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.

where the first sum is over positive energy states and the second over those of negative energy.

where p is the frequency of one allele and q is the frequency of the alternative allele, which necessarily sum to unity.

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

This is particularly so in Sydney and Melbourne, where headways on many lines in the core of the network reach 3 – 5 minutes in peaks and 10 – 20 minutes off peak ( about 18 hours a day ) and enter an underground loop for passenger distribution in the city centre ; and where ridership per capita exceeds the sum of metro and commuter rail in comparable North American urban areas such as Toronto, Boston or the San Francisco Bay Area.

Contrast that to a P2P network, where its aggregated bandwidth actually increases as nodes are added, since the P2P network's overall bandwidth can be roughly computed as the sum of the bandwidths of every node in that network.

where is the vector sum of the physical forces applied to the particle and is the absolute acceleration ( that is, acceleration in a stationary frame ) of the particle, given by:

In fact, the sum of the magnetic pressure and plasma pressure must be constant everywhere on the heliosphere in order to have an equilibrium configuration: where the magnetic field is higher, the plasma must be cooler or less dense.

The size of a grammar is the sum of the sizes of its production rules, where the size of a rule is one plus the length of its right-hand side.

The common notation for the divergence ∇· F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ ( see del ), apply them to the components of F, and sum the results.

Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.

Some ecological principles, however, do exhibit collective properties where the sum of the components explain the properties of the whole, such as birth rates of a population being equal to the sum of individual births over a designated time frame.

where D / Dt is the substantial derivative, which is the sum of local and convective derivatives.

The Dyson series can be alternately rewritten as a sum over Feynman diagrams, where at each interaction vertex both the energy and momentum are conserved, but where the length of the energy momentum four vector is not equal to the mass.

:, where is the sum over any n-sequence of length N.

: where is the sum over any k-sequence of length N.

Coulomb observed that, at failure, a distinct slip plane would form behind a sliding retaining wall and he suggested that the maximum shear stress on the slip plane, for design purposes, was the sum of the soil cohesion,, and friction, where is the normal stress on the slip plane and is the friction angle of the soil.

is an entire function in the variable b for all positive integers a where c < sub > d </ sub >( k ) is Ramanujan's sum.

In sum, Radbruch's formula argues that where statutory law is incompatible with the requirements of justice " to an intolerable degree ", or where statutory law was obviously designed in a way that deliberately negates " the equality that is the core of all justice ", statutory law must be disregarded by a judge in favour of the justice principle.

Now, the next morning, they were anchored at The Elbow and the boat was riding directly over the underwater ledge where the green water turned to deepest blue and the cliff dropped straight down 600 fathoms, with the weighted line beside it ; ;

The fact that he has cast over those materials the light of a skeptical mind does not make him any the less Southern, I rather think, for the South has been no more solid than other regions except in the political and related areas where patronage and force and intimidation and fear may produce a surface uniformity.

it took them over an hour to get back to the station where they should have changed, in order to take the line that went to the Place Redoute.

By making inroads in the name of law enforcement into the protection which Congress has afforded to the marriage relationship, the Court today continues in the path charted by the recent decision in Wyatt v. United States, 362 U.S. 525, where the Court held that, under the circumstances of that case, a wife could be compelled to testify against her husband over her objection.

The word marina was coined by NAEBM originally to describe a waterfront facility where recreational boats could find protection and basic needs to lay over in relative comfort.

She would try to see over the bulge of her cheeks and somewhat under her teeth to the place where she was biting.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

Let Af where the maximization is over all admissible policies Af.

Thus Af where the maximization is over all admissible policies Af, and Af is related to Af by ( 5 ).

There he'd take a compass reading, figure his air speed, and deduce that in a certain number of minutes he'd be over the broad meadows of the Merrimack Valley where it would be safe to let down through the overcast and see the ground before it hit him.

But if there was evidence at the Trial that aimed over Eichmann's head at his collaborators in the societies where he functioned, the press seems to have missed it.

I finally went downstairs to the bar off the main lobby where most of the cast were drowning their sorrows over the untimely passing of Ask Tony.

Even from where he stood he could see the neatly folded blanket that he had spread over Burton, the pillow, the sheet of paper on top of it.

Nick recalled stories that the two best fishing spots in Southern California were over the La Jolla Deep and the Redondo Deep, two spots where the ocean dropped off to fantastic depths almost from the shoreline.

Also the department of justice building is located where J. Edgar Hoover presides over the federal bureau of investigation.

Dick Carroll and his accordion ( which we now refer to as `` Freida '' ) held over at Bahia Cabana where `` Sir '' Judson Smith brings in his calypso capers Oct. 13.

Out of the church and into his big car, it tooling over the road with him driving and the headlights sweeping the pike ahead and after he hit college, his expansiveness, the quaint little pine board tourist courts, cabins really, with a cute naked light bulb in the ceiling ( unfrosted and naked as a streetlight, like the one on the corner where you used to play when you were a kid, where you watched the bats swooping in after the bugs, watching in between your bouts at hopscotch ), a room complete with moths pinging the light and the few casual cockroaches cruising the walls, an insect Highway Patrol with feelers waving.

The Greek alphabet, in its Euboean form, was carried over by Greek colonists to the Italian peninsula, where it gave rise to a variety of alphabets used to write the Italic languages.

The righteous will live in heaven for a thousand years ( the millennium ) where they will sit with God in judgment over the unredeemed and the fallen angels.

Image: Colonnadeparkguell. jpg | Colonnaded pathway where the road projects out from the hillside, with the vaulting forming a retaining wall which curves over to support the road, and transmits the load onto sloping columns.

" Thus Ambrose refused to be drawn into a false conflict over which particular local church had the " right " liturgical form where there was no substantial problem.

In 1880 the school moved to the Hillside Chapel, a building next to the house, where he held conversations and, over the course of successive summers, as he entered his eighties, invited others to give lectures on themes in philosophy, religion and letters.