The aim of this paper is to establish various new inequalities for the operator norm and **numerical** **radius** of sums of bounded linear operators in Hilbert spaces. In particular, two refinements of the generalised triangle inequality for operator norm are obtained. Particular cases of interest for two bounded linear operators and their applications for the Cartesian decomposition of an operator are also considered.

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Since the relative matrix norms on E are given above, it is evident that these determine **numerical** **radius** operator spaces, which we denote by Min E and Max E, respectively. We refer to these **numerical** **radius** operator spaces as the minimal and the maximal quantization of E.

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For other results on **numerical** **radius** inequalities see [1], [3]-[7], [9]-[12], [14] and [18]-[23]. Let X be a linear space over the real or complex number field K and let us denote by H (X) the class of all positive semi-definite Hermitian forms on X, or, for simplicity, nonnegative forms on X, i.e., the mapping (·, ·) : X × X → K belongs to H (X ) if it satisfies the conditions

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The main aim of this paper is to establish other inequalities between the operator norm and its **numerical** **radius**. We employ, amongst others, the Buzano inequality as well as some results for vectors in inner product spaces due to Goldstein-Ryff- Clarke [9], Dragomir-S´ andor [7] and Dragomir [5].

Motivated by the natural question that arise in order to compare the quantity w (AB) with other expressions comprising the norm or the **numerical** **radius** of the involved operators A and B (or certain expressions constructed with these operators), we establish in this paper some natural inequalities of the form

Motivated by the natural connection that exists between the semi-inner products hA, Ii p,n , hA, Ii p,w with p ∈ {i, s} , the **numerical** **radius** w (A) and the operator norm kAk outlined above, the aim of this paper is to establish deeper relationships between these concepts. Amongst others, we show, in fact, that the semi-inner product hA, I i p,n is equal to hA, I i p,w for p ∈ {i, s} and as a consequence the nu- merical **radius** w (A) is bounded below by the maximum of the quantities

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Proof. We may assume that Γ {x α , x ∗ α } α and ϕ α x α 1 for each α, where each ϕ α is a strong peak function in AB X . Notice that if f ∈ AB X : X and vf 0, then vf 0 |x ∗ fx| for any x, x ∗ ∈ ΠX and f attains its **numerical** **radius**. Hence we have only to show that if f ∈ AB X : X, vf 1 and > 0, then there is f ∈ AB X : X such that f attains its

as a lower bound for the **numerical** **radius** v(a). Therefore, it is a natural question to ask how far these quantities are from each other under vari- ous assumptions for the element a in the unital normed algebra A and the scalar β. A number of results answering this question are incorporated in the following theorems.

Utilising the inequality (3.5) we observe that for any complex number located in the closed disc centered in 0 and with **radius** 1 we have jh a; 1 i s j as a lower bound for the **numerical** **radius** v (a) : Therefore, it is a natural question to ask how far these quantities are from each other under various assumptions for the element a in the unital normed algebra A and the scalar : A number of results answering this question are incorporated in the following theorems.

The motivation of this paper is to introduce the notions of **numerical** range and **numerical** **radius** without the inner product structure. In fact, the result extends immediately to the case where the Hilbert space H and inner product h·, ·i, replaced by vector space V and sesquilinear form ϕ, respectively. For the sake of completeness, we reproduce the following definitions and preliminary results, which will be needed in the sequel.

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In this section, we establish a general **numerical** **radius** inequality for Hilbert space operators which yields well known and new **numerical** **radius** inequalities as special cases. To prove our generalized inequality, we need the following basic lemmas. The first lemma is a generalized form of the mixed Schwarz inequality, which has been proved by Kittaneh 13.

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Motivated by the natural questions that arise, in order to compare the quantity wAB with other expressions comprising the norm or the **numerical** **radius** of the involved operators A and B or certain expressions constructed with these operators, we establish in this paper some natural inequalities of the form

6. Davidson, KR, Holbrook, JAR: **Numerical** radii of zero-one matrices. Mich. Math. J. 35, 261-267 (1988) 7. Müller, V: The **numerical** **radius** of a commuting product. Mich. Math. J. 39, 255-260 (1988) 8. Okubo, K, Ando, T: Operator radii of commuting products. Proc. Am. Math. Soc. 56, 203-210 (1976) 9. Dragomir, SS, Sándor, J: Some generalisations of Cauchy-Buniakowski-Schwartz’s inequality. Gaz. Mat. Metod.

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The main aim of this paper is to extend Kittaneh’s result to Euclidean **radius** of two operators and investigate other particular instances of interest. Related results connecting the Euclidean operator **radius**, the usual **numerical** **radius** of a composite operator and the operator norm are also provided.

Abstract. There are many criterion to generalize the concept of **numerical** **radius**; one of the most re- cent interesting generalization is what so called the generalized Euclidean operator **radius**. Simply, it is the **numerical** **radius** of multivariable operators. In this work, several new inequalities, refinements and generalizations are established for this kind of **numerical** **radius**.

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In this paper some generalizations of Buzano inequality for n-tuples of vectors in inner product spaces are given. Applications for norm and **numerical** **radius** inequalities for n-tuples of bounded linear operators and for functions of normal operators defined by power series with nonnegative coefficients are also provided.

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It should be noted that the MLS shape function and its derivatives are dependent on the weight function and the **radius** of influence domain. It’s also required that n$m in the domain of influence so that the matrix A (x) in Eq. 22 can be inverted (Reddy, 1999, 1997). Determine the MLS shape functions N j (x , y ) and its partial derivatives i i

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This paper considers the reflection and transmission characteristics of a Laguerre-Gaussian (LG) beam in a dielectric slab. The fields of the reflected and transmitted beams are described based on plane-wave angular spectrum representation. Using the generalized Fresnel amplitude reflectance and transmittance, the reflected and transmitted fields in each region are expressed. With the Taylor series approximation of reflectance and transmittance, the analytical expressions of the total reflected and transmitted fields in the input and output regions are derived. The effects of the beam-waist **radius** and topological charge on the reflected and transmitted field intensities are simulated and discussed in detail. The centroid shifts of the reflected beam are also presented. It is concluded that the distortion of the intensity distribution including the size of the intensity contour, is influenced by the beam-waist **radius** and the topological charge of the incident beam. The total intensity of the slab, in particular for the case of the transmitted field, is found to be distinguishable from the case of the single interface.

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deep and is surrounded by a bulge deformation. In contrast, for R = 12 mm, the indentation is shallow with barely any bulge deformation. For comparison, the experimental result is shown in Fig. 4. The material of the workpiece used in the experiment is aluminum alloy A5052-H112 (tensile strength: 246 MPa), and a pre-hardened steel (Daido’s special steel NAK55) of HRC 37 to 43 was used for the pin. The pin was dropped from a height of one meter at a free fall speed of 4.42 m·s ¹ 1 . The indentation shape was measured with two orthogonal cross-sections. The indentation of the experimental result is somewhat smaller than that of the analysis result; however, it demonstrates that the analysis can simulate the e ﬀ ect of the tip curvature **radius** R.

In general, the meshing mode for all types of spiral-bevel and hypoid gears is local conjugate contact (or called ‘point contact’). To receive the length crowning between the mating flanks of a bevel gearset, the length curvatures (the curvature is inverse to the **radius**) of the convex flanks have to be larger than the length curvature of the concave flanks, that is, a pair of tooth surfaces with meshing must have a curvature difference. In addition, the normal vector of the two mating flanks at the reference point should be the same, or the two mating flanks should have equal spiral angle and pressure angle at the reference point. The cutter point **radius** is a good measure for the tooth length curvature in the five-cut process. To obtain a suitable curvature for a pair of the mating tooth surfaces, an inside cutter point **radius** for the convex flank must be smaller than an outside cutter point **radius** for the concave flank. The blade angle in the five-cut process is very similar to the pressure angle of the tooth. If there are some small differences, then they are related to the basic setting adjustments which are done in order to fine-tune the rolling performance. With equal pressure angles, the inside blade (IB) **radius** at the blade reference point (not at the tip) has to be larger than the outside blade (OB) **radius** at the reference point, as shown in Fig. 1. In this case, the IB and OB for pinion obviously cannot be mounted on one spread blade head-cutter, which cannot be used in the duplex helical method.

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