Results 1 to 10 of about 162 (63)
Generalized arithmetic subderivative [PDF]
Notes on Number Theory and Discrete Mathematics, 2019 Let ∅ 6 = S ⊆ P. The arithmetic subderivative of n with respect to S is defined as DS(n) = n ∑
Pentti Haukkanen
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Arithmetic subderivatives and Leibniz-additive functions [PDF]
Annales Mathematicae et Informaticae, 2019 We first introduce the arithmetic subderivative of a positive integer with respect to a non-empty set of primes. This notion generalizes the concepts of the arithmetic derivative and arithmetic partial derivative. More generally, we then define that an arithmetic function $f$ is Leibniz-additive if there is a nonzero-valued and completely ...
Jorma K. Merikoski +2 more
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SIAM Journal on Control and Optimization, 1996 Let \(X\) be a real Banach space endowed with a bornology \(\beta\). If \(f:X\to[-\infty,+ \infty]\) is lower semicontinuous and \(f(x)
Jonathan M. Borwein, Qiji Jim Zhu
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Why second-order sufficient conditions are, in a way, easy -- or -- revisiting calculus for second subderivatives [PDF]
, 2022 In this paper, we readdress the classical topic of second-order sufficient optimality conditions for optimization problems with nonsmooth structure. Based on the so-called second subderivative of the objective function and of the indicator function associated with the feasible set, one easily obtains second-order sufficient optimality conditions of ...
Matúš Benko, Patrick Mehlitz
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Subderivative-subdifferential duality formula [PDF]
, 2016 We provide a formula linking the radial subderivative to other subderivatives and subdifferentials for arbitrary extended real-valued lower semicontinuous functions.
Marc Lassonde
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On the Construction of Hölder and Proximal Subderivatives [PDF]
Canadian Mathematical Bulletin, 1998 AbstractWe construct Lipschitz functions such that for all s > 0 they are s-Hölder, and so proximally, subdifferentiable only on dyadic rationals and nowhere else. As applications we construct Lipschitz functions with prescribed Hölder and approximate subderivatives.
Jonathan M. Borwein +2 more
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Second-order conditions for spatio-temporally sparse optimal control via second subderivatives [PDF]
Journal of Nonsmooth Analysis and OptimizationWe address second-order optimality conditions for optimal control problems involving sparsity functionals which induce spatio-temporal sparsity patterns. We employ the notion of (weak) second subderivatives. With this approach, we are able to reproduce the results from Casas, Herzog, and Wachsmuth (ESAIM COCV, 23, 2017, p. 263-295). Our analysis yields
Nicolas Borchard, Gerd Wachsmuth
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Links between subderivatives and subdifferentials
Journal of Mathematical Analysis and Applications, 2017 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Marc Lassonde
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Arithmetic Subderivatives : p-adic Discontinuity and Continuity
, 2020 Summary: In a previous paper, we proved that the arithmetic subderivative \(D_S\) is discontinuous at any rational point with respect to the ordinary absolute value. In the present paper, we study this question with respect to the \(p\)-adic absolute value. In particular, we show that \(D_S\) is in this sense continuous at the origin if \(S\) is finite
Pentti Haukkanen +2 more
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Arithmetic Subderivatives : Discontinuity and Continuity
, 2019 Summary: We first prove that any arithmetic subderivative of a rational number defines a function that is everywhere discontinuous in a very strong sense. Second, we show that although the restriction of this function to the set of integers is continuous (in the relative topology), it is not Lipschitz continuous. Third, we see that its restriction to a
Pentti Haukkanen +2 more
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