 Research group
 Linear Control Systems: A Geometric Approach
 Department
 Department of Applied Mathematics I
 Institute
 Institute of Industrial and Control Engineering (IOC)
 School
 Barcelona School of Industrial Engineering (ETSEIB)
 josep.ferrerupc.edu
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Tangencysaddle singularities of planar bimodal linear systems
Ferrer Llop, Jose; Peña Carrera, Marta; Susin Sanchez, Antonio
Mathematical Models and Methods in Applied Sciences
p. 9092
Presentation's date: 20140923
Presentation of work at congresses
Read the abstract View Share Reference managersWe continue the study of the bifurcations and the structural stability of planar bimodal linear dynamical systems (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here we determine the tangencysaddle singularities in the saddle/spiral case, the only where they can appear. 
Closed orbits in planar bimodal linear systems
Ferrer Llop, Jose; Peña Carrera, Marta; Susin Sanchez, Antonio
International Linear Algebra Society Conference
Presentation's date: 20140808
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Determinant of any matrix that belongs to Z(J)
Ferrer Llop, Jose; Mingueza, David; Montoro López, Maria Eulalia
Meeting on Computer Algebra and Applications
p. 111114
Presentation's date: 20140618
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Differentiable families of planar bimodal linear control systems
Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta
Mathematical problems in engineering
Vol. 2014, num. Article ID 292813, p. 19
DOI: 10.1155/2014/292813
Date of publication: 201406
Journal article
Read the abstract Access to the full text Share Reference managersWe consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along it. For a differentiable family of planar bimodal linear control systems, we obtain its stratification diagram and, if controllability holds for each value of the parameters, we construct a differentiable family of feedbacks which stabilizes both subsystems for each value of the parameters. 
Structural stability of planar bimodal linear systems
Ferrer Llop, Jose; Peña Carrera, Marta; Susin Sanchez, Antonio
Mathematical problems in engineering
Vol. 2014, p. 18
DOI: 10.1155/2014/892948
Date of publication: 20140101
Journal article
Read the abstract Access to the full text Share Reference managersStructural stability ensures that the qualitative behavior of a system is preserved under small perturbations. We study it for planar bimodal linear dynamical systems, that is, systems consisting of two linear dynamics acting on each side of a given hyperplane and assuming continuity along the separating hyperplane. We describe which one of these systems is structurally stable when (real) spiral does not appear and when it does we give necessary and sufficient conditions concerning finite periodic orbits and saddle connections. In particular, we study the finite periodic orbits and the homoclinic orbits in the saddle/spiral case. 
Miniversal deformations of observable marked matrices
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Mathematical methods in the applied sciences
Vol. 37, num. 1, p. 2031
DOI: 10.1002/mma.2780
Date of publication: 2014
Journal article
Read the abstract View Share Reference managersGiven the set of vertical pairs of matrices ${\cal M}\subset M_{m,n}(\mathbb C)\times M_n(\mathbb C)$ keeping the subspace $\mathbb C^d\times\{0\}\subset\mathbb C^n$ invariant,we compute miniversal deformations of a given pair when it is observable, and the subspace $\mathbb C^d\times\{0\}$ is marked. Moreover, we obtain the dimension of the orbit, characterize the structurally stable vertical pairs, and study the effect of each deformation parameter. Copyright © 2013 JohnWiley & Sons, Ltd. 
Determinant of a matrix that commutes with a Jordan matrix
Montoro López, Maria Eulalia; Ferrer Llop, Jose; Mingueza, David
Linear algebra and its applications
Vol. 439, num. 12, p. 39453954
DOI: 10.1016/j.laa.2013.10.023
Date of publication: 20131030
Journal article
Read the abstract View Share Reference managersLet F be an arbitrary field, Mn(F) the set of all matrices n×n over F and J¿Mn(F) a Jordan matrix. In this paper, we obtain an explicit formula for the determinant of any matrix that commutes with J, i.e., the determinant of any element T¿Z(J), the centralizer of J. Our result can also be extended to any T'¿Z(A), where A¿Mn(F), can be reduced to J=S1AS. This is because T=S1T'S¿Z(J), and clearly View the MathML source. If F is algebraically closed, any matrix A can be reduced in this way to a suitable J. In order to achieve our main result, we use an alternative canonical form W¿Mn(F) called the Weyr canonical form. This canonical form has the advantage that all matrices K¿Z(W) are upper block triangular. The permutation similarity of T¿Z(J) and K¿Z(W) is exploited to obtain a formula for the determinant of T. The paper is organized as follows: Section 2 contains some definitions and notations that will be used through all the paper. In Section 3, matrices T¿Z(J) are described and the determinant of T is computed in a particular case. In Section 4, we recall the Weyr canonical form W of a matrix and the corresponding centralizer Z(W). A formula to compute the determinant of any K¿Z(W) is rewritten. Finally, in Section 5 an explicit formula for the determinant of any T¿Z(J) is obtained.
Let F be an arbitrary field, Mn(F) the set of all matrices n×n over F and J∈Mn(F) a Jordan matrix. In this paper, we obtain an explicit formula for the determinant of any matrix that commutes with J, i.e., the determinant of any element T∈Z(J), the centralizer of J. Our result can also be extended to any T′∈Z(A), where A∈Mn(F), can be reduced to J=S−1AS. This is because T=S−1T′S∈Z(J), and clearly View the MathML source. If F is algebraically closed, any matrix A can be reduced in this way to a suitable J. In order to achieve our main result, we use an alternative canonical form W∈Mn(F) called the Weyr canonical form. This canonical form has the advantage that all matrices K∈Z(W) are upper block triangular. The permutation similarity of T∈Z(J) and K∈Z(W) is exploited to obtain a formula for the determinant of T. The paper is organized as follows: Section 2 contains some definitions and notations that will be used through all the paper. In Section 3, matrices T∈Z(J) are described and the determinant of T is computed in a particular case. In Section 4, we recall the Weyr canonical form W of a matrix and the corresponding centralizer Z(W). A formula to compute the determinant of any K∈Z(W) is rewritten. Finally, in Section 5 an explicit formula for the determinant of any T∈Z(J) is obtained. 
Differentiable families of stabilizers for planar bimodal linear control systems
Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta
International Conference on Numerical Analysis and Applied Mathematics
p. 20682071
DOI: 10.1063/1.4825942
Presentation's date: 201309
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersWe consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. For a differentiable family of controllable planar ones, we construct a differentiable family of feedbacks which point wise stabilizes both subsystems
We consider bimodal linear control systems consisting of two subsystems acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. For a differentiable family of controllable planar ones, we construct a differentiable family of feedbacks which pointwise stabilizes both subsystems. 
Structural stability of planar bimodal linear systems
Ferrer Llop, Jose; Peña Carrera, Marta; Susin Sanchez, Antonio
International Conference on Numerical Analysis and Applied Mathematics
p. 22052208
DOI: 10.1063/1.4825976
Presentation's date: 201309
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersWe consider bimodal linear dynamical systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. Focusing in the planar case, we describe which of these systems are structurally stable
We consider bimodal linear dynamical systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. Focusing in the planar case, we describe which of these systems are structurally stable. 
Controllability of continuous bimodal linear systems
Ferrer Llop, Jose; Pacha Andujar, Juan Ramon; Peña Carrera, Marta
Mathematical problems in engineering
Vol. 2013, p. 114
DOI: 10.1155/2013/342548
Date of publication: 20130501
Journal article
Read the abstract Access to the full text Share Reference managersWe consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions
We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced to the unobservable case, and for these ones we obtain a simple explicit characterization of controllability for dimensions 2 and 3, as well as some partial criteria for higher dimensions. 
Classification of monogenic invariant subspaces and uniparametric linear control systems
Compta Creus, Albert; Ferrer Llop, Jose
DOI: MTM 201123892
Date: 201212
Report
Read the abstract Access to the full text Share Reference managersThe classification of the invariant subspaces of an endomorphism has been an open problem for a long time, and it is a ”wild” problem in the general case. Here we obtain a full classification for the monogenic ones. Some applications are derived: in particular, canonical forms for uniparametric linear control systems, non necessarily controllable, with regard to linear changes of state variables 
Learning automation to teach mathematics
Ferrer Llop, Jose; Peña Carrera, Marta; Ortiz Caraballo, Carmen
DOI: DOI: 10.5772/2327
Date of publication: 201207
Book chapter
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Stabilization of controllable planar bimodal linear systems
Peña Carrera, Marta; Ferrer Llop, Jose
SIAM Conference on Applied Linear Algebra
p. 146
Presentation's date: 20120621
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Clasificación de los subespacios invariantes monogenerados
Compta Creus, Albert; Ferrer Llop, Jose
Encuentro de Álgebra Lineal, Análisis Matricial y Aplicaciones
p. 33
Presentation's date: 201206
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Perturbations preserving conditioned invariant subspaces
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Mathematical methods in the applied sciences
Vol. 35, num. 3, p. 249371
DOI: 10.1002/mma.1553
Date of publication: 201202
Journal article
Read the abstract View Share Reference managersGiven the set of matrix pairsM Mm,n.C/ Mn.C/ keeping a subspace S Cn invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset ofM. It generalizes the known results when S is a supplementary subspace of the unobservable one. 
Estructuras geométricas de los sistemas de control lineales, lineales a trozos y sistemas conmutados
Compta Creus, Albert; Clotet Juan, Jose; Garcia Planas, Maria Isabel; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta; Puerta Coll, Francisco Javier; Montoro López, Maria Eulalia; Puerta Sales, Fernando; Ferrer Llop, Jose
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Miniversal deformations of observable marked matrices
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Date: 2012
Report
Read the abstract Access to the full text Share Reference managersGiven the set of vertical pairs of matrices M¿ Mm,n(C)×Mn(C) keeping the subspace Cd×{0} ¿ Cn invariant, we compute miniversal deformations of a given pair when it is observable and the subspace Cd × {0} is marked. Moreover, we obtain the dimension of the orbit, characterize the structurally stable vertical pairs and study the effect of each deformation parameter. 
Geometric structure of single/combined equivalence classes of a controllable pair
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Electronic journal of linear algebra
Vol. 22, p. 11121128
Date of publication: 201111
Journal article
Read the abstract Access to the full text Share Reference managersGiven a pair of matrices representing a controllable linear system, its equivalence classes by the single or combined action of feedbacks, change of state and input variables, as well as their intersection are studied. In particular, it is proved that they are differentiable manifolds and their dimensions are computed. Some remarks concerning the effect of different kinds of feedbacks are derived. 
Contribución al estudio de los subespacios invariantes
Compta Creus, Albert; Ferrer Llop, Jose
Congreso de Ecuaciones Diferenciales y Aplicaciones y Congreso de Matemàtica Aplicada
Presentation's date: 20110906
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Orbit stratification of noncontrollable bimodal systems
Peña Carrera, Marta; Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors
Conference of the International Linear Algebra Society
p. 57
Presentation's date: 20110823
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Estructuras geométricas de los sistemas lineales de control y su extensión a los sistemas conmutados
Clotet Juan, Jose; Montoro López, Maria Eulalia; Peña Carrera, Marta; Compta Creus, Albert; Puerta Coll, Francisco Javier; Puerta Sales, Fernando; Garcia Planas, Maria Isabel; Magret Planas, Maria Dels Dolors; Ferrer Llop, Jose
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On the effect of friend feedbacks
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Mathematical methods in the applied sciences
Vol. 34, num. 3, p. 309314
DOI: 10.1002/mma.1357
Date of publication: 2011
Journal article
Read the abstract View Share Reference managersGiven S an (A,B)invariant subspace, we prove that the set of friend feedbacks is a linear variety, which can be considered as the direct sum of the feedbacks of the restriction to S and the corestriction to S ⊥. In particular, when the natural controllability hypothesis hold, both pole assignments are simultaneously possible bymeans of a convenient friend feedback. Copyright 
Learning engineering to teach mathematics
Ferrer Llop, Jose; Peña Carrera, Marta; Ortiz, Carmen
Annual Frontiers in Education Conference
p. 16
DOI: /fieconference.org/fie2010/papers/1207.pdf
Presentation's date: 20101027
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersThe Bologna process is a good opportunity to bring together firstyear mathematics courses of engineering degrees and technology courses offered in subsequent years. In fact, the Faculty Council has decided that 20% of the credits from basic courses must be related to technological applications. To this end, during the past academic year a mathematical engineering seminar was held with each session dealing with one technological discipline. The main goal of the seminar, which relied on the presence of speakers from both mathematics and engineering departments, was to identify the most commonly used mathematical tools. Furthermore, a set of exercises and some guidelines addressed to faculty lacking an engineering background were created. Here, we present some of this material: first, a summary of the collection of exercises illustrating the use of Linear Algebra in different engineering areas such as Mechanical Engineering, Control and Automation, and second, some exercises 
Estructuras Geométricas de Los Sistemas Lineales de Control y su Extensión a los Sistemas Conmutados
Ferrer Llop, Jose; Garcia Planas, Maria Isabel
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Learning engineering to teach mathematics
Ferrer Llop, Jose; Peña Carrera, Marta; Ortiz Caraballo, Carmen
Frontiers in education conference
num. S1J, p. 16
DOI: /fieconference.org/fie2010/papers/1207.pdf
Date of publication: 201010
Journal article
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Perturbations preserving conditioned invariant subspaces
Peña Carrera, Marta; Compta Creus, Albert; Ferrer Llop, Jose
International Conference on Circuits, Systems, Signals
p. 204210
Presentation's date: 201009
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersGiven the set of matrix pairs M ⊂ Mm,n(C) × Mn(C) keeping a subspace S ⊂ Cn invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset of M. It generalizes the known results when S is a supplementary subspace of the unobservable one. 
Geometric structure of the equivalence classes of a controllable pair
Peña Carrera, Marta; Compta Creus, Albert; Ferrer Llop, Jose
International Conference on Circuits, Systems, Signals
p. 211218
Presentation's date: 201009
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersGiven a pair of matrices representing a controllable linear system, we study its equivalence classes by the single or combined action of feedbacks and change of state and input variables, as well as their intersections. In particular, we prove that they are differentiable manifolds and we compute their dimensions. Some remarks concerning the effect of different kinds of feedbacks are derived. 
A lower bound for the distance from a controllable switched linear system to an uncontrollable one
Clotet Juan, Jose; Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors
Conference of the International Linear Algebra Society
p. 55
Presentation's date: 20100620
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Distance from a controllable switched linear system to an uncontrollable one
Clotet Juan, Jose; Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors
Joint ALAMAGAMM/ANLA Meeting
p. 17
Presentation's date: 20100602
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersThe set of controllable switched linear systems is an open set in the space of all switched linear systems. Then it makes sense to compute the distance from a controllable switched linear system to the set of uncontrollable systems. In this work we obtain an upper bound for such distance. 
Miniversal Deformations of Bimodal Picewise Linear Systems
Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Pacha Andujar, Juan Ramon; Peña Carrera, Marta
Joint ALAMAGAMM/ANLA Meeting
p. 16
Presentation's date: 201006
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersBimodal linear systems are those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane. In this work, we focus on the derivation of (orthogonal) miniversal deformations, by using reduced forms.
Keywords: Bimodal piecewise linear system, miniversal deformations, reduced forms. 
Planar bimodal piecewise linear systems. Bifurcation diagrams
Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Pacha Andujar, Juan Ramon; Peña Carrera, Marta
Boletín de la Sociedad Española de Matemática Aplicada
Vol. 51, p. 5563
Date of publication: 2010
Journal article
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Local differentiable pole assignment
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Linear and multilinear algebra
Vol. 58, num. 5, p. 563569
DOI: 10.1080/03081080902745304
Date of publication: 2010
Journal article
Read the abstract Access to the full text Share Reference managersGiven a general local differentiable family of pairs of matrices, we obtain a local differentiable family of feedbacks solving the pole assignment problem, that is to say, shifting the spectrum into a prefixed one. We point out that no additional hypothesis is needed. In fact, simple approaches work in particular cases (controllable pairs, constancy of the dimension of the controllable subspace, and so on). Here the general case is proved by means of Arnold’s techniques: the key point is to reduce the construction to a versal deformation of the central pair; in fact to a quite singular miniversal one for which the family of feedbacks can be explicitly constructed. As a direct application, a differentiable family of stabilizing feedbacks is obtained, provided that the central pair is stabilizable. 
Output regulation problem for differentiable families of linear systems
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Mathematical problems in engineering
Vol. 2010, num. ID 698548, p. 19
DOI: 10.1155/2010/698548
Date of publication: 2010
Journal article
Read the abstract Access to the full text Share Reference managersGiven a family of linear systems depending on a parameter varying in a differentiable manifold, we obtain sufficient conditions for the existence of a (global or local) differentiable family of controllers solving the output regulation problem for the given family. Moreover, we construct it when these conditions hold. 
Bimodal piecewise linear dynamical systems. Reduced forms
Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta
International journal of bifurcation and chaos
Vol. 20, num. 9, p. 27952808
DOI: 10.1142/S0218127410027362
Date of publication: 2010
Journal article
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Learning technology to teach mathematics
Ferrer Llop, Jose; Peña Carrera, Marta; Ortiz Caraballo, Carmen
American Mathematical Society. National Meeting
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Computation of canonical forms and miniversal deformations of bimodal dynamical systems
Peña Carrera, Marta; Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Pacha Andujar, Juan Ramon
Conference of the International Linear Algebra Society
p. 16
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Switched singular linear systems
Clotet Juan, Jose; Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors
Mediterranean Conference on Control and Automation
p. 13431347
DOI: 10.1109/MED.2009.5164733
Presentation's date: 200906
Presentation of work at congresses
Read the abstract Access to the full text Share Reference managersWe consider switched singular linear systems and determine the set of reachable/controllable states. We derive necessary and sufficient conditions for such a system to be reachable/controllable when an “equisingularity condition” holds. 
Stratification and bifurcation diagrams of bimodal piecewise linear dynamical system
Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Pacha Andujar, Juan Ramon; Peña Carrera, Marta
Int. conference of nonautonomous and stochastic dynamical systems, and multidisciplinary applications (NSDS'09)
p. 30
Presentation's date: 200906
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Use of reduced forms in the disturbance decoupling problem
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Linear algebra and its applications
Vol. 430, num. 56, p. 15741589
DOI: 10.1016/j.laa.2008.04.033
Date of publication: 200903
Journal article
Read the abstract View Share Reference managersSpecific algorithms, such as those involving the supremal of the invariant subspaces contained in a suitable subspace, are known to be able to test whether a disturbance decoupling problem (DDP) is solvable. Here, by reducing the system to its Molinari form, we obtain an alternative description of this supremal object and compute its dimension. Hence we have a general result for solving the decoupling provided that a Molinari basis is known. In particular, a necessary numerical condition for it is derived. The same technique is applied to the DDPS, that is, when stability of the decoupled closed loop system is required. 
Output regulation problem for differentiable families of linear systems
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
AIP Conference proceedings
Vol. 1168, p. 4952
DOI: 10.1063/1.3241507
Date of publication: 2009
Journal article
Read the abstract Access to the full text Share Reference managersGiven a family of linear systems depending on a parameter varying in a differentiable manifold, we obtain sufficient conditions for the existence of a (global or local) differentiable family of controllers solving the output regulation problem for the given family. Moreover, we construct it when these conditions hold. 
On the effect of friend feedbacks
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
AIP Conference proceedings
Vol. 1168, num. 3, p. 564567
DOI: 10.1063/1.3241524
Date of publication: 2009
Journal article
Read the abstract Access to the full text Share Reference managersGiven S an (A;B)invariant subspace, we prove that the set of friend feedbacks is a (nm¡md+dq)dimensional linear variety, which can be considered as the direct sum of the feedbacks of the restriction to S and the corestriction to S?. In particular, if (A;B) is controllable and S is a controllability subspace, both pole assignments are simultaneously possible by means of a convenient friend feedback. 
Use of reduced forms in the disturbance decoupling problem
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Date: 200804
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Geometric Structure of the equivalence classes of a controllable pair
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
International Linear Algebra Society (ILAS). Conference
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Perturbations preserving conditioned invariant subspaces
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
International Linear Algebra Society (ILAS). Conference
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Bimodal Piecewise linear systems. Reduced forms
Ferrer Llop, Jose; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta
International Workshop on dynamical systems and multidisciplinary applications
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Geometría de las perturbaciones de sistemas lineales de control
Puerta Coll, Francisco Javier; Compta Creus, Albert; Magret Planas, Maria Dels Dolors; Peña Carrera, Marta; Clotet Juan, Jose; Ferrer Llop, Jose; Puerta Sales, Fernando
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Geometric structure of the equivalence classes of a controllable pair
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Date: 200705
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Local differentiable pole assignment
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Date: 200702
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Realización de perturbaciones
Compta Creus, Albert; Ferrer Llop, Jose; Peña Carrera, Marta
Séptimo Encuentro Plenario de Teoría de Sistemas
Presentation's date: 20070101
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Uso de formas reducidas en el "disturbance decoupling problem"
Ferrer Llop, Jose
Séptimo Encuentro Plenario de Teoría de Sistemas
Presentation's date: 20070101
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