|
Hilbert–Arnold Problem
In mathematics, particularly in dynamical systems, the Hilbert–Arnold problem is an list of unsolved problems in mathematics, unsolved problem concerning the estimation of limit cycles. It asks whether in a generic property, generic finite-parameter family of smooth function, smooth vector fields on a sphere with a Compact space, compact parameter base, the number of limit cycles is uniformly bounded across all parameter values. The problem is historically related to Hilbert's sixteenth problem and was first formulated by Russians, Russian mathematicians Vladimir Arnold and Yulij Ilyashenko in the 1980s.Ilyashenko, Yu. (1994). "Normal forms for local families and nonlocal bifurcations". ''Astérisque'', Vol. 222, 233-258. It is closely related to the "infinitesimal Hilbert's sixteenth problem", although they are not synonyms. In ''Arnold's Problems'' there are many questions related to the Hilbert–Arnold problem: 1978–6, 1979–16, 1980–1, 1983–11, 1989–17, 1990–24, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Bounded
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of any value of any of the functions in the family. Definition Real line and complex plane Let :\mathcal F=\ be a family of functions indexed by I, where X is an arbitrary set and \mathbb is either the set of real \mathbb or complex numbers \mathbb. We call \mathcal F uniformly bounded if there exists a real number M>0 such that :, f_i(x), \le M \ , \qquad \forall i \in I \ , \quad \forall x \in X. Another way of stating this would be the following: :\sup\limits_ \sup\limits_ , f_i(x), \le M. Metric space In general let Y be a metric space with metric d, then the set :\mathcal F=\ is called uniformly bounded if there exists an element a from Y and a real number M such that :d(f_i(x), a) \leq M \qquad \forall i \in I \quad \forall x \in X. Examples * Every uniformly converge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
