Factorisation of matrix functions: New techniques and applications
Monday 12th August 2019 to Friday 16th 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 WienerHopf factorisation through an intermediate space and applications to diffraction theory An operator factorisation conception is investigated for a general WienerHopf 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, $W \sim P C_{P Z}$. 
INI 1  
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. 
INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 14:00 
Raphael Assier Recent advances in the quarterplane problem using functions of two complex variables 
INI 1  
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 LipschitzHankel
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, 'Wienerhopf analysis of transient phenomenon caused by timevarying resistive screen in waveguide', IEICE transactions on electronics, vol. E85C, 10, pp.18001807, 2002 [2] DS Jones, 'Incomplete Bessel functions. I', proceedings of the Edinburgh Mathematical Society, 50, pp 173183, 2007 [3] MM Agrest, MM Rikenglaz, 'Incomplete LipshitzHankel integrals', USSR Comp. Math. and Math Phys., vol 7, 6, pp.206211, 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 
INI 1  
14:30 to 15:00 
Andrey Shanin Ordered Exponential (OE) equation as an alternative to the WienerHopf method 
INI 1  
15:00 to 15:30 
Anastasia Kisil Generalisation of the WienerHopf pole removal method and application to n by n matrix functions 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 16:30 
Basant Lal Sharma WienerHopf 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 WienerHopf on the unit circle (in an annulus in complex plane).
In some of these examples, the WienerHopf
problem is scalar, while in other cases it is a matrix WienerHopf
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 wellknown WienerHopf equations in
continuum models on the real line (in an strip in complex plane), a few of
which are still open problems.

INI 1  
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 integervalued 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 onetoone 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 1from have first order poles in marked points and removable singularity at infinity. The Fucshian system of equations induced from this 1form 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 1796. 
INI 1  
17:00 to 18:00  Welcome Wine Reception at INI 
09:00 to 10:00 
Ilya Spitkovsky WienerHopf 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
BohrFourier series.
The result for W extends to the matrix setting; not so
for APW. Moreover, the factorability criterion even for 2by2 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.

INI 1  
10:00 to 11:00 
Lasha Ephremidze On JanashiaLagvilava method of matrix spectral factorisation JanashiaLagvilava method is a relatively new algorithm of matrix spectral factorisation which can be applied to compute an approximate spectral factor of any matrix function (nonrational, 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. 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Andreas Kyprianou WienerHopf Factorisations for Levy processes
We give an introduction to the the theory of WienerHopf
factoirsations for Levy processes and discuss some very recent examples which
are stimulated by some remarkable connections with selfsimilar Markov
processes.

INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 14:30 
Sergei Rogosin Factorisation of triangular matrixfunctions of arbitrary order
It will be discussed an efficient method for factorization of square triangular matrixfunctions 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 matrixfunctions 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 matrixfunctions with Hoelder continuous entries. Tough the proposed method could be realised for wider classes of matrixfunctions. Chebotarev's method is extended here to the triangular matrixfunctions 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 matrixfunctions 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). 
INI 1  
14:30 to 15:00 
Cristina Camara A RiemannHilbert approach to Einstein field equations
The field equations of gravitational theories in 4 dimensions are nonlinear PDE's that are difficult to solve in general. By restricting to a subspace of solutions that only depend on two spacetime coordinates, alternative approaches to solving those equations become available. We present here the RiemannHilbert 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 nonlinear gravitational field equations using simple complex analytic methods.

INI 1  
15:00 to 15:30 
Aloknath Chakrabarti Solving WienerHopf Problems by the aid of Fredholm Integral Equations of the Second Kind
A class of WienerHopf 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 WienerHopf problems, for which the WienerHopf factorization of matrices can be avoided. Several examples are taken up, demonstrating the present method of solution of WienerHopf problems.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 16:30 
Victor Adukov On explicit and exact solutions of the WienerHopf 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 WienerHopf 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. 
INI 1  
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
highfrequency 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 boundaryvalue problem for a
special PDE, describing transition from a “modal” to a “scattered”
highfrequency asymptotic behaviour. The latter problem
was first formulated and analysed by M.M. Popov starting from 1970s. 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 nonstandard perturbation analysis at the continuous spectrum end can be performed, ultimately describing the desired map connecting the two asymptotic representations. It also permits a reformulation as a onedimensional 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 WienerHopf 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: http://www.newton.ac.uk/files/webform/587.tex 
INI 1 
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 WienerHopf 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 microbranching instability for dynamic cracks, a connection of friction with selfhealing pulses, and resolution of the energy transport paradox for supersonic cracks.

INI 1  
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 WienerHopf system.The equations retain information on the metamaterial up to twopoint 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 WienerHopf 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.

INI 1  
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: 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:00 
Alexander Movchan Homogenisation and a WienerHopf formulation for a scattering problem around a semiinfinite 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 semiinfinite structured duct. The problem is reduced to a functional equation of the WienerHopf type. The Kernel function reflects on the quasiperiodic 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. 
INI 1  
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 bistable 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 antidissipative. Joint work with N. Gorbushin.

INI 1  
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 
09:00 to 10:00 
Malte Peter Waterwave forcing on submerged plates
We discuss the application of the WienerHopf method to linear waterwave interactions with submerged plates. As the guiding problem, the WienerHopf method is used to derive an explicit expression for the reflection coefficient when a plane wave is obliquely incident upon a submerged semiinfinite 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 complexvalued 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 WienerHopf method for this problem and this is also used to derive an analytical expression for the solution of the corresponding finiteplate problem. Applications to submerged rigid plates and elastic plates are discussed as well.

INI 1  
10:00 to 10:30 
Xun Huang Turbofan noise detection and control studies by the WienerHopf Technique
This talk would focus on one of the main themes of this
workshop: the diverse applications of the WienerHopf
technique for aerospace in general and turbofan noise problems in particular.
First, I will give a theoretical model based on the WienerHopf method (and
matrix kernel
factorisation) to unveil possible noise control
mechanisms due to trailingedge 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 WienerHopf method. The key contribution is the
development of the inverse acoustic scattering approach for a sensor array by
combining compressive sensing in a nonclassical way. Last but not least, I
will demonstrate some of the new aerospace applications of the WienerHopf
technique with recently popular deep neural networks.

INI 1  
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 farfield behaviour of the solutions can be exploited to construct tailormade 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 semiinfinite plane are solved using the new approach. This is joint work with Prof. Stefan G. Llewellyn Smith (UCSD). 
INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:00 
Vito Daniele Fredholm factorization of WienerHopf 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. 
INI 1  
12:00 to 12:30 
Guido Lombardi Complex scattering and radiation problems using the Generalized WienerHopf Technique This talk focuses on the effectiveness of Generalized WienerHopf 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 WienerHopf 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 semianalyticity 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. 
INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 14:00 
Justin Jaworski Owlinspired mechanisms of turbulence noise reduction
Many owl species rely on specialized plumage to mitigate
their aerodynamic noise and achieve functionallysilent flight while hunting.
One such plumage feature, a tattered arrangement of
flexible trailingedge feathers, is idealized as a semiinfinite 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 WienerHopf problem to identify how the noise scales with the flight
velocity, where special attention is paid to the limiting cases of rigidporous
and elasticimpermeable 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 hushkit.

INI 1  
14:00 to 14:30 
Nikolai Gorbushin Steadystate interfacial cracks in bimaterial 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 bimaterials is important as well,
especially in modelling of frictional motion between solids at macroscale and
intergranular fracture in polycrystallines at microscale. The analysis of
fracture in dissimilar materials is the main topic of this research. We present
the analytical model of steadystate cracks in bimaterial square lattices and
show its connection with associated macrolevel fracture problem. We consider a semiinfinite crack propagating
along the interface between two massspring square lattices of different
properties. Assuming the linear interaction between lattice masses, we can
apply integral transforms and obtain the matrix WienerHopf 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 WienerHopf 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 macroscale 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 macrolevel fracture characteristics.

INI 1  
14:30 to 15:00 
Matthew Priddin Using iteration to solve n by n matrix WienerHopf equations involving exponential factors with numerical implementation
WienerHopf 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 WienerHopf 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.

INI 1  
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 quasistatic 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 differentialalgebraic 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 BlochFloquet 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 softrobotics to earthquake protection. Acknowledgments: Financial support from the Marie SklodowskaCurie project 'INSPIRE  Innovative ground interface concepts for structure protection' PITNGA2019813424INSPIRE. [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 
INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 16:30 
Larissa Fradkin Elastic wedge diffraction, with applications to nondestructive evaluation Coauthors: Samar Chehade and Michel Darmon 
INI 1  
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]. References [1] Giarola, D., Capuani, D. Bigoni, D. (2018) The dynamics of a shear band. J. Mech. Phys. Solids, 112, 472490. [2] Giarola, D., Capuani, D. Bigoni, D. (2018) Dynamic interaction of multiple shear bands. Scientific Reports 8 16033 
INI 1 
09:00 to 10:00 
Dmitry Ponomarev Spectral theory of convolution operators on finite intervals: small and large interval asymptotics
Onedimensional 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 onedimensional convolution integral operator on an interval where the kernel is realvalued 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 boundaryvalue problem for secondorder 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 RiemannHilbert approach, of an approximate auxiliary integrodifferential halfline equation of WienerHopf 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 GrenanderSzego, since 1960s, largeinterval 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 finiterank approximation of a compact operator, the auxiliary problems arising in both small and largeinterval 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/LiebLiniger/Gaudin equation. 
INI 1  
10:00 to 10:30 
Michael Nieves Phase transition processes in flexural structured systems with rotational inertia
Failure and phase transition
processes in massspring 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 massspring 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 onedimensional 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 WienerHopf 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 MSCAIF2016747334CATFFLAP. References [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. 9798, 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, https://eur02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fdoi.org%2F10.1093%2Fqjmam%2Fhbz008&data=02%7C01%7C%7Cca24c94f14fb47b2a98908d6f19a9002%7Cd47b090e3f5a4ca084d09f89d269f175%7C0%7C0%7C636962043472937444&sdata=hFcD7qiLBQweKalUwfiI8DE4OoKVDBet7AwngVFgEf0%3D&reserved=0. 
INI 1  
10:30 to 11:00 
Konstantin Ustinov Application of Khrapkov’s technique of 2x2 matrix factorization to solving problems related to interface cracks 
INI 1  
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 semiinfinite 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 WienerHopf 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 WienerHopf 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 WienerHopf 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. References A. N. Norris and Z. Wang. Bendingwave diffraction from strips and cracks on thin plates. Q. J. Mech. Appl. Math., 47:607627, 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, 461:34133434, 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, 463:16151638, 2007. 
INI 1  
12:00 to 12:30 
Pavlos Livasov Two vector WienerHopf equations with 2x2 kernels containing oscillatory terms
In the first part we discuss a steadystate 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 WienerHopf 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 timelag. That, in turn, reflects the impact of both the internal energy accumulated inside the prestressed 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 WienerHopf equation with a kernel containing oscillating terms. We use a perturbation technique to factorise the matrix. Finally, we show similarities and differences of the matrixvalued kernels mentioned above and discuss the chosen approaches for their factorisation. 
INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 14:00 
Alexander Galybin Application of the WienerHopf approach to incorrectly posed BVP of plane elasticity 
INI 1  
14:00 to 14:30 
Matthew Colbrook Solving WienerHopf 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 boundarybased 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
WienerHopf 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.

INI 1  
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 noncanonical 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.

INI 1  
15:00 to 15:30 
Mikhail Lyalinov Functionalintegral 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 SommerfeldMalyuzhinets technique the boundaryvalue problem at hand is reduced
to a nonstandard systems of Malyuzhinetstype functionalintegral 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. 
INI 1  
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 nonsingular 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 nonstable partial indices that is close to the original matrix function. 
INI 1 