Convex set functions in d-space. Periodica Mathematica Hungarica 6, 111-136.Borell, C.: Convex set functions in d-space. Period. Math. Hung. 6, 111-136 (1975)Ch. Borell. Convex set functions in d-space. Period.
In general, for the n-dimensional space, the line segment between any two points x(1) and x(2) can be written as (4.65)x=αx(2)+(1−α)x(1); 0≤α≤1 If the entire line segment of Eq. (4.65) is in the set S, then it is a convex set. Equation (4.65) is a ...
For a matrix X∈PAB, the smallest k such that X∈Sk is called the Schmidt number of X, where PAB (respectively PA and PB) denotes the set of all positive matrices in MA⊗MB (respectively MA and MB). The corresponding convex cone in L(MA,MB) of all maps whose Choi matrices have...
In fact, a closed convex set SS is the intersection of all halfspaces that contain it:S=⋂ {H | H halfspace,S⊆H}S=⋂ {H | H halfspace,S⊆H}3.2 Affine functions A function f:Rn→Rmf:Rn→Rm is affine if it is a sum of a linear function and a constant, i.e., ...
像perspective function一样, linear-fractional functions也维护了凸性。如果C是凸的并且位于f的定义域中( i.e., c^Tx + d \gt 0 对于 x \in C) ,那么它的图像f(C)就是凸的。同样的,如果 C\subseteq R^m 那么f^{-1}(x) 的反图像时也是凸的。
Model (2) is convex with respect touand easy to be solved in the practice. TVL2 model (2) is suitable for Gaussian noise removing. In addition, TV energy does not penalize the discontinuity of the functions along the contours, so the edges can be preserved in the restorationuby model (...
A survey of results concerning a majorization and Schur convex functions may be found in an extensive monograph by Arnold et al. [1]. Now, assume that D is a convex subset of a real linear space. Recall that a function f:D→R is said to be convex if f(tx+(1−t)y)≤tf(x)+...
equality constraint functions. If there are no constraints i.e., m=p=0 we say the problem (0.1) is unconstrained. The set of points for which the objective and all constraint functions are defined: (0.2)D:=⋂i=0mdomfi∩⋂j=1pdomhj ...
Let be a nonempty set, and let be a subspace of the linear space of all real functions on the domain . We assume that contains the unit function 1 defined by 1(x) = 1 for every . Let be an interval, and let be the set containing all functions with the image in . Then, is co...
The set of all functions consistent with this information is a cumulative energy-bound convex model, defined as: (1)Uα=ft:∫0∞ft−f¯tTft−f¯tdt≤α2 We may not know the value of α, and in robust reliability analysis, we do not need to know it. α is the uncertainty ...