Matrix Theory and Linear Algebra (562420), страница 15

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An example of a mathematical proof is the following argument, which proves that the Pythagorean theorem is true. Figure 1 and figure 2 demonstrate that the relationship A² + B² = C² holds in a right-angled triangle with sides A and B and hypotenuse C. Figure 1 shows that a square of side A + B can be divided into four of the right-angled triangles, a square of side A, and a square of side B. Figure 2 shows that a square of side A + B can also be dissected into four of the right-angled triangles and a square of side C. Since the two squares of side A + B have the same area, they must still have the same area once the four triangles are removed from each of them. The total area of the squares that remain on the left side is A ² + B², and the area of the square remaining on the right side is C². Thus A² + B² = C².

The Greek mathematician Euclid laid down some of the conventions central to modern mathematical proofs. His book The Elements, written about 300 bc, contains many proofs in the fields of geometry and algebra. This book illustrates the Greek practice of writing mathematical proofs by first clearly identifying the initial assumptions and then reasoning from them in a logical way in order to obtain a desired conclusion. As part of such an argument, Euclid used results that had already been shown to be true, called theorems, or statements that were explicitly acknowledged to be self-evident, called axioms; this practice continues today.

In the 20th century, proofs have been written that are so complex that no one person understands every argument used in them. In 1976 a computer was used to complete the proof of the four-color theorem. This theorem states that four colors are sufficient to color any map in such a way that regions with a common boundary line have different colors. The use of a computer in this proof inspired considerable debate in the mathematical community. At issue was whether a theorem can be considered proven if human beings have not actually checked every detail of the proof.

Hardy-Weinberg Rule

Hardy-Weinberg Rule, set of algebraic formulas that describe how the proportion of different genes, the hereditary units that determine a particular characteristic in an organism, can remain the same over time in a large population of individuals.

Specifically, this rule indicates how often particular alleles, alternate forms of a particular gene that contain specific information about a trait (eye color, for example), should occur in a population. The rule also reveals how often particular genotypes, the actual combination of genes an organism carries and may pass on to its offspring, should appear in that same population. By studying these allelic and genotypic frequencies, scientists can identify populations that are changing genetically, or evolving. They can also predict the occurrence of genetic defects in populations.

British mathematician Godfrey Harold Hardy and German physician Wilhelm Weinberg independently described the rule in 1908. American mathematician Sewall Wright, British mathematician Sir Ronald Fisher, and British geneticist John B. S. Haldane then used the Hardy-Weinberg rule to develop mathematical theories of evolution, including several based on the concept of natural selection, in which the organisms best adapted to their environment survive and pass on their genetic characteristics. These theories formed the basis for a new branch of science known as population genetics—the study of how genes spread through populations of organisms (see Genetics).

Each individual in a population has two alleles for every gene. These alleles may be the same or different, and one allele may be dominant over the other. For example, in a sample group of 100 individuals from a particular population, the gene for a certain trait has alleles A and a, in which A is dominant over a. Each individual in the group carries two of these alleles in one of the following combinations, or genotypes: AA,Aa, or aa. In the sample group of 100 people, 33 individuals have the AA genotype, or two A alleles; 54 individuals have the Aa genotype, or one A and one a allele; and 13 individuals have the aa genotype, or two a alleles.

The actual frequency of each allele in the sample group—that is, the number of all alleles that are A or a, is determined by dividing the total number of each allele type by the total number of all alleles. For example, the actual frequency of the A allele in the sample group is 0.60, derived by dividing 120, the total number of A alleles (two each from the 33 individuals with the AA genotype and one each from the 54 individuals with the Aa genotype) by 200, the total number of all alleles (two each from the 100 individuals).

The Hardy-Weinberg rule uses the actual allelic frequencies of a population to predict the population’s expected genotypic frequencies—that is, the number of genotypes that should occur in the population. Assuming that a gene has two alleles, A and a (whose frequencies are represented mathematically as p and q, respectively), that can form three genotypes, AA,Aa, and aa, the following formulas can be used to predict expected genotypic frequencies:

Frequency of AA = p × p = p²
Frequency of Aa = 2 × p × q = 2pq
Frequency of aa = q × q = q²

For example, if the frequency of the A allele in a population is equal to 0.60, then the expected frequency of individuals with a AA genotype is 0.36, which is 0.60 multiplied by 0.60.

Scientists compare a population’s expected genotypic frequencies to its actual genotypic frequencies (determined by dividing the total number of each genotype in the group by the total number of individuals in the group) to determine whether the population is maintaining the same ratio, or equilibrium, of genotypes over time.

According to the Hardy-Weinberg rule, this equilibrium remains the same in a population as long as four conditions are met. First, individuals must select mates randomly without regard to visible, or phenotypic, traits. Second, no genotype can be favored in such a way that it will increase in frequency in the population over time. The third condition states that no new alleles can be introduced into the population, either by individuals from outside the population or by alleles that have changed, or mutated, from one form to another. The final condition specifies that the number of individuals and genotypes in the population remain high. A population that meets these conditions maintains the same proportions of different genes over time—the genetic makeup of the population never changes. Rare genes never disappear and common genes remain numerous.

Most populations do not maintain an equilibrium between allelic and genetic frequencies, however, and existing genes are replaced by new or more advantageous genes. This evolution may be due to natural selection—that is, some members of the population producing more or stronger, healthier offspring. Changes can also be caused by genetic mutation, an inheritable change in the character of a gene; genetic drift, the loss of an allele from a population caused when offspring in a generation inherit the alternate form of an allele for a gene; the migration of individuals to and from the population; or a decrease in population size. All of these factors occur naturally over time. Genetic mutations, however, are also caused by exposure to harmful chemicals and radioactive materials.

Determinant

Determinant, mathematical notation consisting of a square array of numbers or other elements between two vertical bars; the value of the expression is determined by its expansion according to certain rules. Determinants were first investigated by the Japanese mathematician Seki Kowa about 1683 and independently by the German philosopher and mathematician Gottfried Wilhelm Leibniz about 1693. These notations are used in almost every branch of mathematics and in the natural sciences.

The symbol is a determinant of the second order, because it is an array of two rows and two columns. Each letter a stands for a number or variable. The determinant itself also represents a number or variable, the value of which is defined as a₁₁ a₂₂-a₁₂a₂₁. For example:

A determinant of the nth order is a square array of n rows and n columns represented by the symbol

The minor, M_ij, of any element a_ij in the array is the determinant formed of the elements remaining after deleting the row i and the column j in which the element a_ij occurs. The cofactor, A_ij, of an element a_ij is equal to (- 1)^i+jM_ij.

The value of any determinant may be expressed in terms of the elements of any row (or column) and their respective cofactors in accordance with the following rule. Each element in the selected row (or column) is multiplied by its corresponding cofactor; the sum of these products is the value of the determinant. Formally, this may be written

if the expansion is in terms of the ith row, or

if it is in terms of the jth column. Thus, to find the value of a third-order determinant using the elements in the first column

These terms may be evaluated in accordance with the definition of the second-order determinant given above. For determinants of higher orders than the third, the process is repeated on the determinants formed by the minors until the determinants can be expanded easily.

Because this method of finding the value of a determinant may be quite laborious, various properties of a determinant are developed and utilized to lessen the amount of calculation needed to evaluate it. Among these properties are the following: (1) a determinant is equal to zero if all the elements in one row (or column) are identical with or proportional to the elements in another row (or column); (2) a determinant is multiplied by a given factor if each element of a row (or column) is multiplied by the same factor; and (3) the value of a determinant is not changed by adding to each element of a row (or column) the corresponding element of another row (or column) multiplied by a constant factor. Hence, through the use of these and other properties, determinants of higher order can be reduced to third-order determinants for simple expansion.

Application of determinants in analytical geometry is illustrated in the following example: If P₁(x₁, y₁), P₂(x₂, y₂), P₃(x₃, y₃) are three distinct points in a rectangular coordinate plane, the area A of triangle P₁P₂P₃, apart from algebraic sign, is given by

When the three points are collinear, the determinant is equal to zero.

An example of the use of determinants in solving linear equations is as follows. Let

be a system of n linear equations in the n unknowns x₁, x, ..., x_n. The determinant Δ given above is the determinant of coefficients; let Δ_k be the determinant obtained by deleting the kth column of Δ and replacing it by the column of constants b₁, b₂, ..., b_n, where k = 1, 2, ..., n. If Δ≠ 0, the equations are consistent—that is, a solution is possible. In this case only one solution is possible; it is given by

If Δ = 0, further investigation is necessary to determine the number and nature of the solutions.

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