The determinant provides important information when the matrix is that of the coefficients of a system of linear equations, or when it corresponds to a linear transformation of a vector space: in the first case the system has a unique solution if and only if the determinant is nonzero, while in the second case that same condition means that the transformation has an inverse operation.

Although new trade theory can explain the growing trend of trade volumes of intermediate goods, Krugman's explanation depends too much on the strict assumption that all firms are symmetrical, meaning that they all have the same production coefficients.

Two terms with the same variables raised to the same powers are called " similar terms " or " like terms ", and they can be combined, using the distributive law, into a single term whose coefficient is the sum of the coefficients of the terms that were combined.

where n is a natural number, the coefficients are elements of R, and X is a formal symbol, whose powers X < sup > i </ sup > are just placeholders for the corresponding coefficients a < sub > i </ sub >, so that the given formal expression is just a way to encode the sequence, where there is an n such that a < sub > i </ sub > = 0 for all i > n. Two polynomials sharing the same value of n are considered equal if and only if the sequences of their coefficients are equal ; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero.

and hence Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully, i. e., involving the same Fourier coefficients multiplying these umbral basis exponentials.

Recall that all the terms in a diagonal going from the upper-left to the lower-right correspond to the same power of x, and that the a-terms are the coefficients of the polynomial ( x + 1 )< sup > n </ sup >, and we are determining the coefficients of ( x + 1 )< sup > n + 1 </ sup >.

Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids.

which sum is effectively finite because of the assumed vanishing of coefficients at sufficiently negative indices, and which sum zero for sufficiently negative k for the same reason.

If R is a given commutative ring, then the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring, denoted R. The same holds true for several variables.

The fundamental law of quantum mechanics is that the evolution is linear, meaning that if state A turns into A ′ and B turns into B ′ after 10 seconds, then after 10 seconds the superposition turns into a mixture of A ′ and B ′ with the same coefficients as A and B.

Very often, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field.

In the same way if y always decreases when x increases, the rank correlation coefficients will be − 1, while the Pearson product-moment correlation coefficient may or may not be close to − 1, depending on how close the points are to a straight line.

If the measures of correlation used are product-moment coefficients, the correlation matrix is the same as the covariance matrix of the standardized random variables X < sub > i </ sub > / σ ( X < sub > i </ sub >) for i = 1, ..., n. This applies to both the matrix of population correlations ( in which case " σ " is the population standard deviation ), and to the matrix of sample correlations ( in which case " σ " denotes the sample standard deviation ).

Since this tends to be a rather large area compared to the projected frontal area, the resulting drag coefficients tend to be low: much lower than for a car with the same drag and frontal area, and at the same speed.

Two objects having the same reference area moving at the same speed through a fluid will experience a drag force proportional to their respective drag coefficients.

The possibility of fabrication was first brought to the attention of the scientific community when Kamin noticed that Burt's correlation coefficients of monozygotic and dizygotic twins ' IQ scores were the same to three decimal places, across articles – even when new data were twice added to the sample of twins.

The method of inspection can be outlined as putting a coefficient of 1 in front of the most complex chemical formula and putting the other coefficients before everything else such that both sides of the arrows have the same number of each atom.

During finishing this breed requires high-energy ( concentrated ) feeds, and will not yield the same results if put on a high-fiber diet.

Sometimes two quite different constructions yield " the same " result ; this is expressed by a natural isomorphism between the two functors.

Specifically, if a judge consistently gives low scores for all divers, or consistently gives high scores for the same divers, the judging will yield fair relative results and will cause divers to place in the correct order.

James's tone delighted Elizabeth, who responded: " So trust I that you will not doubt but that your last letters are so acceptably taken as my thanks cannot be lacking for the same, but yield them to you in grateful sort ".

Since unsaturated fats contain fewer carbon-hydrogen bonds than saturated fats with the same number of carbon atoms, unsaturated fats will yield slightly less energy during metabolism than saturated fats with the same number of carbon atoms.

For such data, one must use a hash function that is compatible with the data equivalence criterion being used: that is, any two inputs that are considered equivalent must yield the same hash value.

These improvements yield a rough doubling in ALU performance over the 386 at the same clock rate.

Informally, word formation rules form " new words " ( that is, new lexemes ), while inflection rules yield variant forms of the " same " word ( lexeme ).

Whereas the TTAPS paper described a 3000 MT counterforce attack on ICBM sites, Michael Altfed of the Michigan State University and Political Scientist Stephen Cimbala of Pennsylvania State University maintained that with the trend towards smaller more accurate warheads in the United States and Soviet Nuclear Arsenals the same counterforce attack could be conducted with 1 / 1000th of that yield.

Since synthetic polymerization techniques typically yield a polymer product including a range of molecular weights, the weight is often expressed statistically to describe the distribution of chain lengths present in the same.

The basic reason is that no productive factor with a non-zero marginal product is left unutilized, and the units of each factor are so allocated as to yield the same indirect marginal utility in all uses, a basic efficiency condition ( if this indirect marginal utility were higher in one use than in other ones, a Pareto improvement could be achieved by transferring a small amount of the factor to the use where it yields a higher marginal utility ).

Thus studies of different populations using the same contraceptive will yield different values for the index.

Such a theory of quantum gravity would yield the same experimental results as ordinary quantum mechanics in conditions of weak gravity ( gravitational potentials much less than c < sup > 2 </ sup >) and the same results as Einsteinian general relativity in phenomena at scales much larger than individual molecules ( action much larger than reduced Planck's constant ), but moreover be able to predict the outcome of situations where both quantum effects and strong-field gravity are important ( at the Planck scale, unless large extra dimension conjectures are correct ).

A neutron bomb releases a much greater number of neutrons than a fission bomb of the same explosive yield.

If one compares explosions with different yields, the EMP at a fixed distance from a nuclear weapon would not increase at the same rate as the explosion yield, but at most only as the square root of the yield ( see the illustration to the right ).

Both definitions yield the same result at offset frequencies well removed from the carrier.

Human development theory reflects both distinctions: it sees labour as the yield of individual capital in the same way that neoclassical macro-economics sees financial capital as the yield of the looser idea of human capital.

Fuzzy logic corresponds to " degrees of truth ", while probabilistic logic corresponds to " probability, likelihood "; as these differ, fuzzy logic and probabilistic logic yield different models of the same real-world situations.

# Uniqueness of Group Rank: The method should yield the same result whenever applied to a set of preferences.

In addition, imprisoning people and threatening to do the same to the court when they failed to yield to the will of said authority would seem to be legally problematic.

A balanced-reinforced beam is one in which both the compressive and tensile zones reach yielding at the same imposed load on the beam, and the concrete will crush and the tensile steel will yield at the same time.

In his view, there were three possibilities: ( 1 ) Korean did not belong with the other three genealogically, but had been influenced by an Altaic substratum ; ( 2 ) Korean was related to the other three at the same level they were related to each other ; ( 3 ) Korean had split off from the other three before they underwent a series of characteristic changes.

* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.

A key characteristic of bandwidth is that a band of a given width can carry the same amount of information, regardless of where that band is located in the frequency spectrum.

The cheek teeth show the same characteristic notches that serve in today's carnivores to hold flesh in place to shear apart with cutting ridges.

A notable characteristic of Linebarger's science fiction is that most of his stories are set in the same universe, with a unified chronology.

Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often ( but not always ) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.

Greuze's most characteristic pictures were the rendering in colour of the same sentiments of domestic virtue and the pathos of common life, which Diderot had attempted to represent upon the stage.

Though modern epigrams are usually thought of as very short, Greek literary epigram was not always as short as later examples, and the divide between " epigram " and " elegy " is sometimes indistinct ( they share a characteristic metre, elegiac couplets ); all the same, the origin of the genre in inscription exerted a residual pressure to keep things concise.

Moreover, orchestral and contrapuntal innovation is characteristic of his style: for instance, the strings producing a rapid ascending scale in Monterone's scene in Rigoletto accentuate the drama, and, in the same opera, the chorus humming six closely grouped notes backstage portrays, very effectively, the brief ominous wails of the approaching tempest.

Though of course hotels have always been built in popular destinations, the defining characteristic of a resort hotel is that it exists purely to serve another attraction, the two having the same owners.

The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results.

Not all of these cultural elements characteristic of the Neolithic appeared everywhere in the same order: the earliest farming societies in the Near East did not use pottery.

) Proposed alternative names such as Mitian, formed from the characteristic Nostratic first-and second-person pronouns mi ' I ' and ti ' you ' ( exactly ' thou '), have not attained the same currency.

To form a concept, one mentally isolates a group of concretes ( of distinct perceptual units ), on the basis of observed similarities which distinguish them from all other known concretes ( similarity is ' the relationship between two or more existents which possess the same characteristic ( s ), but in different measure or degree '); then, by a process of omitting the particular measurements of these concretes, one integrates them into a single new mental unit: the concept, which subsumes all concretes of this kind ( a potentially unlimited number ).

' A concept is a mental integration of two or more units possessing the same distinguishing characteristic ( s ), with their particular measurements omitted.

The smallest subfield is isomorphic to the rationals ( as for any other field of characteristic 0 ), and the order on this rational subfield is the same as the order of the rationals themselves.

Note that that while bound orbits around a point mass or around a spherical body with an Newtonian gravitational field are closed ellipses, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects ( as caused, for example, by the slight oblateness of the Earth, or by relativistic effects, changing the gravitational field's behavior with distance ) will cause the orbit's shape to depart from the closed ellipses characteristic of Newtonian two-body motion.

A general characteristic, however, is that the programming language and the database schema use the same type definitions.

The notable characteristic of stimulated emission compared to everyday light sources ( which depend on spontaneous emission ) is that the emitted photons have the same frequency, phase, polarization, and direction of propagation as the incident photons.

The characteristic of any fluid at rest is that the force exerted on any particle of the fluid is the same at all points at the same depth ( or altitude ) within the fluid.

It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns.

The most obvious characteristic of the flatfish is their asymmetry, with both eyes lying on the same side of the head in the adult fish.

( This is the same as the old use of the terms " ring " and " algebra " in measure theory ) ( Also note that, when a Boolean ring has an identity, then a complement operation becomes definable on it, and a key characteristic of the modern definitions of both Boolean algebra and sigma-algebra is that they have complement operations.