| 42/2001 |
On the differentiability of fuzzy-valued mappings and the stability of a fuzzy differential inclusion Yurilev Chalco-Cano, Marko A. Rojas-Medar We introduce a new concept of differentiability for fuzzy-valued mapping and we study some of its properties. Using this concept, we give a result on stability of the Lyapunov type for fuzzy differential inclusions.  rp-2001-42.pdf |
| 41/2001 |
Mathematical and numerical methods for vertical equilibrium in porous media Maria Cristina C. Cunha, Marcelo M. Santos, J. E. Bonet We consider the motion of two-phases flow in a porous medium under the condition of vertical equilibrium. The mathematical model is deduced and boundary and initial condition are prescribed. The flux function has a bell shape, that is, it changes concavity twice. We solve analytically the corresponding Riemann problem with boundary conditions for short time. Then we design numerical schemes of Godunov type for the mathematical model with and without viscosity. Using these schemes we compute numerically the solutions for long time. These numerical solutions agree very sharply with the analytical ones for short time, what validates our schemes. Furthermore, our work is motivated by a laboratory experiment that presents hysteresis effects. We also present analytical and numerical solutions of our model with hysteresis. rp-2001-41.pdf |
| 40/2001 |
Counting Domains in $\{ p,q\}$ Tessellations Eduardo Brandani Silva, Marcelo Firer, Reginaldo Palazzo Jr. For any given regular $\left\{ p,q\right\} $ tessellation in the hyperbolic plane, we compute the number of vertices and tiles to be found as we distance from a given point, enabling a complete characterization of the asymptotic behavior. rp-2001-40.pdf |
| 39/2001 |
Streamline Diffusion Method for a Nonlinear Equation Governing the Spreading of Oil Spills Márcio Rodolfo Fernandes, Petrônio Pulino, José Luiz Boldrini The paper describes a finite element method that uses a version of the space-time streamline diffusion technique and includes the control of total mass applied to a nonlinear convection--diffusion equation that governs the spreading of oil spills on moving water surfaces. The use of such equation for numerical predictions of the evolution of such spills, although highly desirable to help to lessen their consequences, brings several difficulties. In fact, from the theoretical point of view, the equation presents either parabolic or hyperbolic (in the sense of transport equations) character depending on the solution itself. This is due to the nonlinearity of the diffusion term that can pass from strictly positive to zero and vice-versa depending on the value of the solution. In such {\it a priori} unknown regions, fast transitions may occur, bringing spurious oscillations that may deteriorate the numerical solutions obtained with ordinary algorithms. The performance of the proposed method is compared in controled situations with the corresponding performances of more traditional methods. The results shows clear advantages in its use. rp-2001-39.pdf |
| 38/2001 |
A rank-three condition for invariant $(1,2)$-symplectic almost Hermitian structures on flag manifolds Nir Cohen, Caio J. C. Negreiros, Luiz A. B. San Martin This paper considers invariant $\left( 1,2\right) $-symplectic almost Hermitian structures on the maximal flag manifod associated to a complex semi-simple Lie group $G$. The concept of cone-free invariant almost complex structure is introduced. It involves the rank-three subgroups of $G$, and generalizes the cone-free property for tournaments related to $\mathrm{Sl} \left( n,\Bbb{C}\right) $ case. It is proved that the cone-free property is necessary for an invariant almost-complex structure to take part in an invariant $\left( 1,2\right) $-symplectic almost Hermitian structure. It is also sufficient if the Lie group is not $B_{l}$, $l\geq 3$, $G_{2}$ or $F_{4} $. For $B_{l}$ and $F_{4}$ a close condition turns out to be sufficient. rp-2001-38.pdf |
| 37/2001 |
Grupos Cristalográficos e Empacotamentos de Esferas Marcelo Firer, Trajano Nóbrega Pires rp-2001-37.pdf |
| 36/2001 |
On a $\mathbf{F_{q^2}}$-maximal curve of genus $\mathbf{q(q-3)/6}$} Miriam Abdón, Fernando Torres We show that a $\fq$-maximal curve of genus $q(q-3)/6$ in characteristic three is unique up to $\fq$-isomorphism unless an unexpected situation occurs. rp-2001-36.pdf |
| 35/2001 |
A Mathematical Analysis of a Phase Field Type Model for Solidification with Convection: pure materials in the two dimensional case Cristina Lúcia Dias Vaz, José Luiz Boldrini We investigate the existence and regularity of weak solutions of a phase field type model for pure material solidification in presence of natural convection. We assume that the nostationary solidification process occurs in a bounded domain, which for technical reasons are restricted to be two dimensional. The governing equations of the model are the following: the phase field equation coupled with a nonlinear heat equation and modified Navier-Stokes equations which include buoyancy forces modeled by Boussinesq approximation and a Carman-Koseny term to model the flow in mushy regions. Since this modified Navier-Stokes equations only hold in a priori unknown non-solid regions, we actually have a free boundary value problem. rp-2001-35.pdf |
| 34/2001 |
q-Fibonacci Sequences, Bipartite Numbers and Lattice Paths José Plínio O. Santos, Paulo Mondek We examine a pair of Rogers-Ramanujan type identities given in Slater[7] and give polynomial families from which the original identities can be obtained. The polynomial families term out to be q-analogs of the Fibonacci sequence. Finally we provide combinatorial interpretations for the identities. rp-2001-34.pdf |
| 33/2001 |
Weak Solutions of a Phase-Field Model for Phase Change of an Alloy with Thermal Properties José Luiz Boldrini, Gabriela Planas The phase-field method provides an alternative mathematical description for free-boundary problems corresponding to physical process with phase transitions. It postulates the existence of a function, called the phase-field, whose value identifies the phase at a particular point in space and time, and it is particularly suitable for cases with complex growth strutures occuring during phase transitions. The mathematical model studied in this work describes the solidification process occurring in a binary alloy with temperature dependent properties. It is based on a highly nonlinear degenerate parabolic system of partial differential equations with three independent variables: phase-field, solute concentration and temperature. Existence of weak solutions of such system is obtained via the introduction of a regularized problem, followed by the derivation of suitable estimates and the application of compactness arguments. rp-2001-33.pdf |
