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An Analysis Of Unsolvable Linear Partial Differential Equations Of Order One, Laura J. Fields
An Analysis Of Unsolvable Linear Partial Differential Equations Of Order One, Laura J. Fields
Inquiry: The University of Arkansas Undergraduate Research Journal
It is difficult to underestimate the importance of differential equations in understanding the physical world. These equations, involving not just simple variables like temperature, speed or mass, but also the derivatives, i.e. the rate of change of these variables, are found in nearly every branch of science. Until the mid 20th century, all such equations were thought to be solvable. This was based on the discovery by Leonard Euler that certain differential equations, called ordinary differential equations (ODEs), are indeed always solvable. While ODEs deal with simple conditions, under which some quantity changes with some other quantity and its derivatives, …
Intractability And Undecidability In Small Sets Of Wang Tiles, Adam Delisse
Intractability And Undecidability In Small Sets Of Wang Tiles, Adam Delisse
Inquiry: The University of Arkansas Undergraduate Research Journal
Imagine a never-ending checkerboard, red and black squares alternating forever in every direction. Now close your eyes, wait for a second, and open them again. There is still the checkerboard, but is it different? Has somebody moved the checkerboard over two squares? Four squares? One million squares? It still looks the same. This is the nature of periodic tilings. Wang tiles are squares, much like the red and black ones used on a checkerboard, except Wang tiles have colors on their edges instead of on the whole square. Also, Wang tiles can only be put edge-to-edge with each other where …