The Dimension of Phaseless Near-Field Data by Asymptotic Investigation of the Lifting Operator

In this paper, the question of evaluating the dimension of data space in an inverse source problem from near-field phaseless data is addressed. The study is developed for a 2D scalar geometry made up by a magnetic current strip whose square magnitude of the radiated field is observed in near non-reactive zone on multiple lines parallel to the source. With the aim of estimating the dimension of data space, at first, the lifting technique is exploited to recast the quadratic model as a linear one. After, the singular values decomposition of such linear operator is introduced. Finally, the dimension of data space is evaluated by quantifying the number of “relevant” singular values. In the last part of the article, some numerical simulations that corroborate the analytical estimation of data space dimension are shown.

Backflow from a model fracture network: an asymptotic investigation

We develop a model for predicting the flow resulting from the relaxation of pre-strained, fluid-filled, elastic network structures. This model may be useful for understanding relaxation processes in various systems, e.g. deformable microfluidic systems or by-products from hydraulic fracturing operations. The analysis is aimed at elucidating features that may provide insight on the rate of fluid drainage from fracturing operations. The model structure is a bifurcating network made of fractures with uniform length and elastic modulus, which allows for general self-similar branching and variation in fracture length and rigidity between fractures along the flow path. A late-time $t^{-1/3}$ power law is attained and the physical behaviour can be classified into four distinct regimes that describe the late-time dynamics based on the location of the bulk of the fluid volume (which shifts away from the outlet as branching is increased) and pressure drop (which shifts away from the outlet as rigidity is increased upstream) along the network. We develop asymptotic solutions for each of the regimes, predicting the late-time flux and evolution of the pressure distribution. The effects of the various parameters on the outlet flux and the network’s drainage efficiency are investigated and show that added branching and a decrease in rigidity upstream tend to increase drainage time.

Asymptotic investigation of a nonlinear model of string vibrations

The mathematical model of nonlinear oscillations of a weightless string is analysed. The uniformly valid asymptotic approximation in the long time interval, which is inversely proportional to the small parameter, is constructed. This asymptotic approximation is a solution of averaged along characteristics integro-differential system. A method for constructing special approximations of its solutions is proposed.

A PAIR OF PERFECTLY CONDUCTING DISKS IN AN EXTERNAL FIELD

A pair of non-overlapping perfectly conducting equal disks embedded in a two-dimensional background was investigated by the classic method of images, by Poincar´e series, by use of the bipolar coordinates and by the elliptic functions in the previous works. In particular, successive application of the inversions with respect to circles were applied to obtain the field in the form of a series. For closely placed disks, the previous methods yield slowly convergent series. In this paper, we study the local fields around closely placed disks by the elliptic functions. The problem of small gap is completely investigated since the obtained closed form solution admits a precise asymptotic investigation in terms of the trigonometric functions when the gap between the disks tends to zero. The exact and asymptotic formulae are extended to the case when a prescribed singularity is located in the gap. This extends applications of structural approximations to estimations of the local fields in densely packed fiber composites in various external fields.