Radial Weight Function Research Articles

On heterogeneous radially symmetric semilinear elliptic equations with critical nonlinearities in [formula omitted

This is a continuation of our recent study of the effect of heterogeneity on existence and multiplicity results for elliptic PDEs with critical nonlinearities a la Trudinger–Moser in dimension N=2. Our results here are ultimately connected to the increase in the threshold of compactness that can be achieved in the Trudinger–Moser inequality under radial symmetry and in the presence of a rapidly vanishing radial weight function in H0,r1(BR). Such results are generalization to dimension N=2 of the 1981 seminal result of Ni [1] in dimension N≥3 (cf. other papers also cited in the References).In this work we prove and employ a simple weak continuity result in the subspace of radially symmetric H01(BR) functions for a class of functionals that include the classical Trudinger–Moser functional in the presence of a rapidly vanishing weight functions. As it is well known, such results are crucial in proving suitable compactness conditions for the corresponding energy functionals of the elliptic PDEs under consideration and enables us to prove the existence of a positive, a negative and a sign-changing solution for the equation (0.1)−Δu=a(r)h(u)eαu2inB(R1,R2)u=0on∂B(R1,R2)where B(R1,R2)≔{x∈R2|R1<|x|<R2},0≤R1<R2,without any growth restriction on the lower order terms of the nonlinearity. In addition we also prove existence of Infinitely Many Sign-Changing Solutions in the case of a ball under similar hypotheses. To the best of our knowledge, by highlighting the effect of the heterogeneity in relaxing the usual growth restrictions on the lower order terms of critical nonlinearities, our results complement and generalize a number of existing results in the literature of such problems.

GAMA+KiDS: Alignment of galaxies in galaxy groups and its dependence on galaxy scale

Intrinsic galaxy alignments are a source of bias for weak lensing measurements as well as a tool for understanding galaxy formation and evolution. In this work, we measure the alignment of shapes of satellite galaxies, in galaxy groups, with respect to the brightest group galaxy (BGG), as well as alignments of the BGG shape with the satellite positions, using the highly complete Galaxy And Mass Assembly (GAMA) spectroscopic survey and deep imaging from the Kilo Degree Survey. We control systematic errors with dedicated image simulations and measure accurate shapes using the DEIMOS shape measurement method. We find a significant satellite radial alignment signal, which vanishes at large separations from the BGG. We do not identify any strong trends of the signal with galaxy absolute magnitude or group mass. The alignment signal is dominated by red satellites. We also find that the outer regions of galaxies are aligned more strongly than their inner regions, by varying the radial weight employed during the shape measurement process. This behaviour is evident for both red and blue satellites. BGGs are also found to be aligned with satellite positions, with this alignment being stronger when considering the innermost satellites, using red BGGs and the shape of the outer region of the BGG. Lastly, we measure the global intrinsic alignment signal in the GAMA sample for two different radial weight functions and find no significant difference.

A Note on Hardy Inequalities on Homogeneous Groups

We provide the necessary and sufficient characterizations on a pair of positive radial functions so that the two-weight Hardy inequalities hold true on homogeneous groups, one of most general subclasses of nilpotent Lie groups. We also present a simple condition on a couple of positive radial weight functions for the Lp −Hardy type inequalities to be valid.

Distance Formulas on Weighted Banach Spaces of Analytic Functions

Let v be a radial weight function on the unit disc or on the complex plane. It is shown that for each analytic function $$f_0$$ in the Banach space $$H_v^\infty $$ of all analytic functions f such that v|f| is bounded, the distance of $$f_0$$ to the subspace $$H_v^0$$ of $$H_v^\infty $$ of all the functions g such that v|g| vanishes at infinity is attained at a function $$g_0 \in H_v^0$$ . Moreover a simple, direct proof of the formula of the distance of f to $$H_v^0$$ due to Perfekt is presented. As a consequence the corresponding results for weighted Bloch spaces are obtained.

Expected Number of Real Roots for Random Linear Combinations of Orthogonal Polynomials Associated with Radial Weights

In this note, we obtain asymptotic expected number of real zeros for random polynomials of the form $$f_n(z)=\sum_{j=0}^na^n_jc^n_jz^j$$ where $a^n_j$ are independent and identically distributed real random variables with bounded $(2+\delta)$th absolute moment and the deterministic numbers $c^n_j$ are normalizing constants for the monomials $z^j$ within a weighted $L^2$-space induced by a radial weight function satisfying suitable smoothness and growth conditions.

On methods of estimating cosmological bulk flows

We explore similarities and differences between several estimators of the cosmological bulk flow, $\bf B$, from the observed radial peculiar velocities of galaxies. A distinction is made between two theoretical definitions of $\bf B$ as a dipole moment of the velocity field weighted by a radial window function. One definition involves the three dimensional (3D) peculiar velocity, while the other is based on its radial component alone. Different methods attempt at inferring $\bf B$ for either of these definitions which coincide only for a constant velocity field. We focus on the Wiener Filtering (WF, Hoffman et al. 2015) and the Constrained Minimum Variance (CMV,Feldman et al. 2010) methodologies. Both methodologies require a prior expressed in terms of the radial velocity correlation function. Hoffman et al. compute $\bf B$ in Top-Hat windows from a WF realization of the 3D peculiar velocity field. Feldman et al. infer $\bf B$ directly from the observed velocities for the second definition of $\bf B$. The WF methodology could easily be adapted to the second definition, in which case it will be equivalent to the CMV with the exception of the imposed constraint. For a prior with vanishing correlations or very noisy data, CMV reproduces the standard Maximum Likelihood (ML, Kaiser 1988) estimation for $\bf B$ of the entire sample independent of the radial weighting function. Therefore, this estimator is likely more susceptible to observational biases that could be present in measurements of distant galaxies. Finally, two additional estimators are proposed.

3‐D inversion of gravity data in spherical coordinates with application to the GRAIL data

AbstractThree‐dimensional (3‐D) inversion of gravity data has been widely used to reconstruct the density distributions of ore bodies, basins, crust, lithosphere, and upper mantle. For global model of 3‐D density structures of planetary interior, such as the Earth, the Moon, or Mars, it is necessary to use an inversion algorithm that operates in the spherical coordinates. We develop a 3‐D inversion algorithm formulated with specially designed model objective function and radial weighting function in the spherical coordinates. We present regional and global synthetic examples to illustrate the capability of the algorithm. The inverted results show density distribution features consistent with the true models. We also apply the algorithm to a set of lunar Bouguer gravity anomaly derived from the Gravity Recovery and Interior Laboratory (GRAIL) gravity field and obtain a lunar 3‐D density distribution. High‐density anomalies are clearly identified underlying lunar basins, a wide region of the lateral density heterogeneities that exist beneath the South Pole‐Aitken basin are found, and low‐density anomalies are distributed beneath the Feldspathic Highlands Terrane on the lunar far‐side. The consistency of these results with those obtained independently from other existing methods verifies the newly developed algorithm.

On weighted critical imbeddings of Sobolev spaces

Our concern in this paper lies with two aspects of weighted exponential spaces connected with their role of target spaces for critical imbeddings of Sobolev spaces. We characterize weights which do not change an exponential space up to equivalence of norms. Specifically, we first prove that $${L_{\exp t^{\alpha}}(\chi_B)=L_{\exp t^{\alpha}}(\rho)}$$ if and only if $${\rho^q \in L_q}$$ with some q > 1. Second, we consider the Sobolev space $${W^{1}_{N}(\varOmega),}$$ where Ω is a bounded domain in $${\mathbb{R}^{N}}$$ with a sufficiently smooth boundary, and its imbedding into a weighted exponential Orlicz space $${L_{\exp t^{p'}}(\varOmega,\rho)}$$ , where ρ is a radial and non-increasing weight function. We show that there exists no effective weighted improvement of the standard target $${L_{\exp t^{N'}}(\varOmega)=L_{\exp t^{N'}}(\varOmega,\chi_{\varOmega})}$$ in the sense that if $${W^{1}_{N}(\varOmega)}$$ is imbedded into $${L_{\exp t^{p'}}(\varOmega,\rho),}$$ then $${L_{\exp t^{p'}}(\varOmega,\rho)}$$ and $${L_{\exp t^{N'}}(\varOmega)}$$ coincide up to equivalence of the norms; that is, we show that there exists no effective improvement of the standard target space. The same holds for critical cases of higher-order Sobolev spaces and even Besov and Lizorkin–Triebel spaces.

On the isomorphism classes of weighted spaces of harmonic and holomorphic functions

Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of Hv = { f : Ω → C holomorphic : sup z∈Ω |f(z)|v(z) < ∞ } and investigate some isomorphism classes of hv = { f : Ω → C harmonic : sup z∈Ω |f(z)|v(z) < ∞ } where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either Hv ∼ l∞ or Hv ∼ H∞, and at least two possibilities for hv, again hv ∼ l∞ and hv ∼ H∞. We also discuss many new examples of weights.

A representation formula for radially weighted biharmonic functions in the unit disc

Let $w\colon \mathbb{D}\to(0,\infty)$ be a radial continuous weight function. We consider the weighted biharmonic equation \Delta w^{-1}\Delta u=0\quad\text{in } \mathbb{D} with Dirichlet boundary conditions $u = f_0$ and $\partial_n u = f_1$ on $\mathbb{T} = \partial\mathbb{D}$. Under some extra conditions on the weight function $w$, we establish existence and uniqueness of a distributional solution $u$ of this biharmonic Dirichlet problem. Furthermore, we give a representation formula for the solution $u$. The key to our analysis is a series representation of Almansi type.

Efficient k-space sampling by density-weighted phase-encoding.

Acquisition-weighting improves the localization of MRI experiments. An approach to acquisition-weighting in a purely phase-encoded experiment is presented that is based on a variation of the sampling density in k-space. In contrast to conventional imaging or to accumulation-weighting, where k-space is sampled with uniform increments, density-weighting varies the distance between neighboring sampling points Deltak to approximate a given radial weighting function. A fast, noniterative algorithm has been developed to calculate the sampling matrix in one, two, and three dimensions from a radial weighting function w(k), the desired number of scans NA(tot) and the nominal spatial resolution Deltax(nom). Density-weighted phase-encoding combines the improved shape of the spatial response function and the high SNR of acquisition-weighting with an extended field of view. The artifact energy that results from aliasing due to a small field of view is substantially reduced. The properties of density-weighting are compared to uniform and to accumulation-weighted phase-encoding in simulations and experiments. Density-weighted (31)P 3D chemical shift imaging of the human heart is shown which demonstrates the superior performance of density-weighted metabolic imaging.

On a class of elliptic problems in R2: symmetry and uniqueness results

In the plane R2, we classify all solutions for an elliptic problem of Liouville type involving a (radial) weight function. As a consequence, we clarify the origin of the non-radially symmetric solutions for the given problem, as established by Chanillo and Kiessling.For a more general class of Liouville-type problems, we show that, rather than radial symmetry, the solutions always inherit the invariance of the problem under inversion with respect to suitable circles. This symmetry result is derived with the help of a 'shrinking-sphere' method.

Minimal cubature formulae for a family of radial weight functions

For each fixed n a radial weight function on \(\mathbb{R}^2 \) is given in closed form which admits minimal cubature formulae of both even and odd degree. The connection between the cubature formula and the polynomial interpolation is discussed in detail.

A weighted rotation function.

A simple probability expression is derived to discriminate the intermolecular vectors in the cross rotation function. The radial weighting function introduced takes into account both particle anisometry and crystal packing and can be incorporated into existing molecular replacement packages.

Aperture multipole moments from weak gravitational lensing

The projected mass of a gravitational lens inside (circular) apertures can be derived from the measured shear inside an annulus which is caused by the tidal field of the deflecting mass distribution. Here we show that also the multipoles of the two-dimensional mass distribution can be derived from the shear in annuli. We derive several expressions for these mass multipole moments in terms of the shear, which allow large flexibility in the choice of a radial weight function. In contrast to determining multipole moments from weak-lensing mass reconstructions, this approach allows to quantify the signal-to-noise ratio of the multipole moments directly from the observed galaxy ellipticities, and thus to estimate the significance of the multipole detection. Radial weight functions can therefore be chosen such as to optimize the significance of the detection given an assumed radial mass profile. Application of our formulae to numerically simulated clusters demonstrates that the quadrupole moment of realistic cluster models can be detected with high signal-to-noise ratio S/N; in about 85 per cent of the simulated cluster fields S/N >~ 3. We also show that the shear inside a circular annulus determines multipole moments inside and outside the annulus. This is relevant for clusters whose central region is too bright to allow the observation of the shear of background galaxies, or which extend beyond the CCD. We also generalize the aperture mass equation to the case of `radial' weight functions which are constant on arbitrarily-shaped curves which are not necessarily self-similar.

Splitting of closed ideals in $({\rm DFN})$-algebras of entire functions and the property $({\rm DN})$

For a plurisubharmonic weight function p on cn let Ap(Cn) denote the (DFN)-algebra of all entire functions on cn which do not grow faster than a power of exp(p). We prove that the splitting of many finitely generated closed ideals of a certain type in Ap(Cn), the splitting of a weighted 0-complex related with p, and the linear topological invariant (DN) of the strong dual of Ap(Cn) are equivalent. Moreover, we show that these equivalences can be characterized by convexity properties of p, phrased in terms of greatest plurisubharmonic minorants. For radial weight functions p, this characterization reduces to a covexity property of the inverse of p. Using these criteria, we present a wide range of examples of weights p for which the equivalences stated above hold and also where they fail. For p a nonnegative plurisubharmonic (psh) function on cn) let Ap(Cn) denote the algebra of all entire functions f such that If(Z)l 0 depending on f. Algebras of this type arise at various places in complex analysis and functional analysis, e.g. as Fourier transforms of certain convolution algebras. The structure of their closed ideals has been studied for a long time, primarily in the work of Schwartz [25], Ehrenpreis [9], Malgrange [17], and Palamodov [23] in connection with the existence and approximation of (systems of) convolution equations. The question whether a certain parameter dependence of the right-hand side of such an equation is shared also by its solutions, is closely related with the question of the existence of a continuous linear right inverse. The existence of such a right inverse is equivalent to the splitting of the closed ideal I associated to the corresponding equation. Also, since the quotient space Ap(Cn)/I is quite often identified with the space Ap(V) of holomorphic functions on the variety V of I which satisfy the restricted growth conditions, the latter question is equivalent to the existence of a linear extension operator from Ap(V) to Ap(Cn). Answers to these questions for various algebras have been given e.g. by Grothendieck (see lVeves [28]), Cohoon [7], Djakov and Mityagin [8], and Vogt [33]. The fact that for P(z) = IZl8, s > 1 all closed ideals in Ap(C) are cornplemented, was observed by Taylor [27]. Then Meise [19] extended Taylor's resultsX using a more functional analytic approach. He showed that the structural property (DN) of the strong dual Ap(Cn)b of Ap(Cn) implies that all slowly decreasing ideals Received by the editors July 29, 1986. 1980 Mathematics Subject Classification (1985 Reon). Primary 32E25, 46E25; Secondary 46A12, 32Al .

Properties of the ellipsoidal reflector of the McDonald emissiometer

The non-linear characteristic of the McDonald emissiometer INTEC Model 2145 is investigated. A reflection model yields a characteristic in very good agreement with the experimental data. The areal distribution of the sensitivity of the measuring head shows no azimuthal dependence. The radial weighting function can be approximated by a Gauss function.

Some Characteristics of the Pritchard Photometer

Using the Pritchard photometer as a specific example, this paper introduces a radial weighting function to describe both the in-focus and out-of-focus operation of a photometer. In both cases the central region in the instrument field of view contributes more to the instrument response than equal regions near the edge of the field. However, for out-of- ocus operation the central region of essentially uniform response will be reduced and may vanish. Further, three different fields of view can enter into the operation of a photometer even for in-focus operation. The size of these fields may vary by nearly 2:1. An experimental method of measuring the weighting function is described.