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Search: All articles in the CMB digital archive with keyword convex function

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1. CMB 2016 (vol 59 pp. 225)

Atıcı, Ferhan M.; Yaldız, Hatice
 Convex Functions on Discrete Time Domains In this paper, we introduce the definition of a convex real valued function $f$ defined on the set of integers, ${\mathbb{Z}}$. We prove that $f$ is convex on ${\mathbb{Z}}$ if and only if $\Delta^{2}f \geq 0$ on ${\mathbb{Z}}$. As a first application of this new concept, we state and prove discrete Hermite-Hadamard inequality using the basics of discrete calculus (i.e. the calculus on ${\mathbb{Z}}$). Second, we state and prove the discrete fractional Hermite-Hadamard inequality using the basics of discrete fractional calculus. We close the paper by defining the convexity of a real valued function on any time scale. Keywords:discrete calculus, discrete fractional calculus, convex functions, discrete Hermite-Hadamard inequalityCategories:26B25, 26A33, 39A12, 39A70, 26E70, 26D07, 26D10, 26D15

2. CMB 2014 (vol 58 pp. 44)

Daniilidis, A.; Drusvyatskiy, D.; Lewis, A. S.
 Orbits of Geometric Descent We prove that quasiconvex functions always admit descent trajectories bypassing all non-minimizing critical points. Keywords:differential inclusion, quasiconvex function, self-contracted curve, sweeping processCategories:34A60, 49J99

3. CMB 2012 (vol 57 pp. 178)

Rabier, Patrick J.
 Quasiconvexity and Density Topology We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then $\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while $\inf_{U}f=\operatorname{ess\,inf}_{U}f$ if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second) property is typical of lsc (usc) functions and, even when $U$ is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions. This property ensures that the pointwise extrema of $f$ on any nonempty density open subset can be arbitrarily closely approximated by values of $f$ achieved on large'' subsets, which may be of relevance in a variety of issues. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero. Keywords:density topology, quasiconvex function, approximate continuity, point of continuityCategories:52A41, 26B05

4. CMB 2012 (vol 57 pp. 25)

Bourin, Jean-Christophe; Harada, Tetsuo; Lee, Eun-Young
 Subadditivity Inequalities for Compact Operators Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings. Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalitiesCategories:47A63, 15A45

5. CMB 2011 (vol 55 pp. 697)

Borwein, Jonathan M.; Vanderwerff, Jon
 Constructions of Uniformly Convex Functions We give precise conditions under which the composition of a norm with a convex function yields a uniformly convex function on a Banach space. Various applications are given to functions of power type. The results are dualized to study uniform smoothness and several examples are provided. Keywords:convex function, uniformly convex function, uniformly smooth function, power type, Fenchel conjugate, composition, normCategories:52A41, 46G05, 46N10, 49J50, 90C25

6. CMB 2005 (vol 48 pp. 283)

Thibault, Lionel; Zagrodny, Dariusz
 Enlarged Inclusion of Subdifferentials This paper studies the integration of inclusion of subdifferentials. Under various verifiable conditions, we obtain that if two proper lower semicontinuous functions $f$ and $g$ have the subdifferential of $f$ included in the $\gamma$-enlargement of the subdifferential of $g$, then the difference of those functions is $\gamma$-Lipschitz over their effective domain. Keywords:subdifferential,, directionally regular function,, approximate convex function,, subdifferentially and directionally stable functionCategories:49J52, 46N10, 58C20
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