A new method for constructing Cliﬀord algebra-valued orthogonal polynomials in the open **unit** **ball** of Euclidean space is presented. In earlier research, we only dealt with scalar-valued weight functions. Now the class of weight functions involved is enlarged to encompass Clif- ford algebra-valued functions. The method consists in transforming the orthogonality re- lation on the open **unit** **ball** into an orthogonality relation on the real axis by means of the so-called Cliﬀord-Heaviside functions. Consequently, appropriate orthogonal polynomi- als on the real axis give rise to Cliﬀord algebra-valued orthogonal polynomials in the **unit** **ball**. Three speciﬁc examples of such orthogonal polynomials in the **unit** **ball** are discussed, namely, the generalized Cliﬀord-Jacobi polynomials, the generalized Cliﬀord-Gegenbauer polynomials, and the shifted Cliﬀord-Jacobi polynomials.

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Through this paper, B is the **unit** **ball** of the n-dimensional complex Euclidean space C n , S is the boundary of B . We denote the class of all holomorphic functions, with the compact-open topology on the **unit** **ball** B by H( B ). For any z = (z 1 , z 2 , . . . , z n ), w = (w 1 , w 2 , . . . , w n ) ∈ C n , the inner product is defined by hz, wi = z 1 w 1 + . . . + z n w n , and write |z| =

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∆ = , where Q x ( ) is an arbitrary polynomial are constructed. On this base in [9] polynomial solution of the Dirichlet boundary value problem and a generalized third boundary value problem for the Poisson's equation are constructed. In [10] the Dirichlet boundary value problem for the biharmonic equation in the **unit** **ball** Ω is considered. In [16] solvability conditions of the Neumann boundary value problem for the biharmonic equation in the **unit** **ball** are given and its polynomial solution is constructed. The present paper is a continuation of those investigations for the 3 -harmonic equation ∆ 3 u x ( ) = Q x ( ) in the **unit** **ball** Ω . It hopefully allows to solve the Dirichlet problem for the polyharmonic equation in Ω . It is necessary to mention the papers [12-13] devoted to this research area.

If ϕ has a fixed point inside the **ball**, Theorem 2.5 gives the spectrum. Therefore, we compute the spectrum when ϕ has no fixed point inside the **unit** **ball**. We will denote the composition of ϕ with itself n times by ϕ n , that is, ϕ n = ϕ ◦ ϕ ◦ ··· ◦ ϕ (n times). Now, we give the last theorem of this paper.

(4.4) when r < | z | < 1 and | ϕ(z) | ≤ δ. From (3.1), (4.3), and (4.4), the result follows. Theorem 4.2. Suppose that ϕ is an analytic self-map of the **unit** **ball**, u ∈ H(B), 0 < p, q < ∞ , φ is normal on [0, 1), and uC ϕ : H(p,q,φ) → H α ∞ is bounded. Then uC ϕ : H(p,q,φ) → H α,0 ∞

[3] I. C. Gohberg and M. G. Kre˘ın, Introduction to the Theory of Linear Nonselfadjoint Oper- ators, translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, vol. 18, American Mathematical Society, Rhode Island, 1969. [4] R. Grza´ slewicz, Exposed points of the **unit** **ball** of ᏸ(H), Math. Z. 193 (1986), no. 4, 595–596. [5] J. R. Holub, On the metric geometry of ideals of operators on Hilbert space, Math. Ann. 201

The space Fp, q, s, introduced by Zhao in 1, is known as the general family of function spaces. For appropriate parameter values p,q, and s, Fp, q, s coincides with several classical function spaces. For instance, let D be the **unit** disk in C, Fp, q, s B q2/p if 1 < s < ∞ see 2, where B α , 0 < α < ∞, consists of those analytic functions f in D for

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As is well known, in any infinite-dimensional Banach space one may find fixed point free self-maps of the **unit** **ball**, retractions of the **unit** **ball** onto its boundary, contractions of the **unit** sphere, and nonzero maps without positive eigenvalues and normalized eigen- vectors. In this paper, we give upper and lower estimates, or even explicit formulas, for the minimal Lipschitz constant and measure of noncompactness of such maps.

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In the next section, we will prove that there is an equivalence between the kernel convergence and the convergence on compact sets of biholomorphic mappings on the **unit** **ball** B which satisfy the growth result (1.1). In the last section, we will obtain some consequences of this result in the case of the ker- nel convergence and the convergence on compact sets of normalized starlike and normalized convex mappings on B. Also, we will prove that there is an equivalence between the notions of a Loewner chain, which satisﬁes a certain normality condition, and kernel convergence.

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Several important non-linear optimization problems were investigated with very successful out- come by studying properties of subspaces whose **unit** **ball** is a proximinal or densely **ball** remotal set [16-18]. It turns out that Banach spaces whose dual is isometric to L 1 (µ) for some positive measure µ is a rich domain to solve various deep questions arising here. In such spaces any M -deal is **ball** proximinal and they are also densely **ball** remotal in several natural situations. These questions are more amenable when the subspace under consideration is of finite codimension. A deep analysis in terms of norm attaining functionals led to the result that if the bi-annihilator of a subspace of finite codimension is strongly **ball** proximinal only at the points of the given space, then it is strongly **ball** proximinal [66].

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The asymmetric **unit** of the title compound consists of two crystallographic independent molecules (Fig. 1) with nearly identical geometrical parameters (Tab. 1). These values correspond very well with those of phenyl phosphonic acid that was determined by Mahmoudkhani et al. (2002) at T = 183 (2) K [d(P—O) = 1.536 (2) Å, 1.555 (2) Å; d(P=O) = 1.506 (2) Å and d(P—C) = 1.782 (3) Å] and Weakley (1976) at ambient temperature [d(P—O) = 1.539 (3) Å, 1.550 (4) Å; d(P=O) = 1.496 (4) Å and d(P—C) = 1.773 (5) Å]. With respect to bond angles at phosphorous, tetrahedral

In this paper, we unambiguously show that “sub-atomic like” features can arise from the back bonding configuration of a surface atom being imaged during DFM. This is done by utilising the change in bonding configuration of the surface adatoms of the Si(111) **unit** cell between the faulted and unfaulted half. Due to this change in symmetry across the **unit** cell, the features we observe cannot be assigned to any tip, or feedback artefacts. At the same time, they suggest caution should be used when interpreting “sub-atomic like” features, as our data cannot be interpreted as arising from within a single atom.

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The use of a generic movement or skill test for the purpose of identifying talent is becoming more prevalent in the literature, due to minimising the effect of sport-specific training on results (Faber et al., 2014; Vaeyens et al., 2008). Furthermore, motor skills are relatively stable beyond the age of 6 (Vandorpe, Vandendriessche, Vaeyens, Pion, Matthys, et al., 2012). In gymnastics, the KTK was a better predictor of future performance than sprint, agility and even coach’s ranking (Vandorpe, Vandendriessche, Vaeyens, Pion, Lefevre, et al., 2012). The KTK is a test of locomotive stability (Cools et al., 2008) and therefore clearly has some relevance to gymnastics whilst not being sport specific. A study examining a similar theory has been conducted in the field of table tennis (Faber et al., 2014). In this study participants (aged 7-12 years) were required to throw a **ball** against a wall with one hand and catch with the opposite hand as many times as possible in 30 seconds. The task was designed to be similar to table tennis but not practiced in training. This test was able to discriminate between local, regional and national players. The TRT and CRT operate in a similar vein, in that, the skills and movement pattern required for success maintain a relationship to tennis whilst not being sport specific.

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curve and having the same **unit** normal vector fields along that curve must be the same (see, for example, [5], Chapter 3, Section 4). On the final steps of our proof, we have shown that, possibly after a rotation, the free boundary minimal surface Σ contains the great circle C = {(x, y, 0); x 2 + y 2 = R} and its normal **unit** vector field along C is

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wavelet. These operators are constructed using the procedure proposed in Items 1, 2, and 3 in Section VI, respectively. For the wavelet case, we construct a 3072 ˆ 1024 Daubechies wavelet transform where we only retain its high-pass coefficients (of size 2048 ˆ 1). The number of decomposition levels in the wavelet transformation is two. Except for Random 1*, we construct cosparse signals according to the procedure explained in Section VI. For Random 1* we use x “ P nullpΩ S q c, where c is uniformly distributed on the **unit** sphere S n´1 . In this table, Num E denotes the numerator of E p p and is equal to 2 sup sPBfpxq }s} 2 .

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However, when analyzing precision bearings operat- ing in favourable conditions, we can neglect some of these sources and save only the two first ones. Balmont and Zverev consider the theory of bearing having these inperfections in [2]. Some of the results obtained are presented in Table 1, where the spectrum of radial vi- brodisturbances generated by imperfect **ball** bearing is presented.

The adjustable **unit** uses a **ball**-in-cone valve system. The tension of the spring holding the **ball** in place can be adjusted by turning the rotor (torsion bar) using the exter- nal magnetic adjustment tool, thus changing the operat- ing pressure. The valve has a brake system that holds the rotor in place to prevent unwanted re-adjustment when the shunt is exposed to an external magnetic field. To release the brake, a downward force (800 to 1600 gram- force) is applied to the **unit** using the adjustment tool (Figure 1). The valve has a diameter of 18 mm. It has a rel- atively large internal volume compared to other models, which is intended to minimize the risk of obstruction. The gravitational **unit** (shunt assistant) increases the opening pressure of the shunt when the patient is vertical by blocking the inlet flow using a gravity-assisted **ball** bearing. If the patient is horizontal, the **ball** no longer blocks the opening and the **unit** provides substantially less resistance. When in the vertical position, the opening pressure of the ProGAV is the sum of the opening pressure of the adjustable **unit** and the gravitational **unit**. The grav- itational **unit** is available with a graded cross-sectional area of aperture supporting the **ball**, which is intended to compensate for the 'siphoning' that occurs in CSF-filled tubing of different lengths. The setting of the shunt assist- ant, expressed in cm should be equivalent to the distance between the right atrium of the heart and the outlet of the peritoneal catheter.

Let suppose 𝑑 2 𝑑𝑡 𝑦(𝑡) 2 = 0, and the gravity acting on the steel **ball** is constant as 𝑀𝑔 without being changed proportion to the weight of the steel **ball**, thus if the distance spaced from the center of the **ball** is y* and an electric current required for an equilibrium between the gravity and magnetic force is i*, from the eq. (3), i* should be agreed with

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