This is indeed true. A summary of the answers I obtained from opt-net are given in The Frank-Wolfe Theorem
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.
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.
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.
see Matrix tests for stability.
min ||a-tb|| , t 1 min ||a-tb|| , t infand the related problem
min max (a -tb ) ? t t tMore 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).
I found an answer in: J.M. Carnicer, Adv. Comput. Math. 3 (1995), 395-404.
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}.
The answers I obtained from NA-net are given in History of Nabla
The current state of affairs is given in Set theory notation
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.
A response by Paul Littlefield
A link provided by Zhiwen Chong
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.
Arnold Neumaier (Arnold.Neumaier@univie.ac.at)