Matrix Theory and Linear Algebra (Несколько текстов для зачёта), страница 10

Studying the intersection of quantum theory and classical physics requires developing a theory that can predict how quantum systems will behave as they get larger or as the number of particles involved approaches the size of problems described by classical physics. The mathematics involved is extremely difficult, but physicists continue to advance in their research. The constantly increasing power of computers should continue to help scientists with these calculations.

New research in quantum theory also promises new applications and improvements to known applications. One of the most potentially powerful applications is quantum computing. In quantum computing, scientists make use of the behavior of subatomic particles to perform calculations. Making calculations on the atomic level, a quantum computer could theoretically investigate all the possible answers to a query at the same time and make many calculations in parallel. This ability would make quantum computers thousands or even millions of time faster than current computers. Advancements in quantum theory also hold promise for the fields of optics, chemistry, and atomic theory.

Boolean Algebra

Boolean Algebra, branch of mathematics having laws and properties similar to, but different from, those of ordinary high school algebra. Formally a Boolean algebra is a mathematical system consisting of a set of elements, which may be called B, together with two binary operations, which may be denoted by the symbols  and . These operations are defined on the set B and satisfy the following axioms:

1.  and  are both commutative operations. That is, for any elements x, y of the set B, it is true that xY = yx and xy = yx.

2. Each of the operations  and  distributes over the other. That is, for any elements x, y, and z of the set B, it is true that x (yz) = (xy) (xz), and x (yz) = (xy)  (xz).

3. There exists in the set B a distinct identity element for each of the operations  and . These elements are usually denoted by the symbols 0 and 1 such that 0 ≠ 1, and have the property that 0 x = x and 1 x = x for any element x in the set B.

4. For each element x in the set B there exists a distinct corresponding element called the complement of x, usually denoted by the symbol x’. With respect to the operations  and , the element x’ has the property that xx’ = 1 and x x’ = 0.

A Boolean algebra may have other sets of axioms, all of which may be shown to be equivalent to those just given. The axioms given here are essentially those first published by the American mathematician Edward Huntington in Postulates for the Algebra of Logic (1904). The first treatment of the subject was given in 1854 by the English mathematician George Boole. It is possible to denote the operations  and  by any two symbols; +, , and  are sometimes used instead of , and ×, ^, , ·, and O instead of .

As an example of a Boolean algebra, consider any set X and let P(X) stand for the collection of all possible subsets of the set X. P(X) is sometimes called the power set of the set X. P(X), together with ordinary set union () and set intersection (), forms a Boolean algebra. In fact, every Boolean algebra may be represented as an algebra of sets.

From the symmetry of the axioms with respect to the two operations and their respective identities, one is able to prove the so-called principle of duality. This principle asserts that any algebraic statement deducible from the axioms of Boolean algebra remains true if the operations  and  and the identities 1 and 0 are interchanged throughout the statement. Of the many theorems that can be deduced from the axioms of a Boolean algebra, De Morgan's laws, that (xy)’ = x’y’ and that (xy)’ = x’y’, are particularly noteworthy.

The elements that are contained in the set B of a Boolean algebra may be abstract objects, or concrete things such as numbers, propositions, sets, or electrical networks. In Boole's original development, the elements of a Boolean algebra were a collection of propositions, or simple declarative sentences having the property that they were either true or false but not both. The operations were essentially conjunction and disjunction, denoted by the symbols ^ and  respectively. If x and y represent two propositions, then the expression xy (read x or y) would be true if and only if either x or y or both were true. The statement x ^ y (read x and y) would be true if and only if both x and y were true. In this type of Boolean algebra, the complement of an element or proposition is simply the negation of the statement.

A Boolean algebra of propositions and a Boolean algebra of sets are closely connected. For example, let p be the statement, “The ball is blue,” and let P be the set of all elements for which the statement p is true, that is, the set of all blue balls. P is called the truth set for the proposition p. Indeed, if P and Q are the truth sets for statements p and q, then the truth set for the statement pq is clearly P  Q and for p ^ q the truth set is P  Q.

Boolean algebra has many practical applications in the physical sciences, in electric-circuit theory and particularly in the field of computers.

As an example of an application of Boolean algebra in electrical-circuit theory, let p and q denote two propositions, that is, declarative sentences that are either true or false but not both. If each of the propositions p and q is associated with a switch that will be closed if the proposition is true, and open if the proposition is false, then the statement p ^ q may be represented by connecting the switches in series. The current will flow in this circuit if and only if both switches are closed, that is, if both p and q are true. Similarly, a circuit with switches connected in parallel can be used to represent the statement pq. In this case the current will flow if either p or q or both are true and the respective switches are closed. More complicated statements give rise to more complex switching circuits.

Arithmetic Progression

Arithmetic Progression, sequence of numbers that increase or diminish by a common difference so that any number in the sequence is the arithmetic mean, or average, of the numbers preceding and following it (See also Geometric Progression; Sequence and Series). The numbers 7, 10, 13, 16, 19, 22 form an arithmetic progression, as do the numbers 12, 10, 9, 7, 6. The natural numbers 1, 2, 3, 4 form an arithmetic progression in which the difference is 1. To find the sum of any arithmetic progression, multiply the sum of the first and last terms by half the number of terms. Thus, the sum of the first ten natural numbers is (1 + 10) × (10 ÷ 2) = 55.

The general arithmetic progression is a, a + d, a + 2d, a + 3d,... where the first term, a, and the common difference, d, are arbitrary numbers. The nth term of this progression (often denoted by the term a_n) is given by the formula a_n = a + (n - 1) d, and the sum of the first n terms is n[2a + (n - 1)d], or n(a + a_n).

Geometric Progression

Geometric Progression, in mathematics, sequence of numbers in which the ratio of any term, after the first, to the preceding term is a fixed number, called the common ratio. For example, the sequence of numbers 2, 4, 8, 16, 32, 64, 128 is a geometric progression in which the common ratio is 2, and 1, , , , , , ...i, ... is a geometric progression in which the common ratio is . The first is a finite geometric progression with seven terms; the second is an infinite geometric progression. In general, a geometric progression may be described by denoting the first term in the progression by a, the common ratio by r, and, in a finite progression, the number of terms by n. A finite geometric progression may then be written formally as

and an infinite geometric progression as

In general, if the nth term of a geometric progression is denoted by a_n, it follows from the definition that

If the symbol S_n denotes the sum of the first n terms of a geometric progression, it can be proved that

The terms in a geometric progression between a_i, and a_j, i<j, are called geometric means. The geometric mean between two positive numbers x and y is the same as the mean proportional  between the two numbers. In particular, a_n is the geometric mean or mean proportional between a_{n - 1} and a_{n + 1}.

The formal sum of the terms of an infinite geometric progression, written as

is called a geometric series (see Sequence and Series). In analysis it can be proved that a geometric series converges if the absolute value of the common ratio is less than 1; otherwise, the series diverges. If the series does converge, the limit, S, can be shown to be

The symbol is read “the limit of S_n as n increases without bound.”

Geometric series and geometric progressions have many applications in the physical, biological, and social sciences, as well as in investments and banking. Many problems in compound interest and annuities are easily solved using these concepts.

Mathematical Symbols

Mathematical Symbols, various signs and abbreviations used in mathematics to indicate entities, relations, or operations.

The origin and development of mathematical symbols are not entirely clear. For the probable origin of the remarkable digits 1 through 9, see Numerals. The origin of zero is unknown, because no authentic record exists of its history before ad400. The extension of the decimal position system below unity is attributed to the Dutch mathematician Simon Stevin, who called tenths, hundredths, and thousandths primes, sekondes, and terzes and circled digits to denote the orders; thus, 4.628 was written as 4 ⓪ 6 ① 2 2 8 3. A period was used to set off the decimal part of a number as early as 1492, and later a bar was also used. In the Exempelbüchlein of 1530 by the German mathematician Christoff Rudolf, a problem in compound interest is solved, and some use is made of the decimal fraction. The German astronomer Johannes Kepler used the comma to set off the decimal orders, and the Swiss mathematician Justus Byrgius used the decimal fraction in such forms as 3.2.

Although the early Egyptians had symbols for addition and equality, and the Greeks, Hindus, and Arabs had symbols for equality and the unknown quantity, from earliest times mathematical processes were cumbersome because proper symbols of operation were lacking. The expressions for such processes were either written out in full or denoted by word abbreviations. The later Greeks, the Hindus, and the German-born mathematician Nemorarius Jordanus indicated addition by juxtaposition; the Italians usually denoted it by the letter P or p with a line drawn through it, but their symbols were not uniform. Some mathematicians used p, some e, and the mathematician Niccolò Tartaglia commonly expressed the operation by . German and English algebraists introduced the sign +, but spoke of it as signum additorum and first used it only to indicate excess. The Greek mathematician Diophantus indicated subtraction by the symbol ↗. The Hindus used a dot, and the Italian algebraists denoted it by M or m with a line drawn through the letter. The German and English algebraists were the first to use the present symbol and described it as Signum subtractorum. The symbols + and - were first shown in 1489 by the German Johann Widman.

The English mathematician William Oughtred first used the symbol × for “times.” The German mathematician Gottfried Wilhelm Leibniz used a period to indicate multiplication, and in 1637 the French mathematician René Descartes used juxtaposition. In 1688 Leibniz employed the sign  to denote multiplication and  to denote division. The Hindus wrote the divisor under the dividend. Leibniz used the familiar form a:b. Descartes made popular the notation aⁿ for involution; the English mathematician John Wallis defined the negative exponent and first used the symbol (∞) for infinity.

The symbol of equality, =, was originated by the English mathematician Robert Recorde, and the symbols > and < for “greater than” and “less than” originated with Thomas Harriot, also an Englishman. The French mathematician François Viète introduced various symbols of aggregation. The symbols of differentiation, dx, and integration, ∫, as used in calculus, originated with Leibniz as did the symbol ~ for similarity, as used in geometry. The Swiss mathematician Leonhard Euler was largely responsible for the symbols , f, F, as used in the theory of functions.

The hierarchy of numbers is the following: million, billion, trillion, quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, undecillion, duodecillion, tredecillion, quat(t)uordecillion, quindecillion, sexdecillion, septendecillion, octodecillion, novemdecillion, vigintillion.