Open problem

In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is known). In the history of science, some of these supposed open problems were "solved" by means of showing that they were not well-defined. In mathematics, many open problems are concerned with the question of whether a certain definition is or is not consistent. Two notable examples in mathematics that have been solved and closed by researchers in the late twentieth century are Fermat's Last Theorem and the four-color theorem. An important open mathematics problem solved in the early 21st century is the Poincaré conjecture. Open problems exist in all scientific fields. For example, one of the most important open problems in biochemistry is the protein structure prediction problem – how to predict a protein's structure from its sequenc

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On the power of combinatorial and spectral invariants

We extend the traditional spectral invariants (spectrum and angles) by a stronger polynomial time computable graph invariant based on the angles between projections of standard basis vectors into the eigenspaces (in addition to the usual angles between standard basis vectors and eigenspaces). The exact power of the new invariant is still an open problem. We also define combinatorial invariants based on standard graph isomorphism heuristics and compare their strengths with the spectral invariants. In particular, we show that a simple edge coloring invariant is at least as powerful as all these spectral invariants. (C) 2009 Elsevier Inc. All rights reserved.

2010

Hitting Simplices with Points in $R^3$

Abdul Basit

The so-called first selection lemma states the following: given any set P of n points in a"e (d) , there exists a point in a"e (d) contained in at least c (d) n (d+1)-O(n (d) ) simplices spanned by P, where the constant c (d) depends on d. We present improved bounds on the first selection lemma in a"e(3). In particular, we prove that c (3)a parts per thousand yen0.00227, improving the previous best result of c (3)a parts per thousand yen0.00162 by Wagner (On k-sets and applications. Ph.D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c (3)a parts per thousand currency sign1/4(4)a parts per thousand 0.00390) and Boros and Furedi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69-77, 1984) (where the two-dimensional case was settled).

2010

Normal generation of locally compact groups

Amichai Netanel Eisenmann, Nicolas Monod

It is a well-known open problem since the 1970s whether a finitely generated perfect group can be normally generated by a single element or not. We prove that the topological version of this problem has an affirmative answer as long as we exclude infinite discrete quotients (which is probably a necessary restriction).

Oxford Univ Press2013

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