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세미나

세미나

원드라이브

∂u∂x+∂v∂y=0∂u∂t+u∂u∂x+v∂u∂y=∂2u∂x2+∂2u∂y2−u−uu2+v2∂v∂t+u∂v∂x+v∂v∂y=∂2v∂x2+∂2v∂y2−v−vu2+v2∂θ∂t+u∂θ∂x+v∂θ∂y=∂2θ∂x2+∂2θ∂y2 \begin{align*} {\frac{ \partial u }{ \partial x }} + {\frac{ \partial v }{ \partial y }} =& 0 \\ {\frac{ \partial u }{ \partial t }} + u {\frac{ \partial u }{ \partial x }} + v {\frac{ \partial u }{ \partial y }} =& {\frac{ \partial^{2} u }{ \partial x^{2} }} + {\frac{ \partial^{2} u }{ \partial y^{2} }} - u - u \sqrt{u^{2} + v^{2}} \\ {\frac{ \partial v }{ \partial t }} + u {\frac{ \partial v }{ \partial x }} + v {\frac{ \partial v }{ \partial y }} =& {\frac{ \partial^{2} v }{ \partial x^{2} }} + {\frac{ \partial^{2} v }{ \partial y^{2} }} - v - v \sqrt{u^{2} + v^{2}} \\ {\frac{ \partial \theta }{ \partial t }} + u {\frac{ \partial \theta }{ \partial x }} + v {\frac{ \partial \theta }{ \partial y }} =& {\frac{ \partial^{2} \theta }{ \partial x^{2} }} + {\frac{ \partial^{2} \theta }{ \partial y^{2} }} \end{align*}

Mathematical analysis of Mahjong

tile

Definition

Let non-empty finite set C\mathcal{C} be a color set and N⊂N\mathcal{N} \subset \mathbb{N} be a suit set. A tile xk∈C×Nx_{k} \in \mathcal{C} \times \mathcal{N} is defined by a pair of color and integer.

Example

For ordinary mahjong, C={b,c,d}\mathcal{C} = \left\{ b , c , d \right\} is consists three color. bb, cc, dd stand for bamboo, character, dot, respectively. More commonly, C\mathcal{C} also has wind ww and dragon δ\delta.

  • b3b_{3}-b6b_{6} are mutually suzi, unlike b3b_{3}-b4b_{4} or b3b_{3}-c3c_{3}.
  • Usually, xkx_{k} for k>9k > 9 doesn’t exist.
  • We say wind tile and dragon tile as honor. The honor tile has no suit, so subscript may refer specific identifier, like north wind wNw_{N} and green δG\delta_{G}.
  • Non-terminal and non-honor tile is called by simple.

taz

Definition

A multiset of nn tiles X={x1,⋯ ,xn}X = \left\{ x_{1} , \cdots , x_{n} \right\} is called by nn-taz. The suit range of taz XX, R(X)R(X) is defined by difference of maximal suit and minimal suit of all simple tile in XX.

Example

  • A 1414-taz is called by hand.
  • A 22-taz with two same tiles is called by toiz.
  • A 33-taz is called by menz or meld.
    • A menz with all same tiles is called by kutz.
    • A menz with serial tiles is called by shunz, {Xk−1,Xk,Xk+1}\left\{ X_{k-1} , X_{k} , X_{k+1} \right\}.
  • Here is called “Nine Gates” hand: {d1,d1,d1,d2,d3,d4,d5,d6,d7,d8,d9,d9,d9} \left\{ d_{1}, d_{1}, d_{1}, d_{2}, d_{3}, d_{4}, d_{5}, d_{6}, d_{7}, d_{8}, d_{9}, d_{9}, d_{9} \right\}

wait and better taz

The wait of nn-taz XX, W(X)W(X) is a number of (n+1)(n+1)-taz including XX and the difference of suit range is less than 3. That is, W(X)=card⁡{Y⊃X:∣Y∣=n+1∧R(Y)<R(X)+3}. W ( X ) = \operatorname{card} \left\{ Y \supset X : |Y| = n + 1 \land R(Y) < R(X) + 3 \right\} . For two nn-tazs AA and BB, we say AA is better than BB and denote A>BA > B if W(A)>W(B)W(A) > W(B). Similiary, the equality A=BA = B holds if W(A)=W(B)W(A) = W(B).

Theorem

The terminal tile is the worst. {xk}>{x1}={x9},∀k∈[2,8] \left\{ x_{k} \right\} > \left\{ x_{1} \right\} = \left\{ x_{9} \right\} \qquad , \forall k \in [2, 8]

proof

Without loss of generality, we only have to observe x1,x2,x3x_{1}, x_{2}, x_{3}.

There are only three 22-taz including x1x_{1}: {x1,x1},{x1,x2},{x1,x3} \left\{ x_{1} , x_{1} \right\} , \left\{ x_{1} , x_{2} \right\} , \left\{ x_{1} , x_{3} \right\}

There are four 22-taz including x2x_{2}: {x1,x2},{x2,x2},{x2,x3},{x2,x4} \left\{ x_{1} , x_{2} \right\} , \left\{ x_{2} , x_{2} \right\} , \left\{ x_{2} , x_{3} \right\} , \left\{ x_{2} , x_{4} \right\}

There are five 22-taz including x3x_{3}: {x1,x3},{x2,x3},{x3,x3},{x3,x4},{x3,x5} \left\{ x_{1} , x_{3} \right\} , \left\{ x_{2} , x_{3} \right\}, \left\{ x_{3} , x_{3} \right\} , \left\{ x_{3} , x_{4} \right\} , \left\{ x_{3} , x_{5} \right\}

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