distribution

In mathematical analysis, distributions are generalized functions that permit the derivative to be extended to things which are not necessarily smooth. The theory of distributions was founded in the middle of the 20th century by the French analyst Laurent Schwartz, and independently by I. M. Gelfand and the Russian school. In Russian they are still called generalized functions. Distributions on a manifold X are defined as continuous linear functionals on the space of smooth functions having compact support. In distribution theory this space is conventionally denoted by D(X) or C^∞_c(X), and the distributions therefore by D′(X) or C^-∞(X). Because D(X) is not metrizable (unless X is compact) but only an LF space, the continuity condition is a little technical to state. (An LF space is the inductive limit of an ascending sequence of Fréchet spaces: roughly, the completeness condition is as good as that of a Fréchet space but you only have a uniformity with which to state it rather than a metric.)

• All integrable functions (with respect to a fixed measure) on X are distributions: the function f corresponds to the distribution sending φ ∈ D(X) to ∫_X fφ dx.
• All regular measures on X are distributions: the measure μ corresponds to the distribution sending φ ∈ D to ∫_X φ dμ. For instance, the Dirac delta function beloved of physicists is a distribution, derived from the measure consisting of a unit point mass at 0; δ(φ) = φ(0).
• There are rougher examples, however. For instance, the derivative of the delta function sends φ to -φ′(0), and this distribution is neither a function nor a measure.

The advantage of distributions is that they can be differentiated indefinitely even though they may not be "smooth". We simply imagine that the integration by parts formula is valid, and define the derivative of a distribution u to be u′(φ) = u(-&phi′). This notion of differentiation turns out to yield a calculus with the correct properties.

Note that the word distribution is used by statisticians to mean something very different.

For more information, consult a textbook on partial differential equations for pure mathematicians. I like the first volume of Michael Taylor's three-volume work.

A "distribution" is a transaction in which a corporation pays money to its shareholders. There are two basic types of distributions:

• Dividends, made without consideration
• Repurchases of stock, made in consideration of outstanding shares

The board of directors has the sole power to authorize a distribution.

From a shareholder's perspective, the two types of distribution are very different. Dividends are essentially a freebie—the shareholder gets cash, and they don't have to give anything in return. (Note that "stock dividends," also known as splits, are not distributions since the corporation is not paying anything.) Stock repurchases, to a shareholder, are the same as selling stock on the open market—they get paid in exchange for giving up their interest in the company.

Distributions take on a simpler face when you look at them from the perspective of a creditor, or a holder of preferred stock. If a corporation pays a dividend or buys back outstanding shares, the transaction reduces the total assets of the corporation, and therefore makes it less likely that the creditor or preferred shareholder will get their money back in the future. People in such positions will therefore be wary of distributions.

Since directors are usually elected by holders of common stock, there is a real danger of distributions that screw over creditors and preferred shareholders. So there are specific rules to tell corporations when they can make distributions.

The "traditional" rules

"Traditional" rules apply under corporate laws that recognize par value—a legal "floor" on the value of stock. The most important contemporary examples are the Delaware General Corporation Law, which governs many big American companies, and the New York Business Corporation Law.

Under par value regimes, distributions can only be made from the "surplus" of the corporation. The exact definition of "surplus" varies from place to place, but Delaware and New York both define it as the assets that exceed the legal capital of the corporation—the aggregate par value of its shares. This means that in such states, creditors can only rely upon recovering a fraction of the corporation's assets: the rest are free to be distributed to shareholders at any time.

A few other states impose a further restriction: that distributions have to come from earned surplus, the money that comes from corporate profits. This means that companies cannot pay dividends that would bring their assets below the amount of paid-in capital, the total amount that has been directly paid in by shareholders since the company's incorporation.

The problem with the traditional rules is that par value is "squishy." It's usually kept low to begin with. Even if it isn't, a corporation can usually reduce the par value of its stock by amending its articles of incorporation. So traditional rules don't generally do a whole lot to stop companies from giving all of their assets to shareholders, leaving creditors high and dry in the process.

The "contemporary" rules

Under the Model Business Corporation Act, which has inspired the corporate laws of at least 35 U.S. states, stock does not have par value unless the corporation decides to include a value in its articles of incorporation. Since par value gets in the way, companies like to avoid it if they can. This means that a new standard is needed.

The Official Comments to the MBCA, drafted by the American Bar Association, suggest that a corporation must meet two tests before it makes a distribution. These are:

1. Equity insolvency test - After the distribution, the corporation must be able to pay its debts as they become due in the usual course of its business.
2. Balance sheet test - After the distribution, the corporation's total assets must be greater than or equal to its liabilities plus the amount needed to buy out all of the holders of preferred stock.

Some states have adopted these tests verbatim. Others have gone further: in California, for instance, the balance sheet test requires the corporation to have assets greater than 125% of its liabilities after the distribution.

Compliance with these rules falls upon the directors themselves. If they break the rules, they can be individually liable to the company's creditors. The Official Comments indicate that the decision to make a distribution should be treated as any other business decision, and therefore should be subject to the business judgment rule. This means that as long as the directors make the decision in good faith, on an informed basis, and absent a conflict of interest, they will usually be safe.

Sources

• Bauman et al., Corporations: Law and Policy (5th ed., West 2003)

Dissociated Press = D = distro

distribution n.

1. A software source tree packaged for distribution; but see kit. Since about 1996 unqualified use of this term often implies `Linux distribution'. The short form distro is often used for this sense. 2. A vague term encompassing mailing lists and Usenet newsgroups (but not BBS fora); any topic-oriented message channel with multiple recipients. 3. An information-space domain (usually loosely correlated with geography) to which propagation of a Usenet message is restricted; a much-underutilized feature.

--The Jargon File version 4.3.1, ed. ESR, autonoded by rescdsk.

Distribution is a rule of inference used in propositional logic. Also written as 'Dist.' for short. It goes like this:

{P∧(Q∨R)} = {(P∧Q)∨(P∧R)}
And
{P∨(Q∧R)} = {(P∨Q)∧(P∨R)}

Or in English: "P and either Q or R" is the same as "either P and Q or P and R" and "either P or Q, and also R" is the same as "either P or Q and also either P or R". If you think about it, it really does make sense, but you may find it useful to make a truth table.

Back up to Rules of Inference
Review your Logic symbols

Dis`tri*bu"tion (?), n. [L. distributio: cf. F. distribution.]

1.

The act of distributing or dispensing; the act of dividing or apportioning among several or many; apportionment; as, the distribution of an estate among heirs or children.

The phenomena of geological distribution are exactly analogous to those of geography. A. R. Wallace.

2.

Separation into parts or classes; arrangement of anything into parts; disposition; classification.

3.

That which is distributed.

"Our charitable distributions."

Atterbury.

4. Logic

A resolving a whole into its parts.

5. Print.

The sorting of types and placing them in their proper boxes in the cases.

6. Steam Engine

The steps or operations by which steam is supplied to and withdrawn from the cylinder at each stroke of the piston; viz., admission, suppression or cutting off, release or exhaust, and compression of exhaust steam prior to the next admission.

Geographical distribution, the natural arrangements of animals and plants in particular regions or districts.

Syn. -- Apportionments; allotment; dispensation; disposal; dispersion; classification; arrangement.

 

© Webster 1913.