Suppose that the minimal polynomial for T decomposes over F into a product of linear polynomials.

Suppose that A, B, and C are the matrices representing the transformations T, S, and ST with respect to the given bases.

Suppose we have an n-dimensional oriented Riemannian manifold, M and a target manifold T. Let be the configuration space of smooth functions from M to T. ( More generally, we can have smooth sections of a fiber bundle over M .)

Suppose the rod rotates at a constant rate so that the mass moves at speed v. Then the kinetic energy T of the mass is:

Suppose that x < sup > i </ sup > are local coordinates on the base manifold M. In terms of these base coordinates, there are fibre coordinates p < sub > i </ sub >: a one-form at a particular point of T * M has the form p < sub > i </ sub > dx < sup > i </ sup > ( Einstein summation convention implied ).

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.

Suppose some theory T implies an observation O ( observation meaning here the result of the observation, rather than the process of observation per se ):

First proof: Suppose forms a basis of ker T. We can extend this to form a basis of V:.

Suppose the set of objects is T =

Suppose G is a finitely generated group ; and T is a finite symmetric set of generators

Suppose that an L-formula True ( n ) defines T *.

Suppose that ( X, T ) is a topological space and that Σ is at least as fine as the Borel σ-algebra σ ( T ) on X.

Suppose X is a Tychonoff space, also called a T < sub > 3. 5 </ sub > space, and C ( X ) is the algebra of continuous real-valued functions on X.

If is closed, densely defined and continuous on its domain, then it is defined on B < sub > 1 </ sub >.< ref > Suppose f < sub > j </ sub > is a sequence in the domain of T that converges to.

Suppose X is the T × K matrix of explanatory variables resulting from T observations on K variables.

Suppose a pendulum is swinging with a particular period T. For such a system, it is advantageous to perform calculations relating to the swinging relative to T. In some sense, this is normalizing the measurement with respect to the period.

Suppose the available data consists of T iid observations

Suppose that S and T are two first-order theories.

Suppose that it takes twice as much capital per unit of output to produce trucks than it does to produce lasers, so that the capital cost per unit equals $ 20, 000 for trucks ( T ) and $ 10, 000 for lasers ( L ), where these coefficients are initially assumed not to change.

Suppose that T is a bounded operator on the normed vector space X.

It is frequently stated in the following equivalent form: Suppose that is continuous and that u is a real number satisfying or Then for some c ∈ b, f ( c ) = u.

Suppose a line runs through two points: P = ( 1, 2 ) and Q = ( 13, 8 ).

Proof: Suppose G = ⟨ H | R ⟩ is a finitely presented, residually finite group.

Suppose the entries of Q are differentiable functions of t, and that t = 0 gives Q = I. Differentiating the orthogonality condition

Suppose V is a subset of R < sup > n </ sup > ( in the case of n = 3, V represents a volume in 3D space ) which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have

Suppose p ( x ) = x < sup > 3 </ sup >+ x < sup > 2 </ sup >− 5x + 3

Suppose we have a connected graph G = ( V, E ), The following statements are equivalent:

Suppose we wish to determine if n = 221 is prime.

where g ( w ) = f ( w ) v. Suppose that V is finite dimensional.

Suppose a stock price follows a Geometric Brownian motion given by the stochastic differential equation dS = S ( σdB + μ dt ).

Suppose that N = N < sub > 1 </ sub > N < sub > 2 </ sub >, where N < sub > 1 </ sub > and N < sub > 2 </ sub > are relatively prime.

* Suppose Q ( x ) =

* Suppose we want to differentiate ƒ ( x ) = x < sup > 2 </ sup > sin ( x ).

Suppose the curve is approximated by y = Cx < sup > p / q </ sup > near the origin.

Suppose we are given an element e < sub > 0 </ sub > ∈ E < sub > P </ sub > at P = γ ( 0 ) ∈ M, rather than a section.

Suppose that H is a locally compact Hausdorff group with a compact subgroup K. Then H acts on the quotient space X = H / K.

Suppose that one is summing n values x < sub > i </ sub >, for i = 1 ,..., n. The exact sum is:

Suppose that λ = 0.

Suppose w = z < sup > 1 / 2 </ sup >, and z starts at 4 and moves along a circle of radius 4 in the complex plane centered at 0.

Suppose that R is an algebra over the field C of complex numbers and M = N is a finite-dimensional simple module over R. Then Schur's lemma says that the endomorphism ring of the module M is a division ring ; this division ring contains C in its center, is finite-dimensional over C and is therefore equal to C. Thus the endomorphism ring of the module M is " as small as possible ".

) Then X < sub > i </ sub > is the value ( or realization ) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean μ < sub > i </ sub > and variance σ < sub > i </ sub >< sup > 2 </ sup > for all times i. Then the definition of the autocorrelation between times s and t is

Suppose that the car is ascending at 2. 5 km / h.

Suppose that u and v satisfy the Cauchy – Riemann equations in an open subset of R < sup > 2 </ sup >, and consider the vector field

Suppose n < sub > 1 </ sub >, n < sub > 2 </ sub >, …, n < sub > k </ sub > are positive integers which are pairwise coprime.

Suppose random variable X can take value x < sub > 1 </ sub > with probability p < sub > 1 </ sub >, value x < sub > 2 </ sub > with probability p < sub > 2 </ sub >, and so on, up to value x < sub > k </ sub > with probability p < sub > k </ sub >.

Player 1 moves first and chooses either F or U. Player 2 sees Player 1s move and then chooses A or R. Suppose that Player 1 chooses U and then Player 2 chooses A, then Player 1 gets 8 and Player 2 gets 2.

Suppose that a speaker can have the concept of water we do only if the speaker lives in a world that contains H < sub > 2 </ sub > O.

Suppose that one particle is in the state n < sub > 1 </ sub >, and another is in the state n < sub > 2 </ sub >.

Suppose we have N particles with quantum numbers n < sub > 1 </ sub >, n < sub > 2 </ sub >, ..., n < sub > N </ sub >.

Suppose, for concreteness, that we have an algorithm for examining a program p and determining infallibly whether p is an implementation of the squaring function, which takes an integer d and returns d < sup > 2 </ sup >.

Suppose that you add blue, then the blue – red – black tree defined like red – black trees but with the additional constraint that no two successive nodes in the hierarchy will be blue and all blue nodes will be children of a red node, then it becomes equivalent to a B-tree whose clusters will have at most 7 values in the following colors: blue, red, blue, black, blue, red, blue ( For each cluster, there will be at most 1 black node, 2 red nodes, and 4 blue nodes ).

Suppose two curves γ < sub > 1 </ sub >: (- 1, 1 ) → M and γ < sub > 2 </ sub >: (- 1, 1 ) → M with γ < sub > 1 </ sub >( 0 )

Suppose ( A < sub > 1 </ sub >, φ < sub > 1 </ sub >) is an initial morphism from X < sub > 1 </ sub > to U and ( A < sub > 2 </ sub >, φ < sub > 2 </ sub >) is an initial morphism from X < sub > 2 </ sub > to U. By the initial property, given any morphism h: X < sub > 1 </ sub > → X < sub > 2 </ sub > there exists a unique morphism g: A < sub > 1 </ sub > → A < sub > 2 </ sub > such that the following diagram commutes:

Suppose the thimble were screwed out so that graduation 2, and three additional sub-divisions, were visible ( as shown in the image ), and that graduation 1 on the thimble coincided with the axial line on the frame.

Suppose the angular velocity with respect to O < sub > 1 </ sub > and O < sub > 2 </ sub > is and respectively.