# Questions I'd like to see answered

More questions will come with time...
Write me your answer. Links to (partial) answers are also appreciated.

## Science

• Do bounded quadratic programs have finite minimizers? I'd like to know a (nice?) proof that every quadratic program whose objective function is bounded below on the feasible set has a finite global minimizer. I can currently prove this only for the convex case, but the statement should be true in general.

This is indeed true. A summary of the answers I obtained from opt-net are given in The Frank-Wolfe Theorem

• How to find large low rank submatrices (unsolved). Given a m by n matrix A, suppose there is a submatrix formed by k rows of A that is close to a matrix of small rank p, in the sense that the sum of squares of differences of the entries is bounded by some tolerance delta multiplied by k. Is there an efficient way for finding the set of rows defining the submatrix?

In case the problem turns out to be NP hard, several related questions approximating this problem are also of interest. Moreover, in the application I have in mind, k is unknown (and p<=3) and one wants to find the largest set of rows with the stated property.

• Derivative of noisy curve (unsolved). I am looking for a good algorithm for differentiating noisy scattered data to get a smooth, visually pleasing curve for the derivative.

The problem is that none of the currently available techniques work uniformly well on data with various kinds of difficulties such as sudden change of curvature, very flat and very steep regions, gaps in the data, data-dependent noise level, visual monotony or convexity, oscillations at various scales, etc. Thus, usually, a lot of hand tuning is needed on difficult data. What I'd like to see is a black box algorithm that automatically makes all decisions involved in a way a human would find agreeable.

• A test for stability. Is there a matrix version of the Routh-Hurwitz test for the stability of polynomials p(x), that works for the characteristic polynomial p(x)=det(xB-A) of arbitrary matrix pencils (or at least for the case B=I) but (in order to improve numerical stability) avoids first computing the characteristic polynomial?

What I am looking for is a criterion that gives a finite number of inequalities, rational in the matrix coefficients, that are equivalent to the inequalities given by the Routh-Hurwitz test for checking that a polynomial in power form has all its zeros in the left complex half plane. For B=I and A a companion matrix, the inequalities should reduce to the Routh-Hurwitz inequalities.

• A local variational principle (unsolved). Has anyone seen any work related to the following `local' variational principle?
• Find x such that z=x is a local minimizer of f(x,z), where f(x,z) is C^2 and bounded below for each fixed x.
Many dissipative problems have a presentation in the above form, e.g., fluid dynamics equations (P. Glansdorff and I. Prigogine, Thermodynamic theory of structure, stability and fluctuations, Wiley 1971, Chapter 10). Using the optimality conditions one can rephrase the problem to get the nearly equivalent problem
• Find x such that f_z(x,x)=0 and f_{zz}(x,x) is positive semidefinite.
It would be interesting to have numerical methods that do not require explicit second derivative information but can approximate it in a quasi-Newton manner.

• Nearest multiple of a vector (partly solved). Are there fast algorithms for computing the multiple of a vector b nearest to a vector a in the 1-norm and the max norm,
```   min ||a-tb|| ,
t          1
min ||a-tb||   ,
t          inf
```
and the related problem
```   min max (a -tb ) ?
t        t   t
```
More precisely, I would like to know the corresponding active sets after O(n) operations, if possible with a constant independent of the length of the data.

I edited the most useful replies, giving partial answers. The answer to all three cases is positive (reduction to a 2-variable linear program, solvable in O(n) by a result of Meggido), though nice O(n) algorithms haven't been explicitly described.

Thus there is further scope for research (the above link refers to interesting WWW pages and references that might be useful).

• A convexity test (solved). Given m data points (x_i,f_i) in R^n x R, are there necessary and sufficient conditions that there is a convex function f with f(x_i)=f_i for i=1,...,m? (For n=1, it suffices to check that the second divided differences of adjacent triples of points are nonnegative.)

I found an answer in: J.M. Carnicer, Adv. Comput. Math. 3 (1995), 395-404.

• Limit problem for analytic functions (solved). Let f,g be nonnegative real analytic functions on (0,infinity). Suppose that f(x)g(x) converges to 0 for x to infinity. Does this imply that either f(x) to 0 or g(x) to 0?

The answer is no. Counterexamples were give by Nick Jacobs (nick.jacobs@ubs.com),

```   f(x) = (0.5*(1+sin(x))^x,
g(x) = (0.5*(1+cos(x))^x,
```
and Herman Rubin (hrubin@stat.purdue.edu),
```   f(x) = (sin^2(x)+.1)^{1+x^2},
g(x) = (cos^2(x)+.1)^{1+x^2}.
```

• History of nabla (partly solved). Does anyone know the history of using the symbol \nabla for the gradient, and the meaning of the symbol outside of mathematics?

The answers I obtained from NA-net are given in History of Nabla

• Set theory notation (partly solved). Does anyone know who invented the notations {x in A | P(x)} and {x in A : P(x)} for sets defined by a property P(x)?

The current state of affairs is given in Set theory notation

## Other

• I am looking for pointers to web pages (and ordinary books, too) on science and faith. I am not much interested in discussions about Genesis and evolution, bible contradictions, etc., but rather about ways to open the eyes of scientists to the spiritual side of life.

I am looking for texts that have a chance to enable them to see the gospel as a worthwhile challenge and Good News for them, without raising the widespread but unfounded fear that living for God means having to stop thinking, having to become fanatic, or having to become narrow-minded.

I am looking for texts that either avoid or explain in modern terms the standard Christian vocabulary, assuming no familiarity with Christianity (except perhaps with negative stereotypes known to all of us); texts that give life to terms or attitudes often perceived as dead or out-of-date.

I am looking for texts that assume as given the scientist's quest for Truth (and the standards they are used to for checking out Truth), and try to show this to be part of the search for God, the author of Truth, and that living for God is the fulfilment of living for Truth.

If you are a scientist and Christian, loving God but hating dogma, having learnt to communicate with God with a scientific mind, I'd like to encourage you to write something along the above lines that agrees with your personal experience; something that may be a help for scientists who look for a greater purpose in life than just adding a few leafs to the huge tree of knowledge.

My own attempts over time to do the same can be found in my views on the Christian way of life.

Did Einstein's religious views change towards the end of his life? So far, I have conflicting information about this. Any clarifying information is appreciated.

my home page (http://www.mat.univie.ac.at/~neum)

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)