In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U → R be a real analytic function on an open set U in Rn, and let Z be the zero locus of ƒ. Assume that Z is not empty. Then for any compact set K in U, there exist positive constants α and C such that, for all x in K Here α can be large. A special case of the Łojasiewicz inequality, due to , is commonly used to prove linear convergence of gradient descent algorithms.
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| - Lojasiewicz-Ungleichung (de)
- Неравенство Лоясевича (ru)
- Łojasiewicz inequality (en)
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| - Неравенство Лоясевича — неравенство, установленное польским математиком Станисловом Лоясевичем (польск. Stanisław Łojasiewicz), дающее верхнюю оценку для расстояния от точки произвольного компакта до множества нулевого уровня вещественной аналитической функции многих переменных. Это неравенство нашло применения в различных разделах математики, в том числе, в вещественной алгебраической геометрии, в анализе, в теории дифференциальных уравнений . (ru)
- Die Łojasiewicz-Ungleichung (in deutschsprachiger Literatur meist: Lojasiewicz-Ungleichung; nach Stanisław Łojasiewicz) ist eine Ungleichung der mathematischen Analysis, die vor allem in der Anwendung findet. (de)
- In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U → R be a real analytic function on an open set U in Rn, and let Z be the zero locus of ƒ. Assume that Z is not empty. Then for any compact set K in U, there exist positive constants α and C such that, for all x in K Here α can be large. A special case of the Łojasiewicz inequality, due to , is commonly used to prove linear convergence of gradient descent algorithms. (en)
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| - Die Łojasiewicz-Ungleichung (in deutschsprachiger Literatur meist: Lojasiewicz-Ungleichung; nach Stanisław Łojasiewicz) ist eine Ungleichung der mathematischen Analysis, die vor allem in der Anwendung findet. Anschaulich besagt sie, dass für eine analytische Funktion der Abstand eines Punktes von der Nullstellenmenge der Funktion in Abhängigkeit vom Funktionswert in diesem Punkt abgeschätzt werden kann. Diese Interpretation ist allerdings mit Vorsicht zu betrachten, weil die in der Ungleichung vorkommenden Konstanten von der Funktion abhängen und es je nach Wahl einer Funktion natürlich auch in größerer Entfernung von der Nullstellenmenge kleine Funktionswerte geben kann. (de)
- In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U → R be a real analytic function on an open set U in Rn, and let Z be the zero locus of ƒ. Assume that Z is not empty. Then for any compact set K in U, there exist positive constants α and C such that, for all x in K Here α can be large. The following form of this inequality is often seen in more analytic contexts: with the same assumptions on ƒ, for every p ∈ U there is a possibly smaller open neighborhood W of p and constants θ ∈ (0,1) and c > 0 such that A special case of the Łojasiewicz inequality, due to , is commonly used to prove linear convergence of gradient descent algorithms. (en)
- Неравенство Лоясевича — неравенство, установленное польским математиком Станисловом Лоясевичем (польск. Stanisław Łojasiewicz), дающее верхнюю оценку для расстояния от точки произвольного компакта до множества нулевого уровня вещественной аналитической функции многих переменных. Это неравенство нашло применения в различных разделах математики, в том числе, в вещественной алгебраической геометрии, в анализе, в теории дифференциальных уравнений . (ru)
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