Given the eigenstates of l < sub > 1 </ sub > and l < sub > 2 </ sub >, the construction of eigenstates of l ( which still is conserved ) is the coupling of the angular momenta of electrons 1 and 2.

: Given a weighted directed graph and an integer K, is there a Hamiltonian path ( or Hamiltonian cycle if the salesman must return home ) with total weight less than or equal to K?

For instance, consider the problem " Given a Hamiltonian graph, determine if the graph has a cycle of size 4.

Given the above definitions the interaction Hamiltonian is

Given the military strength of the Great Powers, these interventions were nearly always decisive and quickly ended the civil wars.

Given the high thermal design power of high-speed computer CPUs and components, modern motherboards nearly always include heat sinks and mounting points for fans to dissipate excess heat.

Given the rules of any two-person game with a finite number of positions, one can always trivially construct a minimax algorithm that would exhaustively traverse the game tree.

Given the complex nature of human smuggling and trafficking operations, the difference between these two criminal operations is not always readily apparent.

Given that the businesses in Miami Springs had always relied upon the large disposable incomes of the employees of the large airline carriers, the bankruptcy of both corporations in the same year created a chain reaction, eventually causing many small businesses to close their doors.

Given their sting, however, they must always be treated with caution, and the discovery of men o ' war washed up on a beach may lead to the closure of the whole beach.

Given a set S of matrices, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V, defines a linear functional on the subalgebra U of End ( V ) generated by the set of endomorphisms S ; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This " generalized eigenvalue " is a prototype for the notion of a weight.

Given pressing circumstances, a reasonable person may not always act in a manner similar to how she would have acted in a more relaxed setting.

Given the nature of Arthur's activities, he was always in the spotlight of the local police and crossed paths with several regular and occasional characters:

Given a sufficiently large sample size, a statistical comparison will always show a significant difference unless the population effect size is exactly zero.

Given the time it takes to accomplish multiple shots, it is typically, but not always, used for static subjects.

Another partition of the given interval, Q, is defined as a refinement of the partition, P, when it contains all the points of P and possibly some other points as well ; the partition Q is said to be “ finer ” than P. Given two partitions, P and Q, one can always form their common refinement, denoted P ∨ Q, which consists of all the points of P and Q, re-numbered in order.

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 the slow, methodical manner in which traditional animation is produced, it is almost always easier to synchronize animation to a pre-existing soundtrack than it is to synchronize a soundtrack to pre-existing animation.

is also true in the case of compact manifolds, due to Yau's proof of the Calabi conjecture: Given a compact, Kähler, holomorphically symplectic manifold ( M, I ), it is always equipped with a compatible hyperkähler metric.

Given a particular input, it will always produce the same output, and the underlying machine will always pass through the same sequence of states.

Given the mathematical constraints of these simple rules, the number of lines that change within pair partners will always be even ( either 2, 4, or 6 ).

Given the relation between convexity and duration above, conventional bond convexities must always be positive.

Given that periodic auctions cannot be avoided in the public sector, switching costs are always incurred, and so switching costs would not be a relevant

Given that buying an inventory of G bugles prior to the any-key rule would always be very expensive, the adoption of any-key brass greatly aided the almost explosive increase in the number of all-age corps hitting the field.

Given that the form mixes harangue with self-reflection, it is always clear where the argument tends and the context is brought along with the main thrust, whether the issue is cooking, Lawino's relatives being told they cannot drop in unannounced, or the pretensions and fashions of the urban second wife.

Given the electromechanical principle of the meter, the deviation of the needle is actually approximately proportional to the average of the part of the signal with more than approximately 0. 4V instantaneously because of the two copper-oxide rectifiers always in series, which transfer function curve has a knee around 0. 2V.

Given a coherent such I, the quotient sheaf O < sub > D </ sub > / I is such that / I ( U ) is always a module over O < sub > D </ sub >( U );

Given the absolute magnitude, for objects within our galaxy you can also calculate the apparent magnitude from any distance ( in parsecs ):

Given the absolute magnitude, you can also compute apparent magnitude from its parallax:

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 BASIC's straightforward nature, it was a simple matter to type in the code from the magazine and execute the program.

Love and Erlandson continued to pursue with the band, and released the single " Be A Man "— an outtake from the Celebrity Skin sessions — for the soundtrack of the Oliver Stone film Any Given Sunday ( 1999 ).

Given r, the rate of rotation is easy to infer from the constant angular momentum L, so a 2D solution can be easily reconstructed from a 1D solution of this equation.

Given two manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both

Given the similarities between the rep proteins of the alphasatellites and the nanoviruses, it is likely that the alphasatellites evolved from the nanoviruses.

) Given a smooth Φ < sup > t </ sup >, an autonomous vector field can be derived from it.

Given that the vast majority of all emeralds are treated as described above, and the fact that two stones that appear visually similar may actually be quite far apart in treatment level and therefore in value, a consumer considering a purchase of an expensive emerald is well advised to insist upon a treatment report from a reputable gemological laboratory.

Given that genes are universal to living organisms, genetics can be applied to the study of all living systems, from viruses and bacteria, through plants and domestic animals, to humans ( as in medical genetics ).

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 a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G ( x, y ) ( i. e. the sets of morphisms from x to y ).

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

: Given a point ( in terms of its coordinates ) and the direction ( azimuth ) and distance from that point to a second point, determine ( the coordinates of ) that second point.

On poverty, Hoover said that " Given the chance to go forward with the policies of the last eight years, we shall soon with the help of God, be in sight of the day when poverty will be banished from this nation ", and promised, " We in America today are nearer to the final triumph over poverty than ever before in the history of any land ," but within months, the Stock Market Crash of 1929 occurred, and the world's economy spiraled downward into the Great Depression.

Given the Greek fleet's unpreparedness resulting from the premature outbreak of the war, such an early Ottoman attack might well have been able to achieve a crucial victory.

Given MLB's refusal to consider expansion, Kirksey, Cullinan, Smith, and Hofheinz joined forces with would-be owners from other cities and announced the formation of a new league to compete with the established National and American Leagues.

Given the opportunity to get away from the Sultan and visit new lands, he readily accepted.

Given earlier debates by Christian authors about the existence of Jesus, e. g. in Justin Martyr's 2nd century Dialogue with Trypho, it would have been expected that the passage from Josephus would have been used as a component of the arguments.

Given significant distance from the magnetic poles, one can figure which hand is which using a magnetic compass and the sun.

Given printing practices at the time ( which included type-setting from dictation ), no two editions turned out to be identical, and it is relatively rare to find even two copies that are exactly the same.

Given a year to complete the work, Gibson undertook the actual writing out of " blind animal terror " at the obligation to write an entire novel – a feat which he felt he was " four or five years away from ".