# How To Count Citations If You Must

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 Perhaps the best-known citation index beyond a total citation count is the h-index, see Hirsh (2005). A researcher s h-index is defined to be the largest number, h, of his/her papers, each of which has at least h citations. By design, the h-index limits the effect of a small number of highly cited papers, a feature which, though well intentioned, can produce intuitively implausible rankings. For example, consider two researchers, one with 10 papers, each with 10 citations (h-index =10), and another with 8 papers, each with 100 (or even 1000) citations (h-index =8). Improvements to the h-index have been suggested. Consider, for example, the g-index (Egghe 2006a, 2006b), which is the largest number of papers, g, whose total sum of citations is at least g 2. The g-index is intended to correct for the insensitivity of the h-index to the number of citations received by the h papers with the most citations. Many other variations have been suggested since. Yet, like the h-index, they are ad hoc measures based almost entirely on intuition and rules of thumb with insufficient justification given for choosing them over the infinitely many other unchosen possibilities. But for a novel empirical approach to selecting among a new class of h-indices, see Ellison (2012, 2013). 2 Aformalwaytofind a good index is to take an axiomatic approach. The methodology of this approach is to first select, with care, a number of basic properties that any acceptable index must have. Restricting attention only to indices that satisfy all of these properties is a systematic way of narrowing down the set of potential indices. The advantage of this approach is that it focuses attention on the properties that an index should possess rather than the functional form that it should take. After all, it is the properties we desire an index to possess that should determine its functional form, not the other way around. The approach adopted here is distinct from the literature s common exercise of axiomatizing a given pre-existing index function (e.g, Queseda, 2001; Woeginger, 2008a and 2008b; and Marchant, 2009a). 3 The aim of this latter exercise is to begin with a previously known index function, e.g., the h-index, and identify a set of properties that only that index can satisfy. The given index is then characterized by these properties. While this exercise can usefully identify an index s previously unknown properties, both good and bad, it is not designed to correct any inherent flaws. To find a good index one should instead first begin with well-chosen properties and then allow those properties to lead one to the index, whatever it may turn out to be. Our goal is to provide a method for comparing one citation list to another in the most 2 Ellison (2013) introduces the following class of generalized Hirsch indices. For any a, b > 0 and for any citation list, h (a,b) is defined to be the largest number h such that at least h papers have at least ah b citations. Ellison estimates a and b so that h (a,b) gives the best fit in terms of ranking economists at the top 50 U.S. universities in a manner that is consistent with the observed labor market outcome. 3 An exception is Marchant (2009b) where the consequences of a number of axioms are nicely worked out, even though most are not put forward as necessary for a good index. 2

4 is best to think of these basic properties as pertaining to citation lists whose cited papers differ only in their number of citations but otherwise have common characteristics, e.g., are published in the same year, in the same field of study, even in the same journal, and have the same number of authors, etc. One of the four properties, scale invariance, is motivated with an eye toward the adjustments for differences in characteristics that will be considered in Section 4. There is an essentially unique index that possesses the properties below. This new index has a simple functional form and is distinct from the h-index and all its successors. 2. Four Properties for Citation Indices Citation indices work as follows. First, an individual s record is summarized by a finite list of citations, typically ordered from highest to lowest, so that the i-thnumberinthelistis the number of citations received by the individual s i-th most highly cited paper. The list of citations is then operated upon by some function to produce the individual s index number. In Section 4.2, we argue that one must sometimes rescale one individual s citation list to adequately compare it to another individual s list (for example, when the two individuals are from different fields). 5 Consequently, one must be prepared to consider noninteger lists. Moreover, because the scaling factors will always be rational numbers, our domain of study is the set L consisting of all finite lists of nonnegative rational numbers ordered from highest to lowest. Any element of L is called a citation list. 6 A citation index is any continuous function, ι : L R, that assigns to each citation list a real number, where continuity means that for any citation list x, if a sequence of citation lists x 1,x 2,... each of whose length is the same as the finite length of x, converges to x in the Euclidean sense then ι(x 1 ), ι(x 2 ),... converges to ι(x). 7 Consider now the following four properties. A short discussion of them follows. 1. Monotonicity (MON) For any citation list there is a positive integer m whichcan depend on the list such that the value of the index does not decrease if either, a new paper with m citations is added to the list, or any existing paper in the list receives any number of additional citations. 5 And we also explain (in Section 4.1) why any adjustments should be made to the citation lists themselves and not to the index function. 6 No non-singleton subset of L is connected. While this presents some difficulties, the end result is a theory that takes seriously the integer nature of citations. 7 Note that the index is continuous only on the rationals it is not even defined on the reals. In general, such functions cannot be extended fromtherationalstotherealsinacontinuous manner without additional assumptions. 4

6 Theorem 3.1. A citation index satisfies properties 1-4, if and only if it is equivalent to the index that assigns to each citation list (x 1,..., x n ),thenumber, n 1/σ x σ i, where σ is some fixed number strictly greater than one. 9 According to the theorem, properties 1-4 determine the index (more precisely, its equivalence class) up to the single parameter σ > 1. For a rationale for restricting σ to values no greater than 2 see Section 5. Next, we discuss the interpretation of this important parameter. The value of our index can increase in two different ways: when an existing paper receives more citations in which case we will say that the list becomes deeper and when a new paper is added to the list in which case we will say that the list becomes longer. The parameter σ measures the extent to which depth is considered more important than length. At the one extreme in which σ approaches its lower bound of 1, depth and length are equally important and citation lists are ranked according to their total citation count. At the other extreme in which σ becomes unboundedly large, citation lists are ranked according to which list is lexicographically higher and depth becomes infinitely more important than length. In general, changes in both the depth and the length of a citation list affect its index value and the parameter σ determines the depth/length trade-off. Specifically, each σ% decrease in a citation list s length (i.e., for each k, a σ% decrease in the number of papers with k citations) requires a 1% increase in its depth (i.e., a 1% increase in each paper s number of citations) to restore the index to its original value. That is, σ is the elasticity of substitution of length for depth. 10 In particular, the higher is σ the more weight the index places on depth. Additional intuition for σ is obtained by noting that it is characterized by the answer to the following question. For what value of λ (1, 2) would two papers, each with k citations, be considered equivalent to one paper with λk citations? 9 This index, being in units of citations and homogeneous of degree one, may be more convenient in practice than the equivalent index, n xσ i. Moreover, as defined, our index coincides with the l σ norm from classical mathematical analysis, which is certainly an interesting happenstance. 10 For any citation list x, define v(d, l) :=(l i (dx i) σ ) 1 σ. Then, changes in d correspond to changes in the list s depth and changes in l correspond to changes in the list s length. The index s elasticity of substitution of length for depth is d v/ d l v/ l = σ. 6

7 The answer, of course can be different for different decision-makers. However, if λ is one s answer to this question, then the implied value of σ is ln 2/ ln λ. So, for example if two papers each with 100 citations are considered equivalent to one paper with 140 citations, then the implied value of σ is approximately 2, while if they are instead equivalent to one paper with 160 citations, the implied value of σ is approximately 3/2. Thus, a value of σ between 3/2 and 2 may be suitable for most applications (again, see Section 5 for an argument leading toavalueofσ no greater than 2) An Implied Property Properties 1 through 4 alone completely determine, up to the parameter σ > 1, the ordinal properties of the index. However, we wish now to point out a particularly interesting subsidiary property that holds as a consequence. Consider two individuals, X and Y whose respective citations lists are x 1... x n and y 1... y n, wherewehaveadded0 s if necessary to make the lists equal in length. Suppose that the two lists are related as follows. x 1 y 1 x 1 + x 2 y 1 + y 2. (1) x x n y y n That is, individual X s most cited paper has at least as many citations as Y s most cited paper, and X s two most cited papers have at least as many total citations as Y s two most cited papers,..., and X has at least as many total citations as Y. It would seem rather difficult to argue that individual Y should be ranked above individual X. Yet,withtheh- index, individual Y can easily be ranked above X (e.g., if X has one paper with 100 citations and Y has two papers each with 2 citations). In contrast, we have the following result. If the citation lists of two individuals, X and Y are related as in (1), then any index satisfying properties 1-4 must rank individual X at least as high as individual Y, and must rank X strictly above Y if any of the inequalities in (1) is strict. Note well that the result makes no mention of any particular value of σ. Indeed, in view of our Theorem, an equivalent way to state the result is to say that, for any value of σ > 1, n 1/σ n 1/σ x σ i yi σ, (2) whenever (1) holds, and the inequality is strict if any of the inequalities in (1) is strict (see 7

11 form the figures, namely, that dividing by the average number of citations per paper in one s field corrects for differences in citations across fields in the very strong sense that, after the adjustment, the distributions of citations are virtually identical across fields. 13 In fact, Radicchi et. al. show that the adjustment yields again that same distribution when carried out for years 1990 and So, as noted in their paper, this adjustment corrects not only for differences in field, but also for differencesintheagesofpapers(see Figure 3 in Radicchi et. al. (2008). Let us now apply this adjustment to our index. 13 In Radicchi et al. (2008), the common histogram in Figure 2 is estimated as being log normal with variance 1.3 (the mean must be 1 because of the rescaling). 11

12 Consider a raw citation list (x 1,..., x n ). For each paper i =1,..., n, let a i denote the average number of citations of all papers published in both the same year and field as paper i; and so each a i is a rational number. Note that this permits an individual to have papers in more than one field. To correct for field and paper-age differences, the requisite adjustment is to divide each x i by a i, producing the adjusted citation list (x 1 /a 1,...,x n /a n ); so each entry i is rational, being an integer x i divided by a rational a i. We then simply apply our index to this adjusted list. Consequently, any citation list (x 1,..., x n ) is given the adjusted index value, n xi σ 1/σ. a i 4.3. Scale Invariance Clearly, the above rescaling for field and paper-age that aligns the otherwise disparate distributions of citations across fields and paper-ages is arbitrary up to any positive scaling factor. In particular, a common distribution of citations across fields and paper-ages would also have obtained had Radicchi et. al. (2008) divided any citation list entry x i, not by a i, but instead by 2a i. Consequently, if after rescaling, one s index ranks the adjusted list (x 1 /a 1,..., x n /a n ) above the adjusted list (x 1/a 1,..., x n/a n), then it should also rank (x 1 /2a 1,..., x n /2a n ) above (x 1/2a 1,..., x n/2a n). Indeed, it should similarly rank (x 1 /λa 1,..., x n /λa n ) above (x 1/λa 1,..., x n/λa n) for any (rational) λ > Butthisisprecisely what is ensured by our property 4, scale invariance. 15 When scale invariance is violated, as it is with the h-index, serious difficulties can arise. For instance, within a biology department, biologist B 1 may be ranked above biologist B 2 accordingtotheh-index, but when B 1 and B 2 are each compared to a mathematician, M, where rescaling is required to compare across the two fields, applying the h-index to the rescaled citation lists can result in B 2 being ranked first, M second, and B 1 third! Moreover, the finalrankingcandependonwhetheronefield s lists are scaled up or the other field s lists are scaled down. A typical example follows. Example 4.2. Suppose that B 1 has 15 papers, each with 15 citations; and B 2 has 10 papers, each with 30 citations. Hence, B 1 has an h-index of 15 which is above B 2 s h-index of 10. Suppose that M has 8 papers, each with 7 citations and that to compare biologists to 14 This alignment argument holds of course also for irrational λ. However, there is never any need to consider irrational lists. 15 Of course, once rescaling has aligned all the distributions, any subsequent common-across-fields increasing transformation applied to each citation list entry will maintain that alignment (and these are the only increasing transformations that can do so see fn. 12). But the only such transformations compatible with properties 1-4 are linear. 12

13 mathematicians one must scale down the biologists citations by a factor of 3. Hence, B 1 s rescaled citation list contains 15 papers with 5 citations each and B 2 s rescaled citation list contains 10 papers with 10 citations each. Consequently, when comparing all three individuals, B 2 is ranked first (h =10), M second (h =7)and B 1 third (h =5). On the other hand, had we instead scaled up the mathematician s list by a factor of 3, the final ranking would be B 1 first (h = 15), B 2 second (h =10)and M third (h =8) Other differences and group rankings We now show how to adjust individual citation lists to also account for differences in the scientific ages of individuals and the number of authors, and we also show how our index can be used to rank groups of individuals such as departments, universities, or even countries. 16 Suppose that an individual s raw citation list is (x 1,..., x n ) and that a i denotes, as above, the average number of citations received by all of the papers published in the same field and inthesameyearasthei-th paper in the list. Let k i denote the total number of authors associated with paper i and let t denote the scientific age of the individual. To take into account all of these differences, the individual s citation list should be adjusted by dividing the i-th entry by a i (k i t) 1/σ, where σ > 1 is the elasticity parameter of our index. Consequently, this individual s adjusted index value is, n σ 1/σ x i. (4.3) a i (k i t) 1/σ We now explain why. Since we have already argued above that one must divide each x i by a i to account for differences in fields and the ages of papers, it remains only to adjust for scientific age and number of coauthors. We begin with scientific age. Suppose that two individuals, X and X have citation lists x =(x 1,..., x n ) and x = (x 1,..., x n) and that they differ in fields, ages of papers, and scientific age. Letusfirst eliminate the differences in fields and the ages of their papers by dividing each x i by a i,and dividing each x i by a x i, as above. This first adjustment gives 1 a 1,..., xn a n as X s list and x 1,..., x a n as X s list, and in particular renders all papers the same age. 17 It remains to 1 account for the different scientific agesofx and X. Assume that for each individual and for any given number of citations, the number of papers written per year by that individual and with that number of citations is roughly constant. Hence, if X s (scientific) age is t, then under this assumption, over the next t years we would expect X to match his/her performance of the first t years and so we should append a n 16 Ellison (2012) uses Hirsch-like indices to compare computer science departments. 17 The entries as written here may no longer be in decreasing order. 13

15 above X if or equivalently if n kk k i xi a i t 1/σ σ 1/σ > n kk k i x i σ 1/σ, a i (t ) 1/σ n x i σ 1/σ n > x i σ 1/σ. a i (k i t) 1/σ a i(kit ) 1/σ ) Taken altogether, we obtain the criterion in (4.3). Consider now the problem of ranking groups of individuals. Because our approach is to first adjust individual lists to eliminate differences in important characteristics, it is not unreasonable after doing so to append all of the adjusted lists within a group to obtain a single citation list that represents that group. After adjusting for the size of the group (much like the adjustment for number of authors), our index is applied to the adjusted list. For example, for each of a number of academic departments (which might be from different disciplines) a department citation list is obtained by stringing together all the adjusted citation lists of all of its members. Our index is then applied to this list. Specifically, if the department has m members and member j is of scientific age t j,has citation list (x 1j,...x nj ), and a ij is the average number of citations received by all of the papers published in the same field and in the same year as j s i-th paper, and k ij is the number of authors associated with j s i-th paper, then our index assigns this department the index number, m n σ 1/σ x ij. a ij (mk ij t j ) 1/σ 5.Onthechoiceofσ j=1 While any value of σ that is greater than 1 is consistent with properties 1-4, the following additional, though less basic, property provides a rationale for restricting the value of σ to be no greater than This property, called balancedness, states in particular that when two given lists are equally ranked, the index s value of a new citation list that is a weighted average of the two given lists, is larger when the average is formed by placing the higher, rather than the lower, weight on the more balanced of the two given lists. The formal statement is as follows. Balancedness For any two citation lists x = (x 1,..., x n ) and y = (y 1, 0,..., 0) of equal 18 We are grateful to Faruk Gul and Debraj Ray for discussions that led us to this property. 15

16 length, if ι(x) ι(y), then ι(βx + αy) ι(αx + βy) for all rational α, β such that β α 0 and α + β =1. As the next proposition states, in order for our index to satisfy balancedness, it is necessary and sufficient that the value of σ be no greater than 2. Proposition 5.1. If σ > 1, the index ( n xσ i )1/σ displayed in Theorem 3.1 satisfies balancedness if and only if σ Ranking Journals Related to the topic of this paper is the issue of ranking scholarly journals. However, while the reason for ranking individuals is clear (limited resources must somehow be allocated), thereasonforrankingjournalsislessclear. Onepossibleview,andtheviewtakenhere, is that ranking journals is useful only in so far as it provides information on how to rank individuals Impact factors Consider, for example the problem of assessing young scholars whose papers have not had sufficient time to collect citations. In such cases, one must form an estimate of the impact that their papers will have, i.e., one must estimate, for each paper, its expected number of future citations. This is, in fact, easily done by using a journal s so-called impact factor, which is the average number of citations received per year by papers published in that journal. Once an estimate of the number of citations has been obtained for each paper, our citation index can be applied. This is perhaps one of the most useful roles that impact factors can play in practice Endogenously-weighted citations Palacios-Huerta and Volij (2004) provide an axiomatization of a cardinal index that assigns a numerical score to each journal. A journal s score is a weighted sum of all of the citations it receives from all journals, where the weights are endogenous in the sense that the weight placed on any given citation is proportional to the score of the journal from which it came. 19 One might take this a step further if citations from higher-impact journals lead to more citations due to the increased visibility they generate than citations from lower impact journals. Thus, one might weight citations from different journals differently, and perhaps simply in proportion to journal impact factors, if this leads to a better prediction of future citations something that is presumably testable. Being proportional to impact factors, the weights here are not endogenously determined in the sense of the next section. We thank Pablo Beker for suggesting this additional potential use of journal impact factors. 16

17 One might wonder whether a similarly endogenous approach to assigning weights to citations is possible/desirable when the objective is to rank individuals. While this is certainly an interesting question, we have only two small remarks. First, the endogenous approach allows the potential for over-weighting. For example, suppose that when individual A with many citations cites a paper, then that paper s expected number of additional citations is greater than when individual B with fewer citations cites that paper. Then, even with equally-weighted citations, a paper cited by A will receive a higher score than a paper cited by B because it will receive more citations. There then seems no need/reason to give citations received from A more weight than those from B. And if papers cited by A receive fewer citations than those cited by B, then it is perhaps even less reasonable to give higher weight to citations from A. Of course, what the endogenous-weight approach presumes is that being cited by a highly cited individual is better, for its own sake, than being cited by a less-cited individual. The rationale for this, not uncommon, presumption is not entirely clear to us in the context of evaluating a record as it currently stands. Second, and more important, is the problem of the ordinality of the index for ranking individuals. Without a cardinal index there is no hope of producing a unique set of endogenous weights, since each new monotone transformation of the index can change the weights. Moreover, under ordinality, the trivial set of uniform weights is itself endogenously consistent at least as a limiting solution. And with uniform weights, we are back to a standard index. A. Appendix It is well-known that welfare (or utility) functions defined on various subsets of R n that are continuous, increasing, additively separable and scale invariant must be equivalent to our index, or a closely related functional form (see Bergson, 1936; Roberts, 1980; and Blackorby and Donaldson, 1982). The difficulty in our case is that our domain L is quite fragmented in that no non-singleton subset is connected. Thus, the thrust of the proof is to show that our index (or, more precisely, one that is equivalent to it) can be extended to an index that (i) for each n, is continuous on R n +, and (ii) satisfies all four of our properties. Once this is done we apply the standard results. For the reader s convenience, we restate our main result. Theorem 3.1. A citation index satisfies properties 1-4 if and only if it is equivalent to the index that assigns to each citation list (x 1,..., x n ),thenumber, n 1/σ x σ i, where σ is some fixed number strictly greater than one. Proof. Sufficiency is clear so we focus only on necessity and suppose that the index function ι satisfies properties 1-4. For any natural number n, and to any n nonnegative rational numbers x 1 x 2... x n, ι assigns the real number, ι(x 1,..., x n ). For every n, it will be 17

18 convenient to extend ι to all of Q n + by defining ι(x 1,..., x n ):=ι(x 1,X 2,...,X n ), where X i is i-th order statistic of x 1,..., x n. Thus ι is now defined on the extended domain L := n Q n + in such a way that it is symmetric; i.e., if x L is a permutation of x L, then ι(x) =ι(x ). Clearly, the extended function ι satisfies the associated extensions to L of properties 1-4. We work with the extended index ι from now on. Let us first show that, for every n>0, ι is strongly increasing on Q n + i.e., ι(x i,x i ) > ι(x i,x i ), x Q n +, i, and rational x i >x i. By IND, it suffices to show that ι(x i) > ι(x i ), or by SI, that ι(1) > ι x i x i (A.1). Since x i >x i, it suffices to show that ι(1) > ι(λ) for all rational λ [0, 1). We shall argue by contradiction. If not, then there is a nonnegative rational λ < 1 such that ι(λ) ι(1). Applying scale invariance k times, each time using the scaling factor λ, gives ι(λ k ) ι(λ k 1 ). Since this holds for any positive integer k, we have ι(λ k ) ι(λ k 1 )... ι(λ) ι(1). For any positive rational r<1, we may choose k such that λ k <rsince λ < 1. Hence, ι(r) ι(1) ι(λ k ) ι(r), wherethefirst and last inequalities are by MON. Hence, ι(1) = ι(r) for every positive rational r<1. But then, by SI, also ι(1) = ι( 1 ) (use the scaling factor 1/r). Hence, ι is r constant on Q ++ and, by continuity, on Q +. We next show inductively that, for each n, ι is constant on Q n +, from which we will eventually derive a contradiction. So, suppose that ι is constant on Q n +. We must show that ι is constant on Q n+1 +. But this will follow if we can show that for all (x 1,..., x n+1 ) Q n+1 + ι(1 n+1 )=ι(x 1,..., x n, 1) = ι(x 1,..., x n+1 ), where, for each m, 1 m denotes the vector (1,..., 1) Q m +. But the first equality follows by IND because ι(1 n )=ι(x 1,..., x n ) by the induction hypothesis, and the second equality follows by IND because ι(x 2,..., x n, 1) = ι(x 2,..., x n+1 ) by the induction hypothesis. Hence, ι is constant on Q n + for every n. We next derive the desired contradiction. By DR there is a positive integer k and there is x L such that ι( k, x) > ι(1 k, x),where 1 k =(1,..., 1) is of length k. Hence, by IND, ι( k) > ι(1 k). By MON, there exists m 2 large enough so that ι( k, m 2 ) ι( k) > ι(1 k). Applying MON once again, there exists m 3 large enough so that ι( k, m 2,m 3 ) ι( k, m 2 ) ι( k) > ι(1 k). After k 1 such successive applications of MON, we obtain m 2,..., m k such that ι( k, m 2,..., m k) ι( k) > ι(1 k), implying that ι is not constant on Q k +. This contradiction establishes (A.1). We can now prove the following claim. Claim. There is a continuous function ˆι : R n + R that is equivalent to ι on Q n + i.e., x, y Q n +, ˆι(x) ˆι(y) iff ι(x) ι(y). ProofofClaim.Since n is fixed, we write 1 instead of 1 n. Define a nondecreasing function d : Q n + R + as follows. For every x Q n +, d(x) :=inf{r Q + : ι(r1) ι(x)}. Then d is nondecreasing because ι is nondecreasing. Let us show that, d is equivalent to ι on Q n +. (A.2) 18

19 Since it is obvious that ι(x) =ι(y) implies d(x) =d(y), it suffices to show that ι(x) > ι(y) implies d(x) >d(y). First note that SI implies that d is homogeneous of degree 1 (since SI implies that for any positive rational λ, {r Q + : ι(r 1) ι(λx)} = λ{r Q + : ι(r1) ι(x)}). In particular, d(r1) = r for any nonnegative rational r (d(0) = 0, clearly). Second, note that for x = 0, (A.1) implies that ι(x) > ι(0). Bycontinuitythereisa rational ε (0, 1) such that ι(x) > ι(ε1 n ) and hence, d(x) ε > 0. Suppose now that ι(x) > ι(y). Then, by continuity, there is a positive rational δ < 1 such that ι(y) < ι(δx), andsod(y) d(δx). But, by 1-homogeneity d(δx) =δd(x) < d(x) since d(x) > 0. Hence, d(y) d(δx) <d(x), which proves (A.2). Let x m,y m Q n + be sequences converging to the same point x R n +. We will next establish that, lim d(x m ) = lim d(y m ). (A.3) m m That is, the limits exist and are equal. Let us first dispense with the case in which the common limit point is x =0. Then, for any positive rational, r, x m << r1 for all m large enough. Since d is nondecreasing, 0 d(x m ) d(r1) =r for all m large enough. Since r is arbitrary, we conclude that lim m d(x m )=0;and similarly for lim m d(y m ). So (A.3) holds for x =0and we henceforth assume that x = 0. Evidently, d(x m ) is bounded because, for any z Q n + satisfying z>>x,we have d(0) d(x m ) d(z) for m large enough since d is nondecreasing. Similarly, d(y m ) is bounded. Hence, if either limit fails to exist or if the limits exist but are not equal, the sequences x m and y m would admit subsequences along which d(x m ) converges, to a say, and d(y m ) converges, to b say, and a = b. Restricting attention to these subsequences, reindexing, and supposing that a<b,this would mean that, lim m d(x m )=a<b= lim m d(y m ). (A.4) Thus, to establish (A.3) it suffices to assume(a.4) and derive a contradiction. Choose rationals q, q (a, b) such that q<q and recall that d(r1) =r for any nonnegative rational r. Hence, lim m d(x m )=a<q= d(q1) <d(q 1)=q <b= lim m d(y m ), and so d(x m ) <d(q1) <d(q 1) <d(y m ) for all sufficiently large m. Consequently, since d and ι are equivalent, letting w := q1 and w := q 1, we have ι(x m ) < ι(w) < ι(w ) < ι(y m ), (A.5) for all sufficiently large m. Without loss, we may assume that (A.5) holds for all m. Choose any rational δ (0, 1). Scale invariance applied to (A.5) implies in particular that, ι(δw ) < ι(δy m ), for all m. (A.6) Choose z,z Q n + so that δx i <zi <z i <x i holds for every i such that x i > 0, and such that zi = z i =0for every i such that x i =0. Then both z and z are nonzero (because x is nonzero) and so ι(z ) < ι(z) by (A.1). By continuity, there is a rational ε (0, 1) such that ι(z + ε1) < ι(z). For each i, define zi = zi if z i > 0 and zi = ε if z i =0. The resulting 19

20 vector z Q n ++ is such that all coordinates are positive and z z + ε1. Consequently, by MON, ι(z ) ι(z + ε1) < ι(z). Moreover, (i) δx i <zi = zi <z i <x i holds for every i such that x i > 0, and (ii) zi = ε and z i =0holds for all i such that x i =0. Consequently, δx <<z and z x, with equality in the latter only for coordinates i such that x i =0. Since x m and y m converge to x, there must be m large enough so that δy m << z and z x m. Applying MON to these two vector inequalities and using ι(z ) < ι(z) gives, ι(δy m ) ι(z ) < ι(z) ι(x m ). Combined with (A.5) and (A.6), we obtain ι(δw ) < ι(δy m ) < ι(x m ) < ι(w) < ι(w ) which, in particular implies, ι(δw ) < ι(w) < ι(w ). But since δ is an arbitrary rational in (0, 1), this contradicts the continuity of ι, andwe conclude that (A.3) holds. A consequence of (A.3) is that we may extend d to all of R n + by defining its extension ˆι : R n + R + as follows. For every x R n +, ˆι(x) := lim x m x d(x m), where the limit is taken over any sequence x m Q n + converging to x. The proof of the claim will be complete if we can show that ˆι is continuous on R n + and equivalent to ι on Q n +. For the latter, suppose that x Q n +. Since the limit defining ˆι(x) is independent of the sequence x m Q n + converging to x that is chosen, we can in particular choose the constant sequence x m = x (since x Q n +). Therefore î(x) =d(x) for every x Q n + (and so ˆι is indeed an extension of d). Since d is equivalent to ι on Q n + so too is ˆι. For continuity on R n +,supposethatx m R n + converges to x R n +. For every m, by the definition of ˆι(x m ) we may choose y m Q n + such that x m y m 1 m and ˆι(x m) d(y m ) 1 m. Hence, y m x, and lim m (ˆι(x m ) d(y m )) = 0. But because y m Q n + converges to x, ˆι(x) = lim m d(y m ), by the definition of ˆι(x). Hence, lim m ˆι(x m ) = lim m (ˆι(x m ) d(y m )+ d(y m )) = lim m (ˆι(x m ) d(y m )) + lim m d(y m )=0+ˆι(x) =ˆι(x), proving continuity, and completing the proof of the claim. Let us now establish some additional properties of the continuous function ˆι : R n + R. Since ˆι is equivalent to ι on Q n +, ˆι is a symmetric and strongly increasing function on Q n + and satisfies all four of our properties, MON, IND, SI, and DR on Q n +. We also note that ˆι is nondecreasing on R n +, afactthatisimmediatefromthedefinition because d is nondecreasing on Q n +. We will use these facts, as well as the continuity of ˆι, freely. Fact 1. (Symmetry on R n +)Ifx R n + is a permutation of the coordinates of x, then ˆι(x) =ˆι(x ) follows from the observation that for any x m Q n + converging to x, we have also that x m converges to x, where x m is obtained from x m by applying the same permutation as performed on x to obtain x. But since ˆι is a symmetric function on Q n +, we have ˆι(x m )=ˆι(x m) for every m. Hence, ˆι(x) = lim m ˆι(x m )=lim m ˆι(x m)=ˆι(x ). Fact 2. (Scale Monotonicity on R n +). If x R n +\{0} and δ (0, 1) we wish to show that ˆι(δx) < ˆι(x). Choose z,z Q n + such that δx i <z i <z i <x i for every i such that x i > 0 and set z i = z i =0otherwise. Then z and z are distinct (since x = 0)and δx z z x. Therefore, ˆι(δx) ˆι(z) < ˆι(z ) ˆι(x), where the two weak inequalities follow because ˆι is nondecreasing and the strict inequality follows because ˆι, is strongly increasing on Q n +. 20

### THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages

### Double Sequences and Double Series

Double Sequences and Double Series Eissa D. Habil Islamic University of Gaza P.O. Box 108, Gaza, Palestine E-mail: habil@iugaza.edu Abstract This research considers two traditional important questions,

### 1 Limiting distribution for a Markov chain

Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### 6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

### Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

### 5. Convergence of sequences of random variables

5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

### 1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

### Prime Numbers. Chapter Primes and Composites

Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

### PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.

PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### max cx s.t. Ax c where the matrix A, cost vector c and right hand side b are given and x is a vector of variables. For this example we have x

Linear Programming Linear programming refers to problems stated as maximization or minimization of a linear function subject to constraints that are linear equalities and inequalities. Although the study

### 1. R In this and the next section we are going to study the properties of sequences of real numbers.

+a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

### Math 317 HW #7 Solutions

Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence

### The Ideal Class Group

Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it

### DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS

DEGREES OF ORDERS ON TORSION-FREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsion-free abelian groups, one of isomorphism

### 3. Recurrence Recursive Definitions. To construct a recursively defined function:

3. RECURRENCE 10 3. Recurrence 3.1. Recursive Definitions. To construct a recursively defined function: 1. Initial Condition(s) (or basis): Prescribe initial value(s) of the function.. Recursion: Use a

### MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,

### Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### x if x 0, x if x < 0.

Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

### Appendix F: Mathematical Induction

Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

### Gains from Trade. Christopher P. Chambers and Takashi Hayashi. March 25, 2013. Abstract

Gains from Trade Christopher P. Chambers Takashi Hayashi March 25, 2013 Abstract In a market design context, we ask whether there exists a system of transfers regulations whereby gains from trade can always

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Introduction. Appendix D Mathematical Induction D1

Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

### MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS

1 CHAPTER 6. MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW MATHEMATICAL INDUCTION AND DIFFERENCE EQUATIONS 1 Introduction Recurrence

### Problem Set I: Preferences, W.A.R.P., consumer choice

Problem Set I: Preferences, W.A.R.P., consumer choice Paolo Crosetto paolo.crosetto@unimi.it Exercises solved in class on 18th January 2009 Recap:,, Definition 1. The strict preference relation is x y

### CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE

CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska

Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. Gacovska-Barandovska Smirnov (1949) derived four

### Discrete Mathematics, Chapter 5: Induction and Recursion

Discrete Mathematics, Chapter 5: Induction and Recursion Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 5 1 / 20 Outline 1 Well-founded

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### Weak topologies. David Lecomte. May 23, 2006

Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such

### NOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004

NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σ-algebras 7 1.1 σ-algebras............................... 7 1.2 Generated σ-algebras.........................

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### The Relation between Two Present Value Formulae

James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 61-74. The Relation between Two Present Value Formulae James E. Ciecka,

### How to Gamble If You Must

How to Gamble If You Must Kyle Siegrist Department of Mathematical Sciences University of Alabama in Huntsville Abstract In red and black, a player bets, at even stakes, on a sequence of independent games

### Vector Spaces II: Finite Dimensional Linear Algebra 1

John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition

### Notes: Chapter 2 Section 2.2: Proof by Induction

Notes: Chapter 2 Section 2.2: Proof by Induction Basic Induction. To prove: n, a W, n a, S n. (1) Prove the base case - S a. (2) Let k a and prove that S k S k+1 Example 1. n N, n i = n(n+1) 2. Example

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### CHAPTER 3. Sequences. 1. Basic Properties

CHAPTER 3 Sequences We begin our study of analysis with sequences. There are several reasons for starting here. First, sequences are the simplest way to introduce limits, the central idea of calculus.

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### LOGNORMAL MODEL FOR STOCK PRICES

LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as

### Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### Solving Systems of Linear Equations

LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

### Section 3 Sequences and Limits

Section 3 Sequences and Limits Definition A sequence of real numbers is an infinite ordered list a, a 2, a 3, a 4,... where, for each n N, a n is a real number. We call a n the n-th term of the sequence.

### SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

### Mathematics for Econometrics, Fourth Edition

Mathematics for Econometrics, Fourth Edition Phoebus J. Dhrymes 1 July 2012 1 c Phoebus J. Dhrymes, 2012. Preliminary material; not to be cited or disseminated without the author s permission. 2 Contents

### Math Real Analysis I

Math 431 - Real Analysis I Solutions to Homework due October 1 In class, we learned of the concept of an open cover of a set S R n as a collection F of open sets such that S A. A F We used this concept

### Geometry of Linear Programming

Chapter 2 Geometry of Linear Programming The intent of this chapter is to provide a geometric interpretation of linear programming problems. To conceive fundamental concepts and validity of different algorithms

### 2.5 Gaussian Elimination

page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### On Lexicographic (Dictionary) Preference

MICROECONOMICS LECTURE SUPPLEMENTS Hajime Miyazaki File Name: lexico95.usc/lexico99.dok DEPARTMENT OF ECONOMICS OHIO STATE UNIVERSITY Fall 993/994/995 Miyazaki.@osu.edu On Lexicographic (Dictionary) Preference

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### it is easy to see that α = a

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

### The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

### Duality of linear conic problems

Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least

### Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### Chapter 1. Logic and Proof

Chapter 1. Logic and Proof 1.1 Remark: A little over 100 years ago, it was found that some mathematical proofs contained paradoxes, and these paradoxes could be used to prove statements that were known

### Infinite series, improper integrals, and Taylor series

Chapter Infinite series, improper integrals, and Taylor series. Introduction This chapter has several important and challenging goals. The first of these is to understand how concepts that were discussed

### OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

### Math 317 HW #5 Solutions

Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886)

Chapter 2 Numbers God created the integers and the rest is the work of man. (Leopold Kronecker, in an after-dinner speech at a conference, Berlin, 1886) God created the integers and the rest is the work

### Lecture 3: Finding integer solutions to systems of linear equations

Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture

### 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution.

Name : Instructor: Marius Ionescu Instructions: 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution. 3. If you need extra pages

### This chapter describes set theory, a mathematical theory that underlies all of modern mathematics.

Appendix A Set Theory This chapter describes set theory, a mathematical theory that underlies all of modern mathematics. A.1 Basic Definitions Definition A.1.1. A set is an unordered collection of elements.

### Riesz-Fredhölm Theory

Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

### Course 421: Algebraic Topology Section 1: Topological Spaces

Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............

### Linear Codes. In the V[n,q] setting, the terms word and vector are interchangeable.

Linear Codes Linear Codes In the V[n,q] setting, an important class of codes are the linear codes, these codes are the ones whose code words form a sub-vector space of V[n,q]. If the subspace of V[n,q]

### 15 Limit sets. Lyapunov functions

15 Limit sets. Lyapunov functions At this point, considering the solutions to ẋ = f(x), x U R 2, (1) we were most interested in the behavior of solutions when t (sometimes, this is called asymptotic behavior