derivatives. Our results improve a result of Zhong.

In this paper we introduce the concept of $$\xi $$ -torsion module, $$\xi $$ -torsion-free module and $$\xi $$ -torsionable module. We investigate many properties of these modules. We characterize $$\xi $$ -torsion modules and $$\xi $$ -torsion-free modules using short exact sequences and module homomorphisms.

In this paper, under certain conditions we find a double Jordan derivation near a certain function in a p-Banach algebra. Indeed, we prove the generalized Hyers–Ulam–Rassias stability and Isac-Rassias stability of double Jordan derivations in p-Banach algebras.