On teaching and research

Differential Equations. During the Falls of 2004 and 2005 I was a Teaching fellow for the course Differential Equations MA 226 at the Department of Mathematics at Boston University. During the summer of 2005 I participated in the 3^rd edition (2005) of the book “Differential Equations” by Paul Blanchard, Robert L. Devaney and Dick Hall by producing pictures, doing accuracy checking and formatting of text.

Cellular automata and nonlinear diffusion . A cellular automaton is a uniform array of cells which can each have definite states that change over time according to fixed rules. The dynamics of these automata can often be related to those of certain partial differential equations. With T.J. Kaper, C.E. Wayne, and N. Popovic we use cellular automaton models to study one particular set of rules, called 'Critters,' introduced to mimic quantum mechanical phenomena. We study the relationship of this type of automaton to a class of reaction-diffusion equations with density-dependent diffusivities.

Nonlinear Dynamics: A Mathematica Lab Notebook. David K. Campbell, Andrew Charneski, Gamaliel Lodge, Gary Tam and Sebastian M. Marotta. Unpublished, January 2003 through January 2004. A set of Mathematica notebooks that are meant to accompany the book ‘Nonlinear Science: From Paradigms to Practicalities’ by David K. Campbell (to appear). We created more than 80 Mathematica notebooks to study one and two dimensional maps, ordinary differential equations and partial differential equations, with more than 500 pictures and more that 50 exercises with written answers.

Cellular Automata. Sebastian M. Marotta, December, 2002. This work was presented as a project for the Chaotic Dynamical Systems MA 671 course at Boston University. We introduce some characteristics of cellular automata related to the general behavior of dynamical systems by means of three simple one-dimensional examples. We also analyze one aspect of their evolution that is related to the behavior of continuous functions.

Most of the following work has been done in collaboration and under the direction of Prof. Horacio A. Caruso and can be consulted at the Library of the Departamento de Hidráulica de la Facultad de Ingeniería de la UNLP.

Objects Generated by Deposition, Diffusion and Aggregation (DDA) of Particles by Dichotomous Random Walkers, Part I. Horacio A. Caruso and Sebastián M. Marotta, September 2001. This Part contains the description of the DDA model and the new type of random walker used. All the objects are grown inside a square of seeds. The parameters that define the walkers, i.e., the angle of declination of the branches, the inclination of the whole paths and the probability to select right or left branch are studied. The influence of the size of region where the objects growth and the amount of particles added to the objects are also studied. We present an example of application of the Method of Expectancies which shows its efficiency to fit the Area-Perimeter relationship of the objects.

Objects Generated by Deposition, Diffusion and Aggregation (DDA) of Particles by Dichotomous Random Walkers, Part II. Horacio A. Caruso and Sebastián M. Marotta, September 2001. This Part gives information on objects generated inside the same square of seeds of the previous part but with different initial conditions as, for example, seeds placed at different locations inside the square.

Objects Generated by Deposition, Diffusion and Aggregation (DDA) of Particles by Dichotomous Random Walkers, Part III. Horacio A. Caruso and Sebastián M. Marotta, November 2001. In this Part we study objects that growth inside a circumference of seeds with several initial conditions.

Sites Visited By A Linear Random Walker, Part I. Horacio A. Caruso and Sebastián M. Marotta. December, 2000. A one dimensional random walker may choose one step in a positive direction (to the right of a line) or in a negative direction (to the left). The most common random walker (historical or classical walker) is one with equal possibilities, K = ½, for a move to the right or to the left; he may make the decision with the simple toss of a coin. We herein study the probability distribution of sites visited by a walker along a line but we add the concept that K is not a constant but rather a variable number. If K changes with the direction of the previous step (inertial walkers) we may encourage the walker by increasing his possibilities to make a new step in the same direction he had in the previous step, or we may discourage him by increasing the possibilities to make a new step in the opposite direction of the previous step. We study the probabilities each site of the line has to be visited by classical and inertial walkers.

Sites Visited By A Linear Random Walker, Part II. Horacio A. Caruso and Sebastián M. Marotta. February, 2001. We herein study the reach of inertial walkers, i.e., how far from the origin of their walks they can travel.

Sites Visited By A Linear Random Walker, Part III. Horacio A. Caruso and Sebastián M. Marotta. January, 2001. This Part contains information about the probability distribution of visiting sites for some non-symmetrical initial conditions.

Sites Visited By A Linear Random Walker, Part IV. Horacio A. Caruso and Sebastián M. Marotta. February, 2001. In this Part we explore cases of inertial random walkers in which the inertial term varies along the line where the walkers are rambling on.

Sites Visited By A Linear Random Walker, Part V. Horacio A. Caruso and Sebastián M. Marotta. March, 2001. In this Part we study cases in which the inertial term varies according with the step n of the walker.

Sites Visited By A Linear Random Walker, Part VI. Horacio A. Caruso and Sebastián M. Marotta. March, 2001. This part contains information about the velocity of random walkers as a function of the visited sites in a fashion resembling Poincare’s maps.

Sites Visited By A Linear Random Walker, Part A. Horacio A. Caruso and Sebastián M. Marotta. March, 2001. We herein explore cases of inertial random walkers in which the inertial term varies with the probabilities he had in the previous step. Both theoretical and experimental solutions are drawn.

La Reacción de Belousov-Zhabotinsky. Sebastián M. Marotta. Octubre, 2000. A description of the reaction with photographs of the intriguing patterns and a long and commented bibliographic search are presented.

El conjunto de Mandelbrot, los conjuntos de Julia y otras monstruosidades. ¿Cómo se obtienen?. Sebastián M. Marotta. Septiembre, 2000. The programs that generate the Mandelbrot and Julia sets for the quadratic equation are presented. The same programs are applied to the Julia sets for other functions.

Self-Avoiding Random Walkers With Steps Along Eight Possible Directions (these Octopus Walkers last much more than 71 steps). Horacio A. Caruso and Sebastián M. Marotta. August, 1999. We study the number of steps and the distance the walkers reach before they are trapped.

Dichotomous Path for a Series of Primes and Composite Numbers. Horacio A. Caruso and Ssebastián M. Marotta. March, 2000. This work contains the attractors generated by the branches of a dichotomous tree generated with a sequence of turns coming from a chain of prime numbers.

Links, Chapter VII: Links on a Discrete Cantor Set Chain. Horacio A. Caruso and Sebastián M. Marotta. February, 1999. We apply the tool “Links” to a discrete version of the Cantor Set. A function relating the length of the chain, the amplitude of links and the fractal dimension of the set are obtained. Changes in the diagram of links are studied when the chain is subjected to different number of mutations.

Links, Chapter VIII: Links on an Integer Square Root Chain. Horacio A. Caruso and Sebastián M. Marotta. April, 1999. In this chapter we study links of a symbolic chain that assigns positive elements to a subset of the natural numbers with integer square roots and negative elements to the rest of them. Other roots and mutations are also studied.

Self-avoiding Random Walkers. Horacio A. Caruso and Sebastián M. Marotta. August, 1999. Different types of random walkers who can make steps along different directions are studied. The walkers are not allowed to step on places previously visited. The number of steps and the distance reached before the walkers are trapped are studied.

Links, Chapter XII: Links on a Prime Numbers Chain. Horacio A. Caruso and Sebastián M. Marotta. February, 2000. If in a series of integers (1, 2, 3... JF) we assign the attribute +1 to the primes (2, 3, 5, 7...) and –1 to the composites (4, 6, 8, 9...) we have a chain of symbols that corresponds to a dichotomous tree in which the symbols +1 and –1 are the factors that multiply the angle of declination of each branch of a dichotomy. We analyze the links generated by shifts of the described chain. We present a synthesis of the large amount of experiments done and abundant bibliography.

Links, Chapter XIII: Links on Stair Chains. Horacio A. Caruso and Sebastián M. Marotta. September, 2000. In this chapter we study chains generated with basic stairs which are repeated one after the other. Each step of these stairs is composed by a determined number of elements with a particular attribute and is followed by another step of another group until the basic stair is completed. The diagrams of links with and without mutations are presented. It is possible to resume all the information about these chains in only one diagram of links with a scale change in the horizontal axes.

Análisis Geométrico de un Set de Hipérbolas dado por el Prof. Horacio A. Caruso en un caso particular del Método de las Expectativas. Sebastián M. Marotta. Febrero, 1999. Different solutions of a differential equation that lead to a saddle point are analyzed and simulated with the Method of Expectancies.

On hydraulic and civil engineering:

Most of the following works were done under the direction of Engineer Roberto C. Amarilla in Argentina.

Roads and channel network in Campos de Roca (Construction). March – April 1999.

Topography and soil movement for La Plata Show Center. March 1999.

Urban topography of Dolores and Chascomús. October – December 1998.

System of reservoir and spillway in Campos de Roca (Project). October – December 1998.

Hydraulic Study in Santa Catalina, Esteban Echeverría. November 1998.

Hydraulic essays in a scaled physical model of CODE stability (coast protection system). October 1998.

Plans for the Sistema Inteprovincial Federal (channel). Under the direction of Engineer Pablo Romanazzi. July – October 1998.

Topography of Pinamar coastline. November – December 1996.

Hydraulic and Road Engineering in Campos de Roca (Project). August – October 1997.

Topography of the Riachuelo riversides. First Part. February – May 1997.