We devote a chapter to the binomial distribution not only because it is a mathematical model for an enormous variety of real life phenomena, but also because it has important properties that recur in many other probability models.

For an experiment to qualify as a binomial experiment, it must have four properties: ( 1 ) there must be a fixed number of trials, ( 2 ) each trial must result in a `` success '' or a `` failure '' ( a binomial trial ), ( 3 ) all trials must have identical probabilities of success, ( 4 ) the trials must be independent of each other.

The triangle demonstrates many mathematical properties in addition to showing binomial coefficients.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle.

Forward differences applied to a sequence are sometimes called the binomial transform of the sequence, and have a number of interesting combinatorial properties.

An inductive proof for arithmetic sequences was introduced in the Al-Fakhri ( 1000 ) by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle.

A binomial heap is implemented as a set of binomial trees that satisfy the binomial heap properties:

all of whose non-leading coefficients are divisible by p by properties of binomial coefficients, and whose constant coefficient equal to p, and therefore not divisible by p < sup > 2 </ sup >.

An alternative way to arrive at this conclusion is to use the identity which is valid in characteristic p ( and which is based on the same properties of binomial coefficients, and gives rise to the Frobenius endomorphism ), to compute the reduction modulo p of the quotient of polynomials:

This table summarizes the properties of the trinomial expansion and the trinomial distribution, and it compares them to the binomial and multinomial expansions and distributions:

André Weil's famous Weil conjectures proposed that certain properties of equations with integral coefficients should be understood as geometric properties of the algebraic variety that they define.

The primary coefficients being the physical properties of the line ; R, C, L and G, from which the secondary coefficients may be derived using the telegrapher's equation.

When the algebra can be endowed with a metric, then the structure coefficients are written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations.

In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.

* Binomial coefficients and their properties

Melting temperatures, thermal expansion coefficients, and thermal conductivity can be considered to be secondary properties that are essential in processing, and these properties are related to the bonding in wide bandgap materials.

In physical problems such coefficients represent properties of the medium for each phase.

If the function being expressed as a multipole expansion is real, however, the coefficients must satisfy certain properties.

In zerotree based image compression scheme such as EZW and SPIHT, the intent is to use the statistical properties of the trees in order to efficiently code the locations of the significant coefficients.

where C and p are coefficients which depend on the properties of the plate.

P. Richards and C. Frasier expanded the terms for the reflection and transmission coefficients for a P-wave incident upon a solid-solid interface and simplify the result by assuming only small changes in elastic properties across the interface.

Thus, for an isotropic material only two independent parameters are needed to describe the elastic properties, known as the Lame coefficients.

The two properties of polynomials with integer coefficients can now be formulated formally as follows:

We have entered into an agreement for the sale of the present Cranston properties, effective as soon as we have completed removal to our new plant.

the magnetic moment of an unpaired electron associated nearby may have a tremendous influence on the magnetic resonance properties of nuclei.

The biological and physical properties of infectious particles have been studied intensively during the past fifteen years.

In the present work whole sera have been fractionated by chromatography on DEAE-cellulose using single gradients similar to those described by Sober and Peterson, and certain chemical and serological properties of the fractions containing antibodies of the ABO and Rh systems have been described.

The most important factor in determining what properties the end-product will have is quite naturally the type of hydroxyl-rich compound that is used in its production.

They also will visit properties on which appeals have been made.

Whereas the primary meanings of the Lo Shu diagram seemed to have been based on its inner mathematical properties -- and we shall see that even its secondary meanings rested on some mathematical bases -- the urgent desire to place everything into categories of fives led to other groupings based on other numbers, until an exaggerated emphasis on mere numerology pervaded Chinese thought.

Experiments have been conducted to attempt the synthesis of ununennium ( Uue ), which is likely to be the next member of the group, but they have all met with failure .< ref name =" link "> However, ununennium may not be an alkali metal due to relativistic effects, which are predicted to have a large influence on the chemical properties of superheavy elements.

Besides iodine and tellurium, later several other pairs of elements ( such as argon and potassium, cobalt and nickel ) were known to have nearly identical or reversed atomic weights, leaving their placement in the periodic table by chemical properties to be in violation of known physical properties.

They can also include essential oils that have some antibacterial properties, like phenol, thymol, eugenol, eucalyptol or menthol.

The organosulphur compounds responsible for the characteristic odour are believed to have antioxidant, antibiotic and anticarcinogenic properties, to stimulate the immune system and to be protective of liver functioning.

A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.

The observed properties are consistent with it being a heavier analog of iodine ; many other properties have been estimated based on this resemblance.

Most of the physical properties of astatine have been estimated ( by interpolation or extrapolation ), using various theoretically-grounded or empirically-derived methods.

Despite this controversy, many properties of diatomic astatine have been predicted.

Many chemical properties of astatine have been determined using tracer studies on extremely dilute astatine solutions.

Many plants in this family, such as wild carrot, have estrogenic properties and have been used as a folk medicine for birth control.

Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes.

The new product exhibited chemical properties of an alkali metal ( such as coprecipitating with caesium salts ), which led Perey to believe that it was element 87, caused by the alpha decay of actinium-227.

Following the visit of a diplomatic mission from the Eastern Roman Empire led by Alexander, Huneric restored properties seized by his father from the merchants of Carthage.

These " aether-wind " experiments led to a flurry of efforts to " save " aether by assigning to it ever more complex properties, while only few scientists like Emil Cohn or Alfred Bucherer considered the possibility of the abandonment of the aether concept.

The overlap between physics and materials science has led to the offshoot field of materials physics, which is concerned with the physical properties of materials.

In particular, the field has made great advances in the understanding of the relationship between the atomic-scale structure of minerals and their function ; in nature, prominent examples would be accurate measurement and prediction of the elastic properties of minerals, which has led to new insight into seismological behaviour of rocks and depth-related discontinuities in seismograms of the Earth's mantle.

Establishing verifiable title history is often entangled in legalities relating to the expropriation of 28, 000 properties by the revolutionary government that Ortega led in the 1980s.

A later study published in 2010 revealed that imaging of Rhea from the Cassini mission was inconsistent with the predicted properties of the rings, suggesting that some other mechanism is responsible for the magnetic effects that had led to the ring hypothesis.

This led to intense research to develop barium titanate and later lead zirconate titanate materials with specific properties for particular applications.

This led to a theory of unity between subatomic particles and electromagnetic waves, called wave – particle duality, in which particles and waves were neither one nor the other, but had certain properties of both.

The discovery that particles are discrete packets of energy with wave-like properties led to the branch of physics dealing with atomic and sub-atomic systems which is today called quantum mechanics.

Polychlorinated biphenyls have properties that once favored their use as a dielectic coolant, though concerns over their environmental persistence led to a widespread ban on their use.

The useful physiological properties of yeast have led to their use in the field of biotechnology.

However, a study of Dalton's own laboratory notebooks, discovered in the rooms of the Lit & Phil, concluded that so far from Dalton being led by his search for an explanation of the law of multiple proportions to the idea that chemical combination consists in the interaction of atoms of definite and characteristic weight, the idea of atoms arose in his mind as a purely physical concept, forced upon him by study of the physical properties of the atmosphere and other gases.

For example, the odd metallic color of plastic made by accidentally adding a thousand times too much catalyst led scientists to explore its metal-like properties, inventing electrically conductive plastic and light emitting plastic -— an invention that won the Nobel Prize in 2000 and has led to innovative lighting, display screens, wallpaper and much more ( see conductive polymer, and organic light-emitting diode or OLED ).

Urysohn's lemma has led to the formulation of other topological properties such as the ' Tychonoff property ' and ' completely Hausdorff spaces '.

Later, at the MRC Laboratory of Molecular Biology in Cambridge, England, Altman started the work that led to the discovery of RNase P and the enzymatic properties of the RNA subunit of that enzyme.

During the 1980s a number of folders started systematically studying the mathematical properties of folded forms, which led to a steady increase in the complexity of origami models, which continued well into the 1990s, after which some designers started returning to simpler forms.

Scheele's quick explanation for the vapors led Bergman to suggest that Scheele analyze the properties of manganese dioxide.

Elizabeth bore Sir Thomas three sons, Henry ( who died in early infancy ), Francis ( born in 1540 and took the surname of Darrell ), and Edward, who was later executed for his part in the Wyatt's Rebellion of 1554, led by his legitimate half-brother, Sir Thomas Wyatt, the younger. Sir Thomas left Elizabeth properties in Dorset.

This led to an extremely strong, dense, and durable Kryptonian species with unusual physical properties.

The reported healing properties of these springs led to the creation of a small community ; a post office was established on April 26, 1878, and the city was formally laid out in 1885.