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Concretely and means
Concretely, this means that it is the semidirect product of Δ ( n, F ) by the symmetric group S < sub > n </ sub >:
Concretely, a rational curve of dimension n over F can be parameterized ( except for isolated exceptional points ) by means of n rational functions defined in terms of a single parameter t ; by clearing denominators we can turn this into
Concretely, by a central representation one means a representation such that the center of the Heisenberg group maps into the center of the algebra: for example, if one is studying matrix representations or representations by operators on a Hilbert space, then the center of the matrix algebra or the operator algebra is the scalar matrices.
Concretely, a connected compact 1-manifold with boundary is an interval or a circle ( compactness rules out the open interval and the half-open interval neither of which yields non-trivial embeddings since the open end means that they can be shrunk to a point ), so a possibly disconnected compact 1-manifold is a collection of n intervals and m circles The condition that the boundary of X lies in
Concretely, this can be achieved for instance by defining a topological similarity, by using ontologies to define a distance between words ( a naive metric for terms arranged as nodes in a directed acyclic graph like a hierarchy would be the minimal distance — in separating edges — between the two term nodes ), or using statistical means such as a vector space model to correlate words and textual contexts from a suitable text corpus ( co-occurrence ).
Concretely this means that f satisfies the equation f () = χ ( V ) for every finitely generated R-module V and f is the only group homomorphism that does that.
Concretely, that means that a business rule may do things like detect that a business situation has occurred and raise a business event ( typically carried via a messaging infrastructure ) or create higher level business knowledge ( e. g., evaluating the series of organizational, product, and regulatory-based rules concerning whether or not a loan meets underwriting criteria ).
Concretely, this means the following.

Concretely and if
This yields a convenient way of generating an equivalence relation: given any binary relation R on X, the equivalence relation generated by R is the smallest equivalence relation containing R. Concretely, R generates the equivalence relation a ~ b if and only if there exist elements x < sub > 1 </ sub >, x < sub > 2 </ sub >, ..., x < sub > n </ sub > in X such that a

Concretely and element
Concretely, for a partially ordered set ( S, ≤) one represents each element of S as a vertex in the plane and draws a line segment or curve that goes upward from x to y whenever y covers x ( that is, whenever x < y and there is no z such that x < z < y ).

Concretely and x
Concretely, elements of the cotangent space are linear functionals on T < sub > x </ sub > M.

Concretely and is
Concretely, a stable class is one that does not change when one adds a trivial bundle:.
*: Concretely, the kernel of the action of GL on projective space is exactly the scalar maps, which are quotiented out in PGL.
Concretely, an example of a boundary value ( in one spatial dimension ) is the problem
Concretely, these are most used for learning foreign vocabulary where the foreign pronunciation is not transparent from the foreign writing – in this case the Question is the native word, the Answer is the foreign word ( written ), and the pronunciation is always part of the answer ( Answer *).
Concretely, for the two-dimensional Radon transform, the dual transform is given by
Concretely, this is just an action of, as this generates the algebra, and the operator representing ( the image of i in End ( V )) is denoted J ( to avoid confusion with the identity matrix I ).
where is the universal cover of H. Concretely, the universal covering group of H is the space of homotopy classes of paths in H with pointwise multiplication of paths.
Concretely, this is just an action of i, as this generates the algebra, and the operator representing i ( the image of i in End ( V )) is exactly J.
Concretely, this is because scalar multiplication by i commutes with scalar multiplication by real numbers – – and distributes across vector addition.

Concretely and by
Concretely, when reading an article ( in a web browser ), the reader selects extracts ( similar to underlining or highlighting a paper article ), which are then converted to question-answer format, often by cloze deletion, and then scheduled for learning and review by the learning software.
Concretely, recommendations can be used by the Commission to raze barriers of competition caused by the establishment or the modification of internal norms of a Member State.
Concretely, given a vector space V, it has an underlying affine space A obtained by “ forgetting ” the origin, with V acting by translations, and the affine group of A can be described concretely as the semidirect product of V by GL ( V ), the general linear group of V:
Concretely, any straight line L can be parametrized by

Concretely and which
Concretely: " he long-term average annual profit in the VOC's 1630-70 ' Golden Age ' was 2. 1 million guilders, of which just under half was distributed as dividends and the remainder reinvested.

Concretely and order
Concretely, in Pin < sub >+</ sub >, has order 2, and the preimage of a subgroup

Concretely and are
Concretely, these are the matrices and can be interpreted as hyperbolic rotations, just as the group SO ( 2 ) can be interpreted as circular rotations.
Concretely, in a linear regression where the errors are identically distributed, the variability of residuals of inputs in the middle of the domain will be higher than the variability of residuals at the ends of the domain: linear regressions fit endpoints better than the middle.

Concretely and so
Concretely, given the infinitely branching tree of infinite depth, label the root node the children of the root node the grandchildren and so forth.

means and if
Only by means of an intensive preoccupation with the detailed considerations following from any decision can he ensure attention to the practical details to be dealt with if the implications of immorality in the major decision are effectively to be checked.
But that scarcely means that he was the aloof, forbidding type of student who shared few if any activities with his fellows, the banter of the surviving prolusions providing enough evidence to deny this.
If man is actually the product of his environment and if science can discover the laws of human nature and the ways in which environment determines what people do, then someone -- a someone probably standing outside traditional systems of values -- can turn around and develop completely efficient means for controlling people.
That means that, if we are to achieve a new standard of leadership, we must think in terms of the total context of our situation.
If we fail to develop the means to hunt down and destroy the enemy's military force with extreme care and precision, and if war comes in spite of our most ardent desires for peace, our choice of alternatives will be truly frightening.
This means that if your insured vehicle is laid up for more than 30 days, insurance can be suspended and a proportionate return of your premium made to you.
This, however, is sufficient to show that more or less non-violent resistance and economic conflict ( if both sides are strong enough ) can be war of all against all no less than if other means are used.
Mythological concepts may by all means still be used, but they can be used responsibly only as `` symbols '' or `` ciphers '', that is, only if they are also constantly interpreted in nonmythological ( or existential ) terms.
If death receives more than its share of attention from the theologian and if sin receives less than its share, the gift of the life eternal through Christ begins to look like the divinely appointed means of rescue from temporal, i.e., created, existence.
It means to do it and to know one is doing it, but as only a secondary if certain effect of the good one primarily does and intends.
And he can be on the way to Memphis, your Gratt Shafer can, and you discover you can stay alive and hate him and love him and want him even if it means you want him -- really want him -- dead.
Yet, if he used all of the little means at his disposal, he would be instructing his students wrongly.
However Abdul is a common Arabic prefix meaning " Servant of the " and " Al " is Arabic for " the ", and if " hazra " means " he prohibited ", " he fenced in " or " Great Lord ", then the name would mean " Servant of the Prohibited ", " Servant of the Fenced in ", or " Servant of the Great Lord " which would make sense considering his role, even if it is not a proper Arabic name.
This means that the individual atoms can be treated as if each were in isolation, as the vast majority of the time they are.
" It is ambiguous whether he means the ambrosia itself is rosy-red, or if he is describing a rosy-red nectar Hermes drinks along with the ambrosia.
Other major ideas in the book of Amos include: social justice and concern for the disadvantaged ; the idea that Israel's covenant with God did not exempt them from accountability for sin ; God is God of all nations ; God is judge of all nations ; God is God of moral righteousness ; God made all people ; God elected Israel and then liberated Israel so that He would be known throughout the world ; election by God means that those elected are responsible to live according to the purposes clearly outlined to them in the covenant ; if God destroys the unjust, a remnant will remain ; and God is free to judge whether to redeem Israel.
This means that opaque ( if not minimal ) contrasts can occur in languages like Italian ( with the i-like sound of piede ' foot ', appearing in the nucleus:, and that of piano ' slow ', appearing in the syllable onset: ) and Spanish ( with a near minimal pair being abyecto ' abject ' and abierto ' opened ').
* Knowledge ark, a collection of knowledge preserved in such a way that future generations would have access to said knowledge if current means of access were lost
This means that if one said
xiii, " Cum ad Sacrosanctae Romanae Ecclesiae ") prescribes their work, determines how much they may charge for their labour, fixes a certain tax for an abstract or abridgment of twenty-five words, or their equivalent, 150 letters, forbids them to charge more, even though the abstract goes over twenty-five words but less than fifty words, enacts that the basis of the tax is the labour employed in writing, expediting, etc., the Bulls, and by no means the emoluments accruing to the recipient of the favour or benefice conferred by the Bull, and declares that whoever shall charge more than the tax fixed by him shall be suspended for six months from office, and upon a second violation of the law, shall be deprived of it altogether, and if the delinquent be an abbreviator, he shall be excommunicated.
Austrian Germanist Rudolf Simek says that Bifröst either means " the swaying road to heaven " ( also citing bifa ) or, if Bilröst is the original form of the two ( which Simek says is likely ), " the fleetingly glimpsed rainbow " ( possibly connected to bil, perhaps meaning " moment, weak point ").
When bearing off, a player may also move a lower die roll before the higher even if that means ' the full value of the higher die ' is not fully utilized.
Thus, in the Hindu schools, if a claim was made that could not be substantiated by appeal to the textual canon, it would be considered as ridiculous as a claim that the sky was green and, conversely, a claim which could not be substantiated via conventional means might still be justified through textual reference, differentiating this from the epistemology of modern science.

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