What is Delta Kronecker?

The Delta Kronecker function, marked and _{i, j , is a binary function equal to 1, if i and j are the same and equals 0 differently. Although it is technically the function of two variables, in practice it is used as a notation abbreviation, which allows compact to write complex mathematical statements. Mathematics, physicists and engineers who work in linear algebra, analyze the tensor and digital signal use Delta Kronecker as a purpose, as to say in a single equation that could otherwise take several lines of text. For example, if the company has 30 employees { e 1 , e 2 ... e 30 } and each employee works on a different number of hours { h 1 , h 2 { r 1 , r 2 ... r 30 } e2 + E 3 *h 3 *R 3 + ... E 30 *h 30 *Sub> 30 . Mathematicians can write this briefly as ∑ i i *Sub> i}

in describing physical systems that include more dimensions, physics often use twice as far as more dimensions. Practical scientific applications are very complicated, but a specific example shows how the Konecker Delta function can simplify in these cases.

There are three clothing stores in the mall, each selling a different brand. There are a total of 20 shirt styles: eight offered in the store 1, seven offered 2D five offered in the store 3.for pants. Each combination provides a different outfit.

It is not so easy to calculate the number of ways to choose clothing in which shirts and pants come from different shops. One can choose a T -shirt from shop 1 and pants from shop 2 in 8*3 ways. There are 8*4 Ways to choose a shirt from the store 1 and pants from store 3. Continuation in this way, finds the total number of clothing using cells from different shops is 8*3 + 8*4 + 7*5 + 7*4 + 5*5 + 5*3 = 199.

One could consider the availability of shirts and pants as two sequences, { s _{1 , S 2 , S 3} } = {8, 7, 5} and { p _{1 {5} {8, { { { 1 , p {8, { 1 3, 4}. Then the Konecker Delta function allows you to write this sum as simply ∑ i j i * Sub> j * (1- δ i, j ). Term (1- δ i, j ) eliminates those clothing containing shirts and pants purchased in the same store, because in this case i = , J. delete 0 from the sum.}

The Delta Kronecker function is most often used in multi -dimensional analysis, but can also be used to study unabrome spaces such as real numbers. In this case, one input variant is often used: a ( n ) = 1, if n = 0; Δ ( n ) = 0 otherwise. To see how Delta Kronecker can be used to simplify complex mathematical claims about real numbers, the following two functions can be considered, the inputs of which are simplified fractions:

f (a/b) = and if a = b +1, f (a/b) = -b g (A/B) = a *Δ ( a - b -1)- b

Functions f and g are identical, but the definition for g is more compact and requires no English, so no mathematician can understand it.

As it illustrates these examples, the inputs of the Konecker Delta function are usually integers that are connected to a certain sequence of values. The Delta Dirac distribution is a continuous analogue of the Konecker Delta function used to integrate functions rather than sequence grouping.