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Let R be a prime ring of char R = 0 or char R > 2 n and let d : R → Q m s (R) be an additive map such that d (x 2 n) = d (x n) x n + x n d (x n) for all x ∈ R , where n is a positive integer and Q m s (R) is the maximal symmetric ring of quotients of R. It is shown that there exist a derivation δ : R → Q m s (R) and an additive map μ : R → Q m s (R) with μ (x n) = 0 for all x ∈ R , such that d = δ + μ. This result is a natural generalization of the classic theorem of Herstein for Jordan derivations on prime rings. Moreover, it gives a purely algebraic version of the theorem recently obtained by Kosi-Ulbl, Rodriguez and Vukman for standard operator algebras on Banach spaces. [ABSTRACT FROM AUTHOR]

Let R be a (semi-) prime ring with extended centroid C, let f(X1, ... , Xk) be a multilinear polynomial over C in k noncommutative indeterminates which is not central-valued on R and let g be a generalized derivation of R. In this paper, we completely characterize the form of g and the structure of R such that (g(f(x1, ... , xk))m − γf(x1, ... , xk)n)s = 0 for all x1, ... , xk ∈ R, where γ ∈ C and m, n, s are fixed positive integers. Our results naturally improve and generalize the theorems obtained by Huang and Davvaz in [Generalized derivations of rings and Banach algebras, Comm. Algebra (2013); 43, 1188–1194] and the theorems recently obtained by De Filippis et al. in [Generalized derivations with nilpotent, power-central and invertible values in prime and semiprime rings, Comm. Algebra (2019); 47, 3025–3039]. Moreover, we describe a revised version of the theorem obtained by Huang in [On generalized derivations of prime and semiprime rings, Taiwanese J. Math. (2012); 16, 771–776.] [ABSTRACT FROM AUTHOR]

Smart windows are crucial to dynamic control over light transmission to fulfill various demands in energy saving, privacy, and information display; however, most present technologies still perform a single function (often tint or haze adjustment) and require continuous electricity for operation. In this study, novel self‐assembled ionic liquid crystals (ILCs) doped with negative cholesteric liquid crystals (CLCs) to offer electrically switchable and stable scattering‐mode light modulators are presented. The novel smectic A phase based on the ILCs exhibits high solubility in the adopted nematics, enhancing the LC device's performance in several ways, including improved homogeneity, stable alignment quality, prolonged stability, and simplified fabrication. The LC device can potentially offer a dynamically rapid switching function between stable transparent (imperfect fingerprint textures) states and stable scattering (focal conic textures with small domains) states by using external stimuli and highly maintained multistable states for prolonged periods, even when the external stimuli are removed. The LC device also offers polarization‐independent scattering and transparent‐mode LC light modulators, low operating voltage, excellent contrast, and broad viewing angles. Its versatility and outstanding field‐off stability make it ideal for various applications such as smart lighting, building climate control, energy‐saving displays, and augmented reality (AR) glasses. [ABSTRACT FROM AUTHOR]

Let R be a prime ring with center Z(R), let L be a nonzero left ideal of R and let g,h be two generalized derivations of R. In this paper, we completely characterize the structure of L and all possible forms of g,h such that g(xm)xn − xnh(xm) ∈ Z(R) for all x ∈ L, where m,n are fixed positive integers. With this, several known results can be either deduced or generalized. Moreover, our result can be regarded as the one-sided ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317]. [ABSTRACT FROM AUTHOR]

Let R be a prime ring, let L be a noncentral Lie ideal of R and let g , h be two generalized derivations of R. In this paper, we characterize the structure of R and all possible forms of g and h such that [ g (x m) x n − x s h (x t) , x r ] k = 0 for all x ∈ L , where m , n , s , t , r , k are fixed positive integers. With this, several known results can be either deduced or generalized. In particular, we give a Lie ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317] and describe a more complete version of the theorem recently obtained by Dhara and De Filippis in [Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48 (2020) 154–167]. [ABSTRACT FROM AUTHOR]

We demonstrate polarization-independent holographic gratings (PIHGs) based on azo-dye-doped polymer-dispersed liquid crystal films. The PIHGs are recorded by irradiation under an intensity-modulated interference field, generated by two linearly polarized green optical beams, and by simultaneously applying a suitable AC voltage to the sample. The photoexcited azo dyes are adsorbed onto the UV-cured polymer film with their long axes parallel to the normal of the substrate. When the applied voltage is switched off, the PIHGs are generated with periodic modulation of liquid-crystal structures with transparently homeotropic and randomly scattered alignments. Additionally, the generated PIHGs can be completely switched off by an applied voltage. OCIS codes: 050.1950, 160.3710, 160.5470.