Infinite-dimensional Grassmann Manifold Research Articles

MORPHOLOGICAL ANALYSIS OF CELLS BY MEANS OF AN ELASTIC METRIC IN THE SHAPE SPACE

Shape analysis is of great importance in many fields, such as computer vision, medical imaging, and computational biology. This analysis can be performed considering shapes as closed planar curves in the shape space. This approach has been used for the first time to obtain the morphological classification of erythrocytes in digital images of sickle cell disease considering the shape space S1, which has the property of being isometric to an infinite-dimensional Grassmann manifold of two-dimensional subspaces (Younes et al., 2008), without taking advantage of all the features offered by the elastic metric related to the possibility of stretching and bending of the curves. In this paper, we study this deformation in the shape space, S2, which is based on the representation of closed planar curves by means of the square-root velocity function (SRVF) (Srivastava et al., 2011), using the elastic metric of this space to obtain more efficient geodesics and geodesic lengths between planar curves. Supervised classification with this approach achieved an accuracy of 94.3%, classification using templates achieved 94.2% and unsupervised clustering in three groups achieved 94.7%, considering three classes of erythrocytes: normal, sickle, and with other deformations. These results are better than those previously achieved in the morphological analysis of erythrocytes and the method can be used in different applications related to the treatment of sickle cell disease, even in cases where it is necessary to study the process of evolution of the deformation, something that can not be done in a natural way in the feature space.

Knot types of generalized Kirchhoff rods

Kirchhoff energy is a classical functional on the space of arclength-parameterized framed curves whose critical points approximate configurations of springy elastic rods. We introduce a generalized functional on the space of framed curves of arbitrary parameterization, which model rods with axial stretch or cross-sectional inflation. Our main result gives explicit parameterizations for all periodic critical framed curves for this generalized functional. The main technical tool is a correspondence between the moduli space of shape similarity classes of closed framed curves and an infinite-dimensional Grassmann manifold. The critical framed curves have surprisingly simple parameterizations, but they still exhibit interesting topological features. In particular, we show that for each critical energy level there is a one-parameter family of framed curves whose base curves pass through exactly two torus knot types, echoing a similar result of Ivey and Singer for classical Kirchhoff energy. In contrast to the classical theory, the generalized functional has knotted critical points which are not torus knots.

Shape Analysis of Framed Space Curves

In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed planar shapes, endowed with a natural metric, is isometric to an infinite-dimensional Grassmann manifold via the so-called square root transform. This result facilitates efficient shape comparison by virtue of explicit descriptions of Grassmannian geodesics. In this paper, we extend this shape analysis framework to treat shapes of framed space curves. By considering framed curves, we are able to generalize the square root transform by using quaternionic arithmetic and properties of the Hopf fibration. Under our coordinate transformation, the space of closed framed curves corresponds to an infinite-dimensional complex Grassmannian. This allows us to describe geodesics in framed curve space explicitly. We are also able to produce explicit geodesics between closed, unframed space curves by studying the action of the loop group of the circle on the Grassmann manifold. We apply our results to compute means for collections of space curves and to perform statistical analysis of circular DNA molecule shapes.

Intrinsic derivative, curvature estimates and squeezing function

This survey paper consists of two folds. First of all, we recall the concept of intrinsic derivative which was introduced by Lu (1979) and the related works due to Lu in his last ten years, including the holomorphically isometric embedding into the infinite dimensional Grassmann manifold and the Bergman curvature estimates for bounded domains in ℂn. Inspired by Lu’s idea, we give the lower and upper bounds estimates for the Bergman curvatures in terms of the squeezing function—one concept originally introduced by Deng et al. (2012). Finally, we survey some recent progress on the asymptotic behaviors for Bergman curvatures near the strictly pseudoconvex boundary points and present some open problems on the squeezing functions of bounded domains in ℂn.

GEOMETRIC ANALYSIS OF PLANAR SHAPES WITH APPLICATIONS TO CELL DEFORMATIONS

Shape analysis is of great importance in many fields such as computer vision, medical imaging, and computational biology. In this paper we focus on a shape space in which shapes are represented by means of planar closed curves. In this shape space a new metric was recently introduced with the result that this shape space has the property of being isometric to an infinite-dimensional Grassmann manifold of 2-dimensional subspaces. Using this isometry it is possible, from Younes et al. (2008), to explicitly describe geodesics, a task that previously was not at all easy. Our aim is twofold, namely: to use this general theory in order to show some applications to the study of erythrocytes, using digital images of peripheral blood smears, in the treatment of sickle cell disease; and, since normal erythrocytes are almost circular and many Sickle cells have elliptical shape, to particularize the computation of geodesics and distances between shapes using this metric to planar objects considered as deformations of a template (circle or ellipse). The applications considered include: shape interpolation, shape classification, and shape clustering.

Jacobi inversion on strata of the Jacobian of the $C_{rs}$ curve $y^r = f(x)$, II

Previous work by the authors (this journal, \vol{60} (2008), 1009-1044) produced equations that hold on certain loci of the Jacobian of a cyclic $C_{rs}$ curve. A curve of this type generalizes elliptic curves, and the equations in question are given in terms of (Klein's) generalization of Weierstrass' $\sigma$-function. The key tool is a matrix with entries that are polynomial in the coordinates of the affine plane model of the curve, thus can be expressed in terms of $\sigma$ and its derivatives. The key geometric loci on the Jacobian of the curve give a stratification of Brill-Noether type. The results are of the type of Riemann-Kempf singularity theorem, the methods are germane to those used by J.D. Fay, who gave vanishing tables for Riemann's $\theta$-function and its derivatives. The main objects we use were developed by several contemporary authors, aside from the classical definitions: meromorphic differentials were expressed in terms of the coordinates mainly by V.M. Buchstaber, J.C. Eilbeck, V.Z. Enolski, D.V. Leykin, and Taylor expansions for $\sigma$ in terms of Schur polynomials also contributed by A. Nakayashiki, in terms of Sato's $\tau$-function. Within this framework, following specific results for $\sigma$-derivatives given by Y. \^Onishi, we arrive at our main results, namely statements on the vanishing on given strata of the partial derivatives of $\sigma$ indexed by Young-diagrams subsets that can be worked out in terms of the Weierstrass semigroup of the curve at its point at infinity. The combinatorial statements hold not only for Jacobians but for the stratification of Sato's infinite-dimensional Grassmann manifold as well.

Limit transition between hypergeometric functions of type BC and type A

AbstractLet ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski.

Differential Algebras with Banach-Algebra Coefficients I: from C*-Algebras to the K-Theory of the Spectral Curve

We present an operator-coefficient version of Sato’s infinite-dimensional Grassmann manifold, and τ-function. In this setting the classical Burchnall–Chaundy ring of commuting differential operators can be shown to determine a C*-algebra. For this C*-algebra topological invariants of the spectral ring become readily available, and further, the Brown–Douglas–Fillmore theory of extensions can be applied. We construct KK classes of the spectral curve of the ring and, motivated by the fact that all isospectral Burchnall–Chaundy rings make up the Jacobian of the curve, we compare the (degree-1) K-homology of the curve with that of its Jacobian. We show how the Burchnall–Chaundy C*-algebra extension by the compact operators provides a family of operator τ-functions.

Geometry of infinite dimensional Grassmannians and the Mickelsson–Rajeev cocycle

In their study of the representation theory of loop groups, Pressley and Segal introduced a determinant line bundle over an infinite dimensional Grassmann manifold. Mickelsson and Rajeev subsequently generalized the work of Pressley and Segal to obtain representations of the groups Map ( M , G ) where M is an odd dimensional spin manifold. In the course of their work, Mickelsson and Rajeev introduced for any p ≥ 1 , an infinite dimensional Grassmannian Gr p and a determinant line bundle Det p over it, generalizing the constructions of Pressley and Segal. The definition of the line bundle Det p requires the notion of a regularized determinant for bounded operators. In this paper we specialize to the case when p = 2 (which is relevant for the case when dim M = 3 ) and consider the geometry of the determinant line bundle Det 2 . We construct explicitly a connection on Det 2 and give a simple formula for its curvature. From our results we obtain a geometric derivation of the Mickelsson–Rajeev cocycle.

Holomorphic invariant forms of a bounded domain

Given a complete ortho-normal system ϕ = (ϕ0,ϕ1,ϕ2,…) of L 2 H(\( \mathcal{D} \)), the space of holomorphic and absolutely square integrable functions in the bounded domain \( \mathcal{D} \) of ℂ n , we construct a holomorphic imbedding \( \iota _\phi :\mathcal{D} \to \mathfrak{F}(n,\infty ) \), the complex infinite dimensional Grassmann manifold of rank n. It is known that in \( \mathfrak{F}(n,\infty ) \) there are n closed (μ, μ)-forms τμ (μ = 1,…,n) which are invariant under the holomorphic isometric automorphism of \( \mathfrak{F}(n,\infty ) \) and generate algebraically all the harmonic differential forms of \( \mathfrak{F}(n,\infty ) \). So we obtain in \( \mathcal{D} \) a set of (μ, μ)-forms ι ϕ * τμ (μ = 1,…, n), which are independent of the system ϕ chosen and are invariant under the bi-holomorphic transformations of \( \mathcal{D} \). Especially the differential metric ds 2 1 associated to the Kahler form ι ϕ * τ1 is a Kahler metric which differs from the Bergman metric ds 2 of \( \mathcal{D} \) in general, but in case that the Bergman metric is an Einstein metric, ds 1 2 differs from ds 2 only by a positive constant factor.

Self-duality and Integrable Systems

In his lectures (1984-85) at Kyoto University, Professor M. Sato presented a program for generalizing the soliton theory ([9] ; cf. [10]). The KadomtsevPetviashvili (KP) equation is a typical example of the soliton theory. The KP equation is written in the form of deformation equations of a linear ordinary differential equation. The time evolutions of a solution are interpreted as dynamical motions on an infinite dimensional Grassmann manifold ([7], [9]). The Lie algebra of microdifferential operators of one variable acts on this manifold transitively. He conjectured that any integrable systems can be written in the form of deformation equations of a linear system, and proposed to investigate a deformation of differential equations in higher dimensions. He showed a simple example of a deformation of holonomic systems in higher dimensions ([9]), and its generalization is treated in [4]. In this paper we study a deformation of ^-modules in higher dimensions. First we review the KP equation. We denote by 8 the ring of microdifferential operators of one variable x. We fix a microdifferential operator P, and denote by tP a time variable with respect to P. We study the following evolution equation associated to P:

On spinor representation of the infinite dimensional grassmanian

Fractional powers of the AKNS spectral operator L 3 = σ 3 ∂ x + qσ + − σ − are constructed as an expansion in terms of pseudodifferential operators. We found the existence of a quadruplet of operators L j , ( j = 0, 1, 2, 3), that share the same status as square roots of a common operator, L = L j 2, ( j = 0, 1, 2, 3). Completely integrable systems can then be represented as Lax pairs between L j and the differential operator part of the operator B n = L 0 L n−1 j ( L 0 and L j commute) on the infinite dimensional Grassmann manifold. The representation of the hierarchies of integrable equations can be obtained from an extension of the analysis to a “super algebra.”

Solution space of discrete Wiener-Hopf equations and Grassmann manifold

It is shown that the solution space of a system of discrete Wiener-Hopf equations is a set of points on an infinite-dimensional Grassmann manifold. Fractional transformations acting on the solution space are also discussed.

Infinite-dimensional Lie algebras acting on the solution space of various σ models

Infinite-dimensional Lie algebras of infinitesimal transformations acting on the solution space of various two-dimensional σ models are investigated. The main tools are (i) Takasaki’s interpretation [Commun. Math. Phys. 94, 35 (1984)] of the solutions of the associated linear system in terms of points in an infinite-dimensional Grassmann manifold and (ii) Mikhaïlov’s reduction procedure [Physica D 3, 73 (1981)] for linear systems. Takasaki’s approach leads, for the σ models with values in a Lie group G, to a set of transformations that has the structure of the loop algebra 𝔤⊗R[t,t−1], where 𝔤 is the Lie algebra of G. (This algebra has already been encountered by Dolan [Phys. Rev. Lett. 47, 1371 (1981)] and by Wu [Nucl. Phys. B 211, 160 (1983)] among others.) The σ models with a Wess–Zumino term are also considered; the algebraic structure is found to be the same. Finally, Mikhaïlov’s procedure is used to study the σ models with values in a Riemannian symmetric space (RSS) G/H which is not a Lie group. The algebra in these cases is a subalgebra of the loop algebra found for the principal models but it does not seem to be graded. However, it contains two graded infinite-dimensional subalgebras with the following structure: if 𝔥 and 𝔪 are the two eigenspaces of the involution σ defining the RSS G/H, these two graded subalgebras are 𝔥⊗R[t] and (⊕i∈N𝔥⊗t2i) ⊕(⊕i∈N𝔪⊗t2i+1).

Witten's gauge field equations and an infinite-dimensional Grassmann manifold

Witten's gauge fields are interpreted as motions on an infinite-dimensional Grassmann manifold. Unlike the case of self-dual Yang-Mills equations in Takasaki's work, the initial data must satisfy a system of differential equations since Witten's equations comprise a pair of spectral parameters. Solutions corresponding to (anti-) self-dual Yang-Mills fields are characterized in the space of initial data and in application, some Yang-Mills fields which are not self-dual, anti-self-dual nor abelian can be constructed.

On the support of quasi-invariant measures on infinite-dimensional Grassmann manifolds

One antisymmetric analogue of Gaussian measure on a Hilbert space is a certain measure on an infinite-dimensional Grassmann manifold. The main purpose of this paper is to show that the characteristic function of this measure is continuous in a weighted norm for graph coordinates. As a consequence the measure is supported on a thickened Grassmann manifold. The action of certain unitary transformations, in particular smooth loops S 1 → U ( n , C ) {S^1} \to U(n,{\mathbf {C}}) , extends to this thickened Grassmannian, and the measure is quasiinvariant with respect to these point transformations.