
Review of Short Phrases and Links 
This Review contains major "Sum" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 A sum is the second level administrative subdivision below the Aimags (provinces), roughly comparable to a County in the USA. There are 331 sums in Mongolia.
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 Sum (Siao, Fong SaiYuk) is a well known Cantonese opera star, a woman who has played only male roles for the last twenty years.
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 The sum is finite since p i can only be less than or equal to n for finitely many values of i, and the floor function results in 0 when applied for p i n.
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 This sum is greater than 810.0, its expected value under the null hypothesis of no difference between the two samples Active and Placebo.
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 In sum, the lowincome population in our sample achieved as well in literacy and language as a normative population through the third grade.
 Foundations: fields and vector spaces, subspace, linear independence, basis, ordered basis, dimension, direct sum decomposition.
 Conversely, it can be proven that any semisimple Lie algebra is the direct sum of its minimal ideals, which are canonically determined simple Lie algebras.
 In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e.
 The total energy of the ordinary species is the sum of the energies to remove all of the electrons from the ordinary species.
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 The total number of electrons represented in a Lewis structure is equal to the sum of the numbers of valence electrons on each individual atom.
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 The full scattering amplitude is the sum of all contributions from all possible loops of photons, electrons, positrons, and other available particles.
 The spin of atoms and molecules is the sum of the spins of unpaired electrons.
 The bracket is the scalar product on the Hilbert space; the sum on the right hand side converges in the operator norm.
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 Two (or more) Hilbert spaces can be combined to produce another Hilbert space by taking either their direct sum or their tensor product.
 But if there are only finitely many summands, then the Banach space direct sum is isomorphic to the Hilbert space direct sum.
 In general, the orthogonal complement of a sum of subspaces is the intersection of the orthogonal complements:[ 68].
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 Then, and its orthogonal complement determine a direct sum decomposition of.
 As expected, the sum of the eigenvalues is equal to the number of variables.
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 The trace of a square matrix is the sum of its diagonal entries, which equals the sum of its n eigenvalues.
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 First, the sum over functions with differences of eigenvalues in the denominator resembles the resolvent in Fredholm theory.
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 But that is very easy indeed: given a polynomial we define to be the degree of plus the sum of the absolute values of the coefficients of.
 Of the robust estimators considered in the paper, the one based on minimizing the sum of the absolute values of the residuals performed the best.
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 The infinity norm (or maximum value of the sum of the absolute values of the rows members of a matrix).
 Binding energy  The difference between the total energy of a molecular system and the sum of the energies of its isolated p  and s bonds.
 The difference between the mean and the predicted value of Y. This is the explained part of the deviation, or (Regression Sum of Squares).
 The addition (their sum) and subtraction (their difference) of two integers will always result in an integer.
 The sum of two algebraic integers is an algebraic integer, and so is their difference; their product is too, but not necessarily their ratio.
 The "Chisquare distribution with n degrees of freedom" is therefore the distribution of the sum of n independent squared r.v.
 A graph with n vertices (n ≥ 3) is Hamiltonian if, for each pair of nonadjacent vertices, the sum of their degrees is n or greater (see Ore's theorem).
 The degrees of freedom is equal to the sum of the individual degrees of freedom for each sample.
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 Notice that each Mean Square is just the Sum of Squares divided by its degrees of freedom, and the F value is the ratio of the mean squares.
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 Thus, the sum of all the eigenvalues is equal to the sum squared distance of the points with their mean divided by the number of dimensions.
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 If the M i are actually vector spaces, then the dimension of the direct sum is equal to the sum of the dimensions of the M i.
 The dimension of the space is the sum of the dimensions of the two subspaces, minus the dimension of their intersection.
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 For a square matrix A of order n to be diagonalizable, the sum of the dimensions of the eigenspaces must be equal to n.
 FIG. 14A is a graph showing the sum of hydrophilic peaks as detected by HPLC when a variety of synthetic adsorbents were added to the beer.
 Thus the sum of z 1 and z 2 corresponds to the diagonal OB of the parallelogram shown in Fig.
 The following formulae can be used to find the probability of rolling a sum S using N dice of M faces.
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 The surface area of any prism equals the sum of the areas of its faces, which include the floor, roof and walls.
 In an abelian category, for example the category of abelian groups or a category of modules, the direct sum is the categorical coproduct.
 There is sort of a semiring structure on that category in that vector bundles can be added, using the direct sum, and multiplied, using the direct product.
 Semisimple means that each object in the category is (isomorphic to) the direct sum of (finitely many) simple objects.
 Analogous examples are given by the direct sum of vector spaces and modules, by the free product of groups and by the disjoint union of sets.
 Analogous examples are given by the direct sum of vector space s and modules, by the free product of groups and by the disjoint union of sets.
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 Given such a polyhedron, the sum of the vertices and the faces is always the number of edges plus two.
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 The sum of two points A and B of the complex plane is the point X = A + B such that the triangles with vertices 0, A, B, and X, B, A, are congruent.
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 X A + B: The sum of two points A and B of the complex plane is the point X A + B such that the triangle s with vertices 0, A, B, and X, B, A, are congruent.
 In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold.
 The tensor bundle is the direct sum of all tensor product s of the tangent bundle and the cotangent bundle.
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 Mass Defect and Binding Energy Summary Mass defect is the difference between the mass of the atom and the sum of the masses of its constituent parts.
 An atom or molecule has less mass (by a negligible but real amount) than the sum of the masses of its components taken separately.
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 The formal charge of the atom, the sum of the charge of the proton and the charge of the electron, is zero.
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 So this difference in the actual nuclear mass and the expected nuclear mass (sum of the individual masses of nuclear particles) is called mass defect.
 Properties The direct sum is a submodule of the direct product of the modules M i.
 A direct summand of M is a submodule N such that there is some other submodule N′ of M such that M is the internal direct sum of N and N′.
 A finite direct sum of modules is Noetherian if and only if each summand is Noetherian; it is Artinian if and only if each summand is Artinian.
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 But centuries and centuries elapsed before the sum of human knowledge was equal to what it had been at the fall of the Roman empire.
 Nuclear  Binding energy The sum of the individual masses of various particle in the nucleus must be equal to the nuclear mass.
 To calculate the binding energy of a nucleus, all you have to do is sum the mass of the individual nucleons, and then subtract the mass of the atom itself.
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 The net magnetic moment of an atom is equal to the vector sum of orbital and spin magnetic moments of all electrons and the nucleus.
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 Mass Defect The difference in the mass of a nucleus and the sum of the masses of its constituent particles.
 Composite particles, such as nuclei and atoms, are classified as bosons or fermions based on the sum of the spins of their constituent particles.
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 For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods.
 Average the cosines: Find the cosines of the sum and difference angles using a cosine table and average them.
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 For example, instead of the usual least squares you could request a minimum of the sum of the absolute deviations or possibly the minimum maximum error.
 Any point in the 4 to 7 region will have the same value of 22 for the sum of the absolute deviations.
 Mean(arithmetic mean or average) is the sum of the data in a frequency distribution divided by the number of data elements.
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 In mathematics and statistics, the arithmetic mean of a set of numbers is the sum of all the members of the set divided by the number of items in the set.
 Mean: More accurately called the arithmetic mean, it is defined as the sum of scores divided by the number of scores.
 Different examples include maximising the distance to the nearest point, or using electrons to maximise the sum of all reciprocals of squares of distances.
 If the inputs are error forms, the error is the square root of the reciprocal of the sum of the reciprocals of the squares of the input errors.
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 More precisely, if S(x) denotes the sum of the reciprocals of all prime numbers p with p ≤ x, then S(x) = ln ln x + O(1) for x → ∞.
 A Fourier series is an expansion of a periodic function in terms of an infinite sum of sines and cosines.
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 Suppose that ƒ(x) is periodic function with period 2 π, in this case one can attempt to decompose ƒ(x) as a sum of complex exponentials functions.
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 In mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sines and cosines.
 In the study of Fourier series, complicated periodic functions are written as the sum of simple waves mathematically represented by sines and cosines.
 The spectral method, which represents functions as a sum of particular basis functions, for example using a Fourier series.
 The angle defect at a vertex of a polygon is defined to be minus the sum of the angles at the corners of the faces at that vertex.
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 The conservation of momentum is obtained by adding to the vertices a delta function on the sum of the 4momenta coming into the vertex.
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 The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees.
 A quadratic identity is used by Louis de Lagrange (1706–1783) to show that every positive integer is the sum of four squares of integers.
 In 1770, Lagrange showed that every positive integer could be written as the sum of at most four squares.
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 For instance, it was proven by Lagrange that every positive integer is the sum of four squares.
 In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition.
 A double sum is often the product of two sums, which may be Fourier series.
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 Direct sums are also commutative and associative (up to isomorphism), meaning that it doesn't matter in which order one forms the direct sum.
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 It is remarkable that two random variables, the sum of squares of the residuals and the sample mean, can be shown to be independent of each other.
 Linear regression fits a line to a scatterplot in such a way as to minimize the sum of the squares of the residuals.
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 In statistics, the residual sum of squares (RSS) is the sum of squares of residuals.
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 A square matrix is diagonalizable if the sum of the dimensions of the eigenspaces is the number of rows or columns of the matrix.
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 The sum of the entries on the main diagonal of a square matrix is known as the trace of that matrix.
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 As the trace of a matrix is equal to the sum of its eigenvalues, this implies that the estimated eigenvalues are biased.
 The tr is the trace operator and represents the sum of the diagonal elements of the matrix.
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 Dimension. Sum of diagonal elements.
 Note that the sum of the diagonal elements is conserved; this is the signature of the metric [ +2 ].
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 Principal: The original amount of a debt; a sum of money agreed to by the borrower and the lender to be repaid on a schedule.
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 The borrower may redeem by paying the lender the sum for which the property was sold at foreclosure, plus interest at the same rate as the mortgage.
 When the interest rate is reduced for a specified period of time by depositing a sum of money with the lender.
 Mortgage Note: A written promise to pay a sum of money at a stated interest rate during a specified term.
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 Energy of an Orbit The Total energy of an object in orbit is the sum of kinetic energy (KE) and gravitational potential energy (PE).
 The energy of a volume V at any point is the sum of its kinetic energy and its potential energy (pV). Effects of gravitation and viscosity are neglected.
 One term not listed among manifestations of energy is mechanical energy, which is something different altogether: the sum of potential and kinetic energy.
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 The energy released is equal to the sum of the rest energies of the particles and their kinetic energies.
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 For extended objects composed of many particles, the kinetic energy of the composite body is the sum of the kinetic energies of the particles.
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 In this model the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherical symmetric electronnucleus interactions.
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 Related subjects: Mathematics On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry).
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 Then an elementary calculation of angles shows that the sum of the exterior angles of the polygon is equal to the sum of the face angles at the vertex.
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 Legendre proved that Euclid 's fifth postulate is equivalent to: The sum of the angles of a triangle is equal to two right angles.
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 The sum of interior angles of a geodesic triangle is equal to π plus the total curvature enclosed by the triangle.
 One way in which elliptic geometry differs from Euclidean geometry is that the sum of the interior angles of a triangle is greater than 180 degrees.
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 The sum of the interior angles must be 180 degrees, as with all triangles.
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 Each coordinate in the sum of squares is inverse weighted by the sample variance of that coordinate.
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