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sum and difference
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 >.
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 two
It was possible, however, to decompose the compliance into a sum of a frequency-independent component and two viscoelastic mechanisms, each compatible with the Boltzmann superposition principle and with a consistent set of time-temperature equivalence factors.
If one takes the middle number, 5, and multiplies it by 3 ( the base number of the magic square of three ), the result is 15, which is also the constant sum of all the rows, columns, and two main diagonals.
According to Davis, " It's great triumph was to prove that the sum of two even numbers is even ".
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number.
Each number is the sum of the two directly above it.
The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.
This role as a sink for CO < sub > 2 </ sub > is driven by two processes, the solubility pump and the biological pump .< ref > The former is primarily a function of differential CO < sub > 2 </ sub > solubility in seawater and the thermohaline circulation, while the latter is the sum of a series of biological processes that transport carbon ( in organic and inorganic forms ) from the surface euphotic zone to the ocean's interior.
* < tt > sum </ tt >, a Unix command ( also ported to Win32 ) that generates order-independent sums ; uses two different algorithms for calculating, the SYSV checksum algorithm and the BSD checksum ( default ) algorithm.
The sum of the two values, i. e. the total number of pips, may be referred to as the rank or weight of a tile, and a tile with more pips may be called heavier than a lighter tile with fewer pips.
Similarly, if light consisted strictly of classical particles and we illuminated two parallel slits, the expected pattern on the screen would simply be the sum of the two single-slit patterns.
From the above properties an important feature arises: Looking at two triangles ABD and BCD with the common edge BD ( see figures ), if the sum of the angles α and γ is less than or equal to 180 °, the triangles meet the Delaunay condition.
groups ( G, *) and ( H, ●), denoted by G × H. For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by.
The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PF < sub > 1 </ sub > + PF < sub > 2 </ sub > = 2a ).
The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil.
In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points ( the foci ) is constant.
The parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
Image: Pythagorean. svg | Pythagoras ' theorem: The sum of the areas of the two squares on the legs ( a and b ) of a right triangle equals the area of the square on the hypotenuse ( c ).
The celebrated Pythagorean theorem ( book I, proposition 47 ) states that in any right triangle, the area of the square whose side is the hypotenuse ( the side opposite the right angle ) is equal to the sum of the areas of the squares whose sides are the two legs ( the two sides that meet at a right angle ).

sum and symmetric
Given a circle A, find a circle B such that the area of the intersection of A and B is equal to the area of the symmetric difference of A and B ( the sum of the area of AB and the area of BA ).
Game theory classifies games according to several criteria: whether a game is a symmetric game or an asymmetric one, what a game's " sum " is ( zero-sum, constant sum, and so forth ), whether a game is a sequential game or a simultaneous one, whether a game comprises perfect information or imperfect information, and whether a game is determinate.
Any brightness distribution can be written as the sum of a symmetric component and an anti-symmetric component.
The sum here extends over all elements σ of the symmetric group S < sub > n </ sub >, i. e. over all permutations of the numbers 1, 2, ..., n.
Since the trace of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor.
Technically, the Fock space is ( the Hilbert space completion of ) the direct sum of the symmetric or antisymmetric tensors in the
If the sum of the degrees of x and y in each term is always even or always odd, then the curve is symmetric about the origin and the origin is called a center of the curve.
* If 2 is invertible, then and are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of symmetric and anti-symmetric ( Hermitian and skew Hermitian ) elements.
Observe that when V is a sum of line bundles, the Chern classes of V can be expressed as elementary symmetric polynomials in the,
In mathematics, specifically functional analysis, Mercer's theorem is a representation of a symmetric positive-definite function on a square as a sum of a convergent sequence of product functions.
There are many specific kinds of symmetric polynomials, such as elementary symmetric polynomials, power sum symmetric polynomials, monomial symmetric polynomials, complete homogeneous symmetric polynomials, and Schur polynomials.
It should be noted that the name " symmetric function " for elements of Λ < sub > R </ sub > is a misnomer: in neither construction the elements are functions, and in fact, unlike symmetric polynomials, no function of independent variables can be associated to such elements ( for instance e < sub > 1 </ sub > would be the sum of all infinitely many variables, which is not defined unless restrictions are imposed on the variables ).
In the special case of a symmetric design with λ = 1, that is, a projective plane, the theorem ( which in this case is referred to as the Bruck – Ryser theorem ) can be stated as follows: If a finite projective plane of order q exists and q is congruent to 1 or 2 ( mod 4 ), then q must be the sum of two squares.

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