# Form

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## form

**1.**

*Sport*the previous record of a horse, athlete, etc., esp with regard to fitness

**2.**style, arrangement, or design in the arts, as opposed to content

**3.**a fixed mode of artistic expression or representation in literary, musical, or other artistic works

**4.**

*Education*

*chiefly Brit*a group of children who are taught together; class

**5.**

*Philosophy*

**a.**the structure of anything as opposed to its constitution or content

**b.**essence as opposed to matter

**c.**(in the philosophy of Plato) the ideal universal that exists independently of the particulars which fall under it

**d.**(in the philosophy of Aristotle) the constitution of matter to form a substance; by virtue of this its nature can be understood

**6.**See logical form

**7.**the nest or hollow in which a hare lives

**8.**a group of organisms within a species that differ from similar groups by trivial differences, as of colour

**10.**

*Taxonomy*a group distinguished from other groups by a single characteristic: ranked below a variety

## Form

## Form

assembled composed matter whose surface contains the printing elements and material for blank spaces. (The Russian term *pechatnaia forma* is also used to refer to a flat or cylindrical plate.) It is intended for the production of multiple impressions. The relative placement of the printing and spacing elements determines the method of printing.

The following forms are distinguished, depending on the method of printing, the type of printing press, and the type of materials used: (1) in relief printing—composed matter, cuts, and stereotype; (2) in planographic printing—monometallic (aluminum or zinc), bimetallic, and trimetallic (for example, steel, copper, and chromium), and also glass; (3) in gravure printing—copper or chromed cylinders. A distinction is made among text, illustration, and mixed forms, depending on the nature of the graphic elements and the prints. The materials used in the production of forms include nonferrous metals, alloys, plastics, rubber, wood, and metal or paper-backed foil. Up to 1 million impressions may be produced from a single form, depending on the material used. The form to a great extent determines the quality of printing.

E. M. FARBER

## Form

a grammatical or lexical-grammatical category of the verb in Hamito-Semitic and certain other languages.

Verbs in a given form share a common voice or aspect meaning (reflexive, reciprocal, intensive, etc.) and the same type of affixation or internal inflection. A verbal root can be marked by the indicators of different forms. In Arabic there are more than 15 verb forms; their genetic relationship to the forms of other Semitic, Cushitic, Berber, and Chadic languages suggests a common Hamito-Semitic origin of at least some of the oppositions between forms. Sometimes in Hebrew and Arabic the lexical meaning of verbs with the same root but in different forms does not coincide because of a semantic shift or false etymology.

## Form

one of the infrasubspecies categories in plant and animal systematics. Botanists usually use the term “form” to designate a category lower in rank than a variety; zoologists use the term as a synonym for variety. Sometimes the term “form” is used in the same sense as the term “taxon,” that is, to designate a systematic unit of any rank. In biology the term is used extensively not only in the strictly taxonomic sense but also to note various features associated with the developmental cycle, the type of existence, or the dynamics and formation of a species (for example, holopterous and brachypterous forms of insects; seasonal forms of plants; and ecological, archaic, progressive, or specialized forms of all living organisms).

## Form

in logic, that aspect of reasoning (for example, of a proof, deduction, or argument) which is independent of its content. Logical form in language is established through logical constants and through the individual phrases and combinations of phrases formed by means of such constants—that is, through reasoning schemata; these schemata, which vary in content, are forms of inference expressing the connection between premises and conclusions. Included in the category of logical forms are the laws of logic and rules of logical transition, or inference, applicable to formal and mathematical logic, as well as many of the questions related thereto—including, for example, the problem of refining the concept of logical consequence.

## Form

in mathematics, a polynomial in several variables whose terms are all of the same degree (the degree of the monomial *x ^{α}y^{β}*...

*z*is understood to be the number α + β + . . . + γ). The theory of forms has applications in algebraic geometry, number theory, differential geometry, mechanics, and other fields of pure and applied mathematics.

^{γ}Depending on the number *m* of variables, a form is said to be binary (*m* = 2), *ternary* (*m* = 3), and so forth; depending on the degree *n* of its terms, a form is linear (*n* = 1), quadratic (*n* = 2), cubic (*n* = 3), and so forth. For example, *xy* + 2*y*^{2} + *z*^{2} is a ternary quadratic form. If the variables can be divided into sets such that each term of the form is linearly dependent on the variables of each set, the form is said to be multilinear. An example of a multilinear form is a determinant regarded as a function of its elements; the sets into which the elements are divided in this case are the sets of elements lying in the same row or column. Any form can be obtained from a multilinear form by the identification of certain variables. The reverse process of obtaining a multilinear form from any form is performed by a procedure known as polarization. For example, the form corresponds to the multilinear form *x*_{1}*y*_{1} + *x*_{1}*y*_{2} + *y*_{1}*x*_{2} + *x*_{2}*y _{2}*; by identifying

*y*

_{1}with

*x*

_{1}and

*y*

_{2}with

*x*

_{2}, the multilinear form can be transformed into the given form .

The equation of any algebraic curve in the plane can be written in homogeneous coordinates as *f*(*x*_{1}, *x*_{2}, *x*_{3}) = 0, where *f* is some ternary form. In much the same way, a geometric interpretation may be given to forms in a larger number of variables. In the case of, for example, curved surfaces, geometric properties that are independent of the choice of coordinate system can be expressed in terms of invariants of forms. The theory of invariants is one of the fundamental branches of the algebraic theory of forms and is made use of not only in algebraic geometry but also in a number of other branches of pure and applied mathematics.

Quadratic forms have the most important applications. For example, the square of the length of a vector can be expressed as a quadratic form of the vector’s coordinates. If a mechanical system in motion remains close to its equilibrium position, its kinetic and potential energies (if they do not depend explicitly on time) are expressed by the quadratic forms

respectively. The analysis of the vibrations of these systems is based on the theory of quadratic forms, particularly on the reduction of the forms to a sum of squares. The theory of quadratic forms is closely connected with the theory of second-order curves and surfaces. As in many other cases, the study of forms over the complex numbers (Hermitian forms) is a natural and fruitful extension of the study of forms over the reals.

An extremely important problem in number theory is the representability of integers by forms with integral coefficients where the variables take on integral values. J. Lagrange proved, for example, that any natural number may be represented by an expression of the type *x*^{2} + *y*^{2} + *z*^{2} + *t*^{2}. The question of the representability of integers by an expression of the type *ax*^{2} + 2*bxy* + *cy*^{2}, where *a*, *b*, *c*, *x*, and *y* are integers, was studied by Lagrange and K. Gauss. This problem is closely associated with the theory of algebraic numbers. A. Thue proved that equations of the type *f*(*x*, *y*) = *m*, where the degree of the form *f* is greater than two, have a finite number of integral solutions (*see*DIOPHANTINE EQUATIONS).

Differential geometry and Riemannian geometry make use of differential forms, that is, polynomials in differentials of variables where the terms are all of the same degree with respect to the differentials. The coefficients of differential forms may be arbitrary functions of the variables. Multilinear differential forms are also considered. Examples of differential forms are the first and second fundamental quadratic forms in the theory of surfaces. An important role in differential geometry is played by differential invariants: entire rational functions of the coefficients, and of the derivatives of the coefficients, of quadratic forms are said to be differential invariants if they remain unchanged under any nonsingular differentiable transformation of the variables. For example, the total, or Gaussian, curvature of a surface is a differential invariant of the first fundamental quadratic form. Research in the theory of differential invariants played an important role in the development of tensor calculus. The theory of differential invariants has found extensive application in physics because it permits physical laws to be expressed in invariant formulations, that is, formulations independent of the choice of coordinate system.

Many theorems of integral calculus—such as the theorems of Green, Ostrogradskii, and Stokes—can be regarded as theorems on the relationship between differential forms of various degrees. E. Carton generalized these theorems and formulated the theory of the exterior differentiation of forms, which has played an important role in modern mathematics.

### REFERENCES

Veblen, O.*Invarianty differentsial’nykh kvadratichnykh form*. Moscow, 1948. (Translated from English.)

Gurevich, G. B.

*Osnovy teorii algebraicheskikh invariantov*. Moscow-Leningrad, 1948.

Gantmakher, F. R.

*Teoriia matrits*, 3rd ed. Moscow, 1967.

Borevich, Z. I., and I. R. Shafarevich.

*Teoriia chisel*, 2nd ed. Moscow, 1972.

## form

[fȯrm]## form

## FORM

(mathematics, tool)**ftp://acm.princeton.edu/**,

**ftp://nikhefh.nikhef.nl/**.

Mailing list: <form@can.nl>.

## form

**(1)**A paper form used for printing.

**(2)**A formatted screen display designed for a particular application. See forms software.