* 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 ).

Given a tensor a with rank q and dimensions ( i < sub > 1 </ sub >, ..., i < sub > q </ sub >), and a tensor b with rank r and dimensions ( j < sub > 1 </ sub >, ..., j < sub > r </ sub >), their outer product c has rank q + r and dimensions ( k < sub > 1 </ sub >, ..., k < sub > q + r </ sub >) which are the i dimensions followed by the j dimensions.

Given a short exact sequence with maps q and r:

Given a set M of molecules, chemical reactions can be roughly defined as pairs r =( A, B ) of subsets from M.

#( Affine axiom of parallelism ) Given a point A and a line r, not through A, there is at most one line through A which does not meet r.

Given an m × n matrix with real entries ( or entries from any other field ) and rank r, then there exists at least one non-zero r × r minor, while all larger minors are zero.

: Given r

Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that a = bq + r and 0 ≤ r < | b |, where | b | denotes the absolute value of b.

Given an element a and a non-zero element b in a Euclidean domain R equipped with a Euclidean function d, there exist q and r in R such that and either or.

Given a connected graph G =( V, E ) with V the set of vertices and E the set of edges, and with a root vertex r, the level structure is a partition of the vertices into subsets L < sub > i </ sub > called levels, consisting of the vertices at distance i from r. Equivalently, this set may be defined by setting L < sub > 0 </ sub > =

Given the total mass and the scalars r and v at a single point of the orbit, one can compute r and v at any other point in the orbit.

Given the total mass and the scalars r and v at a single point of the orbit, one can compute the specific orbital energy, allowing an object orbiting a larger object to be classified as having not enough energy to remain in orbit, hence being " suborbital " ( a ballistic missile, for example ), having enough energy to be " orbital ", but without the possibility to complete a full orbit anyway because it eventually collides with the other body, or having enough energy to come from and / or go to infinity ( as a meteor, for example ).

Given m functions r

Given a surface through which a metal wire passes, electrons move in both directions across the surface at an equal rate.

Given the rigorous admission process it is not surprising that CUNY BA boasts a 70 % graduation rate within an average of 2. 2 years and that 60 % graduate with academic honors.

Given that radio communication channels are allocated by agencies such as the Federal Communication Commission giving a prescribed ( maximum ) bandwidth, the advantage of QPSK over BPSK becomes evident: QPSK transmits twice the data rate in a given bandwidth compared to BPSK-at the same BER.

) Given that the telephone system nominal voiceband ( including guardband ) is 4, 000 Hz, the required digital sampling rate is 8, 000 Hz ( see Nyquist rate ).

Given expectations about returns on fixed investment, every level of the real interest rate ( i ) will generate a certain level of planned fixed investment and other interest-sensitive spending: lower interest rates encourage higher fixed investment and the like.

Given a collection of pairs ( time, cash flow ) involved in a project, the internal rate of return follows from the net present value as a function of the rate of return.

Given the ( period, cash flow ) pairs (, ) where is a positive integer, the total number of periods, and the net present value, the internal rate of return is given by in:

Given a 5 % male excess birth rate, there appears to be 3. 15 male SIDS cases per 2 female, for a male fraction of 0. 61.

Given the circumstances, most commentators still rate Kramnik the stronger player in the match.

Given the aforementioned, in the United States, vasectomy is utilized at less than half the rate the alternative female " tubal ligation ".

:: Example: Given this option to reinstate the contract on its original terms when a plan is proposed, it may be anomalous to require the payment of interest at a market rate while a case is pending.

Given the theoretical ' fog ' around this issue, in practice most industry practitioners rely on some form of proxy for the risk free rate, or use other forms of benchmark rate which are presupposed to incorporate the risk free rate plus some risk of default.

Given the circumstances, most commentators rate Kramnik the stronger player in the match.

Given that the lapse rate is not a conservative quantity when the tropopause is considered for stratosphere-troposphere exchanges studies, there exists an alternative definition named dynamic tropopause.

Given the alternative of holding cash, and thus earning 0 %, rather than lending it out, profit-seeking lenders will not lend below 0 %, as that will guarantee a loss, and a bank offering a negative deposit rate will find few takers, as savers will instead hold cash.

Given the ever-accelerating rate of global-scale change, the more critical learning and adaptation become to organization relevance, success, and ultimate survival.

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 an appropriately designed spam filtering algorithm, outbound spam filtering can be implemented with a near zero false positive rate, which keeps customer related issues with blocked legitimate email down to a minimum.

# Given a maximum distortion constraint, minimize the bit rate

# Given a maximum bit rate constraint, minimize the distortion

Given an equilateral triangle, the counterclockwise rotation by 120 ° around the center of the triangle " acts " on the set of vertices of the triangle by mapping every vertex to another one.

Given the irregular shape of the nucleus, Halley's rotation is likely to be complex.

Given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix.

Given a unit vector n in R < sup > 3 </ sup > and an angle φ, let R ( φ, n ) represent a counterclockwise rotation about the axis through n ( with orientation determined by n ).

Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin.

Given that Euler's rotation theorem states that there is always an instantaneous axis of rotation, the angular velocity,, can be given by a vector over this axis

Given a rotation matrix R, a vector u parallel to the rotation axis must satisfy

Given a unit vector u = ( u < sub > x </ sub >, u < sub > y </ sub >, u < sub > z </ sub >), where u < sub > x </ sub >< sup > 2 </ sup > + u < sub > y </ sub >< sup > 2 </ sup > + u < sub > z </ sub >< sup > 2 </ sup > = 1, the matrix for a rotation by an angle of θ about an axis in the direction of u is

Given a triangular orthobicupola, a 60-degree rotation of one cupola before the joining yields a cuboctahedron.

Given a rotation axis z represented by a unit vector k

Given that torques are pseudo-vectors, and hence add linearly, it follows that the rotation speed of the rocket around the aforementioned axis can only increase.