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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 >.
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sum and difference
The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a ring.
One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i. e. given any a and b, with a > b, there exist c and d, all positive and rational, such that
A proposition that says: " The product of the sum and the difference of a and b should give us the difference of the squares of a and b " does express a normative proposition, but this normative statement is based on the theoretical statement "( a + b )( a-b )= a²-b² ".
The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have
If the two input signals are both sinusoids of specified frequencies f < sub > 1 </ sub > and f < sub > 2 </ sub >, then the output of the mixer will contain two new sinsoids that have the sum f < sub > 1 </ sub > + f < sub > 2 </ sub > frequency and the difference frequency absolute value | f < sub > 1 </ sub >-f < sub > 2 </ sub >|.
A multiplier ( which is a nonlinear device ) will generate ideally only the sum and difference frequencies, whereas an arbitrary nonlinear block would generate also signals at e. g. 2 · f < sub > 1 </ sub >- 3 · f < sub > 2 </ sub >, etc.
In many typical circuits, the single output signal actually contains multiple waveforms, namely those at the sum and difference of the two input frequencies and harmonic waveforms.
States where this is a sum are known as symmetric ; states involving the difference are called antisymmetric.
A hierarchical version of this technique takes neighboring pairs of data points, stores their difference and sum, and on a higher level with lower resolution continues with the sums.
These three-wave mixing processes correspond to the nonlinear effects known as second harmonic generation, sum frequency generation, difference frequency generation and optical rectification respectively.
Many questions around prime numbers remain open, such as Goldbach's conjecture, which asserts that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, which says that there are infinitely many pairs of primes whose difference is 2.
The simplest case would be two electrons starting at A and B ending at C and D. The amplitude would be calculated as the " difference ",, where we would expect, from our everyday idea of probabilities, that it would be a sum.
The mixer uses a non-linear component to produce both sum and difference beat frequencies signals, each one containing the modulation contained in the desired signal.
Every resultant is either a sum or a difference of the co-operant forces ; their sum, when their directions are the same -- their difference, when their directions are contrary.
The emergent is unlike its components insofar as these are incommensurable, and it cannot be reduced to their sum or their difference.
The inverse operator of the forward difference operator, so then the umbral integral, is the indefinite sum or antidifference operator.
An infinite difference is a further generalization, where the finite sum above is replaced by an infinite series.
In the most common application, two signals at frequencies f < sub > 1 </ sub > and f < sub > 2 </ sub > are mixed, creating two new signals, one at the sum f < sub > 1 </ sub > + f < sub > 2 </ sub > of the two frequencies, and the other at the difference f < sub > 1 </ sub > − f < sub > 2 </ sub >.
The right hand side shows that the resulting signal is the difference of two sinusoidal terms, one at the sum of the two original frequencies, and one at the difference, which can be considered to be separate signals.
sum and product
** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.
( The reason for the term " colloquially ", is that the sum or product of a " sequence " of cardinals cannot be defined without some aspect of the axiom of choice.
and mean that the sum or product is over all prime numbers:
Similarly, and mean that the sum or product is over all prime powers with strictly positive exponent ( so 1 is not counted ):
and mean that the sum or product is over all positive divisors of n, including 1 and n.
The notations can be combined: and mean that the sum or product is over all prime divisors of n.
and similarly and mean that the sum or product is over all prime powers dividing n.
The great utility in creating CPUs that deal with vectors of data lies in optimizing tasks that tend to require the same operation ( for example, a sum or a dot product ) to be performed on a large set of data.
The common notation for the divergence ∇· F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ ( see del ), apply them to the components of F, and sum the results.
Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.
Functors are often defined by universal properties ; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases.
For N interacting particles, i. e. particles which interact mutually and constitute a many-body situation, the potential energy function V is not simply a sum of the separate potentials ( and certainly not a product, as this is dimensionally incorrect ).
The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum ( as is the case with the arithmetic mean ) e. g. rates of growth.
The volume of a cylinder was taken as the product of the area of the base and the height ; however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases.
The acceleration of each body is equal to the sum of the gravitational forces on it, divided by its mass, and the gravitational force between each pair of bodies is proportional to the product of their masses and decreases inversely with the square of the distance between them.
To work out the product of two polynomials into a sum of terms, the distributive law is repeatedly applied, which results in each term of one polynomial being multiplied by every term of the other.
The sum or product of two polynomials is always a polynomial.
sum and quotient
Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.
This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over K are again algebraic over K. The set of all elements of L which are algebraic over K is a field that sits in between L and K.
Finding a generator of an abelian category allows one to express every object as a quotient of a direct sum of copies of the generator.
In the last representation we can see that the Rayleigh quotient is the sum of the squared cosines of the angles formed by the vector x and each eigenvector, weighted by corresponding eigenvalues.
The sum, product, or quotient ( excepting division by the zero polynomial ) of two rational functions is itself a rational function: however, the process of reduction to standard form may inadvertently result in the removing of such discontinuities unless care is taken.
A quotient verdict is a special case of a compromise verdict in which jurors determine the damages to be paid to the prevailing party by agreeing that each juror will write down his or her assessment of the damages, and the sum of these numbers is divided by the number of jurors.
Arithmetic operators are,,, and ( binary operators, which pop two elements from the stack and push ( respectively ) their sum, difference, product, or quotient ) and the ( underscore ) is unary negation ( which pops one element and pushes its negation ).
The quotient of that sum by σ < sup > 2 </ sup > has a chi-squared distribution with only n − 1 degrees of freedom:
For example, truncating before the 4th partial quotient, we obtain the partial sum, which approximates Champernowne's constant with an error of about 1 × 10 < sup >- 9 </ sup >, while truncating just before the 18th partial quotient, we get
The pushout of these maps is the direct sum of A and B. Generalizing to the case where f and g are arbitrary homomorphisms from a common domain Z, one obtains for the pushout a quotient group of the direct sum ; namely, we mod out by the subgroup consisting of pairs ( f ( z ),- g ( z )).
Specifically, if X and Y are pointed spaces ( i. e. topological spaces with distinguished basepoints x < sub > 0 </ sub > and y < sub > 0 </ sub >) the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x < sub > 0 </ sub > ∼ y < sub > 0 </ sub >:
Weights are assigned so that any if a neighborhood of a point is the quotient space described above, then the sum of the weights of the two unidentified hyperplanes of that neighborhood is the weight of the identified hyperplane space.
If a number which is a sum of two squares is divisible by a prime which is a sum of two squares, then the quotient is a sum of two squares.
:: and thus expresses the quotient as a sum of two squares, as claimed.
If a number which can be written as a sum of two squares is divisible by a number which is not a sum of two squares, then the quotient has a factor which is not a sum of two squares.
If all factors can be written as sums of two squares, then we can divide successively by,, etc., and applying the previous step we deduce that each quotient is a sum of two squares.
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