Compact group - From compact to non-compact groups

The influence of the compact group theory on non-compact groups was formulated by Weyl in his unitarian trick. Inside a general semisimple Lie group there is a maximal compact subgroup, and the representation theory of such groups, developed largely by Harish-Chandra, uses intensively the restriction of a representation to such a subgroup, and also the model of Weyl's character theory.

Portsmouth Compact - Compact of Loyalty

The Compact of Loyalty [4] was written and signed April 30, 1639.

Compact space - Ordered compact spaces

Let X be a simply ordered set endowed with the order topology. Then X is compact if and only if X is a complete lattice (i.e. all subsets have suprema and infima).

Portsmouth Compact - Compact of Loyalty

The names of the signatories above were copied verbatim from the Compact of Loyalty. Note that the only name in common with the signatories of the Portsmouth compact is that of William Hutchinson.

Compact space - Ordered compact spaces

A nonempty compact subset of the real numbers has a greatest element and a least element.

National Compact Stellarator Experiment - Compact stellarators

When one considers the magnet layout needed to achieve both goals, a twisted path around the circumference of the device as well as many smaller twists and mixes along the way, the design becomes extremely complex, well beyond the abilities of conventional design tools. It was only through the use of massively parallel computers that the designs could be studied in depth, and suitable magnet designs created. The result was a very compact device, significantly smaller outside than a classical design for any given volume of plasma, with a low aspect ratio. Lower aspect ratios are highly desirable, because they allow a machine of any given power to be smaller, which lowers construction costs.

Compact city - Compact cities and sustainability

Another selling point of compact cities is that they are supposed to be sustainable developments. However, recent studies suggest that these developments are not as green as promised. For example, in the study by Rérat, the author discussed three criticism of the compact city model. The first criticism was that supply and demand alignments are not feasible in some cases. What this means is that there are limits to how many people can fit into a space because of resource and design limitations. Additionally, while a high-density may be achievable, it likely will counter the demands and expectations of residents who want more privacy and more personal space. The second criticism of the compact city is that to produce highly desirable living conditions and a high quality of life, the cost of living needs to increase. Consequently, integrating desirable traits into compact cities makes the living spaces expensive, pricing out lower-income families and pushing them to the outskirts of town. This stimulates urban sprawl and places the burden of longer commutes on low-income workers, further expanding disparities in wealth and quality of life. The third criticism of the compact city is that its environmental impacts are significant. Dense populations mean waste and pollution are also dense. This concentrates the impact of communities and necessitates expensive control mechanism.

Compact city - Compact cities and sustainability

Rérat is not the only researcher showing evidence that compact cities are not as sustainable as promised. The work by Westernick et al. also show that there are sustainability trade-offs found within compact city designs. Westernick et al. compared compact cities to dispersed cities to see where sustainability factors differed. The findings showed that compact cities excelled in efficient land consumption, more flexible land use patterns, cost efficiency of development and maintenance, and reduced reliance on fossil fuels and motor vehicles. In contrast, however, the sustainability disadvantages included higher vulnerability to disaster impacts, less personal space, less green space where people live, and higher environmental impact because of density. These drawbacks make compact cities less sustainable and justify the need for design improvements.

Compact space - Functions and compact spaces

A continuous image of a compact space is compact. This implies the extreme value theorem: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum. (Slightly more generally, this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of a compact space under a proper map is compact.

Mayflower Compact - Reasons for the Compact

The Mayflower was originally bound for the Colony of Virginia, financed by the Company of Merchant Adventurers of London. Storms forced them to anchor at the hook of Cape Cod in Massachusetts, however, as it was unwise to continue with provisions running short. This inspired some of the non-Puritan passengers (whom the Puritans referred to as "Strangers") to proclaim that they "would use their own liberty; for none had power to command them" since they would not be settling in the agreed-upon Virginia territory. To prevent this, the Pilgrims determined to establish their own government, while still affirming their allegiance to the Crown of England. Thus, the Mayflower Compact was based simultaneously upon a majoritarian model and the settlers' allegiance to the king. It was in essence a social contract in which the settlers consented to follow the community's rules and regulations for the sake of order and survival.

Compact theory - Compact theory in Canada

In the Canadian context, compact theory posits that Confederation was an agreement between the two founding peoples, French and English, and that therefore Quebec should have special veto powers relating to its position in the federal structure. However, compact theory was rejected by the Supreme Court of Canada in the 1981 Patriation Reference. Compact theory is often advanced by Quebec nationalists.

The Compact - Commission for the Compact

Established in 2007, the Commission for the Compact was an independent public body responsible for overseeing and promoting use of the Compact. The Commission closed in April 2010.

Compact closed category - Dagger compact category

A dagger symmetric monoidal category which is compact closed is a dagger compact category.

Hickson Compact Group - Hickson Compact Groups

According to Hickson: “Most compact groups contain a high fraction of galaxies having morphological or kinematical peculiarities, nuclear radio and infrared emission, and starburst or active galactic nuclei (AGN) activity. They contain large quantities of diffuse gas and are dynamically dominated by dark matter. They most likely form as subsystems within looser associations and evolve by gravitational processes. Strong galaxy interactions result and merging is expected to lead to the ultimate demise of the group. Compact groups are surprisingly numerous, and may play a significant role in galaxy evolution.”

Locally compact space - Locally compact groups

The notion of local compactness is important in the study of topological groups mainly because every Hausdorff locally compact group G carries natural measures called the Haar measures which allow one to integrate measurable functions defined on G. The Lebesgue measure on the real line R is a special case of this.

Compact quantum group - Compact Quantum Groups

A compact quantum group is defined as a pair

Locally compact space - Locally compact groups

The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. More precisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups. The study of locally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non-abelian locally compact groups.

Compact city - The compact city and proximity

Compact cities are intended to provide everything a person needs to live in one community, including work opportunities. When a person works inside the compact city, he/she can walk or bike to work instead of driving. This reduces fossil fuel consumption and helps to reduce emissions and traffic density on the highways between cities. However, not everyone finds work inside the compact city, and subsequently, many people commute to neighboring cities to access their employers. The need to commute creates two concerns for the compact city dweller, commuting time and the impact of commuting.

Compact operator - Compact operator on Hilbert spaces

An important subclass of compact operators is the trace-class or nuclear operators.

Compact space - Compact spaces and set operations

The product of any collection of compact spaces is compact. (This is Tychonoff's theorem, which is equivalent to the axiom of choice.)