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# Linear Algebra MA 26500

Purdue

GPA 3.97

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This 54 page Class Notes was uploaded by Dorothea Bode on Saturday September 19, 2015. The Class Notes belongs to MA 26500 at Purdue University taught by Saugata Basu in Fall. Since its upload, it has received 6 views. For similar materials see /class/208111/ma-26500-purdue-university in Mathematics (M) at Purdue University.

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Date Created: 09/19/15

Saugata Basu School of Mathematics amp College of Computing Georgia Institute of Technology 1 Plan the motion of a robot with several degrees of freedom amidst obstacles 2 Find the possible geometric conformations of a molecule given the bond lengths and bond angles 3 Given two ordered sets of n points in the plane is it possible to change the first set continuously into the second maintaining the order type Subsets of Rk defined by a formula involving a finite number of polynomial equalities and inequalities o Subsets of Rk defined by a formula involving a finite number of polynomial equalities and inequalities o A basic semi algebraic set is one defined by a conjunction of weak inequalities of the form P 2 0 o Subsets of Rk defined by a formula involving a finite number of polynomial equalities and inequalities o A basic semi algebraic set is one defined by a conjunction of weak inequalities of the form P 2 0 0 They arise as configurations spaces in robotic motion planning molecular chemistry etc CAD models and many other applications in computational geometry 0 Closed under union intersection complementation and projection 0 Most sets in Rk that arise in practice can be closely approximated by semi algebraic sets witness splines 0 Compact semi algebraic sets are finitely triangulable a First order theory of the reals is decidable Given a description of a semi algebraic set S C Bk 1 given two points 963 6 S decide if they are in the same connected component of S and if so output a semi algebraic path in Sjoining them 2 compute semi algebraic descriptions of the connected components of S 3 compute topological invariants of S eg its Euler characteristic homology groups etc 1 Algorithms a Deciding connectivity questions 2 Quantitative bounds on the complexity of semi algebraic sets a Bounds on Betti numbers b Complexity of single cells and connections to computational geometry The complexity of an algorithm is measured in terms of the following three parameters 0 the number of polynomials 71 used to define the input semi algebraic set S o the maximum degree d of these polynomials and o the number of variables kt 0 Consider the special case when all the input polynomials are linear and thus the given set is semilinear 0 Algorithms for computing properties of semi linear sets are widely studied in computational geometry 0 Typically the complexities of these algorithms are of the order of 001quot where n is the number of linear polynomials in the input o Motivates designing algorithms for semi algebraic sets such that the combinatorial complexity the part depending on 71 matches that for the corresponding semi linear problem 0 In the semi algebraic case there is usually an additional algebraic overhead algebraic complexity of the order of dOlk or dOlkZ o Introduced by Collins 1976 Used by Schwartz and Sharir for solving the piano mover s problem 200 0 Complexity is nd doubly exponential because of iterated projections A roadmap of S RS is a semi algebraic set of dimension at most one satisfying 1 for every semi algebraically connected component C of S C H RS is non empty and semi algebraically connected for every 70 E R and for every semi algebraically connected component C of ST C RS is not empty Grigor ev Vorobjov Canny Gournay Risler Heintz Roy Solerno BPollackRoy 1995 We give an algorithm to solve both problems for semi algebraic set restricted to a variety of dimension k in time 4 2 nk 1d0k In case of a compact smooth algebraic hypersurface S one can obtain the roadmap by 1 Follow the Xg extremal points in the X1 direction 2 Recurse at certain special slices corresponding to the critical values of the projection map onto the X1 co ordinate 11115 In our algorithms whenever we compute a point 1 1mk what we actually compute is 1 A univariate polynomial ft 2 A root say oz of f which is characterized by f and the sign vector signf04a signf3904a asignfldeglfl1l06 3 k 1 polynomials V such that For a general algebraic set ZQ one can obtain the roadmap by 1 Parametrizing a procedure for computing a set of points guaranteed to meet every connected component of an algebraic set treating X1 as a parameter 2 Recurse at certain special slices corresponding to the pseudo critical values mm k Q k m WC For a general semi algebraic set S we obtain the roadmap by 1 Make perturbations such that no k of the input polynomials have a common real zero 2 Computing roadmaps for all possible non empty algebraic sets 3 Recurse at certain special slices corresponding to the special values 1 Arrangement of 71 lines in R2 0 Total combinatorial complexity 0n2 o Combinatorial complexity of a single cell 2 Arrangement of n hyperplanes in Bk 0 Total combinatorial complexity Combinatorial complexity of a single cell 0nlgl Consequence of the Upper Bound Theorem 0 Each surface patch Sl is a closed semi algebraic set contained in a hypersurface ZQZ and defined by a first order quantifier free formula involving a family of polynomials PL71 PM o A cell is a maximal connected subset of the intersection of a fixed possibly empty subset of surface patches that avoids all other surface patches o The combinatorial complexity of an 6 dimensional cell C is the number of cells of dimension less than 6 which are contained in the relative boundary of C 1 For k 2 0 Complexity of the whole arrangement 0n2 0 Complexity of a single cell 0010401 Guibas Sharir Sifrony 2 For k 3 0 Complexity of the whole arrangement 0n3 0 Complexity of a single cell 0012 Halperin and Sharir 3 Conjecture Combinatorial complexity of a single cell is bounded by 0nk1ozn a An important measure of the topological complexity of a set S are the Betti numbers ZS o lntuitively S measures the number of i dimensional holes in S o For example if T is topologically a hollow torus then 50T 17 1T 27 2T 1 iT 02 gt 2 a As a measure computational difficulty of semi algebraic sets eg lower bounds for membership testing in terms of the sum of the Betti numbers Yao et al a As a measure computational difficulty of semi algebraic sets eg lower bounds for membership testing in terms of the sum of the Betti numbers Yao et al a In studying the complexity of arrangements and their substructures in computational geometry a As a measure computational difficulty of semi algebraic sets eg lower bounds for membership testing in terms of the sum of the Betti numbers Yao et al a In studying the complexity of arrangements and their substructures in computational geometry Oleinik and Petrovsky 1949 Thom 1964 and Milnor 1965 proved that the sum of the Betti numbers of a semi algebraic set S C Rk defined by P120Pn20 degPZ g d 1 g i g n is bounded by 0ndk This bound is tight as 0S could be as large OIeinik Petrovsky Thom Milnor technique does not give anything better o Oleinik Petrovsky Thom Milnor technique does not give anything better o In analogy to the single cell results computational geometry one might conjecture that the sum of the Betti numbers of a single connected component of a basic semi algebraic set is bounded by nk10dk o Oleinik Petrovsky Thom Milnor technique does not give anything better o In analogy to the single cell results computational geometry one might conjecture that the sum of the Betti numbers of a single connected component of a basic semi algebraic set is bounded by nk10dk o It is easy to construct a basic semi algebraic set such that it has one connected component whose other Betti numbers sum to 9ndk1 0 Let Pi XI L221quot39Xll Lad2 67 where the LZj E RX1Xk1 are generic linear polynomials and e gt 0 and sufficiently small The set S defined by P1 2 0PS Z 0 has one connected component with S Qndk1 Theorem 1 898 Let C be a kJdimensiona cell in an arrangement of 71 surface patches 81 Sn in Rk Then the combinatorial complexity of C is bounded by 0nk1 for every 6 gt 0 Theorem 2 898 Let 01Cm C Rk be m different connected components of a basic semi algebraic set defined by P1 2 0Pn Z 0 With the degrees of the polynomials PZ bounded by d Then Cj is bounded by m kil0dk a Proof used Morse theory for stratified spaces Consider the union of n compact convex sa sets in Rk The nerve lemma gives us a bound on the individual Betti numbers of the union 0 Consider the union of n compact convex sa sets in Rk The nerve lemma gives us a bound on the individual Betti numbers of the union 0 The homology groups of the union is isomorphic to the homology groups of the nerve complex The nerve complex has n vertices and thus the i th Betti number is bounded by 0 Consider the union of n compact convex sa sets in Rk The nerve lemma gives us a bound on the individual Betti numbers of the union 0 The homology groups of the union is isomorphic to the homology groups of the nerve complex The nerve complex has n vertices and thus the i th Betti number is bounded by What if the intersections are not acyclic but have bounded topology Theorem 3 Let S C Rk be the set defined by the disjunction of n inequalities Then Theorem 4 Let S C Rk be the set defined by the conjunction of n inequalities Then Let SC Rk be defined by P1207quot397Pn207 degB s 21 g i g n 0 Let SC Rk be defined by P1207quot397Pn207 degB s 21 g i g n a They arise in many applications eg as the configuration space of sets of points with pair wise distance constraints etc 0 Let SC Rk be defined by P1207quot397Pn207 degB s 21 g i g n a They arise in many applications eg as the configuration space of sets of points with pair wise distance constraints etc 0 Can be topologically quite complicated If S is defined by X1X1 Z Z 07 2 then clearly 0S k exponential in the dimension Theorem 5 Let E be any fixed number and let S C Rk be defined by P120Pn20 with degPZ g 2 Then kgS 3 WWW Note that this bound is polynomial in the dimension The proofs use the spectral sequence associated with the Mayer Vietoris double complex

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