Triangulating Smooth Submanifolds with Light Scaffolding

We propose an algorithm to sample and mesh a k-submanifold M{\mathcal{M}} of positive reach embedded in \mathbbR^d{\mathbb{R}^{d}} . The algorithm first constructs a crude sample of M{\mathcal{M}} . It then refines the sample according to a prescribed parameter e{\varepsilon} , and builds a mesh that approximates M{\mathcal{M}} . Differently from most algorithms that have been developed for meshing surfaces of \mathbbR ³{\mathbb{R} ^3} , the refinement phase does not rely on a subdivision of \mathbbR ^d{\mathbb{R} ^d} (such as a grid or a triangulation of the sample points) since the size of such scaffoldings depends exponentially on the ambient dimension d. Instead, we only compute local stars consisting of k-dimensional simplices around each sample point. By refining the sample, we can ensure that all stars become coherent leading to a k-dimensional triangulated manifold [^(M)]{\hat{\mathcal{M}}} . The algorithm uses only simple numerical operations. We show that the size of the sample is O(e^-k){O(\varepsilon ^{-k})} and that [^(M)]{\hat{\mathcal{M}}} is a good triangulation of M{\mathcal{M}} . More specifically, we show that M{\mathcal{M}} and [^(M)]{\hat{\mathcal{M}}} are isotopic, that their Hausdorff distance is O(e²){O(\varepsilon ^{2})} and that the maximum angle between their tangent bundles is O(e){O(\varepsilon )} . The asymptotic complexity of the algorithm is T(e) = O(e^-k2-k){T(\varepsilon) = O(\varepsilon ^{-k^2-k})} (for fixed M, d{\mathcal{M}, d} and k).     

The Spectrum of Tetrahedral Quadruple Systems

An ordered analogue of quadruple systems is tetrahedral quadruple systems. A tetrahedral quadruple system of order v and index λ, TQS(v, λ), is a pair (S, T){(S, \mathcal{T})} where S is a finite set of v elements and T{\mathcal{T}} is a family of oriented tetrahedrons of elements of S called blocks, such that every directed 3-cycle on S is contained in exactly λ blocks of T{\mathcal{T}} . When λ = 1, the spectrum problem of TQS(v, 1) has been completely determined. It is proved that a TQS(v, λ) exists if and only if λ(v − 1)(v − 2) ≡ 0 (mod 3), λv(v − 1)(v − 2) ≡ 0 (mod 4) and v ≥ 4.     

One-way monotonicity as a form of strategy-proofness

Suppose that a vote consists of a linear ranking of alternatives, and that in a certain profile some single pivotal voter v is able to change the outcome of an election from s alone to t alone, by changing her vote from P _v to P￠_v{P^\prime_{v}} . A voting rule F{\mathcal{F}} is two-way monotonic if such an effect is only possible when v moves t from below s (according to P _v to above s (according to P￠_v{P^\prime_{v}} . One-way monotonicity is the strictly weaker requirement forbidding this effect when v makes the opposite switch, by moving s from below t to above t. Two-way monotonicity is very strong—equivalent over any domain to strategy proofness. One-way monotonicity holds for all sensible voting rules, a broad class including the scoring rules, but no Condorcet extension for four or more alternatives is one-way monotonic. These monotonicities have interpretations in terms of strategy-proofness. For a one-way monotonic rule F{\mathcal{F}} , each manipulation is paired with a positive response, in which F{\mathcal{F}} offers the pivotal voter a strictly better result when she votes sincerely.     

Resolvable minimum coverings with quadruples

Let V be a finite set of v elements. A covering of the pairs of V by k-subsets is a family F of k-subsets of V, called blocks, such that each pair in V occurs in at least one member of F. For fixed v and k, the covering problem is to determine the number of blocks in any minimum covering. A minimum covering is resolvable if we can partition the blocks into classes (called resolution classes) such that every element is contained in precisely one block of each class. A resolvable minimum covering of the pairs of V by k-subsets is denoted by RC(v, k). In this article, we show that there exist RC(v, 4) for v ≡ 0 (mod 4) except for v = 12 and possibly for v ∈ {104, 108, 116, 132, 156, 164, 204, 212, 228, 276}. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 431–450, 1998     

Cartan-Dieudonné Theorem for {\mathcal {A}}-Modules

Like the classical Cartan-Dieudonné theorem, the sheaf-theoretic version shows that A{\mathcal {A}}-isometries on a convenient A{\mathcal {A}}-module E{\mathcal {E}} of rank n can be decomposed in at most n orthogonal symmetries (reflections) with respect to non-isotropic hyperplanes. However, the coefficient sheaf of \mathbb C{\mathbb {C}}-algebras A{\mathcal {A}} is assumed to be a PID \mathbb C{\mathbb {C}}-algebra sheaf and, if (E,f){(\mathcal {E},\phi)} is a pairing with f{\phi} a non-degenerate A{\mathcal {A}}-bilinear morphism, we assume that E{\mathcal {E}} has nowhere-zero (local) isotropic sections; but, for Riemannian sheaves of A{\mathcal {A}}-modules, this is not necessarily required.     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

Repeated and Continuous Interactions in Open Quantum Systems

We consider a finite quantum system S{\mathcal {S}} coupled to two environments of different nature. One is a heat reservoir R{\mathcal {R}} (continuous interaction) and the other one is a chain C{\mathcal {C}} of independent quantum systems E{\mathcal {E}} (repeated interaction). The interactions of S{\mathcal {S}} with R{\mathcal {R}} and C{\mathcal {C}} lead to two simultaneous dynamical processes. We show that for generic such systems, any initial state approaches an asymptotic state in the limit of large times. We express the latter in terms of the resonance data of a reduced propagator of S+R{\mathcal {S}+\mathcal {R}} and show that it satisfies a second law of thermodynamics. We analyze a model where both S{\mathcal {S}} and E{\mathcal {E}} are two-level systems and obtain the asymptotic state explicitly (at lowest order in the interaction strength). Even though R{\mathcal {R}} and C{\mathcal {C}} are not directly coupled, we show that they exchange energy, and we find the dependence of this exchange in terms of the thermodynamic parameters. We formulate the problem in the framework of W ^*-dynamical systems and base the analysis on a combination of spectral deformation methods and repeated interaction model techniques. We analyze the full system via rigorous perturbation theory in the coupling strength, and do not resort to any scaling limit, like e.g. weak coupling limits, or any other approximations in order to derive some master equation.     

On the coordinate ring of spherical conjugacy classes

Let G be a simple algebraic group over an algebraically closed field k of characteristic zero and O{\mathcal{O}} be a spherical conjugacy class of G. We determine the decomposition of the coordinate ring k[O]{k[\mathcal{O}]} of O{\mathcal{O}} into simple G-modules.     

Unstable structures definable in o-minimal theories

Let M{\mathcal {M}} be a dense o-minimal structure, N{\mathcal {N}} an unstable structure interpretable in M{\mathcal {M}}. Then there exists X, definable in N^eq{\mathcal {N}^{eq}}, such that X, with the induced N{\mathcal {N}}-structure, is linearly ordered and o-minimal with respect to that ordering. As a consequence we obtain a classification, along the lines of Zilber’s trichotomy, of unstable t-minimal types in structures interpretable in o-minimal theories.     

Non-orthogonal Fusion Frames and the Sparsity of Fusion Frame Operators

Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusion frame applications in distributed systems is reflected in the efficiency of the end fusion process. This in turn is reflected in the efficiency of the inversion of the fusion frame operator S_WS_{\mathcal{W}}, which in turn is heavily dependent on the sparsity of S_WS_{\mathcal{W}}. We will show that sparsity of the fusion frame operator naturally exists by introducing a notion of non-orthogonal fusion frames. We show that for a fusion frame {W _i ,v _i } _i∈I , if dim(W _i )=k _i , then the matrix of the non-orthogonal fusion frame operator S_W{\mathcal{S}}_{{\mathcal{W}}} has in its corresponding location at most a k _i ×k _i block matrix. We provide necessary and sufficient conditions for which the new fusion frame operator S_W{\mathcal{S}}_{{\mathcal{W}}} is diagonal and/or a multiple of an identity. A set of other critical questions are also addressed. A scheme of multiple fusion frames whose corresponding fusion frame operator becomes an diagonal operator is also examined.     

Noncommutative Positivstellensätze for pairs representation-vector

We study non-commutative real algebraic geometry for a unital associative *-algebra A{\mathcal {A}} viewing the points as pairs (π, v) where π is an unbounded *-representation of A{\mathcal A} on an inner product space which contains the vector v. We first consider the *-algebras of matrices of usual and free real multivariate polynomials with their natural subsets of points. If all points are allowed then we can obtain results for general A{\mathcal {A}}. Finally, we compare our results with their analogues in the usual (i.e. Schmüdgen’s) non-commutative real algebraic geometry where the points are unbounded *-representation of A{\mathcal {A}}.     

Quasilinear varieties of semigroups

We characterize all quasilinear varieties of semigroups, i.e. semigroup varieties V{\mathcal{V}} with the property that for each word w there is a linear word w′ such that V{\mathcal{V}} satisfies w≈w′.     

Nilpotent algebras and affinely homogeneous surfaces

To every nilpotent commutative algebra N{\mathcal{N}} of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety S ì N{S\subset\mathcal{N}} can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator A{\mathcal{A}} of N{\mathcal{N}}. In case N{\mathcal{N}} admits a grading, the surface S is affinely homogeneous. More can be said if A{\mathcal{A}} has dimension 1, that is, if N{\mathcal{N}} is the maximal ideal of a Gorenstein algebra. In this case two such algebras N{\mathcal{N}}, [(N)\tilde]{\tilde{\mathcal{N}}} are isomorphic if and only if the associated hypersurfaces S, [(S)\tilde]{\tilde S} are affinely equivalent. If one of S, [(S)\tilde]{\tilde S} even is affinely homogeneous, ‘affinely equivalent’ can be replaced by ‘linearly equivalent’. In case the nil-index of N{\mathcal{N}} does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.     

Normal families of meromorphic functions and shared values

Let k be a positive integer, b ≠ 0 be a finite complex number, let P be a polynomial with either deg P ≥ 3 or deg P = 2 and P having only one distinct zero, and let F{\mathcal{F}} be a family of functions meromorphic in a domain D, all of whose zeros have multiplicities at least k. If, each pair of functions f and g in F, P(f)f^(k){\mathcal{F}, P(f)f^{(k)}} and P(g)g ^(k) share b in D, then F{\mathcal{F}} is normal in D.     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Induced Graph Packing Problems

Let H{\mathcal{H}} be a set of undirected graphs. The induced H{\mathcal{H}} -packing problem in an input graph G is to find a subgraph Q of G of maximum size such that each connected component of Q is an induced subgraph of G and is isomorphic to some member of H{\mathcal{H}} . In this paper we focus on the case when H{\mathcal{H}} consists of factor-critical graphs and a certain family of ‘propellers’. Clarifying the methods developed in the related theory of non-induced graph packings, we show a Gallai–Edmonds type structure theorem and a Berge–Tutte type minimax formula. We also give an Edmonds type alternating forest algorithm for the case when H{\mathcal{H}} consists of a sequential set of stars and factor-critical graphs. This simplifies the related result of Egawa, Kano and Kelmans.     

Intertwining the geodesic flow and the Schr?dinger group on hyperbolic surfaces

We construct an explicit intertwining operator L{\mathcal L} between the Schr?dinger group e^{it \frac\triangle2}{e^{it \frac\triangle2}} and the geodesic flow on certain Hilbert spaces of symbols on the cotangent bundle T*X _Γ of a compact hyperbolic surface X _Γ = Γ\D. We also define Γ-invariant eigendistributions of the geodesic flow PS_{j, k, nj,-nk}{PS_{j, k, \nu_j,-\nu_k}} (Patterson-Sullivan distributions) out of pairs of \triangle{\triangle} -eigenfunctions, generalizing the diagonal case j = k treated in Anantharaman and Zelditch (Ann. Henri Poincaré 8(2):361–426, 2007). The operator L{\mathcal L} maps PS_{j, k, nj,-nk}{PS_{j, k, \nu_j,-\nu_k}} to the Wigner distribution W^G_j,k{W^{\Gamma}_{j,k}} studied in quantum chaos. We define Hilbert spaces H_PS{\mathcal H_{PS}} (whose dual is spanned by {PS_{j, k, nj,-nk}{PS_{j, k, \nu_j,-\nu_k}}}), resp. H_W{\mathcal H_W} (whose dual is spanned by {W^G_j,k}{\{W^{\Gamma}_{j,k}\}}), and show that L{\mathcal L} is a unitary isomorphism from H_W ? H_PS.{\mathcal H_{W} \to \mathcal H_{PS}.}     

Unique Ergodicity of Harmonic Currents On Singular Foliations of {\mathbb{P}^2}

Let F{\mathcal{F}} be a holomorphic foliation of \mathbbP²{\mathbb{P}^2} by Riemann surfaces. Assume all the singular points of F{\mathcal{F}} are hyperbolic. If F{\mathcal{F}} has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F{\mathcal{F}}. This implies strong ergodic properties for the foliation F{\mathcal{F}}. We also study the harmonic flow associated to the current T.     

Multisorted dualisability: change of base

We prove that if a quasivariety A{\mathcal{A}} generated by a finite family M{\mathcal{M}} of finite algebras has a multisorted duality based on M{\mathcal{M}}, then A{\mathcal{A}} has a multisorted duality based on any finite family of finite algebras that generates it.