Dimensional Affine Research Articles

A new class of 3-dimensional piecewise affine systems with homoclinic orbits

Based on mathematical analysis, this paper proves the existence ofhomoclinic orbits in a new class of 3-dimensional piecewise affinesystems, and gives an example to illustrate the effectiveness of the method.

Equivariant deformations of algebraic varieties with an action of an algebraic torus of complexity 1

Let $X$ be a 3-dimensional affine variety with a faithful action of a 2-dimensional torus $T$. Then the space of first order infinitesimal deformations $T^1(X)$ is graded by the characters of $T$, and the zeroth graded component $T^1(X)_0$ consists of all equivariant first order (infinitesimal) deformations. Suppose that using the construction of such varieties from [1], one can obtain $X$ from a proper polyhedral divisor $\mathscr D$ on $\mathbb P^1$ such that the tail cone of (any of) the used polyhedra is pointed and full-dimensional, and all vertices of all polyhedra are lattice points. Then we compute $\dim T^1(X)_0$ and find a formally versal equivariant deformation of $X$. We also establish a connection between our formula for $\dim T^1(X)_0$ and known formulas for the dimensions of the graded components of $T^1$ of toric varieties.

Affine hypersurfaces with self congruent center map

In this paper, we study locally strictly convex affine hypersurfaces for which the center map is centroaffine congruent with the original hypersurface. By the equiaffine support function ρ, we show that the hypersurface is locally isometric to a warped product R×|ρ|N, where the gradient direction of ρ is along R. As a main result, we complete the classification when gradρ is the eigenvector of affine shape operator, which shows how to explicitly construct such hypersurfaces starting from one (or two) low dimensional affine hypersphere(s).

CIRCLE-FOLIATED MINIMAL SURFACES IN 4-DIMENSIONAL SPACE FORMS

Abstract. Catenoid and Riemann’s minimal surface are foliated by cir-cles, that is, they are union of circles. In R 3 , there is no other nonpla-nar example of circle-foliated minimal surfaces. In R 4 , the graph G c ofw = c/z for real constant c and ζ ∈C\{0}is also foliated by circles. In thispaper, we show that every circle-foliated minimal surface in R n is eithera catenoid or Riemann’s minimal surface in some 3-dimensional Affinesubspace or a graph surface G c in some 4-dimensional Affine subspace.We use the property that G c is circle-foliated to construct circle-foliatedminimal surfaces in S 4 and H . 1. IntroductionA surface M ∈ R n is said to be circle-foliated if there is a one-parameterfamily of planes whose intersection with M are circles. The catenoid andRiemann’s minimal surface are examples of circle-foliated minimal surfaces inR 3 . Enneper proved that the planes containing the circles of a circle-foliatedminimal surface in R 3 should be parallel [2] and [7]. Then it is easy to see thatthe plane, catenoid and Riemann’s minimal surface are the only circle-foliatedminimal surfaces in R

On the dimensions of a family of overlapping self-affine carpets

We consider the dimensions of a family of self-affine sets related to the Bedford–McMullen carpets. In particular, we fix a Bedford–McMullen system and then randomize the translation vectors with the stipulation that the column structure is preserved. As such, we maintain one of the key features in the Bedford–McMullen set-up in that alignment causes the dimensions to drop from the affinity dimension. We compute the Hausdorff, packing and box dimensions outside of a small set of exceptional translations, and also for some explicit translations even in the presence of overlapping. Our results rely on, and can be seen as a partial extension of, Hochman’s recent work on the dimensions of self-similar sets and measures.

Joint Covariance Estimation With Mutual Linear Structure

We consider the problem of joint estimation of structured covariance matrices. Assuming the structure is unknown, estimation is achieved using heterogeneous training sets. Namely, given groups of measurements coming from centered populations with different covariances, our aim is to determine the mutual structure of these covariance matrices and estimate them. Supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices, we develop a new efficient algorithm discovering the structure and using it to improve the estimation. Our technique is based on the application of principal component analysis in the matrix space. We also derive an upper performance bound of the proposed algorithm in the Gaussian scenario and compare it with the Cramer-Rao lower bound. Numerical simulations are presented to illustrate the performance benefits of the proposed method.

Total diameter and area of closed submanifolds

The total diameter of a closed planar curve $$C\subset \mathbf {R}^2$$ is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of $$C$$ . Furthermore, when $$C$$ is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds in the convex case only when $$C$$ is a circle. We also generalize these results to $$m$$ dimensional submanifolds of $$\mathbf {R}^n$$ , where the “area” will be defined in terms of the mod $$2$$ winding numbers of the submanifold about the $$n-m-1$$ dimensional affine subspaces of $$\mathbf {R}^n$$ .

Conformal immersions of prescribed mean curvature in [formula omitted

We prove the existence of (branched) conformal immersions F:S2→R3 with mean curvature H>0 arbitrarily prescribed up to a 3-dimensional affine indeterminacy. A similar result is proved for the space forms S3, H3 and partial results for surfaces of higher genus.

An inductive construction of minimally rigid body–hinge simple graphs

A d-dimensional body–hinge framework is a collection of d-dimensional rigid bodies connected by hinges, where a hinge is a (d−2)-dimensional affine subspace, i.e., pin-joints in 2-space, line-hinges in 3-space, plane-hinges in 4-space and etc. Bodies are allowed to move continuously in Rd so that the relative motion of any two bodies connected by a hinge is a rotation around it and the framework is called rigid if every motion provides a framework isometric to the original one. A body–hinge framework is expressed as a pair (G,p) of a multigraph G=(V,E) and a mapping p from e∈E to a (d−2)-dimensional affine subspace p(e) in Rd. Namely, v∈V corresponds to a body and uv∈E corresponds to a hinge p(uv) which joins the two bodies corresponding to u and v. Then, G is said to be realized as a body–hinge framework (G,p) in Rd, and is called a body–hinge graph. It is known [9,12] that the infinitesimal rigidity of a generic body–hinge framework (G,p) is determined only by its underlying graph G. So, a graph G is called (minimally) rigid if G can be realized as a (minimally) infinitesimally rigid body–hinge framework in d-dimension.In this paper, we shall present an inductive construction for minimally rigid body–hinge simple graphs in d-dimension with d≥3.

Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

Metric dimension or location number is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs Gn : F = (Gn)n≥1 depending on n as follows: the order |V (G)| = φ(n) and lim n→∞ φ(n) = ∞. If there exists a constant C > 0 such that dim(Gn) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension, otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), F is called a family with constant metric dimension. In this paper, we study the metric dimension of quasi flower snarks, generalized antiprism and cartesian product of square cycle and path. We prove that these classes of graphs have constant or bounded metric dimension. It is natural to ask for characterization of graphs classes with respect to the nature of their metric dimension. It is also shown that the exchange property of the bases in a vector space does not hold for minimal resolving sets of quasi flower snarks, generalized prism and generalized antiprism.

TOA sensor network self-calibration for receiver and transmitter spaces with difference in dimension

We study and solve the previously unstudied problem of finding both transmitter and receiver positions using only time of arrival (TOA) measurements when there is a difference in dimensionality between the affine subspaces spanned by receivers and transmitters. Anchor-free TOA network calibration has uses both in radio, radio strength and sound applications, such as calibrating ad hoc microphone arrays. Using linear techniques and requiring only minimal number of receivers and transmitters, an algorithm is constructed for general dimension p for the lower dimensional subspace. Degenerate cases are determined and partially characterized as when receivers or transmitters inhabit a lower dimensional affine subspace than was given as input. The algorithm is further extended to overdetermined cases in a straightforward manner. Utilizing the minimal solver, an algorithm using the Random Sample Consensus (RANSAC) paradigm has been constructed to simultaneously solve the calibration problem and remove severe outliers, a common problem in TOA applications. Simulated experiments show good performance for the minimal solver and the RANSAC-like algorithm under noisy measurements. Two indoor environment experiments using microphones and speakers give a RMSE of 2.35cm and 3.95cm on receiver and transmitter positions compared to computer vision reconstructions.

J-Tangent Affine Hyperspheres

In this paper we study J-tangent affine hyperspheres. Under some additional conditions we give a local characterization of 3-dimensional J-tangent affine hyperspheres.

Stationary probability vectors of higher-order Markov chains

We consider the higher-order Markov chain, and characterize the second order Markov chains admitting every probability distribution vector as a stationary vector. The result is used to construct Markov chains of higher-order with the same property. We also study conditions under which the set of stationary vectors of the Markov chain has a certain affine dimension.

Tangent-based manifold approximation with locally linear models

In this paper, we consider the problem of manifold approximation with affine subspaces. Our objective is to discover a set of low dimensional affine subspaces that represent manifold data accurately while preserving the manifold׳s structure. For this purpose, we employ a greedy technique that partitions manifold samples into groups, which are approximated by low dimensional subspaces. We start by considering each manifold sample as a different group and we use the difference of local tangents to determine appropriate group mergings. We repeat this procedure until we reach the desired number of sample groups. The best low dimensional affine subspaces corresponding to the final groups constitute our approximate manifold representation. Our experiments verify the effectiveness of the proposed scheme and show its superior performance compared to state-of-the-art methods for manifold approximation.

Algebraic vector bundles on punctured affine spaces and smooth quadrics

There is a canonical An-fibration from a (2n+1)-dimensional smooth affine quadric Yn to an (n+1)-dimensional punctured affine space Xn. For each rank r at least n, we construct a nontrivial algebraic vector bundle of rank r on Xn with trivial pullback to Yn. Moreover, we construct continuous families of arbitrarily large dimensional pairwise non-isomorphic rank n bundles on Xn with trivial pullbacks to Yn.

SYLVESTER–GALLAI TYPE THEOREMS FOR APPROXIMATE COLLINEARITY

AbstractWe study questions in incidence geometry where the precise position of points is ‘blurry’ (for example due to noise, inaccuracy or error). Thus lines are replaced by narrow tubes, and more generally affine subspaces are replaced by their small neighborhood. We show that the presence of a sufficiently large number of approximately collinear triples in a set of points in ${\mathbb{C}}^d$ implies that the points are close to a low dimensional affine subspace. This can be viewed as a stable variant of the Sylvester–Gallai theorem and its extensions. Building on the recently found connection between Sylvester–Gallai type theorems and complex locally correctable codes (LCCs), we define the new notion of stable LCCs, in which the (local) correction procedure can also handle small perturbations in the Euclidean metric. We prove that such stable codes with constant query complexity do not exist. No impossibility results were known in any such local setting for more than two queries.

Lagrangian Curves in a 4-Dimensional Affine Symplectic Space

Lagrangian curves in \(\mathbb {R}^{4}\) entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in \(\mathbb {R}^{4}\) and determine Lagrangian geodesics.

Randomized large distortion dimension reduction

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\). Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\)-dimensional affine subspaces of \({\mathbb {R}}^{n}\). We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\), $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\), which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\).

On the non-injectivity of the Vaserstein symbol in dimension three

L.N. Vaserstein proved over a two dimensional ring that the orbit space of unimodular rows of length three modulo elementary action has a Witt group structure. R.A. Rao and W. van der Kallen showed that the Vaserstein symbol need not be injective over three dimensional affine algebras over the real field, but is injective over three dimensional affine algebras over a field of cohomological dimension one whose characteristic ≠2,3. R.G. Swan, R.A. Rao and J. Fasel gave another example of a real affine algebra of dimension three for which the Vaserstein symbol is not injective. We demonstrate an uncountably infinite family of non-isomorphic affine algebras of dimension three over the real field for which the Vaserstein symbol is not injective.

Fundamental domains for properly discontinuous affine groups

We construct a fundamental region for the action on the $2d+1$-dimensional affine space of some free, discrete, properly discontinuous groups of affine transformations preserving a quadratic form of signature $(d+1, d)$, where $d$ is any odd positive integer.