Volume 32, Issue 6

Let L be a non-central Lie ideal of a prime ring R . The primary objective of this paper is to characterize generalized derivations ℋ {\mathcal{H}} , ℱ {\mathcal{F}} and 𝒢 {\mathcal{G}} of R satisfying the following algebraic identity: x ⊥ ℋ ⁢ ( y ) = ℱ ⁢ ( x ) ⁢ y + y ⁢ 𝒢 ⁢ ( x )   for all  ⁢ x , y ∈ L , x\bot\mathcal{H}(y)=\mathcal{F}(x)y+y\mathcal{G}(x)\quad\text{for all }x,y\in L, where ⊥ {\bot} represents either the Lie product [ ⋅ , ⋅ ] {[\,\cdot\,,\cdot\,]} , or the Jordan product ∘ {\circ} . Furthermore, we extend our results to Banach algebras.

Let C ⁢ [ 0 , T ] {C[0,T]} denote the space of real-valued continuous functions on [ 0 , T ] {[0,T]} and let Z e → , ∞ ⁢ ( x ) = ( x ⁢ ( 0 ) , ∫ 0 T e 1 ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) , ∫ 0 T e 2 ⁢ ( t ) ⁢ 𝑑 x ⁢ ( t ) , … )   for  x ∈ C ⁢ [ 0 , T ] , Z_{\vec{e},\infty}(x)=\Bigg{(}x(0),\int_{0}^{T}e_{1}(t)\,dx(t),\int_{0}^{T}e_{% 2}(t)\,dx(t),\dots\Bigg{)}\quad\text{for ${x\in C[0,T]}$,} where { e j } j = 1 ∞ {\{e_{j}\}_{j=1}^{\infty}} is a sequence of appropriate functions on [ 0 , T ] {[0,T]} . In this paper, we derive a simple evaluation formula for calculating Radon–Nikodym derivatives similar to the conditional Wiener integrals of functions on C ⁢ [ 0 , T ] {C[0,T]} given Z e → , ∞ {Z_{\vec{e},\infty}} which has an initial weight and a kind of drift. As applications of the formula, we evaluate the derivatives of various functions containing the time integral which is of interest in quantum mechanics, especially in Feynman integration theory.

We present a simple and general formula for the generating function of the product of two polynomial sequences, based on lesser-known results. This approach is largely of a matricial nature. We describe how to derive and simplify recent products involving the k -Fibonacci, k -Pell, and Chebyshev polynomials. Several relevant references are provided.

An integral condition for the existence of at least n zeros for any solution of a second order linear equation is established.

In the present article, we examine the approximation properties of certain Gamma-type operators. Precisely, we consider the compositions of Rathore and Lupaş operators. We estimate the moments of the new composition operators and provide their probabilistic interpretation as well. Then, we provide some convergence results and their difference estimates, while focusing on the preservation of the moduli of smoothness of first- and second-order, using a stochastic approach. Additionally, Korovkin-type theorems and Voronovskaja-type asymptotic formulas are established for these operators. Finally, we consider graphical examples and their numerical interpretations to compare the speed of convergence for the composition operators and their components.

Criteria for a pair of measurable functions ( ω 1 , ω 2 ) {(\omega_{1},\omega_{2})} ensuring the boundedness of the k -th order commutators J G b , α , k {J_{G}^{b,\alpha,k}} ( k = 1 , 2 , … {k=1,2,\dots} ) of the G -fractional integral from one generalized G -Morrey space ℳ p , γ , ω 1 ⁢ ( ℝ + ) {\mathcal{M}_{p,\gamma,\omega_{1}}(\mathbb{R}_{+})} to another one ℳ q , γ , ω 2 ⁢ ( ℝ + ) {\mathcal{M}_{q,\gamma,\omega_{2}}(\mathbb{R}_{+})} (resp. from ℳ 1 , γ , ω 1 ⁢ ( ℝ + ) {\mathcal{M}_{1,\gamma,\omega_{1}}(\mathbb{R}_{+})} to the weak space W ⁢ ℳ q , γ , ω 2 ⁢ ( ℝ + ) {W\mathcal{M}_{q,\gamma,\omega_{2}}(\mathbb{R}_{+})} ) are established. In the case when b ∈ BMO G ⁢ ( ℝ + ) {b\in\mathrm{BMO}_{G}(\mathbb{R}_{+})} and 1 < p < q < ∞ {1<p<q<\infty} , necessary and sufficient conditions on the pair ( ω 1 , ω 2 ) {(\omega_{1},\omega_{2})} are found, ensuring the boundedness of the operators J G b , α , k {J_{G}^{b,\alpha,k}} from ℳ p , γ , ω 1 ⁢ ( ℝ + ) {\mathcal{M}_{p,\gamma,\omega_{1}}(\mathbb{R}_{+})} to ℳ q , γ , ω 2 ⁢ ( ℝ + ) {\mathcal{M}_{q,\gamma,\omega_{2}}(\mathbb{R}_{+})} with 1 p - 1 q = α γ {\frac{1}{p}-\frac{1}{q}=\frac{\alpha}{\gamma}} . Necessary and sufficient conditions on the function ω are also found to guarantee the boundedness of the operator J G b , α {J_{G}^{b,\alpha}} from ℳ p , γ , ω ⁢ ( ℝ + ) {\mathcal{M}_{p,\gamma,\omega}(\mathbb{R}_{+})} to ℳ q , γ , ω p q ⁢ ( ℝ + ) {\mathcal{M}_{q,\gamma,\omega^{\frac{p}{q}}}(\mathbb{R}_{+})} provided that 1 < p < q < ∞ {1<p<q<\infty} .

We consider the Emden–Fowler differential equation with variable exponents and establish necessary, sufficient and both necessary and sufficient conditions for the existence of nonoscillatory solutions of this equation with three different patterns of asymptotic behavior at infinity. The Schauder–Tikhonov fixed point theorem and duality principle are used for this purpose.

In this paper, we investigate the problem of when a multiplicative Jordan n -derivation is additive in the case of alternative rings. More precisely, let ℛ {\mathscr{R}} be an alternative ring containing a non-trivial idempotent d 1 {d_{1}} and ℛ = ℛ 11 + ℛ 12 + ℛ 21 + ℛ 22 {\mathscr{R}=\mathscr{R}_{11}+\mathscr{R}_{12}+\mathscr{R}_{21}+\mathscr{R}_{2% 2}} , the Pierce decomposition of ℛ {\mathscr{R}} relative to d 1 {d_{1}} satisfies: (i) if ( d i ⁢ u ⁢ d j ) ⁢ z j ⁢ k = 0 {(d_{i}ud_{j})z_{jk}=0} for all z j ⁢ k ∈ ℛ j ⁢ k {z_{jk}\in\mathscr{R}_{jk}} , then ( d i ⁢ u ⁢ d j ) = 0 {(d_{i}ud_{j})=0} , (ii) if z i ⁢ j ⁢ ( d j ⁢ u ⁢ d k ) = 0 {z_{ij}(d_{j}ud_{k})=0} for all z i ⁢ j ∈ ℛ i ⁢ j {z_{ij}\in\mathscr{R}_{ij}} , then ( d j ⁢ u ⁢ d k ) = 0 {(d_{j}ud_{k})=0} , (iii) if ( d i ⁢ u ⁢ d i ) ⁢ z i ⁢ i + z i ⁢ i ⁢ ( d i ⁢ u ⁢ d i ) = 0 {(d_{i}ud_{i})z_{ii}+z_{ii}(d_{i}ud_{i})=0} for all z i ⁢ i ∈ ℛ i ⁢ i {z_{ii}\in\mathscr{R}_{ii}} , then ( d i ⁢ u ⁢ d i ) = 0 {(d_{i}ud_{i})=0} for i , j , k = 1 , 2 {i,j,k=1,2} . Further, if ξ is a mapping from ℛ {\mathscr{R}} into itself such that ξ is a multiplicative Jordan n -derivation, then ξ is additive. As a consequence of our main theorem, some known results can be either generalized or deduced. (cf. [8] and [9]).

Let R and S be rings and let C S R {{}_{R}C_{S}} be a semidualizing bimodule. We investigate the behavior of the C -flat dimension of injective left R -modules, and establish the relation among the supremum of these C -flat dimensions and the supremum of C -injective dimensions of projective left S -modules and the supremum of C -projective dimensions of injective left R -modules.

The paper deals with the problem of calculating Sylvester’s denumerant D ⁢ ( b ; a 1 , a 2 , … , a n ) {D(b;a_{1},a_{2},\dots,a_{n})} , i.e., the number D ⁢ ( b ) {D(b)} of integer non-negative solutions of an equation ∑ i = 1 n a i ⁢ x i = b {\sum_{i=1}^{n}a_{i}x_{i}=b} , where n ≥ 2 {n\geq 2} , a 1 , a 2 , … , a n {a_{1},a_{2},\dots,a_{n}} are natural numbers and b is a non-negative integer. Unlike traditional methods of finding denumerants, in our study we employ only elementary methods, mainly the properties of the Kronecker function and the elements of combinatorics. The formula is derived to express D ⁢ ( b ) {D(b)} , for an integer non-negative b , coprime a 1 , a 2 , … , a n {a_{1},a_{2},\dots,a_{n}} and s ≠ 0 {s\neq 0} , via D ⁢ ( r ) , D ⁢ ( r + M ) , … , D ⁢ ( r + ( s - 1 ) ⁢ M ) D(r),D(r+M),\dots,D(r+(s-1)M) , where s = [ n - ∑ i = 1 n a i + r M ] {s=[n-\frac{\sum_{i=1}^{n}a_{i}+r}{M}]} , M is the least common multiple of the numbers a 1 , a 2 , … , a n {a_{1},a_{2},\dots,a_{n}} , and r is the remainder of b modulo M . Note that s takes quite small values in some particular cases. Moreover, the recurrent formulas are derived to calculate D ⁢ ( b ) {D(b)} for any non-negative integer b . These formulas, in particular, are used in finding D ⁢ ( r ) , D ⁢ ( r + M ) , … , D ⁢ ( r + ( s - 1 ) ⁢ M ) {D(r),D(r+M),\dots,D(r+(s-1)M)} . For the case where s = 0 {s=0} and a 1 , a 2 , … , a n {a_{1},a_{2},\dots,a_{n}} are coprime, the explicit formula D ⁢ ( b , a 1 , a 2 , … , a n ) = M n - 1 a 1 ⁢ a 2 ⁢ ⋯ ⁢ a n ⁢ ( b ′ + n - 1 n - 1 ) {D(b,a_{1},a_{2},\dots,a_{n})=\frac{M^{n-1}}{a_{1}a_{2}\cdots a_{n}}{\bigl{(}% \begin{smallmatrix}b^{\prime}+n-1\\ n-1\\ \end{smallmatrix}\bigr{)}}} is given (here b ′ = [ b M ] {b^{\prime}=[\frac{b}{M}]} ). To illustrate the proposed method, examples of finding the function D ⁢ ( b ) {D(b)} for linear Diophantine equations with 2 , 3 , 4 , 7 , 10 {2,3,4,7,10} and n variables are given.

The aim of this paper is to prove the boundedness of Marcinkiewicz integral operators on grand variable Herz–Hardy spaces using the atomic decomposition of grand variable Herz–Hardy spaces already established.

In this paper, we discuss many new characterizations of normal matrices by using the representations of the solutions to constructed equations.

In the present paper, we consider problems of convergence of a special series of moduli of the difference of the general Fourier coefficients of functions with derivatives of finite variation with respect to general orthonormal systems (ONS). It is shown that such series with respect to the Haar system or ( 2 ⁢ cos ⁡ 2 ⁢ π ⁢ n ⁢ x ) {(\sqrt{2}\cos 2\pi nx)} system are convergent for functions with derivatives of finite variation. Furthermore, it is proved that, with respect to general ONS, this series, in general, does not converge. Also, we find the optimal condition that needs to be imposed on the functions of the ONS so that the general orthonormal systems, possessing the properties similar to the Haar system, converge for functions with derivatives of finite variation.

In this paper, we first give a sufficient condition for precompactness in the matrix-weighted Lebesgue spaces with variable exponent by translation. Then we obtain a criterion for precompactness in the matrix-weighted Lebesgue spaces with variable exponent by average. Next, we give a criterion for precompactness in the matrix-weighted Lebesgue spaces with variable exponent by approximate identity. Finally, precompactness in the matrix-weighted Sobolev spaces with variable exponent is also considered.

Let R be a ring and let S be a multiplicative subset of R . In this note, we study the localization of S -injective modules and u - S -injective modules under S -Noetherian rings and u - S -Noetherian rings, respectively. The u - S -absolutely pure property is showed to be preserved under localizations over u - S -coherent rings. Besides, we give an example to show the difference between S -injective modules and u - S -injective modules, and some counter-examples to deny questions proposed in [J. Baeck, S -injective modules, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 118 2024, 1, Paper No. 20].

We correct a mistake in the proof of Proposition 2.2 and add explanatory sentences in the proof of Theorem 3.2 of our paper.