If God and Nature were distinct, then it must be the case that Nature had some qualities that God lacked, and the idea of a supreme being lacking anything was incoherent.
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"When you have to draw light 24 times in every second of a two hour movie, it must be the case that you gain a new appreciation of how it can tell a story of its own," he says.
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For a federal habeas court to intervene, it must be the case that the state court adjudication was inconsistent not just with some general understanding of the "clearly established law" at the time, but rather with law that was at the time clearly established by the U.S. Supreme Court.
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A functionalist would argue that it must be the case that qualia would not be created in such a scenario, thus demonstrating that qualia cannot exist without conflicting with the fundamental tenets of Functionalism.
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This person must have gained this knowledge in a prior life, and is now merely recalling it from memory. Since the person in Socrates' story is able to provide correct answers to his interrogator, it must be the case that his answers arose from recollections of knowledge gained during a previous life.
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But this is a contradiction, and thus it must be the case that at least one of the coefficients is transcendental. The non-computable numbers are a strict subset of the transcendental numbers. All Liouville numbers are transcendental, but not vice versa. Any Liouville number must have unbounded partial quotients in its continued fraction expansion.
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1171-1182, July 1992. The proof involves a reduction to the n-graph colorability problem. In other words, solving the SLE RWA problem is as complex as finding the chromatic number of a general graph. Given that dynamic RWA is more complex than static RWA, it must be the case that dynamic RWA is also NP-complete.
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In particular, they should rent for more. As Figure 3 shows, this is true. Renting a square foot in a new building is much more expensive than renting a square foot in a building forty years old. So it must be the case that you are paying for a nicer, more functional and maybe even safer building.
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An M-matrix is commonly defined as follows: Definition: Let be a real Z-matrix. That is, where for all . Then matrix A is also an M-matrix if it can be expressed in the form , where with , for all , where is at least as large as the maximum of the moduli of the eigenvalues of , and is an identity matrix. For the non-singularity of , according to the Perron-Frobenius theorem, it must be the case that .
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For, a perpendicular to the midpoint of each polygon side is a radius, of length r. And since the total side length is greater than the circumference, the polygon consists of n identical triangles with total area greater than T. Again we have a contradiction, so our supposition that C might be less than T must be wrong as well. Therefore, it must be the case that the area enclosed by the circle is precisely the same as the area of the triangle. This concludes the proof.
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In this vein, he has published anthropological work on the cargo cults of the South Pacific and has contributed anthropological studies on the media. His adherence to functionalism in the study of the social differs from that of Durkheim (and his followers) in holding that knowledge and ideas must be presented as causal variables. Further, Jarvie contends, it must be the case that a functionalist framework with an active role for explanatory ideas requires a conception of rationality towards ideas. Politically, Jarvie is a liberal.
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It must be the case that the transformed output g(z) is the correct output for the original input x. That is, if the input-output relations of X and Y are expressed as functions, then their function composition must obey the identity X=g\circ Y\circ f. Alternatively, expressed in terms of algorithms, one possible algorithm for solving X would be to apply f to transform the problem into an instance of Y, solve that instance, and then apply g to transform the output of Y into the correct answer for X.
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Where this risk is material the regulator has no choice but to ensure the regulated firm is adequately compensated on average ex ante. As a general rule, where the regulatory framework ensures that the regulated firm is adequately compensated on average it must be the case that if there is some positive probability that the out-turn return of the firm is below the allowed cost of capital there must be a positive probability that the out-turn return of the firm is above the allowed cost of capital so that, on average, the firm expects to receive the allowed cost of capital.
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A better approach is known as the "grace hash join", after the GRACE database machine for which it was first implemented. This algorithm avoids rescanning the entire S relation by first partitioning both R and S via a hash function, and writing these partitions out to disk. The algorithm then loads pairs of partitions into memory, builds a hash table for the smaller partitioned relation, and probes the other relation for matches with the current hash table. Because the partitions were formed by hashing on the join key, it must be the case that any join output tuples must belong to the same partition.
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Since each point in an image corresponds to a line in 3D space, all points on the line in 3D are projected to the point in the image. If a pair of corresponding points in two, or more images, can be found it must be the case that they are the projection of a common 3D point x. The set of lines generated by the image points must intersect at x (3D point) and the algebraic formulation of the coordinates of x (3D point) can be computed in a variety of ways, as is presented below. In practice, however, the coordinates of image points cannot be measured with arbitrary accuracy.
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This means that over a field K with char(K) ot= 2 every hyperelliptic curve of genus g is isomorphic to one given by an equation of the form C : y^2 = f (x) where f is a monic polynomial of degree 2g + 1 and the curve has no singular points. Note that for curves of this form it is easy to check whether the non-singularity criterion is met. A point P = (a,b) on the curve is singular if and only if b = 0 and f'(a) = 0. As b = 0 and b^2 = f(a) , it must be the case that f(a) = 0 and thus a is a multiple root of f .
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Intuitively, then, this axiom states that since, at any point of verifying that A holds of \alpha, we will only have verified that A holds for a finite initial sequence of \alpha; thus, it must be the case that A also holds for any lawless sequence \beta sharing this initial sequence. This is so because, at any point in the procedure of verifying A(\alpha), for any such \beta sharing the initial prefix of \alpha encoded by n that we have already examined, if we run the identical procedure on \beta, we will get the same result. The axiom can be generalized for any predicate taking an arbitrary number of arguments. Another axiom is required for lawless sequences.
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In lattice theory, a bounded lattice L is called a 0,1-simple lattice if nonconstant lattice homomorphisms of L preserve the identity of its top and bottom elements. That is, if L is 0,1-simple and ƒ is a function from L to some other lattice that preserves joins and meets and does not map every element of L to a single element of the image, then it must be the case that ƒ−1(ƒ(0)) = {0} and ƒ−1(ƒ(1)) = {1}. For instance, let Ln be a lattice with n atoms a1, a2, ..., an, top and bottom elements 1 and 0, and no other elements. Then for n ≥ 3, Ln is 0,1-simple.
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For the algorithm to be correct, it must be the case that popping and merging the top two clusters from the algorithm's stack preserves the property that the remaining clusters on the stack form a chain of nearest neighbors. Additionally, it should be the case that all of the clusters produced during the algorithm are the same as the clusters produced by a greedy algorithm that always merges the closest two clusters, even though the greedy algorithm will in general perform its merges in a different order than the nearest-neighbor chain algorithm. Both of these properties depend on the specific choice of how to measure the distance between clusters. The correctness of this algorithm relies on a property of its distance function called reducibility.
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To prove the reverse direction, let x \in A^\complement \cup B^\complement, and for contradiction assume x ot\in (A\cap B)^\complement. Under that assumption, it must be the case that x \in A\cap B, so it follows that x \in A and x \in B, and thus x ot\in A^\complement and x ot\in B^\complement. However, that means x ot\in A^\complement \cup B^\complement, in contradiction to the hypothesis that x \in A^\complement \cup B^\complement, therefore, the assumption x ot\in (A\cap B)^\complement must not be the case, meaning that x \in (A\cap B)^\complement. Hence, \forall x( x \in A^\complement \cup B^\complement \rarr x \in (A\cap B)^\complement), that is, A^\complement \cup B^\complement \subseteq (A\cap B)^\complement.
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A function is equivariant under the shift map if the transformation on configurations described by commutes with the shift map; that is, for every configuration , it must be the case that . Intuitively, this means that every position of the configuration is updated by using the same rule as every other position. A cellular automaton is defined by a rule for computing the new value of each position in a configuration based only on the values of cells in a prior-fixed finite neighborhood surrounding the position, with all positions of the configuration being updated simultaneously based on the same update rule. That is, the new value of a position is a function only of the values of the cells in its neighborhood rather than depending more generally on an unbounded number of cells of the previous configuration.
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For instance, as the Cartesian skeptic will point out, all of my perceptual experiences are compatible with a skeptical scenario in which I am completely deceived about the existence of the external world, in which case most (if not all) of my beliefs would be false. The typical conclusion to draw from this is that it is possible to doubt most (if not all) of my everyday beliefs, meaning that if I am indeed justified in holding those beliefs, that justification is not infallible. For the justification to be infallable, my reasons for holding my everyday beliefs would need to completely exclude the possibility that those beliefs were false. Consequently, if a belief must be infallibly justified in order to constitute knowledge, then it must be the case that we are mistaken in most (if not all) instances in which we claim to have knowledge in everyday situations.
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The special feature that both Y chromosomes and mtDNA display is that mutations can accrue along a certain segment of both molecules and these mutations remain fixed in place on the DNA. Furthermore, the historical sequence of these mutations can also be inferred. For example, if a set of ten Y chromosomes (derived from ten different men) contains a mutation, A, but only five of these chromosomes contain a second mutation, B, then it must be the case that mutation B occurred after mutation A. Furthermore, all ten men who carry the chromosome with mutation A are the direct male line descendants of the same man who was the first person to carry this mutation. The first man to carry mutation B was also a direct male line descendant of this man, but is also the direct male line ancestor of all men carrying mutation B. Series of mutations such as this form molecular lineages.
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Speculators on the other hand, are interested in making a profit, and will hence only enter the contracts if they expect to make money. Thus, if speculators are holding a net long position, it must be the case that the expected future spot price is greater than the forward price. In other words, the expected payoff to the speculator at maturity is: :E(S_{T}-K) = E(S_{T}) - K, where K is the delivery price at maturity Thus, if the speculators expect to profit, :E(S_{T}) - K > 0 :E(S_{T}) > K :E(S_{T}) > F_{0}, as K = F_{0} when they enter the contract This market situation, where E(S_{T}) > F_{0}, is referred to as normal backwardation. Forward/futures prices converge with the spot price at maturity, as can be seen from the previous relationships by letting T go to 0 (see also basis); then normal backwardation implies that futures prices for a certain maturity are increasing over time.
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Mathematically, the estimated probabilities of each color ball can be represented as: R, Y, and B. If you strictly prefer Gamble A to Gamble B, by utility theory, it is presumed this preference is reflected by the expected utilities of the two gambles: specifically, it must be the case that : R \cdot U(\$100) + (1-R) \cdot U(\$0) > B\cdot U(\$100) + (1-B) \cdot U(\$0) where U( ) is your utility function. If U($100) > U($0) (you strictly prefer $100 to nothing), this simplifies to: : R [U(\$100) - U(\$0)] > B [U(\$100) - U(\$0)] \Longleftrightarrow R > B \; If you also strictly prefer Gamble D to Gamble C, the following inequality is similarly obtained: : B\cdot U(\$100) + Y\cdot U(\$100) + R \cdot U(\$0) > R \cdot U(\$100) + Y\cdot U(\$100) + B \cdot U(\$0) This simplifies to: : B [U(\$100) - U(\$0)] > R [U(\$100) - U(\$0)] \Longleftrightarrow B > R \; This contradiction indicates that your preferences are inconsistent with expected- utility theory.
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The original definition by McCarthy was syntactical rather than semantical. Given a formula T and a predicate P, circumscribing P in T is the following second-order formula: :T(P) \wedge \forall p eg (T(p) \wedge p In this formula p is a predicate of the same arity as P. This is a second-order formula because it contains a quantification over a predicate. The subformula p is a shorthand for: :\forall x (p(x) \rightarrow P(x)) \wedge eg \forall x (P(x) \rightarrow p(x)) In this formula, x is a n-tuple of terms, where n is the arity of P. This formula states that extension minimization has to be done: in order for a truth evaluation on P of a model being considered, it must be the case that no other predicate p can assign to false every tuple that P assigns to false and yet being different from P. This definition only allows circumscribing a single predicate. While the extension to more than one predicate is trivial, minimizing the extension of a single predicate has an important application: capturing the idea that things are usually as expected.
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He says that if motorcycle racers, or even non-professional advanced riders who ride modern sport bikes near their performance limits, were approaching the limits of traction blindly, they would be like a group of blind men wandering around the top of a building, and most of them would wander off the edge and fall. In fact, Spiegel says, crashes among skilled high speed riders are so infrequent that it must be the case that they can feel where the limit of traction is as they approach the limit, before they lose traction. Spiegel's physiological and psychological experiments helped explore how it is possible for a good rider to extend his perception beyond the controls of his motorcycle out to the interface between the contact patches of his motorcycle and the road surface. Those subscribing to the first and fourth of Packer's risk categories are likely to believe no rider can sense when he is near the traction limit, while the second and third risk categories include those who share Spiegel's view that a rider need not lose traction and start to skid to know where the limit is.
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