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Transposition Theorems for Cone-Convex Functions
SIAM Journal on Applied Mathematics, 1973Some transposition theorems for real convex functions on real finite-dimensional spaces, with inequality ordering, are extended to convex functions mapping real Banach spaces into Banach spaces, with partial orderings and convexity defined by closed convex cones.Applications to optimization and optimal control are discussed.
Craven, B. D., Mond, B.
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Inclusion Relations of Gårding’s Cones and k-Convex Cones
Bulletin of the Malaysian Mathematical Sciences Society, 2018The authors prove different properties of Gårding's cones defined as \(\Gamma_{k}=\{\lambda \in \mathbb{R}^{n}\mid S_{j}(\lambda)>0\), \(1\leq j\leq k\}\), for \(k=1,\dots ,n\), \(\lambda =(\lambda_{1},\dots\lambda_{n})\), where \(S_{j}(\lambda)\) is the \(j\)th-order elementary symmetric polynomial given by \(S_{j}(\lambda)=\sum_{i_1,\dots,i_j}\prod_ ...
Gang Li, Saihua Cui, Feida Jiang
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Journal of Optimization Theory and Applications, 1988
We point out in this note that the class of cones in a locally convex topological vector space satisfying property (\(\Pi)\) or piecewise relatively weakly compact cones is exactly the class of cones admitting weakly compact bases or the class of cones whose closures admit weakly compact bases.
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We point out in this note that the class of cones in a locally convex topological vector space satisfying property (\(\Pi)\) or piecewise relatively weakly compact cones is exactly the class of cones admitting weakly compact bases or the class of cones whose closures admit weakly compact bases.
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1993
A cone C in R n is a set of points such that if x∈ C, then so is every nonnegative scalar multiple of x, i.e., if x∈C, then λx∈C for 0 ≤ λ∈R, x∈R n (see Figure 4.1.a for C in R2). If we consider the set of points X = {x}, then the cone generated by X is C = {y|y = λx, 0 ≤ λ∈ R, x∈X}. And if 0∉ X and for each y(≠0)∈C there are unique x∈X with λ > 0 such
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A cone C in R n is a set of points such that if x∈ C, then so is every nonnegative scalar multiple of x, i.e., if x∈C, then λx∈C for 0 ≤ λ∈R, x∈R n (see Figure 4.1.a for C in R2). If we consider the set of points X = {x}, then the cone generated by X is C = {y|y = λx, 0 ≤ λ∈ R, x∈X}. And if 0∉ X and for each y(≠0)∈C there are unique x∈X with λ > 0 such
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The trip of the tip: understanding the growth cone machinery
Nature Reviews Molecular Cell Biology, 2009exaly
Filopodia: molecular architecture and cellular functions
Nature Reviews Molecular Cell Biology, 2008Pieta K Mattila, Pekka Lappalainen
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