Let N be a zero-symmetric left near-ring, not necessarily with a multiplicative identity element; and let Z be its multiplicative center. Define N to be 3-prime if for all a, b ∈ N\{0}, aNb / {0}; and call N 2-torsion-free if N, has no elements of order 2. A **derivation** on N is an additive endomorphism D of N such that Dxy xDy Dxy for all x, y ∈ N. A **generalized** **derivation** f with associated **derivation** D is an additive endomorphism f : N → N such that fxy fxy xDy for all x, y ∈ N. In the case of rings, **generalized** derivations have received significant attention in recent years.

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In [4], Bell and Kappe proved that if d R : → R is a **derivation** holds as a homomorphism or an anti-homomorphism on a nonzero right ideal of R which is a prime ring, then d = 0 . In [5], Rehman proved that if F R : → R is a non- zero **generalized** **derivation** with a nonzero **derivation** d R : → R where R is a 2-torsion free prime ring holds as a homomorphism or an anti homomorphism on a nonzero ideal of R , then R is commutative. In [6], Dhara proved some re- sults when a **generalized** **derivation** acting as a homomorphism or an an- ti-homomorphism of a semiprime ring. In [7], Shakir Ali showed that if

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Throughout the paper R will represent an associative ring with center Z(R). For any x,yϵR the symbol [x,y] will denote the commutator xy-yx; while the symbol xᵒy will stand for anti-commutator xy+yx. R is prime if aRb=0 implies a=0 or b=0. An additive map *:R→R is called an involution if * is an anti-automorphism of order 2; that is (x*)*=x for all xϵR. R is *-prime if aRb*=0 implies a=0 or b=0. An element x in a ring R with involution * is said to be hermitian and skew-hermitian elements of R will be denoted by H(R) and S(R), respectively. The involution is said to be of the firest kind if Z(R)⊆H(R), otherwise it is said to be of second kind. In the later case Z(R)⋂S(R)⧧(0). An additive mapping d:R→R is said to be a **derivation** if d(xy)=d(x)y+xd(y) for all x,y ϵR. An additive map F:R→R is a **generalized** **derivation** if their exists a **derivation** d such that F(xy)=F(x)y+xd(y) for all x,yϵR. All derivations are **generalized** derivations.

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Abstract. Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R → R is called a **generalized** **derivation** on R if there exists a **derivation** d : R → R such that F(xy) = F (x)y+xd(y) holds for all x, y ∈ R. In the present paper we describe the action of **generalized** derivations satisfying several conditions on Lie ideals of semiprime rings.

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Proof. We prove only (ii), since (i) is proved in [2]. For all 𝑥, 𝑦, 𝑧 ∈ 𝑁 we have 𝑓((𝑥𝑦)𝑧) = 𝑓(𝑥𝑦)𝑧 + 𝑥𝑦𝑑(𝑧) = (𝑑(𝑥)𝑦+𝑥𝑓(𝑦))𝑧+𝑥𝑦𝑑(𝑧) and 𝑓(𝑥(𝑦𝑧)) = 𝑑(𝑥)𝑦𝑧+𝑥𝑓(𝑦𝑧) = 𝑑(𝑥)𝑦𝑧 + 𝑥𝑓(𝑓(𝑦)𝑧 + 𝑦𝑑(𝑧)) = 𝑑(𝑥)𝑦𝑧 + 𝑥𝑓(𝑦)𝑧 + 𝑥𝑦𝑑(𝑧). Comparing the two expressions for 𝑓(𝑥𝑦𝑧) gives (ii). Lemma 6. Let 𝑁 be a 3-prime near ring and 𝑓 a **generalized** **derivation** with associated **derivation** 𝑑.

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, , % ∈ and , ∈ Γ. If -ring satisfies the assumption (B) = , for all , , ∈ and , ∈ Γ. Let be a -ring. An additive mapping ': → is called a **derivation** on if '(() = '()( + ('() holds for all , ∈ and ( ∈ Γ. An additive mapping : → is called a **generalized** **derivation** if there exists a **derivation** ': → such that (() = ()( + ('() holds for all , ∈ and ( ∈ Γ. An additive mapping : → is called a left **generalized** **derivation** if there exists a **derivation** ': → such that (() = (() + '()( holds for all , ∈ and ( ∈ Γ. A **derivation** of the form → + where , are fixed elements of and ∈ Γ is called **generalized** inner **derivation**. An additive mapping ): → is called a left (right) centralizer if )() = )() ()() = )()) for all , ∈ and ∈ Γ.

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An additive mapping D : A −→ A is called generalization **derivation** if there exists a **derivation** d : A −→ A such that D(xy) = D(x)y + xd(y) for all x, y ∈ A and we say D is a d-**derivation** [1]. These mappings studied by many authors on various rings and algebras (see [2, 7]). A new type of **generalized** **derivation** introduced by Nakajima in [6], that he called it **generalized** 2-cocycle **derivation**. An additive mapping δ : A −→ A is called **generalized** 2-cocycle **derivation** associate with 2-cocycle γ : A × A −→ A, if

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Theorem 3.7 Let ring R be prime with𝑐ℎ𝑎𝑟 𝑅 ≠ 2, (0) ≠ 𝐿 be a square closed Lie ideal of 𝑅 and 0 ≠ 𝐹, 𝐺: 𝑅 → 𝑅 are **generalized** 𝛼 − derivations associated with 𝛼 − derivations 𝑑 and 𝑔 respectively such that 𝑠 ∈ 𝑍(𝐿)|0 ≠ 𝑑 𝑠 , 0 ≠ 𝑔 𝑠 ∈ 𝑍 𝐿 , 𝑑 𝑠 ≠ ∓𝑔 𝑠 ≠ ∅ and 𝛼 𝐿 ⊂ 𝐿. If one of the following conditions is satisfy, then 𝐿 ⊆ 𝑍 𝑅 .

The tree in figure 1 represents the various **derivation** steps (insofar as they are carried out in the example above) from higher-level laws to an equation of the mass M. We will refer to such a tree as a **derivation** tree. A **derivation** tree is a labeled tree in which each node is annotated with a formula; the boxes are only meant as convenient representations of these labels. The formulas at the top of each “vee” (i.e. each pair of binary branches) in the tree can be viewed as premises, and the formula at the bottom of each “vee” can be viewed as a conclusion which is arrived at by simple term substitution. The last **derivation** step in the tree is not formed by a “vee” but consists in a unary branch which solves the directly preceding formula for a certain variable (in the tree above, for the mass M). Thus, in general, a unary branch refers to a mathematical **derivation** step that solves an equation for (a) certain variable(s), while a binary branch refers to a physical **derivation** step which introduces and combines physical laws or conditions (or other knowledge such as phenomenological corrections and coefficients).

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The non-probabilistic DOP model uses a rather different definition of the best parse tree. Instead of computing the most probable parse of a sentence, it computes the parse tree which can be generated by the fewest corpus-subtrees, i.e., by the shortest **derivation** independent of the subtree probabilities. Since subtrees are allowed to be of arbitrary size, the shortest **derivation** typically corresponds to the parse tree which consists of l a r g e s t possible corpus- subtrees, thus maximizing syntactic context. For example, given the corpus in Figure 1, the best parse tree for She saw the dress with the telescope is given in Figure 3, since that parse tree can be generated by a **derivation** of only two corpus-subtrees, while the parse tree in Figure 4 needs at least three corpus- subtrees to be generated. (Interestingly, the parse tree with the shortest **derivation** in Figure 3 is also the most probable parse tree according to probabilistic DOP for this corpus, but this need not always be so. As mentioned, the probabilistic DOP model has already a bias to assign higher probabilities to parse trees that can be generated by shorter derivations. The non-probabilistic DOP model makes this bias absolute.)

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Abstract. To implement the method of adiabatic waveguide modes for modeling the propagation of polarized monochromatic electromagnetic radiation in irregular integrated optics structures it is necessary to expand the desired solution in basic adiabatic waveg- uide modes. This expansion requires the use of the scalar product in the space of waveg- uide vector ﬁelds of integrated optics waveguide. This work solves the ﬁrst stage of this problem – the construction of the scalar product in the space of vector solutions of the eigenmode problem (classical and **generalized**) waveguide modes of an open pla- nar waveguide. In constructing the mentioned sesquilinear form, we used the Lorentz reciprocity principle of waveguide modes and tensor form of the Ostrogradsky-Gauss theorem.

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Biological ion channels are known to be highly selective and the permeation process involves many interactions. Yet the challenge of including the ion-ion, ion-water and ion-channel interactions in a multi-species non-equilibrium scenario has re- mained a long standing and fundamental theoretical problem, as demonstrated by the multi-decade discussion of the famous paradox of selectivity vs. conductivity. The primary focus of the thesis, therefore, was to derive the theory of multi- species ion conduction through narrow biological channels, taking into account ion-ion, ion-water and ion-channel interactions. The approach taken is based on the rst principles **derivation** of statistical and kinetic theory. The process of **derivation** has led to new results that describe multi-species conduction in and far from equilibrium in KcsA. It is anticipated that these results will be applicable for other narrow voltage-gated ion channels, that they can be used to investigate mixed-valence selectivity and that they can describe multi-species conduction of neutral particles through zeolites.

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Barnes [3], Luh [13], Kyuno [11], Hoque and Paul [8] as well as Uddin and Islam [16,17] studied the structure of Г-rings and obtained various generalizations of corresponding parts in ring theory. Note that during the last few decades, many authors have studied derivations in the context of prime and semiprime rings and Γ-rings with involution [1,2,4,10,18].The notion of **derivation** pair and Jordan **derivation** pair on a *- ring R were defined by [12, 14, 19,20].

S: John hopes that Peter is liked by Mary. To control the syntax and the lexicon of the ex- ercices produced, we take a grammar based ap- proach and make use of generation techniques. More specifically, we generate sentences using a Feature-Based Lexicalised Tree Adjoining Gram- mar (FB-LTAG) for French (SemTAG). We show that the rich linguistic information associated with sentences by the generation process naturally sup- ports the identification of sentence pairs related by a syntactic transformation. In particular, we argue that the **derivation** trees of the FB-LTAG gram- mar provide a level of representation that captures both the formal and the content constraints gov- erning transformations. The content words and the grammatical functions labelling the tree nodes permit checking that the two sentences stand in the appropriate semantic relation (i.e., fully iden- tical content or identical content modulo some lo- cal change). Further, the syntactic properties la- belling these nodes (names of FB-LTAG elemen- tary tree names but also some additional informa- tion provided by our generator) permits ensuring that they stand in the appropriate syntactic rela- tion.

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In this paper, a thermodynamic analysis is presented of the earth’s atmos- phere. This analysis consists of three parts: 1) **Derivation** of excess entropy pro- duced in the atmosphere due to increased temperature; 2) **Derivation** of an equ- ation expressing entropy production (in the atmosphere) based on atmospheric processes (wind, precipitation, lightning, chemical reactions and heat transfer); and 3) Proof that there must be an increase in a combination of these atmos- pheric processes due to increased temperature.

Ternary algebraic operations were considered in the XIX-th century by several mathematicians such that as A. Cayley [6] who first introduced in 1840 the notion of ”cubic matrices” and a generalization of the determinant, called the ”hyperde- terminant”, then were found again and **generalized** by M. Kapranov, I. M. Gelfand and A. Zelevinskii in 1990 [19].

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In the literature on multivariate statistical theory, e.g., [2] and [15], we can find the mathe- matical **derivation** showing that the sample covariance determinant or **generalized** variance converges in distribution to a normal distribution. However, this classical result is not suit- able for practical purposes because the distribution parameters are the limits of the of the true parameters when the sample size tends to infinity and in addition, the convergence to this limit is slow. On the other hand, in the literature on multivariate statistical applica- tions, e.g., Alt and Smith [1] and Montgomery [14], the true parameters are considered but not the distribution. In the construction of control region, for example, Montgomery [14] points out that “most of the probability distribution of sample covariance determinant is contained in the interval ± 3 times the standard deviation of sample covariance determinant from its mean” without specifying the distribution. Such a statement is ambiguous and needs formal explanation. This is the main topic of this paper.

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The concept of a symmetric bi-**derivation** has been introduced by Maksa in [6]. In [9], Vukman has proved some results concerning symmetric bi-**derivation** on prime and semiprime rings. Yenigul and Argac [10] studied ideals and symmetric bi-derivations of prime and semiprime rings. Reddy et al. [5] studied symmetric reverse bi-derivations on prime rings. Sapanci et al. [8] studied few results of symmetric bi-**derivation** on prime rings. In this paper, we extended some results of symmetric reverse bi-derivations on prime rings.

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Let N be a near –ring and is a mapping on N . This paper consists of two sections . In section one , we recall some basic definitions and other concepts , which be used in our paper , we explain these concepts by examples and remarks . In section two , we introduce the notion of left **generalized** ( , )-4-**derivation** in near-ring N and we determine some conditions of left **generalized** ( , )-4-**derivation** and semigroup ideals which make prime near-ring commutative ring .

Using **generalized** special relativity a useful expression of the perturbed momentum is found. This expression is used to describe the behavior of the quantum system in the interaction picture. The spatial evolution of the Schrodinger Equation in the interaction picture is similar to that of time evaluation, where the time differential is replaced by the space one and the Hamiltonian by the momentum operator. The same holds for the unitary operator, where the time integral is replaced by the space one and the Hamiltonian with the momentum operator.

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