Radial Case Research Articles

Electron-hole coherence in core-shell nanowires with partial proximity-induced superconductivity

By solving the Bogoliubov--de Gennes Hamiltonian, the electron-hole coherence within a partially proximitized $n$-doped semiconductor shell of a core-shell nanowire heterostructure is investigated numerically and compared with the Andreev reflection interpretation of proximity induced superconductivity. Partial proximitization is considered to quantify the effects of a reduced coherence length. Three cases of partial proximitization of the shell are explored: radial, angular, and longitudinal. For the radial case, it is found that the boundary conditions impose localization probability maxima in the center of the shell in spite of off-center radial proximitization. The induced superconductivity gap is calculated as a function of the ratio between the proximitized shell thickness and the total shell thickness. In the angular case, the lowest-energy state of a hexagonal wire with a single proximitized side is found to display the essence of Andreev reflection, only by lengthwise summation of the localization probability. In the longitudinal case, a clear correspondence with Andreev reflection is seen in the localization probability as a function of length along a half-proximitized wire. The effect of an external magnetic field oriented along the wire is explored.

Entire downward solitons to the scalar curvature flow in Minkowski space

We investigate existence and uniqueness in Minkowski space of entire downward translating solitons with prescribed values at infinity, for a scalar curvature flow equation. The radial case translates into an ordinary differential equation and the general case into a fully non-linear elliptic PDE on \mathbb{R}^n .

A Framework for Generating Radial and Surface-Oriented Regularized Stokeslets.

Error in the method of regularized Stokeslets is highly dependent on the choice of the blob or regularization function that is utilized to handle singularities in the flow. In this work, we develop a general framework to choose regularizations at the level of the vector potential via smoothing factors. We detail the derivation for radial smoothing factors and specify properties which ensure that the solution is a regularized flow satisfying the incompressible Stokes equations. Error analysis is completed for both the far-field flow (away from the location of the forces) as well as at the location of the forces, relating our newly derived smoothing factors to commonly used blob functions and moment conditions. When forces are on a surface, we extend the radial smoothing factor case to the case of non-radial regularizations that are surface-oriented. We illustrate the utility of this framework by computing the forward and inverse problems of a translating sphere using radial and surface-oriented regularizations.

Limit of the blow-up solution for the inhomogeneous nonlinear Schrödinger equation

We study the H1 blow-up profile for the inhomogeneous nonlinear Schrödinger equation i∂tu=−Δu−|x|k|u|2σu,(t,x)∈R×RN,where k∈(−1,2N−2) and N≥3. We develop a new version of Gagliardo–Nirenberg inequality for σ∈[2+kN,2+kN−2] and k∈(−1,2N−2), and show that for the L2-critical exponent σ=2+kN, u(t) has no L2-limit as t→T∗ when ‖u(t)‖H1 blows up at T∗. Moreover, we investigate L2 concentration at the origin in the radial case. Additionally, if 2+kN<σ<min{2,2+kN−2,2(1+k)+N2N}, we show that there exists a unique u∗∈L2(RN) such that Γ(−t)u(t)→Γ(−T∗)u∗inLr(RN) (r∈[2,2∗)) as t→T∗. Our results extend the work for k=0 by F. Merle in earlier time.

On Channels of Energy for the Radial Linearised Energy Critical Wave Equation in the Degenerate Case

Abstract Channels of energy estimates control the energy of an initial data from that which it radiates outside a light cone. For the linearised energy critical wave equation, they have been obtained in the radial case in odd dimensions, first in three dimensions in [8], then for the general case in [10]. We consider even dimensions, for which such estimates are known to fail [2]. We propose a weaker version of these estimates, around a single ground state as well as around a multisoliton. This allows us in [1] to prove the soliton resolution conjecture in six dimensions.

Almost sure scattering at mass regularity for radial Schrödinger equations

We consider the radial nonlinear Schrödinger equation i∂ t u + Δu = |u| p−1 u in dimension d ⩾ 2 for and construct a natural Gaussian measure μ 0 which support is almost and such that μ 0—almost every initial data gives rise to a unique global solution. Furthermore, for and d ⩽ 10, the solutions constructed scatter in a space which is almost L 2. This paper can be viewed as a higher dimensional counterpart of the work of Burq and Thomann (2020 arXiv:2012.13571), in the radial case.

Poincaré and Logarithmic Sobolev Inequalities for Nearly Radial Measures

Poincaré inequality has been studied by Bobkov for radial measures, but few are known about the logarithmic Sobolev inequality in the radial case. We try to fill this gap here using different methods: Bobkov’s argument and super-Poincaré inequalities, direct approach via L1-logarithmic Sobolev inequalities. We also give various examples where the obtained bounds are quite sharp. Recent bounds obtained by Lee—Vempala in the log-concave bounded case are refined for radial measures.

Blow-up of non-radial solutions for the L 2 critical inhomogeneous NLS equation

We consider the L 2 critical inhomogeneous nonlinear Schrödinger equation in where N ⩾ 1 and 0 < b < min{2, N}. We prove that if satisfies E[u 0] < 0, then the corresponding solution blows-up in finite time. This is in sharp contrast to the classical L 2 critical nonlinear Schrödinger equation where this type of result is only known in the radial case for N ⩾ 2.

Kardar-Parisi-Zhang growth on square domains that enlarge nonlinearly in time.

Fundamental properties of an interface evolving on a domain of size L, such as its height distribution (HD) and two-point covariances, are known to assume universal but different forms depending on whether L is fixed (flat geometry) or expands linearly in time (radial growth). The interesting situation where L varies nonlinearly, however, is far less explored and it has never been tackled for two-dimensional (2D) interfaces. Here, we study discrete Kardar-Parisi-Zhang (KPZ) growth models deposited on square lattice substrates, whose (average) lateral size enlarges as L=L_{0}+ωt^{γ}. Our numerical simulations reveal that the competition between the substrate expansion and the increase of the correlation length parallel to the substrate, ξ≃ct^{1/z}, gives rise to a number of interesting results. For instance, when γ<1/z the interface becomes fully correlated, but its squared roughness, W_{2}, keeps increasing as W_{2}∼t^{2αγ}, as previously observed for one-dimensional (1D) systems. A careful analysis of this scaling, accounting for an intrinsic width on it, allows us to estimate the roughness exponent of the 2D KPZ class as α=0.387(1), which is very accurate and robust, once it was obtained averaging the exponents for different models and growth conditions (i.e., for various γ^{'}s and ω^{'}s). In this correlated regime, the HDs and covariances are consistent with those expected for the steady-state regime of the 2D KPZ class for flat geometry. For γ≈1/z, we find a family of distributions and covariances continuously interpolating between those for the steady-state and the growth regime of radial KPZ interfaces, as the ratio ω/c augments. When γ>1/z the system stays forever in the growth regime and the HDs always converge to the same asymptotic distribution, which is the one for the radial case. The spatial covariances, on the other hand, are (γ,ω)-dependent, showing a trend towards the covariance of a random deposition in enlarging substrates as the expansion rate increases. These results considerably generalize our understanding of the height fluctuations in 2D KPZ systems, revealing a scenario very similar to the one previously found in the 1D case.

Perturbed Fourier uniqueness and interpolation results in higher dimensions

We obtain new Fourier interpolation and uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa [11] and the second author [12]. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radii are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an application, we translate our perturbed Fourier uniqueness results to perturbed Heisenberg uniqueness for the hyperbola, using the interrelation between these fields introduced and studied by Bakan, Hedenmalm, Montes-Rodriguez, Radchenko and Viazovska [1].

Vectorial Prabhakar Hardy Type Generalized Fractional Inequalities under Convexity

We present a detailed great variety of Hardy type fractional inequalities under convexity and Lp norm in the setting of generalized Prabhakar and Hilfer fractional calculi of left and right integrals and derivatives. The radial multivariate case of the above over a spherical shell is developed in detail to all directions. Many inequalities are of vectorial splitting rational Lp type or of separating rational Lp type, others involve ratios of functions and of fractional integral operators.

A universal Hölder estimate up to dimension 4 for stable solutions to half-Laplacian semilinear equations

We study stable solutions to the equation (−Δ)1/2u=f(u), posed in a bounded domain of Rn. For nonnegative convex nonlinearities, we prove that stable solutions are smooth in dimensions n≤4. This result, which was known only for n=1, follows from a new interior Hölder estimate that is completely independent of the nonlinearity f.A main ingredient in our proof is a new geometric form of the stability condition. It is still unknown for other fractions of the Laplacian and, surprisingly, it requires convexity of the nonlinearity. From it, we deduce higher order Sobolev estimates that allow us to extend the techniques developed by Cabré, Figalli, Ros-Oton, and Serra for the Laplacian. In this way we obtain, besides the Hölder bound for n≤4, a universal H1/2 estimate in all dimensions.Our L∞ bound is expected to hold for n≤8, but this has been settled only in the radial case or when f(u)=λeu. For other fractions of the Laplacian, the expected optimal dimension for boundedness of stable solutions has been reached only when f(u)=λeu, even in the radial case.

On sharper estimates of Ohsawa–Takegoshi $$L^2$$-extension theorem in higher dimensional case

Hosono obtained sharper estimates of the Ohsawa–Takegoshi $$L^2$$ -extension theorem by allowing the constant depending on the weight function for a domain in $${\mathbb {C}}$$ . In this article, we show the higher dimensional case of sharper estimates of the Ohsawa–Takegoshi $$L^2$$ -extension theorem. To prove the higher dimensional case of them, we establish an analogue of Berndtsson–Lempert type $$L^2$$ -extension theorem by using the pluricomplex Green functions with poles along subvarieties. As a special case, we consider the sharper estimates in terms of the Azukawa pseudometric and show that the higher dimensional case of sharper estimate provides the $$L^2$$ -minimum extension for radial case.

Large deviations of Schramm-Loewner evolutions: A survey

These notes survey the first results on large deviations of Schramm-Loewner evolutions (SLE) with emphasis on interrelations between rate functions and applications to complex analysis. More precisely, we describe the large deviations of SLEκ when the κ parameter goes to zero in the chordal and multichordal case and to infinity in the radial case. The rate functions, namely Loewner and Loewner-Kufarev energies, are closely related to the Weil-Petersson class of quasicircles and real rational functions.

A rare occurrence of ganglion cysts on the posterolateral aspect of the elbow without neurological manifestations: a case series and review of the literature

ABSTRACT BACKGROUND: The occurrence of ganglion cysts around the elbow is rare, and the occurrence of these lesions without any symptoms of compression to the nearby structures is even rarer. Most published cases of elbow ganglions have reported patients with symptoms relating to compression of the radial nerve, or branches thereof secondary to anteriorly located cysts. We present two cases of ganglions occurring on the posterolateral aspect of the elbow with no pressure symptoms to the radial nerve CASE SERIES: The first case is a 33-year-old male, with a seven-month history of a spontaneous, slow-growing mass on the posterolateral aspect of his left elbow. The second case is a 38-year-old female, with a 12-month history of a spontaneous mass on the posterolateral aspect of her left elbow. In both cases, the reason for presentation was the unsightly elbow with an enlarging mass. The lesions were painless and both patients were neurologically intact with no restriction on range of motion of the joint. Both patients underwent excision of the mass for aesthetic reasons DISCUSSION: Patients with elbow ganglions usually have cysts located anterior to the radiocapitellar joint and almost invariably present with an associated motor, or less commonly, a sensory deficit of the radial nerve. Various treatment modalities have been reported; however, the vast majority undergo open surgical excision due to their association with progressive neurological symptoms. This usually leads to resolution of symptoms, and recurrence is rare CONCLUSION: The clinical presentation of the two patients reported in this case series seems to be far less frequent than patients presenting with a neuropathy of the radial nerve due to an anteriorly located elbow ganglion. It cannot, however, be excluded that there is an underreporting of asymptomatic elbow ganglions. According to our review of the English literature, this is only the third report of an asymptomatic elbow ganglion in the lateral compartment of the elbow Level of evidence: Level 5 Keywords: elbow, ganglion, cyst

Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility

Abstract We consider density solutions for gradient flow equations of the form u t = ∇ · (γ(u)∇ N(u)), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ(u) = u α , and α &gt; 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u) = u. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c 1 t −1 supported in a ball that spreads in time like c 2 t 1/d , thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ(u) = u α , with 0 &lt; α &lt; 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.

Classification of solitary wave solutions to Schrödinger equations with potentials

In this paper, we prove the existence and uniqueness of solitary wave solutions of time-dependent nonlinear Schrödinger equations with behaviors tending to zero at infinity under certain conditions on trapping potentials and parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established. Particularly, for constant trapping potentials, we conclude that there is at most one radial ground state under certain conditions on parameters.

An experimental study on the effect of annular and radial fins on thermal performance of a latent heat thermal energy storage unit

Latent heat thermal energy storage (LHTES) has been used to deal with the cyclical nature of energy production through solar means. One problem with LHTES systems is the PCM having a low thermal conductivity. This results in low heat transfer rate prolonging the charging and discharging cycles. In the current study, the thermal characteristics of a latent heat thermal energy storage system enhanced with annular and radial fins are investigated experimentally. Rubitherm RT-55 is used as the phase change material (PCM) and is enclosed within a vertical cylindrical container. Water is used as the heat transfer fluid (HTF) which is circulated in a copper pipe that passes through the center of the container. Different numbers of annular and radial fins attached to the central pipe, all containing the same volume of copper are studied. The effects of these configurations on the thermal performance of the latent heat thermal energy storage system during the charging and discharging processes is monitored. The no-fin benchmark case took 47.87 h to charge the LHTES unit and 42.5 h to discharge. The ten-annular fins charged the system 84.1% faster and discharged the system 68.21% faster. The twenty-annular fin case reduced the charging time by 85.8% and the discharging by 68.58%. The four-fin radial case was found to decrease the charging time by 81.9% and 70.0%; whereas the eight-fin radial case was found to have the greatest decrease the charging and discharging times, being 86.6% and 80.1%, respectively.

Blow-up results for systems of nonlinear Schr\xf6dinger equations with quadratic interaction

We establish blow-up results for systems of NLS equations with quadratic interaction in anisotropic spaces. We precisely show finite time blow-up or grow-up for cylindrical symmetric solutions. With our construction, we moreover prove some polynomial lower bounds on the kinetic energy of global solutions in the mass-critical case, which in turn implies grow-up along any diverging time sequence. Our analysis extends to general NLS systems with quadratic interactions, and it also provides improvements of known results in the radial case.

Hybrid planning of distributed generation and distribution automation to improve reliability and operation indices

This paper intends to give an effective hybrid planning of distributed generation and distribution automation in distribution networks aiming to improve the reliability and operation indices. The distribution automation platform consists of automatic voltage and VAR control and automatic fault management systems. The objective function minimizes the sum of the expected daily investment, operation, energy loss and reliability costs. The scheme is constrained by linearized AC optimal power flow equations and planning model of sources and distribution automation. A stochastic programming approach is also implemented in this paper based on a hybrid method of Monte Carlo simulation and simultaneous backward method to model uncertainty parameters of the understudy model including load, energy price and availability of network equipment. Finally, the proposed strategy is implemented on an IEEE 69-bus radial distribution network and different case studies are presented to demonstrate the economic and technical benefits of the investigated model. By allocating the optimal places for sources and distribution automation across the distribution network and extracting the optimal performance, the proposed scheme can simultaneously enhance economic, operation, and reliability indices in the distribution system compared to power flow studies.