Extensions of the Laws of Inheritance

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17. září 2004 42 ze 412 4 THE LAWS OF THE FIFTH DISCIPLINE 1 1. Today's problems come from yesterday's "solutions." Once there was a rug merchant who saw that his most beautiful carpet had a large bump in its center. 2 He stepped on the bump to flatten it out—and succeeded. But the bump reappeared in a new spot not far away. He jumped on the bump again, and it disappeared —for a moment, until it emerged once more in a new place. Again and again he jumped, scuffing and mangling the rug in his frustration; until finally he lifted one corner of the carpet and an angry snake slithered out. Often we are puzzled by the causes of our problems; when we merely need to look at our own solutions to other problems in the past. A well-established firm may find that this quarter's sales are off sharply. Why? Because the highly successful rebate program last quarter led many customers to buy then rather than now. Or a new manager attacks chronically high inventory costs and "solves" the problem—except that the salesforce is now spending 20 percent more time responding to angry complaints from customers who are 17. září 2004 43 ze 412 still waiting for late shipments, and the rest of its time trying to convince prospective customers that they can have "any color they want so long as it's black." Police enforcement officials will recognize their own version of this law: arresting narcotics dealers on Thirtieth Street, they find that they have simply transferred the crime center to Fortieth Street. Or, even more insidiously, they learn that a new citywide outbreak of drug-related crime is the result of federal officials intercepting a large shipment of narcotics—which reduced the drug supply, drove up the price, and caused more crime by addicts desperate to maintain their habit. Solutions that merely shift problems from one part of a system to another often go undetected because, unlike the rug merchant, those who "solved" the first problem are different from those who inherit the new problem. 2. The harder you push, the harder the system pushes back. In George Orwell's Animal Farm, the horse Boxer always had the same answer to any difficulty: "I will work harder," he said. At first, his well-intentioned diligence inspired everyone, but gradually, his hard work began to backfire in subtle ways. The harder he worked, the more work there was to do. What he didn't know was that the pigs who managed the farm were actually manipulating them all for their own profit. Boxer's diligence actually helped to keep the other animals from seeing what the pigs were doing. 3 Systems thinking has a name for this phenomenon: "Compensating feedback": when well-intentioned interventions call forth responses from the system that offset the benefits of the intervention. We all know what it feels like to be facing compensating feedback—the harder you push, the harder the system pushes back; the more effort you expend trying to improve matters, the more effort seems to be required. Examples of compensating feedback are legion. Many of the best intentioned government interventions fall prey to compensating feedback. In the 1960s there were massive programs to build low- income housing and improve job skills in decrepit inner cities in the United States. Many of these cities were even worse off in the 1970s despite the largesse of government aid. Why? One reason was that low-income people migrated from other cities and from rural areas to those Extensions of the Laws of Inheritance Extensions of the Laws of Inheritance Bởi: OpenStaxCollege Mendel studied traits with only one mode of inheritance in pea plants The inheritance of the traits he studied all followed the relatively simple pattern of dominant and recessive alleles for a single characteristic There are several important modes of inheritance, discovered after Mendel’s work, that not follow the dominant and recessive, singlegene model Alternatives to Dominance and Recessiveness Mendel’s experiments with pea plants suggested that: 1) two types of “units” or alleles exist for every gene; 2) alleles maintain their integrity in each generation (no blending); and 3) in the presence of the dominant allele, the recessive allele is hidden, with no contribution to the phenotype Therefore, recessive alleles can be “carried” and not expressed by individuals Such heterozygous individuals are sometimes referred to as “carriers.” Since then, genetic studies in other organisms have shown that much more complexity exists, but that the fundamental principles of Mendelian genetics still hold true In the sections to follow, we consider some of the extensions of Mendelism Incomplete Dominance Mendel’s results, demonstrating that traits are inherited as dominant and recessive pairs, contradicted the view at that time that offspring exhibited a blend of their parents’ traits However, the heterozygote phenotype occasionally does appear to be intermediate between the two parents For example, in the snapdragon, Antirrhinum majus ([link]), a cross between a homozygous parent with white flowers (CWCW) and a homozygous parent with red flowers (CRCR) will produce offspring with pink flowers (CRCW) (Note that different genotypic abbreviations are used for Mendelian extensions to distinguish these patterns from simple dominance and recessiveness.) This pattern of inheritance is described as incomplete dominance, meaning that one of the alleles appears in the phenotype in the heterozygote, but not to the exclusion of the other, which can also be seen The allele for red flowers is incompletely dominant over the allele for white 1/13 Extensions of the Laws of Inheritance flowers However, the results of a heterozygote self-cross can still be predicted, just as with Mendelian dominant and recessive crosses In this case, the genotypic ratio would be CRCR:2 CRCW:1 CWCW, and the phenotypic ratio would be 1:2:1 for red:pink:white The basis for the intermediate color in the heterozygote is simply that the pigment produced by the red allele (anthocyanin) is diluted in the heterozygote and therefore appears pink because of the white background of the flower petals These pink flowers of a heterozygote snapdragon result from incomplete dominance (credit: "storebukkebruse"/Flickr) Codominance A variation on incomplete dominance is codominance, in which both alleles for the same characteristic are simultaneously expressed in the heterozygote An example of codominance occurs in the ABO blood groups of humans The A and B alleles are expressed in the form of A or B molecules present on the surface of red blood cells Homozygotes (IAIA and IBIB) express either the A or the B phenotype, and heterozygotes (IAIB) express both phenotypes equally The IAIB individual has blood type AB In a self-cross between heterozygotes expressing a codominant trait, the three possible offspring genotypes are phenotypically distinct However, the 1:2:1 genotypic ratio characteristic of a Mendelian monohybrid cross still applies ([link]) 2/13 Extensions of the Laws of Inheritance This Punnet square shows an AB/AB blood type cross Multiple Alleles Mendel implied that only two alleles, one dominant and one recessive, could exist for a given gene We now know that this is an oversimplification Although individual humans (and all diploid organisms) can only have two alleles for a given gene, multiple alleles may exist at the population level, such that many combinations of two alleles are observed Note that when many alleles exist for the same gene, the convention is to denote the most common phenotype or genotype in the natural population as the wild type (often abbreviated “+”) All other phenotypes or genotypes are considered variants (mutants) of this typical form, meaning they deviate from the wild type The variant may be recessive or dominant to the wild-type allele An example of multiple alleles is the ABO blood-type system in humans In this case, there are three alleles circulating in the population The IA allele codes for A molecules on the red blood cells, the IB allele codes for B molecules on the surface of red blood cells, and the i allele codes for no molecules on the red blood cells In this case, the IA and IB alleles are codominant with each other and are both dominant over the i allele Although there are three alleles present in a population, each individual only gets two of the alleles from their parents This produces the genotypes and phenotypes shown ...Annals of Mathematics Norm preserving extensions of holomorphic functions from subvarieties of the bidisk By Jim Agler and John E. McCarthy* Annals of Mathematics, 157 (2003), 289–312 Norm preserving extensions of holomorphic functions from subvarieties of the bidisk By Jim Agler and John E. M c Carthy* 1. Introduction A basic result in the theory of holomorphic functions of several complex variables is the following special case of the work of H. Cartan on the sheaf cohomology on Stein domains ([10], or see [14] or [16] for more modern treat- ments). Theorem 1.1. If V is an analytic variety in a domain of holomorphy Ω and if f is a holomorphic function on V , then there is a holomorphic function g in Ω such that g = f on V . The subject of this paper concerns an add-on to the structure considered in Theorem 1.1 which arose in the authors’ recent investigations of Nevanlinna- Pick interpolation on the bidisk. The definition for a general pair (Ω,V)isas follows. Definition 1.2. Let V be an analytic variety in a domain of holomor- phy Ω. Say V has the extension property if whenever f is a bounded holo- morphic function on V , there is a bounded holomorphic function g on Ω such that (1.3) g| V = f and sup Ω |g| = sup V |f|. More generally, if Hol ∞ (V ) denotes the bounded holomorphic functions on V and A ⊆ Hol ∞ (V ), then we say V has the A-extension property if there is a bounded holomorphic function g on Ω such that (1.3) holds whenever f ∈ A. Before continuing we remark that in Definition 1.2 it is not essential that V beavariety: interpret f to be holomorphic on V if f has a holomorphic extension to a neighborhood of V . Also, in this paper we shall restrict our attention to the case where Ω = 2 . The authors intend to publish their ∗ The first author was partially supported by the National Science Foundation. The second author was partially supported by National Science Foundation grant DMS-0070639. 290 JIM AGLER AND JOHN E. M C CARTHY results on more general cases in a subsequent paper. Finally, we point out that the notion in Definition 1.2 is different but closely related to extension problems studied by the group that worked out the theory of function algebras in the 60’s and early 70’s (see e.g. [19] and [4]). We now describe in some detail how we were led to formulate the notions in Definition 1.2. The classical Nevanlinna-Pick Theory gives an exhaustive analysis of the following extremal problem on the disk. For data λ 1 , ,λ n ∈ and z 1 , ,z n ∈ , consider (1.4) ρ = inf {sup λ∈ |ϕ(λ)| : ϕ : holo −→ ,ϕ(λ i )=z i }. Functions ψ for which (1.4) is attained are referred to as extremal and the most important fact in the whole theory is that there is only one extremal for given data. Once this fact is realized it comes as no surprise that there is a finite algebraic procedure for creating a formula for the extremal in terms of the data and the critical value ρ (as an eigenvalue problem) an important result, not only in function theory [13], but in the model theory for Hilbert space contractions [12] and in the mathematical theory of control [15]. Now, let us consider the associated extremal problem on the bidisk. For data λ i =(λ 1 i ,λ 2 i ) ∈ 2 , 1 ≤ i ≤ n, and z i ∈ , 1 ≤ i ≤ n, let (1.5) ρ = inf { sup λ∈ 2 |ϕ(λ)| : ϕ : 2 holo −→ ,ϕ(λ i )=z i }. Unlike the case of the disk, extremals for (1.5) are not unique. The authors however have discovered the interesting fact that there is a polynomial variety in the Annals of Mathematics The number of extensions of a number field with fixed degree and bounded discriminant By Jordan S. Ellenberg and Akshay Venkatesh* Annals of Mathematics, 163 (2006), 723–741 The number of extensions of a number field with fixed degree and bounded discriminant By Jordan S. Ellenberg and Akshay Venkatesh* Abstract We give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant ≤ X; these bounds improve on work of Schmidt. We also prove various related results, such as lower bounds for the number of extensions and upper bounds for Galois extensions. 1. Introduction Let K be a number field, and let N K,n (X) be the number of number fields L (always considered up to K-isomorphism) such that [L : K]=n and N K Q D L/K <X. Here D L/K is the relative discriminant of L/K, and N K Q is the norm on ideals of K, valued in positive integers. D L = |D L/ Q | will refer to discriminant over Q. A folk conjecture, possibly due to Linnik, asserts that N K,n (X) ∼ c K,n X (n fixed, X →∞). This conjecture is trivial when n = 2; it has been proved for n =3by Davenport and Heilbronn [7] in case K = Q, and by Datskovsky and Wright in general [6]; and for n =4, 5 and K = Q by Bhargava [3], [2]. A weaker version of the conjecture for n = 5 was also recently established by Kable and Yukie [11]. These beautiful results are proved by methods which seem not to extend to higher n. The best upper bound for general n is due to Schmidt [18], who showed N K,n (X)  X (n+2)/4 where the implied constant depends on K and n. We refer to [4] for a survey of results. In many cases, it is easy to show that N K,n (X) is bounded below by a constant multiple of X; for instance, if n is even, simply consider the set of *The first author was partially supported by NSA Young Investigator Grant MDA905- 02-1-0097. The second author was partially supported by NSF Grant DMS-0245606. 724 JORDAN S. ELLENBERG AND AKSHAY VENKATESH quadratic extensions of a fixed L 0 /K of degree n/2. For the study of lower bounds it is therefore more interesting to study the number of number fields L such that [L : K]=n, N K Q D L/K <Xand the Galois closure of L has Galois group S n over K. Denote this number by N  K,n (X). Malle showed [14, Prop. 6.2] that N  Q ,n (X) >c  n X 1/n for some constant c  n . The main result of this paper is to improve these bounds, with particular attention to the “large n limit.” The upper bound lies much deeper than the lower bound. Throughout this paper we will use  and  where the implicit constant depends on n; we will not make this n-dependency explicit (but see our ap- pendix to [1] for results in this direction). Theorem 1.1. For al l n>2 and all number fields K, we have N K,n (X)  (XD n K A [K: Q ] n ) exp(C √ log n) where A n is a constant depending only on n, and C is an absolute constant. Further, X 1/2+1/n 2  K N  K,n (X). In particular, for all ε>0 lim sup X→∞ log N K,n (X) log X  ε n ε , lim inf X→∞ log N  K,n (X) log X ≥ 1 2 + 1 n 2 .(1.1) Linnik’s conjecture claims that the limit in (1.1) is equal to 1; thus, despite its evident imprecision, the upper bound in Theorem 1.1 seems to offer the first serious evidence towards this conjecture for large n. It is also worth observing that de Jong and Katz [9] have [...]... sense, to each of the other members xviii Preface To mathematicians (at least to most of them, as far as I can make out), mathematics is not just a cultural activity that we have ourselves created, but it has a life of its own, and much of it Wnds an amazing harmony with the physical universe We cannot get any deep understanding of the laws that govern the physical world without entering the world of mathematics... clearly recall how the matter was Wnally resolved, but with the hindsight gained from my much later experiences as a mathematics undergraduate, I guess my schoolteacher was making a brave attempt at telling us the deWnition of a fraction in terms of the ubiquitous mathematical notion of an equivalence class What is this notion? How can it be applied in the case of a fraction and tell us what a fraction... noticed There is, indeed, a profound issue that one comes up against again and again in mathematics and in mathematical physics, which one Wrst encounters in the seemingly innocent operation of cancelling a common factor from the numerator and denominator of an ordinary numerical fraction Those for whom the action of cancelling has become second nature, because of repeated familiarity with such operations,... readership that this will entail I have thought seriously about this question, and have come to the conclusion that what I have to say cannot reasonably be conveyed without a certain amount of mathematical notation and the exploration of genuine mathematical concepts The understanding that we have of the principles that actually underlie the behaviour of our physical world indeed depends upon some appreciation... conveying the idea—and the beauty and the magic—inherent in many important mathematical notions The idea of a fraction such as 3 is simply that it is some kind of an entity which has the 8 property that, when added to itself 8 times in all, gives 3 The magic is that the idea of a fraction actually works despite the fact that we do not really directly experience things in the physical world that are exactly... imprecise physical situations to which the precise mathematical notion of a fraction was to be applied; they did not tell us what that clear-cut mathematical notion actually is Other suggestions came forward, such as 3 is ‘something with The Costs of Inheritance in Semantic Networks Rob't F. Simmons The University of Texas, Austin Abstract Questioning texts represented in semantic relations I requires the recognition that synonyms, instances, and hyponyms may all satisfy a questioned term. A basic procedure for accomplishing such loose matching using inheritance from a taxonomic organization of the dictionary is defined in analogy with the unification a!gorithm used for theorem proving, and the costs of its application are analyzed. It is concluded tl,at inherit,~nce logic can profitably be ixiclu.'ted in the basic questioning procedure. AI Handbook Study In studying the pro ~ss of answering questions from fifty pages of the AI tlandbook, it is striking that such subsections as those describing problem representations are organized so as to define conceptual dictionary entries for the terms. First, class definitions are offered and their terms defined; then examples are given and the computational terms of the definitions are instantiated. Finally the technique described is applied to examples and redel'ined mathematical!y. Organizing these texts (by hand) into coherent hierarchic structures of discourse results in very usable conceptual dictionary definitions that are related by taxonomic and partitive relations, leaving gaps only for non-technical terms. For example, in "give snapshots of the state of the problem at various stages in its solution," terms such as "state', 'problem', and "solution" are defined by the text. while • give', "snapshots', and "stages = are not. Our first studies in representing and questioning this text have used semantic networks with a minimal number of case arcs to represent the sentences and Super:~et/Instance and *Of/llas arcs to represent, respectively, taxonomic and partitive relations between concepts. Equivalence arcs are also used to represent certain relations sig~fified by uses of "is" and apposition 1supported by NSF Grant/ST 8200976 and *AND and *OR arcs represent conjunction. Since June 1982, eight question-answering systems have been' written, some in procedural logic and some in compilable EIJSP. Although we have so far studied questioning and data manipulation operations on about 40 pages of the text, the detailed study of inheritance costs discussed in this paper was based on 170 semantic relations (SRs), represented by 733 binary relations each composed of a node-arc-node triple. In this study the only inference rules used were those needed to obtain transitive closure for inheritance, but in other studies of this text a great deal of power is gained by using general inference rules for paraphrasing the question into the terms given by an answering text. The use of paraphrastie inference rules is computationally expensive and is discussed elsewhere [Simmons 1083]. The text-knowledge base is constructed either as a set of triples using subscripted words, or by establishing node-numbers whose values are the complete SR and indexing these by the first element of every SR. The latter form, shown in Figure 1, occupies only about a third of the space that the triples require and neither form is clearly computationally better than the other. The first experiments with this text-knowledge base showed that the cost of following inheritance ares, i.e. obtaining taxonomic closures for concepts, was very high; some questions required as much as a minute of central processor time. As a result it was necessary to analyze the process and to develop ... number of possible phenotypes depends on the dominance relationships between the three alleles 3/13 Extensions of the Laws of Inheritance Inheritance of the ABO blood system in humans is shown... us consider the biological basis of gene linkage and recombination 7/13 Extensions of the Laws of Inheritance Homologous chromosomes possess the same genes in the same order, though the specific... one gene masks 9/13 Extensions of the Laws of Inheritance or interferes with the expression of another “Epistasis” is a word composed of Greek roots meaning “standing upon.” The alleles that are