Formally, a topological space X is called compact if each of its open covers has a finite subcover.

Formally, one has an implication, not an equivalence, so the solution set may get larger.

Formally the " head of state " can also personally be the " head of government " ( ex officio or by ad hoc cumulation such as an absolute monarch nominating himself ) but otherwise has formal precedence over the Head of Government and other ministers, whether he is their actual political superior ( absolute monarch, executive president ) or rather theoretical or ceremonial in character.

Formally described as " Truck, Cargo: 10-Ton, 8x8 ", it has been nicknamed the " Dragon Wagon ".

Formally, we start with a category C with finite products ( i. e. C has a terminal object 1 and any two objects of C have a product ).

Formally it is a part of Novosibirsk city, and has never been a closed city.

Formally, DFS is an uninformed search that progresses by expanding the first child node of the search tree that appears and thus going deeper and deeper until a goal node is found, or until it hits a node that has no children.

Informally, G has the above presentation if it is the " freest group " generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R.

It has been leased from the National Trust since 1974 by the Somerset Wildlife Trust ( Formally Somerset Trust for Nature Conservation ) and is run as a nature reserve and visitor centre.

Formally known as trilina, the armónico has seven strings, two of which form a doubled string.

Formally known as Bootle-cum-Linacre, the town is < span style =" white-space: nowrap "> 4 miles ( 6. 4 km )</ span > to the north of Liverpool city centre, and has a total resident population of 77, 640.

Formally, this means that, for some function f, the image f ( D ) of a directed set D ( i. e. the set of the images of each element of D ) is again directed and has as a least upper bound the image of the least upper bound of D. One could also say that f preserves directed suprema.

Following a merger with the Furnishing Industry Trust ( Formally The Furnishing Trades Benevolent Association ) in 2011, the Company has become the most significant charity in the sector, supporting the needy and promoting welfare and education of those entering and working within the industry, as well as the wider support of the charitable work of the Mayoralty and the Corporation of the City of London.

Formally established in 1475, though in fact much older, it has 75, 000 codices from throughout history.

Formally, the situation has the characteristics of a prisoner's dilemma game that is played repeatedly between the same partners, and in this case " defection "-jumping the queue-would not be advantageous.

Formally, if R is a Noetherian ring and I is a principal, proper ideal of R, then I has height at most one.

* Formally real field, an algebraic field that has the so-called " real " property

Formally, a deterministic algorithm computes a mathematical function ; a function has a unique value for any given input, and the algorithm is a process that produces this particular value as output.

Formally, an antihomomorphism between X and Y is a homomorphism, where equals Y as a set, but has multiplication reversed: denoting the multiplication on Y as and the multiplication on as, we have.

Formally, given two partially ordered sets ( S, ≤) and ( T, ≤), a function f: S → T is an order-embedding if f is both order-preserving and order-reflecting, i. e. for all x and y in S, one has

Formally described in Stephen Gilligan's article: The relational self: the expanding of love beyond desire ( 1996 ), and expanded on in his book: The Courage to Love ( 1997 ), it has become a major new approach in psychotherapy and healing.

Formally launched in 2004, the network currently has 16 members in nine states.

Enbridge currently has offices in Enbridge Place ( Formally ING Tower ), Manulife Place, and Royal Bank Building.

Formally, given a finite set X, a collection C of subsets of X, all of size n, has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z.

Formally named " Olallie ", it has usually been marketed as olallieberry, just as " Marion " is sold as marionberry.

However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

Formally, the system is said to have memory.

Formally known as the " Six and A Half Point Pole " form, this is the second to last form of the system.

Formally, a generalised self reproducing system consists

Formally, HPGCC is a cross-compiler ; it compiles code for the ARM-based HP calculators, but runs on a PC rather than the target system.

Formally, an unlabelled state transition system is a tuple ( S, →) where S is a set ( of states ) and → ⊆ S × S is a binary relation over S ( of transitions ).

Formally, an iterated function system is a finite set of contraction mappings on a complete metric space.

Formally, knowing the step response of a dynamical system gives information on the stability of such a system, and on its ability to reach one stationary state when starting from another.

Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs ( this definition is slanted towards the state space representation ).

Formally, let G be a Coxeter group with reduced root system R and k < sub > v </ sub > a multiplicity function on R ( so k < sub > u </ sub > = k < sub > v </ sub > whenever the reflections σ < sub > u </ sub > and σ < sub > v </ sub > corresponding to the roots u and v are conjugate in G ).

Formally, a squeeze preserves the hyperbolic metric expressed in the form ; in a different coordinate system.

Formally, an Hamiltonian system is a dynamical system completely described by the scalar function, the Hamiltonian.

Formally, a graph rewriting system usually consists of a set of graph rewrite rules of the form, with being called pattern graph ( or left-hand side ) and being called replacement graph ( or right-hand side of the rule ).

Formally, a redistribution of all agents ’ initial endowments of goods among all agents in the system is a core allocation if no group of agents can redistribute their own initial endowments among themselves so as to improve the satisfaction of someone in the group without impairing that of any other in the group.

Formally known as the " Eastern Division of Surrey " or " Surrey Eastern ", it elected two MPs by the bloc vote system.

Formally, a Markov partition is used to provide a finite cover for the smooth system ; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one of the covering sets to another.

Formally system networks correspond to type lattices in formal lattice theory, although they are occasionally erroneously mistaken for flowcharts or directed decision trees.

Formally known as the Federal Reserve Wire Network, Fedwire is a Real Time Gross Settlement Funds Transfer system operated by the United States Federal Reserve Banks that enables financial institutions to electronically transfer funds between its more than 9, 289 participants ( as of March 19, 2009 ).