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1
Gritschacher, Simon. "Commutative K-theory." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:5d5b0e20-20ef-4eec-a032-8bcb5fe59884.
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The bar construction BG of a topological group G has a subcomplex B<sub>com</sub>G ⊂ BG assembled from spaces of commuting elements in G. If G = U;O (the infinite unitary / orthogonal groups) then B<sub>com</sub>U and B<sub>com</sub>O are E<sub>∞</sub>-ring spaces. The corresponding cohomology theory is called commutative K-theory. In this work we study properties of the spaces B<sub>com</sub>G and of infinite loop spaces built from them, with an emphasis on the cases G = U,O. The content of this thesis is organised as follows: In Chapter 1 we consider a family of self-maps of B<sub>com</sub>G and apply these to study the question when the inclusion map B<sub>com</sub>G ⊂ BG admits a section up to homotopy. In Chapter 2 we show that B<sub>com</sub>U is a model for the E<sub>∞</sub>-ring space underlying the ku-group ring of ℂP<sup>∞</sup>. Thus we provide a complete description of complex commutative K-theory. We also study the space B<sub>com</sub>O. Our results include a computation of the torsionfree part of the homotopy groups of B<sub>com</sub>O and a long exact sequence relating real commutative K-theory to singular mod-2 homology. Chapter 3 is self-contained. We prove a result about the acyclicity of the "comparison map" M<sub>∞</sub> → ΩBM in the group-completion theorem and apply this to compare the infinite loop space associated to a commutative 𝕀-monoid with the Quillen plus-construction. Chapter 4 is concerned with a previously known filtration of Ω<sub>0</sub><sup>∞</sup>S<sup>∞</sup> by certain infinite loop spaces {hocolim<sub>𝕀</sub>B(q, Σ_)}<sub>q≥2</sub>. For each term in this filtration we construct another filtration on the spectrum level, whose subquotients we describe. Our set-up is more general, but the space hocolim<sub>𝕀</sub>B(q, Σ_) will serve as our main example. Appendix A is an excerpt from the author's Oxford transfer thesis. There we gave a construction of an infinite loop space associated to certain subspaces B(q, Γ<sub>g,1</sub>) ⊂ BΓ<sub>g;1</sub>, where Γ<sub>g;1</sub> is the mapping class group of a genus g surface with one boundary component.
2
Levikov, Filipp. "L-theory, K-theory and involutions." Thesis, University of Aberdeen, 2013. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=201918.
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In Part 1, we consider two descriptions of L-homology of a (polyhedron of a) simplicial complex X. The classical approach of Ranicki via (Z,X)-modules (cf. [Ran92]) iswell established and is used in Ranicki’s definition of the total surgery obstruction and his formulation of the algebraic surgery exact sequence (cf. [Ran79], [Ran92],[KMM]). This connection between algebraic surgery and geometric surgery has numerous applications in the theory of (highdimensional) manifolds. The approach described in [RW10] uses a category of homotopy complexes of cosheaves to construct for a manifold M a (rational) orientation class [M]L• in symmetric L-homology which is topologically invariant per construction. This is used to reprove the topological invariance of rational Pontryagin classes. The L-theory of the category of homotopy complexes of sheaves over an ENR X can be naturally identified with L-homology of X. If X is a simplicial complex, both definitions give L-homology, there is no direct comparison however. We close this gap by constructing a functor from the category of (Z,X)-modules to the category of homotopy cosheaves of chain complexes of Ranicki-Weiss inducing an equivalence on L-theory. The work undertaken in Part 1 may be considered as an addendum to [RW10] and suggests some translation of ideas of [Ran92] into the language of [RW10]. Without significant alterations, this work may be generalised to the case of X being a △-set. The L-theory of △-sets is considered in [RW12]. Let A be a unital ring and I a category with objects given by natural numbers and two kinds of morphisms mn → n satisfying certain relations (see Ch.3.4). There is an I-diagram, given by n 7→ ˜K (A[x]/xn) where the tilde indicates the homotopy fiber of the projection induced map on algebraic K-theory (of free modules) K(A[x]/xn) → K(A). In Part 2 we consider the following result by Betley and Schlichtkrull [BS05]. After completion there is an equivalence of spectra TC(A)∧ ≃ holim I ˜K(A[x]/xn)∧ where TC(A) is the topological cyclic homology of A. This is a very important invariant of K-theory (cf. [BHM93], [DGM12]) and comes with the cyclotomic trace map tr : K(A) → TC(A). In [BS05], the authors prove that under the above identification the trace map corresponds to a “multiplication” with an element u∞ ∈ holim I ˜K (Z[x]/xn). In this work we are interested in a generalisation of this result. We construct an element u∞ ∈ holim I ˜K(Cn). where Cn can be viewed as the category of freemodules over the nilpotent extension S[x]/xn of the sphere spectrum S. Let G be a discrete group and S[G] its spherical group ring. Using our lift of u∞ we construct a map trBS : K(S[G]) → holim I ˜K (CG n ) where CG n should be interpreted as the category of free modules over the extension S[G][x]/xn. After linearisation this map coincides with the trace map constructed by Betley and Schlichtkrull. We conjecture but do not prove, that after completion the domain coincides with the topological cyclic homology of S[G]. Some indication is given at the end of the final chapter. To construct the element u∞ we rely on a generalisation of a result of Grayson on the K-theory of endomorphisms (cf. [Gra77]). Denote by EndC the category of endomorphisms of finite CW-spectra and by RC the Waldhausen category of free CW-spectra with an action of N, which are finite in the equivariant sense. Cofibrations are given by cellular inclusions and weak equivalences are given bymaps inducing an equivalence of (reduced) cellular chain complexes of Z[x]-modules, after inverting the set {1 + xZ[x]}. In Chapter 5 we prove (5.8) that there is a homotopy equivalence of spectra ˜K (EndC) ≃ ˜K (RC). where tildes indicate that homotopy fibres of the respective projections are considered. Furthermore, we pursue the goal of constructing an involutive tracemap for themodel of [BS05]. We employ the framework ofWaldhausen categories with duality (cf. [WW98]) to introduce for any G involutions on holim I ˜K (CG n ). We give enough indication for our trace map being involutive, in particular in the last three sections of Chapter 5, we sketch how the generalisation of the theoremof Grayson (5.8) can be improved to an involutive version. In the final chapter, we develop this further. Assuming that the element u∞ ∈ holim I ˜K (Cn) is a homotopy fixed point of the introduced involution, we construct a map from quadratic L-theory of S[G] to the Tate homology spectrum of Z/2 acting on the fibre of trBS (see 6.9) : L•(S[G]) → (hofib(trBS))thZ/2 and discuss the connection of this to a conjecture of Rognes andWeiss. The two parts of the thesis are preluded with their own introduction andmay be read independently. The fewcross references are completely neglectible.
3
Takeda, Yuichiro. "Localization theorem in equivariant algebraic K-theory." 京都大学 (Kyoto University), 1997. http://hdl.handle.net/2433/202419.
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Stefański, Bogdan. "String theory, dirichlet branes and K-theory." Thesis, University of Cambridge, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.621023.
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Braun, Volker Friedrich. "K-theory and exceptional holonomy in string theory." Doctoral thesis, [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=965401650.
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Mitchener, Paul David. "K-theory of C*-categories." Thesis, University of Oxford, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.365771.
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Zakharevich, Inna (Inna Ilana). "Scissors congruence and K-theory." Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/73376.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (p. 83-84).<br>In this thesis we develop a version of classical scissors congruence theory from the perspective of algebraic K-theory. Classically, two polytopes in a manifold X are defined to be scissors congruent if they can be decomposed into finite sets of pairwise-congruent polytopes. We generalize this notion to an abstract problem: given a set of objects and decomposition and congruence relations between them, when are two objects in the set scissors congruent? By packaging the scissors congruence information in a Waldhausen category we construct a spectrum whose homotopy groups include information about the scissors congruence problem. We then turn our attention to generalizing constructions from the classical case to these Waldhausen categories, and find constructions for cofibers, suspensions, and products of scissors congruence problems.<br>by Inna Zakharevich.<br>Ph.D.
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Cain, Christopher. "K-theory of Fermat curves." Thesis, University of Cambridge, 2017. https://www.repository.cam.ac.uk/handle/1810/262483.
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I investigate the K_2 groups of the quotients of Fermat curves given in projective coordinates by the equation F_n:X^n+Y^n=Z^n. On any quotient where the number of known elements is equal to the rank predicted by Beilinson’s Conjecture I verify numerically that the determinant of the matrix of regulator values agrees with the leading coefficient of the L-function up to a simple rational number. The main source of K_2 elements are the so-called “symbols with divisorial support at infinity” that were found by Ross in the 1990’s. These consist of symbols of the form f, g where f and g have divisors whose points P all satisfy XY Z(P) = 0. The image of this subgroup under the regulator is computed and is found to be of rank predicted by Beilinson’s Conjecture on eleven nonisomorphic quotients of dimension greater than one. The L-functions of these quotients are computed using Dokchitser’s ComputeL package and Beilinson’s Conjecture is verified numerically to a precision of 200 decimal digits. In chapter five, with careful analysis of a certain 2 × 2 determinant it is shown that a particular hyperelliptic quotient of all the Fermat curves has K_2 group of rank at least two. In the last chapter of the dissertation, a computational method is used in order to discover new elements of K_2. These elements are rigorously proven to be tame and allow for the full verification of Beilinson’s Conjecture on the Fermat curves F_7 and F_9. Also the method allows us to verify Beilinson’s Conjecture on certain hyperelliptic quotients of F_8 and F_10. Quotients where a similar method might be successful are also suggested.
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Bunch, Eric. "K-Theory in categorical geometry." Diss., Kansas State University, 2015. http://hdl.handle.net/2097/20350.
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Doctor of Philosophy<br>Department of Mathematics<br>Zongzhu Lin<br>In the endeavor to study noncommutative algebraic geometry, Alex Rosenberg defined
in [13] the spectrum of an Abelian category. This spectrum generalizes the prime spectrum
of a commutative ring in the sense that the spectrum of the Abelian category R − mod is
homeomorphic to the prime spectrum of R. This spectrum can be seen as the beginning of
“categorical geometry”, and was used in [15] to study noncommutative algebriac geometry.
In this thesis, we are concerned with geometries extending beyond traditional algebraic
geometry coming from the algebraic structure of rings. We consider monoids in a monoidal
category as the appropriate generalization of rings–rings being monoids in the monoidal
category of Abelian groups. Drawing inspiration from the definition of the spectrum of
an Abelian category in [13], and the exploration of it in [15], we define the spectrum of
a monoidal category, which we will call the monoidal spectrum. We prove a descent condition which is the mathematical formalization of the statment “R − mod is the category
of quasi-coherent sheaves on the monoidal spectrum of R − mod”. In addition, we prove
a functoriality condidition for the spectrum, and show that for a commutative Noetherian
ring, the monoidal spectrum of R − mod is homeomorphic to the prime spectrum of the ring
R.
In [1], Paul Balmer defined the prime tensor ideal spectrum of a tensor triangulated cat-
gory; this can be thought of as the beginning of “tensor triangulated categorical geometry”.
This definition is very transparent and digestible, and is the inspiration for the definition in
this thesis of the prime tensor ideal spectrum of an monoidal Abelian category. It it shown
that for a polynomial identity ring R such that the catgory R − mod is monoidal Abelian,
the prime tensor ideal spectrum is homeomorphic to the prime ideal spectrum.
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Hedlund, William. "K-Theory and An-Spaces." Thesis, Uppsala universitet, Algebra och geometri, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-414082.
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Hahn, Rebekah D. "K(1)-local Iwasawa theory /." Thesis, Connect to this title online; UW restricted, 2003. http://hdl.handle.net/1773/5736.
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Millar, Judith Ruth. "K-theory of Azumaya algebras." Thesis, Queen's University Belfast, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.534610.
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Niwa, Masahiko. "THEORY OF G-CATEGORIES TOWARD EQUIVARIANT ALGEBRAIC K-THEORY." 京都大学 (Kyoto University), 1991. http://hdl.handle.net/2433/168801.
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本文データは平成22年度国立国会図書館の学位論文(博士)のデジタル化実施により作成された画像ファイルを基にpdf変換したものである<br>Kyoto University (京都大学)<br>0048<br>新制・論文博士<br>理学博士<br>乙第7383号<br>論理博第1122号<br>新制||理||718(附属図書館)<br>UT51-91-C116<br>(主査)教授 戸田 宏, 教授 土方 弘明, 教授 丸山 正樹<br>学位規則第5条第2項該当
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Schäfer-Nameki, Sakura. "D-branes in boundary field theory and K-theory." Thesis, University of Cambridge, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.620017.
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Piazza, Paolo. "K-theory and index theory on manifolds with boundary." Thesis, Massachusetts Institute of Technology, 1991. http://hdl.handle.net/1721.1/31020.
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Zhang, Zuhong. "Lower K-theory of unitary groups." Thesis, Queen's University Belfast, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.486261.
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The study of the sandwich classification theorem of unitary hyperbolic groups over commutative form ring (R, A) (in the sense of Bak) is naturally inspired by the sandwich classification theorem of general linear groups, which is initialed by Bak in a manuscript in 1967. In Chapter 1, we briefly review the history of the developing of normal and subnormal structure problems in the setting of general linear group and unitary group, as well as the co~nectionwith other problems. The proofs of sandwich. classification theorem and the structure theorem of subnormal subgroups of general linear group are given in details in Chapter 2. The brief introduction of Bak's form ring and unitary groups, and their fundamental properties are reproduced in Chapter 3. In chapter 4, we study the main theorem of the thesis, i.e., sandwich classification theorem for unitary groups. Furthermore the properties of two important classes of subgroups are studied, i.e., the subnormal subgroups of unitary. groups and mixed commutator subgroups of unitary groups. The final Chapter contains a weaker sandwich theorem on non-stable unitary groups and its proof.
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Kerz, Moritz. "Milnor K-theory of local rings." kostenfrei, 2008. http://www.opus-bayern.de/uni-regensburg/volltexte/2008/991/.
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Clausen, Dustin (Dustin Tate). "Arithmetic duality in algebraic K-theory." Thesis, Massachusetts Institute of Technology, 2013. http://hdl.handle.net/1721.1/83692.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references (pages 37-38).<br>Let X be a regular arithmetic curve or point (meaning a regular separated scheme of finite type over Z which is connected and of Krull dimension </= 1). We define a compactly-supported variant Kc(X) of the algebraic K-theory spectrum K(X), and establish the basic functoriality of Kc. Briefly, K, behaves as if it were dual to K. Then we give this duality some grounding: for every prime t invertible on X, we define a natural l-adic pairing between Kc(X) and K(X). This pairing is of an explicit homotopy-theoretic nature, and reflects a simple relation between spheres, tori, and real vector spaces. Surprisingly, it has the following two properties: first (a consequence of work of Rezk), when one tries to compute it the e-adic logarithm inevitably appears; and second, it can be used to give a new description of the global Artin map, one which makes the Artin reciprocity law manifest.<br>by Dustin Clausen.<br>Ph.D.
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Harris, Thomas. "Binary complexes and algebraic K-theory." Thesis, University of Southampton, 2015. https://eprints.soton.ac.uk/383999/.
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Magill, Matthew. "Topological K-theory and Bott Periodicity." Thesis, Uppsala universitet, Algebra och geometri, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-322927.
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Sia, Charmaine Jia Min. "Structures on Forms of K-Theory." Thesis, Harvard University, 2015. http://nrs.harvard.edu/urn-3:HUL.InstRepos:17467390.
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In the early 1970s, Morava studied forms of topological K-theory and observed that they have interesting number theoretic connections. Until very recently, forms of K-theory have not been studied in greater depth and integrated into the modern theory of topological modular forms. In this dissertation, some expected structured ring spectra and locality results are established on forms of K-theory. Forms of algebraic structures are usually classified by Galois cohomology. Based on the structured ring spectra and locality results established, a criterion is given for distinguishing homotopy equivalence classes of forms of K-theory via a computation in the second homotopy group of the spectrum.<br>Mathematics
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Klippenstien, J. "Applications of the universal coefficient theorem for connective k-theory." Thesis, University of Warwick, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.371053.
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Hazrat, Roozbeh. "On K-theory of classical-like groups." [S.l. : s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=969899742.
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Hekmati, Pedram. "Group Extensions, Gerbes and Twisted K-theory." Licentiate thesis, Stockholm : Teoretisk fysik, Kungliga Tekniska högskolan, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4654.
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Lopez, Jose Maria Cantarero. "Equivariant K-theory, groupoids and proper actions." Thesis, University of British Columbia, 2009. http://hdl.handle.net/2429/14707.
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Equivariant K-theory for actions of groupoids is defined and shown to be
a cohomology theory on the category of finite equivariant CW-complexes.
Under some conditions, these theories are representable. We use this fact to
define twisted equivariant K-theory for actions of groupoids. A classification
of possible twistings is given. We also prove a completion theorem for twisted
and untwisted equivariant K-theory. Finally, some applications to proper actions of Lie groups are discussed.
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Yang, Shuhang. "Large N gauge theory and k-strings." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/33648.
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We considered the k-antisymmetric representation of U(N) gauge group on two dimensional lattice space and derived the free energy by saddle point approximation in large N limit. k is a large integer comparable with N. Besides Gross-Witten phase transition[1], which happens as the coupling constant changes, we found a new phase transition in the strong coupling system that happens as k changes. The free energy of the weak coupling system is a smooth function of k under continuous limit. We have carefully selected the right saddle point solution among other possible ones. The
numerical results match our saddle point calculations.
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Kreisel, Michael. "Gabor frames for quasicrystals and K-theory." Thesis, University of Maryland, College Park, 2015. http://pqdtopen.proquest.com/#viewpdf?dispub=3711683.
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<p> We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal, and the K-theory of an associated twisted groupoid algebra. In particular, we construct a finitely generated projective module over this algebra, and multiwindow Gabor frames can be used to construct an idempotent representing the module in <i>K</i>-theory. For lattice subsets in dimension two, this allows us to prove a twisted version of Bellissard's gap labeling theorem. By viewing Gabor frames in this operator algebraic framework, we are also able to show that for certain quasicrystals it is not possible to construct a tight multiwindow Gabor frame.</p>
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Lakos, Gyula 1973. "Smooth K-theory and locally convex algebras." Thesis, Massachusetts Institute of Technology, 2003. http://hdl.handle.net/1721.1/29357.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2003.<br>Includes bibliographical references (p. 121-122).<br>In this thesis, we improve the loop linearization process from the classical article of Atiyah and Bott on Bott periodicity. The linearization process is made explicit in terms of formulae for smooth loops. Using this improvement allows us to extend K-theory (including periodicity) to a class of locally convex algebras vastly larger then the one of Banach algebras. We find various ways to represent periodicity by explicit formulae. For finite Laurent loops formulae yielding finite matrices to represent the associated Ko classes are obtained. The methods used also allow us to reinterpret some recent results of Melrose on smooth classifying spaces for K-theory. The relationship between the universal even and odd Chern characters and periodicity is investigated, giving correspondences between the various representatives in the form of family index theorems for loop groups. In the discussion Ko and the even Chern character are primarily formulated in the language of involutions. The paper also demonstrates the universality of the involution terminology with respect to vector bundles.<br>by Gyula Lakos.<br>Ph.D.
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Dugger, Daniel (Daniel Keith) 1972. "A Postnikov tower for algebraic K-theory." Thesis, Massachusetts Institute of Technology, 1999. http://hdl.handle.net/1721.1/85300.
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Song, Yongjin. "Hermitian algebraic K-theory and dihedral homology /." The Ohio State University, 1990. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487681788252481.
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Savinien, Jean P. X. "Cohomology and K-theory of aperiodic tilings." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/24732.
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Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008.<br>Committee Chair: Prof. Jean Bellissard; Committee Member: Prof. Claude Schochet; Committee Member: Prof. Michael Loss; Committee Member: Prof. Stavros Garoufalidis; Committee Member: Prof. Thang Le.
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Rodtes, Kijti. "The connective K theory of semidihedral groups." Thesis, University of Sheffield, 2010. http://etheses.whiterose.ac.uk/1103/.
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The real connective K-homology of finite groups ko¤(BG), plays a big role in the Gromov-Lawson-Rosenberg (GLR) conjecture. In order to compute them, we can calculate complex connective K-cohomology, ku¤(BG), first and then follow by computing complex connective K-homology, ku¤(BG), or by real connective K-cohomology,ko¤(BG). After we apply the eta-Bockstein spectral sequence to ku¤(BG) or the Greenlees spectral sequence to ko¤(BG), we shall get ko¤(BG). In this thesis, we compute all of them algebraically and explicitly to reduce the di±culties of geometric construction for GLR, especially for semidehedral group of order 16, SD16 , by using the methods developed by Prof.R.R. Bruner and Prof. J.P.C. Greenlees. We also calculate some relations at the stage of connective K-theory between SD16 and its maximal subgroup, (dihedral groups, quaternion groups and cyclic group of order 8).
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Dell'Aiera, Clément. "Controlled K-theory for groupoids and applications." Thesis, Université de Lorraine, 2017. http://www.theses.fr/2017LORR0114/document.
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Dans leur article de 2015 intitulé "On quantitative operator K-theory", H. Oyono-Oyono et G. Yu introduisent un raffinement de la K-théorie opératorielle adapté au cadre desC*-algèbres filtrées, appelé K-théorie quantitative ou contrôlée. Dans cette thèse, nous généralisons la notion de filtration de C_-algèbres. Nous montrons ensuite que ce cadre contient celui déjà traité par G. Yu et H. Oyono-Oyono, tout en se révélant assez souple pour traiter les produits croisés de groupoïdes étalés et de groupes quantiques discrets. Nous construisons ensuite des applications d'assemblage _a valeurs dans les groupes de K-théorie contrôlée associés, pour les C*-algèbres de Roe à coefficients et les produits croisés de groupoïdes étalés. Nous montrons que ces applications factorisent les applications d'assemblage usuelles de Baum-Connes. Nous prouvons ensuite ce que nous appelons des énoncés quantitatifs, et nous montrons qu'une version contrôlée de la conjecture de Baum-Connes est vérifiée pour une large classe de groupoïdes étalés. La fin de la thèse est consacrée à plusieurs applications de ces résultats. Nous montrons que l'application d'assemblage contrôlée coarse est équivalente à son analogue à coefficients pour le groupoïde coarse introduit par G. Skandalis, J-L. Tu et G. Yu. Nous donnons ensuite une preuve que les espaces coarses qui admettent un plongement hilbertien fibré vérifient la version maximale de la conjecture de Baum-Connes coarse contrôlée. Enfin nous étudions les groupoïdes étalés dont toutes les actions propres sont localement induites par des sous-groupoïdes compacts ouverts, dont un exemple est donné par les groupoïdes amples introduits par J. Renault. Nous développons un principe de restriction pour cette classe de groupoïdes, et prouvons que, sous des hypothèses raisonnables, leurs produits croisés vérifient la formule de Künneth en K-théorie contrôlée<br>In their paper entitled "On quantitative operator K-theory", H. Oyono-Oyono and G. Yu introduced a refinement of operator K-theory, called quantitative or controlled K-theory, adapted to the setting of filtered C_-algebras. In this thesis, we generalize filtration of C*-algebras. We show that this setting contains the theory developed by H. Oyono-Oyono and G. Yu, and is general enough to be applied to the setting of crossed products by étale groupoids and discrete quantum groups. We construct controlled assembly maps with values into this controlled K-groups, for Roe C*-algebras and crossed products by étale groupoids. We show that these controlled assembly maps factorize the usual Baum-Connes and coarse Baum-Connes assembly maps. We prove statements called quantitative statements, and we show that a controlled version of the Baum-Connes conjecture is satisfied for a large class of étale groupoids. The end of the thesis is devoted to several applications of these results. We show that the controlled coarse assembly map is equivalent to its analog with coefficients for the coarse groupoid introduced by G. Skandalis, J-L. Tu and G. Yu. We give a proof that coarse spaces which admit a _bred coarse embedding into Hilbert space satisfy the maximal controlled coarse Baum-Connes conjecture. Finally, we study étale groupoids whose proper actions are locally induced by compact open subgroupoids, e.g. ample groupoids introduced by J. Renault. We develop a restriction principle for these groupoids, and prove that under suitable assumptions, their crossed products satisfy the controlled Künneth formula
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Dell'Aiera, Clément. "Controlled K-theory for groupoids and applications." Electronic Thesis or Diss., Université de Lorraine, 2017. http://www.theses.fr/2017LORR0114.
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Dans leur article de 2015 intitulé "On quantitative operator K-theory", H. Oyono-Oyono et G. Yu introduisent un raffinement de la K-théorie opératorielle adapté au cadre desC*-algèbres filtrées, appelé K-théorie quantitative ou contrôlée. Dans cette thèse, nous généralisons la notion de filtration de C_-algèbres. Nous montrons ensuite que ce cadre contient celui déjà traité par G. Yu et H. Oyono-Oyono, tout en se révélant assez souple pour traiter les produits croisés de groupoïdes étalés et de groupes quantiques discrets. Nous construisons ensuite des applications d'assemblage _a valeurs dans les groupes de K-théorie contrôlée associés, pour les C*-algèbres de Roe à coefficients et les produits croisés de groupoïdes étalés. Nous montrons que ces applications factorisent les applications d'assemblage usuelles de Baum-Connes. Nous prouvons ensuite ce que nous appelons des énoncés quantitatifs, et nous montrons qu'une version contrôlée de la conjecture de Baum-Connes est vérifiée pour une large classe de groupoïdes étalés. La fin de la thèse est consacrée à plusieurs applications de ces résultats. Nous montrons que l'application d'assemblage contrôlée coarse est équivalente à son analogue à coefficients pour le groupoïde coarse introduit par G. Skandalis, J-L. Tu et G. Yu. Nous donnons ensuite une preuve que les espaces coarses qui admettent un plongement hilbertien fibré vérifient la version maximale de la conjecture de Baum-Connes coarse contrôlée. Enfin nous étudions les groupoïdes étalés dont toutes les actions propres sont localement induites par des sous-groupoïdes compacts ouverts, dont un exemple est donné par les groupoïdes amples introduits par J. Renault. Nous développons un principe de restriction pour cette classe de groupoïdes, et prouvons que, sous des hypothèses raisonnables, leurs produits croisés vérifient la formule de Künneth en K-théorie contrôlée<br>In their paper entitled "On quantitative operator K-theory", H. Oyono-Oyono and G. Yu introduced a refinement of operator K-theory, called quantitative or controlled K-theory, adapted to the setting of filtered C_-algebras. In this thesis, we generalize filtration of C*-algebras. We show that this setting contains the theory developed by H. Oyono-Oyono and G. Yu, and is general enough to be applied to the setting of crossed products by étale groupoids and discrete quantum groups. We construct controlled assembly maps with values into this controlled K-groups, for Roe C*-algebras and crossed products by étale groupoids. We show that these controlled assembly maps factorize the usual Baum-Connes and coarse Baum-Connes assembly maps. We prove statements called quantitative statements, and we show that a controlled version of the Baum-Connes conjecture is satisfied for a large class of étale groupoids. The end of the thesis is devoted to several applications of these results. We show that the controlled coarse assembly map is equivalent to its analog with coefficients for the coarse groupoid introduced by G. Skandalis, J-L. Tu and G. Yu. We give a proof that coarse spaces which admit a _bred coarse embedding into Hilbert space satisfy the maximal controlled coarse Baum-Connes conjecture. Finally, we study étale groupoids whose proper actions are locally induced by compact open subgroupoids, e.g. ample groupoids introduced by J. Renault. We develop a restriction principle for these groupoids, and prove that under suitable assumptions, their crossed products satisfy the controlled Künneth formula
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Rallis, Nikolaos. "C-K Theory in Practice : C-K Theory in Practice: How can CK Theory serve as a model of reasoning for Startups’ Internationalization?" Thesis, Linköpings universitet, Företagsekonomi, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-160692.
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Background: In the past few decades the world business map has shrunk considerably. Economic unions, tighter cooperation between different countries and across continents is nowadays setting the pace of current economy trends. Moreover, the rise of the internet and technology has interconnected people and markets more than ever. In this dynamic new setting, entrepreneurs and novel ideas have found the ideal ground to flourish. Startups are taking the business world by storm. Moreover, many of them are ambitious enough to engage in International markets right after their conception. It would be interesting to study the process they undergo and revisit it through the application of C-K Design Thinking Theory. Purpose: The purpose of this thesis dissertation is to apply Design Thinking C-K Theory in the Internationalization process of Startups and study how it can serve as model of reasoning for that process. Methodology: Primary data in the form of qualitative interviews were retrieved from three Startups concerning their Internationalization process. They were in turn analyzed by being revisited, with the application of Design Thinking Theory of C- K (Concept – Knowledge) and supported by relevant theory. The results were thought-provoking and will demonstrate how C-K can be used as a model of reasoning for this Process. Results: The study demonstrated that C-K Theory can be used as a model of reasoning for the Internationalization process by strengthening reasoning, improving management and organizing and working synergistically with other theories to generate creativity and problem solving.
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Jia, Bei. "D-branes and K-homology." Thesis, Virginia Tech, 2013. http://hdl.handle.net/10919/32039.
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In this thesis the close relationship between the topological $K$-homology group of the spacetime manifold $X$ of string theory and D-branes in string theory is examined. An element of the $K$-homology group is given by an equivalence class of $K$-cycles $[M,E,\phi]$, where $M$ is a closed spin$^c$ manifold, $E$ is a complex vector bundle over $M$, and $\phi: M\rightarrow X$ is a continuous map. It is proposed that a $K$-cycle $[M,E,\phi]$ represents a D-brane configuration wrapping the subspace $\phi(M)$. As a consequence, the $K$-homology element defined by $[M,E,\phi]$ represents a class of D-brane configurations that have the same physical charge. Furthermore, the $K$-cycle representation of D-branes resembles the modern way of characterizing fundamental strings, in which the strings are represented as two-dimensional surfaces with maps into the spacetime manifold. This classification of D-branes also suggests the possibility of physically interpreting D-branes wrapping singular subspaces of spacetime, enlarging the known types of singularities that string theory can cope with.<br>Master of Science
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Hüttemann, Thomas. "Algebraic K-theory of non-linear projectice spaces." [S.l. : s.n.], 1999. http://deposit.ddb.de/cgi-bin/dokserv?idn=957056230.
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Savin, Anton, and Boris Sternin. "Eta-invariant and Pontrjagin duality in K-theory." Universität Potsdam, 2000. http://opus.kobv.de/ubp/volltexte/2008/2574/.
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The topological significance of the spectral Atiyah-Patodi-Singer η-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.
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Hignett, Anthony James. "Discrete module categories and operations in K-theory." Thesis, University of Sheffield, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.521994.
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Valentino, Alessandro. "K-theory, D-branes and Ramond-Ramond fields." Thesis, Heriot-Watt University, 2008. http://hdl.handle.net/10399/2175.
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This thesis is dedicated to the study of K-theoretical properties of D-branes and Ramond-Ramond fields. We construct abelian groups which define a homology theory on the category CW-complexes, and prove that this homology theory is equivalent to the bordism 3n of KO-homology, the dual theory to KO-theory. We construct an isomorphism between our geometric representation and the SLUdlytic representation of KO-homology, which induces a natural equivalence of homology functors. We apply this framework to describe mathematical properties of D-branes in type I String theory.
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Markett, Simon A. "The Grayson spectral sequence for hermitian K-theory." Thesis, University of Warwick, 2015. http://wrap.warwick.ac.uk/74068/.
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Let R be a regular ring such that 2 is invertible. We construct a spectral sequence converging to the hermitian K-theory, alias the Grothendieck-Witt theory, of R. In particular, we construct a tower for the hermitian K-groups in even shifts, whose terms are given by the hermitian K-theory of automorphisms. The spectral sequence arises as the homotopy spectral sequence of this tower and is analogous to Grayson’s version of the motivic spectral sequence [Gra95]. Further, we construct similar towers for the hermitian K-theory in odd shifts if R is a field of characteristic different from 2. We show by a counter example that the arising spectral sequence does not behave as desired. We proceed by proposing an alternative version for the tower and verify its correctness in weight 1. Finally we give a geometric representation of the (hermitian) K-theory of automorphisms in terms of the general linear group, the orthogonal group, or in terms of e-symmetric matrices, respectively. The K-theory of automorphisms can be identified with motivic cohomology if R is local and of finite type over a field. Therefore the hermitian K-theory of automorphisms as presented in this thesis is a candidate for the analogue of motivic cohomology in the hermitian world.
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Thiang, Guo Chuan. "Topological phases of matter, symmetries, and K-theory." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:53b10289-8b59-46c2-a0e9-5a5fb77aa2a2.
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This thesis contains a study of topological phases of matter, with a strong emphasis on symmetry as a unifying theme. We take the point of view that the "topology" in many examples of what is loosely termed "topological matter", has its origin in the symmetry data of the system in question. From the fundamental work of Wigner, we know that topology resides not only in the group of symmetries, but also in the cohomological data of projective unitary-antiunitary representations. Furthermore, recent ideas from condensed matter physics highlight the fundamental role of charge-conjugation symmetry. With these as physical motivation, we propose to study the topological features of gapped phases of free fermions through a Z<sub>2</sub>-graded C*-algebra encoding the symmetry data of their dynamics. In particular, each combination of time reversal and charge conjugation symmetries can be associated with a Clifford algebra. K-theory is intimately related to topology, representation theory, Clifford algebras, and Z<sub>2</sub>-gradings, so it presents itself as a powerful tool for studying gapped topological phases. Our basic strategy is to use various K</em-theoretic invariants of the symmetry algebra to classify symmetry-compatible gapped phases. The super-representation group of the algebra classifies such gapped phases, while its K-theoretic difference-group classifies the obstructions in passing between two such phases. Our approach is a noncommutative version of the twisted K-theory approach of Freed--Moore, and generalises the K-theoretic classification first suggested by Kitaev. It has the advantage of conceptual simplicity in its uniform treatment of all symmetries. Physically, it encompasses phenomena which require noncommutative algebras in their description; mathematically, it clarifies and provides rigour to the meaning of "homotopic phases", and easily explains the salient features of Kitaev's Periodic Table.
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Schadeck, Laurent. "On the K-theory of tame Artim stacks." Doctoral thesis, Scuola Normale Superiore, 2019. http://hdl.handle.net/11384/85745.
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This thesis pertains to the algebraic K-theory of tame Artin stacks. Building on earlier work of Vezzosi and Vistoli in equivariant K-theory, which we translate in stacky language, we give a description of the algebraic K-groups of tame quotient stacks. Using a strategy of Vistoli, we recover Grothendieck-Riemann-Roch-like formulae for tame quotient stacks that refine Toën’s
Grothendieck-Riemann-Roch formula for Deligne-Mumford stacks (as it was realized that the latter pertains to quotient stacks since it relies on the resolution property). Our formulae differ from Toën’s in that, instead of using the standard inertia stack, we use the cyclotomic inertia stack introduced by Abramovich, Graber and Vistoli in the early 2000s. Our results involve the rational part of the K'-theory of the object considered. We establish a few conjectures, the main one (Conjecture 6.3) pertaining to the covariance of our Lefschetz-Riemann-Roch map for proper morphisms of tame stacks (not necessarily representable). Other future works might be dedicated to the study of torsion in K'-groups as well as more general Artin stacks.
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Sperber, Ron. "A comparison of assembly maps in algebraic K-theory." Diss., Online access via UMI:, 2004. http://wwwlib.umi.com/dissertations/fullcit/3150488.
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Patronas, Dimitrios [Verfasser]. "The Artin Defect in Algebraic K-Theory / Dimitrios Patronas." Berlin : Freie Universität Berlin, 2014. http://d-nb.info/1058105280/34.
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Kolin, David. "k-Space image correlation spectroscopy: theory, verification, and applications." Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=21933.
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This thesis is about the use and development of new fluorescence correlation techniques to measure the dynamics, number density, and aggregation state of fluorescently labelled proteins in living cells. An extensive investigation of the accuracy and precision of temporal image correlation spectroscopy (TICS) is presented first. Using computer simulations of laser scanning microscopy image time series, the effect of spatiotemporal sampling, particle density, noise, and photobleaching of fluorophores on the recovery of transport coefficients and number densities by TICS is investigated. It is shown that photobleaching of the fluorophore can significantly perturb TICS measurements. The theory of k-space image correlation spectroscopy (kICS) is then developed in detail. kICS involves Fourier transforming each image in an image series, and then correlating these transforms, in time. This technique measures the number density, diffusion coefficient, and velocity of fluorescently labelled macromolecules in a cell membrane. In contrast to r-space correlation techniques, we show kICS can recover accurate dynamics even in the presence of complex fluorophore photobleaching and/or "blinking." We use simulations as a proof-of-principle to show that number densities and transport coefficients can be extracted using this technique. We present calibration measurements with fluorescent microspheres imaged on a confocal microscope, which recover Stokes-Einstein diffusion coefficients, and flow velocities that agree with single particle tracking measurements. The wide applicability of the technique is shown by imaging cells transfected with fluorescent protein, and quantum dot (QD) labelled cells on two-photon and total internal reflection fluorescence microscopes. Finally, kICS is used to measure immune T cell receptor (TCR) clustering in live cells using QDs as labels. kICS quantifies the aggregation of TCR by two different approaches. The first uses spatial intensity fluctuat<br>Cette thèse est à propos de l'utilisation et du développement de nouvelles techniques de corrélation de fluorescence afin de mesurer les dynamiques, la densité, et l'état d'agrégation de protéines marquées par fluorescence dans des cellules vivantes. Une vaste recherche de la précision de la spectroscopie temporelle par corrélations d'images (STCI) est premièrement présentée. En utilisant des simulations informatiques à balayage de laser de séries d'images de microscopie, l'effet de l'échantillon spatiotemporelle, densité de particules, le bruit, la fréquence de prises d'échantillons, et le photoblanchiment des fluorophores lors de la mesure des coefficients de transport et la densité par STCI sont examinés. C'est démontré que le photoblanchiment des fluorophores perturbent de manière significative les mesures STCI. La théorie de la spectroscopie de corrélations d'images d'espace-k (CIEk) est développée en détail. CIEk implique la transformation Fourier de chaque image dans une série d'images, et ensuite de faire la corrélation de ces transformations, dans le temps. Cette technique mesure la densité, le coefficient de diffusion, et la vélocité de macromolécules marquées fluorescentes dans une membrane de cellules. Contrairement aux techniques de corrélation espace-r, nous démontrons que CIEk peut mesurer les dynamiques précises, même en présence de complexes photoblanchiments de fluorophores et/ou "clignotement." Nous utilisons des simulations comme une preuve de principes pour démontrer que les densités et les coefficients de transport peuvent être extraits en utilisant cette technique. Nous présentons des mesures d'étalonnage avec des microsphères fluorescentes imagées sur un microscope confocal, qui mesurent la diffusion de coefficients Stokes-Einstein, et les vitesses d'écroulement qui correspondent avec les mesures de suivi de particules uniques. L'application vaste de cette technique est démontrée avec d
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Strong, Mary-Jane Anne. "Additive Unstable Operations in Complex K-Theory and Cobordism." Thesis, University of Westminster, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.500535.
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Kuber, Amit Shekhar. "K-theory of theories of modules and algebraic varieties." Thesis, University of Manchester, 2014. https://www.research.manchester.ac.uk/portal/en/theses/ktheory-of-theories-of-modules-and-algebraic-varieties(5d4387d5-df36-455a-a09d-922d67b0827e).html.
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Melo, S. T., R. Nest, and Elmar Schrohe. "C*-structure and K-theory of Boutet de Monvel's algebra." Universität Potsdam, 2001. http://opus.kobv.de/ubp/volltexte/2008/2616/.
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We consider the norm closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a manifold X with boundary ∂X. We first describe the image and the kernel of the continuous extension of the boundary principal symbol homomorphism to A. If X is connected and ∂X is not empty, we then show that the K-groups of A are topologically determined. In case the manifold, its boundary, and the cotangent space of its interior have torsion free K-theory, we get Ki(A,k) congruent Ki(C(X))⊕Ksub(1-i)(Csub(0)(T*X)),i = 0,1, with k denoting the compact ideal, and T*X denoting the cotangent bundle of the interior. Using Boutet de Monvel's index theorem, we also prove that the above formula holds for i = 1 even without this torsion-free hypothesis. For the case of orientable, two-dimensional X, Ksub(0)(A) congruent Z up(2g+m) and Ksub(1)(A) congruent Z up(2g+m-1), where g is the genus of X and m is the number of connected components of ∂X. We also obtain a composition sequence 0 ⊂ k ⊂ G ⊂ A, with A/G commutative and G/k isomorphic to the algebra of all continuous functions on the cosphere bundle of ∂X with values
in compact operators on L²(R+).
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Narreddy, Naga Sambu Reddy, and Tuğrul Durgun. "Clusters (k) Identification without Triangle Inequality : A newly modelled theory." Thesis, Uppsala universitet, Institutionen för informatik och media, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-183608.
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Cluster analysis characterizes data that are similar enough and useful into meaningful groups (clusters).For example, cluster analysis can be applicable to find group of genes and proteins that are similar, to retrieve information from World Wide Web, and to identify locations that are prone to earthquakes. So the study of clustering has become very important in several fields, which includes psychology and other social sciences, biology, statistics, pattern recognition, information retrieval, machine learning and data mining [1] [2]. Cluster analysis is the one of the widely used technique in the area of data mining. According to complexity and amount of data in a system, we can use variety of cluster analysis algorithms. K-means clustering is one of the most popular and widely used among the ten algorithms in data mining [3]. Like other clustering algorithms, it is not the silver bullet. K-means clustering requires pre analysis and knowledge before the number of clusters and their centroids are determined. Recent studies show a new approach for K-means clustering which does not require any pre knowledge for determining the number of clusters [4]. In this thesis, we propose a new clustering procedure to solve the central problem of identifying the number of clusters (k) by imitating the desired number of clusters with proper properties. The proposed algorithm is validated by investigating different characteristics of the analyzed data with modified theory, analyze parameters efficiency and their relationships. The parameters in this theory include the selection of embryo-size (m), significance level (α), distributions (d), and training set (n), in the identification of clusters (k).