Then and become partially defined operations on G, and will in fact be defined everywhere ; so we define * to be and to be.

Then define

We wish to maximize total value subject to the constraint that total weight is less than or equal to W. Then for each w ≤ W, define m to be the maximum value that can be attained with total weight less than or equal to w. m then is the solution to the problem.

Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule

For f a real polynomial in x, and for any a in such an algebra define f ( a ) to be the element of the algebra resulting from the obvious substitution of a into f. Then for any two such polynomials f and g, we have that ( fg ) ( a )

For f ∈ R, define D < sub > f </ sub > to be the set of ideals of R not containing f. Then each D < sub > f </ sub > is an open subset of Spec ( R ), and is a basis for the Zariski topology.

Then one might define the glass as being 0. 7 empty and 0. 3 full.

# Then, the juxtaposition of a writer's works leads the critic to define symbolical themes.

Then it may be buttered and eaten as a toast ( its most common use as a breakfast dish ), or it may be filled or topped with either doces ( sweet ) or salgados ( savory ) ingredients, which define the kind of meal the tapioca is used for: breakfast, afternoon tea or dessert.

Then, the angle θ around this axis in the X-Y plane can be used to define the trajectory as,

In particular, let p define the coordinates of points in a reference frame M coincident with a fixed frame F. Then, when the origin of M is displaced by the translation vector d relative to the origin of F and rotated by the angle φ relative to the x-axis of F, the new coordinates in F of points in M are given by

Then it is not possible to define, in ZFC, the set of all ( Gödel numbers of ) formulas that define real numbers.

Then define a process Y, such that each state of Y represents a time-interval of states of X.

Then is an intuitionistic Kripke frame, and we define a model in this frame by

For such a subgroup G, define Fix ( G ) to be the field consisting of all elements of L that are held fixed by all elements of G. Then the maps E Gal ( L / E ) and G Fix ( G ) form an antitone Galois connection.

Then we could define, which grows much faster than any for finite k ( here ω is the first infinite ordinal number, representing the limit of all finite numbers k ).

Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ ( x ) ( or more explicitly σ < sub > B </ sub >( x )) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B.

If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d *, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold.

Then we recursively define the truth of a formula in a model:

Then define f < sub > c </ sub > by

For every r ≥ 0, let n ( r, f ) be the number of poles, counting multiplicity, of the meromorphic function f in the disc | z | ≤ r. Then define the Nevanlinna counting function by

) Suppose additionally that a < sub > n </ sub > is the number of elements of A with weight n. Then we define the formal Dirichlet generating series for A with respect to w as follows:

Then define

Then our choice function can choose the least element of every set under our unusual ordering.

Then an arithmetic function a is

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Then is a group whose identity element is The group inverse of an arbitrary group element is the function inverse

Then, using the periodic Bernoulli function P < sub > n </ sub > defined above and repeating the argument on the interval, one can obtain an expression of ƒ ( 1 ).

Then a mathematical procedure is used to find a filter transfer function which can be realized ( within some constraints ) and which approximates the desired response to within some criterion.

Then an " estimator " is a function that maps the sample space to a set of sample estimates.

Let R be a domain and f a Euclidean function on R. Then:

For instance, suppose that each input is an integer z in the range 0 to N − 1, and the output must be an integer h in the range 0 to n − 1, where N is much larger than n. Then the hash function could be h

Then the probability density function f *( x ) of the size biased population is

Then ƒ is invertible if there exists a function g with domain Y and range X, with the property:

Then for a specific value x of X, the function L ( θ | x )

Then,, and the constant function turns into the identity by substitution.

Then, in 1905, to explain the photoelectric effect ( 1839 ), i. e., that shining light on certain materials can function to eject electrons from the material, Albert Einstein postulated, based on Planck ’ s quantum hypothesis, that light itself consists of individual quantum particles, which later came to be called photons ( 1926 ).

The algorithm for deciding this is conceptually simple: it constructs ( the description of ) a new program t taking an argument n which ( 1 ) first executes program a on input i ( both a and i being hard-coded into the definition of t ), and ( 2 ) then returns the square of n. If a ( i ) runs forever, then t will never get to step ( 2 ), regardless of n. Then clearly, t is a function for computing squares if and only if step ( 1 ) terminates.

Then the output is related to the input by the transfer function as

Then because the construction of f was uniform, this is a recursive function of two variables.

Then, its electrical field traces out a helical pattern as a function of time.

Then the function

Then the function

Then, recalling that the likelihood function is defined up to a multiplicative constant, it is just as valid to say that the likelihood function is approximately

Then, on considering the lengths of the intervals to decrease to zero, the likelihood function for an observation from the discrete component is

Then has probability density function.