diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzllte" "b/data_all_eng_slimpj/shuffled/split2/finalzzllte" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzllte" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nTopological insulators are materials that are insulating in the bulk\nbut allow a current to flow on the boundary. These boundary currents\nare protected by topological invariants and thus, in ideal cases, flow\nwithout dissipation. The mathematical description of a topological\ninsulator uses a \\(\\Cst\\)\\nobreakdash-algebra~\\(\\mathcal{A}\\)\nthat contains the resolvent of the Hamiltonian~\\(H\\)\nof the system; this amounts to \\(H\\in\\mathcal{A}\\)\nif~\\(H\\)\nis bounded. To describe an insulator, the spectrum of~\\(H\\)\nshould have a gap at the Fermi energy~\\(E\\).\nDepending on further symmetries of the system such as a time\nreversal, particle--hole or chiral symmetry, the topological phase\nof the material may be classified by a class in the \\(\\K\\)\\nobreakdash-theory\nof~\\(\\mathcal{A}\\)\nassociated to the spectral projection of~\\(H\\)\nat the Fermi energy (see, for instance,\n\\cites{Kellendonk:Cstar_phases, Prodan-Schulz-Baldes:Bulk_boundary}). \nSo the observable algebra~\\(\\mathcal{A}\\)\nor rather its K\\nobreakdash-theory predicts the possible topological\nphases of a material.\n\nAt first, a material is often modelled without disorder and in a tight\nbinding approximation. This gives a translation-invariant Hamiltonian\nacting on \\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\)\n(see, for instance, \\cites{Bernevig-Hughes-Zhang:Quantum,\n Fu-Kane-Mele:Insulators,\n Liu-Qi-Zhang-Dai-Fang-Zhang:Model_Hamiltonian}). Bloch--Floquet\ntheory describes the Fermi projection through a vector bundle over the\n\\(d\\)\\nobreakdash-torus,\nwith extra structure that reflects the symmetries of the system (see,\nfor instance, \\cites{De_Nittis-Gomi:Real_Bloch,\n De_Nittis-Gomi:Quaternionic_Bloch, Kennedy-Zirnbauer:Bott_gapped}).\nThe K\\nobreakdash-theory of the \\(d\\)\\nobreakdash-torus\nis easily computed. Once \\(d\\ge2\\),\nmany of the topological phases that are predicted this way are\nobtained by stacking a lower-dimensional topological insulator in some\ndirection. Such topological phases are called ``weak'' by\nFu--Kane--Mele~\\cite{Fu-Kane-Mele:Insulators}. They claim that\nweak topological phases are not robust under disorder.\n\nOther authors have claimed instead that weak topological insulators\nare also quite robust, see~\\cite{Ringel-Kraus-Stern:Strong_side}.\nTheir proof of robustness, however, is no longer topological.\nRoughly speaking, the idea is that, although disorder may destroy\nthe topological phase, it must be rather special to do this.\n\\emph{Random} disorder will rarely be so special. So in a finite\nvolume approximation, the topological phase will remain intact in\nmost places, and the small area where the randomness destroys it\nwill become negligible in the limit of infinite volume. Such an\nargument may also work for the Hamiltonian of an insulator that is\nhomotopic to a trivial one. Our study is purely topological in\nnature and thus cannot see such phenomena.\n\nWe are going to explain the difference between strong and weak\ntopological phases and the robustness of the former through a\ndifference in the underlying observable algebras. Namely, we shall\nmodel a material with disorder by the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof \\(\\mathbb{R}^d\\)\nor~\\(\\mathbb{Z}^d\\),\nwhich is a central object of coarse geometry.\nRoe~\\cites{Roe:Index_open_I, Roe:Index_open_II} introduced them\nto get index theorems for elliptic operators on non-compact\nRiemannian manifolds.\n\nBefore choosing our observable algebra, we should ask: What is causing\ntopological phases? At first sight, the answer seems to be the\ntranslation invariance of the Hamiltonian. Translation-invariance\nalone is not enough, however. And it is destroyed by disorder. The\nsubalgebra of translation-invariant operators on the Hilbert space\n\\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\)\nis the algebra of \\(N\\times N\\)-matrices\nover the group von Neumann algebra of~\\(\\mathbb{Z}^d\\),\nwhich is isomorphic to \\(L^\\infty(\\mathbb{T}^d, \\mathbb M_N)\\).\nIf topological phases were caused by translation invariance alone,\nthey should be governed by the K\\nobreakdash-theory of \\(L^\\infty(\\mathbb{T}^d, \\mathbb M_N)\\).\nThis is clearly not the case. Instead, we need the group\n\\(\\Cst\\)\\nobreakdash-algebra,\nwhich is isomorphic to \\(\\Cont(\\mathbb{T}^d,\\mathbb M_N)\\).\nThe reason why the spectral projections of the Hamiltonian belong to\nthe group \\(\\Cst\\)\\nobreakdash-algebra\ninstead of the group von Neumann algebra is that the matrix\ncoefficients of the Hamiltonian for \\((x,y)\\in\\mathbb{Z}^d\\)\nare supported in the region \\(\\norm{x-y}\\le R\\)\nfor some \\(R>0\\);\nlet us call such operators \\emph{controlled}. The controlled\noperators do not form a \\(\\Cst\\)\\nobreakdash-algebra,\nand it makes no difference for \\(\\K\\)\\nobreakdash-theory purposes to allow\nthe Hamiltonian to be a limit of controlled operators in the norm\ntopology. We shall see below that this is equivalent to continuity\nwith respect to the action of~\\(\\mathbb{R}^d\\)\non \\(\\mathbb B(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N))\\)\ngenerated by the position observables. This action restricts to\nthe translation action of~\\(\\mathbb{R}^d\\)\non the group von Neumann algebra \\(L^\\infty(\\mathbb{T}^d, \\mathbb M_N)\\),\nso that its continuous elements are the functions in\n\\(\\Cont(\\mathbb{T}^d,\\mathbb M_N)\\).\nHence \\(\\Cont(\\mathbb{T}^d,\\mathbb M_N) \\subseteq \\mathbb B(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N))\\)\nconsists of those operators that are both translation-invariant and\nnorm limits of controlled operators. \n\nSince disorder destroys translation invariance, we should drop this\nassumption to model systems with disorder. The \\(\\Cst\\)\\nobreakdash-algebra\nof all operators on \\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\)\nthat are norm limits of controlled operators is the algebra\nof \\(N\\times N\\)-matrices\nover the \\emph{uniform Roe \\(\\Cst\\)\\nobreakdash-algebra}\nof~\\(\\mathbb{Z}^d\\).\nTo get the Roe \\(\\Cst\\)\\nobreakdash-algebra,\nwe must work in \\(\\ell^2(\\mathbb{Z}^d,\\Hils)\\)\nfor a separable Hilbert space~\\(\\Hils\\)\nand add a local compactness property, namely, that the operators\n\\(\\braketop{x}{(H-\\lambda)^{-1}}{y} \\in\\mathbb B(\\Hils)\\)\nare compact for all \\(x,y\\in\\mathbb{Z}^d\\),\n\\(\\lambda\\in \\mathbb{C}\\setminus\\mathbb{R}\\).\nThis property is automatic for operators on \\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\).\nWorking on \\(\\ell^2(\\mathbb{Z}^d,\\Hils)\\)\nand assuming local compactness means that we include infinitely many\nbands in our model and require only finitely many states with finite\nenergy in each finite volume.\nKubota~\\cite{Kubota:Controlled_bulk-edge} has already used the\nuniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nand the Roe \\(\\Cst\\)\\nobreakdash-algebra\nin the context of topological insulators. He prefers the uniform\nRoe \\(\\Cst\\)\\nobreakdash-algebra. We explain why we consider this a\nmistake.\n\nWorking in the Hilbert space \\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\)\nalready involves an approximation. We ought to work in a continuum\nmodel, that is, in the Hilbert space \\(L^2(\\mathbb{R}^d,\\mathbb{C}^k)\\),\nwhere~\\(k\\)\nis the number of internal degrees of freedom.\nThis Hilbert space is isomorphic to\n\\[\nL^2(\\mathbb{R}^d,\\mathbb{C}^k)\n\\cong L^2(\\mathbb{Z}^d\\times (0,1]^d)\\otimes \\mathbb{C}^k\n\\cong \\ell^2(\\mathbb{Z}^d,\\Hils\\otimes \\mathbb{C}^k)\n\\]\nwhen we cover~\\(\\mathbb{R}^d\\)\nby the disjoint translates of the fundamental domain~\\((0,1]^d\\).\nThis identification preserves both controlled and locally compact\noperators. Thus the Roe \\(\\Cst\\)\\nobreakdash-algebras\nof \\(\\mathbb{Z}^d\\)\nand~\\(\\mathbb{R}^d\\)\nare isomorphic. For the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{R}^d\\),\nit makes no difference to replace \\(L^2(\\mathbb{R}^d)\\)\nby \\(L^2(\\mathbb{R}^d,\\mathbb{C}^k)\\)\nor \\(L^2(\\mathbb{R}^d,\\Hils)\\):\nall these Hilbert spaces give isomorphic \\(\\Cst\\)\\nobreakdash-algebras\nof locally compact, approximately controlled operators. So\nthere is only one Roe \\(\\Cst\\)\\nobreakdash-algebra\nfor~\\(\\mathbb{R}^d\\),\nand it is isomorphic to the non-uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\).\nWe view the appearance of the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nfor~\\(\\mathbb{Z}^d\\)\nas an artefact of simplifying assumptions in tight binding models.\n\nWe describe some interesting elements of the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{R}^d\\)\nin Example~\\ref{exa:Roe_Rn}. In particular, it contains all\n\\(\\Cont_0\\)\\nobreakdash-functions of the impulse operator~\\(P\\)\non \\(L^2(\\mathbb{R}^d)\\) or, equivalently,\n\\begin{equation}\n \\label{eq:f_of_P}\n \\int_{\\mathbb{R}^d} f(x) \\exp(\\ima x P) \\,\\diff x \n\\end{equation}\nfor \\(f\\in \\Cst(\\mathbb{R}^d)\\);\nthis operator is controlled if and only if~\\(f\\)\nhas compact support. If \\(V\\in L^\\infty(\\mathbb{R}^d)\\),\nthen the operator of multiplication by~\\(V\\) on~\\(L^2(\\mathbb{R}^d)\\)\nis controlled, but not locally compact. Its product with an\noperator as in~\\eqref{eq:f_of_P} belongs to the Roe\n\\(\\Cst\\)\\nobreakdash-algebra.\n\nThe real and complex K\\nobreakdash-theory of the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nis well known: up to a dimension shift of~\\(d\\),\nit is the \\(\\K\\)\\nobreakdash-theory\nof \\(\\mathbb{R}\\)\nor~\\(\\mathbb{C}\\),\nrespectively. In particular, the Roe \\(\\Cst\\)\\nobreakdash-algebra\nas an observable algebra is small enough to predict some distinct\ntopological phases. These coincide with Kitaev's periodic\ntable~\\cite{Kitaev:Periodic_table}. This corroborates the choice of\nthe Roe \\(\\Cst\\)\\nobreakdash-algebra\nas the observable algebra for disordered materials.\n\nWhen we disregard disorder, the Roe \\(\\Cst\\)\\nobreakdash-algebra\nmay be replaced by its translation-invariant subalgebra, which is\nisomorphic to\n\\[\n\\Cst(\\mathbb{Z}^d) \\otimes \\mathbb K(\\Hils) \\cong \\Cont(\\mathbb{T}^d,\\mathbb K(\\Hils)),\n\\]\nwhere \\(\\mathbb K(\\Hils)\\)\ndenotes the \\(\\Cst\\)\\nobreakdash-algebra\nof compact operators on an infinite-dimensional separable Hilbert\nspace~\\(\\Hils\\).\nIn the real case, the \\(d\\)\\nobreakdash-torus\nmust be given the real involution by the restriction of complex\nconjugation on \\(\\mathbb{C}^d\\supseteq\\mathbb{T}^d\\).\nThe real or complex \\(\\K\\)\\nobreakdash-theory\ngroups of the ``real'' \\(d\\)\\nobreakdash-torus describe both weak and strong\ntopological phases in the presence of different types of symmetries.\nWe show that the map\n\\begin{equation}\n \\label{eq:comparison_group_Roe}\n \\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}) \\to \\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\n\\end{equation}\nfor \\(\\mathbb{F} = \\mathbb{R}\\)\nor \\(\\mathbb{F} = \\mathbb{C}\\)\nis split surjective and that its kernel is the subgroup generated by\nthe images of \\(\\K_*(\\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F})\\)\nfor all coordinate embeddings \\(\\mathbb{Z}^{d-1} \\to \\mathbb{Z}^d\\).\nThat is, the kernel of the map in~\\eqref{eq:comparison_group_Roe}\nconsists exactly of the \\(\\K\\)\\nobreakdash-theory\nclasses of weak topological insulators as defined by\nFu--Kane--Mele~\\cite{Fu-Kane-Mele:Insulators}. The strong topological\ninsulators are those that remain topologically protected even if the\nobservable algebra is enlarged to the Roe \\(\\Cst\\)\\nobreakdash-algebra,\nallowing rather general disorder.\n\nFollowing Bellissard \\cites{Bellissard:K-theory_solid,\n Bellissard-Elst-Schulz-Baldes:Noncommutative_Hall}, disorder is\nusually modelled by crossed product \\(\\Cst\\)\\nobreakdash-algebras\n\\(\\mathcal{A} = \\Cont(\\Omega)\\rtimes\\mathbb{Z}^d\\),\nwhere~\\(\\Omega\\)\nis the space of disorder configurations. It is more precise to say,\nhowever, that the space~\\(\\Omega\\)\ndescribes \\emph{restricted} disorder. Uncountably many different\nchoices are possible. Such models are only reasonable when the\nphysically relevant objects do not depend on the choice. But the\n\\(\\K\\)\\nobreakdash-theory\nof the crossed product depends on the topology of the\nspace~\\(\\Omega\\).\nTo make it independent of~\\(\\Omega\\),\nthe space~\\(\\Omega\\)\nis assumed to be contractible\nin~\\cite{Prodan-Schulz-Baldes:Bulk_boundary}.\nThis fits well with standard choices of~\\(\\Omega\\)\nsuch as a product space \\(\\prod_{n\\in\\mathbb{Z}^d} [-1,1]\\)\nto model a random potential. The resulting \\(\\K\\)\\nobreakdash-theory\nthen becomes the same as in the system without disorder. So another\nargument must be used to explain the difference between weak and\nstrong topological phases, compare\n\\cite{Prodan-Schulz-Baldes:Bulk_boundary}*{Remark 5.3.5}.\nIf one allows non-metrisable~\\(\\Omega\\), then\nthere is a maximal choice for~\\(\\Omega\\), namely,\nthe Stone--\\v{C}ech compactification of~\\(\\mathbb{Z}^d\\).\nThe resulting crossed product \\(\\ell^\\infty(\\mathbb{Z}^d)\\rtimes\\mathbb{Z}^d\\)\nis isomorphic to the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\),\nsee~\\cite{Kubota:Controlled_bulk-edge}.\nNevertheless, even this maximal choice of~\\(\\Omega\\) still contains\na hidden restriction on disorder: the number of bands for a tight\nbinding model is fixed, and so the disorder is also limited to a\nfixed finite number of bands. The Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nalso removes this hidden restriction on the allowed disorder.\nIt is also a crossed product, namely,\n\\[\n\\Cst_\\Roe(\\mathbb{Z}^d) \\cong \\ell^\\infty(\\mathbb{Z}^d,\\mathbb K(\\Hils)) \\rtimes \\mathbb{Z}^d.\n\\]\nThe \\(\\Cst\\)\\nobreakdash-algebra \\(\\ell^\\infty(\\mathbb{Z}^d,\\mathbb K(\\Hils))\\)\nis not isomorphic to \\(\\ell^\\infty(\\mathbb{Z}^d) \\otimes \\mathbb K(\\Hils)\\):\nit even has different \\(\\K\\)\\nobreakdash-theory.\n\nSince the Roe \\(\\Cst\\)\\nobreakdash-algebra\nhas not been used much in the context of topological insulators, we\nrecall its main properties in Section~\\ref{sec:Roe_Cstar}. We\nhighlight its robustness or even ``universality.'' Roughly\nspeaking, there is only one Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nin each dimension, which describes all kinds of disordered materials\nin that dimension without symmetries. The various symmetries\n(time-reversal, particle-hole, chiral) may be added by tensoring the\nreal or complex Roe \\(\\Cst\\)\\nobreakdash-algebra\nwith Clifford algebras, which replaces \\(\\K_0\\)\nby~\\(\\K_i\\)\nfor some \\(i\\in\\mathbb{Z}\\).\nWe shall not say much about this here. The Roe \\(\\Cst\\)\\nobreakdash-algebra\nof a coarse space is a coarse invariant. In particular, all coarsely\ndense subsets in~\\(\\mathbb{R}^d\\)\ngive isomorphic Roe \\(\\Cst\\)\\nobreakdash-algebras.\nFurthermore, the Roe \\(\\Cst\\)\\nobreakdash-algebras\nof~\\(\\mathbb{Z}^d\\)\nand other coarsely dense subsets of~\\(\\mathbb{R}^d\\)\nare isomorphic to that of~\\(\\mathbb{R}^d\\).\nThus it makes no difference whether we work in a continuum or lattice\nmodel. We also consider the twists of the Roe \\(\\Cst\\)\\nobreakdash-algebra\ndefined by magnetic fields. The resulting twisted Roe\n\\(\\Cst\\)\\nobreakdash-algebras\nare also isomorphic to the untwisted one. This robustness of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nmeans that the same \\emph{strong} topological phases occur for all\nmaterials of a given dimension and symmetry type, even for\nquasi-crystals and aperiodic materials.\n\nWe compute the \\(\\K\\)\\nobreakdash-theory\nof the Roe \\(\\Cst\\)\\nobreakdash-algebra\nin Section~\\ref{sec:coarse_MV} using the coarse Mayer--Vietoris\nprinciple introduced in~\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris}.\nWe prove the Mayer--Vietoris exact sequence in the real and complex\ncase by reducing it to the \\(\\K\\)\\nobreakdash-theory\nexact sequence for \\(\\Cst\\)\\nobreakdash-algebra\nextensions. The computation of \\(\\K_*(\\Cst_\\Roe(X)_\\mathbb{F})\\)\nfor \\(X=\\mathbb{Z}^d\\)\nis based on the vanishing of this invariant for half-spaces\n\\(\\mathbb{N}\\times\\mathbb{Z}^{d-1}\\).\nThis implies\n\\(\\K_{*+d}(\\Cst_\\Roe(X)_\\mathbb{F}) \\cong \\K_*(\\mathbb{F})\\).\nIt shows also that the inclusion\n\\(\\Cst_\\Roe(\\mathbb{Z}^{d-1})_\\mathbb{F} \\to \\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\\)\ninduces the zero map on \\(\\K\\)\\nobreakdash-theory.\nHence the map~\\eqref{eq:comparison_group_Roe} kills all\nelements of \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nthat come from inclusions \\(\\mathbb{Z}^{d-1} \\hookrightarrow \\mathbb{Z}^d\\).\n\nIn Section~\\ref{sec:weak_phases} we compute the real and complex\n\\(\\K\\)\\nobreakdash-theory for the group \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nand the map in~\\eqref{eq:comparison_group_Roe}. More precisely, we\ndescribe the composite of the map in~\\eqref{eq:comparison_group_Roe}\nwith the isomorphism\n\\(\\K_{*+d}(\\Cst_\\Roe(X)_\\mathbb{F}) \\cong \\K_*(\\mathbb{F})\\):\nthis is the pairing with the fundamental class of the ``real''\n\\(d\\)\\nobreakdash-torus~\\(\\mathbb{T}^d\\).\nExcept for an adaptation to ``real'' manifolds, this fundamental\nclass is introduced in~\\cite{Kasparov:Novikov}. We show that the\nfundamental class extends to a \\(\\K\\)\\nobreakdash-homology\nclass on the Roe \\(\\Cst\\)\\nobreakdash-algebra\nand that the pairing with this \\(\\K\\)\\nobreakdash-homology\nclass is an isomorphism\n\\(\\K_{*+d}(\\Cst_\\Roe(X)_\\mathbb{F}) \\cong \\K_*(\\mathbb{F})\\).\n\n\n\\section{Roe \\texorpdfstring{$\\Cst$}{C*}-algebras}\n\\label{sec:Roe_Cstar}\n\nIn this section, we define the real and complex Roe\n\\(\\Cst\\)\\nobreakdash-algebras\nof a proper metric space and prove that they are invariant under\npassing to a coarsely dense subspace and, more generally, under coarse\nequivalence. We prove that the twists used to encode magnetic fields\ndo not change them. And we describe elements of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nof a subset of~\\(\\mathbb{R}^d\\)\nas those locally compact operators that are continuous for the\nrepresentation of~\\(\\mathbb{R}^d\\)\ngenerated by the position operators. We describe the subalgebras of\nsmooth, real-analytic and holomorphic elements of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nfor this action of~\\(\\mathbb{R}^d\\).\nWe show that Roe \\(\\Cst\\)\\nobreakdash-algebras\nhave approximate units of projections, which simplifies the definition\nof their \\(\\K\\)\\nobreakdash-theory.\n\nLet \\((X,d)\\)\nbe a locally compact, second countable, metric space. We assume the\nmetric~\\(d\\)\nto be \\emph{proper}, that is, bounded subsets of~\\(X\\)\nare compact. We shall be mainly interested in \\(\\mathbb{R}^d\\)\nor a discrete subset of~\\(\\mathbb{R}^d\\)\nwith the restriction of the Euclidean metric. (All our results on\ngeneral proper metric spaces extend easily to the more general coarse\nspaces introduced in~\\cite{Roe:Lectures}.) Let~\\(\\Hils\\)\nbe a real or complex separable Hilbert space and let\n\\(\\varrho\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils)\\)\nbe a nondegenerate representation. We are going to define the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nof~\\(X\\)\nwith respect to~\\(\\varrho\\),\nsee also \\cite{Higson-Roe:Analytic_K}*{Section 6.3}. Depending on\nwhether~\\(\\Hils\\)\nis a real or complex Hilbert space, this gives a real or complex\nversion of the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nBoth cases are completely analogous.\n\nLet \\(T\\in\\mathbb B(\\Hils)\\).\nWe call~\\(T\\)\n\\emph{locally compact} (on~\\(X\\))\nif the operators \\(\\varrho(f)T\\)\nand~\\(T\\varrho(f)\\)\nare compact for all \\(f\\in\\Cont_0(X)\\).\nThe \\emph{support} of~\\(T\\)\nis a subset \\(\\supp T\\subseteq X\\times X\\).\nIts complement consists of all \\((x,y)\\in X\\times X\\)\nfor which there are neighbourhoods \\(U_x\\),\n\\(U_y\\)\nin~\\(X\\)\nsuch that \\(\\varrho(f) T \\varrho(g)=0\\)\nfor all \\(f\\in\\Cont_0(U_x)\\),\n\\(g\\in\\Cont_0(U_y)\\).\nThe operator~\\(T\\)\nis \\emph{controlled} (or has \\emph{finite propagation}) if there is\n\\(R>0\\)\nsuch that \\(d(x,y)\\le R\\)\nfor all \\((x,y)\\in \\supp T\\).\nWe sometimes write ``\\(R\\)\\nobreakdash-controlled'' to highlight the control\nparameter~\\(R\\).\nThe locally compact, controlled operators on~\\(\\Hils\\)\nform a $^*$\\nobreakdash-{}algebra. Its closure in \\(\\mathbb B(\\Hils)\\)\nis the \\emph{Roe \\(\\Cst\\)\\nobreakdash-algebra} \\(\\Cst_\\Roe(X,\\varrho)\\).\n\nThe representation~\\(\\varrho\\)\nis called \\emph{ample} if the operator \\(\\varrho(f)\\)\nfor \\(f\\in\\Cont_0(X)\\)\nis only compact for \\(f=0\\).\n\n\\begin{theorem}\n \\label{the:Roe_ample_unique}\n Let \\(\\varrho_i\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils_i)\\)\n for \\(i=1,2\\)\n be ample representations, where \\(\\Hils_1\\)\n and~\\(\\Hils_2\\)\n are both complex or both real. Then\n \\(\\Cst_\\Roe(X,\\varrho_1) \\cong \\Cst_\\Roe(X,\\varrho_2)\\).\n Even more, there is a unitary operator\n \\(U\\colon \\Hils_1 \\xrightarrow\\sim \\Hils_2\\)\n with\n \\[\n U \\Cst_\\Roe(X,\\varrho_1) U^* = \\Cst_\\Roe(X,\\varrho_2).\n \\]\n\\end{theorem}\n\nMany references only assert the weaker statement that the Roe\n\\(\\Cst\\)\\nobreakdash-algebras\nfor all ample representations have canonically isomorphic\n\\(\\K\\)\\nobreakdash-theory,\ncompare \\cite{Higson-Roe:Analytic_K}*{Corollary 6.3.13}. The\nstatement above is\n\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris}*{Lemma~2}, and our proof is\nthe same.\n\n\\begin{proof}\n If~\\(X\\)\n is compact, then \\(\\Cst_\\Roe(X,\\varrho_i) = \\mathbb K(\\Hils_i)\\).\n Since~\\(\\Hils_i\\)\n for \\(i=1,2\\)\n are assumed to be separable, there is a unitary\n \\(U\\colon \\Hils_1 \\xrightarrow\\sim \\Hils_2\\),\n and it will do the job. So we may assume~\\(X\\)\n to be non-compact. Fix \\(R>0\\).\n The open balls \\(B(x,R)\\)\n for \\(x\\in X\\)\n cover~\\(X\\).\n Since~\\(X\\)\n is second countable, there is a subordinate countable, locally\n finite, open covering \\(X= \\bigcup_{n\\in\\mathbb{N}} U_n\\),\n where each~\\(U_n\\)\n is non-empty and has diameter at most~\\(R\\).\n Then there is a countable covering of~\\(X\\)\n by disjoint Borel sets, \\(X=\\bigsqcup_{n\\in\\mathbb{N}} B'_n\\),\n where each~\\(B'_n\\)\n has diameter at most~\\(R\\),\n and such that any relatively compact subset is already covered by\n finitely many of the~\\(B'_n\\):\n simply take \\(B'_n \\mathrel{\\vcentcolon=} U_n \\setminus \\bigcup_{j< n} U_j\\).\n Next we modify the subsets~\\(B'_n\\)\n so that they all have non-empty interior. Let \\(M\\subseteq\\mathbb{N}\\)\n be the set of all \\(n\\in\\mathbb{N}\\)\n for which~\\(B'_n\\)\n has non-empty interior. Let \\(m\\in M\\).\n Let~\\(K_m\\)\n be the set of all \\(k\\in\\mathbb{N}\\)\n for which~\\(B'_k\\)\n has empty interior and \\(U_m\\cap U_k\\neq \\emptyset\\).\n The set~\\(K_m\\)\n is finite because~\\(U_m\\)\n is bounded and hence relatively compact. Let\n \\(K_m^\\circ \\mathrel{\\vcentcolon=} K_m \\setminus \\bigcup_{i\\in M, ik_1>\\dotsc>k_\\ell\\)\n such that \\(k_0,\\dotsc,k_{\\ell-1}\\in \\mathbb{N}\\setminus M\\) and\n \\[\n U_{k_0} \\cap U_{k_1} \\cap \\dotsb \\cap U_{k_\\ell}\n \\neq \\emptyset.\n \\]\n We eventually reach \\(k_\\ell\\in M\\)\n because \\(B'_1= U_1\\)\n is open and so \\(1\\in M\\).\n We have \\(k_0\\in K_{k_\\ell}\\).\n So \\(\\bigcup_{m\\in M} K_m = \\mathbb{N}\\setminus M\\) as asserted.\n\n We have built a covering of~\\(X\\)\n by disjoint Borel sets \\(X = \\bigsqcup_{m\\in M} B_m\\)\n of diameter at most~\\(3 R\\),\n with non-empty interiors, and such that any relatively compact\n subset is already covered by finitely many of the~\\(B_m\\).\n The set~\\(M\\)\n is at most countable, and it cannot be finite because then~\\(X\\)\n would be bounded and hence compact.\n\n Using the Borel functional calculus for the\n representation~\\(\\varrho_i\\),\n we may decompose the Hilbert space~\\(\\Hils_i\\)\n as an orthogonal direct sum,\n \\(\\Hils_i = \\bigoplus_{m\\in M} \\Hils_{i,m}\\),\n where~\\(\\Hils_{i,m}\\)\n is the image of the projection \\(\\varrho_i(1_{B_m})\\).\n Since each~\\(B_m\\)\n has non-empty interior and our representations are ample, there is a\n non-compact operator on each~\\(\\Hils_{i,m}\\).\n So no~\\(\\Hils_{i,m}\\)\n has finite dimension. Hence there is a unitary\n \\(U_m\\colon \\Hils_{1,m}\\xrightarrow\\sim\\Hils_{2,m}\\)\n for each \\(m\\in M\\).\n We combine these into a unitary operator\n \\(U= \\bigoplus_{m\\in M} U_m\\colon \\Hils_1 \\xrightarrow\\sim \\Hils_2\\).\n\n Let \\(T\\in\\mathbb B(\\Hils_1)\\).\n We claim that~\\(U T U^*\\)\n is locally compact or controlled if and only if~\\(T\\)\n is. This implies\n \\(U \\Cst_\\Roe(X,\\varrho_1) U^* = \\Cst_\\Roe(X,\\varrho_2)\\)\n as asserted. First, we claim that~\\(T\\)\n is locally compact if and only if \\(T \\varrho_1(1_{B_m})\\)\n and \\(\\varrho_1(1_{B_m}) T\\)\n are compact for all \\(m\\in M\\).\n In one direction, this uses that there is \\(g\\in \\Cont_0(X)\\)\n with \\(1_{B_m} \\le g\\)\n because~\\(B_m\\)\n has finite diameter. In the other direction, it uses that any\n relatively compact subset of~\\(X\\)\n is already covered by finitely many~\\(B_m\\).\n Since \\(U(\\Hils_{1,m}) = \\Hils_{2,m}\\),\n the criterion above shows that~\\(T\\)\n is locally compact if and only if \\(U T U^*\\)\n is so. Since the diameter of~\\(B_m\\)\n is at most~\\(3 R\\),\n the operator~\\(U_m\\),\n viewed as a partial isometry on \\(\\Hils_1 \\oplus \\Hils_2\\),\n is \\(3 R\\)\\nobreakdash-controlled.\n Thus~\\(U\\)\n is also \\(3 R\\)\\nobreakdash-controlled.\n So \\(U T U^*\\) is controlled if and only if~\\(T\\) is.\n\\end{proof}\n\n\\begin{corollary}\n \\label{cor:Roe_matrix-stable}\n Let~\\(\\varrho\\)\n be an ample representation and let \\(m\\in\\mathbb{N}_{\\ge2}\\). Then\n \\[\n \\Cst_\\Roe(X,\\varrho) \\cong \\mathbb M_m(\\Cst_\\Roe(X,\\varrho)).\n \\]\n\\end{corollary}\n\n\\begin{proof}\n The direct sum representation \\(m\\cdot \\varrho\\)\n is still ample. So\n \\(\\Cst_\\Roe(X,m\\cdot \\varrho) \\cong \\Cst_\\Roe(X,\\varrho)\\).\n An operator on~\\(\\Hils^m\\)\n is locally compact or controlled if and only if its block\n matrix entries in \\(\\mathbb B(\\Hils)\\)\n are so. Thus\n \\(\\Cst_\\Roe(X,m\\cdot \\varrho) = \\mathbb M_m(\\Cst_\\Roe(X,\\varrho))\\).\n\\end{proof}\n\nThe stabilisation \\(\\Cst_\\Roe(X,\\varrho) \\otimes \\mathbb K(\\ell^2\\mathbb{N})\\),\nhowever, is usually not isomorphic to \\(\\Cst_\\Roe(X,\\varrho)\\).\n\n\\begin{example}\n \\label{exa:Roe_discrete}\n Let~\\(X\\)\n be discrete, for instance, \\(X=\\mathbb{Z}^d\\).\n The representation~\\(\\varrho\\)\n of \\(\\Cont_0(X)\\)\n on~\\(\\ell^2(X)\\)\n by multiplication operators is not ample. It defines the\n \\emph{uniform Roe \\(\\Cst\\)\\nobreakdash-algebra}\n of~\\(X\\).\n To get the Roe \\(\\Cst\\)\\nobreakdash-algebra,\n we may take the representation of \\(\\Cont_0(X)\\)\n on \\(\\ell^2(X)\\otimes \\ell^2(\\mathbb{N})\\).\n\n An operator~\\(T\\)\n on \\(\\ell^2(X)\\otimes \\ell^2(\\mathbb{N})\\)\n is determined by its matrix coefficients\n \\(T_{x,y} = \\braketop{x}{T}{y} \\in \\mathbb B(\\ell^2(\\mathbb{N}))\\)\n for \\(x,y\\in X\\).\n It is locally compact if and only if all~\\(T_{x,y}\\)\n are compact. Its support is the set of all \\((x,y)\\in X^2\\)\n with \\(T_{x,y}\\neq0\\).\n So it is controlled if and only if there is \\(R>0\\)\n so that \\(T_{x,y}=0\\)\n for \\(d(x,y)>R\\).\n The Roe \\(\\Cst\\)\\nobreakdash-algebra is the norm closure of these operators.\n\n If~\\(X\\)\n is a discrete group equipped with a translation-invariant metric,\n then \\(\\Cst_\\Roe(X)\\)\n is isomorphic to the reduced crossed product for the translation\n action of~\\(X\\)\n on~\\(\\ell^\\infty(X, \\mathbb K(\\ell^2\\mathbb{N}))\\)\n (compare \\cite{Roe:Lectures}*{Theorem~4.28} for the uniform Roe\n \\(\\Cst\\)\\nobreakdash-algebra).\n\\end{example}\n\n\\begin{example}\n \\label{exa:Roe_Rn}\n Let \\(X=\\mathbb{R}^d\\).\n The representation~\\(\\varrho\\)\n of \\(\\Cont_0(\\mathbb{R}^d)\\)\n on \\(L^2(\\mathbb{R}^d,\\diff x)\\)\n (real or complex) by multiplication operators is ample. Actually,\n all faithful representations of~\\(\\Cont_0(\\mathbb{R}^d)\\)\n are ample. So they all give isomorphic Roe \\(\\Cst\\)\\nobreakdash-algebras\n by Theorem~\\ref{the:Roe_ample_unique}.\n\n Let \\(T\\in\\Contc(\\mathbb{R}^d)\\)\n (with real or complex values) act on~\\(L^2(\\mathbb{R}^d)\\)\n by convolution. Then \\(T\\cdot \\varrho(f)\\)\n and \\(\\varrho(f)\\cdot T\\)\n are compact because they have a compactly supported, continuous\n integral kernel. And \\(T\\)\n is controlled by the supremum of~\\(\\norm{x}\\)\n with \\(T(x)\\neq0\\).\n So \\(T\\in\\Cst_\\Roe(\\mathbb{R}^d)\\).\n Hence \\(\\Cst(\\mathbb{R}^d) \\subseteq \\Cst_\\Roe(\\mathbb{R}^d)\\).\n In particular, the resolvent of the Laplace operator or another\n translation-invariant elliptic differential operator on~\\(\\mathbb{R}^d\\)\n belongs to\n \\(\\Cst_\\Roe(\\mathbb{R}^d)\\).\n Any multiplication operator is controlled. Thus\n multiplication operators are multipliers of \\(\\Cst_\\Roe(\\mathbb{R}^d)\\).\n And \\(L^\\infty(\\mathbb{R}^d) \\cdot \\Cst(\\mathbb{R}^d) \\cdot L^\\infty(\\mathbb{R}^d)\\)\n is contained in \\(\\Cst_\\Roe(\\mathbb{R}^d)\\).\n (Since the translation action of~\\(\\mathbb{R}^d\\)\n on \\(L^\\infty(\\mathbb{R}^d)\\)\n is not continuous, there is no crossed product for this action and\n it is unclear whether the closed linear spans of\n \\(L^\\infty(\\mathbb{R}^d) \\cdot \\Cst(\\mathbb{R}^d)\\)\n and \\(\\Cst(\\mathbb{R}^d) \\cdot L^\\infty(\\mathbb{R}^d)\\)\n are equal and form a \\(\\Cst\\)\\nobreakdash-algebra.)\n\\end{example}\n\n\\begin{proposition}\n \\label{pro:resolvent_in_Roe}\n Let \\(V\\in L^\\infty(\\mathbb{R}^n)\\)\n and let~\\(\\Delta\\)\n be the Laplace operator on~\\(\\mathbb{R}^d\\).\n Then the resolvent of \\(V+\\Delta\\)\n belongs to the Roe \\(\\Cst\\)\\nobreakdash-algebra of~\\(\\mathbb{R}^d\\).\n\\end{proposition}\n\nWe are indebted to Detlev Buchholz for pointing out the following\nsimple proof.\n\n\\begin{proof}\n View \\(V=V(Q)\\)\n as an operator on \\(L^2(\\mathbb{R}^d)\\).\n Then \\(\\norm{(\\ima c + \\Delta)^{-1} V}^2<1\\)\n for sufficiently large \\(c\\in\\mathbb{R}_{>0}\\).\n Hence the Neumann series \\(\\sum (-(\\ima c + \\Delta)^{-1} V)^n\\)\n converges, and\n \\[\n \\sum_{n=0}^\\infty (-(\\ima c + \\Delta)^{-1} V)^n\n \\cdot (\\ima c + \\Delta)^{-1}\n = (1+(\\ima c + \\Delta)^{-1} V)^{-1} \\cdot (\\ima c + \\Delta)^{-1}\n = (\\ima c + \\Delta+V)^{-1}.\n \\]\n We have already seen that \\((\\ima c + \\Delta)^{-1}\\)\n and \\((\\ima c + \\Delta)^{-1} V\\)\n belong to the Roe \\(\\Cst\\)\\nobreakdash-algebra.\n Hence so does \\((\\ima c + \\Delta+V)^{-1}\\).\n\\end{proof}\n\nIf~\\(\\varrho\\)\nis ample, then we often leave out~\\(\\varrho\\)\nand briefly write \\(\\Cst_\\Roe(X)_\\mathbb{R}\\)\nor \\(\\Cst_\\Roe(X)_\\mathbb{C}\\),\ndepending on whether~\\(\\varrho\\)\nacts on a real or complex Hilbert space.\nTheorem~\\ref{the:Roe_ample_unique} justifies this.\n\n\\begin{definition}\n \\label{def:coarsely_dense}\n A closed subset \\(Y\\subseteq X\\)\n is \\emph{coarsely dense} if there is \\(R>0\\)\n such that for any \\(x\\in X\\) there is \\(y\\in Y\\) with \\(d(x,y)\\le R\\).\n\\end{definition}\n\n\\begin{theorem}\n \\label{the:coarsely_dense}\n Let \\(Y\\subseteq X\\)\n be coarsely dense. Then \\(\\Cst_\\Roe(Y)_\\mathbb{R} \\cong \\Cst_\\Roe(X)_\\mathbb{R}\\)\n and \\(\\Cst_\\Roe(Y)_\\mathbb{C} \\cong \\Cst_\\Roe(X)_\\mathbb{C}\\).\n Both isomorphisms are implemented by unitaries between the\n underlying Hilbert spaces.\n\\end{theorem}\n\n\\begin{proof}\n The proofs in the complex and real case are identical. Let\n \\(\\pi\\colon \\Cont_0(X) \\to \\Cont_0(Y)\\)\n be the restriction homomorphism. Let\n \\(\\varrho_Y\\colon \\Cont_0(Y)\\to\\mathbb B(\\Hils_Y)\\)\n and \\(\\varrho_X\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils_X)\\)\n be ample representations. Then\n \\(\\varrho' \\mathrel{\\vcentcolon=} \\varrho_X \\oplus \\varrho_Y\\circ\\pi\\)\n is an ample representation of~\\(\\Cont_0(X)\\)\n on \\(\\Hils'\\mathrel{\\vcentcolon=} \\Hils_X \\oplus \\Hils_Y\\).\n By Theorem~\\ref{the:Roe_ample_unique}, we may use the particular\n representations \\(\\varrho'\\)\n and~\\(\\varrho_Y\\)\n to define \\(\\Cst_\\Roe(X)\\)\n and \\(\\Cst_\\Roe(Y)\\),\n because different ample representations give Roe\n \\(\\Cst\\)\\nobreakdash-algebras\n that are isomorphic through conjugation with a unitary between the\n underlying Hilbert spaces.\n\n Pick \\(R>0\\)\n such that for each \\(x\\in X\\)\n there is \\(y\\in Y\\)\n with \\(d(x,y) \\le R\\).\n We build a Borel map \\(g\\colon X\\to Y\\)\n with \\(g|_Y=\\Id_Y\\) and \\(d(g(x),x)\\le 2 R\\) for all \\(x\\in X\\),\n First, there is a countable cover \\(X=\\bigsqcup_{m\\in\\mathbb{N}} B_m\\)\n by non-empty, disjoint Borel sets of diameter at most~\\(R\\)\n as in the proof of Theorem~\\ref{the:Roe_ample_unique}. For each\n \\(m\\in\\mathbb{N}\\),\n pick \\(x_m\\in B_m\\)\n and \\(y_m\\in Y\\)\n with \\(d(x_m,y_m) \\le R\\).\n Define \\(g(x) \\mathrel{\\vcentcolon=} x\\)\n for \\(x\\in Y\\)\n and \\(g(x) \\mathrel{\\vcentcolon=} y_m\\)\n for \\(x\\in B_m\\setminus Y\\),\n \\(m\\in\\mathbb{N}\\). This map has all the required properties.\n\n The representation\n \\(\\varrho'\\circ g^*\\colon \\Cont_0(Y)\\to\\mathbb B(\\Hils')\\)\n is ample because it contains\n \\(\\varrho_Y\\circ \\pi\\circ g^* = \\varrho_Y\\)\n as a direct summand. An operator on~\\(\\Hils'\\)\n is locally compact or controlled for~\\(\\varrho'\\)\n if and only if it is so for \\(\\varrho'\\circ g^*\\).\n Thus \\(\\Cst_\\Roe(X,\\varrho') = \\Cst_\\Roe(Y,\\varrho'\\circ g^*)\\).\n\\end{proof}\n\nIn particular, Theorem~\\ref{the:coarsely_dense} shows that all\ncoarsely dense subsets of~\\(\\mathbb{R}^d\\)\nhave isomorphic Roe \\(\\Cst\\)\\nobreakdash-algebras.\nThis applies, in particular, to \\(\\mathbb{Z}^d\\)\nand to all Delone subsets of~\\(\\mathbb{R}^d\\).\nThe latter are often used to model the atomic configurations of\nmaterials that are not crystals (see, for instance,\n\\cite{Bellissard-Herrmann-Zarrouati:Hull}). So all kinds of materials\nlead to the same Roe \\(\\Cst\\)\\nobreakdash-algebra,\nwhich depends only on the dimension~\\(d\\).\n\nTheorem~\\ref{the:coarsely_dense} suffices for our purposes, but we\nmention that it extends to arbitrary coarse equivalences, see also\n\\cite{Higson-Roe:Analytic_K}*{Section 6.3}.\n\n\\begin{definition}\n \\label{def:coarse_maps}\n Let \\(X\\)\n and~\\(Y\\)\n be proper metric spaces as above. Two maps \\(f_0,f_1\\colon X\\to Y\\)\n are \\emph{close} if there is \\(R>0\\)\n so that \\(d(f_0(x),f_1(x))0\\)\n there is \\(S>0\\)\n such that \\(d(x,y)\\le R\\)\n for \\(x,y\\in X\\)\n implies \\(d(f(x),f(y))\\le S\\),\n and \\(f^{-1}(B)\\)\n is bounded in~\\(X\\)\n if \\(B\\subseteq Y\\)\n is bounded. A \\emph{coarse equivalence} is a coarse map\n \\(f\\colon X\\to Y\\)\n for which there is another coarse map \\(g\\colon Y\\to X\\),\n called the \\emph{coarse inverse} of~\\(f\\),\n such that \\(g\\circ f\\)\n and \\(f\\circ g\\)\n are close to the identity maps on \\(X\\)\n and~\\(Y\\),\n respectively. We call \\(X\\)\n and~\\(Y\\)\n \\emph{coarsely equivalent} if there is a coarse equivalence between\n them.\n\\end{definition}\n\nFor instance, the inclusion of a coarsely dense subspace is a coarse\nequivalence: the proof of Theorem~\\ref{the:coarsely_dense} builds a\ncoarse inverse for the inclusion map.\n\n\\begin{theorem}\n \\label{the:coarse_equivalence_Roe}\n Let \\(X\\)\n and~\\(Y\\)\n be coarsely equivalent. Then\n \\(\\Cst_\\Roe(X)_\\mathbb{R} \\cong \\Cst_\\Roe(Y)_\\mathbb{R}\\)\n and \\(\\Cst_\\Roe(X)_\\mathbb{C} \\cong \\Cst_\\Roe(Y)_\\mathbb{C}\\).\n\\end{theorem}\n\n\\begin{proof}\n Here it is more convenient to work with coarse spaces. Let\n \\(f\\colon X\\to Y\\)\n be the coarse equivalence. We claim that there is a coarse\n structure on the disjoint union \\(X\\sqcup Y\\)\n such that both \\(X\\)\n and~\\(Y\\)\n are coarsely dense in \\(X\\sqcup Y\\).\n This reduces the result to Theorem~\\ref{the:coarsely_dense}. We\n describe the desired coarse structure on \\(X\\sqcup Y\\).\n A subset~\\(E\\)\n of \\((X\\sqcup Y)^2\\)\n is called controlled if its intersections with \\(X^2\\)\n and~\\(Y^2\\)\n and the set of all \\((f(x),y) \\in Y^2\\)\n for \\((x,y)\\in E\\)\n or \\((y,x)\\in E\\)\n are controlled. This is a coarse structure on \\(X\\sqcup Y\\)\n because~\\(f\\)\n is a coarse equivalence. And the subspaces \\(X\\)\n and~\\(Y\\) are coarsely dense for the same reason.\n\\end{proof}\n\n\n\\subsection{Twists}\n\\label{sec:twists}\n\nWe show that magnetic twists do not change the isomorphism class of\nthe Roe \\(\\Cst\\)\\nobreakdash-algebra.\nWe let~\\(X\\)\nbe a \\emph{discrete} metric space. Let\n\\(\\varrho\\colon \\Cont_0(X)\\to \\mathbb B(\\Hils)\\)\nbe a representation. This is equivalent to a direct sum decomposition\n\\(\\Hils = \\bigoplus_{x\\in X} \\Hils_x\\),\nsuch that \\(f\\in \\Cont_0(X)\\)\nacts by multiplication with \\(f(x)\\)\non the summand~\\(\\Hils_x\\).\nWe assume for simplicity that each~\\(\\Hils_x\\)\nis non-zero. This is weaker than being ample, which means that\neach~\\(\\Hils_x\\)\nis infinite-dimensional. So the following discussion also covers the\nuniform Roe \\(\\Cst\\)\\nobreakdash-algebra of~\\(X\\).\n\nWe describe an operator on~\\(\\Hils\\)\nby a block matrix \\((T_{x,y})_{x,y\\in X}\\)\nwith \\(T_{x,y}\\in \\mathbb B(\\Hils_y,\\Hils_x)\\).\nThese are multiplied by the usual formula,\n\\((S T)_{x,y} = \\sum_{z\\in X} S_{x,z} T_{z,y}\\).\nWe twist this multiplication by a scalar-valued function\n\\(w\\colon X\\times X \\times X\\to \\mathbb{T}\\):\n\\[\n(S *_w T)_{x,y} = \\sum_{z\\in X} w(x,z,y) S_{x,z} T_{z,y}.\n\\]\nThis defines a bounded bilinear map at least on the subalgebra\n\\(A(X,\\varrho)\\subseteq \\mathbb B(\\Hils)\\)\nof locally compact, controlled operators.\n\n\\begin{lemma}\n \\label{lem:twisted_associative}\n The multiplication~\\(*_w\\)\n on \\(A(X,\\varrho)\\)\n is associative if and only if\n \\begin{equation}\n \\label{eq:twist_cocycle_condition}\n w(x,z,y) w(x,t,z) = w(x,t,y) w(t,z,y)\n \\end{equation}\n for all \\(x,t,z,y\\in X\\).\n\\end{lemma}\n\n\\begin{proof}\n For \\(S,T,U\\in A(X,\\varrho)\\), we compute\n \\begin{align*}\n \\bigl((S *_w T) *_w U\\bigr)_{x,y}\n \n = \\sum_{z,t\\in X} w(x,z,y) w(x,t,z) S_{x,t} T_{t,z} U_{z,y},\\\\\n \\bigl(S *_w (T *_w U)\\bigr)_{x,y}\n \n = \\sum_{z,t\\in X} w(x,t,y) w(t,z,y) S_{x,t} T_{t,z} U_{z,y}.\n \\end{align*}\n The condition~\\eqref{eq:twist_cocycle_condition} holds if and only if these are\n equal for all \\(S,T,U\\in A(X,\\varrho)\\)\n because all \\(\\Hils_x\\) are non-zero.\n\\end{proof}\n\n\\begin{proposition}\n \\label{pro:all_twists_trivial}\n If the function~\\(w\\)\n satisfies the cocycle condition in the previous lemma, then there is\n a function \\(v\\colon X\\times X \\to \\mathbb{T}\\) with\n \\[\n w(x,z,y) = v(x,z) v(z,y) v(x,y)^{-1}.\n \\]\n The map \\(\\varphi\\colon (A(X, \\varrho),*_w) \\to (A(X,\\varrho),\\cdot)\\),\n \\((T_{x,y})_{x,y\\in X} \\mapsto (v(x,y)\\cdot T_{x,y})_{x,y\\in X}\\),\n is an algebra isomorphism.\n\\end{proposition}\n\n\\begin{proof}\n Fix a ``base point'' \\(e\\in X\\)\n and let \\(v(x,y) \\mathrel{\\vcentcolon=} w(x,y,e)\\).\n The condition~\\eqref{eq:twist_cocycle_condition} for \\((x,z,y,e)\\)\n says that\n \\[\n w(x,y,e) w(x,z,y) = w(x,z,e) w(z,y,e)\n \\]\n holds for all \\(x,z,y\\in X\\). So\n \\[\n v(x,z) v(z,y) v(x,y)^{-1}\n = w(x,z,e) w(z,y,e) w(x,y,e)^{-1}\n = w(x,z,y).\n \\]\n The map~\\(\\varphi\\)\n is a vector space isomorphism because \\(v(x,y)\\neq0\\)\n for all \\(x,y\\in X\\). The computation\n \\begin{multline*}\n \\varphi (S *_w T)_{x,y}\n \n = v(x,y) \\sum_{z\\in X} w(x,z,y) S_{x,z} T_{z,y}\n \\\\= \\sum_{z\\in X} w(x,z,y) v(x,y) v(x,z)^{-1} v(z,y)^{-1}\n \\varphi(S)_{x,z} \\varphi(T)_{z,y}\n = \\sum_{z\\in X} \\varphi(S)_{x,z} \\varphi(T)_{z,y}\n \\end{multline*}\n shows that it is an algebra isomorphism.\n\\end{proof}\n\nSo the twisted and untwisted versions of \\(A(X,\\varrho)\\)\nare isomorphic algebras. Thus\na magnetic field does not change the isomorphism type of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra.\n\nNow let~\\(X\\)\nbe no longer discrete. Then controlled, locally compact\noperators are not given by matrices any more. To write down the\ntwisted convolution as above, we use the smaller $^*$\\nobreakdash-{}algebra of\ncontrolled, locally \\emph{Hilbert--Schmidt} operators; it is\nstill dense in the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nLet us assume for simplicity that the representation~\\(\\varrho\\)\nfor which we build the Roe \\(\\Cst\\)\\nobreakdash-algebra\nhas constant multiplicity, that is, it is the pointwise multiplication\nrepresentation on \\(L^2(X,\\mu) \\otimes \\Hils\\)\nfor some regular Borel measure~\\(\\mu\\)\non~\\(X\\)\nand some Hilbert space~\\(\\Hils\\).\nControlled, locally Hilbert--Schmidt operators on\n\\(L^2(X,\\mu) \\otimes \\Hils\\)\nare the convolution operators for measurable functions\n\\(T\\colon X\\times X\\to \\ell^2(\\Hils)\\)\nwith controlled support and such that\n\\[\n\\int_{K\\times K} \\norm{T(x,y)}^2 \\,\\diff \\mu(x)\\,\\diff\\mu(y) < \\infty\n\\]\nfor all compact subsets \\(K\\subseteq X\\)\nand such that the resulting convolution operator is bounded. The\nmultiplication of such operators is given by a convolution of their\nintegral kernels. This may be twisted as above, using a Borel\nfunction \\(w\\colon X^3\\to\\mathbb{T}\\)\nthat satisfies the condition~\\eqref{eq:twist_cocycle_condition}. The\nresulting function~\\(v\\)\nin Proposition~\\ref{pro:all_twists_trivial} is again Borel. So the\nisomorphism in Proposition~\\ref{pro:all_twists_trivial} still works.\nThus the twist gives an isomorphic $^*$\\nobreakdash-{}algebra also in the\nnon-discrete case.\n\nFollowing Bellissard~\\cite{Bellissard:K-theory_solid}, a\n\\(d\\)\\nobreakdash-dimensional\nmaterial is often described through a crossed product\n\\(\\Cst\\)\\nobreakdash-algebra\n\\(\\Cont(\\Omega)\\rtimes\\mathbb{Z}^d\\)\nfor a compact space~\\(\\Omega\\)\nwith a \\(\\mathbb{Z}^d\\)\\nobreakdash-action\nby homeomorphisms and with an ergodic invariant measure\non~\\(\\Omega\\).\nTo encode a magnetic field, the crossed product is replaced by the\ncrossed product \\emph{twisted} by a \\(2\\)\\nobreakdash-cocycle\n\\(\\sigma\\colon \\mathbb{Z}^d \\times \\mathbb{Z}^d \\to \\Cont(\\Omega,\\mathbb{T})\\).\nThe space~\\(\\Omega\\)\nmay be built as the ``hull'' of a point set or a fixed Hamiltonian,\nsee~\\cite{Bellissard-Herrmann-Zarrouati:Hull}. In this case, there is\na \\emph{dense} orbit \\(\\mathbb{Z}^d\\cdot\\omega\\)\nin~\\(\\Omega\\)\nby construction. So assuming the existence of a dense orbit is a\nrather mild assumption in the context of Bellissard's theory.\n\nWe briefly explain why all twisted crossed products\n\\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d\\)\nas above are ``contained'' in the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nand hence also in the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nThis observation is due to Kubota~\\cite{Kubota:Controlled_bulk-edge}.\n\nThe main point here is the description of the uniform Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nas a crossed product \\(\\ell^\\infty(\\mathbb{Z}^d)\\rtimes\\mathbb{Z}^d\\),\nsee \\cite{Roe:Lectures}*{Theorem~4.28}. Let \\(\\omega\\in\\Omega\\).\nThen we define a \\(\\mathbb{Z}^d\\)\\nobreakdash-equivariant\n$^*$\\nobreakdash-{}homomorphism\n\\(\\epsilon_\\omega\\colon \\Cont(\\Omega) \\to \\ell^\\infty(\\mathbb{Z}^d)\\)\nby \\((\\epsilon_\\omega f)(n) \\mathrel{\\vcentcolon=} f(n\\cdot \\omega)\\)\nfor all \\(n\\in\\mathbb{Z}^d\\),\n\\(f\\in \\Cont(\\Omega)\\).\nThis induces a $^*$\\nobreakdash-{}homomorphism\n\\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d \\to \\ell^\\infty(\\mathbb{Z}^d)\n\\rtimes_{\\epsilon_\\omega\\circ \\sigma} \\mathbb{Z}^d\\),\nwhere~\\(\\rtimes_\\sigma\\)\ndenotes the crossed product twisted by a \\(2\\)\\nobreakdash-cocyle~\\(\\sigma\\).\nThe same argument that identifies the crossed product\n\\(\\ell^\\infty(\\mathbb{Z}^d) \\rtimes \\mathbb{Z}^d\\)\nwith the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nidentifies\n\\(\\ell^\\infty(\\mathbb{Z}^d) \\rtimes_{\\epsilon_\\omega\\circ \\sigma} \\mathbb{Z}^d\\)\nwith a twist of the Roe \\(\\Cst\\)\\nobreakdash-algebra\nas above. Since all these twists give isomorphic \\(\\Cst\\)\\nobreakdash-algebras\nby Proposition~\\ref{pro:all_twists_trivial}, we get a\n$^*$\\nobreakdash-{}homomorphism\n\\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d \\to \\ell^\\infty(\\mathbb{Z}^d) \\rtimes\n\\mathbb{Z}^d\\).\nIf the orbit of~\\(\\omega\\)\nis dense, then the $^*$\\nobreakdash-{}homomorphism~\\(\\epsilon_\\omega\\)\nabove is injective. Then the induced $^*$\\nobreakdash-{}homomorphism\n\\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d \\to \\ell^\\infty(\\mathbb{Z}^d) \\rtimes\n\\mathbb{Z}^d\\)\nis also injective. Hence the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nreally contains the twisted crossed product algebra.\n\nNow we turn to the continuum version of the above theory.\nLet~\\(\\Omega\\)\nbe a compact space with a continuous action of~\\(\\mathbb{R}^d\\).\nThis leads to crossed products \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{R}^d\\)\ntwisted, say, by Borel measurable \\(2\\)\\nobreakdash-cocycles\n\\(\\sigma\\colon \\mathbb{R}^d \\times \\mathbb{R}^d \\to \\Cont(\\Omega,\\mathbb{T})\\);\nonce again, the twist encodes a magnetic field. Restricting to the\n\\(\\mathbb{R}^d\\)\\nobreakdash-orbit\nof some \\(\\omega\\in\\Omega\\)\nmaps \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{R}^d\\)\nto\n\\(\\Contb^\\mathrm{u}(\\mathbb{R}^d)\\rtimes_{\\epsilon_\\omega\\circ\\sigma} \\mathbb{R}^d\\)\nfor the \\(\\Cst\\)\\nobreakdash-algebra\n\\(\\Contb^\\mathrm{u}(\\mathbb{R}^d)\\)\nof bounded, uniformly continuous functions on~\\(\\mathbb{R}^d\\).\nWe have seen in Example~\\ref{exa:Roe_Rn} that\n\\(\\Contb^\\mathrm{u}(\\mathbb{R}^d)\\rtimes \\mathbb{R}^d \\subseteq L^\\infty(\\mathbb{R}^d) \\cdot\n\\Cst(\\mathbb{R}^d)\\)\nis contained in the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{R}^d\\).\nThis remains the case also in the twisted case because a Borel\nmeasurable \\(2\\)\\nobreakdash-cocycle\n\\(\\sigma\\colon \\mathbb{R}^d \\times \\mathbb{R}^d \\to \\Cont(\\Omega,\\mathbb{T})\\)\ndefines a Borel function \\((\\mathbb{R}^d)^3 \\to\\mathbb{T}\\),\nwhich is untwisted by Proposition~\\ref{pro:all_twists_trivial}. So\nall twisted crossed products \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{R}^d\\)\nmap to the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{R}^d\\).\nAs above, this map is an embedding if the orbit of~\\(\\omega\\)\nis dense in~\\(\\Omega\\).\n\nThe Roe \\(\\Cst\\)\\nobreakdash-algebras\nfor \\(\\mathbb{R}^d\\)\nand~\\(\\mathbb{Z}^d\\)\nare isomorphic by Theorem~\\ref{the:coarsely_dense}. So there is a unique\nRoe \\(\\Cst\\)\\nobreakdash-algebra\nin each dimension that contains all the twisted crossed product\nalgebras that are used as models for disordered materials, both in\ncontinuum models and tight binding models. This fits interpreting\nthe twisted crossed products \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d\\)\nor \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{R}^d\\)\nas models for disorder with built-in \\emph{a priori} restrictions,\nwhereas the Roe \\(\\Cst\\)\\nobreakdash-algebra describes general disorder.\n\n\n\\subsection{Approximation by controlled operators as a continuity property}\n\\label{sec:approximation_continuity}\n\nIn order to belong to the Roe \\(\\Cst\\)\\nobreakdash-algebra,\nan operator has to be a norm limit of locally compact, controlled\noperators. Any such norm limit is again locally compact. The\nproperty of being a norm limit of controlled operators may be hard to\ncheck. A tool for this is Property~A, an approximation property for\ncoarse spaces that ensures that elements of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nmay be approximated in a systematic way by controlled operators, see\n\\cites{Roe:Warped_A, Brodzki-Cave-Li:Exactness}. We also mention the\nrelated Operator Norm Localization Property for subspaces of~\\(\\mathbb{R}^d\\).\n\nWe now specialise to the case where~\\(X\\)\nis a closed subset of~\\(\\mathbb{R}^d\\)\nwith the restriction of the Euclidean metric. Such spaces have\nProperty~A. We use it to define complex Roe \\(\\Cst\\)\\nobreakdash-algebras\nthrough continuity for a certain representation of~\\(\\mathbb{R}^d\\).\nWe fix a representation \\(\\varrho\\colon \\Cont_0(X) \\to \\mathbb B(\\Hils)\\)\non a complex Hilbert space~\\(\\Hils\\).\nLet \\(\\bar\\varrho\\colon \\Contb(X)\\to\\mathbb B(\\Hils)\\)\nbe its unique strictly continuous extension to the multiplier algebra.\nFor \\(t\\in\\mathbb{R}^d\\),\ndefine \\(\\Euler_t\\in\\Contb(X)\\)\nby \\(\\Euler_t(x) \\mathrel{\\vcentcolon=} \\Euler^{\\ima x\\cdot t}\\).\nThe map \\(t\\mapsto \\Euler_t\\)\nis continuous for the strict topology on~\\(\\Contb(X)\\).\nHence the representation~\\(\\sigma\\)\nof~\\(\\mathbb{R}^d\\)\non~\\(\\Hils\\)\ndefined by \\(\\sigma_t(\\xi) \\mathrel{\\vcentcolon=} \\bar\\varrho(\\Euler_t)(\\xi)\\)\nis continuous. This representation is generated by the position\noperators. If \\(X\\subseteq \\mathbb{Z}^d\\),\nthen \\(\\Euler_t=1\\)\nfor \\(t\\in 2\\pi\\mathbb{Z}^d\\),\nso that the representation~\\(\\sigma\\)\ndescends to the torus \\((\\mathbb{R}\/2\\pi \\mathbb{Z})^d\\).\n\nBy conjugation, \\(\\sigma\\)\ninduces an action \\(\\Ad \\sigma\\)\nof~\\(\\mathbb{R}^d\\)\nby automorphisms of \\(\\mathbb B(\\Hils)\\).\nWe call \\(S\\in\\mathbb B(\\Hils)\\)\n\\emph{continuous} with respect to \\(\\Ad \\sigma\\)\nif the map \\(\\mathbb{R}^d \\to \\mathbb B(\\Hils)\\),\n\\(t\\mapsto \\Ad \\sigma_t(S)\\),\nis continuous in the norm topology on~\\(\\mathbb B(\\Hils)\\).\nThe following theorem describes the Roe \\(\\Cst\\)\\nobreakdash-algebra\nthrough this continuity property:\n\n\\begin{theorem}\n \\label{the:controlled_continuity}\n An operator \\(S\\in\\mathbb B(\\Hils)\\)\n is a norm limit of controlled operators if and only if it is\n continuous with respect to \\(\\Ad \\sigma\\).\n And \\(\\Cst_\\Roe(X,\\varrho)\\)\n is the \\(\\Cst\\)\\nobreakdash-subalgebra\n of all operators on~\\(\\Hils\\)\n that are locally compact and continuous with respect to\n \\(\\Ad \\sigma\\).\n\\end{theorem}\n\n\\begin{proof}\n For each \\(S\\in\\mathbb B(\\Hils)\\),\n the map \\(\\mathbb{R}^d\\to\\mathbb B(\\Hils)\\)\n is continuous for the strong topology on \\(\\mathbb B(\\Hils)\\).\n Therefore, the \\(\\mathbb B(\\Hils)\\)-valued integral\n \\[\n f*S \\mathrel{\\vcentcolon=} \\int_{\\mathbb{R}^d} f(t) \\Ad \\sigma_t(S) \\,\\diff t\n \\]\n makes sense for any \\(f\\in L^1(\\mathbb{R}^d)\\).\n Let \\((f_n)_{n\\in\\mathbb{N}}\\)\n be a bounded approximate unit in the Banach algebra \\(L^1(\\mathbb{R}^d)\\).\n We claim that~\\(S\\)\n is continuous if and only if \\((f_n*S)_{n\\in\\mathbb{N}}\\)\n converges in the norm topology to~\\(S\\).\n It is well known that any continuous representation of~\\(\\mathbb{R}^d\\)\n becomes a nondegenerate module over \\(L^1(\\mathbb{R}^d)\\).\n Thus \\((f_n*S)_{n\\in\\mathbb{N}}\\)\n converges in norm to~\\(S\\)\n if~\\(S\\)\n is continuous. Conversely, operators of the form \\(f*S\\)\n are continuous because the action of~\\(\\mathbb{R}^d\\)\n on \\(L^1(\\mathbb{R}^d)\\)\n is continuous. Since the set of continuous operators is closed in\n the norm topology, \\(S\\)\n is continuous if \\((f_n*S)_{n\\in\\mathbb{N}}\\) converges in norm to~\\(S\\).\n\n There is an approximate unit~\\((f_n)_{n\\in\\mathbb{N}}\\)\n for~\\(L^1(\\mathbb{R}^d)\\)\n such that the Fourier transform of each~\\(f_n\\)\n has compact support. For instance, we may use the Fej\u00e9r kernel\n \\[\n \\Lambda(x_1,\\dotsc,x_n) \\mathrel{\\vcentcolon=} \\prod_{j=1}^d \\frac{\\sin^2(\\pi x_j)}{\\pi^2 x_j^2},\n \\]\n which has Fourier transform \\(\\prod_{j=1}^d (1-\\abs{x_j})_+\\),\n and rescale it to produce an approximate unit for \\(L^1(\\mathbb{R}^d)\\).\n We claim that \\(f_n*S\\)\n is controlled for each \\(n\\in\\mathbb{N}\\).\n More precisely, assume that \\(\\widehat{f_n}\\)\n is supported in the ball of radius~\\(R\\)\n in~\\(\\mathbb{R}^d\\).\n We claim that~\\(f_n*S\\)\n is \\(R\\)\\nobreakdash-controlled.\n\n This is easy to prove if~\\(X\\)\n is discrete. Then we may describe operators on~\\(\\Hils\\)\n using matrix coefficients \\(S_{x,y}\\in\\mathbb B(\\Hils_y,\\Hils_x)\\)\n for \\(x,y\\in X\\).\n A direct computation shows that the matrix coefficients of \\(f_n*S\\)\n are \\(\\widehat{f_n}(x-y)\\cdot S_{x,y}\\).\n This vanishes for \\(\\norm{x-y}>R\\).\n So~\\(f_n*S\\)\n is \\(R\\)\\nobreakdash-controlled\n as asserted. The following argument extends this result to the case\n where~\\(X\\) is not discrete, such as \\(X=\\mathbb{R}^d\\).\n\n Let \\(U,V\\subseteq \\mathbb{R}^d\\)\n be two relatively compact, open subsets of distance at least~\\(R\\)\n and let \\(g,h\\in\\Cont^\\infty(\\mathbb{R}^d)\\)\n be smooth functions supported in \\(U\\)\n and~\\(V\\), respectively. Then\n \\begin{equation}\n \\label{eq:Roe_through_continuity_integral}\n \\int_{\\mathbb{R}^d} g(y) \\Euler^{\\ima y\\cdot t}\\cdot h(x) \\Euler^{-\\ima x\\cdot t} f_n(t) \\,\\diff t\n \n = g(y) h(x) \\widehat{f_n}(y-x)\n = 0\n \\end{equation}\n for all \\(x,y\\in \\mathbb{R}^d\\). We restrict \\(g,h\\) to~\\(X\\) and compute\n \\begin{align*}\n \\varrho(g) (f_n* S) \\varrho(h)\n &\\mathrel{\\vcentcolon=} \\int_{\\mathbb{R}^d} \\varrho(g) \\bar\\varrho(\\Euler_t) S\n \\bar\\varrho(\\Euler_{-t}) \\varrho(h) f_n(t) \\,\\diff t\n \\\\&= \\int_{\\mathbb{R}^d} \\varrho(g\\cdot \\Euler_t) S\n \\varrho(h\\cdot \\Euler_{-t}) f_n(t) \\,\\diff t.\n \\end{align*}\n Let \\(\\Cont_0^\\infty(U\\times V)\\)\n denote the Fr\u00e9chet space of smooth function on~\\(\\mathbb{R}^{2d}\\)\n supported in \\(U\\times V\\).\n We may identify this with the complete projective tensor product of\n \\(\\Cont^\\infty_0(U)\\)\n and \\(\\Cont^\\infty_0(V)\\).\n Hence there is a continuous linear map\n \\begin{equation}\n \\label{eq:function_on_both_side}\n \\Cont^\\infty_0(U\\times V)\\to \\mathbb B(\\Hils),\\qquad\n g\\otimes h\\mapsto \\varrho(g) S \\varrho(h).\n \\end{equation}\n The integral in~\\eqref{eq:Roe_through_continuity_integral} converges\n to~\\(0\\)\n in the Fr\u00e9chet topology of \\(\\Cont^\\infty_0(U\\times V)\\).\n Hence~\\eqref{eq:Roe_through_continuity_integral} implies\n \\(\\varrho(g) (f_n*S) \\varrho(h)=0\\).\n\n The continuous map~\\eqref{eq:function_on_both_side} still exists if\n \\(U\\)\n and~\\(V\\)\n are not of distance~\\(R\\).\n If~\\(S\\)\n is locally compact, then \\(\\varrho(g) S \\varrho(h)\\)\n is a compact operator on~\\(\\Hils\\)\n for all \\(g\\in\\Cont^\\infty_0(U)\\),\n \\(h\\in\\Cont^\\infty_0(V)\\).\n This remains so for all operators in the image\n of~\\eqref{eq:function_on_both_side} by continuity. Therefore,\n \\(\\varrho(g) (f_n*S)\\varrho(h)\\)\n is compact for all \\(g,h\\)\n as above. Choosing~\\(U\\)\n large enough, we may take~\\(g\\)\n to be constant equal to~\\(1\\)\n on the \\(R\\)\\nobreakdash-neighbourhood\n of~\\(V\\).\n Then \\(\\varrho(g) (f_n*S)\\varrho(h) =(f_n*S)\\varrho(h)\\)\n because~\\(f_n*S\\)\n is \\(R\\)\\nobreakdash-controlled.\n So operators of the form \\((f_n*S)\\varrho(h)\\)\n with smooth, compactly supported~\\(h\\)\n are compact. Since any continuous, compactly supported function is\n dominated by a smooth, compactly supported function, we get the same\n for all \\(h\\in\\Contc(X)\\).\n A similar argument shows that \\(\\varrho(g)(f_n*S)\\)\n is compact for all \\(g\\in\\Contc(X)\\).\n Hence the operators \\(f_n*S\\)\n are locally compact, controlled operators if~\\(S\\)\n is locally compact.\n\\end{proof}\n\nProperty~A is equivalent to the ``Operator Norm Localization\nProperty'' for metric spaces with bounded geometry,\nsee~\\cite{Sako:A_operator_localization}. Roughly speaking, this\nproperty says that the operator norm of a controlled operator may be\ncomputed using vectors in the Hilbert space with bounded support. The\nsupport of a vector \\(\\xi\\in\\Hils\\)\nis the set of all \\(x\\in X\\)\nsuch that \\(f\\cdot \\xi\\neq0\\)\nfor all \\(f\\in\\Cont_0(X)\\)\nwith \\(f(x)\\neq0\\).\nWe formulate this property for subspaces of~\\(\\mathbb{R}^d\\):\n\n\\begin{theorem}\n \\label{the:ONL}\n Let \\(X\\subseteq \\mathbb{R}^d\\)\n and let \\(\\varrho\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils)\\)\n be a representation. Pick scalars \\(R>0\\)\n and \\(c\\in(0,1)\\).\n Then there is a scalar \\(S>0\\)\n such that for any \\(R\\)\\nobreakdash-controlled\n operator \\(T\\in\\mathbb B(\\Hils)\\),\n there is \\(\\xi\\in\\Hils\\)\n with \\(\\norm{\\xi}=1\\)\n such that the support of~\\(\\xi\\)\n has diameter at most~\\(S\\)\n and \\(\\norm{T(\\xi)} \\ge \\norm{T} \\ge c\\cdot \\norm{T(\\xi)}\\).\n\\end{theorem}\n\n\\begin{proof}\n The statement of the theorem is that the space~\\(X\\) has the\n ``Operator Norm Localisation Property'' defined\n in~\\cite{Chen-Tessera-Wang:Operator_localization}. This property\n is invariant under coarse equivalence and passes to subspaces by\n \\cite{Chen-Tessera-Wang:Operator_localization}*{Propositions 2.5\n and~2.6}.\n \\cite{Chen-Tessera-Wang:Operator_localization}*{Theorem~3.11 and\n Proposition~4.1} show that solvable Lie groups such as~\\(\\mathbb{R}^d\\)\n have this property, and hence also all subspaces of~\\(\\mathbb{R}^d\\).\n\\end{proof}\n\n\n\\subsection{Dense subalgebras with isomorphic K-theory}\n\\label{sec:dense_same_K}\n\nLet~\\(A\\)\nbe a \\(\\Cst\\)\\nobreakdash-algebra\nwith a continuous \\(\\mathbb{R}^d\\)\\nobreakdash-action\n\\(\\alpha\\colon \\mathbb{R}^d\\to\\Aut(A)\\).\nThe action defines several canonical $^*$\\nobreakdash-{}subalgebras of~\\(A\\)\nwith the same \\(\\K\\)\\nobreakdash-theory.\nThe $^*$\\nobreakdash-{}subalgebra of \\emph{smooth elements} is\n\\[\nA^\\infty \\mathrel{\\vcentcolon=} \\setgiven{a\\in A}\n{t\\mapsto \\alpha_t(a) \\text{ is a smooth function } \\mathbb{R}^d\\to A}.\n\\]\nThis Fr\u00e9chet $^*$\\nobreakdash-{}subalgebra is closed under holomorphic functional\ncalculus and also under smooth functional calculus for normal\nelements, see~\\cite{Blackadar-Cuntz:Differential}.\n\nLet \\(F \\subseteq \\mathbb{R}^d\\)\nbe a compact convex subset with non-empty interior and\ncontaining~\\(0\\).\nLet \\(\\mathcal{O}(A,\\alpha,F) \\subseteq A\\)\nbe the set of all \\(a\\in A\\)\nfor which the function \\(\\mathbb{R}^d\\ni t\\mapsto \\alpha_t(a)\\)\nextends to a continuous function on \\(\\mathbb{R}^d + \\ima F\\)\nthat is holomorphic on the interior of \\(\\mathbb{R}^d + \\ima F\\).\nThis is a dense Banach subalgebra in~\\(A\\),\nand the inclusion \\(\\mathcal{O}(A,\\alpha,F) \\hookrightarrow A\\)\ninduces an isomorphism on topological \\(\\K\\)\\nobreakdash-theory\nby \\cite{Bost:Principe_Oka}*{Th\u00e9or\u00e8me~2.2.1}. Let\n\\(\\mathcal{O}^\\infty(A,\\alpha,F) \\subseteq A\\)\nbe the set of those \\(a\\in A\\)\nfor which the function \\(\\mathbb{R}^d\\ni t\\mapsto \\alpha_t(a)\\)\nextends to a smooth function on \\(\\mathbb{R}^d + \\ima F\\)\nthat is holomorphic on the interior of \\(\\mathbb{R}^d + \\ima F\\).\nThe inclusion \\(\\mathcal{O}^\\infty(A,\\alpha,F) \\hookrightarrow A\\)\ninduces an isomorphism on topological \\(\\K\\)\\nobreakdash-theory\nas well. If \\(F_1\\subseteq F_2\\),\nthen \\(\\mathcal{O}(A,\\alpha,F_2) \\hookrightarrow \\mathcal{O}(A,\\alpha,F_1)\\).\nThere are two important limiting cases of the subalgebras\n\\(\\mathcal{O}(A,\\alpha,F)\\).\n\nFirst, let~\\(F\\)\nrun through a neighbourhood basis of~\\(0\\)\nin~\\(\\mathbb{R}^d\\).\nThen the dense Banach subalgebras \\(\\mathcal{O}(A,\\alpha,F)\\)\nform an inductive system, whose colimit is the dense $^*$\\nobreakdash-{}subalgebra\n\\(A^\\omega \\subseteq A\\)\nof all \\emph{real-analytic elements} of~\\(A\\),\nthat is, those \\(a\\in A\\)\nwith the property that each \\(t\\in\\mathbb{R}^d\\)\nhas a neighbourhood on which \\(s\\mapsto \\alpha_s(a)\\)\nis given by a convergent power series with coefficients in~\\(A\\).\nThe subalgebra~\\(A^\\omega\\)\nis still closed under holomorphic functional calculus by\n\\cite{Meyer:HLHA}*{Proposition~3.46}. This gives an easier\nexplanation than Bost's Oka principle why~\\(A^\\omega\\)\nhas the same topological \\(\\K\\)\\nobreakdash-theory as~\\(A\\).\n\nSecondly, let~\\(F\\)\nrun through an increasing sequence whose union is~\\(\\mathbb{R}^d\\).\nThen the dense Banach subalgebras \\(\\mathcal{O}(A,\\alpha,F)\\)\nform a projective system, whose limit is the dense $^*$\\nobreakdash-{}subalgebra\n\\(\\mathcal{O}(A,\\alpha)\\)\nof all \\emph{holomorphic elements} \\(a\\in A\\),\nthat is, those elements for which the map\n\\(\\mathbb{R}^d \\ni t\\mapsto \\alpha_t(a)\\)\nextends to a holomorphic function on~\\(\\mathbb{C}^d\\).\nThis is a locally multiplicatively convex Fr\u00e9chet algebra.\nPhillips~\\cite{Phillips:K_Frechet} has extended\ntopological \\(\\K\\)\\nobreakdash-theory\nto such algebras. The Milnor \\(\\varprojlim^1\\)-sequence\nin \\cite{Phillips:K_Frechet}*{Theorem 6.5} shows that the\ninclusion \\(\\mathcal{O}(A,\\alpha)\\hookrightarrow A\\)\ninduces an isomorphism in topological \\(\\K\\)\\nobreakdash-theory.\n\nWe apply all this to the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nand the continuous \\(\\mathbb{R}^d\\)\\nobreakdash-action~\\(\\sigma\\)\ndefined in Section~\\ref{sec:approximation_continuity}. Here this\naction descends to the torus~\\(\\mathbb{T}^d\\),\nwhich simplifies the study of the dense subalgebras above. We\ndescribe the dense subalgebras of smooth, real-analytic and\nholomorphic elements in \\(\\Cst_\\Roe(\\mathbb{Z}^d)\\).\nAll these have the same topological \\(\\K\\)\\nobreakdash-theory.\nLet \\(\\varrho\\colon \\Cont_0(\\mathbb{Z}^d)\\to\\mathbb B(\\Hils)\\)\nbe a representation on a separable Hilbert space, not necessarily\nample. Let~\\(\\Hils_x\\)\nfor \\(x\\in X\\)\nbe the fibres of~\\(\\Hils\\)\nwith respect to~\\(\\varrho\\).\nDescribe operators on~\\(\\Hils\\)\nby block matrices \\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\nwith \\(T_{x,y}\\in\\mathbb B(\\Hils_y,\\Hils_x)\\)\nfor all \\(x,y\\in\\mathbb{Z}^d\\). Then\n\\[\n\\sigma_t(T_{x,y}) = (\\exp(\\ima t\\cdot (x-y)) T_{x,y})_{x,y\\in\\mathbb{Z}^d}.\n\\]\n\n\\begin{proposition}\n \\label{pro:smooth_analytic_in_Roe_Zd}\n A block matrix \\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n as above gives a smooth element for the\n \\(\\mathbb{R}^d\\)\\nobreakdash-action~\\(\\sigma\\) on~\\(\\Cst_\\Roe(\\mathbb{Z}^d)\\) if and only if the function\n \\[\n \\mathbb{Z}^d \\ni k \\mapsto \\sup_{n\\in\\mathbb{Z}^d} \\{ \\norm{T_{n,n+k}} \\}\n \\]\n has rapid decay, that is, for each \\(a>0\\)\n there is a constant~\\(C_a>0\\)\n such that \\(\\norm{T_{n,n+k}} \\le C_a (1+ \\norm{k})^{-a}\\)\n for all \\(n,k\\in\\mathbb{Z}^d\\).\n It gives a real-analytic element for~\\(\\sigma\\)\n if and only if there are \\(a>0\\) and \\(C_a>0\\) such that\n \\[\n \\norm{T_{n,n+k}} \\le C_a \\cdot \\exp(-a \\norm{k})\n \\]\n for all \\(n,k\\in\\mathbb{Z}^d\\).\n It gives a holomorphic element for~\\(\\sigma\\)\n if and only if for each \\(a>0\\) there is \\(C_a>0\\) such that\n \\[\n \\norm{T_{n,n+k}} \\le C_a \\cdot \\exp(-a \\norm{k}).\n \\]\n\\end{proposition}\n\n\\begin{proof}\n The \\(j\\)th\n generator of the \\(\\mathbb{R}^d\\)\\nobreakdash-action~\\(\\sigma\\)\n maps a block matrix~\\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n to\n \\[\n \\lim_{t\\to0} \\frac{1}{t} (\\sigma_{t e_j}(T_{x,y}) - (T_{x,y}))_{x,y\\in\\mathbb{Z}^d}\n = ((x_j-y_j)T_{x,y})_{x,y\\in\\mathbb{Z}^d}.\n \\]\n Hence polynomials in these generators multiply the\n entries~\\(T_{x,y}\\)\n with polynomials in \\(x-y\\in\\mathbb{Z}^d\\).\n So~\\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n belongs to a smooth element of~\\(\\Cst_\\Roe(\\mathbb{Z}^d)\\)\n if and only if \\((p(x-y)\\cdot T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n belongs to a bounded operator for each polynomial~\\(p\\)\n in \\(d\\)~variables.\n It suffices to consider the polynomials \\(1+\\norm{x-y}_2^{2 b}\\)\n for \\(b\\in\\mathbb{N}\\).\n Since the operator norm for diagonal block matrices is the supremum\n of the operator norms of the entries, we see that the boundedness of\n \\(((1+ \\norm{x-y}_2^{2 b})\\cdot T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n for all \\(b\\in\\mathbb{N}\\)\n is equivalent to the boundedness of\n \\(\\sup_{k,n\\in\\mathbb{Z}^d} \\norm{T_{n,n+k}} (1+ \\norm{k}_2^{2 b})\\)\n for all \\(b\\in\\mathbb{N}\\).\n This proves the claim about the smooth elements. The analytic\n extension of~\\(\\sigma\\)\n to \\(\\ima z\\in \\mathbb{C}^d\\)\n must map~\\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n to \\(((\\exp(z\\cdot (x-y))T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\).\n Thus~\\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n describes an element of\n \\(\\mathcal{O}^\\infty(\\Cst_\\Roe(\\mathbb{Z}^d),\\sigma,F)\\) if and only if\n \\begin{equation}\n \\label{eq:O_F_estimate}\n \\sup_{k,n\\in\\mathbb{Z}^d} \\norm{T_{n,n+k}} (1+ \\norm{k}_2^{2 b}) \\exp(z\\cdot k)<\\infty \n \\end{equation}\n for all \\(z\\in F\\),\n \\(b\\in\\mathbb{N}\\).\n When we let \\(F\\searrow\\{0\\}\\)\n or \\(F\\nearrow \\mathbb{C}^d\\),\n we may leave out the polynomial factors because they are dominated\n by \\(\\exp(z\\cdot k)\\).\n This proves the claims about the\n real-analytic elements and holomorphic elements.\n\\end{proof}\n\nEstimates of the form\n\\(\\norm{T_{x,y}} \\le C_a \\exp(-a\\cdot \\norm{x-y})\\)\nfor some \\(a>0\\),\n\\(C_a>0\\)\nplay an important role in the study of Anderson localisation; see, for\ninstance,\n\\cite{Aizenman-Molchanov:Localization_elementary}*{Equation~(2.3)}.\n\n\n\\subsection{Approximate unit of projections}\n\\label{sec:apprid_projection}\n\nUnlike the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra,\nthe Roe \\(\\Cst\\)\\nobreakdash-algebra\nof a proper metric space is never unital. Instead, it has an\napproximate unit of projections:\n\n\\begin{proposition}\n \\label{pro:Roe_apprid}\n Let~\\(X\\)\n be a proper metric space and let\n \\(\\varrho\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils)\\)\n be a representation. The Roe \\(\\Cst\\)\\nobreakdash-algebra\n \\(\\Cst_\\Roe(X,\\varrho)\\) has an approximate unit of projections.\n\\end{proposition}\n\n\\begin{proof}\n Any proper metric space contains a coarsely dense, discrete\n subspace. By Theorem~\\ref{the:coarsely_dense}, we may assume\n that~\\(X\\)\n itself is discrete. By Theorem~\\ref{the:Roe_ample_unique}, we may\n further assume that the Roe \\(\\Cst\\)\\nobreakdash-algebra\n is built using the obvious representation of~\\(\\Cont_0(X)\\)\n on \\(\\ell^2(X,\\ell^2(\\mathbb{N}))\\).\n Then the Roe \\(\\Cst\\)\\nobreakdash-algebra\n contains \\(\\ell^\\infty(X, \\Cont_0(\\mathbb{N}))\\)\n as multiplication operators. Any function \\(h\\colon X\\to\\mathbb{N}\\)\n defines a projection in \\(\\ell^\\infty(X, \\Cont_0(\\mathbb{N}))\\),\n namely, the characteristic function of\n \\(\\setgiven{(x,n)\\in X\\times\\mathbb{N}}{n0\\) with\n \\[\n \\supp(T) \\subseteq \\setgiven{(x,y)\\in X \\times X}\n {d(x,Y)0\\)\n and let~\\(T\\)\n be supported in the \\(R\\)\\nobreakdash-neighbourhood\n of~\\(Y\\).\n This \\(R\\)\\nobreakdash-neighbourhood\n is a closed subspace~\\(Y_R\\)\n of~\\(X\\),\n and \\(Y\\subseteq Y_R\\)\n is coarsely dense by construction. The restriction of~\\(\\varrho'\\)\n to the Hilbert subspace\n \\(\\Hils_{Y,R} \\mathrel{\\vcentcolon=} \\varrho'(1_{Y_R})(\\Hils_Y\\oplus \\Hils_X)\\)\n is an ample representation of~\\(\\Cont_0(Y_R)\\).\n Hence it defines \\(\\Cst_\\Roe(Y_R)\\)\n by Theorem~\\ref{the:Roe_ample_unique}. This \\(\\Cst\\)\\nobreakdash-algebra\n is simply the corner in \\(\\Cst_\\Roe(X)\\)\n generated by the projection onto~\\(\\Hils_{Y,R}\\).\n Theorem~\\ref{the:coarsely_dense} gives a unitary\n \\(U\\colon \\Hils_Y \\xrightarrow\\sim \\Hils_{Y,R}\\)\n such that \\(U \\Cst_\\Roe(Y) U^* = \\Cst_\\Roe(Y_R)\\).\n The unitary~\\(U\\)\n is built in the proof of Theorem~\\ref{the:Roe_ample_unique}, and the\n construction there shows that it is controlled as an operator on\n \\(\\Hils_Y\\oplus \\Hils_X\\).\n So it is a multiplier of~\\(\\Cst_\\Roe(X)\\),\n where it is no longer unitary but a partial isometry. The operator\n \\(U^* T U\\)\n belongs to \\(\\Cst_\\Roe(Y)\\)\n because \\(T\\in\\Cst_\\Roe(Y_R)\\).\n Since multipliers of \\(\\Cst_\\Roe(X)\\)\n are also multipliers of any ideal in~\\(\\Cst_\\Roe(X)\\),\n the operator \\(T = U (U^* T U) U^*\\)\n belongs to the ideal in~\\(\\Cst_\\Roe(X)\\)\n generated by \\(\\Cst_\\Roe(Y) = P \\Cst_\\Roe(X) P\\).\n Thus \\(\\Cst_\\Roe(Y_R) \\subseteq \\Cst_\\Roe(X) P \\Cst_\\Roe(X)\\).\n\\end{proof}\n\n\n\\subsection{The coarse Mayer--Vietoris sequence}\n\\label{sec:coarse_MV_subsection}\n\n\\begin{proposition}[\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris}]\n \\label{pro:omega-excisiv}\n Let~\\(X\\)\n be a proper metric space and let \\(Y_1,Y_2\\subseteq X\\)\n be closed subspaces with \\(Y_1 \\cup Y_2 = X\\).\n Let \\(Z \\mathrel{\\vcentcolon=} Y_1\\cap Y_2\\).\n Then\n \\[\n \\Cst_\\Roe(Y_1\\subseteq X) + \\Cst_\\Roe(Y_2\\subseteq X) = \\Cst_\\Roe(X).\n \\]\n We have\n \\(\\Cst_\\Roe(Y_1\\subseteq X) \\cap \\Cst_\\Roe(Y_2\\subseteq X) =\n \\Cst_\\Roe(Z\\subseteq X)\\)\n if and only if the following coarse transversality condition holds:\n for any \\(R>0\\)\n there is \\(S(R)>0\\)\n such that if \\(x\\in X\\)\n satisfies \\(d(x,Y_1)0\\)\n of~\\(Y_1\\)\n and~\\(U\\)\n within distance \\(P_U>0\\)\n of~\\(Y_2\\).\n Let \\(R \\mathrel{\\vcentcolon=} R_T+R_U+P_T+P_U\\).\n If \\((x,y)\\in\\supp(T U)\\),\n then there is \\(z\\in X\\)\n with \\((x,z)\\in \\supp(T)\\) and \\((z,y)\\in \\supp(U)\\). Then\n \\begin{align*}\n d(x,Y_1)&1\\),\nsee \\cite{Spakula:Thesis}*{Example II.3.4}.\n\nWhen we consider Hamiltonians with symmetries, then we should tensor\nthe real Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nwith a real or complex Clifford algebra. This gives a\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n\\(\\Cst\\)\\nobreakdash-algebra.\nUp to Morita equivalence, there are ten\ndifferent real or complex Clifford algebras. So we get ten\ndifferent observable algebras in each dimension. The resulting\nreal or complex \\(\\K\\)\\nobreakdash-groups\nagree with those in Kitaev's periodic\ntable~\\cite{Kitaev:Periodic_table}. Hence the latter agrees with\nthe \\(\\K\\)\\nobreakdash-theory\nof the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nWe interpret it as saying that Kitaev's table gives only the\n\\emph{strong} topological phases.\n\nThe real and complex \\(\\K\\)\\nobreakdash-groups\nof the point form a graded commutative, graded ring in a\nnatural way, and the \\(\\K\\)\\nobreakdash-theory\nof any real or complex \\(\\Cst\\)\\nobreakdash-algebra\nis a graded module over this ring. The boundary map for an extension\nof real or complex \\(\\Cst\\)\\nobreakdash-algebras\nautomatically preserves this module structure. In the\ncomplex case, the relevant ring is the ring of Laurent\npolynomials \\(\\mathbb{Z}[\\beta,\\beta^{-1}]\\)\nin \\(\\beta\\in \\K_2(\\mathbb{C})\\)\nthat describes Bott periodicity. That a map on \\(\\K\\)\\nobreakdash-theory\nis a \\(\\K_*(\\mathbb{C})\\)-module\nhomomorphism only says that it is obtained by the maps on \\(\\K_0\\)\nand~\\(\\K_1\\)\nand Bott periodicity. In other words, it is a homomorphism of\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-graded groups.\nIn the real case, the relevant ring is more complicated, and so the\nmodule structure contains more useful information.\nOne way to get the \\(\\K_*(\\mathbb{R})\\)-module structure on \\(\\K_*(A)\\)\nfor a real \\(\\Cst\\)\\nobreakdash-algebra~\\(A\\)\nis to identify~\\(\\K_j(A)\\)\nwith the bivariant Kasparov groups\n\\(\\K_j(A) \\cong \\KK_0(\\mathbb{R},A \\otimes \\Cliff_j)\\).\nThe exterior product in Kasparov theory provides both the graded\ncommutative ring structure on\n\\(\\bigoplus_{j\\in\\mathbb{Z}} \\KK_0(\\mathbb{R},\\Cliff_j)\\)\nand the module structure on\n\\(\\bigoplus_{j\\in\\mathbb{Z}} \\KK_0(\\mathbb{R},A\\otimes \\Cliff_j)\\).\nThese structures are compatible with Kasparov products, and the\nboundary map in an extension may be written as such a Kasparov\nproduct.\n\nThe isomorphism\n\\(\\K_{*+d}(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{R}) \\cong \\K_*(\\mathbb{R})\\)\nof \\(\\mathbb{Z}\\)\\nobreakdash-graded groups\nin Corollary~\\ref{cor:K_Roe_Zd} is a \\(\\K_*(\\mathbb{R})\\)-module\nisomorphism. So \\(\\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{R})\\)\nis a free \\(\\K_*(\\mathbb{R})\\)-module\nof rank~\\(1\\),\nshifted in degree by~\\(d\\).\nAnd the boundary map~\\(\\partial_\\mathrm{MV}\\)\nis a module isomorphism. Thus it is determined by a single sign,\ndescribing whether the ``standard'' generator of\n\\(\\K_d(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{R})\\)\ngoes to the ``standard'' generator of\n\\(\\K_{d-1}(\\Cst_\\Roe(\\mathbb{Z}^{d-1})_\\mathbb{R})\\)\nor its negative. This sign is, in fact, a matter of convention: it\nchanges when we change the role of the left and right half-spaces in\nthe Mayer--Vietoris sequence. So there is not much need to\n``compute'' the boundary map for the Roe \\(\\Cst\\)\\nobreakdash-algebras\nbecause the \\(\\K\\)\\nobreakdash-theory\ngroups in question are so small, even in the real case.\n\nThe boundary map~\\(\\partial_\\mathrm{MV}\\)\nis the incarnation of the bulk--edge correspondence in our\nRoe \\(\\Cst\\)\\nobreakdash-algebra\ncontext. It is shown by Kubota~\\cite{Kubota:Controlled_bulk-edge}\nthat the boundary maps in the Toeplitz extension, which is used by\nmany authors to describe the bulk--edge correspondence, and the\ncoarse Mayer--Vietoris sequence are compatible.\n\n\\begin{proposition}\n \\label{pro:low_dimensional_killed_Roe}\n Let \\(\\varphi\\colon \\mathbb{Z}^{d-1}\\to \\mathbb{Z}^d\\)\n be an injective group homomorphism. Then the induced map\n \\(\\varphi_*\\colon \\Cst_\\Roe(\\mathbb{Z}^{d-1}) \\to \\Cst_\\Roe(\\mathbb{Z}^d)\\)\n induces the zero map in \\(\\K\\)\\nobreakdash-theory, both in the real and\n complex cases.\n\\end{proposition}\n\n\\begin{proof}\n Since~\\(\\varphi\\)\n is an injective group homomorphism, it is a coarse equivalence\n from~\\(\\mathbb{Z}^{d-1}\\)\n onto a subspace of~\\(\\mathbb{Z}^d\\).\n This explains the definition of\n \\(\\varphi_*\\colon \\Cst_\\Roe(\\mathbb{Z}^{d-1}) \\to \\Cst_\\Roe(\\mathbb{Z}^d)\\).\n There is \\(x\\in\\mathbb{Z}^d\\)\n so that the map \\(\\mathbb{Z}^{d-1}\\times\\mathbb{Z} \\to \\mathbb{Z}^d\\),\n \\((a,b) \\mapsto \\varphi(a)+b\\cdot x\\),\n is injective. So the map\n \\(\\varphi_*\\colon \\Cst_\\Roe(\\mathbb{Z}^{d-1}) \\to \\Cst_\\Roe(\\mathbb{Z}^d)\\)\n factors through \\(\\Cst_\\Roe(\\mathbb{Z}^{d-1}\\times\\mathbb{N})\\).\n Since the \\(\\K\\)\\nobreakdash-theory\n of \\(\\Cst_\\Roe(\\mathbb{Z}^{d-1}\\times\\mathbb{N})\\)\n vanishes by Proposition~\\ref{vanish},\n the map~\\(\\varphi\\)\n induces the zero map on \\(\\K\\)\\nobreakdash-theory.\n\\end{proof}\n\n\n\\section{Comparison with the periodic case}\n\\label{sec:weak_phases}\n\nLet \\(\\mathbb{F} \\in \\{\\mathbb{R},\\mathbb{C}\\}\\).\nThe observable algebra \\(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}\\)\nor a matrix algebra over it describes periodic observables in the\nlimiting case of no disorder, in the tight-binding approximation.\nThis is contained in the corresponding Roe \\(\\Cst\\)\\nobreakdash-algebra\n\\(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\\).\nIn this section, we recall how to compute the \\(\\K\\)\\nobreakdash-theory\nof \\(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\\)\nand we describe the map in \\(\\K\\)\\nobreakdash-theory\ninduced by the inclusion\n\\(\\Cst(\\mathbb{Z}^d)_\\mathbb{F} \\hookrightarrow \\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\\).\nIn particular, we show that this map is split surjective and that its\nkernel is generated by those elements that come from the\n\\(\\K\\)\\nobreakdash-theory\nof \\(\\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F}\\)\nfor a coordinate embedding \\(\\mathbb{Z}^{d-1} \\to \\mathbb{Z}^d\\).\nSo its kernel consists of those topological phases that are obtained\nby stacking lower-dimensional topological insulators in a coordinate\ndirection. These are called ``weak topological phases''\nin~\\cite{Fu-Kane-Mele:Insulators}. In the end, we argue that stable\nhomotopy instead of homotopy is the physically reasonable equivalence\nrelation on Hamiltonians.\n\nThe following arguments are easier and more standard in the complex\ncase. Hence we only discuss the real case. It is convenient to\nreplace real \\(\\Cst\\)\\nobreakdash-algebras by ``real'' ones, that is, complex\n\\(\\Cst\\)\\nobreakdash-algebras equipped with a real involution. We first\nrecall some basic facts and definitions about ``real'' and real\n\\(\\Cst\\)\\nobreakdash-algebras and then describe the relevant ``real''\n\\(d\\)\\nobreakdash-torus.\n\nA real \\(\\Cst\\)\\nobreakdash-algebra~\\(A\\)\ncorresponds to the ``real'' \\(\\Cst\\)\\nobreakdash-algebra\n\\(A\\otimes_\\mathbb{R} \\mathbb{C}\\)\nwith the real involution\n\\(\\conj{a\\otimes z} \\mathrel{\\vcentcolon=} a\\otimes \\conj{z}\\).\nA ``real'' \\(\\Cst\\)\\nobreakdash-algebra~\\(A\\)\ncorresponds to the real \\(\\Cst\\)\\nobreakdash-algebra\n\\[\nA_\\mathbb{R} \\mathrel{\\vcentcolon=} \\setgiven{a\\in A}{\\conj{a}=a}.\n\\]\nA ``real'' locally compact space~\\(X\\)\nis a locally compact space with an involutive homeomorphism\n\\(X\\to X\\),\n\\(x\\mapsto \\conj{x}\\).\nThen we turn \\(\\Cont_0(X)\\)\ninto a ``real'' \\(\\Cst\\)\\nobreakdash-algebra\nusing the real involution \\(\\conj{f}(x) \\mathrel{\\vcentcolon=} \\conj{f(\\conj{x})}\\)\nfor all \\(x\\in X\\), \\(f\\in\\Cont_0(X)\\). So\n\\[\n\\Cont_0(X)_\\mathbb{R} = \\setgiven*{f\\in\\Cont_0(X)}\n{f(\\conj{x}) = \\conj{f(x)}\\text{ for all }x\\in X}.\n\\]\nFor a ``real'' \\(\\Cst\\)\\nobreakdash-algebra~\\(A\\), we define\n\\[\n\\KR_*(A) \\mathrel{\\vcentcolon=} \\K_*(A_\\mathbb{R}).\n\\]\nFor a ``real'' locally compact space~\\(X\\), we let\n\\[\n\\KR^*(X)\n\\mathrel{\\vcentcolon=} \\KR_{-*}(\\Cont_0(X))\n= \\K_{-*}(\\Cont_0(X)_\\mathbb{R}).\n\\]\nNote the grading convention here, which is analogous to the numbering\nconvention when a chain complex is treated as a cochain complex.\n\nFrom now on, \\(\\Cst(\\mathbb{Z}^d)\\)\ndenotes the ``real'' \\(\\Cst\\)\\nobreakdash-algebra\nthat corresponds to the real \\(\\Cst\\)\\nobreakdash-algebra\n\\(\\Cst(\\mathbb{Z}^d)_\\mathbb{R}\\).\nThat is, the real involution acts on \\(f\\colon \\mathbb{Z}^d\\to\\mathbb{C}\\)\nby pointwise complex conjugation. We give the \\(d\\)\\nobreakdash-torus\n\\(\\mathbb{T}^d\\subseteq \\mathbb{C}^d\\)\nthe real involution by complex conjugation. So \\(\\Cont(\\mathbb{T}^d)_\\mathbb{R}\\)\nis the closed \\(\\mathbb{R}\\)\\nobreakdash-linear\nspan of the functions \\(z^k \\mathrel{\\vcentcolon=} z_1^{k_1} \\dotsm z_d^{k_d}\\)\non~\\(\\mathbb{T}^d\\)\nfor \\(k_1,\\dotsc,k_d\\in\\mathbb{Z}\\).\nThis is the unique real structure on~\\(\\mathbb{T}^d\\)\nfor which the Fourier isomorphism \\(\\Cst(\\mathbb{Z}^d) \\cong \\Cont(\\mathbb{T}^d)\\)\nis an isomorphism of ``real'' \\(\\Cst\\)\\nobreakdash-algebras. Thus\n\\[\n\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{R}) \\cong \\KR_*(\\Cont(\\mathbb{T}^d)) = \\KR^{-*}(\\mathbb{T}^d).\n\\]\nWe shall also use the ``real'' manifolds~\\(\\mathbb{R}^{p,q}\\)\nfor \\(p,q\\in\\mathbb{N}\\);\nthis is~\\(\\mathbb{R}^{p+q}\\)\nwith the real involution \\(\\conj{(x,y)} \\mathrel{\\vcentcolon=} (x,-y)\\)\nfor \\(x\\in\\mathbb{R}^p\\),\n\\(y\\in\\mathbb{R}^q\\).\nWe may also realise this as\n\\(\\mathbb{R}^p \\times (\\ima \\mathbb{R})^q \\subseteq \\mathbb{C}^{p+q}\\)\nwith complex conjugation as real involution.\n\n\\begin{proposition}\n \\label{pro:Cst_Zd_in_KK}\n The ``real'' \\(\\Cst\\)\\nobreakdash-algebra\n \\(\\Cont(\\mathbb{T})\\)\n is \\(\\KK\\)\\nobreakdash-equivalent\n to \\(\\mathbb{C}\\oplus \\Cont_0(\\mathbb{R}^{0,1})\\).\n And \\(\\Cont(\\mathbb{T}^d)\\)\n is \\(\\KK\\)\\nobreakdash-equivalent\n to a direct sum of copies of \\(\\Cont_0(\\mathbb{R}^{0,j})\\)\n for \\(j=0,\\dotsc,d\\),\n where the summand \\(\\Cont_0(\\mathbb{R}^{0,j})\\)\n appears \\(\\binom{d}{j}\\) times.\n The \\(\\K\\)\\nobreakdash-theory\n of \\(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}\\)\n is a free \\(\\K_*(\\mathbb{F})\\)-module\n of rank~\\(2^d\\),\n with \\(\\binom{d}{j}\\) generators of degree \\(-j \\bmod 8\\).\n\\end{proposition}\n\n\\begin{proof}\n The points \\(\\pm 1\\in\\mathbb{T}\\)\n are real, that is, fixed by the real involution. The complement\n \\(\\mathbb{T}\\setminus\\{1\\}\\)\n is diffeomorphic as a ``real'' manifold to~\\(\\mathbb{R}^{0,1}\\),\n say, by stereographic projection at~\\(1\\).\n Hence we get an extension of ``real'' \\(\\Cst\\)\\nobreakdash-algebras\n \\[\n \\Cont_0(\\mathbb{R}^{0,1}) \\rightarrowtail \\Cont(\\mathbb{T})\n \\twoheadrightarrow \\mathbb{C},\n \\]\n where the quotient map is evaluation at~\\(1\\).\n This extension splits by embedding~\\(\\mathbb{C}\\)\n as constant functions in \\(\\Cont(\\mathbb{T})\\).\n Since Kasparov theory is split-exact, also for ``real''\n \\(\\Cst\\)\\nobreakdash-algebras,\n \\(\\Cont(\\mathbb{T})\\)\n is \\(\\KK\\)\\nobreakdash-equivalent to \\(\\Cont_0(\\mathbb{R}^{0,1}) \\oplus \\mathbb{C}\\).\n\n We may get \\(\\Cst(\\mathbb{Z}^d)\\)\n by tensoring \\(d\\)~copies\n of \\(\\Cst(\\mathbb{Z})\\).\n The tensor product of \\(\\Cst\\)\\nobreakdash-algebras\n descends to a bifunctor in \\(\\KK\\)\\nobreakdash-theory,\n also in the ``real'' case. So \\(\\Cst(\\mathbb{Z}^d)\\)\n is \\(\\KK\\)\\nobreakdash-equivalent\n to the \\(d\\)\\nobreakdash-fold\n tensor power of \\(\\mathbb{C} \\oplus \\Cont_0(\\mathbb{R}^{0,1})\\).\n The tensor product of \\(\\Cst\\)\\nobreakdash-algebras\n is additive in each variable, and\n \\(\\Cont_0(\\mathbb{R}^{p,q})\\otimes \\Cont_0(\\mathbb{R}^{r,s}) \\cong\n \\Cont_0(\\mathbb{R}^{p+r,q+s})\\).\n A variant of the binomial formula now gives.\n \\[\n \\Cst(\\mathbb{Z}^d)\n \\sim_\\KK (\\mathbb{R} \\oplus \\Cont_0(\\mathbb{R}^{0,1}))^{\\otimes d}\n \\cong \\bigoplus_{j=0}^d \\binom{d}{j} \\Cont_0(\\mathbb{R}^{0,j}).\n \\]\n A Bott periodicity theorem by Kasparov shows that\n \\(\\Cont_0(\\mathbb{R}^{p,q})\\)\n is \\(\\KK\\)\\nobreakdash-equivalent\n to~\\(\\Cont_0(\\mathbb{R}^{0,0})\\)\n with a dimension shift of \\(p-q\\),\n see \\cite{Kasparov:Operator_K}*{Theorem~7}. This implies the claim\n about \\(\\K\\)\\nobreakdash-theory.\n\\end{proof}\n\n\\begin{proposition}\n \\label{pro:low_dimensional_killed}\n Let \\(\\varphi\\colon \\mathbb{Z}^{d-1}\\to \\mathbb{Z}^d\\)\n be an injective group homomorphism. It induces an injective\n $^*$\\nobreakdash-{}homomorphism\n \\(\\varphi_*\\colon \\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F} \\to\n \\Cst(\\mathbb{Z}^d)_\\mathbb{F}\\)\n and a grading-preserving \\(\\K_*(\\mathbb{F})\\)-module\n homomorphism\n \\(\\K_*(\\varphi_*)\\colon \\K_*(\\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F}) \\to\n \\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\).\n The map\n \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}) \\to \\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\)\n vanishes on the image of~\\(\\K_*(\\varphi_*)\\).\n\\end{proposition}\n\n\\begin{proof}\n Proposition~\\ref{pro:low_dimensional_killed_Roe} shows\n that~\\(\\varphi\\)\n induces the zero map on the \\(\\K\\)\\nobreakdash-theory\n of the Roe \\(\\Cst\\)\\nobreakdash-algebra.\n The canonical map \\(\\Cred(G) \\hookrightarrow \\Cst_\\Roe(G)\\)\n for a group~\\(G\\)\n is a natural transformation with respect to injective group\n homomorphisms. So there is a commuting square\n \\[\n \\begin{tikzcd}\n \\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F}\n \\arrow[r, hookrightarrow] \\arrow[d, hookrightarrow, \"\\varphi_*\"]&\n \\Cst_\\Roe(\\mathbb{Z}^{d-1})_\\mathbb{F} \\arrow[d, hookrightarrow, \"\\varphi_*\"]\\\\\n \\Cst(\\mathbb{Z}^d)_\\mathbb{F}\n \\arrow[r, hookrightarrow]&\n \\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\n \\end{tikzcd}\n \\]\n This implies the statement.\n\\end{proof}\n\nThe coordinate embeddings\n\\[\n\\iota_k\\colon \\mathbb{Z}^{d-1}\\to \\mathbb{Z}^d,\\qquad\n(x_1,\\dotsc,x_{d-1}) \\mapsto\n(x_1,\\dotsc,x_{k-1},0,x_k,\\dotsc,x_{d-1}),\n\\]\nare injective group homomorphisms and induce injective\n$^*$\\nobreakdash-{}homomorphisms\n\\[\n\\iota_k\\colon \\Cst(\\mathbb{Z}^{d-1}) \\to \\Cst(\\mathbb{Z}^d).\n\\]\nThe Fourier transform maps \\(\\iota_k(\\Cst(\\mathbb{Z}^{d-1}))\\)\nonto the \\(\\Cst\\)\\nobreakdash-subalgebra\nof \\(\\Cont(\\mathbb{T}^d)\\)\nconsisting of all functions that are constant equal to~\\(1\\)\nin the \\(k\\)th\ncoordinate direction. We have seen that\n\\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nis a free \\(\\K_*(\\mathbb{F})\\)-module\nof rank~\\(2^d\\)\n(with generators in different degrees). Now compute\n\\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nas in Proposition~\\ref{pro:Cst_Zd_in_KK}. The inclusion of functions\nthat are constant in the \\(k\\)th\ndirection corresponds in \\(\\K\\)\\nobreakdash-theory\nto the inclusion of those~\\(2^{d-1}\\)\nof the~\\(2^d\\)\nfree \\(\\K_*(\\mathbb{R})\\)-module\nsummands in \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nwhere we take the summand~\\(\\mathbb{R}\\)\nin the \\(k\\)th\nfactor. The summands in the image of~\\(\\K_*(\\iota_k)\\)\ncorrespond to topological insulators that are built by stacking\ncopies of a \\(d-1\\)-dimensional\ninsulator in the \\(k\\)th\ndirection. Such topological insulators are considered weak by\nFu--Kane--Mele~\\cite{Fu-Kane-Mele:Insulators}. So the map\n\\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}) \\to \\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\)\nkills the \\(\\K\\)\\nobreakdash-theory classes of weak topological insulators.\n\nIf~\\(k\\)\nvaries, then all but one of the~\\(2^d\\)\nsummands \\(\\K_{*-j}(\\mathbb{F})\\)\nin \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nare in the image of \\(\\K_*(\\iota_k)\\)\nfor some \\(k\\in\\{1,\\dotsc,d\\}\\).\nAll these summands are mapped to~\\(0\\)\nin \\(\\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\)\nby Proposition~\\ref{pro:low_dimensional_killed}. The remaining\nsummand is the \\(\\K\\)\\nobreakdash-theory\nof the ideal \\(\\Cont_0(\\mathbb{R}^{0,d}) \\idealin \\Cont(\\mathbb{T}^d)\\).\nHere we identify~\\(\\mathbb{R}^{0,d}\\)\nwith an open subset of~\\(\\mathbb{T}^d\\)\nusing the stereographic projection in each variable, compare the proof\nof Proposition~\\ref{pro:Cst_Zd_in_KK}. Its ``real'' or complex\n\\(\\K\\)\\nobreakdash-theory\nis identified with \\(\\K_{*-d}(\\mathbb{R})\\)\nor \\(\\K_{*-d}(\\mathbb{C})\\)\nby Bott periodicity. Kasparov proves Bott periodicity isomorphisms\n\\(\\KR_*(\\Cont_0(\\mathbb{R}^{p,q})) \\cong \\K_{*+p-q}(\\mathbb{R})\\)\nusing a canonical generator~\\(\\alpha_{p,q}\\)\nfor the \\(\\K\\)\\nobreakdash-homology group\n\\[\n\\KK_{q-p}^\\mathbb{R}(\\Cont_0(\\mathbb{R}^{p,q}),\\mathbb{C})\n\\cong \\KK_0^\\mathbb{R}(\\Cont_0(\\mathbb{R}^{p,q}, \\Cliff_{p,q}),\\mathbb{C});\n\\]\nhere~\\(\\Cliff_{p,q}\\)\nis the Clifford algebra with \\(p+q\\)~anti-commuting,\nodd, self-adjoint generators \\(\\gamma_1,\\dotsc,\\gamma_{p+q}\\)\nwith \\(\\conj{\\gamma_i} = \\gamma_i\\)\nfor \\(1\\le i\\le p\\)\nand \\(\\conj{\\gamma_i} = -\\gamma_i\\)\nfor \\(p+1\\le i \\le p+q\\).\nAnd we write~\\(\\KK^\\mathbb{R}\\)\nto highlight that the entries are treated as ``real''\n\\(\\Cst\\)\\nobreakdash-algebras.\n\nThe Bott generators~\\(\\alpha_{p,0}\\)\nare generalised by Kasparov in \\cite{Kasparov:Novikov}*{Definition\n and Lemma 4.2} to build a ``fundamental class''\n\\[\n\\alpha_X \\in \\KK^\\mathbb{R}_0(\\Cont_0(X,\\Cliff X),\\mathbb{C})\n\\]\nfor any complete Riemannian manifold~\\(X\\)\n(without boundary). Here~\\(\\Cliff X\\)\nis the bundle of ``real'' \\(\\Cst\\)\\nobreakdash-algebras\nover~\\(X\\)\nwhose fibre at \\(x\\in X\\)\nis the \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n``real'' Clifford algebra of the cotangent space~\\(T_x^* X\\)\nfor the positive definite quadratic form induced by the Riemannian\nmetric, and \\(\\Cont_0(X,\\Cliff X)\\)\nmeans the \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n``real'' \\(\\Cst\\)\\nobreakdash-algebra\nof \\(\\Cont_0\\)\\nobreakdash-sections\nof this Clifford algebra bundle. We now adapt Kasparov's fundamental\nclass to the case where~\\(X\\)\nis a ``real'' complete Riemannian manifold, in such a way that the\nfundamental class for~\\(\\mathbb{R}^{p,q}\\)\nis the generator~\\(\\alpha_{p,q}\\)\nof Bott periodicity from~\\cite{Kasparov:Operator_K}. The only changes\nare in the real structure. In particular, all the analysis needed to\nproduce cycles for Kasparov theory is already done\nin~\\cite{Kasparov:Novikov}.\n\nRecall that the real involution on \\(\\Cont_0(X)\\)\nis defined by \\(\\conj{f}(x) \\mathrel{\\vcentcolon=} \\conj{f(\\conj{x})}\\)\nfor \\(f\\in\\Cont_0(X)\\).\nThere is a unique conjugate-linear involution on the space of\ncomplex \\(1\\)\\nobreakdash-forms on~\\(X\\)\nsuch that \\(\\conj{\\diff f} = \\diff{\\conj{f}}\\)\nfor all smooth \\(f\\in\\Cont_0(X)\\).\nThere is a unique conjugate-linear involution on\n\\(\\Cont_0(X,\\Cliff X)\\) with\n\\[\n\\conj{\\omega_1 \\dotsm \\omega_m}\n= \\conj{\\omega_1} \\dotsm \\conj{\\omega_m}\n\\]\nfor all sections \\(\\omega_1,\\dotsc,\\omega_m\\)\nof \\(T^* X \\otimes\\mathbb{C}\\).\nThis involution is also compatible with the multiplication and the\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-grading. So it turns \\(\\Cont_0(X,\\Cliff X)\\)\ninto a \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded ``real'' \\(\\Cst\\)\\nobreakdash-algebra.\n\nLet~\\(L^2(\\Lambda^*(X))\\)\nbe the Hilbert space of square-integrable complex differential forms\non~\\(X\\).\nThis is the underlying Hilbert space of Kasparov's fundamental\nclass. It is \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\nso that sections of~\\(\\Lambda^{2\\ell}(X)\\)\nare even and sections of~\\(\\Lambda^{2\\ell+1}(X)\\)\nare odd. There is a unique conjugate-linear, isometric involution\non \\(L^2(\\Lambda^*(X))\\) with\n\\[\n\\conj{\\omega_1 \\wedge \\dotsb \\wedge \\omega_\\ell} =\n\\conj{\\omega_1} \\wedge \\dotsb \\wedge \\conj{\\omega_\\ell}\n\\]\nfor all complex \\(1\\)\\nobreakdash-forms \\(\\omega_1,\\dotsc,\\omega_\\ell\\).\nIt commutes with the \\(\\mathbb{Z}\/2\\)\\nobreakdash-grading,\nso that \\(L^2(\\Lambda^*(X))\\)\nbecomes a \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n``real'' Hilbert space.\n\nGiven a complex \\(1\\)\\nobreakdash-form~\\(\\omega\\)\nand a differential form~\\(\\eta\\),\nlet \\(\\lambda_\\omega(\\eta) \\mathrel{\\vcentcolon=} \\omega \\wedge \\eta\\).\nThese operators satisfy the relations\n\\begin{equation}\n \\label{eq:lambda_and_star}\n \\lambda_\\omega \\lambda_\\eta + \\lambda_\\eta \\lambda_\\omega=0,\\qquad\n \\lambda_\\omega^* \\lambda_\\eta + \\lambda_\\eta \\lambda_\\omega^*\n = \\braket{\\omega}{\\eta}\n\\end{equation}\nfor all complex \\(1\\)\\nobreakdash-forms \\(\\omega,\\eta\\),\nwhere \\(\\braket{\\omega}{\\eta}\\in\\Cont_0(X)\\)\ndenotes the pointwise inner product, which acts on\n\\(L^2(\\Lambda^*(X))\\)\nby pointwise multiplication.\nThe representation of \\(\\Cont_0(X,\\Cliff X)\\)\non \\(L^2(\\Lambda^*(X))\\)\nis defined by letting a complex \\(1\\)\\nobreakdash-form~\\(\\omega\\),\nviewed as an element of \\(\\Cont_0(X,\\Cliff X)\\),\nact by \\(\\lambda_\\omega + \\lambda_{\\omega^*}^*\\).\nHere~\\(\\omega^*\\)\nis the adjoint of~\\(\\omega\\)\nin the \\(\\Cst\\)\\nobreakdash-algebra \\(\\Cont_0(X,\\Cliff X)\\),\nthat is,\n\\(\\omega^*(x) = \\omega(x)^*\\)\nfor all \\(x\\in X\\),\nwhere \\(\\omega(x)^* \\in T^*_x X \\otimes \\mathbb{C}\\)\nis the pointwise complex conjugation in the second tensor\nfactor~\\(\\mathbb{C}\\).\nThis defines a $^*$\\nobreakdash-{}representation of \\(\\Cont_0(X,\\Cliff X)\\)\nby~\\eqref{eq:lambda_and_star}. It is grading-preserving and real as\nwell.\n\nLet~\\(d\\) be the de Rham differential, defined on smooth\nsections of~\\(\\Lambda^*(X)\\)\nwith compact support, and let~\\(d^*\\)\nbe its adjoint. The unbounded operator \\(\\mathcal{D} \\mathrel{\\vcentcolon=} d+d^*\\)\nis essentially self-adjoint because~\\(X\\)\nis complete. So\n\\[\nF \\mathrel{\\vcentcolon=} (1+\\mathcal{D}^2)^{-1\/2} \\mathcal{D}\n\\]\nis a well defined self-adjoint operator. The operator~\\(d\\) is odd\nand real. This is inherited by~\\(\\mathcal{D}\\)\nand~\\(F\\).\nKasparov shows that \\((1-F^2)\\cdot a\\)\nand \\([F,a]\\)\nare compact for all \\(a\\in \\Cont_0(X,\\Cliff X)\\).\nThus \\(\\alpha_X \\mathrel{\\vcentcolon=} (L^2(\\Lambda^*(X)),F)\\)\nis a cycle for the ``real'' Kasparov group\n\\(\\KK^\\mathbb{R}_0(\\Cont_0(X,\\Cliff X),\\mathbb{C})\\).\nWe call this the \\emph{fundamental class} of the ``real''\nmanifold~\\(X\\).\n(Kasparov calls it ``Dirac element'' instead.)\n\nIn particular, the fundamental class of the ``real''\nmanifold~\\(\\mathbb{R}^{p,q}\\)\nbecomes the Bott periodicity generator~\\(\\alpha_{p,q}\\)\nfrom~\\cite{Kasparov:Operator_K} when we trivialise the Clifford\nalgebra bundle on~\\(\\mathbb{R}^{p,q}\\)\nin the obvious way. So\n\\(\\alpha_{\\mathbb{R}^{p,q}}\\in \\KK^\\mathbb{R}_0(\\Cont_0(\\mathbb{R}^{p,q}) \\otimes\n\\Cliff_{p,q},\\mathbb{C})\\)\nis invertible.\n\nWe give~\\(\\mathbb{T}^d\\)\nthe \\(\\mathbb{T}^d\\)\\nobreakdash-invariant\nRiemannian metric to build its fundamental class.\nThe torus~\\(\\mathbb{T}^d\\)\nis parallelisable as a ``real'' manifold: its tangent bundle is\nisomorphic to \\(\\mathbb{T}^d \\times \\mathbb{R}^{0,d}\\).\nThis induces an isomorphism\n\\(\\Cont(\\mathbb{T}^d, \\Cliff \\mathbb{T}^d) \\cong \\Cont(\\mathbb{T}^d) \\otimes \\Cliff_{0,d}\\).\nSo the fundamental class~\\(\\alpha_{\\mathbb{T}^d}\\)\nalso gives an element in \\(\\KK^\\mathbb{R}_d(\\Cont(\\mathbb{T}^d),\\mathbb{C})\\).\n\nLet~\\(\\Hils[L]\\)\nbe a separable ``real'' Hilbert space and build \\(\\Cst_\\Roe(\\mathbb{Z}^d)\\)\non the ``real'' Hilbert space \\(\\ell^2(\\mathbb{Z}^d,\\Hils[L])\\).\nThere is an obvious embedding\n\\(\\Cst(\\mathbb{Z}^d) \\otimes \\mathbb K(\\Hils[L]) \\subseteq \\Cst_\\Roe(\\mathbb{Z}^d)\\). Let\n\\[\n\\alpha_{\\mathbb{T}^d}^{\\Hils[L]} \\in\n\\KK^\\mathbb{R}_0(\\Cont(\\mathbb{T}^d) \\otimes \\Cliff_{0,d} \\otimes \\mathbb K(\\Hils[L]), \\mathbb{C})\n\\cong\n\\KK^\\mathbb{R}_0(\\Cst(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\otimes \\mathbb K(\\Hils[L]), \\mathbb{C})\n\\]\nbe the exterior product of the fundamental class~\\(\\alpha_{\\mathbb{T}^d}\\)\nand the Morita equivalence \\(\\mathbb K(\\Hils[L]) \\sim \\mathbb{C}\\).\nThis is the Kasparov cycle with underlying \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n``real'' Hilbert space\n\\(L^2(\\mathbb{T}^d, \\Lambda^*(\\mathbb{C}^d)) \\otimes \\Hils[L]\\)\nwith the operator \\(\\tilde{F} \\mathrel{\\vcentcolon=} F\\otimes 1\\)\nwith~\\(F\\)\nas above for the manifold \\(X=\\mathbb{T}^d\\).\nSo~\\(\\tilde{F}\\)\nis an odd, self-adjoint, real bounded operator with\n\\begin{equation}\n \\label{eq:Kasparov_cycle}\n [\\tilde{F},T],(1-\\tilde{F}^2)\\cdot T\\in\n \\mathbb K(\\ell^2(\\mathbb{Z}) \\otimes \\Lambda^*(\\mathbb{C}^d) \\otimes \\Hils[L])\n\\end{equation}\nfor all\n\\(T\\in\\Cst(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\otimes \\mathbb K(\\Hils[L])\\)\n(the commutator is the graded one).\n\n\\begin{theorem}\n \\label{the:fundamental_class}\n Equation~\\eqref{eq:Kasparov_cycle} still holds for\n \\(T\\in\\Cst_\\Roe(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d}\\).\n This gives\n \\[\n \\alpha'_{\\mathbb{T}^d}\n \\mathrel{\\vcentcolon=} [(L^2(\\mathbb{T}^d, \\Lambda^*(\\mathbb{C}^d)) \\otimes \\Hils[L], \\tilde{F})]\n \\in \\KK^\\mathbb{R}_0(\\Cst_\\Roe(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d}, \\mathbb{C}).\n \\]\n The following diagram in~\\(\\KK^\\mathbb{R}\\) commutes:\n \\[\n \\begin{tikzcd}[column sep=2.3em]\n \n \\Cont_0(\\mathbb{R}^{0,d}, \\Cliff_{0,d}) \\arrow[r, \"\\textup{incl.}\"]\n \\arrow[dr, \"\\alpha_{\\mathbb{R}^{0,d}}\", \"\\cong\"'] &\n \\Cont(\\mathbb{T}^d, \\Cliff_{0,d}) \\arrow[r, \"\\textup{Fourier}\"]\n \\arrow[d, \"\\alpha_{\\mathbb{T}^d}\"] &\n \\Cst(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\arrow[r, \"\\textup{incl.}\"] &\n \\Cst_\\Roe(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\arrow[dll, \"\\alpha'_{\\mathbb{T}^d}\"]\n \\\\ &\\mathbb{C}\n \\end{tikzcd}\n \\]\n\\end{theorem}\n\n\\begin{corollary}\n \\label{cor:K_periodic_split-injective}\n The inclusion \\(\\Cont_0(\\mathbb{R}^{0,d}) \\to \\Cst_\\Roe(\\mathbb{Z}^d)\\)\n induces a split injective map\n \\(\\KR_*(\\Cont_0(\\mathbb{R}^{0,d})) \\to \\KR_*(\\Cst_\\Roe(\\mathbb{Z}^d))\\).\n The map\n \\(\\K_{*+d}(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}) \\to \\K_*(\\mathbb{F})\\)\n induced by~\\(\\alpha'_{\\mathbb{T}^d}\\)\n is an isomorphism. Analogous statements hold in complex\n \\(\\K\\)\\nobreakdash-theory.\n\\end{corollary}\n\n\\begin{proof}[Proof of the corollary]\n Both \\(\\KR_*(\\Cont_0(\\mathbb{R}^{0,d}))\\)\n and \\(\\KR_*(\\Cst_\\Roe(\\mathbb{Z}^d))\\)\n are isomorphic to free \\(\\K_*(\\mathbb{R})\\)-modules\n with a generator in degree~\\(-d\\).\n The Bott periodicity generator~\\(\\alpha_{\\mathbb{R}^{0,d}}\\)\n maps the generator of \\(\\KR_{-d}(\\Cont_0(\\mathbb{R}^{0,d}))\\)\n onto a generator of \\(\\K_0(\\mathbb{R})\\).\n The commuting diagram in Theorem~\\ref{the:fundamental_class} shows\n that its image in \\(\\KR_{-d}(\\Cst_\\Roe(\\mathbb{Z}^d))\\)\n must be a generator as well. So~\\(\\alpha'_{\\mathbb{T}^d}\\)\n acts by multiplication with~\\(\\pm1\\)\n on a generator. Since~\\(\\alpha'_{\\mathbb{T}^d}\\)\n is a \\(\\K\\)\\nobreakdash-homology\n class, the map on \\(\\K\\)\\nobreakdash-theory\n that it induces is a \\(\\K_*(\\mathbb{R})\\)-module\n homomorphism. Hence it is multiplication by~\\(\\pm1\\)\n everywhere once this happens on a generator. So the map\n on~\\(\\KR_*\\)\n induced by~\\(\\alpha'_{\\mathbb{T}^d}\\)\n is invertible. The same proof works for complex \\(\\K\\)\\nobreakdash-theory.\n\\end{proof}\n\nWe have already shown that all but one of the free\n\\(\\K_*(\\mathbb{F})\\)-module\nsummands in \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nare killed by the map to \\(\\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\).\nWhen we combine this with the above corollary, it follows that the\nkernel of the map from \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nto \\(\\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\)\nis exactly the sum of the images of \\(\\K_*(\\iota_k)\\)\nfor \\(k=1,\\dotsc,d\\),\nthat is, the subgroup generated by the \\(\\K\\)\\nobreakdash-theory\nclasses of weak topological insulators.\n\nWe still have to prove Theorem~\\ref{the:fundamental_class}. The\nleft triangle in the diagram in Theorem~\\ref{the:fundamental_class}\ncommutes because of the following general fact:\n\n\\begin{proposition}\n \\label{pro:fundamental_class_open_restrict}\n Let \\(U\\subseteq X\\)\n be an open subset of a ``real'' manifold~\\(X\\)\n that is invariant under the real involution.\n Give \\(U\\)\n and~\\(X\\)\n some complete Riemannian metrics.\n The Kasparov product of the ideal inclusion\n \\(j\\colon \\Cont_0(U,\\Cliff U) \\hookrightarrow \\Cont_0(X,\\Cliff X)\\)\n and the fundamental class\n \\(\\alpha_X\\in \\KK_0^\\mathbb{R}(\\Cont_0(X,\\Cliff X), \\mathbb{C})\\)\n is the fundamental class\n \\(\\alpha_U\\in \\KK_0^\\mathbb{R}(\\Cont_0(U,\\Cliff U), \\mathbb{C})\\).\n In particular, the fundamental class does not depend on the choice\n of the Riemannian metric.\n\\end{proposition}\n\n\\begin{proof}\n Both Kasparov cycles \\(j^*(\\alpha_X)\\)\n and~\\(\\alpha_U\\)\n live on Hilbert spaces of \\(L^2\\)\\nobreakdash-differential\n forms on~\\(U\\).\n Here square-integrability is with respect to different metrics.\n The resulting Hilbert spaces are isomorphic by pointwise\n application of a suitable strictly positive, smooth function\n \\(X\\to \\mathbb B(\\Lambda^* X)\\).\n This isomorphism also respects the \\(\\mathbb{Z}\/2\\)\\nobreakdash-grading\n and the real involution. The operator~\\(\\mathcal{D}\\)\n used to construct~\\(\\alpha_X\\)\n is a first-order differential operator. Hence \\(F \\mathrel{\\vcentcolon=}\n (1+\\mathcal{D}^2)^{-1\/2} \\mathcal{D}\\)\n is an order-zero pseudodifferential operator, and it has the same\n symbol as~\\(\\mathcal{D}\\).\n This is the function\n \\[\n S^* X \\to \\mathbb B(\\Lambda^*(X)),\\qquad\n (x,\\xi) \\mapsto \\lambda_\\xi + \\lambda_\\xi^*.\n \\]\n The symbol of the operator~\\(F\\)\n of~\\(\\alpha_U\\)\n is given by the same formula, except that the adjoint is for\n another Riemannian metric. So the unitary\n between the spaces of \\(L^2\\)\\nobreakdash-forms\n will also identify these symbols. The class of the Kasparov\n cycle defined by an order-zero pseudodifferential operator~\\(F\\)\n depends only on the symbol of~\\(F\\).\n So \\(j^*(\\alpha_X)\\)\n and~\\(\\alpha_U\\)\n have the same class in \\(\\KK_0^\\mathbb{R}(\\Cont_0(U,\\Cliff U), \\mathbb{C})\\).\n The last statement is the case \\(U=X\\)\n of the proposition.\n\\end{proof}\n\nNow we build the Kasparov cycle\n\\(\\alpha'_{\\mathbb{T}^d}\\in \\KK_0^\\mathbb{R}(\\Cst_\\Roe(\\mathbb{Z}^d)\\otimes \\Cliff_{0,d},\\mathbb{C})\\).\nLet \\(z_j\\colon \\mathbb{T}^d\\to\\mathbb{C}\\)\nbe the \\(j\\)th coordinate function and let\n\\[\nz^k \\mathrel{\\vcentcolon=} z_1^{k_1} \\dotsm z_d^{k_d}\n\\qquad\\text{for }\nk=(k_1,\\dotsc,k_d)\\in\\mathbb{Z}^d.\n\\]\nThe real involution on~\\(\\mathbb{T}^d\\)\nis defined so that these are real elements of \\(\\Cont(\\mathbb{T}^d)\\).\nHence the \\(1\\)\\nobreakdash-forms \\(z_j^{-1} \\diff z_j\\)\nfor \\(j=1,\\dotsc,d\\)\nare real. They form a basis of the space of \\(1\\)\\nobreakdash-forms\nas a \\(\\Cont(\\mathbb{T}^d)\\)-module.\nThe differential forms\n\\[\nz^k\\cdot (z_{i_1} \\dotsm z_{i_\\ell})^{-1}\n\\diff z_{i_1} \\wedge \\dotsb \\wedge \\diff z_{i_\\ell}\n\\]\nfor \\(k\\in\\mathbb{Z}^d\\)\nand \\(1\\le i_1 < i_2 < \\dotsb < i_\\ell\\le d\\)\nform a real, orthonormal basis of the Hilbert space\n\\(L^2(\\Lambda^*(\\mathbb{T}^d))\\).\nHence there is a unitary operator\n\\begin{multline*}\n U\\colon\n \\ell^2(\\mathbb{Z}^d) \\otimes \\Lambda^*(\\mathbb{C}^d) \\xrightarrow\\sim\n L^2(\\Lambda^*(\\mathbb{T}^d)),\\\\\n \\delta_k \\otimes e_{i_1} \\wedge \\dotsb \\wedge e_{i_\\ell} \\mapsto\n z^k\\cdot (z_{i_1} \\dotsm z_{i_\\ell})^{-1}\n \\diff z_{i_1} \\wedge \\dotsb \\wedge \\diff z_{i_\\ell}.\n\\end{multline*}\nThis unitary is grading-preserving and real for the\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-grading and real structure on\n\\(\\ell^2(\\mathbb{Z}^d) \\otimes \\Lambda^\\ell(\\mathbb{C}^d)\\)\nwhere the standard basis vector\n\\(\\delta_k \\otimes e_{i_1} \\wedge \\dotsb \\wedge e_{i_\\ell}\\)\nis real and is even or odd depending on the parity of~\\(\\ell\\).\n\nThe above trivialisation of the cotangent bundle of~\\(\\mathbb{T}^d\\)\ngives the isomorphism\n\\[\n\\Cont(\\mathbb{T}^d) \\otimes \\Cliff_{0,d} \\xrightarrow\\sim \\Cont(\\mathbb{T}^d,\\Cliff \\mathbb{T}^d),\n\\qquad\n\\gamma_j\\mapsto \\ima z_j^{-1}\\,\\diff z_j;\n\\]\nrecall that \\(\\gamma_1,\\dotsc,\\gamma_d\\)\nare the odd, self-adjoint, anti-commuting unitaries that\ngenerate~\\(\\Cliff_{0,d}\\).\nThe action of \\(\\Cont(\\mathbb{T}^d,\\Cliff \\mathbb{T}^d)\\)\non \\(L^2(\\Lambda^* \\mathbb{T}^d)\\)\nnow translates to an action of\n\\(\\Cont(\\mathbb{T}^d) \\otimes \\Cliff_{0,d}\\)\non \\(\\ell^2(\\mathbb{Z}^d) \\otimes \\Lambda^*(\\mathbb{C}^d)\n\\cong \\ell^2(\\mathbb{Z}^d, \\Lambda^*(\\mathbb{C}^d))\\).\nNamely, the scalar-valued function \\(z^k\\in\\Cont(\\mathbb{T}^d)\\)\nacts by the shift \\((\\tau_k f)(n) \\mathrel{\\vcentcolon=} f(n-k)\\)\nfor all \\(k,n\\in\\mathbb{Z}^d\\),\n\\(f\\in \\ell^2(\\mathbb{Z}^d,\\Lambda^*(\\mathbb{C}^d))\\).\nAnd the Clifford generator \\(\\gamma_j\\in\\Cliff_{0,d}\\)\nacts by\n\\[\n(\\gamma_j f)(n)\n= \\ima \\lambda_{e_j}\\bigl(f(n)\\bigr)\n- \\ima \\lambda_{e_j}^*\\bigl(f(n)\\bigr).\n\\]\nThe unitary~\\(U^*\\)\nmaps the domain of~\\(d\\)\nto the space of rapidly decreasing functions\n\\(\\mathbb{Z}^d\\to\\Lambda^*(\\mathbb{C}^d)\\),\nwhere~\\(U^* d U\\)\nacts by pointwise application of the function\n\\[\nA\\colon \\mathbb{Z}^d\\to \\mathbb B(\\Lambda^*(\\mathbb{C}^d)),\\qquad\nn \\mapsto \\lambda_n\n= \\sum_{j=1}^d n_j\\cdot \\lambda_{e_j},\n\\]\nbecause\n\\[\nd\\left(z^k\\cdot \\frac{\\diff z_{i_1}}{z_{i_1}} \\wedge \\dotsc\n\\wedge \\frac{\\diff z_{i_\\ell}}{z_{i_\\ell}}\\right)\n= \\sum_{j=1}^d k_j z^k \\cdot \\frac{\\diff z_j}{z_j}\n\\wedge \\frac{\\diff z_{i_1}}{z_{i_1}} \\wedge \\dotsc\n\\wedge \\frac{\\diff z_{i_\\ell}}{z_{i_\\ell}}.\n\\]\nSo~\\(U^* \\mathcal{D} U\\)\nacts by pointwise application of the matrix-valued function \\(A+A^*\\)\non the space of rapidly decreasing functions\n\\(\\mathbb{Z}^d\\to\\Lambda^*(\\mathbb{C}^d)\\).\nWe compute\n\\[\n(A+A^*)^2(n)\n= \\lambda_n \\lambda_n^* + \\lambda_n^* \\lambda_n\n= \\norm{n}^2.\n\\]\nSo~\\(U^* F U\\)\nacts by pointwise application of the matrix-valued function\n\\[\n\\hat\\alpha_{\\mathbb{Z}^d}\\colon\n\\mathbb{Z}^d \\to \\mathbb B(\\Lambda^*(\\mathbb{C}^d)),\\qquad\nn \\mapsto (1+\\norm{n}^2)^{-1\/2} (\\lambda_n + \\lambda_n^*).\n\\]\n\nNext we take the exterior product with the Morita equivalence\nbetween \\(\\mathbb K(\\Hils[L])\\)\nand~\\(\\mathbb{C}\\).\nThis simply gives the Hilbert space\n\\(\\ell^2(\\mathbb{Z}^d,\\Lambda^*(\\mathbb{C}^d)) \\otimes \\Hils[L]\\)\nwith the induced \\(\\mathbb{Z}\/2\\)\\nobreakdash-grading\nand ``real'' structure, the exterior tensor product representation\nof \\(\\Cont(\\mathbb{T}^d)\\otimes\\Cliff_{0,d}\\otimes \\mathbb K(\\Hils[L])\\),\nand with the operator \\(F\\otimes 1_{\\Hils[L]}\\).\nThis is a Kasparov cycle for\n\\(\\KK^\\mathbb{R}_0(\\Cst(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\otimes \\mathbb K(\\Hils[L]),\n\\mathbb{C})\\).\nIn particular, the operator \\(\\tilde{F} \\mathrel{\\vcentcolon=} U^* F U\\otimes 1_{\\Hils[L]}\\)\nis real, odd, and self-adjoint.\nLet \\(T\\in \\Cst_\\Roe(\\mathbb{Z}^d) \\subseteq \\mathbb B(\\ell^2(\\mathbb{Z}^d,\\Hils[L]))\\)\nand \\(S\\in\\Cliff_{0,d}\\).\nWe must show that \\((1-\\tilde{F}^2) \\cdot (T\\otimes S)\\)\nand \\([\\tilde{F}^2, T\\otimes S]\\)\nare compact operators.\nThe operator \\(1 - \\tilde{F}^2\\)\nacts by pointwise multiplication with \\((1+\\norm{n}^2)^{-1}\\).\nSince~\\(T\\)\nis locally compact and~\\(\\Lambda^* \\mathbb{C}^d\\)\nhas finite dimension, the operator\n\\((1-\\tilde{F}^2) \\cdot (T\\otimes S)\\)\nis compact. Describe~\\(T\\)\nas a block matrix \\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\nwith \\(T_{x,y}\\in\\mathbb B(\\Hils[L])\\).\nThe operator~\\(\\tilde{F}\\)\nanti-commutes with \\(1\\otimes S\\).\nSo the graded commutator\n\\([A+A^*,T\\otimes S] = [A+A^*,T\\otimes 1]\\cdot (1\\otimes S)\\)\ncorresponds to the block matrix with \\((x,y)\\)-entry\n\\[\nT_{x,y} \\otimes (\\lambda_{x-y} + \\lambda_{x-y}^*) S\n\\in \\mathbb B(\\Hils[L]\\otimes \\Lambda^* \\mathbb{C}^d).\n\\]\nAssume that~\\(T\\)\nis \\(R\\)\\nobreakdash-controlled,\nthat is, \\(T_{x,y}=0\\)\nif \\(\\norm{x-y}>R\\),\nand that \\(\\sup_x \\sum_y \\norm{T_{x,y}}\\)\nand \\(\\sup_y \\sum_x \\norm{T_{x,y}}\\)\nare bounded; block matrices with these two properties give bounded\noperators, and these are dense in the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nFor such~\\(T\\),\nthe commutator \\([A+A^*,T\\otimes S]\\)\nsatisfies analogous bounds because\n\\(\\norm{\\lambda_{x-y}+\\lambda_{x-y}^*} \\le 2 \\norm{x-y} \\le 2 R\\)\nwhenever \\(T_{x,y}\\neq0\\).\nSo the set of \\(T\\in \\Cst_\\Roe(X)\\)\nfor which \\([A+A^*,T \\otimes S]\\)\nis bounded is dense in \\(\\Cst_\\Roe(X)\\).\nThus~\\(A+A^*\\)\ndefines a spectral triple over\n\\(\\Cst_\\Roe(X) \\otimes \\Cliff_{0,d}\\).\nAs a consequence, \\([\\tilde{F}, T\\otimes 1]\\)\nis compact for all \\(T\\in\\Cst_\\Roe(X) \\otimes \\Cliff_{0,d}\\).\nThis finishes the proof of Theorem~\\ref{the:fundamental_class}.\n\n\n\\subsection{Another topological artefact of the tight binding\n approximation}\n\\label{sec:artefact_tight_binding}\n\nWe already argued in the introduction that the tight binding\napproximation may produce topological artefacts. Namely, it suggests\nto use the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\ninstead of the Roe \\(\\Cst\\)\\nobreakdash-algebra,\nwhose \\(\\K\\)\\nobreakdash-theory\nis much larger. We briefly mention another artefact caused by the\ntight binding approximation.\n\nWe work in Bloch--Floquet theory for greater clarity. The\nFermi projection of a Hamiltonian is described by a vector bundle\n\\(V\\twoheadrightarrow \\mathbb{T}^d\\)\nover the \\(d\\)\\nobreakdash-torus,\nmaybe with extra symmetries. Here~\\(d\\)\nis the dimension of the material, which is \\(2\\)\nor~\\(3\\)\nin the most relevant cases. In \\(\\K\\)\\nobreakdash-theory,\ntwo vector bundles \\(\\xi_1,\\xi_2\\)\nare identified if they are \\emph{stably isomorphic}, that is, there is\na trivial vector bundle~\\(\\vartheta\\)\nwith \\(\\xi_1\\oplus\\vartheta \\cong \\xi_2\\oplus\\vartheta\\).\nSeveral authors put in extra work to refine the classification of\nvector bundles (with symmetries) provided by \\(\\K\\)\\nobreakdash-theory\nto a classification up to isomorphism, see\n\\cites{De_Nittis-Gomi:Real_Bloch, De_Nittis-Gomi:Quaternionic_Bloch,\n Kennedy:Thesis, Kennedy-Zirnbauer:Bott_gapped}. Here we argue that\nsuch a refinement of the classification is of little physical\nsignificance. The tight binding approximation leaves out energy bands\nthat are sufficiently far below the Fermi level. Their inclusion only\nadds a trivial vector bundle -- but this is the difference\nbetween stable isomorphism and isomorphism.\n\n\\begin{theorem}[\\cite{Husemoller:Fibre_bundles}*{Chapter 8, Theorem 1.5}]\n \\label{the:K_stable_range}\n Let~\\(X\\)\n be an \\(n\\)\\nobreakdash-dimensional\n CW-complex and let \\(\\xi_1\\)\n and~\\(\\xi_2\\)\n be two \\(k\\)\\nobreakdash-dimensional\n vector bundles. Let \\(c=1,2,4\\)\n depending on whether the vector bundles are real, complex or\n quaternionic. Assume \\(k \\ge \\lceil (n+2)\/c \\rceil - 1\\).\n If \\(\\xi_1\\)\n and~\\(\\xi_2\\) are stably isomorphic, then they are isomorphic.\n\\end{theorem}\n\nSo for a \\(3\\)\\nobreakdash-dimensional\nspace~\\(X\\),\nthe isomorphism and stable isomorphism classification agree for real\nvector bundles of dimension at least~\\(4\\),\nfor complex vector bundles of dimension at least~\\(2\\),\nand for all quaternionic vector bundles.\nFor instance, consider the material Bi$_2$Se$_3$ studied in\n\\cites{Zhang-Liu-Qi-Adi-Fang-Zhang:Insulators_BiSe,\n Liu-Qi-Zhang-Dai-Fang-Zhang:Model_Hamiltonian}. The model\nHamiltonian in \\cites{Zhang-Liu-Qi-Adi-Fang-Zhang:Insulators_BiSe,\n Liu-Qi-Zhang-Dai-Fang-Zhang:Model_Hamiltonian} focuses on four\nbands, of which half are below and half above the Fermi energy. But\nthe dimension of the physically relevant vector bundle is\n\\(2\\cdot 83 + 3\\cdot 34 = 268\\),\nthe number of electrons per unit cell of the crystal; each atom of\nBismuth has 83~electrons and each atom of Se has 34~electrons.\n\n\nThe theorem cited above does not take into account a real involution\non the space~\\(X\\).\nThe proof of Theorem~\\ref{the:K_stable_range} is elementary enough,\nhowever, to extend to ``real'' vector bundles over ``real'' manifolds.\nTo see this, one first describes a ``real'' manifold as a\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-CW-complex.\nThe main step in the proof of Theorem~\\ref{the:K_stable_range} is to\nbuild nowhere vanishing sections of vector bundles, assuming that the\nfibre dimension is large enough. This allows to split off a trivial\nrank-\\(1\\)\nvector bundle as a direct summand. Similarly, if two vector bundles\nwith nowhere vanishing sections are homotopic, then there is a nowhere\nvanishing section for the homotopy if the dimension of the fibres is\nlarge enough. The only change in the ``real'' case is that we need\na \\(\\mathbb{Z}\/2\\)\\nobreakdash-equivariant\nnowhere vanishing section of a ``real'' vector bundle to split off\ntrivial summands. Such sections are built by induction over the\ncells of the \\(\\mathbb{Z}\/2\\)\\nobreakdash-CW-complex.\nThe \\(\\mathbb{Z}\/2\\)\\nobreakdash-action\non the interior of such a cell is either free or trivial. In the\nfirst case, a \\(\\mathbb{Z}\/2\\)\\nobreakdash-equivariant\nsection is simply a section on one half of the cell. In the second\ncase, the cell is contained in the fixed-point submanifold, and we\nneed a nowhere vanishing section of a real vector bundle in the usual\nsense. So the argument in~\\cite{Husemoller:Fibre_bundles} allows to\nbuild nowhere vanishing real sections of ``real'' vector bundles\nunder the same assumptions on the dimension as for real vector bundles.\n\n\\begin{bibdiv}\n \\begin{biblist}\n \\bibselect{references}\n \\end{biblist}\n\\end{bibdiv}\n\\end{document}\n\nWe propose the Roe C*-algebra from coarse geometry as a model for topological phases of disordered materials. We explain the robustness of this C*-algebra and formulate the bulk-edge correspondence in this framework. We describe the map from the K-theory of the group C*-algebra of Z^d to the K-theory of the Roe C*-algebra, both for real and complex K-theory.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nClustering of inertial particles in turbulent flows is relevant for meteorology\nand engineering, as well as fundamental research. It is believed to play a\ncrucial role in rain-drop formation \\cite{falkovich_nature02}, as well as in the aggregation of\nproto-planetesimals in Keplerian accretion disks \\cite{bracco_pof99}. The physical mechanism which\noriginates such clustering is indeed rather simple: particles heavier than the\nfluid in which they are transported experience inertial forces which expel\nthem from vortices; particles lighter than the fluid are attracted into\nvortical structures, for similar reasons \\cite{Squires1991, cencini_jot06, Bec2005}. In realistic flows, however,\nparticles are advected by the small scale vortical structures of turbulent\nflows: these have highly non-trivial statistical features, resulting in a\ncomplex clustering process which is still far from being completely understood.\nFrom the point of view of applications, the properties of concentration and\ndistribution of inertial particles play a crucial role in engineering and\nfor the design of industrial processes involving combustion and mixing \\cite{Warnatz2006,Rouson2001, Sbrizzai2006}. Suspensions of particles in viscoelastic fluids are used in many products of commercial and industrial relevance \\cite{barnes2003}.\n\n\nIn this paper we investigate, by means of direct numerical simulations \nof a turbulent flow, how the clustering properties of a dilute \nsuspension of inertial particles can be affected by the addition of \nsmall amounts of polymer additives. \nThe effects induced by polymers on turbulent flows \nare themselves of enormous relevance. It is enough to mention the celebrated \ndrag reduction effect which occurs in pipe flows \\cite{Lumley1969},\nor the recently discovered elastic turbulence regime \\cite{gs_nature00}.\nPolymers have striking effects also on Lagrangian properties of the flow. \nIn particular it has been shown that polymer addition in turbulent flows \nreduces the chaoticity of Lagrangian trajectories \\cite{bcm_prl03} \nand affects acceleration of fluid tracers \\cite{cmxb_njp08}.\nConversely in the elastic turbulence regime polymers are \nable to generate Lagrangian chaos in flows at vanishing \nReynolds number, which would be non chaotic in the Newtonian case\n\\cite{gs_nature01,bcm_prl03}.\n\nHere we show that the addition of polymers in a turbulent flow has \nimportant effects on the statistical properties of inertial particles \nwhich can result in both an increase or a decrease of the clustering.\nAn example of the effect of polymers on clustering \nis shown in Fig.~\\ref{fig1} which represents the distribution \nof an ensemble of inertial particles in a turbulent flow\nbefore and after the introduction of polymers. It is evident, already\nat the qualitative level of Fig.~\\ref{fig1}, that polymers are able\nto change the statistical distribution of particles.\nWe show that these effects can be understood and quantified \nin terms of the Lyapunov exponents of inertial particles, \nwhich are very sensitive to the presence of polymers.\nPrevious systematic investigations of inertial particle dynamics \nin Newtonian turbulent flows \\cite{calzavarini_jfm08} \nand stochastic flows \\cite{bec_pof03}\nhave shown that clustering (quantified by means of the Lyapunov Dimension of particle attractor) is maximum when the particle relaxation time\nis of the order of the shortest characteristic time of the flow. \n\n\n\n\\begin{figure}\n\\includegraphics[width=4.2cm]{fig1a.eps}\n\\includegraphics[width=4.2cm]{fig1b.eps}\n\\includegraphics[width=4.2cm]{fig1c.eps}\n\\includegraphics[width=4.2cm]{fig1d.eps}\n\\caption{Section on plane $z=0$ of the distribution of heavy particles\nwith $\\tau_S=0.035$ (upper panels) and light particles with $\\tau_S=0.03$ (lower panels) in statistically stationary conditions in a Newtonian \nflow (left) and a viscoelastic flow at $\\rm Wi=1$ (right).\nBoth flows are forced with the same \nforcing ${\\bf f}({\\bf x},t)$ $\\delta$-correlated in time and localized\non large scales. Numerical simulations are done by a pseudo-spectral,\nfully dealiased code at resolution $256^3$. For the viscoelastic simulations,\na small diffusive term is added to (\\ref{eq:4}) to prevent numerical\ninstabilities \\cite{sb_jnnfm95}.}\n\\label{fig1}\n\\end{figure}\n\n\nWe consider the case of a dilute suspension of small inertial particles, \nin which the effects of the disturbance flow induced by the particles \ncan be neglected. The dynamics of the suspension is hence \nmodeled by an ensemble of non-interacting point particles, \nwhich experience viscous drag and added mass forces. \nThe equation of motion of each particle reads \\cite{maxey_pof83}: \n\\begin{eqnarray}\n{d {\\bm x} \\over dt}&=& \\bm v \n\\label{eq:1} \\\\\n{d {\\bm v} \\over dt}&=&-{1 \\over \\tau_S}\\left[\\bm v - \\bm u(\\bm x(t),t)\\right]+\n\\beta {d {\\bm u} \\over dt}\n\\label{eq:2}\n\\end{eqnarray}\nwhere $\\tau_S=a^2\/(3\\beta\\nu)$ is the Stokes relaxation time,\n$a$ is the particle radius, $\\beta=3\\rho_f\/(\\rho_f+2\\rho_p)$ \n($\\rho_p$ and $\\rho_f$ representing particle and fluid densities\nrespectively) and $\\nu$ is the kinematic viscosity of the fluid \n(replaced by the total viscosity $\\nu_T$ in a viscoelastic fluid, see below).\nLight (heavy) particles correspond to $\\beta>1$ ($\\beta<1$). \nIn this work we consider the two extreme cases of\nvery light particles (e.g. air bubble in water) for which $\\beta=3$\nand very heavy particles with $\\beta=0$. \nWe define the Stokes number as $\\rm St=\\tau_S \\lambda^ 0_1$, where\n$\\lambda^0_1$ is the maximum Lyapunov exponent of neutral Lagrangian tracers\n(i.e. $\\rm St=0$ particles) in the flow. With this definition, \nmaximum clustering is obtained for $\\rm St \\simeq 0.1$ \n\\cite{bec_pof03, calzavarini_jfm08}.\n\nThe viscoelastic flow ${\\bf u}({\\bf x},t)$ in which the particles are \nsuspended can be described by standard viscoelastic models,\nsuch as the Oldroyd-B model or the nonlinear FENE-P model, \nwhich accounts for the finite extensibility of polymers. \nIn spite of their simplicity, these models are able to reproduce\nmany relevant properties of dilute polymer solutions, including\nturbulent drag reduction~\\cite{sbh_pof97,bcm_pre05} and elastic turbulence\nphenomenology~\\cite{bbbcm_pre08}. \nHere we choose the Oldroyd-B model~\\cite{bird87}, in which the \ncoupled dynamics of the velocity field ${\\bf u}({\\bf x},t)$ \nand the polymer conformation tensor $\\sigma({\\bf x},t)$\n(which is proportional to local square polymer elongation) \nreads: \n\\begin{eqnarray}\n{\\partial \\bm u \\over \\partial t}+\\bm u\\cdot\\bm\\nabla\\bm u &=&\n-\\bm\\nabla p+\\nu\\nabla^2\\bm u+{2\\nu\\gamma \\over \\tau_p}\\bm\\nabla\\cdot\\sigma\n+{\\bm f}\n\\label{eq:3}\\\\\n{\\partial \\sigma \\over \\partial t}+\\bm u\\cdot\\nabla\\sigma &=&\n(\\nabla\\bm u)^T\\cdot\\sigma+\\sigma\\cdot(\\nabla\\bm u)-\n{2 \\over \\tau_p}(\\sigma-\\mathbb{I})\n\\label{eq:4}\n\\end{eqnarray}\nThe total viscosity of the solution $\\nu_T=\\nu (1+\\gamma)$ \nis written in terms of the kinematic viscosity of the solvent $\\nu$\nand the zero-shear contribution of the polymer $\\gamma$ which \nis proportional to the polymer concentration.\nThe polymer time $\\tau_p$ represents the longest relaxation time to the equilibrium configuration ($\\sigma=\\mathbb{I}$ in dimensionless units).\nViscoelasticity of the turbulent flow is parametrized by the \nWeissenberg number $\\rm Wi$, the ratio between $\\tau_p$ and a characteristic\ntime of the flow. Here we use $\\rm Wi=\\tau_p \\lambda_1^N$ \nwhere $\\lambda_1^N$ is the Lagrangian Lyapunov exponent of the Newtonian flow,\nbefore the addition of polymers (i.e. (\\ref{eq:3}) with $\\gamma=0$). We\nstress that $\\lambda_1^0$ introduced above refers instead to the specific flow\nthat carries the suspension and it clearly depends on $\\rm Wi$. \nTherefore $\\lambda_1^N\\equiv\\lambda_1^0|_{Wi=0}$.\n\n\\begin{table}[b]\n\\label{tab:1}\n\\begin{tabular}{c c c c c}\n$\\rm Wi$\t&$\\varepsilon_f$& $\\varepsilon_\\nu$&$u_{\\rm rms}$\t&$\\lambda^0_1$\t\\\\\n\\hline\n0\t&\t0.28\t&\t0.28\t&\t0.76\t&\t1.36\t\\\\\n0.5\t&\t0.28\t&\t0.18\t&\t0.73\t&\t1.08\t\\\\\n1\t&\t0.28\t&\t0.092\t&\t0.68\t&\t0.75\t\\\\\n\\end{tabular}\n\\caption{Parameters for the Newtonian and viscoelastic simulations. \nThe Weissenberg number $\\rm Wi$, energy input $\\varepsilon_f$, viscous\ndissipation rate $\\varepsilon_\\nu$, rms velocity $u_{rms}$ and Lagrangian \nLyapunov exponent $\\lambda_1^0$ of the carrier flow are shown. \nIn both viscoelastic runs an additional dissipative term was added on \npolymers (see text), with coefficient $\\nu_p=2.3\\times 10^3$ }\n\\label{table1}\n\\end{table}\n\nIn the following we discuss results obtained by integrating numerically\nthe viscoelastic model (\\ref{eq:3}-\\ref{eq:4}) at high resolution for different \nvalues of $\\rm Wi$ (see Table~\\ref{table1}). The flow is sustained by a \nstochastic Gaussian forcing ${\\bf f}({\\bf x},t)$ \n$\\delta$-correlated in time and localized on large scales. Fluid equations were integrated by means of a standard, fully dealiased, pseudo spectral code, on a cubic, triple-periodic domain with 256 grid points per side.\nWhen the flow reaches a turbulent, statistically stationary state, different \nfamilies (i.e. with different values of parameters $\\beta$ and $\\tau_S$) \nof inertial particles are injected, with initial homogeneous \ndistribution in space, and their motion integrated according to \n(\\ref{eq:1}-\\ref{eq:2}). For each value of $\\rm Wi$, we \nintegrated the motion of $1024$ particles for each of $21$ values\nof $\\tau_S$ and two values of $\\beta$, namely very heavy particles with \n$\\beta=0$ and \"bubbles\" with $\\beta=3$. \n\nAs an effect of inertia the distribution of particles does not remain \nhomogeneous and evolves to a fractal set dynamically evolving with \nthe flow, such as the examples shown in Fig.~\\ref{fig1}. In the language\nof dynamical systems, the equations (\\ref{eq:1}-\\ref{eq:2}) for particle motion\nrepresent a dissipative system whose chaotic trajectories evolve to \na fractal attractor (which evolves in time following the flow).\nA quantitative measure of clustering at small scales \nis therefore obtained by measuring the fractal dimension of the attractor \n(for each family of particles) using the Lyapunov dimension \n\\cite{bec_pof03,bec_pof06} defined in terms of Lyapunov exponents as \n$D_L=K+\\sum_{i=1}^{K} \\lambda_i\/|\\lambda_{K+1}|$ where $K$ is the \nlargest integer for which $\\sum_{i=1}^{K} \\lambda_i \\ge 0$ \\cite{ccv2010}.\nSince the space distribution of the particles is the projection \nof the attractor on the sub-space of particle positions, \nthe fractal dimension of clusters \nis given by $\\min(D_L, 3)$~\\cite{sy97,hk97}, \nprovided that the projection is generic \n(for a discussion on this issue see e.g. \\cite{bch07}). \nThis implies that $D_L<3$ gives fractal \ndistributions of dimension $D_L$, while $D_L>3$ corresponds to space-filling\nconfigurations, which however can be non-homogeneous. \n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig2a.eps}\n\\includegraphics[width=8.0cm]{fig2b.eps}\n\\caption{\nLyapunov dimension for light (upper panel) and heavy (lower panel) particles\nplotted as a function of $\\tau_S$. Different lines correspond\nto the different Weissenberg numbers: $\\rm Wi=0$ (squares), $\\rm Wi=0.5$ (circles) and\n$\\rm Wi=1.0$ (triangles). \n}\n\\label{fig2}\n\\end{figure}\n\nIn Fig.~\\ref{fig2} we plot the fractal dimensions for both heavy\nand light particles as a function of $\\tau_S$ for the three simulations \nat different $\\rm Wi$.\nIt is evident that the addition of polymer changes substantially \nthe clustering properties of the particles, both increasing \n$D_L$ and reducing $D_L$ depending on value of $\\tau_S$. \nFigure~\\ref{fig1} shows examples of clustering reduction, for heavy and light particles respectively. The upper panels refer to heavy particles ($\\beta=0$) with $\\tau_S=0.035$, while the bottom ones are extracted from a simulation with $\\beta=3$ and $\\tau_S=0.03$. Both values of Stokes time are, for the Newtonian flow, on the left of the minimum in $D_L$. As a consequence, polymers produce a reduction of clustering. Such effect is more visible for light particles. A possible reason for this difference will be discussed further on.\n\nThe mechanism at the basis of this effect is not trivial and\nis a consequence of the change induced by the polymers on the \nsmall-scale properties of the turbulent flow. \nIn Fig.\\ref{fig3} we plot the energy spectra for the different $\\rm Wi$ numbers.\nThe effect of polymers is evident in the high-wavenumber range \nwhere velocity fluctuations are clearly suppressed, resulting in a \ndepletion of the energy spectrum, while large-scale fluctuations are \nunaffected.\n\nIndeed one can expect that only the fastest eddies of the flow, \ni.e. those whose eddy turn-over time $\\tau_\\ell$ is shorter that \nthe polymer relaxation time $\\tau_p$, \ncan produce a significant elongation of polymers. \nThe elastic feedback therefore affects only small scales $\\ell$ \nwith $\\tau_\\ell < \\tau_p$. \nConversely, large scales exhibit the same phenomenology of a \nNewtonian flow, characterized by a turbulent cascade with a \nconstant energy flux equal to the energy input rate $\\varepsilon_f$. \nThe turbulent cascade proceeds almost unaffected by the presence \nof polymers down to the Lumley scale $\\ell_L$,\nwhose eddy turn-over time equals the polymer relaxation time. \nA dimensional estimate, based on the Kolmogorov scaling for \nthe typical velocity $u_\\ell\\sim\\varepsilon_f^{1\/3}\\ell^{1\/3}$ \nand turn-over time $\\tau_\\ell=\\ell\/u_\\ell \\sim\\varepsilon_f^{-1\/3}\\ell^{2\/3}$ \nof an eddy of size $\\ell$, gives $\\ell_L=\\tau_p^{3\/2}\\varepsilon_f^{1\/2}$. \nPolymers would therefore affect only the small scales $\\ell < \\ell_L$.\nOur results are in qualitative agreement with this picture: the\n$\\rm Wi=0.5$ spectrum differs from the Newtonian \none only for $k\\gtrsim 8$, while\nat $\\rm Wi=1$ polymers are active over a larger range of scales.\nThe reduction of kinetic energy at small scales, due to the transfer of \nenergy to the polymers, is accompanied by a reduction of the viscous \ndissipation $\\varepsilon_\\nu=\\nu\\langle (\\nabla u)^2\\rangle$ \nat fixed energy input $\\varepsilon_f$, \nas can be seen from Table~\\ref{table1} and in the inset of \nFig.\\ref{fig3}. \nThis phenomenon has been previously observed both in forced and decaying\nsimulations of statistically homogeneous and isotropic turbulence\n(see, e.g., ~\\cite{dcbp05,pmp06}). \n\n\\begin{figure}\n\\includegraphics[width=9.0cm]{fig3.eps}\n\\caption{\nEnergy spectra for the Newtonian case ${\\rm Wi}=0$ (squares) and for the\nviscoelastic ones ${\\rm Wi}=0.5$ (circles) and ${\\rm Wi}=1$ (triangles). The\ndepletion due to polymer feed-back is evident on large wavenumbers, while the\nlarger scales are unaffected. The effect of polymers extends at lower\nwave-numbers as $\\rm Wi$ increases. Inset: viscous energy dissipation $\\varepsilon_\\nu$ during\na typical time interval in the stationary simulations, for the Newtonian (solid\nline), ${\\rm Wi}=0.5$ (dashed line) and ${\\rm Wi}=1$ (dash-dot) flows. The\ndecrease in $\\varepsilon_\\nu$ with ${\\rm Wi}$ is evident, as well as the reduction in\nfluctuations.\n}\n\\label{fig3}\n\\end{figure}\n\nThe suppression of small-scale motions caused by polymers \nhas major consequences also on the Lagrangian statistics. \nIt is responsible of the reduction of chaoticity \nof Lagrangian trajectories~\\cite{boffetta_prl03}.\nIndeed the chaoticity of the flow is directly related to its \nstretching efficiency via the Lyapunov exponents. \nWhen polymers are stretched, the elastic stress tensor produces a negative\nfeed-back on small scale stretching, thus reducing the degree of chaoticity of\nthe flow \\cite{boffetta_prl03,balkovsky_pre01}. This effect is clearly\nobservable in the decrease of the Lagrangian Lyapunov exponent of the flow \nat increasing polymer elasticity (see the inset of Fig.\\ref{fig4}). \n\nIt is worth to notice that, because of polymers counteraction, \nthe Lyapunov exponent of the resulting viscoelastic flow is smaller than $\\tau_p^{-1}$. \nIn other words, the $\\rm Wi$ number computed a posteriori (i.e. after polymer injection) \nis always smaller than unity. \nThis is not in contrast with the hypothesis that polymers have a strong active effect on the flow \nmainly when they are stretched, i.e. above the so-called coil-stretch transition, which is\nexpected to happen around $\\rm Wi\\simeq 1$ \\cite{balkovsky_prl00}.\nIndeed, the Lyapunov exponent simply provides a measure of the \naverage stretching in a chaotic flow. One should bear in mind that large fluctuations \nof the stretching rates (and therefore strong viscoelastic effects) \ncan occur also when $\\rm Wi \\lesssim 1$. \n\n\\begin{figure}\n\\includegraphics[clip=true,keepaspectratio,width=8.0cm]{fig4.eps}\n\\caption{Comparison between the Cram\\'er functions of the stretching rate\n$\\gamma_1$ computed at ${\\rm Wi}=0$ (solid line),${\\rm Wi}=0.5$ (dashed\nline),${\\rm Wi}=1$ (dash-dot). Inset: first Lagrangian Lyapunov exponent\n$\\lambda^0_1$ (circles) and width $\\mu$ (squares) of the Cram\\'er function (see\ntext) as a function of $\\rm Wi$. The Lyapunov exponents are compared with the\nNewtonian value $\\lambda_1^N$. \n}\n\\label{fig4}\n\\end{figure}\n\nDetailed information on the fluctuations of the stretching rates can be \nobtained from the statistics of the Finite Time Lyapunov Exponents (FTLE)\n$\\gamma_i$. \nThe FTLE are defined via the exponential growth rate during a finite time\n$T$ of an infinitesimal $M$-dimensional volume as\n$\\sum_{i=1}^M\\gamma_i=(1\/T)\\ln[V^M(T)\/V^M(0)]$ \\cite{ccv2010}.\nFrom the definition of the Lyapunov exponents it follows that \n$\\lim_{T\\rightarrow\\infty}\\gamma^T_i=\\lambda_i$. A large deviation approach\nsuggests that the probability density function (PDF) of the largest \nstretching rate $\\gamma_1$ measured over a long time \n$T\\gg 1\/\\lambda_1$ takes the asymptotic form \n$P_{T}(\\gamma_1)\\sim N(t)\\exp[-H(\\gamma_1)T]$ where the Cram\\'er function \n$H(\\gamma_1)$ is convex and obeys the conditions \n$H(\\lambda_1)=0$, $H^\\prime(\\lambda_1)=0$. \nWe computed the Cram\\'er function for the Lagrangian FTLE for the Newtonian case\nand the two viscoelastic cases. \nIn the inset of Fig.~\\ref{fig4} we plotted the average of the stretching rates\n(i.e., the first Lagrangian Lyapunov exponent of the flow $\\lambda_1^0$) and\nthe rescaled variance $\\mu=T\\langle\\gamma_1^2\\rangle$, for the three values of\n$\\rm Wi$ that we considered. The decrease of the Lyapunov exponent (rescaled\nwith the Newtonian value $\\lambda_1^N$ for comparison) gives a measure of the\ndecrease in the chaoticity of the flow, due to the action of Polymers. On the\nother hand, we also observe a decrease in the relative variance\n$\\mu\/\\Lambda_1^0$, which implies that polymer feedback induces also a reduction\nof the fluctuations of of stretching rates. Inspection of the main panel of\nFig.~\\ref{fig4}, however, shows that fluctuations are not reduced uniformly.\nIndeed, the shape of $P(\\gamma_1)$ changes when polymers are added.\nAs is evident in Fig.\\ref{fig4}, elasticity has the effect of raising the \nright branch of the Cram\\'er function, while the left one is comparatively \nless affected. \nGiven the definition of $H(\\gamma_1)$, this amounts to a relative\nsuppression of positive fluctuations in the stretching rate: as one could\nexpect, polymers have a larger (negative) feedback on events of larger\nstretching.\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig5a.eps}\n\\includegraphics[width=8.0cm]{fig5b.eps}\n\\caption{\nLyapunov dimension for light (upper panel) and heavy (lower panel) particles\nplotted as a function of ${\\rm St=\\tau_S \\lambda^0_1}$. Different lines correspond\nto the different Weissenberg numbers with symbols as in Fig.~\\ref{fig2}.\n}\n\\label{fig5}\n\\end{figure}\n\nThe effect of polymers on Lyapunov exponents and the Lagrangian nature of the latter suggests to introduce the dimensionless Stokes\nnumber defined as $\\rm St=\\tau_S \\lambda^0_1$ which depends on $\\rm Wi$ by the\ndependence of $\\lambda^0_1$ shown in Fig.~\\ref{fig4}. Figure~\\ref{fig5} \nshows the Lyapunov dimension $D_L$ for both heavy and light particles \nas a function of $\\rm St$.\nIt is evident that, with respect to Fig.~\\ref{fig2},\nthe collapse of the curves at different $\\rm Wi$ is improved. \nIn particular, the minimum \nof the fractal dimension (which corresponds to maximum clustering) occurs \nalmost for the same $\\rm St$ number. \nStill, some differences are observable, in particular for small $\\rm St$ in the case\nof light particles. \nThis can be understood by the following argument. \nBubbles, at variance with heavy particles, \nhave tendency to concentrate on filaments of high vorticity. \nIndeed, while the minimal dimension for heavy particles is about $2.5$\n(at $\\rm St \\simeq 0.1$), for light particles at maximal clustering \nit becomes as small as $1.26$.\nVortex filaments correspond to quasi-one-dimensional\nregions of intense stretching, in the direction\nlongitudinal to the vortex, which give major contributions to the right\ntail of the Cram\\'er function. \nAs shown in Fig.~\\ref{fig4}, the effects of polymers on the distribution\nof Lyapunov exponent is more evident in this region of strong fluctuations,\nwhere the distribution does not rescale with $\\lambda^0_1$. It is therefore\nnot surprising that also the effects on clustering of light particles\ncannot be completely absorbed in the rescaling of $\\tau_S$ with the mean\nstretching rate $\\lambda^0_1$.\n\nAs the fractal dimension is given by a combination of the Lyapunov exponents,\nin order to better understand the differences on light and heavy particles,\nin Fig.\\ref{fig6} we show the first three Lyapunov exponents as a function\nof $\\rm St$. The first observation is that bubbles, at variance with heavy \nparticles, exhibit negative values of $\\lambda_2$, consistently with the \nlower value of $D_L$ and the tendency of light particles to concentrate \ntowards vortex filaments.\n\nThe first Lyapunov exponent decreases with $\\rm Wi$ for any value of $\\rm St$,\nthus indicating that the phenomenon of chaos reduction, already discussed \nfor the case of Lagrangian tracers, is generic also for inertial particles. \nOn the contrary, the second Lyapunov exponent shows a different behavior\nfor light and heavy particles: it increases for the former but slightly \ndecreases for the latter. Figure~\\ref{fig6} shows that the effect of \npolymers is not a simple rescaling of the Lyapunov spectrum, which \nwould trivially keep the dimension $D_L$ unchanged. From this point\nof view, the almost perfect rescaling of the Lyapunov dimensions\nshown in Fig.~\\ref{fig5} is quite surprising and arises as the result\nof compensations of different effects.\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig6a.eps}\n\\includegraphics[width=8.0cm]{fig6b.eps}\n\\caption{The first three Lyapunov exponents for light ($\\beta=3$, upper panel)\nand heavy ($\\beta=0$, lower panel) particles, at different $\\rm Wi$.\nContinuous, dashed and dotted lines represent the first, second and\nthird Lyapunov exponents, while symbols correspond to different $\\rm Wi$ \nas in Fig.~\\ref{fig2}.\n}\n\\label{fig6}\n\\end{figure}\n\nIn conclusion, we investigated the clustering properties of inertial \n(heavy and light) particles in a turbulent viscoelastic fluid. \nThe main effect of polymers on turbulent\nflows is to counteract small-scale fluctuations and to reduce its chaoticity.\nQuantitatively, this results in a decrease in the first Lyapunov exponent of \nthe flow, which, in turn, affects clustering of inertial particles. \nThe latter can be quantified by means of the fractal (Lyapunov) dimension of\nparticle distributions. Although the effects of polymers on the particle\nLyapunov exponents are complex and qualitatively different for light \nand heavy particles, the overall effect on fractal dimension is relatively\nsimple and can be rephrased in the rescaling of the characteristic\ntime of the flow. \nIndeed, when particle inertia is parametrized by the Stokes number $\\rm St$ \ndefined with the Lyapunov time of the flow, one can approximately rescale \nthe curves $D_L(\\rm St)$ at all $\\rm Wi$. \nIn contrast, as polymers do not affect large scale properties of the \nflow, a parametrization of particle inertia based on integral time scales\nwould not show a collapse of the curves $D_L(\\rm St)$ at different $\\rm Wi$. \nAs a consequence, any prediction of\nparticle clustering in turbulent polymeric solutions requires an accurate\nestimate of small scale stretching rates.\n\nWe acknowledge support from the the EU COST Action MP0806.\n\n\\input{paper.bbl}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nA $0\/1$-simplex is an $n$-dimensional $0\/1$-polytope \\cite{KaZi} with $n+1$ vertices. \nEquivalently, it is the convex hull of $n+1$ of the $2^n$ elements of the set $\\BB^n$ of \nvertices of the unit $n$-cube $I^n$ whenever this hull has dimension $n$. Throughout \nthis paper, we will study $0\/1$-simplices modulo the action of the hyperocthedral group \n$\\Bn$ of symmetries of $I^n$. As a consequence, we may assume without loss of \ngenerality that a $0\/1$-simplex $S$ has the origin as a vertex. This makes it possible \nto represent $S$ by a non-singular $n\\times n$ matrix $P$ whose columns are the \nremaining $n$ vertices of $S$. Of course, this representation is far from unique, as is \nillustrated by the $0\/1$-tetrahedron in Figure~\\ref{Nfigure1}. First of all, there is a \nchoice which vertex of $S$ is located at the origin. Secondly, column permutations of $P$ \ncorrespond to relabeling of the nonzero vertices of $S$, and thirdly, row \npermutations correspond to relabeling of the coordinate axis.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.75, every node\/.style={scale=0.75}]\n\\begin{scope}[shift={(0,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(0.5,3)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (2,0)--(0.5,1);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,3) circle [radius=0.05];\n\\node[scale=0.8] at (3.4,0.5) {$\\left[\\begin{array}{rrr} 0 & 0 & 1 \\\\ 1 & 1 & 0\\\\ 1 & 0 & 0\\end{array}\\right]$};\n\\end{scope}\n\\begin{scope}[shift={(4.5,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(2.5,1)--(0,2)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (2,0)--(0,2);\n\\draw (0,0)--(2.5,1);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (0,2) circle [radius=0.05];\n\\node[scale=0.8] at (3.4,0.5) {$\\left[\\begin{array}{rrr} 1 & 1 & 0 \\\\ 0 & 1 & 0\\\\ 0 & 0 & 1\\end{array}\\right]$};\n\\end{scope}\n\\begin{scope}[shift={(9,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(2,2)--(0.5,1)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (2,0)--(0.5,1);\n\\draw (0,0)--(2,2);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\node[scale=0.8] at (3.4,0.5) {$\\left[\\begin{array}{rrr} 1 & 0 & 1 \\\\ 0 & 1 & 0\\\\ 0 & 0 & 1\\end{array}\\right]$};\n\\end{scope}\n\\begin{scope}[shift={(13.5,0)}]\n\\draw[fill=gray!20!white] (0,0)--(0,2)--(2.5,3)--(2,2)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (0,0)--(2,2);\n\\draw (0,0)--(2.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0,2) circle [radius=0.05];\n\\draw[fill=black] (2.5,3) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\node[scale=0.8] at (3.4,0.5) {$\\left[\\begin{array}{rrr} 0 & 1 & 1 \\\\0 & 1 & 0\\\\ 1 & 1 & 1\\end{array}\\right]$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Matrix representations of the same $0\/1$-tetrahedron modulo the action of $\\mathcal{B}_3$.}}\n\\label{Nfigure1}\n\\end{figure} \n\n\\smallskip\n \nWe will be studying $0\/1$-simplices with certain geometric properties. These will be \ninvariant under congruence, and in particular invariant under the action of $\\Bn$, \nwhich forms a subset of the congruences of $I^n$. Thus, each of the matrix representations \ncarries the required geometric information of the $0\/1$-simplex it represents. To be more \nspecific, we will study the set of {\\em acute} $0\/1$-simplices, whose dihedral angles are \nall acute, the {\\em nonobtuse} $0\/1$-simplices, none of whose dihedral angles is obtuse, \nand the set of {\\em orthogonal} simplices. An orthogonal simplex is a nonobtuse simplex \nwith exactly $n$ acute dihedral angles and $\\half n(n-1)$ right dihedral angles.\\\\[2mm]\nIt is not difficult to establish that a $0\/1$-simplex $S$ is nonobtuse if and only if the \ninverse $\\ptpi$ of the Gramian of any matrix representation $P$ of $S$ is a diagonally \ndominant Stieltjes matrix. This Gramian is {\\em strictly} diagonally dominant and has \neven negative off-diagonal entries if and only if $S$ is acute. See \\cite{BrCi,BrCi2,BrKoKr} \nfor details. In this paper we will study the properties of the $0\/1$-matrices \nthat represent acute, nonobtuse, and orthogonal simplices.\n\n\\subsection{Motivation} \nThe motivation to study nonobtuse simplices goes back to their appearance in \nfinite element methods \\cite{Bra,Bre} to approximate solutions of PDEs, in \nwhich triangulations consisting of nonobtuse simplices can be used to guarantee \ndiscrete maximum and comparison principles \\cite{BrKoKr2}. We then found that \nthey figure in other applications, see \\cite{BrKoKrSo} and the references therein. \nIn the context of $0\/1$-simplices and $0\/1$-matrices, it is well known \\cite{Gr} \nthat the {\\em Hadamard conjecture} \\cite{Hada} is equivalent to the existence \nof a {\\em regular $0\/1$-simplex} in each $n$ cube with $n-3$ divisible by $4$. \nNote that a regular simplex is always acute. Thus, studying acute $0\/1$-simplices \ncan be seen as an attempt to study the Hadamard conjecture in new context, which \nis wider, but not too wide. Indeed, acute $0\/1$-simplices, although present in any \ndimension, are still very rare in comparison to {\\em all} $0\/1$-simplices. \nSee \\cite{BrCi2}, in which we describe the computational generation of \nacute $0\/1$-simplices, as well as several mathematical properties. This \npaper can be seen as a continuation of \\cite{BrCi2}, in which some new \nresults on acute $0\/1$-simplices are presented, as well as on the again \nslightly larger class of nononbtuse $0\/1$-simplices.\n\n\n\\subsection{Outline}\nWe start in Section~\\ref{Sect-2} with some preliminaries related to the hyperoctahedral group \nof cube symmetries, to combinatorics, and to the linear algebra behind the geometry of \nnonobtuse and acute simplices. We refer to \\cite{BrCi2} for much more detailed information \non the hyperoctahedral group and combinatorical aspects, and to \\cite{BrKoKrSo} for \napplications of nonobtuse simplices. In Section~\\ref{Sect-3} we present our new results \nconcerning {\\em sign properties} of the {\\em transposed inverses} $P^{-\\top}$ of \nmatrix representations $P$ of acute $0\/1$-simplices $S$. These results imply that the \nmatrices $P$ are {\\em fully indecomposable} with {\\em doubly stochastic pattern} \\cite{Bru1}. \nFrom this follows the so-called {\\em one neighbor theorem}, which states that \nall $(\\nmo)$-facets $F$ of $S$ are {\\em interior} to the cube, and that each is \nshared by at most one other acute $0\/1$-simplex in $I^n$. See \\cite{BrDiHaKr} \nfor an alternative proof of that fact. If $S$ is merely a nonobtuse $0\/1$-simplex, \nthe support of $P$ only {\\em contains} a doubly stochastic pattern, and moreover, \n$P$ can be {\\em partly decomposable}. In Section~\\ref{Sect-4} we study the matrix \nrepresentations of such nonobtuse $0\/1$-simplices with partly decomposable matrix \nrepresentations. The main conclusion is that each of them consists of $p$ with \n$2\\leq p \\leq n$ mutually orthogonal facets $F_1,\\dots,F_p$ of respective \ndimensions $k_1,\\dots,k_p$ that add up to $n$. Moreover, each $k_j\\times k_j$ matrix \nrepresentation of each facet $F_j$ is fully indecomposable. In case all facets \n$F_1,\\dots,F_n$ are one-dimensional, the corresponding $0\/1$-simplex is a \nso-called {\\em orthogonal} simplex, as it has a spanning tree of mutually \northogonal edges. Orthogonal $0\/1$-simplices played an important role in \nthe nonobtuse cube triangulation problem, solved in \\cite{BrDiHaKr}. \nWe briefly recall them in Section~\\ref{Sect-5} and put them into the \nnovel context of Section~\\ref{Sect-4}. Finally, in Section~\\ref{Sect-6} \nwe use the insights developed so far to prove a one neighbor \ntheorem for a wider class of nonobtuse simplices.\n\n\\section{Preliminaries}\\label{Sect-2}\nLet $\\BB=\\{0,1\\}$, and write $\\BB^n=\\BB^{n\\times 1}$ for the \nset of vertices of the unit $n$-cube $I^n=[0,1]^n$, which also \ncontains the standard basis vectors $e_1^n,\\dots,e_n^n$ and their \nsum $e^n$, the {\\em all-ones vector}. The $0\/1$-matrices of size \n$n\\times k$ we denote by $\\BB^{n\\times k}$. For each \n$X\\in\\BB^{n\\times k}$, define its {\\em antipode} $\\ol{X}$ by\n\\be \\ol{X}=e^n(e^k)^\\top-X,\\ee\nand write\n\\be \\ones(X) = (e^n)^\\top X e^k \\und \\zeros(X) = \\ones(\\ol{X})\\ee\nfor the number of entries of $X$ equal to one, and equal to zero, \nrespectively. For any $n\\times k$ matrix $X$ define its {\\em support} $\\supp(X)$ by\n\\be \\supp(X) = \\{(i,j)\\in\\{1,\\dots,n\\}\\times\\{1,\\dots,k\\} \\sth (e_i^n)^\\top X e_j^k \\not=0\\}.\\ee\nWe now recall some concepts from combinatorial matrix theory \\cite{Bru1,BrRy}.\n\n\\begin{Def}{\\rm A nonnegative matrix $A$ has a {\\em doubly stochastic pattern} \nif there exists a doubly stochastic matrix $D=(d_{ij})$ such that $\\supp(D)=\\supp(A)$}.\n\\end{Def}\n\n\\begin{Def}{\\rm A matrix $A\\in\\Bnn$ is {\\em partly decomposable} if there \nexists a $k\\in\\{1,\\dots,n-1\\}$ and permutation matrices $\\Pi_1,\\Pi_2$ such that \n\\be\\label{one-1} \\Pi_1^\\top A \\Pi_2 = \\left[\\begin{array}{cc}A_{11} & A_{12} \\\\ 0 & A_{22}\\end{array}\\right], \\ee\nwhere $A_{11}$ is a $k\\times k$ matrix and $A_{22}$ an $(n-k)\\times(n-k)$ \nmatrix. If $\\Pi_1$ and $\\Pi_2$ can be taken equal in (\\ref{one-1}) then $A$ \nis called {\\em reducible}. If $A$ is not partly decomposable it is called \n{\\em fully indecomposable}. If $A$ is not reducible it is called {\\em irreducible}.}\n\\end{Def} \nNote that $A\\in\\Bnn$ is partly decomposable if and only if there exist \nnonzero $v,w\\in\\BB^n$ with $\\ones(v)+\\ones(w)=n$ such that \n$v^\\top A w=0$, and that $A$ is reducible if additionally, $w=\\ol{v}$.\n \n\\begin{Le}\\label{lem-1} Let $X\\in\\Bnn$ be nonsingular and $X=[\\,X_1\\,|\\,X_2\\,]$ \na block partition of $X$ where $X_1\\in\\BB^{n\\times k}$ and $X_2\\in\\BB^{n\\times(n-k)}$ \nfor some $1\\leq k] (2.5,0)--(0.5,1);\n\\draw[->] (2.5,0)--(0,2.5);\n\\draw[->] (2.5,0)--(3,3.5);\n\\draw[fill] (2.5,0) circle [radius=0.07];\n\\node at (2.8,0.1) {$v_0$};\n\\node at (1.3,2.5) {$F_0$};\n\\end{scope}\n\\begin{scope}[shift={(3.7,0)}]\n\\draw[gray!10!white, fill=gray!10!white] (2.5,0)--(0.5,1)--(0,2.5)--(3,3.5)--cycle;\n\\draw[gray!30!white, fill=gray!30!white] (2.5,0)--(0,2.5)--(3,3.5)--cycle;\n\\draw[->] (0.5,1)--(2.5,0);\n\\draw[->] (0.5,1)--(0,2.5);\n\\draw[->] (0.5,1)--(3,3.5);\n\\draw[fill] (0.5,1) circle [radius=0.07];\n\\node at (0.3,0.7) {$v_1$};\n\\node at (1.8,1.5) {$F_1$};\n\\end{scope}\n\\begin{scope}[shift={(7.4,0)}]\n\\draw[gray!10!white, fill=gray!10!white] (2.5,0)--(0.5,1)--(0,2.5)--(3,3.5)--cycle;\\\n\\draw[gray!30!white, fill=gray!30!white] (0.5,1)--(2.5,0)--(3,3.5)--cycle;\n\\draw[->] (0,2.5)--(2.5,0);\n\\draw[->] (0,2.5)--(0.5,1);\n\\draw[->] (0,2.5)--(3,3.5);\n\\draw[fill] (0,2.5) circle [radius=0.07];\n\\node at (-0.3,2.2) {$v_2$};\n\\node at (2.1,1.2) {$F_2$};\n\\end{scope}\n\\begin{scope}[shift={(11.1,0)}]\n\\draw[gray!10!white, fill=gray!10!white] (2.5,0)--(0.5,1)--(0,2.5)--(3,3.5)--cycle;\n\\draw[gray!30!white, fill=gray!30!white] (0.5,1)--(0,2.5)--(2.5,0)--cycle;\n\\draw[->] (3,3.5)--(2.5,0);\n\\draw[->] (3,3.5)--(0.5,1);\n\\draw[->] (3,3.5)--(0,2.5);\n\\draw[fill] (3,3.5) circle [radius=0.07];\n\\node at (3.3,3.2) {$v_3$};\n\\node at (1,1) {$F_3$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Dihedral angles are present in different ways in different matrix representations.}}\n\\label{Nfigure4}\n\\end{figure} \n\n\\smallskip \n\nFor each $j\\in\\{0,1,2,3\\}$, let $P_j$ be a matrix representations of a tetrahedron $S\\in\\SS^3$ with its vertex $v_j$ located at the origin. If for \ninstance $(P_j^\\top P_j)^{-1}e^n \\geq 0$ for $j\\in\\{1,2,3\\}$ then $F_1,F_2$ and $F_3$ make only nonobtuse dihedral angles. These include \n{\\em all} the six dihedral angles of $S$.\n\n\\begin{rem}\\label{rem-2} {\\rm To characterize acute simplices similarly, replace $\\geq$ in (\\ref{eq-1.1}) by $>$ and $\\leq$ in (\\ref{eq-1.2}) by \n$<$. Moreover, replace {\\em onto} by {\\em into the interior} of its opposite facet.}\n\\end{rem}\nThe following two simple combinatorial lemmas will be used further on in this paper.\n \n\\begin{Le}\\label{lem-3} Let $P\\in\\Bnn$ represent a nonobtuse $0\/1$-simplex $S$. Then for all $v\\in\\BB^n$, \n\\be v^\\top(P^\\top P)^{-1}\\ol{v} \\leq 0.\\ee\nIf $S$ is even acute then \n\\be v^\\top(P^\\top P)^{-1}\\ol{v} < 0 \\ee\nfor all $v\\in\\BB^n$ with $v\\not\\in\\{0,e^n\\}$.\n\\end{Le} \n{\\bf Proof. } In fact, $v^\\top(P^\\top P)^{-1}\\ol{v}$ is the sum of $\\ones(v)\\ones(\\ol{v})$ of the off-diagonal entries of $(P^\\top P)^{-1}$. \nAccording to Proposition \\ref{pro-1} these are nonpositive if $S$ is nonobtuse. According to Remark \\ref{rem-2} they are negative if $S$ is \nacute. \\hfill $\\Box$ \n \n\\begin{Le}\\label{lem-4} Let $P\\in\\Bnn$ represent a nonobtuse $0\/1$-simplex $S$. If for some $v\\in\\BB^n$,\n\\be v^\\top (P^\\top P)^{-1} \\ol{v} = 0, \\ee\nthen $v=0$ or $\\ol{v}=0$ or $P^\\top P$ is reducible.\n\\end{Le}\n{\\bf Proof. } Let $v\\not=0\\not=\\ol{v}$ and write $k=\\zeros(v)$. Then $1\\leq k \\leq n-1$. Let $\\Pi$ be a permutation such that \n$\\supp(\\Pi\\ol{v})=\\{1,\\dots,k\\}$. Writing $w=\\Pi v$, we have that $\\ol{w}=\\Pi\\ol{v}$ and\n\\be 0 = v^\\top (P^\\top P)^{-1} \\ol{v} = v^\\top\\Pi^\\top\\Pi (P^\\top P)^{-1} \\Pi^\\top\\Pi \\ol{v} = w\\Pi (P^\\top P)^{-1} \\Pi^\\top\\ol{w},\\ee\nwhich is the sum of the entries of $\\Pi (P^\\top P)^{-1}\\Pi^\\top$ with indices $(i,j)$ with $k+1\\leq i\\leq n$ and $1\\leq j \\leq k$. Since these \nentries are non-positive and their sum equals zero, they are all zero, leading to an $(n-k)\\times k$ bottom left block of zeros in \n$\\Pi (P^\\top P)^{-1}\\Pi^\\top$. Thus, $(P^\\top P)^{-1}$ is reducible, and hence, so is its inverse $P^\\top P$. \\hfill $\\Box$\\\\[3mm]\nA final important observation is the following classical result by Fiedler.\n\n\\begin{Le}[\\cite{Fie}]\\label{lem-5} All $k$-facets of a nonobtuse simplex are nonobtuse and all $k$-facets of an acute simplex are acute.\n\\end{Le}\nIt is well known that the converse does not hold. Simplices whose facets are all nonobtuse or acute were studied recently in \\cite{BrCi}. \n \n\\section{Doubly stochastic patterns and full indecomposability}\\label{Sect-3}\nWe start our investigations with some results on matrix representations of acute $0\/1$-simplices. The first one gives a remarkable connection \nwith doubly stochastic matrices. It follows from the observation in Remark \\ref{rem-1} that for an acute $0\/1$-simplex, the altitude from each \nvertex points into the interior of $I^n$. This, in turn, defines the signs of the entries of the normals to its facets in terms of the supports of \ntheir corresponding vertices.\n\n\\begin{Th}\\label{th-1} Let $P\\in\\Bnn$ be a matrix representation of an acute $0\/1$-simplex $S\\in\\SS^n$, and write $Q=P^{-\\top}$. Then \n\\be\\label{eq-3.1} q_{ij}>0 \\Leftrightarrow p_{ij} = 1 \\und q_{ij}<0 \\Leftrightarrow p_{ij}=0. \\ee\nDefining $0\\leq C=(c_{ij})$ and $0\\leq D=(d_{ij})$ by \n\\be\\label{eq-3.2} C = \\half\\left( |Q|-Q\\right) \\und D = \\half\\left(|Q|+Q\\right), \\ee\nwhere $|Q|$ is the matrix whose entries are the moduli of the entries of $Q$, we have that \n\\be Q = D-C, \\ee\nwhere $D$ is doubly stochastic and $C$ row-substochastic. \n\\end{Th} \n{\\bf Proof}. The $j$th column $q_j$ of $Q$ is an inward normal to the facet $F_j$ of $S$ opposite the $j$th column $p_j$ of $P$. Thus, \n$p_j-\\alpha q_j$ is an element of the interior of $I^n$ for $\\alpha>0$ small enough. From this, (\\ref{eq-3.1}) immediately follows. Combining \nthis with the fact that the inner product between $p_j$ and $q_j$ equals one, the positive elements in each column of $Q$ add to one. But \nsince $Q^\\top P=I$, also the inner products between corresponding rows of $P$ and $Q$ equals one, and thus also the positive elements in \neach row of $Q$ add to one. Finally, $Qe^n$ is the outward normal to the facet of $S$ opposite the origin and thus it points into the interior of \n$I^n$. Consequently, $Qe^n>0$, hence $De^n>Ce^n \\geq 0$, which shows that $C$ is row-substochastic. \\hfill $\\Box$ \n \n\\begin{rem}\\label{rem-3}{\\em For $n\\geq 7$ there exist matrix representations $P\\in\\BB^{n\\times n}$ of acute $0\/1$-simplices in $I^n$ for \nwhich the matrix $C$ in (\\ref{eq-3.2}) is not column-substochastic. This shows that Theorem \\ref{th-1} cannot be strengthened in this \ndirection. It also proves that if $P$ represents an acute $0\/1$-simplex, its transpose $P^\\top$ may not do the same. See Figure~\\ref{Nfigure5} for an \nexample.}\\end{rem}\n \n\\begin{Co}\\label{co-1} Let $P\\in\\Bnn$ be a matrix representation of an acute $0\/1$-simplex $S\\in\\SS^n$. Then $P$ has a fully \nindecomposable doubly stochastic pattern. \n\\end{Co}\n{\\bf Proof. } Due to (\\ref{eq-3.1}), the matrix $D$ in (\\ref{eq-3.2}) has the same support as $P$, hence $P$ has a doubly stochastic pattern. \nNext, assume to the contrary that $P$ is partly decomposable, then there exist permutations $\\Pi_1,\\Pi_2$ such that\n\\[ \\Pi_1^\\top P \\Pi_2 = \\left[\\begin{array}{cc}P_{11} & P_{12} \\\\ 0 & P_{22}\\end{array}\\right], \\]\nwhere $P_{11}$ is a $k\\times k$ matrix and $P_{22}$ an $(n-k)\\times(n-k)$ matrix for some $k\\in\\{1,\\dots,n-1\\}$. But then $Q=P^{-\\top}$ \nhas entries equal to zero, which contradicts (\\ref{eq-3.1}) in Theorem \\ref{th-1}. \\hfill $\\Box$\\\\[3mm] \nTheorem \\ref{th-1}, Corollary \\ref{co-1}, and Remark \\ref{rem-3} are all illustrated in Figure~\\ref{Nfigure5}.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!20!white] (0,0)--(2.5,1)--(2,2)--cycle;\n\\draw[gray] (0,0)--(0.5,3)--(2,2);\n\\draw[thick,->] (1.5,1.2)--(0.6,2.75);\n\\node[scale=0.9] at (0.2,3) {$p$};\n\\node[scale=0.9] at (1,1.5) {$q$};\n\\node[scale=0.8] at (1.2,0.7) {$F_p$};\n\\node[scale=0.9] at (1.7,1.2) {$\\pi$};\n\\draw[fill=white] (1.5,1.2) circle [radius=0.05];\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (0,0)--(2,2);\n\\draw[gray] (2.5,1)--(0.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,3) circle [radius=0.05]; \n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\node[scale=0.73] at (8.5,1.5) {$P=\\left[\\begin{array}{rrrrrrr}\n \\fbox{1} & \\fbox{1} & \\fbox{1} & 0 & 0 & \\fbox{1} & \\fbox{1}\\\\\n \\fbox{1} & 0 & 0 & \\fbox{1} & \\fbox{1} & 0 & 0\\\\\n 0 & \\fbox{1} & 0 & \\fbox{1} & \\fbox{1} & 0 & 0\\\\\n 0 & 0 & \\fbox{1} & \\fbox{1} & \\fbox{1} & \\fbox{1} & 0\\\\\n 0 & 0 & \\fbox{1} & \\fbox{1} & \\fbox{1} & 0 & \\fbox{1}\\\\\n 0 & 0 & 0 & \\fbox{1} & 0 & \\fbox{1} & \\fbox{1}\\\\\n 0 & 0 & 0 & 0 & \\fbox{1} & \\fbox{1} & \\fbox{1}\n \\end{array}\\right], \\hspace{3mm} P^{-\\top}=\\dfrac{1}{13}\\left[\\begin{array}{ccccccc} \n \\fbox{4} & \\fbox{4} & \\fbox{3} & -2 & -2 & \\fbox{1} & \\fbox{1}\\\\\n \\fbox{9} & -4 & -3 & \\fbox{2} & \\fbox{2} & -1 & -1\\\\\n -4 & \\fbox{9} & -3 & \\fbox{2} & \\fbox{2} & -1 & -1\\\\\n -2 & -2 & \\fbox{5} & \\fbox{1} & \\fbox{1} & \\fbox{6} & -7\\\\\n -2 & -2 & \\fbox{5} & \\fbox{1} & \\fbox{1} & -7 & \\fbox{6}\\\\\n -1 & -1 & -4 & \\fbox{7} & -6 & \\fbox{3} & \\fbox{3}\\\\\n -1 & -1 & -4 & -6 & \\fbox{7} & \\fbox{3} & \\fbox{3}\n \\end{array}\\right]$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{In an acute $0\/1$-simplex, each vertex $p$ following the inward normal $q$ to its opposite facet $F_p$ (in converse direction) \nprojects as $\\pi$ in the interior of $F_p$ and hence in the interior of $I^n$. This fixes the signs of the entries of $q$ in terms of those of $p$. \nThe matrices $P$ and $P^{-\\top}$ (not related to the depicted tetrahedron) constitute an example of the linear algebraic consequences. The \npositions of the positive entries (boxed) of $P^{-\\top}$ and $P$ coincide. The positive part $D$ of $P^{-\\top}$ is doubly-stochastic. The \nnegated negative part $C$ of $P^{-\\top}$ is a row-substochastic. It is not column-substochastic because the third column of $C$ adds to \n$\\frac{14}{13}$. Thus, even though also $P^\\top$ has a doubly stochastic pattern and is fully indecomposable, it is not a matrix \nrepresentation of an acute binary simplex.}}\n\\label{Nfigure5}\n\\end{figure} \n\\begin{rem}{\\rm Because {\\em each} matrix representation $P$ of an acute $0\/1$-simplex has a fully indecomposable doubly stochastic \npattern, applying to such a matrix $P$ any operation of type (X), as described below Figure~\\ref{Nfigure3}, results in another matrix with a fully \nindecomposable doubly stochastic pattern. From the linear algebraic point of view, this is remarkable because generally, both the $0\/1$-matrix \nproperties of full indecomposability and of having a doubly stochastic pattern are destroyed under operations of type (X). See for instance\n\\be \\small \\left[\\begin{array}{rrr} 1 & 1 & 1 \\\\ 1 & 1 & 1 \\\\ 1 & 1 & 1\\end{array}\\right] \\overset{\\mbox{\\rm(X)}}{\\longrightarrow} \\left[\\begin{array}{rrr} 1 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 1 & 0 & 0\\end{array}\\right] \\und \\left[\\begin{array}{rrr} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{array}\\right] \\overset{\\mbox{\\rm(X)}}{\\longrightarrow} \\left[\\begin{array}{rrr} 1 & 1 & 1 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{array}\\right]. \\ee\nFrom the geometric point of view, this is easy to understand, as the fact that each altitude from each vertex of $S$ points into the interior of \n$I^n$ is invariant under the action of $\\Bn$.}\n\\end{rem} \nThe geometric translation of Corollary \\ref{co-1} is that if $S\\in\\SS^n$ is acute, none of its $k$-dimensional facets is contained in a $k$-\ndimensional facet of $I^n$ for $k\\in\\{1,\\dots,n-1\\}$. The geometric {\\em proof} of this is to note that, given a $k$-facet $C$ of $I^n$, no \nvertex of $I^n$ orthogonally projects into {\\em the interior} of $C$. In fact, each vertex of $I^n$ projects on a {\\em vertex} of $C$. See \nFigure~\\ref{Nfigure6}. Thus, if an arbitrary $0\/1$-simplex $S$ has a $k$-facet $K$ contained in $C$, each remaining vertex of $S$ projects on a vertex of \n$C$. Remark \\ref{rem-2} now shows that $S$ cannot be acute.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(2.5,1)--(0.5,1)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\node[scale=0.8] at (1.5,0.3) {$C$};\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=white] (0,2) circle [radius=0.05];\n\\draw[fill=white] (2,2) circle [radius=0.05];\n\\draw[fill=white] (0.5,3) circle [radius=0.05];\n\\draw[fill=white] (2.5,3) circle [radius=0.05];\n\\draw[thick,->] (0,1.95)--(0,1.3);\n\\draw[thick,->] (2,1.95)--(2,1.3);\n\\draw[thick,->] (0.5,2.95)--(0.5,2.3);\n\\draw[thick,->] (2.5,2.95)--(2.5,2.3);\n\\begin{scope}[shift={(4,0)}]\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw[gray,very thick] (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw[fill=white] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=white] (0.5,1) circle [radius=0.05];\n\\draw[fill=white] (0,2) circle [radius=0.05];\n\\draw[fill=white] (2,2) circle [radius=0.05];\n\\draw[fill=white] (0.5,3) circle [radius=0.05];\n\\draw[fill=white] (2.5,3) circle [radius=0.05];\n\\draw[thick,->] (0,1.95)--(0.7,1.3);\n\\draw[thick,->] (2,1.95)--(2,1.3);\n\\draw[thick,->] (0.5,2.95)--(1.2,2.3);\n\\draw[thick,->] (2.5,2.95)--(2.5,2.3);\n\\draw[thick,->] (0.05,0)--(0.8,0);\n\\draw[thick,->] (0.55,1)--(1.2,1);\n\\node[scale=0.8] at (2.5,0.3) {$C$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{For each facet $C$ of $I^n$, each vertex of $I^n$ projects onto \na vertex of $C$. Consequently, no acute $0\/1$-simplex has a \nfacet contained in a facet of $I^n$.}}\n\\label{Nfigure6}\n\\end{figure} \n\n\\smallskip\n\nContrary to an acute $0\/1$-simplex, a {\\em nonobtuse} $0\/1$-simplex $S$ may indeed have a $k$-facet $K$ that is contained in a cube facet \n$C$ of $I^n$. If this is the case, then each remaining vertex of $S$ projects on a vertex of $K$. Moreover, $S$ has a partly decomposable \nmatrix representation. Before discussing this structure, we first formulate the equivalent of Theorem \\ref{th-1} for nonobtuse simplices and \ndiscuss some of the differences with Theorem \\ref{th-1} using an example.\n\n\\begin{Th}\\label{th-2} Let $P\\in\\Bnn$ be a matrix representation of a nonobtuse $0\/1$-simplex $S\\in\\SS^n$, and write $Q=P^{-\\top}$. Then \n\\be\\label{eq-3.1-n} q_{ij}\\geq 0 \\Leftrightarrow p_{ij} = 1 \\und q_{ij}\\leq 0 \\Leftrightarrow p_{ij}=0. \\ee\nDefining $0\\leq C=(c_{ij})$ and $0\\leq D=(d_{ij})$ by \n\\be\\label{eq-3.2-n} C = \\half\\left( |Q|-Q\\right) \\und D = \\half\\left(|Q|+Q\\right), \\ee\nwhere $|Q|$ is the matrix whose entries are the moduli of the entries of $Q$, we have that \n\\be Q = D-C, \\ee\nwhere $D$ is doubly stochastic and $C$ row-substochastic. \n\\end{Th} \n{\\bf Proof}. The proof only differs from the proof of Theorem \\ref{th-1} in the sense that $p_j-\\alpha q_j$ is now an element of $I^n$ \nincluding its boundary. This accounts for the $\\geq$ and $\\leq$ signs in (\\ref{eq-3.1-n}) in comparison to the $>$ and $<$ signs in \n(\\ref{eq-3.1}). \\hfill $\\Box$\\\\[3mm]\nTheorem \\ref{th-2} is rather weaker than Theorem \\ref{th-1}. First of all, the matrix $P^{-\\top}$ can have entries equal to zero. Moreover, it \ncannot anymore be concluded that $P$ has a doubly stochastic pattern, only that it {\\em contains} a doubly stochastic pattern,\n\\be \\supp(D)\\subset \\supp(P). \\ee\nA typical example of this is the following matrix representation $P$ of a nonobtuse $0\/1$-simplex,\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!20!white] (0,0)--(2.5,1)--(2.5,3)--cycle;\n\\draw[gray] (2,0)--(2.5,3);\n\\draw[thick,->] (1.25,0.5)--(1.9,0.1);\n\\node[scale=0.9] at (2.2,0) {$p$};\n\\node[scale=0.9] at (1.3,0.2) {$q$};\n\\node[scale=0.8] at (1.6,1.3) {$F_p$};\n\\node[scale=0.9] at (1.4,0.7) {$\\pi$};\n\\draw[fill=white] (1.25,0.5) circle [radius=0.05];\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05]; \n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (2.5,3) circle [radius=0.05];\n\\node[scale=0.73] at (8.5,1.5) {$P=\\left[\\begin{array}{ccccccc}\n \\fbox{1} & \\fbox{1} & 0 & 0 & \\fbox{1} & \\fbox{1} & \\fbox{1} \\\\\n 0 & \\fbox{1} & 0 & 0 & \\fbox{1} & \\fbox{1} & \\fbox{1}\\\\\n 0 & 0 & \\fbox{1} & \\fbox{1} & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & \\fbox{1} & 0 & 0 & 0\\\\\n 0 & 0 & 0 & 0 & \\fbox{1} & \\fbox{1} & 0\\\\ \n 0 & 0 & 0 & 0 & \\fbox{1} & 0 & \\fbox{1} \\\\\n 0 & 0 & 0 & 0 & 0 & \\fbox{1} & \\fbox{1} \n \\end{array}\\right], \\hspace{3mm} P^{-\\top} = \\dfrac{1}{2}\n \\left[\\begin{array}{rrrrrrr}\n \\fbox{2} & 0 & 0 & 0 & 0 & 0 & 0\\\\\n -2 & \\fbox{2} & 0 & 0 & 0 & 0 & 0\\\\\n 0 & 0 & \\fbox{2} & 0 & 0 & 0 & 0\\\\\n 0 & 0 & -2 & \\fbox{2} & 0 & 0 & 0\\\\\n 0 & -1 & 0 & 0 & \\fbox{1} & \\fbox{1} & -1\\\\\n 0 & -1 & 0 & 0 & \\fbox{1} & -1 & \\fbox{1} \\\\\n 0 & -1 & 0 & 0 & -1 & \\fbox{1} & \\fbox{1} \n \\end{array}\\right]$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Analogue of Figure~\\ref{Nfigure5} for matrix representations of nonobtuse $0\/1$-simplices.}}\n\\label{Nfigure7}\n\\end{figure} \n\n\\smallskip\n\nObviously, $P$ is partly decomposable and has no doubly stochastic pattern. It is only valid that the support of the doubly stochastic matrix $D$ \nis {\\em contained} in the support of $P$. \n\\section{Partly decomposable matrix representations}\\label{Sect-4}\nWe will continue to study nonobtuse $0\/1$-simplices having a partly decomposable matrix representation $P$. Without loss of generality, we \nmay assume that $P$ is nontrivially block partitioned as\n\\be\\label{eq-5} P = \\left[\\begin{array}{r|r} N & R \\\\ \\hline 0 & A \\end{array}\\right], \\ee\nand that $A$ is fully indecomposable.\n\n\\begin{Th}\\label{th-3} Let $S\\in\\SS^n$ be nonobtuse with a matrix representation $P$ as in (\\ref{eq-5}) with $N\\in\\BB^{k\\times k}$ with \n$k\\in\\{1,\\dots,n-1\\}$ and with $A$ fully indecomposable. Then:\\\\[3mm]\n$\\bullet$ $N$ is a matrix representation of a nonobtuse simplex in $I^k$;\\\\[3mm]\n$\\bullet$ $A$ is a matrix representation of a nonobtuse simplex in $I^{n-k}$;\\\\[3mm]\n$\\bullet$ $R=\\nu(e^k)^\\top$, where $\\nu=0$ or $\\nu$ is a column of $N$.\n\\end{Th}\n{\\bf Proof.} Lemma \\ref{lem-5} proves that the first $k$ columns of $P$ together with the origin form a nonobtuse $k$-simplex, and obviously \nits vertices all lie in a $k$-facet of $I^n$. This proves the first item in the list of statements. Next, we compute \n\\be\\label{eq-6a} P^{-\\top} = \\left[\\begin{array}{rc} N^{-\\top} & 0 \\\\ -A^{-\\top}R^\\top N^{-\\top} & A^{-\\top}\\end{array}\\right]. \\ee \nand thus,\n\\be\\label{eq-6} (P^\\top P)^{-1} = \\left[\\begin{array}{rr} (N^\\top N)^{-1} + N^{-1}R(A^\\top A)^{-1}R^\\top N^{-\\top} & -N^{-1}R(A^\\top A)^{-1} \\\\-(A^\\top A)^{-1}R^\\top N^{-\\top} & (A^\\top A)^{-1}\\end{array}\\right]. \\ee \nDue to Proposition \\ref{pro-1}, the matrix $(P^\\top P)^{-1}$ has nonpositive off-diagonal entries (\\ref{eq-1.2}) and nonnegative row sums \n(\\ref{eq-1.1}). Both properties are clearly inherited by its trailing submatrix $(A^\\top A)^{-1}$, possibly even with larger row sums. This \nproves the second statement of the theorem. Next, due to (\\ref{eq-1.2}), the top-right block in (\\ref{eq-6}) satisfies \n\\be\\label{eq-7} -N^{-1}R(A^\\top A)^{-1} \\leq 0. \\ee\nMultiplication of this block from the left with $N\\geq 0$ and from the right with $\\ol{R}\\geq 0$ gives that\n\\be R(A^\\top A)^{-1}\\ol{R}^\\top \\geq 0.\\ee\nHowever, by Lemma \\ref{lem-3}, the diagonal entries of $R(A^\\top A)^{-1}\\ol{R}^\\top$ are also non{\\em positive}, and thus, they all equal \nzero. We can therefore apply Lemma \\ref{lem-4}. Note that because $A$ is assumed fully indecomposable, $A^\\top A$ is irreducible by Lemma \n\\ref{lem-1}. Thus, row-by-row application of Lemma \\ref{lem-4} proves that each row of $R$ contains only zeros or only ones. This proves that \nthere exists an $r\\in\\BB^n$ such that\n\\be\\label{eq-8} R=r(e^{n-k})^\\top.\\ee\nWe will proceed to show that $r$ is a column of $N$, or zero. Substituting (\\ref{eq-8}) back into (\\ref{eq-7}) yields that \n\\be\\label{eq-9} wu^\\top \\geq 0, \\hdrie \\mbox{\\rm where } w=N^{-1}r \\und u^\\top = (e^{n-k})^\\top (A^\\top A)^{-1}. \\ee \nDue to (\\ref{eq-1.1}) we have $u\\geq 0$. Because $A^\\top A$ is non-singular, $u$ has at least one positive entry. Thus, also $w$ is \nnonnegative. This turns $r=Nw$ into a nonnegative linear combination of columns of $N$. We continue to prove that it is a {\\em convex} \ncombination. For this, observe that the sums of the last $k$ rows of $(P^\\top P)^{-1}$ are nonnegative due to (\\ref{eq-1.1}). Thus,\n\\[ 0 \\leq -(A^\\top A)^{-1}R^\\top N^{-\\top}e^k + (A^\\top A)^{-1}e^{n-k} = u\\left(1-w^\\top e^k\\right) \\]\nwith $u,w$ as in (\\ref{eq-9}) and where we have used that $R^\\top N^{-\\top}=e^{n-k}r^\\top N^{-\\top} = e^{n-k}w^\\top$. As we showed \nalready that $u\\geq 0$ has at least one positive entry, we conclude that \\[ w^\\top e^k \\leq 1.\\] \nTherefore we now have that $Nw=r\\in\\BB^k$ for some $w\\geq 0$ with $w^\\top e^k \\leq 1$. Thus also\n\\be [0\\,|\\,N] \\left[\\begin{array}{c} 1-w^\\top e^k \\\\ w\\end{array}\\right] = r. \\ee\nAccording to Lemma \\ref{lem-2}, this implies that $r$ is a column of $[0\\,|\\,N]$. This proves the third item in the list of statements to \nprove.~\\hfill $\\Box$\\\\[3mm]\nIt is worthwhile to stress a number of facts concerning Theorem \\ref{th-3} and its nontrivial proof.\n\n\\begin{rem}\\label{rem-4}{\\rm The assumption that $A$ in (\\ref{eq-5}) is fully indecomposable is very natural, as each partly decomposable matrix can be put in the form (\\ref{eq-5}) using operations of type (C) and (R). But in the proof of Theorem \\ref{th-3} we only needed that $A^\\top A$ is irreducible. This is {\\em implied} by the full indecomposability of $A$ due to Lemma \\ref{lem-1}, but is not {\\em equivalent} to it. In fact, if $A$ is fully indecomposable, then $A^\\top A\\geq e^n(e^n)^\\top + I$. See Corollary \\ref{co-5}.}\n\\end{rem} \n\n\\begin{rem}\\label{rem-5}{\\rm The result proved in the third bullet of Theorem \\ref{th-3} that $R$ consists of $n-k$ copies of {\\em the same} column of $N$ is stronger than the geometrical observation that each of the last $n-k$ columns of $P$ should project on {\\em any} vertex of the $k$-simplex represented by $N$. It is the {\\em irreducibility} of $A^\\top A$ that forces the equality of all columns of $R$.}\n\\end{rem}\n\n\\begin{rem}\\label{rem-6}{\\rm Permuting rows and columns of the block upper triangular matrix in (\\ref{eq-5}) show that also for\n\\be \\left[\\begin{array}{r|r} A & 0 \\\\ \\hline R & N \\end{array}\\right] \\ee\nwith $A$ and $N$ as in Theorem \\ref{th-3}, similar conclusions can be drawn for $R$}.\n\\end{rem}\nSome details in the above remarks will turn out to be of central importance in Section \\ref{Sect-6}.\n\n\\begin{Co}\\label{co-3} Let $S\\in\\SS^n$ be a nonobtuse $0\/1$-simplex with matrix representation $P$. Then the following statements are \nequivalent:\\\\[3mm]\n$\\bullet$ $P$ is partly decomposable;\\\\[3mm]\n$\\bullet$ $S$ has a block diagonal matrix representation with at least one fully indecomposable block;\\\\[3mm]\n$\\bullet$ each matrix representation of $S$ is partly decomposable.\n\\end{Co} \n{\\bf Proof}. Suppose that $P$ is partly decomposable. Then Theorem \\ref{th-1} shows that $P$ is of the form\n\\be\\label{eq-10} P = \\left[\\begin{array}{r|c} N & \\nu \\left(e^{n-k}\\right)^\\top \\\\[1mm] \\hline 0 & A \\end{array}\\right],\\ee\nand $\\nu=0$ or $\\nu$ is a column of $N$. If $\\nu=0$ then $P$ itself is block diagonal. If $\\nu\\not=0$, apply to $P$ the operation of type (X) \nas described below Figure~\\ref{Nfigure3} with column $c$ equal to the column $(\\nu,0)^\\top$ of $P$. The simple observation that $\\nu+\\nu$ equal zero \nmodulo $2$ proves that the resulting matrix $\\tilde{P}$ is block diagonal. As the bottom right block of $\\tilde{P}$ equals $A$, this shows that \nat least one block is fully indecomposable. To show that each matrix representation of $S$ is partly decomposable, simply note that each \noperation of type (X) applied to the block-diagonal matrix representation will leave one of the two off-diagonal zero blocks \ninvariant.~\\hfill $\\Box$\n\n\\begin{rem}{\\rm The converse of Theorem \\ref{th-3} is also valid. Indeed, suppose that $N$ and $A$ are matrix representations of nonobtuse \nsimplices. Then it is trivially true that the block diagonal matrix $P$ having $N$ and $A$ as diagonal blocks represents a nonobtuse simplex. \nApplying operations of type (X) to $P$ proves that all matrices of the form (\\ref{eq-10}) then represent nonobtuse simplices. Note that this also \nholds without the assumption that $A$ is fully indecomposable.}\n\\end{rem}\nAnother corollary of Theorem \\ref{th-3} concerns its implications for the structure of the transposed inverse $P^{-\\top}$ of a partly \ndecomposable matrix representation of a nonobtuse $0\/1$-simplex.\n\n\\begin{Co}\\label{co-4} If $\\nu=Ne_j^k$ in (\\ref{eq-10}) for some $j\\in\\{1,\\dots,k\\}$, then\n\\be P^{-\\top} = \\left[\\begin{array}{c|c} N^{-\\top} & 0 \\\\\\hline\\\\[-4mm] ae_j^\\top & A^{-\\top}\\end{array}\\right], \\ee\nwhere $a$ is the inward normal to the facet opposite the origin of the simplex represented by $A$. As a consequence, the sums of the \nlast $n-k$ rows of $P^{-\\top}$ all add to zero.\n\\end{Co}\n{\\bf Proof. } Substitute $R=\\nu e^{n-k}$ with $\\nu=Ne_j^k$ into the expression for $P^{-\\top}$ in (\\ref{eq-6a}). \\hfill $\\Box$\\\\[3mm]\nTo illustrate Corollary \\ref{co-3}, consider again the matrix $P\\in\\BB^{7\\times7}$ from Figure~\\ref{Nfigure7}, now displayed not as $0\/1$-matrix but as \ncheckerboard black-white pattern, at the top in Figure~\\ref{Nfigure8}. Applying an operation of type (X) with the second column results in the matrix to its \nright, which is block diagonal. Swapping the first two rows of that matrix yields one in which the top left block is now in its most reduced form. \nApplying an operation of type (X) with the sixth column results in a matrix in which the bottom left $3\\times 4$ zero block has been destroyed. \nFinally, swapping columns $2$ and $6$, and swapping rows $1$ and $2$, results in the matrix that could also have been obtained by applying \noperation (X) with column $6$ directly to $P$.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\n\n\\node[xscale=0.6,yscale=0.73] at (5,1) {$\\left[\\begin{array}{ccccccc}\n \\blacksquare & \\blacksquare & \\square & \\square & \\blacksquare & \\blacksquare & \\blacksquare \\\\\n \\square & \\blacksquare & \\square & \\square & \\blacksquare & \\blacksquare & \\blacksquare\\\\\n \\square & \\square & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\square & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\square & \\square & \\blacksquare & \\blacksquare & \\square\\\\ \n \\square & \\square & \\square & \\square & \\blacksquare & \\square & \\blacksquare \\\\\n \\square & \\square & \\square & \\square & \\square & \\blacksquare & \\blacksquare \n \\end{array}\\right]$};\n \n\\draw[<->] (6.8,1)--(7.8,1);\n\\draw[<->] (3.3,1)--(2.3,1);\n\\node[scale=0.9] at (7.3,1.3) {(X)};\n\\node[scale=0.9] at (2.8,1.3) {(X)};\n\\node[scale=0.9] at (7.3,0.7) {$c=2$};\n\\node[scale=0.9] at (2.8,0.7) {$c=6$};\n\n\\draw[<->] (10,-1.3)--(9,-2.3);\n\\draw[<->] (0,-1.3)--(1,-2.3);\n\\node[scale=0.9] at (10.5,-2) {(R), $1\\leftrightarrow 2$};\n\\node[scale=0.9] at (-0.5,-2) {(C), $2\\leftrightarrow 6$};\n\\node[scale=0.9] at (-0.5,-2.5) {(R), $1\\leftrightarrow 2$};\n\n\\draw[<->] (4.5,-3)--(5.5,-3);\n\\node[scale=0.9] at (5,-2.7) {(X)};\n\\node[scale=0.9] at (5,-3.3) {$c=6$};\n \n\\node[xscale=0.6,yscale=0.73] at (9.5,0) {$\\left[\\begin{array}{ccccccc}\n \\square& \\blacksquare & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\square & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\square & \\square & \\blacksquare & \\blacksquare & \\square\\\\ \n \\square & \\square & \\square & \\square & \\blacksquare & \\square & \\blacksquare \\\\\n \\square & \\square & \\square & \\square & \\square & \\blacksquare & \\blacksquare \n \\end{array}\\right]$};\n \n \\node[xscale=0.6,yscale=0.73] at (7.2,-3) {$\\left[\\begin{array}{ccccccc}\n \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square& \\blacksquare & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\square & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\square & \\square & \\blacksquare & \\blacksquare & \\square\\\\ \n \\square & \\square & \\square & \\square & \\blacksquare & \\square & \\blacksquare \\\\\n \\square & \\square & \\square & \\square & \\square & \\blacksquare & \\blacksquare \n \\end{array}\\right]$};\n\n\\node[xscale=0.6,yscale=0.73] at (0.5,0) {$\\left[\\begin{array}{ccccccc}\n \\square & \\square & \\blacksquare & \\blacksquare & \\square & \\blacksquare & \\square\\\\\n \\blacksquare & \\square & \\blacksquare & \\blacksquare & \\square & \\blacksquare & \\square\\\\\n \\square & \\square & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\square & \\blacksquare & \\square & \\square & \\square\\\\\n \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\square & \\blacksquare & \\blacksquare\\\\ \n \\square & \\square & \\square & \\square & \\blacksquare & \\square & \\blacksquare \\\\\n \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\square\n \\end{array}\\right]$};\n \n\\node[xscale=0.6,yscale=0.73] at (2.8,-3) {$\\left[\\begin{array}{ccccccc}\n \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\blacksquare & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\blacksquare & \\blacksquare & \\square & \\square & \\square\\\\\n \\square & \\square & \\square & \\blacksquare & \\square & \\square & \\square\\\\\n \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\square & \\blacksquare & \\blacksquare\\\\ \n \\square & \\square & \\square & \\square & \\blacksquare & \\square & \\blacksquare \\\\\n \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare & \\square\n \\end{array}\\right]$};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Illustration of Corollary \\ref{co-3} using the matrix $P\\in\\BB^{7\\times7}$ from Figure~\\ref{Nfigure7}.}}\n\\label{Nfigure8}\n\\end{figure} \n\n\\smallskip\n\nCorollary \\ref{co-3} shows that the matrix representations of a nonobtuse $0\/1$-simplex are either all partly decomposable, or all fully \nindecomposable. This motivates to the following definition.\n \n\\begin{Def} {\\rm A nonobtuse simplex is called partly decomposable if it has a partly decomposable matrix representation, and fully \nindecomposable if it has not}.\n\\end{Def}\nWe will now investigate to what structure the recursive application of Theorem \\ref{th-3} leads. For this, assume again that $P$ is a partly \ndecomposable matrix representation of a nonobtuse $0\/1$-simplex $S\\in\\SS^n$. Then by Corollary \\ref{co-3}, $S$ has a matrix representation \nof the form\n\\be\\label{eq-11} P = \\left[\\begin{array}{r|c} N_1 & R_1 \\\\ \\hline 0 & A_1 \\end{array}\\right],\\ee\nin which $A_1$ is fully indecomposable. According to Theorem \\ref{th-3}, the $k\\times k$ matrix $N_1$ represents a nonobtuse $k$-simplex \n$K$ in $I^k$. If also $K$ is partly decomposable, we can block-partition $N_1$ using row (R) and column (C) permutations $\\Pi_1$ and \n$\\Pi_2$, such that \n\\be\\label{eq-12} \\tilde{P}=\\Pi_1 P\\Pi_2 = \\left[\\begin{array}{c|c|c} N_2 & R_{12} & R_{13} \\\\\\hline 0 & A_2 & R_{23}\\\\ \\hline 0 & 0 & A_1 \\end{array}\\right],\\ee\nwith $A_2$ fully indecomposable and $N_2$ possibly partly decomposable. Theorem \\ref{th-3} shows that\n\\[ R_{12} \\hdrie \\mbox{\\rm consists of copies of a column $\\nu$ of} \\hdrie [0|N_2], \\]\nand\n\\[ \\left[\\begin{array}{c}R_{13}\\\\\\hline R_{23}\\end{array}\\right]\\hdrie\\mbox{\\rm consists of copies of a column of} \\hdrie \\left[\\begin{array}{r|c} N_2 & R_{12} \\\\ \\hline 0 & A_2 \\end{array}\\right]. \\]\nThis means that if $R_{23}$ is nonzero, then $R_{13}$ consists of copies of the same column $\\nu$ of $N$ as does $R_{12}$. Thus the whole \nblock $[R_{12}\\,|\\,R_{23}]$ consists of copies of a column $\\nu$ of $[0|N_2]$. On the other hand, if $R_{23}$ is zero, then $R_{13}$ can \neither be zero, or consist of copies of {\\em any} column of $N_2$, including $\\nu$.\\\\[3mm]\nBy including operations of type (X) it is possible to map the entire strip above one of the fully indecomposable diagonal blocks to zero. Although \nthis will in general destroy the block upper triangular form of the square submatrix to the left of that strip, Corollary \\ref{co-4} shows that this \nsubmatrix remains partly decomposable. Therefore, its block upper triangular structure can be restored using only operations of type (R) and \n(C), which leave the zero strip intact.\n\n\\begin{rem}{\\rm It is generally not possible to transform $\\tilde{P}$ to block diagonal form with more than two diagonal blocks. See the \ntetrahedron $S$ in $I^3$ in Figure~\\ref{Nfigure9}. It is possible to put a vertex of $S$ at the origin such that facets $A$ and $N$ are orthogonal, and \nhence the corresponding matrix representation is block diagonal with two blocks. The triangular facet however requires a {\\em different} vertex \nat the origin for its $2\\times 2$ matrix representation to be diagonal.}\n\\end{rem}\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}] \n\\begin{scope}[shift={(0,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(2.5,1)--(0,0)--cycle;\n\\draw (2,0)--(0,2);\n\\draw (0,0)--(2.5,1);\n\\draw (0,2)--(2.5,1);\n\\draw[gray,very thick] (0,0)--(0,2);\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05]; \n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (0,2) circle [radius=0.05];\n\\node[scale=0.9] at (-0.3,1) {$A$};\n\\node[scale=0.9] at (1.2,0.2) {$N$};\n\\end{scope}\n\\node[scale=0.85] at (4.3,1.5) {$\\left[\\begin{array}{rr|r} 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\\\hline 0 & 0 & 1\\end{array}\\right]$};\n\\draw[<->] (5.5,1.5)--(6,1.5);\n\\node[scale=0.85] at (7.2,1.5) {$\\left[\\begin{array}{r|rr} 1 & 0 & 0 \\\\\\hline 0 & 1 & 1 \\\\ 0 & 0 & 1\\end{array}\\right]$};\n\\begin{scope}[shift={(9,0)}]\n\\draw[fill=gray!20!white] (0,0)--(0.5,1)--(0.5,3)--cycle;\n\\draw (2,0)--(0.5,1);\n\\draw (2,0)--(0.5,3);\n\\draw[gray,very thick] (0,0)--(2,0);\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05]; \n\\draw[fill=black] (0.5,3) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\node[scale=0.9] at (0.7,0.3) {$A$};\n\\node[scale=0.9] at (0.35,1.3) {$N$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Any simplex $S$ with a partly decomposable matrix representation has a pair of facets $A$ and $N$ of dimensions adding to \n$n$ are orthogonal to one another.}}\n\\label{Nfigure9}\n\\end{figure} \n\n\\smallskip\n\nSummarizing, the above discussion shows that each nonobtuse $0\/1$-simplex $S$ has a matrix representation that is block upper triangular, \nwith fully indecomposable diagonal blocks (possibly only one). The strip above each diagonal block is of rank one and consists only of copies of \na column to the left of the strip. Any matrix representation $P$ of $S$ can be brought into this form using only operations of type (C) and (R). \nUsing an additional reflection of type (X), it is possible to transform an entire strip above one of the diagonal blocks to zero using a column to \nthe left of the strip. Although this may destroy the block upper triangular form to the left of the strip, this form can be restored using operations \nof type (R) and (C) only.\\\\[3mm]\nIn view of Corollary \\ref{co-4}, the transposed inverse $P^{-\\top}$ of a partly decomposable matrix representation $P$ of a nonobtuse $0\/1$-\nsimplex $S$ is perhaps even simpler in structure than $P$ itself. On the diagonal it has the transposed inverses of the fully indecomposable \ndiagonal blocks $A_1,\\dots,A_p$ of $P$, and each {\\em horizontal} strip to the left of such a diagonal block $(A_j)^{-\\top}$ has \n{\\em at most} one nonpositive column that is not identically zero. This columns has two interesting features. The first is that it nullifies the \nsums of the \nrows in its strip. The second is that its position clearly indicates to which vertex of which other block $A_i$ the block $A_j$ is related. To \nillustrate what we mean by this, see Figure~\\ref{Nfigure7} for an example. Above the bottom right $3\\times 3$ block $A$ of $P$ we see copies of the second \ncolumn of $P$. This fact can also be read from the position of the nonzero entries to the left of the corresponding block $A^{-\\top}$ in \n$P^{-\\top}$.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!10!white] (0,4.5)--(4.5,0)--(4.5,4.5)--cycle;\n\\draw[fill=gray!90!white] (3,0)--(4.5,0)--(4.5,1.5)--(3,1.5)--cycle;\n\\draw[fill=gray!70!white] (2,1.5)--(3,1.5)--(3,2.5)--(2,2.5)--cycle;\n\\draw[fill=gray!40!white] (1,2.5)--(2,2.5)--(2,3.5)--(1,3.5)--cycle;\n\\draw[fill=gray!30!white] (0.5,3.5)--(1,3.5)--(1,4)--(0.5,4)--cycle;\n\\draw[fill=gray!20!white] (0,4)--(0.5,4)--(0.5,4.5)--(0,4.5)--cycle;\n\\node at (0.7,2.2) {$0$};\n\\node at (1.5,0.7) {$0$};\n\\node at (3.7,0.7) {$A$}; \n\\draw (0,4)--(0,0)--(3,0);\n\\draw (3,0)--(4.5,0);\n\\draw (0,1.5)--(4.5,1.5);\n\\draw (1,2.5)--(4.5,2.5);\n\\draw (0.5,3.5)--(4.5,3.5);\n\\draw (0,4)--(4.5,4);\n\\draw (0,4.5)--(4.5,4.5);\n\\draw (0,4)--(0,4.5);\n\\draw (0.5,3.5)--(0.5,4.5);\n\\draw (1,2.5)--(1,4.5);\n\\draw (2,1.5)--(2,4.5);\n\\draw (3,0)--(3,4.5);\n\\draw (4.5,0)--(4.5,4.5);\n\\node at (7,2.5) {\\fbox{$P$}};\n\\draw[->] (7,3.5)--(5,3.5);\n\\draw[->] (7,1.5)--(9.5,1.5);\n\\node at (6.3,4) {(C)+(R)};\n\\node at (8.2,1) {(C)+(R)+(X)};\n\\begin{scope}[shift={(10,0)}]\n\\draw[fill=gray!10!white] (0,4.5)--(3,1.5)--(3,4.5)--cycle;\n\\draw[fill=gray!90!white] (3,0)--(4.5,0)--(4.5,1.5)--(3,1.5)--cycle;\n\\draw[fill=gray!70!white] (2,1.5)--(3,1.5)--(3,2.5)--(2,2.5)--cycle;\n\\draw[fill=gray!40!white] (1,2.5)--(2,2.5)--(2,3.5)--(1,3.5)--cycle;\n\\draw[fill=gray!30!white] (0.5,3.5)--(1,3.5)--(1,4)--(0.5,4)--cycle;\n\\draw[fill=gray!20!white] (0,4)--(0.5,4)--(0.5,4.5)--(0,4.5)--cycle;\n\\node at (1.5,0.7) {$0$};\n\\node at (3.7,3) {$0$};\n\\node at (3.7,0.7) {$A$};\n\\node at (0.7,2.2) {$0$};\n\\draw (0,4)--(0,0)--(3,0);\n\\draw (3,0)--(4.5,0);\n\\draw (0,1.5)--(4.5,1.5);\n\\draw (1,2.5)--(3,2.5);\n\\draw (0.5,3.5)--(3,3.5);\n\\draw (0,4)--(3,4);\n\\draw (0,4.5)--(4.5,4.5);\n\\draw (0,4)--(0,4.5);\n\\draw (0.5,3.5)--(0.5,4.5);\n\\draw (1,2.5)--(1,4.5);\n\\draw (2,1.5)--(2,4.5);\n\\draw (3,0)--(3,4.5);\n\\draw (4.5,0)--(4.5,4.5);\n\\end{scope}\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{A partly decomposable matrix representation $P$ of a nonobtuse $0\/1$-simplex $S$ can be brought in the left form using \noperations of type (C) and (R) only, and in the right form if using additional operations of type (X). The diagonal blocks are fully \nindecomposable.}}\n\\label{Nfigure10}\n\\end{figure} \n\n\\begin{rem}\\label{rem-10}{\\rm The above shows that to each nonobtuse $0\/1$-simplex $S$ of dimension $n$ we can associate a special type \nof simplicial complex $C_p$ consisting of $p$ mutually orthogonal fully indecomposable simplicial facets $S_1,\\dots, S_p$ with respective \ndimensions $k_1,\\dots,k_p$ adding to $n$, where each facet $S_j$ lies in in its own $k_j$-facet of $I^n$. Explicitly, let $C_1=S_1$. The \ncomplex $C_{j+1}$ is obtained by attaching a vertex of $S_{j+1}$ to a vertex $v$ of $C_j$, such that the orthogonal projection of $S_{j+1}$ \nonto the $(k_1+\\dots+k_j)$-dimensional ambient space of $C_j$ equals $v$.}\n\\end{rem}\nRemark \\ref{rem-10} is illustrated by the tetrahedron in Figure~\\ref{Nfigure9}. It can be built from three $1$-simplices $S_1,S_2,S_3$ simply by first \nattaching $S_2$ with a vertex to a vertex of $S_1$ orthogonally to $S_1$, giving a right triangle $C_2$. Then attaching $S_3$ to the correct \nvertex $v$ of $C_2$ such that the projection of $S_3$ onto $C_2$ equals $v$ gives the tetrahedron. In Section~\\ref{Sect-5} we will pay special \nattention to the nonobtuse simplices whose fully indecomposable components are $n$-cube edges.\n\n\\begin{rem}{\\rm The $1$-simplex in $I^n$ has a fully indecomposable matrix representation with doubly stochastic pattern. It is formally an \nacute simplex. Indeed, the normals to its $0$-dimensional facets $0$ and $1$ point in opposite directions. Hence, its only dihedral angle equals \nzero. Since there does not exist a fully indecomposable triangle in $I^2$, matrix representations of a partly decomposable simplex do not have \n$2\\times 2$ fully indecomposable diagonal blocks.}\n\\end{rem} \nWe would like to stress that although each partly decomposable matrix representation of a nonobtuse $0\/1$-simplex can be transformed into \nblock diagonal form by operations of types (C),(R) and (X) as depicted in the right in Figure~\\ref{Nfigure10}, the bottom right block cannot be {\\em any} of \nthe fully indecomposable diagonal blocks $A_j$. This is only possible if the corresponding simplex $S_j$ is attached to the remainder of the \ncomplex at exactly one vertex. For example, in Figure~\\ref{Nfigure11}, with $0$ as the origin, the matrix $A_3$ representing a regular tetrahedron in $I^3$ \nis a block of a block diagonal matrix. After mapping vertex $4$ to the origin with a reflection of type (X), the block $A_4$ representing a so-\ncalled antipodal $4$-simplex in $I^4$ is. However, the block $A_2$ representing the $1$-simplex in $I^1$ is never a block of a block diagonal \nmatrix representation of $S$. The only configurations of the three building blocks $S_1,S_2,S_3$ having a matrix representation that can be \ntransformed by operations of type (C),(R) and (X) onto block diagonal form with three diagonal blocks, are those in which $S_1,S_2$ and \n$S_3$ have a common vertex.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\begin{scope}[scale=1.1, every node\/.style={scale=1.1}]\n\\draw[fill=gray!30!white] (2,2)--(3.8,0.5)--(5,1.3)--(5,2.7)--(3.8,3.5)--cycle;\n\\draw[fill=gray!30!white] (1,2)--(-1,1)--(-1.5,2.5)--(-0.2,3.4)--cycle;\n\\draw (2,2)--(1,2);\n\\node[scale=0.9] at (3.2,2) {$S_1$};\n\\node[scale=0.9] at (-0.3,1.9) {$S_3$};\n\\node[scale=0.9] at (1.5,1.7) {$S_2$};\n\\draw (3.8,0.5)--(5,2.7)--(2,2); \n\\draw (2,2)--(5,1.3)--(3.8,3.5); \n\\draw (3.8,0.5)--(3.8,3.5);\n\\draw (-1,1)--(-0.2,3.4);\n\\draw[gray] (1,2)--(-1.5,2.5);\n\\draw[fill=black] (-1,1) circle [radius=0.05];\n\\draw[fill=white] (1,2) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\draw[fill=black] (-1.5,2.5) circle [radius=0.05]; \n\\draw[fill=black] (-0.2,3.4) circle [radius=0.05];\n\\draw[fill=black] (3.8,0.5) circle [radius=0.05];\n\\draw[fill=black] (5,1.3) circle [radius=0.05];\n\\draw[fill=black] (5,2.7) circle [radius=0.05];\n\\draw[fill=black] (3.8,3.5) circle [radius=0.05];\n\\node[scale=0.8] at (0,3.6) {$1$};\n\\node[scale=0.8] at (1.2,2.2) {$0$};\n\\node[scale=0.8] at (1.8,2.2) {$4$};\n\\node[scale=0.8] at (-1,0.7) {$3$};\n\\node[scale=0.8] at (-1.7,2.5) {$2$};\n\\node[scale=0.8] at (4,0.4) {$5$};\n\\node[scale=0.8] at (5.2,1.4) {$6$};\n\\node[scale=0.8] at (5.2,2.6) {$7$};\n\\node[scale=0.8] at (4,3.6) {$8$};\n\\end{scope}\n\\begin{scope}[shift={(8.5,2.3)}]\n\\node[scale=0.9] at (0,-1.5) {$1\\,\\,\\,\\,2\\,\\,\\,\\,3\\,\\,\\,\\,4\\,\\,\\,\\,5\\,\\,\\,\\,6\\,\\,\\,\\,7\\,\\,\\,8$}; \n\\node[xscale=0.6,yscale=0.73] at (0,0.1) {$\\left[\\begin{array}{ccc|c|cccc}\n \\blacksquare & \\blacksquare & \\square & \\square & \\square & \\square & \\square & \\square\\\\\n \\blacksquare & \\square & \\blacksquare & \\square & \\square & \\square & \\square& \\square\\\\\n \\square & \\blacksquare & \\blacksquare & \\square & \\square & \\square & \\square& \\square\\\\\\hline\n \\square & \\square & \\square & \\blacksquare & \\blacksquare & \\blacksquare & \\blacksquare& \\blacksquare\\\\\\hline\n \\square & \\square & \\square & \\square & \\blacksquare & \\blacksquare & \\blacksquare & \\square\\\\ \n \\square & \\square & \\square & \\square & \\blacksquare & \\blacksquare & \\square & \\blacksquare \\\\\n \\square & \\square & \\square & \\square & \\blacksquare & \\square & \\blacksquare & \\blacksquare \\\\\n \\square & \\square & \\square & \\square & \\square & \\blacksquare & \\blacksquare & \\blacksquare \\\\\n \\end{array}\\right]$}; \n \\node[scale=0.9] at (3.3,0.1) {$=\\left[\\begin{array}{c|c|c} A_3 & 0 & 0 \\\\\\hline 0 & A_2 & R \\\\\\hline 0 & 0 & A_1\\end{array}\\right]$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{A simplicial complex of three mutually orthogonal simplices $S_1,S_2,S_3$. The given matrix representation corresponds to \nchoosing the vertex $0$ as the origin. Reflecting vertex $4$ to the origin decouples, alternatively, the bottom right $4\\times 4$ block. It is not \npossible to transform the matrix to block diagonal form with $A_2=[\\,1\\,]$ as one of the diagonal blocks. Reflecting any other vertex to the \norigin does not even lead to a block diagonal matrix representation.}}\n\\label{Nfigure11}\n\\end{figure} \n\n\\smallskip\n\nIn general, the decomposability structure of a nonobtuse $0\/1$-simplex can be well visualized as a special type of planar graph, at the cost of \nthe geometrical structure. For this, assign to each $p\\times p$ fully indecomposable diagonal block a regular $p$-gon, and to attach these to \none another at the common vertex of the simplices they represent. See Figure~\\ref{Nfigure12} for an example. At the white vertices it is indicated how many \n$p$-gons meet. This number equals the number of diagonal blocks in the matrix representation when this vertex is reflected onto the origin. \n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!10!white] (0,0)--(1,-1)--(2,0)--(1,1)--cycle;\n\\draw (2,0)--(3,0);\n\\draw[fill=gray!30!white] (3,0)--(2,-0.7)--(2.4,-1.7)--(3.6,-1.7)--(4,-0.7)--cycle;\n\\draw (3,0)--(4,0);\n\\draw (4,0)--(5,1);\n\\draw (4,0)--(3,1);\n\\draw[fill=gray!50!white] (5,1)--(5.5,1.7)--(6.2,1.7)--(6.7,1)--(6.2,0.3)--(5.5,0.3)--cycle;\n\\draw[fill=gray!20!white] (5,0.3)--(6,-0.7)--(5,-1.7)--(4,-0.7)--cycle;\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=white] (2,0) circle [radius=0.05];\n\\draw[fill=white] (3,0) circle [radius=0.05];\n\\draw[fill=white] (4,0) circle [radius=0.05];\n\\draw[fill=white] (5,1) circle [radius=0.05];\n\\draw[fill=black] (3,1) circle [radius=0.05];\n\\draw[fill=black] (1,1) circle [radius=0.05];\n\\draw[fill=black] (1,0-1) circle [radius=0.05];\n\\draw[fill=black] (2,-0.7) circle [radius=0.05];\n\\draw[fill=black] (2.4,-1.7) circle [radius=0.05];\n\\draw[fill=black] (3.6,-1.7) circle [radius=0.05];\n\\draw[fill=black] (4,-0.7) circle [radius=0.05];\n\\draw[fill=black] (5.5,1.7) circle [radius=0.05];\n\\draw[fill=black] (6.2,1.7) circle [radius=0.05];\n\\draw[fill=black] (6.2,0.3) circle [radius=0.05];\n\\draw[fill=black] (5.5,0.3) circle [radius=0.05];\n\\draw[fill=black] (6.7,1) circle [radius=0.05];\n\\draw[fill=black] (6,-0.7) circle [radius=0.05];\n\\draw[fill=black] (5,-1.7) circle [radius=0.05];\n\\draw[fill=black] (5,0.3) circle [radius=0.05];\n\\draw[fill=white] (4,-0.7) circle [radius=0.05];\n\\node[scale=0.8] at (2.1,0.2) {$2$};\n\\node[scale=0.8] at (3,0.2) {$3$};\n\\node[scale=0.8] at (4,0.3) {$3$};\n\\node[scale=0.8] at (4.7,1) {$2$};\n\\node[scale=0.8] at (4,-0.4) {$2$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Schematic representation of a simplicial complex, built from a $5$-simplex, a $4$-simplex, two tetrahedra, and four edges, of \ntotal dimension $19$. With the origin at a white vertex, the matrix representation decouples into the indicated number of diagonal blocks. If the \nvertex is located at another vertex, the matrix representation does not decouple.}}\n\\label{Nfigure12}\n\\end{figure} \n\n\\smallskip\n\nBefore studying further properties of partly decomposable nonobtuse $0\/1$-simplices in terms of their fully indecomposable components, we \nwill pay special attention to {\\em orthogonal} simplices.\n\\section{Orthogonal simplices and their matrix representations}\\label{Sect-5}\nThe simplest class of nonobtuse simplices is formed by the {\\em orthogonal simplices}. These are nonobtuse simplices with $\\binom{n}{2}-n$ \nright dihedral angles. Note that this is the {\\em maximum} number of right dihedral angles a simplex can have, as Fiedler proved in \\cite{Fie} \nthat any simplex has at least $n$ acute dihedral angles. Orthogonal simplices are useful in many applications, see \\cite{BrDiHaKr,BrKoKrSo} \nand the references therein. We will restrict our attention to orthogonal $0\/1$-simplices.\\\\[3mm]\nThe orthogonal $0\/1$-simplices can be defined recursively as follows \\cite{BrDiHaKr}. The cube edge $I^1$ is an orthogonal simplex. Now, a \nnonobtuse $0\/1$-simplex $S$ in $I^n$ is orthogonal if it has an $(\\nmo)$-facet $F$ with the properties that:\\\\[2mm]\n$\\bullet$ $F$ is contained in an $(\\nmo)$-facet of $I^n$;\\\\[2mm]\n$\\bullet$ $F$ is an orthogonal $(\\nmo)$-simplex.\\\\[2mm]\nClearly, this way to construct orthogonal $0\/1$-simplices is a special case of how nonobtuse $0\/1$-simplices were constructed from their fully \nindecomposable parts in Section~\\ref{Sect-4}. This is because a vertex $v$ forms an orthogonal simplex $S$ together with an $(\\nmo)$-facet \n$F$ that is contained in an $(\\nmo)$-facet of $I^n$ if and only if $v$ projects orthogonally on a vertex of $F$. This shows in particular that all \nfully indecomposable components of any matrix representation of $S$ equal $[\\,1\\,]$, which limits the number of their upper triangular matrix \nrepresentations.\n\n\\begin{Pro} There exist $n!$ distinct upper triangular $0\/1$ matrices that represent orthogonal $0\/1$-simplices in $I^n$. \n\\end{Pro}\n{\\bf Proof. } Let $P\\in\\BB^{n\\times n}$ be an upper triangular matrix representing an orthogonal $n$-simplex. Then the matrix\n\\[ \\tilde{P} = \\left[\\begin{array}{r|r} P & r \\\\\\hline 0 & 1\\end{array}\\right] \\]\nis upper triangular, and according to Theorem \\ref{th-3} it represents a nonobtuse simplex if and only if $r$ equals one of the $n+1$ distinct \ncolumns of $[0\\,|\\,P]$. Thus there are $\\npo$ times as many matrix representations of orthogonal $(\\npo)$-simplices in $I^{\\npo}$ as of \northogonal $n$-simplices in $I^n$, whereas $[\\,1\\,]$ is the only one in $I^1$.\\hfill $\\Box$\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[->] (6,-0.2)--(4,-0.8);\n\\draw[->] (6,-0.2)--(8,-0.8);\n\\draw[->] (3,-1.3)--(1.6,-2.3);\n\\draw[->] (3,-1.3)--(4.4,-2.3);\n\\draw[->] (3,-1.3)--(3,-1.8);\n\\draw[->] (9,-1.3)--(7.6,-2.3);\n\\draw[->] (9,-1.3)--(10.4,-2.3);\n\\draw[->] (9,-1.3)--(9,-1.8);\n\\draw[->] (7,-2.8)--(5,-3.8);\n\\draw[->] (7,-2.8)--(6.5,-3.8);\n\\draw[->] (7,-2.8)--(7.5,-3.8);\n\\draw[->] (7,-2.8)--(9,-3.8);\n\\node[xscale=0.4, yscale=0.5] at (6,0) {$[\\,\\blacksquare\\,]$};\n\\node[xscale=0.4, yscale=0.5] at (3,-1) {$\\left[\\begin{array}{c|c} \\blacksquare & \\square \\\\ \\hline \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (9,-1) {$\\left[\\begin{array}{c|c} \\blacksquare & \\blacksquare \\\\ \\hline \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (1,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\square & \\square \\\\ \\square & \\blacksquare & \\square \\\\\\hline \\square & \\square & \\blacksquare \\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (3,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\square & \\blacksquare \\\\ \\square & \\blacksquare & \\square \\\\ \\hline \\square & \\square & \\blacksquare \\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (5,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\square & \\square \\\\ \\square & \\blacksquare & \\blacksquare \\\\\\hline \\square & \\square & \\blacksquare \\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (7,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\blacksquare &\\square\\\\ \\square & \\blacksquare &\\square \\\\\\hline \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (9,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\blacksquare &\\blacksquare\\\\ \\square & \\blacksquare &\\square \\\\\\hline \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (11,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\blacksquare & \\blacksquare\\\\ \\square & \\blacksquare &\\blacksquare \\\\\\hline \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (4.8,-4.5) {$\\left[\\begin{array}{ccc|c} \\blacksquare & \\blacksquare & \\square& \\square\\\\ \\square & \\blacksquare &\\square & \\square\\\\ \\square & \\square & \\blacksquare& \\square \\\\\\hline \\square & \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (6.3,-4.5) {$\\left[\\begin{array}{ccc|c} \\blacksquare & \\blacksquare &\\square& \\blacksquare\\\\ \\square & \\blacksquare &\\square & \\square\\\\\\square & \\square & \\blacksquare& \\square\\\\\\hline \\square & \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (7.7,-4.5) {$\\left[\\begin{array}{ccc|c} \\blacksquare & \\blacksquare &\\square& \\blacksquare\\\\ \\square & \\blacksquare &\\square & \\blacksquare\\\\ \\square & \\square & \\blacksquare& \\square\\\\ \\hline\\square & \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5] at (9.2,-4.5) {$\\left[\\begin{array}{ccc|c} \\blacksquare & \\blacksquare &\\square&\\square\\\\ \\square & \\blacksquare &\\square &\\square\\\\ \\square & \\square & \\blacksquare&\\blacksquare\\\\ \\hline\\square & \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{There are $n!$ upper triangular matrix representations of orthogonal $0\/1$-simplices.}}\n\\label{Nfigure13}\n\\end{figure} \n\\begin{rem}{\\rm Modulo the action of the hyperoctahedral group, there remain as many as the number of unlabeled trees on $n+1$ vertices. \nIndeed, it is not hard to verify that two matrices $P$ and $R$ representing orthogonal $0\/1$-simplices can be transformed into one another \nusing operations of type (R),(C) and (X) if and only if the spanning trees of orthogonal edges of the simplices corresponding to $P$ and $R$ are \nisomorphic as graphs.}\n\\end{rem}\n\n \n\\section{One Neighbor Theorem for a class of nonobtuse simplices}\\label{Sect-6}\nIn this section we will discuss the one neighbor theorem for acute simplices \\cite{BrDiHaKr} and generalize it to a larger class of nonobtuse \nsimplices. This appears a very nontrivial matter, which can be compared with the complications that arise when generalizing the Perron-\nFrobenius theory for positive matrices to nonnegative matrices \\cite{BaRa,BePl}.\n\n\\subsection{The acute case revisited}\nThe one neighor theorem for acute $0\/1$-simplices reads as follows. We present a alternative proof to the proof in \\cite{BrDiHaKr}, based in \nTheorem \\ref{th-1}.\n\n\\begin{Th}[One Neighbor Theorem] \\label{th-5} Let $S$ be an acute $0\/1$-simplex in $I^n$, and $F$ an $(\\nmo)$-facet of $S$ opposite the \nvertex $v$ of $S$. Write $\\hat{S}$ for the convex hull of $F$ and $\\ol{v}$. Then:\\\\[3mm]\n$\\bullet$ $F$ does not lie in an $(\\nmo)$-facet of $I^n$;\\\\[3mm]\n$\\bullet$ $\\hat{S}$ is the only $0\/1$-simplex having $F$ as a facet that may be acute, too.\\\\[3mm]\nIn words, an acute $0\/1$-simplex has at most one acute face-to-face neighbor at each facet.\n\\end{Th}\n{\\bf Proof. } Let $q$ be a normal vector to a facet of $F$ opposite a vertex $p$ of an acute $0\/1$-simplex $S$. Then due to \nTheorem \\ref{th-1}, $q$ has no zero entries. Therefore, the line $v+\\alpha q$ parametrized by $\\alpha\\in\\RR$ intersect the interior of $I^n$ if \nand only if $v\\in\\{p,\\ol{p}\\}$. Thus, for other vertices $v$ of $I^n$, the altitude from $v$ to the ambient hyperplane of $F$ does not land in \n$I^n$ and thus not in $F$. This is however a necessary condition for the convex hull of $F$ and $v$ to be acute. \\hfill $\\Box$\\\\[3mm]\nSee the left picture in Figure~\\ref{Nfigure14} for an illustration of Theorem \\ref{th-5} in $I^3$. Only the pair of white vertices end up inside $I^3$ when \nfollowing the normal direction $q$ of the facet $F$. All six other vertices, when projected on the plane containing $F$ end up outside $I^3$, or \non themselves.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!20!white] (0,0)--(2.5,1)--(2,2)--(0.5,3)--cycle;\n\\draw[fill=gray!50!white] (0,0)--(2.5,1)--(2,2)--cycle;\n\\draw[gray,thick,<->] (-0.35,2.8)--(0.35,1.2);\n\\draw[gray,thick,<->] (2.15,3.5)--(2.85,2.5);\n\\node at (2.3,0) {$\\overline{p}$};\n\\node at (0.2,3) {$p$};\n\\node at (0.9,0.6) {$F$};\n\\node at (0.9,1.7) {$q$};\n\\node at (1.5,0.3) {$-q$};\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (0,0)--(2,2);\n\\draw (2.5,1)--(0.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\draw[fill=black] (2.5,3) circle [radius=0.05];\n\\draw[fill=black] (0,2) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=white] (1.45,1.1) circle [radius=0.05];\n\\draw[thick,->] (2,0)--(1.55,0.9);\n\\draw[thick,->] (0.5,3)--(1.4,1.2);\n\\draw[fill=white] (0.5,3) circle [radius=0.07];\n\\draw[fill=white] (2,0) circle [radius=0.07];\n\\begin{scope}[shift={(5,0)}]\n\\node at (-0.3,0) {$p$};\n\\node at (2.8,1) {$\\hat{p}$};\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(0.5,3)--cycle;\n\\draw[fill=gray!50!white] (2,0)--(0.5,3)--(0.5,1)--cycle;\n\\node at (0.6,0.5) {$q$};\n\\node at (0.9,1.5) {$F$}; \n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (2,0)--(0.5,1);\n\\draw[fill=white] (0,0) circle [radius=0.07];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,3) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\draw[fill=white] (2.5,1) circle [radius=0.07];\n\\draw[fill=white] (0,2) circle [radius=0.07];\n\\draw[fill=white] (2.5,3) circle [radius=0.07];\n\\draw[fill=white] (1.25,0.5) circle [radius=0.05];\n\\draw[fill=white] (1.25,2.5) circle [radius=0.05];\n\\draw[thick,->] (0.1,0.05)--(1.2,0.47);\n\\draw[thick,->] (2.4,0.95)--(1.3,0.53);\n\\draw[thick,->] (0.1,2.05)--(1.2,2.47);\n\\draw[thick,->] (2.4,2.95)--(1.3,2.53);\n\\draw[gray,thick,<->] (1.3,1.7)--(2.7,2.3);\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Left: for any facet $F$ of an acute $0\/1$-simplex, there is only one pair of antipodal vertices $p,\\ol{p}$ that may project onto \n$F$. All others end up outside $I^n$ when following the normal $q$ in either direction. Right: in a nonobtuse simplex, there can be more than \ntwo vertices that remain in $I^n$ when following the normal direction to an interior facet.}}\n\\label{Nfigure14}\n\\end{figure} \n\n\\smallskip\n\nThe translation of Theorem \\ref{th-5} in terms of linear algebra is as follows. \n\\begin{Co} Let $P\\in\\BB^{n\\times(\\nmo)}$. The matrix $[P\\,|\\,v] \\in\\Bnn$ is a matrix representation of an acute $0\/1$-simplex for at most \none pair of antipodal points $v\\in\\{p,\\ol{p}\\}\\subset\\BB^n$.\n\\end{Co}\nThe one neighbor theorem dramatically restricts the number of $0\/1$-polytopes that can be face-to face triangulated by acute simplices. For \ninstance, only from dimension $n=7$ onwards there exists a pair of face-to-face acute simplices in $I^n$. In $I^7$ it is the Hadamard regular \nsimplex \\cite{Gr} and its face-to-face neighbor, which is unique modulo the action of the hyperoctahedral group, and which has the one-but-\nlargest volume in $I^7$ over all acute $0\/1$-simplices \\cite{BrCi2}. Also in \\cite{BrDiHaKr} the theorem turned out useful in constructing all \npossible face-to-face triangulations of $I^n$ consisting on nonobtuse simplices only, due to the following sharpening of the statement.\n\n\\begin{Co} Each acute $0\/1$-simplex $S$ in $I^n$ has at most one face-to-face nonobtuse neighbor at each of its facets.\n\\end{Co}\n{\\bf Proof.} This follows from the fact that in the proof of Theorem \\ref{th-5}, the altitudes from $v\\not=\\{p,\\ol{p}\\}$ intersect $I^n$ only in \n$v$ itself.\\hfill $\\Box$\\\\[3mm]\nA natural question is what can be proved for nonobtuse-$0\/1$ simplices. Theorem \\ref{th-2} showed that a normal to a facet of a nonobtuse \n$0\/1$-simplex $S$ can have entries equal to zero. Writing $\\zeros(q)$ for the number of entries of $q$ equal to zero, there \nare$2^{\\zeros(q)+1}$ vertices $v$ of $I^n$ from which the altitudes starting at $v$ onto the plane containing $F$ do not leave $I^n$. This is \nillustrated in the right picture in Figure~\\ref{Nfigure14}. The normal vector $q$ to the facet $F$ has one zero entry: $\\zeros(q)=1$. The altitudes from the \n$2^2$ white vertices of $I^3$ onto the plane containing $F$ lie in $I^3$.\\\\[3mm] \nNevertheless, only the altitudes from $p$ and $\\hat{p}$ land on $F$ itself, and we see that the interior facet $F$ of $S$ in $I^3$ has exactly \none nonobtuse neighbor. It is tempting to conjecture that the one neighbor theorem holds also for nonobtuse $0\/1$-simplices. The only \nadaption to make is then, based on the example in Figure~\\ref{Nfigure14}, that instead of the antipodal $\\ol{p}$ of $p$ in $I^n$, the second vertex \n$\\hat{p}$ such that the convex hull of $F$ with $\\hat{p}$ is a nonobtuse simplex would satisfy \n\\be\\label{restr-ant} \\hat{p}_j=1-p_j \\hdrie \\Leftrightarrow \\hdrie q_j\\not=0 \\und \\hat{p}_j = 0 \\hdrie \\Leftrightarrow \\hdrie q_j=0. \\ee\nIn words, $\\hat{p}$ is the antipodal of $p$ restricted to the $(n-\\zeros(q))$-dimensional $n$-cube facet that contains both $p$ and $q$. In \nFigure~\\ref{Nfigure14}, $\\hat{p}$ is the antipodal of $p$ in the bottom square facet of $I^3$.\\\\[3mm]\nAs an attempt to prove this conjecture, one may try to demonstrate that the remaining white vertices in the top square facet of $I^3$, although \ntheir altitudes lie in $I^3$, cannot fall onto $F$. Although we did not succeed in doing so, the nonobtusity of $S$ is a necessary condition. To \nsee this, consider the $0\/1$-simplex $S$ in $I^5$ with matrix representation\n\\be \\left[\\begin{array}{ccccc} 1 & 1 & 1 & 0 & 0\\\\ 1 & 1 & 0 & 1 & 0\\\\ 1 & 0 & 0 & 0 & 1\\\\ 0 & 0 & 1 & 1 & 0 \\\\ 0 & 1 & 1 & 0 & 1\\end{array}\\right] \\hdrie \\hdrie\\mbox{\\rm with} \\hdrie q = \\frac{1}{2}\\left[\\begin{array}{c} 0 \\\\ 1 \\\\ 1 \\\\ 1 \\\\ 1\\end{array}\\right],\\ee\nand $q$ is the normal to the facet $F$ of $S$ opposite the origin. Observe that the line from the origin to the vertex $2q$ in $I^5$ intersects \n$F$ in the midpoint of its edge between the two vertices of $S$ in the last two columns of $P$. This shows that both the origin and its \nantipodal in the bottom $4$-facet of $I^5$ as defined in (\\ref{restr-ant}) land in $F$ when following their respective altitudes. But also the line \nbetween $e_1^5$ and $e^5$ intersects $F$ in the midpoint of its edge between the two vertices in the first and third column of $P$. Thus, \nalso both the vertices $e_1^5$ and $e^5$ in the top $4$-facet of $I^5$ land in $F$ when following their altitudes. Thus, for a $0\/1$-simplex \nthat is not nonobtuse, it can occur that more than two vertices of $I^n$ project orthogonally onto $F$.\n\n\\subsection{More on fully indecomposable nonobtuse simplices}\nIn Section~\\ref{Sect-6.3} we will study the one neighbor theorem in the context of partly decomposable \nnonobtuse simplices. For this, but also for its own interest, we derive two auxiliary results on fully indecomposable nonobtuse simplices.\n\n\\begin{Le}\\label{lem-6} Each representation $P\\in\\Bnn$ of a fully indecomposable $0\/1$-simplex $S$ satisfies\n\\be P^\\top P \\geq I+e^n\\left(e^n\\right)^\\top. \\ee\nIn geometric terms this implies that all triangular facets of $S$ are acute.\n\\end{Le}\n{\\bf Proof. } A standard type of argument is the following. Write $D$ for the diagonal matrix having the same diagonal \nentries as $B=(P^\\top P)^{-1}$, and let $C=D-B$. Then $C\\geq 0$, and\n\\be B = D-C = D(I-D^{-1}C) \\und P^\\top P =B^{-1} = (I-D^{-1}C)^{-1}D^{-1}. \\ee\nBecause $B$ is an M-matrix \\cite{Joh,JoSm}, the spectral radius of $D^{-1}C$ is less than one, and the following Neumann series converges:\n\\be (I-D^{-1}C)^{-1} = \\sum_{j=0}^\\infty \\left(D^{-1}C\\right)^j. \\ee\nSince $P$ is fully indecomposable, $P^\\top P$ is irreducible, and thus $B$ is irreducible. But then, so are $C$ and $D^{-1}C$. \nBecause of the latter, for each pair $k,\\ell$ there is a $j$ such that $e_k^\\top(D^{-1}C)^j e_\\ell >0$. This proves that \n$B^{-1}=P^\\top P>0$. Since its entries are integers, $P^\\top P \\geq e^n(e^n)^\\top$. Thus, each pair of edges of $S$ that meet \nat the origin makes an acute angle. As by Theorem \\ref{th-1} all matrix representations of $S$ are fully indecomposable, we conclude \nthat all triangular facets of $S$ are acute. This implies that any diagonal entry of $P^\\top P$ is greater than the remaining entries in \nthe same row. Indeed, if two entries in the same row would be equal, then $p_j^\\top (p_j-p_i)=0$, which corresponds to two edges of $S$ making a right angle.\\hfill $\\Box$\n\n\\begin{Co}\\label{co-5} Let $P$ be a fully indecomposable matrix representation of a nonobtuse $0\/1$-simplex. \nIf $\\hat{P}$ equals $P$ with one column replaced by its antipodal, then $\\hat{P}^\\top\\hat{P}>0$.\n\\end{Co}\n{\\bf Proof. } Without loss of generality, assume that\n\\be P = [p\\,|\\,P_1] \\und \\hat{P} = [\\ol{p}\\,|\\,P_1] \\ee\nwith $p\\in\\BB^n$. Then\n\\be \\hat{P}^\\top\\hat{P} = \\left[\\begin{array}{cc} \\ol{p}^\\top\\ol{p} & \\ol{p}^\\top P_1 \\\\ P_1\\ol{p} & P_1^\\top P_1 \\end{array}\\right], \\ee\nand $P_1^\\top P_1>0$ because $P^\\top P>0$ as proved in Lemma \\ref{lem-5}. Due to the fact that for all $a,b\\in\\BB^n$,\n\\be a^\\top \\ol{b} = a^\\top (e^n-b) = a^\\top(a-b),\\ee\nwe see that also $P_1^\\top\\ol{p}>0$. Indeed, a zero entry would contradict that the diagonal entries of $P^\\top P$ are greater than its \noff-diagonal entries, as proved in Lemma \\ref{lem-6}.\\hfill $\\Box$\n \n\\begin{rem}{\\rm The matrix $\\hat{P}$ in Corollary \\ref{co-5} is not always fully indecomposable. See\n\\be P=\\left[\\begin{array}{ccc} 0 & 1 & 1 \\\\ 1 & 0 & 1\\\\ 1 & 1 & 0\\end{array}\\right] \\und \\hat{P}=\\left[\\begin{array}{ccc} 1 & 1 & 1 \\\\ 0 & 0 & 1\\\\ 0 & 1 & 0\\end{array}\\right], \\ee \nwhere the first columns of both matrices are each other's antipodal. This example also shows that the top left diagonal entry of \n$\\hat{P}^\\top\\hat{P}$ need not be greater than the other entries in its row.}\n\\end{rem}\nWe end this section with a theorem which was proved by inspection of a finite number of cases. We refer to \\cite{BrCi2} for details \non how to computationally generate the necessary data.\n\n\\begin{Th}\\label{th-6} Each fully indecomposable nonobtuse $0\/1$-simplex in $\\SS^n$ with $n\\leq 8$ is acute. There exist fully \nindecomposable nonobtuse $0\/1$-simplices in $\\SS^n$ with $n\\geq 9$ that are not acute.\n\\end{Th}\n{\\bf Proof. } See \\cite{BrCi2} for details on an algorithm to compute $0\/1$-matrix representations of $0\/1$-simplices modulo the action \nof the hyperoctahedral group. By inspection of all $0\/1$-simplices of dimensions less than or equal to $8$, we conclude the first \nstatement. For the second statement, we give an example. The $9\\times 9$ matrix in Figure~\\ref{Nfigure15} represents a nonobtuse simplex \nthat is not acute, but $P$ is fully indecomposable. \\hfill $\\Box$\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\node[scale=0.7] at (0,0) {$P = \\left[\\begin{array}{rrrrrrrrr}\n 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 0\\\\\n 1 & 0 & 1 & 1 & 1 & 0 & 0 & 1 & 1\\\\ \n 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1\\\\\n 0 & 1 & 1 & 1 & 1 & 0 & 1 & 0 & 1\\\\\n 0 & 1 & 1 & 1 & 0 & 1 & 0 & 1 & 1\\\\\n 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 0\\\\\n 0 & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1\\\\\n 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1\\\\\n 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1\n \\end{array}\\right], \\hspace{3mm}P^{-\\top}= \\dfrac{1}{20} \n \\left[\\begin{array}{rrrrrrrrr} \n 6 & 6 & -2 & -6 & 2 & 2 & 2 & 2 & -2\\\\\n 7 & -3 & 1 & 3 & 4 & -6 & -6 & 4 & 1\\\\\n 7 & -3 & 1 & 3 & -6 & 4 & 4 & -6 & 1\\\\\n -3 & 7 & 1 & 3 & 4 & -6 & 4 & -6 & 1\\\\\n -3 & 7 & 1 & 3 & -6 & 4 & -6 & 4 & 1\\\\\n -4 & -4 & 8 & 4 & 2 & 2 & 2 & 2 & -12\\\\\n -2 & -2 & 4 & -8 & 6 & 6 & -4 & -4 & 4\\\\\n -2 & -2 & 4 & -8 & -4 & -4 & 6 & 6 & 4\\\\\n -4 & -4 & -12 & 4 & 2 & 2 & 2 & 2 & 8\n \\end{array}\\right], \\hspace{3mm}\\dfrac{1}{20} \n\\left[\\begin{array}{r} 10 \\\\ 5 \\\\ 5 \\\\ 5 \\\\ 5 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array}\\right]$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Example of a fully indecomposable matrix representation $P$ of a nonobtuse simplex $S$ that is not acute. The vector \non the right is its normal $q$ to the facet $F$ opposite the origin. Since $q$ has entries equal to zero, $S$ cannot be acute. But \n$(P^\\top P)^{-1}$ satisfies (\\ref{eq-1.1}) and (\\ref{eq-1.2}), hence $S$ is nonobtuse. Note that none of the other normals has a zero entry.}}\n\\label{Nfigure15}\n\\end{figure} \n\n\\smallskip\n\nThus, Theorem \\ref{th-6} and Figure~\\ref{Nfigure14} prove that the {\\em indecomposability} of a matrix representation of a nonobtuse $0\/1$-simplex $S$ is, in \nfact, a {\\em weaker} property than the {\\em acuteness} of $S$. \\\\[3mm]\nEspecially since the two concepts coincide up to dimension eight, this came as a surprise. Citing G\\\"unther Ziegler in \nChapter 1 of {\\em Lectures on $0\/1$-Polytopes} \\cite{KaZi}: ``{\\em Low-dimensional intuition does not work!}\\,''. \nSee \\cite{BrCi2} for more such examples in the context of $0\/1$-simplices. \n \n\n\\subsection{A One Neighbor Theorem for partly decomposable simplices}\\label{Sect-6.3}\nLet $S$ be a partly decomposable nonobtuse simplex. Then according to Corollary \\ref{co-3}, $S$ has a matrix representation\n\\be P = \\left[\\begin{array}{cc} N & 0 \\\\ 0 & A\\end{array}\\right] \\ee \nin which $A$ is fully indecomposable. We will discuss some cases in which a modified version of the One Neighbor Theorem \\ref{th-5} \nholds also for nonobtuse simplices that are not acute.\\\\[3mm]\n{\\bf Case I.} To illustrate the main line of argumentation, consider first the simplest case, which is that $S\\in\\SS^n$ can be represented by\n\\be\\label{eq-21} P = \\left[\\begin{array}{cc} A_2 & 0 \\\\ 0 & A_1\\end{array}\\right],\\hdrie\\mbox{\\rm with} \\hdrie A_1\\in\\BB^{k\\times k} \\und A_2\\in\\BB^{(n-k)\\times(n-k)} \\ee\nand in which $A_1$ and $A_2$ represent acute simplices $S_1$ and $S_2$. Then $A_1$ and $A_2$ are fully indecomposable, \nand $S_1$ and $S_2$ satisfy the One Neighbor Theorem \\ref{th-5}. Assume first that neither $A_1$ or $A_2$ equals the $1\\times 1$ matrix $[\\,1\\,]$.\\\\[3mm] \n{\\bf Notation. } We will write $X^j(y)$ for the matrix $X$ with column $j$ replaced by $y$.\\\\[3mm]\nNow, let $v\\in\\BB^n$, partitioned as $v^\\top = (v_1^\\top \\,\\,v_2^\\top)$ with $v_1\\in\\BB^k$. Assume that the block lower \ntriangular matrix $P^1(v)$ represents a nonobtuse simplex. This implies that its top-left diagonal block $A_2^1(v_1)$ does \nso, too, hence $v_1=a_1=Ae_1^k$ or $v_1=\\ol{a_1}$ by Theorem \\ref{th-5}. Corollary \\ref{co-5} shows that in both cases \n$A_2^1(v_1)^\\top A_2^1(v_1) > 0$. But then Theorem \\ref{th-3} in combination with the observations in Remarks \\ref{rem-4} \nand \\ref{rem-6} proves that $v_2=0$, because all columns in the off-diagonal block must be copies of one and the same column \nof $A_1$. Hence, there is at most one $v\\in\\BB^n$ other than $Pe_1^n$ such that $P^1(v)$ is nonobtuse. See Figure~\\ref{Nfigure16} for a sketch of the proof.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw (0,0)--(3,0)--(3,3)--(0,3)--cycle;\n\n\\draw[fill=gray!20!white] (2,0)--(3,0)--(3,1)--(2,1)--cycle;\n\\draw[fill=gray!50!white] (0,1)--(2,1)--(2,3)--(0,3)--cycle;\n\n\\draw (0.5,1)--(0.5,3);\n\n\\node at (2.5,0.5) {$A_1$};\n\\node at (1,0.5) {$0$};\n\\node at (2.5,2) {$0$};\n\\node at (0.25,2) {$a_1$};\n\n\\draw[->] (3.2,1.5)--(3.8,1.5);\n\n\\begin{scope}[shift={(4,0))}]\n\n\\draw (0,0)--(3,0)--(3,3)--(0,3)--cycle;\n\n\\draw[fill=gray!20!white] (2,0)--(3,0)--(3,1)--(2,1)--cycle;\n\\draw[fill=gray!50!white] (0.5,1)--(2,1)--(2,3)--(0.5,3)--cycle;\n\n\\draw (0.5,0)--(0.5,3);\n\\draw (0,1)--(0.5,1);\n\n\\node at (2.5,0.5) {$A_1$};\n\\node at (1,0.5) {$0$};\n\\node at (2.5,2) {$0$};\n\n\\node at (0.25,2) {$v_1$};\n\\node at (0.25,0.5) {$v_2$};\n\n\\draw[->] (3.2,1.5)--(3.8,1.5);\n\n\\end{scope}\n\n\\begin{scope}[shift={(8,0))}]\n\n\\draw (0,0)--(3,0)--(3,3)--(0,3)--cycle;\n\n\\draw[fill=gray!20!white] (2,0)--(3,0)--(3,1)--(2,1)--cycle;\n\\draw[fill=gray!50!white] (0.5,1)--(2,1)--(2,3)--(0.5,3)--cycle;\n\\draw[fill=gray!30!white] (0,1)--(0.5,1)--(0.5,3)--(0,3)--cycle;\n\n\\draw (0.5,0)--(0.5,3);\n\\draw (0,1)--(0.5,1);\n\n\\node at (2.5,0.5) {$A_1$};\n\\node at (1,0.5) {$0$};\n\\node at (2.5,2) {$0$};\n\n\\node at (0.25,2.4) {$a_1$};\n\\node at (0.25,2) {or};\n\\node at (0.25,1.6) {$\\overline{a_1}$};\n\\node at (0.25,0.5) {$v_2$};\n\n\\draw[->] (3.2,1.5)--(3.8,1.5);\n\n\\end{scope}\n\n\\begin{scope}[shift={(12,0))}]\n\n\\draw (0,0)--(3,0)--(3,3)--(0,3)--cycle;\n\n\\draw[fill=gray!20!white] (2,0)--(3,0)--(3,1)--(2,1)--cycle;\n\\draw[fill=gray!50!white] (0.5,1)--(2,1)--(2,3)--(0.5,3)--cycle;\n\\draw[fill=gray!30!white] (0,1)--(0.5,1)--(0.5,3)--(0,3)--cycle;\n\n\\draw (0.5,0)--(0.5,3);\n\\draw (0,1)--(0.5,1);\n\n\\node at (2.5,0.5) {$A_1$};\n\\node at (1,0.5) {$0$};\n\\node at (2.5,2) {$0$};\n\n\\node at (0.25,2.4) {$a_1$};\n\\node at (0.25,2) {or};\n\\node at (0.25,1.6) {$\\overline{a_1}$};\n\\node at (0.25,0.5) {$0$};\n\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Steps in proving a one neighbor theorem for partly decomposable nonobtuse simplices with two fully indecomposable \nblocks representing acute simplices.}}\n\\label{Nfigure16}\n\\end{figure} \n\n\\smallskip\n\nClearly, the same argument can be applied to prove that for all $j\\in\\{1,\\dots,n\\}$, the matrix $P^j(v)$ represents a nonobtuse $0\/1$-simplex \nfor at most one $v\\in\\BB^n$ other than $Pe_j^n$. This proves that for each column $p$ of $P$, the facet $F_p$ of $S$ opposite $p$ has at \nmost one nonobtuse neighbor. It remains to prove the same for the facet $F_0$ of $S$ opposite the origin. But because $S_1$ and $S_2$ are \nby assumption acute, the normals $q_1$ and $q_2$ to their respective facets opposite the origin, which satisfy $A_1^\\top q_1 = e_k^k$ and \n$A_2^\\top q_2 = e_{n-k}^{n-k}$, are both positive. But then so is the normal $q$ of $F_0$, which satisfies $P^\\top q = e_n^n$, and \nhence $q^\\top = (q_1^\\top\\,\\,q_2^\\top)>0$. And thus, apart from the origin, only $e_n^n$ can form a nonobtuse simplex together with $F_0$.\n\n\\begin{rem}{\\rm Note that this last argument does not hold if $A_1$ and $A_2$ are merely assumed to represent fully indecomposable \nnonobtuse simplices: the $9\\times 9$ matrix in Figure~\\ref{Nfigure15} shows that the normal of the facet opposite the origin may contain entries equal to zero.}\n\\end{rem}\nTo finish the case in which $S$ has a matrix representation as in (\\ref{eq-21}), assume without loss of generality that $A_1=[\\,1\\,]$ and \n$A_2\\not=[\\,1\\,]$. Then the facet $F$ of $S$ opposite the last column of $P$ lies in a cube facet, and thus it cannot have a nonobtuse \nface-to-face neighbor. For the remaining $n$ facets of $S$, arguments as above apply, and we conclude that $S$ has at most one nonobtuse \nneighbor at each of its facets. The remaining case that $A_1=A_2=[\\,1\\,]$ is trivial.\\\\[3mm]\nNote that if a nonobtuse $0\/1$-simplex has a block diagonal matrix representation with $p>2$ blocks, each representing an acute simplex, the \nresult remains valid, based on a similar proof.\\\\[3mm]\n{\\bf Case II.} Assume now that the matrix representation $P$ of a nonobtuse $0\/1$-simplex $S$ has the form\n\\be\\label{case-iii} P = \\left[\\begin{array}{cc} N_1 & 0 \\\\ 0 & A_1\\end{array}\\right], \\ee \nwhere $A_1\\in\\BB^{(n-k)\\times(n-k)}$ represents an acute simplex and $N_1$ a merely nonobtuse simplex. Using similar \narguments as in Case I it is easily seen that the only two choices of $v\\in\\BB^n$ such that $P^j(v)$ with $k+1\\leq j \\leq n$ is nonobtuse are\n\\be v = Pe_j^n = \\left[\\begin{array}{c} 0 \\\\ a_j \\end{array}\\right] \\und v = \\left[\\begin{array}{c} 0 \\\\ \\ol{a_j} \\end{array}\\right], \\ee \nas no additional properties of $N_1$ need to be known. This changes if we examine the matrix $P^j(v)$ with $1\\leq j\\leq k$, as it is generally not \ntrue that $N_1^\\top N_1>0$. A way out is the following. Assume that also $N_1$ is partly decomposable, then using only row and column permutations, \nwe can first transform $N_1$ into the form\n\\be N_1 \\overset{(C)+(R)}{\\longrightarrow } \\left[\\begin{array}{cc} N_2 & R \\\\ 0 & A_2\\end{array}\\right] \\ee \nwhere we assume that $A_2$ represents an acute simplex. Then reflecting the vertex to the origin such that the block above $A_2$ becomes zero, we find that\n\\be\\label{eq-22} P \\sim \\tilde{P} = \\left[\\begin{array}{c|c|c} N_2 & R & 0 \\\\ \\hline 0 & A_2 & 0 \\\\\\hline 0 & 0 & A_1\\end{array}\\right] \\sim \\left[\\begin{array}{c|c|c} \\tilde{N}_2 & 0 & \\tilde{R} \\\\\\hline 0 & A_2 & 0 \\\\\\hline 0 & 0 & A_1\\end{array}\\right] = \\hat{P},\\ee\nwhere $\\tilde {R}$ has the same columns as $R$ but possibly a different number of them. Now, select a column of $\\hat{P}$ that contains entries of $A_2$ and \nreplace it by $v$, partitioned as $v^\\top = (v_1^\\top \\,\\,v_2^\\top\\,\\,v_3^\\top)$. Because the bottom right $2\\times 2$ block part of $\\hat{P}$ is a matrix \nrepresentation of a nonobtuse simplex as considered in Case I, we conclude that $v_3=0$. Because the top left $2\\times 2$ block part of $\\tilde{P}$ is a \nmatrix representation as in (\\ref{case-iii}), we conclude that $v_3=0$ and $v_2$ is a column of $A_2$ or its antipodal. Thus, also the facets of the \nvertices of $S$ corresponding to its indecomposable part $A_2$ all have at most one nonobtuse neighbor.\\\\[3mm] \nNow, this process can be inductively repeated in case $\\tilde{N_2}$ is partly decomposable with a fully indecomposable part that represents an acute \nsimplex, and so on, until a fully indecomposable top left block $A_p$ remains. This block presents vertices for which it still needs to be proved that \ntheir opposite facets have at most one nonobtuse neighbor. To illustrate how to do this, consider the case $p=3$. Or, in other words, assume that \n$N_2$ in (\\ref{eq-22}) represents an acute simplex. Replace one of the corresponding columns of $\\tilde{P}$ by $v$ partitioned as \n$v^\\top = (v_1^\\top \\,\\,v_2^\\top\\,\\,v_3^\\top)$. Then $v_3=0$ because the $(1,3)$ block of $\\tilde{P}$ equals zero. Similarly, because \nthe $(1,2)$-block in $\\hat{P}$ equals zero, we find that $v_2=0$. And thus, $v_3$ is a column of $N_2$ or its antipodal. For $p>3$, we \ncan do the same one by one for the blocks at positions $(1,p),\\dots,(1,2)$.\\\\[3mm]\nThe analysis in this section can be summarized in the following theorem. \n\n\\begin{Th} Let $S$ be a nonobtuse $0\/1$-simplex whose fully indecomposable components are all acute. Then $S$ has at most one face-to-face neighbor at each of its interior facets.\n\\end{Th}\n\n\n\\subsection*{Acknowledgments}\nJan Brandts and Apo Cihangir acknowledge the support by Research Project 613.001.019 of the Netherlands Organisation for Scientific Research (NWO), and are grateful to Michal K\\v{r}\\'{\\i}\\v{z}ek for comments and discussions on earlier versions of the manuscript.\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe mechanical properties of amorphous solids have been harnessed\nextensively in designing materials which are ubiquitous in our everyday\nlife. However, a complete microscopic understanding of the mechanisms\nleading to the macroscopic response of these materials is still missing.\nIn order to develop materials with specific functions, it is necessary to\nhave an improved knowledge of these underlying processes. This remains\na challenging task.\n\nIt is known that the material properties of amorphous solids, such as\ncolloidal or metallic glasses, depend on their history of production,\ne.g.~the cooling rate by which they were quenched from a fluid phase\n\\cite{glassbook}. This dependence on the history, i.e.~the age of the\namorphous solid, is an important issue in computer simulations of glasses,\nespecially because the accessible cooling rates in simulations are many\norders of magnitudes larger than those accessible in experiments of real\nsystems. With respect to the comparison between simulation and experiment,\nit is therefore crucial to systematically understand the dependence of\nstructural and dynamic properties on the age of the glassy solid.\n\nIt is thus expected that the response of a glass to an external mechanical\nloading is affected by the age of the glass. If one shears an amorphous\nsolid under a constant strain rate, it is in general transformed\ninto a flowing fluid \\cite{rodneyrev2011,barratlemaitrerev}. While\nat sufficiently high strains, the flowing fluid reaches a steady state\nwithout any memory of the initial unsheared state, the transient response\nto the shear is affected by the history of the initial glass state. The\ncharacteristic stress-strain relation of a glassy system, in response\nto an externally applied shear rate, exhibits typically a maximum at a\nstrain of the order of 0.1 \\cite{zausch08}. In numerical simulations\nof sheared thermal glasses, the amplitude of this maximum is observed\nto depend on the age of the material, typically growing logarithmically\nwith increasing age \\cite{varnik04,robbinsprl05}.\n\nMoreover, the transient response of glasses to an external shear\nfield is often associated with the occurrence of shear bands,\ni.e.~band-like structures with strain or mobility higher than other\nregions, observed both in experiments\n\\cite{schuhrev,mb08,divouxrev15,fs14,vp11,bp10,wilde11,divouxprl10} and\nnumerical simulations\n\\cite{vb03,ch13,ir14,gps15,sf06,bailey06,chboc12,ratul-procaccia-12}. Such\nspatially localised structures are seen to emerge after the occurrence\nof the stress overshoot, as the stress relaxes to the steady state\nvalue. Also, the formation of these transient shearbands have been\nobserved to be influenced by the thermal history of the glassy\nstate, with states which are obtained by faster cooling being less\nsusceptible to shearband formation. Further, it has been noted that\nsuch a spatially heterogeneous response is more likely to occur\nin the transient regime beyond the stress overshoot, at any given temperature.\n\nThe focus of our study are thermal glasses which are often characterized\nas simple yield stress fluids, e.g.~colloids, emulsions. For such\nmaterials, the steady state flow curve (i.e.~stress vs.~imposed shear\nrate) is a monotonic function \\cite{bp10,nordstrom2010, BBDM15}.. Thus there are no persistent shear-bands,\nwhich would be the case for non-monotonic flow curves \\cite{fs14,coussot2010,mb12,ir14}. In the case of\nsimple yield stress fluids, the transient shearband that emerges are seen\nto broaden with time and eventually the entire material is fluidized, with\nthe timescale of fluidization depending on the imposed shear-rate or stress \n\\cite{divouxprl10,divoux12,ch13}. Such\nspatio-temporal fluctuations are also visible during steady flow, both\nin experiments and simulations \\cite{vb03,tsamados10,bp10}. \nHowever, in this work, our objective is\nto characterize the transient spatial heterogeneities, prior to onset of\nsteady flow. \n\nThe formation of transient shearbands has been addressed within\nthe scope of various theoretical models. Within the framework of\nspatially-resolved fluidity and soft-glass-rheology (SGR) models\n\\cite{fs14, moorcroft-cates-fielding-11, moorcroft-fielding-13},\nthe age-dependent spatially heterogeneous\nresponse has been obtained, with the occurrence of the stress-overshoot\nunder an applied shear-rate being associated with an instability\nleading to the formation of these transient heterogeneities. The\nmodel recovers the observation that more pronounced and long-lived\nshearbanding occurs for more aged glassy samples. Similarly, Manning et\nal.~\\cite{manningpre2007,manningpre2009} also observe various transient\nheterogeneous states, by analysing a shear-transformation-zone (STZ)\nmodel of glassy materials, which depend upon the initial state of the\nsystem (characterised by an initial effective temperature) and the\nimposed shear-rate. Further, they were able to map their results to\nthose obtained from numerical simulations \\cite{shiprl2007}. The same\nphenemenology has also been reproduced by other mesoscopic models\n\\cite{jaglajstat,damienroux2011}.\n\nIn this work, we address the question how the combination of ambient\ntemperature, applied shear-rate and age of the glass affects the\ntransient response, specifically the observation of spatio-temporal\nheterogeneities. This has not been systematically studied in earlier\nnumerical simulations. Consequently, we also compare our observations\nwith those from the theoretical models.\n\nTo this end, we perform molecular dynamics computer simulations of\nthe Kob-Andersen binary Lennard-Jones (KABLJ) model \\cite{ka94},\na well-studied glass former. Amorphous states are prepared by quenching\na supercooled liquid to different temperatures below the mode coupling\ntemperature, followed by a relaxation of the sample over a waiting time\n$t_{\\rm w}$. Then, the resulting glass samples are sheared with different\nshear rates $\\dot{\\gamma}$. The onset of plastic flow occurs around\nthe location of $\\sigma^{\\rm max}$, i.e.~at a strain $\\gamma^{\\rm max} =\n\\dot{\\gamma} t^\\star \\approx 0.1$ (with $t^\\star$ the time at which the\nmaximum is obtained), with the appearance of a peak in the stress-strain\nresponse. We demonstrate that at the different temperatures $T$, the\npeak height, $\\sigma_{\\rm max}$, for all ages and shear-rates, obeys the\nfunctional behavior $C(\\dot{\\gamma}, T) + A(T) {\\rm ln}(\\dot{\\gamma}t_{\\rm\nw})$ (with $C$ a function depending on $\\dot{\\gamma}$ and $T$ and $A$ a\ntemperature-dependent amplitude). Note that this finding is consistent\nwith earlier studies \\cite{varnik04,robbinsprl05}. Further, as we have\nshown recently \\cite{gps15}, transient (but long-lived) shear bands\nare formed for $\\gamma > \\gamma^{\\rm max}$, provided that shear rate\nis sufficiently low. We quantify the contrast in spatial mobilities\nand demonstrate that the extent of spatial heterogeneities is not only\ndependent on the age of the glass, but also on the ambient temperature.\n\nThe rest of the paper is organized as follows. In Sec.~\\ref{sec2} we\ndescribe the KABLJ model and the details of the simulation. Then, we\npresent the results for the stress-strain relations and the analysis in\nterms of mobility maps in Sec.~\\ref{sec3}. Finally, in Sec.~\\ref{sec4},\nwe summarize the results and draw some conclusions.\n\n\\section{Model and Methods}\n\\label{sec2}\nWe consider a binary mixture of Lennard-Jones (LJ) particles (say A and B)\nwith 80:20 ratio. This is a well-studied glass former. Particles interact\nvia LJ potential which is defined as:\n\\begin{eqnarray}\n\\label{LJ1}\n\\textrm{U}^{\\textrm{LJ}}_{\\alpha\\beta}(r) &=& \n\\phi_{\\alpha\\beta}(r)-\\phi_{\\alpha\\beta}(R_{c})-\\left(r-R_{c}\\right)\\left. \n\\frac{d\\phi_{\\alpha\\beta}}{dr}\\right|_{r=R_{c}},\\nonumber\\\\\n\\phi_{\\alpha\\beta}(r) &=& \n4\\epsilon_{\\alpha\\beta}\\left[\\left(\\sigma_{\\alpha\\beta}\/r\\right)^{12}-\n\\left(\\sigma_{\\alpha\\beta}\/r\\right)^{6}\\right]\\: r