If a transaction ( T < sub > i </ sub >) wants to write to an object, and if there is another transaction ( T < sub > k </ sub >), the timestamp of T < sub > i </ sub > must precede the timestamp of T < sub > k </ sub > ( i. e., TS ( T < sub > i </ sub >) < TS ( T < sub > k </ sub >)) for the object write operation to succeed.

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.

If we try to study T using characteristic values, we are confronted with two problems.

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

If Af is the operator induced on Af by T, then evidently Af, because by definition Af is 0 on the subspace Af.

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If T is the total `` length '' of the process, its feed state may be denoted by a vector p(T) and the product state by p(Q).

* If S and T are in M then so are S ∪ T and S ∩ T, and also a ( S ∪ T )

* If S and T are in M with S ⊆ T then T − S is in M and a ( T − S ) =

* If a set S is in M and S is congruent to T then T is also in M and a ( S )

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If T is a good symmetry of the dynamics, then

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

* If a * b is defined, then ( a * b )< sup >− 1 </ sup >

If a large, b-bit number is the product of two primes that are roughly the same size, then no algorithm has been published that can factor in polynomial time, i. e., that can factor it in time O ( b < sup > k </ sup >) for some constant k. There are published algorithms that are faster than O (( 1 + ε )< sup > b </ sup >) for all positive ε, i. e., sub-exponential.

A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

If the hypothetical monkey has a typewriter with 90 equally likely keys that include numerals and punctuation, then the first typed keys might be " 3. 14 " ( the first three digits of pi ) with a probability of ( 1 / 90 )< sup > 4 </ sup >, which is 1 / 65, 610, 000.

* If You Love Me, Let Me Know ( 1974 )< sup > b </ sup >

If, instead of 1 matching entry, there are k matching entries, the same algorithm works but the number of iterations must be π ( N / k )< sup > 1 / 2 </ sup >/ 4 instead of πN < sup > 1 / 2 </ sup >/ 4.

: If, then the column vectors are v < sub > 1 </ sub > = ( 1, 0, 2 )< sup > T </ sup > and v < sub > 2 </ sub > = ( 0, 1, 0 )< sup > T </ sup >.

If the " opposite " side has length x − 4 and the " adjacent " side has length 3, then the Pythagorean theorem tells us that the hypotenuse has length √(( x − 4 )< sup > 2 </ sup > + 3 < sup > 2 </ sup >)

If GRH is true, then every proper subgroup of the multiplicative group omits a number less than 2 ( ln n )< sup > 2 </ sup >, as well as a number coprime to n less than 3 ( ln n )< sup > 2 </ sup >.

If the tested number n is composite, the strong liars a coprime to n are contained in a proper subgroup of the group, which means that if we test all a from a set which generates, one of them must be a witness for the compositeness of n. Assuming the truth of the generalized Riemann hypothesis ( GRH ), it is known that the group is generated by its elements smaller than O (( log n )< sup > 2 </ sup >), which was already noted by Miller.

If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square ( e. g., 4 / 9 = ( 2 / 3 )< sup > 2 </ sup >).

* If A is a square matrix, then det ( A < sup >*</ sup >) = ( det A )< sup >*</ sup > and tr ( A < sup >*</ sup >) = ( tr A )< sup >*</ sup >

If the technifermions transform under ⊗ U ( 1 )< sub > EW </ sub > as left-handed doublets and right-handed singlets, three linear combinations of these Goldstone bosons couple to three of the electroweak gauge currents.

If R ( x ) is the remainder when P ( x ) is divided by ( x – r )< sup > 2 </ sup > — that is, by x < sup > 2 </ sup > – 2rx + r < sup > 2 </ sup > — then the equation of the tangent line to P ( x ) at x

If P is a d-dimensional unit hypercube whose vertices are the integer lattice points all of whose coordinates are 0 or 1, then the t-fold dilation of P is a cube with side length t, containing ( t + 1 )< sup > d </ sup > integer points.

If U is the unit matrix in the direction of the Higgs field, then the index is proportional to the integral of U ( dU )< sup > n − 1 </ sup > over the ( n − 1 )- sphere at infinity.

If K is a field of characteristic p, and if V is an algebraic variety over K of dimension greater than zero, the function field K ( V ) is a purely inseparable extension over the subfield K ( V )< sup > p </ sup > of pth powers ( this follows from condition 2 above ).

If the functions f and g are analytic and f is not identically zero in a neighbourhood of c on the complex plane, then the limit of f ( z )< sup > g ( z )</ sup > will always be 1.

If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.

If each node additionally records the size of its subtree ( including itself and its descendants ), then the nodes can be retrieved by index in O ( log n ) time as well.

If there is refraction at a collective spherical surface, or through a thin positive lens, O ' 2 will lie in front of O ' 1 so long as the angle u2 is greater than u1 ( under correction ); and conversely with a dispersive surface or lenses ( over correction ).

If the angle u1 is very small, O ' 1 is the Gaussian image ; and O ' 1 O ' 2 is termed the longitudinal aberration, and O ' 1R the lateral aberration of the pencils with aperture u2.

If the pencil with the angle u2 is that of the maximum aberration of all the pencils transmitted, then in a plane perpendicular to the axis at O ' 1 there is a circular disk of confusion of radius O ' 1R, and in a parallel plane at O ' 2 another one of radius O ' 2R2 ; between these two is situated the disk of least confusion.

If rays issuing from O ( fig.

If the object point O is infinitely distant, u1 and u2 are to be replaced by h1 and h2, the perpendicular heights of incidence ; the sine condition then becomes sin u ' 1 / h1 = sin u ' 2 / h2.

If evolutionary processes are blind to the difference between function F being performed by conscious organism O and non-conscious organism O *, it is unclear what adaptive advantage consciousness could provide.

If a diatomic molecule consists of two atoms of the same element, such as H < sub > 2 </ sub > and O < sub > 2 </ sub >, then it is said to be homonuclear, but otherwise it is heteronuclear.

If we insert vertices in random order, it turns out ( by a somewhat intricate proof ) that each insertion will flip, on average, only O ( 1 ) triangles – although sometimes it will flip many more.

If a skip list is used, the insertion time is brought down to O ( log n ), and swaps are not needed because the skip list is implemented on a linked list structure.

If the items are stored in a linked list, then the list can be sorted with O ( 1 ) additional space.

If the table size n is large enough, linear search will be faster than binary search, whose cost is O ( log n ).

Anderson is best remembered as a film maker for his " Mick Travis trilogy ", all of which star Malcolm McDowell as the title character: If .... ( 1968 ), a satire on public schools ; O Lucky Man!

If the carbonyl is at position 1 ( that is, n or m is zero ), the molecule begins with an formyl group H ( C = O )-, and is technically an aldehyde.