NEWTON FORWARD INTERPOLATION ON EQUISPACED POINTS
• Lagrange Interpolation has a number of disadvantages
• The amount of computation required is large
• Interpolation for additional values of requires the same amount of effort as the
first value (i.e. no part of the previous calculation can be used)
• When the number of interpolation points are changed (increased/decreased), the
results of the previous computations can not be used
• Error estimation is difficult (at least may not be convenient)
Numerical Methods was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
Share with it ur friends & Follow me for more updates.!
This paper investigates fuzzy triangle and similarity of fuzzy triangles. Five rules to determine similarity of
fuzzy triangles are studied. Extension principle and concept of same and inverse points in fuzzy geometry
are used to analyze the proposed concepts.
Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Orde...IOSR Journals
In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on
Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
(RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This
method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
well to find the numerical solution of Nth – order FIVPs.
NEWTON FORWARD INTERPOLATION ON EQUISPACED POINTS
• Lagrange Interpolation has a number of disadvantages
• The amount of computation required is large
• Interpolation for additional values of requires the same amount of effort as the
first value (i.e. no part of the previous calculation can be used)
• When the number of interpolation points are changed (increased/decreased), the
results of the previous computations can not be used
• Error estimation is difficult (at least may not be convenient)
Numerical Methods was a core subject for Electrical & Electronics Engineering, Based On Anna University Syllabus. The Whole Subject was there in this document.
Share with it ur friends & Follow me for more updates.!
This paper investigates fuzzy triangle and similarity of fuzzy triangles. Five rules to determine similarity of
fuzzy triangles are studied. Extension principle and concept of same and inverse points in fuzzy geometry
are used to analyze the proposed concepts.
Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Orde...IOSR Journals
In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on
Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
(RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This
method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
well to find the numerical solution of Nth – order FIVPs.
Eucluidian and Non eucluidian space in Tensor analysis.Non Euclidian space AJAY CHETRI
Eucluidian and Non eucluidian space in Tensor analysis.
Introduction to type of system in sphere.Benefit and advantage of using Tensor analysis.EUCLID’S GEOMETRY
VS.
NON-EUCLIDEAN GEOMETRY
Chapter summary and solutions to end-of-chapter exercises for "Data Visualization: Principles and Practice" book by Alexandru C. Telea
Chapter provides an overview of a number of methods for visualizing tensor data. It explains principal component analysis as a technique used to process a tensor matrix and extract from it information that can directly be used in its visualization. It forms a fundamental part of many tensor data processing and visualization algorithms. Section 7.4 shows how the results of the principal component analysis can be visualized using the simple color-mapping techniques. Next parts of the chapter explain how same data can be visualized using tensor glyphs, and streamline-like visualization techniques.
In contrast to Slicer, which is a more general framework for analyzing and visualizing 3D slice-based data volumes, the Diffusion Toolkit focuses on DT-MRI datasets, and thus offers more extensive and easier to use options for fiber tracking.
A COMPARATIVE STUDY ON DISTANCE MEASURING APPROACHES FOR CLUSTERINGIJORCS
Clustering plays a vital role in the various areas of research like Data Mining, Image Retrieval, Bio-computing and many a lot. Distance measure plays an important role in clustering data points. Choosing the right distance measure for a given dataset is a biggest challenge. In this paper, we study various distance measures and their effect on different clustering. This paper surveys existing distance measures for clustering and present a comparison between them based on application domain, efficiency, benefits and drawbacks. This comparison helps the researchers to take quick decision about which distance measure to use for clustering. We conclude this work by identifying trends and challenges of research and development towards clustering.
Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogramClustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Clustering and dendogram Cluster
ON FINDING MINIMUM AND MAXIMUM PATH LENGTH IN GRID-BASED WIRELESS NETWORKSijwmn
In this paper, we obtain the minimum and maximum hop counts between any pair of cells in the 3D gridbased wireless networks. We start by determining the minimum path length between any two points in a 2D grid coordinate system. We establish that the minimum path length is the maximum difference between the
corresponding coordinates of the two points. We then extend the result to derive the minimum and maximum hop counts for the 3D grid-based wireless networks. We establish that the maximum path length is the sum of the differences between the corresponding coordinates of the two cells. Whilst the minimum path length depends on the positions of the two cells; it does not exceed the maximum difference between the corresponding coordinates of the two cells.
Tim Maudlin: New Foundations for Physical GeometryArun Gupta
New Foundations for Physical Geometry
Original URL: http://www.unil.ch/webdav/site/philo/shared/summer_school_2013/NYU.ppt
Tim Maudlin
NYU
Physics & Philosophy of Time
July 25, 2013
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to …
2.01 Identify that if all parts of an object move in the same direction and at the same rate, we can treat the object as if it
were a (point-like) particle. (This chapter is about the motion of such objects.)
2.02 Identify that the position of a particle is its location as
read on a scaled axis, such as an x-axis.
2.03 Apply the relationship between a particle’s
displacement and its initial and final positions.
2.04 Apply the relationship between a particle’s average
velocity, its displacement, and the time interval for that
displacement.
2.05 Apply the relationship between a particle’s average
speed, the total distance it moves, and the time interval for
the motion.
2.06 Given a graph of a particle’s position versus time,
determine the average velocity between any two particular
times.
2-1 POSITION, DISPLACEMENT, AND AVERAGE VELOCITY
After reading this module, you should be able to . . .
2.07 Given a particle’s position as a function of time,
calculate the instantaneous velocity for any particular time.
2.08 Given a graph of a particle’s position versus time, determine the instantaneous velocity for any particular time.
2.09 Identify speed as the magnitude of the instantaneous
velocity.
etc......
Connectivity-Based Clustering for Mixed Discrete and Continuous DataIJCI JOURNAL
This paper introduces a density-based clustering procedure for datasets with variables of mixed type. The proposed procedure, which is closely related to the concept of shared neighbourhoods, works particularly well in cases where the individual clusters differ greatly in terms of the average pairwise distance of the associated objects. Using a number of concrete examples, it is shown that the proposed clustering algorithm succeeds in allowing the identification of subgroups of objects with statistically significant distributional characteristics.
On algorithmic problems concerning graphs of higher degree of symmetrygraphhoc
Since the ancient determination of the five platonic solids the study of symmetry and regularity has always
been one of the most fascinating aspects of mathematics. One intriguing phenomenon of studies in graph
theory is the fact that quite often arithmetic regularity properties of a graph imply the existence of many
symmetries, i.e. large automorphism group G. In some important special situation higher degree of
regularity means that G is an automorphism group of finite geometry. For example, a glance through the
list of distance regular graphs of diameter d < 3 reveals the fact that most of them are connected with
classical Lie geometry. Theory of distance regular graphs is an important part of algebraic combinatorics
and its applications such as coding theory, communication networks, and block design. An important tool
for investigation of such graphs is their spectra, which is the set of eigenvalues of adjacency matrix of a
graph. Let G be a finite simple group of Lie type and X be the set homogeneous elements of the associated
geometry. The complexity of computing the adjacency matrices of a graph Gr on the vertices X such that
Aut GR = G depends very much on the description of the geometry with which one starts. For example, we
can represent the geometry as the totality of 1 cosets of parabolic subgroups 2 chains of embedded
subspaces (case of linear groups), or totally isotropic subspaces (case of the remaining classical groups), 3
special subspaces of minimal module for G which are defined in terms of a G invariant multilinear form.
The aim of this research is to develop an effective method for generation of graphs connected with classical
geometry and evaluation of its spectra, which is the set of eigenvalues of adjacency matrix of a graph. The
main approach is to avoid manual drawing and to calculate graph layout automatically according to its
formal structure. This is a simple task in a case of a tree like graph with a strict hierarchy of entities but it
becomes more complicated for graphs of geometrical nature. There are two main reasons for the
investigations of spectra: (1) very often spectra carry much more useful information about the graph than a
corresponding list of entities and relationships (2) graphs with special spectra, satisfying so called
Ramanujan property or simply Ramanujan graphs (by name of Indian genius mathematician) are important
for real life applications (see [13]). There is a motivated suspicion that among geometrical graphs one
could find some new Ramanujan graphs.
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
I have heard many times that architecture is not important for the front-end. Also, many times I have seen how developers implement features on the front-end just following the standard rules for a framework and think that this is enough to successfully launch the project, and then the project fails. How to prevent this and what approach to choose? I have launched dozens of complex projects and during the talk we will analyze which approaches have worked for me and which have not.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
JMeter webinar - integration with InfluxDB and GrafanaRTTS
Watch this recorded webinar about real-time monitoring of application performance. See how to integrate Apache JMeter, the open-source leader in performance testing, with InfluxDB, the open-source time-series database, and Grafana, the open-source analytics and visualization application.
In this webinar, we will review the benefits of leveraging InfluxDB and Grafana when executing load tests and demonstrate how these tools are used to visualize performance metrics.
Length: 30 minutes
Session Overview
-------------------------------------------
During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
To view the webinar recording, go to:
https://www.rttsweb.com/jmeter-integration-webinar
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
State of ICS and IoT Cyber Threat Landscape Report 2024 previewPrayukth K V
The IoT and OT threat landscape report has been prepared by the Threat Research Team at Sectrio using data from Sectrio, cyber threat intelligence farming facilities spread across over 85 cities around the world. In addition, Sectrio also runs AI-based advanced threat and payload engagement facilities that serve as sinks to attract and engage sophisticated threat actors, and newer malware including new variants and latent threats that are at an earlier stage of development.
The latest edition of the OT/ICS and IoT security Threat Landscape Report 2024 also covers:
State of global ICS asset and network exposure
Sectoral targets and attacks as well as the cost of ransom
Global APT activity, AI usage, actor and tactic profiles, and implications
Rise in volumes of AI-powered cyberattacks
Major cyber events in 2024
Malware and malicious payload trends
Cyberattack types and targets
Vulnerability exploit attempts on CVEs
Attacks on counties – USA
Expansion of bot farms – how, where, and why
In-depth analysis of the cyber threat landscape across North America, South America, Europe, APAC, and the Middle East
Why are attacks on smart factories rising?
Cyber risk predictions
Axis of attacks – Europe
Systemic attacks in the Middle East
Download the full report from here:
https://sectrio.com/resources/ot-threat-landscape-reports/sectrio-releases-ot-ics-and-iot-security-threat-landscape-report-2024/
When stars align: studies in data quality, knowledge graphs, and machine lear...
Distance
1. Distance (or farness) is a numerical description of how far apart objects are. In physics or everyday
discussion, distance may refer to a physical length, or an estimation based on other criteria (e.g. "two
counties over"). In mathematics, a distance function or metric is a generalization of the concept of
physical distance. A metric is a function that behaves according to a specific set of rules, and is a
concrete way of describing what it means for elements of some space to be "close to" or "far away from"
each other. In most cases, "distance from A to B" is interchangeable with "distance between B and A".
Contents
[hide]
1 Mathematics
o
1.1 Geometry
o
1.2 Distance in Euclidean space
o
1.3 Variational formulation of distance
o
1.4 Generalization to higher-dimensional objects
o
1.5 Algebraic distance
o
1.6 General case
o
1.7 Distances between sets and between a point
and a set
o
1.8 Graph theory
2 Distance versus directed distance and displacement
o
2.1 Directed distance
3 Other "distances"
4 See also
5 References
[edit]Mathematics
See also: Metric (mathematics)
[edit]Geometry
2. In analytic geometry, the distance between two points of the xy-plane can be found using the distance
formula. The distance between (x1, y1) and (x2, y2) is given by:
Similarly, given points (x1, y1, z1) and (x2, y2, z2) in three-space, the distance between them is:
These formulae are easily derived by constructing a right triangle with a leg on the hypotenuse of
another (with the other leg orthogonal to the plane that contains the 1st triangle) and applying
thePythagorean theorem.
In the study of complicated geometries, we call this (most common) type of distance Euclidean
distance, as it is derived from the Pythagorean theorem, which does not hold in Non-Euclidean
geometries. This distance formula can also be expanded into the arc-length formula.
[edit]Distance
in Euclidean space
In the Euclidean space Rn, the distance between two points is usually given by the Euclidean
distance (2-norm distance). Other distances, based on other norms, are sometimes used
instead.
For a point (x1, x2, ...,xn) and a point (y1, y2, ...,yn), the Minkowski distance of order p (p-norm
distance) is defined as:
1-norm distance
2-norm distance
p-norm distance
infinity norm distance
p need not be an integer, but it cannot be less than 1, because otherwise the triangle
inequality does not hold.
The 2-norm distance is the Euclidean distance, a generalization of the Pythagorean theorem to
more than two coordinates. It is what would be obtained if the distance between two points were
measured with a ruler: the "intuitive" idea of distance.
The 1-norm distance is more colourfully called the taxicab norm or Manhattan distance, because
it is the distance a car would drive in a city laid out in square blocks (if there are no one-way
streets).
3. The infinity norm distance is also called Chebyshev distance. In 2D, it is the minimum number of
moves kings require to travel between two squares on a chessboard.
The p-norm is rarely used for values of p other than 1, 2, and infinity, but see super ellipse.
In physical space the Euclidean distance is in a way the most natural one, because in this case
the length of a rigid body does not change with rotation.
[edit]Variational
formulation of distance
The Euclidean distance between two points in space (
and
) may be
written in a variational form where the distance is the minimum value of an integral:
Here
is the trajectory (path) between the two points. The value of the integral (D)
represents the length of this trajectory. The distance is the minimal value of this integral and
is obtained when
where
is the optimal trajectory. In the familiar Euclidean case
(the above integral) this optimal trajectory is simply a straight line. It is well known that the
shortest path between two points is a straight line. Straight lines can formally be obtained by
solving the Euler-Lagrange equations for the above functional. In non-Euclidean manifolds
(curved spaces) where the nature of the space is represented by a metric
the integrand
has be to modified to
used.
, where Einstein summation convention has been
[edit]Generalization
to higher-dimensional objects
The Euclidean distance between two objects may also be generalized to the case where the
objects are no longer points but are higher-dimensional manifolds, such as space curves, so
in addition to talking about distance between two points one can discuss concepts of
distance between two strings. Since the new objects that are dealt with are extended objects
(not points anymore) additional concepts such as non-extensibility, curvature constraints,
and non-local interactions that enforce non-crossing become central to the notion of
distance. The distance between the two manifolds is the scalar quantity that results from
minimizing the generalized distance functional, which represents a transformation between
the two manifolds:
The above double integral is the generalized distance functional between two plymer
conformation. is a spatial parameter and is pseudo-time. This means
that
is the polymer/string conformation at time
and is parameterized
along the string length by . Similarly
is the trajectory of an infinitesimal
segment of the string during transformation of the entire string from
4. conformation
to conformation
. The term with cofactor is
a Lagrange multiplier and its role is to ensure that the length of the polymer remains the
same during the transformation. If two discrete polymers are inextensible, then the
minimal-distance transformation between them no longer involves purely straight-line
motion, even on a Euclidean metric. There is a potential application of such generalized
distance to the problem of protein folding[1][2] This generalized distance is analogous to
the Nambu-Goto action in string theory, however there is no exact correspondence
because the Euclidean distance in 3-space is inequivalent to the space-time distance
minimized for the classical relativistic string.
[edit]Algebraic
distance
This section requires expansion.
(December 2008)
This is a metric often used in computer vision that can be minimized by least
squares estimation. [1][2] For curves or surfaces given by the equation
(such as a conic in homogeneous coordinates), the algebraic distance from the point
to the curve is simply
. It may serve as an "initial guess" for geometric
distance to refine estimations of the curve by more accurate methods, such as nonlinear least squares.
[edit]General
case
In mathematics, in particular geometry, a distance function on a given set M is
a function d: M×M → R, where R denotes the set of real numbers, that satisfies the
following conditions:
•
d(x,y) ≥ 0, and d(x,y) = 0 if and only if x = y. (Distance is positive between two
•
different points, and is zero precisely from a point to itself.)
It is symmetric: d(x,y) = d(y,x). (The distance between x and y is the same in either
direction.)
•
It satisfies the triangle inequality: d(x,z) ≤ d(x,y) + d(y,z). (The distance between two
points is the shortest distance along any path).
Such a distance function is known as a metric. Together with the set, it makes up
a metric space.
For example, the usual definition of distance between two real
numbers x and y is: d(x,y) = |x − y|. This definition satisfies the three conditions above,
and corresponds to the standard topology of the real line. But distance on a given set is
a definitional choice. Another possible choice is to define: d(x,y) = 0 if x = y, and 1
otherwise. This also defines a metric, but gives a completely different topology, the
"discrete topology"; with this definition numbers cannot be arbitrarily close.
[edit]Distances
between sets and between a point and a set
5. d(A, B) > d(A, C) + d(C, B)
Various distance definitions are possible between objects. For example, between
celestial bodies one should not confuse the surface-to-surface distance and the centerto-center distance. If the former is much less than the latter, as for a LEO, the first tends
to be quoted (altitude), otherwise, e.g. for the Earth-Moon distance, the latter.
There are two common definitions for the distance between two non-empty subsets of a
given set:
•
One version of distance between two non-empty sets is the infimum of the
distances between any two of their respective points, which is the every-day
meaning of the word, i.e.
This is a symmetric premetric. On a collection of sets of which some touch or overlap each other,
it is not "separating", because the distance between two different but touching or overlapping sets
is zero. Also it is not hemimetric, i.e., the triangle inequality does not hold, except in special
cases. Therefore only in special cases this distance makes a collection of sets a metric space.
•
The Hausdorff distance is the larger of two values, one being
the supremum, for a point ranging over one set, of the infimum, for a
second point ranging over the other set, of the distance between the points,
and the other value being likewise defined but with the roles of the two sets
swapped. This distance makes the set of non-empty compact subsets of a
metric space itself a metric space.
The distance between a point and a set is the infimum of the distances between
the point and those in the set. This corresponds to the distance, according to
the first-mentioned definition above of the distance between sets, from the set
containing only this point to the other set.
6. In terms of this, the definition of the Hausdorff distance can be simplified: it is
the larger of two values, one being the supremum, for a point ranging over one
set, of the distance between the point and the set, and the other value being
likewise defined but with the roles of the two sets swapped.
[edit]Graph
theory
In graph theory the distance between two vertices is the length of the
shortest path between those vertices.
[edit]Distance
versus directed distance and
displacement
Distance along a path compared with displacement
Distance cannot be negative and distance travelled never decreases. Distance
is a scalar quantity or a magnitude, whereas displacement is a vectorquantity
with both magnitude and direction.
The distance covered by a vehicle (for example as recorded by an odometer),
person, animal, or object along a curved path from a point A to a point Bshould
be distinguished from the straight line distance from A to B. For example
whatever the distance covered during a round trip from A to B and back to A,
the displacement is zero as start and end points coincide. In general the
straight line distance does not equal distance travelled, except for journeys in a
straight line.
[edit]Directed
distance
Directed distances are distances with a direction or sense. They can be
determined along straight lines and along curved lines. A directed distance
along a straight line from A to B is a vector joining any two points in a ndimensional Euclidean vector space. A directed distance along a curved line is
not a vector and is represented by a segment of that curved line defined by
endpoints A and B, with some specific information indicating the sense (or
direction) of an ideal or real motion from one endpoint of the segment to the
other (see figure). For instance, just labelling the two endpoints as A and B can
indicate the sense, if the ordered sequence (A, B) is assumed, which implies
that A is the starting point.
7. A displacement (see above) is a special kind of directed distance defined
in mechanics. A directed distance is called displacement when it is the distance
along a straight line (minimum distance) from A and B, and when A and B are
positions occupied by the same particle at two different instants of time. This
implies motion of the particle. displace is a vector quantity.
Another kind of directed distance is that between two different particles or point
masses at a given time. For instance, the distance from the center of gravity of
the Earth A and the center of gravityof the Moon B (which does not strictly imply
motion from A to B).Shortest path length may be equal to displacement or may
not be equal to.Distance from starting point is always equal to magnitude of
displacement. For same particle distance travelled is always greater than or
equal to magnitude of displacement. Shortest path length is not necessary
always displacement. Displacement may increase or decrease but distance
travelled never decreases.
[edit]Other
"distances"
•
E-statistics, or energy statistics, are functions of distances between
•
statistical observations.
Mahalanobis distance is used in statistics.
•
Hamming distance and Lee distance are used in coding theory.
•
Levenshtein distance
•
Chebyshev distance
•
Canberra distance
Circular distance is the distance traveled by a wheel. The circumference of the
wheel is 2π × radius, and assuming the radius to be 1, then each revolution of
the wheel is equivalent of the distance 2π radians. In engineering ω = 2πƒ is
often used, where ƒ is the frequency.
[edit]See
also
•
Taxicab geometry
•
Astronomical units of length
•
Cosmic distance ladder
•
L
•
Distance measures (cosmology)
•
M
•
Comoving distance
•
M
8. •
Distance geometry
•
M
•
Distance (graph theory)
•
O
•
Dijkstra's algorithm
•
P
•
Distance-based road exit numbers
•
D
•
Distance measuring equipment (DME)
•
H
•
Engineering tolerance
•
L
•
Great-circle distance
•
P
•
M
[edit]References
1.
^ SS Plotkin, PNAS.2007; 104: 14899–14904,
2.
^ AR Mohazab, SS Plotkin,"Minimal Folding Pathways for Coarse-Grained
Biopolymer Fragments" Biophysical Journal, Volume 95, Issue 12, Pages
5496–5507
•
Deza, E.; Deza, M. (2006), Dictionary of Distances, Elsevier, ISBN 0-44452087-2.