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function and G
Among his staff was Isham G. Harris, the Governor of Tennessee, who had ceased to make any real effort to function as governor after learning that Abraham Lincoln had appointed Andrew Johnson as military governor of Tennessee.
If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.
The latter notation corresponds to viewing R as the characteristic function on " X " x " Y " for the set of pairs of G.
Second, let G = G ( s ) be the external force per unit length acting on a small segment of a chain as a function of s. The forces acting on the segment of the chain between s and s + Δs are the force of tension T ( s + Δs ) at one end of the segment, the nearly opposite force − T ( s ) at the other end, and the external force acting on the segment which is
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.
The identity function, I ( x )= x, is an obvious element of G ;
Hence G is also a transformation group ( and an automorphism group ) because function composition preserves the partitioning of A.
According to Manolo Sanlúcar E is here the tonic, F has the harmonic function of dominant while Am and G assume the functions of subdominant and mediant respectively.
In mathematics, given two groups ( G, *) and ( H, ·), a group homomorphism from ( G, *) to ( H, ·) is a function h: G → H such that for all u and v in G it holds that
An equivalent definition of group homomorphism is: The function h: G → H is a group homomorphism if whenever we have.
Spelled out, this means that a group isomorphism is a bijective function such that for all < var > u </ var > and < var > v </ var > in < var > G </ var > it holds that
A groupoid is a set G with a unary operation and a partial function Here * is not a binary operation because it is not necessarily defined for all possible pairs of G-elements.
* For each triple of objects x, y, and z, there exists the function G ( x, y ) G ( y, z ) → G ( x, z ).
* There exists the function G ( x, y ) → G ( y, x ).

function and such
We first define a function b{t} as follows: given the set of squares such that each has three corners on C and vertex at t, b{t} is the corresponding set of positive parametric differences between T and the backward corner points.
Moreover, whereas in Interstate Commerce Commission parlance `` variable cost '' means a cost deemed to vary in direct proportion to changes in rate of output, in the type of analysis now under review `` variable cost '' has been used more broadly, so as to cover costs which, while a function of some one variable ( such as output of energy, or number of customers ), are not necessarily a linear function.
This is specified by a distribution function Af such that the probability that Af lies in some region D of the stage space is Af.
Discussion of minority housing necessarily involves such basic issues as the intensity of one's democratic conviction and religious belief concerning equality of opportunity, the function and limitations of government in the securing of such equality, and the spotlight that world opinion plays upon local incidents of racial agitation and strife.
Her brain and intelligence controlled every function from navigation to such loading as a scout ship of her class needed.
A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:
: For any set A there is a function f such that for any non-empty subset B of A, f ( B ) lies in B.
Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.
* There exists a model of ZF ¬ C in which there is a function f from the real numbers to the real numbers such that f is not continuous at a, but f is sequentially continuous at a, i. e., for any sequence
No neuron ever has more than one axon ; however in invertebrates such as insects or leeches the axon sometimes consists of several regions that function more or less independently of each other.
It can be proved that such a function exists.
It can be proved that such an area function actually exists.
Some scholars believe that the apologetic view of Luke ’ s work is overemphasized and that it should not be regarded as a “ major aim of the Lucan writings .” While Munck believes that purpose of Luke ’ s work is not that clear-cut and sympathizes with other claims, he believes that Luke ’ s work can function as an apology only in the sense that it “ presents a defense of Christianity and Paul ” and may serve to “ clarify the position of Christianity within Jewry and within the Roman Empire .” Pervo disagrees that Luke ’ s work is an apology and even that it could possibly be addressed to Rome because he believes that “ Luke and Acts speak to insiders, believers in Jesus .” Freedman believes that Luke is writing an apology but that his goal is “ not to defend the Christian movement as such but to defend God ’ s ways in history .”
* Despite its heritage, Applesoft lacked commands common to most other Microsoft BASIC interpreters, such as INSTR ( which searched for a substring in a given string ; this had to be done manually with loops and the MID $ function ), PRINT USING ( which formatted numbers with commas and currency signs according to a format string ), and INKEY $ ( which checked for a keypress without stopping the program as Applesoft's GET command — analogous to the INPUT $ function — did, although a PEEK location did provide this functionality ).
Limited possibilities have been explored by artists as an outlet to artistic expression, and has been cultivated to become a positive style and tradition, emphasizing the decorative function of art, or its religious functions via non-representational forms such as Geometric patterns, floral patterns, and arabesques.
These antidepressants typically have fewer adverse effects than the tricyclics or the MAOIs, although such effects as drowsiness, dry mouth, nervousness, anxiety, insomnia, decreased appetite, long-term weight gain and decreased ability to function sexually may occur.
The adenovirus E1B-55K protein and the hepatitis B virus HBx protein are examples of viral proteins that can perform such a function.
Its function is to control the expulsion of feces, unwanted semi-solid matter produced during digestion, which, depending on the type of animal, may be one or more of: matter which the animal cannot digest, such as bones ; food material after all the nutrients have been extracted, for example cellulose or lignin ; ingested matter which would be toxic if it remained in the digestive tract ; and dead or excess gut bacteria and other endosymbionts.

function and Af
If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.
The concept of the strain energy as a Gibbs function difference Af and exerting a force normal to the shearing face is compatible with the information obtained from optical birefringence studies of fluids undergoing shear.
A proton magnetic resonance study of polycrystalline Af as a function of magnetic field and temperature is presented.
Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.
When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.
We consider now the graph of the function f{t} on Af.
Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.
The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.
In some neighborhood of an isolated tangent point in the f-plane, say Af, the function Af is either double-valued or has no values defined, except at the tangent point itself, where it is single-valued.
In the second category the function Af has no values defined in a neighborhood Af.
For it is clear that the total number of ordinary intersections of C and Af must be even ( otherwise, starting in the interior of C, Af could not finally return to the interior ), and the center of rotation at T is the argument of the function, not a value.

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