Add holes in top, forming `` S '' for salt and `` P '' for pepper.

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.

Since Af and P divides Af for Af, we have Af.

It is clear that each vector in the range of Af is in Af for if **ya is in the range of Af, then Af and so Af because Af is divisible by the minimal polynomial P.

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

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 D denotes the differentiation operator and P is the polynomial Af then V is the null space of the operator p (, ), because Af simply says Af.

Ten students act as managers for a high-school football team, and of these managers a proportion P are licensed drivers.

The word `` binomial '' means `` of two names '' or `` of two terms '', and both usages apply in our work: the first to the names of the two outcomes of a binomial trial, and the second to the terms P and Af that represent the probabilities of `` success '' and `` failure ''.

The constancy of P and the independence are the conditions most likely to give trouble in practice.

Obviously, very slight changes in P do not change the probabilities much, and a slight lack of independence may not make an appreciable difference.

We shall find a formula for the probability of exactly X successes for given values of P and N.

Therefore each binomial trial can be thought of as producing a value of a random variable associated with that trial and taking the values 0 and 1, with probabilities Q and P respectively.

On C, from the point P at Af to the point Q at Af, we construct the chord, and upon the chord as a side erect a square in such a way that as S approaches zero the square is inside C.

As S increases we consider the two free corner points of the square, Af and Af, adjacent to P and Q respectively.

We again consider a fixed point P at Af and a variable point Q at Af on C.

We erect a square with PQ as a side and with free corners Af and Af adjacent to P and Q respectively.

Now with P fixed at Af, Af-values occur when the corner Af crosses C, and are among the values of S such that Af.

Moreover, if Af and Af are two planes intersecting in a line l, tangent to Q at a point P, the two free intersections of the image curves Af and Af must coincide at P', the image of P, and at this point Af and Af must have a common tangent l'.

To do this we must first show that every line which meets **zg in a point P meets its image at P.

But if this is the case, then an arbitrary pencil of lines having a point, P, of **zg as vertex is transformed into a ruled surface of order Af having Af generators concurrent at P.

Propionibacterium acnes ( P. acnes ) is the anaerobic bacterium species that is widely concluded to cause acne, though Staphylococcus epidermidis has been universally discovered to play some role since normal pores appear colonized only by P. acnes.

Regardless, there are specific clonal sub-strains of P. acnes associated with normal skin health and others with long-term acne problems.

In vitro, resistance of P. acnes to commonly used antibiotics has been increasing, as well.

It works against the " P. acnes " bacterium, and normally causes just dryness of the skin, slight redness, and occasional peeling when side-effects occur.

With increasing resistance of P. acnes worldwide, they are becoming less effective.

The mechanism appears to be that a porphyrin ( Coproporphyrin III ) produced within P. acnes generates free radicals when irradiated by 420 nm and shorter wavelengths of light.

In addition, basic science and clinical work by dermatologists Yoram Harth and Alan Shalita and others have produced evidence that intense blue / violet light ( 405 – 425 nanometer ) can decrease the number of inflammatory acne lesion by 60 – 70 % in four weeks of therapy, in particular, when the P. acnes is pretreated with delta-aminolevulinic acid ( ALA ), which increases the production of porphyrins.

* Tea tree oil ( melaleuca oil ) has been used with some success, where it is comparable to benzoyl peroxide but without excessive drying, kills P. acnes, and has been shown to be an effective anti-inflammatory in skin infections.

Electron micrographs of siphovirus from P. acnes.

P. acnes bacteria live deep within follicles and pores, away from the surface of the skin.

In these follicles, P. acnes bacteria use sebum, cellular debris and metabolic byproducts from the surrounding skin tissue as their primary sources of energy and nutrients.

Elevated production of sebum by hyperactive sebaceous glands ( sebaceous hyperplasia ) or blockage of the follicle can cause P. acnes bacteria to grow and multiply.

P. acnes bacteria secrete many proteins, including several digestive enzymes.

The cellular damage, metabolic byproducts and bacterial debris produced by the rapid growth of P. acnes in follicles can trigger inflammation.

The damage caused by P. acnes and the associated inflammation make the affected tissue more susceptible to colonization by opportunistic bacteria, such as Staphylococcus aureus.

Preliminary research shows healthy pores are only colonized by P. acnes, while unhealthy ones universally include the nonpore-resident Staphylococcus epidermidis, amongst other bacterial contaminants.

Whether this is a root causality, just opportunistic and a side effect, or a more complex pathological duality between P. acnes and this particular Staphylococcus species is not known.

P. acnes has also been found in corneal ulcers, and is a common cause of chronic endophthalmitis following cataract surgery.

The subspecies of P. acnes that cause these infections of otherwise sterile tissues ( prior to medical procedures ), however, are the same subspecies found on the skin of individuals who do not have acne-prone skin, so are likely local contaminants.

P. acnes is an opportunistic pathogen, causing a range of postoperative and device-related infections e. g., surgery, post-neurosurgical infection, joint prostheses, shunts and prosthetic heart valves.