ON GEOMETRY OF FRAMELESS BINOCULAR PERCEPTUAL SPACE

Alleys in visual space

An approach to geometry of visual space with no a priori mapping functions: Multidimensional mapping according to Riemannian metrics

Integrable systems associated with separation of the variables in real Riemannian spaces of constant curvature are considered. An isomorphism between all such systems and the hyperbolic Gaudin magnet is established. This isomorphism is used in a classification of all coordinate systems that admit separation of the variables, the basis of which is the classification of the correspondingL operators of the Gaudin magnet.

We compare temporal growth of the mean magnetic energy ℰ driven by a small-scale dynamo in Euclidean and Lobachevsky spaces. The governing parameters of the dynamo, such as the rms turbulent velocity and the correlation scale of turbulence, are presumed to vary randomly in space so the dynamo growth rate is a Gaussian random field. Since such a field is unbounded in unbounded space and can achieve, with a low probability, very large values of ℰ, it can grow super-exponentially in both cases. The super-exponential growth of ℰ in Euclidian space, known since the 1980s, can be considered as a statement that the mean energy growth rate is determined up to a weakly growing factor proportional to . We demonstrate that the super-exponential growth of ℰ in Lobachevsky space is a much more radical phenomenon, where ℰ grows as exp (const × t 5/3). We stress that extrapolating the properties of small-scale dynamos in Euclidean space to curved geometries such Lobachevsky space is not straightforward and requires some care. The effects under discussion becomes however important only if the spatial scale of domain in which the small-scale magnetic field is excited exceeds the radius of curvature.

Hypersurfaces in spaces of constant curvature satisfying a particular Roter type equation

A new formulation concerning the Riemannian space of constant curvature is preposed. Our formula, which is based on the fundamental quantity γ·lλ in place of the fundamental Riemannian metric tensor, is applied to discuss the wave equations for particles with low spin values. It is shown that the theory concerning the Riemannian space of constant curvature can be constructed in a unified manner in terms of the quantity γ·lλ.

Visual space is, according to Luneburg, a non-Euclidean hyperbolic space of constant negative curvature. Two personal constants enter into the geometrical description of binocular subjective space. One is σ indicating the degree of depth perception characteristic to the observer. The second is K, the curvature; a value which denotes by how much the visual space of an individual departs from Euclidean space. The constants σ and K were determined experimentally for 30 untrained observers by Luneburg's 3- and 4-point tests. All values found for K were negative and support the hypothesis that the geometry of binocular space is hyperbolic. Some discrepancies between the theory and findings were observed, while the sensitivity of the experiments to the lack of training and even a slight measure of aniseikonia is evident.

Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances between two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W_1 metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. However, the W_2 metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric. In this paper we take the first step towards explaining the lack of efficient algorithms for the W_2 metric, even over the three-dimensional Euclidean space R^3. We prove that there are no meaningful embeddings of W_2 over R^3 into a wide class of normed spaces, as well as that there are no efficient sketching algorithms for W_2 over R^3 achieving constant approximation. For example, our results imply that: 1) any embedding into L1 must incur a distortion of Omega(sqrt(log(n))) for pointsets of size n equipped with the W_2 metric; and 2) any sketching algorithm of size s must incur Omega(sqrt(log(n))/sqrt(s)) approximation. Our results follow from a more general statement, asserting that W_2 over R^3 contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W_2.

A geometric method for obtaining two-sided estimates for general quasilinear elliptic equations and its applications to problems of the calculus of variations and the problem of recovering a hypersurface from its mean curvature in spaces of constant curvature.

The hydrostatic mapping function, m(ε), describes the dependence of the hydrostatic path delay on the elevation angle ε. We have developed a simple mapping function, where the only free parameter is an “effective height” of the atmosphere, Hatm, corresponding to about the first two scale heights above the surface. The value of m(ε) is given by the ratio of the straight‐line ray path length within height Hatm to height Hatm itself. We used simulated delays at GPS (Global Positioning System) frequencies derived from high resolution ECMWF (European Centre for Medium‐Range Weather Forecasts) atmospheric analysis fields to assess the performance of the “geometric” mapping function relative to well established ones (the Davis, Niell, and Herring mapping functions, respectively). At elevations >6° the new mapping function generally exhibits, based on a single parameter, a performance comparable to or better than the other mapping functions.

In each of five experiments Ss made direct assessments of interpoint distances within a stimulus configuration. A multidimensional scaling method was applied to make explicit the form of the subjective configuration. The stimulus configurations consisted of small light points arranged in a two- or three-dimensional array in a dark room, and of real stars in the sky. All the data were adequately accounted for by a configuration constructed in Euclidean space of the appropriate dimensionality. That was true even in the situation where alley experiments with the same Ss gave the result that is usually regarded as evidence for a hyperbolic binocular space. The Euclidean interpretation entails a more complicated form of correspondence between physical and visual spaces than the hyperbolic interpretation.

Functional mapping has emerged as a powerful tool for mapping quantitative trait loci (QTL) that control developmental patterns of complex dynamic traits. Original functional mapping has been constructed within the context of simple interval mapping, without consideration of separate multiple linked QTL for a dynamic trait. In this article, we present a statistical framework for mapping QTL that affect dynamic traits by capitalizing on the strengths of functional mapping and composite interval mapping. Within this so-called composite functional-mapping framework, functional mapping models the time-dependent genetic effects of a QTL tested within a marker interval using a biologically meaningful parametric function, whereas composite interval mapping models the time-dependent genetic effects of the markers outside the test interval to control the genome background using a flexible nonparametric approach based on Legendre polynomials. Such a semiparametric framework was formulated by a maximum-likelihood model and implemented with the EM algorithm, allowing for the estimation and the test of the mathematical parameters that define the QTL effects and the regression coefficients of the Legendre polynomials that describe the marker effects. Simulation studies were performed to investigate the statistical behavior of composite functional mapping and compare its advantage in separating multiple linked QTL as compared to functional mapping. We used the new mapping approach to analyze a genetic mapping example in rice, leading to the identification of multiple QTL, some of which are linked on the same chromosome, that control the developmental trajectory of leaf age.

A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be faithfully preserved by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of $$k$$k points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set $$S = \{v_1, \ldots , v_k \}$$S={v1,?,vk} of points in a general metric space: the product of the distances from $$v_i$$vi to $$\{ v_1, \dots , v_{i-1} \}$${v1,?,vi-1}, normalized by $$\tfrac{1}{(k-1)!}$$1(k-1)!, where the ordering of the points is that given by Prim's minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of $$S$$S into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige's definition: the use of any other order in the product affects volume$$^{1/(k-1)}$$1/(k-1) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embedding.

In this article we explore the space of constant curvature. We consider the principal bundle over pseudoconformal plane. The elements of differential geometry are found for a surface of pseudo- Euclidean space. The elements of the matrix of the metric tensor, as well as the coefficients of the Riemannian connection, are calculated.

Hypersurfaces of constant $2$-mean curvature in spaces of constant sectional curvature are known to be solutions to a variational problem. We extend this characterization to ambient spaces which are Einstein. We then estimate the $2$-mean curvature of certain hypersurfaces in Einstein manifolds. A consequence of our estimates is a generalization of a result, first proved by Chern, showing that there are no complete graphs in the Euclidean space with positive constant $2$-mean curvature.