If a poorly organized low pressure weather system is present, virga or weak intermittent precipitation may fall from those stratocumuliform clouds that form mostly in the low and lower-middle height ranges of the troposphere.

If the error ranges from-1 to + 1, with the analog-to-digital converter used having a resolution of 0. 25, then the input variable's fuzzy set ( which, in this case, also applies to the output variable ) can be described very simply as a table, with the error / delta / output values in the top row and the truth values for each membership function arranged in rows beneath:

If there had been no Government program, if the old order had obtained in 1933 and 1934, that drought on the cattle ranges of America and in the corn belt would have resulted in the marketing of thin cattle, immature hogs and the death of these animals on the range and on the farm, and if the old order had been in effect those years, we would have had a vastly greater shortage than we face today.

If is a formula with a free set variable X and free number variables then the set is the intersection of the sets of the form as n ranges over the set of natural numbers.

If the compound is more concentrated more light will be absorbed by the sample ; within small ranges, the Beer-Lambert law holds and the absorbance between samples vary with concentration linearly.

If instead the ranges are, and then there is no interval consistent with all these values but is consistent with the largest number of sources — namely, two of them.

If the Cold War had continued, then many of these systems could have seen deployment: the United States got as far as developing working railguns, and a laser that could destroy missiles at range, though the power requirements of both were phenomenal, and the ranges and firing cycles utterly impractical.

If colored dither is used at these intermediate processing stages then the frequency content can " bleed " into other, more noticeable frequency ranges and become distractingly audible.

If the operator is using acoustic ranges to position items in unknown locations they will need to use more than the single range example shown above.

If the object semi-major axis is outside these narrow ranges, the orbit becomes chaotic, with widely changing orbital elements.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

If f: A < sub > 1 </ sub > → A < sub > 2 </ sub > and g: B < sub > 1 </ sub > → B < sub > 2 </ sub > are morphisms in Ab, then the group homomorphism Hom ( f, g ): Hom ( A < sub > 2 </ sub >, B < sub > 1 </ sub >) → Hom ( A < sub > 1 </ sub >, B < sub > 2 </ sub >) is given by φ g o φ o f. See Hom functor.

If f: X < sub > 1 </ sub > → X < sub > 2 </ sub > and g: Y < sub > 1 </ sub > → Y < sub > 2 </ sub > are morphisms in C, then the group homomorphism Hom ( f, g ): Hom ( X < sub > 2 </ sub >, Y < sub > 1 </ sub >) → Hom ( X < sub > 1 </ sub >, Y < sub > 2 </ sub >) is given by φ g o φ o f.

If I is ordered ( not simply partially ordered ) and countable, and C is the category Ab of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms f < sub > ij </ sub > that ensures the exactness of.

Similar statements apply to the dual situation of terminal morphisms from U. If such morphisms exist for every X in C one obtains a functor V: C → D which is right-adjoint to U ( so U is left-adjoint to V ).

Any collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

If J is a category with two objects and two parallel morphisms from object 1 to object 2 then a diagram of type J is a pair of parallel morphisms in C. The limit L of such a diagram is called an equalizer of those morphisms.

If the categories C and D have all limits of type J then lim is a functor and the morphisms τ < sub > F </ sub > form the components of a natural transformation

If C is any category and I is a small category, we can form the functor category C < sup > I </ sup > having as objects all functors from I to C and as morphisms the natural transformations between those functors.

If ( E, M ) is a factorization system, then the morphisms in M may be regarded as the embeddings, especially when the category is well powered with respect to M. Concrete theories often have a factorization system in which M consists of the embeddings in the previous sense.

If C is a complete category, then the functors with left adjoints can be characterized by the adjoint functor theorem of Peter J. Freyd: G has a left adjoint if and only if it is continuous and a certain smallness condition is satisfied: for every object Y of D there exists a family of morphisms

* If is a topological space ( viewed as a category as above ) and is some small category, we can form the category of all contravariant functors from to, using natural transformations as morphisms.

If we do this, however, we need to presuppose that the category C has zero morphisms, or equivalently that C is enriched over the category of pointed sets.

If we denote the-fold product of with itself by, then morphisms from to are m-by-n matrices with entries from the ring.

If it does, however, it is unique in a strong sense: given another direct limit X ′ there exists a unique isomorphism X ′ → X commuting with the canonical morphisms.

* If C is a small category, then the functor category Set < sup > C </ sup > consisting of all covariant functors from C into the category of sets, with natural transformations as morphisms, is cartesian closed.

If G is any group, then the set Ch ( G ) of these morphisms forms an abelian group under pointwise multiplication.

If C has a zero object 0, given two objects X and Y in C, there are canonical morphisms f: 0 → X and g: Y → 0.

If C is a category with zero morphisms, then the collection of 0 < sub > XY </ sub > is unique.

If one assumes that the average flux did not change between measurements, a mass-distribution curve is obtained which relates the flux of particles larger than a given radius to the inverse 7/2 power of the radius.

If and are invertible then is also invertible with inverse.

If the condition number is close to one, the matrix is well conditioned which means its inverse can be computed with good accuracy.

If production of one good increases along the curve, production of the other good decreases, an inverse relationship.

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa.

If we think of composition as a kind of multiplication of functions, this identity says that the inverse of a function is analogous to a multiplicative inverse.

If an inverse function exists for a given function ƒ, it is unique: it must be the inverse relation.

If it does, however, it is unique in a strong sense: given any other inverse limit X ′ there exists a unique isomorphism X ′ → X commuting with the projection maps.

If is another measurable space then a function is called measurable if for every Y-measurable set, the inverse image is X-measurable i. e..

If the orbiting periods were in this relation, the mean motions ( inverse of periods, often expressed in degrees per day ) would satisfy the following

* If a has a multiplicative inverse in R, then f ( a ) has a multiplicative inverse in S and we have f ( a < sup >− 1 </ sup >) = ( f ( a ))< sup >− 1 </ sup >.

If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index.

which is natural in the variables N and F. The counit of this adjunction is simply the universal cone from lim F to F. If the index category J is connected ( and nonempty ) then the unit of the adjunction is an isomorphism so that lim is a left inverse of Δ.

The unit of this adjunction is the universal cocone from F to colim F. If J is connected ( and nonempty ) then the counit is an isomorphism, so that colim is a left inverse of Δ.

If is an identity element of ( i. e., S is a unital magma ) and, then is called a left inverse of and is called a right inverse of.

If an element is both a left inverse and a right inverse of, then is called a two-sided inverse, or simply an inverse, of.