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Formally and styled
* Formally, a Laird is styled as " The Much Honoured of " or " The Much Honoured The Laird of " or " The Much Honoured, Laird of

Formally and e
Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map
Formally, a frame is defined to be a lattice L in which finite meets distribute over arbitrary joins, i. e. every ( even infinite ) subset
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, an analytic function ƒ ( z ) of the real or complex variables z < sub > 1 </ sub >,…, z < sub > n </ sub > is transcendental if z < sub > 1 </ sub >, …, z < sub > n </ sub >, ƒ ( z ) are algebraically independent, i. e., if ƒ is transcendental over the field C ( z < sub > 1 </ sub >, …, z < sub > n </ sub >).
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.
Formally most of these approaches are similar to an artificial neural network, as inputs to a node are summed up and the result serves as input to a sigmoid function, e. g., but proteins do often control gene expression in a synergistic, i. e. non-linear, way.
Formally, the group was called Studiengesellschaft für Geistesurgeschichte ‚ Deutsches Ahnenerbe e. V.
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, the person who directly draws the funds (" the payee ") instructs his or her bank to collect ( i. e., debit ) an amount directly from another's (" the payer's ") bank account designated by the payer and pay those funds into a bank account designated by the payee.

Formally and N
Formally, the discrete cosine transform is a linear, invertible function ( where denotes the set of real numbers ), or equivalently an invertible N × N square matrix.
Formally, the discrete sine transform is a linear, invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup > ( where R denotes the set of real numbers ), or equivalently an N × N square matrix.
Formally Twist N ' Go, TNG is a motor scooter brand of CMSI, an importer, assembler, and distributor in the United States with more than 150 dealers and service centers.
Formally a congruence subgroup is one that contains Γ ( N ) for some N, and the least such N is the level or Stufe of the subgroup.
Formally, the normal bundle to N in M is a quotient bundle of the tangent bundle on M: one has the short exact sequence of vector bundles on N:

Formally and de
Formally known as the Organisation internationale de la Francophonie ( OIF ) or the International Organization of the Francophonie, the organization comprises 56 member states and governments, 3 associate members, and 19 observers.
Formally, all of them are entitled El Camarón de la Isla con la colaboración especial de Paco de Lucía with the exception of Canastera.

Formally and X
Formally, a topological space X is called compact if each of its open covers has a finite subcover.
Formally, the convex hull may be defined as the intersection of all convex sets containing X or as the set of all convex combinations of points in X.
If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is " not too large ", in the sense that if R is Noetherian, the same must be true for R. Formally,
Formally, the sequence of partial sums of some infinite summation converges if for every fixed power of X the coefficient stabilizes: there is a point beyond which all further partial sums have the same coefficient.
Formally, a statistic s is a measurable function of X ; thus, a statistic s is evaluated on a random variable X, taking the value s ( X ), which is itself a random variable.
Formally, we start with a metric space M and a subset X.
Formally, a ringed space ( X, O < sub > X </ sub >) is a topological space X together with a sheaf of rings O < sub > X </ sub > on X.
Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.
Formally, the mutual information of two discrete random variables X and Y can be defined as:
Formally, we begin by considering some family of distributions for a random variable X, that is indexed by some θ.
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, sending X to and acting as the identity on maps is a functor ( indeed, an involution ).
Formally, f < sub > X, Y </ sub >( x, y ) is the probability density function of ( X, Y ) with respect to the product measure on the respective supports of X and Y.

Formally and Lord
Formally, the appointment remains a royal one made on the advice of the Lord Chancellor, but he / she no longer comments on individual applications, and merely supervises the process and reviews the panel's recommendations in general terms ( satisfying himself / herself that the process as operated is fair and efficient ).
Formally, under Henry IV, Arundel served twice as Lord Chancellor, first in 1399 and again from 1407 to 1410.

Formally and ),
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, arbitrage transactions have negative skew – prices can get a small amount closer ( but often no closer than 0 ), while they can get very far apart.
Formally given to bishop Philip of Senj, the permission to use the Glagolitic liturgy ( the Roman Rite conducted in Slavic language instead of Latin, not the Byzantine rite ), actually extended to all Croatian lands, mostly along the Adriatic coast.
* Nuclear bunker buster: Formally known as the Robust Nuclear Earth Penetrator ( RNEP ), this program aimed to modify an existing gravity bomb to penetrate into soil and rock in order to destroy underground targets.
Formally, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with ( in other words, not parallel ), such that and for all.
Formally, we have for the approximation to the full solution A, a series in the small parameter ( here called ), like the following:
Formally a census-designated place ( CDP ), Laguna West-Lakeside was annexed by Elk Grove in 2003.
Formally, however, there is an examination board consisting of a candidate's teachers / examiners, the headmaster / headmistress and one external Vorsitzende ( r ) ( head ), usually a high-ranking school official or the head of another school.
* Blades ( hip-hop group ) ( Formally Blades Of Hades ), Australian hip hop group from Newcastle, New South Wales
Formally, a frieze group is a class of infinite discrete symmetry groups for patterns on a strip ( infinitely wide rectangle ), hence a class of groups of isometries of the plane, or of a strip.
Formally known as the World Commission on Environment and Development ( WCED ), the Brundtland Commission's mission is to unite countries to pursue sustainable development together.
Formally, a mortgage lender ( mortgagee ), or other lien holder, obtains a termination of a mortgage borrower ( mortgagor )' s equitable right of redemption, either by court order or by operation of law ( after following a specific statutory procedure ).
Other amenities in Saltney include the Corner Pin public house, a post office, Saltney Business Centre, St David's Retail Park ( Farmfoods / Poundstretchers / Ableworld ), an Asda Supermarket ( Formally a Netto store ) and a Morrisons supermarket.
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.
Enbridge currently has offices in Enbridge Place ( Formally ING Tower ), Manulife Place, and Royal Bank Building.
Formally, we define a bad field as a structure of the form ( K, T ), where K is an algebraically closed field and T is an infinite, proper, distinguished subgroup of K, such that ( K, T ) is of finite Morley rank in its full language.
Formally, given a G-bundle B and a map H → G ( which need not be an inclusion ),
Formally described, it alternates slow sections in a modal F with faster sections, " Neue Kraft fühlend " ( with renewed strength ), in D. The slow sections each have two elements, ( 1 ) a passage reminiscent of the opening of the first movement in which the instruments overlap each other with a brief motive ; ( 2 ) a chorale, the actual song.
Formally, however, there is an examination board consisting of a candidate ’ s teachers / examiners, the headmaster / headmistress and a Vorsitzende ( r ) ( head ), usually a high-ranking school official or the head of another school.
* 2004: Newsboys, Relient K, Christafari, The Tribe ( Formally The World Wide Message Tribe ), Pillar, Paul Colman Trio, Magnify, Darlene Zschech, Hillsong United

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