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Timetable (WHTW01)

Factorisation of matrix functions: New techniques and applications

Monday 12th August 2019 to Friday 16th August 2019

Monday 12th August 2019
09:20 to 09:50 Registration
09:50 to 10:00 Welcome from David Abrahams (Isaac Newton Institute)
10:00 to 11:00 Frank Speck
Wiener-Hopf factorisation through an intermediate space and applications to diffraction theory

An operator factorisation conception is investigated for a general Wiener-Hopf operator $W = P_2 A |_{P_1 X}$ where $X,Y$ are Banach spaces, $P_1 \in \mathcal{L}(X), P_2 \in \mathcal{L}(Y)$ are projectors and $A \in \mathcal{L}(X,Y)$ is invertible. Namely we study a particular factorisation of $A = A_- C A_+$ where $A_+ : X \rightarrow Z$ and $A_- : Z \rightarrow Y$ have certain invariance properties and the cross factor $C : Z \rightarrow Z$ splits the "intermediate space" $Z$ into complemented subspaces closely related to the kernel and cokernel of $W$, such that $W$ is equivalent to a "simpler" operator, $W \sim P C|_{P Z}$.  

The main result shows equivalence between the generalised invertibility of the Wiener-Hopf operator and this kind of factorisation (provided $P_1 \sim P_2$) which implies a formula for a generalised inverse of $W$. It embraces I.B. Simonenko's generalised factorisation of matrix measurable functions in $L^p$ spaces and various other factorisation approaches.  

As applications we consider interface problems in weak formulation for the n-dimensional Helmholtz equation in $\Omega = \mathbb{R}^n_+ \cup \mathbb{R}^n_-$ (due to $x_n > 0$ or $x_n < 0$, respectively), where the interface $\Gamma = \partial \Omega$ is identified with $\mathbb{R}^{n-1}$ and divided into two parts, $\Sigma$ and $\Sigma'$, with different transmission conditions of first and second kind. These two parts are half-spaces of $\mathbb{R}^{n-1}$ (half-planes for $n = 3$). We construct explicitly resolvent operators acting from the interface data into the energy space $H^1(\Omega)$. The approach is based upon the present factorisation conception and avoids an interpretation of the factors as unbounded operators. In a natural way, we meet anisotropic Sobolev spaces which reflect the edge asymptotic of diffracted waves.

11:00 to 11:30 Morning Coffee
11:30 to 12:30 Eugene Shargorodsky
Quantitative results on continuity of the spectral factorisation mapping

It is well known that  the matrix spectral factorisation mapping is continuous from the Lebesgue space $L^1$ to  the Hardy space $H^2$ under the additional assumption of uniform integrability of the logarithms of the spectral densities to be factorised (S. Barclay; G. Janashia, E. Lagvilava, and L. Ephremidze). The talk will report on a joint project with Lasha Epremidze and Ilya Spitkovsky, which aims at obtaining quantitative results characterising this continuity.

12:30 to 13:30 Lunch at Churchill College
13:30 to 14:00 Raphael Assier
Recent advances in the quarter-plane problem using functions of two complex variables
14:00 to 14:30 J.M.L. Bernard
Novel exact and asymptotic series with error functions, for a function involved in diffraction theory: the incomplete Bessel function
The incomplete Bessel function, closely related to incomplete Lipschitz-Hankel integrals, is a well known known special function commonly encountered in many problems of physics, in particular in wave propagation and diffraction [1]-[5]. We present here novel exact and asymptotic series with error functions, for arbitrary complex arguments and integer order, derived from our recent publication [5].  

[1] Shimoda M, Iwaki R, Miyoshi M, Tretyakov OA, 'Wiener-hopf analysis of transient phenomenon caused by time-varying resistive screen in waveguide', IEICE transactions on electronics, vol. E85C, 10, pp.1800-1807, 2002
[2] DS Jones, 'Incomplete Bessel functions. I', proceedings of the Edinburgh Mathematical Society, 50, pp 173-183, 2007
[3] MM Agrest, MM Rikenglaz, 'Incomplete Lipshitz-Hankel integrals', USSR Comp. Math. and Math Phys., vol 7, 6, pp.206-211, 1967
[4] MM Agrest amd MS Maksimov, 'Theory of incomplete cylinder functions and their applications', Springer, 1971.
[5] JML Bernard, 'Propagation over a constant impedance plane: arbitrary primary sources and impedance, analysis of cut in active case, exact series, and complete asymptotics', IEEE TAP, vol. 66, 12, 2018
14:30 to 15:00 Andrey Shanin
Ordered Exponential (OE) equation as an alternative to the Wiener-Hopf method
15:00 to 15:30 Anastasia Kisil
Generalisation of the Wiener-Hopf pole removal method and application to n by n matrix functions
15:30 to 16:00 Afternoon Tea
16:00 to 16:30 Basant Lal Sharma
Wiener-Hopf factorisation on the unit circle: some examples of discrete scattering problems
I will provide certain examples of scattering problems, motivated by lattice waves (phonons), electronic waves under certain assumptions, nanoscale effects, etc in crystals. The mathematical formulation is posed on lattices and involves difference equations that can be reduced to the problem of Wiener-Hopf on the unit circle (in an annulus in complex plane). In some of these examples, the Wiener-Hopf  problem is scalar, while in other cases it is a matrix Wiener-Hopf problem. For the latter, in a few cases it may be reduced to a scalar problem but it appears to be not the case in others. Some of these problems can be considered as discrete analogues of well-known Wiener-Hopf equations in continuum models on the real line (in an strip in complex plane), a few of which are still open problems.
16:30 to 17:00 Grigori Giorgadze
On the partial indices of piecewise constant matrix functions
Every holomorphic vector bundle    on Riemann sphere  splits into the direct sum of line bundles and the total Chern number of this vector bundle  is equal to sum of Chern numbers of line bundles. The integer-valued vector with components Chern number of line bundles is called splitting type of holomorphic vector bundle and is analytic invariant of complex vector bundles.  
There exists a one-to-one correspondence between the H\"older continues matrix function and the holomorphic vector bundles described above, wherein the splitting type of vector bundles coincides with partial indices of matrix functions. It is known that every holomorphic vector bundle equipped with meromorphic (in general) connection  with logarithmic singularities at finite set of marked points and corresponding meromorphic 1-from  have first order poles in marked points and removable singularity at infinity.  
The Fucshian system of equations induced from this 1-form gives the monodromy representation of the fundamental group of Riemann sphere without marked points. The monodromy representation induces trivial holomorphic vector bundles  with connection. The extension of the pair (\texttt{bundle, connection}) on the Riemann sphere is not unique and defines a family of holomorphically nontrivial vector bundles.    
In the talk we present about the following statements:     
1. From the solvability condition (in the sense Galois differential theory) of the Fuchsian
   system  follows formula for computation of partial indices of piecewise constant matrix function.     
2. All extensions of  vector bundle on noncompact Riemann surface correspond to
   rational matrix functions  algorithmically computable by monodromy matrices of Fucshian system.

This work was supported, in part, by the Shota Rustaveli National Science Foundation under Grant No 17-96.
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 13th August 2019
09:00 to 10:00 Ilya Spitkovsky
Wiener-Hopf factorization: the peculiarities of the matrix almost periodic case
For several classes  of  functions invertibility and factorability are equivalent; such is the case, e.g., for the Wiener class W or the algebra APW of almost periodic functions with absolutely convergent Bohr-Fourier series. The result for W extends to the matrix setting; not so for APW. Moreover, the factorability criterion even for 2-by-2 triangular matrix functions with APW entries and constant determinant remains a mystery. We will discuss some known results in this direction, and more specific open problems.
10:00 to 11:00 Lasha Ephremidze
On Janashia-Lagvilava method of matrix spectral factorisation

Janashia-Lagvilava method is a relatively new algorithm of matrix spectral factorisation which can be applied to compute an approximate spectral factor of any matrix function (non-rational, large scale, singular) which satisfies the necessary and sufficient condition for the existence of spectral factorisation. The numerical properties of the method strongly depend on the way it is algorithmised and we propose its efficient algorithmisation. The method has already been successfully used in connectivity analysis of complex networks. The algorithm has the potential to be used in control system design and implementation for the required optimal controller computations by using frequency response data directly from measurements on real systems. It also provides a robust way of Granger causality computation for noisy singular data.

11:00 to 11:30 Morning Coffee
11:30 to 12:30 Andreas Kyprianou
Wiener-Hopf Factorisations for Levy processes
We give an introduction to the the theory of Wiener-Hopf factoirsations for Levy processes and discuss some very recent examples which are stimulated by some remarkable connections with self-similar Markov processes.
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:30 Sergei Rogosin
Factorisation of triangular matrix-functions of arbitrary order
It will be discussed an efficient method for factorization of square triangular matrix-functions of arbitrary order which was recently proposed in [1]. The idea goes back to the paper by G. N. Chebotarev [2] who constructed factorisation of 2x2 triangular matrix-functions by using representation of the certain functions related to entries of the initial matrix into continuous fraction. In order to avoid additional technical difficulties, we consider matrix-functions with Hoelder continuous entries. Tough the proposed method could be realised for wider classes of matrix-functions. Chebotarev's method is extended here to the triangular matrix-functions of arbitrary order. An inductive consideration which allows to obtain such an extension is based on an auxiliary statement.
Theoretical construction is illustrated by a number of examples.
The talk is based on a joint work with Dr. L. Primachuk and Dr. M.Dubatovskaya.

1. Primachuk, L., Rogosin, S.: Factorization of triangular matrix-functions of an arbitrary order, Lobachevsky J. Math., 39 (6), 809–817 (2018)
2. Chebotarev, G. N.: Partial indices of the Riemann boundary value problem with a triangular matrix of the second order, Uspekhi Mat. Nauk, XI (3(69)), 192_202 (1956) (in Russian).
14:30 to 15:00 Cristina Camara
A Riemann-Hilbert approach to Einstein field equations
The field equations of gravitational theories in 4 dimensions are non-linear PDE's that are difficult to solve in general. By restricting to a subspace of solutions that only depend on two space-time coordinates, alternative approaches to solving those equations become available. We present here the Riemann-Hilbert approach, looking at the dimensionally reduced field equations as an integrable system associated to a certain Lax pair, whose solutions can be obtained by factorizing a so called monodromy matrix. This approach allows for the explicit construction of solutions to the non-linear gravitational field equations using simple complex analytic methods.
15:00 to 15:30 Aloknath Chakrabarti
Solving Wiener-Hopf Problems by the aid of Fredholm Integral Equations of the Second Kind
A class of Wiener-Hopf problems is shown to be solvable by reducing the original problems to Fredholm integral equations of the second kind.  The resulting Fredholm integral equations are shown to be finally solvable,  numerically, by using standard techniques.  The present method is found to be applicable to systems of Wiener-Hopf problems, for which the Wiener-Hopf factorization of matrices can be avoided.  Several examples are taken up, demonstrating the present method of solution of Wiener-Hopf problems. 
15:30 to 16:00 Afternoon Tea
16:00 to 16:30 Victor Adukov
On explicit and exact solutions of the Wiener-Hopf factorization problem for some matrix functions

By an explicit solution of the factorization problem we mean the solution that can be found by finite number of some steps which we call "explicit". When we solve a specific factorization problem we must rigorously define these steps. In this talk we will do this for matrix polynomials, rational matrix functions, analytic matrix functions, meromorphic matrix functions, triangular matrix functions and others. For these classes we describe the data and procedures that are necessary for the explicit solution of the factorization problem. Since the factorization problem is unstable, the explicit solvability of the problem does not mean that we can get its numerical solution. This is the principal obstacle to use the Wiener-Hopf techniques in applied problems. For the above mentioned classes the main reason of the instability is the instability of the rank of a matrix. Numerical experiments show that the use of SVD for computation of the ranks often allows us to correctly find the partial indices for matrix polynomials. To create a test case set for numerical experiments we have to solve the problem exactly. By the exact solutions of the factorization problem we mean those solutions that can be found by symbolic computation. In the talk we obtain necessary and sufficient conditions for the existence of the exact solution to the problem for matrix polynomials and propose an algorithm for constructing of the exact solution. The solver modules in SymPy and in Maple that implement this algorithm are designed.

16:30 to 17:00 Valery Smyshlyaev
Whispering gallery waves diffraction by boundary inflection: an unsolved canonical problem
The problem of interest is that of a whispering gallery high-frequency asymptotic mode propagating along a concave part of a boundary and approaching a boundary inflection point. Like Airy ODE and associated Airy function are fundamental for describing transition from oscillatory to exponentially decaying asymptotic behaviors, the boundary inflection problem leads to an arguably equally fundamental canonical boundary-value problem for a special PDE, describing transition from a “modal” to a “scattered” high-frequency asymptotic behaviour. The latter problem was first formulated and analysed by M.M. Popov starting from 1970-s. The associated solutions have asymptotic behaviors of a modal type (hence with a discrete spectrum) at one end and of a scattering type (with a continuous spectrum) at the other end. Of central interest is to find the map connecting the above two asymptotic regimes. The problem however lacks separation of variables, except in the asymptotical sense at both of the above ends.  
Nevertheless, the problem asymptotically admits certain complex contour integral solutions, see [1] and further references therein. Further, a non-standard perturbation analysis at the continuous spectrum end can be performed, ultimately describing the desired map connecting the two asymptotic representations. It also permits a re-formulation as a one-dimensional boundary integral equation, whose regularization allows its further asymptotic and numerical analysis. We briefly review all the above, with an interesting open question being whether the presence of an ‘incoming’ and an ‘outgoing’ parts in the sought complex integral solution implies relevance of factorization techniques of Wiener-Hopf type.  

[1] D. P. Hewett, J. R. Ockendon, V. P. Smyshlyaev, Contour integral solutions of the parabolic wave equation, Wave Motion, 84, 90–109 (2019) Preformatted version:
Wednesday 14th August 2019
09:00 to 10:00 Michael Marder
Analytical solutions of dynamic fracture and friction at the atomic scale
Following an example set by Slepyan, it proves possible to employ the Wiener-Hopf method to obtain exact solutions for fracture and friction problems at the atomic scale. I will describe a number of physical phenomena that have been analyzed in this way. These include the velocity gap and micro-branching instability for dynamic cracks, a connection of friction with self-healing pulses, and resolution of the energy transport paradox for supersonic cracks.
10:00 to 10:30 John Raymond Willis
Transmission and reflection at an interface between metamaterial and ordinary material
A contribution to the subject in the title is made, in the case that the metamaterial has random microstructure. A variational approach permits the development of a system of integral equations which can be replaced by a Wiener-Hopf system.The equations retain information on the metamaterial up to two-point probabilities. The formulation will be developed in detail for a configuration of particular simplicity -- acoustic materials, all with the same modulus but different densities. A special case, for which the problem reduces to a very simple scalar Wiener-Hopf problem, has been solved, giving explicit formulae for transmission and reflection coefficients. It should be possible to develop the analysis further and obtain more general solutions... It is likely that the audience will be able to provide useful input.
10:30 to 11:00 Leonid Slepyan
Greater generality brings simplicity

In this talk, I will discuss listed below problems with attendant circumstances and the results following straightforwardly from the formulation:

Mechanical wave momentum from the first principles. Wave Motion, 2016, 68, 283-290.
On the energy partition in oscillations and waves. Proc. R. Soc. A, 2015, 471: 20140838.
Betty Theorem and Orthogonality Relations for Eigenfunctions. Mechanics of Solids, 1979, 14, 74-77.
On a displacement of a deformable body in an acoustic medium. J. Appl. Math. Mech., 1963, 27, 1402-1411,
and possibly some others.

11:00 to 11:30 Morning Coffee
11:30 to 12:00 Alexander Movchan
Homogenisation and a Wiener-Hopf formulation for a scattering problem around a semi-infinite elastic structured duct
Authors: I.S. Jones, N.V. Movchan, A.B. Movchan

Abstract: The lecture will cover analysis of elastic waves in a flexural plate, which contains a semi-infinite structured duct.
The problem is reduced to a functional equation of the Wiener-Hopf type. The Kernel function reflects on the quasi-periodic
Green's function for an infinite periodic structure. Analysis of the Kernel function enables us to identify localised waveguide modes.
Homogenisation approximation has been derived to explain the modulation of the wave trapped within the structured duct.
Analytical findings are accompanied by numerical examples and simulations.
12:00 to 12:30 Lev Truskinovsky
Supersonic kinks in active solids
To show that steadily propagating nonlinear waves in active matter can be driven internally, we develop a prototypical model of a topological kink moving with a constant supersonic speed in a discrete bi-stable FPU chain capable of generating active stress. In contrast to subsonic kinks in such systems, that are necessarily dissipative, the obtained supersonic solutions are purely anti-dissipative. Joint work with N. Gorbushin.
12:30 to 13:30 Lunch at Churchill College
13:30 to 18:30 Free afternoon
19:30 to 22:00 Formal Dinner at Christ's College

Christ's College
St Andrew's Street, CB2 3BU - map

Smart casual
Thursday 15th August 2019
09:00 to 10:00 Malte Peter
Water-wave forcing on submerged plates
We discuss the application of the Wiener-Hopf method to linear water-wave interactions with submerged plates. As the guiding problem, the Wiener-Hopf method is used to derive an explicit expression for the reflection coefficient when a plane wave is obliquely incident upon a submerged semi-infinite porous plate in water of finite depth. Having used the Cauchy Integral Method in the factorisation, the expression does not rely on knowledge of any of the complex-valued eigenvalues or corresponding vertical eigenfunctions in the region occupied by the plate. It is shown that the Residue Calculus technique yields the same result as the Wiener-Hopf method for this problem and this is also used to derive an analytical expression for the solution of the corresponding finite-plate problem. Applications to submerged rigid plates and elastic plates are discussed as well.
10:00 to 10:30 Xun Huang
Turbofan noise detection and control studies by the Wiener-Hopf Technique
This talk would focus on one of the main themes of this workshop: the diverse applications of the Wiener-Hopf technique for aerospace in general and turbofan noise problems in particular. First, I will give a theoretical model based on the Wiener-Hopf method (and matrix kernel factorisation) to unveil possible noise control mechanisms due to trailing-edge chevrons on the bypass duct of aircraft engine. Next, I will propose a new testing approach that relies on the forward propagation model based on the Wiener-Hopf method. The key contribution is the development of the inverse acoustic scattering approach for a sensor array by combining compressive sensing in a non-classical way. Last but not least, I will demonstrate some of the new aerospace applications of the Wiener-Hopf technique with recently popular deep neural networks.
10:30 to 11:00 Elena Luca
Numerical solution of matrix Wiener–Hopf problems via a Riemann–Hilbert formulation
In this talk, we present a fast and accurate numerical method for the solution of scalar and matrix Wiener–Hopf problems. The Wiener–Hopf problems are formulated as Riemann–Hilbert problems on the real line, and the numerical approach for such problems of e.g. Trogdon & Olver (2015) is employed. It is shown that the known far-field behaviour of the solutions can be exploited to construct tailor-made numerical schemes providing accurate results. A number of scalar and matrix Wiener–Hopf problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the new approach.

This is joint work with Prof. Stefan G. Llewellyn Smith (UCSD).
11:00 to 11:30 Morning Coffee
11:30 to 12:00 Vito Daniele
Fredholm factorization of Wiener-Hopf equations (presented by Guido Lombardi)

In spite of the great efforts by many  studies, there have been little progresses towards a general method of constructive factorizations to get exact solution of vector WH equations.   The aim of  this talk  is the presentation of an alternative solution technique that is based to the reduction of the WH equations to  Fredholm equations of second kind (Fredholm factorization). The presentation will focus to the applications of the  Fredholm factorization  to  WH equations occurring in diffraction problem. In particular it   is based on five steps:1) Deduction of the WH equations of the problem,2) Reduction of the WH equations to Fredholm integral equations (FIE) ,3) Solution of the Fredholm integral equations , 4)Analytical continuation of the numerical solution of the FIE,5) Evaluation of the physical field components if present: reflected and refracted plane waves, diffracted fields, surface waves, lateral waves, leaky waves, mode excitations, near field. A characteristic  example of problem will be presented in the following talk.

12:00 to 12:30 Guido Lombardi
Complex scattering and radiation problems using the Generalized Wiener-Hopf Technique

This talk focuses on the effectiveness of Generalized Wiener-Hopf Technique (GWHT) in studying complex scattering and radiation problems constituted of planar and angular regions made by impenetrable and/or composite penetrable materials. First, we present theoretical models in the spectral domain using Generalized Wiener-Hopf equations (GWHEs). Next,  we apply the novel and effective Fredholm factorization technique to get semi analytical solution of the problem by using integral equation representations. The semi-analyticity of the GWHT solution allows physical insights in terms of spectral component of fields. The case study presented in the talk is the electromagnetic field scattering and radiation of a perfectly electrically conducting wedge over a grounded dielectric slab.

Authors: V. Daniele, G. Lombardi, R.S. Zich, Politecnico di Torino, Torino, Italy

12:30 to 13:30 Lunch at Churchill College
13:30 to 14:00 Justin Jaworski
Owl-inspired mechanisms of turbulence noise reduction
Many owl species rely on specialized plumage to mitigate their aerodynamic noise and achieve functionally-silent flight while hunting. One such plumage feature, a tattered arrangement of flexible trailing-edge feathers, is idealized as a semi-infinite poroelastic plate to model the effects that edge compliance and flow seepage have on the noise production. The interaction of the poroelastic edge with a turbulent eddy is examined analytically with respect to how efficiently the edge scatters the eddy as aerodynamic noise. The scattering event is formulated and solved as a scalar Wiener-Hopf problem to identify how the noise scales with the flight velocity, where special attention is paid to the limiting cases of rigid-porous and elastic-impermeable plate conditions. Results from this analysis identify new parameter spaces where the porous and/or elastic properties of a trailing edge may be tailored to diminish or effectively eliminate the edge scattering effect and may contribute to the owl hush-kit.
14:00 to 14:30 Nikolai Gorbushin
Steady-state interfacial cracks in bi-material elastic lattices
Fracture mechanics serves both engineering and science in various ways, such as studies of material integrity and physics of earthquakes. Its main object is to analyse crack nucleation and growth depending on features of a particular application. It is common to study cracks in homogeneous materials, however analysis of cracks in bi-materials is important as well, especially in modelling of frictional motion between solids at macro-scale and inter-granular fracture in polycrystallines at micro-scale. The analysis of fracture in dissimilar materials is the main topic of this research. We present the analytical model of steady-state cracks in bi-material square lattices and show its connection with associated macro-level fracture problem.  We consider a semi-infinite crack propagating along the interface between two mass-spring square lattices of different properties. Assuming the linear interaction between lattice masses, we can apply integral transforms and obtain the matrix Wiener-Hopf problem from original equations of motion. In this particular case, the kernel matrix is triangular which significantly simplifies the factorisation procedure and even makes possible to reduce to the scalar Wiener-Hopf problem. The discreteness of the problem, however, does not allow to derive factorisation analytically and numerical factorisation was performed. We show that the problem discreteness reveals microscopic radiation in form of decaying elastic waves emanating from a crack tip. These waves are invisible at macro-scale but their energy contributes to the global energy dissipation during the fracture process. We also demonstrate effects of the material properties mismatch and link the microscopic parameters with the macro-level fracture characteristics.
14:30 to 15:00 Matthew Priddin
Using iteration to solve n by n matrix Wiener-Hopf equations involving exponential factors with numerical implementation
Wiener-Hopf equations involving $n\times n$ matrices can arise when solving mixed boundary value problems with $n$ junctions at which the boundary condition to be imposed changes form.  The offset Fourier transforms of the unknown boundary values lead to exponential factors which require careful consideration when applying the Wiener-Hopf technique. We consider the generalisation of an iterative method introduced recently (Kisil 2018) from $2\times 2$ to $n\times n$ problems. This may be effectively implemented numerically by employing a spectral method to compute Cauchy transforms. We illustrate the approach through various examples of scattering from collinear rigid plates and consider the merits of the iterative method relative to alternative approaches to similar problems.
15:00 to 15:30 Francesco Dal corso
Moving boundary value problems in the dynamics of structures
The dynamics of structures partially inserted into a frictionless sliding sleeve defines a moving boundary value problem revealing the presence of an outward configurational force at the constraint, parallel to the sliding direction. The configurational force, differing from that obtained the quasi-static case only for a negligible proportionality coefficient, strongly affects the motion and introduces intriguing structural dynamic response. This will be shown through the two following problems:
- The sudden release of a rod with a concentrated weight attached at one end [1]. The solution of a differential-algebraic equation (DAE) system provides the evolution, where the elastic rod may slip alternatively in and out from the sliding sleeve. The nonlinear dynamics eventually ends with the rod completely injected into or completely ejected from the constraint;
-  The vibrations of a periodic and infinite structural system [2]. Through Bloch-Floquet analysis it is shown that the band gap structure for purely longitudinal vibration is broken so that axial propagation may occur at frequencies that are forbidden in the absence of a transverse oscillation.
Moreover, conditions for which flexural oscillation may induce axial resonance are disclosed.  
The results represent innovative concepts ready to be used in advanced applications, ranging from soft-robotics to earthquake protection.  

Acknowledgments: Financial support from the Marie Sklodowska-Curie project 'INSPIRE - Innovative ground interface concepts for structure  protection'

[1]  Armanini, Dal Corso, Misseroni, Bigoni (2019). Configurational forces and nonlinear structural dynamics. J. Mech. Phys. Solids, doi: 10.1016/j.jmps.2019.05.009
[2] Dal Corso, Tallarico, Movchan, Movchan, Bigoni, (2019). Nested Bloch waves in elastic structures with configurational forces. Phil. Trans. R. Soc. A, doi: 10.1098/rsta.2019.0101
15:30 to 16:00 Afternoon Tea
16:00 to 16:30 Larissa Fradkin
Elastic wedge diffraction, with applications to non-destructive evaluation

Co-authors: Samar Chehade and Michel Darmon

Diffraction of the elastic plane wave by an infinite straight-edged 2D or 3D wedge made of an isotropic solid is a canonical problem that has no analytical solution. We review three major semi-analytical approaches to this problem and discuss their application in non-destructive evaluation as well as testing, cross-validation and experimental validation. We draw attention to high sensitivity of the backscatter diffraction coefficients to the Poisson ratio.

16:30 to 17:00 Davide Bigoni
Shear band dynamics
When a ductile material is subject to severe strain, failure is preluded by the emergence of shear bands, which initially nucleate  in  a  small  area,  but  quickly  extend  rectilinearly  and  accumulate  damage,  until  they  degenerate  into  fractures.  Therefore, research on shear bands yields a fundamental understanding of the intimate rules of failure, so that it may be important in the design of new materials with superior mechanical performances.A shear band of finite length, formed inside a ductile material at a certain stage of a continued homogeneous strain, provides  a  dynamic  perturbation to an incident wave field, which strongly influences the dynamics of the material and affects  its  path  to  failure.  The  investigation  of  this  perturbation  is  presented  for  a  ductile  metal,  with  reference  to  the  incremental mechanics of a material obeying the J2–deformation theory of plasticity. The treatment originates from the derivation of integral representations relating the incremental mechanical fields at every point of the medium to the incremental  displacement  jump  across  the  shear  band  faces,  generated  by  an  impinging  wave.  The  boundary  integral  equations are numerically approached through a collocation technique, which keeps into account the singularity at the shear band tips and permits the analysis of an incident wave impinging a shear band. It is shown that the presence of the shear band induces a resonance, visible in the incremental displacement field and in the stress intensity factor at the shear band tips, which promotes shear band growth. Moreover, the waves scattered by the shear band are shown to generate a fine texture of vibrations, parallel to the shear band line and propagating at a long distance from it, but leaving a sort of conical shadow zone, which emanates from the tips of the shear band [1,2].
[1] Giarola, D., Capuani, D. Bigoni, D. (2018) The dynamics of a shear band. J. Mech. Phys. Solids, 112, 472-490.
[2] Giarola, D., Capuani, D. Bigoni, D. (2018) Dynamic interaction of multiple shear bands. Scientific Reports 8 16033
Friday 16th August 2019
09:00 to 10:00 Dmitry Ponomarev
Spectral theory of convolution operators on finite intervals: small and large interval asymptotics
One-dimensional convolution integral operators play a crucial role in a variety of different contexts such as approximation and probability
theory, signal processing, physical problems in radiation transfer, neutron transport, diffraction problems, geological prospecting issues and quantum gases statistics,.
Motivated by this, we consider a generic eigenvalue problem for one-dimensional convolution integral operator on an interval where the kernel is
real-valued even $C^1$-smooth function which (in case of large interval) is absolutely integrable on the real line.
We show how this spectral problem can be solved by two different asymptotic techniques that take advantage of the
size of the interval.
In case of small interval, this is done by approximation with an integral operator for which there exists a commuting
differential operator thereby reducing the problem to a boundary-value problem for second-order ODE, and often
giving the solution in terms of explicitly available special functions such as prolate spheroidal harmonics.
In case of large interval, the solution hinges on solvability, by Riemann-Hilbert approach, of an approximate auxiliary
integro-differential half-line equation of Wiener-Hopf type, and culminates in simple characteristic equations for
eigenvalues, and, with such an approximation to eigenvalues, approximate eigenfunctions are given in an explicit form.
Besides the crude periodic approximation of Grenander-Szego, since 1960s, large-interval spectral results were available
only for integral operators with kernels of a rapid (typically exponential) decay at infinity or those whose symbols
are rational functions. We assume the symbol of the kernel, on the real line, to be continuous and, for the sake of
simplicity, strictly monotonically decreasing with distance from the origin. Contrary to other approaches, the proposed
method thus relies solely on the behavior of the kernel's symbol on the real line rather than the entire complex plane
which makes it a powerful tool to constructively deal with a wide range of integral operators.
We note that, unlike finite-rank approximation of a compact operator, the auxiliary problems arising in both small-
and large-interval cases admit infinitely many solutions (eigenfunctions) and hence structurally better represent
the original integral operator.
The present talk covers an extension and significant simplification of the previous author's result on
Love/Lieb-Liniger/Gaudin equation.
10:00 to 10:30 Michael Nieves
Phase transition processes in flexural structured systems with rotational inertia
Failure and phase transition processes in mass-spring systems have been extensively studied in the literature, based on the approach developed in [1]. Only a few attempts at characterising these processes in flexural systems exist, see for instance [2, 3, 4, 5]. In comparison with mass-spring systems, flexural structures have a larger range  of applicability. They can describe phenomena in systems at various scales, including microlevel waves in materials and  dynamic processes in civil engineering assemblies such as bridges and buildings found in society. Flexural systems also provide a greater variety of modelling tools, related to loading configurations and physical parameters, that can be used to achieve a particular response.

Here we consider the role of rotational inertia in the process of phase transition in a one-dimensional flexural system, that may represent a simplified model of the  failure of a bridge exposed to hazardous vibrations. The phase transition process is assumed to occur with a uniform speed that is driven by feeding waves carrying energy produced by an applied oscillating moment and force. We show that the problem can be reduced to a functional equation via the Fourier transform which is solved using the Wiener-Hopf technique. From the solution we identify the dynamic behaviour of the system during the transition process. The minimum energy required to initiate the phase transition process with a given speed is determined and it is shown there exist parameter domains defined by the force and moment amplitudes where the phase transition can occur. The influence of the rotational inertia of the system on the wave radiation phenomenon connected with the phase transition is also discussed. All results are supplied with numerical illustrations confirming the analytical predictions.

Acknowledgement: M.J.N. and M.B. gratefully acknowledge the support of the EU H2020 grant MSCA-IF-2016-747334-CAT-FFLAP.

[1] Slepyan, L.I.: Models and Phenomena in Fracture Mechanics, Foundations of Engineering Mechanics, Springer, (2002).
[2] Brun, M., Movchan, A.B. and Slepyan, L.I.: Transition wave in a supported heavy beam, J. Mech. Phys. Solids 61, no. 10, pages 2067–2085, (2013).
[3] Brun, M., Giaccu, G.F., Movchan, A., B., and Slepyan, L. I.. Transition wave in the collapse of the San Saba Bridge. Front. Mater. 1:12, (2014). doi: 10.3389/fmats.2014.00012.
[4] Nieves, M.J., Mishuris, G.S., Slepyan, L.I.: Analysis of dynamic damage propagation in discrete beam structures, Int. J. Solids Struct. 97-98, pages 699–713, (2016).
[5] Garau, M., Nieves, M.J. and Jones, I.S. (2019): Alternating strain regimes for failure propagation in flexural systems, Q. J. Mech. Appl. Math., hbz008,
10:30 to 11:00 Konstantin Ustinov
Application of Khrapkov’s technique of 2x2 matrix factorization to solving problems related to interface cracks
11:00 to 11:30 Morning Coffee
11:30 to 12:00 Ian Thompson
Diffraction in Mindlin plates
Plate theory is important for modelling thin components used in engineering applications, such as metal panels used in aeroplane wings and submarine hulls. A typical application is nondestructive testing, where a wave is transmitted into a panel, and analysis of the scattered response is used to determine the existence, size and location of cracks and other defects. To use this technique, one must first develop a clear theoretical understanding the diffraction patterns that occur when a wave strikes the tip of a fixed or free boundary. Diffraction by semi-infinite rigid strips and cracks in isotropic plates modelled by Kirchhoff theory was considered by Norris & Wang(1994). Although both problems require the application of two boundary conditions on the rigid or free boundary, the resulting Wiener-Hopf equations can be decoupled, leading to a pair of scalar problems. Later, Thompson & Abrahams (2005 & 2007) considered diffraction caused by a crack in a fibre reinforced Kirchhoff plate. The resulting problem is much more complicated than the corresponding isotropic case, but again leads to two separate, scalar Wiener-Hopf equations. In this presentation, we consider diffraction by rigid strips and cracks in plates modelled by Mindlin theory. This is a more accurate model, which captures physics that is neglected by Kirchhoff theory, and is valid at higher frequencies. However, it requires three boundary conditions at an interface. The crack problem and the rigid strip problem each lead to one scalar Wiener-Hopf equation and one 2x2 matrix equation (four problems in total). The scalar problems can be solved in a relatively straightforward manner, but the matrix problems (particularly the problem for the crack) are complicated. However, the kernels have some interesting properties that suggest the possibility of accurate approximate factorisations.

A. N. Norris and Z. Wang. Bending-wave diffraction from strips and cracks on thin plates. Q. J. Mech. Appl. Math., 47:607-627, 1994.
I. Thompson and I. D. Abrahams. Diffraction of flexural waves by cracks in orthotropic thin elastic plates.I Formal solution. Proc. Roy. Soc. Lond., A, 461:3413-3434, 2005.
I. Thompson and I. D. Abrahams. Diffraction of flexural waves by cracks in orthotropic thin elastic plates.II. Far field analysis. Proc. Roy. Soc. Lond., A, 463:1615-1638, 2007.
12:00 to 12:30 Pavlos Livasov
Two vector Wiener-Hopf equations with 2x2 kernels containing oscillatory terms
In the first part we discuss a steady-state problem for an interface crack between two dissimilar elastic materials. We consider a model of the process zone described by imperfect transmission conditions that reflect the bridging effect along a finite part of the interface in front of the crack. By means of Fourier transform, the problem is reduced into a Wiener-Hopf equation with a 2x2 matrix, containing oscillatory terms. We factorize the kernel following an existing numerical method and analyse its performance for various parameters of the problem. We show that the model under consideration leads to the classic stress singularity at the crack tip. Finally, we derive conditions for the existence of an equilibrium state and compute admissible length of the process zone.  
For the second part of the talk, we consider propagation of a dynamic crack in a periodic structure with internal energy. The structural interface is formed by a discrete set of uniformly distributed alternating compressed and stretched bonds. In such a structure, the fracture of the initially stretched bonds is followed by that of the compressed ones with an unspecified time-lag. That, in turn, reflects the impact of both the internal energy accumulated inside the pre-stressed interface and the energy brought into the system by external loading. The application to the original problem of continuous (with respect to time) and  selective discrete (with respect to spatial coordinate) Fourier transforms yields another vector Wiener-Hopf equation with a kernel containing oscillating terms. We use a perturbation technique to factorise the matrix.   Finally, we show similarities and differences of the matrix-valued kernels mentioned above and discuss the chosen approaches for their factorisation.
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:00 Alexander Galybin
Application of the Wiener-Hopf approach to incorrectly posed BVP of plane elasticity
14:00 to 14:30 Matthew Colbrook
Solving Wiener-Hopf type problems numerically: a spectral method approach
The unified transform is typically associated with the solution of integrable nonlinear PDEs. However, after an appropriate linearisation, one can also apply the method to linear PDEs and develop a spectral boundary-based method. I will discuss recent advances of this method, in particular, the application of the method to problems in unbounded domains with solutions having corner singularities. Consequently, a wide variety of mixed boundary condition problems can be solved without the need for the Wiener-Hopf technique. Such problems arise frequently in acoustic scattering or in the calculation of electric fields in geometries involving finite and/or multiple plates. The new approach constructs a global relation that relates known boundary data, such as the scattered normal velocity on a rigid plate, to unknown boundary values, such as the jump in pressure upstream of the plate. This can be viewed formally as a domain dependent Fourier transform of the boundary integral equations. By approximating the unknown boundary functions in a suitable basis expansion and evaluating the global relation at collocation points, one can accurately obtain the expansion coefficients of the unknown boundary values. The local choice of basis functions is flexible, allowing the user to deal with singularities and complicated boundary conditions such as those occurring in elasticity models or spatially variant Robin boundary conditions modelling porosity.
14:30 to 15:00 Ivan Argatov
Application of the Wiener–Hopf technique in contact problems
Problems involving the contact interaction between two elastic bodies, or between an elastic body (called substrate) and a rigid body (called indenter), have occupied the attention of engineering researchers for well over a century. In recent years much attention has been paid to mechanical aspects of contact and adhesion in biological systems, which has resulted in formulating new contact problems, in particular, for a thin elastic layer on a substrate being indented by an indenter of non-canonical shape. Since problems in contact mechanics belong to the class of mixed boundary value problems and can be usually reduced to solving integral equations, it is natural to expect that the Wiener–Hopf method will one of the powerful analytical tools for their investigation. The Wiener–Hopf technique in combination with asymptotic methods has the advantage of universality in obtaining solutions in the analytical form as well as of simplicity for further qualitative analysis. In the present talk we briefly overview the application of the Wiener–Hopf technique to a representative range of contact problems, emphasizing the need of using complementarity asymptotic techniques to cover a larger space of the problem parameters.
15:00 to 15:30 Mikhail Lyalinov
Functional-integral equations and diffraction by a truncated wedge
In this work we study diffraction of a plane incident wave in a complex 2D domain composed by two shifted angular domains having a part of their common boundary. The perfect (Dirichlet or Neumann) boundary conditions are postulated on the polygonal boundary of such compound domain. By means of the Sommerfeld-Malyuzhinets technique the boundary-value problem at hand is reduced to a non-standard systems of Malyuzhinets-type functional-integral equations and then to a Fredholm integral equation of the second kind. Existence and uniqueness of the solution for the diffraction problem is studied and is based on the Fredholm alternative for the integral equation. The far field asymptotics of the wave field is also addressed.

15:30 to 16:00 Gennady Mishuris
Comments on the approximate factorisation of matrix functions with unstable sets of partial indices

It is well known for more than 60 years that the set of partial indices of a non-singular matrix function may be unstable under small perturbations of the matrix [1]. This happens when the difference between the largest and the smallest indices is larger than unity. Although the total index of the matrix preserves its value, this former makes it extremely difficult to use this very powerful method for solving practical problems in this particular case. Moreover, since there does not exist a general constructive technique for matrix factorisation or even for the determination of the partial indices of the matrix, this fact looks like an unavoidable obstacle. Following [2], in this talk, we try to answer a less ambitious question focusing on the determination of the conditions allowing one to construct a family of matrix functions preserving a majority of the properties of the original matrix with non-stable partial indices that is close to the original matrix function.

This work was partially supported by a grant from the Simons Foundation. GM is also acknowledge Royal Society for the Wolfson Research Merit Award.

[1] Gohberg I. & Krein M. 1958 Uspekhi Mat. Nauk.XIII, 3–72 (in Russian).
[2] Mishuris G, Rogosin S. 2018 Regular approximate factorization of a class of matrix-function with an unstable set of partial indices. Proc.R.Soc.A 474:20170279.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons