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چكيده انگليسي :

Some properly critical subsets of Euclidean spaces Amaneh Amiryouse aa math@yahoo com August 29 2012 Master of Science Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor DR Mansour Aghasi m aghasi@cc iut ac irAdvisor DR Azam Etemad ae110math@cc iut ac ir2000 MSC 51F05 51F20Key words Critical points critical sets properly critical sets Whitney s theorem AbstractMany nonlinear problems in physics engineering biology and social sciences can be reduced to ndingcritical points The rst class of critical points to be studied were minima and maxima and much ofthe activity in the calculus of variations has been devoted to nding such points So far we may say to some extend that there is an organized procedure for producing such criticalpoints and these methods are called global variational and topological methods Roughly speaking themodern variational and topological methods consist of the following two parts Minimax methodsLjusternik and Schnirelman in 1929 mark the beginning of global analysis by which some earliermathematicians no longer consider only the minima or maxima of variational integrals In 1934 Ljusternik and schnirelman developed a method that seeks to get information concerning the numberof critical points of a functional from topological data These ideas are referred to as the Ljusternik Schnirelman theory One celebrated and important result in the last 30 years has been the mountain pass theorem due toAmbrosetti and Rabinowitz in 1973 Since then a series of new theorems in the form of minimax haveappeared via variouse linking category and index theories Now these results in fact become a wonderfultool in studying the existence of solutions to di erential equations with variational Structures Morse theoryThis approach towards a global theory of critical points was pursued by Morse in 1934 It reveals a deeprelation between the topology of spaces the number and types of critical points of any function de nedon it This theory was highly successful in topology in the 1950s due to the e orts of Milnor and Smale In the works of Palais male and Rothe Morse theory was generalized to in nite dimensional spaces Bythen it was recognized as a useful approach in dealing with di erential equations and in particular in nding existence of multiple solutions The critical group and Morse index also can be derived in somecases Although there are some profound works on Morse theory and related topics the applicationsare some what limited by the smoothness and nondegeneracy assumptions on the functionals However both minimax theory and Morse theory essentialy give answers on the existence of multiple critical points of a functional They usually cannot provide many more additional properties of thecritical point except some special pro les such as the Morse index critical groups and so on In this thesis we rst characterize smooth functions on R with prescribed zero sets and prove thatthe closed subsets of R are all critical but not all of them are properly critical Then we provide acaractrization of properly critical subsets of the real line and use it to produce some properly criticalsubsets of higher dimensional Euclidean spaces We particularly get the proper criticality of C n Rn where C 0 1 is the middle third Cantor set Finally we characterize the critical sets of the sphereand of the closed cylinder

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زير مجموعه هاي كاملا بحراني از فضاهاي اقليدسي

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