We could, alternatively, choose an encoding for Turing machines, where an encoding is a function which associates to each Turing Machine M a bitstring < M >.

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

He currently works at Texas A & M University, United States, as a Distinguished Professor where he holds the College of Engineering Endowed Chair in Computer Science.

For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

The name originated from the Basic Input Output System used in the CP / M operating system ( released in 1976 ), where the BIOS was loaded from disk, with only a small boot loader program stored in read-only memory.

The unit MBtu was defined as one thousand Btu, presumably from the Roman numeral system where " M " stands for one thousand ( 1, 000 ).

In large churches where they were celebrated the services were usually grouped ; e. g. Matins and Lauds ( about 7. 30 A. M .); Prime, Terce ( High Mass ), Sext, and None ( about 10 A. M .); Vespers and Compline ( 4 P. M .); and from four to eight hours ( depending on the amount of music and the number of high masses ) are thus spent in choir.

He also often employed inanimate objects in his films, often transforming them into other objects in an almost surreal way, such as in The Pawnshop ( 1916 ) and One A. M. ( 1916 ), where Chaplin is the only actor aside Chester Conklin's brief appearance in the very first scene.

HTML (< u > H </ u > yper < u > t </ u > ext < u > M </ u > arkup < u > L </ u > anguage ) has been in use since 1991, but HTML 4. 0 ( December 1997 ) was the first standardized version where international characters were given reasonably complete treatment.

Let M be a smooth manifold and let x be a point in M. Let I < sub > x </ sub > be the ideal of all functions in C < sup >∞</ sup >( M ) vanishing at x, and let I < sub > x </ sub >< sup > 2 </ sup > be the set of functions of the form, where f < sub > i </ sub >, g < sub > i </ sub > ∈ I < sub > x </ sub >.

where θ ∈ T < sub > f ( x )</ sub >< sup >*</ sup > N and X < sub > x </ sub > ∈ T < sub > x </ sub > M.

In cells with a nucleus ( eukaryotes ), the cell cycle can be divided in two periods: interphase — during which the cell grows, accumulating nutrients needed for mitosis and duplicating its DNA — and the mitosis ( M ) phase, during which the cell splits itself into two distinct cells, often called " daughter cells " and the final phase, cytokinesis, where the new cell is completely divided.

The town behind the front where George spends much of his time is referred to as M ---.

For example an electrochemical cell, where two copper electrodes are submerged in two copper ( II ) sulfate solutions, whose concentrations are 0. 05 M and 2. 0 M, connected through a salt bridge.

where p is pressure, ρ is density, R < sub > u </ sub > is the gas constant, M is the molar mass and T is temperature.

Other games which feature graffiti include Bomb the World ( 2004 ), an online graffiti simulation created by graffiti artist Klark Kent where users can virtually paint trains at 20 locations worldwide, and Super Mario Sunshine ( 2002 ), in which the hero, Mario must clean the city of graffiti left by the villain, Bowser Jr. in a plotline which evokes the successes of the Anti-Graffiti Task Force of New York's Mayor Rudolph Giuliani ( a manifestation of " broken window theory ") or those of the " Graffiti Blasters " of Chicago's Mayor Richard M. Daley.

All Mersenne primes are in the form M < sub > q </ sub >, where q is the ( prime ) exponent.

where ' is the rest mass of the electron, and M is the mass of the atomic nucleus.

He also offers a conversion table ( see Cohen, 1988, p. 283 ) for eta squared ( η < sup > 2 </ sup >) where 0. 0099 constitutes a small effect, 0. 0588 a medium effect and 0. 1379 a large effect.

Reactions of acids are often generalized in the form HA H < sup >+</ sup > + A < sup >−</ sup >, where HA represents the acid and A < sup >−</ sup > is the conjugate base.

The numerical value of K < sub > a </ sub > is equal to the concentration of the products divided by the concentration of the reactants, where the reactant is the acid ( HA ) and the products are the conjugate base and H < sup >+</ sup >.

Because the range of possible values for K < sub > a </ sub > spans many orders of magnitude, a more manageable constant, pK < sub > a </ sub > is more frequently used, where pK < sub > a </ sub >

These nodes are areas where the action potential is amplified using a high density of sodium ( Na < big >+</ big >) ions and is subsequently passed along the axon.

functions as real combinations of spherical harmonics Y < sub > lm </ sub >( θ, φ ) ( where l and m are quantum numbers ).

Within a shell where n is some integer n < sub > 0 </ sub >, ranges across all ( integer ) values satisfying the relation.

It lacks the NH < sub > 2 </ sub > group because of the cyclization of the side-chain and is known as an imino acid ; it falls under the category of special structured amino acids .</ ref > where R is an organic substituent known as a " side-chain "); often the term " amino acid " is used to refer specifically to these.

½ ( pKa < sub > 1 </ sub > + pKa < sub > R </ sub >), where pKa < sub > R </ sub > is the side-chain pKa.

## < tt > SubBytes </ tt >— a non-linear substitution step where each byte is replaced with another according to a lookup table.

## < tt > ShiftRows </ tt >— a transposition step where each row of the state is shifted cyclically a certain number of steps.

The < tt > MixColumns </ tt > function takes four bytes as input and outputs four bytes, where each input byte affects all four output bytes.

where n < sup > c </ sup > denotes the charge conjugate state, i. e., the antiparticle.

where we use the symbol k to denote the quantum numbers p and σ of the previous section and the sign of the energy, E ( k ), and a < sub > k </ sub > denotes the corresponding annihilation operators.

where E < sub > 0 </ sub > is an infinite negative constant.

The Battle of Qarqar is mentioned in extra-biblical records, and was perhaps at Apamea where Shalmaneser III of Assyria fought a great confederation of princes from Cilicia, Northern Syria, Israel, Ammon, and the tribes of the Syrian desert ( 853 BC ), including Ahab ( A-ha-ab-bu < sup > mat </ sup >) ( Adad -' idri ).

where < big ></ big > is the number of degrees of freedom divided by two, R is the universal gas constant and n is the number of moles in the system ( a constant ).

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.

auxiliary regression is TR < sup > 2 </ sup >, where T is the sample size and R < sup > 2 </ sup > is the coefficient of determination.

Thus, all the antiderivatives of x < sup > 2 </ sup > can be obtained by changing the value of C in F ( x ) = ( x < sup > 3 </ sup >/ 3 ) + C ; where C is an arbitrary constant known as the constant of integration.

where n > 1 is an integer and p, q, and r are prime numbers, then 2 < sup > n </ sup >× p × q and 2 < sup > n </ sup >× r are a pair of amicable numbers.

: where M is any Hermitian positive-definite matrix, and x < sup >*</ sup > the conjugate transpose of x.

rk ( A < sup >*</ sup >), where A < sup >*</ sup > is the conjugate transpose or hermitian transpose of A.

where γ < sup >*</ sup > is the conjugate of γ, and the product is Clifford multiplication.

A subbase for the weak topology is the collection of sets of the form φ < sup >- 1 </ sup >( U ) where φ ∈ X < sup >*</ sup > and U is an open subset of the base field K. In other words, a subset of X is open in the weak topology if and only if it can be written as a union of ( possibly infinitely many ) sets, each of which is an intersection of finitely many sets of the form φ < sup >- 1 </ sup >( U ).

More generally, an complex matrix M is said to be positive definite if z < sup >*</ sup > Mz is real and positive for all non-zero complex vectors z ; where z < sup >*</ sup > denotes the conjugate transpose of z.

where x < sub >*</ sub > can be any point in interval j.

Generally, unless both the objective function and the feasible region are convex in a minimization problem, there may be several local minima, where a local minimum x < sup >*</ sup > is defined as a point for which there exists some δ > 0 so that for all x such that

This can be seen by looking at the diagonal entries of A < sup >*</ sup > A and AA < sup >*</ sup >, where A is a normal, triangular matrix.

Thus MR pulse sequences sensitive to T < sub > 2 </ sub >< sup >*</ sup > show more MR signal where blood is highly oxygenated and less where it is not.

The hard part to deal with is more or less a group H < sup > 1 </ sup >( P < sup > 1 </ sup >, j < sub >*</ sub > E ) = H ( U, E ), where U is the points the projective line with non-singular fibers, and j is the inclusion of U into the projective line, and E is the sheaf with fibers the spaces E < sub > x </ sub > of vanishing cycles.

If we specifically choose the Euclidean norm on both R < sup > n </ sup > and R < sup > m </ sup >, then we obtain the matrix norm which to a given matrix A assigns the square root of the largest eigenvalue of the matrix A < sup >*</ sup > A ( where A < sup >*</ sup > denotes the conjugate transpose of A ).

where z and b are complex numbers, z < sup >*</ sup > and b < sup >*</ sup > are the complex conjugates of z and b, respectively, and c is a real number.

where M < sub >*</ sub >< sup > n-1 </ sup > is the ( n-1 )- dimensional Minkowski content, L < sup > n </ sup > is the n-dimensional Lebesgue measure, and ω < sub > n </ sub > is the volume of the unit ball in R < sup > n </ sup >.

where GL ( V, F ) is the general linear group of invertible linear transformations of V over F and F < sup >*</ sup > here is the normal subgroup consisting of multiplications of vectors in V by nonzero elements of F ( that is, scalar multiples of the identity ; scalar transformations ).

where ƒ < sup >*</ sup >( x ) ƒ ( x ) = ƒ ( x ) ƒ < sup >*</ sup >( x ) = x.

This can be seen by looking at the diagonal entries of A < sup >*</ sup > A and AA < sup >*</ sup >, where A is a normal, triangular matrix.