Academic literature on the topic 'Cantor sets'

The paper is devoted to an investigation of the genus of an orientable closed surface $M^{2}$ which admits $A$-endomorphisms whose nonwandering set contains a one-dimensional strictly invariant contracting repeller $\Lambda_{r}$ with a uniquely defined unstable bundle and with an admissible boundary of finite type. First, we prove that, if $M^{2}$ is a torus or a sphere, then $M^{2}$ admits such an endomorphism. We also show that, if $\Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^{2}\to M^{2}$ of a closed orientable surface $M^{2}$ and $f$ is not a diffeomorphism, then $\Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^{2}\to M^{2}$ is an $A$-endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a one-dimensional strictly invariant contracting repeller of Cantor type $\Omega_{r}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_{r}$ is regular, then $M^{2}$ is a two-dimensional torus $\mathbb{T}^{2}$ or a two-dimensional sphere $\mathbb{S}^{2}$.

AbstractLet Ca be the central Cantor set obtained by removing a central interval of length 1−2a from the unit interval, and then continuing this process inductively on each of the remaining two intervals. We prove that if log b/log a is irrational, then where dim is Hausdorff dimension. More generally, given two self-similar sets K,K′ in ℝ and a scaling parameter s>0, if the dimension of the arithmetic sum K+sK′ is strictly smaller than dim (K)+dim (K′)≤1 (‘geometric resonance’), then there exists r<1 such that all contraction ratios of the similitudes defining K and K′ are powers of r (‘algebraic resonance’). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.

We find conditions on the ratios of dissection of a Cantor set so that the group it generates under addition has positive Lebesgue measure. In particular, we answer affirmatively a special case of a conjecture posed by J. Palis.

We examine the formation of sets homeomorphic to the ternary Cantor set by continued radicals. We determine properties of bridges and gaps and calculate the thickness of the Cantor set. From this we apply information from continued fractions to continued radicals to obtain new results. We also consider the measure of several Cantor sets.

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008.
Committee Chair: Bellissard, Jean; Committee Member: Baker, Matt; Committee Member: Bakhtin, Yuri; Committee Member: Garoufalidis, Stavros; Committee Member: Putnam, Ian

This paper examines self-similar sets and some of their properties, including the natural equivalence relation found in bilipschitz equivalence. Both dimension and preservation of paths are determined to be invariant under this equivalence. Also, sophisticated techniques, one involving the use of directed graphs, show the equivalence of two spaces.
Master of Science

In this paper we present a topological epistemic logic, with modalities for knowledge (modeled as the universal modality), knowability (represented by the topological interior operator), and unknowability of the actual world. The last notion has a non-self-referential reading (modeled by Cantor derivative: the set of limit points of a given set) and a self-referential one (modeled by Cantor's perfect core of a given set: its largest subset without isolated points). We completely axiomatize this logic, showing that it is decidable and PSPACE-complete, and we apply it to the analysis of a famous epistemic puzzle: the Surprise Exam Paradox.

Grand Canyon National Park is all about time and timescales. Time is the currency of our daily life, of history, and of biological evolution. Grand Canyon’s beauty has inspired explorers, artists, and poets. Behind it all, Grand Canyon’s geology and sense of timelessness are among its most prominent and important resources. Grand Canyon has an exceptionally complete and well-exposed rock record of Earth’s history. It is an ideal place to gain a sense of geologic (or deep) time. A visit to the South or North rims, a hike into the canyon of any length, or a trip through the 277-mile (446-km) length of Grand Canyon are awe-inspiring experiences for many reasons, and they often motivate us to look deeper to understand how our human timescales of hundreds and thousands of years overlap with Earth’s many timescales reaching back millions and billions of years. This report summarizes how geologists tell time at Grand Canyon, and the resultant “best” numeric ages for the canyon’s strata based on recent scientific research. By best, we mean the most accurate and precise ages available, given the dating techniques used, geologic constraints, the availability of datable material, and the fossil record of Grand Canyon rock units. This paper updates a previously-published compilation of best numeric ages (Mathis and Bowman 2005a; 2005b; 2007) to incorporate recent revisions in the canyon’s stratigraphic nomenclature and additional numeric age determinations published in the scientific literature. From bottom to top, Grand Canyon’s rocks can be ordered into three “sets” (or primary packages), each with an overarching story. The Vishnu Basement Rocks were once tens of miles deep as North America’s crust formed via collisions of volcanic island chains with the pre-existing continent between 1,840 and 1,375 million years ago. The Grand Canyon Supergroup contains evidence for early single-celled life and represents basins that record the assembly and breakup of an early supercontinent between 729 and 1,255 million years ago. The Layered Paleozoic Rocks encode stories, layer by layer, of dramatic geologic changes and the evolution of animal life during the Paleozoic Era (period of ancient life) between 270 and 530 million years ago. In addition to characterizing the ages and geology of the three sets of rocks, we provide numeric ages for all the groups and formations within each set. Nine tables list the best ages along with information on each unit’s tectonic or depositional environment, and specific information explaining why revisions were made to previously published numeric ages. Photographs, line drawings, and diagrams of the different rock formations are included, as well as an extensive glossary of geologic terms to help define important scientific concepts. The three sets of rocks are separated by rock contacts called unconformities formed during long periods of erosion. This report unravels the Great Unconformity, named by John Wesley Powell 150 years ago, and shows that it is made up of several distinct erosion surfaces. The Great Nonconformity is between the Vishnu Basement Rocks and the Grand Canyon Supergroup. The Great Angular Unconformity is between the Grand Canyon Supergroup and the Layered Paleozoic Rocks. Powell’s term, the Great Unconformity, is used for contacts where the Vishnu Basement Rocks are directly overlain by the Layered Paleozoic Rocks. The time missing at these and other unconformities within the sets is also summarized in this paper—a topic that can be as interesting as the time recorded. Our goal is to provide a single up-to-date reference that summarizes the main facets of when the rocks exposed in the canyon’s walls were formed and their geologic history. This authoritative and readable summary of the age of Grand Canyon rocks will hopefully be helpful to National Park Service staff including resource managers and park interpreters at many levels of geologic understandings...

In the last decade large data sets have become available, both in terms of the number of time series and with higher frequencies (weekly, daily and even higher). All series may suffer from seasonality, which hides other important fluctuations. Therefore time series are typically seasonally adjusted. However, standard seasonal adjustment methods cannot handle series with higher than monthly frequencies. Recently, Abeln et al. (2019) presented CAMPLET, a new seasonal adjustment method, which does not produce revisions when new observations become available. The aim of this paper is to show the attractiveness of CAMPLET for seasonal adjustment of daily time series. We apply CAMPLET to daily data on the gas system in the Netherlands.

This report describes publicly available data sets and quantitative analysis that local communities can use to evaluate environmental justice concerns associated with pipeline projects. We applied these data and analytical methods to two counties in North Carolina (Northampton and Robeson counties) that would be affected by the proposed Atlantic Coast Pipeline (ACP). We compared demographic and vulnerability characteristics of census blocks, census block groups, and census tracts that lie within 1 mile of the proposed pipeline route with corresponding census geographies that lie outside of the 1-mile zone. Finally, we present results of a county-level analysis of race and ethnicity data for the entire North Carolina segment of the proposed ACP route. Statistical analyses of race and ethnicity data (US Census Bureau) and Social Vulnerability Index scores (University of South Carolina’s Hazards & Vulnerability Research Institute) yielded evidence of significant differences between the areas crossed by the pipeline and reference geographies. No significant differences were found in our analyses of household income and cancer risk data.

We investigate the computational complexity of several decision, enumeration and counting problems related to pseudo-intents. We show that given a formal context and a set of its pseudo-intents, checking whether this context has an additional pseudo-intent is in conp and it is at least as hard as checking whether a given simple hypergraph is saturated. We also show that recognizing the set of pseudo-intents is also in conp and it is at least as hard as checking whether a given hypergraph is the transversal hypergraph of another given hypergraph. Moreover, we show that if any of these two problems turns out to be conp-hard, then unless p = np, pseudo-intents cannot be enumerated in output polynomial time. We also investigate the complexity of finding subsets of a given Duquenne-Guigues Base from which a given implication follows. We show that checking the existence of such a subset within a specified cardinality bound is np-complete, and counting all such minimal subsets is #p-complete.

Background Lung cancer is the number one cause of cancer death worldwide.(1) It is the fifth most commonly diagnosed cancer in Australia (12,741 cases diagnosed in 2018) and the leading cause of cancer death.(2) The number of years of potential life lost to lung cancer in Australia is estimated to be 58,450, similar to that of colorectal and breast cancer combined.(3) While tobacco control strategies are most effective for disease prevention in the general population, early detection via low dose computed tomography (LDCT) screening in high-risk populations is a viable option for detecting asymptomatic disease in current (13%) and former (24%) Australian smokers.(4) The purpose of this Evidence Check review is to identify and analyse existing and emerging evidence for LDCT lung cancer screening in high-risk individuals to guide future program and policy planning. Evidence Check questions This review aimed to address the following questions: 1. What is the evidence for the effectiveness of lung cancer screening for higher-risk individuals? 2. What is the evidence of potential harms from lung cancer screening for higher-risk individuals? 3. What are the main components of recent major lung cancer screening programs or trials? 4. What is the cost-effectiveness of lung cancer screening programs (include studies of cost–utility)? Summary of methods The authors searched the peer-reviewed literature across three databases (MEDLINE, PsycINFO and Embase) for existing systematic reviews and original studies published between 1 January 2009 and 8 August 2019. Fifteen systematic reviews (of which 8 were contemporary) and 64 original publications met the inclusion criteria set across the four questions. Key findings Question 1: What is the evidence for the effectiveness of lung cancer screening for higher-risk individuals? There is sufficient evidence from systematic reviews and meta-analyses of combined (pooled) data from screening trials (of high-risk individuals) to indicate that LDCT examination is clinically effective in reducing lung cancer mortality. In 2011, the landmark National Lung Cancer Screening Trial (NLST, a large-scale randomised controlled trial [RCT] conducted in the US) reported a 20% (95% CI 6.8% – 26.7%; P=0.004) relative reduction in mortality among long-term heavy smokers over three rounds of annual screening. High-risk eligibility criteria was defined as people aged 55–74 years with a smoking history of ≥30 pack-years (years in which a smoker has consumed 20-plus cigarettes each day) and, for former smokers, ≥30 pack-years and have quit within the past 15 years.(5) All-cause mortality was reduced by 6.7% (95% CI, 1.2% – 13.6%; P=0.02). Initial data from the second landmark RCT, the NEderlands-Leuvens Longkanker Screenings ONderzoek (known as the NELSON trial), have found an even greater reduction of 26% (95% CI, 9% – 41%) in lung cancer mortality, with full trial results yet to be published.(6, 7) Pooled analyses, including several smaller-scale European LDCT screening trials insufficiently powered in their own right, collectively demonstrate a statistically significant reduction in lung cancer mortality (RR 0.82, 95% CI 0.73–0.91).(8) Despite the reduction in all-cause mortality found in the NLST, pooled analyses of seven trials found no statistically significant difference in all-cause mortality (RR 0.95, 95% CI 0.90–1.00).(8) However, cancer-specific mortality is currently the most relevant outcome in cancer screening trials. These seven trials demonstrated a significantly greater proportion of early stage cancers in LDCT groups compared with controls (RR 2.08, 95% CI 1.43–3.03). Thus, when considering results across mortality outcomes and early stage cancers diagnosed, LDCT screening is considered to be clinically effective. Question 2: What is the evidence of potential harms from lung cancer screening for higher-risk individuals? The harms of LDCT lung cancer screening include false positive tests and the consequences of unnecessary invasive follow-up procedures for conditions that are eventually diagnosed as benign. While LDCT screening leads to an increased frequency of invasive procedures, it does not result in greater mortality soon after an invasive procedure (in trial settings when compared with the control arm).(8) Overdiagnosis, exposure to radiation, psychological distress and an impact on quality of life are other known harms. Systematic review evidence indicates the benefits of LDCT screening are likely to outweigh the harms. The potential harms are likely to be reduced as refinements are made to LDCT screening protocols through: i) the application of risk predication models (e.g. the PLCOm2012), which enable a more accurate selection of the high-risk population through the use of specific criteria (beyond age and smoking history); ii) the use of nodule management algorithms (e.g. Lung-RADS, PanCan), which assist in the diagnostic evaluation of screen-detected nodules and cancers (e.g. more precise volumetric assessment of nodules); and, iii) more judicious selection of patients for invasive procedures. Recent evidence suggests a positive LDCT result may transiently increase psychological distress but does not have long-term adverse effects on psychological distress or health-related quality of life (HRQoL). With regards to smoking cessation, there is no evidence to suggest screening participation invokes a false sense of assurance in smokers, nor a reduction in motivation to quit. The NELSON and Danish trials found no difference in smoking cessation rates between LDCT screening and control groups. Higher net cessation rates, compared with general population, suggest those who participate in screening trials may already be motivated to quit. Question 3: What are the main components of recent major lung cancer screening programs or trials? There are no systematic reviews that capture the main components of recent major lung cancer screening trials and programs. We extracted evidence from original studies and clinical guidance documents and organised this into key groups to form a concise set of components for potential implementation of a national lung cancer screening program in Australia: 1. Identifying the high-risk population: recruitment, eligibility, selection and referral 2. Educating the public, people at high risk and healthcare providers; this includes creating awareness of lung cancer, the benefits and harms of LDCT screening, and shared decision-making 3. Components necessary for health services to deliver a screening program: a. Planning phase: e.g. human resources to coordinate the program, electronic data systems that integrate medical records information and link to an established national registry b. Implementation phase: e.g. human and technological resources required to conduct LDCT examinations, interpretation of reports and communication of results to participants c. Monitoring and evaluation phase: e.g. monitoring outcomes across patients, radiological reporting, compliance with established standards and a quality assurance program 4. Data reporting and research, e.g. audit and feedback to multidisciplinary teams, reporting outcomes to enhance international research into LDCT screening 5. Incorporation of smoking cessation interventions, e.g. specific programs designed for LDCT screening or referral to existing community or hospital-based services that deliver cessation interventions. Most original studies are single-institution evaluations that contain descriptive data about the processes required to establish and implement a high-risk population-based screening program. Across all studies there is a consistent message as to the challenges and complexities of establishing LDCT screening programs to attract people at high risk who will receive the greatest benefits from participation. With regards to smoking cessation, evidence from one systematic review indicates the optimal strategy for incorporating smoking cessation interventions into a LDCT screening program is unclear. There is widespread agreement that LDCT screening attendance presents a ‘teachable moment’ for cessation advice, especially among those people who receive a positive scan result. Smoking cessation is an area of significant research investment; for instance, eight US-based clinical trials are now underway that aim to address how best to design and deliver cessation programs within large-scale LDCT screening programs.(9) Question 4: What is the cost-effectiveness of lung cancer screening programs (include studies of cost–utility)? Assessing the value or cost-effectiveness of LDCT screening involves a complex interplay of factors including data on effectiveness and costs, and institutional context. A key input is data about the effectiveness of potential and current screening programs with respect to case detection, and the likely outcomes of treating those cases sooner (in the presence of LDCT screening) as opposed to later (in the absence of LDCT screening). Evidence about the cost-effectiveness of LDCT screening programs has been summarised in two systematic reviews. We identified a further 13 studies—five modelling studies, one discrete choice experiment and seven articles—that used a variety of methods to assess cost-effectiveness. Three modelling studies indicated LDCT screening was cost-effective in the settings of the US and Europe. Two studies—one from Australia and one from New Zealand—reported LDCT screening would not be cost-effective using NLST-like protocols. We anticipate that, following the full publication of the NELSON trial, cost-effectiveness studies will likely be updated with new data that reduce uncertainty about factors that influence modelling outcomes, including the findings of indeterminate nodules. Gaps in the evidence There is a large and accessible body of evidence as to the effectiveness (Q1) and harms (Q2) of LDCT screening for lung cancer. Nevertheless, there are significant gaps in the evidence about the program components that are required to implement an effective LDCT screening program (Q3). Questions about LDCT screening acceptability and feasibility were not explicitly included in the scope. However, as the evidence is based primarily on US programs and UK pilot studies, the relevance to the local setting requires careful consideration. The Queensland Lung Cancer Screening Study provides feasibility data about clinical aspects of LDCT screening but little about program design. The International Lung Screening Trial is still in the recruitment phase and findings are not yet available for inclusion in this Evidence Check. The Australian Population Based Screening Framework was developed to “inform decision-makers on the key issues to be considered when assessing potential screening programs in Australia”.(10) As the Framework is specific to population-based, rather than high-risk, screening programs, there is a lack of clarity about transferability of criteria. However, the Framework criteria do stipulate that a screening program must be acceptable to “important subgroups such as target participants who are from culturally and linguistically diverse backgrounds, Aboriginal and Torres Strait Islander people, people from disadvantaged groups and people with a disability”.(10) An extensive search of the literature highlighted that there is very little information about the acceptability of LDCT screening to these population groups in Australia. Yet they are part of the high-risk population.(10) There are also considerable gaps in the evidence about the cost-effectiveness of LDCT screening in different settings, including Australia. The evidence base in this area is rapidly evolving and is likely to include new data from the NELSON trial and incorporate data about the costs of targeted- and immuno-therapies as these treatments become more widely available in Australia.

Description logic knowledge bases can be used to represent knowledge about a particular domain in a formal and unambiguous manner. Their practical relevance has been shown in many research areas, especially in biology and the semantic web. However, the tasks of constructing knowledge bases itself, often performed by human experts, is difficult, time-consuming and expensive. In particular the synthesis of terminological knowledge is a challenge every expert has to face. Because human experts cannot be omitted completely from the construction of knowledge bases, it would therefore be desirable to at least get some support from machines during this process. To this end, we shall investigate in this work an approach which shall allow us to extract terminological knowledge in the form of general concept inclusions from factual data, where the data is given in the form of vertex and edge labeled graphs. As such graphs appear naturally within the scope of the Semantic Web in the form of sets of RDF triples, the presented approach opens up the possibility to extract terminological knowledge from the Linked Open Data Cloud. We shall also present first experimental results showing that our approach has the potential to be useful for practical applications.

Women’s schooling has long been regarded as one of the best investments in development. Using two different cross-nationally comparable data sets which both contain measures of schooling, assessments of literacy, and life outcomes for more than 50 countries, we show the association of women’s education (defined as schooling and the acquisition of literacy) with four life outcomes (fertility, child mortality, empowerment, and financial practices) is much larger than the standard estimates of the gains from schooling alone. First, estimates of the association of outcomes with schooling alone cannot distinguish between the association of outcomes with schooling that actually produces increased learning and schooling that does not. Second, typical estimates do not address attenuation bias from measurement error. Using the new data on literacy to partially address these deficiencies, we find that the associations of women’s basic education (completing primary schooling and attaining literacy) with child mortality, fertility, women’s empowerment and the associations of men’s and women’s basic education with positive financial practices are three to five times larger than standard estimates. For instance, our country aggregated OLS estimate of the association of women’s empowerment with primary schooling versus no schooling is 0.15 of a standard deviation of the index, but the estimated association for women with primary schooling and literacy, using IV to correct for attenuation bias, is 0.68, 4.6 times bigger. Our findings raise two conceptual points. First, if the causal pathway through which schooling affects life outcomes is, even partially, through learning then estimates of the impact of schooling will underestimate the impact of education. Second, decisions about how to invest to improve life outcomes necessarily depend on estimates of the relative impacts and relative costs of schooling (e.g., grade completion) versus learning (e.g., literacy) on life outcomes. Our results do share the limitation of all previous observational results that the associations cannot be given causal interpretation and much more work will be needed to be able to make reliable claims about causal pathways.