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Page "Laurent polynomial" ¶ 10
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More and rigorously
More rigorously, the divergence of a vector field F at a point p is defined as the limit of the net flow of F across the smooth boundary of a three dimensional region V divided by the volume of V as V shrinks to p. Formally,
More rigorously, if a surface S is a union of finitely many pieces S < sub > 1 </ sub >, …, S < sub > r </ sub > which do not overlap except at their boundaries then
More rigorously, an interpretation of quantum mechanics satisfies CFD if it includes in the statistical population of measurement results, those measurements which are counterfactual by virtue of their being excluded by the quantum mechanical prohibition on simultaneous measurement of certain pairs of properties.
More rigorously, metonymy and synecdoche may be considered as sub-species of metaphor, intending metaphor as a type of conceptual substitution ( as Quintilian does in Institutio oratoria Book VIII ).
More rigorously, since the phases are in thermodynamic equilibrium with each other, the chemical potentials of the phases must be equal.
More rigorously, the mathematical operation involved in classical limits is a group contraction, approximating physical systems where the relevant action is much larger than Planck's constant ħ, so the " deformation parameter " ħ / S can be effectively taken to be zero.
More rigorously, any summation method that possesses these properties and which assigns a finite value to the geometric series must assign this value.
More rigorously, they are said to be isomorphic or “ structure preserving ”.
More serious consultation with a range of internationally adopted people from various professional and community-work based backgrounds needs to be included before the field of adoption study is more truly representative and rigorously informed.
More rigorously the effective temperature corresponds to the temperature at the radius that is defined by the Rosseland optical depth.
The minimal separators also form an algebraic structure: For two fixed vertices a and b of a given graph G, an ( a, b )- separator S can be regarded as a predecessor of another ( a, b )- separator T, if every path from a to b meets S before it meets T. More rigorously, the predecessor relation is defined as follows: Let S and T be two ( a, b )- separators in ' G '.
More rigorously
More rigorously, Pickover's " biomorphs " encompass the class of organismic morphologies created by small changes to traditional convergence tests in the field of " Julia set " theory.

More and is
More profound and more disturbing, however, is the moral isolation of Raymond Chandler's Philip Marlowe.
More important is the simple human point that all men suffer, and that it is a kind of anthropological-religious pride on the part of the Jew to believe that his suffering is more poignant than mine or anyone else's.
More than that, Sam Rayburn is the very living symbol of an iron-clad integrity so powerful in his nature and so constantly demonstrated that he can count some of his best friends in the opposition.
More than 50% of all lumber is unitized ; ;
More unitized lumber is being shipped on flat cars, and NLRDA studies show that flat cars loaded with the new Type 6-B floating-load method can be unloaded for as little as $.054/mbf.
More than anything, it is the therapist's intuitive sensing of these latent meanings in the stereotype which helps these meanings to become revealed, something like a spread-out deck of cards, on sporadic occasions over the passage of the patient's and his months of work together.
But to return to the main line of our inquiry, it is doubtful that Utopia is still widely read because More was medieval or even because he was a martyr -- indeed, it is likely that these days many who read Utopia with interest do not even know that its author was a martyr.
Utopia is still widely read because in a sense More stood on the margin of modernity.
we are concerned with Utopian communism -- that is, simply communism as it appears in the imaginary commonwealth of Utopia, as More conceived it.
More rarely, the hymen is so sturdy that it does not yield to penetration.
More typical is the case of a suburban Long Island housewife described by a marriage counselor.
More often, though, he is so accustomed to submitting to authority on the job without argument that he lives by the same rule at home.
More than 10 companies maintain facilities in Dallas and one large manufacturer is located to the north at Sherman.
More than 700 employees make gin machinery that's sold anywhere cotton is grown.
More than 200 are expected at the autumn event which is matched in the spring.
More than 300 teenagers last Sunday proved there is and as many more are expected to prove it again for Jim Kern and his wife Lynn from 4 to 8 p.m. Sunday at First Presbyterian Church.
More than a beautiful visualization of the illustrious adventures and escapades of the tragi-comic knight-errant and his squire, Sancho Panza, in seventeenth-century Spain, this inevitably abbreviated rendering of the classic satire on chivalry is an affectingly warm and human exposition of character.
More recently, Alasdair MacIntyre has attempted to reform what he calls the Aristotelian tradition in a way that is anti-elitist and capable of disputing the claims of both liberals and Nietzscheans.
More rarely, a script may have separate letters for tones, as is the case for Hmong and Zhuang.
More precisely, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function.
More explicitly, it is stating that for every indexed family of nonempty sets there exists an indexed family of elements such that for every.

More and polynomial
More precisely, these proofs have to be verifiable in polynomial time by a deterministic Turing machine.
More formally, # P is the class of function problems of the form " compute ƒ ( x )," where ƒ is the number of accepting paths of a nondeterministic Turing machine running in polynomial time.
More generally, if is the characteristic polynomial of a matrix A, then
More precisely, replacing by X, and likewise for other variables ( formally this is done by a Fourier transform ), converts a constant-coefficient PDE into a polynomial of the same degree, with the top degree ( a homogeneous polynomial, here a quadratic form ) being most significant for the classification.
More specifically, let be the set of all polynomials with integer coefficients ( such a polynomial is also referred to as a polynomial over ), with addition and multiplication defined to be natural polynomial addition and multiplication.
* The Laurent polynomial ring RX < sup >− 1 </ sup > is isomorphic to the group ring of the group Z of integers over R. More generally, the Laurent polynomial ring in n variables is isomorphic to the group ring of the free abelian group of rank n. It follows that the Laurent polynomial ring can be endowed with a structure of a commutative, cocommutative Hopf algebra.
More specifically, one includes additional first-order properties, such as the existence of a root for every odd-degree polynomial.
More precisely, ζ ( X, s ) can be written as a finite alternating product < dl >< dd ></ dl > where each P < sub > i </ sub >( T ) is an integral polynomial.
More formally, a language L belongs to UP if there exists a two input polynomial time algorithm A and a constant c such that
More specifically, the running time for testing whether H is a minor of G in this case is O ( n < sup > 3 </ sup >), where n is the number of vertices in G and the big O notation hides a constant that depends superexponentially on H. Thus, by applying the polynomial time algorithm for testing whether a given graph contains any of the forbidden minors, it is possible to recognize the members of any minor-closed family in polynomial time.
Another example: P being as above, a resolvent R for a group G is a polynomial whose coefficients are polynomials in the coefficients of p, which provides some information on the Galois group of P. More precisely, if R is separable and has a rational root then the Galois group of P is contained in G. For example, if D is the discriminant of P then is a resolvent for the alternating group.
More generally, if an irreducible ( non-zero ) polynomial f in F does not have distinct roots, not only must the characteristic of F be a ( non-zero ) prime number p, but also f ( X )= g ( X < sup > p </ sup >) for some irreducible polynomial g in F. By repeated application of this property, it follows that in fact, for a non-negative integer n and some separable irreducible polynomial g in F ( where F is assumed to have prime characteristic p ).
More generally, the Chevalley – Shephard – Todd theorem characterizes finite groups whose algebra of invariants is a polynomial ring.
More precisely, only totally ramified primes have a chance of being Eisenstein primes for the polynomial.
More recently, however, it was demonstrated that, using standard polynomial approximations to the normal cdf, the normal-ogive model is no more computationally demanding than logistic models.
More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime p is the inverse of the Hecke polynomial, a quadratic polynomial in p < sup >− s </ sup >.

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