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Page "Dynamical system" ¶ 14
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Given and smooth
Given a function f ∈ I < sub > x </ sub > ( a smooth function vanishing at x ) we can form the linear functional df < sub > x </ sub > as above.
Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).
Given a vector v in R < sup > n </ sup > one defines the directional derivative of a smooth map ƒ: R < sup > n </ sup >→ R at a point x by
Given a smooth curve γ on ( M, g ) and a vector field V along γ its derivative is defined by
Given two tensor bundles E → M and F → M, a map A: Γ ( E ) → Γ ( F ) from the space of sections of E to sections of F can be considered itself as a tensor section of if and only if it satisfies A ( fs ,...) = fA ( s ,...) in each argument, where f is a smooth function on M. Thus a tensor is not only a linear map on the vector space of sections, but a C < sup >∞</ sup >( M )- linear map on the module of sections.
Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M,
Given a manifold M representing ( continuous / smooth / with certain boundary conditions / etc.
Given a smooth 4n-dimensional manifold M and a collection of natural numbers
Given a local smooth frame ( e < sub > 1 </ sub >, …, e < sub > k </ sub >) of E over U, any section σ of E can be written as ( Einstein notation assumed ).
Given a cobordism there exists a smooth function such that.
* Given a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing set of sections.
Let E be a rank k vector bundle over a smooth manifold M and let ∇ be a connection on E. Given a piecewise smooth loop γ: → M based at x in M, the connection defines a parallel transport map P < sub > γ </ sub >: E < sub > x </ sub > → E < sub > x </ sub >.
Given a smooth map φ: M → N and a vector field X on M, it is not usually possible to define a pushforward of X by φ as a vector field on N. For example, if the map φ is not surjective, there is no natural way to define such a pushforward outside of the image of φ.

Given and Φ
Given a natural transformation Φ from h < sup > A </ sup > to F, the corresponding element of F ( A ) is.
Given a dynamical system ( T, M, Φ ) with T a group, M a set and Φ the evolution function
Given a real dynamical system ( R, M, Φ ), I ( x ) is an open interval in the real numbers, that is.
Given a vector field E, its scalar potential Φ can be calculated to be
Given a natural transformation Φ: Hom ( A ,–) → F the corresponding element u ∈ F ( A ) is given by

Given and <
* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).
After Christians in Ephesus first wrote to their counterparts recommending Apollos to them, he went to Achaia where Paul names him as an apostle ( 1 Cor 4: 6, 9-13 ) Given that Paul only saw himself as an apostle ' untimely born ' ( 1 Cor 15: 8 ) it is certain that Apollos became an apostle in the regular way ( as a witness to the risen Lord and commissioned by Jesus-1 Cor 15: 5-9 ; 1 Cor 9: 1 ).< ref > So the Alexandrian recension ; the text in < sup > 38 </ sup > and Codex Bezae indicate that Apollos went to Corinth.
Given any element x of X, there is a function f < sup > x </ sup >, or f ( x ,·), from Y to Z, given by f < sup > x </ sup >( y ) := f ( x, y ).
Given points P < sub > 0 </ sub > and P < sub > 1 </ sub >, a linear Bézier curve is simply a straight line between those two points.
Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.
Given two groups (< var > G </ var >, *) and (< var > H </ var >, ), a group isomorphism from (< var > G </ var >, *) to (< var > H </ var >, ) is a bijective group homomorphism from < var > G </ var > to < var > H </ var >.
Given f ∈ G ( x * x < sup >- 1 </ sup >, y * y < sup >-1 </ sup >) and g ∈ G ( y * y < sup >-1 </ sup >, z * z < sup >-1 </ sup >), their composite is defined as g * f ∈ G ( x * x < sup >-1 </ sup >, z * z < sup >-1 </ sup >).
Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.
Given a topological space X, let G < sub > 0 </ sub > be the set X.
Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z < sub > 0 </ sub > in its domain is defined by the limit
Given a polynomial of degree with zeros < math > z_n < z_
Given a ( random ) sample the relation between the observations Y < sub > i </ sub > and the independent variables X < sub > ij </ sub > is formulated as

Given and sup
Given the need for such fine regulation of Ca < sup > 2 +</ sup > signaling, it is perhaps unsurprising that dysregulated mitochondrial Ca < sup > 2 +</ sup > has been implicated in several neurodegenerative diseases, while the catalogue of tumor suppressors includes a few that are enriched at the MAM.

Given and t
Given the state at some initial time ( t = 0 ), we can solve it to obtain the state at any subsequent time.
Given m real values t < sub > i </ sub >, called knots, with
Given that both A and not-A are seen to be “ true ,” Kant concludes that it ’ s not that “ God doesn ’ t exist ” but that there is something wrong with how we are asking questions about God and how we have been using our rational faculties to talk about universals ever since Plato got us started on this track!
Given time t, the source produces ωt oscillations.
Given the orientation matrix A ( t ) of a frame, we can obtain its instant angular velocity tensor W as follows.
Given a linear homogeneous recurrence relation with constant coefficients of order d, let p ( t ) be the characteristic polynomial ( also " auxiliary polynomial ")
Given a function w on U × Y, with finite integral of its modulus for any input function u and initial state x ( 0 ) over any finite time t, called the " supply rate ", a system is said to be dissipative if there exist a continuous nonnegative function V ( x ), with x ( 0 ) = 0, called the storage function, such that for any input u and initial state x ( 0 ) the difference V ( x ( t )) − V ( x ( 0 )) does not exceed the integral of the supply over ( 0, t ) for any t ( dissipation inequality ).
" Given his fantasies involving the busty tobacconist, the sensual math teacher, the fat-bottomed peasant women on bicycles, Volpina the man-eater and Gradisca whom he tried to grope at the Cinema Fulgor, Titta complains that it cant be helped.
: Given that the particle begins at position x < sub > 1 </ sub > at time t < sub > 1 </ sub > and ends at position x < sub > 2 </ sub > at time t < sub > 2 </ sub >, the physical trajectory that connects these two endpoints is an extremum of the action integral.
Given a circle k, with a center O, and a point P outside of the circle, we want to construct the ( red ) tangent ( s ) to k that pass through P. Suppose the ( as yet unknown ) tangent t touches the circle in the point T. From symmetry, it is clear that the radius OT is orthogonal to the tangent.
Given the kernel K ( t, s ), and the function, the problem is typically to find the function.
Given this, the eagles would not have been among the spirits summoned by Yavanna in that paragraph, suggesting that Tolkien didn ’ t change from the view that the eagles are animals, and have no fëar.
Given the width and definition of the window function w ( t ), we initially require the height of the window function to be scaled so that
Given our current estimate of the parameters θ < sup >( t )</ sup >, the conditional distribution of the Z < sub > i </ sub > is determined by Bayes theorem to be the proportional height of the normal density weighted by τ:
Given approximations of A from three distinct step sizes h, h / t, and h / s, the exact relationship
The problem in more mathematical terms is: Given a needle of length dropped on a plane ruled with parallel lines t units apart, what is the probability that the needle will cross a line?
Given a perfect graph G, Lovász forms a graph G * by replacing each vertex v by a clique of t < sub > v </ sub > vertices, where t < sub > v </ sub > is the number of distinct maximum independent sets in G that contain v. It is possible to correspond each of the distinct maximum independent sets in G with one of the maximum independent sets in G *, in such a way that the chosen maximum independent sets in G * are all disjoint and each vertex of G * appears in a single chosen set ; that is, G * has a coloring in which each color class is a maximum independent set.
Given a time series of data X < sub > t </ sub >, the ARMA model is a tool for understanding and, perhaps, predicting future values in this series.
: Given any n formal power series f < sub > 1 </ sub >,..., f < sub > n </ sub > in tC < nowiki ></ nowiki > t < nowiki ></ nowiki > which are linearly independent over Q, then the field extension C ( t, f < sub > 1 </ sub >,..., f < sub > n </ sub >, exp ( f < sub > 1 </ sub >),..., exp ( f < sub > n </ sub >)) has transcendence degree at least n over C ( t ).

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