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<h2><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><!WA54><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec3-title.au">Section 3 </A> Knuth's examples of fractions and exponents.</h2>These examples are taken verbatim from the TeX Book, by Donald Knuth.They are used in the TeX Book to demonstrate the power of the TeX layoutoperators.Notice that all of these examples comprise of the same 6 symbols, but are verydifferent!AsTeR can render these as unambiguously as TeX can.<OL><LI><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><!WA55><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec3-ex1.au"> audio </A><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><!WA56><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx13.tex"> LaTeX </A><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><!WA57><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx13.ps"> Postscript </A><LI><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><!WA58><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec3-ex2.au"> audio </A><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><!WA59><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx14.tex"> LaTeX </A><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><!WA60><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx14.ps"> Postscript </A><LI><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><!WA61><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec3-ex3.au"> audio </A><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><!WA62><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx15.tex"> LaTeX </A><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><!WA63><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx15.ps"> Postscript </A><LI><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><!WA64><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec3-ex4.au"> audio </A><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><!WA65><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx16.tex"> LaTeX </A><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><!WA66><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx16.ps"> Postscript </A><LI><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><!WA67><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec3-ex5.au"> audio </A><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><!WA68><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx17.tex"> LaTeX </A><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><!WA69><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx17.ps"> Postscript </A><LI><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><!WA70><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec3-ex6.au"> audio </A><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><!WA71><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx18.tex"> LaTeX </A><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><!WA72><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx18.ps"> Postscript </A><LI><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><!WA73><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec3-ex7.au"> audio </A><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><!WA74><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx19.tex"> LaTeX </A><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><!WA75><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx19.ps"> Postscript </A></OL><h2><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><!WA76><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec4-title.au">Section 4 </A> A continued fraction.</h2>Moving along a dimension in audio space defines a perceptibly monotonicchange. This notion of perceptible monotonicity is vital in conveying nesting.<p><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><!WA77><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec4-ex1.au"> audio </A><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><!WA78><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx20.tex"> LaTeX </A><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><!WA79><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx20.ps"> Postscript </A><h2><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><!WA80><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec5-title.au">Section 5 </A> Simple School algebra.</h2><OL><LI><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><!WA81><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec5-ex1.au"> audio </A><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><!WA82><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx21.tex"> LaTeX </A><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><!WA83><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx21.ps"> Postscript </A><LI><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><!WA84><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec5-ex2.au"> audio </A><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><!WA85><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx22.tex"> LaTeX </A><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><!WA86><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx22.ps"> Postscript </A><LI><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><!WA87><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec5-ex3.au"> audio </A><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><!WA88><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx23.tex"> LaTeX </A><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><!WA89><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx23.ps"> Postscript </A></OL><h2><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><!WA90><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec6-title.au">Section 6 </A> square roots.</h2>Notice the choice of unambiguous renderings for the following expressions: <OL><LI><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><!WA91><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec6-ex1.au"> audio </A><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><!WA92><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx24.tex"> LaTeX </A><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><!WA93><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx24.ps"> Postscript </A><LI><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><!WA94><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec6-ex2.au"> audio </A><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><!WA95><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx25.tex"> LaTeX </A><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><!WA96><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx25.ps"> Postscript </A><LI><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><!WA97><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec6-ex3.au"> audio </A><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><!WA98><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx26.tex"> LaTeX </A><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><!WA99><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx26.ps"> Postscript </A></OL><h2><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><!WA100><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec7-title.au">Section 7 </A> Trigonometric identities.</h2>Written mathematical notation can be ambiguous and hard to recognize.Notice the complete absence of parenthesis in some of the examples below.AsTeR uses several heuristics to construct the correct tree structure forthese expressions.<OL><LI><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><!WA101><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec7-ex1.au"> audio </A><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><!WA102><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx27.tex"> LaTeX </A><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><!WA103><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx27.ps"> Postscript </A><LI><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><!WA104><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec7-ex2.au"> audio </A><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><!WA105><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx28.tex"> LaTeX </A><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><!WA106><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx28.ps"> Postscript </A><LI><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><!WA107><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec7-ex3.au"> audio </A><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><!WA108><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx29.tex"> LaTeX </A><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><!WA109><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx29.ps"> Postscript </A><LI><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><!WA110><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec7-ex4.au"> audio </A><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><!WA111><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx30.tex"> LaTeX </A><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><!WA112><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx30.ps"> Postscript </A><LI><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><!WA113><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec7-ex5.au"> audio </A><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><!WA114><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx31.tex"> LaTeX </A><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><!WA115><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx31.ps"> Postscript </A><LI><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><!WA116><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec7-ex6.au"> audio </A><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><!WA117><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx32.tex"> LaTeX </A><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><!WA118><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx32.ps"> Postscript </A><LI><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><!WA119><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec7-ex7.au"> audio </A><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><!WA120><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx33.tex"> LaTeX </A><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><!WA121><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/latex/xx33.ps"> Postscript </A></OL><h2><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><!WA122><A HREF = "http://www.cs.cornell.edu/Info/People/raman/aster/math-examples/sec8-title.au">Section 8 </A> Logarithms.</h2>Notice the context-specific rendering when speaking the base of the logarithm.The renderings are chosen to reduce cognitive load; <blockquote>log base a of

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