Dr. Bernardo Abrego

Teaching (Spring 2008)

Problem of the Week

Not running while I am away on Sabbatical. However the database of problems is still available.

The Problem of the Week is a contest open to any enrolled undergraduate or graduate student at CSUN. Every Monday a new problem will be available on our web page for you to try. The first correct and complete solution receives a prize. For more information (including the rules and the current problem of the week) visit:

www.csun.edu/math/probweek

Interested in starting a research project?

We have research projects linked to some "Problems of the Week". Take a look! (use the link above). If interested please contact me. You could

• Develop your research as part of a reading course
• Prepare a presentation or create a poster (for a local or national MAA meeting, or the CSUN Research Symposium, the Math Club...)
• Start a Master thesis.
• Even get your research published in a professional journal!

Putnam Exam

The William Lowell Putnam Competition is a prestigiouos nationwide mathematics exam for undergraduate students. The exam usually takes place the first Saturday of December each year. If you are an undergraduate student at CSUN and would like to take the exam in 2006 please contact me.

Problem Solving

If you like puzzles or the challenge of solving an interesting nontraditional mathematics problem then be part of the "Problem Solving Group".
This can give you the opportunity to see your name on one of the Mathematical Association of America journals (the Mathematics Magazine or the College Mathematics Journal). For more information please contact me or Dr. Bernardo Abrego. For a list of CSUN student whose name appear in such journals click here.

Research

Combinatorics, Discrete and Computational Geometry, Graph Theory.

Publications

• B. M. Ábrego, S. Fernández-Merchant, M. G. Neubauer, and W. Watkins. Sum of squares of degrees in a graph. Submitted to Journal of Graph Theory (2008).
• B. M. Ábrego, S. Fernández-Merchant, and B. Llano. On the maximum number of translates. Submitted to Discrete and Computational Geometry (2008).
• B. M. Ábrego, S. Fernández-Merchant, and B. Llano. An inequality for Kruskal-Macaulay Functions. Submitted to European Journal of Combinatorics (2007).
• B. M. Ábrego, S. Fernández-Merchant, J. Leaños, and G. Salazar. A central approach to bound the number of crossings in a generalized configuration. Electronic Notes in Discrete Mathematics 30 (2008) 273-278.
• B. M. Ábrego, S. Fernández-Merchant, J. Leaños, and G. Salazar. The maximum number of halving lines and the rectilinear crossing number of Kn for n<=27 . Electronic Notes in Discrete Mathematics 30 (2008), 261-266.
• B. M. Ábrego, J. Balogh, S. Fernández-Merchant, J. Leaños, and G. Salazar. An extended lower bound on the number of (<=k)-edges to generalized configurations of points and the pseudolinear crossing number of Kn. To appear in Journal of Combinatorial Theory, Series A. Available online as article in press.
• B. M. Ábrego, J. Balogh, S. Fernández-Merchant, J. Leaños, and G. Salazar. On (<=k)–pseudoedges in generalized configurations and the pseudolinear crossing number of Kn. Preprint (2006).
• B. M. Ábrego and S. Fernández-Merchant. Geometric drawings of Kn with few crossings. Journal of Combinatorial Theory, Series A 114 (2007) 373-379.
  
• B. M. Ábrego, S. Fernández-Merchant, M. G. Neubauer, and W. Watkins. Trace-minimal graphs and D-optimal weighing designs, Linear Algebra and its Applications 412 (2006), 161-221.
  
• B. M. Ábrego and S. Fernández-Merchant. A lower bound for the rectilinear crossing number. Graphs and Combinatorics, Springer-Verlag Tokyo Inc., Vol. 21, Num 3, (2005), p. 293 - 300.
  
• B. M. Ábrego, E. Arkin, S. Fernández-Merchant, F. Hurtado, M. Kano, J. Mitchell, J. Urrutia. Matching points with geometric objects: combinatorial results. Discrete and Computational Geometry Japanese Conference, JCDCG 2004, Tokyo, Japan, October 8-11, 2004. Springer-Verlag, Lecture Notes in Computer Science Series, Vol. 3742, Akiyama, Jin; Kano, Mikio; Tan, Xuehou (Eds.) 2005, VIII, p. 1-15, ISBN: 3-540-30467-3.
  
• B. M. Ábrego, S. Fernández-Merchant, and Gy. Elekes. Structural results for planar sets with many similar subsets. Combinatorica 24 (2004), no. 4, 541-554.
  
• B. M. Ábrego, S. Fernández-Merchant, M. G. Neubauer, and W. Watkins. D-optimal weighing designs for n´ -1 (mod 4) objects and a large number of weighings, Linear Algebra and its Applications 374 (2003), 175-218.
  
• B. M. Ábrego, S. Fernández-Merchant, M. G. Neubauer, and W. Watkins. Maximum volume simplices and trace-minimal graphs, Proceedings of X Encuentros de Geometria Computacional, University of Sevilla, Sevilla Spain, June 16-17, (2003), 51-56.
  
• B. M. Ábrego and S. Fernández-Merchant. The unit distance problem for centrally symmetric convex polygons. Discrete and computational geometry and graph drawing (Columbia, SC, 2001). Discrete and Computational Geometry 28 (2002), no 4, 467-473.
  
• B. M. Ábrego and S. Fernández-Merchant. Convex Polyhedra in R^3 Spanning W(n^(4/3)) Congruent Triangles, Journal of Combinatorial Theory, Series A 98, 406-409 (2002).
  
• B. M. Ábrego and S. Fernández-Merchant. On the Maximum Number of Equilateral Triangles I. Discrete and Computational Geometry 23, (2000), no. 1, 129-115.
  
• B. M. Ábrego, J. L. Arocha, S. Fernández-Merchant, and V. Neumann-Lara. Tightness Problems in the Plane. Discrete Mathematics 194 (1999), no. 1-3, 1-11.
• B. M. Ábrego, Problemas Combinatorios Sobre Conjuntos Finitos de Puntos (Combinatorial Problems on Finite Sets). Aportaciones Matemáticas: Comunicaciones 19, SMM, Mexico (1997), ii+177pp, (1997) ISBN 968-36-5771-0.
• B. M. Ábrego, El Teorema de Sylvester (Sylvester's Theorem), Miscelánea Matemática, 23 (1996), 17-26.