Geometric Generalisations of shake and rattle

A geometric analysis of the shake and rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises shake and rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting. In order for shake and rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given.     

Universal geometric cluster algebras

We consider, for each exchange matrix $B$ , a category of geometric cluster algebras over $B$ and coefficient specializations between the cluster algebras. The category also depends on an underlying ring $R$ , usually $\mathbb {Z},\,\mathbb {Q}$ , or $\mathbb {R}$ . We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over $B$ with universal geometric coefficients, or the universal geometric cluster algebra over $B$ . Constructing universal geometric coefficients is equivalent to finding an $R$ -basis for $B$ (a “mutation-linear” analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan ${\mathcal {F}}_B$ , which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between ${\mathcal {F}}_B$ and $\mathbf{g}$ -vectors. We construct universal geometric coefficients in rank $2$ and in finite type and discuss the construction in affine type.     

Deitmar’s versus To?n-Vaquié’s schemes over {\mathbb{F}_{1}}

Deitmar introduced schemes over ${\mathbb {F}_{1}}$ , the so-called “field with one element”, as certain spaces with an attached sheaf of monoids, generalizing the definition of schemes as ringed spaces. On the other hand, To?n and Vaquié defined them as particular Zariski sheaves over the opposite category of monoids, generalizing the definition of schemes as functors of points. We show the equivalence between Deitmar’s and To?n-Vaquiés notions and establish an analog of the classical case of schemes over ${\mathbb {Z}}$ . This result has been assumed by the leading experts on ${\mathbb {F}_{1}}$ , but no proof was given. During the proof, we also conclude some new basic results on commutative algebra of monoids, such as a characterization of local flat epimorphisms and of flat epimorphisms of finite presentation. We also inspect the base-change functors from the category of schemes over ${\mathbb {F}_{1}}$ to the category of schemes over ${\mathbb {Z}}$ .     

Highly Incident Configurations with Chiral Symmetry

A geometric $k$ -configuration is a collection of points and straight lines in the plane so that $k$ points lie on each line and $k$ lines pass through this point. We introduce a new construction method for constructing $k$ -configurations with non-trivial dihedral or chiral (i.e., purely rotational) symmetry, for any $k \ge 3$ ; the configurations produced have $2^{k-2}$ symmetry classes of points and lines. The construction method produces the only known infinite class of symmetric geometric 7-configurations, the second known infinite class of symmetric geometric 6-configurations, and the only known 6-configurations with chiral symmetry.     

The Noether number of the non-abelian group of order 3p

It is proven that for any representation over a field of characteristic \(0\) of the non-abelian semidirect product of a cyclic group of prime order \(p\) and the group of order \(3\) the corresponding algebra of polynomial invariants is generated by elements of degree at most \(p+2\) . We also determine the exact universal degree bound for separating systems of polynomial invariants of this group in characteristic not dividing \(3p\) .     

Sharp geometric rigidity of isometries on Heisenberg groups

We prove sharp geometric rigidity estimates for isometries on Heisenberg groups. Our main result asserts that every $(1+\varepsilon )$ -quasi-isometry on a John domain of the Heisenberg group $\mathbb H ^n, n>1,$ is close to some isometry up to proximity order $\sqrt{\varepsilon }+\varepsilon $ in the uniform norm, and up to proximity order $\varepsilon $ in the $L_p^1$ -norm. We give examples showing the asymptotic sharpness of our results.     

C^{0}-nonconforming tetrahedral and cuboid elements for the three-dimensional fourth order elliptic problem

In this paper, a theoretical framework is constructed on how to develop $C^0$ -nonconforming elements for the fourth order elliptic problem. By using the bubble functions, a simple practical method is presented to construct one tetrahedral $C^{0}$ -nonconforming element and two cuboid $C^{0}$ -nonconforming elements for the fourth order elliptic problem in three spacial dimensions. It is also proved that one element is of first order convergence and other two are of second order convergence. From the best knowledge of us, this is the first success in constructing the second-order convergent nonconforming element for the fourth order elliptic problem.     

Notes on Schneider’s stability estimates for convex sets

In 2009 Schneider obtained stability estimates in terms of the Banach–Mazur distance for several geometric inequalities for convex bodies in an n-dimensional normed space ${\mathbb{E}^n}$ . A unique feature of his approach is to express fundamental geometric quantities in terms of a single function ${\rho:\mathfrak{B} \times \mathfrak{B} \to \mathbb{R}}$ defined on the family of all convex bodies ${\mathfrak{B}}$ in ${\mathbb{E}^n}$ . In this paper we show that (the logarithm of) the symmetrized ρ gives rise to a pseudo-metric d _D on ${\mathfrak{B}}$ inducing, from our point of view, a finer topology than Banach–Mazur’s d _BM . Further, d _D induces a metric on the quotient ${\mathfrak{B}/{\rm Dil}^+}$ of ${\mathfrak{B}}$ by the relation of positive dilatation (homothety). Unlike its compact Banach–Mazur counterpart, d _D is only “boundedly compact,” in particular, complete and locally compact. The general linear group ${{\rm GL}(\mathbb{E}^n)}$ acts on ${\mathfrak{B}/{\rm Dil}^+}$ by isometries with respect to d _D , and the orbit space is naturally identified with the Banach–Mazur compactum ${\mathfrak{B}/{\rm Aff}}$ via the natural projection ${\pi:\mathfrak{B}/{\rm Dil}^+\to\mathfrak{B}/{\rm Aff}}$ , where Aff is the affine group of ${\mathbb{E}^n}$ . The metric d _D has the advantage that many geometric quantities are explicitly computable. We show that d _D provides a simpler and more fitting environment for the study of stability; in particular, all the estimates of Schneider turn out to be valid with d _BM replaced by d _D .     

Naive configurations

We describe a new way to construct finite geometric objects. For every \(k\) we obtain a symmetric configuration \(\mathcal{E }(k-1)\) with \(k\) points on a line. In particular, we have a constructive existence proof for such configurations. The method is very simple and purely geometric. It also produces interesting periodic matrices.     

Linear Drift and Entropy for Regular Covers

We consider a regular Riemannian cover ${\widetilde{M}}$ of a compact Riemannian manifold. The linear drift ? and the Kaimanovich entropy h are geometric invariants defined by asymptotic properties of the Brownian motion on ${\widetilde{M}}$ . We show that ? ≤ h.     

On the Stretch Factor of Randomly Embedded Random Graphs

We consider a random graph $\mathcal{G}(n,p)$ whose vertex set $V,$ of cardinality $n,$ has been randomly embedded in the unit square and whose edges, which occur independently with probability $p,$ are given weight equal to the geometric distance between their end vertices. Then each pair $\{u,v\}$ of vertices has a distance in the weighted graph, and a Euclidean distance. The stretch factor of the embedded graph is defined as the maximum ratio of these two distances, over all $\{u,v\}\subseteq V.$ We give upper and lower bounds on the stretch factor (holding asymptotically almost surely), and show that for $p$ not too close to 0 or 1, these bounds are the best possible in a certain sense. Our results imply that the stretch factor is bounded with probability tending to 1 if and only if $n(1-p)$ tends to 0, answering a question of O’Rourke.     

Approximation of the Laplace and Stokes operators with Dirichlet boundary conditions through volume penalization: a spectral viewpoint

We report the results of a study on the spectral properties of Laplace and Stokes operators modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, \(\eta \) , tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of \(\eta \) , both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed \(\eta \) , we find that only the part of the spectrum corresponding to eigenvalues \(\lambda \lesssim \eta ^{-1}\) approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of \(\eta \) and \(\lambda \) . Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision \(O(\eta )\) , Navier slip boundary conditions with slip length equal to \(\sqrt{\eta }\) . Moreover, for a given discretization, we show that there exists a value of \(\eta \) , corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier–Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases.     

On the stability of \varphi -uniform domains

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \subset {\mathbb R}^n$ . In the sequel, we investigate a class of domains, so called $\varphi $ -uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\varphi $ from $[0,\infty )$ to itself. Finally, we discuss a number of stability properties of $\varphi $ -uniform domains. In particular, we show that the class of $\varphi $ -uniform domains is stable in the sense that removal of a geometric sequence of points from a $\varphi $ -uniform domain yields a $\varphi _1$ -uniform domain.     

Square root 3 -Subdivision Schemes: Maximal Sum Rule Orders

Abstract. Subdivision with finitely supported masks is an efficient method to create discrete multiscale representations of smooth surfaces for CAGD applications. Recently a new subdivision scheme for triangular meshes, called $\sqrt 3$ -subdivision , has been studied. In comparison to dyadic subdivision, which is based on the dilation matrix 2I , $\sqrt 3$ -subdivision is based on a dilation M with det M=3 . This has certain advantages, for example, a slower growth for the number of control points. This paper concerns the problem of achieving maximal sum rule orders for stationary $\sqrt 3$ -subdivision schemes with given mask support, which is important because the sum rule order characterizes the order of the polynomial reproduction, and provides an upper bound on the Sobolev smoothness of the surface. We study both interpolating and approximating schemes for a natural family of symmetric mask support sets related to squares of sidelength 2n in Z ² , and obtain exact formulas for the maximal sum rule order for arbitrary n . For approximating schemes, the solution is simple, and schemes with maximal sum rule order are realized by an explicit family of schemes based on repeated averaging [15]. In the interpolating case, we use properties of multivariate Lagrange polynomial interpolation to prove the existence of interpolating schemes with maximal sum rule orders. These can be found by solving a linear system which can be reduced in size by using symmetries. From this, we construct some new examples of smooth (C ² ,C ³ ) interpolating $\sqrt 3$ -subdivision schemes with maximal sum rule order and symmetric masks. The construction of associated dual schemes is also discussed.     

Approximation faible et principe de Hasse pour des espaces homogènes à stabilisateur fini résoluble

Let \(K\) be a global field and \(G\) a finite solvable \(K\) -group. Under certain hypotheses concerning the extension splitting \(G\) , we show that the homogeneous space \(V=G'/G\) with \(G'\) a semi-simple simply connected \(K\) -group has the weak approximation property. We use a more precise version of this result to prove the Hasse principle for homogeneous spaces \(X\) under a semi-simple simply connected \(K\) -group \(G'\) with finite solvable geometric stabilizer \({\bar{G}}\) , under certain hypotheses concerning the \(K\) -kernel (or \(K\) -lien) \(({\bar{G}},\kappa )\) defined by \(X\) .     

Michael–Simon inequalities for k-th mean curvatures

This paper continues the study of Alexandrov–Fenchel inequalities for quermassintegrals for \(k\) -convex domains. It focuses on the application to the Michael–Simon type inequalities for \(k\) -curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a domain.     

Okounkov bodies and toric degenerations

Let $\varDelta $ be the Okounkov body of a divisor $D$ on a projective variety $X$ . We describe a geometric criterion for $\varDelta $ to be a lattice polytope, and show that in this situation $X$ admits a flat degeneration to the corresponding toric variety. This degeneration is functorial in an appropriate sense.     

Uniqueness for Inverse Boundary Value Problems by Dirichlet-to-Neumann Map on Subboundaries

We consider inverse boundary value problems for elliptic equations of second order of determining coefficients by Dirichlet-to-Neumann map on subboundaries, that is, the mapping from Dirichlet data supported on subboundary ${\partial \Omega \setminus \Gamma_{-}}$ to Neumann data on subboundary ${\partial \Omega \setminus \Gamma_{+}}$ . First we prove uniqueness results in three dimensions under some conditions such as ${\overline{\Gamma_{+}\cup\Gamma_{-}}= \partial\Omega}$ Next we survey uniqueness results in two dimensions for various elliptic systems for arbitrarily given ${\Gamma_{-} = \Gamma_{+}}$ Our proof is based on complex geometric optics solutions which are constructed by a Carleman estimate.     

Sporadic Reinhardt Polygons

Let $n$ be a positive integer, not a power of two. A Reinhardt polygon is a convex $n$ -gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter. For almost all $n$ , there are many Reinhardt polygons with $n$ sides, and many of them exhibit a particular periodic structure. While these periodic polygons are well understood, for certain values of $n$ , additional Reinhardt polygons exist, which do not possess this structured form. We call these polygons sporadic. We completely characterize the integers $n$ for which sporadic Reinhardt polygons exist, showing that these polygons occur precisely when $n=pqr$ with $p$ and $q$ distinct odd primes and $r\ge 2$ . We also prove that a positive proportion of the Reinhardt polygons with $n$ sides is sporadic for almost all integers $n$ , and we investigate the precise number of sporadic Reinhardt polygons that are produced for several values of $n$ by a construction that we introduce.     

An efficient polynomial time approximation scheme for load balancing on uniformly related machines

We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, \(C_i\) denotes the completion time of machine i. Our goal is to find a schedule that minimizes or maximizes \(\sum _{i=1}^m C_i^p\) for a fixed value of p such that \(0 . For \(p>1\) the minimization problem is equivalent to the well-known problem of minimizing the \(\ell _p\) norm of the vector of the completion times of the machines, and for \(0 , the maximization problem is of interest. Our main result is an efficient polynomial time approximation scheme (EPTAS) for each one of these problems. Our schemes use a non-standard application of the so-called shifting technique. We focus on the work (total size of jobs) assigned to each machine and introduce intervals of work that are forbidden. These intervals are defined so that the resulting effect on the goal function is sufficiently small. This allows the partition of the problem into sub-problems (with subsets of machines and jobs) whose solutions are combined into the final solution using dynamic programming. Our results are the first EPTAS’s for this natural class of load balancing problems.