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Topic: Determinacy

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Determinacy Maximum Projective determinacy only claims determinacy for games where positions can be coded by integers and payoff sets definable in a simple way from real numbers. By claiming existence of complicated sets, determinacy hypotheses also have large proof theoretical strength.  A theory cannot prove propositions of larger proof theoretical strength than itself.  By limiting oneself to ZFC, one is cut off from such propositions—most of which are accessible through determinacy hypotheses. Since, however, determinacy hypotheses are less obvious than the separation axioms or even the axiom of choice, one must examine their consequences before accepting them as axioms.  Projective determinacy provides a very nice and practically complete theory of projective sets.  Every projective set is measurable, has Baire property, and has perfect subset property.  Pointclasses Π web.mit.edu /dmytro/www/DeterminacyMaximum.htm

Projective determinacy You have come across our Projective determinacy article. Unfortunately, work on Projective determinacy has not finished and we are still working on the contents. Please come back to this page in the future when more definitive content is published on Projective determinacy. www.bambooweb.com /articles/p/r/Projective_determinacy.html

Statical determinacy The degree of statical determinacy of a plane frame indicates the category of methods that could be applied to determine the magnitude and type of force in each member of the frame. This learning outcome deals with frames, the degree of statical determinacy of which is zero; such frames are said to be "statically determinate." The degree of statical determinacy, i, of a plane frame could be determined using the following equation: floti.bell.ac.uk /kingr/wbtcontent/determinacy.html

Kinematic determinacy - Wikipedia, the free encyclopedia Kinematical determinacy can be loosely used to classify an arrangement of structural members as a structure (stable) instead of a mechanism (unstable). Kinematic determinacy is a term used in structural mechanics to describe a structure where material compatibility conditions alone can be utilized to calculate deflections. A kinematically determinate structure can be defined as a structure where, if it is possible to find nodal displacements compatible with member extensions, those nodal displacements are unique. en.wikipedia.org /wiki/Kinematic_determinacy

Projective Determinacy -- Martin and Steel 85 (18): 6582 -- Proceedings of the National Academy of Sciences Projective Determinacy -- Martin and Steel 85 (18): 6582 -- Proceedings of the National Academy of Sciences It is shown that projective determinacy follows from large cardinal axioms weaker than the assertion that supercompact cardinals exist. intl.pnas.org /cgi/content/abstract/85/18/6582

Axiom of determinacy - Wikipedia, the free encyclopedia It is possible that the axiom of determinacy can be proved false without the use of the axiom of choice. In mathematics, the axiom of determinacy (abbreviated as AD) is an axiom in set theory. Not all games require the axiom of determinacy to prove them determined. en.wikipedia.org /wiki/Determinacy_axiom

Set Theory In view of this result one must consider the possibility that a mathematical conjecture that resists a proof might be an example of such an unprovable statement, and Gödel immediately embarked on the project of showing that the Continuum Hypothesis might be undecidable in the axiomatic set theory. His Incompleteness Theorem states that every axiomatic system that purports to describe mathematics as we know it must be incomplete, in the sense that one can find a true statement expressible in the system that cannot be formally proved from the axioms. Several years after proving the Incompleteness Theorem, Gödel proved another groundbreaking result: he showed that both the Axiom of Choice and the Continuum Hypothesis are consistent with the axioms of set theory, that is that neither can be refuted by using those axioms. adt.library.usyd.edu.au /stanford/entries/set-theory

Active structure - Wikipedia, the free encyclopedia In other words, a structure that has both statical determinacy and kinematic determinacy is optimal for actuation. The advantage of active structures is that they can be far more massive than a traditional static structure: an example would be a space fountain, which is a building which reaches into orbit. The result of the activity is a structure more suited for the type and magnitude of the load it is carrying. www.wikipedia.org /wiki/Active_structure

Ernest Schimmerling They are important in inner model theory and are also used in the proof by Martin and Steel that if there are infinitely many Woodin cardinals then projective determinacy is true. I will develop some of the theory of iteration trees from scratch and show how to use a Woodin cardinal to construct a special kind of iteration tree, called an alternating chain, that is related to the proof of projective determinacy. www.math.cmu.edu /~eschimme/seminar/reaber1.html

LC '98 abstract: Kai Hauser The last problem from the list in [1] asks whether Projective Determinacy (PD) can be derived in ZFC from its regularity consequences for the projective sets. The projective sets form the smallest class of sets containing the Borel sets which is closed under complementation and the formation of continuous images. In 1981 Woodin [2] conjectured that (in ZFC) these regularity properties conversely imply PD. By a recent theorem of Steel this is not the case, however, effective versions of the question are still open and suggest an intimate connection with the theory of canonical models for large cardinal axioms. www.math.cas.cz /~lc98/abstracts/Hauser.html

angles3d.topic Kinematic data, obtained from landmarks placed at arbitrary points on the segments, can be used for dynamic analysis if the positions of these points with respect to the segmental axes are known. However, during the process of data acquisition and data processing or in the formulations of kinematic models, one is free to use any of the conventions as long as it meets the criteria for the data processing and the mathematics of the model. In 2-D kinematics, these might be just the current position (2) and attitude (1), adding up to 3 d.o.f.'s, but one might also consider instantaneous changes, and take along the velocities, accelerations, etc., thus increasing the nummber of free and independent state parameters to 6, 9, etc. isb.ri.ccf.org /biomch-l/files/angles3d.topic

University of Windsor, Windsor, Ontario Canada Stability and determinacy of trusses and frames; analysis of statistically determinate trusses and frames; influence lines and moving loads. Statically indeterminate problems and inelastic response are also studied. Engineering Mechanics of Deformable Bodies II A study of flexure and deflection of beams, eccentric loads, connections, experimental determination of principal stresses, buckling of columns, and additional topics. cronus.uwindsor.ca /units/registrar/calendars/under2004.nsf/printerFriendlyView/62C2A06E4F09D47A85256D39005E7A2B

BE 50 Headrick Syllabus Although statics can be considered a special case of dynamics (zero acceleration), statics deserves special treatment in engineering education, since most structures are designed with the intention that they remain in equilibrium. Statics deals with bodies in equilibrium; whereas, dynamics is concerned with the accelerated motion of bodies. Statics is presently the first course of a three course sequence in Engineering Mechanics: Statics, Dynamics, and Mechanics of Materials. web.umr.edu /~be50-5/policy

Help about static analysis Could you help me to find a little software or a little piece of code to teaching the concepts of stability, determinacy and indeterminacy in a structural analysis laboratory for a high graduate school. = Could you help me to find a little software or a little piece of code to = teaching = = the concepts of stability, determinacy and indeterminacy in a structural = analysis laboratory for a high graduate school. Since different techniques are used analyze statically determinate versus statically indeterminate structures it is important that students be able to identify whether a structure is stable, determinate or indeterminate. www.ee-one.com /new-4938804-4305.html

Analysis of Statically Determinate Trusses This section presents methods for assessing the stability and degree of statical indeterminacy of plane trusses, as well as graphical and analytical methods for calculating forces in truss members. One common case of statically indeterminate trusses that can be analyzed using the methods presented in this section is the case of redundant X bracing. This type of design is not used for structures in which fatigue is an important consideration. www.ecf.utoronto.ca /apsc/courses/civ214/2003%20and%202004/2003%20CIV%20214S%20Website/Lectures/Analysis%20of%20Statically%20Determinate%20Trusses/analysis_of_statically_determinate_trusses.htm

U of M Faculty of Engineering: 023.376 - Structural Analysis Statical determinacy and indeterminacy; bending moment and shear force diagrams for frames and curved members; influence lines for trusses and beams The objective of the course is to introduce students to load-deformation behavior of statically determinate and statically indeterminate structures through an understanding of classical and modern methods of structural analysis. Different structural forms and load distribution; statically determinate and indeterminate beams and frames; influence lines; deflection calculations; flexibility and stiffness methods; computer-aided structural analysis; introduction to structural dynamics. www.umanitoba.ca /cgi-bin/faculties/engineering/courses/print.cgi?dept_no=023&course_no=376

Countable dense homogeneity of definable spaces by Michael Hru\v{s}\'ak and Beatriz Zamora~Avil\'es Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. at.yorku.ca /p/a/a/o/38.htm

Transactions of the American Mathematical Society This, together with a construction of W. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. Determinacy and weakly Ramsey sets in Banach spaces www.ams.org /tran/2002-354-04/S0002-9947-01-02926-9/home.html

leung.abstract.doc The pin-jointed kagome repetitive truss demonstrates both static and kinematic determinacy and therefore a single member actuation of the daughter rigid-jointed structure has low actuation resistance while retaining structural integrity. Earlier linear finite element results have shown that this unique behaviour of the kagome lattice is dependent on the shape of the deformed structure. www-civ.eng.cam.ac.uk /Seminarabstracts/leung.abstract.doc

AnyBody Technology: Version 0.1.0 (1st beta) It now checks for determinacy in each time step, and a number of numerical improvements have been implemented. In fact, the accelerations of the driven kinematic measures are zero, and this means that the analysis is only valid for cases with insignificant acceleration levels. The measurement of orientation in the kinematic measure AnyKinRotational is known to have minor bugs, which you may encounter in special cases. www.anybodytech.com /49.0.html

Untitled Document There is a proper, generalized interpretation of Maxwell's rule establishing that the determinacy measure actually expresses a quantity (s-m), where (s) is the number of possible independent states of prestress, and (m) is the number of possible independent (infinitesimal) kinematic mechanisms. The T-3 prism yields a determinacy measure of 0, which conventionally describes a determinate truss. When member lengths can be varied or when large elastic elongations are admitted tensegrity structures present a new family of machine kinematics and a new family of shape-adaptable structures, with the advantage that actuation is required only for tension members, requiring a far simpler technology than general bi-directional translational actuators. www.ri.cmu.edu /~isarc/abstract_htmls/Oppenheim-tensegrityPrisms.html

University of Bath Computing Services Static and manoeuvre margins, both stick fixed and stick free. To introduce the fundamental principles of statics, kinematics and dynamics as applied in an engineering context. To give an understanding of the principles of aircraft stability and the significance of the permitted centre of gravity limits which must be considered when loading an aircraft. www.bath.ac.uk /catalogues/2001-2002/Mech/MECHCAT.html

Untitled Document There is a proper, generalized interpretation of Maxwell's rule establishing that the determinacy measure actually expresses a quantity (s-m), where (s) is the number of possible independent states of prestress, and (m) is the number of possible independent (infinitesimal) kinematic mechanisms. The full mathematics of tensegrity geometry, statics, and kinematics have not been formulated, and such mathematical results must be developed and assembled before applications can be undertaken. The T-3 consists of two equilateral triangular end faces, parallel to one another and normal to the axis joining their centers; in such a geometry there is a relative twist angle between the end faces of 30 degrees. www.ri.cmu.edu /~isarc/abstract_htmls/Oppenheim-tensegrityPrisms.html

Geometry.Net - Pure_And_Applied_Math: Statical Mechanics Extractions: The degree of statical determinacy of a plane frame indicates the category of methods that could be applied to determine the magnitude and type of force in each member of the frame. Exploratory exercise(s) Calculate the degree of statical determinacy of the following trusses: (i) Kingpost truss. The analysis of statically indeterminate frames is complex and beyond the scope of the present learning outcome. www4.geometry.net /pure_and_applied_math/statical_mechanics_page_no_2.html

Atlas: Blackwell games and projective ordinals by Benedikt Lowe under Blackwell Determinacy), we investigate further ordinals under the assumption of Blackwell Determinacy. We show that the omega-club filters on the projective ordinals are indeed normal ultrafilters and hence the projective ordinals are measurable cardinals. Atlas: Blackwell games and projective ordinals by Benedikt Lowe atlas-conferences.com /c/a/d/v/05.htm

Patent 4437680: Adjustable two-bar linkage for a four-wheel trailer From this statical determinacy, it follows that the forces on individual links are statically determinate whenever appropriate axes exist so that the link assembly is unlocked and uncollapsed. Statical determinacy follows because the summed moments of all loads and reactions on either vehicle, considered as a free body about that hinge, must equal zero. The statical determinacy of link forces for four link systems requires that all the links not be coplanar or parallel. www.freepatentsonline.com /4437680.html

S002076830000230.abstract.en The similarity between this method and equilibrium requirements in a plane pin-jointed truss is highlighted, and an analogy between yield-line mechanism compatibility and statical determinacy in pin-jointed trusses is established. A systematic procedure is described for checking the compatibility of postulated yield-line mechanisms. www.elsevier.com /cdweb/journals/00207683/articles/38/18/S002076830000230.abstract.en

CE 241 Mechanics of Materials It is also anticipated that the theory and design approach for the mechanics of deformable bodies will help prepare the students for complex systems that will be encountered in advanced design courses. This course is designed to introduce basic principles of statics for rigid and deformable bodies. This course introduces the basic principles of mechanics with direct implications and applications to design of structures. www.ce.boun.edu.tr /eng/people/faculty/lus/CE%20241%20Mechanics%20of%20Materials.htm

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