A new invariant and decompositions of manifolds

2018 ◽  
Vol 27 (02) ◽  
pp. 1850019
Author(s):  
Eiji Ogasa
Keyword(s):  
Topological Invariant ◽  
The Body ◽  
Precise Definition ◽  
Closed Manifold ◽  
Compact Manifolds ◽  
Connected Components ◽  
Fixed Base

We introduce a new topological invariant [Formula: see text] of compact manifolds-with-boundaries [Formula: see text] which is much connected with boundary-unions. A boundary-union is a kind of decomposition of compact manifolds-with-boundaries. See the body of the paper for the precise definition. Let [Formula: see text] and [Formula: see text] be [Formula: see text]-dimensional compact manifolds-with-boundaries. Let [Formula: see text] be a boundary-union of [Formula: see text] and [Formula: see text]. Then we have [Formula: see text] We define [Formula: see text] as follows: First, define an invariant of [Formula: see text]-closed manifolds. Take the maximum of the invariant of all connected-components of the boundary of each handle-body of an ordered-handle-decomposition with a fixed base [Formula: see text], where we impose the condition that the base [Formula: see text] is a (not necessarily connected) closed manifold. Take the minimum of the maximum for all ordered-handle-decompositions with the base [Formula: see text]. It is our another invariant [Formula: see text]. Take the maximum of the minimum, [Formula: see text], for all basis to satisfy the above condition. It is [Formula: see text]. See the body of the paper for the precise definition.

scholarly journals Topological entropy and partially hyperbolic diffeomorphisms

2008 ◽  
Vol 28 (3) ◽  
pp. 843-862 ◽  
Author(s):  
YONGXIA HUA ◽  
RADU SAGHIN ◽  
ZHIHONG XIA
Keyword(s):  
Topological Entropy ◽  
Topological Invariant ◽  
Compact Manifolds ◽  
Two Dimensional ◽  
One Dimensional ◽  
Hyperbolic Diffeomorphisms ◽  
Lower Semi ◽  
Partially Hyperbolic ◽  
Partially Hyperbolic Diffeomorphisms

AbstractWe consider partially hyperbolic diffeomorphisms on compact manifolds. We define the notion of the unstable and stable foliations stably carrying some unique non-trivial homologies. Under this topological assumption, we prove the following two results: if the center foliation is one-dimensional, then the topological entropy is locally a constant; and if the center foliation is two-dimensional, then the topological entropy is continuous on the set of all $C^{\infty }$ diffeomorphisms. The proof uses a topological invariant we introduced, Yomdin’s theorem on upper semi-continuity, Katok’s theorem on lower semi-continuity for two-dimensional systems, and a refined Pesin–Ruelle inequality we proved for partially hyperbolic diffeomorphisms.

The Mechanics of Parallel Mechanisms and Walking Machines

1994 ◽  
Vol 208 (6) ◽  
pp. 367-377 ◽  
Author(s):  
S J Zhang ◽  
D J Sanger ◽  
D Howard
Keyword(s):  
Parallel Mechanism ◽  
Virtual Work ◽  
The Body ◽  
Position Vector ◽  
Parallel Mechanisms ◽  
Active Joint ◽  
End Effector ◽  
Walking Machine ◽  
Walking Machines ◽  
Fixed Base

A parallel mechanism is one whose links and joints form two or more serially connected chains which join the fixed base and the end effector The mechanism of a multi-legged walking machine can be considered as a parallel mechanism whose base is not fixed and whose configuration changes during different phases of its gait. This paper presents methods for analysing the mechanics of parallel mechanisms and walking machines using vector and screw algebra Firstly, displacement analysis is covered; this includes general methods for deriving the position vector of any joint in any leg and for calculating the active joint displacements in any leg. Secondly, velocity analysis is covered which tackles the problem of calculating active joint velocities given the velocity, position and the orientation of the body and the positions of the feet. Thirdly, the static analysis of these classes of mechanisms using the principle of virtual work and screw algebra is given. Expressions are derived for the actuator forces and torques required to balance a given end effector (or body) wrench and, in the case of a walking machine, the ground reactions at the feet. Numerical examples are given to demonstrate the application of these methods.

Complexity of shadows and traversing flows in terms of the simplicial volume

2016 ◽  
Vol 08 (03) ◽  
pp. 501-543 ◽  
Author(s):  
Gabriel Katz
Keyword(s):  
Compact Manifolds ◽  
Connected Components ◽  
Fundamental Groups ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Simplicial Volume ◽  
Localization Technique ◽  
Close Relatives ◽  
Universal Bounds

We combine Gromov’s amenable localization technique with the Poincaré duality to study the traversally generic vector flows on smooth compact manifolds [Formula: see text] with boundary. Such flows generate well-understood stratifications of [Formula: see text] by the trajectories that are tangent to the boundary in a particular canonical fashion. Specifically, we get lower estimates of the numbers of connected components of these flow-generated strata of any given codimension. These universal bounds are basically expressed in terms of the normed homology of the fundamental groups [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the double of [Formula: see text]. The norm here is the Gromov simplicial semi-norm in homology. It turns out that some close relatives of the normed homology spaces [Formula: see text], [Formula: see text] form obstructions to the existence of [Formula: see text]-convex traversally generic vector flows on [Formula: see text].

scholarly journals Automated Detection of Working Area of Peripheral Blood Smears Using Mathematical Morphology

2003 ◽  
Vol 25 (1) ◽  
pp. 37-49 ◽  
Author(s):  
Jesús Angulo ◽  
Georges Flandrin
Keyword(s):  
Mathematical Morphology ◽  
Peripheral Blood ◽  
Central Zone ◽  
The Body ◽  
Connected Components ◽  
Blood Smears ◽  
Optimal Area ◽  
Two Stages ◽  
Peripheral Blood Smears ◽  
Edge Side

The paper presents a technique to automatically detect the working area of peripheral blood smears stained with May‐Grünwuald Giemsa. The optimal area is defined as the well spread part of the smear. This zone starts when the erythrocytes stop overlapping (on the body film side) and finishes when the erythrocytes start losing their clear central zone (on the feather edge side). The approach yields a quick detection of this area in images scanned under low magnifying power (immersion objective ×25 or ×16). The algorithm consists of two stages. First, an image analysis procedure using mathematical morphology is applied for extracting the erythrocytes, the centers of erythrocytes and the erythrocytes with center. Second, the number of connected components from the three kinds of particles is counted and the coefficient of spreading ρs and the coefficient of overlapping ρo are calculated. The data from fourteen smears illustrate how the technique is used and its performance. Colour figures can be viewed onhttp://www.esacp.org/acp/2003/25‐1/angulo.htm.

Micro-Scale Simulation of Adhesive Effects During Normal Impact

1997 ◽  
Author(s):  
Jeffrey L. Streator
Keyword(s):  
Equations Of Motion ◽  
Interaction Strength ◽  
The Body ◽  
Normal Impact ◽  
Dynamical Equations ◽  
Fixed Base ◽  
Type Interaction ◽  
Approach Velocity ◽  
The Impact

Abstract A numerical simulation is developed to investigate the role of adhesive forces during the normal impact of a rigid body against an elastic surface. The model consists of a rigid, 2D translating body and a surface that is composed of a flexible linear array of particles, coupled to a fixed base via linear springs. Adhesive effects are incorporated in the model by ascribing a Lennard-Jones type interaction potential between the surface and the body. Dynamical equations of motion for the interface are integrated numerically during an impact event. It is found that the coefficient of restitution decreases with increasing interaction strength and decreasing approach velocity. Above a certain interaction strength for given velocity and below a certain velocity for given interaction strength, the body is found not to rebound, but sticks to the surface. The simulation results are found to be in qualitative agreement with the impact model of Dahneke (1973, 1975). Quantitative discrepancies are explained by the effects of surface “plucking.”

scholarly journals THE EFFECTS OF LOSS OF GASTRIC AND PANCREATIC SECRETIONS AND THE METHODS FOR RESTORATION OF NORMAL CONDITIONS IN THE BODY

1929 ◽  
Vol 50 (3) ◽  
pp. 387-405 ◽  
Author(s):  
Alexis F. Hartmann ◽  
Robert Elman ◽  
Keyword(s):  
Gastric Juice ◽  
Pancreatic Juice ◽  
Body Fluids ◽  
Chloride Ions ◽  
The Body ◽  
Fixed Base ◽  
Combined Solution ◽  
Fixed Acid

The composition of gastric and pancreatic juices and the effects of their loss on the composition of the body fluids were studied. Loss of gastric juice by removing water and chloride ions only partly neutralized by fixed base results in dehydration and alkalosis. Loss of pancreatic juice by removing water and a relative excess of fixed base results in dehydration and acidosis. Normal conditions in the body may be restored after the loss of either gastric or pancreatic juice by the administration of a combined solution, which provides (1) water in abundance because of its hypotonicity (2) an adequate source of the fixed anion Cl' and of the cations Na+, K+, and Ca++ in proper physiological ratio and (3) an excess of fixed base over fixed acid in the form of B-lactate.

Cobordism of Morse maps and its application to map germs

2009 ◽  
Vol 147 (1) ◽  
pp. 235-254 ◽  
Author(s):  
KAZUICHI IKEGAMI ◽  
OSAMU SAEKI
Keyword(s):  
Critical Points ◽  
Topological Invariant ◽  
Closed Manifold ◽  
General Situation ◽  
Stable Perturbation ◽  
Smooth Map

AbstractLetf:M→S1be a Morse map of a closed manifoldMinto the circle, where a Morse map is a smooth map with only nondegenerate critical points. In this paper, we classify such maps up to fold cobordism. In the course of the classification, we get several fold cobordism invariants for such Morse maps. We also consider a slightly general situation where the source manifoldMhas boundary and the mapfrestricted to the boundary has no critical points. Letg: (Rm, 0) → (R2, 0),m≥ 2, be a generic smooth map germ, where the targetR2is oriented. Using the above-mentioned fold cobordism invariants, we show that the number of cusps with a prescribed index appearing in aC∞stable perturbation ofg, counted with signs, gives a topological invariant ofg.

scholarly journals Multimodal Integrated Technique for Wrist Fracture Identification

2020 ◽  
Vol 9 (9) ◽  
pp. 573-582
Keyword(s):  
Detection System ◽  
Principal Component ◽  
Clinical Analysis ◽  
Wrist Fracture ◽  
The Body ◽  
Connected Components ◽  
Fracture Detection ◽  
Modal System ◽  
Analysis Technique ◽  
Integrated Technique

Medical image processing is one of the fastest growing fields in Computer Science. It is a technique used to obtain the images of various parts of the body for clinical analysis to identify and treat diseases. Medical Imaging helps in detecting fractures, lesions present in the images like X-ray, CT-scan, and MRI. Fracture detections are difficult and sometimes may lead to the misjudgment. Existing fracture detection system is complex and accuracy of the detection is low. Hence, the proposed paper focuses on single and multi-modal system that helps radiologists in detecting the wrist fractures. The proposed system uses the multimodal system to detect the fracture in the wrist bone. It uses the combination of the Hierarchical centroid, Principal Component Analysis and Connected components analysis technique to identify the fracture in the wrist bones. This paper also elaborates on the various segmentation techniques used in the multimodal and single modal system.

scholarly journals КІНЕМАТИКА РУХУ РОБОТА З ТРЬОМА РОЛИКОНЕСУЧИМИ КОЛЕСАМИ

2018 ◽  
pp. 60-71
Author(s):  
Анатолий Максимович Суббота ◽  
Елена Юрьевна Костерная
Keyword(s):  
Position Control ◽  
Center Of Mass ◽  
Roller Bearing ◽  
Vertical Axis ◽  
The Body ◽  
Additional Time ◽  
Starting Position ◽  
Advantages And Disadvantages ◽  
Fire Extinguishing ◽  
Fixed Base

The article introduction provides an overview of the historical development of mechatronic devices, from ancient times to the present. It is emphasized that the development of modern robotics in relation to work in aggressive environments is a very urgent task. Especially important is the creation of autonomous functioning robots to work in high radiation areas, chemically contaminated areas, demining, fire extinguishing, etc. Then this article presents material on the physics of a mechanical system motion, which is a mobile platform with three roller-bearing wheels, or so-called omni-wheels. This question is revealed on the basis of the derivation of the kinematic equations for the platform motion, based on transformation matrices, which allow to obtain the total dependences of the projections of the linear velocities of the roller-bearing wheels on the axis of the fixed (base) coordinate system. It is indicated that the transition to movement from the position at to the position at  can be carried out in two ways. In the first method, the robot turns around  on the center of mass by creating a torque about the vertical axis  of the robot, followed by movement parallel to the axis . In the second method, by creating such a state of the wheels, i.e. the magnitude of the linear velocity and its direction, which will ensure a linear movement of the center of mass of the robot in a given direction without first rotating the body about the vertical axis. It is noted that the first method in relation to the second has both advantages and disadvantages. The advantages include ease of management and the ability to rigidly fix the camera of the review on the platform body. However, this method is more energy consuming and requires additional time for the implementation of the camera turn to a given direction. The second method is not deprived of these drawbacks, and the overview camera may have a turning mechanism, which ensures its independent functioning from the platform position control system. Given this, the kinematics of the movement of the platform according to the second method are considered. As an example, it is shown that by jointly solving the obtained kinematic equations, for example, for selected mutually perpendicular directions of the platform mass center movement, characterized by angles  or , it is easy to explain the physics of platform moving in a given direction from any starting position without first turning to a given direction.

scholarly journals LOOP HOMOLOGY AS FIBREWISE HOMOLOGY

2008 ◽  
Vol 51 (1) ◽  
pp. 27-44 ◽  
Author(s):  
M. C. Crabb
Keyword(s):  
Basic Structure ◽  
Closed Manifold ◽  
Compact Manifolds ◽  
Homotopy Invariance

AbstractThe loop homology ring of an oriented closed manifold, defined by Chas and Sullivan, is interpreted as a fibrewise homology Pontrjagin ring. The basic structure, particularly the commutativity of the loop multiplication and the homotopy invariance, is explained from the viewpoint of the fibrewise theory, and the definition is extended to arbitrary compact manifolds.

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