**Mary Aprahamian**

The University of Manchester

School of Mathematics

Alan Turing Building, 2.111

M13 9PL, Manchester, UK

+44 (0)161 306 3670 (office)

Mary.Aprahamian[at]manchester.ac.uk

The University of Manchester

School of Mathematics

Alan Turing Building, 2.111

M13 9PL, Manchester, UK

+44 (0)161 306 3670 (office)

Mary.Aprahamian[at]manchester.ac.uk

Here is a brief overview of my experience so far. My full CV offers more detail.

Download my CV

I am a postdoc in the Numerical Linear Algebra group at the School of Mathematics, The University of Manchester. I am working with Prof. Nick Higham on the project Functions of Matrices: Theory and Computation. I have worked on some special matrix functions and have a strong interest in network analysis.

My research

I recently defended my thesis * Theory and Algorithms for Periodic Functions of Matrices, with Applications *, which was completed under the supervision of Prof. Nick Higham at The University of Manchester.
I completed my undergraduate degree in Mathematics at The University of Edinburgh in 2011.
Prior to this I attended the Red Cross Nordic UWC in Norway, where I studied for an International Baccalaureate Diploma.

I was born and raised in Varna, at the Black Sea coast of Bulgaria and I am ethnically Armenian. In my spare time I enjoy reading Maths papers at different locations in the world.

Theoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions are studied. Conditions for existence are given, all possible values are characterized, and principal values are defined and shown to be unique primary matrix functions. Various functional identities are derived, some of which are new even in the scalar case, with care taken to specify precisely the choices of signs and branches. Key tools used in the derivations are the matrix unwinding function and the matrix sign function. A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree rational approximation is derived.

The relative importance of nodes in a network can be quantified via functions of the adjacency matrix. Two popular choices of function are the exponential, which is parameter-free, and the resolvent function, which yields the Katz centrality measure. Katz centrality can be the more computationally efficient, especially for large directed networks, and has the benefit of generalizing naturally to time-dependent network sequences, but it depends on a parameter. We give a prescription for selecting the Katz parameter based on the objective of matching the centralities of the exponential counterpart. For our new choice of parameter the resolvent can be very ill conditioned, but we argue that the centralities computed in floating point arithmetic can nevertheless reliably be used for ranking. Experiments on six real networks show that the new choice of Katz parameter leads to rankings of nodes that generally match those from the exponential centralities well in practice.

A new matrix function corresponding to the scalar unwinding number of Corless, Hare, and Jeffrey is introduced. This matrix unwinding function, is shown to be a valuable tool for deriving identities involving the matrix logarithm and fractional matrix powers. The unwinding function is also shown to be closely connected with the matrix sign function. An algorithm for computing the unwinding function based on the Schur–Parlett method with a special reordering is proposed. It is shown that matrix argument reduction using the matrix unwinding function can give significant computational savings in the evaluation of the exponential by scaling and squaring algorithms.

Here you can find some of the meetings I have attended recently and some I am planning to attend in the near future.