Let $R$ be a prime ring and $I$ a nonzero ideal of $R.$ The purpose of this paper is to classify generalized derivations of $R$ satisfying some algebraic identities with power values on $I.$ More precisely, we consider two generalized derivations $F$ and $H$ of $R$ satisfying one of the following identities: (1) ...

This paper is concerned with the isomorphic structure of the Banach space ${\ell }_{\infty }/c₀$ and how it depends on combinatorial tools whose existence is consistent with but not provable from the usual axioms of ZFC. Our main global result is that it is consistent that ${\ell }_{\infty }/c₀$ does not have an orthogonal ${\ell }_{\infty }$-decomposition, that is, it is not of the form ${\ell }_{\infty }\left(X\right)$ for any Banach space X. The main local result is that it is consistent that ${\ell }_{\infty }\left(c₀\left(\right)\right)$ does not embed isomorphically into ${\ell }_{\infty }/c₀$, where is the cardinality of the continuum,...

In this paper, we define the direct sum ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the H-property if and only if each $X_i$ has the H-property, and ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ has the Schur property if and only if each $X_i$ has the Schur property. Moreover, we also show that ${\left({⨁}_{i=1}^n{X}_i\right)}_{ce{s}_p}$ is rotund if and only if each $X_i$ is rotund.

For the integral equation (1) below we prove the existence on an interval $J=[0,a]$ of a solution $x$ with values in a Banach space $E$, belonging to the class $L^p(J,E)$, $p>1$. Further, the set of solutions is shown to be a compact one in the sense of Aronszajn.

For Banach spaces $X$ and $Y$, let $K_{w^{*}}(X^{*},Y)$ denote the space of all $w^{*}-w$ continuous compact operators from $X^{*}$ to $Y$ endowed with the operator norm. A Banach space $X$ has the $wGP$ property if every Grothendieck subset of $X$ is relatively weakly compact. In this paper we study Banach spaces with property $wGP$. We investigate whether the spaces $K_{w^{*}}(X^{*},Y)$ and $X{\otimes }_{ϵ}Y$ have the $wGP$ property, when $X$ and $Y$ have the $wGP$ property.

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies ${\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\partial }_j{f}_j{\left(\epsilon \right)\left|\right|}^{p}d\mu \left(\epsilon \right){\le }^p{\int }_{-1,1ⁿ}{\int }_{-1,1ⁿ}\left|\right|{\sum }_{j=1}^n{\delta }_j\Delta f_j\left(\epsilon \right){\left|\right|}^{p}d\mu \left(\epsilon \right)d\mu \left(\delta \right)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${{\partial }_j}_{j=1}^n$ and $\Delta ={\sum }_{j=1}^n{\partial }_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by ${ⁿ}_p\left(X\right)$, we show that ${ⁿ}_p\left(X\right)\le {\sum }_{k=1}^{n}1/k$ for every Banach space (X,||·||). This extends the classical Pisier inequality, which corresponds to the special...

By using the concepts of limited $p$-converging operators between two Banach spaces $X$ and $Y$, $L_p$-sets and $L_p$-limited sets in Banach spaces, we obtain some characterizations of these concepts relative to some well-known geometric properties of Banach spaces, such as $*$-Dunford–Pettis property of order $p$ and Pelczyński’s property of order $p$, $1\le p<\infty$.

Let $r(G_1,G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V\left(S(n_1,n_2)\right)=\{v_0,v_1,...,v_{n_1},w_0$, $w_1,...,w_{n_2}\}$ and $E\left(S(n_1,n_2)\right)=\{v_0{v}_1,...,v_0{v}_{n_1},v_0{w}_0,w_0{w}_1,...,w_0{w}_{n_2}\}$. We determine $r(K_{1,m-1},$ $S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{\text{'}\text{'}}=(V,E_2)$ and $T_n^{\text{'}\text{'}\text{'}}=(V,E_3)$, where $V=\{v_0,v_1,...,v_{n-1}\}$, $E_2=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3}$, $v_1{v}_{n-2},v_2{v}_{n-1}\}$ and $E_3=\{v_0{v}_1,...,v_0{v}_{n-4},v_1{v}_{n-3},$ $v_2{v}_{n-2},v_3{v}_{n-1}\}$. We also obtain explicit formulas for $r(K_{1,m-1},T_n)$, $r(T_m^{\text{'}},T_n)$ $(n\ge m+3)$, $r(T_n,T_n)$, $r(T_n^{\text{'}},T_n)$ and $r(P_n,T_n)$, where $T_n\in \{T_n^{\text{'}\text{'}},T_n^{\text{'}\text{'}\text{'}},T_n^3\}$, $P_n$ is the path on $n$ vertices and $T_n^{\text{'}}$ is the unique tree with $n$ vertices and maximal degree $n-2$.

We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces $X_i$ for which $\left(X_i\right)/ℐn\left(X_i\right)$ is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces $X_i$ can be divided into subsets in such a way that if $X_i$ and $X_j$ are in different...

Given an integer $n\ge 3$, let $u_1,...,u_n$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

Let $G$ be a group and $\omega \left(G\right)$ be the set of element orders of $G$. Let $k\in \omega \left(G\right)$ and $m_k\left(G\right)$ be the number of elements of order $k$ in $G$. Let nse$\left(G\right)=\{m_k\left(G\right):k\in \omega \left(G\right)\}$. Assume $r$ is a prime number and let $G$ be a group such that nse$\left(G\right)=$ nse$\left(S_r\right)$, where $S_r$ is the symmetric group of degree $r$. In this paper we prove that $G\cong S_r$, if $r$ divides the order of $G$ and $r^2$ does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

We discuss the validity of the Helmholtz decomposition of the Muckenhoupt $A_p$-weighted $L^p$-space ${\left(L_w^p\left(\Omega \right)\right)}^n$ for any domain $\Omega$ in ${ℝ}^n$, $n\in \mathbb{Z}$, $n\ge 2$, $1<p<\infty$ and Muckenhoupt $A_p$-weight $w\in A_p$. Set $p^{\text{'}}:=p/(p-1)$ and $w^{\text{'}}:=w^{-1/(p-1)}$. Then the Helmholtz decomposition of ${\left(L_w^p\left(\Omega \right)\right)}^n$ and ${\left(L_{w^{\text{'}}}^{p^{\text{'}}}\left(\Omega \right)\right)}^n$ and the variational estimate of $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\pi }^{p^{\text{'}}}\left(\Omega \right)$ are equivalent. Furthermore, we can replace $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\pi }^{p^{\text{'}}}\left(\Omega \right)$ by $L_{w,\sigma }^p\left(\Omega \right)$ and $L_{w^{\text{'}},\sigma }^{p^{\text{'}}}\left(\Omega \right)$, respectively. The proof is based on the reflexivity and orthogonality of $L_{w,\pi }^p\left(\Omega \right)$ and $L_{w,\sigma }^p\left(\Omega \right)$ and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation...

Let $G$ be a finite group and $k$ a field of characteristic $p>0$. In this paper, we obtain several equivalent conditions to determine whether the principal block $B_0$ of a finite $p$-solvable group $G$ is $p$-radical, which means that $B_0$ has the property that $e_0{\left(k_P\right)}^G$ is semisimple as a $kG$-module, where $P$ is a Sylow $p$-subgroup of $G$, $k_P$ is the trivial $kP$-module, ${\left(k_P\right)}^G$ is the induced module, and $e_0$ is the block idempotent of $B_0$. We also give the complete classification of a finite $p$-solvable group $G$ which has not more...

It is proved that real functions on $ℝ$ which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower $C^1$-functions, or of two strongly paraconvex functions) coincide with semismooth functions on $ℝ$ (i.e. those locally Lipschitz functions on $ℝ$ for which $f_{+}^{\text{'}}\left(x\right)={lim}_{t\to x+}{f}_{+}^{\text{'}}\left(t\right)$ and $f_{-}^{\text{'}}\left(x\right)={lim}_{t\to x-}{f}_{-}^{\text{'}}\left(t\right)$ for each $x$). Further, for each modulus $\omega$, we characterize the class $DS{C}_{\omega }$ of functions on $ℝ$ which can be written as $f=g-h$, where $g$ and $h$ are semiconvex with modulus $C\omega$ (for some $C>0$) using a new...

Let $R$ be a prime ring with center $Z\left(R\right)$ and $I$ a nonzero right ideal of $R$. Suppose that $R$ admits a generalized reverse derivation $(F,d)$ such that $d\left(Z\right(R\left)\right)\ne 0$. In the present paper, we shall prove that if one of the following conditions holds: (i) $F\left(xy\right)\pm xy\in Z\left(R\right)$, (ii) $F\left(\right[x,y\left]\right)\pm \left[F\right(x),y]\in Z\left(R\right)$, (iii) $F\left(\right[x,y\left]\right)\pm \left[F\right(x),F(y\left)\right]\in Z\left(R\right)$, (iv) $F(x\circ y)\pm F\left(x\right)\circ F\left(y\right)\in Z\left(R\right)$, (v) $\left[F\right(x),y]\pm [x,F(y\left)\right]\in Z\left(R\right)$, (vi) $F\left(x\right)\circ y\pm x\circ F\left(y\right)\in Z\left(R\right)$ for all $x,y\in I$, then $R$ is commutative.

We consider the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y(x-\varphi \left(x\right))=f\left(x\right),x\in ℝ,$$ where $\varphi$ and $q$ ($q\ge 1$) are positive continuous functions for all $x\in ℝ$ and $f\in C\left(ℝ\right)$. By a solution of the equation we mean any function $y$, continuously differentiable everywhere in $ℝ$, which satisfies the equation for all $x\in ℝ$. We show that under certain additional conditions on the functions $\varphi$ and $q$, the above equation has a unique solution $y$, satisfying the inequality $$\parallel y^{\text{'}}{\parallel }_{C\left(ℝ\right)}+{\parallel qy\parallel }_{C\left(ℝ\right)}\le c{\parallel f\parallel }_{C\left(ℝ\right)},$$ where the constant $c\in (0,\infty )$ does not depend on the choice of $f$.

Let $K$ be a number field defined by an irreducible polynomial $F\left(X\right)\in \mathbb{Z}\left[X\right]$ and ${\mathbb{Z}}_K$ its ring of integers. For every prime integer $p$, we give sufficient and necessary conditions on $F\left(X\right)$ that guarantee the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$, where $\overline{F}\left(X\right)$ factors into powers of $r$ monic irreducible polynomials in ${𝔽}_p\left[X\right]$. The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly $r$ prime ideals of ${\mathbb{Z}}_K$ lying above $p$....