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Author: Grafiati
Published: 11 March 2023

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1

Duduchava, R., and D. Kapanadze. "On the Prandtl Equation." gmj 6, no. 6 (December 1999): 525–36. http://dx.doi.org/10.1515/gmj.1999.525.

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Abstract The unique solvability of the airfoil (Prandtl) integro-differential equation on the semi-axis is proved in the Sobolev space and Bessel potential spaces under certain restrictions on 𝑝 and 𝑠.
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2

Duduchava, Roland, and Medea Tsaava. "Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain." Georgian Mathematical Journal 27, no. 2 (June 1, 2020): 211–31. http://dx.doi.org/10.1515/gmj-2019-2031.

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AbstractThe purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces {\mathbb{H}^{s}_{p}(\Omega_{\alpha})}, {s>\frac{1}{p}}, {1<p<\infty}. The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev–Slobodečkii space on a semi-infinite axis {\mathbb{W}^{{s-1/p}}_{p}(\mathbb{R}^{+})}, which is of Mellin convolution type. Applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731], explicit conditions of the unique solvability of this BIE in the Sobolev–Slobodečkii {\mathbb{W}^{r}_{p}(\mathbb{R}^{+})} and Bessel potential {\mathbb{H}^{r}_{p}(\mathbb{R}^{+})} spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 2013, 3, 439–467], but there were made fatal errors. In the present paper, we correct these results.
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3

Treumann, Rudolf A., and Wolfgang Baumjohann. "Generalised partition functions: inferences on phase space distributions." Annales Geophysicae 34, no. 6 (June 2, 2016): 557–64. http://dx.doi.org/10.5194/angeo-34-557-2016.

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Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs–Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1∕|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ ≠ ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel–Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs–Boltzmann partition function is fundamental not only to Gibbs–Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.
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4

Narimani, Ghassem. "Smooth pointwise multipliers of modulation spaces." Analele Universitatii "Ovidius" Constanta - Seria Matematica 20, no. 1 (May 1, 2012): 317–28. http://dx.doi.org/10.2478/v10309-012-0021-8.

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Abstract Let 1 < p, q < ∞ and s, r ∈ ℝ. It is proved that any function in the amalgam space W(Hrp(ℝd), ℓ∞), where p' is the conjugate exponent to p and Hrp′ (ℝd) is the Bessel potential space, defines a bounded pointwise multiplication operator in the modulation space Msp,q(ℝd), whenever r > |s| + d
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5

Bragg, L. R. "Derivative-type ascent formulas for kernels of some half-space Dirichlet problems." ANZIAM Journal 42, no. 2 (October 2000): 185–94. http://dx.doi.org/10.1017/s144618110001186x.

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AbstractDerivative-type ascent formulas are deduced for the kernels of certain half-space Dirichlet problems. These have the character of differentiation formulas for the Bessel functions but involve modifying variables after completing the differentiations. The Laplace equation and the equation of generalized axially-symmetric potential theory (GASPT) are considered in these. The methods employed also permit treating abstract versions of Dirichlet problems.
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6

Waphare, B. B., and S. G. Gajbhiv. "Pseudo-differential type operators and Gevrey spaces." Asian-European Journal of Mathematics 13, no. 01 (October 1, 2018): 2050027. http://dx.doi.org/10.1142/s1793557120500278.

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In this paper, the pseudo-differential type operator [Formula: see text] associated with the Bessel type operator [Formula: see text] defined by (2.3) involving the symbol [Formula: see text] whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain Gevrey spaces. It is shown that the operator [Formula: see text] is a continuous linear map of one Gevrey space into another Gevrey space. A special pseudo-differential type operator called the Gevrey–Hankel type potential is defined and some of its properties are investigated. A variant of [Formula: see text] is also studied.
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7

HAMZA, Boutebba, Hakim LAKHAL, Slimani KAMEL, and Belhadi TAHAR. "The nontrivial solutions for nonlinear fractional Schrödinger-Poisson system involving new fractional operator." Advances in the Theory of Nonlinear Analysis and its Application 7, no. 1 (March 31, 2023): 121–32. http://dx.doi.org/10.31197/atnaa.1141136.

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In this paper, we investigate the existence of nontrivial solutions in the Bessel Potential space for nonlinearfractional Schrödinger-Poisson system involving distributional Riesz fractional derivative. By using themountain pass theorem in combination with the perturbation method, we prove the existence of solutions.
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8

Ševčovič, Daniel, and Cyril Izuchukwu Udeani. "Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing." Mathematics 9, no. 13 (June 22, 2021): 1463. http://dx.doi.org/10.3390/math9131463.

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The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.
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9

Chen, Wei, and Chao Zhang. "Regularity Properties and Lipschitz Spaces Adapted to High-Order Schrödinger Operators." Mathematics 10, no. 15 (July 26, 2022): 2600. http://dx.doi.org/10.3390/math10152600.

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Let be the high-order Schrödinger operator (−Δ)2+V2, where V is a non-negative potential satisfying the reverse Hölder inequality (RHq), with q>n/2 and n≥5. In this paper, we prove that when 0<α≤2−n/q, the adapted Lipschitz spaces Λα/4 we considered are equivalent to the Lipschitz space CLα associated to the Schrödinger operator L=−Δ+V. In order to obtain this characterization, we should make use of some of the results associated to (−Δ)2. We also prove the regularity properties of fractional powers (positive and negative) of the operator , Schrödinger Riesz transforms, Bessel potentials and multipliers of the Laplace transforms type associated to the high-order Schrödinger operators.
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10

Behzadan, A., and M. Holst. "On the Space of Locally Sobolev-Slobodeckij Functions." Journal of Function Spaces 2022 (July 18, 2022): 1–30. http://dx.doi.org/10.1155/2022/9094502.

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The study of certain differential operators between Sobolev spaces of sections of vector bundles on compact manifolds equipped with rough metric is closely related to the study of locally Sobolev functions on domains in the Euclidean space. In this paper, we present a coherent rigorous study of some of the properties of locally Sobolev-Slobodeckij functions that are especially useful in the study of differential operators between sections of vector bundles on compact manifolds with rough metric. The results of this type in published literature generally can be found only for integer order Sobolev spaces W m , p or Bessel potential spaces H s . Here, we have presented the relevant results and their detailed proofs for Sobolev-Slobodeckij spaces W s , p where s does not need to be an integer. We also develop a number of results needed in the study of differential operators on manifolds that do not appear to be in the literature.
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11

Ike, CC, HN Onah, and CU Nwoji. "BESSEL FUNCTIONS FOR AXISYMMETRIC ELASTICITY PROBLEMS OF THE ELASTIC HALF SPACE SOIL: A POTENTIAL FUNCTION METHOD." Nigerian Journal of Technology 36, no. 3 (June 30, 2017): 773–81. http://dx.doi.org/10.4314/njt.v36i3.16.

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Elasticity problems are formulated using displacement methods or stress methods. In this paper a displacement formulation of axisymmetric elasticity problem is presented. The formulation uses the Boussinesq– Papkovich – Neuber potential function. The problem is then solved by assuming Boussinesq – Papkovich - Neuber potential functions in the form of Bessel functions of order zero and of the first kind. The potential functions are then made to satisfy the governing field equations and the associated boundary conditions for the particular problem of a point load at the origin of the semi-infinite linear elastic isotropic soil mass. The unknown parameters of the function are thus determined and used to find the stresses, strains and displacement fields in the loaded soil. The results obtained were identical with the results obtained by Boussinesq. http://dx.doi.org/10.4314/njt.v36i3.16
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12

Farwig, Reinhard, and Andreas Schmidt. "An $$L^2$$ approach to viscous flow in the half space with free elastic surface." Journal of Elliptic and Parabolic Equations 7, no. 2 (September 29, 2021): 601–21. http://dx.doi.org/10.1007/s41808-021-00111-2.

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AbstractWe consider a linearized fluid-structure interaction problem, namely the flow of an incompressible viscous fluid in the half space $${\mathbb {R}}^n_+$$ R + n such that on the lower boundary a function h satisfying an undamped Kirchhoff-type plate equation is coupled to the flow field. Originally, h describes the height of an underlying nonlinear free surface problem. Since the plate equation contains no damping term, this article uses $$L^2$$ L 2 -theory yielding the existence of strong solutions on finite time intervals in the framework of homogeneous Bessel potential spaces. The proof is based on $$L^2$$ L 2 -Fourier analysis and also deals with inhomogeneous boundary data of the velocity field on the “free boundary” $$x_n=0$$ x n = 0 .
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13

Chowdhury, K. L. "AXISYMMETRIC CHARGE DISTRIBUTIONS ON AN ELASTIC DIELECTRIC HALF SPACE." Transactions of the Canadian Society for Mechanical Engineering 22, no. 4B (December 1998): 485–99. http://dx.doi.org/10.1139/tcsme-1998-0028.

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The solution of the axisymmetric boundary value problem of an isotropic elastic dielectric half space subjected to charge distribution on its rigid polarization free surface is constructed by Hankel transforms. For the problem of an electric point dipole applied at origin, exact expressions for the components of displacement and polarization vectors and the potential fields are obtained in terms of Bessel function and fundamental solutions 1/R and e-mR/R, R being the distance from the source point. The electric field is determined both inside and outside the polarized region. In the particular case of a continuous electric charge distribution with density of the form l/(r2+h2)1/2, the mechanical and electric stresses on the surface of the semi-space are derived. The MathematicsTM software is used to present the numerical results on graphs depicting the variation of surface stresses for the particular charge distributions.
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14

Golard, A., P. Witkovsky, and D. Tranchina. "Membrane currents of horizontal cells isolated from turtle retina." Journal of Neurophysiology 68, no. 2 (August 1, 1992): 351–61. http://dx.doi.org/10.1152/jn.1992.68.2.351.

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1. Membrane currents of horizontal cells isolated from the retina of the turtle, Pseudemys, were characterized by the whole-cell patch-clamp technique. 2. Four membrane currents were identified: an anomalous rectifier blocked by barium, a transient A-current, a sustained L-type calcium current enhanced by Bay K 8644, and a fast, tetrodotoxin-sensitive sodium current. Each of these four currents was found in both horizontal cell somata and axon terminals. 3. The current-voltage relations of axon terminals and somata were similar, but, in the normal operating range of the cell (-30 to -50 mV), the mean slope resistance of the axon terminal was higher (1.38 G omega) than that of the soma (0.26 G omega). 4. Exposure to either glutamate, kainate, or quisqualate induced a sustained inward current in horizontal cell axon terminals. The reversal potential for this current was -3 mV when tested with voltage steps and +9.1 mV when measured by a voltage ramp. The same horizontal cells were insensitive to N-methyl-D-aspartate. 5. A continuum model was developed to compute the degree of signal transfer between a horizontal cell body and its axon terminal. The model consisted of a network of electrically coupled somata that communicates with a network of electrically coupled axon terminals through the connecting axons. The specific membrane resistances used for the model derived from the patch-clamp measures. 6. We computed the voltage change elicited in either the layer of somata or of axon terminals by a static light stimulus of arbitrary dimensions. The amplitude of a spot response as a function of its radius was given by the weighted sum of two Bessel functions with different space constants. 7. The computed responses of the cell body were dominated by the Bessel function with the smaller space constant, whereas those of the axon terminal depended primarily on the Bessel function with the larger space constant. 8. The model predicts that, in contrast to the findings in teleost retina, there is little signal transfer between the somata and axon terminals of horizontal cell in the turtle retina.
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15

SALIG, J. B., M. V. CARPIO-BERNIDO, C. C. BERNIDO, and J. B. BORNALES. "ON NEURON MEMBRANE POTENTIAL DISTRIBUTIONS FOR VOLTAGE AND TIME DEPENDENT CURRENT MODULATION." International Journal of Modern Physics: Conference Series 17 (January 2012): 19–22. http://dx.doi.org/10.1142/s2010194512007891.

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Tracking variations of neuronal membrane potential in response to multiple synaptic inputs remains an important open field of investigation since information about neural network behavior and higher brain functions can be inferred from such studies. Much experimental work has been done, with recent advances in multi-electrode recordings and imaging technology giving exciting results. However, experiments have also raised questions of compatibility with available theoretical models. Here we show how methods of modern infinite dimensional analysis allow closed form expressions for important quantities rich in information such as the conditional probability density (cpd). In particular, we use a Feynman integral approach where fluctuations in the dynamical variable are parametrized with Hida white noise variables. The stochastic process described then gives variations in time of the relative membrane potential defined as the difference between the neuron membrane and firing threshold potentials. We obtain the cpd for several forms of current modulation coefficients reflecting the flow of synaptic currents, and which are analogous to drift coefficients in the configuration space Fokker-Planck equation. In particular, we consider cases of voltage and time dependence for current modulation for periodic and non-periodic oscillatory current modulation described by sinusoidal and Bessel functions.
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16

Hewett, D. P., and A. Moiola. "On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space." Analysis and Applications 15, no. 05 (December 8, 2016): 731–70. http://dx.doi.org/10.1142/s021953051650024x.

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This paper concerns the following question: given a subset [Formula: see text] of [Formula: see text] with empty interior and an integrability parameter [Formula: see text], what is the maximal regularity [Formula: see text] for which there exists a non-zero distribution in the Bessel potential Sobolev space [Formula: see text] that is supported in [Formula: see text]? For sets of zero Lebesgue measure, we apply well-known results on set capacities from potential theory to characterize the maximal regularity in terms of the Hausdorff dimension of [Formula: see text], sharpening previous results. Furthermore, we provide a full classification of all possible maximal regularities, as functions of [Formula: see text], together with the sets of values of [Formula: see text] for which the maximal regularity is attained, and construct concrete examples for each case. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterizing the regularity that can be achieved on certain special classes of sets, such as [Formula: see text]-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations.
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17

Pak, Ronald Y. S., and Xiaoyong Bai. "Analytic resolution of time-domain half-space Green's functions for internal loads by a displacement potential-Laplace-Hankel-Cagniard transform method." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2235 (March 2020): 20190610. http://dx.doi.org/10.1098/rspa.2019.0610.

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A refined yet compact analytical formulation is presented for the time-domain elastodynamic response of a three-dimensional half-space subject to an arbitrary internal or surface force distribution. By integrating Laplace and Hankel transforms into a method of displacement potentials and Cagniard's inversion concept, it is shown that the solution can be derived in a straightforward manner for the generalized classical wave propagation problem. For the canonical case of a buried point load with a step time function, the response is proved to be naturally reducible with the aid of a parametrized Bessel function integral representation to six wave-group integrals on finite contours in the complex plane that stay away from all branch points and the Rayleigh pole except possibly at the starting point of the contours. On the latter occasions, the possible singularities of the integrals can be rigorously extracted by an extended method of asymptotic decomposition, rendering the residual numerical computation a simple exercise. With the new solution format, the arrival time of each wave group is derivable by simple criteria on the contour. Typical results for the time-domain response for an internal point force as well as the degenerate case of a surface point source are included for comparison and illustrations.
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18

Reid, M. J. "On the Accuracy of Three-dimensional Kinematic Distances." Astronomical Journal 164, no. 4 (September 9, 2022): 133. http://dx.doi.org/10.3847/1538-3881/ac80bb.

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Abstract Over the past decade, the BeSSeL Survey and the VERA project have measured trigonometric parallaxes to ≈250 massive, young stars using VLBI techniques. These sources trace spiral arms over nearly half of the Milky Way. What is now needed are accurate distances to such stars that are well past the Galactic center. Here we analyze the potential for addressing this need by combining line-of-sight velocities and proper motions to yield three-dimenensional (3D) kinematic distance estimates. For sources within about 10 kpc of the Sun, significant systematic uncertainties can occur, and trigonometric parallaxes are generally superior. However, for sources well past the Galactic center, 3D kinematic distances are robust and more accurate than can usually be achieved by trigonometic parallaxes.
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19

Kotlařík, P., D. Kofroň, and O. Semerák. "Static Thin Disks with Power-law Density Profiles *." Astrophysical Journal 931, no. 2 (June 1, 2022): 161. http://dx.doi.org/10.3847/1538-4357/ac6027.

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Abstract The task of finding the potential of a thin circular disk with power-law radial density profile is revisited. The result, given in terms of infinite Legendre-type series in the above reference, has now been obtained in closed form thanks to the method of Conway employing Bessel functions. Starting from a closed-form expression for the potential generated by the elementary density term ρ 2l , we cover more generic—finite solid or infinite annular—thin disks using superposition and/or inversion with respect to the rim. We check several specific cases against the series-expansion form by numerical evaluation at particular locations. Finally, we add a method to obtain a closed-form solution for finite annular disks whose density is of “bump” radial shape, as modeled by a suitable combination of several powers of radius. Density and azimuthal pressure of the disks are illustrated on several plots, together with radial profiles of free circular velocity.
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20

Shaposhnikova, Tatjana Olegovna. "On the spectrum of multipliers in Bessel potential spaces." Časopis pro pěstování matematiky 110, no. 2 (1985): 197–206. http://dx.doi.org/10.21136/cpm.1985.108588.

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21

DuDuchava, R., M. Tsaava, and T. Tsutsunava. "Mixed Boundary Value Problem on Hypersurfaces." International Journal of Differential Equations 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/245350.

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The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equationdivC(A∇Cφ)=fon a smooth hypersurfaceCwith the boundaryΓ=∂CinRn.A(x)is ann×nbounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected partsΓ=ΓD∪ΓNand onΓDthe Dirichlet boundary conditions are prescribed, while onΓNthe Neumann conditions. The unique solvability of the mixed BVP is proved, based upon the Green formulae and Lax-Milgram Lemma. Further, the existence of the fundamental solution todivS(A∇S)is proved, which is interpreted as the invertibility of this operator in the settingHp,#s(S)→Hp,#s-2(S), whereHp,#s(S)is a subspace of the Bessel potential space and consists of functions with mean value zero.
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22

Desjacques, Vincent, Yonadav Barry Ginat, and Robert Reischke. "Statistics of a single sky: constrained random fields and the imprint of Bardeen potentials on galaxy clustering." Monthly Notices of the Royal Astronomical Society 504, no. 4 (May 4, 2021): 5612–20. http://dx.doi.org/10.1093/mnras/stab1228.

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ABSTRACT We explore the implications of a single observer’s viewpoint on measurements of galaxy clustering statistics. We focus on the Bardeen potentials, which imprint characteristic scale-dependent signatures in the observed galaxy power spectrum. The existence of an observer breaks homogeneity as it singles out particular field values at her/his position, like a spontaneous symmetry breaking. As a result, spatial averaging of the data must be performed while holding the Bardeen potentials fixed at the observer’s position. In practice, this can be implemented with the formalism of constrained random fields. In the traditional Cartesian Fourier decomposition, this constraint imprints a signature in the observed galaxy power spectrum at wavenumbers comparable to the fundamental mode of the survey. This effect, which is well within the cosmic variance, is the same for all observers regardless of their local environment because differences of potential solely are observable. In a spherical Bessel Fourier decomposition, this constraint affects the monopole of the observed galaxy distribution solely, like in CMB data. As a corollary, the scale dependence of the non-Gaussian bias induced by a local primordial non-Gaussianity is not significantly affected by the observer’s viewpoint.
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23

Immer, K., J. Li, L. H. Quiroga-Nuñez, M. J. Reid, B. Zhang, L. Moscadelli, and K. L. J. Rygl. "Anomalous peculiar motions of high-mass young stars in the Scutum spiral arm." Astronomy & Astrophysics 632 (December 2019): A123. http://dx.doi.org/10.1051/0004-6361/201834208.

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We present trigonometric parallax and proper motion measurements toward 22 GHz water and 6.7 GHz methanol masers in 16 high-mass star-forming regions. These sources are all located in the Scutum spiral arm of the Milky Way. The observations were conducted as part of the Bar and Spiral Structure Legacy (BeSSeL) survey. A combination of 14 sources from a forthcoming study and 14 sources from the literature, we now have a sample of 44 sources in the Scutum spiral arm, covering a Galactic longitude range from 0° to 33°. A group of 16 sources shows large peculiar motions of which 13 are oriented toward the inner Galaxy. A likely explanation for these high peculiar motions is the combined gravitational potential of the spiral arm and the Galactic bar.
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24

Osada, Hirofumi. "Stochastic geometry and dynamics of infinitely many particle systems—random matrices and interacting Brownian motions in infinite dimensions." Sugaku Expositions 34, no. 2 (October 12, 2021): 141–73. http://dx.doi.org/10.1090/suga/461.

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We explain the general theories involved in solving an infinite-dimensional stochastic differential equation (ISDE) for interacting Brownian motions in infinite dimensions related to random matrices. Typical examples are the stochastic dynamics of infinite particle systems with logarithmic interaction potentials such as the sine, Airy, Bessel, and also for the Ginibre interacting Brownian motions. The first three are infinite-dimensional stochastic dynamics in one-dimensional space related to random matrices called Gaussian ensembles. They are the stationary distributions of interacting Brownian motions and given by the limit point processes of the distributions of eigenvalues of these random matrices. The sine, Airy, and Bessel point processes and interacting Brownian motions are thought to be geometrically and dynamically universal as the limits of bulk, soft edge, and hard edge scaling. The Ginibre point process is a rotation- and translation-invariant point process on R 2 \mathbb {R}^2 , and an equilibrium state of the Ginibre interacting Brownian motions. It is the bulk limit of the distributions of eigenvalues of non-Hermitian Gaussian random matrices. When the interacting Brownian motions constitute a one-dimensional system interacting with each other through the logarithmic potential with inverse temperature β = 2 \beta = 2 , an algebraic construction is known in which the stochastic dynamics are defined by the space-time correlation function. The approach based on the stochastic analysis (called the analytic approach) can be applied to an extremely wide class. If we apply the analytic approach to this system, we see that these two constructions give the same stochastic dynamics. From the algebraic construction, despite being an infinite interacting particle system, it is possible to represent and calculate various quantities such as moments by the correlation functions. We can thus obtain quantitative information. From the analytic construction, it is possible to represent the dynamics as a solution of an ISDE. We can obtain qualitative information such as semi-martingale properties, continuity, and non-collision properties of each particle, and the strong Markov property of the infinite particle system as a whole. Ginibre interacting Brownian motions constitute a two-dimensional infinite particle system related to non-Hermitian Gaussian random matrices. It has a logarithmic interaction potential with β = 2 \beta = 2 , but no algebraic configurations are known.The present result is the only construction.
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25

Scheid, Werner, Aurelian Isar, and Aurel Sandulescu. "Diffusion and dissipation by linear momentum in spherical environment." International Journal of Modern Physics B 28, no. 11 (March 26, 2014): 1450077. http://dx.doi.org/10.1142/s0217979214500775.

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An open quantum system is studied consisting of a particle moving in a spherical space with an infinite wall. With the theory of Lindblad the system is described by a density matrix which gets affected by operators with diffusive and dissipative properties depending on the linear momentum and density matrix only. It is shown that an infinite number of basis states leads to an infinite energy because of the infinite unsteadiness of the potential energy at the infinite wall. Therefore only a solution with a finite number of basis states can be performed. A slight approximation is introduced into the equation of motion in order that the trace of the density matrix remains constant in time. The equation of motion is solved by the method of searching eigenvalues. As a side-product two sums over the zeros of spherical Bessel functions are found.
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26

Zhongyuan, Li. "A Storage Process of the Magnetic Energy in the Space Active Regions and Stellar Atmosphere." Symposium - International Astronomical Union 140 (1990): 17–19. http://dx.doi.org/10.1017/s0074180900189405.

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A few authors (Barnes and Sturrock, 1972; Ma, 1977; Svestka, 1977) have calculated the quantitative relationship between the static force-free field connecting the magnetic field and the twisting processes. They pointed out that the potential magnetic field without the current may be twisted into the force-free field with the enhanced current produced by the plasma rotation. Li et al. (1982) and Li and Hu (1984) have stated that the processes should be unsteady, and especially that they should not be static. The magnetic Reynold number is usually much larger than 100 in stellar atmosphere (Li et al., 1982). We adopt the following MHD equations: where the force - free factor α (t, r) depends on both, t and r. According to t h e kinematical momentum conservation, the following constraint is easily obtained: where V = (u, v, w) is the velocity field in the cylindrical coordinates. When studying the evolution of the kinematical force - free field, the in fluence of a reasonable flow on the variations of the magnetic field should be taken into account. After some reasonable simplification we deduce the specific expression of the variation law of the toroidal magnetic energy where J1 is the Bessel function of the first order. In the active region, magnetic energy including the term of a twisted effect f(t) is larger than that of the static force - free field.
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27

Alhaidari, A. D. "Progressive approximation of bound states by finite series of square-integrable functions." Journal of Mathematical Physics 63, no. 8 (August 1, 2022): 082102. http://dx.doi.org/10.1063/5.0093014.

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We use the “tridiagonal representation approach” to solve the time-independent Schrödinger equation for bound states in a basis set of finite size. We obtain two classes of solutions written as a finite series of square integrable functions that support a tridiagonal matrix representation of the wave operator. The differential wave equation becomes an algebraic three-term recursion relation for the expansion coefficients of the series, which is solved in terms of finite polynomials in the energy and/or potential parameters. These orthogonal polynomials contain all physical information about the system. The basis elements in configuration space are written in terms of either the Romanovski–Bessel polynomial or the Romanovski–Jacobi polynomial. The maximum degree of both polynomials is limited by the polynomial parameter(s). This makes the size of the basis set finite but sufficient to give a very good approximation of the bound state wavefunctions that improves with an increase in the basis size.
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28

Bokayev, N. A., M. L. Goldman, and G. Zh Karshygina. "CRITERIA FOR EMBEDDING OF GENERALIZED BESSEL AND RIESZ POTENTIAL SPACES IN REARRANGEMENT INVARIANT SPACES." Eurasian Mathematical Journal 10, no. 2 (2019): 8–29. http://dx.doi.org/10.32523/2077-9879-2019-10-2-08-29.

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29

Kuzmin, Yuri, and Stanislav Proshkin. "Diffraction of the Field of a Grounded Cable on an Elongated Dielectric Spheroid in a Conducting Layer." Applied Sciences 13, no. 3 (February 3, 2023): 2012. http://dx.doi.org/10.3390/app13032012.

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Based on a rigorous solution to the problem, analytical expressions are obtained for calculating the diffraction of the electromagnetic field of a grounded cable on an elongated dielectric spheroid in a conductive layer. The field of a grounded AC cable in a conductive layer is determined by solving the Helmholtz equation for the vector potential by using the method of integral Fourier–Bessel transformations, taking into account the boundary conditions at the bottom and surface of the conductive layer. The process of finding the secondary field of an elongated dielectric spheroid on an alternating current in a conducting layer is divided into two stages. First, we find an exact solution to the problem of an elongated dielectric spheroid at a constant current in a homogeneous field, in free space, decomposing this solution into a Taylor series and retaining the first term, which is a dipole approximation. In the second stage, the resulting field as the sum of the fields of the horizontal and vertical dipoles is analytically continued into the frequency domain. The field of the horizontal and vertical dipoles in the conducting layer is obtained by using the method of integral Fourier–Bessel transformations, taking into account the boundary conditions at the bottom and surface of the conducting layer. The resulting solution is presented in a closed form in elementary functions and has an accuracy level acceptable for the practice. Graphs showing the flow characteristics of an elongated dielectric spheroid modeling a swimmer in a light diving suit are given. The influence of the water–air boundary on the increase in the secondary field of the dielectric spheroid, which leads to an increase in the reliability of object detection, is revealed. The practical implementation of the described device protected by a patent and the experimental data of testing the device layout on the Gulf of Finland are given. A good agreement between the theoretical and experimental flow characteristics of a dielectric object both in shape, amplitude, and phase, is revealed.
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30

Almeida, Alexandre, and Stefan Samko. "Characterization of Riesz and Bessel potentials on variable Lebesgue spaces." Journal of Function Spaces and Applications 4, no. 2 (2006): 113–44. http://dx.doi.org/10.1155/2006/610535.

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Riesz and Bessel potential spaces are studied within the framework of the Lebesgue spaces with variable exponent. It is shown that the spaces of these potentials can be characterized in terms of convergence of hypersingular integrals, if one assumes that the exponent satisfies natural regularity conditions. As a consequence of this characterization, we describe a relation between the spaces of Riesz or Bessel potentials and the variable Sobolev spaces.
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31

Gogatishvili, Amiran, Bohumír Opic, and Júlio S. Neves. "Optimality of embeddings of Bessel-potential-type spaces into Lorentz–Karamata spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 134, no. 6 (December 2004): 1127–47. http://dx.doi.org/10.1017/s0308210500003668.

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We establish sharpness of embedding theorems for Bessel-potential spaces modelled upon Lorentz–Karamata (LK) spaces and we prove the non-compactness of such embeddings. Target spaces in our embeddings are LK spaces. As a consequence of our results, we get growth envelopes of Bessel-potential spaces modelled upon LK spaces.
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32

Moritoh, Shinya, and Yumi Tanaka. "An integral representation formula for logarithmic potentials and embeddings of Bessel-potential spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 3 (May 26, 2009): 541–49. http://dx.doi.org/10.1017/s0308210508000103.

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We give an integral representation formula for logarithmic Riesz potentials. This plays an essential role in proving the sharpness of the embeddings of Bessel-potential spaces, which have logarithmic exponents both in the smoothness and in the underlying Lorentz—Zygmund spaces. These results are natural extensions of those obtained by Edmunds, Gurka, Opic and Trebels.
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33

Goldman, M. L., and D. Haroske. "Optimal calderon space for bessel potentials." Doklady Mathematics 90, no. 2 (September 2014): 599–602. http://dx.doi.org/10.1134/s106456241406026x.

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34

Gurka, Petr, Petteri Harjulehto, and Aleš Nekvinda. "Bessel potential spaces with variable exponent." Mathematical Inequalities & Applications, no. 3 (2007): 661–76. http://dx.doi.org/10.7153/mia-10-61.

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35

Roy, Archisman, and Syed Khalid Mustafa. "Distribution of Charges on a Flat Disk as the Application of Far-Ranging Mathematics." Trends in Sciences 19, no. 16 (August 15, 2022): 5691. http://dx.doi.org/10.48048/tis.2022.5691.

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The foremost grail of this academic indagation is to delineate a mathematical expression of the normalised charge density over a flat disk. Aiming to ensue, firstly, 2 different frameworks have been dealt with to formulate the potential distribution which allows stability of a non-uniform charge distribution. At first, a logical but mathematically toilsome integral method has been approached. Out of the unyielding territory, we reduced the expression into algebraical functions using the Bessel coefficient and Green’s theorem, eventually inferring a new mathematical equivalence. Subsequently, this paper explores beta function as a solving tool of complete elliptic integral so that the normalization of charge apportion leads to 0 gradients of potential. Finally, the article deduces an integral equation whose implicit solution brings into the required charge distribution. The write-up also encounters finding a proximate graphical illustration of the assortment following the CAS system and direction fields. Beyond the conventional approach of real analysis, it facilitates proving the convergence of an acclaimed series. Consequently, it conceives a discussion on image charges for a flat disk. Even a short view of the article’s impact on practical fields of biology and engineering sciences has been included as the denouement. So, it might be of interest to the wide-ranged audience of research scholars in both the fields of physical and mathematical sciences. HIGHLIGHTS Expression of potential at any arbitrary 3D space point due to uniformly charged disk Natural unconstrained normalisation of charge density Implicating trigonometric solution in complete elliptic integral Proving the convergence of a bivariate divergent series Results can be of interest to electrostatic and condensed matter physicist including mathematicians of real analysis GRAPHICAL ABSTRACT
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36

Jaupart, Étienne, Étienne Parizot, and Denis Allard. "Contribution of the Galactic centre to the local cosmic-ray flux." Astronomy & Astrophysics 619 (November 2018): A12. http://dx.doi.org/10.1051/0004-6361/201833683.

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Context. Recent observations of unexpected structures in the Galactic cosmic ray (GCR) spectrum and composition, as well as growing evidence for episodes of intense dynamical activity in the inner regions of the Galaxy, call for an evaluation of the high-energy particle acceleration associated with such activity and its potential impact on the global GCR phenomenology. Aims. We investigate whether particles accelerated during high-power episodes around the Galactic centre can account for a significant fraction of the observed GCRs, or, conversely, what constraints can be derived regarding their Galactic transport if their contributions are negligible. Methods. Particle transport in the Galaxy is described with a two-zone analytical model. We solved for the contribution of a Galactic centre cosmic-Ray (GCCR) source using Green functions and Bessel expansion, and discussed the required injection power for these GCCRs to influence the global GCR phenomenology at Earth. Results. We find that, with standard parameters for particle propagation in the galactic disk and halo, the GCCRs can make a significant or even dominant contribution to the total CR flux observed at Earth. Depending on the parameters, such a source can account for both the observed proton flux and boron-to-carbon ratio (in the case of a Kraichnan-like scaling of the diffusion coefficient), or potentially produce spectral and composition features. Conclusions. Our results show that the contribution of GCCRs cannot be neglected a priori, and that they can influence the global GCR phenomenology significantly, thereby calling for a reassessement of the standard inferences from a scenario where GCRs are entirely dominated by a single type of sources distributed throughout the Galactic disk.
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37

Dappa, H., and W. Trebels. "On anisotropic Besov and Bessel potential spaces." Banach Center Publications 22, no. 1 (1989): 69–87. http://dx.doi.org/10.4064/-22-1-69-87.

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38

Didenko, Victor D., and Roland Duduchava. "Mellin convolution operators in Bessel potential spaces." Journal of Mathematical Analysis and Applications 443, no. 2 (November 2016): 707–31. http://dx.doi.org/10.1016/j.jmaa.2016.05.043.

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39

Edmunds, David E., Petr Gurka, and Bohumı́r Opic. "On Embeddings of Logarithmic Bessel Potential Spaces." Journal of Functional Analysis 146, no. 1 (May 1997): 116–50. http://dx.doi.org/10.1006/jfan.1996.3037.

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40

Ombe, Hitoshi. "Besov spaces and Bessel potential spaces on certain groups." Proceedings of the Japan Academy, Series A, Mathematical Sciences 67, no. 1 (1991): 6–10. http://dx.doi.org/10.3792/pjaa.67.6.

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41

Belyaev, A. A. "Characterization of spaces of multipliers for Bessel potential spaces." Mathematical Notes 96, no. 5-6 (November 2014): 634–46. http://dx.doi.org/10.1134/s0001434614110029.

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42

Bakhtigareeva, Elza G., Mikhail L. Goldman, and Dorothee D. Haroske. "Optimal Calderón Spaces for Generalized Bessel Potentials." Proceedings of the Steklov Institute of Mathematics 312, no. 1 (March 2021): 37–75. http://dx.doi.org/10.1134/s008154382101003x.

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43

Goldman, M. L., and D. Haroske. "Optimal Calderon spaces for generalized Bessel potentials." Doklady Mathematics 92, no. 1 (July 2015): 404–7. http://dx.doi.org/10.1134/s1064562415040031.

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44

Marcos, Miguel Andrés. "Bessel Potentials in Ahlfors Regular Metric Spaces." Potential Analysis 45, no. 2 (March 22, 2016): 201–27. http://dx.doi.org/10.1007/s11118-016-9543-4.

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45

Iaffei, Bibiana. "Generalized Bessel potentials on Lipschitz type spaces." Mathematische Nachrichten 278, no. 4 (March 2005): 421–36. http://dx.doi.org/10.1002/mana.200310250.

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46

Castro, Luis P., and David Kapanadze. "DIFFRACTION BY A UNION OF STRIPS WITH IMPEDANCE CONDITIONS IN BESOV AND BESSEL POTENTIAL SPACES." Mathematical Modelling and Analysis 13, no. 2 (June 30, 2008): 183–94. http://dx.doi.org/10.3846/1392-6292.2008.13.183-194.

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We consider an impedance boundary‐value problem for the Helmholtz equation which models a wave diffraction problem with imperfect conductivity on a union of strips. Pseudo‐differential operators acting between Bessel potential spaces and Besov spaces are used to deal with this wave diffraction problem. In particular, these operators allow a reformulation of the problem into a system of integral equations. The main result presents impedance parameters which ensure the well-posedness of the problem in scales of Bessel potential spaces and Besov spaces.
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47

CAICEDO, ALEJANDRO, and ARLÚCIO VIANA. "A DIFFUSIVE LOGISTIC EQUATION WITH MEMORY IN BESSEL POTENTIAL SPACES." Bulletin of the Australian Mathematical Society 92, no. 2 (June 16, 2015): 251–58. http://dx.doi.org/10.1017/s0004972715000581.

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48

Edmunds, D. "Optimality of embeddings of logarithmic Bessel potential spaces." Quarterly Journal of Mathematics 51, no. 2 (June 1, 2000): 185–209. http://dx.doi.org/10.1093/qjmath/51.2.185.

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49

Edmunds, David E., Petr Gurka, and Bohumír Opic. "Norms of embeddings of logarithmic Bessel potential spaces." Proceedings of the American Mathematical Society 126, no. 8 (1998): 2417–25. http://dx.doi.org/10.1090/s0002-9939-98-04327-5.

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50

Edmunds, David E., Petr Gurka, and Bohumír Opic. "Sharpness of embeddings in logarithmic Bessel-potential spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 5 (1996): 995–1009. http://dx.doi.org/10.1017/s0308210500023210.

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This paper is a continuation of [4], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalised Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [8].
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