In game theory, a null move or pass is a decision by a player to not make a move when it is that player's turn to move. Even though null moves are against the rules of many games, they are often useful to consider when analyzing these games. Examples of this include the analysis of zugzwang (a situation in chess or other games in which a null move, if it were allowed, would be better than any other move), and the null-move heuristic in game tree analysis (a method of pruning game trees involving making a null move and then searching to a lower depth). The reason a reduced-depth null move is effective in game tree alpha-beta search reduction is that tactical threats tend to show up very quickly, in just one or two moves. If the opponent has no tactical threats revealed by null move search, the position may be good enough to exceed the best result obtainable in another branch of the tree (i.e. "beta"), so that no further search need be done from the current node, and the result from the null move can be returned as the search value. Even if the null move search value doesn't exceed beta, the returned value may set a higher floor on the valuation of the position than the present alpha, so more cutoffs will occur at descendant sibling nodes from the position. The underlying assumption is that at least some legal move available to the player on move at the node is better than no move at all. In the case of the player on move being in zugzwang, that assumption is false, and the null move result is invalid (in that case, it actually sets a ceiling on the value of the position). Therefore it is necessary to have logic to exclude null moves at nodes in the tree where zugzwang is possible. In chess, zugzwang positions can occur in king and pawn endgames, and sometimes in end games that include other pieces as well. (Wikipedia).
In this video I start to discuss the idea of the null space of a matrix. In these situations, the right-hand side of all the equations in the linear system is equal to zero. There is the trivial solution, where all the elements of the solution is zero. We are more interested in the spec
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In today's lecture I work through an example to show you a well-known pitfall when it comes to the null space of a matrix. In the example I show you how to create the special cases and how to use them to represent the null space. There is also a quick look at the NullSpace function in Ma
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In this video we finish with three more example of calculating the null space of a matrix. These three example help us gain an even deeper insight into the null space by consider how many special cases we will get. Remember that the linear combinations of the special cases give us the nu
From playlist Introducing linear algebra
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In this video I give a brief overview of the null space of a matrix and how to calculate (y hand) the column vectors that will make up the null space of the column. This becomes much easier in sympy, by simply using the nullspace function. If you want to learn more about the null space o
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Given a matrix A(ie a linear transformation) there are several important related subspaces. In this video we investigate the Nullspace of A and the column space of A. The null space is the vectors that are "killed" by the transformation - ie sent to zero. The column space will be the image
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