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Page "Standard Model (mathematical formulation)" ¶ 17
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If and abstract
If we take in our hand any volume ; of divinity or school metaphysics, for instance ; let us ask, Does it contain any abstract reasoning concerning quantity or number?
If, in some future experiment, one of the mass-related phenomena is shown to not be proportional to the others, then that specific phenomenon will no longer be considered a part of the abstract concept of mass.
If Prout's hypothesis had proven accurate, then the abstract concept of mass, as we now know it, might never have evolved, since mass could always be defined in terms of amounts of the hydrogen atomic mass.
If properly done, though, abstract interpretation is sound ( every property true of the abstract system can be mapped to a true property of the original system ).
Some students of universal grammar study a variety of grammars to abstract generalizations called linguistic universals, often in the form of " If X holds true, then Y occurs.
If one applies Occam's Razor only concretely, then it may not be used on abstract concepts ( this route, however, has serious consequences for selecting between hypotheses about the abstract ).
If you have an abstract issued from our office, please contact us to discuss replacement.
If the center is non-trivial ( and not the whole group ) then the simple Lie group is not simple as an abstract group.
If abstract training is given, learners must also absorb the money and time costs of obtaining supplemental training for each distinct application.
If X and Y are algebraic structures of some fixed type ( such as groups, rings, or vector spaces ), and if the function f from X to Y is a homomorphism, then ker f will be a subalgebra of the direct product X × X. Subalgebras of X × X that are also equivalence relations ( called congruence relations ) are important in abstract algebra, because they define the most general notion of quotient algebra.
: If we take in our hand any volume ; of divinity or school metaphysics, for instance ; let us ask, Does it contain any abstract reasoning concerning quantity or number?
If we consider the rewriting relation as an indexed set of relations, as some authors do, then a labeled state transition system is equivalent with an abstract rewriting system with the indices being the labels.
If the operations are associative, each one defines the structure of a monoid on, and the conditions above are equivalent to the more abstract condition that * is a monoid homomorphism ( or vice versa ).
An even more abstract way of stating the theorem is: If is a monoid object in the monoidal category of monoids, then is in fact a commutative monoid.
If data were a journal article meta-information would be the abstract and conclusion.
Darwin received it on 18 June 1858, and wrote to Lyell that " your words have come true with a vengeance ,... forestalled " and " If Wallace had my MS. sketch written out in 1842, he could not have made a better short abstract!
During the Vietnam War the first rock protest songs were heard, inspired by the songs of folk musicians such as Woody Guthrie and Bob Dylan, which ranged from abstract evocations of peace Peter, Paul and Mary's " If I Had a Hammer " to blunt anti-establishment diatribes Crosby, Stills, Nash & Young's " Ohio ".
If a codex could not be obtained by fair means, he was ready to use fraud, as when he bribed a monk to abstract a Livy and an Ammianus from the library of Hersfeld Abbey.
If it were, mathematicians would work only with F < sup > n </ sup > rather than with abstract vector spaces.
: If, in page after page of The Fountainhead and Atlas Shrugged, you show someone being heroic by ruthlessly setting feelings aside, and if you show someone being rotten and depraved by, in effect, diving headlong into his feelings and emotions, and if that is one of your dominant methods of characterization, repeated again and again, then it doesn't matter what you profess, in abstract philosophy, about the relationship of reason and emotion.

If and index
If the one- or two-digit address of an index word or electronic switch is used or is included in the operand of an XRESERVE or SRESERVE statement ( see page 99 ), the corresponding index word or electronic switch is reserved.
If a statement has been assigned an address in the index word area
If the symbolic name or actual address of an index word or electronic switch appears or is included in the operand of an XRELEASE or SRELEASE statement ( see page 101 ), the specified index word or electronic switch will again be made available, regardless of the method by which it was reserved.
If, at any time during the assignment pass, the compiler finds that there are no more index words available for assignment, the warning message `` No More Index Words Available '' will be placed in the object program listing, the table will be altered to show that index words 1 through 96 are available, and the assignment will continue as before.
If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.
If each node additionally records the size of its subtree ( including itself and its descendants ), then the nodes can be retrieved by index in O ( log n ) time as well.
If the keys match, then a matching element has been found so its index, or position, is returned.
If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative directions of the x -, y -, and z-axes in a right-handed system.
If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.
If the list is stored as an array data structure, the location may be the index of the item found ( usually between 1 and n, or 0 and n − 1 ).
If the photons reside mainly in the higher index material, the confinement is due to total internal reflection.
If the reflected and direct image of the horizon are in line there is no index error.
If one is above the other adjust the index mirror until the two horizons merge.
If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index.
If the refractive index is lower on the other side of the boundary and the incident angle is greater than the critical angle, no light can pass through and all of the light is reflected.
If for example, visible light were traveling through acrylic glass ( with an index of refraction of 1. 50 ) into air ( with an index of refraction of 1. 00 ), the calculation would give the critical angle for light from acrylic into air, which is
If an array is used to represent a cycle, it is convenient to obtain the index with a modulo operator, which can result in zero.

If and notation
If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).
If and, then this abbreviated form is written in Leibniz notation as:
If certain coordinate systems are used, for instance, polar-toroidal coordinates ( common in plasma physics ) using the notation ∇ × F will yield an incorrect result.
If the datum to be hashed is small enough, one can use the datum itself ( reinterpreted as an integer in binary notation ) as the hashed value.
If one wishes to employ the notation for a different purpose ( such as denoting open intervals on the real number line ) the ordered pair may be denoted by the variant notation
If, e. g., u is the speed of a particle, then the straightforward notation for its velocity is u, < u > u </ u >, or.
If there are multiple operations, the operator is given immediately after its second operand ; so the expression written " 3 − 4 + 5 " in conventional infix notation would be written " 3 4 − 5 +" in RPN: first subtract 4 from 3, then add 5 to that.
If the center is a distinguished point considered as origin of E, as in a normed space, it is not mentioned in the definition and notation.
If the symbol "#" stands for a word boundary ( initial or final ), the notation "/ __ #"
If and are any complex numbers, the inner product is the standard inner product and the bar notation is used for complex conjugation then the inequality may be restated in more explicitly as
If composed before being performed, music can be performed from memory, through written musical notation, or through a combination of both.
If a rook is unmoved and can still castle, yet there is more than one rook on that side, FEN notation as traditionally interpreted is ambiguous.
If the Greek letter μ is not available, the notation uF is often used as a substitute for μF in electronics literature.
If instead the objects are groups not known to be abelian, then multiplicative rather than additive notation is traditional, and the identity element -- as well as the trivial group -- is often written as " 1 " instead of " 0 ".
If f ∈ L < sup > p </ sup >( μ ) and g ∈ L < sup > q </ sup >( μ ), then the notation is adequate.
If you apply this to vectors written in index notation, then this becomes
No notation is used to describe the set of all λ, which satisfy this condition, but for a subset: If λ-T does not have dense range but is injective, λ is said to be in the residual spectrum of T, denoted by σ < sub > r </ sub >( T ).
If a bipartite graph is not connected, it may have more than one bipartition ; in this case, the notation is helpful in specifying one particular bipartition that may be of importance in an application.
If the notation He is used for these Hermite polynomials, and H for those above, then these may be characterized by
If the variables being used form an indexed family like,,, ..., then multi-index notation is helpful: if we write
If for some constant ( using Big O notation )
If E is the standard basis, the transformation from B to E can be represented with the following simplified notation:
If a piece of music for a two-handed instrument has a passage where only one hand is notated, lines of numbered notation without score brackets at the left can be used for this passage.
If present, this will be identical to the metronome marks used in Western music ( this is the only place in numbered notation where Western symbols for note values such as quarter-notes and eighth-notes are used ).

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