Orthomodular Lattices and Quantales

Let L be a complete orthomodular lattice. There is a one to one correspondence between complete Boolean subalgebras of L contained in the center of L and endomorphisms j of L satisfying the Borceux–van den Bossche conditions. This article is dedicated to Raquel Hernández. SUBJCLASS: 0210.Ab, 0210.De, 03.65.-ca     

Characterization of Morita Equivalence Pairs of Quantales

We characterize the pairs of sup-lattices which occur as pairs of Morita equivalence bimodules between quantales in terms of the mutual relation between the sup-lattices.     

Some Remarks on Special Conformal and Special Projective Symmetries in General Relativity

Those space-times admitting special conformal vector fields and those admitting special projective vector fields have recently been studied. In this paper these two classes of space times are shown to be very closely related to each other. Certain uniqueness features of (and necessary extra symmetries contained in) the associated Lie algebras are discussed and the dimensionality of each of the algebras is computed.     

Morita Contexts and their Lattices of Relations

We study the interaction between the lattices of relations of members of a general Morita context. The pairs of reversing-order maps are defined, which determine the dualities between the lattices of ‘closed’ relations. Under rather weak conditions, these dualities can be composed obtaining the projectivities defined by simple maps. PACS: 02.10.De,02.10.Hh.     

Projective differential geometry and geodesic conservation laws in general relativity. I: Projective actions

The Lagrangian structure of the geodesic equation allows an extension of classical projective geometry to one-parameter projective group actions onR×TM (whereM is the spacetime). We determine all those projective actions which arise by prolongation of oneparameter group actions onR×M. The relation between projective actions onR×TM and the equation of geodesic deviation is developed.     

A Quantization on Riemann Surfaces with Projective Structure

Let X be a Riemann surface equipped with a projective structure. Let be a square-root of the holomorphic cotangent bundle K _X . Consider the symplectic form on the complement of the zero section of obtained by pulling back the symplectic form on K _X using the map ². We show that this symplectic form admits a natural quantization. This quantization also gives a quantization of the complement of the zero section in K _X equipped with the natural symplectic form.     

Projective Relativity: Present Status and Outlook

We give a critical analysis of projective relativity theory. Examining Kaluza's own intention and the following development by Klein, Jordan, Pauli, Thiry, Ludwig and others, we conclude that projective relativity was abused in its own terms and much more in the case of newer higher dimensional Kaluza–Klein theories with non-Abelian gauge groups. Reviewing the projective formulation of the Jordan isomorphy theorem yields some hints how one can proceed in a different direction. We can interpret the condition not as a field equation in a 5-dimensional Riemannian space, e.g. as vacuum Einstein-Hilbert equation, but can (or should) interpret it as a geometrical object, a null-quadric. Projective aspects of quantum (field) theory are discussed under this viewpoint.     

Projective quantum spaces

Associated to the standard SU _q (n) R-matrices, we introduce quantum spheresS _q ^2n-1 , projective quantum spaces _q ^n-1 , and quantum Grassmann manifoldsG _k( _q ⁿ ). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziski and S. Majid.     

Disentangling q-Exponentials: A General Approach

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson q-exponential of the sum of two non-q-commuting operators as an (in general) infinite product of q-exponential operators involving repeated q-commutators of increasing order, E_q(A+B) = E_q0(A)E_q1 (B) _i=2 E_qi. By systematically transforming the q-exponentials into exponentials of series and using the conventional Baker–Campbell–Hausdorff formula, we prove that one can make any choice for the bases ^qi, i=0, 1, 2, ..., of the q-exponentials in the infinite product. An explicit calculation of the operators C _i in the successive factors, carried out up to sixth order, also shows that the simplest q-Zassenhaus formula is obtained for ₀ = ₁ =1, and ₂ = 2, and ₃ = 3. This confirms and reinforces a result of Sridhar and Jagannathan, on the basis of fourth-order calculations.     

A Criterion for Compatibility of Conformal and Projective Structures

In a space-time M, a conformal structure is defined by the distribution of light-cones. Geodesics are traced by freely falling particles, and the collection of all unparameterized geodesics determines the projective structure of M. The article contains a formulation of the necessary and sufficient conditions for these structures to be compatible, i.e. to come from a metric tensor which is then unique up to a constant factor. The theorem applies to all dimensions and signatures.     

Projective versus metric structures

We present a number of conditions which are necessary for an n-dimensional projective structure (M,[∇]) to include the Levi-Civita connection ∇ of some metric on M. We provide an algorithm, which effectively checks whether a Levi-Civita connection is in the projective class and, which finds this connection and the metric, when it is possible. The article also provides basic information on invariants of projective structures, including the treatment via the Cartan normal projective connection. In particular we show that there are a number of Fefferman-like conformal structures, defined on a subbundle of the Cartan bundle of the projective structure, which encode the projectively invariant information about (M,[∇]).     

50°视场角投影式头盔光学系统设计及逆反射屏研究

设计了一款适于医疗培训用投影式头盔物镜系统，出瞳直径8mm，视场50°，能够满足对角0．9英寸微显示器对超高分辨SXGA模式的显示要求．其结构紧凑，口径10mm，长度12mm，重量仅有5g．本物镜系统场曲仅0．35D，畸变仅0．64％．为考察本投影式头盔系统的成像特性，研究了不同“逆反射材料”的特性．实验得到高强级逆反射材料在±40°视场范围内反射光强无明显变化，适合做为投影屏幕材料．定量分析了逆反射屏放置位置对成像的影响．     

用射影不变性纠正鱼眼镜头畸变

计算机视觉通常采用针孔摄像机模型，但对于存在较大畸变的鱼眼镜头或广角镜头，必须首先纠正镜头畸变才能应用针孔模型。为了从同时存在透视变形和鱼眼畸变的图像中纠正鱼眼畸变，采用了以下射影不变性：空间中直线的投影为直线；平行直线束的投影平行或相交于一点；直线段投影的交比不变。首先，用待纠正的鱼眼镜头对一幅等间隔正交网格图成像，提取网格结点作为控制点，然后用射影不变性求解畸变模型，达到纠正畸变的目的。该算法巧妙地运用三次方程的Cardan解法。大大提高了速度。结果表明，运用该方法纠正鱼眼镜头畸变速度快、精度高。     

Anti-Selfdual Connections on the Quantum Projective Plane: Monopoles

We present several results on the geometry of the quantum projective plane. They include: explicit generators for the K-theory and the K-homology; a real calculus with a Hodge star operator; anti-selfdual connections on line bundles with explicit computation of the corresponding ‘classical’ characteristic classes (via Fredholm modules); complete diagonalization of gauged Laplacians on these line bundles; ‘quantum’ characteristic classes via equivariant K-theory and q-indices.     

Personal View: Synchrotron Radiation Instrumentation: A Personal Experience

The scope of this two-day meeting (held June 25–26, 2007), organized by the French MELUSYN (Medicine and Synchrotron Light) work group at the new synchrotron radiation facility SOLEIL (Saint-Aubin, France), was to create a timely forum for multi-disciplinary discussions on modern radiation biology. The meeting brought together physicists of diluted and condensed matter, chemists, biochemists, biologists, as well as physicians having a common interest in using synchrotron radiation in the UV, VUV, soft and hard X-ray ranges and the related techniques in order to explore at molecular, cellular or tissular levels various aspects of radiation effects on living systems and the medical consequences for radiotherapy.     

Projective geometry from Poisson algebras

By analogy with the Poisson algebra of quadratic forms on the symplectic plane and with the concept of duality in the projective plane introduced by Arnold (2005) [1], where the concurrence of the triangle altitudes is deduced from the Jacobi identity, we consider the Poisson algebras of the first degree harmonics on the sphere, on the pseudo-sphere and on the hyperboloid, to obtain analogous duality concepts and similar results for spherical, pseudo-spherical and hyperbolic geometry. Such algebras, including the algebra of quadratic forms, are isomorphic either to the Lie algebra of the vectors in R³${\mathbb{R}}^3$, with the vector product, or to algebra s_l2(R)$s{l}_2\left(\mathbb{R}\right)$. The Tomihisa identity, introduced in (Tomihisa, 2009) [3] for the algebra of quadratic forms, holds for all these Poisson algebras and has a geometrical interpretation. The relationships between the different definitions of duality in projective geometry inherited by these structures are shown here.     

A General No-Cloning Theorem

A simple theorem is proved which has most of the known no-cloning and no-broadcasting results as corollaries. It also implies the standard restrictions on measuring non-commuting observables.