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Showing papers by "École Polytechnique de Montréal published in 1993"


Journal ArticleDOI
TL;DR: In this article, all available thermodynamic and phase diagram data have been critically assessed for all phases in the CaO-Al 2O3, Al2O3-SiO2, and CaO al2O 3-Si2 systems at 1 bar pressure from 298 K to above the liquidus temperatures.
Abstract: All available thermodynamic and phase diagram data have been critically assessed for all phases in the CaO-Al2O3, Al2O3-SiO2, and CaO-Al2O3-SiO2 systems at 1 bar pressure from 298 K to above the liquidus temperatures. All reliable data for the binary systems have been simultaneously optimized to obtain, for each system, one set of model equations for the Gibbs energy of the liquid slag and all solid phases as functions of composition and temperature. The modified quasichemical model was used for the slag. With these binary parameters and those from the optimization of the CaO-SiO2 system reported previously, the quasichemical model was used to predict the thermodynamic properties of the ternary slag. Two additional small ternary parameters were required to reproduce the ternary phase diagram and ternary activity data to within experimental error limits. The calculated optimized phase diagram and thermodynamic properties are self-consistent and are the most reliable currently available estimates of the true values.

263 citations


Journal ArticleDOI
TL;DR: The dual kriging technique presented here simplifies considerably the handling of interpolated data and is especially useful for 3-D applications.
Abstract: For contouring very large data sets, such as those arising from 3-D finite element computations, for instance, or in numerical cartography, a computer program based on dual kriging interpolation was developed at Ecole Polytechnique for the Castor project (a multidisciplinary research and development work in computational software for hydroelectric projects). The dual kriging technique presented here simplifies considerably the handling of interpolated data and is especially useful for 3-D applications. It also containsspline interpolation as a particular case and theleast squares method as a limit case. In order to minimize the computational effort, several original features have been incorporated in this program: (1) the concept ofdistance of influence was introduced to allow the algorithm, when evaluating the interpolation value at a given location, to discard data points that are situated too far apart; (2)arbitrary geometric domains are decomposed into simpler regions, inside which the requested contours are drawn directly by scanning vertical slices from left to right, instead of building each contour line sequentially as in direct contouring; (3) contour lines may also be stored in arandom access database (this last feature was added to enable the automatic assembly of isovalue surfaces in 3-D applications such as stress analysis); and (4) contours can be smoothed by interpolating separately the sequences ofX andY coordinates for each contour line. This process, calledparametric kriging, permits the efficient compression of the number of data points necessary to record contours.

231 citations


Journal ArticleDOI
TL;DR: The initial secant stiffness of the interface at 75% of the maximum resistance load is found to be larger for the bone cubes from the lateral and medial regions and for the metal plate with smooth surface, while the smooth-surfaced metal plate has significantly smaller friction resistance than porous-coated ones.
Abstract: An apparatus was developed to measure load-displacement friction properties at the cancellous bone/porous-coated metal plate interface. Bone cubes were obtained from different proximal regions of four resurfaced cadaveric tibiae. Three different porous-surfaced metal plates (one fiber mesh and two bead) and a smooth-surface metal plate were used. In the presence of a constant normal contact pressure (0.10, 0.15, or 0.25 MPa), a variable tangential load up to the maximum resistance of the interface was applied and both relative normal and tangential displacements were recorded. Repetitive and fatigue loadings were also considered. Measured results show that the interface friction curve is highly nonlinear, exhibiting large relative tangential displacements in the range of 50-400 microns before the maximum load is reached. Relative displacements in the normal direction remain below 10 microns. The maximum resistance in friction is independent of the bone excision site, type of porous-surfaced metal plate, magnitude of normal load, placement of bone cubes on metal plates or vice versa, repetition of applied load, and conservation period of bone cubes in saline solution. The smooth-surfaced metal plate has significantly smaller friction resistance than porous-coated ones. The fatigue loading of up to 4000 cycles at 1 Hz, in the presence of 0.25 MPa contact pressure, slightly decreases the interface friction coefficient. Finally, the initial secant stiffness of the interface at 75% of the maximum resistance load is found to be larger for the bone cubes from the lateral and medial regions and for the metal plate with smooth surface.

228 citations


Journal ArticleDOI
TL;DR: In this paper, the analysis of carbon fiber/epoxy composites with fiber volume fractions of 12, 17, 38 and 61 vol% were subjected to flexural deformation on a Dupont DMA 983 instrument.
Abstract: The application of dynamic mechanical analysis (DMA) for quantifying interfacial interactions in composites is briefly reviewed. Carbon fiber/epoxy composites with fiber volume fractions of 12, 17, 38 and 61 vol% were subjected to flexural deformation on a Dupont DMA 983 instrument. The dependencies of dynamic mechanical properties of the composites on experimental parameters such as oscillation mode, amplitude, frequency, and temperature were investigate. As opposed to the storage modulus, the loss modulus is found to be sensitive to all parameters. In a fixed multiple frequency mode, the loss modulus of the composites increases with oscillation amplitude and decreases with frequency and the number of tests. The information produced in the resonant mode is more reproducible. An additional damping at the interfaces, apart from those of the constituents, suggests a poor interface adhesion in these composites. A linear relationship between the excess damping at the interfaces and the fiber volume fraction shows a similar interface quality for these composites having different fiber volume fractions. The detection of interfacial properities was found to be more sensitive in the flexural deformation mode than in the torsional mode. At temperatures higher than the glass transition temperature of the matrix, the effective volume fraction of the matrix is reduced. Such a reduction can be interpreted from the mismatch of thermal expansion of the matrix and the fibers.

185 citations


Journal ArticleDOI
TL;DR: In this paper, a theoretical study of antiresonant waveguides, in cylindrical and planar geometry, using leaky mode solutions of the vector wave equation, is presented.
Abstract: The authors present a theoretical study of antiresonant waveguides, in cylindrical and planar geometry, using leaky mode solutions of the vector wave equation. General analytical expressions are derived to describe the attenuation coefficients of TE, TM, HE, and EH modes of multilayered leaky waveguides, taking into account material absorption. Results obtained with these expressions are compared to numerical solutions and found to be accurate when the leakage loss is low. >

156 citations


Journal ArticleDOI
TL;DR: In this paper, all available thermodynamic and phase diagram data have been critically assessed for all phases in the MnO-TiO2, MgO-MgO,TiO 2, FeO-FeO 2, Na2O-Na2O 2 and NaO-K2O 3 systems at 1 bar pressure from 298 K to above the liquidus temperatures, and the modified quasichemical model was used for the molten slag phases.
Abstract: All available thermodynamic and phase diagram data have been critically assessed for all phases in the MnO-TiO2, MgO-TiO2, FeO-TiO2, Ti2O3-TiO2, Na2O-TiO2, and K2O-TiO2 systems at 1 bar pressure from 298 K to above the liquidus temperatures All reliable thermodynamic and phase diagram data have been simultaneously optimized to obtain, for each system, one set of model equations for the Gibbs energy of the liquid slag as a function of composition and temperature and equations for the Gibbs energies of all compounds as functions of temperature The modified quasichemical model was used for the molten slag phases

138 citations


Journal ArticleDOI
TL;DR: The effects of long exposure of epoxy resin surfaces to partial discharges under AC field stress were examined in this article, where the use of electrical pulse measuring techniques in conjunction with light emission measurements using a photomultiplier tube, which allowed simultaneous detection of both spark and glow discharges, permitted the establishment of a definite transition from spark to glow discharging.
Abstract: The effects of long exposure of epoxy resin surfaces to partial discharges under AC field stress were examined. The use of electrical pulse measuring techniques in conjunction with light emission measurements using a photomultiplier tube, which allowed simultaneous detection of both spark and glow discharges, permitted the establishment of a definite transition from spark to glow discharges. Spark discharges were found to be responsible for the initial surface modification leading to increased surface conductivity, which favored the transition to a pulseless glow type discharge within the cavity. The latter form of discharge was found to prevail over the major portion of the remainder of the exposure period. >

137 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proposed a method for improving the heterogeneous diffusion procedure for reactor design and operating calculations based on the use of pin-by-pin properties for the assemblies.
Abstract: Proposals are made for improving the heterogeneous diffusion procedure for reactor design and operating calculations. The procedure is based on the use of pin-by-pin properties for the assemblies a...

136 citations


Journal ArticleDOI
TL;DR: In this article, the surface properties of cellulose fibers have been modified by heat treatment, by silane coupling agents, and by maleated polypropylene grafts, which yielded information on the fibers' acid/base interaction potential.
Abstract: The surface properties of cellulose fibers have been modified by heat treatment, by silane coupling agents, and by maleated polypropylene grafts. The effectiveness of these methods has been evaluated by electron spectroscopy (ESCA), by contact angle measurements, and by inverse gas chromatography. The latter analyses yielded information on the fibers' acid/base interaction potential. Cellulose was found to be amphoteric, with prevalent acidic properties. Heat and chloro-silane treatments accentuated acidity, while amino-silane treatment produced net basicity in the fiber surface. Modification with maleated polypropylene reduced specific interactions and converted the fiber to a predominantly dispersion-force solid. The modified fibers were used in composites with polypropylene (neutral), polystyrene (base), and chlorinated polyethylene (acid) as matrix. Stress/strain and dynamic mechanical parameters were found to vary with acid/base interactions between polymer and fiber, significant improvements being noted in elastic and storage moduli, in tensile strength and elongation. In polypropylene, properties were unaffected by acid/base considerations. Acid/base forces, not necessarily dominant, merit consideration in the design of surface modification strategies intended to optimize composite mechanical properties.

134 citations


Journal ArticleDOI
TL;DR: In this paper, a finite element method for seismic fracture analysis of concrete gravity dams is presented, based on the non-linear fracture behaviour of concrete, and the following features have been considered in the development of the model: (i) the strain softening of concrete due to microcracking; (ii) the rotation of the fracture band with the progressive evolution of micro-crack damage in finite elements; (iii) the conservation of fracture energy; (iv) strain-rate sensitivity of concrete fracture parameters; (v) the softening initiation criterion under biaxial
Abstract: A finite element method for seismic fracture analysis of concrete gravity dams is presented. The proposed smeared crack analysis model is based on the non-linear fracture behaviour of concrete. The following features have been considered in the development of the model: (i) the strain softening of concrete due to microcracking; (ii) the rotation of the fracture band with the progressive evolution of microcrack damage in finite elements; (iii) the conservation of fracture energy; (iv) the strain-rate sensitivity of concrete fracture parameters; (v) the softening initiation criterion under biaxial loading conditions; (vi) the closing-reopening of cracks under cyclic loading conditions. The seismic fracture and energy response of dams and the significance of viscous damping models to take account of non-cracking structural energy dissipation mechanisms are discussed. The influences of global or local degradation of the material fracture resistance on the seismic cracking response of concrete dams were also studied. Two-dimensional seismic response analyses of Koyna Dam were performed to demonstrate the application of the proposed non-linear fracture mechanics model.

117 citations


Journal ArticleDOI
TL;DR: In this paper, all available thermodynamic and phase diagram data at 1 bar pressure have been critically assessed for the liquid and solid (rock salt structure) phases of the six binary systems formed among the components CaO, FeO, MgO, and MnO.
Abstract: All available thermodynamic and phase diagram data at 1 bar pressure have been critically assessed for the liquid and solid (rock salt structure) phases of the six binary systems formed among the components CaO, FeO, MgO, and MnO. In the case of FeO-containing systems, data were evaluated at Fe saturation. All reliable data for each system were optimized simultaneously to give one set of model equations for the liquid and solid solutions. In systems for which the liq-uidus or solidus curves have not been measured, they were calculated. The thermodynamic properties and phase diagrams calculated from these equations are self-consistent and are the most reliable currently available estimates of the true values.

Journal ArticleDOI
TL;DR: A mixed microbial culture required more than 20 days to degrade 150-microns-thick samples of 100% P(HB-co-HV), whereas samples containing 50% (wt/wt) starch disappeared in fewer than 8 days.
Abstract: Wheat starch granules and poly-(beta-hydroxybutyrate-co-beta-hydroxyvalerate) [P(HB-co-HV), (19.1 mol% HV)] were blended at 160 degrees C. Increasing the starch content from 0 to 50% (wt/wt) decreased the tensile strength of P(HB-co-HV) from 18 MPa to 8 MPa and diminished flexibility as Young's modulus increased from 1,525 MPa to 2,498 MPa, but overall mechanical properties of the polymer remained in a useful range. A mixed microbial culture required more than 20 days to degrade 150-microns-thick samples of 100% P(HB-co-HV), whereas samples containing 50% (wt/wt) starch disappeared in fewer than 8 days. Starch granules degraded before P(HB-co-HV) did. Aerobic degradation proceeded more rapidly than anaerobic degradation.

Journal ArticleDOI
TL;DR: Load-strain and load-time curves obtained from the hysteresis and stress-relaxation tests point out three different phenomena, which may be explained by the possibility that muscle fibres capable of contracting spontaneously could be present in lumbodorsal fascia ligaments.

Patent
17 Mar 1993
TL;DR: In this paper, a medication dispensing device for implantation into an animal or human body, comprises a plurality of compartments each containing a dose of medicine to be dispensed and having a delivery opening permitting delivery of the medicine; a rupturable membrane sealing the delivery opening of each compartment, the membrane having a predetermined elastic deformation limit and a predetermined rupture point; and a membrane rupturing system associated with each compartment for rupturing the membrane thereof in response to an electrical signal.
Abstract: A medication dispensing device for implantation into an animal or human body, comprises a plurality of compartments each containing a dose of medicine to be dispensed and having a delivery opening permitting delivery of the medicine; a rupturable membrane sealing the delivery opening of each compartment, the membrane having a predetermined elastic deformation limit and a predetermined rupture point; and a membrane rupturing system associated with each compartment for rupturing the membrane thereof in response to an electrical signal. The membrane rupturing system includes a stress-inducing member maintaining the membrane stressed to substantially the elastic deformation limit thereof, and a piezoelectric transducer responsive to the electrical signal for applying to the membrane additional stress sufficient to exceed the rupture point of the membrane, thereby causing the membrane to rupture and allowing body fluids to enter into the compartment for mixing with the medicine contained therein so that the medicine is released in admixture with the body fluids through the delivery opening into the animal or human body. The medication dispensing device of the invention further comprises a control circuit connected to a power source for supplying the electrical signal to a respective piezoelectric transducer of each membrane rupturing system to activate the respective piezoelectric transducer.

Journal ArticleDOI
TL;DR: In this paper, the Gibbs energies of all phases in Na2OSiO2 and K2O-SiO 2 systems were evaluated at 1 bar pressure from 298 K to above the liq-uidus temperatures.
Abstract: Available thermodynamic and phase diagram data have been evaluated for all phases in the Na2O-SiO2 and K2O-SiO2 systems at 1 bar pressure from 298 K to above the liq-uidus temperatures. All reliable thermodynamic and phase diagram data have been simultaneously optimized in order to obtain one set of model equations for the Gibbs energies of all phases as functions of temperature and composition. The thermodynamic properties and phase diagrams calculated from these parameters are self-consistent. The modified quasi-chemical model was used to represent the Gibbs energies of the molten slag phases.

Journal ArticleDOI
TL;DR: An overview is presented of the principal on-line techniques for handwriting preprocessing and word normalization, covering the major difficulties encountered and the various approaches usually used to resolve these problems.

Journal ArticleDOI
01 Oct 1993-Networks
TL;DR: In this paper, a two-commodity flow formulation for the traveling salesman problem is presented, where each commodity corresponds to a resource that is either distributed or picked up along the tour of all nodes.
Abstract: We present a new two-commodity flow formulation for the traveling salesman problem. Each commodity corresponds to a resource that is either distributed or picked up along the tour of all nodes. This formulation is partcularly well suited to handling time window constraints; the resource used is then the time. This formulation can be extended to the makespan problem. For a n-node problem, the linear relaxation of the formulation involves only O(n) constraints and O(n2) variables. Implementation issues are discussed and numerical experimentations have been realized for problems of up to 60 nodes. © 1993 by John Wiley & Sons, Inc.

Journal ArticleDOI
TL;DR: In this paper, the morphology of the dispersed phase in immiscible polymer blends is investigated and factors that influence the final state of deformation of a dispersed phase, and in particular, the formation of fibers and lamellae, are considered.
Abstract: The morphology of the dispersed phase in immiscible polymer blends plays an important role in the determination of the final physical properties. This paper considers factors that influence the final state of deformation of the dispersed phase, and in particular, the formation of fibers and lamellae. Blends of polyethylene and nylon-6 were extruded by ribbon extrusion at different draw ratios. Prior to single-screw extrusion the materials were blended in a co-rotating twin-screw extruder, and the size of the dispersed phase was studied as a function of the viscosity ratio. As the blends are extruded into ribbons and drawn through the calender rolls, the morphology of the dispersed phase undergoes drastic transformations. The fiber formation is enhanced by increasing the draw ratio. At high draw ratios, long thin fibers are observed. Some biaxial deformation is obtained for the noncompatibilized systems when the extruded materials enter the calender with the maximum closing pressure applied to the rolls. The same effect is observed for the compatibilized systems with lower values of the viscosity ratio. As a general rule, it has been observed that the final dispersed phase deformation is diminished in interfacially compatibilized systems.

Journal ArticleDOI
TL;DR: The construction of a new and detailed model of the human heart is described, based on 132 transverse sections obtained following a computed tomography scan of a frozen human heart whose chambers were inflated with pressurized air.

Journal ArticleDOI
TL;DR: Fiber tapering reduces the effective area and increases the Kerr nonlinear phase shift of the fundamental mode and the numerical simulation of a second-order soliton entering the taper gives a spectral widening that matches the observed spectrum.
Abstract: Fiber tapering reduces the effective area and increases the Kerr nonlinear phase shift of the fundamental mode. A significant nonlinear effect was observed in a micrometer-diameter centimeter-long section of tapered fiber at a wavelength of 1550 nm. The numerical simulation of a second-order soliton entering the taper gives a spectral widening that matches the observed spectrum.

Journal ArticleDOI
TL;DR: In this paper, a core-annular flow model was used to describe the hydrodynamics of a circulating fluidized bed (CFB) using radioactive argon as a tracer.

Journal ArticleDOI
TL;DR: In this article, the influence of shear thinning and viscoelasticity on the power required for the mixing of viscous liquids using six different helical ribbon agitators has been investigated.
Abstract: The influence of shear thinning and viscoelasticity on the power required for the mixing of viscous liquids using six different helical ribbon agitators has been investigated. Four Newtonian and 12 non-Newtonian fluids prepared using several polymers dissolved in varying concentrations in different solvents cover a wide range of rheological properties. By a careful choice of test media, the specific and combined effects of shear thinning and viscoelasticity on the power requirement have been examined. Simple models are proposed to predict the effective shear rate in the tank from the knowledge of the torque or power number. The effective shear rate predictions compared with the effective shear rate estimated using the scheme of Metzner and Otto (1957) show that they slightly depend on the shear thinning properties. Fluid's elasticity increases appreciably the power requirement, and departures from the generalized Newtonian power curve in the laminar regime are observed at smaller Reynolds numbers for viscoelastic fluids. Bottom wall resistance of the mixing vessel makes a negligible contribution to the power consumption.

Journal ArticleDOI
01 Feb 1993-Spine
TL;DR: An indirect horseradish peroxidase immunohistochemical technique was used to identify neurofilaments protein-immunoreactive fibers in lumbar spine ligaments, from patients who underwent spinal surgery for disc herniation, and revealed no recognizable sensory corpuscles in these ligaments.
Abstract: An indirect horseradish peroxidase immunohistochemical technique was used to identify neurofilaments protein-immunoreactive fibers in lumbar spine ligaments, from patients who underwent spinal surgery for disc herniation. Histologically, neural elements were abundant in all ligaments examined. In the ligamentum flavum the neurofilaments protein-immunoreactive nerve fibers were located especially close to blood vessels and fat globules. Bundles of nerve fibers were seen in all ligaments specimens except those from the ligamentum flavum. Supraspinous ligament and lumbodorsal fascia show also individual axons and free nerve endings. Contrary to the gold chloride impregnation method, immunohistochemical staining revealed no recognizable sensory corpuscles in these ligaments. This can be explained by the lack of immunoreaction of both the capsule of sensory corpuscles and the perineural sheaths of nerve fibers.

Journal ArticleDOI
TL;DR: Two versions of the arc-length method (Crisfield and Ramm) are presented and compared using a geometrical interpretation, and a new method, called the modified Crisfield-Ramm method, is proposed.

Journal ArticleDOI
TL;DR: The general formulation of thermodynamics introduced in this paper is illustrated by the example of a k-component system undergoing a chemical reaction.
Abstract: A geometrical formulation of thermodynamics is carried in this paper to a domain that includes dynamics and driven systems. Let it be known from experience that the behavior of an externally unforced or a driven system is well described in a state space N. The thermodynamics of the system is a geometrical structure in N that arises from an analysis of the time evolution in a more microscopic (i.e., depicting more details) state space M. If in particular the system is externally unforced then the state space N can be chosen to be the state space ${\mathit{N}}_{\mathrm{ET}}$ of equilibrium thermodynamics. The geometrical structure in ${\mathit{N}}_{\mathrm{ET}}$, obtained by analyzing the time evolution in a more microscopic state space M, appears to be the geometrical formulation of classical equilibrium thermodynamics. The general formulation of thermodynamics introduced in this paper is illustrated by the example of a k-component system undergoing a chemical reaction.

Journal ArticleDOI
TL;DR: In this paper, the complexity of bounded vertex coloring is discussed and several classes of graphs for which this problem is polynomially solvable are identified, and upper and lower bounds on the bounded chromatic number of a graph G are given in terms of k, the number n of vertices, the usual chromatic numbers of G, and the maximum degree of G.

Journal ArticleDOI
01 Jan 1993-Spine
TL;DR: It is postulated that the anterior location of the line of gravity of the upper-body weight is regulated to provide the required combined loads on the lumbar spine so higher compression can be tolerated by the spine at minimal energetic cost.
Abstract: Basic questions regarding how extreme compressive loads can be tolerated by the spine without experiencing abnormal motions or instabilities remain unresolved. Two finite element models of the human lumbar spine were generated. The detailed model accounted for the three-dimensional irregular geometry, material and geometric nonlinearities, nonhomogeneous fiber-matrix nature of the discs, ligaments, and articulation at the facet joints. The nonlinear stability response of the model was predicted under an axial compression force (200 N to 700 N) applied at the L1 while the S1 was fixed. The effect of the presence of a combined flexion moment and a horizontal support on the response was investigated. Another nonlinear model using rigid bodies interconnected by deformable beam elements was also considered. The computed results under the axial compression loads indicated that the response is highly nonlinear with no bifurcation or limit point (critical load). The unconstrained lumbar spine is most flexible in the sagittal plane (least stiff plane). The existence of the horizontal support and the combined flexion moment significantly increased the load-bearing capacity of the lumbar spine; the lumbar spine resisted the axial compression force of 400 N with minimal displacements. Under axial compression force, the flexion moment tends to restrict the posterior translational movement of the lordotic lumbar spine, whereas the horizontal support constrains the coupled lateral motion. A slight decrease in the lordosis was predicted for the compression load of 400 N. It is postulated that the anterior location of the line of gravity of the upper-body weight is regulated to provide the required combined loads on the lumbar spine so higher compression can be tolerated by the spine at minimal energetic cost. In vivo experimental results support the validity of the model predictions.

Journal ArticleDOI
TL;DR: A series of cobalt and nickel based perovskite type catalysts with high specific surface area (8-20 m2 /g) was prepared by spray-freezing/freeze-drying method as discussed by the authors.
Abstract: A series of cobalt and nickel based perovskite type catalysts with high specific surface area (8–20 m2 /g) was prepared by spray-freezing/freeze-drying method. The catalytic activity of all samples in methane combustion was evaluated by measuring the light-off temperature, the conversion at 823 K and the temperature of the end of the reaction. The experimental data suggest higher activity than reported in literature for similar or other perovskites, and confirm its strong dependence on the specific surface area. Among eleven tested catalysts, including seven new compositions four of which showed excellent activity, La0.66Sr0.34Ni0.3Co0.7O3 was the best performing.

Journal ArticleDOI
TL;DR: The Ni, Co and Fe composite-coatings were prepared by electrolytic composite-deposition as discussed by the authors and the characterizations for the hydrogen evolution reaction (HER) were carried out in a 5 M KOH solution.

Journal ArticleDOI
TL;DR: In the case where (N/n, q)=1, all quasi cyclic codes are shown to be decomposable into the direct sum of a fixed number of indecomposable components called irreducible cyclicFq[λ]-submodules providing for the complete characterisation and enumeration of some subclasses of quasicyclic codes.
Abstract: Given any finite fieldF q , an (N, K) quasi cyclic code is defined as aK dimensional linear subspace ofF which is invariant underT n for some integern, 0