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We provide a concise analysis about what is known regarding when the closure of the domain of a maximally monotone operator on an arbitrary real Banach space is convex. In doing so, we also provide an affirmative answer to a problem posed by Simons. 相似文献
4.
The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm
over a compact set in ℂ
d
. We study this problem on general sets but devote special attention to product sets such as cube and polydisk. We also establish
a multivariate analog of the Hilbert–Fekete upper bound for the integer Chebyshev constant, which depends on the dimension
of space. In the case of single-variable polynomials in the complex plane, our estimate coincides with the Hilbert–Fekete
result.
相似文献
5.
Jonathan M. Borwein 《Optimization Letters》2010,4(4):473-490
6.
Summary LetP be a finite set of three or more noncollinear points in the plane. A line which contains two or more points ofP is called aconnecting line (determined byP), and we call a connecting lineordinary if it contains precisely two points ofP. Almost a century ago, Sylvester posed the disarmingly simple question:Must every set P determine at least one ordinary line? No solution was offered at that time and the problem seemed to have been forgotten. Forty years later it was independently rediscovered by Erdös, and solved by Gallai. In 1943 Erdös proposed the problem in the American Mathematical Monthly, still unaware that it had been asked fifty years earlier, and the following year Gallai's solution appeared in print. Since then there has appeared a substantial literature on the problem and its generalizations.In this survey we review, in the first two sections, Sylvester's problem and its generalization to higher dimension. Then we gather results about the connecting lines, that is, the lines containing two or more of the points. Following this we look at the generalization to finite collections of sets of points. Finally, the points will be colored and the search will be for monochromatic connecting lines. 相似文献
7.
David Borwein Jonathan M. Borwein David M. Bradley 《Journal of Mathematical Analysis and Applications》2006,316(1):328-338
We consider some parametrized classes of multiple sums first studied by Euler. Identities between meromorphic functions of one or more variables in many cases account for reduction formulae for these sums. 相似文献
8.
The speeds of convergence of best rational approximations, best polynomial approximations, and the modulus of continuity on the unit disc are compared. We show that, in a Baire category sense, it is expected that subsequences of these approximants will converge at the same rate. Similar problems on the interval [−1, 1] are also examined. A problem raised by P. Turán (J. Approx. Theory29, 1980, 23-89) concerning rational approximation to non-analytically continuable ƒ on the unit circle is negated as an application. 相似文献
9.
Jonathan M. Borwein Jay S. Treiman Qiji J. Zhu 《Transactions of the American Mathematical Society》1998,350(6):2409-2429
We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions without constraint qualification in terms of smooth subderivatives and normal cones. These results, in different versions, are set in reflexive and smooth Banach spaces.
10.
Jonathan M. Borwein 《Optimization Letters》2007,1(1):21-32
This paper is a companion to a lecture given at the Prague Spring School in Analysis in April 2006. It highlights four distinct variational methods of proving that a finite dimensional Chebyshev set is convex and hopes to inspire renewed work on the open question of whether every Chebyshev set in Hilbert space is convex. 相似文献