Associative Cones and Integrable Systems

Abstract We identify ℝ⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S⁶. It is known that a cone over a surface M in S⁶ is an associative submanifold of ℝ⁷ if and only if M is almost complex in S⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S⁶ are the equation for primitive maps associated to the 6-symmetric space G₂=T², and use this to explain some of the known results. Moreover, the equation for S¹-symmetric almost complex curves in S⁶ is the periodic Toda lattice, and a discussion of periodic solutions is given. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0529756.     

A Berger type normal holonomy theorem for complex submanifolds

We prove a Berger type theorem for the normal holonomy F^{^}{\Phi^\perp} (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space \mathbbC Pⁿ{\mathbb{C} P^n}. Namely, if F^{^}{\Phi^\perp} does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of \mathbbCⁿ{\mathbb{C}^n} the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the \mathbbC Pⁿ{\mathbb{C} P^n} case) and basic facts of complex submanifolds.     

On the asymptotic existence of complex Hadamard matrices

Let N = N(q) be the number of nonzero digits in the binary expansion of the odd integer q. A construction method is presented which produces, among other results, a block circulant complex Hadamard matrix of order 2^αq, where α ≥ 2N - 1. This improves a recent result of Craigen regarding the asymptotic existence of Hadamard matrices. We also present a method that gives complex orthogonal designs of order 2^α+1q from complex orthogonal designs of order 2^α. We also demonstrate the existence of a block circulant complex Hadamard matrix of order 2^βq, where © 1997 John Wiley & Sons, Inc. J Combin Designs 5:319–327, 1997     

Why Gaussian quadrature in the complex plane?

This paper synthesizes formally orthogonal polynomials, Gaussian quadrature in the complex plane and the bi-conjugate gradient method together with an application. Classical Gaussian quadrature approximates an integral over (a region of) the real line. We present an extension of Gaussian quadrature over an arc in the complex plane, which we call complex Gaussian quadrature. Since there has not been any particular interest in the numerical evaluation of integrals over the long history of complex function theory, complex Gaussian quadrature is in need of motivation. Gaussian quadrature in the complex plane yields approximations of certain sums connected with the bi-conjugate gradient method. The scattering amplitude c ^T A ^–1 b is an example where A is a discretization of a differential–integral operator corresponding to the scattering problem and b and c are given vectors. The usual method to estimate this is to use c ^T x ^(k). A result of Warnick is that this is identically equal to the complex Gaussian quadrature estimate of 1/. Complex Gaussian quadrature thereby replaces this particular inner product in the estimate of the scattering amplitude.     

Complexes of Gorenstein Flat Modules and Gorenstein Cotorsion Modules

A complex (C, δ) is called strongly Gorenstein flat if C is exact and Ker δ ⁿ is Gorenstein flat in R-Mod for all n ∈ ?. Let 𝒮𝒢 stand for the class of strongly Gorenstein flat complexes. We show that a complex C of left R-modules over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ? and Hom^.(G, C) is exact for any strongly Gorenstein flat complex G. Furthermore, a bounded below complex C over a right coherent ring R is in the right orthogonal class of 𝒮𝒢 if and only if C ⁿ is Gorenstein cotorsion in R-Mod for all n ∈ ?. Finally, strongly Gorenstein flat covers and 𝒮𝒢^⊥-envelopes of complexes are considered. For a right coherent ring R, we show that every bounded below complex has a 𝒮𝒢^⊥-envelope.     

Maslovian Lagrangian surfaces of constant curvature in complex projective or complex hyperbolic planes

A Lagrangian submanifold is called Maslovian if its mean curvature vector H is nowhere zero and its Maslov vector field JH is a principal direction of A_H . In this article we classify Maslovian Lagrangian surfaces of constant curvature in complex projective plane CP ² as well as in complex hyperbolic plane CH ². We prove that there exist 14 families of Maslovian Lagrangian surfaces of constant curvature in CP ² and 41 families in CH ². All of the Lagrangian surfaces of constant curvature obtained from these families admit a unit length Killing vector field whose integral curves are geodesics of the Lagrangian surfaces. Conversely, locally (in a neighborhood of each point belonging to an open dense subset) every Maslovian Lagrangian surface of constant curvature in CP ² or in CH ² is a surface obtained from these 55 families. As an immediate by‐product, we provide new methods to construct explicitly many new examples of Lagrangian surfaces of constant curvature in complex projective and complex hyperbolic planes which admit a unit length Killing vector field. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Plemelj Formula of Higher Order Partial Derivatives of the Bochner-Martinelli Type Integral

In this paper, by using the technique of integral transformation, we obtain the Plemelj formulas with the Cauchy principal value and the Hadamard principal value of mixed higher order partial derivatives for integral of the Bochner-Martinelli type on a closed smooth manifold ∂D in Cⁿ. From the Plemelj formulas and using the theory of complex partial differential equation, we prove that the problem of higher order boundary value D^κΦ⁺(t) = D^κΦ⁻(t) + f(t) is equivalent to a complex linear higher order partial differential equation. Moreover, given a proper condition of the Cauchy boundary value problem, the problem of higher order boundary value has a unique branch complex harmonic solution satisfying Φ⁻(∞) = 0 in Cⁿ\∂D.     

The fixed point property in -triples and preduals of -triples

We prove that for every member X in the class of real or complex JB^*-triples or preduals of JBW^*-triples, the following assertions are equivalent:

(1) X has the fixed point property.

(2) X has the super fixed point property.

(3) X has normal structure.

(4) X has uniform normal structure.

(5) The Banach space of X is reflexive.

As a consequence, a real or complex C^*-algebra or the predual of a real or complex W^*-algebra having the fixed point property must be finite-dimensional.

A Note on Aluthge Transforms of Complex Symmetric Operators and Applications

In this paper, we reprove that: (i) the Aluthge transform of a complex symmetric operator [(T)\tilde] = |T|^\frac12 U|T|^\frac12\tilde{T} = |T|^{\frac{1}{2}} U|T|^{\frac{1}{2}} is complex symmetric, (ii) if T is a complex symmetric operator, then ([(T)\tilde])^*(\tilde{T})^{*} and [(T^*)\tilde]\widetilde{T^{*}} are unitarily equivalent. And we also prove that: (iii) if T is a complex symmetric operator, then [((T^*))\tilde]_s,t\widetilde{(T^{*})}_{s,t} and ([(T)\tilde]_t,s)^*(\tilde{T}_{t,s})^{*} are unitarily equivalent for s, t > 0, (iv) if a complex symmetric operator T belongs to class wA(t, t), then T is normal.     

Complex analytic curves and maximal surfaces

Maximal immersions of a surfaceM ² inton-dimensional Lorentz space which are isometric to a fixed holomorphic mapping ofM ² into complex Lorentz space are determined. The main tool is an adaption of Calabi's Rigidity Theorem. Such an adaption is necessary because of the existence of degenerate hyperplanes in complex Lorentz space.Partially supported by a grant from Wellesley College.     

Vietoris–Rips Complexes of Planar Point Sets

Fix a finite set of points in Euclidean n-space \mathbbEⁿ\mathbb{E}^{n} , thought of as a point-cloud sampling of a certain domain D ì \mathbbEⁿD\subset\mathbb{E}^{n} . The Vietoris–Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of D. There is a natural “shadow” projection map from the Vietoris–Rips complex to \mathbbEⁿ\mathbb{E}^{n} that has as its image a more accurate n-dimensional approximation to the homotopy type of D.     

On the Compactness Theorem for Differential Forms

Kichenassamy found conditions under which the space W _p ^k of differential forms on a closed manifold M embeds compactly in the space F _p ^k of currents on M. We give a version of Kichenassamy's theorem for an arbitrary Banach complex and, in particular, for an elliptic differential complex on a closed manifold.     

Laplacians on the holomorphic tangent bundle of a Kaehler manifold

Let M be a connected complex manifold endowed with a Hermitian metric g. In this paper, the complex horizontal and vertical Laplacians associated with the induced Hermitian metric 〈·, ·〉 on the holomorphic tangent bundle T ^1,0 M of M are defined, and their explicit expressions are obtained. Using the complex horizontal and vertical Laplacians associated with the Hermitian metric 〈·, ·〉 on T ^1,0 M, we obtain a vanishing theorem of holomorphic horizontal p forms which are compactly supported in T ^1,0 M under the condition that g is a Kaehler metric on M.     

Characterization and properties of extreme operators intoC(Y)

LetE be a real (or complex) Banach space,Y a compact Hausdorff space, andC(Y) the space of real (or complex) valued continuous functions onY. IfT is an extreme point in the unit ball of bounded linear operators fromE intoC(Y), then it is shown thatT ^* maps (the natural imbedding inC(Y) ^* of)Y into the weak ^*-closure of extS(E ^*), provided thatY is extremally disconnected, orE=C(X), whereX is a dispersed compact Hausdorff space.     

Complex Euler’s groups and values of Euler’s function at complex integer Gauss points

The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z, relatively prime to z. This group is calculated for z=(3+0i) ⁿ : it is isomorphic to the product of three cyclic group or orders (8, 3 ⁿ⁻¹ and 3 ⁿ⁻¹).     

Biharmonic submanifolds of {\mathbb{C}P^n}

We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold [`(M)]{\bar{M}} in \mathbbCPⁿ{\mathbb{C}P^n} and the bitension field of the inclusion of the corresponding Hopf-tube in \mathbbS²ⁿ⁺¹{\mathbb{S}^{2n+1}}. Using this relation we produce new families of proper-biharmonic submanifolds of \mathbbCPⁿ{\mathbb{C}P^n}. We study the geometry of biharmonic curves of \mathbbCPⁿ{\mathbb{C}P^n} and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.     

Classification of torus manifolds with codimension one extended actions

The purpose of this paper is to classify torus manifolds (M ²ⁿ , T ⁿ ) with codimension one extended G-actions (M ²ⁿ , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T ⁿ . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.     

Laplacians on quotients of Cauchy-Riemann complexes and Szegö map for L-harmonic forms

We compute the spectra of the Tanaka type Laplacians on the Rumin complex Q, a quotient of the tangential Cauchy-Riemann complex on the unit sphere S²ⁿ⁻¹ in ⁿ. We prove that Szegö map is a unitary operator from a subspace of (p, q −1)-forms on the sphere defined by the operators and the normal vector field onto the space of L²-harmonic (p, q)-forms on the unit ball. Our results generalize earlier result of Folland.     

Complex Dirichlet Forms: Non Symmetric Diffusion Processes and Schrödinger Operators

The concept of complex Dirichlet forms ^c resp. operators L _c in complex weighted L ²-spaces is introduced. Perturbations of classical Dirichlet forms by forms associated with complex first-order differential operators provide examples of complex Dirichlet forms.Complex Dirichlet operators L _c are unitarily equivalent with (a family of) Schrödinger operators with electromagnetic potentials.To ^c there is associated a pair of real-valued non symmetric Dirichlet forms on the corresponding real weighted L ²-spaces, which in turn are associated with (non-symmetric) diffusion processes.Results by Stannat on non symmetric Dirichlet forms and their perturbations can be used for discussing the essential self-adjointness of L _c .New closability criteria for (perturbation of) non symmetric Dirichlet forms are obtained.     

On Partial Tilting Complexes

Let R be a commutative noetherian ring and A a noetherian R-algebra. Let P ^? ∈ 𝒦^b(𝒫 _A ) with Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i > 0. We will provide a sufficient condition for P ^? to be a direct summand of a silting complex. Also, in case Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i ≠ 0, we will provide a sufficient condition for P ^? to be a direct summand of a tilting complex.