Geometric Property of Off Resonance Error Robust Composite Pulse

The precision of quantum operations is affected by unavoidable systematic errors. A composite pulse (CP), which has been well investigated in nuclear magnetic resonance (NMR), is a technique that suppresses the influence of systematic errors by replacing a single operation with a sequence of operations. In NMR, there are two typical systematic errors, Pulse Length Error (PLE) and Off Resonance Error (ORE). Recently, it was found that PLE robust CPs have a clear geometric property. In this study, we show that ORE robust CPs also have a simple geometric property, which is associated with trajectories on the Bloch sphere of the corresponding operations. We discuss the geometric property of ORE robust CPs using two examples.

Geometric Invariants of Surjective Isometries between Unit Spheres

In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX→SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(−F)=−T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.

Measures of noncompactness in modular spaces and fixed point theorems for multivalued nonexpansive mappings

This paper is devoted to state some fixed point results for multivalued mappings in modular vector spaces. For this purpose, we study the uniform noncompact convexity, a geometric property for modular spaces which is similar to nearly uniform convexity in the Banach spaces setting. Using this property, we state several new fixed point theorems for multivalued nonexpansive mappings in modular spaces.

Research on Tree Pith Location in Radial Direction Based on Terrestrial Laser Scanning

Obtaining the direction of a diameter line through the tree pith is the basis of effective sampling by a micro-drill resistance instrument. In order to implement non-destructive tree pith location in the radial direction, the geometric property of tree pith, the longest chord through the tree pith on the cross-section will bisect outer contour circumference, as first proposed and proven in this paper. Based on this property, a non-destructive tree pith radial location method based on terrestrial laser scanning was developed. The experiments of pith radial location were made on the tree discs and the error of location is less than 1.5% for cross-section shape closed to ellipse on four tree species. The geometric property and location method of the tree pith in this research would play an important role in studying the growth process of standing trees, obtaining processed wood properties, and estimating tree age.

FUNGSI ELEMEN KELAS STUMMEL MODULUS TERBATAS TETAPI BUKAN ELEMEN DARI KELAS STUMMEL

In the paper [1], it was given a function which belongs to the bounded Stummel modulus classes but not in Stummel classes. The given proof of this function properties in that paper was not obvious and very concise. By using the countable linearity property of integral, polar coordinate of integration, other properties of Lebesgue measure and integration, and some observation on the geometric property of the open ball in Euclidean spaces, we prove in detail the properties of this function.

L-interval spaces and some of their properties

In this paper, notions of L-interval spaces and L-2-arity convex spaces are introduced. It is showed that there is a Galois’s connection between the category of L-convex spaces and the category of L-interval spaces. In particular, the category of L-2-arity convex spaces can be embedded in the category of L-interval spaces as a coreflective subcategory. Further, some properties of L-interval spaces are introduced including L-geometric (resp. L-Peano, L-Pasch and L-sand-glass) property. It is proved that an L-2-arity convex space is an L-JHC convex space iff its segment operator has L-Peano property. It is also proved that an L-JHC convex space with an L-idempotent segment operator has L-sand-glass property. Further, it is also proved that an L-idempotent interval space having L-Peano+L-Pasch property has L-geometric property and L-sand-glass property.

Geometric property-based convolutional neural network for indoor object detection

Indoor object detection is a very demanding and important task for robot applications. Object knowledge, such as two-dimensional (2D) shape and depth information, may be helpful for detection. In this article, we focus on region-based convolutional neural network (CNN) detector and propose a geometric property-based Faster R-CNN method (GP-Faster) for indoor object detection. GP-Faster incorporates geometric property in Faster R-CNN to improve the detection performance. In detail, we first use mesh grids that are the intersections of direct and inverse proportion functions to generate appropriate anchors for indoor objects. After the anchors are regressed to the regions of interest produced by a region proposal network (RPN-RoIs), we then use 2D geometric constraints to refine the RPN-RoIs, in which the 2D constraint of every classification is a convex hull region enclosing the width and height coordinates of the ground-truth boxes on the training set. Comparison experiments are implemented on two indoor datasets SUN2012 and NYUv2. Since the depth information is available in NYUv2, we involve depth constraints in GP-Faster and propose 3D geometric property-based Faster R-CNN (DGP-Faster) on NYUv2. The experimental results show that both GP-Faster and DGP-Faster increase the performance of the mean average precision.