3-point functions of scalar and vector double-trace operators with approximate dimensions 4 and 5 respectively. We also find that the conformal dimensions of certain towers of double-trace operators in the 105, 84 and 175 irreps are non-renormalized. We show that, despite the absence of a non-renormalization theorem for the double-trace **operator** in the 20 irrep, its anomalous dimension vanishes. As by-products of our investigation, we derive explicit expressions for the conformal block of the stress tensor, and for the conformal partial wave amplitudes of a conserved current and of a stress tensor in d dimensions.

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QCD sum rules [1] are relations between features of hadrons—the bound states governed by the strong interactions—and the parameters of their underlying quantum ﬁeld theory—QCD. Such relations may be established (rather straightforwardly) by analyzing vacuum expectation values of nonlocal products of interpolating operators—speciﬁcally, of appropriate quark currents—at both QCD and hadron level. Upon application of Wilson’s **operator** **product** **expansion** (OPE) for casting, at QCD level, any arising nonlocal **operator** **product** into the form of a series of local operators, contributions of both perturbative as well as nonperturbative (NP) origin enter: the former are usually represented by dispersion integrals of certain spectral densities while the latter—also called the “power” corrections—involve the vacuum expectation values of all local OPE operators—crucial quantities going, in this context, under the name of “vacuum condensates.” Then, performing a Borel transformation from one’s momentum variable to a new variable, the Borel parameter τ, lessens the importance of hadronic excited and continuum states for such “Borelized” sum rules and removes potential subtraction terms. Our lack of knowledge about higher states is dealt with by postulating quark–hadron duality: all contributions of hadron excited and continuum states roughly cancel against those of perturbative QCD above an eﬀective threshold s e ﬀ (τ).

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In the present paper the advection of the passive scalar tracer ﬁeld by the Navier-Stokes velocity ensemble has been examined. The ﬂuid was assumed to be compressible and the space dimension was close to d = 4. The problem has been investigated by means of renormalization group and **operator** **product** **expansion**; the double **expansion** in y [see (4)] and ε = 4 − d was constructed. The present study has been aimed at the investigation of the anomalous scaling in the equal-time structure functions for the tracer ﬁeld.

In this section we give a brief introduction to the notion of p-**operator** spaces, defined by Daws in [5] and studied in [1]. Let n ∈ N , p ∈ (1, ∞), and let E be a vector space. We denote the vector space of n × m matrices with entries from E by M n,m (E). We put M n,m := M n,m ( C ).

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The elliptic theory of differential operators and pseudodifferential calculus operate perfectly in noncommutative settings as developed in [12], and analyzed in further detail for noncommutative tori in [41, 42, 67]. Moreover, there is a crucial need for local index formulas for twisted spectral triples as we explained in §2.3. Therefore, in the present paper we take the heat **expansion** approach to find a local formula for the index of the Dirac **operator** of the twisted spectral triple described in §2.3 with coefficients (or twisted by) an auxiliary finitely generated projective module on the noncommutative two torus, playing the role of a general vector bundle, cf. [56, 63]. We follow closely the setup provided in [21, 60], and the work has intimate connections with [49].

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In Table 2 we report the calculated 0νββ NME ob- tained using an e ff ective **operator** derived by second order MBPT and taking into account all the intermediate states up to an excitation energy of 14 ω, which are enough to provide convergent NME values. To clarify the role of Pauli blocking diagrams we have derived two di ff erent op- erators with and without taking into account diagrams (a) and (b) of Fig. 2.

Abstract. In this paper, the concept of a family of local atoms in a 2-inner **product** space is introduced and then this concept is generalized to an atomic system for an **operator**. Next a characterization of atomic systems is proved. This characterization lead us to obtain a new frame which is a generalization of frames in 2-inner **product** spaces.

The continuous drive for more is the major impetus for **product** development and industrial **expansion**. As long as there is a need the trend for more will accelerate. Creating sales frequency has its effect on **product** quality and availability. Because in order to induce sales products and services must be both, available and a fordable. A perpetual repetitive motion without the prospect of progress is unsustainable.

In the present paper we use a fractional differential **operator** and **product** of general class of multivariable polynomials, Riemann Zeta function and multivariable Gimel-function. On account of the kernel, due to the general class of multivariable polynomials and multivariable Gimel- function our findings provide interesting unification and extension of a number of results. Some new special cases of our main result are mentioned briefly.

In this article, we would like to give some brief introduction to Operator Production Expansion and Factorization, and Section 3 gives some applications to deduce the deca[r]

Since the weighted Bergman-Nevanlinna space is a Fréchet space and not a Banach space, it is necessary to introduce several deﬁnitions needed in this paper. Let X and Y be topological vector spaces whose topologies are given by translation invariant metrics d X and d Y , respectively, and let L : X → Y be a linear **operator**. It is said that L is metrically

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Esmaeili and Lindström in [] investigated weighted composition operators between Zygmund-type spaces. Ramos Fernández in [] studied the boundedness and compact- ness of composition operators on Bloch-Orlicz spaces. Li and Stević in [] investigated products of Volterra-type **operator** and composition **operator** from H ∞ and Bloch spaces to Zygmund spaces, and they in [] studied products of composition and diﬀerentiation operators from Zygmund spaces to Bloch spaces and Bers spaces. Liu and Yu in [] char- acterized the boundedness and compactness of products of composition, multiplication and radial derivative operators from logarithmic Bloch spaces to weighted-type spaces on the unit ball. Sharma in [] studied the boundedness and compactness of products of composition multiplication and diﬀerentiation between Bergman and Bloch-type spaces. In [], Stević investigated the properties of weighted diﬀerentiation composition oper- ators from mixed-norm spaces to weighted-type spaces. Stević in [] studied weighted radial operators from the mixed-norm space to the nth weighted-type space on the unit ball. Stević et al. in [] characterized the boundedness and compactness of products of multiplication composition and diﬀerentiation operators on weighted Bergman spaces. Zhu in [] studied extended Cesàro operators from mixed-norm spaces to Zygmund- type spaces.

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The main result in [18] could fill up the gap in Theorem 1.1. In this paper, we shall give a new result about the reverse order law for {1, 2, 3}- and {1, 2, 4}-reverses by the relationship of the range conclusion. In section 2, we shall give some preliminaries. Some necessary and sufficient conditions for an **operator** G ∈ B ( K , H ) to be in A{1, 2, 3} and A{1, 2, 4} are pointed. In section 3, we will derive a new sufficient and necessary conditions for B{1, 2, i}A{1, 2, i} ⊆ (AB){1, 2, i}(i ∈ {3,4}) respectively, when R(A), R(B), R(AB) are closed. And also our result will fill up the gap in Theorem 1.1.

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1. Introduction. Had we the uniform estimates of Grauert and Lieb for the Cauchy- Riemann equation [5] on polycylinders, we would have improved Weinstock’s result [6, Theorem 1.1] on polycylinders a long time ago. On the other hand, we have had several estimates for the ∂-**operator** in Sobolev spaces [2, 3, 4] on polycylinders, which we have been trying to improve. At the same time, we noticed that because some Sobolev norms dominate the uniform norm, if we did the approximation in those Sobolev norms, we would get the desired improvement of the above-mentioned theorem of Weinstock on polycylinders. We therefore went ahead here and jazzed up our previous estimates to get new estimates. We also applied our results to solve the Sobolev-Corona problem.

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The findings of this study showed that steel mill workers are exposed to heat stress in the poker furnace **operator**, lift **operator**, ruffing **operator**, wrench **operator**, rolling work **operator**, scissors **operator**, and lathe **operator** units. Thus, heat conservation planning and interventions should be conducted to reduce exposure to heat stress. Measures such as reducing the exposure time, using cooling vests and mechanical ventilation systems, providing workers with cool water, designing a radiation absorber around the heat sources, creating sufficient space between heat source and person, and enclosing individuals must be taken.

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We let η denote the category of normed spaces, in which the objects are the normed spaces and the morphisms are the bounded linear mappings. Similarly, we let D be the category of numerical radius **operator** spaces with the mor- phisms being the -completely bounded mappings. We have a natural “for- getful” functor N : D → η which maps a numerical radius into its underlying normed space. We say that a functor Q : η → D is a strict quantization if for each normed space E, N Q E ( ) = E , and for each bounded linear mapping of normed space ϕ : E → F , the corresponding mapping Q ( ) ( ) ϕ : Q E → Q F ( ) satisfies ( Q ( ) ϕ ) cb = ( ) ϕ .

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Abstract. In our paper, we study a class W R (λ, β, α, µ, θ ), which consists of analytic and univalent functions with negative coefficients in the open unit disk U = {z ∈ C : |z| < 1} defined by Hadamard **product** (or convolution) with Rafid - **Operator**, we obtain coefficient bounds and extreme points for this class. Also distortion theorem using fractional calculus techniques and some results for this class are obtained.

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connected with the Riemann-Liouville operator, we give a new characterization of the dual space M 𝑝 ′ ( ℝ 2 ) and we describe its bounded subsets. Submitted September, 2009.. In particul[r]

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At ﬁrst, the experts studied **operator**-theoretic properties of these operators on spaces of holomorphic functions in terms of their symbols separately. A systematic study of prod- ucts of concrete linear operators between spaces of holomorphic functions started approx- imately a decade ago; see, for example, [–], and the related references therein. The ﬁrst **product**-type operators diﬀerent from weighted composition operators, which have been considerably studied, are the products of composition and diﬀerentiation operators (see, e.g., [, , , , , , , , , ] and the references therein). Quite recently, there appeared some more complex **product**-type operators that include some of classical oper- ators, such as composition, diﬀerentiation, multiplication, or integral-type operators (see, e.g., [–, , , –] and the related references therein).

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The Pommiez **operator** ( ∆ f )(z) = ( f (z) − f (0))/z is considered in the space Ᏼ(G) of the holomorphic functions in an arbitrary finite Runge domain G. A new proof of a repre- sentation formula of Linchuk of the commutant of ∆ in Ᏼ(G) is given. The main result is a representation formula of the commutant of the Pommiez **operator** in an arbitrary invariant hyperplane of Ᏼ(G). It uses an explicit convolution **product** for an arbitrary right inverse **operator** of ∆ or of a perturbation ∆ − λI of it. A relation between these two types of commutants is found.

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