and General Motors and Du Pont were to be ordered to terminate any agreement that provided for the purchase by General Motors of any specified percentage of its requirements of any Du Pont manufactured product, or for the grant of exclusive patent rights, or for a grant by General Motors to Du Pont of a preferential right to make or sell any chemical discovery of General Motors, or for the maintenance of any joint commercial enterprise by the two companies.

In any given time period, the aggregate demand for the industry's product is determined by two things: the price charged by the industry, and the level of Aj.

The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.

In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula

Ada Lovelace reported in her notes on the Analytical Engine: " Mr. Babbage believes he can, by his engine, form the product of two numbers, each containing twenty figures, in three minutes ".

The following year the Amiga product line was expanded with the introduction of two new models: the Amiga 2000 for high-end graphics and business use, and the Amiga 500 for home use.

These two product lines were the very first acrylic emulsion artists paints.

It is not known whether a pair of coprime amicable numbers exists, though if any does, the product of the two must be greater than 10 < sup > 67 </ sup >.

It is so called because the inner product ( or dot product ) of two states is denoted by a bra | c | ket ;

The inner product of two vectors is a complex number.

Bra-ket notation splits this inner product ( also called a " bracket ") into two pieces, the " bra " and the " ket ":

If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.

If is a ket in V and is a ket in W, the direct product of the two kets is a ket in.

In quantum mechanics it often occurs that little or no information about the inner product of two arbitrary ( state ) kets is present, while it is possible to say something about the expansion coefficients and of those vectors with respect to a chosen ( orthonormalized ) basis.

are just different notations for expressing an inner product between two elements in a Hilbert space ( or for the first three, in any inner product space ).

* Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution.

These categories surely have some objects that are " special " in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i. e., we do not know whether an object A is a set, a topology, or any other abstract concept – hence, the challenge is to define special objects without referring to the internal structure of those objects.

Within chemical engineering, two broad subgroups include 1 ) design, manufacture, and operation of plants and machinery in industrial chemical and related processes (" chemical process engineers "); and 2 ) development of new or adapted substances for products ranging from foods and beverages to cosmetics to cleaners to pharmaceutical ingredients, among many other products (" chemical product engineers ").

Then the previous expression is equal to the product of two factors:

When this happens, the limit of the product of these two factors will equal the product of the limits of the factors.

The dot product of the two vector quantities ( A and J ) is a scalar that represents the electric current.

With the above product, the set of all homotopy classes of loops with base point x < sub > 0 </ sub > forms the fundamental group of X at the point x < sub > 0 </ sub > and is denoted

** the smash product and wedge sum ( sometimes called the wedge product ) in homotopy.

The Whitehead product on homotopy groups gives many examples of Lie superalgebras over the integers.

The Whitehead product is an operation in homotopy theory.

All these cohomology theories share common properties, e. g. existence of Mayer-Vietoris-sequences, homotopy invariance ( H < sup >∗</ sup >( X )≅ H < sup >∗</ sup >( X × A < sup > 1 </ sup >), the product of X with the affine line ) and others.

The subject of stable homotopy theory was conceived as a simplification, by introducing the suspension ( smash product with a circle ) operation, and seeing what ( roughly speaking ) remained of homotopy theory once one was allowed to suspend both sides of an equation, as many times as one wished.

It makes the stable homotopy category into a monoidal category ; in other words it behaves like the tensor product of abelian groups.

A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy.

on homotopy groups, where Ω denotes the loop functor and ∧ denotes the smash product.

For example, in homotopy theory, the smash product X ∧ Y of spaces X and Y is used.

use a Massey product to show that the homotopy type of the configuration space of two points in a lens space depends non-trivially on the simple homotopy type of the lens space.

A binary relation R is usually defined as an ordered triple ( X, Y, G ) where X and Y are arbitrary sets ( or classes ), and G is a subset of the Cartesian product X × Y.

Biologics / Biopharmaceuticals ( e. g., vaccines, stored blood product ), genetic engineering, and various agricultural applications are some major classes of biotechnology.

It is now denoted by He showed that this ring has the four units ± 1 and ± i, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that ( except for order ), the composites have unique factorization as a product of primes.

Cafiero explains in Anarchy and Communism ( 1880 ) that private property in the product of labor will lead to unequal accumulation of capital and, therefore, the reappearance of social classes and their antagonisms, and thus the resurrection of the state: " If we preserve the individual appropriation of the products of labour, we would be forced to preserve money, leaving more or less accumulation of wealth according to more or less merit rather than need of individuals.

It was a product of the desire of Hamburg's patrician classes to defend their political privileges in the face of dramatic social change and attendant demands for political reform.

Originally a product available to only the Italian upper classes, a cheaper form of balsamic vinegar became widely known and available around the world in the late 20th century.

It is possible to develop " our own abstract and concrete product classes ", because " all window systems do generally the same thing " ( p. 52 ).

Hence, only one set of classes are needed for each type of, and only one set of classes are needed for each platform ( rather than the Cartesian product of all available types and platforms ).

Common Criteria is very generic ; it does not directly provide a list of product security requirements or features for specific ( classes of ) products: this follows the approach taken by ITSEC, but has been a source of debate to those used to the more prescriptive approach of other earlier standards such as TCSEC and FIPS 140-2.

Furthermore, in some emerging economies, western fast food represents an expensive niche product price well above the price of traditional staples — i. e. the Big Mac is not a mainstream ' cheap ' meal as it is in the West, but a luxury import for the middle classes and foreigners.

However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal ( except R ) is a product of prime ideals.

* Surplus-value can be viewed as a social relation of production, or as the monetary valuation of surplus-labour-a sort of " index " of the balance of power between social classes or nations in the process of the division of the social product.

For example, if a personal computer is the core product, then whole product would include software applications, training classes, peripheral devices ( mouse, keyboard, printer, etc.

Two common classes of algebraic type are product types, i. e. tuples and records, and sum types, also called tagged unions or variant types.

Long-term mean precipitation by monthIn meteorology, precipitation ( also known as one of the classes of hydrometeors, which are atmospheric water phenomena ) is any product of the condensation of atmospheric water vapor that falls under gravity.

If we work on an oriented manifold of dimension 2n, then any product of Chern classes of total degree 2n can be paired with the orientation homology class ( or " integrated over the manifold ") to give an integer, a Chern number of the vector bundle.

The formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not ( ordinary ) addition, but rather a formal group law.

The licensed Japanese teacher is normally in charge of junior high school and high school classes regardless of where the ALT originates ; however in the case of elementary school classes, the ALT is normally responsible for the entire class, with the Japanese teacher either providing limited input or in some cases not being present in the classroom, and for that reason the continuity of school management is sometimes maintained with the school principal in compliance with any legal requirements, as the product being contracted itself is an abstract, education, and the contract basis for private language teaching corporations is to provide education and educational services.