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Let f f be an inner function in the unit ball B n ⊂ C n B_n \subset \mathbb {C}^n , n ≥ 1 n\ge 1 . Assume that \[ sup z ∈ B n | R f ( z ) | ( 1 − | z | 2 ) 1 + β ( 1 − | f ( z ) | 2 ) 2 > ∞ , \sup _{z\in B_n} \frac {|\mathcal {R} f(z)|(1-|z|^2)^{1+\beta }}{\left (1-|f(z)|^2 \right )^2} > \infty , \] where β ∈ ( 0 , 1 ) \beta \in (0,1) and R f \mathcal {R} f is the radial derivative. Then, for all α ∈ ∂ B 1 \alpha \in \partial B_1 , the set { ζ ∈ ∂ B n : f ∗ ( ζ ) = α } \{\zeta \in \partial B_n:\, f^*(\zeta ) =\alpha \} has a non-zero real Hausdorff t 2 n − 1 − β t^{2n-1-\beta } -content, and it has a non-zero complex Hausdorff t n − β t^{n-\beta } -content.

In this paper, the relations between D p and B n p , M( D p ) and B n p are discussed by means of high order radial derivative and random series. The main results of this paper are: (1) Let −∞ (i) if 0 B n q ⊂ D p ; (ii) if n−p/2⩽q B n q and D p cannot included each other; (iii) if q⩾(3n−p)/2, then D p ⊂ B n q . (2) Let 0 (i) if 0 B n q ⊂M( D p ) ; (ii) if n−p/2⩽q B n q and M( D p ) cannot included each other; (iii) if q⩾(3n−p)/2, then M( D p )⊂ B n q . (3) Let (n−1)/n M( D n(1−p) )⊂Q p .

The paper defines an extended Cesaro operator Tg with holomorphic symbol g in the unit ball B of Cn as T g ( f ) ( z ) = ∫ 0 1 f ( t z ) ℜ g ( t z ) d t t , f ∈ H ( B ) , z ∈ B . Where ℜ g ( z ) = ∑ z j ∂ g ∂ z j is the radial derivative of g. In this paper, the author characterizes g for which Tg is bounded (or compact) on the Bloch space B and the little Bloch space B0.

We consider weighted Bergman spaces and radial derivatives on the spaces. We also prove that for each element f in Bp, r/ there is a unique f in Bp, r/ such that f is the radial derivative of f and for each fBr/(i), f is the radial derivative of some element of Br/(i) if and only if, lim f(tz)= 0 for all zH.

We establish weighted Hardy{Littlewood inequalities for radial derivative and fractional radial derivatives on bounded symmetric domains.

The thermodynamics of electrowetting is treated. A general equation of electrowetting is derived from the first principles. It is demonstrated that the well-known Lippmann Equation describes a particular case of electrowetting when the radial derivative of the capacitance of the double layer is constant. The apparent contact angle of electrowetting depends on the gradient of capacity of a double layer in the vicinity of the triple line. The role of the area adjacent to the triple line in constituting the equilibrium apparent contact angle of electrowetting is emphasized., Comment: 11 pages, 3 Figures

We define an extended Cesaro operator Tg with holomorphic symbol g in the unit ball B of C n as T g (f)(z) = ∫ 1 0 f(tz)R g (tz)dt/t, f ∈ H(B), z ∈ B, where R g (z) = Σ n j=1 z j ∂ f /∂ zj is the radial derivative of g. In this paper we characterize those g for which T g is bounded (or compact) on the mixed norm space H p,q (ω).

In this paper, The integral characterizations of alpha-Bloch (little alpha-Bloch) axe given in terms of higher radial derivative, and their characterizations of Caxleson type measure are obtained.

In the setting of the half-space of the euclidean n-space, we show that every harmonic Bergman function is the radial derivative of a Bergman function with an appropriate norm bound.

Let D be a bounded symmetric domain and Σ be the Shilov boundary of D. For λ∈W, the Wallach set, and a nonnegative integer l, we study the weighted Bergman space Aλ2(D) and the weighted Bergman–Sobolev space A2,λ,l(D). For 0

In this paper we propose a version of Newton method for finding zeros of a quaternion function of a quaternion variable, based on the concept of quaternion radial derivative. Several numerical examples involving elementary functions are presented.

In this paper a BMO-type characterization of Dirichlet type spaces D p on the unit ball of Cn is given.

A formula of a radial derivative @ @r Uf(r;i;u) is obtained with the aid of derivatives with respect to i and to u of the functions closely connected with the spherical Poisson integral Uf(r;i;u) and the boundary values are determined for @ @r Uf(r;i;u). The bound- ary values are also found for partial derivatives with respect to the Cartesian coordinates @ @x UF, @ @y UF and @ @z UF.

It is shown that the magnetorotational instability (MRI) can be driven in nonmagnetized plasma. In this case, in contrast to the case of strongly magnetized plasma, radial derivative of plasma rotation frequency should be positive for such a driving. The characteristic wave lengths of MRI in nonmagnetized plasma are of the same order of magnitude as those for the Weibel instability.

The recent development in this laboratory of an automated data capture system, for refractometric optics on the analytical ultracentrifuge has removed the requirement for tedious and time consuming manual acquisition which had led to a decline in the use of schlieren optics. At the same time this system has increased the amount of data easily available from such an optical system with maintained or increased precision. From the advent of such a system has arisen the need for a package to facilitate the analysis of these data and to extend the range of analytical methods used. Using the improved data sets now available has also enabled us to successfully use methods which have lapsed in popularity over the last two decades. We have also been able to successfully apply radial derivative methods (Bridgman 1942) which have not routinely been applied to the analysis of sedimentation velocity experiments using schlieren optics. In this paper we describe the methods we have so far used to analyse data and present results for previously well defined molecules to demonstrate that the results obtained are reliable.

In this paper, the authors characterize the boundedness and compactness of the following Riemann-Stieltjes operator \vspace{-0.1cm} $$ L_g(f)(z)=\int_0^1{\cal R}f(tz)g(tz)\,\frac{dt}{t}, z\in B, $$ where ${\cal R}f(z)$ is the radial derivative of function $f$ at $z$, from mixed norm spaces $H(p,\, q,\, \phi)$ to Zygmund-type spaces on the unit ball.

We apply J Mather's theory (1991) on minimizing measures to the case of positive definite autonomous Lagrangians L:TM to R. We show that the minimal action function beta (h)=min( integral Ld mu mod rho ( mu )=h), has radial derivative, rho ( mu ) in H1(M,R) is the homology vector of the invariant probability mu . As one consequence, for any supporting domain S of beta , the closure of the union of the support of any minimizing measures with homology in S is contained in a fixed energy level. It is also shown that there is a one-to-one correspondence between minimizing measures for Lagrangians associated to mechanical systems (kinetic minus potential energy) and minimizing measures for the geodesic problem on a fixed energy level.

Obtaining the radial derivatives of wavefronts in projections is a critical step for volume optical computerized tomography. In this paper, the moire effect of two identical circular gratings in acquiring the first-order radial derivative of a wavefront is analyzed. Based on scalar diffraction theory, the formation mechanism of circular gratings’ moire fringes is derived. A more explicit analytical relation between moire fringes of different diffraction orders and tested wavefront is obtained. The involved results will be useful for extracting the projection information that is used in three-dimensional reconstruction by volume OCT.

We introduce the generic expressions for computing the gravitational field (potential and its radial derivative) generated by an arbitrary density (contrast) layer with a variable depth and thickness having a laterally-distributed radial density variation. The information on the geometry and density distribution of a volumetric mass layer is described by means of spherical harmonics. These generic expressions can uniformly be applied to model all major known density structures within the Earth’s interior using methods for a spherical harmonic analysis and synthesis of the gravitational field. This is demonstrated on specific examples given for various density models commonly adopted for the approximation of crust density structures.

Let B denote the unit ball in C n with boundary S and let σ( v ) be the standard normalized measure on S ( B ). For fixed 1 ≤ p ≤ ∞, R ≥ 1 let BH p ( B R ) ( BA p ( B R )) denote the unit ball of the Hardy space H p (resp. the Bergman space A p ) in B R ≔ RB and for l ∈ N let H R ( l , p , n ) (resp. A R ( l , p , n )) denote the class of those functions which have the l th radial derivative belonging to BH p ( B R ) ( BA p ( B R )) for l = 0, let H R (0, p , n ) ≔ BH p ( B R ) ( A R (0, p , n ) ≔ BA p ( B R )). The values of Kolmogorov, Gel′fand, and Bernstein and linear N -widths of classes H R ( l , p , n ) and A R ( l , p , n ) in the metrics L p (σ) and L p ( v ) (except for A R ( l , p , n ) in L p (σ)) are found. For all 1≤ p , q ≤ ∞, R > 1 the asymptotic estimates of N -widths for classes H R ( l , p , n ) and A R ( l , p , n ) in the spaces L q (σ) and L q ( v ) are also obtained.

We answer a question posed by B. Sims in 1972, by exhibiting an example of a Banach spaceX such that the numerical radius attaining operators onX are not dense. Actually,X is an old example used by J. Lindenstrauss to solve the analogous problem for norm attaining operators, but the proof for the numerical radius seems to be much more difficult. Our result was conjectured by C. Cardassi in 1985.

We consider 3d Schrodinger operator with long-range potential that has short-range radial derivative. The long-time asymptotics of non-stationary problem is studied and existence of modified wave operators is proved. It turns out, the standard WKB correction should be replaced by the solution to certain evolution equation., Comment: This is a preprint of an article whose final and definitive form has been published in Comm. Partial Differential Equations, available online at http://www.informaworld.com

We consider two-parameter Besov spaces of holomorphic functions on the unit ball of CN: We obtain various exclusions between Besov spaces of di®erent parameters using gap series. We estimate the growth near the boundary and the growth of Taylor coe±cients of functions in these spaces. We ¯nd the unique function with maximum value at each point of the ball in each Besov space. We base our proofs on Bergman projections and imbeddings between Lebesgue classes and Besov spaces. Special cases apply to the Hardy space H2, the Arveson space, the Dirichlet space, and the Bloch space.

A spontaneous transition phenomena between two states of a plasma with an internal transport barrier (ITB) is observed in the steady-state phase of the magnetic shear in the negative magnetic shear plasma in the JT-60U tokamak. These two ITB states are characterized by different profiles of the second radial derivative of the ion temperature inside the ITB region (one has a weak concave shape and the other has a strong convex shape) and by different degrees of sharpness of the interfaces between the L mode and the ITB region, which is determined by the turbulence penetration into the ITB region.

The paper defines an extended Cesaro operator Tg with holomorphic symbol g in the unit ball B of Cn as T g ( f ) ( z ) = ∫ 0 1 f ( t z ) ℜ g ( t z ) d t t , f ∈ H ( B ) , z ∈ B . Where ℜ g ( z ) = ∑ z j ∂ g ∂ z j is the radial derivative of g. In this paper, the author characterizes g for which Tg is bounded (or compact) on the Bloch space B and the little Bloch space B0.

Maximum-J (J is the second adiabatic invariant) capability, i.e., the radial derivative of J has the same sign as that of pressure, is investigated in a quasiaxisymmetric (QA) stellarator to investigate improved confinement. Due to the existence of nonaxisymmetry of the magnetic field strength, a local maximum of J is created to cause the drift reversal. External controllability of the maximum-J condition is also demonstrated, by which the impact of magnetic configuration on turbulent transport can be studied.

In this note we obtain a sharp estimate for a radial derivative of bounded harmonic functions in the ball.

The invariantly harmonic functions in the unit ball ${\Bbb B}^n$ in ${\Bbb C}^n$ are those annihilated by the Bergman Laplacian $\Delta$. The Poisson-Szegö kernel $P(z,\zeta)$ solves the Dirichlet problem for $\Delta$: if $f\in C(S^n)$, the Poisson-Szegö transform of $f$, $P[f](z)=\int_{S^n}P(z,\zeta) f(\zeta)\,d\sigma(\zeta),$ where $d\sigma$ is the normalized Lebesgue measure on $S^n$, is the unique invariantly harmonic function $u$ in ${\Bbb B}^n$, continuous up to the boundary, such that $u=f$ on $S^n$. The Poisson-Szegö transform establishes, loosely speaking, a one-to-one correspondence between function theory in $S^n$ and invariantly harmonic function theory in ${\Bbb B}^n$. When $n\geq 2$, it is natural to consider on $S^n$ function spaces related to its natural non-isotropic metric, for these are the spaces arising from complex analysis. In the paper, different characterizations of such spaces of smooth functions are given in terms of their invariantly harmonic extensions, using maximal functions and area integrals, as in the corresponding Euclidean theory. Particular attention is given to characterization in terms of purely radial or purely tangential derivatives. The smoothness is measured in two different scales: that of Sobolev spaces and that of Lipschitz spaces, including BMO and Besov spaces.

This paper is concerned with bounded functions h on the unit ball B 3 in ℝ3 admitting a generalized Poisson type representation. Estimates for h(x) in terms of ∥x∥ and h(0), as well as estimates for the radial derivative ∂ r h(x) at x ≠ 0 in terms of ∥x ∥ and h(x) are presented. In both cases we determine the functions for which the inequalities become equalities.

The “radial” polynomiality criterion for entire functions of several complex variables is proved.

Purpose: To determine the accuracy and variability of an automated sonographic breast lesion segmentation algorithm when it is used across different patient populations. Method and Materials: Two sonographic databases containing images of breast lesions were collected: one consisting of 456 lesions (145 malignant, 311 benign) from patients in Asia(database A), and one consisting of 433 lesions (127 malignant, 306 benign) from patients in the United States (database B). The same model of ultrasoundscanner was used to generate the images in both databases. Our average radial derivative‐based segmentation method was used to segment all the lesions in each database by using the center of each lesion as the starting seed point. The amount of overlap between the automated segmentation and an outline drawn by a radiologist was calculated for each lesion using O = area(S∩R)/area(S∪R) where S is the automated segmentation and R is the radiologist‐drawn outline. An overlap value of less than 0.4 is considered to be poor. Results: In database A, 85% of the lesions had an overlap value of 0.4 or greater. In database B, 80% of the lesions had an overlap value of 0.4 or greater. The automated segmentation performed better on the Asian database (p‐value=0.003). Conclusion: While the automated segmentation algorithm performs well on both databases, the disparity in performance raises interesting questions. The difference in performance could be the result of variation due to different radiologist‐drawn truths, differences in clinical imaging protocol, or even differences between the breast anatomy of the American and Asian populations. Further investigation is necessary to determine both the cause and magnitude of the variability. Conflict of Interest: Research supported in part by NIH. Some authors receive royalties, research funding, and/or are stock holders in Hologic.

A method for obtaining a detailed analysis of the apparent response of a photocathode is proposed. This involves an application of the self-derivative method and gives the radial derivative of sensitivity experimentally. Three different photomultipliers have been analyzed; their characteristics have been tentatively described by means of the function R(r) = [S(r)]−1 dS(r)/dr, where S(r) is the photomultiplier response in function of the radius of the photocathode.

1 Summary A second-order approximation of the tesseroid method has been presented in the paper “A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling” (Heck and Seitz 2007) for the gravitational potential and its first radial derivative. In the paper “Optimizedformulasforthegravitationalfieldofatesseroid” (Grombein et al. 2013) this analytical approach was optimized and extended to all first- and second-order derivatives of the potential. In both papers the general expression of the Taylor series expansion contains a formal error and needs to be corrected. As will be shown, this correction or rather erratum has no impact on the published and widely used second-order tesseroid formulas.