Academic literature on the topic 'R/Z K-Theory'

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.

We prove that the Novikov assembly map for a group $\Gamma$ factorizes, in ‘low homological degree’, through the algebraic K-theory of its integral group ring. In homological degree 2, this answers a question posed by N. Higson and P. Julg. As a direct application, we prove that if $\Gamma$ is torsion-free and satisfies the Baum-Connes conjecture, then the homology group $H_{1}(\Gamma;\,\mathbb{Z})$ injects in $K_{1}(C^{*}_{r}\Gamma)$ and in $K_{1}^{\rm alg}(A)$, for any ring $A$ such that $\mathbb{Z}\Gamma\subseteq A\subseteq C^{*}_{r}\Gamma$. If moreover $B\Gamma$ is of dimension less than or equal to 4, then we show that $H_{2}(\Gamma;\,\mathbb{Z})$ injects in $K_{0}(C^{*}_{r}\Gamma)$ and in $K_{2}^{\rm alg}(A)/\Delta_{2}$, where $A$ is as before, and $\Delta_{2}$ is generated by the Steinberg symbols $\{\gamma,\,\gamma\}$, for $\gamma\in\Gamma$. 2000 Mathematical Subject Classification: primary 19D55, 19Kxx, 58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.

AbstractWe consider a generalization {K_{0}^{\operatorname{gr}}(R)} of the standard Grothendieck group {K_{0}(R)} of a graded ring R with involution. If Γ is an abelian group, we show that {K_{0}^{\operatorname{gr}}} completely classifies graded ultramatricial {*}-algebras over a Γ-graded {*}-field A such that (1) each nontrivial graded component of A has a unitary element in which case we say that A has enough unitaries, and (2) the zero-component {A_{0}} is 2-proper ({aa^{*}+bb^{*}=0} implies {a=b=0} for any {a,b\in A_{0}}) and {*}-pythagorean (for any {a,b\in A_{0}} one has {aa^{*}+bb^{*}=cc^{*}} for some {c\in A_{0}}). If the involutive structure is not considered, our result implies that {K_{0}^{\operatorname{gr}}} completely classifies graded ultramatricial algebras over any graded field A. If the grading is trivial and the involutive structure is not considered, we obtain some well-known results as corollaries. If R and S are graded matricial {*}-algebras over a Γ-graded {*}-field A with enough unitaries and {f:K_{0}^{\operatorname{gr}}(R)\to K_{0}^{\operatorname{gr}}(S)} is a contractive {\mathbb{Z}[\Gamma]}-module homomorphism, we present a specific formula for a graded {*}-homomorphism {\phi:R\to S} with {K_{0}^{\operatorname{gr}}(\phi)=f}. If the grading is trivial and the involutive structure is not considered, our constructive proof implies the known results with existential proofs. If {A_{0}} is 2-proper and {*}-pythagorean, we also show that two graded {*}-homomorphisms {\phi,\psi:R\to S} are such that {K_{0}^{\operatorname{gr}}(\phi)=K_{0}^{\operatorname{gr}}(\psi)} if and only if there is a unitary element u of degree zero in S such that {\phi(r)=u\psi(r)u^{*}} for any {r\in R}. As an application of our results, we show that the graded version of the Isomorphism Conjecture holds for a class of Leavitt path algebras: if E and F are countable, row-finite, no-exit graphs in which every infinite path ends in a sink or a cycle and K is a 2-proper and {*}-pythagorean field, then the Leavitt path algebras {L_{K}(E)} and {L_{K}(F)} are isomorphic as graded rings if any only if they are isomorphic as graded {*}-algebras. We also present examples which illustrate that {K_{0}^{\operatorname{gr}}} produces a finer invariant than {K_{0}}.

In the paper we prove for the first time an analogue of the Wiman inequality in the class of analytic functions f∈A0p(G) in an arbitrary complete Reinhard domain G⊂Cp, p∈N represented by the power series of the form f(z)=f(z1,⋯,zp)=∑‖n‖=0+∞anzn with the domain of convergence G. We have proven the following statement: If f∈Ap(G) and h∈Hp, then for a given ε=(ε1,…,εp)∈R+p and arbitrary δ>0 there exists a set E⊂|G| such that ∫E∩Δεh(r)dr1⋯drpr1⋯rp<+∞ and for all r∈Δε∖E we have Mf(r)≤μf(r)(h(r))p+12lnp2+δh(r)lnp2+δ{μf(r)h(r)}∏j=1p(lnerjεj)p−12+δ. Note, that this assertion at p=1,G=C,h(r)≡const implies the classical Wiman–Valiron theorem for entire functions and at p=1, the G=D:={z∈C:|z|<1},h(r)≡1/(1−r) theorem about the Kővari-type inequality for analytic functions in the unit disc D; p>1 implies some Wiman’s type inequalities for analytic functions of several variables in Cn×Dk, n,k∈Z+,n+k∈N.

The chalcogenides LaxTbyErzPbSi2S8 were obtained by synthesizing the elementary components in vacuum quartz containers at 1320 K. The synthesized alloys were homogenized by annealing at 770 K during 500 hours. The cell parameters of synthesized sulfides are: a = 0,89576(3) nm, c = 2,65646(8) nm – La1,2Tb0,4Er0,4PbSi2S8; a = 0,89209(1) nm, c = 2,63466(5) nm – La0,9Tb0,2Er0,9PbSi2S8; a = 0,89002(3) nm, c = 2,62714(7) nm – La0,67Tb0,67Er0,67PbSi2S8; a = 0,88993(1) nm, c = 2,62973(4) nm – La0,6Tb1,2Er0,2PbSi2S8; a = 0,885161(7) nm, c = 2,60445(3) nm – La0,2Tb0,9Er0,9PbSi2S8 respectively. The atoms of statistical mixture (La,Tb,Er,Pb) occupy the site 18e (x y 1/4), and the atoms of Si occupy the site 12c (1/3 2/3 z) in the structure of the obtained chalcogenides. Coordinating polyhedra of atoms of the statistical mixture (La, Tb, Er, Pb) are trigonal prism with two additional atoms (CN = 8), and the atoms of Si occupying the crystallographic point system 12c describes with the tetrahedron. According to the results of the experiment, the synthesized chalcogenides crystallize in the structure type of La2PbSi2S8 (hR26,167). The structure of La2PbSi2S8 is described by using the theory of second anion coordination (SAC).

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

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.