Singular matrix conjugacy problem with rapidly oscillating off-diagonal entries. Asymptotics of the solution in the case when a diagonal entry vanishes at a stationary point

2021 ◽  
Vol 32 (5) ◽  
pp. 847-864
Author(s):  
A. Budylin
Keyword(s):  
Stationary Point ◽  
Hilbert Problem ◽  
Conjugacy Problem ◽  
Diagonal Entry ◽  
Content Type ◽  
Quadratic Phase ◽  
Shock Region ◽  
The Cauchy Problem ◽  
Leading Term ◽  
Power Orders

The ( 2 × 2 ) (2\times 2) matrix conjugacy problem (the Riemann–Hilbert problem) with rapidly oscillating off-diagonal entries and quadratic phase function is considered, specifically, the case when one of the diagonal entries vanishes at a stationary point. For solutions of this problem, the leading term of the asymptotics is found. However, the method allows us to construct complete expansions in power orders. These asymptotics can be used, for example, to construct the asymptotics of solutions of the Cauchy problem for the nonlinear Schrödinger equation for large times in the case of the so-called collisionless shock region.

scholarly journals Lattice of infinite bending-resistant fibers

2016 ◽  
Vol 12 (2) ◽  
pp. 397-422
Author(s):  
Vladimir Kobelev
Keyword(s):  
Fracture Behavior ◽  
Hilbert Problem ◽  
Crack Tip ◽  
Elastic Solid ◽  
Periodic Lattice ◽  
Infinite Length ◽  
Two Dimensional ◽  
Content Type ◽  
Dimensional Lattice ◽  
Riemann Hilbert Problem

Purpose – The purpose of this paper is to introduce the double-periodic lattice, composed of bending-resistant fibers. The essence of the model is that the filaments are of infinite length and withstand tension and bending. The constitutive equations of the lattice in discrete and differential formulations are derived. Two complementary systems of loads, which cause different deformation two orthogonal families of fibers, occur in the lattice. The fracture behavior of the material containing a semi-infinite crack is investigated. The crack problem reduces to the exactly solvable Riemann-Hilbert problem. The solution demonstrates that the behavior of material cardinally depends upon the tension in the orthogonal family of fibers. If tension in fibers exists, opening of the crack under action of loads in two-dimensional lattice is similar to those in elastic solid. In the absence of tension, contrarily, there is a finite angle between edges at the crack tip. Design/methodology/approach – The description of stress state in the crack vicinity is reduced to the solution of mixed boundary value problem for simultaneous difference equations. In terms of Fourier images for unknown functions the problem is equivalent to a certain Riemann-Hilbert problem. Findings – The analytical solution of the problem shows that fracture behavior of the material depends upon the presence of stabilizing tension in fibers, parallel to crack direction. In the presence of tension in parallel fibers fracture character of two-dimensional lattice is similar to behavior of elastic solid. In this case the condition of crack grows can be formulated in terms of critical stress intensity factor. Otherwise, in the absence of stabilizing tension, the crack surfaces form a finite angle at the tip. Research limitations/implications – Linear behavior of fibers until rupture. Small deflections. Perfect two-dimensional lattice. Practical implications – The model provides exact analytical estimation of stresses on the crack tip as the function of fibers’ stiffness. Originality/value – The model is the extension of known lattice models, taking into account the semi-infinite crack in the lattice. This is the first known closed form solution for an infinite lattice model with the crack.

Rotating- and translating-coil magnetometers for extracting pseudo-multipoles in accelerator magnets

2017 ◽  
Vol 36 (5) ◽  
pp. 1552-1567 ◽  
Author(s):  
Stephan Russenschuck
Keyword(s):  
Magnetic Measurement ◽  
Scaling Law ◽  
Field Measurements ◽  
Expansion Method ◽  
Induction Coil ◽  
Orthogonal Expansion ◽  
Content Type ◽  
Leading Term ◽  
Accelerator Magnets ◽  
Longitudinal Position

Purpose The purpose of this paper is to establish the mathematical foundations of magnetic measurement methods based on translating-coil and rotating-coil magnetometers for accelerator magnets and solenoids. These field transducers allow a longitudinal scanning of the field distribution, but require a sophisticated post-processing step to extract the coefficients of the Fourier–Bessel series (known as pseudo-multipoles or generalized gradients) as well as a novel design of the rotating coil magnetometers. Design/methodology/approach Calculating the transversal field harmonics as a function of the longitudinal position in the magnet, or measuring these harmonics with a very short, rotating induction-coil scanner, allows the extraction of the coefficients of a Fourier–Bessel series, which can then be used in the thin lens approximation of the end regions of accelerator magnets. Findings The extraction of the leading term in the Fourier–Bessel series requires the solution of a differential equation by means of a Fourier transform. This yields a natural way to de-convolute the measured distribution of the multipole content. The author has shown that the measurement technique requires iso-parametric coils to avoid interception of the longitudinal field component. The compensation of the main signal cannot be done with the classical arrangement of search coils at different radii, because no easy scaling law exists. A new design of an iso-perimetric induction coil has been found. Research limitations/implications In the literature, it is stated that the pseudo-multipoles can be extracted from field computations or measurements. While this is true for computations, the author shows that the measurement of the field harmonics must be done with iso-parametric coils because otherwise the leading term in the Fourier–Bessel series cannot be extracted. Practical implications The author has now established the theory behind a number of field transducers, such as the moving fluxmeter, the rotational coil scanner and the solenoidal field transducer. Originality/value This paper brought together the known theory of the orthogonal expansion method with the methods and tools for magnetic field measurements to establish a field description in accelerator magnets.

Cauchy type nonlinear inverse problem in a two-layer area

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Michał Ciałkowski ◽  
Aleksander Olejnik ◽  
Magda Joachimiak ◽  
Krzysztof Grysa ◽  
Andrzej Frąckowiak
Keyword(s):  
Inverse Problem ◽  
Cauchy Problem ◽  
Heat Flow ◽  
Gas Turbine ◽  
Cauchy Type ◽  
Content Type ◽  
Temperature Changes ◽  
The Cauchy Problem ◽  
Changes Over Time ◽  
Over Time

Purpose To reduce the heat load of a gas turbine blade, its surface is covered with an outer layer of ceramics with high thermal resistance. The purpose of this paper is the selection of ceramics with such a low heat conduction coefficient and thickness, so that the permissible metal temperature is not exceeded on the metal-ceramics interface due to the loss ofmechanical properties. Design/methodology/approach Therefore, for given temperature changes over time on the metal-ceramics interface, temperature changes over time on the inner side of the blade and the assumed initial temperature, the temperature change over time on the outer surface of the ceramics should be determined. The problem presented in this way is a Cauchy type problem. When analyzing the problem, it is taken into account that thermophysical properties of metal and ceramics may depend on temperature. Due to the thin layer of ceramics in relation to the wall thickness, the problem is considered in the area in the flat layer. Thus, a one-dimensional non-stationary heat flow is considered. Findings The range of stability of the Cauchy problem as a function of time step, thickness of ceramics and thermophysical properties of metal and ceramics are examined. The numerical computations also involved the influence of disturbances in the temperature on metal-ceramics interface on the solution to the inverse problem. Practical implications The computational model can be used to analyze the heat flow in gas turbine blades with thermal barrier. Originality/value A number of inverse problems of the type considered in the paper are presented in the literature. Inverse problems, especially those Cauchy-type, are ill-conditioned numerically, which means that a small change in the inputs may result in significant errors of the solution. In such a case, regularization of the inverse problem is needed. However, the Cauchy problem presented in the paper does not require regularization.

Stable method for solving the Cauchy problem with the use of Chebyshev polynomials

2019 ◽  
Vol 30 (3) ◽  
pp. 1441-1456 ◽  
Author(s):  
Magda Joachimiak ◽  
Michał Ciałkowski ◽  
Andrzej Frąckowiak
Keyword(s):  
Heat Flux ◽  
Cauchy Problem ◽  
Chebyshev Polynomials ◽  
Heat Flux Density ◽  
Rectangular Domain ◽  
Cauchy Type ◽  
Laplace’S Equation ◽  
Content Type ◽  
Laplace's Equation ◽  
The Cauchy Problem

Purpose The purpose of this paper is to present the method for solving the inverse Cauchy-type problem for the Laplace’s equation. Calculations were made for the rectangular domain with the target temperature on three boundaries and, additionally, on one of the boundaries, the heat flux distribution was selected. The purpose of consideration was to determine the distribution of temperature on a section of the boundary of the investigated domain (boundary Γ1) and find proper method for the problem regularization. Design/methodology/approach The solution of the direct and the inverse (Cauchy-type) problems for the Laplace’s equation is presented in the paper. The form of the solution is noted as the linear combination of the Chebyshev polynomials. The collocation method in which collocation points had been determined based on the Chebyshev nodes was applied. To solve the Cauchy problem, the minimum of functional describing differences between the target and the calculated values of temperature and the heat flux on a section of the domain’s boundary was sought. Various types of the inverse problem regularization, based on Tikhonov and Tikhonov–Philips regularizations, were analysed. Regularization parameter α was chosen with the use of the Morozov discrepancy principle. Findings Calculations were performed for random disturbances to the heat flux density of up to 0.01, 0.02 and 0.05 of the target value. The quality of obtained results was next estimated by means of the norm. Effect of the disturbance to the heat flux density and the type of regularization on the sought temperature distribution on the boundary Γ1 was investigated. Presented methods of regularization are considerably less sensitive to disturbances to measurement data than to Tikhonov regularization. Practical implications Discussed in this paper is an example of solution of the Cauchy problem for the Laplace’s equation in the rectangular domain that can be applied for determination of the temperature distribution on the boundary of the heated element where it is impossible to measure temperature or the measurement is subject to a great error, for instance on the inner wall of the boiler. Authors investigated numerical examples for functions with singularities outside the domain, for which values of gradients change significantly within the calculation domain what corresponds to significant changes in temperature on the wall of the boiler during the fuel combustion. Originality/value In this paper, a new method for solving the Cauchy problem for the Laplace’s equation is described. To solve this problem, the Chebyshev polynomials and nodes were used. Various types of regularization of this problem were considered.

The More Control, the Better?

2014 ◽  
Vol 26 (2) ◽  
pp. 81-91 ◽  
Author(s):  
Jeeyun Oh ◽  
Mun-Young Chung ◽  
Sangyong Han
Keyword(s):  
Mixed Design ◽  
Negative Effects ◽  
Story Structure ◽  
Content Type ◽  
High User ◽  
User Control ◽  
Rigorous Research ◽  
Control Versus ◽  
The Media ◽  
The Impact

Despite of the popularity of interactive movie trailers, rigorous research on one of the most apparent features of these interfaces – the level of user control – has been scarce. This study explored the effects of user control on users’ immersion and enjoyment of the movie trailers, moderated by the content type. We conducted a 2 (high user control versus low user control) × 2 (drama film trailer versus documentary film trailer) mixed-design factorial experiment. The results showed that the level of user control over movie trailer interfaces decreased users’ immersion when the trailer had an element of traditional story structure, such as a drama film trailer. Participants in the high user control condition answered that they were less fascinated with, absorbed in, focused on, mentally involved with, and emotionally affected by the movie trailer than participants in the low user control condition only with the drama movie trailer. The negative effects of user control on the level of immersion for the drama trailer translated into users’ enjoyment. The impact of user control over interfaces on immersion and enjoyment varies depending on the nature of the media content, which suggests a possible trade-off between the level of user control and entertainment outcomes.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

GREEN OPEN ACCESS IN KENYA: A REVIEW OF THE CONTENT, POLICIES AND USAGE OF INSTITUTIONAL REPOSITORIES

Mousaion ◽  
2016 ◽  
Vol 33 (3) ◽  
pp. 25-54
Author(s):  
Wanyenda Leonard Chilimo
Keyword(s):  
Developing Countries ◽  
Open Access ◽  
Descriptive Study ◽  
Google Scholar ◽  
Research Institutions ◽  
Journal Articles ◽  
Content Type ◽  
Institutional Repositories ◽  
Current State ◽  
Theses And Dissertations

 There is scant research-based evidence on the development and adoption of open access (OA) and institutional repositories (IRs) in Africa, and in Kenya in particular. This article reports on a study that attempted to fill that gap and provide feedback on the various OA projects and advocacy work currently underway in universities and research institutions in Kenya and in other developing countries. The article presents the findings of a descriptive study that set out to evaluate the current state of IRs in Kenya. Webometric approaches and interviews with IR managers were used to collect the data for the study. The findings showed that Kenya has made some progress in adopting OA with a total of 12 IRs currently listed in the Directory of Open Access Repositories (OpenDOAR) and five mandatory self-archiving policies listed in the Registry of Open Access Repositories Mandatory Archiving Policies (ROARMAP). Most of the IRs are owned by universities where theses and dissertations constitute the majority of the content type followed by journal articles. The results on the usage and impact of materials deposited in Kenyan IRs indicated that the most viewed publications in the repositories also received citations in Google Scholar, thereby signifying their impact and importance. The results also showed that there was a considerable interest in Swahili language publications among users of the repositories in Kenya.

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