arXiv subjects

Amie Albrecht

Publications and source records attributed to Amie Albrecht.

4 recordsLinked to original sources

arXiv2026

The fundamental equations for inversion of operator pencils on Banach space

We prove that the resolvent of a linear operator pencil is analytic on an open annulus if and only if the coefficients of the Laurent series satisfy a system of fundamental equations and are geometrically bounded. Our analysis extends earlier work on the fundamental equations to include the case where the resolvent has an isolated essential singularity. We find a closed form for the resolvent and use the fundamental equations to establish key spectral separation properties when the resolvent has only a finite number of isolated singularities. Finally we show that our results can also be applied to polynomial pencils.

math.FA↗

arXiv2022

Optimal driving strategies for a fleet of trains on level track with prescribed intermediate signal times and safe separation

We propose an analytic solution to the problem of finding optimal driving strategies that minimize total tractive energy consumption for a fleet of trains travelling on the same track in the same direction subject to clearance-time equality constraints that ensure safe separation and compress the line-occupancy timespan. We assume the track is divided into sections by a set of trackside signals at fixed locations. For each intermediate signal there is a signal-location segment consisting of the two adjacent sections. Successive trains are safely separated only if the leading train leaves each signal-location segment before the following train enters. The fleet can be safely separated by a complete set of clearance times and associated clearance-time inequality constraints. The problem of finding optimal schedules with safe separation has been solved for two trains but for larger fleets the problem rapidly becomes intractable as the number of trains and signals increases. The main difficulty is in distinguishing between active equality constraints and inactive inequality constraints. The curse of dimensionality means it is not feasible to check every different combination of active constraints, optimize the corresponding prescribed times and calculate the cost. Nevertheless we can formulate and solve an alternative problem with active clearance-time equality constraints for successive trains on every signal-location segment. We show that this problem can be formulated as an unconstrained convex optimization and we propose a viable solution algorithm that finds the optimal schedule and the associated optimal strategies for each train. Finally we use our solution to find optimal schedules for a busy inter-city shuttle service.

math.OC↗

arXiv2020

Optimal splitting of Parseval frames using Walsh matrices

In 2014 Adam Marcus, Daniel Spielman and Nikhil Srivastava used random vectors to prove a key discrepancy theorem and in so doing gave a positive answer to the long-standing Kadison-Singer Problem. In this paper we use Walsh matrices to construct a class of natural frames in Euclidean space and discuss how these frames relate to the key discrepancy theorem.

math.FA↗

arXiv2020

Inversion of operator pencils on Banach space using Jordan chains when the generalized resolvent has an isolated essential singularity

We assume that the generalized resolvent for a bounded linear operator pencil mapping one Banach space onto another has an isolated essential singularity at the origin and is analytic on some annular region of the complex plane centred at the origin. In such cases the resolvent operator can be represented on the annulus by a convergent Laurent series and the spectral set has two components---a bounded component inside the inner boundary of the annulus and an unbounded component outside the outer boundary. In this paper we prove that the complementary spectral separation projections on the domain space are uniquely determined by the respective generating subspaces for the associated infinite-length generalized Jordan chains of vectors and that the domain space is the direct sum of these two subspaces. We show that the images of the generating subspaces under the mapping defined by the pencil provide a corresponding direct sum decomposition for the range space and that this is simply the decomposition defined by the complementary spectral separation projections on the range space. If the domain space has a Schauder basis we show that the separated systems of fundamental equations are reduced to two semi-infinite systems of matrix equations which can be solved recursively to obtain a basic solution and thereby determine the Laurent series coefficients for the resolvent operator on the given annular region.

math.FA↗