Synthesis, as Opposed to Separation, of Variables

Abstract. Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e., a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore, it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for nonvanishing boundary conditions), which renders such expressions unsuitable for numerical computations. This paper describes a recently introduced transform method that can be applied to certain nonseparable and non-self-adjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The startin

Influence of solvent choice on the optimisation of a reaction–separation operation : application to a Beckmann rearrangement reaction

In pharmaceutical syntheses, the solvent choice generally represents a complex design step. Traditionally, this choice is operated according to criteria connected with the reaction step and without any consideration on the following separation steps. The purpose of this study is to highlight the benefits of a global approach of optimisation for the solvent determination. In this way, an optimisation framework dedicated to global synthesis is applied to a simple reaction–separation operation integrating a Beckmann rearrangement reaction, leading to interesting solvent choices

Tipping points in complex coupled life-environment systems

Simple models of complex phenomena provide powerful insights and suggest low-level mechanistic descriptions. The Earth system arises from the interaction of subsystems with multi-scale temporal and spatial variability; from the microbial to continental scales, operating over the course of days to geological time. System-level homeostasis has been demonstrated in a number of conceptual, artificial life, models which share the advantage of a thorough and transparent analysis. We reintroduce a general model for a coupled life-environment model, concentrating on a minimal set of assumptions, and explore the consequences of interaction between simple life elements and their shared, multidimensional environment. In particular stability, criticality and transitions are of great relevance to understanding the history, and future of the Earth system. The model is shown to share salient features with other abstract systems such as Ashby's Homeostat and Watson and Lovelock's Daisyworld. Our generic description is free to explore high-dimensional, complex environments, and in doing so we show that even a small increase in the environmental complexity gives rise to very complex attractor landscapes which require a much richer conception of critical transitions and hysteresi

Synthesizing Short-Circuiting Validation of Data Structure Invariants

This paper presents incremental verification-validation, a novel approach for checking rich data structure invariants expressed as separation logic assertions. Incremental verification-validation combines static verification of separation properties with efficient, short-circuiting dynamic validation of arbitrarily rich data constraints. A data structure invariant checker is an inductive predicate in separation logic with an executable interpretation; a short-circuiting checker is an invariant checker that stops checking whenever it detects at run time that an assertion for some sub-structure has been fully proven statically. At a high level, our approach does two things: it statically proves the separation properties of data structure invariants using a static shape analysis in a standard way but then leverages this proof in a novel manner to synthesize short-circuiting dynamic validation of the data properties. As a consequence, we enable dynamic validation to make up for imprecision in sound static analysis while simultaneously leveraging the static verification to make the remaining dynamic validation efficient. We show empirically that short-circuiting can yield asymptotic improvements in dynamic validation, with low overhead over no validation, even in cases where static verification is incomplete

Entrepreneurial discovery and exploitation processes: sequence or symbiosis?

This study examined the effect that temporal order within the entrepreneurial discovery-exploitation process has on the outcomes of venture creation. Consistent with sequential theories of discovery-exploitation, the general flow of venture creation was found to be directed from discovery toward exploitation in a random sample of nascent ventures. However, venture creation attempts which specifically follow this sequence derive poor outcomes. Moreover, simultaneous discovery-exploitation was the most prevalent temporal order observed, and venture attempts that proceed in this manner more likely become operational. These findings suggest that venture creation is a multi-scale phenomenon that is at once directional in time, and simultaneously driven by symbiotically coupled discovery and exploitation

Populating the Galaxy with low-mass X-ray binaries

We perform binary population synthesis calculations to investigate the incidence of low-mass X-ray binaries and their birth rate in the Galaxy. We use a binary evolution algorithm that models all the relevant processes including tidal circularization and synchronization. Parameters in the evolution algorithm that are uncertain and may affect X-ray binary formation are allowed to vary during the investigation. We agree with previous studies that under standard assumptions of binary evolution the formation rate and number of black-hole low-mass X-ray binaries predicted by the model are more than an order of magnitude less than what is indicated by observations. We find that the common-envelope process cannot be manipulated to produce significant numbers of black-hole low-mass X-ray binaries. However, by simply reducing the mass-loss rate from helium stars adopted in the standard model, to a rate that agrees with the latest data, we produce a good match to the observations. Including low-mass X-ray binaries that evolve from intermediate-mass systems also leads to favourable results. We stress that constraints on the X-ray binary population provided by observations are used here merely as a guide as surveys suffer from incompleteness and much uncertainty is involved in the interpretation of results.Comment: 17 pages and 9 figures; accepted by MNRA