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Contributions to the Founding of the Theory of Transfinite Numbers
Abstract:Dear readers, when you are hunting the new book collection to read this day, contributions to the founding of the theory of transfinite numbers dover books on mathematics can be your referred book. Yeah, even many books are offered, this book can steal the reader heart so much. The content and theme of this book really will touch your heart. You can find more and more experience and knowledge how the life is undergone.
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Exploring the uncharted territories of networked control systems : by scavenging for structure in dynamics and communication
2019
摘要
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.
Quasi-ordres généralisés et représentation numérique
1978
摘要
© Centre d’analyse et de mathématiques sociales de l’EHESS, 1978, tous droits réservés. L’accès aux archives de la revue « Mathématiques et sciences humaines » (http:// msh.revues.org/) implique l’accord avec les conditions générales d’utilisation (http://www. numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
Sets and Their Sizes Sets and Their Sizes
2001
摘要
x) is to be read as the result of abstracting from the element x, Unit (x) as x is a unit, and |M | as the cardinal number of M . So (1.1) gives a definition of cardinal number, in terms of the operation of abstraction, from which Cantor proves both ONE-ONEa and ONE-ONEb. M ∼ N → |M | = |N | (ONE-ONEa) |M | = |N | → M ∼ N (ONE-ONEb) ONE-ONEa is true, says Cantor, because the cardinal number |M | remains unaltered if in the place of one or many or even all elements m of M other things are substituted. [Cantor, p.80] and so, if f is a one-one mapping fromM ontoN , then in replacing each element, m, of M with f(m) M transforms into N without change of cardinal number. (p.88) 1.2. CANTOR’S ARGUMENT 5 In its weakest form, the principle Cantor cites says that if a single element of M is replaced by an arbitrary element not in M then the cardinal number of the set will remain the same. That is, (1.4) (a ∈ M ∧ b / ∈ M) ∧N = {x | (x ∈ M ∧ x / ∈ a) ∨ x ∈ b } → |M | = |N | The reasoning is clear: so far as the cardinal number of a set is concerned, one element is much the same as another. It is not the elements of a set, but only their abstractions, that enter into the cardinal number of a set. But abstractions of elements are just units; so one is much the same as another. ONE-ONEb is true, Cantor says, because . . . |M | grows, so to speak, out of M in such a way that from every element m of M a special unit of |M | arises. Thus we can say that M ∼ |M |. So, since a set is similar to its cardinal number, and similarity is an equivalence relation, two sets with same cardinal number are similar. Unless each element of a set abstracts to a ‘special’, i.e. distinct, unit, the correspondence from M to its cardinal number will be many-one and not one-one. A weak version of this principle is: M = {a, b} ∧ a 6= b → |M | = {Abstract(a),Abstract(b)} ∧ Abstract(a) 6= Abstract(b) (1.5) These two arguments do one another in. (1.4) says that replacing an element of a set with any element not in the set does not affect the cardinality. But, by the definition of |M |, (1.1), this means that (1.6) ∀x∀y(Abstract(x) = Abstract(y)) For, consider an arbitrary pair of elements, a and b. Let M = {a} and let N = {b}. So, the conditions of (1.4) are met and |M | = |N |. But |M | = {Abstract(a)} and |N | = {Abstract(b)}, by (1.1). So Abstract(a) = Abstract(b). Generalizing this argument yields (1.6). So Cantor’s argument for ONE-ONEa only works by assigning all nonempty sets the same, one-membered, cardinal number. But, this contradicts ONE-ONEb. Conversely, the argument that a set is similar to its cardinal number relies on (1.5), which entails (1.7) ∀x∀y(x 6= y → Abstract(x) 6= Abstract(y))
Set Theory from Cantor to Cohen
Sets and Extensions in the Twentieth Century
2012
摘要
ion, from a set M to its order type M and then to its cardinality M. Almost two decades after his [1874] result that the reals are uncountable, Cantor in a short note [1891] subsumed it via his celebrated diagonal argument. With it, he estab15After describing the similarity between ω and √ 2 as limits of sequences, Cantor [1887: 99] interestingly correlated the creation of the transfinite numbers to the creation of the irrational numbers, beyond merely breaking new ground in different number contexts: “The transfinite numbers are in a certain sense new irrationalities, and in my opinion the best method of defining the finite irrational numbers [via Cantor’s fundamental sequences] is wholly similar to, and I might even say in principle the same as, my method of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers: they are like each other in their innermost being [Wesen]; for the former like the latter are definite delimited forms or modifications of the actual infinite.” 16Ferreiros [1995] suggests how the formulation of the second number class as a completed totality with a succeeding transfinite number emerged directly from Cantor’s work on the operation P′, drawing Cantor’s transfinite numbers even closer to his earlier work on trigonometric series.
Autonomic Model for Self-Configuring C#.NET Applications
ArXiv
2012
摘要
With the advances in computational technologies over the last decade, large organizations have been investing in Information Technology to automate their internal processes to cut costs and efficiently support their business projects. However, this comes to a price. Business requirements always change. Likewise, IT systems constantly evolves as developers make new versions of them, which require endless administrative manual work to customize and configure them, especially if they are being used in different contexts, by different types of users, and for different requirements. Autonomic computing was conceived to provide an answer to these ever-changing requirements. Essentially, autonomic systems are self-configuring, self-healing, self-optimizing, and self-protecting; hence, they can automate all complex IT processes without human intervention. This paper proposes an autonomic model based on Venn diagram and set theory for self-configuring C#.NET applications, namely the self-customization of their GUI, event-handlers, and security permissions. The proposed model does not require altering the source-code of the original application; rather, it uses an XML-based customization file to turn on and off the internal attributes of the application. Experiments conducted on the proposed model, showed a successful automatic customization for C# applications and an effective self-adaption based on dynamic business requirements. As future work, other programming languages such as Java and C++ are to be supported, in addition to other operating systems such as Linux and Mac so as to provide a standard platform-independent autonomic self-configuring model.
Context and Semantic Aware Location Privacy
2016
摘要
With ever-increasing computational power, and improved sensing and communication capabilities, smart devices have altered and enhanced the way we process, perceive and interact with information. Personal and contextual data is tracked and stored extensively on these devices and, oftentimes, ubiquitously sent to online service providers. This routine is proving to be quite privacy-invasive, since these service providers mine the data they collect in order to infer more and more personal information about users. Protecting privacy in the rise of mobile applications is a critical challenge. The continuous tracking of users with location- and time-stamps expose their private lives at an alarming level. Location traces can be used to infer intimate aspects of users' lives such as interests, political orientation, religious beliefs, and even more. Traditional approaches to protecting privacy fail to meet users' expectations due to simplistic adversary models and the lack of a multi-dimensional awareness. In this thesis, the development of privacy-protection approaches is pushed further by (i) adapting to concrete adversary capabilities and (ii) investigating the threat of strong adversaries that exploit location semantics. We first study user mobility and spatio-temporal correlations in continuous disclosure scenarios (e.g., sensing applications), where the more frequently a user discloses her location, the more difficult it becomes to protect. To counter this threat, we develop adversary- and mobility-aware privacy protection mechanisms that aim to minimize an adversary's exploitation of user mobility. We demonstrate that a privacy protection mechanism must actively evaluate privacy risks in order to adapt its protection parameters. We further develop an Android library that provides on-device location privacy evaluation and enables any location-based application to support privacy-preserving services. We also implement an adversary-aware protection mechanism in this library with semantic-based privacy settings. Furthermore, we study the effects of an adversary that exploits location semantics in order to strengthen his attacks on user traces. Such extensive information is available to an adversary via maps of points of interest, but also from users themselves. Typically, users of online social networks want to announce their whereabouts to their circles. They do so mostly, if not always, by sharing the type of their location along with the geographical coordinates. We formalize this setting and by using Bayesian inference show that if location semantics of traces is disclosed, users' privacy levels drop considerably. Moreover, we study the time-of-day information and its relation to location semantics. We reveal that an adversary can breach privacy further by exploiting time-dependency of semantics. We implement and evaluate a sensitivity-aware protection mechanism in this setting as well. The battle for privacy requires social awareness and will to win. However, the slow progress on the front of law and regulations pushes the need for technological solutions. This thesis concludes that we have a long way to cover in order to establish privacy-enhancing technologies in our age of information. Our findings opens up new venues for a more expeditious understanding of privacy risks and thus their prevention.
Time Hierarchies: A Survey
Electron. Colloquium Comput. Complex.
2007
摘要
We survey time hierarchies, with an emphasis on recent work on hierarchies for semantic classes.
Ordinal Arithmetic with List Structures
LFCS
1992
摘要
We provide a set of “natural” requirements for well-orderings of (binary) list structures. We show that the resultant order-type is the successor of the first critical epsilon number.
EagerMerge: an optimistic technique for efficient points-to analysis
ISSTA
2016
摘要
We present an information-merging technique for efficient computation of points-to information for C programs. Invalid use of pointers can lead to hard-to-find bugs and may expose security vulnerabilities. Thus, analyzing them is critical for software analysis as well as optimization. Pointer analysis is a key step during compilation, and the computed points-to information is useful for client analyses from varied domains such as bug finding, data-flow analysis, identifying security vulnerabilities, and parallelization, to name a few. Former research on pointer analysis has indicated that the main bottleneck towards scalability is large propagation of points-to information in the constraint graph. To reduce the propagation cost, we present a technique based on temporal similarity of points-to sets. The method tracks pointers whose dynamically changing points-to information remains equal for a while. Based on the optimism gained by observing the points-to sets over time, the analysis decides to merge the corresponding nodes. Using the notion of merge and split, we build a family of points-to analyses, and compare their relative precisions in the context of existing analyses. We illustrate the effectiveness of our approach using a suite of sixteen SPEC 2000 benchmarks and three large open-source programs, and show that the technique improves the analysis time over BDD and bitmap based Hybrid Cycle Detection, well-known Andersen's analysis, and Deep Propagation, affecting minimal precision (precision is 96.4% on an average). Specifically, it is faster than Deep Propagation by 45%.
Specification and Verification of Concurrent Programs Through Refinements
Journal of Automated Reasoning
2012
摘要
We present a framework for the specification and verification of reactive concurrent programs using general-purpose mechanical theorem proving. We define specifications for concurrent programs by formalizing a notion of refinements analogous to stuttering trace containment. The formalization supports the definition of intuitive specifications of the intended behavior of a program. We present a collection of proof rules that can be effectively orchestrated by a theorem prover to reason about complex programs using refinements. The proof rules systematically reduce the correctness proof for a concurrent program to the definition and proof of an invariant. We include automated support for discharging this invariant proof with a predicate abstraction tool that leverages the existing theorems proven about the components of the concurrent programs. The framework is integrated with the ACL2 theorem prover and we demonstrate its use in the verification of several concurrent programs in ACL2.
Separable Banach space theory needs strong set existence axioms
1996
摘要
We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, Π1 comprehension, is needed to prove such basic facts as the existence of the weak-∗ closure of any norm-closed subspace of `1 = c0. This is in contrast to earlier work [5, 7, 6, 25, 22] in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for Π2 sentences. En route to our main results, we prove the Krein-Smulian theorem in ACA0, and we give a new, elementary proof of a result of McGehee on weak-∗ sequential closure ordinals.
Contextual Considerations in Probabilistic Situations: An Aid or a Hindrance?
2014
摘要
We examine the responses of secondary school teachers to a probability task with an infinite sample space. Specifically, the participants were asked to comment on a potential disagreement between two students when evaluating the probability of picking a particular real number from a given interval of real numbers. Their responses were analyzed via the theoretical lens of reducing abstraction. The results show a strong dependence on a contextualized interpretation of the task, even when formal mathematical knowledge is evidenced in the responses.
Efficient execution in an automated reasoning environment
J. Funct. Program.
2008
摘要
We describe a method that permits the user of a mechanized mathematical logic to write elegant logical definitions while allowing sound and efficient execution. In particular, the features supporting this method allow the user to install, in a logically sound way, alternative executable counterparts for logically defined functions. These alternatives are often much more efficient than the logically equivalent terms they replace. These features have been implemented in the ACL2 theorem prover, and we discuss several applications of the features in ACL2.
Problem Theory
ArXiv
2014
摘要
We define a problem theory from first principles. We investigate the objects of this theory: problems, resolutions, and solutions. We relate problem theory with set theory and with computing theory. We find taxonomies for resolutions and for problems. We build a hierarchy of resolvers: mechanism, adaptor, internalizer, learner, and subject. We show that the problem theory is complete, that is, that there are just three ways to resolve any problem: routine, trial, and analogy. Finally, we propose a thesis: We are Turing complete subjects because we are the result of an evolution of resolvers of the survival problem.
PHILOSOPHICAL METHOD AND GALILEO'S PARADOX OF INFINITY
2008
摘要
We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to re-engineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by one-to-one correspondence, and yet less numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that one-to-one correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Godel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Godel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding.
Measuring fractals by infinite and infinitesimal numbers
2008
摘要
Traditional mathematical tools used for analysis of fractals allow one to distinguish results of self-similarity processes after a finite number of iterations. For example, the result of procedure of construction of Cantor’s set after two steps is different from that obtained after three steps. However, we are not able to make such a distinction at infinity. It is shown in this paper that infinite and infinitesimal numbers proposed recently allow one to measure results of fractal processes at different iterations at infinity too. First, the new technique is used to measure at infinity sets being results of Cantor’s procedure. Second, it is applied to calculate the lengths of polygonal geometric spirals at different points of
Numerical computations and mathematical modelling with infinite and infinitesimal numbers
ArXiv
2012
摘要
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer—the Infinity Computer—able to work with all these types of numbers. The new computational tools both give possibilities to execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given.
A New Computational Methodology Using Infinite and Infinitesimal Numbers
ACRI
2010
摘要
Traditional computers work numerically only with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this lecture, a new computational methodology (that is not related to non-standard analysis approaches) is described. It is based on the principle 'The part is less than the whole' applied to all quantities (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The new methodology has allowed the author to introduce the Infinity Computer working numerically with infinite and infinitesimal numbers. The new computational paradigm both gives possibilities to execute computations of a new type and simplifies fields of Mathematics and Computer Science where infinity and/or infinitesimals are required. Examples of the usage of the introduced computational tools are given during the lecture.
The Garden of Knowledge as a Knowledge Manifold A Conceptual Framework for Computer Supported Subjective Education
1997
摘要
This work presents a unified pattern-based epistemological framework,called a Knowledge Manifold, for the description and extraction of knowledge from information. Within this framework it also presents the metaphor of the Garden Of Knowledge as a constructive example. Any type of KM is defined in terms of its objective calibration protocols procedures that are implemented on top of the participating subjective knowledge-patches. They are the procedures of agreement and obedience that characterize the coherence of any type of interaction, and which are used here in order to formalize the concept of participator consciousness in terms of the inverse-direct limit duality of Category Theory.
Representation and estimation of stochastic populations
2015
摘要
This work is concerned with the representation and the estimation of populations composed of an uncertain and varying number of individuals which can randomly evolve in time. The existing solutions that address this type of problems make the assumption that all or none of the individuals are distinguishable. In other words, the focus is either on specific individuals or on the population as a whole. These approaches have complimentary advantages and drawbacks and the main objective in this work is to introduce a suitable representation for partially-indistinguishable populations. In order to fulfil this objective, a sufficiently versatile way of quantifying different types of uncertainties has to be studied. It is demonstrated that this can be achieved within a measure-theoretic Bayesian paradigm. The proposed representation of stochastic populations is then used for the introduction of various filtering algorithms from the most general to the most specific. The modelling possibilities and the accuracy of one of these filters are then demonstrated in different situations.