Ambarzumian-type Problems for Discrete Schrödinger Operators

We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.

Unique heliophysics science opportunities along the Interstellar Probe journey up to 1000 AU from the Sun

<p>The Interstellar Probe is a space mission to discover physical interactions shaping globally the boundary of our Sun`s heliosphere and its dynamics and for the first time directly sample the properties of the local interstellar medium (LISM). Interstellar Probe will go through the boundary of the heliosphere to the LISM enabling for the first time to explore the boundary with a dedicated instrumentation, to take the image of the global heliosphere by looking back and explore in-situ the unknown LISM. The pragmatic concept study of such mission with a lifetime 50 years that can be implemented by 2030 was funded by NASA and has been led by the Johns Hopkins University Applied Physics Laboratory (APL). The study brought together a diverse community of more than 400 scientists and engineers spanning a wide range of science disciplines across the world.</p><p>Compelling science questions for the Interstellar Probe mission have been with us for many decades. Recent discoveries from a number of space missions exploring the heliosphere raised new questions strengthening the science case. The very shape of the heliosphere, a manifestation of complex global interactions between the solar wind and the LISM, remains the biggest mystery. Interpretations of imaging the heliosphere in energetic neutral atoms (ENAs) in different energy ranges on IBEX and Cassini/INCA from inside show contradictory pictures. Global physics-based models also do not agree on the global shape. Interstellar Probe on outbound trajectory will image the heliosphere from outside for the first time and will provide a unique determination of the global shape.</p><p>The LISM is a completely new area for exploration and discovery. We have a crude understanding of the LISM inferred from in-situ measurements inside the heliosphere of interstellar helium, pick-up-ions, ENAs, remote observations of solar backscattered Lyman-alpha emission and absorption line spectroscopy in the lines of sight of stars. We have no in-situ measurements of most LISM properties, e.g. ionization, plasma and neutral gas, magnetic field, composition, dust, and scales of possible inhomogeneities. Voyagers with limited capabilities have explored 30 AU beyond the heliosphere which appeared to be a region of significant heliospheric influence. The LISM properties are among the key unknowns to understand the Sun`s galactic neighborhood and how it shapes our heliosphere. Interstellar Probe will be the first NASA mission to discover the very nature of the LISM and shed light on whether the Sun enters a new region in the LISM in the near future.</p><p>In this presentation we give an overview of heliophysics science for the Interstellar Probe mission focusing on the critical science questions of the three objectives for the mission. We will discuss in more details a need for direct measurements in the LISM uniquely enabled by the Interstellar Probe.</p>

Imaging of bi-anisotropic periodic structures from electromagnetic near-field data

This paper is concerned with the inverse scattering problem for the three-dimensional Maxwell equations in bi-anisotropic periodic structures. The inverse scattering problem aims to determine the shape of bi-anisotropic periodic scatterers from electromagnetic near-field data at a fixed frequency. The factorization method is studied as an analytical and numerical tool for solving the inverse problem. We provide a rigorous justification of the factorization method which results in the unique determination and a fast imaging algorithm for the periodic scatterer. Numerical examples for imaging three-dimensional periodic structures are presented to examine the efficiency of the method.

Analysis of nonideality: insights from high concentration simulations of sedimentation velocity data

The Aviv fluorescence detection system (Aviv-FDS) has allowed the performance of sedimentation velocity experiments on therapeutic antibodies in highly concentrated environments like formulation buffers and serum. Methods were implemented in the software package SEDANAL for the analysis of nonideal, weakly associating AUC data acquired on therapeutic antibodies and proteins (Wright et al. Eur Biophys J 47:709–722, 2018, Anal Biochem 550:72–83, 2018). This involved fitting both hydrodynamic, ks, and thermodynamic, BM1, nonideality where concentration dependence is expressed as s = so/(1 + ksc) and D = Do(1 + 2BM1c)/(1 + ksc) and so and Do are values extrapolated to c = 0 (mg/ml). To gain insight into the consequences of these phenomenological parameters, we performed simulations with SEDANAL of a monoclonal antibody as a function of ks (0–100 ml/g) and BM1 (0–100 ml/g). This provides a visual understanding of the separate and joint impact of ks and BM1 on the shape of high-concentration sedimentation velocity boundaries and the challenge of their unique determination by finite element methods. In addition, mAbs undergo weak self- and hetero-association (Yang et al. Prot Sci 27:1334–1348, 2018) and thus we have simulated examples of nonideal weak association over a wide range of concentrations (1–120 mg/ml). Here we demonstrate these data are best analyzed by direct boundary global fitting to models that account for ks, BM1 and weak association. Because a typical clinical dose of mAb is 50–200 mg/ml, these results have relevance for biophysical understanding of concentrated therapeutic proteins.

NONLOCAL PROBLEM FOR A MIXED TYPE FOURTH-ORDER DIFFERENTIAL EQUATION WITH HILFER FRACTIONAL OPERATOR

In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

Unique Determination of the Shape of a Scattering Screen from a Passive Measurement

We consider the problem of fixed frequency acoustic scattering from a sound-soft flat screen. More precisely, the obstacle is restricted to a two-dimensional plane and interacting with an arbitrary incident wave, it scatters acoustic waves to three-dimensional space. The model is particularly relevant in the study and design of reflecting sonars and antennas, cases where one cannot assume that the incident wave is a plane wave. Our main result is that given the plane where the screen is located, the far-field pattern produced by any single arbitrary incident wave determines the exact shape of the screen, as long as it is not antisymmetric with respect to the plane. This holds even for screens whose shape is an arbitrary simply connected smooth domain. This is in contrast to earlier work where the incident wave had to be a plane wave, or more recent work where only polygonal scatterers are determined.