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The proof that both regions have the same eigenvalues uses the symmetries of the Laplacian. This idea has been generalized by Buser, Conway, Doyle, and Semmler [4] who constructed numerous similar examples. So, the answer to Kac's question is: for many shapes, one cannot hear the shape of the drum completely. However, some information can be ...
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A function [10] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x 2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x").
The undergraduate course of mathematics at Cambridge still reflects a historically broad approach; and problem-solving skills are tested in examinations, though the setting of excessively taxing questions has been discouraged for many years. Example questions from 1881, before the reforms, are quoted in A Mathematician's Miscellany:
Sometimes written in the form: m × 10 n. Or more compactly as: 10 n. This is generally used to denote powers of 10. Where n is positive, this indicates the number of zeros after the number, and where the n is negative, this indicates the number of decimal places before the number. As an example: 10 5 = 100,000 [1] 10 −5 = 0.00001 [2]
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. [1]
For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as = +. [ 27 ] Variables allow one to describe general problems, [ 4 ] without specifying the values of the quantities that are involved.
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