Добірка наукової літератури з теми "Degree of irrationality"

We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to lower bounds on these invariants. In particular, we prove that if $X\subseteq \mathbf{P}^{n+1}$ is a very general smooth hypersurface of dimension $n$ and degree $d\geqslant 2n+1$, then any dominant rational mapping $f:X{\dashrightarrow}\mathbf{P}^{n}$ must have degree at least $d-1$. We also propose a number of open problems, and we show how our methods lead to simple new proofs of results of Ran and Beheshti–Eisenbud concerning varieties of multi-secant lines.

Noether’s problem asks whether, for a given field [Formula: see text] and finite group [Formula: see text], the fixed field [Formula: see text] is a purely transcendental extension of [Formula: see text], where [Formula: see text] acts on the [Formula: see text] by [Formula: see text]. The field [Formula: see text] is naturally the function field for a quotient variety [Formula: see text]. We study the degree of irrationality [Formula: see text] of [Formula: see text] for an abelian group [Formula: see text], which is defined to be the minimal degree of a dominant rational map from [Formula: see text] to projective space. In particular, we give bounds for [Formula: see text] in terms of the arithmetic of cyclotomic extensions [Formula: see text].

Structural synthesis creates a structural scheme according to the given structural characteristics - the degree of abnormality of the structure S and the degree of its irrationality s. Structural scheme is created at the first stage of synthesis, containing only rotational kinematic pairs and corresponds to a given degree of abnormality, which is ensured by the ratio of structural units – plus monads, minus monads and null monads. At the second stage, the independence of structural characteristics makes it possible to bring the obtained scheme to a given degree of irrationality by downgrading the class of kinematic pairs. The proposed algorithm makes it possible to synthesize structural schemes of both normal structure and adaptive (with redundant motions), as well as indifferent (with redundant links). Monadic approach to structure research allows to formalize the structural features of the circuit and to create a knowledge base for automated design of structural circuits with given structural properties.

Cette thèse est consacrée à l'étude des cycles algébriques dans les variétés hyper-Kähleriennes projectives et les variétés de Calabi-Yau strictes. Elle contribue à la compréhension des conjectures de Beauville et de Voisin sur les anneaux de Chow des variétés hyper-kählériennes projectives et des variétés de Calabi-Yau strictes. Elle étudie également certains invariants birationnels des variétés hyper-kählériennes projectives.La première partie de la thèse, parue dans Mathematische Zeitschrift [C. Bai, On Abel-Jacobi maps of Lagrangian families, Math. Z. 304, 34 (2023)] et présentée dans le chapitre 2, étudie si les sous-variétés lagrangiennes dans une variété hyper-kählérienne partageant la même classe cohomologique ont également la même classe de Chow. Nous étudions la notion de familles lagrangiennes et ses applications aux applications d'Abel-Jacobi associées. Nous adoptons une approche infinitésimale pour donner un critère de trivialité de l'application d'Abel-Jacobi d'une famille lagrangienne, et utilisons ce critère pour donner une réponse négative à la question précédente, ajoutant aux subtilités d'une conjecture de Voisin. Nous explorons également comment la maximalité de la variation des structures de Hodge sur la cohomologie de degré 1 de la famille lagrangienne implique la trivialité de l'application d'Abel-Jacobi. La deuxième partie de la thèse, à paraître dans International Mathematics Research Notices [C. Bai, On some birational invariants of hyper-Kähler manifolds, ArXiv: 2210.12455, to appear in International Mathematics Research Notices, 2024] et présentée dans le chapitre 3, étudie le degré d'irrationalité, la gonalité fibrante et le genre fibrant des variétés hyper-kählériennes projectives. Nous commençons par donner une légère amélioration d'un résultat de Voisin sur la borne inférieure du degré d'irrationalité des variétés hyper-kählériennes générales de Mumford-Tate. Nous étudions ensuite la relation entre les trois invariants birationnels susmentionnés pour les surfaces K3 projectives de nombre de Picard 1, rajoutant la compréhension sur une conjecture de Bastianelli, De Poi, Ein, Lazarsfeld, Ullery sur le comportement asymptotique du degré d'irrationalité des surfaces K3 projectives très générales. La troisième partie de la thèse, présentée dans le chapitre 4, étudie les applications de Voisin de dimension supérieure sur les variétés de Calabi-Yau strictes. Voisin a construit des applications auto-rationnelles de variétés de Calabi-Yau obtenues comme des variétés de r-plans dans des hypersurfaces cubiques de dimension adéquate. Cette application a été largement étudiée dans le cas r=1, qui est le cas de Beauville-Donagi. Dans les cas de dimensions supérieures, nous étudions d'abord l'action de l'application de Voisin sur les formes holomorphes. Nous démontrons ensuite la conjecture de Bloch généralisée pour l'action des applications de Voisin sur les groupes de Chow dans le cas de r=2. Enfin, via l'étude de l'application de Voisin, nous apportons des éléments de preuve à une conjecture de Voisin sur l'existence d'un 0-cycle spécial sur les variétés de Calabi-Yau strictes
This thesis is devoted to the study of algebraic cycles in projective hyper-Kähler manifolds and strict Calabi-Yau manifolds. It contributes to the understanding of Beauville's and Voisin's conjectures on the Chow rings of projective hyper-Kähler manifolds and strict Calabi-Yau manifolds. It also studies some birational invariants of projective hyper-Kähler manifolds.The first part of the thesis, appeared in Mathematische Zeitschrift [C. Bai, On Abel-Jacobi maps of Lagrangian families, Math. Z. 304, 34 (2023)] and presented in Chapter 2, studies whether the Lagrangian subvarieties in a hyper-Kähler manifold sharing the same cohomological class have the same Chow class as well. We study the notion of Lagrangian families and its associated Abel-Jacobi maps. We take an infinitesimal approach to give a criterion for the triviality of the Abel-Jacobi map of a Lagrangian family, and use this criterion to give a negative answer to the above question, adding to the subtleties of a conjecture of Voisin. We also explore how the maximality of the variation of the Hodge structures on the degree 1 cohomology the Lagrangian family implies the triviality of the Abel-Jacobi map. The second part of the thesis, to appear in International Mathematics Research Notices [C. Bai, On some birational invariants of hyper-Kähler manifolds, ArXiv: 2210.12455, to appear in International Mathematics Research Notices, 2024] and presented in Chapter 3, studies the degree of irrationality, the fibering gonality and the fibering genus of projective hyper-Kähler manifolds, with emphasis on the K3 surfaces case, en mettant l'accent sur le cas des surfaces K3. We first give a slight improvement of a result of Voisin on the lower bound of the degree of irrationality of Mumford-Tate general hyper-Kähler manifolds. We then study the relation of the above three birational invariants for projective K3 surfaces of Picard number 1, adding the understandinf of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld, Ullery on the asymptotic behavior of the degree of irrationality of very general projective K3 surfaces. The third part of the thesis, presented in Chapter 4, studies the higher dimensional Voisin maps on strict Calabi-Yau manifolds. Voisin constructed self-rational maps of Calabi-Yau manifolds obtained as varieties of r-planes in cubic hypersurfaces of adequate dimension. This map has been thoroughly studied in the case r=1, which is the Beauville-Donagi case. For higher dimensional cases, we first study the action of the Voisin map on the holomorphic forms. We then prove the generalized Bloch conjecture for the action of the Voisin maps on Chow groups for the case of r=2. Finally, via the study of the Voisin map, we provide evidence for a conjecture of Voisin on the existence of a special 0-cycle on strict Calabi-Yau manifolds

Rationality is a central concept for epistemology, ethics, and the study of practical reason. But what sort of concept is it? It is argued here that—contrary to objections that have recently been raised—rationality is a normative concept. In general, normative concepts cannot be explained in terms of the concepts expressed by ‘reasons’ or ‘ought’. Instead, normative concepts are best understood in terms of values. Thus, for a mental state or a process of reasoning to be rational is for it to be in a certain way good. Specifically, rationality is a virtue, while irrationality is a vice. What rationality requires of you at a time is whatever is necessary for your thinking at that time to be as rational as possible; this makes ‘rationally required’ equivalent to a kind of ‘ought’. Moreover, rationality is an “internalist” normative concept: what it is rational for you to think at a time depends purely on what is in your mind at that time. Nonetheless, rationality has an external goal—namely, getting things right in your thinking, or thinking correctly. The connection between rationality and correctness is probabilistic: if your thinking is irrational, that is bad news about your thinking’s degree of correctness; and the more irrational your thinking is, the worse the news is about your thinking’s degree of correctness. This account of the concept of rationality indicates how we should set about giving a substantive theory of what it is for beliefs and choices to be rational.

Abstract Chapter 9 turns to explaining what is here being referred to as the “rational probability function” for each case. According to the account that is being developed here, the degree of irrationality of the belief-system that the thinker has in each case is in effect determined by the belief-system’s distance from the rational probability function for the case. But are the degrees of irrationality of all belief-systems that are available to the thinker at the time determined by how distant they are from the same probability function? A consideration of so-called “Moore-paradoxical” propositions reveals that the answer to this question is: No. The internal mental facts “given” to the thinker at the time determine an appropriate prior probability function; and the degree of irrationality of each of these belief-systems is determined by how distant it is from the result of conditionalizing this prior probability on the supposition that the thinker has that very belief-system. This appropriate prior probability can be conceived by imagining an “evaluating angel”. This angel knows all the internal mental facts about what is given to the thinker at the time, and relies solely on these facts to form a body of opinions about what the world is like; the appropriate prior probability function may be conceived as this angel’s body of opinions about the world.

Abstract Chapter 3 proposes a simple evolutionary explanation for irrational behaviour using the binary choice framework introduced in Chapter 2. This framework shows that irrational behaviours are, in fact, necessary for evolution in stochastic environments because they provide much-needed diversity that confers survival benefits when certain environments arise. Furthermore, there is an optimal degree of irrationality in the population, depending on the degree of environmental randomness. Therefore, mutation provides an important link between rational and irrational behaviours, and hence a diversity of traits in any given generation. These results reconcile the debate between rational and irrational behaviour—both can and do emerge through the forces of natural selection operating on behaviours, as well as genes.

Abstract Obsessive-compulsive disorder (OCD) is a complex and debilitating disorder that has a high degree of comorbidity with mood and anxiety disorders. Strategies for differential diagnosis of OCD and these disorders, as well as the interactions of the diagnoses in treatment settings, are discussed. The chapter also explores the relationships between OCD and personality disorders, as well as between OCD and the OCD-related disorders. OCD is also characterized by impaired quality of life, family involvement, and varying insight. The presence of OCD is associated with high levels of functional impairment, especially in the social and emotional domains. Relationships with important others can be affected by the disorder. Family members are often involved in OCD symptoms, and family variables like symptom accommodation and hostility are related to poorer treatment outcomes. Treatment outcomes can also be influenced by level of insight held by the individual with OCD; those with poor insight into the irrationality of their symptoms often face challenges with treatment engagement. The chapter aims to provide a brief overview of comorbidity, family, insight, and quality-of-life issues and to review assessment and treatment implications of those issues. Measures for assessing relevant constructs are described.

This chapter explores irrationality, the second ground for judicial review identified by Lord Diplock in Council of Civil Service Unions and Others v Minister for the Civil Service. It examines the meaning of this principle, its foundation upon the test of unreasonableness, and the approach that the courts have adopted since that case. Irrationality, and the notion of unreasonableness upon which it is based, is a particularly vague and ambiguous term, with a range of possible interpretations and meanings. This has meant that the courts have often considered judicial review claims, brought on the basis of irrationality, with varying degrees of caution, often employing the necessary tests with notable stringency. In part as a result of this, and in part also due to the increasing influence of European legal practices on the UK system, the test of proportionality has developed as a substantive ground for judicial review, often overlapping and sometimes conflicting with application of the irrationality doctrine.

This chapter explores irrationality, the second ground for judicial review identified by Lord Diplock in Council of Civil Service Unions and Others v Minister for the Civil Service. It examines the meaning of this principle, its foundation upon the test of unreasonableness, and the approach that the courts have adopted since that case. Irrationality, and the notion of unreasonableness upon which it is based, is a particularly vague and ambiguous term, with a range of possible interpretations and meanings. This has meant that the courts have often considered judicial review claims, brought on the basis of irrationality, with varying degrees of caution, often employing the necessary tests with notable stringency. In part as a result of this, and in part also due to the increasing influence of European legal practices on the UK system, the test of proportionality has developed as a substantive ground for judicial review, often overlapping and sometimes conflicting with application of the irrationality doctrine.

This chapter explores irrationality, the second ground for judicial review identified by Lord Diplock in Council of Civil Service Unions and Others v Minister for the Civil Service. It examines the meaning of this principle, its foundation upon the test of unreasonableness, and the approach that the courts have adopted since that case. Irrationality, and the notion of unreasonableness upon which it is based, is a particularly vague and ambiguous term, with a range of possible interpretations and meanings. This has meant that the courts have often considered judicial review claims, brought on the basis of irrationality, with varying degrees of caution, often employing the necessary tests with notable stringency. In part as a result of this, and in part also due to the increasing influence of European legal practices on the UK system, the test of proportionality has developed as a substantive ground for judicial review, often overlapping and sometimes conflicting with application of the irrationality doctrine.

Abstract Chapter 10 turns to a range of issues concerning the determinacy of rational belief. First, we consider cases in which the opinions of the “evaluating angel” do not consist in a unique probability function, but only in a big set of such functions. Here, it is argued that, even in these cases, a perfectly rational thinker would in effect pick one of the probability functions in the set that constitutes the angel’s opinions, and would have a belief-system that matches the result of conditionalizing that probability function on the supposition that she has that belief-system. Secondly, we consider whether rational agents would sometimes have imprecise or indeterminate credences. Tentatively, it is argued that our account implies that perfectly rational thinkers never have imprecise credences of this sort: it is only imperfectly rational thinkers who ever have such imprecise credences. Thirdly, it is argued that degrees of irrationality are sometimes incommensurable: the degrees of irrationality sometimes fail to rank alternative belief-systems in relation to each other. However, these incommensurabilities only affect irrational belief-systems: for every thinker and time, some available belief-systems are perfectly rational; these perfectly rational belief-systems are all equally rational, and determinately more rational than all irrational systems.

Abstract Most legal theorists, of course, would deny that Robert Bork’s intellectual instincts and style resemble theirs and that this similarity was an important, legitimate reason for keeping him from the Supreme Court. However, in at least one respect, Bork and his critics are undeniably alike. To the extent that they can influence the content of the dialogue that formed the public aspect of the confirmation process, Bork, Tribe, and most other academics encourage the jurisprudential and doctrinal focus that I criticized in the last chapter. Bork’s complaint about the process is that it became too political, and Dworkin’s defense is that the Judiciary Committee in fact conducted an ‘extended seminar.’ Here, at least, is common ground on what confirmation hearings should be. This agreement is remarkable because, while our constitutional system contemplates considerable insulation for federal judges, the nomination and confirmation provisions explicitly provide the opportunity for political influence. The belief that even this opportunity should be utilized as a seminar suggests how deep the consensus is within the mainstream about the primacy of intellect in constitutional interpretation. It follows powerfully that the irrationality and emotionality of politics should be excluded from the decisionmaking of judges after they are appointed. Near the beginning of The Tempting of America, Bork appealed to this consensus by evoking a ‘disturbing’ image: 56 Judicial Power and American Character the legal force of that amendment arises from the extraordinary nature of the public debate, from the transformative power of focused and inspired public opinion. One problem with the distinction based on informal authority is that it is likely to be wildly inaccurate. Debate in the ABA may be full and thoughtful or it may be posed or even fraudulent; a single letter may reflect a foolish impulse or the most careful internal moral dialogue. Moreover, the same decisionmaking process can be complex enough to justify very different characterizations. Ackerman emphasizes how the Civil War amendments resulted from deep intellectual and moral deliberations, but they also resulted from (as he acknowledges) brute military and political force. To shift the example a great deal, law review articles might be given low weight because they are typically written by cloistered eccentrics and edited by inexperienced students, but they also can be characterized as the product of a focused, thoughtful collegial discourse. Given the room for error and choice in characterization, emphasis on informal authority is treacherous. To some degree, it inevitably reflects judgments about how correct, not how deliberative, a form of political pressure is.Justice Blackmun, being only human, is inclined to reject the informal authority of state legislatures that oppose his conclusions about abortion; he sees their decisions, like armed resistance to desegregation, as illegitimate expressions of emotion and hostility. Then again, it would be only natural for him to welcome supportive demonstrations by advocates of freedom of choice. Legalism, Realism, and Edwin Meese’s Heresy The confused and ambivalent state of our attitudes toward political pressure on the courts was vividly captured by the vociferous reaction to Attorney General Edwin Meese’s notorious speech, ‘The Law of the Constitution,’ which he delivered in the fall of 1986.

Q. Which comes first, the sites or the patients? A. The patients. Find the patients and the sites will follow. The main objective is to select only those sites that have definitively identified the required number of study-specific patients. To ensure that objective, whenever it is feasible, these patients should be lined up and ready to be enrolled before the sites are initiated. Screening is the means for identifying potential volunteers for a trial by assessing whether or not they satisfy the inclusion, exclusion, and other relevant criteria for a given study. Screening and identification can be carried out long before a study is scheduled to start. Recruitment is the formal process of getting agreement from patients that they will take part in a study and getting their informed consent. Patients who give informed consent can then be enrolled in the study. Some procedures to determine eligibility may need to be carried out immediately before enrollment, so the fact that a patient has been recruited does not automatically indicate that the patient will be enrolled. Enrollment is the process of entering a patient who has given consent into the study by carrying out whatever initial procedure is needed. The initial procedure may be randomization, a run-in phase, or dosing if the study is open label. . . .If the matter is one that can be settled by observation, make the observation yourself. Aristotle could have avoided the mistake of thinking that women have fewer teeth than men by the simple device of asking Mrs. Aristotle to keep her mouth open while he counted. . . . . . . —Bertrand Russell, philosopher. . . Everyone is aware of the uncertainty and inefficiency that characterize clinical research. It is therefore hard to explain why those few aspects that can be quantified with some degree of accuracy and certainty continue to be the very ones that are usually approached subjectively and irrationally. No two aspects illustrate this paradox more vividly (and no two aspects are more responsible for time delays, wasted money, and poor data) than poor clinical trial site selection and inadequate patient screening and identification before the study starts.