Given a topological space X, a subset A of X is meagre if it can be expressed as the union of countably many nowhere dense subsets of X.

Given the countably infinite number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the precision of approximate equality might be the most obvious way to assess mathematical coincidences ; but there is no standard, and the strong law of small numbers is the sort of thing one has to appeal to with no formal opposing mathematical guidance.

Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms.

Given, for countably many special values of, called characteristic values, the Mathieu equation admits solutions which are periodic with period.

Given objects and in an additive category, we can represent morphisms as-by-matrices

Given Planck's quantization rule for the harmonic oscillator, either condition determines the correct classical quantity to quantize in a general system up to an additive constant.

Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

noted, " Given that the bandwidth for conducting crawls is neither infinite nor free, it is becoming essential to crawl the Web in not only a scalable, but efficient way, if some reasonable measure of quality or freshness is to be maintained.

Given a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure.

Given the potential for sampling biases of individual survey reports, researchers and investors try sometimes to average the values of different index reports into a single aggregated measure of consumer confidence.

When the measure was sent in May to the parliament they said " Given the damage it produces on those rules which allow the life in community, ensure the dignity of the person and equality between sexes, this practice, even if it is voluntary, cannot be tolerated in any public place ".

Given that deviance is a measure of the difference between a given model and the saturated model, smaller values indicate better fit.

Given a finite observation set S, one can simply select the measure for all.

Given a subset A of G, the measure can be thought of as answering the question: what is the probability that a random element of G is in A?

Having a measure on G allows us to define integration of bounded functions on G. Given a bounded function, the integral

Given a left-invariant measure, the function is a right-invariant measure.

Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of and its analytic capacity, it might be

Given a σ-finite measure space ( S, Σ, μ ), consider the Banach space L < sup > p </ sup >( μ ).

Given a sequence ( f < sub > n </ sub >) of M-valued measurable functions on some measure space ( X, Σ, μ ), and a measurable subset A of finite μ-measure such that ( f < sub > n </ sub >) converges μ-almost everywhere on A to a limit function f, the following result holds: for every ε > 0, there exists a measurable subset B of A such that μ ( B ) < ε, and ( f < sub > n </ sub >) converges to f uniformly on the relative complement A

Given two objects, A and B, each with n binary attributes, the Jaccard coefficient is a useful measure of the overlap that A and B share with their attributes.

The most recent measure of social dominance orientation ( see SDO-6 above ) focuses on the “ general desire for unequal relations among social groups, regardless of whether this means ingroup domination or ingroup subordination ” ( p. 312 ) Given these changes, Rubin and Hewstone believe that evidence for social dominance theory should be considered “ as supporting three separate SDO hypotheses, rather than one single theory ” ( p. 22 ).

Given a fixed total concentration of one or more species over the measurement time, the scattering signal is a direct measure of the weight-averaged molar mass of the solution, which will vary as complexes form or dissociate.

Given a characteristic functional on a nuclear space A, the Bochner – Minlos theorem ( after Salomon Bochner and Robert Adol ' fovich Minlos ) guarantees the existence and uniqueness of the corresponding probability measure on the dual space, given by

Given a Blum complexity measure and a total computable function with two parameters, then there exists a total computable predicate ( a boolean valued computable function ) so that for every program for, there exists a program for so that for almost all

Given μ, define a function by

# Given A and B, μ ( A ∪ B ) + μ ( A ∩ B )

* 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.

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 this, the bidding is said to start at the one-level when contracting for a total of seven tricks, at the two-level for eight tricks and so on to the seven-level to contract to take all thirteen tricks.

Given the federal nature of the holiday, celebrating Canada Day can be a cause of friction in the province of Quebec, where the holiday is overshadowed by Quebec's National Holiday, on June 24.

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 an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

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.

* Frederick Douglass lecture on Haiti – Given at the World's Fair in Chicago, January 1893.

Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold ( usually defined by giving the metric in specific coordinates ), and specific matter fields defined on that manifold.

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 an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given these, a new term Anthropocene, is specifically proposed and used informally for the latest part of this epoch since approximately synchronous lithospheric evidence, or more recently atmospheric evidence, of human impacts have been found on the Earth and its ecosystems ; these impacts may be considered of global significance for future evolution of living species.

Given that this etymology does not appear in the usual scholarly works on etymology, this claim is usually dismissed as a false etymology.

Given the dual status of science as objective knowledge and as a human construct, good historiography of science draws on the historical methods of both intellectual history and social history.

Given a binary operation ★ on a set S, an element x is said to be idempotent ( with respect to ★) if

Given its heavy dependence on imported energy, Japan has aimed to diversify its sources.

Along with a hand-picked escort led by Given Campbell, Davis and his wife were captured on May 10, 1865, at Irwinville in Irwin County, Georgia.

Given the repeated failures and frustrations of workers ' revolutions and movements, Marx also sought to understand capitalism, and spent a great deal of time in the reading room of the British Museum studying and reflecting on the works of political economists and on economic data.

Given that estimation is undertaken on the basis of a least squares analysis, estimates of the unknown parameters β < sub > j </ sub > are determined by minimising a sum of squares function

Given that spinning a Dyson Sphere would result in the atmosphere pooling around the equator, the Ringworld removes all the extraneous parts of the structure, leaving a spinning band landscaped on the sun-facing side, with the atmosphere and inhabitants kept in place through centrifugal force and 1000 mile high perimeter walls ( rim walls ).