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http://hdl.handle.net/2117/3917
Mon, 25 May 2015 19:49:29 GMT2015-05-25T19:49:29Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoNonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
http://hdl.handle.net/2117/22391
Title: Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
Authors: Cabré Vilagut, Xavier; Sire, Yannick
Abstract: This is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.http://hdl.handle.net/2117/22391Cabré Vilagut, Xavier; Sire, YannicknoThis is the first of two articles dealing with the equation (-)sv = f (v) in Rn, with s ¿ (0,1), where (-)s stands for the fractional Laplacian — the in¿nitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in Rn+1+ together with a nonlinear Neumann boundary condition on ¿Rn+1 + =Rn.
In this ¿rst article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of R. These necessary conditions (which will be proven in a follow-up paper to be also suficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform ass ¿1, establishing in the limit the corresponding known results for the Laplacian.
In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.Estructuras A-infinito en la opérada de cactus
http://hdl.handle.net/2117/22097
Title: Estructuras A-infinito en la opérada de cactus
Authors: Gálvez Carrillo, Maria Immaculada; Lombardi, Leandro; Tonks, Andrew
Abstract: Diversas versiones de la opérada de cactus inicialmente definida por Voronov han sido estudiadas. Es conocida su equivalencia débil con la opérada de pequeños discos. Se conoce pues que la opérada de cactus admite una acción de la opérada de Gerstenhaber salvo homotopía. En este proyecto, nuestro objetivo es obtener una realización explícita de dicha acción. Por el momento, hemos construido una acción explícita de la opérada A8 en la opérada de cactus, que presentamos en este pósterhttp://hdl.handle.net/2117/22097Gálvez Carrillo, Maria Immaculada; Lombardi, Leandro; Tonks, AndrewnoDiversas versiones de la opérada de cactus inicialmente definida por Voronov han sido estudiadas. Es conocida su equivalencia débil con la opérada de pequeños discos. Se conoce pues que la opérada de cactus admite una acción de la opérada de Gerstenhaber salvo homotopía. En este proyecto, nuestro objetivo es obtener una realización explícita de dicha acción. Por el momento, hemos construido una acción explícita de la opérada A8 en la opérada de cactus, que presentamos en este pósterJ2 effect and elliptic inclined periodic orbits in the collision three-body problem
http://hdl.handle.net/2117/21117
Title: J2 effect and elliptic inclined periodic orbits in the collision three-body problem
Authors: Barrabes, Esther; Cors Iglesias, Josep Maria; Pinyol, Conxita; Soler Villanueva, Jaume
Abstract: The existence of a new class of inclined periodic orbits of the collision restricted
three{body problem is shown. The symmetric periodic solutions found are perturbations of elliptic
kepler orbits and they exist only for special values of the inclination and are related to the motion
of a satellite around an oblate planet.http://hdl.handle.net/2117/21117Barrabes, Esther; Cors Iglesias, Josep Maria; Pinyol, Conxita; Soler Villanueva, Jaumenocollision restricted three-body problem, periodic orbits, symmetric orbits, critical inclination, continuation methodThe existence of a new class of inclined periodic orbits of the collision restricted
three{body problem is shown. The symmetric periodic solutions found are perturbations of elliptic
kepler orbits and they exist only for special values of the inclination and are related to the motion
of a satellite around an oblate planet.Arbitrary lagrangian-eulerian finite element analysis of strain localization in transient problems
http://hdl.handle.net/2117/28021
Title: Arbitrary lagrangian-eulerian finite element analysis of strain localization in transient problems
Authors: Pijaudier-Cabot, Gilles; Bode, L; Huerta, Antonio
Abstract: Non-local models guaranty that finite element computations on strain softening materials remain sound up to failure from a theoretical and computational viewpoint. The non-locality prevents strain localization with zero global dissipation of energy, and consequently finite element calculations converge upon mesh refinements to non-zero width localization zones. One of the major drawbacks of these models is that the element size needed in order to capture the localization zone must be smaller than the intemallength. Hence, the total number of degrees of freedom becomes rapidly prohibitive for most engineering applications and there is an obvious need for mesh adaptivity. This paper deals with the application of the arbitrary Lagrangian-Eulerian (ALE) formulation, well known in hydrodynamics and fluid-structure interaction problems, to transient strain localization in a non-local damageable material. It is shown that the ALE formulation which is employed in large boundary motion problems can also be well suited for non-linear transient analysis of softening materials where localization bands appear. The remeshing strategy is based on the equidistribution of an indicator that quantifies the interelement jump of a selected state variable. Two well known one-dimensional examples illustrate the capabilities of this technique: the first one deals with localization due to a propagating wave in a bar, and the second one studies the wave propagation in a hollow sphere.Mon, 25 May 2015 08:38:50 GMThttp://hdl.handle.net/2117/280212015-05-25T08:38:50ZPijaudier-Cabot, Gilles; Bode, L; Huerta, Antoniononon-linear computational mechanics, arbitrary Lagrangian-Eulerian, mesh adaptivity, strain-softening, localization, damage mechanics, wave propagationNon-local models guaranty that finite element computations on strain softening materials remain sound up to failure from a theoretical and computational viewpoint. The non-locality prevents strain localization with zero global dissipation of energy, and consequently finite element calculations converge upon mesh refinements to non-zero width localization zones. One of the major drawbacks of these models is that the element size needed in order to capture the localization zone must be smaller than the intemallength. Hence, the total number of degrees of freedom becomes rapidly prohibitive for most engineering applications and there is an obvious need for mesh adaptivity. This paper deals with the application of the arbitrary Lagrangian-Eulerian (ALE) formulation, well known in hydrodynamics and fluid-structure interaction problems, to transient strain localization in a non-local damageable material. It is shown that the ALE formulation which is employed in large boundary motion problems can also be well suited for non-linear transient analysis of softening materials where localization bands appear. The remeshing strategy is based on the equidistribution of an indicator that quantifies the interelement jump of a selected state variable. Two well known one-dimensional examples illustrate the capabilities of this technique: the first one deals with localization due to a propagating wave in a bar, and the second one studies the wave propagation in a hollow sphere.Static output-feedback control for vehicle suspensions: a single-step linear matrix inequality approach
http://hdl.handle.net/2117/28020
Title: Static output-feedback control for vehicle suspensions: a single-step linear matrix inequality approach
Authors: Rubió Massegú, Josep; Palacios Quiñonero, Francisco; Rossell Garriga, Josep Maria; Karimi, Hamid Reza
Abstract: In this paper, a new strategy to design static output-feedback controllers for a class of vehicle suspension systems is presented. A theoretical background on recent advances in output-feedback control is first provided, which makes possible an effective synthesis of static output-feedback controllers by solving a single linear matrix inequality optimization problem. Next, a simplified model of a quarter-car suspension system is proposed, taking the ride comfort, suspension stroke, road holding ability, and control effort as the main performance criteria in the vehicle suspension design. The new approach is then used to design a static output-feedback controller that only uses the suspension deflection and the sprung mass velocity as feedback information. Numerical simulations indicate that, despite the restricted feedback information, this static output-feedback controller exhibits an excellent behavior in terms of both frequency and time responses, when compared with the corresponding state-feedback controller.Fri, 22 May 2015 15:49:52 GMThttp://hdl.handle.net/2117/280202015-05-22T15:49:52ZRubió Massegú, Josep; Palacios Quiñonero, Francisco; Rossell Garriga, Josep Maria; Karimi, Hamid RezanoIn this paper, a new strategy to design static output-feedback controllers for a class of vehicle suspension systems is presented. A theoretical background on recent advances in output-feedback control is first provided, which makes possible an effective synthesis of static output-feedback controllers by solving a single linear matrix inequality optimization problem. Next, a simplified model of a quarter-car suspension system is proposed, taking the ride comfort, suspension stroke, road holding ability, and control effort as the main performance criteria in the vehicle suspension design. The new approach is then used to design a static output-feedback controller that only uses the suspension deflection and the sprung mass velocity as feedback information. Numerical simulations indicate that, despite the restricted feedback information, this static output-feedback controller exhibits an excellent behavior in terms of both frequency and time responses, when compared with the corresponding state-feedback controller.Bifurcation of relative equilibria of the (1+3)-body problem
http://hdl.handle.net/2117/28007
Title: Bifurcation of relative equilibria of the (1+3)-body problem
Authors: Corbera, M.; Cors Iglesias, Josep Maria; Llibre Saló, Jaume; Moeckel, Richard
Abstract: We study the relative equilibria of the limit case of the planar Newtonian 4-body problem when three masses tend to zero, the so-called (1+3)-body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the others are concave. Each convex relative equilibrium of the (1+3)-body problem can be continued to a unique family of relative equilibria of the general 4-body problem when three of the masses are sufficiently small and every convex relative equilibrium for these masses belongs to one of these six families.Thu, 21 May 2015 15:13:18 GMThttp://hdl.handle.net/2117/280072015-05-21T15:13:18ZCorbera, M.; Cors Iglesias, Josep Maria; Llibre Saló, Jaume; Moeckel, RichardnoCelestial mechanics, Relative equilibria, (1+n)-body problemWe study the relative equilibria of the limit case of the planar Newtonian 4-body problem when three masses tend to zero, the so-called (1+3)-body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the others are concave. Each convex relative equilibrium of the (1+3)-body problem can be continued to a unique family of relative equilibria of the general 4-body problem when three of the masses are sufficiently small and every convex relative equilibrium for these masses belongs to one of these six families.Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy
http://hdl.handle.net/2117/28004
Title: Phase-field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy
Authors: Li, Bin; Peco, Christian; Millán, Daniel; Arias Vicente, Irene; Arroyo Balaguer, Marino
Abstract: Crack propagation in brittle materials with anisotropic surface energy is important in applications involving single crystals, extruded polymers, or geological and organic materials. Furthermore, when this anisotropy is strong, the phenomenology of crack propagation becomes very rich, with forbidden crack propagation directions or complex sawtooth crack patterns. This problem interrogates fundamental issues in fracture mechanics, including the principles behind the selection of crack direction. Here, we propose a variational phase-field model for strongly anisotropic fracture, which resorts to the extended Cahn-Hilliard framework proposed in the context of crystal growth. Previous phase-field models for anisotropic fracture were formulated in a framework only allowing for weak anisotropy. We implement numerically our higher-order phase-field model with smooth local maximum entropy approximants in a direct Galerkin method. The numerical results exhibit all the features of strongly anisotropic fracture and reproduce strikingly well recent experimental observations.Thu, 21 May 2015 12:33:42 GMThttp://hdl.handle.net/2117/280042015-05-21T12:33:42ZLi, Bin; Peco, Christian; Millán, Daniel; Arias Vicente, Irene; Arroyo Balaguer, Marinonofracture, meshfree methods, phase-field models, strongly anisotropic surface energy, local maximum entropy approximantsCrack propagation in brittle materials with anisotropic surface energy is important in applications involving single crystals, extruded polymers, or geological and organic materials. Furthermore, when this anisotropy is strong, the phenomenology of crack propagation becomes very rich, with forbidden crack propagation directions or complex sawtooth crack patterns. This problem interrogates fundamental issues in fracture mechanics, including the principles behind the selection of crack direction. Here, we propose a variational phase-field model for strongly anisotropic fracture, which resorts to the extended Cahn-Hilliard framework proposed in the context of crystal growth. Previous phase-field models for anisotropic fracture were formulated in a framework only allowing for weak anisotropy. We implement numerically our higher-order phase-field model with smooth local maximum entropy approximants in a direct Galerkin method. The numerical results exhibit all the features of strongly anisotropic fracture and reproduce strikingly well recent experimental observations.Phase-field modeling of fracture in ferroelectric materials
http://hdl.handle.net/2117/28003
Title: Phase-field modeling of fracture in ferroelectric materials
Authors: Abdollahi Hosnijeh, Amir; Arias Vicente, Irene
Abstract: This paper presents a family of phase-field models for the coupled simulation of the microstructure formation and evolution, and the nucleation and propagation of cracks in single and polycrystalline ferroelectric materials. The first objective is to introduce a phase-field model for ferroelectric single crystals. The model naturally couples two existing energetic phase-field approaches for brittle fracture and ferroelectric domain formation and evolution. Simulations show the interactions between the microstructure and the crack under mechanical and electromechanical loadings. Another objective of this paper is to encode different crack face boundary conditions into the phase-field framework since these conditions strongly affect the fracture behavior of ferroelectrics. The smeared imposition of these conditions are discussed and the results are compared with that of sharp crack models to validate the proposed approaches. Simulations show the effects of different conditions and electromechanical loadings on the crack propagation. In a third step, the model is modified by introducing a crack non-interpenetration condition in the variational approach to fracture accounting for the asymmetric behavior in tension and compression. The modified model makes it possible to explain anisotropic crack growth in ferroelectrics under the Vickers indentation loading. This model is also employed for the fracture analysis of multilayer ferroelectric actuators, which shows the potential of the model for future applications. The coupled phase-field model is also extended to polycrystals by introducing realistic polycrystalline microstructures in the model. Inter- and trans-granular crack propagation modes are observed in the simulations. Finally, and for completeness, the phase-field theory is extended to the simulation of the propagation of conducting cracks under purely electrical loading and to the three-dimensional simulation of crack propagation in ferroelectric single crystals. Salient features of the crack propagation phenomenon predicted by the simulations of this paper are directly compared with experimental observations.Thu, 21 May 2015 12:12:44 GMThttp://hdl.handle.net/2117/280032015-05-21T12:12:44ZAbdollahi Hosnijeh, Amir; Arias Vicente, IrenenoFerroelectricity, Piezoelectricity, Fracture, Phase-field models, Polycrystals, Finite element analysis, Domain switchingThis paper presents a family of phase-field models for the coupled simulation of the microstructure formation and evolution, and the nucleation and propagation of cracks in single and polycrystalline ferroelectric materials. The first objective is to introduce a phase-field model for ferroelectric single crystals. The model naturally couples two existing energetic phase-field approaches for brittle fracture and ferroelectric domain formation and evolution. Simulations show the interactions between the microstructure and the crack under mechanical and electromechanical loadings. Another objective of this paper is to encode different crack face boundary conditions into the phase-field framework since these conditions strongly affect the fracture behavior of ferroelectrics. The smeared imposition of these conditions are discussed and the results are compared with that of sharp crack models to validate the proposed approaches. Simulations show the effects of different conditions and electromechanical loadings on the crack propagation. In a third step, the model is modified by introducing a crack non-interpenetration condition in the variational approach to fracture accounting for the asymmetric behavior in tension and compression. The modified model makes it possible to explain anisotropic crack growth in ferroelectrics under the Vickers indentation loading. This model is also employed for the fracture analysis of multilayer ferroelectric actuators, which shows the potential of the model for future applications. The coupled phase-field model is also extended to polycrystals by introducing realistic polycrystalline microstructures in the model. Inter- and trans-granular crack propagation modes are observed in the simulations. Finally, and for completeness, the phase-field theory is extended to the simulation of the propagation of conducting cracks under purely electrical loading and to the three-dimensional simulation of crack propagation in ferroelectric single crystals. Salient features of the crack propagation phenomenon predicted by the simulations of this paper are directly compared with experimental observations.On computing flat outputs through Goursat normal form
http://hdl.handle.net/2117/27990
Title: On computing flat outputs through Goursat normal form
Authors: Franch Bullich, Jaume; Manzanera, Ana; Valero, Gemma
Abstract: This paper is devoted to computation of flat outputs by means of the Goursat normal form of the Pfaffian system associated to any control system in state space form. The algorithm explained here consists in three steps: i) Transformation of the system into its Pfaffian equivalent. ii) Computation of the Goursat normal form for the Pfaffian system. iii) Rewriting of the state space equations in the new variables. At this stage, a feedback law simplifies the equations and, therefore, the flat outputs can be easily computed. As an example, the algorithm is applied to a system consisting in a car with expanding wheels. Point to point trajectories are obtained thanks to the property of differential flatness. The paper also includes a control law that minimizes errors in the initial conditions.Wed, 20 May 2015 17:10:42 GMThttp://hdl.handle.net/2117/279902015-05-20T17:10:42ZFranch Bullich, Jaume; Manzanera, Ana; Valero, GemmanoControl theory, Differential equations, State space methods, System theory
Control laws, Differential flatness, Feedback laws, Initial conditions, Normal form, Point to point, State space equation, State space formThis paper is devoted to computation of flat outputs by means of the Goursat normal form of the Pfaffian system associated to any control system in state space form. The algorithm explained here consists in three steps: i) Transformation of the system into its Pfaffian equivalent. ii) Computation of the Goursat normal form for the Pfaffian system. iii) Rewriting of the state space equations in the new variables. At this stage, a feedback law simplifies the equations and, therefore, the flat outputs can be easily computed. As an example, the algorithm is applied to a system consisting in a car with expanding wheels. Point to point trajectories are obtained thanks to the property of differential flatness. The paper also includes a control law that minimizes errors in the initial conditions.Quaternion algebras and quadratic forms towards Shimura curves
http://hdl.handle.net/2117/27989
Title: Quaternion algebras and quadratic forms towards Shimura curves
Authors: Alsina Aubach, MontserratWed, 20 May 2015 15:20:32 GMThttp://hdl.handle.net/2117/279892015-05-20T15:20:32ZAlsina Aubach, MontserratnoA methodology to assess ionospheric models for GNSS
http://hdl.handle.net/2117/27981
Title: A methodology to assess ionospheric models for GNSS
Authors: Rovira Garcia, Adrià; Juan Zornoza, José Miguel; Sanz Subirana, Jaume; González Casado, Guillermo; Ibáñez Segura, Marcos - DeimosWed, 20 May 2015 12:32:18 GMThttp://hdl.handle.net/2117/279812015-05-20T12:32:18ZRovira Garcia, Adrià; Juan Zornoza, José Miguel; Sanz Subirana, Jaume; González Casado, Guillermo; Ibáñez Segura, Marcos - DeimosnoComparison of the microbial dynamics and biochemistry of laboratory sourdoughs prepared with grape, apple and yogurt
http://hdl.handle.net/2117/27975
Title: Comparison of the microbial dynamics and biochemistry of laboratory sourdoughs prepared with grape, apple and yogurt
Authors: Gordún Quiles, Elena; Valle Mendoza, Luis Javier del; Ginovart Gisbert, Marta; Carbó Moliner, Rosa
Abstract: The microbiological culture-dependent characterization and physicochemical characteristics of laboratory sourdough prepared with grape (GS) were evaluated and compared with apple (AS) and yogurt (YS), which are the usual Spanish sourdough ingredients. Ripe GS took longer than AS and YS to reach the appropriate acidity and achieved lower values of lactic acidWed, 20 May 2015 10:22:58 GMThttp://hdl.handle.net/2117/279752015-05-20T10:22:58ZGordún Quiles, Elena; Valle Mendoza, Luis Javier del; Ginovart Gisbert, Marta; Carbó Moliner, RosanoSourdough, grape, ingredients, lactic acid bacteria, yeastThe microbiological culture-dependent characterization and physicochemical characteristics of laboratory sourdough prepared with grape (GS) were evaluated and compared with apple (AS) and yogurt (YS), which are the usual Spanish sourdough ingredients. Ripe GS took longer than AS and YS to reach the appropriate acidity and achieved lower values of lactic acidThe proportional partitional Shapley value
http://hdl.handle.net/2117/27972
Title: The proportional partitional Shapley value
Authors: Alonso Meijide, José María; Carreras Escobar, Francisco; Costa Bouzas, Julián; García Jurado, Ignacio
Abstract: A new coalitional value is proposed under the hypothesis of isolated unions. The main
difference between this value and the Aumann–Drèze value is that the allocations within
each union are not given by the Shapley value of the restricted game but proportionally
to the Shapley value of the original game. Axiomatic characterizations of the new value,
examples illustrating its application and a comparative discussion are provided.Tue, 19 May 2015 17:50:13 GMThttp://hdl.handle.net/2117/279722015-05-19T17:50:13ZAlonso Meijide, José María; Carreras Escobar, Francisco; Costa Bouzas, Julián; García Jurado, IgnacionoGame theory, (TU) cooperative game, Shapley value, Coalition structure, Aumann–Drèze valueA new coalitional value is proposed under the hypothesis of isolated unions. The main
difference between this value and the Aumann–Drèze value is that the allocations within
each union are not given by the Shapley value of the restricted game but proportionally
to the Shapley value of the original game. Axiomatic characterizations of the new value,
examples illustrating its application and a comparative discussion are provided.Empty non-convex and convex four-gons in random point sets
http://hdl.handle.net/2117/27964
Title: Empty non-convex and convex four-gons in random point sets
Authors: Fabila Monroy, Ruy; Huemer, Clemens; Mitsche, Dieter
Abstract: Let S be a set of n points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of S in its interior. We show that the expected number of empty non-convex four-gons with vertices from S is 12 n(2) log n + o(n(2) log n) and the expected number of empty convex four-gons with vertices from S is Theta(n(2)).Tue, 19 May 2015 11:34:59 GMThttp://hdl.handle.net/2117/279642015-05-19T11:34:59ZFabila Monroy, Ruy; Huemer, Clemens; Mitsche, Dieternorandom point set, empty four-gon, polygon, geometric probability, N-RANDOM POINTS, PROBABILITY, TRIANGLES, POSITION, POLYGONS, NUMBER, HOLESLet S be a set of n points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of S in its interior. We show that the expected number of empty non-convex four-gons with vertices from S is 12 n(2) log n + o(n(2) log n) and the expected number of empty convex four-gons with vertices from S is Theta(n(2)).Geometric classification of monogenic subspaces and uniparametric linear control systems
http://hdl.handle.net/2117/27960
Title: Geometric classification of monogenic subspaces and uniparametric linear control systems
Authors: Compta Creus, Albert; Ferrer Llop, Josep
Abstract: We present a geometric approach to the classification of monogenic invariant subspaces, alternative to the classical algebraic one, which allows us to obtain several matricial canonical forms for each class. Some applications are derived: canonical coordinates of a vector with regard to an endomorphism, and a canonical form for uniparametric linear control systems, not necessarily controllable, with regard to linear changes of state variables. Moreover, the pointwise construction
can be extended to differentiable families of changes of basis when differentiable families of equivalent monogenic subspaces are considered.Tue, 19 May 2015 08:05:40 GMThttp://hdl.handle.net/2117/279602015-05-19T08:05:40ZCompta Creus, Albert; Ferrer Llop, Josepnoendomorphism, invariant subspaces, monogenic subspaces, marked
matrices, uniparametric control system, bimodal dynamical systemWe present a geometric approach to the classification of monogenic invariant subspaces, alternative to the classical algebraic one, which allows us to obtain several matricial canonical forms for each class. Some applications are derived: canonical coordinates of a vector with regard to an endomorphism, and a canonical form for uniparametric linear control systems, not necessarily controllable, with regard to linear changes of state variables. Moreover, the pointwise construction
can be extended to differentiable families of changes of basis when differentiable families of equivalent monogenic subspaces are considered.