Gauss               Hilbert

•   Research topics:  approximation theory, numerical analysis, potential theory

•   My pearl idea is given in the 4th edition of the popular book "Proofs from the BOOK" (see "Pearl Lemma" on page 54). My paper about Hilbert's third problem is referenced on page 61. Pearls are called "basic points" in that paper (PDF), and are called pearls in a magazine article.

•   During 2012-2015 I will represent AMS at the Committee on American Mathematics Competitions.

•   Erdõs Number:  1.33  (that is, 2 with multiplicipy 3, 1+1/3=1.33)

•   Wiles Number in tennis:  2 - this means that I played tennis with someone who played tennis with Andrew Wiles. 

•   János Bolyai (see Bolyai-Lobachevsky geometry) may be a distant relative of mine. His mother was Zsuzsanna Árkosi Benkõ, and my father Pal Benko (a former world chess champion candidate) is coming from the Árkosi Benkõ family.

Publications:


16.)  (with P. D. Dragnev and V. Totik) Convexity of harmonic densities, Revista Mat. Iberoam. (to appear)


15.)  (with P. D. Dragnev) Balayage ping-pong: a convexity of equilibrium measures, Constr. Approx. 36 (2012), no. 2, 191–214.


14.)  The Basel problem as a telescoping series,  College Math. J. 43 (2012), no. 3, 244–250.


13.)  (with S. B. Damelin and P. D. Dragnev) On supports of equilibrium measures with concave signed equilibria,  J. Comput. Anal. Appl. 14 (2012), no. 4, 752–766.


12.)  Weighted polynomial approximation for weak convex external fields,  J. Math. Anal. Appl. 385 (2012), no. 1, 447–457.


11.)  (with C. Ernst and D. Lanphier) Asymptotic bounds on the integrity of graphs and separator theorems for graphs. SIAM J. Discrete Math. 23 (2008/09), no. 1, 265–277 --- PDF


10.)  (with A. Kroó) A Weierstrass-type theorem for homogeneous polynomials, Trans. Amer. Math. Soc., 361 (2009), 1645-1665 --- PDF


9.)  (with D. Biles, M. P. Robinson, J. Spraker) Numerical Approximation for Singular Second Order Differential Equations (2009), Mathematical and Computer Modelling, 49, 1109-1114 --- PDF


8.)  (with D. Biles, M. P. Robinson, J. Spraker) Nyström methods and singular second-order differential equations, Comput. Math. Appl. 56 (2008), no. 8, 1975–1980. --- PDF


7.)  A new approach to Hilbert's third problem, Amer. Math. Monthly  114  (2007),  no. 8, 665-676 --- PDF


6.)  (with S. B. Damelin and P. D. Dragnev) On the support of the equilibrium measure for arcs of the unit circle and for real intervals, Electron. Trans. Numer. Anal. 25 (2006), 27-40 --- PDF


5.)  The support of the equilibrium measure, Acta Sci. Math. (Szeged) 70 (2004), no. 1-2, 35-55 --- PDF


4.)  (with T. Erdélyi, and J. Szabados) The full Markov-Newman inequality for Müntz polynomials on positive intervals, Proc. Amer. Math. Soc. 131 (2003), no. 8, 2385-2391 --- PDF


3.)  (with T. Erdélyi) Markov inequality for polynomials of degree n with m distinct zeros, J. Approx. Theory 122 (2003), no. 2, 241-248 --- PDF


2.)  Approximation by weighted polynomials, J. Approx. Theory 120 (2003), no. 1, 153-182 --- PDF


1.)  (with V. Totik) Sets with interior extremal points for the Markoff inequality, J. Approx. Theory 110 (2001), no. 2, 261-265 --- PDF


Other works:


The Riemann zeta function and the striped anaconda, Polygon (to appear)


Get rich slowly, almost surely, The Mathematical Gazette (to appear)


A lottery-like stock market, Math Horizons, (2011 Febr.) 24-28


The equilibrium measure and the Saff conjecture, Ph.D. dissertation (2006),
University of Szeged, Hungary


Approximation by weighted polynomials, Ph.D. dissertation (2001), 
University of South Florida


On Slowly Diverging Series, Polygon (1995), V.2, 89-100


On the Number of Legal Chess Positions, Alpha (1995), no.1, 10-11


Fast decreasing polynomials, Master thesis (1995), University of Szeged


On the generalization of the Fundamental Theorem of Algebra, University of Szeged,
Research Competition for Students (1995)


On Slowly Converging Series, Polygon (1994), IV.2, 95-108


Collaborators:

  •  D. Biles

  •  S. Damelin

  •  P. Dragnev

  •  T. Erdélyi

  •  C. Ernst

  •  A. Kroó

  •  D. Lanphier

  •  M. Robinson

  •  J. Spraker

  •  J. Szabados

  •  V. Totik

 

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Problems:

I have some challenging math problems which have been used in math contests. For example:

(a)     The following problem of mine was one of the problems at the prestigious Schweitzer competition. (This is an annual math competition in Hungary. Students get 10-12 hard problems to solve; they can take them home and they have 10 days for thinking. They can even use the library.)

Let C denote the set of all convergent series with strictly positive terms. Let D denote the set of all divergent series with strictly positive terms. Does a bijection between C and D exist which satisfies the following property? :

If an and bn are two elements of C and An and Bn are the corresponding elements in D, then

an / bn  tends to zero if and only if  An / Bn  tends to infinity.

(b)     The following was given in the KÖMAL mathematics journal in the hardest problems category:

Does a three variable real polynomial P(x,y,z) exist such that P(x,y,z) is positive if and only if we can construct a triangle from three line segments whose lengths are |x|, |y| and |z|?