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Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.
In mathematics, convex conjugation is a generalization of the Legendre transformation. It is also known as Legendre–Fenchel transformation or Fenchel transformation (after Adrien-Marie Legendre and Werner Fenchel).
For a functional
taking values on the extended real number line, the convex conjugate
is defined in terms of the supremum by
or, equivalently, in terms of the infimum by
The convex conjugate of an affine function
The convex conjugate of a power function
The convex conjugate of the absolute value function
The convex conjugate of the exponential function is
Convex conjugate and Legendre transform of the exponential function agree except that the domain of the convex conjugate is strictly larger as the Legendre transform is only defined for positive real numbers.
has the convex conjugate
A particular interpretation has the transform
as this is a nondecreasing rearrangement of the initial function f; in particular, for ƒ nondecreasing.
The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function.
Convex-conjugation is order-reversing: if then . Here
The convex conjugate of a function is always lower semi-continuous. The biconjugate (the convex conjugate of the convex conjugate) is also the closed convex hull, i.e. the largest lower semi-continuous convex function with . For proper functions f, f = f** if and only if f is convex and lower semi-continuous by Fenchel–Moreau theorem.
For any function f and its convex conjugate f* Fenchel's inequality (also known as the Fenchel–Young inequality) holds for every and :
It is interesting to observe that the derivative of the function is the maximizing argument to compute the convex conjugate:
If, for some , , then
In case of an additional parameter (α, say) moreover
where is chosen to be the maximizing argument.
Let A be a bounded linear operator from X to Y. For any convex function f on X, one has
A closed convex function f is symmetric with respect to a given set G of orthogonal linear transformations,
if and only if its convex conjugate f* is symmetric with respect to G.
The infimal convolution of two functions f and g is defined as
Let f1, …, fm be proper convex functions on Rn. Then
The infimal convolution of two functions has a geometric interpretation: The (strict) epigraph of the infimal convolution of two functions is the Minkowski sum of the (strict) epigraphs of those functions.
The following table provides Legendre transforms for many common functions as well as a few useful properties.
|(where )||(where )|
|(where )||(where )|