Smooth approximations by continuous choice-functions

Smooth approximations by continuous choice-functions | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Smooth approximations by continuous choice-functions Mihai Prunescu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-341149/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract We explore the existence of rational-valued approximation processes by continuous functions of two variables, such that the output continuously depends of the imposed error-bound. To this sake we prove that the theory of densely ordered sets with generic predicates is ℵ0- categorical. A model of the theory and a particular continuous choice-function are constructed. This function transfers to all other models by the respective isomorphisms. If some common-sense conditions are fulfilled, the processes are computable. As a byproduct, other functions with surprising properties can be constructed. Software Engineering choice densely ordered sets order topology continuous functions roots algebraic and transcendental numbers ℵ0-categorical theories Full Text Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 22 Jul, 2021 Reviewers invited by journal 22 Jul, 2021 Editor assigned by journal 20 Jul, 2021 First submitted to journal 20 Jul, 2021 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-341149","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":41154840,"identity":"ea2bd397-5df3-44fa-9210-7f34ea8f4a2e","order_by":0,"name":"Mihai Prunescu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBElEQVRIiWNgGAWjYDACZh4GhgQE14afgRlEGxwgWkuaZANBLQw8KLzDkg0QBm4tuu28Bz88YLiTb3C7+fCHn23nJXTbGdikCwruJPY38D58gEWL2WG+ZIkEhmeWG+4cS5PsbbstYXYYqGWGwbPEGQfYjQ2wauExAGo5bGBwI8eMmbHtdh1YC4/B4cQNDGxsEti1GP+AaMn//Jmx7ZwEMVrMYLYwSDO2HSBOi0WCwTMDyRtpZpI955KBWhibrXkMnhnPOMzGjNUv588Y3/xRcceA70by4w8/yuwkzM4fPnib588d2f72NkZsIQYBqBHH2AChmXGqB4EDeGVHwSgYBaNghAMArRdgiSiYEZ4AAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0003-1627-050X","institution":"Universitatea din Bucuresti","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Mihai","middleName":"","lastName":"Prunescu","suffix":""}],"badges":[],"createdAt":"2021-03-18 07:04:21","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-341149/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-341149/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":13662246,"identity":"8f58f612-7bb9-469a-b46e-a2ad06c469a2","added_by":"auto","created_at":"2021-09-17 10:31:59","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":206034,"visible":true,"origin":"","legend":"","description":"","filename":"SOCOD2000498R2reviewer.pdf","url":"https://assets-eu.researchsquare.com/files/rs-341149/v1_covered.pdf"},{"id":11740828,"identity":"6ac23bc6-3cff-4c61-a82e-8b8b78a14d5e","added_by":"auto","created_at":"2021-07-23 12:16:30","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":202444,"visible":true,"origin":"","legend":"","description":"","filename":"SOCOD2000498R2reviewer.pdf","url":"https://assets-eu.researchsquare.com/files/rs-341149/v1_covered.pdf"}],"financialInterests":"","formattedTitle":"Smooth approximations by continuous choice-functions","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-341149/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"soft-computing","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"soco","sideBox":"Learn more about [Soft Computing](https://www.springer.com/journal/500)","snPcode":"500","submissionUrl":"https://submission.nature.com/new-submission/500/3","title":"Soft Computing","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"choice, densely ordered sets, order topology, continuous functions, roots, algebraic and transcendental numbers, ℵ0-categorical theories","lastPublishedDoi":"10.21203/rs.3.rs-341149/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-341149/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We explore the existence of rational-valued approximation processes by continuous functions of two variables, such that the output continuously depends of the imposed error-bound. 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