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Relevant bibliographies by topics / Z K-Theory

Academic literature on the topic 'Z K-Theory'

Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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Journal articles on the topic "Z K-Theory"

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Elmrabty, Adnane. "Some Connections Between Bunke-Schick Differential K-theory and Topological $\mathbb{Z}/k\mathbb{Z}$ K-theory." Missouri Journal of Mathematical Sciences 30, no. 1 (May 2018): 32–44. http://dx.doi.org/10.35834/mjms/1534384951.

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Elmrabty, Adnane. "Some geometric results on K-theory with $${\mathbb{Z}}/k{\mathbb{Z}}$$-coefficients." São Paulo Journal of Mathematical Sciences 14, no. 2 (July 6, 2020): 562–79. http://dx.doi.org/10.1007/s40863-020-00179-z.

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Arlettaz, Dominique. "On the algebraic K-theory of Z." Journal of Pure and Applied Algebra 51, no. 1-2 (March 1988): 53–64. http://dx.doi.org/10.1016/0022-4049(88)90077-1.

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Bárcenas, Noé, and Mario Velásquez. "Equivariant $K$-theory of central extensions and twisted equivariant $K$-theory: $SL_{3}\mathbb{Z}$ and $St_{3}{\mathbb{Z}}$." Homology, Homotopy and Applications 18, no. 1 (2016): 49–70. http://dx.doi.org/10.4310/hha.2016.v18.n1.a4.

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Boyde, Guy. "p-Hyperbolicity of homotopy groups via K-theory." Mathematische Zeitschrift 301, no. 1 (January 7, 2022): 977–1009. http://dx.doi.org/10.1007/s00209-021-02917-1.

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AbstractWe show that $$S^n \vee S^m$$ S n ∨ S m is $${\mathbb {Z}}/p^r$$ Z / p r -hyperbolic for all primes p and all $$r \in {\mathbb {Z}}^+$$ r ∈ Z + , provided $$n,m \ge 2$$ n , m ≥ 2 , and consequently that various spaces containing $$S^n \vee S^m$$ S n ∨ S m as a p-local retract are $${\mathbb {Z}}/p^r$$ Z / p r -hyperbolic. We then give a K-theory criterion for a suspension $$\Sigma X$$ Σ X to be p-hyperbolic, and use it to deduce that the suspension of a complex Grassmannian $$\Sigma Gr_{k,n}$$ Σ G r k , n is p-hyperbolic for all odd primes p when $$n \ge 3$$ n ≥ 3 and $$0<k<n$$ 0 < k < n . We obtain similar results for some related spaces.

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Gutierrez, Julio. "Bivariant K-theory of locally convex Z-graded algebras." Selecciones Matemáticas 9, no. 01 (June 30, 2022): 167–72. http://dx.doi.org/10.17268/sel.mat.2022.01.14.

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In the present work, we describe some results about the K-theory of Z-graded algebras. First, in the context of C* algebras, we begin with the Pimsner-Voiculescu sequence for crossed products and its generalizations. We will see that there are results analog to these in the context of locally convex algebras and we conclude with results for generalized Weyl algebras.

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Lee, Jae Min, and Byungdo Park. "A Superbundle Description of Differential K-Theory." Axioms 12, no. 1 (January 12, 2023): 82. http://dx.doi.org/10.3390/axioms12010082.

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We construct a model of differential K-theory using superbundles with a Z/2Z-graded connection and a differential form on the base manifold and prove that our model is isomorphic to the Freed–Lott–Klonoff model of differential K-theory.

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Orlitzky, Michael. "Positive and Z-operators on Closed Convex Cones." Electronic Journal of Linear Algebra 34 (February 21, 2018): 444–58. http://dx.doi.org/10.13001/1081-3810.3782.

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Let $K$ be a closed convex cone with dual $\dual{K}$ in a finite-dimensional real Hilbert space. A \emph{positive operator} on $K$ is a linear operator $L$ such that $L\of{K} \subseteq K$. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. It is said that $L$ is a \emph{\textbf{Z}-operator} on $K$ if % \begin{equation*} \ip{L\of{x}}{s} \le 0 \;\text{ for all } \pair{x}{s} \in \cartprod{K}{\dual{K}} \text{ such that } \ip{x}{s} = 0. \end{equation*} % The \textbf{Z}-operators are generalizations of \textbf{Z}-matrices (whose off-diagonal elements are nonpositive) and they arise in dynamical systems, economics, game theory, and elsewhere. In this paper, the positive and \textbf{Z}-operators are connected. This extends the work of Schneider, Vidyasagar, and Tam on proper cones, and reveals some interesting similarities between the two families.

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Snaith, Victor. "l-adic and Z/l∞-algebraic and topological K-theory." Proceedings of the Edinburgh Mathematical Society 28, no. 1 (February 1985): 73–90. http://dx.doi.org/10.1017/s0013091500003217.

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Let l be an odd prime and let A be a commutative ring containing 1/l. Let K*(A;Z/lv) denote the mod lv algebraic K-theory of A [3]. As explained in [4] there exists a “Bott element” βv∈K21v–1(l–1)(Z[1/l];Z/lv) and, using the K-theory product we may, following [16, Part IV], formwhich is defined as the direct limit of iterated multiplication by βv. There is a canonical localisation map

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Karoubi, Max. "Clifford Modules and Twisted K-Theory." Advances in Applied Clifford Algebras 18, no. 3-4 (May 27, 2008): 765–69. http://dx.doi.org/10.1007/s00006-008-0101-z.

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Dissertations / Theses on the topic "Z K-Theory"

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Bräuer, Jonas Verfasser], and Christopher [Akademischer Betreuer] [Deninger. "Entropies of algebraic Z d -actions and K-theory / Jonas Bräuer. Betreuer: Christopher Deninger." Münster : Universitäts- und Landesbibliothek der Westfälischen Wilhelms-Universität, 2011. http://d-nb.info/1027018017/34.

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Bräuer, Jonas [Verfasser], and Christopher [Akademischer Betreuer] Deninger. "Entropies of algebraic Z d -actions and K-theory / Jonas Bräuer. Betreuer: Christopher Deninger." Münster : Universitäts- und Landesbibliothek der Westfälischen Wilhelms-Universität, 2011. http://d-nb.info/1027018017/34.

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Winges, Christoph [Verfasser], and Arthur [Akademischer Betreuer] Bartels. "Filtering the assembly map in algebraic K-theory and transfer reducibility of Z n Z / Christoph Winges ; Betreuer: Arthur Bartels." Münster : Universitäts- und Landesbibliothek Münster, 2014. http://d-nb.info/1141681617/34.

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Barlak, Selçuk [Verfasser], and Joachim [Akademischer Betreuer] Cuntz. "On the K-theory of crossed product C*-algebras by actions of Z n / Selçuk Barlak ; Betreuer: Joachim Cuntz." Münster : Universitäts- und Landesbibliothek Münster, 2014. http://d-nb.info/1138282782/34.

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Lim, Johnny. "Analytic Pontryagin Duality." Thesis, 2019. http://hdl.handle.net/2440/124554.

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Let X be a smooth compact manifold. We propose a geometric model for the group K⁰(X,R/Z): We study a well-defined and non-degenerate analytic duality pairing between K⁰(X,R/Z) and its Pontryagin dual group, the Baum-Douglas geometric K-homology K₀(X); whose pairing formula comprises of an analytic term involving the Dai-Zhang eta-invariant associated to a twisted Dirac-type operator and a topological term involving a differential form and some characteristic forms. This yields a robust R/Z-valued invariant. We also study two special cases of the analytic pairing of this form in the cohomology groups H¹(X,R/Z) and H²(X,R/Z):
Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Sciences, 2019

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Deeley, Robin. "Geometric K-homology with coefficients." Thesis, 2010. http://hdl.handle.net/1828/2911.

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We construct geometric models for K-homology with coefficients based on the theory of Z/k-manifolds. To do so, we generalize the operations and relations Baum and Douglas put on spinc-manifolds to spinc Z/kZ-manifolds. We then de fine a model for K-homology with coefficients in Z/k using cycles of the form ((Q,P), (E,F), f) where (Q, P) is a spinc Z/k-manifold, (E, F) is a Z/k-vector bundle over (Q, P) and f is a continuous map from (Q, P) into the space whose K-homology we are modelling. Using results of Rosenberg and Schochet, we then construct an analytic model for K-homology with coefficients in Z/k and a natural map from our geometric model to this analytic model. We show that this map is an isomorphism in the case of finite CW-complexes. Finally, using direct limits, we produced geometric models for K-homology with coefficients in any countable abelian group.

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Štěpánková, Helena. "Komparace vývoje stranických systémů ČR, Slovenska a Polska z hlediska teorie cleavages." Master's thesis, 2013. http://www.nusl.cz/ntk/nusl-321952.

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This diploma thesis deals with a development of the socioeconomic cleavage in the party systems of the Czech Republic, Slovakia and Poland in 2006 - 2012. At the beginning the theory of cleavages by S. Rokkan and S. M. Lipset and the enlargement to post-communist countries of Klaus von Beyme, are presented. There are also mentioned concepts of other authors. The thesis tries to answer a few questions. Firstly, what form did this line have in the monitored period (transformation line vs. classical socioeconomic line). Secondly, how big was an impact on the political right-left orientation of the political parties, and thirdly, how significant was this line for the party-political system of selected countries in this period. There is also a brief development of the party system after 1989 as well as the ideological profiles of parliamentary parties given in each chapter focused on particular country. One part of it deals also with the economic and social climate in the monitored period. Then the work is based mainly on the analysis of party's agenda, with an emphasis on the socioeconomic line.

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Pasanen, Trevor Lee. "Irreducible characters of SL[k](Z/p[n]Z)." 2009. http://hdl.handle.net/10048/760.

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Thesis (M. Sc.)--University of Alberta, 2009.
In the title, the character [k] is in subscript and the character [n] is in superscript. Title from PDF file main screen (viewed on Oct. 19, 2009). "A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Science in Pure Mathematics, Department of Mathematical and Statistical Sciences, University of Alberta." Includes bibliographical references.

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Rambousek, Jiří. "K počátkům a vývoji českého překladu dětské literatury z angličtiny." Doctoral thesis, 2015. http://www.nusl.cz/ntk/nusl-352279.

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The dissertation deals with various aspects of the translation of children's literature from English into Czech, focusing on its earliest stages. The methodology of the research is based on the polysystem theory. It indicates - in agreement with the hypotheses of the theory - that the polysystem of Czech children's literature developed under a strong influence of translation. In the earliest stages, original English texts played a marginal role; they were mostly translated into Czech indirectly via German mediating texts. With the commencement of direct translation at the turn of the 19th and the 20th centuries, the influence of literary models of English origin became stronger; they enriched Czech children's literature by specific thematic genres, as well as new stimuli of the new conception of imaginative children's literature, which replaced older didactically oriented texts rooted in the tradition of philanthropism. The dissertation also examines bibliographic issues of the analyzed subsystem. It illustrates by examples how the customary enumerative bibliography can be extended to the sphere of textology and yield new findings concerning translated texts (e.g. the identification of the mediating text or the correct form of the translation). The dissertation pays attention especially to texts...

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Langhammer, Pavel. "Teorie sociální paměti jako alternativa k přístupu z hlediska kritérií autenticity v ježíšovském bádání." Master's thesis, 2018. http://www.nusl.cz/ntk/nusl-384027.

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The topic of social memory has been discussed for almost 20 years in the international field, but it hasn't been reflected in the Czech environment yet. This master thesis tries to introduce the topic of social memory theory to the Czech theological audience. It concentrates on the critique of the "criteria approach" in the historical Jesus research from the social memory theorist, esp. Chris Keith. It also tries to find roots of this way of thinking either among theorist of social memory like its "father" late Maurice Halbwachs and our contemporary Jan Assmann, or among New Testament scholars like Birger Gerhardsson and Werner Kelber who challenged the development model of the gospel tradition presented by Rudolf Bultmann. The critique of Bultmann is the basis of Keith's critique of the "criteria approach". This thesis presents Keith's concepts of "Jesus-memory approach" and "New historiography" and presents critiques of it both from the international scholars and its own. This thesis is a contribution to the field of hermeneutics and the development of methodology.

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Books on the topic "Z K-Theory"

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Emilio, Lluis, and Snaith V. P. 1944-, eds. On K*̳(Z/n) and K*̳(F̳q̳[t]/(t²). Providence, R.I., U.S.A: American Mathematical Society, 1985.

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Aisbett, Janet E. On K[subscript *](Z/n) and K[subscript *](F[subscript q][t]/(t[superscript 2)). Providence, R.I: American Mathematical Society, 1985.

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Lange, Elena Louisa. Value Without Fetish: Uno Kōzō's Theory of 'Pure Capitalism' in Light of Marx's Critique of Political Economy. BRILL, 2021.

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Green, Mark L. The Spread Philosophy in the Study of Algebraic Cycles. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0010.

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This chapter discusses the spread philosophy in the study of algebraic cycles, in order to make use of a geometry by considering a variation of Hodge structure where D is the Hodge domain (or the appropriate Mumford–Tate domain) and Γ‎ is the group of automorphisms of the integral lattice preserving the intersection pairing. If we have an algebraic cycle Z on X, taking spreads yields a cycle Ƶ on X. Applying Hodge theory to Ƶ on X gives invariants of the cycle. Another related situation is algebraic K-theory. For example, to study Kₚsuperscript Milnor(k), the geometry of S can be used to construct invariants.

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Voisin, Claire. Decomposition of the Diagonal. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691160504.003.0003.

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This chapter explains the method initiated by Bloch and Srinivas, which leads to statements of the following: if a smooth projective variety has trivial Chow groups of k-cycles homologous to 0 for k ≤ c − 1, then its transcendental cohomology has geometric coniveau ≤ c. This result is a vast generalization of Mumford's theorem. A major open problem is the converse of this result. It turns out that statements of this kind are a consequence of a general spreading principle for rational equivalence. Consider a smooth projective family X → B and a cycle Z → B, everything defined over C; then, if at the very general point b ∈ B, the restricted cycle Z𝒳b ⊂ X𝒳b is rationally equivalent to 0, there exist a dense Zariski open set U ⊂ B and an integer N such that NZsubscript U is rationally equivalent to 0 on Xsubscript U.

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Book chapters on the topic "Z K-Theory"

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Farley, Daniel Scott, and Ivonne Johanna Ortiz. "The Homology Groups $$H_{n}^{\varGamma }(E_{\mathcal{F}\mathcal{I}\mathcal{N}}(\varGamma ); \mathbb{K}\mathbb{Z}^{-\infty })$$." In Algebraic K-theory of Crystallographic Groups, 81–98. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-08153-3_7.

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Guaschi, John, Daniel Juan-Pineda, and Silvia Millán López. "Lower Algebraic K-Theory Groups of the Group Ring $$\mathbb Z[B_4(\mathbb S^{2})]$$." In SpringerBriefs in Mathematics, 63–72. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-99489-5_4.

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Abspoel, Mark, Ronald Cramer, Ivan Damgård, Daniel Escudero, and Chen Yuan. "Efficient Information-Theoretic Secure Multiparty Computation over $$\mathbb {Z}/p^k\mathbb {Z}$$ via Galois Rings." In Theory of Cryptography, 471–501. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-36030-6_19.

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Benschop, Nico F. "Additive Structure of Z(.) mod m k (Squarefree) and Goldbach’s Conjecture." In Associative Digital Network Theory, 123–35. Dordrecht: Springer Netherlands, 2009. http://dx.doi.org/10.1007/978-1-4020-9829-1_9.

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Huang, Huajun, and Ming-Cheng Tsai. "Linear K-Power Preservers and Trace of Power-Product Preservers." In Matrix Theory - Classics and Advances [Working Title]. IntechOpen, 2022. http://dx.doi.org/10.5772/intechopen.103713.

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Let V be the set of n×n complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix k∈Z\\01. We characterize linear maps ψ:V→V that satisfy ψAk=ψAk on an open neighborhood S of In in V. The k-power preservers are necessarily k-potent preservers, and k=2 corresponds to Jordan homomorphisms. Applying the results, we characterize maps ϕ,ψ:V→V that satisfy “trϕAψBk=trABk for all A∈V, B∈S, and ψ is linear” or “trϕAψBk=trABk for all A,B∈S and both ϕ and ψ are linear.” The characterizations systematically extend existing results in literature, and they have many applications in areas like quantum information theory. Some structural theorems and power series over matrices are widely used in our characterizations.

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Huang, Huajun, and Ming-Cheng Tsai. "Linear K-Power Preservers and Trace of Power-Product Preservers." In Matrix Theory - Classics and Advances [Working Title]. IntechOpen, 2022. http://dx.doi.org/10.5772/intechopen.103713.

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Let V be the set of n×n complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix k∈Z\\01. We characterize linear maps ψ:V→V that satisfy ψAk=ψAk on an open neighborhood S of In in V. The k-power preservers are necessarily k-potent preservers, and k=2 corresponds to Jordan homomorphisms. Applying the results, we characterize maps ϕ,ψ:V→V that satisfy “trϕAψBk=trABk for all A∈V, B∈S, and ψ is linear” or “trϕAψBk=trABk for all A,B∈S and both ϕ and ψ are linear.” The characterizations systematically extend existing results in literature, and they have many applications in areas like quantum information theory. Some structural theorems and power series over matrices are widely used in our characterizations.

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Smullyan, Raymond M. "Symmetric and Double Recursion Theorems." In Recursion Theory for Metamathematics. Oxford University Press, 1993. http://dx.doi.org/10.1093/oso/9780195082326.003.0013.

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We have proved that every completely E.I. pair of r.e. sets is D.U. In the next chapter we will show the stronger result that every E.I. pair of r.e. sets is D.U. The proof of this is based on the double recursion theorem of this chapter. Our original formulation of the double recursion theorem (T.F.S.) required the recursive pairing function J(x,y), not only for its proof, but for its very statement. In this chapter, we give an improved version whose statement and proof are independent of J, K and L. In Part III of this chapter, we compare our new version with the original J, K and L version and show that they are easily interderivable. §1. The Weak Double Recursion Theorem. Consider two r.e. relations, M1(x,y, z) and M2(x, y, z). For any number b, the relation M1(x,y,b) is an r.e. relation between x and y, and so by the weak recursion theorem (Theorem 1, Chapter 8), there is a number a such that ωa = x : M1(x, a, b). Likewise, for any number a, the relation M2(x,a,y) is an r.e. relation between x and y, and so there is a number b such that ωb, = x : M2(x,a,b). Our next theorem tells us that we can choose a and b so that both these conditions hold simultaneously.

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"Expectancy-Value Theory ALLAN WIGFIELD, STEPHEN M . TO NKS , AND SUSAN LU T Z K L AU DA." In Handbook of Motivation at School, 67–86. Routledge, 2016. http://dx.doi.org/10.4324/9781315773384-10.

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Conference papers on the topic "Z K-Theory"

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Camacho, Erika, Richard Rand, and Howard Howland. "Dynamics of Two Van Der Pol Oscillators Coupled via a Bath." In ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/detc2003/vib-48593.

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In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

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Etgen, John T. "V(z) F‐K prestack migration of common‐offset common‐azimuth data volumes, Part I: Theory." In SEG Technical Program Expanded Abstracts 1998. Society of Exploration Geophysicists, 1998. http://dx.doi.org/10.1190/1.1820290.

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Panta Pazos, Rube´n. "Krylov Techniques for 3D Problems in Transport Theory." In 14th International Conference on Nuclear Engineering. ASMEDC, 2006. http://dx.doi.org/10.1115/icone14-89562.

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When solving integro-differential equations by means of numerical methods one has to deal with large systems of linear equations, such as happens in transport theory [10]. Many iterative techniques are now used in Transport Theory in order to solve problems of 2D and 3D dimensions. In this paper, we choose two problems to solve the following transport equation, μ∂ψ∂x(x,μ)+h(x,μ)ψ(x,μ)==∫−11k(x,μ,μ')ψ(x,μ')dμ'+q(x,μ)(1) where x: represents the spatial variable, μ: the cosine of the angle, ψ: the angular flux, h(x, μ): is the collision frequency, k(x, μ, μ'): the scattering kernel, q(x, μ): the source. The aim of this work is the straightforward application of the Krylov spaces technique [2] to the governing equation or to its discretizations derived of the discrete ordinates method (choosing a finite number of directions and then approximating the integral term by means of a proper sum). The equation (1) can be written in functional form as Aψ=q(2) with ψ in the Hilbert space L2 ([0,a] × [-1,1])., and q is the source function. The operator A˜ derived from a discrete ordinates scheme that approximates the operator Aψ=μ∂ψ∂x(x,μ)++h(x,μ)ψ(x,μ)−∫−11k(x,μ,μ')ψ(x,μ')dμ'(3) generates the following subspace K'''(A˜,q)=Span{q,Aq˜,A˜2q,…,A˜m−1q}(4)i.e. the subspace generated by the iterations of order 0, 1, 2, ... , m−1 of the source function q. Two methods are specially outstanding, the Lanczos method to solve the problem given by equation (2) with certain boundary conditions, and the conjugate gradient method to solve the same problem with identical boundary conditions. We discuss and accelerate the basic iterative method [8]. An important conclusion is the generation of these methods to solve linear systems in Hilbert spaces, if verify the convergence conditions, which are outlined in this work. The first problem is a cubic domain with two regions, one with a source near the vertex at the origin and the shield region. In this case, the Cartesian planes (specifically 0<x<L, 0<y<L, 0<z<L) are reflexive boundaries and the rest faces of the cube are vacuum boundaries. The second problem has a geometry with three regions, one of the source, other is the void region and the third one is a shield domain.

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Yuanpeng Liu and Elza Erkip. "On the sum capacity of K-user cascade Gaussian Z-interference channel." In 2011 IEEE International Symposium on Information Theory - ISIT. IEEE, 2011. http://dx.doi.org/10.1109/isit.2011.6033765.

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Tang, Xin, and Taher Saif. "Loss of Cell Adhesion in Colon Cancer Cells During In Vitro Metastasis Measured by Bio-MEMS Force Sensor." In ASME 2012 Summer Bioengineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/sbc2012-80936.

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Human colon carcinoma (HCT-8) cells show metastatic phenotype when cultured on appropriately soft substrates. Here, we studied the surface non-specific adhesion in HCT-8 cells throughout the in vitro metastasis process. A novel bio-MEMS force sensor was used to measure the cell-probe non-specific adhesion. The adhesion characteristics are analyzed using classical Johnson-Kendall-Roberts (JKR) theory. Our results indicate that the post-metastatic HCT-8 cells (dissociated R cells) display remarkably diminished surface adhesion and are potentially more invasive than original pre-metastatic HCT-8 cells (E cells). To the best of our knowledge, this is the first quantitative data on cancer cells adhesion change as in vitro metastasis proceeds. It is well known that, during in vivo cancer metastasis, malignant cancer cells reduce their surface adhesion (both specific and non-specific) [1] as well as modify their extracellular matrix (ECM) ligands [2] to detach from primary tumor and enhance successful invasion into distant healthy organs. Simultaneously, cancer cells down-regulate their surface cell-cell adhesion molecules, i.e. E-Cadherin, to escape from tumor and initiate metastasis [1]. However, there is no quantitative report on cancer cell adhesion throughout the entire metastasis process, since in vivo metastasis is nearly impossible to detect [3]. We had discovered [4] that human colon cancer cells (HCT-8) can consistently display an in vitro metastasis-like phenotype (MLP) within only 7 days of culture on soft hydrogel substrates with appropriate mechanical stiffness (Poly-acrylamide gels with Elastic modulus: 21∼ 47 kPa [14, 15]). We found that MLP is consistent, repeatable and irreversible (Fig. 1a-1c). In addition, the post MLP cancer cells (referred to here as R cells meaning round-shaped in contrast to the E-cells, i.e., the original HCT-8 cells that are epithelial in nature) up-regulate a number of in vivo tissue-destructive proteinases, such as, MMPs [4]. R cells also express remarkably diminished E-Cadherin patterns compared to HCT8 E cells (Fig. 1d, 1e). Using this model system, we are able to study the kinetics of non-specific and specific surface adhesion change on HCT-8 cancer cells. In this paper, we measure the non-specific adhesion of both pre and post metastatic HCT-8 cells (E and R cells respectively) using a novel bio-MEMS force sensor. The adhesion energy and other mechanical properties are analyzed using classical Johnson-Kendall-Roberts (JKR) theory [5]. We find that after undergoing metastasis (or MLP), the dissociated HCT-8 cells (R cells) down-regulate non-specific adhesion, in contrast to their ancestors, HCT-8 E cells. The reduction of non-specific adhesion is coincident with the immuno-fluorescent staining data of cell-cell specific adhesion molecule E-Cadherin, which shows 4 ∼ 6 times down-regulation after MLP (Fig. 1d-1e). The bio-MEMS sensor consists of a micro cantilever beam with spring constant k = 3.48 nN/ μm. A flat probe is attached with the beam which forms adhesive contact with cells. The sensor is made from single crystal silicon, and is coated with a thin layer of native silicon oxide (SiO2). The probe and the sensor are not functionalized. The sensor is manipulated with an x-y-z piezo stage. To measure the cell adhesion, the flat probe is brought in contact with cells’ lateral convex surface at the boundary. After a 2-minute contact, force sensor is pulled away horizontally from the cell island at a constant quasi-static speed of 2.1 ± 0.4 μm/s (Fig. 2a). Due to the cell-probe adhesion, the sensor beam deforms during retraction. Corresponding restoring force of the cell island is given by F = kδ (Fig. 2a-c). Note the probe is non-functionalized (free of any extra-cellular matrix proteins), and only has a coating of SiO2 on the surface due to air exposure. During probe retraction, the cell is continuously stretched while the cell-probe contact area radius Rc remains unchanged (Fig. 3b-e) and the contact angle θ increases (Fig. 3b). At critical value of force, Fc, the cell suddenly detaches from probe (Fig. 3d). The critical Fc at detachment is optically recorded by video camera and was determined as 27.8 ± 2.2 nN. A similar experiment on cells after MLP shows so measurable adhesion, i.e, the force to detach was zero for all the cells tested. Figure shows the measured adhesion in pre and post metastatic cells.

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