There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

‡We describe the use of singular value decomposition in transforming genome-wide expression data from genes 3 arrays space to reduced diagonalized ‘‘eigengenes’’ 3 ‘‘eigenarrays’’ space, where the eigengenes (or eigenarrays) are unique orthonormal superpositions of the genes (or arrays). Normalizing the data by filtering out the eigengenes (and eigenarrays) that are inferred to represent noise or experimental artifacts enables meaningful comparison of the expression of different genes across different arrays in different experiments. Sorting the data according to the eigengenes and eigenarrays gives a global picture of the dynamics of gene expression, in which individual genes and arrays appear to be classified into groups of similar regulation and function, or similar cellular state and biological phenotype, respectively. After normalization and sorting, the significant eigengenes and eigenarrays can be associated with observed genome-wide effects of regulators, or with measured samples, in which these regulators are overactive or underactive, respectively.

Singular Value Decomposition for Genome-Wide Expression Data Processing and Modeling

Computational Analysis of Microarray Data

Stability in Aviation. By Prof. G. H. Bryan, F.E.S. Pp. X + 192. 5S. net. 1911. (Macmillan.)

From Introduction: "The subject of this paper is so broad in scope that a large volume might be devoted to it. In a short paper of this kind it is possible simply to sketch in the high lights of aircraft engine design showing the development to date, the possibilities of the future, and the underlying fundamental principles. Summarizing this development and referring to the graph (Fig.1), we that there is now a water-cooled engine in every power from 150 to 800 HP. and an air-cooled engine in the 200 to 400 HP. classes."

/pdf/aircraft-engine-design-3v346pwxhw.pdf

Aircraft Engine Design

DESIGN of maneuvers for carefree access of an aircraft to its complete flight envelope including poststall regimes is useful not only from a combat strategy point of view, but also for devising recovery strategies from an accident scenario. Maneuvers for an aircraft can be efficiently designed if a priori knowledge of its maneuverability characteristics is available to the control designers.
Different types of agility metrics that characterize aircraft maneuvering capabilities have been proposed in literature based on different criteria [1,2]. A recent approach to define maneuverability characteristics is based on computing “attainable equilibrium sets,” as suggested in [3] and [4]. This approach involves computing a two dimensional (2-D) section of attainable equilibrium sets of a particular maneuver using an inverse trimming formulation.
Construction of maneuvers based on attainable equilibrium sets involves accessing desired aircraft states in the attainable equilibrium set from a normal flying condition, such as a level flight trim condition. Computing an attainable equilibrium set for a given aircraft model and developing control algorithms to switch aircraft states between different operating points lying within the accessible region defined by attainable equilibrium sets are thus essential ingredients of aircraft maneuver design. For aircraft models, which are inherently nonlinear due to nonlinear aerodynamics in the poststall regimes, and because of various couplings, use of nonlinear control design techniques based on dynamic inversion (DI) or
sliding-mode control (SMC) have been proposed for control
prototyping to design maneuvers [3,5–7]. Using bifurcation theory and continuation methods, Raghavendra et al. [5] computed spin solutions for the F18/high-alpha research vehicle (HARV) model, and demonstrated use of the DI controller to recover the aircraft from a flat oscillatory spin motion. Recently, the authors have proposed a
systematic approach using bifurcation analysis and continuation methods in conjunction with a SMC technique to design maneuvers for a nonlinear aircraft model

Accessible Regions for Controlled Aircraft Maneuvering

Tools based on the bifurcation and continuation method have been found to be extremely useful for studying multiparameter nonlinear dynamical systems under state and parameter-constrained condition

https://arc.aiaa.org/doi/pdf/10.2514/1.C034264

Direct Methodology for Constrained System Analysis with Applications to Aircraft Dynamics

The extended bifurcation analysis (EBA) technique has been recently proposed for the bifurcation analysis of dynamical systems with constrained equilibrium states. In this paper, the EBA te...

Bifurcation analysis of inertia coupled roll manoeuvres of airplanes

Over the past two decades, bifurcation and continuation methods have emerged as efficient tools for prediction, and control of flight instabilities. Bifurcation phenomena have been associated with nonlinear behaviour of aircraft in actual flight tests, and the critical control combinations, which signify onset of instabilities, have been identified for almost all generations of modern fighter aircraft. A standard bifurcation analysis procedure has been used in the past. In this paper, the bifurcation theory, relevant to preliminary bifurcation analysis of nonlinear aircraft dynamics, has been introduced, and a stepswise methodology used in a standard bifurcation analysis procedure has been illustrated with an application to open-loop dynamics of an F- 18/HARV model in landing configuration. Further, an example manoeuvre is constructed, and numerical time simulations of an F-18/HARV model in this manoeuvre is carried out to validate the predictions from the bifurcation analysis. Numerical time simulation results confirm the onset of nonlinear behaviour at critical control combinations identified in bifurcation analysis of the aircraft model. Thus, bifurcation methods, in conjunction with selective numerical simulations, can be extremely useful in the design, development, evaluation, and flight training phases of a fighter aircraft development programme.

Nandan K. Sinha

Papers

Accessible Regions for Controlled Aircraft Maneuvering

Direct Methodology for Constrained System Analysis with Applications to Aircraft Dynamics

Bifurcation analysis of inertia coupled roll manoeuvres of airplanes

Applications of Bifurcation Methods to F-181HARV Open-loop Dynamics in Landing Configuration

Advanced Flight Dynamics with Elements of Flight Control