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The plate then deflects downward to medications and grapefruit buy keppra 500 mg on-line form a "dish" of depth b symptoms hypoglycemia cheap 250mg keppra with mastercard, then a "dome" of height b2 medications like prozac purchase keppra 500 mg without a prescription, and so on medicine 54 092 generic 250 mg keppra fast delivery. The value an measures the amplitude "lost" at the nth oscillation (due to friction, say). Show that zero, that is, lim,b,+, Cr=,a, = bo, and explain why this is physically obvious. The East and West crews were setting track 12 miles apart, the East crew working at 5 miles per hour, the West crew working at 7 miles per hour. The official with the Golden Spike travelled feverishly by carriage back and forth between the crews until the rails joined. If we can prove that a given series converges, we can approximate its sum to any desired accuracy by adding up enough terms. One way to tell whether a series converges or diverges is to compare it with a series which we already know to converge or diverge. As a fringe benefit of such a "comparison test," we sometimes get an estimate of the difference between the nth partial sum and the exact sum. Thus if we want to find the sum with a given accuracy, we know how many terms to take. The following fact is true for any series: by A careful proof of this fact is given at the end of this section; for now we implies that the absolute simply observe that the convergence of ~F=",a,l values la,l approach zero quickly, and the possibility of varying signs in the aiYs can only help in convergence. Therefore, if 0 < la,/ < b, and Cb, converges, then Cla,l converges, and therefore so does Cai. If the terms of two series C a i and Cb, "resemble" one another, we may expect that one of the series converges if the other does. This is the case when the ratio ai/bi approaches a limit, as can be deduced from the comparison test. For instance, suppose that lim,+,(la,//b,) = M < oo, with all bi > 0 Then for large enough i, we have lai[/ b, < M I, or /a,/ < (M l)b. Now if C bi converges, so does C (M l)bi, by the constant multiple rule for series, and hence Ca, converges by the comparison test. A the following tests can both be similarly justified using the original comparison test. To choose bi in applying the ratio comparison test, you should look for the "dominant terms" in the expression for a. Strictly speaking, to apply the comparison test we should have lai/ < (M + l)bi for all i, not just sufficiently large i; but, as we saw earlier, the convergence or divergence of a series C a, is not affected by the values of its "early" terms, but only the behavior of ai for large i. The next example illustrates how one may estimate the difference between a partial sum and the full series. We sometimes refer to this difference as a tail of the series; it is equal to the sum of all the terms not included in the partial sum. Example 5 (see Example 2) and estimate the difference i=1 13 between this partial sum and the sum of the entire series. A + the second kind of series which we will treat in this section is called an alternating series. If we put a minus sign in front of every other term to obtain the series l - + + + - L 4. 0 This argument also shows that the sum S lies between any two successive partial sums, so that the tail corresponding to the partial sum Snis less than lan+11,the size of the first omitted term. If Cy= is a series such that the ai alternate in sign, are decreasing in absolute value, and tend to zero, then it converges. Thus the series is not an alternating one and the alternating series test does not apply. To make the comparison between la,l and b, precise, look at the ratios: limi+,(lail/bi) = lim,+,[i/(i 4)] = 1, so C? To see whether the absolute values lafl form a decreassign and lim,a, ing sequence, it is convenient to look at the function f(x) = &/(x 4). The derivative is + + + which is negative for x > 4, so f(x) is decreasing for x > 4.
In the experiment medications in spanish cheap 500 mg keppra visa, 47 regular coffee drinkers consumed 500 milligrams of caffeine in a pill form (equivalent to medicine for pink eye 250 mg keppra sale four 8-oz cups) during one workday medicine 751 buy generic keppra 500mg line, and a placebo pill during another workday medicine for anxiety keppra 500mg generic. Identify the response variable(s), explanatory variable, experimental units, and the treatments. Each subject who suffers from hay fever and volunteers for the study is observed on three separate days, with a different treatment used each day. There are two days between treatments, so that a treatment does not have a carry-over effect for the next treatment assigned. How can an experiment explore this possibility in a study to compare bupropion with placebo? Suppose the researchers split the subjects into two groups: those who live with another smoker, and those who do not live with smokers. Within each group, the subjects are randomly assigned to take bupropion or a placebo. The figure shows a flow chart of this design, when 250 of the 429 study subjects live with nonsmokers and 179 live with another smoker. Experimental Units: 429 Smokers Block the subjects according to whether they live with a smoker. Treatment 1 Bupropion (125 subjects) Treatment 2 Placebo (125 subjects) Treatment 1 Bupropion (90 subjects) Treatment 2 Placebo (89 subjects) Compare the two treatment percentages of those who have relapsed. An experiment assigns subjects to experimental conditions (such as drug or placebo) called treatments. An observational study is a type of nonexperimental study that observes subjects on the response and explanatory variables. The study samples the population of interest, but merely observes rather than applies treatments to those subjects. Since association does not imply causation, with observational studies we must be aware of potential lurking variables that influence the association. In an experiment, a researcher uses randomization in assigning experimental units (the subjects) to the treatments. To reduce bias, experiments should be double-blind, with neither the subject nor the data collector knowing to which treatment a subject was assigned. A sample survey is a type of nonexperimental study that takes a sample from a population. A simple random sample of n subjects from a population is one in which each possible sample of size n has the same chance of being selected. A cluster random sample takes a simple random sample of clusters (such as city blocks) and uses subjects in those clusters as the sample. A stratified random sample divides the population into separate groups, called strata, and then selects a simple random sample from each stratum. Be cautious of results from studies that use a convenience sample, such as a volunteer sample, which Internet polls use. Prospective studies follow subjects into the future, as is true with many experiments. The cases and controls are compared on an explanatory variable, such as whether they had been smokers. With a completely randomized design, subjects are assigned randomly to categories of each explanatory variable. Often this is the same subject observed for each of two treatments, such as in a crossover study. Observational versus experimental study Without using technical language, explain the difference between observational and experimental studies to someone who has not studied statistics. Illustrate with an example, using it also to explain the possible weaknesses of an observational study. Unethical experimentation Give an example of a scientific question of interest for which it would be unethical to conduct an experiment. In trying to identify the population under consideration, we must recognize that different organizations have different definitions of the term unemployment. However, she also knows about a teacher who died recently who may have had asbestos exposure. In deciding whether to leave teaching, should she give more weight to the study or to the story she heard about the teacher who died? It was reported that 44% of the more than 3300 Indiana resident respondents believed Butler would win the regional, and 78% of the more than 5600 Wisconsin resident respondents believed Wisconsin would win.
The relativity o f points o f v ie w o f space treatment croup cheap keppra 500mg, tim e and energy do not symptoms after miscarriage discount keppra 250mg line, fo r all that medications reactions 500 mg keppra amex, absorb the real into the dream medicine 6 year order 250mg keppra mastercard. The category o f Tim e dissolves into cosm o logical reflections on the B ig Bang even as the category o f ir reversibility is affirm ed. Residual o b jectivity is w hat resists scanning by the infinite variation o f points o f v ie w constitutable upon it. A d ifferen t panorama, an- 168 Fashionable Nonsense other ontological consistency. Th e m ecanosphere draws out and actualises configurations w hich exist am ongst an infinity o f others in fields o f virtuality. Existential machines are at the same level as being in its intrinsic multiplicity. T h ey are not m ediated by transcendent signifiers and subsumed by a uni vo ca l on tological foundation. Existence, as a process o f deterritorialisation, is a specific inter-machinic operation w hich superim poses itself on the prom otion o f singularised existential intensities. Besides, the article of Guattari (1988) on tensor calculus applied to psychology is a real gem. With an astonishing erudition, which mixes spacedistances and time-distances, this researcher opens up an important field o f philosophical questions that he calls "dromocracy" (from the Greek dromos: speed). T h e y c o n ta in a p le t h o r a o f r e fe r e n c e s t o p h y sic s, p a rtic u la rly th e the o r y o f re la tiv ity. F u r th e r m o re, h is a n a lo g ie s b e t w e e n p h y s ic s a n d s o c ia l q u estio n s a re th e m o s t a rb itra ry im a g in a b le, w h e n h e d o e s n o t s im p ly b e c o m e in to x ic a the d with h is o w n w o r d s. L e t us s ta rt w it h a m in o r e x a m p le o f th e a s to n is h in g e ru d i tio n v a u n the d b y L e M o n d e: Recent m e g a l o p o u t a n hyperconcentration (Mexico City, Tokyo. Drawing inspiration from the theory of relativity, Virilio con tinues: H o w can w e fully take in such a situation w ithout enlist ing the aid o f a n ew type o f interval, k in d the in t e r v a l o f the l ig h t (neutral sign)? Th e relativistic innovation o f this third interval is actually in itself a sort o f unremarked cultural re v elation. If the interval o f space t im e (p ositive sign) and the interval o f (n egative sign) have laid out the geography and history o f the w orld through the geom etric design o f agrarian areas (fragm entation into plots o f land) and urban areas (the cadas tral system), the organization o f calendars and the measure m ent o f tim e (c lo c k s) have also presided o v e r a vast ch ronopolitical regulation o f human societies. The very re cent em ergence o f an interval o f the third type thus signals a sudden qualitative leap, a profound mutation in the relation ship betw een man and his surroundings. See note 221 below for a critique o f the published English 2 translations (Virilio 1993, p. In English physics usage, "speed" designates the length o f the velocity vector and thus can never be negative. However, these are inter vals in space-time, which do not coincide with what we habit ually call "space" and "time". The "very recent emer gence of an interval of the third type" is nothing but a pedantic allusion to modem telecommunications. In this passage, Virilio shows perfectly how to package a banal observation in sophis ticated terminology. What comes next is even more surprising: Listen to the physicist speaking about the lo gic o f par ticles: "A representation is defin ed by a com p lete set o f com m uting observables. It has nothing to do with "real time" or with any "macroscopic logic" (quite the 22 2 the book by Taylor and Wheeler (1966) gives a beautiful introduction to the notion o f space-time interval. See note 224 below for a critique o f the published English translations (Virilio 1993, p. But above all, in order to un derstand the precise meaning of this sentence, one needs to have studied physics and mathematics seriously for several years. We find it incredible that Virilio could consciously copy a sentence that he manifestly does not understand, add to it a completely arbitrary comment, and still be taken seriously by editors, commentators, and readers. Transparence is no longer com posed o f light rays (so la r or electric) but instead o f elem ental particles (electrons and pho ton s) that are transmitted at the speed o f light. One rendered it as "A representation is defined by a sum o f observables that are flickering back and forth" (Virilio 1993, p. It will definitely contradict those who still think that postmodemity is merely a fashionable term or an empty fad. The nagging opinion that cultural and critical theory is "too abstract," hopelessly removed from reality, devoid o f ethical values and above all incompatible with erudition, systematic thinking, intellectual rig o r and creative criticism, will simply be pulverized. This collection assembles some of the most recent and fresh work by leading culture critics and theoreticians o f the arts and sciences, such as Paul Virilio, Felix Guattari.
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Since then a considerable number of further decision problems have been shown to symptoms 9 days post ovulation order keppra 250 mg line be unsolvable medications 4 less canada buy 250mg keppra with visa. They are all proved to treatment 7th feb cardiff purchase keppra 250 mg free shipping be unsolvable by showing that if they were solvable one could use the solution to treatment 6th nerve palsy buy keppra 250 mg fast delivery provide a solution of the original one. No attempt is made to explain the technical terms used, as most readers will be familiar with some of them, and the space required for the explanation would be quite out of proportion to its usefulness in this context. No adequately complete proof is yet available, but if it is correct this is a considerable step forward. Although quite a number of decision problems are now known to be unsolvable, we are still very far from being in a position to say of a given decision problem, whether it is solvable or not. Indeed, we shall never be quite in that position, for the question whether a given decision problem is solvable is itself one of the undecidable decision problems. The results which have been found are on the whole ones which have fallen into our laps rather than ones which have positively been searched for. Considerable efforts have however been made over the word problem in groups (see (5) above). This is something like one of the problems we have already mentioned, that concerning the twisted wire puzzles. Given a number of interlacing pieces of plasticine one may be asked to transform them in this way into another given form. A similar decision problem which might well be unsolvable is the one concerning knots which has already been mentioned. The results which have been described in this article are mainly of a negative character, setting certain bounds to what we can hope to achieve purely by reasoning. Turing presents his ideas on computability from an unusual perspective and puts them in a distinctive light. The essay is playful, yet serious; strange, yet familiar; informal, yet proves rigorously the unsolvability of a decision problem. Substitution puzzles If we consider, as Turing does, ordinary puzzles like those concerning sliding squares or knots, then a particular type of question can be raised immediately, namely, is it possible to transform a given initial position into another configuration via the finitely many moves the puzzle allows? In the case of sliding squares, take this configuration as the initial one: 1 3 5 7 2 4 6 8 15 13 11 9 14 12 10 Is it possible to rearrange this configuration, exploiting the empty square and sliding the numbered squares, into the following one? Examining the Work and Its Later Impact 333 Concerning puzzles for knots, Turing asks (on p. For these puzzles, one has an unlimited supply of counters, possibly only of two distinct kinds, say B (lack) and W (hite). An initial configuration is a finite sequence of such counters, and the puzzle task is to transform a given configuration into another one by substitutions from a fixed finite list. When facing a seemingly intractable puzzle, frustration might suggest the natural and general question, how one can tell whether this or any other puzzle is solvable. There is no systematic procedure for determining whether a puzzle is solvable or not. How are the concepts in the statement of the theorem defined so that it can be proved by a rigorous argument? In Section 2, I try to fill in some conceptually significant parts and show how the 1954-presentation wonderfully dovetails with the 1936-analysis of (mechanical) calculability. Turing views this not-quite-definite statement as being situated between a theorem and a definition: In so far as we know a priori what is a puzzle and what is not, the statement is a theorem. In so far as we do not know what puzzles are, the statement is a definition that tells us something about what they are. Turing summarises the status of the methodological discussion among logicians when asserting, `. Let me briefly recall what a (two-letter) Turing machine is; in doing so I follow Post and Davis rather than Turing. The tape is divided into squares, and each square may carry a symbol from a finite alphabet, say, just the twoletter alphabet consisting of 0 and 1 or, to emphasise the connection to substitution puzzles, B and W. The machine is able to scan one square at a time and perform, depending on the content of the observed square and its own internal state, one of four operations: print 0, print 1, or shift attention to one of the two immediately adjacent squares. The operation of the machine is fixed by a finite list of commands in the form of quadruples qi sk cl qm that express the following: If the machine is in 2 That is, I am following the development in Post (1947) and Davis (1958). For example, to realise that proving a theorem in an axiomatic system can be viewed as solving a substitution puzzle is an easier task than to grasp the connection between theorem proving and computations of a two-letter machine! First, the reduction of substitution puzzles to Turing machines is not crucial for the central observations (and can easily be established).