Celso M Doria | Universidade Federal de Santa Catarina - UFSC (Federal University of Santa Catarina) (original) (raw)

Papers by Celso M Doria

ABSTRACT Let X 4 be a smooth closed 4-manifold. We consider on X the coupling of the Einstein equ... more ABSTRACT Let X 4 be a smooth closed 4-manifold. We consider on X the coupling of the Einstein equation with the Seiberg-Witten equation through a variational formulation. Although the SW equation does not have a natural variational formulation, the most trivial functional, which Euler-Lagrange equations are satisfied by the solutions of SW-equation, is used to perform the coupling with the Einstein equation. Since the Einstein equation has a variational formulation given by the Einstein-Hubert functional, we simply compute the tensor energy-momentum of the coupled theory.

Springer eBooks, 2021

Applications are widespread in many topics of Pure and Applied Mathematics. To apply the formalis... more Applications are widespread in many topics of Pure and Applied Mathematics. To apply the formalism of differential forms and the Stokes Theorem, we will discuss the topics on Harmonic Functions and the geometric formulation of Electromagnetism without delving into the contents.

Springer eBooks, 2021

The analysis of the behavior of a function is efficiently carried out when we study the way in wh... more The analysis of the behavior of a function is efficiently carried out when we study the way in which the function varies. In this chapter, techniques used in studying functions of one real variable \( f:I\rightarrow \mathbb {R}\), defined in an open interval \(I\subset \mathbb {R}\), are extended for functions of several real variables \(f:U\rightarrow \mathbb {R}^{m}\) defined over an open subset \(U\subset \mathbb {R}^{n}\). Several real variables is understood to mean a finite number of variables \((x_{1},\ldots ,x_{n})\in \mathbb {R}^{n}\). The simple topological nature of \(\mathbb {R}^{n}\) allows the theory to be more easily understood as all the concepts and techniques. The same concepts and techniques will be studied in the chapters ahead within the framework of Banach spaces.

L e t M be a surface with dM * 0 and N a n-m anifold , also consider S a embedded surface in N. T... more L e t M be a surface with dM * 0 and N a n-m anifold , also consider S a embedded surface in N. The problem treated in this thesis is the existence of a smooth map <p:M-* N satisfying the following conditions : (i) AM<p + Nr(<p)(d<p,d<p) « 0 , AM-Laplace-Beltrami operator on (M,y) Nr-Christoffel symbols o f (N,h) (ii) There exist a strictly positive function X :M-» R such that the pull-back metric <p*h on each fiber of the pull-back vector bundle <p''(TN) over M satisfies the relation 9 h ■ X.y (iii) 9(9M) c S (iv) |2 (w) x T<p(w)S for all w e 9M ; dtp.n where n is the normal direction along 9M induced by the orientation on M. T h e technique used is based on Critical Point Theory applied to Variational Analysis. Instead of finding a solution <p, o f the elliptic system in (i) , as a solution for the Euler-Lagrange equations o f the energy functional E(«p)-M , the solution is found by considering 9 as the limit when o-» l o f solutions for the Euler-Lagange equations associated with the a-en erg y functionals Ea (<p)-I J[ 1+ Idepl2 f 1 dM for a > 1. T h e condition in (ii) is proved assuming a condition known as Douglas C on d itio n , which statement guarantees that the minimal surface in the homotopy class of 9 h a s the same number o f boundaries components and the same genus (as a topological space) as M. CHAPTER 1.

Given a symmetry: before looking for its possible extensions one should first search for its corr... more Given a symmetry: before looking for its possible extensions one should first search for its corresponding asymmetry. This means to find out the opportunities that such symmetry offers. Then our effort here is to study an asymmetry possibility by introducing different potential fields rotating under a same single group. In this work asymmetric classical properties are studied for the non-abelian case. The presence of asymmetry is first derived through new strength field tensors and collective aspects. Then field equations, Bianchi identities, Noether theorem and conserved currents are obtained. Diversity and connectivity are the main results from asymmetry. Diversity can be seen through the various quanta carrying different masses, spin and coupling constants obtained through equations of motion and their corresponding conserved currents. Connectivity through new inductive relationships derived from new Bianchi identities and coupled equations. It is a gauge theory that emphasizes the meaning of entanglement.

Communications in Theoretical Physics, Jun 1, 1992

ABSTRACT We adopt the analytic approach to connections on a general vector bundle to discuss the ... more ABSTRACT We adopt the analytic approach to connections on a general vector bundle to discuss the independence of vector gauge potentials introduced simultaneously in association with a single simple group.

It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witte... more It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witten equation on a compact 4-manifold X admit a regular solution once the nonhomogeneous Palais-Smale condition Ᏼ is satisfied. The approach consists in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation. The gauge invariance of the functional allows to restrict the problem to the Coulomb subspace Ꮿ C α of configuration space. The coercivity of the ᐃ α-functional, when restricted into the Coulomb subspace, imply the existence of a weak solution. The regularity then follows from the boundedness of L ∞-norms of spinor solutions and the gauge fixing lemma.

Springer eBooks, 2021

We will review some operations in vector calculus that will motivate using differential forms whe... more We will review some operations in vector calculus that will motivate using differential forms when integrating vector fields. The differential forms formalism allows us to generalize the Stokes Theorem to describe the conditions of integrability (Frobenius Theorem), and to write Maxwell’s equations succinctly to obtain topological invariants using differentiable tools and many other applications.

Differentiability in Banach Spaces, Differential Forms and Applications, 2021

Applications are widespread in many topics of Pure and Applied Mathematics. To apply the formalis... more Applications are widespread in many topics of Pure and Applied Mathematics. To apply the formalism of differential forms and the Stokes Theorem, we will discuss the topics on Harmonic Functions and the geometric formulation of Electromagnetism without delving into the contents.

Proceedings of Fifth International Conference on Mathematical Methods in Physics — PoS(IC2006), 2007

Given a symmetry: before looking for its possible extensions one should first search for its corr... more Given a symmetry: before looking for its possible extensions one should first search for its corresponding asymmetry. This means to find out the opportunities that such symmetry offers. Then our effort here is to study an asymmetry possibility by introducing different potential fields rotating under a same single group. In this work asymmetric classical properties are studied for the non-abelian case. The presence of asymmetry is first derived through new strength field tensors and collective aspects. Then field equations, Bianchi identities, Noether theorem and conserved currents are obtained. Diversity and connectivity are the main results from asymmetry. Diversity can be seen through the various quanta carrying different masses, spin and coupling constants obtained through equations of motion and their corresponding conserved currents. Connectivity through new inductive relationships derived from new Bianchi identities and coupled equations. It is a gauge theory that emphasizes the meaning of entanglement.

In the Theory of the Seiberg-Witten Equations, the configuration space is Cα = Aα × Γ (S + α ), w... more In the Theory of the Seiberg-Witten Equations, the configuration space is Cα = Aα × Γ (S + α ), where Aα is a space of u1-connections and Γ (S + α ) is the space of sections of the complex spinor bundle over X. Since the SWequation fits in a variational approach, invariant by the action of the Gauge Group Gα = Map(X, U1), which satisfies the Palais-Smale Condition, our aim is to obtain existence of solutions to the 2-order SW-equations by describing the weak homotopy type of the space Aα ×Gα Γ (S + α ). The 2 -order SWequations are the Euler-Lagrange equation of a functional, they generalise the 1-order SW-equations

Proceedings of 7th International Conference on Mathematical Methods in Physics — PoS(ICMP 2012), 2013

The instability of the reducible critical points (A, 0) of the Seiberg-Witten functional is studi... more The instability of the reducible critical points (A, 0) of the Seiberg-Witten functional is studied by analysing the lowest eigenvalue of the elliptic operator L A = A + k g 4. A short resume about the existence and relevance of SW-monopoles is given and also how geometrical structures on a 4-manifold X plays an important role to the theory.

Robotica, 2012

SUMMARYThis paper presents applications of group theory tools to simplify the analysis of kinemat... more SUMMARYThis paper presents applications of group theory tools to simplify the analysis of kinematic chains associated with mechanisms and parallel manipulators. For the purpose of this analysis, a kinematic chain is described by its properties, i.e. degrees-of-control, connectivity and redundancy matrices. In number synthesis, kinematic chains are represented by graphs, and thus the symmetry of a kinematic chain is the same as the symmetry of its graph. We present a formal definition of symmetry in kinematic chains based on the automorphism group of its associated graph. The symmetry group of the graph is associated with the graph symmetry. By using the group structure induced by the symmetry of the kinematic chain, we prove that degrees-of-control, connectivity and redundancy are invariants by the action of the automorphism group of the graph. Consequently, it is shown that it is possible to reduce the size of these matrices and thus reduce the complexity of the kinematic analysis ...

ABSTRACT Let X 4 be a smooth closed 4-manifold. We consider on X the coupling of the Einstein equ... more ABSTRACT Let X 4 be a smooth closed 4-manifold. We consider on X the coupling of the Einstein equation with the Seiberg-Witten equation through a variational formulation. Although the SW equation does not have a natural variational formulation, the most trivial functional, which Euler-Lagrange equations are satisfied by the solutions of SW-equation, is used to perform the coupling with the Einstein equation. Since the Einstein equation has a variational formulation given by the Einstein-Hubert functional, we simply compute the tensor energy-momentum of the coupled theory.

Springer eBooks, 2021

Applications are widespread in many topics of Pure and Applied Mathematics. To apply the formalis... more Applications are widespread in many topics of Pure and Applied Mathematics. To apply the formalism of differential forms and the Stokes Theorem, we will discuss the topics on Harmonic Functions and the geometric formulation of Electromagnetism without delving into the contents.

Springer eBooks, 2021

The analysis of the behavior of a function is efficiently carried out when we study the way in wh... more The analysis of the behavior of a function is efficiently carried out when we study the way in which the function varies. In this chapter, techniques used in studying functions of one real variable \( f:I\rightarrow \mathbb {R}\), defined in an open interval \(I\subset \mathbb {R}\), are extended for functions of several real variables \(f:U\rightarrow \mathbb {R}^{m}\) defined over an open subset \(U\subset \mathbb {R}^{n}\). Several real variables is understood to mean a finite number of variables \((x_{1},\ldots ,x_{n})\in \mathbb {R}^{n}\). The simple topological nature of \(\mathbb {R}^{n}\) allows the theory to be more easily understood as all the concepts and techniques. The same concepts and techniques will be studied in the chapters ahead within the framework of Banach spaces.

L e t M be a surface with dM * 0 and N a n-m anifold , also consider S a embedded surface in N. T... more L e t M be a surface with dM * 0 and N a n-m anifold , also consider S a embedded surface in N. The problem treated in this thesis is the existence of a smooth map <p:M-* N satisfying the following conditions : (i) AM<p + Nr(<p)(d<p,d<p) « 0 , AM-Laplace-Beltrami operator on (M,y) Nr-Christoffel symbols o f (N,h) (ii) There exist a strictly positive function X :M-» R such that the pull-back metric <p*h on each fiber of the pull-back vector bundle <p''(TN) over M satisfies the relation 9 h ■ X.y (iii) 9(9M) c S (iv) |2 (w) x T<p(w)S for all w e 9M ; dtp.n where n is the normal direction along 9M induced by the orientation on M. T h e technique used is based on Critical Point Theory applied to Variational Analysis. Instead of finding a solution <p, o f the elliptic system in (i) , as a solution for the Euler-Lagrange equations o f the energy functional E(«p)-M , the solution is found by considering 9 as the limit when o-» l o f solutions for the Euler-Lagange equations associated with the a-en erg y functionals Ea (<p)-I J[ 1+ Idepl2 f 1 dM for a > 1. T h e condition in (ii) is proved assuming a condition known as Douglas C on d itio n , which statement guarantees that the minimal surface in the homotopy class of 9 h a s the same number o f boundaries components and the same genus (as a topological space) as M. CHAPTER 1.

Given a symmetry: before looking for its possible extensions one should first search for its corr... more Given a symmetry: before looking for its possible extensions one should first search for its corresponding asymmetry. This means to find out the opportunities that such symmetry offers. Then our effort here is to study an asymmetry possibility by introducing different potential fields rotating under a same single group. In this work asymmetric classical properties are studied for the non-abelian case. The presence of asymmetry is first derived through new strength field tensors and collective aspects. Then field equations, Bianchi identities, Noether theorem and conserved currents are obtained. Diversity and connectivity are the main results from asymmetry. Diversity can be seen through the various quanta carrying different masses, spin and coupling constants obtained through equations of motion and their corresponding conserved currents. Connectivity through new inductive relationships derived from new Bianchi identities and coupled equations. It is a gauge theory that emphasizes the meaning of entanglement.

Communications in Theoretical Physics, Jun 1, 1992

ABSTRACT We adopt the analytic approach to connections on a general vector bundle to discuss the ... more ABSTRACT We adopt the analytic approach to connections on a general vector bundle to discuss the independence of vector gauge potentials introduced simultaneously in association with a single simple group.

It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witte... more It is shown that the nonhomogeneous Dirichlet and Neuman problems for the 2nd-order Seiberg-Witten equation on a compact 4-manifold X admit a regular solution once the nonhomogeneous Palais-Smale condition Ᏼ is satisfied. The approach consists in applying the elliptic techniques to the variational setting of the Seiberg-Witten equation. The gauge invariance of the functional allows to restrict the problem to the Coulomb subspace Ꮿ C α of configuration space. The coercivity of the ᐃ α-functional, when restricted into the Coulomb subspace, imply the existence of a weak solution. The regularity then follows from the boundedness of L ∞-norms of spinor solutions and the gauge fixing lemma.

Springer eBooks, 2021

We will review some operations in vector calculus that will motivate using differential forms whe... more We will review some operations in vector calculus that will motivate using differential forms when integrating vector fields. The differential forms formalism allows us to generalize the Stokes Theorem to describe the conditions of integrability (Frobenius Theorem), and to write Maxwell’s equations succinctly to obtain topological invariants using differentiable tools and many other applications.

Differentiability in Banach Spaces, Differential Forms and Applications, 2021

Applications are widespread in many topics of Pure and Applied Mathematics. To apply the formalis... more Applications are widespread in many topics of Pure and Applied Mathematics. To apply the formalism of differential forms and the Stokes Theorem, we will discuss the topics on Harmonic Functions and the geometric formulation of Electromagnetism without delving into the contents.

Proceedings of Fifth International Conference on Mathematical Methods in Physics — PoS(IC2006), 2007

Given a symmetry: before looking for its possible extensions one should first search for its corr... more Given a symmetry: before looking for its possible extensions one should first search for its corresponding asymmetry. This means to find out the opportunities that such symmetry offers. Then our effort here is to study an asymmetry possibility by introducing different potential fields rotating under a same single group. In this work asymmetric classical properties are studied for the non-abelian case. The presence of asymmetry is first derived through new strength field tensors and collective aspects. Then field equations, Bianchi identities, Noether theorem and conserved currents are obtained. Diversity and connectivity are the main results from asymmetry. Diversity can be seen through the various quanta carrying different masses, spin and coupling constants obtained through equations of motion and their corresponding conserved currents. Connectivity through new inductive relationships derived from new Bianchi identities and coupled equations. It is a gauge theory that emphasizes the meaning of entanglement.

In the Theory of the Seiberg-Witten Equations, the configuration space is Cα = Aα × Γ (S + α ), w... more In the Theory of the Seiberg-Witten Equations, the configuration space is Cα = Aα × Γ (S + α ), where Aα is a space of u1-connections and Γ (S + α ) is the space of sections of the complex spinor bundle over X. Since the SWequation fits in a variational approach, invariant by the action of the Gauge Group Gα = Map(X, U1), which satisfies the Palais-Smale Condition, our aim is to obtain existence of solutions to the 2-order SW-equations by describing the weak homotopy type of the space Aα ×Gα Γ (S + α ). The 2 -order SWequations are the Euler-Lagrange equation of a functional, they generalise the 1-order SW-equations

Proceedings of 7th International Conference on Mathematical Methods in Physics — PoS(ICMP 2012), 2013

The instability of the reducible critical points (A, 0) of the Seiberg-Witten functional is studi... more The instability of the reducible critical points (A, 0) of the Seiberg-Witten functional is studied by analysing the lowest eigenvalue of the elliptic operator L A = A + k g 4. A short resume about the existence and relevance of SW-monopoles is given and also how geometrical structures on a 4-manifold X plays an important role to the theory.

Robotica, 2012

SUMMARYThis paper presents applications of group theory tools to simplify the analysis of kinemat... more SUMMARYThis paper presents applications of group theory tools to simplify the analysis of kinematic chains associated with mechanisms and parallel manipulators. For the purpose of this analysis, a kinematic chain is described by its properties, i.e. degrees-of-control, connectivity and redundancy matrices. In number synthesis, kinematic chains are represented by graphs, and thus the symmetry of a kinematic chain is the same as the symmetry of its graph. We present a formal definition of symmetry in kinematic chains based on the automorphism group of its associated graph. The symmetry group of the graph is associated with the graph symmetry. By using the group structure induced by the symmetry of the kinematic chain, we prove that degrees-of-control, connectivity and redundancy are invariants by the action of the automorphism group of the graph. Consequently, it is shown that it is possible to reduce the size of these matrices and thus reduce the complexity of the kinematic analysis ...