361 \label{fzo3} |
363 \label{fzo3} |
362 \end{figure} |
364 \end{figure} |
363 In the first step we have inserted a copy of $(x\times I)\times I$. |
365 In the first step we have inserted a copy of $(x\times I)\times I$. |
364 Figure \ref{fzo4} shows the other case. |
366 Figure \ref{fzo4} shows the other case. |
365 \begin{figure}[t] |
367 \begin{figure}[t] |
366 \begin{equation*} |
368 \begin{center} |
367 \mathfig{.83}{tempkw/zo4} |
369 \begin{tikzpicture} |
368 \end{equation*} |
370 |
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371 \newcommand{\vertex}{node[circle,fill=black,inner sep=1pt] {}} |
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372 \newcommand{\nsep}{1.8} |
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373 |
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374 \node[outer sep=\nsep](A) at (0,0) { |
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375 \begin{tikzpicture} |
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376 \draw (0,0) coordinate (p1); |
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377 \draw (4,0) coordinate (p2); |
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378 \draw (2.4,0) coordinate (p2a); |
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379 \draw (2,1.2) coordinate (pu); |
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380 \draw (2,-1.2) coordinate (pd); |
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381 |
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382 \begin{scope} |
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383 \clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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384 \foreach \t in {0,.065,...,1} { |
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385 \draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(90 - \t*90 + \t*6 : 4); |
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386 \draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(-90 + \t*90 - \t*6 : 4); |
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387 } |
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388 \draw[dashed] ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
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389 \end{scope} |
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390 |
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391 \draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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392 \draw (p1)--(p2); |
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393 |
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394 \draw (p1) \vertex; |
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395 \draw (p2) \vertex; |
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396 \draw (p2a) \vertex; |
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397 \end{tikzpicture} |
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398 }; |
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399 |
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400 \node[outer sep=\nsep](B) at (5,0) { |
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401 \begin{tikzpicture} |
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402 \draw (0,0) coordinate (p1); |
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403 \draw (4,0) coordinate (p2); |
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404 \draw (2.4,0) coordinate (p2a); |
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405 \draw (2,1.2) coordinate (pu); |
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406 \draw (2,-1.2) coordinate (pd); |
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407 |
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408 \begin{scope} |
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409 \clip (-.1,3)--($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$)--(-.1,-3)--cycle; |
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410 |
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411 \begin{scope} |
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412 \clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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413 \foreach \t in {0,.065,...,1} { |
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414 \draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(90 - \t*90 + \t*6 : 4); |
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415 \draw[green!50!brown] ($(p1)!\t!(p2a)$) -- +(-90 + \t*90 - \t*6 : 4); |
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416 } |
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417 \draw ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
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418 \end{scope} |
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419 |
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420 |
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421 \draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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422 %\draw (p1)--(p2); |
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423 \end{scope} |
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424 |
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425 \begin{scope} |
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426 \clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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427 \draw ($(p2a) + (-.6,3)$)--(p2a)--($(p2a) + (-.6,-3)$); |
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428 \end{scope} |
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429 |
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430 \draw (p1) \vertex; |
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431 \draw (p2a) \vertex; |
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432 \end{tikzpicture} |
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433 }; |
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434 |
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435 \node[outer sep=\nsep](C) at (9,0) { |
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436 \begin{tikzpicture} |
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437 \draw (0,0) coordinate (p1); |
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438 \draw (4,0) coordinate (p2); |
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439 \draw (2,1.2) coordinate (pu); |
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440 \draw (2,-1.2) coordinate (pd); |
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441 |
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442 \begin{scope} |
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443 \clip (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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444 \foreach \t in {0,.045,...,1} { |
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445 \draw[green!50!brown] ($(p1)!\t!(p2) + (0,2)$) -- +(0,-4); |
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446 } |
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447 \end{scope} |
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448 |
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449 \draw (p1) .. controls (pu) .. (p2) .. controls (pd) .. (p1); |
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450 |
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451 \draw (p1) \vertex; |
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452 \draw (p2) \vertex; |
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453 \end{tikzpicture} |
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454 }; |
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455 |
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456 \draw[->, thick, blue!50!green] (A) -- (B); |
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457 \draw[->, thick, blue!50!green] ($(B) + (1,0)$) -- node[black, above] {$=$} (C); |
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458 |
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459 \end{tikzpicture} |
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460 \end{center} |
369 \caption{Composition of weak identities, 2} |
461 \caption{Composition of weak identities, 2} |
370 \label{fzo4} |
462 \label{fzo4} |
371 \end{figure} |
463 \end{figure} |
372 We identify a product region and remove it. |
464 We identify a product region and remove it. |
373 |
465 |