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In this article we provide a complete description of the set of all solutions to an autoregressive law of motion in a finite-dimensional complex vector space. Every solution is shown to be the sum of three parts, each corresponding to a directed flow of time. One part flows forward from the arbitrarily distant past; one flows backward from the arbitrarily distant future; and one flows outward from time zero. The three parts are obtained by applying three complementary spectral projections to the solution, these corresponding to a separation of the eigenvalues of the autoregressive operator according to whether they are inside, outside or on the unit circle. We provide a finite-dimensional parametrization of the set of all solutions.

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., Comment: 18 pages: correction to published article

We prove an extended Granger-Johansen representation theorem (GJRT) for finite or infinite order integrated autoregressive time series on Banach space. We assume only that the resolvent of the autoregressive polynomial for the series is analytic on and inside the unit circle except for an isolated singularity at unity. If the singularity is a pole of finite order the time series is integrated of the same order. If the singularity is an essential singularity the time series is integrated of order infinity. When there is no deterministic forcing the value of the series at each time is the sum of an almost surely convergent stochastic trend, a deterministic term depending on the initial conditions and a finite sum of embedded white noise terms in the prior observations. This is the extended GJRT. In each case the original series is the sum of two separate autoregressive time series on complementary subspaces--a singular component which is integrated of the same order as the original series and a regular component which is not integrated. The extended GJRT applies to all integrated autoregressive processes irrespective of the spatial dimension, the number of stochastic trends and cointegrating relations in the system, and the order of integration., Comment: 29 pages, two figures, Preprint for article submitted for presentation at ANZESG 2023

It is well known that the non-autonomous scalar differential equation of evolution has a unique solution given by an elementary exponential function. In general there is no such analogous solution to the corresponding non-autonomous evolution equation for square matrices. In this paper we propose and justify an explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation for bounded linear operators on Banach space., Comment: 30 pages. No figures. Corrected version of article published in Applied Mathematics letters

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., Comment: 44 pages, 9 figures, revision of earlier submission to correct typographical errors

Let ${\mbox{$\mbox{\boldmath $f$}$}}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space $H$, and let ${\mbox{$\mbox{\boldmath $g$}$}}$ be an associated square-integrable, zero-mean, random vector with realizations, which are not observable, in a Hilbert space $K$. We seek an optimal filter in the form of a closed linear operator $X$ acting on the observable realizations of a proximate vector ${\mbox{$\mbox{\boldmath $f$}$}}_{\epsilon} \approx {\mbox{$\mbox{\boldmath $f$}$}}$ that provides the best estimate $\widehat{{\mbox{$\mbox{\boldmath $g$}$}}}_{\epsilon} = X {\mbox{$\mbox{\boldmath $f$}$}}_{\epsilon}$ of the vector ${\mbox{$\mbox{\boldmath $f$}$}}$. We assume the required covariance operators are known. The results are illustrated with a typical example.

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., Comment: 21 pages

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., Comment: 22 pages

We propose and justify a new approach to constructing optimal nonlinear transforms of random vectors. We show that the proposed transform improves such characteristics of rank-reduced transforms as compression ratio, accuracy of decompression and reduces required computational work. The proposed transform ${\mathcal T}_p$ is presented in the form of a sum with $p$ terms where each term is interpreted as a particular rank-reduced transform. Moreover, terms in ${\mathcal T}_p$ are represented as a combination of three operations ${\mathcal F}_k$, ${\mathcal Q}_k$ and ${\boldsymbol{\phi}}_k$ with $k=1,...,p$. The prime idea is to determine ${\mathcal F}_k$ separately, for each $k=1,...,p$, from an associated rank-constrained minimization problem similar to that used in the Karhunen--Lo\`{e}ve transform. The operations ${\mathcal Q}_k$ and ${\boldsymbol{\phi}}_k$ are auxiliary for finding ${\mathcal F}_k$. The contribution of each term in ${\mathcal T}_p$ improves the entire transform performance. A corresponding unconstrained nonlinear optimal transform is also considered. Such a transform is important in its own right because it is treated as an optimal filter without signal compression. A rigorous analysis of errors associated with the proposed transforms is given., Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/