We prove a Fatou-type theorem and its converse for certain positive eigenfunctions of the Laplace-Beltrami operator $mathcal{L}$ on a Harmonic $NA$ group. We show that a positive eigenfunction $u$ of $mathcal{L}$ with eigenvalue $beta^2-rho^2$, $beta
in (0,infty)$, has an admissible limit in the sense of Koranyi, precisely at those boundary points where the strong derivative of the boundary measure of $u$ exists. Moreover, the admissible limit and the strong derivative are the same. This extends a result of Ramey and Ullrich regarding nontangential convergence of positive harmonic functions on the Euclidean upper half space.
We have measured magnetoresistance of suspended graphene in the Corbino geometry at magnetic fields up to $B=0.15,$T, i.e., in a regime uninfluenced by Shubnikov-de Haas oscillations. The low-temperature relative magnetotoresistance $[R(B)-R(0)]/R(0)
$ amounts to $4000 B^2% $ at the Dirac point ($B$ in Tesla), with a quite weak temperature dependence below $30,$K. A decrease in the relative magnetoresistance by a factor of two is found when charge carrier density is increased to $|n| simeq 3 times 10^{-10}$ cm$^{-2}$. The gate dependence of the magnetoresistance allows us to characterize the role of scattering on long-range (Coulomb impurities, ripples) and short-range potential, as well as to separate the bulk resistance from the contact one. Furthermore, we find a shift in the position of the charge neutrality point with increasing magnetic field, which suggests that magnetic field changes the screening of Coulomb impurities around the Dirac point. The current noise of our device amounts to $10^{-23}$ A$^2$/$sqrt{textrm{Hz}}$ at $1,$kHz at $4,$K, which corresponds to a magnetic field sensitivity of $60$ nT/$sqrt{textrm{Hz}}$ in a background field of $0.15,$T.
In this article, we generalize a theorem of Victor L. Shapiro concerning nontangential convergence of the Poisson integral of a $L^p$-function. We introduce the notion of $sigma$-points of a locally finite measure and consider a wide class of convolu
tion kernels. We show that convolution integrals of a measure have nontangential limits at $sigma$-points of the measure. We also investigate the relationship between $sigma$-point and the notion of the strong derivative introduced by Ramey and Ullrich. In one dimension, these two notions are the same.
In this article, we are concerned with a certain type of boundary behavior of positive solutions of the heat equation on a stratified Lie group at a given boundary point. We prove that a necessary and sufficient condition for the existence of the par
abolic limit of a positive solution $u$ at a point on the boundary is the existence of the strong derivative of the boundary measure of $u$ at that point. Moreover, the parabolic limit and the strong derivative are equal.
In this article, we study certain type of boundary behaviour of positive solutions of the heat equation on the upper half-space of $R^{n+1}$. We prove that the existence of the parabolic limit of a positive solution of the heat equation at a point in
the boundary is equivalent to the existence of the strong derivative of the boundary measure of the solution at that point. Moreover, the parabolic limit and the strong derivative are equal.
In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space $R_+^{n+1}$. We show that similar results remain valid for more general approximate identities. We apply
this result to prove a result regarding boundary behavior of nonnegative eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $mathbb H^n$. We shall also prove a generalization of a result regarding large time behavior of solution of the heat equation proved in cite{Re}. We use this result to prove a result regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space $mathbb H^n$.
We prove an analogue of Chernoffs theorem for the sublaplacian on the Heisenberg group and use it prove a version of Inghams theorem for the Fourier transform on the same group.
Scanning nanoscale superconducting quantum interference devices (SQUIDs) are gaining interest as highly sensitive microscopic magnetic and thermal characterization tools of quantum and topological states of matter and devices. Here we introduce a nov
el technique of collimated differential-pressure magnetron sputtering for versatile self aligned fabrication of SQUID on tip (SOT) nanodevices, which cannot be produced by conventional sputtering methods due to their diffusive, rather than the required directional point-source, deposition. The new technique provides access to a broad range of superconducting materials and alloys beyond the elemental superconductors employed in the existing thermal deposition methods, opening the route to greatly enhanced SOT characteristics and functionalities. Utilizing this method, we have developed MoRe SOT devices with sub-50 nm diameter, magnetic flux sensitivity of 1.2 $muPhi_0/Hz^{1/2}$ up to 3 T at 4.2 K, and thermal sensitivity better than 4 $mu K/Hz^{1/2}$ up to 5 T, about five times higher than any previous report, paving the way to nanoscale imaging of magnetic and spintronic phenomena and of dissipation mechanisms in previously inaccessible quantum states of matter.
We investigate the current-current correlations in a four-terminal Al-AlOx-Al tunnel junction where shot noise dominates. We demonstrate that cross-correlations in the presence of two biasing sources of the Hanbury-Brown and Twiss type are much stron
ger (approximately twice) than an incoherent sum of correlations generated by single sources. The difference is due to voltage fluctuations of the central island that give rise to current-current correlations in the four contacts of the junction. Our measurements are in close agreement with results obtained using a simple theoretical model based on the theory of shot noise in multi-terminal conductors, generalized here to arbitrary contacts.
We examine a Bloch Oscillating Transistor pair as a differential stage for cryogenic low-noise measurements. Using two oppositely biased, nearly symmetric Bloch Oscillating Transistors, we measured the sum and difference signals in the current gain a
nd transconductance modes while changing the common mode signal, either voltage or current. From the common mode rejection ratio we find values $sim 20$ dB even under non-optimal conditions. We also characterize the noise properties and obtain excellent noise performance for measurements having source impedances in the M$Omega$ range.
