Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

Given the derivatives, wave – particle duality can be assessed as follows.

Given such estimates of first-order derivatives, the gradient magnitude is then computed as:

Given this equivalence, it is valid to assume “ risk neutrality ” when pricing derivatives.

Given a field tensor, a scalar called the Lagrangian density can be constructed from and its derivatives.

Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.

Given a parameterized family of probability density functions ( or probability mass functions in the case of discrete distributions )

Given its importance, < tt > dvips </ tt > is a standard part of most TeX distributions, such as teTeX, and TeX Live.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

Idealists are skeptics about the physical world, maintaining either: 1 ) that nothing exists outside the mind, or 2 ) that we would have no access to a mind-independent reality even if it may exist ; the latter case often takes the form of a denial of the idea that we can have unconceptualised experiences ( see Myth of the Given ).

Given the complexity of the calculations involved and the convoluted structure that a convertible bond can have, an arbitrageur often relies on sophisticated quantitative models in order to identify bonds that are trading cheap versus their theoretical value.

Given that Bozizé accuses Sudan of supporting the UFDR rebels who are actively fighting the Central African Government, the relation between the two countries has remained good.

Given this popular understanding of the term, it can also be used illegitimately and inappropriately, as a means to dismiss what are in fact substantial and well-evidenced accusations.

Given that many journeys are for relatively short distances, there is considerable scope to replace car use with walking or cycling, though in many settings this may require some infrastructure modification, particularly to attract the less experienced and confident.

Given the rugged terrain of Ngazidja and Nzwani, and the dedication of extensive tracts to agriculture on all three islands, population pressures on Comoros are becoming increasingly critical.

Given this basic definition, further criteria are needed to establish a dividing line between postlexical clitics and morphological affixes, since both are characterized by a lack of prosodic autonomy.

Given that a father's age is 1 less than twice that of his son, and that the digits AB making up the father's age are reversed in the son's age ( i. e. BA ), leads to the equation 19B-8A

Given its different forms, there are various ways of representing uncertainty and modelling economic agents ' responses to it.

* Given a partition of A, G is a transformation group under composition, whose orbits are the cells of the partition ‡;

* Given a transformation group G over A, there exists an equivalence relation ~ over A, whose equivalence classes are the orbits of G.

Raymond also coined an aphorism he dubbed " Linus ' Law ", inspired by Linus Torvalds: " Given enough eyeballs, all bugs are shallow ", that first appeared in The Cathedral and the Bazaar.

Given the great historical link between Persian and Jewish history, modern day Persian Jews are called " Esther's Children ".

Given that the vast majority of all emeralds are treated as described above, and the fact that two stones that appear visually similar may actually be quite far apart in treatment level and therefore in value, a consumer considering a purchase of an expensive emerald is well advised to insist upon a treatment report from a reputable gemological laboratory.

Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by considering small perturbations of exact solutions.

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given that genes are universal to living organisms, genetics can be applied to the study of all living systems, from viruses and bacteria, through plants and domestic animals, to humans ( as in medical genetics ).

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given that the coiled texture is the natural state of some African Americans ' hair, or perceived as being more " African ", this simple style is now often seen as a sign of self-acceptance and an affirmation that the beauty norms of the ( eurocentric ) dominant culture are not absolute.

Given these two assumptions, the coordinates of the same event ( a point in space and time ) described in two inertial reference frames are related by a Galilean transformation.

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 A − B and the area of B − A ).

So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.

Given a Boolean algebra B, we turn B into a Boolean ring by using the symmetric difference as addition and the meet operation as multiplication.

: Given a matrix-valued function A ( x ) which is symmetric and positive definite for every x, having components a < sup > ij </ sup >, the operator

Given a vector space V, the polynomials on this space are S ( V *), the symmetric algebra of the dual space: a polynomial on a space evaluates vectors on the space, via the pairing.

Given this complexity, early games using the system tended to be quite simple in layout, using the TIA to create simple symmetric playfields with players on top.

Given a symmetric matrix we visualize the matrix as the adjacency matrix of a graph.

Given a symplectic form ω and a linear complex structure J, one may define an associated symmetric bilinear form g < sub > J </ sub > on V < sub > J </ sub >

Given a finite symmetric group S < sub > n </ sub > and specific Young tableau λ corresponding to a numbered partition of n, define two permutation subgroups and of S < sub > n </ sub > as follows:

Given this kind of business climate, what competitive marketing problems will your company face in the next 10 yrs.??

Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education.

Given our formula φ, we group strings of quantifiers of one kind together in blocks:

Given the structure of the experiment, it may or may not be possible to actually perform it, and, in the case that it is possible for it to be performed, there need be no intention of any kind to actually perform the experiment in question.

Given that he refers to clothes as food and considers them a kind of delicacy, it may be that Decapodians themselves consume clothes.

Given the massing of the Muslim armies at Yarmouk, however, Vahan had little choice but to respond in kind.

Given a Riemannian manifold with metric tensor, we can compute the Ricci tensor, which collects averages of sectional curvatures into a kind of " trace " of the Riemann curvature tensor.

Given a set of data and the desire to produce some kind of “ model ” of those data ( model, in this case, meaning a function fitted through the data ), there are a variety of functions that can be chosen for the fit.

Given that their orders were to capture the ship intact at all costs, that kind of opposition proved most frustrating.

Given two integer parameters P and Q, the Lucas sequences of the first kind U < sub > n </ sub >( P, Q ) and of the second kind V < sub > n </ sub >( P, Q ) are defined by the recurrence relations:

Given the controversy surrounding her grave site, it is possible that she may have been initially interred at Kaithal and then later at Delhi but this remains purely a speculation with no evidence of any kind.

Given the nature of most Assassins ' Guild traditions, this could be seen as a kind of impromptu " pass or fail " examination ( see bitter almonds for the reason why ).

Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic ; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf Ω < sup > 1 </ sup > of Kähler differentials.

Given a graph, one may write for the largest book ( of the kind being considered ) contained within.