text
stringlengths 256
16.4k
|
|---|
A\mathrm{sin}\left(x\right)
A
x
t
n
p
p
p
The trace=n option specifies that a number of previous frames of the animation be kept visible. When
n
n+1
n=5
When
is a list of integers, then the frames in those positions are the frames that remain visible. Each integer in
n=0
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3\right)
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3,\mathrm{trace}=5,\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[A{x}^{2},x=-4..4],A=-3..3,\mathrm{trace}=[30,35,40,45,50],\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot3d},[A\left({x}^{2}+{y}^{2}\right),x=-3..3,y=-3..3],A=-2..2,\mathrm{style}=\mathrm{patchcontour}\right)
\mathrm{animate}\left(\mathrm{implicitplot},[{x}^{2}+{y}^{2}={r}^{2},x=-3..3,y=-3..3],r=1..3,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{implicitplot},[{x}^{2}+Axy-{y}^{2}=1,x=-2..2,y=-3..3],A=-2..2,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{plot},[[\mathrm{sin}\left(t\right),\mathrm{sin}\left(t\right)\mathrm{exp}\left(-\frac{t}{5}\right)],t=0..x],x=0..6\mathrm{\pi },\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),t=0..A]],A=0..2\mathrm{\pi },\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=50\right)
\mathrm{animate}\left(\mathrm{plot},[[\frac{1-{t}^{2}}{1+{t}^{2}},\frac{2t}{1+{t}^{2}},t=-10..A]],A=-10..10,\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=50,\mathrm{view}=[-1..1,-1..1]\right)
\mathrm{opts}≔\mathrm{thickness}=5,\mathrm{numpoints}=100,\mathrm{color}=\mathrm{black}:
\mathrm{animate}\left(\mathrm{spacecurve},[[\mathrm{cos}\left(t\right),\mathrm{sin}\left(t\right),\left(2+\mathrm{sin}\left(A\right)\right)t],t=0..20,\mathrm{opts}],A=0..2\mathrm{\pi }\right)
B≔\mathrm{plot3d}\left(1-{x}^{2}-{y}^{2},x=-1..1,y=-1..1,\mathrm{style}=\mathrm{patchcontour}\right):
\mathrm{opts}≔\mathrm{thickness}=5,\mathrm{color}=\mathrm{black}:
\mathrm{animate}\left(\mathrm{spacecurve},[[t,t,1-2{t}^{2}],t=-1..A,\mathrm{opts}],A=-1..1,\mathrm{frames}=11,\mathrm{background}=B\right)
\mathrm{animate}\left(\mathrm{ball},[0,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{scaling}=\mathrm{constrained},\mathrm{frames}=100\right)
\mathrm{sinewave}≔\mathrm{plot}\left(\mathrm{sin}\left(x\right),x=0..4\mathrm{\pi }\right):
\mathrm{animate}\left(\mathrm{ball},[t,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{frames}=50,\mathrm{background}=\mathrm{sinewave},\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(\mathrm{ball},[t,\mathrm{sin}\left(t\right)],t=0..4\mathrm{\pi },\mathrm{frames}=50,\mathrm{trace}=10,\mathrm{scaling}=\mathrm{constrained}\right)
\mathrm{animate}\left(F,[\mathrm{\theta }],\mathrm{\theta }=0..2\mathrm{\pi },\mathrm{background}=\mathrm{plot}\left([\mathrm{cos}\left(t\right)-2,\mathrm{sin}\left(t\right),t=0..2\mathrm{\pi }]\right),\mathrm{scaling}=\mathrm{constrained},\mathrm{axes}=\mathrm{none}\right)
|
While on vacation with your family, you will be driving from Los Angeles to San Diego. If you take the freeway it is a distance of
144
miles and the average freeway speed is
75
miles per hour. If you take the scenic route along the coast, the drive will be
122
miles at an average speed of 65 miles per hour. Which is the better route to take if you want to get to San Diego in the shortest amount of time? Show your work using Giant Ones.
Review the Math Notes box in Lesson 2.3.1 for help on dimensional analysis using Giant Ones.
\text{Taking the freeway: }\frac{\text{144 miles}}{1}\cdot\frac{\text{1 hour}}{\text{75 miles}}=\frac{144}{75}\text{ hours} =1.92 \text{ hours}
Now calculate the scenic route's driving time.
|
Geometric Topology and Connections with Quantum Field Theory | EMS Press
The workshop \emph{Geometric Topology and Connections with Quantum Field Theory}, organised by Stephan Stolz (Notre Dame) and Peter Teichner (La Jolla) was held June 12th--June 18th, 2005. As mentioned above, this workshop was intendend to bring together people working in the fields of traditional geometric topology and theoretical physics. For that purpose, the organizers asked several well known expositors like Dror Bar-Natan, Dan Freed and Graeme Segal to give survey lectures in some of the most exciting connections between topology and QFT. In what follows we give a brief survey of the three most central topics. {\em Elliptic Cohomology}: The elliptic cohomology of a space
X
is the home of the `Witten genusÕ of a family of `stringÕ manifolds parametrized by
X
, similarly to the role of the K-theory of
X
as the home of the family version of the A-roof genus of a family of spin manifolds parametrized by
X
(i.e., a fiber bundle over
X
with spin manifold fibers). There is now a homotopy theoretic construction of elliptic cohomology (the `topological modular form theoryÕ
TMF
of Hopkins and Miller) and one of the highlights of the workshop was Jacob Lurie's lecture about his new interpretation (and construction) of
TMF
via `derived algebraic geometry'. This approach makes many aspects of
TMF
more transparent and allows for things like an equivariant version to be defined. There is also a family version of the Witten genus with values in
TMF
due to Ando, Hopkins, Reszk and Strickland, however, a geometric/analytic construction of elliptic cohomology and the Witten genus (say analogous to the description of
K(X)
as families of Fredholm operators parametrized by
X
) is still missing. This is despite a two decade old proposal of Graeme Segal to interpret elements of the elliptic cohomology of
X
essentially as families of conformal field theories parametrized by
X
. Segal gave the opening lecture in the workshop, surveying some of the progress along the lines of his proposal. During the last two years various more precise candidates for a geometric definition of elliptic cohomology were developed. These were represented by the talks of Rognes and Hu-Kriz and an informal evening session by Stolz-Teichner. Antony Wasserman presented an approach to construct outer representations of Lie groups which may well be the starting point of an equivariant version of one of these geometric theories. {\em Differential K-Theory}: The differential K-Theory of a manifold is a crossover between differential forms and K-theory. For example, Freed and Hopkins have recently announced a version of the Atiyah-Singer Index Theorem which is expressed as an equality of an `analyticalÕ and a `topologicalÕ index, both of which live in a `differential K-theory groupÕ. This version of the Index Theorem represents a common generalization of the K-theory version of the Index Theorem and the Local Index Theorem. This way, differential K-theory groups can be seen as the common place where geometric aspects of the manifold, encoded as index densities of associated geometric operators meet topological aspects encoded in the K-theory class represented by the principal symbol of these operators. Moreover, what physicists call `abelian Gauge fields' also fits naturally into this context, where, depending on the phycisal setting, K-theory may have to be replaced by another generalized cohomology theory. Talks by Ulrich Bunke, Dan Freed, Mike Hopkins and Greg Moore represented this aspect of the workshop. Closely related is the geometry of `gerbesÕ; isomorphism classes of gerbes over a space
X
are classified by elements of
H^3(X)
(like complex line bundle are classified by
H^2(X)
. As connections on complex line bundles show up in physics (as `electromagnetic potentialÕ) so do geometric structures on gerbes (`B-fieldsÕ) which in particular give 3-forms representing the deRham cohomology class of the gerbe. There are also nonabelian version of such gerbes, explained in lectures of Aschieri-Jurco and related to Andre Henriques' talk. {\em Topological quantum field theory}: Recently, there has been much advance in this area of low-dimensional topology. Khovanov homology is a categorification of the Jones polynomial of knots in 3-space and it was used to distinguish the smooth and topological 4-genus of certain Alexander polynomial one knots. This is the first combinatorial, non Gauge theoretic, argument that smooth and topological
4
-manifold are very distinct. Dror Bar-Natan gave a survey of this theory and then Sergej Gukov explained an approach that could possibly lead to a categorification of the two variable Homfly-polynomial. Another very exciting conjecture in this area is the volume conjecture, relating a certain asymptotic behavior of the Jones polynomial at special values to the hyperbolic volume of a knot. Stavros Garoufalidis gave the closing lecture of the workshop on some progress that he and Thang Le made in this area. The format of the workshop was 4 lectures per day, except for Wednesday afternoon when we arranged a hike as well as a soccer game. We were very happy with the large amount of interaction that was going on inside and across the various groups of researchers. In addition, the relatively large number of young participants contributed a lot of activity through their curiosity and insistence. Finally, we are particularly thankful to the physicists Aschieri, Jurco, Gukov and Moore, as well as to the semi-physicists Freed and Segal for their participation. They were certainly among the most looked after discussion partners.
Peter Teichner, Stephan Stolz, Geometric Topology and Connections with Quantum Field Theory. Oberwolfach Rep. 2 (2005), no. 2, pp. 1505–1546
|
Topological Sort · USACO Guide
HomeGoldTopological Sort
Topological SortDFSFinding a CycleBFSCourse Schedule SolutionDynamic ProgrammingProblems
Authors: Benjamin Qi, Nathan Chen
Contributors: Michael Cao, Andi Qu, Andrew Wang, Qi Wang, Maggie Liu
An ordering of vertices in a directed acyclic graph that ensures that a node is visited before every node it has a directed edge to.
Gold - Breadth First Search (BFS)
Gold - Introduction to DP
To review, a directed graph consists of edges that can only be traversed in one direction. Additionally, an acyclic graph defines a graph which does not contain cycles, meaning you are unable to traverse across one or more edges and return to the node you started on. Putting these definitions together, a directed acyclic graph, sometimes abbreviated as DAG, is a graph which has edges which can only be traversed in one direction and does not contain cycles.
A topological sort of a directed acyclic graph is a linear ordering of its vertices such that for every directed edge
u\to v
from vertex
u
v
u
v
in the ordering.
There are two common ways to topologically sort, one involving DFS and the other involving BFS.
interactive, both versions
16.1 - Topological Sort
4.2.5 - Topological Sort
cp-algo
vector<int> graph[100000], top_sort; // Assume that this graph is a DAG
public class CourseSchedule {
static List<Integer>[] graph;
static List<Integer> topo = new ArrayList<Integer>();
Finding a Cycle
We can modify the algorithm above to return a directed cycle in the case where a topological sort does not exist. To find the cycle, we add each node we visit onto the stack until we detect a node already on the stack.
For example, suppose that our stack currently consists of
s_1,s_2,\ldots,s_i
and we then visit
u=s_j
j\le i
s_j\to s_{j+1}\to \cdots\to s_i\to s_j
is a cycle. We can reconstruct the cycle without explicitly storing the stack by marking
u
as not part of the stack and recursively backtracking until
u
is reached again.
bool visited[(int)1e5 + 5], on_stack[(int)1e5 + 5];
vector<int> adj[(int)1e5 + 5];
static List<Integer>[] adj;
static boolean[] visited, onStack;
static List<Integer> cycle;
This code assumes that there are no self-loops.
The BFS version is known as Kahn's Algorithm.
Course Schedule Solution
public class TopoSort {
static int[] inDegree;
static List<Integer>[] edge; // adjacency list
static int N; // number of nodes
static void topological_sort() {
# The code is in a function so it can run faster.
If the length of the array containing the end order does not equal the number of elements that need to be sorted, then there is a cycle in the graph.
We can also use Kahn's algorithm to extract the lexicographically minimum topological sort by breaking ties lexographically.
Although the above code does not do this, one can simply replace the queue with a priority_queue to implement this extension.
16.2 - Dynamic Programming
One useful property of directed acyclic graphs is, as the name suggests, that no cycles exist. If we consider each node in the graph as a state, we can perform dynamic programming on the graph if we process the states in an order that guarantees for every edge
u\to v
u
is processed before
v
. Fortunately, this is the exact definition of a topological sort!
Longest Flight Route
In this task, we must find the longest path in a DAG.
Easy Show Tags TopoSort
Normal Show Tags Binary Search, TopoSort
Hard Show Tags TopoSort
|
Longman Panorma Geography Solutions for Class 6 Social science Chapter 1 - The Earth In The Solar System
Longman Panorma Geography Solutions Solutions for Class 6 Social science Chapter 1 The Earth In The Solar System are provided here with simple step-by-step explanations. These solutions for The Earth In The Solar System are extremely popular among Class 6 students for Social science The Earth In The Solar System Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Longman Panorma Geography Solutions Book of Class 6 Social science Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Longman Panorma Geography Solutions Solutions. All Longman Panorma Geography Solutions Solutions for class Class 6 Social science are prepared by experts and are 100% accurate.
Whate are celestial bodies?
All the objects that we see in the sky are called celestial bodies or heavenly bodies. These celestial bodies include stars, planets, satellites, asteroids, meteorids and comets. The celestial bodies that do not have their own light and heat are called planets. On the other hand, stars are the celestial bodies that are made up of gases and have their own heat and light, which they emit in large amounts.
To which galaxy does our solar system belong?
The Milky Way is the galaxy that contains our solar system. It is also known as Akash Ganga. The word 'milky' signifies a bright band of light in the night sky where it is almost impossible to distinguish between stars. Akash Ganga can be seen as a faint band of light in a clear night sky.
The solar system consists of the Sun, eight planets, satellites and other celestial bodies known as asteroids and meteoroids. We often call it a solar family, with the Sun as its head. All the planets and celestial bodies revolve around the Sun in their respective orbits and, in turn, get influenced by the gravitational pull and heat of the Sun.
Write the names of the eight planets in order of their distances from the sun.
(In Million Kilometres)
Which are the two types of movement of the planets?
Planets have two types of movement: rotation and revolution.
1. Rotation is the movement caused when a planet rotates on its axis; for example, the Earth rotates from west to east on its axis.
2. Revolution is the movement when a planet revolves around the Sun in a fixed path. The path of revolution of each planet is either egg-shaped or elliptical; this path is known as an orbit.
What are the favourable conditions that make life possible on earth?
The Earth is the only planet where conditions are favourable to support life. It is neither too hot nor too cold and thus have an optimum temperature to support human existence. It has water and air, which very much essential for the survival of humans. The air is composed of life-supporting gases like oxygen, carbon dioxide and ozone. Because of these factors, the Earth is regarded as a unique planet of our solar system.
What are asteroids? Where are they found?
Apart from stars, planets and satellites, there are several other tiny bodies that move around the Sun. These bodies are known as asteroids. They are found between the orbits of Mars and Jupiter. The largest known asteroid is named Ceres.
What are periodical comets? Give an example.
Periodical comets are the comets that orbit around the Sun and return to the innermost point of their orbits after a regular interval of time. A good example of a periodical comet is Halley's Comet, which returns to its original position after 75
-
Stars Planets
Stars are celestial bodies made up of gases; they are huge in size and have very high temperatures. Planets are the celestial bodies that do not have their own heat and light.
They have their own heat and light, which they emit in large amounts. They are lit by the light of stars.
For example, the Sun is a star. For example, the Earth on which we live is a planet.
Planets Satellites
These are celestial bodies that do not have their own heat and light; they lit by the light of stars. Satellites are celestial bodies that move around planets in the same way planets move around the Sun.
They are made up of solid materials and gases. They do not have their own light.
For example, the Earth is a planet. For example, the Moon is a natural satellite of the Earth.
Comet and meteor
Comets Meteors
Comets are heavenly bodies that revolve around the Sun in an elongated orbit. Meteoroids are small rock pieces that revolve around the Sun.
They are usually made up of dust, ice particles and gases. Some meteoroids enter the Earth's atmosphere with a great velocity.
Most comets have a head, a nucleus and a tail. When they approach the Sun, the gases get heated up; as a result they glow. They get heated up due to the friction of the atmosphere. This causes the meteoroids to glow.
Example: Halley Example: Leonid
Mercury is the hottest planet while Neptune is the coldest.
Mercury is the hottest planet because it is nearest to the Sun and stands at a distance of 58 million kilometres from the Sun. Hence, Mercury receives the maximum amount of sunlight.
On the other hand, Neptune is the coldest planet because it is farthest from the Sun and stands at a distance of 4,497 million kilometres from the Sun. Hence, Neptune receives the least amount of sunlight among all planets in our solar system.
The Moon takes 27 days and 8 hours to revolve around the Earth. This period coincides with the rotation period of the Earth on its axis, i.e. 24 hours. This is the reason we always see the same side of the Moon.
There is no possibility of life on the moon.
There is no possibility of life on the Moon because of the absence of the essentials of life on it, i.e. air and water. Also, there is no atmosphere around it, due to which days are very hot and nights are very cold. This makes the conditions unfavourable for the existence of life on the Moon.
The temperature of the surface of the sun is about
b. 5,500°C
c. 6,000°C
d. 6,500°C
Explanation: The Sun is composed of extremely hot and burning gases like helium and hydrogen. The fusion of hydrogen molecules into helium molecules results in the release of a huge amount of energy and heat that raise the temperature of the Sun up to 6,000 degree Centigrade.
Explanation: Mars has a reddish appearance because of the presence of iron oxides. Mars is generally referred to as the Red Planet of our solar system.
Which of the following planets rotates in a clockwise direction?
The correct answer is option (b).
Explanation: Venus is the only planet in our solar system that rotates clockwise. All other planets rotate anticlockwise.
Halley's Comet appears after every
The correct answer is option (d).
Explanation: Halley's Comet appears after a regular interval of 76 years. It last appeared in the solar system in 1986.
All heavenly bodies are called cerebral bodies.
All heavenly bodies are called celestial bodies.
The light of the sun takes about eight seconds to reach the earth.
The light of the Sun takes about eight minutes to reach the Earth.
Explanation: Since the Earth is very far from the Sun, it takes light around eight minutes to reach the Earth's surface.
The star nearest to the sun is Ceres.
The star nearest to the sun is Proxima Centauri.
Explanation: The Sun is the nearest star to the Earth, but Proxima Centauri is the second nearest star to the Earth and the nearest star to the Sun.
Moon is not the satellite of earth.
Explanation: Satellites are the bodies that revolve around planets. The Moon is the only natural satellite of our planet Earth.
The largest planet in the solar system. __________
The largest planet in our solar system is Jupiter.
Explanation: The mass of Jupiter is almost 300 times that of Earth. The diameter of Jupiter is 11 times that of Earth. It is the largest planet of our solar system.
The path taken by the planets to go around the sun. ____________
The path taken by the planets to go around the Sun is known as the orbit.
Explanation: The path of revolution of each planet around the Sun is either egg-shaped or elliptical. This path is known as the orbit.
The first man in space. ____________
The first man to visit space was Yuri Alekseyevich Gagarin.
Explanation: Yuri Alekseyevich Gagarin was the Russian Soviet cosmonaut who travelled in the outer space in the spacecraft Vostok in 1961.
The planet with a system of rings. ____________
The planet with a notable system of rings is Saturn.
Explanation: Large and colourful rings form the orbiting disc of Saturn. Apart from Saturn, there are other planets that possess planetary rings. These are Jupiter, Uranus and Neptune, but their rings have less visibility and they are not very prominent.
The number of planets that rotate in a clockwise direction. ____________
There are two planets that rotate clockwise: Venus and Uranus.
Explanation: Except Venus and Uranus, all planets in our solar system rotate anticlockwise.
Mercury, Venus, Earth and Mars are collectively called this. ____________
Mercury, Venus, Earth and Mars are collectively known as inner planets in our solar system.
Explanation: They are named inner planets because they are the nearest to the sun. There is huge difference in planetary conditions of these planets and the rest of the planets of our solar system.
The largest asteroid. ____________
The largest asteroid known is Vesta.
Explanation: Though Ceres is way larger in mass than any other asteroid, since it has been listed among dwarf planets, the largest asteroid is Vesta.
The fictitious planet where Superman came from. ____________
Krypton is a fictitious planet of the DC Universe from where the fictitious character Superman has come.
Diagram of the solar system.
|
Home : Support : Online Help : System : Information : Updates : Maple 9.5 : New Packages
New Packages in Maple 9.5
Maple 9.5 contains many new packages.
For information on enhancements and improvements to existing packages, see Enhanced Packages in Maple 9.5.
MmaTranslator Package
Student[MultivariateCalculus] Package
The Cache package is a set of functions for manipulating the new Cache data structure. A Cache can be used like a table; however, a Cache stores two types of elements, permanent and temporary. Permanent elements are like those of a table, once inserted they remain in the table until removed. Temporary elements are stored as long as the Cache has space to store them. When a new temporary element is inserted, an older one may be removed.
Caches are primarily intended to be used as remember tables. Elements that are added automatically are added as temporary. This restricts the remember table to store only recently accessed elements. Important remember table entries can be stored as permanent entries, assuring that they are not automatically removed.
The ContextMenu package provides tools to control and customize Maple context-sensitive menus. It supersedes the context package. A context-sensitive menu is generated when a user right-clicks (Control-click on Macintosh platforms) a Maple expression.
The information necessary to build context-sensitive menus is encapsulated in context menu modules. The module corresponding to the built-in context menu is available as ContextMenu[CurrentContext]. To build a new context menu module, which replaces the default, see ContextMenu[New].
Within a context menu module, you can add new entries to the context menu or alter the criteria under which entries are displayed in a menu. For more information, see ContextMenu[CurrentContext].
The Logic package is a collection of commands for manipulating and transforming expressions using two-valued Boolean logic.
With the Logic package, you can simplify logical expressions, test two expressions for equivalence, convert logical expressions to algebraic expressions modulo 2, and perform a variety of other logical operations. In the following example, the dual of a logical expression is computed using the Logic[Dual] command.
with(Logic):
Dual( (x &and y) &implies z );
\textcolor[rgb]{0,0,1}{¬}\left(\textcolor[rgb]{0,0,1}{z}\textcolor[rgb]{0,0,1}{⇒}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{\vee }\textcolor[rgb]{0,0,1}{y}\right)\right)
The MmaTranslator package converts Mathematica commands and notebooks to Maple commands and worksheets.
You can use the FromMma and FromMmaNotebook commands or the Mathematica to Maple Translator Maplet application for the translation.
For more information, see MmaTranslator.
The Optimization package is a collection of commands for numerically solving optimization problems, which involve minimization or maximization of an objective function possibly subject to constraints. The package takes advantage of built-in library routines provided by the Numerical Algorithms Group (NAG). Key features are described as follows.
The package provides the ability to solve linear programs, quadratic programs, nonlinear programs, and both linear and nonlinear least-squares problems. For non-convex problems, local solutions are computed. Both unconstrained and constrained problems are accepted.
An Interactive Maplet application provides and easy-to-use interface to all of the computation routines in the Optimization package, including the Minimize and Maximize commands. Alternatively, users can take full advantage of the solvers' capabilities by calling the specialized routines: LPSolve for linear programs, QPSolve for quadratic programs, NLPSolve for nonlinear programs, and LSSolve for least-squares problems. The following examples demonstrate the Minimize and QPSolve commands. The solution consists of the final objective function value followed by a point at which this value is attained.
[\textcolor[rgb]{0,0,1}{\mathrm{ImportMPS}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Interactive}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LPSolve}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{LSSolve}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Maximize}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{Minimize}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{NLPSolve}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{QPSolve}}]
Minimize(sin(x)/x, x=1..10);
[\textcolor[rgb]{0,0,1}{-0.217233628211222}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4.49340945753529}]]
QPSolve(2*x+5*y+3*x^2+3*x*y+2*y^2, {x-y>=2});
[\textcolor[rgb]{0,0,1}{-3.53333333333333}\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.466666666666667}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-1.60000000000000}]]
The Optimization package commands allow a simple and natural way to express the objective function and the constraints using algebraic expressions. Alternative forms of input using procedures, Vectors, and Matrices are available, which provide greater flexibility and efficiency. For a summary of the various ways in which the optimization problem can be expressed, see Optimization[InputForms]. In addition, the package features the Optimization[ImportMPS] command for importing linear programs in MPS format.
Computations can be performed using hardware floating-point data or arbitrary-precision software floating-point data. The Optimization commands automatically select the most appropriate floating-point computation environment and attempt to solve the problem as efficiently as possible. For details concerning numeric computation in the Optimization package, see the Optimization[Computation] help page.
For a brief introduction to the Optimization package, see the examples[Optimization] worksheet.
The RootFinding package enhances Maple's ability to compute and locate roots numerically. The package contains four commands. The RootFinding[Analytic] and RootFinding[AnalyticZerosFound] commands compute the zeros of an analytic univariate function. The RootFinding[BivariatePolynomial] command computes the solutions of two or more bivariate polynomials. The RootFinding[Homotopy] command finds numerical approximations to roots of systems of polynomial equations.
A new package for multivariate calculus is available in the Student package. The Student[MultivariateCalculus] package assists with the teaching and learning of the calculus of functions from
{R}^{n}
R
. Interactive tutors are provided covering the basic concepts, and a range of commands are included that allow exploration of these and other concepts in greater depth.
For more information, see Student[MultivariateCalculus].
|
Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities | EMS Press
Klaus Deckelnick
We consider the Hamilton-Jacobi equation of eikonal type
H(\nabla u) = f(x), \quad x \in \Omega,
H
f
is allowed to be discontinuous. Under a suitable assumption on
f
we prove a comparison principle for viscosity sub- and supersolutions in the sense of Ishii. Furthermore, we develop an error analysis for a class of finite difference schemes, which are monotone, consistent and satisfy a suitable stability condition.
Charles M. Elliott, Klaus Deckelnick, Uniqueness and error analysis for Hamilton-Jacobi equations with discontinuities. Interfaces Free Bound. 6 (2004), no. 3, pp. 329–349
|
Analysis on the Evaluation of Suburban Real Estate Investment Environment
—Taking Guangzhou as an Example
―Taking Guangzhou as an Example
\left\{\begin{array}{l}{F}_{1}={a}_{11}{X}_{1}+{a}_{21}{X}_{2}+\cdots +{a}_{p1}{X}_{p}\\ {F}_{2}={a}_{12}{X}_{1}+{a}_{22}{X}_{2}+\cdots +{a}_{p2}{X}_{p}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ {F}_{p}={a}_{1p}{X}_{1}+{a}_{2p}{X}_{2}+\cdots +{a}_{pp}{X}_{p}\end{array}
Z{X}_{ij}=\frac{{X}_{ij}-{\stackrel{¯}{X}}_{i}}{{\sigma }_{i}}
\begin{array}{l}{F}_{11}=0.291Z{X}_{1}+0.221Z{X}_{2}+0.158Z{X}_{3}+0.302Z{X}_{4}+0.297Z{X}_{5}\\ {F}_{12}=0.279Z{X}_{1}-0.452Z{X}_{2}+0.575Z{X}_{3}-0.025Z{X}_{4}-0.217Z{X}_{5}\end{array}
Z{X}_{1},\cdots ,Z{X}_{5}
Wu, S.L. (2019) Analysis on the Evaluation of Suburban Real Estate Investment Environment. Modern Economy, 10, 914-930. https://doi.org/10.4236/me.2019.103061
1. Xiu, D., Lu, P. and Yan, B. (2013) The Suburbanization Characteristics of Real Estate and Its Development Model—Taking Beijing as an Example. Real Estate Development, 4, 24-30. https://doi.org/10.1080/028418501127346846
2. Fielding, A.J. (1989) Migration and Urbanization in Western Europe since 1950. The Geographical Journal, 155, 60-69. https://doi.org/10.2307/635381
3. Lawrence, T. and Yezer, A.M.J. (1994) Causality in the Suburbanization of Population and Employment. Journal of Urban Economics, 35, 105-118. https://doi.org/10.1006/juec.1994.1006
4. Walker, R. (2001) Industry Builds the City: The Suburbanization of Manufacturing in the San Francisco Bay Area, 1850-1940. Journal of Historical Geography, 27, 57. https://doi.org/10.1006/jhge.2000.0268
5. Kellerman, A. (1985) The Suburbanization of Retail Trade: A U.S. Nationwide View. Geoforum, 16, 15-23. https://doi.org/10.1016/0016-7185(85)90003-X
6. Buckner, H.T. and Palen, J.J. (1976) The Urban World. Contemporary Sociology, 5, 832. https://doi.org/10.2307/2063174
7. Muller, E.K. (2001) Industrial Suburbs and the Growth of Metropolitan Pittsburgh, 1870-1920. Journal of Historical Geography, 27, 58-73. https://doi.org/10.1006/jhge.2000.0269
8. Mills, E.S. and Price, R. (1984) Metropolitan Suburbanization and Central City Problems. Journal of Urban Economics, 15, 1-17. https://doi.org/10.1016/0094-1190(84)90019-6
9. Oron, Y., Pines, D. and Sheshinski, E. (1974) The Effect of Nuisances Associated with Urban Traffic on Suburbanization and Land Values. Journal of Urban Economics, 1, 382-394. https://doi.org/10.1016/0094-1190(74)90002-3
10. Zhou, Y. (1999) It Is Necessary to Take Advantage of the Suburbanization of the City. Urban Planning, 4, 13-18.
11. Liu, B. and Zheng, L. (2004) Characteristics and Dynamic Mechanism of Suburbanization of Chinese Cities. Journal of Theory, 10, 68-70.
12. Wu, G. and Liu, J. (2000) Comparison of Suburbanization of Chinese and Foreign Cities. Urban Planning, 8, 36-39.
13. Shi, C. (2006) Discuss the Suburbanization of Chinese Cities. Theory, No. 2, 11-12.
14. Liu, X. and Ai, G. (2016) Does FDI Promote the Suburbanization of Chinese Cities—An Empirical Test Based on Satellite Night Light Data. Financial Research, No. 6, 52-62.
15. Chen, Z. and Fang, L. (2002) The Suburban Real Estate Industry Needs to Develop Moderately in the Process of Urban Suburbanization. Economic Frontiers, No. 3, 40-42.
16. Jia, Y. and Xu, Q. (2007) Discuss the Relationship between Suburbanization and Suburban Real Estate Development. Business Economy, No. 2, 103-105.
17. Wheeler, D. and Mody, A. (1992) International Investment Location Decisions: The Case of U.S. Firms. Journal of International Economics, 33, 57-76. https://doi.org/10.1016/0022-1996(92)90050-T
18. Florida, R. and Kenney, M. (1992) Restrictions in Place, Japanese Investment, Production Organization, and the Geography of Steel. Economic Geography, No. 8, 80-92.
19. Bai, C., Lu, J. and Tao, Z. (2004) The Impact of Investment Environment on the Benefits of Foreign-Funded Enterprises—Evidence from Enterprise Level. Economic Research Journal, No. 9, 82-89.
20. Zhou, P., Zhang, H., Xie, N. and Zheng, J. (2010) A Comprehensive Evaluation System of Real Estate Investment Environment Based on Principal Component Analysis and Delphi Method. China Land Science, 12, 58-63.
21. Lu, X., Gui, T. and Wan, K. (2013) A Regional Real Estate Investment Environment Evaluation System Based on Principal Component Analysis—Taking Wuhan City Circle as an Example. Real Estate Market, No. 1, 62-72.
22. Wang, B. (2004) Research on Systematic Evaluation of China’s Real Estate Regional Investment Environment. M.Ec. Thesis, Chongqing University, Chongqing.
23. Liu, H. (2014) Analysis of Investment Environment in Chongqing. M.Ec. Thesis, Southwestern University of Finance and Economics.
|
The new JSON package allows import and export of files and strings in the JSON format, a popular format used by many modern applications for exchanging structured data.
Example: Import JSON data encoding the mailing address of Maplesoft headquarters.
\mathrm{currentdir}\left(\mathrm{FileTools}:-\mathrm{JoinPath}\left(\left["example"\right], \mathrm{base}=\mathrm{datadir}\right)\right):
T ≔ \mathrm{JSON}:-\mathrm{ParseFile}\left("address.json"\right)
\textcolor[rgb]{0,0,1}{\mathrm{table}}\left(\left[\textcolor[rgb]{0,0,1}{"address"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{table}}\left(\left[\textcolor[rgb]{0,0,1}{"streetAddress"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"615 Kumpf Drive"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"city"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Waterloo"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"postalCode"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"N2V 1K8"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"country"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Canada"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"province"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"ON"}\right]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"founded"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1988}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"phoneNumbers"}\textcolor[rgb]{0,0,1}{=}\left[\textcolor[rgb]{0,0,1}{\mathrm{table}}\left(\left[\textcolor[rgb]{0,0,1}{"type"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"local"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"number"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"+1 \left(519\right) 747-2373"}\right]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{table}}\left(\left[\textcolor[rgb]{0,0,1}{"type"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"tollfree"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"number"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"+1 \left(800\right) 267-6583"}\right]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{table}}\left(\left[\textcolor[rgb]{0,0,1}{"type"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"fax"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"number"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"+1 \left(519\right) 747-5284"}\right]\right)\right]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"companyName"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Maplesoft"}\right]\right)
T\left["companyName"\right]
\textcolor[rgb]{0,0,1}{"Maplesoft"}
T\left["address"\right]\left["city"\right], T\left["address"\right]\left["country"\right]
\textcolor[rgb]{0,0,1}{"Waterloo"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Canada"}
The KML file format and the related KMZ compressed format are popular XML-based map data formats which are used by many mapping applications.
KML permits geographic areas and contours to be defined using line and polygon primitives.
The new Import command, described in more detail below, can import a KML map file and represent it as a 2-D plot.
In addition to KML and KMZ, Import also supports the GPX and SHP cartographical formats.
Example: Import a four-colour KMZ map of the 48 contiguous states of the United States of America.
\mathrm{Import}\left("http://www.maplesoft.com/data/examples/kmz/CONUS.kmz", \mathrm{title} = "The Contiguous United States", \mathrm{titlefont} = \left[\mathrm{Times}, \mathrm{Bold}, 20\right], \mathrm{size} = \left[800, 400\right]\right)
The Import and Export commands also support three popular text-based formats for representing DNA and protein sequences: FASTA, FASTQ, and GenBank.
You can import genetic or proteinomic data into your session and use text processing tools to analyze the imported sequence.
Example: Import a DNA sequence from a FASTA file.
\mathrm{mtDNASequence}≔\mathrm{Import}\left("humanmtDNA.fasta"\right):
Study the metadata for the first sequence in the file:
\mathrm{mtDNASequence}\left[1,1\right]
\textcolor[rgb]{0,0,1}{"Human mitochondrial genome,HVR2,CR,HVR1"}
Examine the nucleotide codes at positions 16200 through 16250:
\mathrm{mtDNASequence}\left[1,2\right]\left[16200..16250\right]
\textcolor[rgb]{0,0,1}{"TTACAAGCAAGTACAGCAATCAACCCTCAACTATCACACATCAACTGCAAC"}
Count the frequency of each of nucleotide base (A, C, G, or T) in the sequence:
\mathrm{frequencies} ≔ \left[\mathrm{StringTools}:-\mathrm{CharacterFrequencies}\left(\mathrm{mtDNASequence}\left[1,2\right]\right)\right]
\left[\textcolor[rgb]{0,0,1}{"A"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5118}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"C"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{5185}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"G"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2175}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"N"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"T"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4092}\right]
\mathrm{Statistics}:-\mathrm{ColumnGraph}\left( \mathrm{frequencies}\right)
The GraphTheory package now supports six new formats for import and export: DGML , Graphlet , GraphML , GXL , Pajek , and TGF.
For more information on supported graph-theoretic formats, see Formats.
\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{Import}\left("petersen.graphml"\right)\right)
\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{Import}\left("draco.net"\right),\mathrm{color}=\mathrm{red}\right)
The new Import and Export commands provide a generic command-based mechanism to move data between Maple and the operating environment.
The Import command provides a ubiquitous mechanism for importing data with a single command.
Import is agnostic of the type of data: it can handle numeric and tabular data, images, cartographic data, specialized text file formats like XML and JSON, and special-purpose formats for graph theory and linear optimization. Several examples appear below.
\mathrm{Import}\left("dodecahedron.stl"\right)
M ≔ \mathrm{Import}\left("timedata.csv"\right)
\left[\begin{array}{c}\textcolor[rgb]{0,0,1}{\mathrm{28 x 6}}\textcolor[rgb]{0,0,1}{\mathrm{Matrix}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Data Type:}}\textcolor[rgb]{0,0,1}{\mathrm{anything}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Storage:}}\textcolor[rgb]{0,0,1}{\mathrm{rectangular}}\\ \textcolor[rgb]{0,0,1}{\mathrm{Order:}}\textcolor[rgb]{0,0,1}{\mathrm{Fortran_order}}\end{array}\right]
M\left[1..5,1..2\right]
\left[\begin{array}{cc}\textcolor[rgb]{0,0,1}{"Mar 06 01:16"}& \textcolor[rgb]{0,0,1}{"3/7/2005"}\\ \textcolor[rgb]{0,0,1}{"Mar 06 20:12"}& \textcolor[rgb]{0,0,1}{"3/7/2005"}\\ \textcolor[rgb]{0,0,1}{"Mar 06 20:43"}& \textcolor[rgb]{0,0,1}{"3/7/2005"}\\ \textcolor[rgb]{0,0,1}{"Mar 07 00:25"}& \textcolor[rgb]{0,0,1}{"3/7/2005"}\\ \textcolor[rgb]{0,0,1}{"Mar 07 00:44"}& \textcolor[rgb]{0,0,1}{"3/7/2005"}\end{array}\right]
\mathrm{Company} ≔\mathrm{Import}\left("address.json"\right):
\mathrm{Company}\left["companyName"\right]
\textcolor[rgb]{0,0,1}{"Maplesoft"}
\mathrm{Company}\left["founded"\right]
\textcolor[rgb]{0,0,1}{1988}
\mathrm{Company}\left["address"\right]\left["streetAddress"\right]
\textcolor[rgb]{0,0,1}{"615 Kumpf Drive"}
\mathrm{Company}\left["address"\right]\left["city"\right]
\textcolor[rgb]{0,0,1}{"Waterloo"}
\mathrm{Company}\left["address"\right]\left["country"\right]
\textcolor[rgb]{0,0,1}{"Canada"}
\mathrm{Company} ≔\mathrm{Import}\left("coxeter.col"\right);
\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: an undirected unweighted graph with 28 vertices and 42 edge\left(s\right)}}
\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(,\mathrm{style}=\mathrm{spring}\right)
The Export command provides an equally generic and powerful mechanism for exporting data from Maple with a single command.
\mathrm{OutputFile} ≔ \mathrm{FileTools}:-\mathrm{JoinPath}\left(\left["graphic.png"\right], \mathrm{base}=\mathrm{homedir}\right)
\textcolor[rgb]{0,0,1}{"C:\Users\JohnSmith\graph.png"}
E:=\left[\mathrm{seq}\left(\mathrm{Array}\left(\mathrm{evalf}\left(\left[\mathrm{seq}\left(1+\mathrm{sin}\left(\frac{10\mathrm{\pi }i}{15j}\right),i=1..15\right)\right]\right)\right),j=1..15\right)\right]:
\mathrm{MyGraphic} ≔ \mathrm{dataplot}\left(E,\mathrm{bar},\mathrm{format}=\mathrm{stacked},\mathrm{color}="Maroon".."RoyalBlue",\mathrm{gridlines}\right)
\mathrm{Export}\left(\mathrm{OutputFile}, \mathrm{MyGraphic}\right)
\textcolor[rgb]{0,0,1}{22685}
\mathrm{OutputFile} ≔ \mathrm{FileTools}:-\mathrm{JoinPath}\left(\left["integral.mml"\right], \mathrm{base}=\mathrm{homedir}\right)
\textcolor[rgb]{0,0,1}{"C:\Users\JohnSmith\integral.mml"}
Compute an integral in Maple and export it to a MathML file.
\mathrm{MyIntegral} ≔ ∫\mathrm{sin}\left({x}^{2}+x\right) ⅆ\textcolor[rgb]{0.784313725490196,0,0.784313725490196}{x}
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{π}}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{FresnelS}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{π}}}}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{FresnelC}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{π}}}}\right)\right)
\mathrm{Export}\left(\mathrm{OutputFile}, \mathrm{MyIntegral}\right)
\textcolor[rgb]{0,0,1}{3088}
Demonstrate that the export was successful by re-importing the data and retrieving the original expression.
\mathrm{Import}\left(\mathrm{OutputFile}\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{π}}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{FresnelS}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{π}}}}\right)\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{FresnelC}}\textcolor[rgb]{0,0,1}{}\left(\frac{\sqrt{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\right)}{\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{π}}}}\right)\right)
|
MTI | Free Full-Text | Displays for Productive Non-Driving Related Tasks: Visual Behavior and Its Impact in Conditionally Automated Driving
Supporting User Onboarding in Automated Vehicles through Multimodal Augmented Reality Tutorials
Resisting Resolution: Enterprise Civic Systems Meet Community Organizing
Context-Adaptive Availability Notifications for an SAE Level 3 Automation
Schartmüller, C.
Weigl, K.
Löcken, A.
Wintersberger, P.
Displays for Productive Non-Driving Related Tasks: Visual Behavior and Its Impact in Conditionally Automated Driving
CARISSMA Institute of Automated Driving, Technische Hochschule Ingolstadt (THI), 85049 Ingolstadt, Germany
Institute for Pervasive Computing, Johannes Kepler University Linz (JKU), 4020 Linz, Austria
Department of Psychology, Catholic University of Eichstätt-Ingolstadt, 85072 Eichstätt, Germany
Institute of Visual Computing and Human-Centered Technology, TU Wien, 1040 Vienna, Austria
Academic Editor: Mu-Chun Su
(This article belongs to the Special Issue Interface and Experience Design for Future Mobility)
(1) Background: Primary driving tasks are increasingly being handled by vehicle automation so that support for non-driving related tasks (NDRTs) is becoming more and more important. In SAE L3 automation, vehicles can require the driver-passenger to take over driving controls, though. Interfaces for NDRTs must therefore guarantee safe operation and should also support productive work. (2) Method: We conducted a within-subjects driving simulator study (
N=53
) comparing Heads-Up Displays (HUDs) and Auditory Speech Displays (ASDs) for productive NDRT engagement. In this article, we assess the NDRT displays’ effectiveness by evaluating eye-tracking measures and setting them into relation to workload measures, self-ratings, and NDRT/take-over performance. (3) Results: Our data highlights substantially higher gaze dispersion but more extensive glances on the road center in the auditory condition than the HUD condition during automated driving. We further observed potentially safety-critical glance deviations from the road during take-overs after a HUD was used. These differences are reflected in self-ratings, workload indicators and take-over reaction times, but not in driving performance. (4) Conclusion: NDRT interfaces can influence visual attention even beyond their usage during automated driving. In particular, the HUD has resulted in safety-critical glances during manual driving after take-overs. We found this impacted workload and productivity but not driving performance. View Full-Text
Keywords: automated driving; take-over requests; conditional automation; non-driving related tasks; eye-tracking; performance; behavior; displays; productivity; visual attention automated driving; take-over requests; conditional automation; non-driving related tasks; eye-tracking; performance; behavior; displays; productivity; visual attention
Schartmüller, C.; Weigl, K.; Löcken, A.; Wintersberger, P.; Steinhauser, M.; Riener, A. Displays for Productive Non-Driving Related Tasks: Visual Behavior and Its Impact in Conditionally Automated Driving. Multimodal Technol. Interact. 2021, 5, 21. https://doi.org/10.3390/mti5040021
Schartmüller C, Weigl K, Löcken A, Wintersberger P, Steinhauser M, Riener A. Displays for Productive Non-Driving Related Tasks: Visual Behavior and Its Impact in Conditionally Automated Driving. Multimodal Technologies and Interaction. 2021; 5(4):21. https://doi.org/10.3390/mti5040021
Schartmüller, Clemens, Klemens Weigl, Andreas Löcken, Philipp Wintersberger, Marco Steinhauser, and Andreas Riener. 2021. "Displays for Productive Non-Driving Related Tasks: Visual Behavior and Its Impact in Conditionally Automated Driving" Multimodal Technologies and Interaction 5, no. 4: 21. https://doi.org/10.3390/mti5040021
|
Revision as of 15:59, 14 March 2019 by Nikolay (talk | contribs) (Created page with "CCZ-inequivalent APN Functions over <math>\mathbb{F}_{2^n}</math> from the Known APN Classes for <math>6\leqslant n \leqslant 11</math> <table> <tr> <th>Dimension</th> <th>F...")
{\displaystyle \mathbb {F} _{2^{n}}}
{\displaystyle 6\leqslant n\leqslant 11}
{\displaystyle 6}
{\displaystyle x^{24}+ax^{17}+a^{8}x^{10}+ax^{9}+x^{3}}
{\displaystyle C3}
{\displaystyle ax^{3}+x^{17}+a^{4}x^{24}}
{\displaystyle C7-C9}
{\displaystyle 7}
{\displaystyle x^{3}+Tr_{7}(x^{9})}
{\displaystyle C4}
{\displaystyle 8}
{\displaystyle x^{3}+x^{17}+p^{48}x^{18}+p^{3}x^{33}+px^{34}+x^{48}}
{\displaystyle C3}
{\displaystyle x^{3}+Tr_{8}(x^{9})}
{\displaystyle C4}
{\displaystyle x^{3}+a^{-1}Tr_{8}(a^{3}x^{9})}
{\displaystyle C4}
{\displaystyle a(x+x^{16})(ax+a^{16}x^{16})+a^{17}(ax+a^{16}x^{16})^{12}}
{\displaystyle C10}
{\displaystyle 9}
{\displaystyle x^{3}+Tr_{9}(x^{9})}
{\displaystyle C4}
{\displaystyle x^{3}+Tr_{9}^{3}(x^{9}+x^{18})}
{\displaystyle C5}
{\displaystyle x^{3}+Tr_{9}^{3}(x^{18}+x^{36})}
{\displaystyle C6}
{\displaystyle x^{3}+a^{246}x^{10}+a^{47}x^{17}+a^{181}x^{66}+a^{428}x^{129}}
{\displaystyle C11}
{\displaystyle 10}
{\displaystyle x^{6}+x^{33}+p^{31}x^{192}}
{\displaystyle C3}
{\displaystyle x^{3}+x^{72}+p^{31}x^{258}}
{\displaystyle C3}
{\displaystyle x^{3}+Tr_{10}(x^{9})}
{\displaystyle C4}
{\displaystyle x^{3}+a^{-1}Tr_{10}(a^{3}x^{9})}
{\displaystyle C4}
{\displaystyle 11}
{\displaystyle x^{3}+Tr_{11}(x^{9})}
{\displaystyle C4}
|
Ratna Sagar History Solutions for Class 6 Social science Chapter 8 - The First Empire The Mauryas
the first empire the mauryas
Ratna Sagar History Solutions Solutions for Class 6 Social science Chapter 8 The First Empire The Mauryas are provided here with simple step-by-step explanations. These solutions for The First Empire The Mauryas are extremely popular among Class 6 students for Social science The First Empire The Mauryas Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Ratna Sagar History Solutions Book of Class 6 Social science Chapter 8 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Ratna Sagar History Solutions Solutions. All Ratna Sagar History Solutions Solutions for class Class 6 Social science are prepared by experts and are 100% accurate.
The literary sources for the Mauryan period include the Indika and the
a. rock edicts
d. pillar edicts.
Explanation: Indika was written by Megasthenes, a Greek ethnographer and explorer in the Hellenistic period. He was the ambassador of Seleucus I of the Seleucid dynasty to Chandragupta Maurya in Pataliputra.
The Arthashastra is an ancient Indian treatise on statecraft, economic policy and military strategy, written in Sanskrit by Chanakya.
________________ defeated Seleucus Nicator, a general of the Greek king Alexander.
b. Chanakya
Explanation: Chandragupta Maurya defeated Seleucus Nikator, a general of Alexander. Chandragupta fought to regain Alexander's satrapies. Seleucus I Nicator fought to defend these territories but both sides made peace in 303 BC. The treaty ended the Seleucid–Mauryan war and allowed Chandragupta to control the regions it was warring for.
Under _______________, the Mauryan Empire spread across the whole of the Indian subcontinent, except for Kalinga and few kingdoms in the south.
Explanation: Bindusara inherited a large empire that consisted of what is now the northern, central and eastern parts of India along with parts of Afghanistan and Balochistan from his father, Chandragupta. Bindusara extended his empire further as far as south Mysore. He conquered 16 states and extended the empire from sea to sea. His empire comprised of the whole of India except Kalinga. Kalinga was conquered by his son Ashoka.
Ashoka gave up the policy of conquest through war and began to follow a policy of conquest through
d. sea routes
Explanation: After the Kalinga war in 261 BC, Ashoka gave up his policy of conquest through war and adopted a policy of conquest through dharma. In other words, the policy of dig-vijay was replaced with dhamma-vijay.
Most of the edicts were in ______________ script.
d. Kharoshthi
Explanation: Most Ashokan edicts were written in Brahmi script and in Prakrit language. This language was adopted by Ashoka because it was the language that was spoken by the common man, by the masses; in order to increase the spread and efficacy of his edicts, Ashoka used this language for inscribing his edicts, so that the message of dhamma could spread far and wide in his people.
The ______________ was the head of the district.
a. Yukta
b. Rajuka
d. Pradeshta
Explanation: Provinces were divided into districts. At the district level, the Pradeshta was the head of administration. He was assisted by yuktas and rajukas. They measured the land, collected taxes and maintained law and order.
Land revenue was fixed between one-fourth and ______________ of the produce.
c. one-sixth
d. one-eighth
Explanation: Land revenue was the main source of revenue for the state. It was fixed between one-fourth and one-sixth of the produce, depending upon the fertility of the soil.
1. Respect only your own religion.
\overline{) }
2. Treat your servants and slaves harshly.
\overline{) }
3. What is a problem in itself; it is not a solution to any problem.
\overline{) }
4. Never give money to a poor man.
\overline{) }
5. Others have as much equal right to live as you.
\overline{) }
Explanation: Ashoka asked his people to respect all religions. He asked his people to follow the principles of dhamma, which included secularism and respect for all religions.
Explanation: Ashoka asked his people to follow the principles of dhamma, which included following the principles of treating one’s slaves and servants with kindness.
Explanation: After the Kalinga War of 261 BC, Ashoka realised the futility of war and started treating war as a problem, rather than as a solution. Ashoka gave up his policy of conquest through war and adopted a policy of conquest through dharma.
Explanation: Ashoka asked his people to follow the principles of dhamma, which included charity and giving money to the poor.
Explanation: Ashoka asked his people to follow the principles of dhamma, which included treating others equally and accepting the fact that everyone has equal rights to live.
Seven virtues, recommended by Emperor Ashoka, are hidden in this word search. Find them.
D E F N V D Z E M K F S
L T L U N A T X D I F X
W O B J C K J B K N M P
Z L A G I C U V Y D Y T
P E A C E I E Z L N Q E
T R U T H F U L N E S S
R A B P G A R O J S W K
H N Q O R A H I M S A H
S C H A R I T Y G N C M
R E S P E C T Q I D W H
Following are the words hidden in the word search:
What do the Indika and the Arthashastra tell us about the Mauryas?
The main source of information on the Mauryas are through two books:
1. Indika, written by Megasthenes, tells us about the social, political and economic life of the people during the Mauryan times.
2. Arthashastra by Kautilya deals with governance of an empire. It describes the administration of the Mauryas.
When was the Kalinga War fought? Why did Ashoka attack Kalinga?
When Ashoka became the king, Kalinga was the only kingdom that was not under Mauryan control. Kalinga was important because it controlled the land and sea routes to south India and south-east Asia. In 261 BC, Ashoka attacked Kalinga and conquered it after a fierce battle.
What was Dhamma?
Dhamma is a Prakrit word, which is derived from the Sanskrit word ‘dharma’ meaning religious duty. Dhamma did not involve worship of Gods or the performance of sacrifices. Instead, Dhamma was a code of conduct and morals such as charity, kindness, benevolence and tolerance, to be followed.
What do you know about the central administration of the Mauryas?
Mauryan administration can be divided into four divisions: central, provincial, district and village.
At the central level, the king was the supreme authority. He took all the important decisions of the empire. In this task, he was aided and advised by a council of ministers.
When Ashoka became the king, Kalinga was the only kingdom that was not under Mauryan control. Ashoka attacked Kalinga and conquered it after fighting a tough war.
But this battle proved to be a turning point in Ashoka’s life. He was saddened by the death and suffering caused by the war. He realised that war was a futile affair that only led to death and sadness.
After the war, Ashoka gave up his policy of conquest through war (dig-vijaya) and began to follow a policy of conquest through dharma (dharma-vijaya). The spread of dharma became the goal of Ashoka’s life.
What were the welfare measures adopted by Ashoka?
For Ashoka, the citizen were just like his own children. The well-being of his children was his responsibility. He took a number of measures to promote welfare of his citizen. Some of these are as follows:
Good roads were built and trees were planted on both sides of the roads.
Rest-houses were constructed for travellers, along the roads.
A large number of wells were dug.
Hospitals, for both people and animals, were constructed.
Write any two steps taken by Ashoka to spread Dhamma.
Dhamma means religious duty. It was Ashoka's desire that his citizen should understand the concept of rightful living and practise it to the fullest. In order to spread dhamma, he took the following steps:
a. Edicts containing the principles of dhamma were issued by Ashoka. These edicts were engraved on rocks and pillars and were placed throughout the kingdom at public places like markets and temples.
b. Ashoka appointed officials known as dharma mahamatras to spread dhamma. These officials went from place to place and propagated the message of dhamma. Some even went outside the country to places like Sri Lanka, Myanmar, South-East and central Asia, etc.
Write short notes on – Administration of Pataliputra, Sources of revenue.
1) The administration of Patliputra, the capital of the Mauryan Empire, was carried out through a 30-member committee. This committee was divided into six boards, which catered to specific departments of
a. comfort and safety of foreigners
b. registration of births and deaths
e. inspection of goods
f. collection of taxes
2) The most important source of income was the land revenue. Usually it was fixed at one-fourth to one-sixth of the produce. However, this revenue was fixed according to the fertility of the soil. Apart from land tax, trade was also an important source of revenue. The Mauryans flourished due to the practise of both inland and overseas trade. Mines, custom duties, gifts and water tax were also significant revenue sources.
It is said that Ashoka was the first king who spoke directly to his people. How did he do this
Ashoka was the first king to speak directly to his people. He achieved this through his edicts. These edicts contained the various principles of dhamma that he wanted his people to learn and apply in their lives.
Why do you think the Mauryan kings employed spies?
Every kingdom faces the danger of revolts from the opponents. This could have been possible during the Mauryan times too, where the kingdom was so far-stretched. To control irrational dissent, first hand information of such plotting was essential. Thus, there was a need of spies.
|
Biological half-life - wikidoc
The biological half-life of a substance is the time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacologic, physiologic, or radiologic activity, as per the MeSH definition.
2.1 Zero-order elimination
The biological half-life of water in a human is about 7 to 10 days. It can be altered by behavior. Drinking large amounts of alcohol will reduce the biological half-life of water in the body. This has been used to decontaminate humans who are internally contaminated with tritiated water (tritium). Drinking the same amount of water would have a similar effect, but many would find it difficult to drink a large volume of water. The basis of this decontamination method (used at Harwell) is to increase the rate at which the water in the body is replaced with new water.
The removal of ethanol (alcohol) through oxidation by alcohol dehydrogenase in the liver from the human body is limited. Hence the removal of a large concentration of alcohol from blood may follow zero-order kinetics. Also the rate-limiting steps for one substance may be in common with other substances. For instance, the blood alcohol concentration can be used to modify the biochemistry of methanol and ethylene glycol. In this way the oxidation of methanol to the toxic formaldehyde and formic acid in the human body can be prevented by giving an appropriate amount of ethanol to a person who has ingested methanol. Note that methanol is very toxic and causes blindness and death. A person who has ingested ethylene glycol can be treated in the same way.
Cisplatin 20 to 30 minutes
Chlorambucil 1.53 hours
Digoxin 24 to 36 hours
Methadone 15 to 60 hours, in rare cases up to 190 hours.[2]
Salbutamol 7 hours
For some substances, it is important to think of the human or animal body as being made up of several parts, each with their own affinity for the substance, and each part with a different biological half-life. Attempts to remove a substance from the whole organism may have the effect of increasing the burden present in one part of the organism. For instance, if a person who is contaminated with lead is given EDTA in a chelation therapy, then while the rate at which lead is lost from the body will be increased, the lead within the body tends to relocate into the brain where it can do the most harm.
There are circumstances where the half-life varies with the concentration of the drug. For example, ethanol may be consumed in sufficient quantity to saturate the metabolic enzymes in the liver, and so is eliminated from the body at an approximately constant rate (zero-order elimination). Thus the half-life, under these circumstances, is proportional to the initial concentration of the drug A0 and inversely proportional to the zero-order rate constant k0 where:
{\displaystyle t_{1/2}={\frac {0.5A_{0}}{k_{0}}}\,}
This process is usually a logarithmic process - that is, a constant proportion of the agent is eliminated per unit time (Birkett, 2002). Thus the fall in plasma concentration after the administration of a single dose is described by the following equation:
{\displaystyle C_{t}=C_{0}e^{-kt}\,}
{\displaystyle k={\frac {\ln 2}{t_{1/2}}}\,}
{\displaystyle t_{1/2}={\frac {{\ln 2}.{V_{D}}}{CL}}\,}
In clinical practice, this means that it takes just over 4.7 times the half-life for a drug's serum concentration to reach steady state after regular dosing is started, stopped, or the dose changed. So, for example, digoxin has a half-life (or t½) of 24-36 hours; this means that a change in the dose will take the best part of a week to take full effect. For this reason, drugs with a long half-life (e.g. amiodarone, elimination t½ of about 90 days) are usually started with a loading dose to achieve their desired clinical effect more quickly.
Birkett DJ (2002). Pharmacokinetics Made Easy (Revised Edition). Sydney: McGraw-Hill Australia. ISBN 0-07-471072-9.
↑ Template:GoldBookRef
↑ Manfredonia, John (March 2005). "Prescribing Methadone for Pain Management in End-of-Life Care". JAOA—The Journal of the American Osteopathic Association. Retrieved 2007-01-29.
↑ Ehrsson, Hans; et al. (Winter 2002). "Pharmacokinetics of oxaliplatin in humans". Medical Oncology. Retrieved 2007-03-28. CS1 maint: Explicit use of et al. (link)
Small Molecule Drug Half-life prediction software
de:Plasmahalbwertszeit gl:Semivida (medicina) it:Emivita (farmacologia) fi:Biologinen puoliintumisaika
Retrieved from "https://www.wikidoc.org/index.php?title=Biological_half-life&oldid=669835"
|
Blood urea nitrogen - wikidoc
WikiDoc Resources for Blood urea nitrogen
Most recent articles on Blood urea nitrogen
Most cited articles on Blood urea nitrogen
Review articles on Blood urea nitrogen
Articles on Blood urea nitrogen in N Eng J Med, Lancet, BMJ
Powerpoint slides on Blood urea nitrogen
Images of Blood urea nitrogen
Photos of Blood urea nitrogen
Podcasts & MP3s on Blood urea nitrogen
Videos on Blood urea nitrogen
Cochrane Collaboration on Blood urea nitrogen
Bandolier on Blood urea nitrogen
TRIP on Blood urea nitrogen
Ongoing Trials on Blood urea nitrogen at Clinical Trials.gov
Trial results on Blood urea nitrogen
Clinical Trials on Blood urea nitrogen at Google
US National Guidelines Clearinghouse on Blood urea nitrogen
NICE Guidance on Blood urea nitrogen
FDA on Blood urea nitrogen
CDC on Blood urea nitrogen
Books on Blood urea nitrogen
Blood urea nitrogen in the news
Be alerted to news on Blood urea nitrogen
News trends on Blood urea nitrogen
Blogs on Blood urea nitrogen
Definitions of Blood urea nitrogen
Patient resources on Blood urea nitrogen
Discussion groups on Blood urea nitrogen
Patient Handouts on Blood urea nitrogen
Directions to Hospitals Treating Blood urea nitrogen
Risk calculators and risk factors for Blood urea nitrogen
Symptoms of Blood urea nitrogen
Causes & Risk Factors for Blood urea nitrogen
Diagnostic studies for Blood urea nitrogen
Treatment of Blood urea nitrogen
CME Programs on Blood urea nitrogen
Blood urea nitrogen en Espanol
Blood urea nitrogen en Francais
Blood urea nitrogen in the Marketplace
Patents on Blood urea nitrogen
List of terms related to Blood urea nitrogen
The blood urea nitrogen (BUN) test is a measure of the amount of nitrogen in the blood that comes from urea. Urea is a substance secreted by the liver, and removed from the blood by the kidneys.
The liver produces urea in the urea cycle as a waste product of the digestion of protein. Normal human adult blood should contain between 7 and 25 mg of urea nitrogen per 100 ml (7-25 mg/dL) of blood. Individual laboratories may have different reference ranges, and this is because the procedure may vary.
The most common cause of an elevated BUN, azotemia, is poor kidney function, although a serum creatinine level is a somewhat more specific measure of renal function.
A greatly elevated BUN (>60 mg/dl) generally indicates a moderate-to-severe degree of renal failure. Impaired renal excretion of urea may be due to temporary conditions such as dehydration or shock, or may be due to either acute or chronic disease of the kidneys themselves.
Elevated BUN in the setting of a relatively normal creatinine may reflect a physiological response to a relative decrease of blood flow to the kidney (as seen in heart failure or dehydration) without indicating any true injury to the kidney. However, an isolated elevation of BUN may also reflect excessive formation of urea without any compromise to the kidneys.
Increased production of urea is seen in cases of moderate or heavy bleeding in the upper gastrointestinal tract (e.g. from ulcers). The nitrogenous compounds from the blood are resorbed as they pass through the rest of the GI tract and then broken down to urea by the liver. Enhanced metabolism of proteins will also increase urea production, as may be seen with high protein diets, steroid use, burns, or fevers.
A low BUN usually has little significance, but its causes include liver problems, malnutrition (insufficient dietary protein), or excessive alcohol consumption. Overhydration from intravenous fluids can result in a low BUN. Normal changes in renal bloodflow during pregnancy will also lower BUN.
Urea itself is not toxic. This was demonstrated by Johnson et al. by adding large amounts of urea to the dialysate of hemodialysis patients for several months and finding no ill effects.[2]. However, BUN is a marker for other nitrogenous waste. Thus, when renal failure leads to a buildup of urea and other nitrogenous wastes (uremia), an individual may suffer neurological disturbances such as altered cognitive function (encephalopathy), impaired taste (dysgeusia) or loss of appetite (anorexia). The individual may also suffer from nausea and vomiting, or bleeding from dysfunctional platelets. Prolonged periods of severe uremia may result in the skin taking on a grey discolouration or even forming frank urea crystals ("uremic frost") on the skin.
Because multiple variables can interfere with the interpretation of a BUN value, GFR and creatinine clearance are more accurate markers of kidney function. Age, sex, and weight will alter the "normal" range for each individual, including race. In renal failure or chronic kidney disease (CKD), BUN will only be elevated outside "normal" when more than 60% of kidney cells are no longer functioning. Hence, more accurate measures of renal function are generally preferred to assess the clearance for purposes of medication dosing.
BUN is reported as mg/dL in the United States. Elsewhere, the concentration of urea is reported as mmol/L. To convert from mg/dL of blood urea nitrogen to mmol/L of urea, one multiplies by 0.357
(10 dL/1 L)/(28 mg of nitrogen/mmol of urea) = 0.357
The test as originally carried out was by flame photometry; now chemical colorimetric tests are more widely used. Three methods are shown below: Diacetyl Monoxime, Urograph and Modified Berthelot Enzymatic methods:
DIACETYL MONOXIME Method
This method utilizes the Fearon Reaction, wherein urea in hot solution of diacetyl monoxime condenses to form diazine derivative.
Reagents to use
BUN Std (usually 15 mg/dl);
Serum or Plasma (Patient's Sample);
Acid Reagent (Diacetyl monoxime);
Color Reagent (Ferrithiocyanate).
Boiling water bath;
Iced cold water bath;
Cuvet;
Prepare 3 to 10 mL test tubes
Blank - 0.05 mL Distilled water
Standard - 0.05 mL Std Solution
Test - 0.05 mL Serum or Plasma
To all tubes add 3.0 mL of Acid Reagent and mix by lateral tapping;
Then add Color reagent and mix well again;
Place all tubes into an already boiling water bath for 15 minutes;
Cool in iced water bath for 2 minutes;
Let stand at room temperature for 1 minute;
Read against Blank at 520 nm in Spectrophotometer
{\displaystyle (OpticalDensityofTest/OpticalDensityofStd)*(15mg/dlasConcentrationofStd)*(0.357)=BUNinmmol/L}
0.357 is the Conversion Factor for BUN to mmol/L
2.86 to 7.14 mmol/L (or 8 to 20 mg/dL)
UROGRAPH method
This method utilizes Physical and Chemical means via Paper Chromatography and Conway Microdiffusion respectively. Conway Microdiffusion involves the hydrolysis of Urea with buffered Urease, the liberation of Ammonia gas and reaction with indicator system.
Urastrat strip;
Test tubes w/ Test Tube Rack;
Ruler or Caliper.
Place 0.2 mL of Serum or Plasma at the bottom of a 5.0 mL test tube;
Place Urastrat strip in the test tube with the lower portion touching the serum;
Place the tube in a rack in straight position and let stand for 30 minutes;
Read the blue colored band produced using the millimeter graduation of a ruler or a caliper.
{\displaystyle (Heightofbluebandinmm)*5+10=(BUNinmg/dL)*(0.357)=BUNinmmol/L}
Modified Berthelot enzymatic method
This method utilizes the principle of Urea Hydrolysis. Urea is hydrolyzed in the presence of Urease to Carbon Dioxide and Ammonium Ions.
Patient's Serum;
Urea N Standard (25 mg/dL);
Urea N-Zyme reagent (Buffered Urease);
Urea N Color reagent (Sodium Salicylates and Sodium Nitroferricyanide);
Urea N Base reagent (NaOH and Sodium Hypochlorite).
37°C water bath.
Prepare 3 to 10 mL test tubes labeled as Blank, Test and Standard;
Place 0.5 mL Urea N Color reagent to all tubes and add sample as follows:
Blank - 0.5 mL Distilled water
Standard - 0.5 mL Std Solution
Test - 0.5 mL Serum or Plasma
Mix by gentle swirling and pre-warm at 37°C water bath for 5 minutes;
Add 2.0 mL Urea N Base Reagent to all tubes, mix and return to 37°C water bath for 5 minutes;
Read at 630 nm against Reagent Blank in Spectrophotometer.
{\displaystyle (OpticalDensityofTest/OpticalDensityofStd)*(15mg/dLasConcentrationofStd)=BUNinmg/dL*(0.357)=BUNinmmol/L}
^ Johnson WJ, Hagge WW, Wagoner RD, Dinapoli RP, Rosevear JW. Effects of urea loading in patients with far-advanced renal failure. Mayo Clin Proc. 1972 Jan;47(1):21-9. PMID 5008253
de:Blood urea nitrogen it:Azoto ureico
Retrieved from "https://www.wikidoc.org/index.php?title=Blood_urea_nitrogen&oldid=1045649"
This page was last edited 15:21, 8 December 2014 by wikidoc user Kiran Singh. Based on work by wikidoc user WikiBot.
|
find the distance between two points, or a point and a line
distance(P1, l)
The routine computes the distance between two points P1 and P2 or between a point P1 and a line l.
The command with(geometry,distance) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{geometry}\right):
\mathrm{point}\left(A,a,b\right),\mathrm{point}\left(B,c,d\right):
\mathrm{distance}\left(A,B\right)
\sqrt{{\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{c}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\left(\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{d}\right)}^{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{assume}\left(m\ne 0\right):
\mathrm{line}\left(l,mx+ny=w,[x,y]\right):
\mathrm{distance}\left(A,l\right)
\frac{|\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{m~}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{w}|}{\sqrt{{\textcolor[rgb]{0,0,1}{\mathrm{m~}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}}}
\mathrm{distance}\left(B,l\right)
\frac{|\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{m~}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{d}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{n}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{w}|}{\sqrt{{\textcolor[rgb]{0,0,1}{\mathrm{m~}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{n}}^{\textcolor[rgb]{0,0,1}{2}}}}
|
In figure, a particle is placed at the highest point a of a smo-Turito
In figure, a particle is placed at the highest point of a smooth sphere of radius . It is given slight push, and it leaves the sphere at , at a depth vertically below such that is equal to
Answer:The correct answer is: If velocity acquired at , then
The particle will leave the sphere at , when
h
h
A
B
t,
AB
A
{v}_{H}=u\mathrm{cos}60°=\frac{u}{2} \therefore AC={u}_{H}×t=\frac{ut}{2}
AB=AC\mathrm{sec}30°=\frac{ut}{2}×\frac{2}{\sqrt{3}}=\frac{ut}{2}
A
B
t,
AB
A
{v}_{H}=u\mathrm{cos}60°=\frac{u}{2} \therefore AC={u}_{H}×t=\frac{ut}{2}
AB=AC\mathrm{sec}30°=\frac{ut}{2}×\frac{2}{\sqrt{3}}=\frac{ut}{2}
{\mathrm{Lt}}_{x\to 0}\frac{{e}^{\mathrm{tan} x}-{e}^{x}}{\mathrm{tan} x-x}
{\mathrm{Lt}}_{x\to 0}\frac{{e}^{\mathrm{tan} x}-{e}^{x}}{\mathrm{tan} x-x}
\stackrel{^}{k}
\stackrel{^}{k}
{\mathrm{Lt}}_{x\to 0} \frac{x\cdot {2}^{x}-x}{1-\mathrm{cos} x}
{\mathrm{Lt}}_{x\to 0} \frac{x\cdot {2}^{x}-x}{1-\mathrm{cos} x}
{\mathrm{Lt}}_{x\to 3}\frac{{x}^{3}-8{x}^{2}+45}{2{x}^{2}-3x-9}
{\mathrm{Lt}}_{x\to 3}\frac{{x}^{3}-8{x}^{2}+45}{2{x}^{2}-3x-9}
C-O-C
C-O-C
\stackrel{\to }{B}=2T\text{ in }X-Y
\stackrel{\to }{B}=2T\text{ in }X-Y
L{t}_{x\to 0}\frac{\sqrt{\left(1+x+{x}^{2}\right)}-1}{x}
L{t}_{x\to 0}\frac{\sqrt{\left(1+x+{x}^{2}\right)}-1}{x}
l
l
W=T
cos\theta +sin\theta
<T
P+Q=T
cos\theta +sin\theta
<T
W=T
cos\theta +sin\theta
<T
P+Q=T
cos\theta +sin\theta
<T
|
Solid axle suspension with coil spring - Simulink - MathWorks Nordic
Solid Axle Suspension - Coil Spring
Solid axle suspension with coil spring
The Solid Axle Suspension - Coil Spring block implements a solid axle suspension with a coil spring for multiple axles with multiple tracks per axle.
\begin{array}{l}\left[\begin{array}{c}{\stackrel{¨}{x}}_{a}\\ {\stackrel{¨}{y}}_{a}\\ {\stackrel{¨}{z}}_{a}\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}{F}_{xa}\\ {F}_{ya}\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}{\stackrel{˙}{x}}_{a}\\ {\stackrel{˙}{y}}_{a}\\ {\stackrel{˙}{z}}_{a}\end{array}\right]×\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\frac{1}{{M}_{a}}\left[\begin{array}{c}0\\ 0\\ {F}_{za}\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ {\stackrel{˙}{z}}_{a}\end{array}\right]×\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ g\end{array}\right]=\left[\begin{array}{c}0\\ p{\stackrel{˙}{z}}_{a}\\ \frac{{F}_{za}}{{M}_{a}}+g\end{array}\right]\\ \\ \left[\begin{array}{c}\stackrel{˙}{p}\\ \stackrel{˙}{q}\\ \stackrel{˙}{r}\end{array}\right]=\left[\left[\begin{array}{c}{M}_{x}\\ {M}_{y}\\ {M}_{z}\end{array}\right]-\left[\begin{array}{c}p\\ q\\ r\end{array}\right]×\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{-1}\\ =\left[\left[\begin{array}{c}{M}_{x}\\ 0\\ 0\end{array}\right]-\left[\begin{array}{c}p\\ q\\ 0\end{array}\right]×\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]\left[\begin{array}{c}p\\ 0\\ 0\end{array}\right]\right]{\left[\begin{array}{ccc}{I}_{xx}& 0& 0\\ 0& {I}_{yy}& 0\\ 0& 0& {I}_{zz}\end{array}\right]}^{-1}=\left[\begin{array}{c}\frac{{M}_{x}}{{I}_{xx}}\\ 0\\ 0\end{array}\right]\end{array}
{F}_{za}=\sum _{t=1}^{Nta}\left({F}_{w{z}_{a,t}}+{F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{s}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{c}_{{z}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)\right)
{M}_{x}=\sum _{t=1}^{Nta}\left({F}_{w{z}_{a,t}}{y}_{{w}_{t}}+\left({F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{s}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{c}_{{z}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)\right){y}_{{s}_{t}}+{M}_{w{x}_{a,t}}\frac{{I}_{xx}}{{I}_{xx}+{M}_{a}{y}_{{w}_{t}}}\right)
\begin{array}{l} T{c}_{t}=\left[\begin{array}{ccc}{x}_{{w}_{1}}& {x}_{{w}_{2}}& \dots \\ {y}_{{w}_{1}}& {y}_{{w}_{2}}& \dots \\ {z}_{{w}_{1}}& {z}_{{w}_{2}}& \dots \end{array}\right]\\ S{c}_{t}=\left[\begin{array}{ccc}{x}_{{s}_{1}}& {x}_{{s}_{2}}& \dots \\ {y}_{{s}_{1}}& {y}_{{s}_{2}}& \dots \\ {z}_{{s}_{1}}& {z}_{{s}_{2}}& \dots \end{array}\right]\end{array}
{F}_{v{z}_{a,t}=-}\left({F}_{z{0}_{a}}+{k}_{{z}_{a}}\left({z}_{{v}_{a,t}}-{z}_{{s}_{a,t}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{c}_{{z}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)+{F}_{zhsto{p}_{a,t}}\right)
\begin{array}{l}{F}_{v{x}_{a,t}}={F}_{w{x}_{a,t}}\\ {F}_{v{y}_{a,t}}={F}_{w{y}_{a,t}}\\ {F}_{v{z}_{a,t}}=-{F}_{w{z}_{a,t}}\\ \\ {M}_{v{x}_{a,t}}={M}_{w{x}_{a,t}}+{F}_{w{y}_{a,t}}\left(R{e}_{w{y}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{y}_{a,t}}={M}_{w{y}_{a,t}}+{F}_{w{x}_{a,t}}\left(R{e}_{w{x}_{a,t}}+{H}_{a,t}\right)\\ {M}_{v{z}_{a,t}}={M}_{w{z}_{a,t}}\end{array}
{F}_{w{z}_{a,t}}=-Fw{a}_{z0}-kw{a}_{z}\left({z}_{{w}_{a,t}}-{z}_{{s}_{a,t}}\right)-cw{a}_{z}\left({\stackrel{˙}{z}}_{{w}_{a,t}}-{\stackrel{˙}{z}}_{{s}_{a,t}}\right)
\begin{array}{l}{\xi }_{a,t}={\xi }_{0a}+{m}_{hcambe{r}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{camberstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\\ {\eta }_{a,t}={\eta }_{0a}+{m}_{hcaste{r}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{casterstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\\ {\zeta }_{a,t}={\zeta }_{0a}+{m}_{hto{e}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{toestee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\\ \end{array}
{\delta }_{whlstee{r}_{a,t}}={\delta }_{stee{r}_{a,t}}+{m}_{hto{e}_{a}}\left({z}_{{w}_{a,t}}-{z}_{{v}_{a,t}}-{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)+{m}_{toestee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|
{P}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{E}_{sus{p}_{a,t}}={F}_{wzlooku{p}_{a}}\left({\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\stackrel{˙}{z}}_{{v}_{a,t}}-{\stackrel{˙}{z}}_{{w}_{a,t}},{\delta }_{stee{r}_{a,t}}\right)
{H}_{a,t}=-\left({z}_{{v}_{a,t}}-{z}_{{w}_{a,t}}+\frac{{F}_{z{0}_{a}}}{{k}_{{z}_{a}}}+{m}_{hstee{r}_{a}}|{\delta }_{stee{r}_{a,t}}|\right)
{z}_{wt{r}_{a,t}}=R{e}_{{w}_{a,t}}+{H}_{a,t}
\mathrm{WhlPz}={z}_{w}=\left[\begin{array}{cccc}{z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
\mathrm{Whl}\mathrm{Re}=R{e}_{w}=\left[\begin{array}{cccc}R{e}_{{w}_{1,1}}& R{e}_{{w}_{1,2}}& R{e}_{{w}_{2,1}}& R{e}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlVz}={\stackrel{˙}{z}}_{w}=\left[\begin{array}{cccc}{\stackrel{˙}{z}}_{{w}_{1,1}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlFx}={F}_{wx}=\left[\begin{array}{cccc}{F}_{w{x}_{1,1}}& {F}_{w{x}_{1,2}}& {F}_{w{x}_{2,1}}& {F}_{w{x}_{2,2}}\end{array}\right]
\mathrm{WhlFy}={F}_{wy}=\left[\begin{array}{cccc}{F}_{w{y}_{1,1}}& {F}_{w{y}_{1,2}}& {F}_{w{y}_{2,1}}& {F}_{w{y}_{2,2}}\end{array}\right]
\mathrm{WhlM}={M}_{w}=\left[\begin{array}{cccc}{M}_{w{x}_{1,1}}& {M}_{w{x}_{1,2}}& {M}_{w{x}_{2,1}}& {M}_{w{x}_{2,2}}\\ {M}_{w{y}_{1,1}}& {M}_{w{y}_{1,2}}& {M}_{w{y}_{2,1}}& {M}_{w{y}_{2,2}}\\ {M}_{w{z}_{1,1}}& {M}_{w{z}_{1,2}}& {M}_{w{z}_{2,1}}& {M}_{w{z}_{2,2}}\end{array}\right]
\mathrm{VehP}=\left[\begin{array}{c}{x}_{v}\\ {y}_{v}\\ {z}_{v}\end{array}\right]=\left[\begin{array}{cccc}{x}_{v}{}_{{}_{1,1}}& {x}_{v}{}_{{}_{1,2}}& {x}_{v}{}_{{}_{2,1}}& {x}_{v}{}_{{}_{2,2}}\\ {y}_{v}{}_{{}_{1,1}}& {y}_{v}{}_{{}_{1,2}}& {y}_{v}{}_{{}_{2,1}}& {y}_{v}{}_{{}_{2,2}}\\ {z}_{v}{}_{{}_{1,1}}& {z}_{v}{}_{{}_{1,2}}& {z}_{v}{}_{{}_{2,1}}& {z}_{v}{}_{{}_{2,2}}\end{array}\right]
\mathrm{VehV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{v}\\ {\stackrel{˙}{y}}_{v}\\ {\stackrel{˙}{z}}_{v}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{v}_{1,1}}& {\stackrel{˙}{x}}_{{v}_{1,2}}& {\stackrel{˙}{x}}_{{v}_{2,1}}& {\stackrel{˙}{x}}_{{v}_{2,2}}\\ {\stackrel{˙}{y}}_{{v}_{1,1}}& {\stackrel{˙}{y}}_{{v}_{1,2}}& {\stackrel{˙}{y}}_{{v}_{2,1}}& {\stackrel{˙}{y}}_{{v}_{2,2}}\\ {\stackrel{˙}{z}}_{{v}_{1,1}}& {\stackrel{˙}{z}}_{{v}_{1,2}}& {\stackrel{˙}{z}}_{{v}_{2,1}}& {\stackrel{˙}{z}}_{{v}_{2,2}}\end{array}\right]
\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]
\mathrm{WhlAng}\left[1,...\right]=\xi =\left[{\xi }_{a,t}\right]
\mathrm{WhlAng}\left[2,...\right]=\eta =\left[{\eta }_{a,t}\right]
\mathrm{WhlAng}\left[3,...\right]=\zeta =\left[{\zeta }_{a,t}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlP}=\left[\begin{array}{c}{x}_{w}\\ {y}_{w}\\ {z}_{w}\end{array}\right]=\left[\begin{array}{cccc}{x}_{w}{}_{{}_{1,1}}& {x}_{w}{}_{{}_{1,2}}& {x}_{w}{}_{{}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{w}{}_{{}_{1,1}}& {y}_{w}{}_{{}_{1,2}}& {y}_{w}{}_{{}_{2,1}}& {y}_{w}{}_{{y}_{2,2}}\\ {z}_{wtr}{}_{{}_{1,1}}& {z}_{wtr}{}_{{}_{1,2}}& {z}_{wtr}{}_{{}_{2,1}}& {z}_{wt{r}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]
\mathrm{VehF}={F}_{v}=\left[\begin{array}{cccc}{F}_{v}{}_{{x}_{1,1}}& {F}_{v}{}_{{x}_{1,2}}& {F}_{v}{}_{{x}_{2,1}}& {F}_{v}{}_{{x}_{2,2}}\\ {F}_{v}{}_{{y}_{1,1}}& {F}_{v}{}_{{y}_{1,2}}& {F}_{v}{}_{{y}_{2,1}}& {F}_{v}{}_{{y}_{2,2}}\\ {F}_{v}{}_{{z}_{1,1}}& {F}_{v}{}_{{z}_{1,2}}& {F}_{v}{}_{{z}_{2,1}}& {F}_{v}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{VehM}={M}_{v}=\left[\begin{array}{cccc}{M}_{v{x}_{1,1}}& {M}_{v{x}_{1,2}}& {M}_{v{x}_{2,1}}& {M}_{v{x}_{2,2}}\\ {M}_{v{y}_{1,1}}& {M}_{v{y}_{1,2}}& {M}_{v{y}_{2,1}}& {M}_{v{y}_{2,2}}\\ {M}_{v{z}_{1,1}}& {M}_{v{z}_{1,2}}& {M}_{v{z}_{2,1}}& {M}_{v{z}_{2,2}}\end{array}\right]
\mathrm{WhlF}={F}_{w}=\left[\begin{array}{cccc}{F}_{w}{}_{{x}_{1,1}}& {F}_{w}{}_{{x}_{1,2}}& {F}_{w}{}_{{x}_{2,1}}& {F}_{w}{}_{{x}_{2,2}}\\ {F}_{w}{}_{{y}_{1,1}}& {F}_{w}{}_{{y}_{1,2}}& {F}_{w}{}_{{y}_{2,1}}& {F}_{w}{}_{{y}_{2,2}}\\ {F}_{w}{}_{{z}_{1,1}}& {F}_{w}{}_{{z}_{1,2}}& {F}_{w}{}_{{z}_{2,1}}& {F}_{w}{}_{{z}_{2,2}}\end{array}\right]
\mathrm{WhlV}=\left[\begin{array}{c}{\stackrel{˙}{x}}_{w}\\ {\stackrel{˙}{y}}_{w}\\ {\stackrel{˙}{z}}_{w}\end{array}\right]=\left[\begin{array}{cccc}{\stackrel{˙}{x}}_{{w}_{1,1}}& {\stackrel{˙}{x}}_{{w}_{1,2}}& {\stackrel{˙}{x}}_{{w}_{2,1}}& {\stackrel{˙}{x}}_{{w}_{2,2}}\\ {\stackrel{˙}{y}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{y}}_{{w}_{1,2}}& {\stackrel{˙}{y}}_{{w}_{2,1}}& {\stackrel{˙}{y}}_{{w}_{2,2}}\\ {\stackrel{˙}{z}}_{{w}_{{}_{1,1}}}& {\stackrel{˙}{z}}_{{w}_{1,2}}& {\stackrel{˙}{z}}_{{w}_{2,1}}& {\stackrel{˙}{z}}_{{w}_{2,2}}\end{array}\right]
\mathrm{WhlAng}=\left[\begin{array}{c}\xi \\ \eta \\ \zeta \end{array}\right]=\left[\begin{array}{cccc}{\xi }_{1,1}& {\xi }_{1,2}& {\xi }_{2,1}& {\xi }_{2,2}\\ {\eta }_{1,1}& {\eta }_{1,2}& {\eta }_{2,1}& {\eta }_{2,2}\\ {\zeta }_{1,1}& {\zeta }_{1,2}& {\zeta }_{2,1}& {\zeta }_{2,2}\end{array}\right]
\mathrm{StrgAng}={\delta }_{steer}=\left[\begin{array}{cc}{\delta }_{stee{r}_{1,1}}& {\delta }_{stee{r}_{1,2}}\end{array}\right]
T{c}_{t}=\left[\begin{array}{cccc}{x}_{{w}_{1,1}}& {x}_{{w}_{1,2}}& {x}_{{w}_{2,1}}& {x}_{{w}_{2,2}}\\ {y}_{{w}_{1,1}}& {y}_{{w}_{1,2}}& {y}_{{w}_{2,1}}& {y}_{{w}_{2,2}}\\ {z}_{{w}_{1,1}}& {z}_{{w}_{1,2}}& {z}_{{w}_{2,1}}& {z}_{{w}_{2,2}}\end{array}\right]
S{c}_{t}=\left[\begin{array}{cccc}{x}_{{s}_{1,1}}& {x}_{{s}_{1,2}}& {x}_{{s}_{2,1}}& {x}_{{s}_{2,2}}\\ {y}_{{s}_{1,1}}& {y}_{{s}_{1,2}}& {y}_{{s}_{2,1}}& {y}_{{s}_{2,2}}\\ {z}_{{s}_{1,1}}& {z}_{{s}_{1,2}}& {z}_{{s}_{2,1}}& {z}_{{s}_{2,2}}\end{array}\right]
Solid Axle Suspension | Solid Axle Suspension - Leaf Spring | Solid Axle Suspension - Mapped
|
Find the matrix associated with rotating counterclockwise through the given angle measure.
Think about the points
\left(1, 0\right)
\left(0, 1\right)
. If you rotated them through the given angle, where would they end up?
90°
\left(1, 0\right) → \left(0, 1\right)
\left(0, 1\right) → \left(−1, 0\right)
\left[ \begin{array} { c c } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right] \left[ \begin{array} { l l } { a } & { b } \\ { c } & { d } \end{array} \right] = \left[ \begin{array} { c c } { 0 } & { 1 } \\ { - 1 } & { 0 } \end{array} \right]
150°
\left. \begin{array} { l } { ( 1,0 ) \Rightarrow ( - \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } ) } \\ { ( 0,1 ) \Rightarrow ( - \frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } ) } \end{array} \right.
|
Kinect-Based Motion Recognition Tracking Robotic Arm Platform
Jilin University Instrument Science and Engineering Institute, Changchun, China
\{\begin{array}{l}{x}_{r}=\left({x}_{i}-\frac{w}{2}\right)\left({z}_{r}-10\right)P\times \left(\frac{w}{h}\right)\hfill \\ {y}_{\gamma }=\left({y}_{i}-\frac{h}{2}\right)\left({z}_{r}-10\right)P\hfill \\ {z}_{r}=11.48\times \mathrm{tan}\left(H{z}_{i}+1.18\right)-O\hfill \end{array}
{x}_{r},{y}_{r},{z}_{r}
{x}_{i},{y}_{i},{x}_{i}
{y}_{k}=\underset{J=m}{\overset{n}{\sum }}{p}_{i}{y}_{k+j}
k=m+1,m+2,\cdots ,N-n
{\sum }_{i=m}^{n}{p}_{i}=1
{y}_{1}-{y}_{20}
\stackrel{\to }{a},\stackrel{\to }{b},\stackrel{\to }{c}
\theta =〈\stackrel{\to }{a},\stackrel{\to }{b}〉
\beta =〈\stackrel{\to }{b},\stackrel{\to }{c}〉
\theta =\frac{\left[\stackrel{\to }{a}\cdot \stackrel{\to }{b}\right]}{|\stackrel{\to }{a}|\cdot |\stackrel{\to }{b}|}
{\theta }_{i-j}=\frac{{x}_{{i}_{1}}{x}_{{j}_{1}}+{x}_{{i}_{2}}{x}_{{j}_{2}}+{x}_{{i}_{3}}{x}_{{j}_{3}}}{\sqrt{{x}_{{i}_{1}}^{2}+{x}_{{i}_{2}}^{2}+{x}_{{i}_{3}}^{2}}+\sqrt{{x}_{{j}_{1}}^{2}+{x}_{{j}_{2}}^{2}+{x}_{{j}_{3}}^{2}}}
\left\{\begin{array}{l}{x}_{i}={x}_{{i}_{1}}x+{x}_{{i}_{2}}y+{x}_{{i}_{3}}z\hfill \\ {x}_{j}={x}_{{j}_{1}}x+{x}_{{j}_{2}}y+{x}_{{j}_{3}}z\hfill \end{array}
Gao, J.X., Chen, Y.N. and Li, F.H. (2019) Kinect-Based Motion Recognition Tracking Robotic Arm Platform. Intelligent Control and Automation, 10, 79-89 https://doi.org/10.4236/ica.2019.103005
1. Matsen, F., Lauder, A., Rector, K., Keeling, P. and Cherones, A.L. (2016) Measurement of Active Shoulder Motion Using the Kinect, a Commercially Available Infrared Position Detection System. Journal of Shoulder and Elbow Surgery, 25, 216-223. https://doi.org/10.1016/j.jse.2015.07.011
2. Wang, P., Liu, H.Y., Wang, L.H. and Gao, R.X. (2018) Deep Learning-Based Human Motion Recognition for Predictive Context-Aware Human-Robot Collaboration. CIRP Annals—Manufacturing Technology, 67, 17-20. https://doi.org/10.1016/j.cirp.2018.04.066
3. Kumar, S.H. and Sivaprakash, P. (2013) New Approach for Action Recognition Using Motion Based Features. 2013 IEEE Conference on Information & Communication Technologies, Thuckalay, Tamil Nadu, India, 11-12 April 2013, Article No. 13653652. https://doi.org/10.1109/CICT.2013.6558292
4. Zhao, W.B. (2016) A Concise Tutorial on Human Motion Tracking and Recognition with Microsoft Kinect. Science China (Information Sciences), 59, 237-241. https://doi.org/10.1007/s11432-016-5604-y
5. Ning, W.J. (2017) Research and Application of Sports Demonstration Teaching System Based on Kinect. Proceedings of the 6th International Conference on Social Science, Education and Humanities Research (SSEHR 2017), Jinan, China, 18-19 October 2017, 570-574. https://doi.org/10.2991/ssehr-17.2018.126
6. Xiang, C.K., Hsu, H.H., Hwang, W.Y. and Ma, J.H. (2014) Comparing Real-Time Human Motion Capture System Using Inertial Sensors with Microsoft Kinect. Ubi-Media Computing and Workshops (UMEDIA), 7th International Conference, Ulaanbaatar, Mongolia, 12-14 July 2014, 53-58. https://doi.org/10.1109/U-MEDIA.2014.25
7. Faria, D.R., Premebida, C. and Nunes, U. (2014) A Probabilistic Approach for Human Everyday Activities Recognition Using Body Motion from RGB-D Images. The 23rd IEEE International Symposium on Robot and Human Interactive Communication, Edinburgh, UK, 25-29 August 2014, 732-737. https://doi.org/10.1109/ROMAN.2014.6926340
8. Tseng, Y.-W., Li, C.-M., Lee, A. and Wang, G.-C. (2013) Intelligent Robot Motion Control System Part I: System Overview and Image Recognition. 2013 International Symposium on Next-Generation Electronics, Kaohsiung, Taiwan, 25-26 February 2013, 1-5.
9. Vantigodi, S. and Radhakrishnan, V.B. (2014) Action Recognition from Motion Capture Data Using Meta-Cognitive RBF Network Classifier. Intelligent Sensors, Sensor Networks and Information Processing (ISSNIP), 2014 IEEE Ninth International Conference, Singapore, 21-24 April 2014, 1-6. https://doi.org/10.1109/ISSNIP.2014.6827664
|
Archimedes' principle - Wikipedia
This article is about fluid dynamics. For the algebraic axiom, see Archimedean property.
Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially, is equal to the weight of the fluid that the body displaces.[1] Archimedes' principle is a law of physics fundamental to fluid mechanics. It was formulated by Archimedes of Syracuse.[2]
5 Principle of flotation
In On Floating Bodies, Archimedes suggested that (c. 246 BC):
Archimedes' principle allows the buoyancy of any floating object partially or fully immersed in a fluid to be calculated. The downward force on the object is simply its weight. The upward, or buoyant, force on the object is that stated by Archimedes' principle, above. Thus, the net force on the object is the difference between the magnitudes of the buoyant force and its weight. If this net force is positive, the object rises; if negative, the object sinks; and if zero, the object is neutrally buoyant—that is, it remains in place without either rising or sinking. In simple words, Archimedes' principle states that, when a body is partially or completely immersed in a fluid, it experiences an apparent loss in weight that is equal to the weight of the fluid displaced by the immersed part of the body(s).
A floating object's weight Fp and its buoyancy Fa (Fb in the text) must be equal in size.
Consider a cuboid immersed in a fluid, its top and bottom faces orthogonal to the direction of gravity (assumed constant across the cube's stretch). The fluid will exert a normal force on each face, but only the normal forces on top and bottom will contribute to buoyancy. The pressure difference between the bottom and the top face is directly proportional to the height (difference in depth of submersion). Multiplying the pressure difference by the area of a face gives a net force on the cuboid — the buoyancy — equaling in size the weight of the fluid displaced by the cuboid. By summing up sufficiently many arbitrarily small cuboids this reasoning may be extended to irregular shapes, and so, whatever the shape of the submerged body, the buoyant force is equal to the weight of the displaced fluid.
{\displaystyle {\text{ weight of displaced fluid}}={\text{weight of object in vacuum}}-{\text{weight of object in fluid}}\,}
The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). The weight of the object in the fluid is reduced, because of the force acting on it, which is called upthrust. In simple terms, the principle states that the buoyant force (Fb) on an object is equal to the weight of the fluid displaced by the object, or the density (ρ) of the fluid multiplied by the submerged volume (V) times the gravity (g)[1][3]
We can express this relation in the equation:
{\displaystyle F_{a}=\rho gV}
{\displaystyle F_{a}}
denotes the buoyant force applied onto the submerged object,
{\displaystyle \rho }
denotes the density of the fluid,
{\displaystyle V}
represents the volume of the displaced fluid and
{\displaystyle g}
is the acceleration due to gravity. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy.
Suppose a rock's weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting on it. Suppose that, when the rock is lowered into the water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea-floor. It is generally easier to lift an object through the water than it is to pull it out of the water.
For a fully submerged object, Archimedes' principle can be reformulated as follows:
{\displaystyle {\text{apparent immersed weight}}={\text{weight of object}}-{\text{weight of displaced fluid}}\,}
{\displaystyle {\frac {\text{density of object}}{\text{density of fluid}}}={\frac {\text{weight}}{\text{weight of displaced fluid}}}}
yields the formula below. The density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volume is
{\displaystyle {\frac {\text{density of object}}{\text{density of fluid}}}={\frac {\text{weight}}{{\text{weight}}-{\text{apparent immersed weight}}}}.\,}
Example: A helium balloon in a moving car. When increasing speed or driving in a curve, the air moves in the opposite direction to the car's acceleration. However, due to buoyancy, the balloon is pushed "out of the way" by the air and will drift in the same direction as the car's acceleration.
When an object is immersed in a liquid, the liquid exerts an upward force, which is known as the buoyant force, that is proportional to the weight of the displaced liquid. The sum force acting on the object, then, is equal to the difference between the weight of the object ('down' force) and the weight of displaced liquid ('up' force). Equilibrium, or neutral buoyancy, is achieved when these two weights (and thus forces) are equal.
Forces and equilibriumEdit
{\displaystyle \mathbf {f} +\operatorname {div} \,\sigma =0}
{\displaystyle \sigma _{ij}=-p\delta _{ij}.\,}
{\displaystyle \mathbf {f} =\nabla p.\,}
{\displaystyle \mathbf {f} =-\nabla \Phi .\,}
{\displaystyle \nabla (p+\Phi )=0\Longrightarrow p+\Phi ={\text{constant}}.\,}
{\displaystyle p=\rho _{f}gz.\,}
{\displaystyle \mathbf {B} =\oint \sigma \,d\mathbf {A} .}
{\displaystyle \mathbf {B} =\int \operatorname {div} \sigma \,dV=-\int \mathbf {f} \,dV=-\rho _{f}\mathbf {g} \int \,dV=-\rho _{f}\mathbf {g} V}
{\displaystyle B=\rho _{f}V_{\text{disp}}\,g,\,}
{\displaystyle B=\rho _{f}Vg.\,}
{\displaystyle F_{\text{net}}=0=mg-\rho _{f}V_{\text{disp}}g\,}
{\displaystyle mg=\rho _{f}V_{\text{disp}}g,\,}
{\displaystyle m=\rho _{f}V_{\text{disp}}.\,}
{\displaystyle T=\rho _{f}Vg-mg.\,}
{\displaystyle N=mg-\rho _{f}Vg.\,}
Buoyancy force = weight of object in empty space − weight of object immersed in fluid
Simplified modelEdit
Archimedes' principle does not consider the surface tension (capillarity) acting on the body.[4] Moreover, Archimedes' principle has been found to break down in complex fluids.[5]
There is an exception to Archimedes' principle known as the bottom (or side) case. This occurs when a side of the object is touching the bottom (or side) of the vessel it is submerged in, and no liquid seeps in along that side. In this case, the net force has been found to be different from Archimedes' principle, owing to the fact that since no fluid seeps in on that side, the symmetry of pressure is broken.[6]
Principle of flotationEdit
Archimedes' principle shows the buoyant force and displacement of fluid. However, the concept of Archimedes' principle can be applied when considering why objects float. Proposition 5 of Archimedes' treatise On Floating Bodies states that
— Archimedes of Syracuse[7]
In other words, for an object floating on a liquid surface (like a boat) or floating submerged in a fluid (like a submarine in water or dirigible in air) the weight of the displaced liquid equals the weight of the object. Thus, only in the special case of floating does the buoyant force acting on an object equal the objects weight. Consider a 1-ton block of solid iron. As iron is nearly eight times as dense as water, it displaces only 1/8 ton of water when submerged, which is not enough to keep it afloat. Suppose the same iron block is reshaped into a bowl. It still weighs 1 ton, but when it is put in water, it displaces a greater volume of water than when it was a block. The deeper the iron bowl is immersed, the more water it displaces, and the greater the buoyant force acting on it. When the buoyant force equals 1 ton, it will sink no farther.
When any boat displaces a weight of water equal to its own weight, it floats. This is often called the "principle of flotation": A floating object displaces a weight of fluid equal to its own weight. Every ship, submarine, and dirigible must be designed to displace a weight of fluid at least equal to its own weight. A 10,000-ton ship's hull must be built wide enough, long enough and deep enough to displace 10,000 tons of water and still have some hull above the water to prevent it from sinking. It needs extra hull to fight waves that would otherwise fill it and, by increasing its mass, cause it to submerge. The same is true for vessels in air: a dirigible that weighs 100 tons needs to displace 100 tons of air. If it displaces more, it rises; if it displaces less, it falls. If the dirigible displaces exactly its weight, it hovers at a constant altitude.
While they are related to it, the principle of flotation and the concept that a submerged object displaces a volume of fluid equal to its own volume are not Archimedes' principle. Archimedes' principle, as stated above, equates the buoyant force to the weight of the fluid displaced.
One common point of confusion[by whom?] regarding Archimedes' principle is the meaning of displaced volume. Common demonstrations involve measuring the rise in water level when an object floats on the surface in order to calculate the displaced water. This measurement approach fails with a buoyant submerged object because the rise in the water level is directly related to the volume of the object and not the mass (except if the effective density of the object equals exactly the fluid density).[8][9][10]
EurekaEdit
Main article: Eureka (word)
Archimedes reportedly exclaimed "Eureka" after he realized how to detect whether a crown is made of impure gold. While he did not use Archimedes' principle in the widespread tale and used displaced water only for measuring the volume of the crown, there is an alternative approach using the principle: Balance the crown and pure gold on a scale in the air and then put the scale into water. According to Archimedes' principle, if the density of the crown differs from the density of pure gold, the scale will get out of balance under water.[11][12]
^ a b "What is buoyant force?". Khan Academy.
^ Acott, Chris (1999). "The diving "Law-ers": A brief resume of their lives". South Pacific Underwater Medicine Society Journal. 29 (1). ISSN 0813-1988. OCLC 16986801. Archived from the original on 2011-07-27. Retrieved 2009-06-13.
^ http://physics.bu.edu/~duffy/sc527_notes01/buoyant.html[bare URL]
^ "Floater clustering in a standing wave: Capillarity effects drive hydrophilic or hydrophobic particles to congregate at specific points on a wave" (PDF). 2005-06-23.
^ "Archimedes's principle gets updated". R. Mark Wilson, Physics Today 65(9), 15 (2012); doi:10.1063/PT.3.1701
^ Lima, F M S. (2012). "Using surface integrals for checking the Archimedes' law of buoyancy". European Journal of Physics. 33 (1): 101–113. arXiv:1110.5264. Bibcode:2012EJPh...33..101L. doi:10.1088/0143-0807/33/1/009. S2CID 54556860.
^ "The works of Archimedes". Cambridge, University Press. 1897. p. 257. Retrieved 11 March 2010. Any solid lighter than a fluid will, if placed in the fluid, be so far immersed that the weight of the solid will be equal to the weight of the fluid displaced.
^ Mohindroo, K. K. (1997). Basic Principles of Physics. Pitambar Publishing. pp. 76–77. ISBN 978-81-209-0199-5.
^ Redish, Edward F.; Vicentini, Matilde; fisica, Società italiana di (2004). Research on Physics Education. IOS Press. p. 358. ISBN 978-1-58603-425-2.
^ Proof of Concept carpeastra.co.uk
^ "The Golden Crown". physics.weber.edu.
^ "'Eureka!' – The Story of Archimedes and the Golden Crown". Long Long Time Ago. 16 May 2014.
Media related to Archimedes' principle at Wikimedia Commons
Retrieved from "https://en.wikipedia.org/w/index.php?title=Archimedes%27_principle&oldid=1085512448"
|
Error, invalid input: Algebraic:-Resultant expects its 3rd argument, x, to be of type Or(name,And(function,Not(AlgebraicObject))), but received sin(1/7*Pi) - Maple Help
Home : Support : Online Help : Error, invalid input: Algebraic:-Resultant expects its 3rd argument, x, to be of type Or(name,And(function,Not(AlgebraicObject))), but received sin(1/7*Pi)
\mathrm{sin}\left(\left[1,2, 3\right]\right);
\mathrm{whattype}\left(\left[1,2,3\right]\right);
\textcolor[rgb]{0,0,1}{\mathrm{list}}
\left[1,2,3\right]
\mathrm{Describe}\mathit{}\left(\mathrm{sin}\right)
x
\mathrm{sin}
\mathrm{sin}\left(1\right);\mathrm{sin}\left(2\right);\mathrm{sin}\left(3\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)
\mathrm{sin}~\left(\left[1,2,3\right]\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)\right]
|
2017 Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales
We provide an Itô formula for stochastic dynamical equation on general time scales. Based on this Itô’s formula we give a closed-form expression for stochastic exponential on general time scales. We then demonstrate Girsanov’s change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale (
q
-time scale).
Wenqing Hu. "Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales." Abstr. Appl. Anal. 2017 1 - 13, 2017. https://doi.org/10.1155/2017/9140138
Received: 8 December 2016; Revised: 28 March 2017; Accepted: 29 March 2017; Published: 2017
Wenqing Hu "Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales," Abstract and Applied Analysis, Abstr. Appl. Anal. 2017(none), 1-13, (2017)
|
Congratulate me, for I have finished last revise of last sheet of my Book. It has been an awful job,—7
\frac{1}{2}
months correcting the press— the book from much small type does not look big, but is really very big.1 I have had hard work to keep up to the mark, but during last week only few revises came, so that I have rested & feel more myself. Hence, after our long mutual silence, I enjoy myself by writing a note to you, for the sake of exhaling & hearing from you.— On account of Index, I do not suppose that you will receive your copy till middle of next month.—2 I shall be intensely curious to hear what you think about pangenesis; though I can see how fearfully imperfect even in mere conjectural conclusions it is, yet it has been an infinite satisfaction to me somehow to connect the various large groups of facts, which I have long considered, by an intelligible thread.3 I shall not be at all surprised if you attack it & me with unparalleled ferocity.— It will be my endeavour to do as little as possible for some time, but shall soon prepare a paper or two for Linn. Soc.—4 In a short time we shall go to London for 10 days, but the time is not yet fixed.5 Now I have told you a deal about myself; & do let me hear a good deal about your own past & future doings. Can you pay us a visit?6 Early in December Woolner is coming here to make a bust of me for my Brother;7 & a most horrid bore it is; but he being here wd. not interfere with your visit if you could come. I have seen no one for an age & heard no news. I enclose some Himalayan Balsam seed, which my wife collected for you, as you said you wanted it—but the seed does not appear very good.8
About my book, I will give you a bit of advice, skip the whole of Vol I, except last Chapt. (& that need only be skimmed) & skip largely in 2d. vol., & then you will say it is very good book.—9
The reference is to Variation. In his ‘Journal’, CD recorded receiving the first proof on 1 March and finishing revisions on 15 November 1867 (see Correspondence vol. 15, Appendix II). On the size of the book and the use of two different sizes of type, see the letter to John Murray, 8 January [1867].
The index of Variation was being prepared by William Sweetland Dallas, who had expected to complete it by about 18 November 1867 (letter from W. S. Dallas, 4 November 1867). However, the task progressed more slowly than Dallas had predicted (see letter from W. S. Dallas, 20 November 1867). Variation was published on 30 January 1868 (see ‘Journal’ (Correspondence vol. 15, Appendix II)); Hooker mentioned in his letter of 1 February 1868 (Correspondence vol. 16) that he had received his copy.
Chapter 27 of Variation was headed ‘Provisional hypothesis of pangenesis’. On pangenesis, see also the letter to Charles Lyell, 22 August [1867] and n. 4.
CD’s two papers, ‘Illegitimate offspring of dimorphic and trimorphic plants’ and ‘Specific difference in Primula’, were read at the meetings of the Linnean Society on 20 February and 19 March 1868, respectively, and subsequently appeared in the Journal of the Linnean Society.
CD visited London from 28 November to 7 December 1867 (‘Journal’ (Correspondence vol. 15, Appendix II)).
Hooker next visited Down on 21 December 1867 (Emma Darwin’s diary (DAR 242)).
The references are to Thomas Woolner and Erasmus Alvey Darwin.
Numerous species of balsam (Impatiens) are endemic to the Himalayas; for a contemporary list, see J. D. Hooker and Thomson 1859, pp. 117–18. Himalayan balsam is now I. glandulifera (A. Huxley et al. eds. 1992), but was formerly I. roylei, which was described as endemic and very common in the western Himalayas (J. D. Hooker and Thomson 1859, pp. 117, 127–8). Receipt of these seeds at Kew was recorded on 27 November 1867 with the note: ‘The lady who gathered it mixed seeds of white Balsam with it. Dead’ (Royal Botanic Gardens, Kew, Inwards book). No letter from Hooker requesting balsam seed has been found.
The last chapter of the first volume of Variation is headed ‘On bud-variation, and on certain anomalous modes of reproduction and variation’ (Variation 1: 373–411).
Has finished last revise of his book [Variation].
Is curious to know what JDH thinks of Pangenesis. It is fearfully imperfect, yet satisfying, for it connects large groups of facts by an intelligible thread.
Thomas Woolner is coming [to do a bust of CD].
|
A Valuation Model for the Variable Rate Demand Obligation
A Valuation Model for the Variable Rate Demand Obligation
1School of Economics and Management, Xiamen University Malaysia, Kuala Lumpur, Malaysia
In this paper, a valuation framework is developed for the variable rate demand obligation (VRDO). The VRDO is a class of floating rate note whose coupon rate changes on a regular basis and is “puttable” by the bondholder, given a notice of one week to the issuer. We model the coupon rate as a geometric Brownian motion process and assume that the incidence of puts is Poisson distributed, across time. Put events are assumed to be brought about by factors such as a change in the liquidity and consumption preferences of investors or a change in a Municipal issuer’s creditworthiness. This paper is unique because as of July 2019 there exists no attempt at valuing VRDOs in the research literature.
Variable Rate Demand Obligation, Municipal Corporation, Public Policy, Stochastic Calculus
The purpose of this paper is to develop a valuation framework for a variable rate demand obligation (VRDO). There exists no valuation formula in the literature and so an attempt is made to construct a parsimonious framework of valuation for the VRDO. VRDOs exist predominantly in the United States than in other countries. The concepts underlying the VRDO are simple. They are floating rate obligations that have a nominal long-term maturity but have a coupon rate that is reset either daily or less frequently, depending upon the terms and conditions specified in the bond agreement (Dawson [1] [1993]). Investors accrue interest on a continuous basis and may put the bond back with the issuer at any time. When the bondholder exercises the put option they receive the par value of the bond with accrued interest. The major issuers of VRDOs in the US have been electric utilities, colleges and universities and some large water companies. An 1895 court decision, Pollock versus Farmer’s Loan and Trust Co., established Municipal securities’ interest income as tax-free (Walter [2] [1986]).
The unusually short-term interval over which the coupon rate is adjusted means that the valuation of this instrument can be valued within a continuous-time finance context. In this paper, a stochastic model is developed in which the coupon rate follows a geometric Brownian motion process and the arrival time of the put by the investor conforms to negative exponential distribution. The latter is equivalent to the probability of an arrival, within a given time interval, being Poisson distributed.
The advantages of using VRDOs are that they have led to a lowering of the cost of capital by many municipal issuers. Indeed, VRDOs were introduced at the beginning of the 1980s when long-term interest rates were considerably higher than short-term rates. VRDOs provide flexibility for issuers to restructure their debt because advance refunding restrictions do not apply to VRDOs as they apply to fixed-rate debt. However, the issue is rated by Standard and Poors in terms of their financial liquidity and credit-worthiness. A change in the bond rating of a municipal or corporate issuer could cause bondholders to put the VRDO back with the issuer if they suspected possible future financial distress.
The disadvantages of VRDOs from the perspective of the issuer are that in a period of rising interest rates, they will find themselves paying a higher coupon, possibly until maturity. If the bonds are put by the investor when interest rates are high, the issuer faces higher borrowing costs to reissue, as well as transaction costs. In addition, certain market conditions may cause many investors to put back their bonds with the issuer simultaneously.
1.1. Classes of VRDOs
The most common form of VRDO is the “lower floater”, in which the interest rate is adjusted weekly with respect to a specified index. With lower floaters, bond holders must give the issuer seven days notice if they wish to put the bonds ([Moody’s Investors Service Inc. [3] [1987]). Longer-term VRDOs allow the coupon rate to be reset monthly or quarterly. Some floaters can be put at any time by the investor, whereas others can only be put on the interest adjustment date [4] [Epstein, 1992].
The coupon rate for VRDOs is assigned according to a rate on a specified interest rate index that reflects the current short-term rate (Moody’s Investors Service [1987]) [3] . Hence, a potential long-term liability is financed at short-term rates.
1.3. Rating Procedures
Bond rating agencies decompose the risk of VRDOs into two components. The first is a creditworthiness rating which reflects the issuer’s ability to pay the coupon over the life of the bond. The second, the liquidity rating represents the ability of the issuer to make timely payments, if the bond is put by the investors. The latter, depends upon the terms in the credit agreement between the issuer and the commercial bank (Standard and Poors Corporation [5] [1994]).
The credit factors that Standard and Poors focus upon when determining the impact of floating rate debt are the issuer’s debt management strategy; floating rate and short-term exposure as a percentage of total debt; financial flexibility and liquidity resources; and synthetic floating rate risks, such as termination payment risk and counterparty risk. The advantage of issuing floating rate debt is that it helps issuers match assets and liabilities. Increased debt service costs associated with a rise in interest rates are offset by increases in investment income and vice-versa.
1.4. Remarketing of VRDOs after Puts
The structure of long-term variable rate demand obligations involves the issuer engaging in an agreement with a commercial bank to remarket the debt to possible other investors should the bondholder exercise their put option. The remarketing agent resets the coupon rate and attempts to resell the bonds. If after this process some bonds remain unsold, the issuer’s remaining cash flows are satisfied by the credit facility provided by the commercial bank [Dawson [1] , 1983]. The credit agreement between the bond issuer and commercial agreement will normally specify that either the bank will underwrite the reissue or provide the issuer with a loan to match any short fall in the issuer’s cash flows arising from the exercise of the put option by the bond holder (Standard and Poors Corporation [6] [1990]). If the bank chooses to make a loan rather than to purchase the excess bonds, then the interest rate charged will be connected to the bank prime rate (Peterson [7] [1991]).
Factors which cause bondholders to ‘put’ the VRDO back with the issuer may be wide-ranging. These could be a change in credit rating by Standard and Poors, a change in the liquidity or consumption preferences of the bondholder or personal taxation implications.
Finally, there has been some recent litigation in the US relating to VRDO by Baltimore against ten banks for violating antitrust and state law, alleging collusion in market manipulation of VRDOs (Bloomberg Law [8] , 2019).
2. Valuing the Variable Rate Demand Obligation
Consider a domestic bond subject to a variable coupon rate i. The coupon payment during dt is Pidt. When the bondholder puts the bond, the value of the bond is P(1 + h), which includes the premium at the proportionate rate h (normally h = 0 because the bonds are usually repaid at par). Now, the time to the put being exercised, after any initial period within which putting is not allowed is assumed to be negatively exponentially distributed, with mean 1/λ. This is consistent with a Poisson arrival rate of λ. Hence, the probability of a put during dt is λdt. However, given a market value of m, and a put during dt being possible, then if the bond is not put, the expected value at the end of the period will be E(m + dm), with an associated probability of (1 − λdt). By discounting at the rate k during dt, it follows that the value of the bond is:
m=Pi\text{d}t+P\left(\text{1}+h\right)\lambda \text{d}t/\left(\text{1}+k\text{d}t\right)+E\left(m+\text{d}m\right)\left(\text{1}-\lambda \text{d}t\right)/\left(\text{1}+k\text{d}t\right)
This accounts for the coupon flows and the end of period value with or without the bond being put.
Now, let the coupon rate follow a geometric Brownian motion with a growth trend parameter of μ and a standard deviation parameter of σ:
i=\mu i\text{d}t+\sigma i\text{d}z
\text{d}z=n{\left(\text{d}t\right)}^{1/2}
i.e. n is a standard normally distributed variable. From Itô’s [1965] lemma [9] :
E\left(\text{d}m\right)=\frac{\text{d}m}{\text{d}i}\mu i\text{d}t+\frac{1}{2}\frac{{\text{d}}^{2}m}{\text{d}{i}^{2}}{\sigma }^{2}{i}^{2}\text{d}t.
\lambda \text{d}t/\left(\text{1}+k\text{d}t\right)=\lambda \text{d}t\left(\text{1}-k\text{d}t\right)/\left(\text{1}-{k}^{\text{2}}{\left(\text{d}t\right)}^{\text{2}}\right)=\lambda \text{d}t
ignoring terms in (dt)2. Also,
\left(1-\lambda \text{d}t\right)/\left(1+k\text{d}t\right)=\left(1-\lambda \text{d}t\right)\left(1-k\text{d}t\right)/\left(1-{k}^{2}{\left(\text{d}t\right)}^{2}\right)=1-\lambda \text{d}t-k\text{d}t
again ignoring terms in (dt)2. Therefore,
m=Pi\text{d}t+P\left(1+h\right)\lambda \text{d}t+\left(m+\frac{\text{d}m}{\text{d}i}\mu i\text{d}t+\frac{1}{2}\frac{{\text{d}}^{2}m}{\text{d}{i}^{2}}{\sigma }^{2}{i}^{2}\text{d}t\right)\left(1-\lambda \text{d}t-k\text{d}t\right).
Hence, ignoring terms in (dt)2:
\frac{1}{2}\frac{{\text{d}}^{2}m}{\text{d}{i}^{2}}{\sigma }^{2}{i}^{2}+\frac{\text{d}m}{\text{d}i}\mu i-\left(\lambda +k\right)m=P\left(i+\left(1+h\right)\lambda \right).
Since this a second-order linear non-homogeneous differential equation, the solution is given by the sum of the general solution to the corresponding homogeneous version of the equation plus any particular solution to the non-homogeneous equation. Dixit and Pindyck [10] [1994] have demonstrated that by ignoring the possibilities of non-economic solutions and speculation, the general solution to the homogeneous version of an equation of this type is zero. Now the right-hand side of the Equation (8) is a linear function in i. Hence, we can introduce a linear trial function in order to determine any particular solution. Therefore, let
m=ai+b
, where a and b are constants. (9)
Since dm/di = a and d2m/di2 = 0, we can make substitutions accordingly in Equation (8):
0+a\mu i-\left(\lambda +k\right)\left(ai+b\right)=-P\left(i+\left(1+h\right)\lambda \right).
Equating the coefficients of i gives:
a=P/\left(k-\mu +\lambda \right),
And similarly for constant terms:
b=P\left(1+h\right)\lambda /\left(k+\lambda \right).
By substituting these values into Equation (9):
m=P\left\{i/\left(k-\mu +\lambda \right)+\lambda \left(1+h\right)/\left(k+\lambda \right)\right\}.
This represents the value of the bond, when the put is exercised at any time.
Now, if there are restrictions, preventing the bond being put during the first y years, then the present value, I of the coupon flows during this period will be:
I=\underset{0}{\overset{y}{\int }}Pi{\text{e}}^{\mu t}{\text{e}}^{-kt}\text{d}t=Pi\left(1-{\text{e}}^{-\left(k-\mu \right)y}\right)/\left(k-\mu \right).
In Equation (14) we can interpret i as the initial variable coupon rate. Also, in Equation (13), we can replace i by ieμy to reflect the expected coupon rate in y years’ time and then discount the whole expression at the rate k. Therefore, the initial value of the bond, m0, is:
{m}_{0}=Pi\left(1-{\text{e}}^{-\left(k-\mu \right)y}\right)/\left(k-\mu \right)+P{\text{e}}^{-ky}\left[i{\text{e}}^{\mu y}/\left(k-\mu +\lambda \right)+\lambda \left(1+h\right)/\left(k+\lambda \right)\right]
The discrete equivalent is:
{m}_{0}=\frac{Pi}{k-\mu }\left\{1-{\left(\frac{1+\mu }{1+k}\right)}^{y}\right\}+\frac{P}{{\left(1+k\right)}^{y}}\left[\frac{i{\left(1+\mu \right)}^{y}}{k-\mu +\lambda }+\frac{\lambda \left(1+h\right)}{k+\lambda }\right].
Suppose $100 nominal debt is priced at $106.94. The variable coupon rate is 7 per cent with an expected trend of 0.5 per cent per annum. After 5 years the debt can be put. The Poisson event is expected to occur a further 4 years later. The premium on redemption is 3 percent.
Thus for P = 100, m0 = 106.94, i = 0.07, μ = 0.005, y = 5, 1/λ = 4 and h = 0.03, the cost of funds is 6 per cent, which satisfies Equation (16).
This paper has valued the variable rate demand obligation, an important security issued by municipal corporations in the US. The framework assumed that the coupon rate followed a geometric Brownian motion process and that put by investors were Poisson distributed across time. The framework gives rise to a valuation equation that is closed form. Further research may help to refine our model and incorporate the impact of taxation because increasingly, VRDOs are being issued by municipal corporations. This paper is unique because as of July 2019 there exists no attempt at valuing VRDOs in the research literature.
Hooper, V. and Pointon, J. (2019) A Valuation Model for the Variable Rate Demand Obligation. Journal of Mathematical Finance, 9, 388-393. https://doi.org/10.4236/jmf.2019.93022
1. Dawson, W. (1993) Variable Rate Demand Notes. In: Lamb, R., Leighland, J. and Rappaport, S., Eds., The Handbook of Municipal Bonds and Public Finance, New York Institute of Finance, New York, 531-535.
2. Walter, J. (1986) Short-Term Municipal Securities. Economic Review Federal Reserve Bank of Richmond, 72, 25-34.
3. Moody’s Investors Service (1987) Moody’s on Municipals: An Introduction to Issuing Debt. New York, 19-20.
4. Epstein, L. (1992) Basis Points: Put up with Floaters; Put up with the Phones. Corporate Cash Flow, 13, 46-47.
5. Standard and Poors Corporation (1994) Municipal Finance Criteria. New York, 5.
6. Standard and Poors Corporation (1994) Municipal Finance Criteria. New York, 92.
7. Peterson, J.E. (1991) Debt Markets and Instruments. In: Peterson, J.E. and Strachota, D.R., Eds., Local Government Finance: Concepts and Practices, Government Finance Officers Association, Chicago, 308.
8. Bloomberg Law (2019) Baltimore Sues 10 Banks over VRDOs, Alleging Price Collusion. Securities Law News. https://news.bloomberglaw.com/securities-law/baltimore-sues-10-banks-over-vrdos-alleging-price-collusion-2
9. Itô, K. (1965) On Stochastic Differential Equations. Memoirs of the American Mathematical Society, 4, 1-51.
|
Self-adjoint operator - Citizendium
In mathematics, a self-adjoint operator is a densely defined linear operator mapping a complex Hilbert space onto itself and which is invariant under the unary operation of taking the adjoint. That is, if A is an operator with a domain
{\displaystyle \scriptstyle H_{0}}
which is a dense subspace of a complex Hilbert space H then it is self-adjoint if
{\displaystyle \scriptstyle A=A^{*}}
{\displaystyle \scriptstyle A^{*}}
denotes the adjoint operator of A. Note that the adjoint of any densely defined linear operator is always well-defined (in fact, the denseness of the domain of an operator is necessary for the existence of its adjoint) and two operators A and B are said to be equal if they have a common domain and their values coincide on that domain.
On an infinite dimensional Hilbert space, a self-adjoint operator can be thought of as the analogy of a real symmetric matrix (i.e., a matrix which is its own transpose) or a Hermitian matrix in (i.e., a matrix which is its own Hermitian transpose) when these matrices are viewed as (bounded) linear operators on
{\displaystyle \scriptstyle \mathbb {R} ^{n}}
{\displaystyle \scriptstyle \mathbb {C} ^{n}}
Special properties of a self-adjoint operator
The self-adjointness of an operator entails that it has some special properties. Some of these properties include:
1. The eigenvalues of a self-adjoint operator are real. As a special well-known case, all eigenvalues of a real symmetric matrix and a complex Hermitian matrix are real.
2. By the von Neumann’s spectral theorem, any self-adjoint operator X (not necessarily bounded) can be represented as
{\displaystyle X=\int _{-\infty }^{\infty }xE^{X}(dx),}
{\displaystyle \scriptstyle E^{X}}
is the associated spectral measure of X (in particular, a spectral measure is a Hilbert space projection operator-valued measure)
3. By Stone’s Theorem, for any self-adjoint operator X the one parameter unitary group
{\displaystyle \scriptstyle U=\{U_{t}\}_{t\in \mathbb {R} }}
{\displaystyle \scriptstyle U_{t}=\int _{-\infty }^{\infty }e^{-itx}\,E^{X}(dx)}
{\displaystyle \scriptstyle E^{X}}
is the spectral measure of X, satisfies:
{\displaystyle {\frac {dU_{t}}{dt}}u=-iXU_{t}u=U_{t}(-iX)u,}
for all u in the domain of X. One says that the operator -iX is the generator of the group U and writes:
{\displaystyle \scriptstyle U_{t}=e^{-itX},\,\,t\in \mathbb {R} }
Examples of self-adjoint operators
As mentioned above, a simple instance of a self-adjoint operator is a Hermitian matrix.
For a more advanced example consider the complex Hilbert space
{\displaystyle \scriptstyle L^{2}(\mathbb {R} ;\mathbb {C} )}
of all complex-valued square integrable functions on
{\displaystyle \scriptstyle \mathbb {R} }
with the complex inner product
{\displaystyle \scriptstyle \langle f,g\rangle =\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx}
, and the dense subspace
{\displaystyle \scriptstyle C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} )}
{\displaystyle \scriptstyle L^{2}(\mathbb {R} ;\mathbb {C} )}
of all infinitely differentiable complex-valued functions with compact support on
{\displaystyle \scriptstyle \mathbb {R} }
. Define the operators Q, P on
{\displaystyle \scriptstyle C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} )}
{\displaystyle Q(f)(x)=xf(x)\quad \forall f\in C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} )}
{\displaystyle P(f)(x)=i\hbar {\frac {d}{dx}}f(x)\quad \forall f\in C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} ),}
{\displaystyle \scriptstyle \hbar }
is the real valued Planck's constant. Then Q and P are self-adjoint operators satisfying the commutation relation
{\displaystyle \scriptstyle [Q,P]=i\hbar I}
{\displaystyle \scriptstyle C_{0}^{\infty }(\mathbb {R} ;\mathbb {C} )}
, where I denotes the identity operator. In quantum mechanics, the pair Q and P is known as the Schrödinger representation, on the Hilbert space
{\displaystyle \scriptstyle L^{2}(\mathbb {R} ;\mathbb {C} )}
, of canonical conjugate position and momentum operators q and p satisfying the canonical commutation relation (CCR)
{\displaystyle \scriptstyle [q,p]=i\hbar }
K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980.
K. Parthasarathy, An Introduction to Quantum Stochastic Calculus, ser. Monographs in Mathematics, Basel, Boston, Berlin: Birkhauser Verlag, 1992.
Retrieved from "https://citizendium.org/wiki/index.php?title=Self-adjoint_operator&oldid=328297"
|
Home : Support : Online Help : Science and Engineering : Units : Environments : Simple : piecewise
Piecewise Functions in the Simple Units Environment
In the Simple Units environment, the piecewise function is modified so that it verifies that the values it can return have the same dimension.
\mathrm{with}\left(\mathrm{Units}[\mathrm{Simple}]\right):
This is a legal expression, because the two possible values have the same dimension (namely, length).
\mathrm{piecewise}\left(x<1,2\mathrm{Unit}\left(m\right),3\mathrm{Unit}\left(\mathrm{ft}\right)\right)
\left({\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{<}\textcolor[rgb]{0,0,1}{1}\\ \frac{\textcolor[rgb]{0,0,1}{1143}}{\textcolor[rgb]{0,0,1}{1250}}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}\right)\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{m}⟧
This is an illegal expression, because the three possible values do not all have the same dimension: two of them are pressures, whereas the third is a force. Consequently, Maple signals an error.
\mathrm{piecewise}\left(x<20\mathrm{Unit}\left(m\right),2\mathrm{Unit}\left(\mathrm{bar}\right),x<40\mathrm{Unit}\left(m\right),35\mathrm{Unit}\left(\mathrm{inch_mercury}\right),3\mathrm{Unit}\left(\mathrm{gigadyne}\right)\right)
Error, (in Units:-Simple:-piecewise) the following expressions imply incompatible dimensions: {piecewise(x < 20*Units:-Unit(m),2*Units:-Unit(bar),x < 40*Units:-Unit(m),35*Units:-Unit(inHg),3*Units:-Unit(Gdyn))}
|
Difference between revisions of "Vectorial Boolean Functions" - Boolean Functions
Difference between revisions of "Vectorial Boolean Functions"
\max_{j=0,\dots ,2^n-1/\, \delta_j\neq 0}w_2(j).
=== Multi-dimensional Walsh Transform ===
== Basic Properties of Vectorial Boolean Functions ==
<div class="proposition">
An <math>(n,m)</math>-function <math>F</math> is balanced if and only if its component functions are balanced, i.e. if and only if <math>v \cdot F</math> is balanced for every <math>v \in {\mathbb{F}_2^m}^*</math>.
=== Covering Sequences ===
=== Algebraic Immunity ===
1.1 Cryptanalytic attacks
2 Generalities on Boolean functions
2.1 Walsh transform
2.2.1 Algebraic Normal Form
2.2.2 Univariate Representation
2.3 Basic Properties of Vectorial Boolean Functions
{\displaystyle \mathbb {F} _{2}^{n}}
be the vector space of dimensio{\displaystyle n}
{\displaystyle \mathbb {F} _{2}}
with two elements. Functions from
{\displaystyle \mathbb {F} _{2}^{n}}
{\displaystyle \mathbb {F} _{2}^{m}}
{\displaystyle (n,m)}
-functions or simply vectorial Boolean functions when the dimensions of the vector spaces are implicit or irrelevant.
{\displaystyle (n,m)}
{\displaystyle F}
can be written as a vector
{\displaystyle F=(f_{1},f_{2},\ldots f_{n})}
{\displaystyle m}
-dimensional Boolean functions
{\displaystyle f_{1},f_{2},\ldots f_{n}}
which are called the coordinate functions of
{\displaystyle F}
Vectorial Boolean functions, also referred to as "S-boxes", or "Substitution boxes", in the context of cryptography, are a fundamental building block of block ciphers and are crucial to their security: more precisely, the resistance of the block cipher to cryptanalytic attacks directly depends on the properties of the S-boxes used in its construction.
The main types of cryptanalytic attacks that result in the definition of design criteria for S-boxes are the following:
the differential attack introduced by Biham and Shamir; to resist it, an S-box must have low differential uniformity;
the linear attack introduced by Matsui; to resist it, an S-box must have high nonlinearity;
the higher order differential attack; to resist it, an S-box must have high algebraic degree;
the interpolation attack; to resist it, the univariate representation of an S-box must have high degree, and its distance to the set of low univariate degree functions must be large;
algebraic attacks.
Generalities on Boolean functions
{\displaystyle F:\mathbb {F} _{2}^{n}\rightarrow \mathbb {F} _{2}^{m}}
is the integer-valued function
{\displaystyle W_{F}:\mathbb {F} _{2}^{n}\times \mathbb {F} _{2}^{m}}
{\displaystyle W_{F}(u,v)=\sum _{x\in \mathbb {F} _{2}^{n}}(-1)^{v\cdot F(x)+u\cdot x}}
It can be observed that the Walsh transform of some
{\displaystyle F}
is in fact the Fourier transform of the indicator of its graph, i.e. the Fourier transform of the function
{\displaystyle 1_{G_{F}}}
{\displaystyle 1_{G_{F}}(x,y)={\begin{cases}1&F(x)=y\\0&F(x)\neq y.\end{cases}}}
The Walsh spectrum of
{\displaystyle F}
is the multi-set of all the values of its Walsh transform for all pairs
{\displaystyle (u,v)\in \mathbb {F} _{2}^{n}\times {\mathbb {F} _{2}^{m}}^{*}}
. The extended Walsh spectrum of
{\displaystyle F}
is the multi-set of the absolute values of its Walsh transform, and the Walsh support of
{\displaystyle F}
is the set of pairs
{\displaystyle (u,v)}
{\displaystyle W_{F}(u,v)\neq 0}
Vectorial Boolean functions can be represented in a number of different ways.
{\displaystyle (n,m)}
{\displaystyle F}
can be uniquely represented as a polynomial with coefficients in
{\displaystyle \mathbb {F} _{2}^{m}}
{\displaystyle F(x)=\sum _{I\in {\cal {P}}(N)}a_{I}\,\left(\prod _{i\in I}x_{i}\right)=\sum _{I\in {\cal {P}}(N)}a_{I}\,x^{I},}
{\displaystyle {\cal {P}}(N)}
is the power set of
{\displaystyle N=\{1,\ldots ,n\}}
{\displaystyle a_{I}}
{\displaystyle \mathbb {F} _{2}^{m}}
. This representation is known as the algebraic normal form (ANF) of
{\displaystyle F}
. The algebraic degree of
{\displaystyle F}
{\displaystyle d^{\circ }(F)}
is then defined as the global degree of its ANF, i.e.
{\displaystyle d^{\circ }(F)=\ max\{|I|/\,a_{I}\neq (0,\dots ,0);I\in {\cal {P}}(N)\}}
and is equal to the maximal algebraic degree of the coordinate functions of
{\displaystyle F}
Univariate Representation
If we identify the vector space
{\displaystyle \mathbb {F} _{2}^{n}}
with the finite field
{\displaystyle \mathbb {F} _{2^{n}}}
{\displaystyle 2^{n}}
elements, then any
{\displaystyle (n,n)}
-function can be uniquely represented as univariate polynomial over
{\displaystyle \mathbb {F} _{2^{n}}}
of degree at most
{\displaystyle 2^{n}-1}
taking the form
{\displaystyle F(x)=\sum _{j=0}^{2^{n}-1}\delta _{j}x^{j}~,~~\delta _{j}\in {\mathbb {F}}_{2^{n}}.}
This is the univariate representation of
{\displaystyle F}
The algebraic degree of
{\displaystyle F}
can be expressed using the 2-weight of the exponents of the univariate representation. Given a positive integer
{\displaystyle j}
, its 2-weight is the number of summands in its representation as a sum of powers of two; that is, if we write
{\displaystyle j=\sum _{i=0}^{n-1}j_{s}\cdot 2^{s}}
{\displaystyle j_{s}\in \{0,1\}}
, then its 2-weigt is
{\displaystyle w_{2}(j)=\sum _{i=0}^{n-1}j_{s}}
{\displaystyle F}
can the be written as
{\displaystyle \max _{j=0,\dots ,2^{n}-1/\,\delta _{j}\neq 0}w_{2}(j).}
Basic Properties of Vectorial Boolean Functions
Balanced Functions
{\displaystyle (n,m)}
{\displaystyle F}
is called balanced if it takes every value of
{\displaystyle \mathbb {F} _{2}^{m}}
precisely
{\displaystyle 2^{n-m}}
times. In particular, an
{\displaystyle (n,n)}
-function is balanced if and only if it is a permutation.
{\displaystyle (n,m)}
{\displaystyle F}
is balanced if and only if its component functions are balanced, i.e. if and only if
{\displaystyle v\cdot F}
is balanced for every
{\displaystyle v\in {\mathbb {F} _{2}^{m}}^{*}}
Retrieved from "https://boolean.h.uib.no/mediawiki/index.php?title=Vectorial_Boolean_Functions&oldid=182"
|
Taylor the agent2 has found ships for Calcutta, but it is obviously impossible to decide what passage had better be taken except Scott himself comes up & inspects, whether 2d. class at £42 or
\frac{1}{2}
cabin 1st. Class at £65.— so much depends on ship &c &c &c.3 I have sent Scott the documents & written to this effect to him, also offering him opportunity of reading up India at Kew if he inclines to lodge here—4
Answers to your Queries are half done & will go early next week.5
The date is established by the relationship between this letter and the letter from J. D. Hooker, [4–]6 August 1864. The first Friday before 4 August 1864 was 29 July.
Hooker refers to the shipping agent Henry Taylor, of 7 East India Chambers, London (Post Office London directory 1864). See letter to J. D. Hooker, 12 July [1864].
For details of John Scott’s passage to Calcutta, see the letter from J. D. Hooker, [15 August 1864].
Scott visited Hooker at Kew on 13 August 1864 (see letter from J. D. Hooker, [15 August 1864]).
See memorandum to J. D. Hooker, [24 July 1864?].
|
Gauss - The Great Mathematician | Toph
Gauss - The Great Mathematician
By himuhasib · Limits 1s, 1.0 GB
Carl Friedrich Gauss was a great mathematician. When he was a child, his teacher told him to sum up all the numbers from 1 to 100 to keep him busy for some time. To his surprise, Gauss solved the problem in a few moments. He contributed in many fields of science. Some of his contributions are used in Computer Science too! #Respect!
In this problem, we want to keep you busy too! You have to find the sum of an arithmetic sequence. In mathematics, an arithmetic sequence is a sequence of numbers such that the difference between two consecutive terms is constant.
1, 2, 3, 4, 5, 6, \cdots
1,2,3,4,5,6,⋯
2, 7, 12, 17, 22, 27, \cdots
2,7,12,17,22,27,⋯
27, 22, 17, 12, 7, 2, -3, -8, \cdots
27,22,17,12,7,2,−3,−8,⋯
-8, -3, 2, 7, 12, 17, \cdots
−8,−3,2,7,12,17,⋯
But in this problem, we are only interested in increasing arithmetic sequence where each term is greater than the previous term. You will be given the first two terms and the last term of an increasing arithmetic sequence. You have to find the sum of all the terms of the given sequence.
The first line of the input will only contain a single integer
T(1\leq T \leq 10^3)
T(1≤T≤103) denoting the number of test cases. In the next
T lines, there will be 3 integers
A_1, A_2, A_n
A1,A2,An where
A_1
A1 is the first term,
A_2
A2 is the second term and
A_n
An is the last term of the sequence. It is guaranteed that a valid sequence of integers can be formed from the given 3 integers.
1 \leq T \leq 10^3
-10^4 \leq A_1 < A_2 \leq A_n \leq 10^4
−104≤A1<A2≤An≤104
For each test case, print the sum of the sequence in a single line.
fsshakkhorEarliest, Dec '18
fsshakkhorLightest, 131 kB
|
Regularity Criterion for Weak Solution to the 3D Micropolar Fluid Equations
2011 Regularity Criterion for Weak Solution to the 3D Micropolar Fluid Equations
Yu-Zhu Wang, Zigao Chen
Regularity criterion for the 3D micropolar fluid equations is investigated. We prove that, for some
T>0
{\int }_{0}^{T}\parallel {v}_{{x}_{3}}{\parallel }_{{L}^{\varrho }}^{\rho }dt<\infty
3/\varrho +2/\rho \le 1
\varrho \ge 3
, then the solution
\left(v,w\right)
can be extended smoothly beyond
t=T
. The derivative
{v}_{{x}_{3}}
can be substituted with any directional derivative of v.
Yu-Zhu Wang. Zigao Chen. "Regularity Criterion for Weak Solution to the 3D Micropolar Fluid Equations." J. Appl. Math. 2011 1 - 12, 2011. https://doi.org/10.1155/2011/456547
Yu-Zhu Wang, Zigao Chen "Regularity Criterion for Weak Solution to the 3D Micropolar Fluid Equations," Journal of Applied Mathematics, J. Appl. Math. 2011(none), 1-12, (2011)
|
Truth - Everything Wiki
File:Quran, Tunisia.JPG
File:Saint Irenaeus.jpg
File:Time Saving Truth from Falsehood and Envy.jpg
Michael Adams, lexicology professor at North Carolina State University, discussing the neologism "truthiness", defined as "the quality of stating concepts one wishes or believes to be true, rather than the facts" in "Linguists Vote 'Truthiness' Word of 2005", AP via Yahoo! News, (6 January 2006)
File:Geological time spiral.png
They speak falsely who say that truth is the daughter of time; it is the child of eternity, and as old as the Divine mind. The perception of it takes place in the order of time; truth itself knows nothing of the succession of ages. ~ George Bancroft
File:Berdorf (LU), Hohllay -- 2015 -- 6097-101.jpg
File:CARRACCI, Annibale - An allegory of Truth and Time (1584-5).JPG
File:Traunstein 008.JPG
File:BentoXVI-28-10052007.jpg
Truth draws strength from itself and not from the number of votes in its favour. ~ Pope Benedict XVI
File:First demonstrations calling for toppling the regime in Libya (Bayda, Libya, 2011-02-16).jpg
Relations between States and within States are correct to the extent that they respect the truth. When, instead, truth is violated, peace is threatened, law is endangered, then, as a logical consequence, forms of injustice are unleashed. ~ Pope Benedict XVI
File:Yungay6.jpg
Certainly civilization cannot advance without freedom of inquiry. This fact is elf-evident. What seems equally self-evident is that in the process of history certain immutable truths have been revealed and discovered and that their value is not subject to the limitations of time and space. The probing, the relentless debunking, has engendered a skepticism that threatens to pervade and atrophy all our values. In apologizing for our beliefs and our traditions we have bent ovr backwards so far that we have lost our balance, and we see a topsy-turvy world and we say topsy-turvy things, such as that the way to beat Communism is by making our democracy better. What a curious self-examination! Beat the Union of Soviet Socialist Republics by making America socialistic. Beat atheism by denying God. Uphold individual freedom by denying natural rights.
William F. Buckley, Jr., (11 June 1950), "Today We Are Educated Men" oration at Yale University,
Edward Bulwer-Lytton, Caxtoniana.
File:Orvieto Pozzo San Patrizio 5.JPG
They tell me that truth lies somewhere at the bottom of a well, and at virtually the door of our home is a most notable if long dried well. Our location is thus quite favorable, if we but keep patience. ~ James Branch Cabell
File:Rain on grass2.jpg
File:Hans von Aachen 002.jpg
File:Iss007e10807 darker.jpg
Whatever the future may have in store for us, one thing is certain... Human thought will never go backward. When a great truth once gets abroad in the world, no power on earth can imprison it, or prescribe its limits, or suppress it. It is bound to go on till it becomes the thought of the world... Now that it has got fairly fixed in the minds of the few, it is bound to become fixed in the minds of the many, and be supported at last by a great cloud of witnesses, which no man can number and no power can withstand. ~ Frederick Douglass
File:Fiori di mandorlo.jpg
File:Veritas (Mangin) 2013-08-27 4.jpg
Great is truth, and mighty above all things. ~ 1 Esdras
Variant translation: Great is truth, and mighty above all things.
1 Esdras 4:41; this is often quoted in the Latin: Magna est veritas et praevalet.
File:Double slit x-ray simulation monochromatic blue-white.png
File:UN and Banner of Peace (Stamp).jpg
Reason is man's instrument for arriving at the truth, intelligence is man's instrument for manipulating the world more successfully; the former is essentially human, the latter belongs to the animal part of man. ~ Erich Fromm
File:Naruto Whirlpools taken 4-21-2008.jpg
File:Darkness Over Eden 2709.jpg
Truth is the daughter of Time. ~ Aulus Gellius
André Gide, {{#invoke:citation/CS1|citation
File:Hegel portrait by Schlesinger 1831.jpg
File:John William Waterhouse - Ulysses and the Sirens (1891).jpg
File:01. La Vérité de Lucien Pallez - Parc de la Tête d'Or.JPG
File:Thomas Jefferson rev.jpg
File:Thomas-Jefferson.jpg
Truth advances, and error recedes step by step only; and to do to our fellow men the most good in our power, we must lead where we can, follow where we cannot, and still go with them, watching always the favorable moment for helping them to another step. ~ Thomas Jefferson
File:Redentor.jpg
File:Transport restraints 02.jpg
File:What is truth.jpg
File:JFK limousine.png
File:La Vérité - Luc-Olivier Merson - musée d'Orsay.jpg
Love comes first. … What is known is not truth. ~ Jiddu Krishnamurti
File:Yyjpg.svg
Truth cannot be sought; it comes to you. ~ Jiddu Krishnamurti
File:Ancient version of the Taijitu by Lai Zhi-De, sideways.svg
File:Hoag's object.jpg
Irving Kristol, quoted in Bailey, Ronald (July 1997). "Origin of the Specious: Why do neoconservatives doubt Darwin?". Reason. http://www.reason.com/news/show/30329.html.
File:Hong kong bruce lee statue.jpg
File:John Locke by John Greenhill.jpg
File:La Verità scoperta dal Tempo di Guidobono Bartolomeo.jpg
File:TV highquality.jpg
File:La Vérité sortant du puits.jpg
File:The Mystical Nativity.jpg
File:James Russell Lowell, Brady-Handy Photograph Collection.jpg
File:Arco iris circular.JPG
File:Solsort.jpg
File:Under the Milky Way in Black Rock Desert, Nevada.jpg
File:J’accuse.jpg
File:Maimonides bas-relief in the U.S. House of Representatives chamber.jpg
File:Murrow challengeofideas desk.jpg
File:J C W Beyer Veritas KGM 90-434.jpg
File:Everest North Face toward Base Camp Tibet Luca Galuzzi 2006.jpg
File:JUL Soul Iris.png
File:Angels (7809083316).jpg
File:2011 05 17 Thueringer Staatskanzlei (8841-2-3 com).jpg
File:Hs-2001-16-p-full jpg.jpg
Truth can never be confined to time and culture; in history it is known, but it also reaches beyond history. ~ Pope John Paul II
Thomas Paine, The Rights of Man.
{\displaystyle e^{i\pi }+1=0.\,\!}
File:At The End Of The Tunnel.jpg
File:Michael Maier Atalanta Fugiens Emblem 21.jpeg
File:Vahagn2.jpg
File:Trees2.jpg
File:ChancelWindowTruth.jpg
When truth doesn't matter, lies about policy are sure to follow.
Ben Shapiro, Is Truth Becoming Irrelevant to Conservatives?, The Daily Wire (December 5, 2016)
File:Twain in Tesla Lab.jpg
Jules Tygiel, Baseball’s Great Experiment (revised edition 1997), Oxford University Press, ISBN 0-19-510620-2{{#invoke:check isxn|check_isbn|0-19-510620-2|error= Invalid ISBN}}, p. 69
File:Bacchus (painting).jpg
File:The babe in the womb; Leonardo da Vinci (1511).JPG
File:Spike-of-rice-Taiwan.png
File:Willem van Oranje Standbeeld Den Haag, juni 2003.JPG
File:CanadaStatueTruth crop.jpg
Just as soon as any conviction of important truth becomes central and vital, there comes the desire to utter it—a desire which is immediate and irresistible. Sacrifice is gladness, service is joy, when such an idea becomes a commanding power.
Wikipedia has related information at Truth
bs:Pravda bg:Истина — Истинност cs:Pravda de:Wahrheit et:Tõde el:Αλήθεια es:Verdad eo:Vero eu:Egia fr:Vérité hy:Ճշմարտություն hr:Istina is:Sannleikur it:Verità he:אמת ko:진실 ku:Rastî la:Veritas lt:Tiesa hu:Igazság nl:Waarheid ja:真理 nn:Sanning pl:Prawda pt:Verdade ru:Истина sk:Pravda sv:Sanning ta:உண்மை tr:Gerçek uk:Правда zh:真理 fa:حقیقت
This page was moved from wikiquote:en:Truth. Its edit history can be viewed at Truth/edithistory
Retrieved from "https://everything.wiki/index.php?title=Truth&oldid=946797"
|
Error, invalid input: sin expects its 1st argument, x, to be of type algebraic, but received [1, 2, 3] - Maple Help
Home : Support : Online Help : Error, invalid input: sin expects its 1st argument, x, to be of type algebraic, but received [1, 2, 3]
\mathrm{sin}\left(\left[1,2, 3\right]\right);
\mathrm{whattype}\left(\left[1,2,3\right]\right);
\textcolor[rgb]{0,0,1}{\mathrm{list}}
\left[1,2,3\right]
\mathrm{Describe}\mathit{}\left(\mathrm{sin}\right)
x
\mathrm{sin}
\mathrm{sin}\left(1\right);\mathrm{sin}\left(2\right);\mathrm{sin}\left(3\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)
\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)
\mathrm{sin}~\left(\left[1,2,3\right]\right)
\left[\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{3}\right)\right]
|
Estimate ARIMA Models - MATLAB & Simulink - MathWorks Italia
This example shows how to estimate autoregressive integrated moving average (ARIMA) models.
Models of time series containing non-stationary trends (seasonality) are sometimes required. One category of such models are the ARIMA models. These models contain a fixed integrator in the noise source. Thus, if the governing equation of an ARMA model is expressed as A(q)y(t)=Ce(t), where A(q) represents the auto-regressive term and C(q) the moving average term, the corresponding model of an ARIMA model is expressed as
A\left(q\right)y\left(t\right)=\frac{C\left(q\right)}{\left(1-{q}^{-1}\right)}e\left(t\right)
\frac{1}{1-{q}^{-1}}
represents the discrete-time integrator. Similarly, you can formulate the equations for ARI and ARIX models.
Using time-series model estimation commands ar, arx and armax you can introduce integrators into the noise source e(t). You do this by using the IntegrateNoise parameter in the estimation command.
The estimation approach does not account any constant offsets in the time-series data. The ability to introduce noise integrator is not limited to time-series data alone. You can do so also for input-output models where the disturbances might be subject to seasonality. One example is the polynomial models of ARIMAX structure:
A\left(q\right)y\left(t\right)=B\left(q\right)u\left(t\right)+\frac{C\left(q\right)}{\left(1-{q}^{-1}\right)}e\left(t\right)
See the armax reference page for examples.
Estimate an ARI model for a scalar time-series with linear trend.
model = ar(y,4,'ls','Ts',Ts,'IntegrateNoise', true);
% 5 step ahead prediction
compare(y,model,5)
Estimate a multivariate time-series model such that the noise integration is present in only one of the two time series.
y2 = cumsum(y);
% artificially construct a bivariate time series
data = iddata([y, y2],[],Ts); na = [4 0; 0 4];
model1 = armax(data, [na nc], 'IntegrateNoise',[false; true]);
% Forecast the time series 100 steps into future
yf = forecast(model1,data(1:100), 100);
plot(data(1:100),yf)
If the outputs were coupled ( na was not a diagonal matrix), the situation will be more complex and simply adding an integrator to the second noise channel will not work.
|
Real Root Finding - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 2019 : Real Root Finding
Isolate(sqrt(2)*x^2 - Pi*x - exp(2));
[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-1.430647445}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3.652088914}]
In particular, Isolate can now be used to find roots of polynomials with algebraic coefficients. We illustrate this in an example where we manually study the real solutions of a bivariate equation system of the form
{F\left(x,y\right)=0,G\left(x,y\right)=0}
F := (2*x^2*y - 2*x^2 - 3*x + y^3 - 33*y + 32) * ((x-2)^2 + y^2 + 3):
G := (x^2 + y^2 - 23) * (x^2 + y^2 + 2):
plots[implicitplot]([F = 0, G = 0], x=-16..16, y=-7..6, color=["Teal", "Red"], gridrefine=2, scaling=constrained, size=[0.7,0.35]);
Elimination theory for algebraic equation systems tells us that all x-coordinates of the common solutions are roots of the resultant polynomial of
F
G
y
R := resultant(F, G, y):
candidates := sort([RealDomain[solve](R)], key=evalf):
evalf(candidates);
[\textcolor[rgb]{0,0,1}{-4.795738156}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-3.854101966}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1.739664347}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.250000000}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2.854101966}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3.227646598}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4.307755905}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7.500000000}]
However, some of the candidates might be spurious. We can use the new interface for RootFinding:-Isolate to determine the roots along the fibers of
F
G
when we substitute the candidates:
a := candidates[1];
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{27}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{28}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{_Z}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{116}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-4.7957383}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{-4.7957372}\right)
fa := subs(x=a, F):
ga := subs(x=a, G):
Isolate(fa);
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-0.02992556510}]
Isolate(ga);
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-0.02992556510}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.02992556510}]
Indeed, there is a common solution close to
x={\mathrm{evalf}}_{3}\left(a\right)
y=-0.03
is(RootOf(fa, y, -0.03) = RootOf(ga, y, -0.03));
\textcolor[rgb]{0,0,1}{\mathrm{true}}
However, let's look at the candidate at
b=1.25
b := candidates[4];
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{5}}{\textcolor[rgb]{0,0,1}{4}}
fb := subs(x=b, F):
gb := subs(x=b, G):
Isolate(fb);
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-5.845766191}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.8624794396}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4.983286752}]
Isolate(gb);
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-4.630064794}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4.630064794}]
This clearly is a spurious candidate; the roots of
F\left(1.25,y\right)
G\left(1.25,y\right)
Isolate([F,G], [x,y]);
[[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3.227646598}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-3.547153427}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-3.854101966}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-2.854101966}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-4.795738156}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-0.02992556510}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4.307755905}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2.107899206}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2.854101966}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{3.854101966}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-1.739664347}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4.469179786}]]
However, the multivariate polynomial solver requires coefficients of type numeric (that is, rationals or floats). Consider the case where we slightly change
F
3
\mathrm{\pi }
in the last term:
F := (2*x^2*y - 2*x^2 - 3*x + y^3 - 33*y + 32) * ((x-2)^2 + y^2 + Pi):
[\textcolor[rgb]{0,0,1}{-4.795738156}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-3.854101966}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-1.739664347}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1.285398164}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2.854101966}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{3.227646598}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{4.307755905}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{7.535398164}]
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{RootOf}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{27}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{\mathrm{_Z}}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{28}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{_Z}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{116}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-4.795738156}\right)
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-0.02992556510}]
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-0.02992556510}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.02992556510}]
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{≔}\frac{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-5.827352781}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.8577160795}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4.969636701}]
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-4.620362709}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{4.620362709}]
rts_fb, gb_at_rts_fb := Isolate(fb, constraints=[gb], output=interval):
contains_zero := iv -> evalb(iv[1] <= 0 and iv[2] >= 0):
seq(contains_zero(rhs(gb_at_rts_fb[i][1])), i=1..nops(rts_fb));
\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{gb}
evaluated over all isolating intervals for
\mathrm{fb}
does not contain zero, which confirms that
F
G
have no common zero at
x=b
rts_fa, ga_at_rts_fa := Isolate(fa, constraints=[ga], output=interval):
seq(contains_zero(rhs(ga_at_rts_fa[i][1])), i=1..nops(rts_fa));
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{ga}
, evaluated at the isolating intervals for the root of
\mathrm{fa}
, contains zero. This still does not validate the simultaneous zero of both systems, but is a strong hint. Techniques along these lines can serve to filter candidates numerically before trying time-consuming symbolic simplification and zero-testing, and can be used as cornerstones for complete solvers.
apx := evalf(a);
\textcolor[rgb]{0,0,1}{\mathrm{apx}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-4.795738156}
fapx := subs(x=apx, F):
gapx := subs(x=apx, G):
rts_fapx, gapx_at_rts_fapx := Isolate(fapx, constraints=[gapx], output=interval):
seq(contains_zero(rhs(gapx_at_rts_fapx[i][1])), i=1..nops(rts_fapx));
\textcolor[rgb]{0,0,1}{\mathrm{false}}
Even a tiny perturbation of the candidate solution in
x
will produce distinct roots of
F
G
y
. Thus, the direct handling of arbitrary real coefficients is not only convenient, but required for correctness.
The new default algorithm of Isolate also features vastly improved performance for ill-conditioned polynomials with clustered roots. The root finding method eventually converges quadratically to regions containing roots, rather than just linearly. For example, the following class of polynomials has a cluster of roots extremely close to
{2}^{-100}
mig := n -> x^n - (nextprime(2^100)*x^2 - 1)^2:
time(Isolate(mig(10)));
\textcolor[rgb]{0,0,1}{0.015}
time(Isolate(mig(10), method=RS));
\textcolor[rgb]{0,0,1}{0.027}
\textcolor[rgb]{0,0,1}{0.209}
\textcolor[rgb]{0,0,1}{0.877}
time(Isolate(mig(100)));
\textcolor[rgb]{0,0,1}{1.125}
time(Isolate(mig(100), method=RS));
\textcolor[rgb]{0,0,1}{7.609}
\textcolor[rgb]{0,0,1}{7.469}
timelimit(600, Isolate(mig(200), method=RS));
\textcolor[rgb]{0,0,1}{"time expired"}
f := add(rand(-1. .. 1.)() * x^i, i=0..100):
time(Isolate(f, digits=100));
\textcolor[rgb]{0,0,1}{0.056}
time(Isolate(f, digits=100, method=RS));
\textcolor[rgb]{0,0,1}{0.239}
time(Isolate(f, digits=1000));
\textcolor[rgb]{0,0,1}{0.293}
time(Isolate(f, digits=1000, method=RS));
\textcolor[rgb]{0,0,1}{107.328}
time(Isolate(f, digits=10000));
\textcolor[rgb]{0,0,1}{10.824}
timelimit(600, Isolate(f, digits=10000, method=RS));
\textcolor[rgb]{0,0,1}{"time expired"}
RootFinding:-Isolate
|
9 Exercise (2 points)
{\displaystyle {}x,y,z,w}
be elements in a field and suppose that
{\displaystyle {}z}
{\displaystyle {}w}
are not zero. Prove the following fraction rules.
{\displaystyle {\frac {x}{1}}=x,}
{\displaystyle {\frac {1}{-1}}=-1,}
{\displaystyle {\frac {0}{z}}=0,}
{\displaystyle {\frac {z}{z}}=1,}
{\displaystyle {\frac {x}{z}}={\frac {xw}{zw}},}
{\displaystyle {\frac {x}{z}}\cdot {\frac {y}{w}}={\frac {xy}{zw}},}
{\displaystyle {\frac {x}{z}}+{\frac {y}{w}}={\frac {xw+yz}{zw}}.}
Does there exist an analogue of formula (7), which arises when one replaces addition by multiplication (and subtraction by division), that is
{\displaystyle {}(x-z)\cdot (y-w)=(x+w)\cdot (y+z)-(z+w)\,?}
Show that the “popular formula”
{\displaystyle {}{\frac {x}{z}}+{\frac {y}{w}}={\frac {x+y}{z+w}}\,}
Determine which of the two rational numbers
{\displaystyle {}p}
{\displaystyle {}q}
is larger:
{\displaystyle p={\frac {573}{-1234}}{\text{ und }}q={\frac {-2007}{4322}}.}
a) Give an example of rational numbers
{\displaystyle {}a,b,c\in {]0,1[}}
{\displaystyle a^{2}+b^{2}=c^{2}.}
b) Give an example of rational numbers
{\displaystyle {}a,b,c\in {]0,1[}}
{\displaystyle a^{2}+b^{2}\neq c^{2}.}
c) Give an example of irrational numbers
{\displaystyle {}a,b\in {]0,1[}}
and a rational number
{\displaystyle {}c\in {]0,1[}}
{\displaystyle a^{2}+b^{2}=c^{2}.}
The following exercises should only be made with reference to the ordering axioms of the real numbers.
Prove the following properties of real numbers.
{\displaystyle {}1>0}
{\displaystyle {}a\geq b}
{\displaystyle {}c\geq 0}
{\displaystyle {}ac\geq bc}
{\displaystyle {}a\geq b}
{\displaystyle {}c\leq 0}
{\displaystyle {}ac\leq bc}
{\displaystyle {}a^{2}\geq 0}
{\displaystyle {}a\geq b\geq 0}
{\displaystyle {}a^{n}\geq b^{n}}
{\displaystyle {}n\in \mathbb {N} }
{\displaystyle {}a\geq 1}
{\displaystyle {}a^{n}\geq a^{m}}
{\displaystyle {}n\geq m}
{\displaystyle {}a>0}
{\displaystyle {}{\frac {1}{a}}>0}
{\displaystyle {}a>b>0}
{\displaystyle {}{\frac {1}{a}}<{\frac {1}{b}}}
Show that for a real number
{\displaystyle {}x\geq 3}
{\displaystyle x^{2}+(x+1)^{2}\geq (x+2)^{2}}
{\displaystyle {}x<y}
be two real numbers. Show that for the arithmetic mean
{\displaystyle {}{\frac {x+y}{2}}}
{\displaystyle {}x<{\frac {x+y}{2}}<y\,}
Prove the following properties for the absolute value function
{\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto \vert {x}\vert ,}
(here let
{\displaystyle {}x,y}
be arbitrary real numbers).
{\displaystyle {}\vert {x}\vert \geq 0}
{\displaystyle {}\vert {x}\vert =0}
{\displaystyle {}x=0}
{\displaystyle {}\vert {x}\vert =\vert {y}\vert }
{\displaystyle {}x=y}
{\displaystyle {}x=-y}
{\displaystyle {}\vert {y-x}\vert =\vert {x-y}\vert }
{\displaystyle {}\vert {xy}\vert =\vert {x}\vert \vert {y}\vert }
{\displaystyle {}x\neq 0}
{\displaystyle {}\vert {x^{-1}}\vert =\vert {x}\vert ^{-1}}
{\displaystyle {}\vert {x+y}\vert \leq \vert {x}\vert +\vert {y}\vert }
(Triangle inequality for the absolute value).
Sketch the following subsets of
{\displaystyle {}\mathbb {R} ^{2}}
{\displaystyle {}{\left\{(x,y)\mid x=5\right\}}}
{\displaystyle {}{\left\{(x,y)\mid x\geq 4{\text{ und }}y=3\right\}}}
{\displaystyle {}{\left\{(x,y)\mid y^{2}\geq 2\right\}}}
{\displaystyle {}{\left\{(x,y)\mid \vert {x}\vert =3{\text{ und }}\vert {y}\vert \leq 2\right\}}}
{\displaystyle {}{\left\{(x,y)\mid 3x\geq y{\text{ und }}5x\leq 2y\right\}}}
{\displaystyle {}{\left\{(x,y)\mid xy=0\right\}}}
{\displaystyle {}{\left\{(x,y)\mid xy=1\right\}}}
{\displaystyle {}{\left\{(x,y)\mid xy\geq 1{\text{ und }}y\geq x^{3}\right\}}}
{\displaystyle {}{\left\{(x,y)\mid 0=0\right\}}}
{\displaystyle {}{\left\{(x,y)\mid 0=1\right\}}}
{\displaystyle {}x_{1},\ldots ,x_{n}}
be real numbers. Show by induction the following inequality
{\displaystyle \vert {\sum _{i=1}^{n}x_{i}}\vert \leq \sum _{i=1}^{n}\vert {x_{i}}\vert .}
Prove the general distributive property for a field.
{\displaystyle {}\mathbb {R} ^{2}}
{\displaystyle {}{\left\{(x,y)\mid x+y=3\right\}}}
{\displaystyle {}{\left\{(x,y)\mid x+y\leq 3\right\}}}
{\displaystyle {}{\left\{(x,y)\mid (x+y)^{2}\geq 4\right\}}}
{\displaystyle {}{\left\{(x,y)\mid \vert {x+2}\vert \geq 5{\text{ and }}\vert {y-2}\vert \leq 3\right\}}}
{\displaystyle {}{\left\{(x,y)\mid \vert {x}\vert =0{\text{ and }}\vert {y^{4}-2y^{3}+7y-5}\vert \geq -1\right\}}}
{\displaystyle {}{\left\{(x,y)\mid -1\leq x\leq 3{\text{ and }}0\leq y\leq x^{3}\right\}}}
A page has been ripped from a book. The sum of the numbers of the remaining pages is
{\displaystyle {}65000}
. How many pages did the book have?
Hint: Show that it cannot be the last page. From the two statements “A page is missing” and “The last page is not missing” two inequalities can be set up to deliver the (reasonable) upper and lower bound for the number of pages.
|
Regional Math Olympiad 2017 - Problems and Solutions - Cheenta
Regional Math Olympiad 2017
Here are the questions asked in Regional Math Olympiad 2017 and their solutions. Try to solve it first and then see the solutions.
Looking for just the problems? Download the PDF here.
RMO 2017, Problem 1:
Let AOB be a given angle less than \( 180^o \) and let P be an interior point of the angular region determined by \( \angle AOB \) . Show, with proof, how to construct, using only ruler and compass, a line segment CD passing through P such that C lies on the ray OA and D lies on the ray OB and CP:PD = 1:2.
Show that the equation $$ a^3 + (a+1)^3 + (a+2)^3 + (a+3)^3 + (a+4)^3 + (a+5)^3 + (a+6)^3 = b^4 + (b+1)^4 $$ has no solutions in integer a, b.
Let P(x)=x2+12x+b
P(x) = x^2 + \frac {1}{2} x + b
and Q(x)=x2+cx+d
Q(x) = x^2 + cx + d
be two polynomials with real coefficients such that P(x) Q(x) = Q(P(x)) for all real x. Find all real roots of P(Q(x)) = 0
Consider n2
n^2
unit squares in the xy-plane centered at the point (i, j) with integer coordinates, 1≤i≤n
1 \le i \le n
, 1≤j≤n
1 \le j \le n
. It is required to color each unit square in such a way that whenever 1≤ i < j and 1 ≤ k < l ≤ n the three squares with centers at (i, k), (j, k) , (j, l) have distinct colours. What is the least possible colours needed?
Let \Omega
be a circle with a chord AB which is not a diameter. Let Γ1
\Gamma_1
be a circle on one side of AB such that it is tangent to AB at C and internally tangent to \Omega
at D. Likewise let Γ2
\Gamma_2
be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to \Omega
at F. Suppose the line DC intersects \Omega
at X≠D
X \neq D
and the line FE intersects \Omega
at Y≠F
Y \neq F
. Prove that XY is a diameter of Ω
Written solution part 1
Let x, y, z be real numbers, each greater than 1. Prove that
$\frac{x+1}{y+1}$ + $\frac{y+1}{z+1}$ + $\frac{z+1}{x+1}$ $\leq$ $\frac{x-1}{y-1}$ + $\frac{y-1}{z-1}$ + $\frac{z-1}{x-1}$.
~ Discussion by Souvik Mondal & Writabrata Bhattacharya (Associate Faculty - Cheenta)
Practice Previous Year Problems for RMO
Cyclic Pentagon - RMO 2008, Problem 1 - Watch and Learn
P(x) = x^2 + \frac {1}{2} x + b
Q(x) = x^2 + cx + d
n^2
1 \le i \le n
1 \le j \le n
Let \Omega
\Gamma_1
be a circle on one side of AB such that it is tangent to AB at C and internally tangent to \Omega
\Gamma_2
be a circle on the other side of AB such that it is tangent to AB at E and internally tangent to \Omega
at F. Suppose the line DC intersects \Omega
X \neq D
and the line FE intersects \Omega
Y \neq F
|
ConicsTutor - Maple Help
Home : Support : Online Help : Education : Student Packages : Precalculus : Interactive : ConicsTutor
Student[Precalculus][ConicsTutor] - illustrates graphs and information of conic sections
ConicsTutor()
ConicsTutor(f)
(optional) input equation of the conic section in cartesian xy-coordinates or polar rt-coordinates
The ConicsTutor command launches a tutor interface that illustrates the graph and provides information about the related conic section.
The equation f can be one of the following forms:
\mathrm{expr1}=\mathrm{expr2}
\mathrm{expr1}
\mathrm{expr2}
are of degree at most 2, in terms of x and y.
\mathrm{expr}
\mathrm{expr}
is of degree at most
2
in terms of x and y.
r=g\left(t\right)
g\left(t\right)
is the polar form of an equation for a conic section, in terms of the variable t.
g\left(t\right)
g\left(t\right)
g\left(t\right)
can either be a constant or be in the form
\frac{a}{b+cd\left(t\right)}
, where a, b, and c are real numbers, and d is the sine or cosine function.
If f is not specified, ConicsTutor uses a default function.
By default, the tutor returns the plot when you close it. You can instead choose to return nothing, an embedded table (including the plot and summary information), the command used to generate the plot, or summary information.
\mathrm{with}\left(\mathrm{Student}[\mathrm{Precalculus}]\right):
If the ConicsTutor command is run with no arguments, the default expression is
{x}^{2}+{y}^{2}=1
\mathrm{ConicsTutor}\left(\right)
\mathrm{ConicsTutor}\left({x}^{2}+{y}^{2}=1\right)
\mathrm{ConicsTutor}\left({x}^{2}+2xy+{y}^{2}+2x-2y+4\right)
Parabola in polar coordinates:
\mathrm{ConicsTutor}\left(r=\frac{1}{1+\mathrm{cos}\left(t+\frac{\mathrm{\pi }}{3}\right)}\right)
\mathrm{ConicsTutor}\left({x}^{2}+xy+{y}^{2}-3x-1=0\right)
\mathrm{ConicsTutor}\left(8{x}^{2}+24xy+{y}^{2}+x+2y+1\right)
|
Tactic in poker and other card games
In this 1904 cartoon by E. A. Bushnell, the Russian Empire (represented by a bear) and the Empire of Japan (represented by a fox) play poker, with their respective arsenals as stakes. Both wonder if the other is bluffing. The Russo-Japanese War began 17 days later.
In the card game of poker, a bluff is a bet or raise made with a hand which is not thought to be the best hand. To bluff is to make such a bet. The objective of a bluff is to induce a fold by at least one opponent who holds a better hand. The size and frequency of a bluff determines its profitability to the bluffer. By extension, the phrase "calling somebody's bluff" is often used outside the context of poker to describe situations where one person demands that another proves a claim, or proves that they are not being deceptive.[1]
1 Pure bluff
2 Semi-bluff
3 Bluffing circumstances
4 Optimal bluffing frequency
4.1 Example (Texas Hold'em)
5 Bluffing in other games
Pure bluff[edit]
A pure bluff, or stone-cold bluff, is a bet or raise with an inferior hand that has little or no chance of improving. A player making a pure bluff believes they can win the pot only if all opponents fold. The pot odds for a bluff are the ratio of the size of the bluff to the pot. A pure bluff has a positive expectation (will be profitable in the long run) when the probability of being called by an opponent is lower than the pot odds for the bluff.
For example, suppose that after all the cards are out, a player holding a busted drawing hand decides that the only way to win the pot is to make a pure bluff. If the player bets the size of the pot on a pure bluff, the bluff will have a positive expectation if the probability of being called is less than 50%. Note, however, that the opponent may also consider the pot odds when deciding whether to call. In this example, the opponent will be facing 2-to-1 pot odds for the call. The opponent will have a positive expectation for calling the bluff if the opponent believes the probability the player is bluffing is at least 33%.
Semi-bluff[edit]
In games with multiple betting rounds, to bluff on one round with an inferior or drawing hand that might improve in a later round is called a semi-bluff. A player making a semi-bluff can win the pot two different ways: by all opponents folding immediately or by catching a card to improve the player's hand. In some cases a player may be on a draw but with odds strong enough that they are favored to win the hand. In this case their bet is not classified as a semi-bluff even though their bet may force opponents to fold hands with better current strength.
For example, a player in a stud poker game with four spade-suited cards showing (but none among their downcards) on the penultimate round might raise, hoping that their opponents believe the player already has a flush. If their bluff fails and they are called, the player still might be dealt a spade on the final card and win the showdown (or they might be dealt another non-spade and try to bluff again, in which case it is a pure bluff on the final round rather than a semi-bluff).
Bluffing circumstances[edit]
Bluffing may be more effective in some circumstances than others. Bluffs have a higher expectation when the probability of being called decreases. Several game circumstances may decrease the probability of being called (and increase the profitability of the bluff):
Fewer opponents who must fold to the bluff.
The bluff provides less favorable pot odds to opponents for a call.
A scare card comes that increases the number of superior hands that the player may be perceived to have.
The player's betting pattern in the hand has been consistent with the superior hand they are representing with the bluff.
The opponent's betting pattern suggests the opponent may have a marginal hand that is vulnerable to a greater number of potential superior hands.
The opponent's betting pattern suggests the opponent may have a drawing hand and the bluff provides unfavorable pot odds to the opponent for chasing the draw.
Opponents are not irrationally committed to the pot (see sunk cost fallacy).
Opponents are sufficiently skilled and paying sufficient attention.
The opponent's current state of mind should be taken into consideration when bluffing. Under certain circumstances external pressures or events can significantly impact an opponent's decision making skills.
Optimal bluffing frequency[edit]
If a player bluffs too infrequently, observant opponents will recognize that the player is betting for value and will call with very strong hands or with drawing hands only when they are receiving favorable pot odds. If a player bluffs too frequently, observant opponents snap off their bluffs by calling or re-raising. Occasional bluffing disguises not just the hands a player is bluffing with, but also their legitimate hands that opponents may think they may be bluffing with. David Sklansky, in his book The Theory of Poker, states "Mathematically, the optimal bluffing strategy is to bluff in such a way that the chances against your bluffing are identical to the pot odds your opponent is getting."
Optimal bluffing also requires that the bluffs must be performed in such a manner that opponents cannot tell when a player is bluffing or not. To prevent bluffs from occurring in a predictable pattern, game theory suggests the use of a randomizing agent to determine whether to bluff. For example, a player might use the colors of their hidden cards, the second hand on their watch, or some other unpredictable mechanism to determine whether to bluff.
Example (Texas Hold'em)[edit]
Here is an example for the game of Texas Hold'em, from The Theory of Poker:
when I bet my $100, creating a $300 pot, my opponent was getting 3-to-1 odds from the pot. Therefore my optimum strategy was ... [to make] the odds against my bluffing 3-to-1.
Since the dealer will always bet with (nut hands) in this situation, they should bluff with (their) "Weakest hands/bluffing range" 1/3 of the time in order to make the odds 3-to-1 against a bluff.[2]
Ex: On the last betting round (river), Worm has been betting a "semi-bluff" drawing hand with: A♠ K♠ on the board:
10♠ 9♣ 2♠ 4♣ against Mike's A♣ 10♦ hand.
The river comes out:
The pot is currently 30 dollars, and Worm is contemplating a 30-dollar bluff on the river. If Worm does bluff in this situation, they are giving Mike 2-to-1 pot odds to call with their two pair (10's and 2's).
In these hypothetical circumstances, Worm will have the nuts 50% of the time, and be on a busted draw 50% of the time. Worm will bet the nuts 100% of the time, and bet with a bluffing hand (using mixed optimal strategies):
{\displaystyle x=s/(1+s)}
Where s is equal to the percentage of the pot that Worm is bluff betting with and x is equal to the percentage of busted draws Worm should be bluffing with to bluff optimally.
Pot = 30 dollars. Bluff bet = 30 dollars.
s = 30(pot) / 30(bluff bet) = 1.
Worm should be bluffing with their busted draws:
{\displaystyle x=1/(1+s)=50\%}
Where s = 1
Assuming four trials, Worm has the nuts two times, and has a busted draw two times. (EV = expected value)
Worm bets with the nuts (100% of the time)
Worm bets with a busted draw (50% of the time)
Worm checks with a busted draw (50% of the time)
Worm's EV = 60 dollars Worm's EV = 60 dollars Worm's EV = 30 dollars (if Mike folds) and −30 dollars (if Mike calls) Worm's EV = 0 dollars (since they will neither win the pot, nor lose 30 dollars on a bluff)
Mike's EV = −30 dollars (because he would not have won the original pot, but lost to Worm's value bet on the end) Mike's EV = −30 dollars (because he would not have won the original pot, but lost to Worm's value bet on the end) Mike's EV = 60 dollars (if he calls, he'll win the whole pot, which includes Worm's 30-dollar bluff) and 0 dollars (if Mike folds, he can't win the money in the pot) Mike's EV = 30 dollars (assuming Mike checks behind with the winning hand, he will win the 30-dollar pot)
Under the circumstances of this example: Worm will bet their nut hand two times, for every one time they bluff against Mike's hand (assuming Mike's hand would lose to the nuts and beat a bluff). This means that (if Mike called all three bets) Mike would win one time, and lose two times, and would break even against 2-to-1 pot odds. This also means that Worm's odds against bluffing is also 2-to-1 (since they will value bet twice, and bluff once).
Say in this example, Worm decides to use the second hand of their watch to determine when to bluff (50% of the time). If the second hand of the watch is between 1 and 30 seconds, Worm will check their hand down (not bluff). If the second hand of the watch is between 31 and 60 seconds, Worm will bluff their hand. Worm looks down at their watch, and the second hand is at 45 seconds, so Worm decides to bluff. Mike folds his two pair saying, "the way you've been betting your hand, I don't think my two pair on the board will hold up against your hand." Worm takes the pot by using optimal bluffing frequencies.
This example is meant to illustrate how optimal bluffing frequencies work. Because it was an example, we assumed that Worm had the nuts 50% of the time, and a busted draw 50% of the time. In real game situations, this is not usually the case.
The purpose of optimal bluffing frequencies is to make the opponent (mathematically) indifferent between calling and folding. Optimal bluffing frequencies are based upon game theory and the Nash equilibrium, and assist the player using these strategies to become unexploitable. By bluffing in optimal frequencies, you will typically end up breaking even on your bluffs (in other words, optimal bluffing frequencies are not meant to generate positive expected value from the bluffs alone). Rather, optimal bluffing frequencies allow you to gain more value from your value bets, because your opponent is indifferent between calling or folding when you bet (regardless of whether it's a value bet or a bluff bet).[3]
Bluffing in other games[edit]
Although bluffing is most often considered a poker term, similar tactics are useful in other games as well. In these situations, a player makes a play that should not be profitable unless an opponent misjudges it as being made from a position capable of justifying it. Since a successful bluff requires deceiving one's opponent, it occurs only in games in which the players conceal information from each other. In games like chess and backgammon, both players can see the same board and so should simply make the best legal move available. Examples include:
Contract Bridge: Psychic bids and falsecards are attempts to mislead the opponents about the distribution of the cards. A risk (common to all bluffing in partnership games) is that a bluff may also confuse the bluffer's partner. Psychic bids serve to make it harder for the opponents to find a good contract or to accurately place the key missing cards with a defender. Falsecarding (a tactic available in most trick taking card games) is playing a card that would naturally be played from a different hand distribution in hopes that an opponent will wrongly assume that the falsecarder made a natural play from a different hand and misplay a later trick on that assumption.
Stratego: Much of the strategy in Stratego revolves around identifying the ranks of the opposing pieces. Therefore, depriving your opponent of this information is valuable. In particular, the "Shoreline Bluff" involves placing the flag in an unnecessarily-vulnerable location in the hope that the opponent will not look for it there. It is also common to bluff an attack that one would never actually make by initiating pursuit of a piece known to be strong, with an as-yet unidentified but weaker piece. Until the true rank of the pursuing piece is revealed, the player with the stronger piece might retreat if their opponent does not pursue them with a weaker piece. That might buy time for the bluffer to bring in a faraway piece that can actually defend against the bluffed piece.
Spades: In late game situations, it is useful to bid a nil even if it cannot succeed.[4] If the third seat bidder sees that making a natural bid would allow the fourth seat bidder to make an uncontestable bid for game, they may bid nil even if it has no chance of success. The last bidder then must choose whether to make their natural bid (and lose the game if the nil succeeds) or to respect the nil by making a riskier bid that allows their side to win even if the doomed nil is successful. If the player chooses wrong and both teams miss their bids, the game continues.
Scrabble: Scrabble players will sometimes deliberately play a phony word in the hope the opponent does not challenge it. Bluffing in Scrabble is a bit different from the other examples. Scrabble players conceal their tiles but have little opportunity to make significant deductions about their opponent's tiles (except in the endgame) and even less opportunity to spread disinformation about them. Bluffing by playing a phony is instead based on assuming players have imperfect knowledge of the acceptable word list.[citation needed]
Evan Hurwitz and Tshilidzi Marwala developed a software agent that bluffed while playing a poker-like game.[5][6] They used intelligent agents to design agent outlooks. The agent was able to learn to predict its opponents' reactions based on its own cards and the actions of others. By using reinforcement neural networks, the agents were able to learn to bluff without prompting.[7]
In economics, bluffing has been explained as rational equilibrium behavior in games with information asymmetries. For instance, consider the hold-up problem, a central ingredient of the theory of incomplete contracts. There are two players. Today player A can make an investment; tomorrow player B offers how to divide the returns of the investment. If player A rejects the offer, they can realize only a fraction x<1 of these returns on their own. Suppose player A has private information about x. Goldlücke and Schmitz (2014) have shown that player A might make a large investment even if player A is weak (i.e., when they know that x is small). The reason is that a large investment may lead player B to believe that player A is strong (i.e., x is large), so that player B will make a generous offer. Hence, bluffing can be a profitable strategy for player A.[8]
^ "call bluff". The Free Dictionary by Farlex. Retrieved October 22, 2020.
^ Game Theory and Poker
^ a b The Mathematics of Poker, Bill Chen and Jerrod Ankenman
^ Marwala, Tshilidzi; Hurwitz, Evan (May 7, 2007). "Learning to bluff". arXiv:0705.0693 [cs.AI].
^ "Software learns when it pays to deceive". New Scientist. May 30, 2007.
^ "1XBET". Friday, 20 May 2022
^ Goldlücke, Susanne; Schmitz, Patrick W. (2014). "Investments as signals of outside options". Journal of Economic Theory. 150: 683–708. doi:10.1016/j.jet.2013.12.001. ISSN 0022-0531.
David Sklansky (1987). The Theory of Poker. Two Plus Two Publications. ISBN 1-880685-00-0.
David Sklansky (2001). Tournament Poker for Advanced Players. Two Plus Two Publications. ISBN 1-880685-28-0.
David Sklansky and Mason Malmuth (1988). Hold 'em Poker for Advanced Players. Two Plus Two Publications. ISBN 1-880685-22-1.
Dan Harrington and Bill Robertie (2004). Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume I: Strategic Play. Two Plus Two Publications. ISBN 1-880685-33-7.
Dan Harrington and Bill Robertie (2005). Harrington on Hold'em: Expert Strategy For No-Limit Tournaments; Volume II: The Endgame. Two Plus Two Publications. ISBN 1-880685-35-3.
Bill Chen, Jerrod Ankenman. The Mathematics of Poker.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bluff_(poker)&oldid=1088813645"
|
Newton's method - Simple English Wikipedia, the free encyclopedia
algorithm for finding a zero of a function
Newton's method provides a way for finding the real zeros of a function. This algorithm is sometimes called the Newton–Raphson method, named after Sir Isaac Newton and Joseph Raphson.
The method uses the derivative of the function in order to find its roots. An initial "guess value" for the location of the zero must be made. From this value, a new guess is calculated by this formula:
{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}}
Here xn is the initial guess and xn+1 is the next guess. The function f (whose zero is being solved for) has the derivative f'.
By repeatedly applying this formula to the generated guesses (that is by setting the value of xn to the formula's output and recomputing), the value of the guesses will approach a zero of the function.
The function (blue) is being used to calculate the slope of a tangent line (red) at xn.
Newton's method can be explained graphically by looking at intersections of tangent lines with the x-axis. First, a line tangent to the f at xn is calculated. Next, the intersection between this tangent line and the x-axis is found. Finally, the x-position of this intersection is recorded as the next guess, xn+1.
Problems with Newton's MethodEdit
Newton's method can find a solution quickly if the guess value begins sufficiently near the desired root. However, when the initial guess value is not close, and depending on the function, Newton's method may find the answer slowly or not at all.
Fernández, J. A. E., & Verón, M. Á. H. (2017). Newton’s method: An updated approach of Kantorovich’s theory. Birkhäuser.
Peter Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Second printed edition. Series Computational Mathematics 35, Springer (2006)
Kantorovich theorem (Statement about the convergence of Newton's method, found by Leonid Kantorovich)
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Newton%27s_method&oldid=7194482"
|
Active Science for Class 7 Science Chapter 13 - Time And Speed
Active Science Solutions for Class 7 Science Chapter 13 Time And Speed are provided here with simple step-by-step explanations. These solutions for Time And Speed are extremely popular among Class 7 students for Science Time And Speed Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Active Science Book of Class 7 Science Chapter 13 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Active Science Solutions. All Active Science Solutions for class Class 7 Science are prepared by experts and are 100% accurate.
Which of these activities can be measured in hours?
(a) writing one page
(b) eating a sandwich
(c) a journey from Chennai to Mumbai
(d) watching a TV serial
Writing a page and eating a sandwich need a smaller unit than hour. Watching a TV serial can be measured in hours, but the most appropriate option is the journey from Chennai to Mumbai.
The speed of a moving object means
(a) how far it goes.
(b) how long it goes on moving.
(c) how far it goes in a certain time.
(c) how far it goes in a certain time
Speed is the rate at which an object covers a given distance in a particular period of time.
On Saturday night, Prachi spent 18 minutes on her social science homework, 35 minutes on her mathematics homework and 22 minutes on her English homework. How much time did she spend on her homework in total?
Time spent by Prachi on social science homework = 18 min
Time spent by Prachi on mathematics homework = 35 min
Time spent by Prachi on English homework = 22 min
Total time spent by Prachi on her homework = 18 + 35 + 22
=75 min or (60 + 15) min (∵ 1 h = 60 min)
If a moving body covers equal distances in equal intervals of time, it is said
(a) to have the same velocity.
(b) to have uniform motion.
(c) to have non-uniform motion
(b) to have uniform motion
Uniform motion is a motion in which a body covers equal distances in equal intervals of time. This suggests that the body moves at a constant speed.
Study the table given below. Tick the correct statement.
Vikram 144 km 2 h
Tarun 25 m 2 s
Ram 100 m 6 s
(a) Vikram travels fastest.
(b) All three travel at the same speed.
(c) Tarun and Ram travel at the same speed.
(d) Ram travels at the greatest speed.
(a) Vikram travels the fastest.
Distance travelled by Vikram = 144 km
×
= 144000 m (∵ 1 km = 1000 m)
Time taken by him = 2 h
×
×
60) s
= 7200 s (∵ 1 h = 60
×
60 s)
Speed of Vikram = Distance travelled/Time taken
Speed of Tarun = Distance travelled by Tarun/Time taken
Speed of Ram = Distance travelled by Ram/Time taken
It is clear from the above calculation that Vikram travels the fastest among the three.
A. A train runs from New Delhi to Kolkata. It first covers a distance of 400 km in 7 hours and then a distance of 550 km in 8 hours.
B. Ratna takes part in a car race. She drivers a distance of 70 km in the first, second and third hours.
(a) A is an example of uniform motion and B is an example of non-uniform motion.
(b) A is an example of non-uniform motion and B is an example of uniform motion.
(c) A and B are examples of uniform motion.
(d) A and B are example of non-uniform motion.
A. Speed of the train in the first part of the journey = 400/7
= 57.14 km/h
Speed of the train in the second part of the journey= 550/8
= 68.75 km/h (∵ Speed = Distance/Time)
In both the parts, the train has different speeds. Hence, this is an example of non-uniform motion.
B. Ratna travels a distance of 70 km in the first, second and third hours. It means that she is travelling equal distances in equal intervals of time. In other words, she is travelling at a constant speed. Thus, this is an example of uniform motion.
This graph shows−
(a) the motion of your school bus as it picks up students.
(b) the motion of Ravi, who stops at the market on his way back home from school.
(c) the motion of an ant as it collects rice grains.
(d) the motion of an athlete running a 200 m race.
(b) the motion of Ravi, who stops at the market on his way back home from the school.
Ravi stops for (11
-
7) minutes, that is, 4 minutes.
In which of these graphs is the object at rest?
(a) graph 1
(b) graph 2
(c) graph 3
(d) graph 4
Here, the distance of the object is constant with respect to time.
In which of these graphs is the speed of the moving object constant?
The distance–time graph of an object moving at a constant speed is always represented by a straight line.
The graph shows the motion of two vehicles A and B. Which one of them is moving with greater speed?
(c) both A and B are moving with the same speed.
(d) Cannot say from these graphs.
The above relation shows that the speed of a vehicle is greater if it covers maximum distance in a given interval of time. To compare the distances, draw a line perpendicular to the time axis.
It is now evident that for a given time t, the distance covered by vehicle A is more than that covered by vehicle B. Hence, vehicle A is moving with a greater speed.
Express these times according to the 24-hour clock.
(e) 7.05 pm
This is 2 hours after the beginning of a day, so the 24-hour clock will show the time as 0225.
(c) 10:50 p.m.
This is 22 hours (12 + 10) after the beginning of a day, so the 24-hour clock will show the time as 2250.
This is 20 hours (12 + 8) after the beginning of a day, so the 24-hour clock will show the time as 2005.
(e) 7:05 p.m.
(a) 0050 hr
(b) 1650 hr
(c) 1830 hr
(d) 1007 hr
(e) 2345 hr
Express the following speeds in m/s.
1 h = (60
×
60 ) s = 3600 s
(a) 45 km/h = (45
×
1000)/(1
×
(b) 135 km/h = (135
×
×
(c) 90 km/h = (90
×
×
(d) 75 km/h = (75
×
×
Express the following speeds in km/h.
1 m = (1/1000) km
×
60) s or 3600 s = 1 h
1 s = (1/3600) h
= \frac{\left(65/1000\right)}{\left(1/3600\right)}
= 234 \mathrm{km}/\mathrm{h}
(b) 40 m/s
= \frac{\left(40/1000\right)}{\left(1/3600\right)}
= 144 \mathrm{km}/\mathrm{h}
(c) 100 m/s
= \frac{\left(100/1000\right)}{\left(1/3600\right)}
= 360 \mathrm{km}/\mathrm{h}
(d) 10 m /s
= \frac{\left(10/1000\right) }{\left(1/3600\right) }
= 36 \mathrm{km}/\mathrm{h}
An athlete covers 1500 m in 4 minutes. Calculate his speed in m/s and km/h.
×
Distance travelled by the athlete = 1500 m
= (1500/1000) km
Time taken = 4 min
= 4/60 h
= 1/15 h or (4
×
Speed in m/s:
= 1.5/(1/15)
A cheetah runs with a speed of 96 km/h. how long would it take to cover 288000 m.
Speed of the cheetah = 96 km/h
Distance covered by the cheetah = 288000 m
= (288000/1000) km
= 288 km (∵ 1 km = 1000 m)
∴ Time = Distance/Speed
What is the speed of a swimmer if she covers 100 m in 60 seconds?
Distance covered by the swimmer = 100 m
Time taken by the swimmer = 60 s
∴ Speed = Distance/Time
How far would a car travel in 4 seconds at a speed of 45 km/h?
= \frac{45×1000}{1 ×3600}
= 12.5 m/s (∵ 1 km = 1000 m and 1 h = 3600 s)
Time taken by the car = 4 s
Using the formula, we get:
×
×
Prashant cycled 100 m in 20 seconds. At this rate how long will it take him to go 1 km? What is his speed in km/h? What distance will he cover in 40 minutes?
Distance travelled by Prashant in 20 s = 100 m
∴ Distance travelled by Prashant in 1 s = 100/20
Speed of Prashant = 5 m/s
Time taken by Prashant to travel a distance of 5 m = 1 s
∴ Time taken by him to travel a distance of 1 km or 1000 m
= \frac{1}{5}×1000
= 200 \mathrm{s}
= \frac{\left(5/1000\right)}{1/3600}
= 18 \mathrm{km}/\mathrm{h}
(∵ 1 m = 1/1000 km and 1 s = 1/3600 h)
Distance covered in 1 s = 5 m
∴ Distance covered in 40 min or 2400 s = 5
×
Therefore, Prashant will cover 12 kilometres in 40 minutes.
Point out the mistakes in the following cases.
(a) Two cars move for 5 minutes and 2 minutes respectively. The second car is faster because it takes less time.
(b) Two cars move at a speed of 45 km/h. Their velocities are the same.
(c) Trains moving in the same direction have the same velocity.
(d) A motorist travels 600 km. Another motorist travels only 560 km. the second car is slower than the first.
(a) The distances travelled by the two cars are not given; these are required to calculate the speeds of the cars and to evaluate which car is faster. Hence, we cannot say that the second car is faster than the first car.
(b) If both the cars are moving at a speed of 45 km/h, then it is not necessary that their velocities are also the same because the directions of motion are not given.
(c) For the same velocity, the magnitudes and directions must be the same, but only direction is given in the statement.
(d) Time, which is required to calculate the speeds of the cars and to evaluate which motorist is slower, is not given. Hence, we cannot say that the second car is slower than the first.
Sarang wanted to study how fast snails can move. To do this he placed four snails next to each other and marked their trails. He put a cross (x) where each snail had reached after 20 seconds.
(a) Which snail went fastest?
(b) If snail C went on at the same speed for another 10 seconds how far would it go beyond point X?
Distance travelled by snail A = 70 mm
= (70/1000) m
∴ Speed of snail A = Distance travelled/Time taken
Speed of snail B = 0.05/20
Speed of snail C = 0.06/20
Speed of snail D = 0.04/20
(a) Snail A moved the fastest.
(b) Speed of snail C = 0.0030 m/s
×
×
Thus, snail C would go a distance of 0.03 m or 30 mm beyond point X.
Study the table given below and the answer the questions that follow.
City Distance From Delhi Time Taken Aircraft
Agra 360 km _______h Pushpak
Nagpur 765 km 3 h 45 m Dakota
Mumbai ___________km 1 h 50 m Airbus
Kolkata 1035 Km 1 h 55 m Boeing 737
Chennai 1860 m __________ MIG
(a) Calculate the speeds of a Boeing 737 and a Dakota.
(b) Calculate how long a Pushpak travelling at 200km/h will take to cover the distance between Delhi and Agra?
(c) An Airbus travels at the same speed as s Boeing 737. What is the distance between Delhi and Mumbai?
(d) How long will a MIG with the speed of 1800 km/h take to travel from Delhi to Chennai?
(a) For Boeing 737:
Time = 1 h 55 min
= 1 h + (55/60) h
= 23/12 h
Speed of Boeing 737 = Distance travelled/Time taken
= 955.4 km/h
For Dakota:
=15/4 h
Speed of Dakota = 765/(15/4)
(b) Speed of Pushpak = 200 km/h
Time taken by Pushpak = Distance/Speed
(c) Speed of the airbus = Speed of Boeing 737 = 955.4 km/h
Time taken = 1 h 50 min
= 11/6 h
∴ Required distance = Speed
×
×
(d) Speed of MIG = 1800 km/h
Distance travelled = 1860 m
= 1.860/1800
The graph shows the journeys of Raj and Suraj.
(a) What was the speed of Raj in the first five minutes?
(b) Both of them felt hungry and stopped at Ramji's Mithaiwala. For how long did they stop at the shop?
(c) How far from the start did Raj meet Suraj?
(d) How long did they walk together?
(a) Distance travelled by Raj = 500 m
×
(b) Time that they spent at the shop = 35
-
They stopped at the shop for 20 minutes.
(c) Thousand metres far from the start
(d) Distance that they walked together = 1500
-
They walked together a distance of 500 metres.
|
Develop a Model That Complies with the IEC 61508 Standard - MATLAB & Simulink - MathWorks Benelux
Run the IEC 61508 Modeling Standard Checks
Review Check Results
This example shows how to use the Model Advisor to check that a model complies with the IEC 61508 safety standard.
The IEC 61508 Model Advisor checks identify issues with a model that impede deployment in safety-related applications or limit traceability.
Model rtwdemo_iec61508 check whether the 1-norm distance between points (x1,x2) and (y1,y2) is less than or equal to a given threshold thr. For two points (x1,x2) and (y1,y2), the 1-norm distance is given as:
\sum _{i=1}^{2}|{x}_{i}-{y}_{i}|
Open and review the model.
model='rtwdemo_iec61508';
To deploy the model in a safety-related software component that must comply with the IEC 61508 safety standard, check the model for issues that might impede deployment in such an environment or limit traceability between the model and generated source code.
To open the Model Advisor, in the Simulink® editor, click the Modeling tab and select Model Advisor. A System Selector - Model Advisor dialog box opens. Select the model or system that you want to review and click OK. Or enter modeladvisor('rtwdemo_IEC61508') at the MATLAB® command line.
In the left pane of the Model Advisor, expand By Task > Modeling Standards for IEC 61508. If the By Task folder is not displayed in the Model Advisor window, open Settings > Preferences and select Show By Task Folder.
Select the checks that you want to include in your model analysis.
Click on the folder that contains the checks and, on the right pane of the Model Advisor, select Show report after run to automatically generate and display the report in HTML format.
Click Run Selected Checks to execute the analysis.
The Model Advisor processes the IEC 61508 checks and displays the results.
After the analysis is complete, review the aggregate results in the right pane of the Model Advisor. You can see the total number of checks that passed, failed, were flagged as warnings, and did not execute.
To review the results for a specific check, in the By Task > Modeling Standards for IEC 61508 folder, select a check. For more information about the check and how to resolve reported issues, click Help.
Address the reported issues and rerun the checks.
To review the generated HTML report of the results, select the By Task > Modeling Standards for IEC 61508 folder and click the link in the Report box.
Print the generated HTML report. You can use the report as evidence in the IEC 61508 compliance process.
If desired, click the Generate Code Using Embedded Coder button in the model to inspect the generated code and the traceability report.
For descriptions of the IEC 61508 checks, see IEC 61508, IEC 62304, ISO 26262, ISO 25119, and EN 50128/EN 50657 Checks (Simulink Check) in the Simulink Check™ documentation.
For more information on using Model Advisor, see Run Model Advisor Checks and Review Results in the Simulink documentation.
For more information on resolving issues, see Address Model Check Results in the Simulink documentation.
|
Bijective function - Citizendium
(Redirected from Bijection)
3 Bijections and the concept of cardinality
In mathematics, an invertible function, also known as a bijective function or simply a bijection is a function that establishes a one-to-one correspondence between elements of two given sets. Loosely speaking, all elements of the sets can be matched up in pairs so that each element of one set has its unique counterpart in the second set. A bijective function from a set X to itself is also called a permutation of the set X.
More formally, a function
{\displaystyle f}rom set
{\displaystyle X}
{\displaystyle Y}
is called a bijection if and only if for each
{\displaystyle y}
{\displaystyle Y}
{\displaystyle x}
{\displaystyle X}
{\displaystyle f(x)=y}
The most important property of a bijective function is the existence of an inverse function which undoes the operation of the function. These functions can then be viewed as dictionaries by which one can translate information from the domain to the codomain and back again. The existence of an inverse function often forces the domain and codomain to have common properties.
The function from set
{\displaystyle \{1,2,3,4\}}
{\displaystyle \{10,11,12,13\}}
{\displaystyle f(x)=x+9}
A less obvious example is the function
{\displaystyle f}rom the set
{\displaystyle X=\{(x,y)\}}
of all pairs (x,y) of positive integers to the set of all positive integers given by formula
{\displaystyle f(x,y)=2^{x-1}\cdot (2y-1)}
{\displaystyle \tan \colon (-{\frac {\pi }{2}},{\frac {\pi }{2}})\to R}
{\displaystyle f\colon X\to Y}
{\displaystyle g\colon Y\to Z}
are bijections than so is their composition
{\displaystyle g\circ f\colon X\to Z}
{\displaystyle f\colon X\to Y}
is a bijective function if and only if there exists function
{\displaystyle g\colon Y\to X}
such that their compositions
{\displaystyle g\circ f}
{\displaystyle f\circ g}
are identity functions on relevant sets. In this case we call function
{\displaystyle g}
an inverse function o{\displaystyle f}
and denote it by
{\displaystyle f^{-1}}
Bijections and the concept of cardinality
Two finite sets have the same number of elements if and only if there exists a bijection from one set to another. Georg Cantor generalized this simple observation to infinite sets and introduced the concept of cardinality of a set. We say that two set are equinumerous (sometimes also equipotent or equipollent) if there exists a bijection from one set to another. If this is the case, we say the sets have the same cardinality or the same cardinal number. Cardinality can be thought of as a generalization of number of elements of finite sets.
A function is a bijection iff it is both an injection and a surjection.
The quadratic function
{\displaystyle R\to R:x\mapsto x^{2}}
is neither injection nor surjection, hence is not bijection. However if we change its domain and codomain to the set
{\displaystyle [0,+\infty )}
than the function becomes bijective and the inverse function
{\displaystyle {\sqrt {\colon }}[0,+\infty )\to [0,+\infty ),\ x\mapsto {\sqrt {x}}}
exists. This procedure is very common in mathematics, especially in calculus.
A continuous function from the closed interval
{\displaystyle [a,b]}
in the real line to closed interval
{\displaystyle [c,d]}
is bijection if and only if is monotonic function with f(a) = c and f(b) = d.
Retrieved from "https://citizendium.org/wiki/index.php?title=Bijective_function&oldid=506379"
|
Memory efficient spearman correlation on sparse matrices
GoI Newsletter
Utilizing a simple property of covariances to minimize memory usage
An important property of the covariance function is that it is invariant under shifts, i.e., for any two random variables
\mathbf{X}
\mathbf{Y}
, you get the same covariance if you add constant quantitie to either $X$ or $Y$:
\begin{aligned} \text{Cov}(\mathbf{X} + a, \mathbf{Y} + b) &= \text{Cov}(\mathbf{X}, \mathbf{Y}) \end{aligned}
\text{Cov}(\mathbf{X}, \mathbf{Y}) = \mathbb{E}[(\mathbf{X} - \mathbb{E}[\mathbf{X}])(\mathbf{X} - \mathbb{E}[\mathbf{X}])]
a,b
are real valued quantities. Essentially
\text{Cov}(\mathbf{X}, \mathbf{Y})
is a measure of the product of how much $\mathbf{X}$ and $\mathbf{Y}$ are deviating from their respective means so adding a constant does not change anything (because the deviation from the mean remains the same).
Two commonly used correlations are pearson and spearman . A pearson correlation is essentially a normalized measure of covariance, which tries to measure how “linearly dependent” are $\textit{X}$ and $\textit{Y}$:
\begin{aligned} \text{Cor}(\mathbf{X} + a, \mathbf{Y} + b) &= \frac{\text{Cov}(\mathbf{X}, \mathbf{Y})}{\sigma_X \sigma_Y},\\ \sigma^2_X &= \mathbb{E}[(X-\mathbb{E}[X])^2),\\ \sigma^2_Y &= \mathbb{E}[(Y-\mathbb{E}[Y])^2).\\ \end{aligned}
The spearman correlation on the other hand asseses if the relationship between $\textit{X,Y}$ is monotonic (either increasing or decreasing). It is equivalent to running pearson correlation between the ranks of values in $X$ and $Y$ instead of the actual value themselves. So it essentially asks if $X$ is increasing (decreasing) would values in $Y$ would be increasing (decreasing) as well? A perfect score of 1 (-1) would result in a yes (no). Both the types of correlation are often employed in genomics to assess relationship between two variables of interest.
One particular context, where correlations are employed is in multi-omics experiments, say where we are profiling RNA and open chromatin regions (ATAC) in the same cells. For example, a recent study used correlations to find potential gene-enhancer links (Ma et al., 2020). The idea is simple: we have a bunch of cells in which we simultaneously profiled both the transcriptome (RNA) and the open chromatin regions (ATAC). We then ask, for each gene, which open chromatin regions are highly correlated (after necessary adjustment for background) to predict potential gene-enhancer links. The default correlation function in R cor(RNA, ATAC, method="pearson") or cor(RNA, ATAC, method="spearman") would ideally be sufficient to do this. Here, RNA and ATAC are vectors of equal length with entries summarizing the transcriptome signal and ATAC signal at a gene and potential enhancer, respectively.
However, both RNA and ATAC matrices are often sparse matrices, i.e. they have lots of entries that zeroes, which are not explicitly stored to save space. The default cor() method does not work on sparse matrices. The problem here is a simple one then: convert the RNA and ATAC sparse matrices to a usual (dense) matrix using as.matrix() and run the correlation function. However, converting to denser matrix format will take loads of memory, especially if you are searching for link between 10,000 genes and say only about 5,000 potential enhancers in around 10,000 cells all at once, parallely.
Sparsity makes it easier
The solution to avoid this is rather easy and has been previously discussed for pearson correlation. A detailed description is available in the documentation of qlcMatrix::corSparse(). But in short, the idea is to utilize the sparsity in a vector and avoid doing operations that would make a sparse matrix dense. For example, the variance calulation for a sparse vect the essential idea here is that we do not want to lose the sparsity structure during our calculations. For example, for a sparse vector, if we are interested in calculating the variance $Var(X) = \mathbb{E}[(X-\mathbb{E}[X])^2]$, if we do the $X-\mathbb{E}[X]$ operation first, the sparsity structure of X is now destroyed and we land up with a dense matrix. Instead, we can use the fact that the variance can equivalentyl be written as $\text{Var}(X) = \mathbb{E}[X^2] - E[X]^2$, retaining the sparity throughout. That solves our problem of calculating pearson correlation on sparse vectors (or matrices).
The next question is then, what about sparse matrices and spearman correlation? cor(X, Y, kind="spearman") does not work for sparse matrices and we do not want to convert them to dense form. The solution is again simple, but took me a while to figure out. A naive idea would be to use the definition of spearman correlation - we calculate ranks of $X$ and $Y$ and then run it through cor() with method="spearman" as the ranks are not sparse. The problem however is again the same - the rank matrix is not sparse. But if you think about ranks in a sparse matrix, it does have some interesting properties that we can utilize to make it sparse.
We can look at a sparse vector for an example. Consider a vector y <- c(0,0,0,42,21,10) with 3 non zero entries. We will use $n_z$ to denote the number of non-zero entries in a vector. But if we know the number of non-zero entries, we also know what these ranks are going to be - they are fixed. For a vector with $n_z$ entries, the rank(ties.method="average") method will set them all to $\frac{1}{n_z}\sum_{i=1}^{n_z} i = \frac{(n_z+1)}{2}$. We also know that the lowest non-zero entry in such a vector would have a rank of $(n_z+1)$. For example, rank vector rank(y) = c(2,2,2,6,5,4) - by default the ranks of tied entries are averaged. So the rank of 0s is $\frac{1+2+3}{3}= \frac{(n_z+1)}{2}$. Our rank vector is not sparse, but we can retain its sparsity if we were to subtract $\frac{(n_z+1)}{2}$ from each of the entries. Since a shift operation will not change the (co)variance, the variance of c(0,0,0,4,3,2) which we called the “sparsified rank vector” is the same as original rank vector c(2,2,2,6,5,4). So we should aim to get our “sparsified rank” vector somehow.
The trick to arrive at “sparsified rank” vector is to use calculate ranks on the non-zero entries in our vector. We will forget about the zero entries in such a vector and only focus on the non-zero entries - they are few and it is fast to calculate ranks of just these. In this version of the vector (where there are no zeros) the lowest non-zero entry has a rank of $1$ (assuming there are no ties, but the following arguments hold without loss of generality). To arrive at the “sparsified rank” vector, we subtracted $\frac{(n_z+1)}{2}$ from the original rank vector, so the non-zero entry’s rank will now be $n_z + 1 - \frac{(n_z+1)}{2} = 1 + \frac{n_z}{2} - \frac{1}{2}$ which is equivalent to adding $\frac{(n_z-1)}{2}$ to the rank of the non-zero entries! By this way, we retain the sparsity in ranks and can then just use corSparse() to calculate pearson correlation on sparsified rank vectors, resulting in spearman correlation.
While this approach is memory efficient, it unfortunately is not always the fastest. See this notebook for some time benchmarks. I did not explictly perform memory benchmarks.
Update: The approach is both memory efficient and fast. See an updated post and associated notebook
y <- c(0,0,0,42,21,10)
rank(y) = c(2,2,2,6,5,4)
sparsified_rank(y) <- c(0,0,0,4,3,2) (Subtract $\frac{(n_z+1)}{2}=2$ from all entries to make the previous vector a sparse vector)
rank(y[y!=0]) = rank(c(42,21,10)) = c(3,2,1) .
If we now add $\frac{(n_z-1)}{2} = \frac{(3-1)}{2}$ to all the entries of the last vector, we get c(4,3,2) which are the non-zero ranks from our <code<sparisifed_rank</code> vector which will be the input to corSparse.
Ma, S., Zhang, B., LaFave, L. M., Earl, A. S., Chiang, Z., Hu, Y., Ding, J., Brack, A., Kartha, V. K., Tay, T., & others. (2020). Chromatin potential identified by shared single-cell profiling of RNA and chromatin. Cell, 183(4), 1103–1116.
|
Hyperbolic equations and SBV functions
In this article we survey some recent results in the regularity theory of admissible solutions to hyperbolic conservation laws and Hamilton-Jacobi equations.
title = {Hyperbolic equations and {SBV} functions},
TI - Hyperbolic equations and SBV functions
De Lellis, Camillo. Hyperbolic equations and SBV functions. Journées équations aux dérivées partielles (2010), article no. 6, 10 p. doi : 10.5802/jedp.63. http://www.numdam.org/articles/10.5802/jedp.63/
[1] Alberti, G., and Ambrosio, L. A geometrical approach to monotone functions in
{\mathbf{R}}^{n}
. Math. Z. 230, 2 (1999), 259–316. | MR 1676726 | Zbl 0934.49025
[2] Ambrosio, L., and De Lellis, C. A note on admissible solutions of 1D scalar conservation laws and 2D Hamilton-Jacobi equations. J. Hyperbolic Differ. Equ. 1, 4 (2004), 813–826. | MR 2111584 | Zbl 1071.35032
[3] Ambrosio, L., De Lellis, C., and Malý, J. On the chain rule for the divergence of BV-like vector fields: applications, partial results, open problems. In Perspectives in nonlinear partial differential equations, vol. 446 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2007, pp. 31–67. | MR 2373724
[4] Ambrosio, L., Fusco, N., and Pallara, D. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000. | MR 1857292 | Zbl 0957.49001
[5] Ancona, F., and Coclite, G. M. On the attainable set for Temple class systems with boundary controls. SIAM J. Control Optim. 43, 6 (2005), 2166–2190 (electronic). | MR 2179483 | Zbl 1087.93010
[6] Ancona, F., and Marson, A. On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optim. 36, 1 (1998), 290–312 (electronic). | MR 1616586 | Zbl 0919.35082
[7] Ancona, F., and Marson, A. Asymptotic stabilization of systems of conservation laws by controls acting at a single boundary point. In Control methods in PDE-dynamical systems, vol. 426 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2007, pp. 1–43. | MR 2311519 | Zbl 1128.35069
[8] Ancona, F., and Nguyen, K. T. SBV regularity for solutions to genuinely nonlinear Temple systems of balance laws. In preparation.
[9] Bianchini, S., and Caravenna, L. SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws. In preparation.
[10] Bianchini, S., De Lellis, C., and Robyr, R. SBV regularity for Hamilton-Jacobi equations in
{R}^{n}
. To appear in Arch. Rat. Mech. Anal. (2010).
[11] Bressan, A. Hyperbolic systems of conservation laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. | MR 1816648 | Zbl 0997.35002
[12] Bressan, A., and Marson, A. A maximum principle for optimally controlled systems of conservation laws. Rend. Sem. Mat. Univ. Padova 94 (1995), 79–94. | Numdam | Zbl 0935.49012
[13] Bressan, A., and Shen, W. Optimality conditions for solutions to hyperbolic balance laws. In Control methods in PDE-dynamical systems, vol. 426 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2007, pp. 129–152. | MR 2311524
[14] Cannarsa, P., and Sinestrari, C. Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston Inc., Boston, MA, 2004. | MR 2041617 | Zbl 1095.49003
[15] Dafermos, C. M. Hyperbolic conservation laws in continuum physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2000. | MR 1763936 | Zbl 0940.35002
[16] Dafermos, C. M. Wave fans are special. Acta Math. Appl. Sin. Engl. Ser. 24, 3 (2008), 369–374. | MR 2433867 | Zbl 1170.35478
[17] De Giorgi, E., and Ambrosio, L. New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82, 2 (1988), 199–210 (1989). | MR 1152641 | Zbl 0715.49014
[18] Evans, L. C. Partial differential equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998. | MR 1625845 | Zbl 0902.35002
[19] Robyr, R. SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function. J. Hyperbolic Differ. Equ. 5, 2 (2008), 449–475. | MR 2420006 | Zbl 1152.35074
[20] Tonon, D. Personal communication..
|
Derived Hall algebras
1 December 2006 Derived Hall algebras
Bertrand Toën1
1Laboratoire Emile Picard, UMR CNRS 5580, Université Paul Sabatier
The purpose of this work is to define a derived Hall algebra
\mathrm{DH}\left(T\right)
, associated to any differential graded (DG) category
T
(under some finiteness conditions), generalizing the Hall algebra of an abelian category. Our main theorem states that
\mathrm{DH}\left(T\right)
is associative and unital. When the associated triangulated category
\left[T\right]
is endowed with a t-structure with heart
A
\mathrm{DH}\left(T\right)
contains the usual Hall algebra
H\left(A\right)
. We also prove an explicit formula for the derived Hall numbers purely in terms of invariants of the triangulated category associated to
T
. As an example, we describe the derived Hall algebra of a hereditary abelian category
Bertrand Toën. "Derived Hall algebras." Duke Math. J. 135 (3) 587 - 615, 1 December 2006. https://doi.org/10.1215/S0012-7094-06-13536-6
Bertrand Toën "Derived Hall algebras," Duke Mathematical Journal, Duke Math. J. 135(3), 587-615, (1 December 2006)
|
Small_stellated_dodecahedron Knowpia
Type Kepler–Poinsot polyhedron
Stellation core regular dodecahedron
V = 12 (χ = -6)
Faces by sides 12 5
Schläfli symbol {5⁄2,5}
Face configuration V(55)/2
Wythoff symbol 5 | 2 5⁄2
Properties Regular nonconvex
(5⁄2)5
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {5⁄2,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex.
3D model of a small stellated dodecahedron
If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the height of the pentagonal pyramids adjusted so that the five triangles in the pentagram become coplanar. The critical angle is atan(2) above the dodecahedron face.
If we regard it as having 12 pentagrams as faces, with these pentagrams meeting at 30 edges and 12 vertices, we can compute its genus using Euler's formula
{\displaystyle V-E+F=2-2g}
and conclude that the small stellated dodecahedron has genus 4. This observation, made by Louis Poinsot, was initially confusing, but Felix Klein showed in 1877 that the small stellated dodecahedron could be seen as a branched covering of the Riemann sphere by a Riemann surface of genus 4, with branch points at the center of each pentagram. In fact this Riemann surface, called Bring's curve, has the greatest number of symmetries of any Riemann surface of genus 4: the symmetric group
{\displaystyle S_{5}}
acts as automorphisms[1]
(See also: animated)
This polyhedron also represents a spherical tiling with a density of 3. (One spherical pentagram face, outlined in blue, filled in yellow)
It can also be constructed as the first of three stellations of the dodecahedron, and referenced as Wenninger model [W20].
Floor mosaic by Paolo Uccello, 1430
A small stellated dodecahedron can be seen in a floor mosaic in St Mark's Basilica, Venice by Paolo Uccello circa 1430.[2] The same shape is central to two lithographs by M. C. Escher: Contrast (Order and Chaos) (1950) and Gravitation (1952).[3]
Animated truncation sequence from {5⁄2, 5} to {5, 5⁄2}
Its convex hull is the regular convex icosahedron. It also shares its edges with the great icosahedron; the compound with both is the great complex icosidodecahedron.
There are four related uniform polyhedra, constructed as degrees of truncation. The dual is a great dodecahedron. The dodecadodecahedron is a rectification, where edges are truncated down to points.
The truncated small stellated dodecahedron can be considered a degenerate uniform polyhedron since edges and vertices coincide, but it is included for completeness. Visually, it looks like a regular dodecahedron on the surface, but it has 24 faces in overlapping pairs. The spikes are truncated until they reach the plane of the pentagram beneath them. The 24 faces are 12 pentagons from the truncated vertices and 12 decagons taking the form of doubly-wound pentagons overlapping the first 12 pentagons. The latter faces are formed by truncating the original pentagrams. When an {n⁄d}-gon is truncated, it becomes a {2n⁄d}-gon. For example, a truncated pentagon {5⁄1} becomes a decagon {10⁄1}, so truncating a pentagram {5⁄2} becomes a doubly-wound pentagon {10⁄2} (the common factor between 10 and 2 mean we visit each vertex twice to complete the polygon).
Stellations of the dodecahedron
Platonic solid Kepler–Poinsot solids
Compound of small stellated dodecahedron and great dodecahedron
^ Weber, Matthias (2005). "Kepler's small stellated dodecahedron as a Riemann surface". Pacific J. Math. Vol. 220. pp. 167–182. pdf
^ Coxeter, H. S. M. (2013). "Regular and semiregular polyhedra". In Senechal, Marjorie (ed.). Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination (2nd ed.). Springer. pp. 41–52. doi:10.1007/978-0-387-92714-5_3. See in particular p. 42.
^ Barnes, John (2012). Gems of Geometry (2nd ed.). Springer. p. 46.
Eric W. Weisstein, Small stellated dodecahedron (Uniform polyhedron) at MathWorld.
|
Biswa and Picnic | Toph
Biswa and Picnic
Biswa’s department is going on a picnic and as usual, he is a member of the organizing committee. This time, he is in charge of transportation. He has decided to allocate a bus for each class of the department.
All the buses are of same configuration. There are 40 rows in a bus and each row has exactly 6 seats. There is an aisle in the middle to move inside the bus. So in each row, there are 3 adjacent seats on both sides of the aisle. Rows are numbered from 1 to 40 and seats are labeled as A, B, C, D, E, F on each row. A, B, C are adjacent on one half of the row whereas D, E, F are adjacent on the other.
Here is a picture of the first 8 rows of the bus:
At the beginning, there is a queue in front of all the buses. All the students from a particular class are waiting in the queue in front of their allocated bus.
Everyone in a queue starts moving at the 0-th second. It takes exactly one second to move:
from the front of the queue to the first row in the aisle
between adjacent positions in the queue
between adjacent seats on same row
between adjacent rows in the aisle
from a row in the aisle to any aisle adjacent seat on the same row
It can be safely assumed that the aisle is divided into 40 rows as well.
Biswa has to assign seats for everyone in a way so that the time when everyone has reached their assigned seat is minimized. That is, the time, when the latest person (not necessarily the last person on the queue) reaches his or her allocated seat, is as little as possible.
First line of the test case contains the number
T (
1 ≤ T ≤ 240
1≤T≤240), denoting the number of classes in the department. Then
T lines follow and in each line
i (
1 ≤ i ≤ T
1≤i≤T), there’s a number
N (
1 ≤ N ≤ 240
1≤N≤240), denoting the number of students in
i-th class).
For each class, you have to print the minimum possible time when everyone can reach their assigned seat.
One possible assignment that minimizes the required time is, assigning first person in the queue to seat D in the first row and assigning second person in the queue to seat C in the first row.
At the end of the 1st second, first person in the queue moves to first row in the aisle, second person in the queue moves to the front of the queue.
At the end of 2nd second, first person moves from first row in the aisle to seat D in the first row, second person moves from front of the queue to first row in the aisle.
At the end of 3rd second, second person moves from first row in the queue to seat C in the first row.
mdyaminEarliest, Apr '19
mdyaminFastest, 0.0s
pabonsahaLightest, 0 B
The time to reach a particular seat from the front of the queue can be calculated as the summation o...
IUT 10th ICT Fest Programming Contest
Replay of IUT 10th ICT Fest Programming Contest
|
Broly and His Respect | Toph
Broly and His Respect
By rakibahmed · Limits 500ms, 512 MB
Broly is the legendary warrior who loves his planet and always prepares himself to protect the planet from his enemies. Frieza is the warrior from another planet who is the enemy of Broly. To take the control of Broly’s planet, Frieza sends his Army to fight with Broly.
The fighters are numbered from
1 to
N and each fighter comes sequentially one after another to fight with Broly. The
i^{th}
ith fighter have
i amount of strength. After destroying
i^{th}
ith fighter Broly gains
i^{3}
i3 of respect, and initially, Broly doesn’t have any respect. After completing the fight, Broly needs to report to his master how much respect he earned after destroying
N number of fighters and the total strength of Frieza’s Army. During the fight, he was able to calculate his respect after destroying each fighter but somehow he forgot that for which fighter
j, he earned total
X respect.
Formally, if Broly destroys a total of
9 fighters and his master asks him after destroying which fighter Broly gains
36 respect then the answer will be
3 because after destroying the
3^{rd}
3rd fighter Broly gains a total of
36 respect.
There were a total of
j^{2}
j2 fighters in the Frieza's Army. Now, your task is to find out the total strength of the Frieza’s Army.
Each test contains multiple test cases. The first line contains
t (
1 \leq t \leq 1000
The only line of each test case will contain an integer
X (
1 \leq X \leq 10^6
1≤X≤106) — The respect earned by Broly after destroying
j^{th}
jth fighter.
For each test case, output the total strength of the Frieza’s Army.
Being_GoromEarliest, 11M ago
ADRI0777Fastest, 0.0s
Being_GoromLightest, 131 kB
Being_GoromShortest, 232B
ULAB Take-off Programming Contest - Summer 2021
|
Spoiler Alert | Toph
By rezaulhsagar · Limits 10s, 32 MB
This problem contains Breaking Bad TV series spoilers. If you haven't watched season 5 so far, you can skip this problem to avoid spoilers.
We're done when I say we're done. - Walter White to Saul Goodman
After killing Gus Fring, Walter White and Jesse Pinkman are planning to run their own Methamphetamine business.
One month, they produced a batch of Methamphetamine which has an interesting property. It contains all amount of packets from
1 to
n distinctly. As the business is risky, their lawyer Saul Goodman wants to know the oddness of the batch.
To calculate the oddness of the batch, first you have to arrange all meth packets from
1 to
n serially in an array. The oddness of the batch is the number of the subarrays which contains an odd sum in this array. For example, there are
4 meth packets in the batch. So, the array will be like
\{1,2,3,4\}. There are 6 subarrays which contains an odd sum:
\{1\}, \{3\}, \{1,2\}, \{2, 3\}, \{3, 4\}, \{2, 3, 4\}
{1},{3},{1,2},{2,3},{3,4},{2,3,4}. So, the oddness of the batch is
6.
Can you help Saul Goodman?
t
(1 \leq t \leq 5000)
(1≤t≤5000) — the number of test cases.
Each of the next
t lines contains an integer
n
(1 \leq n \leq 10^{7})
(1≤n≤107) — the number of packets.
In a single line, output the oddness of the batch.
sunkuet02Earliest, Jul '20
KabbyaFastest, 0.0s
sunkuet02Lightest, 131 kB
Replay of Intra KUET Programming Contest 2020
|
Formats/Text - Maple Help
Home : Support : Online Help : Formats/Text
Text (.txt) File Format
A plain text file is a universal format for textual information.
The Export as Plain Text command in the File menu can be used to export a document to a text file.
The Import and Export commands can be used to read and write to text files.
The FileTools[Text] package can be used for more advanced processing of text files.
Import a text file into Maple as a string
\mathrm{story}≔\mathrm{Import}\left("example/UnderTheMaples.txt",\mathrm{base}=\mathrm{datadir}\right)
\textcolor[rgb]{0,0,1}{\mathrm{story}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{"Under The Maples, by John Burroughs Chapter I: The Falling Leaves The time of the falling of leaves has come again. Once more in our morning walk we tread upon carpets of gold and crimson, of brown and bronze, woven by the winds or the rains out of these delicate textures while we slept."}
\mathrm{StringTools}:-\mathrm{WordCount}\left(\mathrm{story}\right)
\textcolor[rgb]{0,0,1}{54}
|
Modeling Mutual Coupling in Large Arrays Using Embedded Element Pattern - MATLAB & Simulink Example - MathWorks América Latina
Create Full-Wave Model of the 11 X 11 Array
Comparison with 25 X 25 Array
This example demonstrates the embedded element pattern approach to model large finite arrays. Such an approach is only good for very large arrays so that the edge effects may be ignored. It is common to consider an infinite array analysis as a first step for such kind of analysis. This approach is presented in Modeling Mutual Coupling in Large Arrays Using Infinite Array Analysis; Modeling Mutual Coupling in Large Arrays Using Infinite Array Analysis>. The embedded element pattern refers to the pattern of a single element embedded in the finite array, that is calculated by driving the central element in the array and terminating all other elements into a reference impedance [1]-[3]. The pattern of the driven element, referred to as the embedded element, incorporates the effect of coupling with the neighboring elements. It is common to choose the central region/element of the array for the embedded element, depending on whether the array has an even or odd number of elements(for large arrays it does not matter). The pattern of the isolated element (the radiator located in space by itself) changes when it is placed in an array due to the presence of mutual coupling. This invalidates the use of pattern multiplication, which assumes that all elements have the same pattern. To use pattern multiplication to calculate the total array radiation pattern, and improve the fidelity of the analysis, we replace the isolated element pattern with the embedded element pattern.
As mentioned in the introduction, it is the aim of this example to illustrate the use of the embedded element pattern when modeling large finite arrays. To do so we will model 2 arrays: first using the pattern of the isolated element, second with the embedded element pattern and compare the results of the two with the full-wave Method of Moments (MoM) based solution of the array. The array performance for scanning at broadside, and for scanning off broadside is established. Finally, we adjust the array spacing to investigate the occurrence of scan blindness and compare against reference results [3]. For this example we choose the center of the X-band as our design frequency.
In [4], it was discussed that the central element of a 5
\lambda
\lambda
array starts to behave like it is in an infinite array. Such an aperture would correspond to a 10 X 10 array of half-wavelength spaced radiators. We choose to slightly exceed this limit and consider a 11 X 11 array of
\lambda /2
dipoles.
Dipole as Antenna Element
The individual element we choose is a dipole. Choose its length to be slightly lower than
\lambda /2
and radius of approximately
\lambda /150
URA with an Isolated Dipole
Create a 11 X 11 URA and assign the isolated dipole as its element. Adjust the spacing to be half-wavelength at 10 GHz. The dipole tilt is now set to zero, so that its orientation matches the array geometry in the Y-Z plane.
myURA2.Element = mydipole;
Use the Antenna Toolbox™ to create a full-wave model of the 11 x 11 array of resonant dipoles. Since the default orientation of the dipole element in the library is along z-axis, we tilt it so that the array is initially formed in the X-Y plane and then tilt the array to match the array axis of the URA.
myFullWaveArray = rectangularArray;
myFullWaveArray.Element = mydipole;
myFullWaveArray.Element.Tilt = 90;
myFullWaveArray.Element.TiltAxis = [0 1 0];
myFullWaveArray.Size = [Nrow Ncol];
myFullWaveArray.RowSpacing = drow;
myFullWaveArray.ColumnSpacing = dcol;
myFullWaveArray.Tilt = 90;
myFullWaveArray.TiltAxis = 'Y';
show(myFullWaveArray)
Calculate Embedded Element Pattern
Calculate the full 3D embedded element pattern in terms of the electric field magnitude. In [3], the scan resistance and scan reactance for an infinite array of resonant dipoles spaced
\lambda /2
apart is provided. Choose the resistance at broadside as the termination for all elements. To calculate the embedded element pattern, use the pattern function and pass in additional input parameters of the element number(index of the center element) and termination resistance.
ElemCenter = (prod(myFullWaveArray.Size)-1)/2 + 1;
h = waitbar(0,'Calculating center element embedded pattern....');
embpattern = pattern(myFullWaveArray,freq,az,el, ...
'ElementNumber',ElemCenter, ...
'Termination',real(Zinf), ...
waitbar(1,h,'Pattern computation complete');
URA with Embedded Element Pattern
Import this embedded element pattern into the custom antenna element.
embpattern = 20*log10(embpattern);
EmbAnt = phased.CustomAntennaElement('FrequencyVector',freqVector,...
Create a uniform rectangular array(URA), with the custom antenna element, which has the embedded element pattern.
myURA1.Element = EmbAnt;
Calculate the pattern in the elevation plane (specified by azimuth = 0 deg and also called the E-plane) and azimuth plane (specified by elevation = 0 deg and called the H-plane) for the three arrays : based on the isolated element pattern, based on the embedded element pattern, and based on the full-wave model.
Eplane1 = pattern(myURA1,freq,0,el);
[Eplane3,~,el3e] = pattern(myFullWaveArray,freq,0,el);
plot(el,Eplane2,el,Eplane1,el3e,Eplane3,'LineWidth',1.5);
axis([min(el) max(el) -60 30])
xlabel('Elevation Angle (deg.)');
title('E-plane Array Directivity Comparison')
legend('With Isolated Pattern','With Embedded Pattern','Full Wave Solution')
Hplane1 = pattern(myURA1,freq,az/2,0);
Hplane3 = pattern(myFullWaveArray,freq,az/2,0);
plot(az/2,Hplane2,az/2,Hplane1,az/2,Hplane3,'LineWidth',1.5);
axis([min(az/2) max(az/2) -60 30])
xlabel('Azimuth Angle (deg.)');
title('H-plane Array Directivity Comparison')
The array directivity is approximately 23 dBi. This result is close to the theoretical calculation for the peak directivity [5] after taking into account the absence of a reflector, D = 4
\pi
A
{\lambda }^{2}
NrowNcol
A=drow*dcol
Normalize the directivity for the three arrays and plot it for comparison.
Eplanenormlz1 = Eplane1 - max(Eplane1);
plot(el,Eplanenormlz2,el,Eplanenormlz1,el,Eplanenormlz3,'LineWidth',1.5);
axis([min(el) max(el) -60 0])
ylabel('Directivity (dB)');
title('Normalized E-plane Array Directivity Comparison')
Hplanenormlz1 = Hplane1 - max(Hplane1);
plot(az/2,Hplanenormlz2,az/2,Hplanenormlz1,az/2,Hplanenormlz3,'LineWidth',1.5);
title('Normalized H-plane Array Directivity Comparison')
The pattern comparison suggests that the main beam and the first sidelobes are aligned for all three cases. Moving away from the main beam shows the increasing effect of coupling on the sidelobe level. As expected,the embedded element pattern approach suggests a coupling level in between the full-wave simulation model and the isolated element pattern approach.
The behavior of the array pattern is intimately linked to the embedded element pattern. To understand how our choice of a 11 X 11 array impacts the center element behavior, we increase the array size to a 25 X 25 array (12.5
\lambda
X 12.5
\lambda
aperture size). Note that the triangular mesh size for the full wave Method of Moments (MoM) analysis with 625 elements increases to 25000 triangles (40 triangles per dipole) and the computation for the embedded element pattern takes approximately 12 minutes on a 2.4 GHz machine with 32 GB memory. This time can be reduced by lowering the mesh size per element by meshing manually using a maximum edge length of
\lambda /20
load dipolearray
embpattern = 20*log10(DipoleArrayPatData.ElemPat);
EmbAnt2 = clone(EmbAnt);
EmbAnt2.AzimuthAngles = DipoleArrayPatData.AzAngles;
EmbAnt2.ElevationAngles = DipoleArrayPatData.ElAngles;
EmbAnt2.MagnitudePattern = embpattern;
Eplane1 = pattern(EmbAnt2,freq,0,el);
Eplane1 = Eplane1 - max(Eplane1);
Eplane2 = pattern(mydipole,freq,0,el);
embpatE = pattern(EmbAnt,freq,0,el);
embpatE = embpatE-max(embpatE);
plot(el,Eplane2,el,embpatE,el,Eplane1,'LineWidth',1.5);
title('Normalized E-plane Element Directivity Comparison')
legend('IsolatedPattern','Embedded Pattern - 11 X 11','Embedded Pattern - 25 X 25','location', 'best')
Hplane1 = pattern(EmbAnt2,freq,0,az/2);
Hplane1 = Hplane1 - max(Hplane1);
Hplane2 = pattern(mydipole,freq,0,az/2);
embpatH = pattern(EmbAnt,freq,az/2,0);
embpatH = embpatH-max(embpatH);
plot(az/2,Hplane2,az/2,embpatH,az/2,Hplane1,'LineWidth',1.5);
title('Normalized H-plane Element Directivity Comparison')
Scan the array based on the embedded element pattern in the elevation plane defined by azimuth = 0 deg and plot the normalized directivity. Also, overlay the normalized embedded element pattern. Note the overall shape of the normalized array pattern approximately follows the normalized embedded element pattern. This is also predicted by the pattern multiplication principle.
hsv = phased.SteeringVector;
hsv.SensorArray = myURA1;
hsv.IncludeElementResponse = true;
weights = step(hsv,freq,scanEplane);
legend_string1 = cell(1,numel(scan_el1)+1);
legend_string1{end} = 'Embedded element';
scanEPat(:,i) = pattern(myURA1,freq,scan_az1(i),el,'Weights',weights(:,i)); % -23.13;
scanEPat = scanEPat - max(max(scanEPat));
plot(el,scanEPat,'LineWidth',1.5);
plot(el,embpatE,'-.','LineWidth',1.5);
xlabel('Elevation (deg.)')
title('E-plane Scan Comparison')
legend(legend_string1,'Location','southeast')
In large arrays, it is possible that the array directivity will reduce drastically at certain scan angles. At these scan angle, referred to as the blind angles, the array does not radiate the power supplied at its input terminals [3]. Two common mechanisms under which blindness conditions occur are
It is possible to detect scan blindness in large finite arrays by studying the embedded element pattern (also known as array element pattern in the infinite array analysis). The array being investigated in this example does not have a dielectric substrate/ground plane, and therefore the surface waves are eliminated. However we can investigate the second mechanism, i.e. the grating lobe excitation. To do so, let us increase the spacing across rows and columns of the array to be 0.7
\lambda
. Since this spacing is greater than the half-wavelength limit we should expect grating lobes in the visible space beyond a specific scan angle. As pointed out in [3], to accurately predict the depth of grating lobe blind angles in the finite array of dipoles, we need to have an array of the size 41 X 41 or higher. We will compare 3 cases, namely the 11 X 11, 25 X 25 and the 41 X 41 size arrays and check if the existence of blind angles can at least be observed in the 11 X 11 array. As mentioned earlier, the results were precomputed in Antenna Toolbox™ and saved in a MAT file. To reduce the computational time, the elements were meshed with maximum edge length of
\lambda /20
load dipolearrayblindness.mat
The normalized H-plane embedded element pattern for arrays of three sizes. Notice the blind angle around 24-26 deg.
The embedded element pattern approach is one possible way of performing the analysis of large finite arrays. They need to be so large that the edge effects can be ignored. The isolated element pattern is replaced with the embedded element pattern which includes the effect of mutual coupling.
|
Sept. 23d
There never was such a man as you. I did not in the least expect to hear about the Brit. Assoc., & so many things you have told me that I liked to hear.2 What splendid news about Scott, i.e. if he gets it; I do not know when I have been so much pleased & I am delighted that I paid his passage & I fully believe that he will justify all your extraordinary kindness: you have just made the fortune of an able & I am convinced worthy man.3 I shall be disappointed if hereafter he does not do some good work in science. Please remember & let me hear how I must propose him as Assoc. for Linn. Soc.—4 To hear how pleasant Bath was makes me a little envious; but I must try & be contented, for I begin pretty plainly to see that the best I can hope for is not to be worse.—
I thank you sincerely for your previous letter: your openness towards me gratifies me deeply, & you must know that you have my entire sympathy.5 I never remember dates for good or evil, & I do believe I have thus escaped many a bitter day.— Do not be in a hurry about the operation;6 for I distinctly remember some very good authority being against it on such occasions.— How many anxieties & sorrows there are, great & small, as life advances, & nothing to be done but bear them as well as one can, & that I cannot do at all well.—
I enclose a note for Ray Soc. which I hope will do.—7 I am prodigal of suggestions, & it has occurred to me that a Royal Medal might before long be well bestowed on Wallace.—8
I am glad to hear that the Lyells are so well pleased; I think I quite agree to what you say about his Address.9 I regretted most the confined view which he took on change of temperature during Glacial period, with not the slightest allusion to New Zealand or S. America; & he knows well that all our continents are old as continents. I can never believe that change of land & water will suffice.—10
I sent on A. Gray’s note about Orchids direct to Masters,11 as I had a few monstrous plants which I thought he would like to see: he was very glad to get Asa Gray’s note.—
I hardly know what to think about Bentham’s address.12 A man sometimes uses such expressions as “life without renewal or break” in some non-natural sense. I shd. be pleased if he were to give up successive creations. How many have gone thus far within the few last years!—
Remember to give me name of the climbing Nepenthes.—13
I have begun looking over my old M.S.14 & it is as fresh as if I had never written it: parts are astonishingly dull, but yet worth printing I think; & other parts strike me as very good. I am a complete millionaire in odd & curious little facts & I have been really astounded at my own industry whilst reading my chapters on Inheritance & Selection.15 God knows when the Book will ever be completed,16 for I find that I am very weak & on my best days cannot do more than 1 or 1
\frac{1}{2}
hours work. It is a good deal harder than writing about my dear climbing plants.17
GoodBye my dear old fellow & with thanks for your two charming letters,18 farewell; but do not write soon again
Do you object to my putting this sentence from old note from you?19
“Annual plants sometimes become perennial under a different climate, as I hear from Dr. Hooker is the case with the stock & migniotte in Tasmania”.
(say yes or no)
I know the case is nothing wonderful, & I want only just thus to allude to it—
[Draft]20 Down My dear Hooker
Would you propose or suggest for me to the Council of the Ray Society, the translation of Gärtners great work “Versuche & Beobachtungen ueber die Bastarderzeugung 1849” in 790 pages.21 I believe I have read with attention everything that has been published on hybridisation & worked a little practically on the subject, & I do not hesitate to affirm that there is more useful & trust worthy matter in Gärtners work than in all others combined even including Kölreuter perhaps.22
This work is very little known in England & apparently even less in France. I am convinced that the Ray Soc. would confer an essential benefit on natural science by its translation—
My dear Hooker | Yours sincerely
The year is established by the relationship between this letter and the letters from J. D. Hooker, 16 September 1864 and [19 September 1864].
See letter from J. D. Hooker, [19 September 1864].
CD refers to Hooker’s recommendation of John Scott for a new post at a botanic garden in Darjeeling (see letter from J. D. Hooker, [19 September 1864] and n. 22). Hooker had also assisted Scott in making arrangements for the journey to India, and CD had paid for Scott’s passage (see letters from J. D. Hooker, [29 July 1864] and [15 August 1864], and letter to Asa Gray, 13 September [1864] and n. 9).
George Bentham, the president of the Linnean Society, had been impressed by Scott’s paper on reproduction in the Primulaceae (Scott 1864a), and proposed that he be elected an associate of the Society. See letter from J. D. Hooker, [before 9 February 1864], and letter to John Scott, 9 February [1864] and nn. 6–9.
CD refers to Hooker’s discussion of his feelings on the anniversary of his daughter’s death (see letter from J. D. Hooker, 16 September 1864 and n. 24).
The reference to an operation may have been in the missing portion of Hooker’s letter of 16 September 1864.
See letters from J. D. Hooker, 16 September 1864 and [19 September 1864]. CD refers to Charles and Mary Elizabeth Lyell, and to C. Lyell 1864.
In his presidential address to the British Association for the Advancement of Science (C. Lyell 1864, pp. lxx–lxxiii), Lyell confined his discussion of glaciation to central Europe. Lyell first proposed that alterations in physical geography might have given rise to changes in climate in 1830 (see C. Lyell 1830–3, 1: 140, and Ospovat 1977). CD had long been opposed to this theory (see Correspondence vol. 7, letters to J. D. Hooker, 15 March [1859] and 30 March [1859], Correspondence vol. 10, letter to A. C. Ramsay, 5 September [1862], and this volume, letter to J. D. Hooker, 5 April [1864] and n. 8). CD also refers to Lyell’s failure to discuss glaciation in the southern hemisphere; he had been critical of Lyell for not discussing this in Antiquity of man (C. Lyell 1863a); see Correspondence vol. 11, letter to Charles Lyell, 6 March [1863]. CD examined the geological evidence of a past glacial epoch in New Zealand and South America in Origin, and used the hypothesis of a simultaneous glacial period in the northern and southern hemispheres to explain the modern distribution of plant and animal species (see Origin, pp. 373–82). CD incorporated additional information on glaciation in the southern hemisphere in later editions of Origin (see Peckham ed. 1959, pp. 592–3).
See letter from M. T. Masters, 19 September 1864 and nn. 1, 8, and 10.
See letter from J. D. Hooker, 16 September 1864 and n. 18.
See letter to J. D. Hooker, 13 September [1864] and n. 13, and letters from J. D. Hooker, 16 September 1864 and [28 September 1864] and n. 2.
According to his journal, CD had not worked on the manuscript of Variation since 20 July 1863 (see Correspondence vol. 11, Appendix II, and this volume, Appendix II). See also letter to Asa Gray, 13 September [1864]. CD began writing Variation in 1860 (see Correspondence vol. 8, Appendix II).
CD refers to his draft chapter on inheritance, published as chapters 12 to 14 of Variation (Variation 2: 1–84), which was finished on 1 April 1863, and to his draft chapter on selection, which became chapters 20 and 21 of Variation (Variation 2: 192–249), and was completed on 20 July 1863 (see Correspondence vol. 11, Appendix II).
It had taken CD four months to write ‘Climbing plants’ (see ‘Journal’ (Correspondence vol. 12, Appendix II)). See also letter to Asa Gray, 13 September [1864] and n. 3.
Letters from J. D. Hooker, 16 September 1864 and [19 September 1864].
This postscript was written on a separate sheet and was returned to CD. On the original, below the question mark, is written ‘No JH’. The letter in which Hooker made this statement has not been identified, and CD does not appear to have made use of the information. Hooker visited Tasmania in 1840, during his period of service as assistant surgeon and naturalist on HMS Erebus (see R. Desmond 1999, pp. 44–6). For CD’s early interest in the nature of annual plants and the possibility of their becoming perennial, see Correspondence vol. 4, letter from Abraham Clapham, 8 March 1850 and n. 1.
The enclosure to this letter has not been found; however, this is evidently a draft of it. See also letter to Ray Society, [before 4 November 1964].
CD refers to Versuche und Beobachtungen über die Bastarderzeugung im Pflanzenreich (Experiments and observations on the production of hybrids in the plant kingdom: Gärtner 1849). CD had expressed a wish to see this work translated and Hooker had offered to raise the matter with the Ray Society, with the suggestion that CD put his request in writing (see letter to J. D. Hooker, 13 September [1864], and letter from J. D. Hooker, 16 September 1864). For a discussion of the importance of Gärtner 1849 for CD’s work on hybridisation, see the letter to J. D. Hooker, 13 September [1864] and n. 6.
CD refers to Joseph Gottlieb Kölreuter’s work on plant hybridisation (Kölreuter 1761–6). CD’s heavily annotated copy of this work is in the Darwin Library–CUL (see Marginalia 1: 458–71); it is cited in Origin, pp. 271–2, and is also frequently cited in Natural selection and Variation. For the significance attached by CD to Kölreuter’s experiments see, for example, Correspondence vols. 10 and 11. For a discussion of Kölreuter’s and Gärtner’s respective contributions to the understanding of hybridisation, see Olby 1985, pp. 1–39, and Mayr 1986.
Lyell, Charles. 1864. Presidential address. Report of the thirty-fourth meeting of the British Association for the Advancement of Science; held at Bath, pp. lx–lxxv.
Ospovat, Dov. 1977. Lyell’s theory of climate. Journal of the History of Biology 10: 317–39.
DAR 96: 14; DAR 115: 250a–c
ALS 7pp encl (Adraft, 2pp)
|
Superellipsoid - Wikipedia
Superellipsoid collection with exponent parameters, created using POV-Ray. Here, e = 2/r, and n = 2/t (equivalently, r = 2/e and t = 2/n).[1]
In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same exponent r, and whose vertical sections through the center are superellipses with the same exponent t.
Superellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids).[2][3] However, while some superellipsoids are superquadrics, neither family is contained in the other.
2.1 Basic shape
A handful of notable mathematical figures can arise as special cases of superellipsoids given the correct set of values, which are depicted in the above graphic:
Cube, as a limiting case where the exponents tend to infinity
Piet Hein's supereggs are also special cases of superellipsoids.
Basic shape[edit]
The basic superellipsoid is defined by the implicit inequality
{\displaystyle \left(\left|x\right|^{r}+\left|y\right|^{r}\right)^{t/r}+\left|z\right|^{t}\leq 1.}
The parameters r and t are positive real numbers that control the amount of flattening at the tips and at the equator. Note that the formula becomes a special case of the superquadric's equation if (and only if) t = r.
Any "parallel of latitude" of the superellipsoid (a horizontal section at any constant z between -1 and +1) is a Lamé curve with exponent r, scaled by
{\displaystyle a=(1-\left|z\right|^{t})^{1/t}}
{\displaystyle \left|{\frac {x}{a}}\right|^{r}+\left|{\frac {y}{a}}\right|^{r}\leq 1.}
Any "meridian of longitude" (a section by any vertical plane through the origin) is a Lamé curve with exponent t, stretched horizontally by a factor w that depends on the sectioning plane. Namely, if x = u cos θ and y = u sin θ, for a fixed θ, then
{\displaystyle \left|{\frac {u}{w}}\right|^{t}+\left|z\right|^{t}\leq 1,}
{\displaystyle w=(\left|\cos \theta \right|^{r}+\left|\sin \theta \right|^{r})^{-1/r}.}
In particular, if r is 2, the horizontal cross-sections are circles, and the horizontal stretching w of the vertical sections is 1 for all planes. In that case, the superellipsoid is a solid of revolution, obtained by rotating the Lamé curve with exponent t around the vertical axis.
The basic shape above extends from −1 to +1 along each coordinate axis. The general superellipsoid is obtained by scaling the basic shape along each axis by factors A, B, C, the semi-diameters of the resulting solid. The implicit inequality is
{\displaystyle \left(\left|{\frac {x}{A}}\right|^{r}+\left|{\frac {y}{B}}\right|^{r}\right)^{t/r}+\left|{\frac {z}{C}}\right|^{t}\leq 1.}
Setting r = 2, t = 2.5, A = B = 3, C = 4 one obtains Piet Hein's superegg.
The general superellipsoid has a parametric representation in terms of surface parameters -π/2 < v < π/2, -π < u < π.[3]
{\displaystyle x(u,v)=Ac\left(v,{\frac {2}{t}}\right)c\left(u,{\frac {2}{r}}\right)}
{\displaystyle y(u,v)=Bc\left(v,{\frac {2}{t}}\right)s\left(u,{\frac {2}{r}}\right)}
{\displaystyle z(u,v)=Cs\left(v,{\frac {2}{t}}\right)}
where the auxiliary functions are
{\displaystyle c(\omega ,m)=\operatorname {sgn}(\cos \omega )|\cos \omega |^{m}}
{\displaystyle s(\omega ,m)=\operatorname {sgn}(\sin \omega )|\sin \omega |^{m}}
and the sign function sgn(x) is
{\displaystyle \operatorname {sgn}(x)={\begin{cases}-1,&x<0\\0,&x=0\\+1,&x>0.\end{cases}}}
The volume inside this surface can be expressed in terms of beta functions (and Gamma functions, because β(m,n) = Γ(m)Γ(n) / Γ(m + n) ), as:
{\displaystyle V={\frac {2}{3}}ABC{\frac {4}{rt}}\beta \left({\frac {1}{r}},{\frac {1}{r}}\right)\beta \left({\frac {2}{t}},{\frac {1}{t}}\right).}
^ "POV-Ray: Documentation: 2.4.1.11 Superquadric Ellipsoid".
^ Barr, A.H. (January 1981), Superquadrics and Angle-Preserving Transformations. IEEE_CGA vol. 1 no. 1, pp. 11–23
^ a b Barr, A.H. (1992), Rigid Physically Based Superquadrics. Chapter III.8 of Graphics Gems III, edited by D. Kirk, pp. 137–159
Aleš Jaklič, Aleš Leonardis, Franc Solina, Segmentation and Recovery of Superquadrics. Kluwer Academic Publishers, Dordrecht, 2000.
Aleš Jaklič, Franc Solina (2003) Moments of Superellipsoids and their Application to Range Image Registration. IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, 33 (4). pp. 648–657
Bibliography: SuperQuadric Representations
SuperQuadric Ellipsoids and Toroids, OpenGL Lighting, and Timing
Superquadratics by Robert Kragler, The Wolfram Demonstrations Project.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Superellipsoid&oldid=1087556454"
|
Sideband - Wikipedia
In radio communications, a band of frequencies higher or lower than the carrier frequency
This article is about radio communications. For participation in other music projects, see Side project.
The power of an AM radio signal plotted against frequency. fc is the carrier frequency, fm is the maximum modulation frequency
In radio communications, a sideband is a band of frequencies higher than or lower than the carrier frequency, that are the result of the modulation process. The sidebands carry the information transmitted by the radio signal. The sidebands comprise all the spectral components of the modulated signal except the carrier. The signal components above the carrier frequency constitute the upper sideband (USB), and those below the carrier frequency constitute the lower sideband (LSB). All forms of modulation produce sidebands.
1 Sideband creation
1.1 Sideband Characterization
Sideband creation[edit]
We can illustrate the creation of sidebands with one trigonometric identity:
{\displaystyle \cos(A)\cdot \cos(B)\equiv {\tfrac {1}{2}}\cos(A+B)+{\tfrac {1}{2}}\cos(A-B)}
{\displaystyle \cos(A)}
{\displaystyle \cos(A)\cdot [1+\cos(B)]={\tfrac {1}{2}}\cos(A+B)+\cos(A)+{\tfrac {1}{2}}\cos(A-B)}
Substituting (for instance)
{\displaystyle A\triangleq 1000\cdot t}
{\displaystyle B\triangleq 100\cdot t,}
{\displaystyle t}
represents time:
{\displaystyle \underbrace {\cos(1000\ t)} _{\text{carrier wave}}\cdot \underbrace {[1+\cos(100\ t)]} _{\text{amplitude modulation}}=\underbrace {{\tfrac {1}{2}}\cos(1100\ t)} _{\text{upper sideband}}+\underbrace {\cos(1000\ t)} _{\text{carrier wave}}+\underbrace {{\tfrac {1}{2}}\cos(900\ t)} _{\text{lower sideband}}.}
Adding more complexity and time-variation to the amplitude modulation also adds it to the sidebands, causing them to widen in bandwidth and change with time. In effect, the sidebands "carry" the information content of the signal.[1]
Sideband Characterization[edit]
In the example above, a cross-correlation of the modulated signal with a pure sinusoid,
{\displaystyle \cos(\omega t),}
is zero at all values of
{\displaystyle \omega }
except 1100, 1000, and 900. And the non-zero values reflect the relative strengths of the three components. A graph of that concept, called a Fourier transform (or spectrum), is the customary way of visualizing sidebands and defining their parameters.
Frequency spectrum of a typical modulated AM or FM radio signal.
Amplitude modulation[edit]
Amplitude modulation of a carrier signal normally results in two mirror-image sidebands. The signal components above the carrier frequency constitute the upper sideband (USB), and those below the carrier frequency constitute the lower sideband (LSB). For example, if a 900 kHz carrier is amplitude modulated by a 1 kHz audio signal, there will be components at 899 kHz and 901 kHz as well as 900 kHz in the generated radio frequency spectrum; so an audio bandwidth of (say) 7 kHz will require a radio spectrum bandwidth of 14 kHz. In conventional AM transmission, as used by broadcast band AM stations, the original audio signal can be recovered ("detected") by either synchronous detector circuits or by simple envelope detectors because the carrier and both sidebands are present. This is sometimes called double sideband amplitude modulation (DSB-AM), but not all variants of DSB are compatible with envelope detectors.
In some forms of AM, the carrier may be reduced, to save power. The term DSB reduced-carrier normally implies enough carrier remains in the transmission to enable a receiver circuit to regenerate a strong carrier or at least synchronise a phase-locked loop but there are forms where the carrier is removed completely, producing double sideband with suppressed carrier (DSB-SC). Suppressed carrier systems require more sophisticated circuits in the receiver and some other method of deducing the original carrier frequency. An example is the stereophonic difference (L-R) information transmitted in stereo FM broadcasting on a 38 kHz subcarrier where a low-power signal at half the 38-kHz carrier frequency is inserted between the monaural signal frequencies (up to 15 kHz) and the bottom of the stereo information sub-carrier (down to 38–15 kHz, i.e. 23 kHz). The receiver locally regenerates the subcarrier by doubling a special 19 kHz pilot tone. In another example, the quadrature modulation used historically for chroma information in PAL television broadcasts, the synchronising signal is a short burst of a few cycles of carrier during the "back porch" part of each scan line when no image is transmitted. But in other DSB-SC systems, the carrier may be regenerated directly from the sidebands by a Costas loop or squaring loop. This is common in digital transmission systems such as BPSK where the signal is continually present.
If part of one sideband and all of the other remain, it is called vestigial sideband, used mostly with television broadcasting, which would otherwise take up an unacceptable amount of bandwidth. Transmission in which only one sideband is transmitted is called single-sideband modulation or SSB. SSB is the predominant voice mode on shortwave radio other than shortwave broadcasting. Since the sidebands are mirror images, which sideband is used is a matter of convention.
In SSB, the carrier is suppressed, significantly reducing the electrical power (by up to 12 dB) without affecting the information in the sideband. This makes for more efficient use of transmitter power and RF bandwidth, but a beat frequency oscillator must be used at the receiver to reconstitute the carrier. If the reconstituted carrier frequency is wrong then the output of the receiver will have the wrong frequencies, but for speech small frequency errors are no problem for intelligibility. Another way to look at an SSB receiver is as an RF-to-audio frequency transposer: in USB mode, the dial frequency is subtracted from each radio frequency component to produce a corresponding audio component, while in LSB mode each incoming radio frequency component is subtracted from the dial frequency.
Frequency modulation[edit]
Frequency modulation also generates sidebands, the bandwidth consumed depending on the modulation index - often requiring significantly more bandwidth than DSB. Bessel functions can be used to calculate the bandwidth requirements of FM transmissions. Carson's rule is a useful approximation of bandwidth in several applications.
Sidebands can interfere with adjacent channels. The part of the sideband that would overlap the neighboring channel must be suppressed by filters, before or after modulation (often both). In broadcast band frequency modulation (FM), subcarriers above 75 kHz are limited to a small percentage of modulation and are prohibited above 99 kHz altogether to protect the ±75 kHz normal deviation and ±100 kHz channel boundaries. Amateur radio and public service FM transmitters generally utilize ±5 kHz deviation.
To accurately reproduce the modulating waveform, the entire signal processing path of the system of transmitter, propagation path, and receiver must have enough bandwidth so that enough of the sidebands can be used to recreate the modulated signal to the desired degree of accuracy.
In a non-linear system such as an amplifier, sidebands of the original signal frequency components may be generated due to distortion. This is generally minimized but may be intentionally done for the fuzzbox musical effect.
Out-of-band communications involve a channel other than the main communication channel.
Sideband computing is a distributed computing method using a channel separate from the main communication channel.
^ Tony Dorbuck (ed.), The Radio Amateur's Handbook, Fifty-Fifth Edition, American Radio Relay League, 1977, p. 368
Department of The Army Technical Manual TM 11-685 "Fundamentals of Single Sideband Communications"
Retrieved from "https://en.wikipedia.org/w/index.php?title=Sideband&oldid=1070745054"
|
Natural number/Related Articles - Citizendium
Natural number/Related Articles
< Natural number
A list of Citizendium articles, and planned articles, about Natural number.
See also changes related to Natural number, or pages that link to Natural number or to this page or whose text contains "Natural number".
Numeral system [r]: Add brief definition or description
Peano axioms [r]: A set of axioms that completely describes the natural numbers. [e]
Elementary arithmetic [r]: Add brief definition or description
Cardinal number [r]: The generalization of natural numbers (as means to count the elements of a set) to infinite sets. [e]
{\displaystyle i^{2}=-1}
Abacus [r]: A mechanical aid to performing arithmetic which dates back many centuries and is still used in modern times. [e]
Finger [r]: Add brief definition or description
Retrieved from "https://citizendium.org/wiki/index.php?title=Natural_number/Related_Articles&oldid=627521"
|
Self Organizing Map and Multi-attribute Analysis - SEG Wiki
Self Organizing Map and Multi-attribute Analysis
Self-organizing map (SOM) is an artificial neural network which is trained using unsupervised learning algorithm to produce a low dimensional map to reduce dimensionality non-linearly. [1][2][3]Self-organizing map has been proven as a useful tool in seismic interpretation and multi-attribute analysis by a machine learning approach. By exploring big data, self-organizing map reveals patterns of the samples clustering and classifying them into different subsets. These subsets provide information of seismic facies, petrophysical properties, and geological features. With modern visualization capabilities and the application of 2D color maps, SOM routinely identifies meaningful geologic patterns.[4]
2 Multi-attribute analysis
3 Enhance Seismic Resolution Using SOM
Self-organizing map of World Bank quality of life data. Resource: World Bank [5]
Self-organizing map applies competitive learning that no supervision is needed. Opposed to supervise learning algorithm which minimizes the error between real data and predictive data by gradient descent and backpropagation, self-organizing map trains and classifies sample to find the winning neuron that is closest to the sample by Euclidean distance. This competitive learning includes the following step:[6]
Each neuron’s weight of the map is randomized.
Select one of the input vector in the training dataset.
Traverse every neuron in the training dataset and examine which one’s weights are most similar to the input vector by using Euclidean distance. The neuron that has the smallest distance to the input vector is known as the winning neuron and it is also called Best Matching Unit (BMU).
Update the weight vectors of the neurons that are adjacent to BMU (including the BMU itself) by moving them closer to the input vector. The neuron adjusts the weights itself by the following recursion for n iterations:
{\displaystyle w_{j}(n+1)=w_{j}(n)+\theta (n)\cdot \alpha (n)\cdot (x_{i}-w_{j}(n))}
Repeat the procedure from step 2 to step 4 for all input vectors.
The input vectors that are close to each neuron are classified into different groups which have certain colors. This implies that every neuron is associated with a given set of samples.
{\displaystyle w_{j}(n)}
is the weight factor of neuron j at step n.
{\displaystyle \theta (n)}
is a restraint due to distance from BMU that is also called neighborhood function.
{\displaystyle \alpha (n)}
is a learning restraint due to iteration progress
{\displaystyle x_{i}-w_{j}(n)}
is the distance between input vector and neuron j at step n.
Multi-attribute analysis
Offshore West Africa 2D seismic line processed by SOM analysis. In the figure insert, each neuron is shown as a unique color in the 2D color map. After training, each multiattribute seismic sample was classified by finding the neuron closest to the sample by Euclidean distance. The color of the neuron is assigned to the seismic sample in the display. A great deal of geologic detail is evident in classification by SOM neurons. Resource: Geophysical Insights.[4]
Seismic attributes are derived from seismic data in order to delineate geological or geophysical features. Seismic attributes include categories of time, amplitude, frequency, and attenuation with both pre-stack and post-stack seismic data. Common used attributes include Dip and Azimuth maps, Amplitude extraction, Coherence, Spectral decomposition, and Seismic Inversion.
In an effort to improve the interpretation of seismic attributes, combine two or more attributes together could better visualize features. The goal can be achieved by a Self-organizing map which includes multiple information of the features with low dimension. First, a subset of attributes that are extracted from the 3D seismic survey are selected as the input training data, and then SOM is able to run competitive leaning algorithm. The input samples are assigned to the classes or colors of the closest winning neurons. In order to identify the quality of learning, interpreters can examine the proportional reduction of error between initial and final epochs. The error is measured by summing distances as a measure of how near the winning neurons are to their respective data samples. The largest reduction in error is the indicator of best learning.
Enhance Seismic Resolution Using SOM
SOM results: (a) original stacked amplitude, (b) SOM results with associated Embedded Image color map, and (c) SOM results with color map showing two neurons that highlight flat spots in the data. The hydrocarbon contacts (flat spots) in the field were confirmed by well control. Resource: Geophysical Insights.[4]
What a Self-organizing map can do is to detect detail in formation by multiple seismic attributes. When applied to an SOM, two or more attributes will help to distinguish different types of anomalous features which may be just one same anomaly without using SOM. The combination of seismic attribute provide by SOM analysis has better resolution images in the reservoir than any one of the seismic attributes or the original amplitude volume individually.
With higher resolution, it can be clearer to identify thin beds in the reservoir and Direct Hydrocarbon Indicator (DHI).
The development of unconventional plays is becoming a trend in exploration. This type of play requires detail information in facies and stratigraphy changing for horizontal wells drilling. When the target shale or mud rock is too thin to detect by seismic, SOM can be used to improve the resolution laterally and vertically compared to the seismic amplitude line.[7]
Direct Hydrocarbon Indicator (DHI) is defined due to seismic amplitude anomalies. Common DHI includes bright spot, flat spot, polarity reversal, and AVO analysis. Containing subtle facies changing information, SOM helps to verify whether the DHI is real and helps to detect the edge of reservoir. When better understanding the existing reservoir, interpreters can calculate the volume of the target more accurately and can decrease risk for future exploration in the area.[8]
SOM results: SOM highlights the reservoir above the oil/water and gas/oil contacts and the hydrocarbon contacts (flat spots). Resource: Geophysical Insights.[8]
Geophysical Insights - Self-Organizing Map
↑ Kohonen, Teuvo; Honkela, Timo (2007). "Kohonen Network". Scholarpedia. http://www.scholarpedia.org/article/Kohonen_network.
↑ Kohonen, Teuvo (1982). "Self-Organized Formation of Topologically Correct Feature Maps". Biological Cybernetics 43 (1): 59–69. doi:10.1007/bf00337288.
↑ Smith, T. (2010). Unsupervised neural networks-disruptive technology for seismic interpretation. Oil & Gas Journal, 108(37), 42-47. https://www.geoinsights.com/unsupervised-neural-networks-disruptive-technology-for-seismic-interpretation/
↑ 4.0 4.1 4.2 Roden, R., Smith, T., & Sacrey, D. (2015). Geologic pattern recognition from seismic attributes: Principal component analysis and self-organizing maps. https://www.geoinsights.com/geologic-pattern-recognition-from-seismic-attributes-principal-component-analysis-and-self-organizing-maps/
↑ World Poverty Map, SOM research page, Univ. of Helsinki, http://www.cis.hut.fi/research/som-research/worldmap.html
↑ Kohonen, Teuvo (2005). "Intro to SOM". SOM Toolbox. http://www.cis.hut.fi/projects/somtoolbox/theory/somalgorithm.shtml. Retrieved 2006-06-18.
↑ Roden, R., Smith, T. A., Santogrossi, P., Sacrey, D., & Jones, G. (2017). Seismic interpretation below tuning with multiattribute analysis. The Leading Edge, 36(4), 330-339. https://geoinsights.com/seismic-interpretation-below-tuning-with-multiattribute-analysis/
↑ 8.0 8.1 Roden, R., & Chen, C. W. (2017). Interpretation of DHI characteristics with machine learning. First Break, 35(5), 55-63.https://www.geoinsights.com/interpretation-of-dhi-characteristics-with-machine-learning/
Retrieved from "https://wiki.seg.org/index.php?title=Self_Organizing_Map_and_Multi-attribute_Analysis&oldid=118257"
|
Interfacial Fracture Toughness Measurement of a Ti/Si Interface | J. Electron. Packag. | ASME Digital Collection
Mitul Modi,
Computer-Aided Simulation of Packaging Reliability (CASPaR) Laboratory, The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405
Contributed by the Electronic and Photonic Packaging Division for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received February 2003; final revision, February 2004. Associate Editor: A. Y-H. Hung.
Modi , M., and Sitaraman, S. K. (October 6, 2004). "Interfacial Fracture Toughness Measurement of a Ti/Si Interface ." ASME. J. Electron. Packag. September 2004; 126(3): 301–307. https://doi.org/10.1115/1.1772410
Titanium adhesive layers are commonly used in microelectronic and MEMS applications to help improve the adhesion of other metal layers to a silicon substrate. Such Ti/Si interfaces could potentially delaminate under externally applied mechanical loads, thermally induced stresses, or process-induced intrinsic stresses or a combination of these different loads. In order to design against delamination, knowledge of the interfacial fracture toughness of the Ti/Si interface is necessary. However, interfacial fracture toughness data for such interfaces is not widely available in the open literature, in part due to the difficulty in measuring the strength of thin film interfaces. The Modified Decohesion Test (MDT), a new test developed by the authors, has been used to characterize the mode mix dependent interfacial fracture toughness of a Ti/Si interface. In this approach, a highly stressed super layer is used to drive delamination and generate any mode mix at the crack tip. MDT uses the change in crack surface area to vary the available energy per unit area for crack growth and thus to bound the interfacial fracture toughness. Therefore, this technique uses a single sample to measure the interfacial fracture toughness. Since the deformations remain elastic, a mechanics-based solution can be used to correlate test parameters to the energy release rate. Common IC fabrication techniques are used to prepare the sample and execute the test, thereby making the test compatible with current microelectronic or MEMS facilities. Using the MDT, interfacial fracture toughness (Γ) bounds were found for a Ti/Si interface at three mode mixes. At a mode mix of 19.5 deg,
5.97J/m2⩽Γ⩽7.87J/m2,
at a mode mix of 23 deg,
9.32J/m2⩽Γ⩽10.42J/m2,
and at a mode mix of 30 deg,
12.70J/m2⩽Γ⩽17.02J/m2.
titanium, metallic thin films, fracture toughness, internal stresses, delamination, surface cracks, adhesion, silicon, elemental semiconductors, semiconductor-metal boundaries, fracture toughness testing
Fracture toughness, Stress, Titanium, Silicon, Fracture (Materials), Delamination, Metals, Adhesion, Strips, Adhesives
A Test Specimen for Determining the Fracture Resistance of Bimaterial Interfaces
Metal-Ceramic Interfacial Fracture Resistance Using the Continuous Microscratch Technique
Hohlfelder, R. J., Luo, H., Vlassak, J. J., Chidsey, C. E. D., and Nix, W. D., 1997, “Measuring Interfacial Fracture Toughness with the Blister Test,” Materials Research Society Symposium Proceedings, Vol. 436, pp. 115–120.
Shaffer, E. O., McGarry, F. J., and Trussel, F., 1994, “The Edge Delamination Test: Measuring the Critical Adhesion Energy of Thin-Film Coatings. II. Mode Mixity and Application,” Material Research Society Symposium Proceedings, Vol. 338, pp. 541–551.
Kim, K.-S., 1988, “Mechanics of the Peel Test for Thin Film Adhesion,” Material Research Society Symposium Proceedings, Vol. 119, pp. 31–41.
A New Procedure for Measuring the Decohesion Energy for Thin Ductile Films on Substrates
Modi, M., and Sitaraman, S. K., 2002, “Modified Decohesion Test (MDT) for Interfacial Fracture Toughness Measurement in Microelectronics/MEMS Applications,” Proceedings of 2002 ASME International Mechanical Engineering Congress and Exposition, New Orleans, LA.
Smith, D. L., Fork, D. K., Thornton, R. L., and Alimonda, A. S., Chua, C. L., Dunnrowicz, C., and Ho, J., 1998, “Flip-Chip Bonding on 6-μm Pitch using Thin-Film Microspring Technology,” Proceedings of the 48th Electronics Components and Technology Conference (ECTC), Seattle, Washington, pp. 325–329.
Aksyuk, V. A., et al., 2000, “Lucent MicrostarTM Micromirror Array Technology for Large Optical Crossconnects,” Proceedings of MOEMS and Miniaturized Systems, SPIE Vol. 4178, Santa Clara, CA, pp. 320–4.
Modi, M., Sitaraman, S. K., and Fork, D. K., 2001, “Numerical Approximation of the ERR in Intrinsically Stressed Micro-Springs,” Proceedings of IPACK’01, Kauai, Hawaii.
Geometrically Nonlinear Stress-Deflection Relations for Thin Film/Substrate Systems
Dalgeish
The Fracture Energy of Bimaterial Interfaces
Effect of Moisture on the Interfacial Adhesion of the Underfill/Solder Mask Interface
Evaluation and Improvement of the Adhesive Fracture Toughness of CVD Diamond on Silicon Substrate
Use of a Trilayer Shell Model to Determine Intrinsic Stress Within Titanium-Silicon Carbonitride Coating
|
Without using a calculator, simplify and evaluate each of the following log expressions. Your result should be an expression without logarithms.
(4) +
(25)
These have the same base.
How does that help you to simplify?
\frac { 7 ^ { \operatorname { log } _ { 7 } ( 3 ) } } { \operatorname { log } _ { 5 } 5 ^ { 4 } }
\frac{3}{4}
|
5 Bertie Terrace | Leamington
Your letter saying that your father will kindly take care of the 8 little rabbits, was most grateful. If you will fill up the enclosed note addressed to him & fix a day not earlier than Tuesday, not later than Thursday next week, I am sure that Fraser will pack them off safely.1
There is one thing, note, to attend to, that is exceptional with this breed of rabbits; namely, a strong tendency to scurf (eczema) on the nose, muzzle, eye-lids, and legs. This is apt to become a most serious evil but is infallibly checked by tobacco-water (an infusion of strong pig-tail tobacco) rubbed well in. I use a soft tooth brush. They squeal very much, sometimes, and they are usually narcotised afterwards, but it all goes off;—pain, narcotism & scurf.
These 8 rabbits are, 6 of them by one doe, & 2 by another—both thoroughly x circulated. Their father was the same &, in addition he is the son of a pair who had both been injected with defibrinized blood.2 My paper will come out in the next no. of the R Soc: Proceedings & I will send your father a copy, with their pedigree, marked.3
I shd. propose that these rabbits be kept to grow & strengthen till the end of July (when they will all be more than 6
\frac{1}{2}
months old) Then I wd. put all the does to the bucks, and get perhaps 20 young from them, and I would not put the does again to the bucks but let them thoroughly recover & let them & the bucks be operated on early in Septr; thus—
July 30 (say) does put to bucks
Aug 30 litters born
Sept 30 litters weaned
Oct 7–11 does & bucks operated on.
As regards the 20(?) young which are to be expected, I should be inclined to operate on 10(?) of them in a new way, (into jugular vein from a supplying carotid artery) while they are quite young, say,
Oct 14–21 operate on 10(?) of the young
and the remaining 10 to be kept for further proceedings,—breeding or operations, as may be then thought advisable.
Though I shall not have my old excellent assistant Fraser, who sails this day week for Calcutta, I shall have the run of the University Coll: Physiolog: Laboratory & shall be able, I believe, to conduct all the operations there with convenience, greater than hitherto.4
With kindest remembrance, | Ever your sincerely yours | Francis Galton
4.6 July … on. 4.9] outlined red crayon
5.4 Oct … young] outlined red crayon
Oscar Louis Fraser was Galton’s assistant in his experiments on rabbits (Galton 1871, p. 395). The enclosure has not been found.
For more on Galton’s experiments, see the letter from Francis Galton, 9 January 1871, n. 1.
No copy of Galton 1871 has been found in the Darwin Archive–CUL.
In 1871, Fraser left England to take up a position as osteologist to the museum at Calcutta (Galton 1871, p. 395).
|
Albedo - Marspedia
The albedo of a planet is the ratio between the amount of incident light on that planet to how much light is reflected.
{\displaystyle {\alpha }={{I} \over {R}}}
{\displaystyle {\alpha }}
is albedo,
{\displaystyle I}
is the measure of incident light and
{\displaystyle R}
is the measure of reflected light.
The albedo is a strong factor in the regulation of a planet's atmosphere. Albedo may be increased by surface features (i.e. ice or snow in polar regions) or atmospherics (i.e. clouds), so any changes in climate strongly influences albedo. In the case of an ice age, much of a planet's surface may be covered in snow, increasing the reflectivity, hindering the greenhouse effect acting on the atmosphere. This would therefore lead to further cooling. Likewise, if all the ice melts, the albedo will decrease, enhancing the greenhouse effect, amplifying global warming.
The albedo of Mars is .16 or 16%[1].
↑ http://www.astronomynotes.com/solarsys/plantblb.htm
Retrieved from "https://marspedia.org/index.php?title=Albedo&oldid=125944"
|
We consider the 8180 APN functions in dimension 8 constructed by the matrix method <ref name="yu">Yu, Yuyin, Mingsheng Wang, and Yongqiang Li. "A Matrix Approach for Constructing Quadratic APN Functions."</ref> We enumerate the functions in the order in which they appear in <ref name="yu" />.
The [[Walsh spectra of quadratic APN functions over GF(2^8)]] can take three distinct values.
All of these 8180 functions have one of the 21 values 11818, 12370, 13200, 13800, 13804, 13842, 13848, 14024, 14028, 14030, 14032, 14034, 14036, 14038, 14040, 14042, 14044, 14046, 14048, 14050, 15358 as their <math>\Gamma</math>-rank. The following table lists the indices of the functions having each of the given <math>\Gamma</math>-rank.
{\displaystyle \Gamma }
{\displaystyle \Gamma }
|
trig functions - Maple Help
Home : Support : Online Help : Mathematics : Basic Mathematics : Exponential, Trig, and Hyperbolic Functions : Conversion : trig functions
convert exponentials and special functions into trigonometric functions
convert(expr, trig)
convert/trig converts the exponentials in an expression as well as the special functions when possible into trigonometric functions, that is, into any of
\mathrm{sin},\mathrm{cos},\mathrm{tan},\mathrm{sec},\mathrm{csc},\mathrm{cot},\mathrm{sinh},\mathrm{cosh},\mathrm{tanh},\mathrm{sech},\mathrm{csch},\mathrm{coth}
\frac{1}{2}\mathrm{exp}\left(x\right)+\frac{1}{2}\mathrm{exp}\left(-x\right)
\frac{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\frac{{\textcolor[rgb]{0,0,1}{ⅇ}}^{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}}}{\textcolor[rgb]{0,0,1}{2}}
\mathrm{convert}\left(,\mathrm{trig}\right)
\textcolor[rgb]{0,0,1}{\mathrm{cosh}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
{\mathrm{\pi }}^{\frac{1}{2}}\mathrm{MeijerG}\left([[],[]],[[\frac{1}{2}],[0]],\frac{1}{4}{x}^{4}\right)\mathrm{MeijerG}\left([[],[]],[[0],[]],-x\right)
\sqrt{\textcolor[rgb]{0,0,1}{\mathrm{\pi }}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{MeijerG}}\textcolor[rgb]{0,0,1}{}\left([[]\textcolor[rgb]{0,0,1}{,}[]]\textcolor[rgb]{0,0,1}{,}[[\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{0}]]\textcolor[rgb]{0,0,1}{,}\frac{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}}{\textcolor[rgb]{0,0,1}{4}}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{MeijerG}}\textcolor[rgb]{0,0,1}{}\left([[]\textcolor[rgb]{0,0,1}{,}[]]\textcolor[rgb]{0,0,1}{,}[[\textcolor[rgb]{0,0,1}{0}]\textcolor[rgb]{0,0,1}{,}[]]\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{convert}\left(,\mathrm{trig}\right)
\frac{\sqrt{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{}\sqrt{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{\mathrm{cosh}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{\mathrm{sinh}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)}{\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}
|
Jong-Young Kwak, Mizuko Mamura, Jana Barlic-Dicen, Evelin Grage-Griebenow, "Pathophysiological Roles of Cytokine-Chemokine Immune Network", Journal of Immunology Research, vol. 2014, Article ID 615130, 2 pages, 2014. https://doi.org/10.1155/2014/615130
Jong-Young Kwak ,1 Mizuko Mamura,2,3 Jana Barlic-Dicen,4 and Evelin Grage-Griebenow5
1Department of Biochemistry, School of Medicine and Immune-Network Pioneer Research Center, Dong-A University, 32 Daesin-gongwon-ro, Seo-gu, Busan 602-714, Republic of Korea
2Department of Molecular Pathology, Tokyo Medical University, 6-1-1 Shinjuku, Shinjuku-ku, Tokyo 160-8402, Japan
3Department of Internal Medicine, Kyungpook National University, 50 Samduk-2ga, Jun-gu, Daegu 700-721, Republic of Korea
4Cardiovascular Biology, Oklahoma Medical Research Foundation and Department of Cell Biology, University of Oklahoma Health Sciences Center, 825 N.E. 13th Street, Oklahoma City, OK 73104, USA
5Institute for Experimental Medicine, Inflammatory Carcinogenesis, UK S-H, Kiel Campus, Arnold-Heller-Straße 3, Building 17, 24105 Kiel, Germany
Cytokines are small proteins or glycoproteins, and chemokines form a group of smaller cytokines with chemotactic properties that are produced by a variety of cells. Cytokines and chemokines exert crucial roles in the development, homeostasis, activation, differentiation, regulation, and functions of innate and adaptive immunity. Excessive and/or inappropriate production and actions of cytokines and chemokines are involved in the pathogenesis of infection, inflammation, allergy, autoimmune diseases, and immune-related diseases, such as diabetes, atherosclerosis, rheumatic arthritis, and cancer. Understanding the underlying mechanisms by which cytokines and chemokines exert their functions will lead to development of therapeutics for immune-related diseases. In this special issue, the articles and reviews discuss recent findings that highlight the pathophysiological role of cytokine and chemokines.
Effector and memory T cells are phenotypically and functionally heterogeneous. Upon antigenic stimulation, naïve CD4+ T cells become effector Th1, Th2, or Th17 cells or regulatory T cells (Tregs). Effector and regulatory T cell subsets are generated in the presence of specific cytokines such as IL-12 and TGF-β, respectively. Transcription factors initiate and stabilize commitment toward the Th1 or Th2 lineage. T-bet (the T-box protein expressed in T cells) is the master regulator of Th1 differentiation [1]. A review article by S. Oh and E. S. Hwang summarizes the current state of knowledge regarding the molecular mechanisms that underlie the multiple roles played by T-bet in T helper cell development and fine-modulation of IL-2 production in Th1 cells. Interestingly, IFN-γ stimulation of dendritic cells (DCs) might have an equally important role in generating different effector T cells [2]. In this special issue, A. Visperas et al. describe several cytokines that are able to influence the generation of effector T cells by directly affecting T cells as well as targeting non-T cells. Inflammatory cytokine signaling also plays an important role in the pathogenic conversion of natural Tregs. The article by R. Takahashi and A. Yoshimura is a review in which the possibility is proposed that suppressor of cytokine signaling 1 (SOCS1) may protect Tregs from harmful effects of inflammatory cytokines, and SOCS1 upregulation maintains Treg functions. SOCS1 may be important in the pathogenesis of systemic lupus erythematosus through Treg plasticity, because SOCS1-deficient T cells induce lupus-like autoimmunity [3].
Both tumor-antagonizing and tumor-promoting inflammatory cells can be found in most neoplastic lesions. Inflammation can increase the risk of cancer by providing bioactive molecules from cells that infiltrate the tumor microenvironment, including cytokines and chemokines [4]. G. Landskron et al. review the roles of several inflammatory mediators, including TNF-α, IL-6, TGF-β, and IL-10, in events of carcinogenesis, such as their capacity to generate reactive oxygen and nitrogen species; their potential mutagenic effect; and involvement in mechanisms for epithelial mesenchymal transition, angiogenesis, and metastasis. They also provide an in-depth analysis of the participation of cytokines in cancer that is attributable to chronic inflammatory diseases, such as colitis-associated colorectal cancer and cholangiocarcinoma.
Recent studies have shown that the circadian clock is responsible for the temporal dynamics of the immune system [5]. For example, joint stiffness and secretion of inflammatory cytokines in rheumatoid arthritis (RA) patients are influenced by the diurnal rhythm and peaks in the morning. RA is a devastating autoimmune disease that is characterized by progressive bone destruction and it was found that the circadian clock not only impacts arthritic symptoms but is also involved in the pathogenesis of RA [6]. Proinflammatory cytokines, such as IL-1, IL-6, IL-8, IL-11, IL-17, and TNF-α, are known to be osteoclastogenic [7]. In this special issue, S. M. Jung et al. discuss the osteoclastogenic role of the proinflammatory cytokines and immune cells in the pathophysiology of RA. Moreover, a review by A. Nakao summarizes recent advances regarding the emerging role of the circadian clock as a novel regulator of cytokines. K. Yoshida et al. describe the link between the circadian clock and inflammation, focusing on the interactions of various clock genes with the immune-pathways underlying the pathology of RA. In another article, Y. Nakamura et al. demonstrate that mechanical disruption of the suprachiasmatic nucleus (SCN) of the hypothalamus resulted in the absence of time of day-dependent variation in anaphylactic reaction in mice. These articles provide evidences that daily variations in cytokine levels are related to the pathophysiology of immune diseases.
Obesity, insulin resistance, and atherosclerosis are chronic inflammatory processes that are affected by the activation of innate and adaptive immunity [8, 9]. With regard to the role of inflammation of adipose tissue in obesity and obesity-related diseases, aberrant production of adipokines, cytokines, and chemokines in adipose tissue leads to inflammation in the tissue. A review article on this topic by L. Yao et al. covers the chemokine system and signaling in the development of obesity, insulin resistance, and plaque formation.
A clinical study by K. Theodoraki et al. demonstrates that IL-10 levels were elevated in patients that were subjected to major abdominal surgery procedures with a more liberal red blood cell transfusion strategy. Their results indicate that IL-10 may be an important factor in transfusion-associated immunomodulation. Expression of proinflammatory cytokines and chemokines has also been reported in patients with chronic venous insufficiency, such as varicose veins [10]. V. Tisato et al. demonstrate that EGF, PDGF, and RANTES were increased in varicose veins compared with general circulation, and a patient who exhibited recurrence of the disease 6 months after surgery showed higher levels of these factors compared with nonrecurrent patients. Therefore, V. Tisato et al. suggest that EGF, PDGF, and RANTES can be used as sensitive biomarkers of chronic venous insufficiency. Finally, J. Sturgill et al. examine the correlation of cytokine production in plasma of patients with fibromyalgia and they suggest that suppression of Th2 cytokines is related to immune dysregulation in patients with fibromyalgia.
We wish to thank the authors and reviewers for their contributions to this special issue.
Evelin Grage-Griebenow
S. J. Szabo, S. T. Kim, G. L. Costa, X. Zhang, C. G. Fathman, and L. H. Glimcher, “A novel transcription factor, T-bet, directs Th1 lineage commitment,” Cell, vol. 100, no. 6, pp. 655–669, 2000. View at: Publisher Site | Google Scholar
J. S. Do, K. Asosingh, W. M. Baldwin III, and B. Min, “Cutting edge: IFN-
\gamma
R signaling in non-T cell targets regulates T cell-mediated intestinal inflammation through multiple mechanisms,” Journal of Immunololy, vol. 192, no. 6, pp. 2537–2541, 2014. View at: Google Scholar
C. Scheiermann, Y. Kunisaki, and P. S. Frenette, “Circadian control of the immune system,” Nature Reviews Immunology, vol. 13, no. 3, pp. 190–198, 2013. View at: Publisher Site | Google Scholar
J. E. Gibbs and D. W. Ray, “The role of the circadian clock in rheumatoid arthritis,” Arthritis Research & Therapy, vol. 15, no. 1, article 205, 2013. View at: Publisher Site | Google Scholar
S. Guo, “Insulin signaling, resistance, and the metabolic syndrome: insights from mouse models to disease mechanisms,” Journal of Endocrinology, vol. 220, no. 2, pp. T1–T23, 2014. View at: Google Scholar
H. E. Bays, P. P. Toth, P. M. Kris-Etherton et al., “Obesity, adiposity, and dyslipidemia: a consensus statement from the National Lipid Association,” Journal of Clinical Lipidology, vol. 7, no. 4, pp. 304–383, 2013. View at: Publisher Site | Google Scholar
L. del Rio Solá, M. Aceves, A. I. Dueñas et al., “Varicose veins show enhanced chemokine expression,” European Journal of Vascular and Endovascular Surgery, vol. 38, no. 5, pp. 635–641, 2009. View at: Publisher Site | Google Scholar
Copyright © 2014 Jong-Young Kwak et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
|
UHPC 101 Part 5 - Coulombic Efficiency: the Breakdown - Novonix
UHPC 101 Part 5 – Coulombic Efficiency: the Breakdown
The UHPC 101 series has thus far been all about Coulombic Efficiency (CE); how it relates to cell cycle life, how the CE of cells can be normalized to account for different cycle times due to either different cycling rates or different states of charge (CIE/hr) and the two degradation metrics that make up CE: capacity fade and charge end-point capacity slippage.
In this post, the intention is to demonstrate, both graphically and mathematically, how capacity fade and charge end-point capacity slippage add up to give the CE of a cell. The first part of this post derives the CE expression explicitly in terms of capacity fade and charge end-point capacity slippage, while the second part uses a fictional case study to demonstrate how a decomposed CE measurement can be interpreted.
The goal is to show the wealth of information that high fidelity CE measurements hold, well beyond predicting which cell will have the longest lifetime.
The graph above shows fictional data that has been modified to exaggerate both capacity fade and charge end-point capacity slippage. For a given cycle
n
n
, the charge end-point is labelled
{q}_{c}^{n}
q_c^n
, and the discharge end-point is labelled
{q}_{d}^{n}
q_d^n
. The discharge and charge end-point capacity slippages are the differences between the end-points of cycle
n
n
and cycle
n-1
n-1
\mathrm{\Delta }{q}_{c}^{n}={q}_{c}^{n}-{q}_{c}^{n-1}
\Delta q_c^n = q_c^n - q_c^{n-1}
\mathrm{\Delta }{q}_{d}^{n}={q}_{d}^{n}-{q}_{d}^{n-1}
\Delta q_d^n = q_d^n - q_d^{n-1}
, respectively. The capacity fade,
{Q}_{f}
Q_f
, for cycle
n
n
is the difference between the discharge capacity of cycle
n-1
n-1
n
n
{Q}_{f}={Q}_{d}^{n-1}-{Q}_{d}^{n}
Q_f = Q_d^{n-1}- Q_d^n
Interestingly,
{Q}_{f}
Q_f
can also be written as the difference between the discharge and charge end-point capacity slippages:
{Q}_{f}=\mathrm{\Delta }-\mathrm{\Delta }
Q_f = \Delta q_d^n-\Delta q_c^n
. To see this, imagine sliding the discharge curve of cycle
n
n
to the left by an amount
\mathrm{\Delta }{q}_{c}^{n}
\Delta q_c^n
such that the charge end-points of cycle
n
n
n-1
n-1
lie on top of each other. The difference in discharge end-points then gives the capacity fade,
{Q}_{f}
Q_f
, and is the same as the difference between the discharge and charge end-point capacity slippages. The CE for cycle
n
n
, which is the discharge to charge capacity ratio, can now be written in terms of the capacity fade,
{Q}_{f}^{n}
Q_f^n
, and the charge end-point capacity slippage,
\mathrm{\Delta }{q}_{c}^{n}
\Delta q_c^n
, by noting that the cycle
n
n
discharge capacity is the cycle
n
n
charge capacity minus the discharge capacity slippage,
Q_d^n = Q_c^n - \Delta q_d^n
And, as described above, the discharge end-point capacity slippage can be expressed in terms of the capacity fade and charge end-point capacity slippage:
C{E}^{n}=1-\frac{\mathrm{\Delta }{q}_{d}^{n}}{{Q}_{c}^{n}}=1-\frac{{Q}_{f}^{n}+\mathrm{\Delta }{q}_{c}^{n}}{{Q}_{c}^{n}}
This is ultimately a very useful expression because the values
{Q}_{f}^{n}
Q_f^n
\mathrm{\Delta }{q}_{c}^{n}
\Delta q_c^n
can easily be extracted from cycling data for each cycle number and immediately gives insight into how the cell is degrading, whether electrolyte reduction or oxidation, for example. Before considering an example, it is worth showing how the corresponding expression for CIE/hr is obtained. The CIE is defined as:
CIE=1–CE
CIE = 1 – CE
, and the CIE/hr is obtained by dividing by the time, in hours, per cycle. Thus,
CIE/hr=\left(1-CE\right)/hr=\left(\frac{{Q}_{f}^{n}+\mathrm{\Delta }{q}_{c}^{n}}{{Q}_{c}^{n}}\right)/hr=\frac{{Q}_{f}^{n}/hr+\mathrm{\Delta }{q}_{c}^{n}/hr}{{Q}_{c}^{n}}
This expression shows that the CIE/hr can easily be obtained from the capacity fade and charge end-point capacity slippage simply by normalizing each of them by the time per cycle. This allows comparison between cells that were cycled at different rates and/or to different states of charge causing the cycle time to be different.
The following graph shows fictional data to demonstrate the importance of breaking up a CE measurement. Consider a hypothetical situation where two cells have identical CE, as shown in the top panel. If no further analysis was done, one would conclude that these cells have identical electrochemical performance.
However, by differentiating capacity fade and charge end-point capacity slippage, it is revealed that these two cells are achieving the same CE in different ways; the cell in black has more capacity fade (center panel), the one in red has more charge end-point capacity slippage (bottom panel).
This begs the question: is one of these two cells better than the other even though they have identical CE*? The answer to that question is generally not straightforward, however, examining other cycle metrics can give clues. For example, if the cell with larger charge end-point capacity slippage (red) also had a comparably increased impedance, it is possible that salt from the electrolyte was being consumed during electrolyte oxidation, which would likely lead to worse performance over time compared to the cell with larger capacity fade (black).
Breaking down the CE this way and cross-referencing with other cycle metrics (e.g. impedance growth – to be the subject of a future post), provides additional information that can be used to make informed decisions not only about which cell is better compared to the other, but also about how cells degrade and how chemistries can be improved.
*Even though this is a fictional scenario – cells would typically have at least very small differences in CE – this example nonetheless represents a real challenge that arises when testing cells carefully, especially when comparing cells that have very similar electrochemical performance.
← UHPC 101 Part 4 – Charge End-point Capacity Slippage
|
Country of Alphabet | Toph
Country of Alphabet
By rakibahmed · Limits 1s, 512 MB
In an imaginary galaxy, there is a planet where two Countries are very unique for their Houses. In those countries, people make their houses in Uppercase Alphabetic shape. Levi and Eren are the presidents of those countries. There are total
N number of houses in Levi’s Country and
M number of houses in Eren’s Country. Levi is a very strict president so he declared that in his country all the houses should be aligned sequentially, on the other hand, Eren doesn’t care about the sequence because he is a wise and flexible president.
ADPRW is sequentially aligned whereas
PADWR is not sequentially aligned.
For Levi’s strict and mad decision, he’s the headline of the World News. On the other hand, for Eren’s flexibility, he earned the prestigious Nobel Peace. Since the houses should be aligned sequentially in Levi’s country, some houses might shift from one position to another with a very long distance. After sequentially aligning the houses of Levi’s country, if we compare Levi and Eren’s country’s houses then there might be some houses that look the same in terms of their shape. If there are any of the houses
L_{i}
Li of Levi’s country look the same as any of the houses
E_{j}
Ej of Eren’s country then you have to calculate the ratio of their Position.
For example, if the houses of Levi’s country become
ADPRW after sequentially aligned and if the houses of Eren’s country are aligned in this way —
KWLA. Then,
W and
A are the common houses among them. In Levi’s country,
W is located at
5^{th}
5th position, and in Eren’s country,
W is located at
2^{nd}
2nd position. In Levi’s country,
A is located at
1^{st}
1st position, and in Eren’s country,
A is located at
4^{th}
4th position. For
W shaped houses, the ratio is
5:2, and for
A shaped houses, the ratio of their position is
1:4.
Now, If there are any of the houses of Levi’s country look the same as the houses of Eren’s country then, your task is to calculate the ratio of the house’s position. If not found then, print “unmatched“ without quotes.
If there are same-shaped houses in the same country or another country appear more than once, then consider only for the first house.
N (
1 \leq N \leq 100
1≤N≤100) — The number of houses in Levi’s country.
The second line of the input contains a string — The sequence of the
N number of Houses of Levi’s Country.
The third line of the input contains
M (
1 \leq M \leq 100
1≤M≤100) — The number of houses in Eren’s country.
The fourth line of the input contains a string — The initial sequence of the
M number of Houses of Eren’s Country.
For each input, If there are any of the houses of Levi’s country look the same as the houses of Eren’s country then, you have to print all the ratios of the house’s position. If not found then, print “unmatched“ without quotes.
ADPRW
HDKAP
XBZYI
BDAPOATBD
VDFABA
MursaleenEarliest, 10M ago
n3wb13_223Fastest, 0.0s
MursaleenLightest, 131 kB
|
A Ridiculous Problem | Toph
Let's face it. This problem is here just to spite you off. So, I will not beat around the bush.
You will be given a grid; a grid with dots and asterisks. The dots are white spaces. The asterisks are, well, on their own, they mean nothing. But if you look at the grid, squint your eyes and try to observe the shape that the asterisks make in the grid you will notice that they look like the English alphabet.
Your task here is to write a program that will take a grid as an input, and print the text that it represents.
The grid will be composed of blocks. Each block,
5×7 in size, will have a character in it represented by the shape made by the asterisks. Between two blocks with a character each, will be an empty column of dots. The grid will only have letters from the uppercase English alphabet. There will be no lower case letters, spaces, punctuation, numbers, or special characters.
Here is an image showing what the characters will look like:
And, here is an example of what the character "I" would look like with asterisks in a grid:
There will always be 7 lines of characters in the input. The number of characters in each line will be the same and they will never exceed 250. The lines will only have dots (".") or asterisks ("*") and the grid will represent valid text only (i.e. text consisting of the 26 uppercase English alphabets).
Print the text that is represented by the grid.
.***...***...***..****.
*...*.*...*.*...*.*...*
*.....*...*.*...*.*...*
*..**.*...*.*...*.*...*
*.....*...*..***..*...*
*.....*...*.*...*.*..*.
*.....*...*.*.....*.*..
*.....*...*.*.....**...
*****..***...***..*...*
Mushfiq_4513Earliest, Sep '19
Mushfiq_4513Fastest, 0.0s
Riz1ahmedLightest, 0 B
rummanrakib11Shortest, 856B
|
Matching Brackets | Toph
You will be given a sequence of opening and closing brackets of different types ((, ), [, ], {, and }). You will have to determine if the sequence is a valid one.
A sequence of brackets is considered valid if every opened bracket of a type has a closing bracket of an equivalent type appearing after it. And, there are no unpaired brackets in the sequence.
The input will contain a string of opening and closing brackets. The string will be no longer than 25 characters.
\texttt{Yes}
Yes if the input string contains a valid sequence of parentheses. Otherwise, print
\texttt{No}
[[(){]}]
(([]{}[{}]){})[]
DataStructure, Implementation, Stack
md_jakariyaEarliest, Sep '19
md_jakariyaFastest, 0.0s
JUNIORHMMC_CODLightest, 0 B
|
2015 New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions
Ahmed Alsaedi, Sotiris K. Ntouyas, Bashir Ahmad
We consider a boundary value problem of fractional integrodifferential equations with new nonlocal integral boundary conditions of the form:
x\left(0\right)=\beta x\left(\theta \right), x\left(\xi \right)=\alpha {\int }_{\eta }^{1}\mathrm{}x\left(s\right)ds
0<\theta <\xi <\eta <1
. According to these conditions, the value of the unknown function at the left end point
t=0
is proportional to its value at a nonlocal point
\theta
while the value at an arbitrary (local) point
\xi
is proportional to the contribution due to a substrip of arbitrary length
\left(1-\eta \right)
. These conditions appear in the mathematical modelling of physical problems when different parts (nonlocal points and substrips of arbitrary length) of the domain are involved in the input data for the process under consideration. We discuss the existence of solutions for the given problem by means of the Sadovski fixed point theorem for condensing maps and a fixed point theorem due to O’Regan. Some illustrative examples are also presented.
Ahmed Alsaedi. Sotiris K. Ntouyas. Bashir Ahmad. "New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions." Abstr. Appl. Anal. 2015 (SI05) 1 - 10, 2015. https://doi.org/10.1155/2015/205452
Ahmed Alsaedi, Sotiris K. Ntouyas, Bashir Ahmad "New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions," Abstract and Applied Analysis, Abstr. Appl. Anal. 2015(SI05), 1-10, (2015)
|
Method of stationary phase - SEG Wiki
Many useful results in mathematical physics are the result of asymptotic approximations. That is, the are formulae and results that are obtained by deriving an asymptotic series and keeping the leading order term as the approximation, and using the next order term as an error estimate.[1] [2][3]
On such result is called The method of stationary phase, which applies to integrals that resemble the Fourier transform, but have a more general phase function. The method of stationary phase applies to one dimensional Fourier-like integrals as well as to multi-dimensional Fourier integrals.
Many wave phenomena are the result of the preferential constructive interference of the wavefield. These phenomena are examples of result that would be described by extrema of the phase function of a Fourier representation.
1 The method of stationary phase in 1-dimension
1.1 Simple critical point
1.1.1 Stationary phase formula -- stationary point at the lower endpoint of integration
{\displaystyle t=a}
1.1.2 Stationary phase formula -- upper endpoint of integration
{\displaystyle t=b}
1.1.3 Stationary phase formula -- interior stationary point
{\displaystyle t=c}
{\displaystyle a<c<b}
2 What about the endpoint contribution?
3 Higher order stationary points in 1D
3.1 Higher order stationary point at the lower endpoint of integration at
{\displaystyle t=a}
3.2 Higher order stationary point at the upper endpoint of integration at
{\displaystyle t=b}
4 Stationary phase analysis
4.1 Example, asymptotic form of the Bessel function
{\displaystyle J_{0}(\lambda r)}
{\displaystyle \lambda r>0}
5 Multidimensional stationary phase
5.1 The Multidimensional stationary phase formula
The method of stationary phase in 1-dimension
We consider integrals of the form
{\displaystyle I(\lambda )=\int _{a}^{b}f(t)e^{i\lambda \phi (t)}\;dt}
{\displaystyle \lambda >0}
is a large parameter, which may be frequency or wave number in problems of wave propagation. The function
{\displaystyle f(x)}
is called the amplitude and the real-valued function
{\displaystyle \phi (x)}
is called the phase. As the name of the technique implies our points of interest are places where the phase function is slowly varying.
Simple critical point
The endpoints of integration, places where the derivatives of
{\displaystyle f(t)}
fail to be continuous, and places where the derivatives of
{\displaystyle \phi (t)}
vanish are called critical points.
A simple critical point is a point
{\displaystyle t=a}
{\displaystyle \phi ^{\prime }(a)=0}
{\displaystyle \phi ^{\prime \prime }(a)\neq 0}
. Such a simple critical point is also called a stationary point because this is the place where the phase function has a minimum or a maximum and is thus, stationary.
Through a number of coordinate transformations, the Fourier-like integral can be repetitively integrated by parts to yield an asymptotic series with large parameter
{\displaystyle \lambda }
. The leading order term of that series is called the stationary phase formula and the second term or third term is the estimate of the asymptotic error of the approximation.
We consider three possibilities, that the stationary point is on the lower endpoint of integration
{\displaystyle t=a}
, the upper endpoint of integration
{\displaystyle t=b}
, or an interior stationary point
{\displaystyle t=c}
{\displaystyle a<c<b}
Stationary phase formula -- stationary point at the lower endpoint of integration
{\displaystyle t=a}
{\displaystyle I_{a}(\lambda )\sim {\frac {1}{2}}e^{i\lambda \phi (a)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(a)\right)\pi /4}\left\{f(a){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(a)|}}}+{\frac {2}{\lambda |\phi ^{\prime \prime }(a)|}}\left[f^{\prime }(a)-{\frac {\phi ^{\prime \prime \prime }(a)}{3|\phi ^{\prime \prime }(a)|}}\right]e^{i\lambda \operatorname {sgn}(\phi ^{\prime \prime }(a))\pi /4}\right\}+O(\lambda ^{-3/2})}
In practice the stationary phase formula for a stationary point at the lower limit of integration is the leading order term in inverse powers of
{\displaystyle \lambda }
{\displaystyle I_{a}(\lambda )\sim {\frac {1}{2}}e^{i\lambda \phi (a)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(a)\right)\pi /4}f(a){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(a)|}}}+O(\lambda ^{-1})}
{\displaystyle \lambda \rightarrow \infty }
Stationary phase formula -- upper endpoint of integration
{\displaystyle t=b}
{\displaystyle I_{b}(\lambda )\sim {\frac {1}{2}}e^{i\lambda \phi (b)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(b)\right)\pi /4}\left\{f(b){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(b)|}}}-{\frac {2}{\lambda |\phi ^{\prime \prime }(b)|}}\left[f^{\prime }(b)-{\frac {\phi ^{\prime \prime \prime }(b)}{3|\phi ^{\prime \prime }(b)|}}\right]e^{i\lambda \operatorname {sgn}(\phi ^{\prime \prime }(b))\pi /4}\right\}+O(\lambda ^{-3/2})}
In practice the stationary phase formula for a stationary point at the upper limit of integration is the leading order term in inverse powers of
{\displaystyle \lambda }
{\displaystyle I_{b}(\lambda )\sim {\frac {1}{2}}e^{i\lambda \phi (b)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(b)\right)\pi /4}f(b){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(b)|}}}+O(\lambda ^{-1})}
{\displaystyle \lambda \rightarrow \infty }
Stationary phase formula -- interior stationary point
{\displaystyle t=c}
{\displaystyle a<c<b}
The stationary phase expansion at an interior stationary point is found by combining the upper and lower limit stationary phase series expansions, which causes the terms of order
{\displaystyle O(\lambda ^{-1})}
yielding the familiar result
{\displaystyle I_{c}(\lambda )\sim e^{i\lambda \phi (c)+i\operatorname {sgn} \left(\phi ^{\prime \prime }(c)\right)\pi /4}f(a){\sqrt {\frac {2\pi }{\lambda |\phi ^{\prime \prime }(c)|}}}+O(\lambda ^{-3/2})}
{\displaystyle \lambda \rightarrow \infty }
What about the endpoint contribution?
When we learned to compute definite integrals in undergraduate calculus, we were evaluating the integral at the limits of integration. We call these limits of integration the endpoints. Thus, we are accustomed to evaluating the endpoint contribution. What is asymptotic order of the endpoint contribution compared with the stationarity contribution?
We can answer that question by performing repetitive integration by parts formally to yield the following series representation. Here, we assume that there are no singularities of the amplitude or the phase of our Fourier-like integral
{\displaystyle I(\lambda )=\int _{a}^{b}f(t)e^{i\lambda \phi (t)}\;dt}
and apply integration by parts repetitively so as to bring down factors of the large parameter
{\displaystyle \lambda }
in the denominators of the resulting terms. To do this we multiply and divide the integrant by
{\displaystyle i\lambda \phi ^{\prime }(t)}
. The first application of integration by parts (integrating the exponential) yields
{\displaystyle I(\lambda )=\int _{a}^{b}{\frac {f(t)}{(i\lambda \phi ^{\prime }(t))}}(i\lambda \phi ^{\prime }(t))e^{i\lambda \phi (t)}\;dt=\left.{\frac {f(t)}{(i\lambda \phi ^{\prime }(t))}}e^{i\lambda \phi (t)}\right|_{a}^{b}-{\frac {1}{i\lambda }}\int _{a}^{b}{\frac {d}{dt}}\left[{\frac {f(t)}{\phi ^{\prime }(t)}}\right]e^{i\lambda \phi (t)}\;dt}
Applying this operation repetitively yields
{\displaystyle I(\lambda )=\left.\sum _{n=0}^{N-1}{\frac {(-1)^{n}e^{i\lambda \phi (t)}}{(i\lambda )^{n+1}}}\left[{\frac {1}{\phi ^{\prime }(t)}}{\frac {d}{dt}}\right]^{n}\left[{\frac {f(t)}{\phi ^{\prime }(t)}}\right]\right|_{a}^{b}+{\frac {(-1)^{N}}{(i\lambda )^{N}}}\int _{a}^{b}e^{i\lambda \phi (t)}{\frac {d}{dt}}\left[{\frac {1}{\phi ^{\prime }(t)}}{\frac {d}{dt}}\right]^{N-1}\left[{\frac {f(t)}{\phi ^{\prime }(t)}}\right]\;dt}
{\displaystyle \lambda \rightarrow \infty }
This formal result assumes that all of the parts are sufficiently differentiable, and there are no divisions by zero. The first term of the summation is
{\displaystyle O(\lambda ^{-1})}
{\displaystyle \lambda \rightarrow \infty }
, whereas a stationary point is of an asymptotically more slowly decaying contribution of
{\displaystyle O(\lambda ^{-1/2})}
Endpoint contributions occur wherever there are discontinuities in the data. These constitute such phenomena as ringing, or diffraction smiles seen in processed seismic data.
Higher order stationary points in 1D
Similar formulas may be derived for higher order stationary points. In general, such a stationary point would be represented as critical points in the amplitude
{\displaystyle f(t)}
and in the phase
{\displaystyle \phi (t)}
Higher order stationary point at the lower endpoint of integration at
{\displaystyle t=a}
In general, we may consider the amplitude factor
{\displaystyle f(t)}
to represented by the more general power series representations
{\displaystyle f(t)=f_{a}(t-a)^{(\gamma -1)}+o((t-a)^{(\gamma -1).}\qquad }
{\displaystyle \qquad \gamma >0}
and the phase is represented as
{\displaystyle \phi (t)-\phi (a)=\phi _{a}(t-a)^{\alpha }+o((t-a)^{\alpha })\qquad }
{\displaystyle \qquad \alpha >0}
{\displaystyle f_{a}}
{\displaystyle \phi _{a}}
are the first non vanishing coefficients of the power series representations of the functions.
The resulting stationary phase formula is
{\displaystyle I_{a}(\lambda )\sim {\frac {f_{a}\Gamma (\gamma /\alpha )}{\alpha (\lambda |\phi _{a}|)^{\gamma /\alpha }}}e^{i\lambda \phi _{a}+i\pi \operatorname {sgn}(\phi _{a})\gamma /2\alpha }+o(\lambda ^{\gamma /\alpha })}
{\displaystyle \lambda \rightarrow \infty }
Higher order stationary point at the upper endpoint of integration at
{\displaystyle t=b}
{\displaystyle f(t)}
{\displaystyle f(t)=f_{b}(b-t)^{\beta }+o((b-t)^{\beta })\qquad }
{\displaystyle \qquad \beta >0}
{\displaystyle \phi (t)-\phi (b)=\phi _{b}(b-t)^{(\delta -1)}+o((b-t)^{(\delta -1)})\qquad }
{\displaystyle \qquad \delta >0}
{\displaystyle I_{b}(\lambda )\sim {\frac {f_{b}\Gamma (\delta /\beta )}{\beta (\lambda |\phi _{b}|)^{\delta /\beta }}}e^{i\lambda \phi _{b}+i\pi \operatorname {sgn}(\phi _{b})\delta /2\beta }+o(\lambda ^{\delta /\beta })}
{\displaystyle \lambda \rightarrow \infty }
Here, the Gamma function is
{\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\;dt.}
Stationary phase analysis
A common misunderstanding of the method of stationary phase is that this merely a formula lookup. A better approach is to consider that applying the method of stationary phase is a method of analysis. To perform this analysis, given a Fourier-like integral, the following steps must be applied
identify the large parameter
identify the phase function, it's derivative, and next highest order non-vanishing derivative at the zeros of the first derivative of the phase. If this is the second derivative of the phase, then this is a simple critical point, also known as a stationary point.
find the stationary point(s)
apply the appropriate stationary phase formula.
Example, asymptotic form of the Bessel function
{\displaystyle J_{0}(\lambda r)}
{\displaystyle \lambda r>0}
We consider the integral representation of a Bessel function
{\displaystyle {\displaystyle J_{0}(\lambda r)={\frac {1}{2\pi }}\int _{-\pi }^{\pi }e^{i\lambda r\sin(\theta )}\,d\theta .}}
{\displaystyle \lambda r>0}
is the large parameter. Here, both
{\displaystyle \lambda >0}
{\displaystyle r>0}
{\displaystyle \phi (\theta )=\sin(\theta )}
{\displaystyle \phi ^{\prime }(\theta )=\cos(\theta )}
{\displaystyle \phi ^{\prime \prime }(\theta )=-\sin(\theta )}
the critical points are simple interior stationary points at
{\displaystyle \theta =\pm \pi /2}
we apply the stationary phase formula for simple interior stationary points to each stationary point and combine the results
{\displaystyle J_{0}(\lambda r)\sim {\frac {1}{2\pi }}{\sqrt {\frac {2\pi }{\lambda r}}}\left[e^{i\lambda r-i\pi /4}+e^{-i\lambda r+i\pi /4}\right]+O((\lambda r)^{-3/2})}
Simplifying, yields the common asymptotic form of the Bessel function for large
{\displaystyle \lambda r>0}
{\displaystyle J_{0}(\lambda r)\sim {\sqrt {\frac {2}{\pi \lambda r}}}\cos(\lambda r-\pi /4)+O((\lambda r)^{-3/2})}
Thus, at the end of this analysis, we have the asymptotic approximation as well a an order estimate of the error. In this case, the product
{\displaystyle \lambda r}
might further be reduced into a large
{\displaystyle r}
approximation, or a large
{\displaystyle \lambda }
approximation, where either the
{\displaystyle \lambda }
{\displaystyle r}
, respectively, become part of the phase.
Multidimensional stationary phase
For problems involving Fourier-like integrals in higher dimensions
{\displaystyle I(\lambda )=\int _{D}f({\boldsymbol {x}})e^{i\lambda \phi ({\boldsymbol {x}})}\;d{\boldsymbol {x}}}
{\displaystyle \lambda \rightarrow \infty }
{\displaystyle D}
is a subdomain in
{\displaystyle n-}
dimensions, and
{\displaystyle d{\boldsymbol {x}}}
{\displaystyle n-}
dimensional hypervolume (or hypersurface element).
The Multidimensional stationary phase formula
For an interior stationary point, the asymptotic representation of the multidimensional Fourier-like integral, for a simple stationary point, which is to say a point
{\displaystyle {\boldsymbol {x}}={\boldsymbol {x}}_{0}}
{\displaystyle \nabla \phi ({\boldsymbol {x}}_{0})=0}
but for which the determinant of the Hessian matrix
{\displaystyle \det A\equiv \det {\frac {\partial ^{2}\phi ({\boldsymbol {x}}_{0})}{\partial x^{\alpha }\partial x^{\beta }}}\neq 0}
{\displaystyle \alpha ,\beta =1,2,...,n}
. As with other discussions, the large parameter
{\displaystyle \lambda >0}
{\displaystyle I(\lambda )\sim \left[{\frac {2\pi }{\lambda }}\right]^{n/2}{\frac {f({\boldsymbol {x}}_{0})}{\left.{\sqrt {|\det A|}}\right|_{{\boldsymbol {x}}={\boldsymbol {x}}_{0}}}}e^{i\lambda \phi ({\boldsymbol {x}}_{0})+{\frac {i\pi }{4}}{\mbox{sig}}(A)}+O(\lambda ^{-n})}
{\displaystyle \lambda \rightarrow \infty }
{\displaystyle {\mbox{sig}}(A)}
is the signature of the matrix
{\displaystyle A}
, which is the number of positive eigenvalues of
{\displaystyle A}
minus the number of its negative eigenvalues.
↑ Bleistein, N. and Handelsman, R.A., (1986). Asymptotic expansions of integrals. Courier Corporation.
↑ Bleistein, N. (1984). Mathematical methods for wave phenomena. Academic Press.
↑ Erdélyi, A. (1956). Asymptotic expansions (No. 3). Courier Corporation.
Retrieved from "https://wiki.seg.org/index.php?title=Method_of_stationary_phase&oldid=161854"
|
A simple test for determining if a matrix is full rank is to calculate its [http://mathworld.wolfram.com/Determinant.html determinant]. If the determinant is zero, there are linearly dependent columns and the matrix is not full rank. Prof. John Doyle also mentioned during lecture that one can perform the [http://mathworld.wolfram.com/SingularValueDecomposition.html singular value decomposition] of a matrix, and if the lowest singular value
A simple test for determining if a square matrix is full rank is to calculate its [http://mathworld.wolfram.com/Determinant.html determinant]. If the determinant is zero, there are linearly dependent columns and the matrix is not full rank. Prof. John Doyle also mentioned during lecture that one can perform the [http://mathworld.wolfram.com/SingularValueDecomposition.html singular value decomposition] of a matrix, and if the lowest singular value
For Single-input-single-output (SISO) systems, which are the focus of this course, the reachability matrix will always be square; more inputs make it wider (because the width <math>B</math> is equal to the number of inputs). In the case of non-square matrices, full rank means that the number of independent vectors is as large as possible.
{\displaystyle A}
{\displaystyle A}
{\displaystyle v_{i}}
{\displaystyle A}
{\displaystyle v_{j}\neq \Sigma _{i=0,i\neq j}^{n}a_{i}v_{i}}
{\displaystyle A}
{\displaystyle B}
|
feasible - Maple Help
Home : Support : Online Help : Mathematics : Optimization : Simplex Linear Optimization : feasible
determine if system is feasible or not
feasible(C, vartype)
feasible(C, vartype, 'NewC', 'Transform')
set of linear constraints
The function feasible returns true if a feasible solution to the linear system C exists, and false otherwise.
Non-negativity constraints on all the variables can be indicated by use of a second argument, NONNEGATIVE, or by explicitly listing the constraints. No restriction on the signs of the variable may be indicated by using UNRESTRICTED as the second argument to feasible.
The final two arguments are used to return, as sets, the final system found by feasible, and any variable transformations which occurred. The new system may have global artificial and slack variables present (such as _AR or _SL1).
The command with(simplex,feasible) allows the use of the abbreviated form of this command.
\mathrm{with}\left(\mathrm{simplex}\right):
\mathrm{feasible}\left({4x+3y\le 5,3x+4y=4},\mathrm{NONNEGATIVE}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{feasible}\left({0\le x,0\le y,4x-3y\le 5,3x-4y=4}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
|
XControlLimits - Maple Help
Home : Support : Online Help : Statistics and Data Analysis : ProcessControl Package : XControlLimits
XControlLimits(X, n, options)
(optional) equation(s) of the form option=value where option is one of confidencelevel, ignore, rbar, or xbar; specify options for computing the control limits
The XControlLimits command computes the upper and lower control limits for the X chart. Unless explicitly given, the mean and the average of the moving ranges of two observations of the underlying quality characteristic are computed based on the data.
ignore=truefalse -- This option controls how missing values are handled by the XControlLimits command. Missing values are represented by undefined or Float(undefined). So, if ignore=false and X contains missing data, the XControlLimits command returns undefined. If ignore=true, all missing items in X are ignored. The default value is true.
\mathrm{with}\left(\mathrm{ProcessControl}\right):
\mathrm{infolevel}[\mathrm{ProcessControl}]≔1:
A≔[33.75,33.05,34.00,33.81,33.46,34.02,33.68,33.27,33.49,33.20,33.62,33.00,33.54,33.12,33.84]:
\mathrm{XControlLimits}\left(A\right)
[\textcolor[rgb]{0,0,1}{32.2448378926038}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{34.8018287740628}]
l≔\mathrm{XControlLimits}\left(A,\mathrm{confidencelevel}=0.95\right)
\textcolor[rgb]{0,0,1}{l}\textcolor[rgb]{0,0,1}{≔}[\textcolor[rgb]{0,0,1}{32.2448378926038}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{34.8018287740628}]
|
Gate Turn-Off Thyristor - MATLAB - MathWorks Italia
Forced commutation switch-off loss
Switch-off loss, Eoff(Tj,Iak)
The GTO block models a gate turn-off thyristor (GTO). The I-V characteristic of a GTO is such that if the gate-cathode voltage exceeds the specified gate trigger voltage, the GTO turns on. If the gate-cathode voltage falls below the specified gate turn-off voltage value, or if the load current falls below the specified holding-current value, the device turns off .
To define the I-V characteristic of the GTO, set the On-state behaviour and switching losses parameter to either Specify constant values or Tabulate with temperature and current. The Tabulate with temperature and current option is available only if you expose the thermal port of the block.
In the off state, the anode-cathode path behaves like a linear resistor with a low off-state conductance value, Goff.
if ((v > Vf)&&((G>Vgt)||(i>Ih)))&&(G>Vgt_off)
Vgt_off is the gate turn-off voltage.
Using the Integral Diode parameters, you can include an integral cathode-anode diode. A GTO that includes an integral cathode-anode diode is known as an asymmetrical GTO (A-GTO) or reverse-conducting GTO (RCGTO). An integral diode protects the semiconductor device by providing a conduction path for reverse current. An inductive load can produce a high reverse-voltage spike when the semiconductor device suddenly switches off the voltage supply to the load.
Switching losses are one of the main sources of thermal loss in semiconductors. During each on-off switching transition, the GTO parasitics store and then dissipate energy.
Switching losses depend on the off-state voltage and the on-state current. When the switching device is turned on, the power losses depend on the initial off-state voltage across the device and the final on-state current once the device is fully in its on state. Similarly, when the switching device is force commutated off, the power losses depend on the initial on-state current through the device and the final off-state voltage once the device is fully in its off state. The switch on and force commutated switch off losses are either fixed or dependent on junction temperature and drain-source current, depending on how you specify the On-state behavior and switching losses parameter. In both cases, the losses are scaled by the off-state voltage prior to the latest device turn-on event.
When the current falls below the holding current and the device is naturally commutated off, the losses are set by the Natural commutation rectification loss parameter. Because it’s not possible to know when to measure the starting current or final voltage for the rectification loss, it is not possible to scale it by the off-state voltage or on-state current.
In this block, switching losses are applied by stepping up the junction temperature with a value equal to the switching loss divided by the total thermal mass at the junction.
As the final current after a switching event is not known during the simulation, if you use the GTO block as a fully-controlled device, the block records the on-state current at the point that the device is commanded off. If you use the GTO block as a partially-controlled device, the block records the on-state current once the current is greater than the holding current for a time longer than the value of the Wait time before switch-on current measurement parameter. Similarly, the block records the off-state voltage at the point that the device is commanded on. For this reason, the simlog does not report the switching losses to the thermal network until one switching cycle later
Gate trigger voltage, Vgt Gate trigger voltage, Vgt
Forward voltage, Vf On-state behaviour and switching losses
On-state resistance Forward voltage, Vf On-state voltage, Vak(Tj,Iak)
Off-state conductance On-state resistance Temperature vector, Tj
Rate of change of voltage versus current above the forward voltage. The default value is 0.001.
Anode-cathode conductance when the device is off. The value must be less than 1/R, where R is the value of On-state resistance. The default value is 1e-5.
Forced commutation switch-off loss — Forced commutation switch-off loss
600 V (default) | 300 V
Switch-off loss, Eoff(Tj,Iak) — Switch-off loss
[0, .0007, .0066, .033, .066, .17, .33, .83, 1.5; 0, .001, .01, .05, .1, .25, .5, 1.2, 2.2] * 1e-3 J (default)
Anode-cathode currents for which the switch-on loss and switch-off- loss are defined. The first element must be zero. Specify this parameter using a vector quantity.
Wait time before switch-on current measurement — Wait time before switch-on current measurement
Time to wait before recording the on-state current if you use the GTO block as a partially-controlled device. If you use the GTO block as a fully-controlled device, set this parameter to 0.
-\frac{{i}^{2}{}_{RM}}{2a},
From R2021a forward, the Energy dissipation time constant parameter of the GTO (Ideal, Switching) block is no longer used. A step in junction temperature now reflects the switching losses. If your model contains a thermal mass directly connected to this block thermal port, remove it and model the thermal mass inside the component itself.
From R2020b forward, the GTO (Ideal, Switching) block has improved losses and thermal modelling options.
As a result from these changes:
Diode | Ideal Semiconductor Switch | IGBT (Ideal, Switching) | MOSFET (Ideal, Switching) | N-Channel MOSFET | P-Channel MOSFET | Thyristor (Piecewise Linear)
|
To Edward Cresy 2 November [1860]1
15 Marine Parade, Eastbourne
Nov: 2nd.
I return by this post Dr. Taylor’s pamphlet which is very interesting.—2
This morning I vowed to myself that I would give you no more trouble; this evening I break my vow! Do read over Hofman’s note,3 it seems to me clear that he asserts that he has easily distinguished 0.002 of milligram of iodine and this equals, as I make out, 3/100,000 of a grain! Whereas Taylor, in your copy, speaks of only
\frac{1}{1400}
of a grain.4 Can you understand this? Pray do not trouble Hofman again, for I shall see him in course of early winter. Now I am on weights, can you assist me in another point. I have used ordinary apothecary measures (which I shall state in my paper), viz, fluid oz. divided into 8 drachms and each drachm into 60 minims,—so that each oz. contains 480 minims.5 But when I looked into such books as I possessed I found the most ridiculous discrepancies and misprints of weight of one such oz. of distilled water. I applied to good and clever druggist and he says positively one such oz. weighs 437.5 grains and this agreed with one of my printed authorities; so I have provisionally used it in calculating proportions. Can you anyhow aid me in telling me whether this is correct?6
God forgive me for being so troublesome, and believe me, | Yours very sincerely | C. Darwin
Please return Hofman’s note.
The letter was written in the evening, after the preceding letter was posted.
Taylor 1860. See preceding letter.
Letter from A. W. von Hofmann to Edward Cresy, 27 October 1860. See preceding letter.
Taylor 1860. There is a note on this pamphlet in DAR 60.1: 69 listing the amounts of arsenic and antimony that could be detected.
A note giving these calculations and apparently subsequent emendations is in DAR 60.1: 70.
See letter from Edward Cresy, 10 November 1860.
Taylor, Alfred Swaine. 1860. Facts and fallacies connected with the research for arsenic and antimony; with suggestions for a method of separating these poisons from organic matter. Guy’s Hospital Reports 3d ser. 6: 201–65.
Discusses pamphlet by A. S. Taylor
and note by A. W. v. Hofmann concerning iodine solution.
|
A sharply 2-transitive group without a non-trivial abelian normal subgroup
Eliyahu RipsYoav SegevKatrin Tent
SRB measures for partially hyperbolic systems whose central direction is weakly expanding
José F. AlvesCarla L. DiasStefano LuzzattoVilton Pinheiro
Superscars in the Šeba billiard
Pär KurlbergHenrik Ueberschär
Topological groups,
\mu
-types and their stabilizers
Ya'acov PeterzilSergei Starchenko
A new proof of Savin's theorem on Allen–Cahn equations
Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials
A model-theoretic study of right-angled buildings
Andreas BaudischAmador Martin-PizarroMartin Ziegler
Cocenters and representations of affine Hecke algebras
Dan CiubotaruXuhua He
Geometric Eisenstein series: twisted setting
|
The rank of a matrix <math>A</math> is the number of independent columns of <math>A</math>. A square matrix is full rank if all of its columns are independent. That is, a square full rank matrix has no column vector <math>v_i</math> of <math>A</math> that can be expressed as a linear combination of the other column vectors <math>v_j \neq \Sigma_{i = 0, i\neq j}^{n} a_i v_i</math>. For example, if one column of <math>A</math> is equal to twice another one, then those two columns are linearly dependent (with a scaling factor 2) and thus the matrix would not be full rank.
The rank of a matrix <math>A</math> is the number of independent columns of <math>A</math>. A square matrix is full rank if all of its columns are independent. That is, a square full rank matrix has no column vector <math>v_i</math> of <math>A</math> that can be expressed as a linear combination of the other column vectors. That is, <math>v_j \neq \Sigma_{i = 1, i\neq j}^{n} a_i v_i</math> for any set of <math>a_i</math>. For example, if one column of <math>A</math> is equal to twice another one, then those two columns are linearly dependent (with a scaling factor 2) and thus the matrix would not be full rank.
modified by [[User:Fuller|Sawyer Fuller]] 18:12, 3 November 2007 (PDT) to be more specific about square and non-square matrices
{\displaystyle A}
{\displaystyle A}
{\displaystyle v_{i}}
{\displaystyle A}
{\displaystyle v_{j}\neq \Sigma _{i=1,i\neq j}^{n}a_{i}v_{i}}
{\displaystyle a_{i}}
{\displaystyle A}
{\displaystyle B}
|
Mertens' theorems - Wikipedia
Mertens' theorems
For Mertens's theorem on convergence of Cauchy products of series, see Cauchy product § Convergence and Mertens' theorem.
In number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.[1] "Mertens' theorem" may also refer to his theorem in analysis.
1.1 Changes in sign
1.2 Mertens' second theorem and the prime number theorem
1.3 Mertens' third theorem and sieve theory
{\displaystyle p\leq n}
mean all primes not exceeding n.
Mertens' first theorem:
{\displaystyle \sum _{p\leq n}{\frac {\ln p}{p}}-\ln n}
does not exceed 2 in absolute value for any
{\displaystyle n\geq 2}
. (A083343)
Mertens' second theorem:
{\displaystyle \lim _{n\to \infty }\left(\sum _{p\leq n}{\frac {1}{p}}-\ln \ln n-M\right)=0,}
where M is the Meissel–Mertens constant (A077761). More precisely, Mertens[1] proves that the expression under the limit does not in absolute value exceed
{\displaystyle {\frac {4}{\ln(n+1)}}+{\frac {2}{n\ln n}}}
{\displaystyle n\geq 2}
Mertens' third theorem:
{\displaystyle \lim _{n\to \infty }\ln n\prod _{p\leq n}\left(1-{\frac {1}{p}}\right)=e^{-\gamma },}
where γ is the Euler–Mascheroni constant (A001620).
Changes in sign[edit]
In a paper [2] on the growth rate of the sum-of-divisors function published in 1983, Guy Robin proved that in Mertens' 2nd theorem the difference
{\displaystyle \sum _{p\leq n}{\frac {1}{p}}-\ln \ln n-M}
changes sign infinitely often, and that in Mertens' 3rd theorem the difference
{\displaystyle \ln n\prod _{p\leq n}\left(1-{\frac {1}{p}}\right)-e^{-\gamma }}
changes sign infinitely often. Robin's results are analogous to Littlewood's famous theorem that the difference π(x) − li(x) changes sign infinitely often. No analog of the Skewes number (an upper bound on the first natural number x for which π(x) > li(x)) is known in the case of Mertens' 2nd and 3rd theorems.
Mertens' second theorem and the prime number theorem[edit]
Regarding this asymptotic formula Mertens refers in his paper to "two curious formula of Legendre",[1] the first one being Mertens' second theorem's prototype (and the second one being Mertens' third theorem's prototype: see the very first lines of the paper). He recalls that it is contained in Legendre's third edition of his "Théorie des nombres" (1830; it is in fact already mentioned in the second edition, 1808), and also that a more elaborate version was proved by Chebyshev in 1851.[3] Note that, already in 1737, Euler knew the asymptotic behaviour of this sum.
Mertens diplomatically describes his proof as more precise and rigorous. In reality none of the previous proofs are acceptable by modern standards: Euler's computations involve the infinity (and the hyperbolic logarithm of infinity, and the logarithm of the logarithm of infinity!); Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre-Gauss conjecture, which was not proved until 1896 and became better known as the prime number theorem.
Mertens' proof does not appeal to any unproved hypothesis (in 1874), and only to elementary real analysis. It comes 22 years before the first proof of the prime number theorem which, by contrast, relies on a careful analysis of the behavior of the Riemann zeta function as a function of a complex variable. Mertens' proof is in that respect remarkable. Indeed, with modern notation it yields
{\displaystyle \sum _{p\leq x}{\frac {1}{p}}=\log \log x+M+O(1/\log x)}
whereas the prime number theorem (in its simplest form, without error estimate), can be shown to be equivalent to[4]
{\displaystyle \sum _{p\leq x}{\frac {1}{p}}=\log \log x+M+o(1/\log x).}
In 1909 Edmund Landau, by using the best version of the prime number theorem then at his disposition, proved[5] that
{\displaystyle \sum _{p\leq x}{\frac {1}{p}}=\log \log x+M+O(e^{-(\log x)^{1/14}})}
holds; in particular the error term is smaller than
{\displaystyle 1/(\log x)^{k}}
for any fixed integer k. A simple summation by parts exploiting the strongest form known of the prime number theorem improves this to
{\displaystyle \sum _{p\leq x}{\frac {1}{p}}=\log \log x+M+O(e^{-c(\log x)^{3/5}(\log \log x)^{-1/5}})}
{\displaystyle c>0}
Similarly a partial summation shows that
{\displaystyle \sum _{p\leq x}{\frac {\log p}{p}}=\log x+C+o(1)}
is equivalent to the PNT.
Mertens' third theorem and sieve theory[edit]
An estimate of the probability of
{\displaystyle X}
{\displaystyle X\gg n}
) having no factor
{\displaystyle \leq n}
{\displaystyle \prod _{p\leq n}\left(1-{\frac {1}{p}}\right)}
This is closely related to Mertens' third theorem which gives an asymptotic approximation of
{\displaystyle P(p\nmid X\ \forall p\leq n)={\frac {1}{e^{\gamma }\ln n}}}
The main step in the proof of Mertens' second theorem is
{\displaystyle O(n)+n\log n=\log n!=\sum _{p^{k}\leq n}\lfloor n/p^{k}\rfloor \log p=\sum _{p^{k}\leq n}\left({\frac {n}{p^{k}}}+O(1)\right)\log p=n\sum _{p^{k}\leq n}{\frac {\log p}{p^{k}}}\ +O(n)}
where the last equality needs
{\displaystyle \sum _{p^{k}\leq n}\log p=O(n)}
{\displaystyle \sum _{p\in (n,2n]}\log p\leq \log {2n \choose n}=O(n)}
Thus, we have proved that
{\displaystyle \sum _{p\leq n}{\frac {\log p}{p}}=\log n+O(1)}
A partial summation yields
{\displaystyle \sum _{p\leq n}{\frac {1}{p}}=\log \log n+M+O(1/\log n)}
^ a b c F. Mertens. J. reine angew. Math. 78 (1874), 46–62 Ein Beitrag zur analytischen Zahlentheorie
^ Robin, G. (1983). "Sur l'ordre maximum de la fonction somme des diviseurs". Séminaire Delange–Pisot–Poitou, Théorie des nombres (1981–1982). Progress in Mathematics. 38: 233–244.
^ P.L. Tchebychev. Sur la fonction qui détermine la totalité des nombres premiers. Mémoires présentés à l'Académie Impériale des Sciences de St-Pétersbourg par divers savants, VI 1851, 141–157
^ Although this equivalence is not explicitly mentioned there, it can for instance be easily derived from the material in chapter I.3 of: G. Tenenbaum. Introduction to analytic and probabilistic number theory. Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge,1995.
^ Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909, Repr. Chelsea New York 1953, § 55, p. 197-203.
Yaglom and Yaglom Challenging mathematical problems with elementary solutions Vol 2, problems 171, 173, 174
Weisstein, Eric W. "Mertens Constant". MathWorld.
Sondow, Jonathan & Weisstein, Eric W. "Mertens Theorem". MathWorld.
Weisstein, Eric W. "Mertens' Second Theorem". MathWorld.
Varun Rajkumar, π(x) and the Sieve of Eratosthenes
Retrieved from "https://en.wikipedia.org/w/index.php?title=Mertens%27_theorems&oldid=1080598647"
|
System Study on Partial Gasification Combined Cycle With CO2 Recovery | J. Eng. Gas Turbines Power | ASME Digital Collection
System Study on Partial Gasification Combined Cycle With
CO2
Yujie Xu,
, Chinese Academy of Sciences, Beijing 100190, P.R.C.;
Hongguang Jin,
Hongguang Jin
e-mail: hgjin@mail.etp.ac.cn
Rumou Lin,
Rumou Lin
Xu, Y., Jin, H., Lin, R., and Han, W. (June 17, 2008). "System Study on Partial Gasification Combined Cycle With
CO2
Recovery." ASME. J. Eng. Gas Turbines Power. September 2008; 130(5): 051801. https://doi.org/10.1115/1.2938273
A partial gasification combined cycle with
CO2
recovery is proposed in this paper. Partial gasification adopts cascade conversion of the composition of coal. Active composition of coal is simply gasified, while inactive composition, that is char, is burnt in a boiler. Oxy-fuel combustion of syngas produces only
CO2
H2O
, so the
CO2
can be separated through cooling the working fluid. This decreases the amount of energy consumption to separate
CO2
compared with conventional methods. The novel system integrates the above two key technologies by injecting steam from a steam turbine into the combustion chamber of a gas turbine to combine the Rankine cycle with the Brayton cycle. The thermal efficiency of this system will be higher based on the cascade utilization of energy level. Compared with the conventional integrated gasification combined cycle (IGCC), the compressor of the gas turbine, heat recovery steam generator (HRSG) and gasifier are substituted for a pump, reheater, and partial gasifier, so the system is simplified obviously. Furthermore, the novel system is investigated by means of energy-utilization diagram methodology and provides a simple analysis of their economic and environmental performance. As a result, the thermal efficiency of this system may be expected to be 45%, with
CO2
recovery of 41.2%, which is 1.5–3.5% higher than that of an IGCC system. At the same time, the total investment cost of the new system is about 16% lower than that of an IGCC. The comparison between the partial gasification technology and the IGCC technology is based on the two representative cases to identify the specific feature of the proposed system. The promising results obtained here with higher thermal efficiency, lower cost, and less environmental impact provide an attractive option for clean-coal utilization technology.
boilers, Brayton cycle, coal gasification, combined cycle power stations, combustion equipment, gas turbines, steam turbines
Boilers, Coal, Combined cycles, Combustion, Fuel gasification, Gas turbines, Integrated gasification combined cycle power stations, Syngas, Temperature, Thermal efficiency, Combustion chambers, Steam, Steam turbines, Exergy
U.S. DOE Assistant Secretary for Fossil Energy
,” DOE/ASME Report No. 9309152.
Coal-Fired Combined Cycle Power Generation Technology With High Efficiency, Low Pollution, and Low Water Consumption
UNESCO, Senior Conference of Cleaning Coal-Fired Technology
Performance Analysis and Comparison of Three Coal-Fired Combined Cycle Systems
Gas Turbine Technology in China
Combined-Cycle Power Stations Using “Clean-Coal Technologies:” Thermodynamic Analysis of Full Gasification Versus Fluidised Bed Combustion With Partial Gasification
Foster Wheeler Development Corporation
,” Department of Energy, Contract No. DE-FC26-00NT40972.
Development of Advanced PFBC Technology
Proceedings of the Eighth SCEJ Symposium on Fluidization
Second-Generation PFBC Systems Research and Development—Phase 2, Circulating PFBC Test Results
,” Report No. DOE/MC/21023-94/C0352.
Exergy Analysis of Integration Between Air Separation Process and IGCC
CO2 Emission Abatement in IGCC Power Plants by Semiclosed Cycles—Part A: With Oxygen-Blown Combustion
Leading Options for the Capture of CO2 at Power Stations: IEA Greenhouse Gas R&D Programme
Proceedings of the Fifth International Conference on Greenhouse Gas Control Technologies
Thermodynamics Made Comprehensible
Moritsuka
Graphic Energy Analysis for Coal Gasification—Combined Power Cycle Based on Energy Utilization Diagram
The Multi-Objective and Unified Scale Evaluation Criterion for IGCC System
Journal of Engineering Thermophysics in China
CO2 Capture in Coal-Based IGCC Power Plants
Proceedings of the Seventh International Conference on Greenhouse Gas Control Technologies
, Sep.
Gas Turbine World 2001-02 GTW Handbook for Project Planning, Design and Construction
Pequot Publishing, Inc.
Co-Production of Hydrogen, Electricity and CO2 From Coal With Commercially Ready Technology. Part B: Economic Analysis
The Thermodynamic Performance of Two Combined Cycle Power Plants Integrated With Two Coal Gasification Systems
System Study on Partial Gasification Combined Cycle With CO 2 Recovery
Split Stream Boilers for High Temperature/High Pressure Topping Steam Turbine Combined Cycles
|
Home : Support : Online Help : Science and Engineering : Units : Environments : Simple : verify
The verify Function in the Simple Units Environment
In the Simple Units environment, the global verify function is replaced by a verify function that converts any unevaluated arithmetic operators, equalities, or inequalities to their global equivalents. This prevents problems that would otherwise arise because the global verify command expects to find the global versions, not the versions from the Simple Units environment.
\mathrm{verify}\left(3.50000003=a,3.499999997=a,'\mathrm{float}\left(100\right)=\mathrm{boolean}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
The following result is unexpectedly false due to the Simple Units version of an equation being passed to the global version of verify.
\mathrm{with}\left(\mathrm{Units}[\mathrm{Simple}]\right):
:-\mathrm{verify}\left(3.50000003=a,3.499999997=a,'\mathrm{float}\left(100\right)=\mathrm{boolean}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
By using the Simple Units version of verify, we make sure this equation is converted to its global version, and we get the expected answer.
\mathrm{verify}\left(3.50000003=a,3.499999997=a,'\mathrm{float}\left(100\right)=\mathrm{boolean}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
|
Heat capacity/Related Articles - Citizendium
Heat capacity/Related Articles
< Heat capacity
A list of Citizendium articles, and planned articles, about Heat capacity.
See also changes related to Heat capacity, or pages that link to Heat capacity or to this page or whose text contains "Heat capacity".
Auto-populated based on Special:WhatLinksHere/Heat capacity. Needs checking by a human.
Count Rumford [r]: (1753–1814) An American born soldier, statesman, scientist, inventor and social reformer. [e]
Joule [r]: The SI unit of energy (symbol: J) which is a measure of the capacity to do work or generate heat. [e]
Nuclear fuel [r]: Material that can be consumed to derive nuclear energy, usually heavy fissile elements that can be made to undergo nuclear fission chain reactions in a nuclear fission reactor. [e]
{\displaystyle C_{p}}
{\displaystyle C_{v}}
Specific heat [r]: The ratio of the quantity of heat required to raise the temperature of a body one degree to that required to raise the temperature of an equal mass of water one degree Celsius. [e]
Temperature [r]: A fundamental quantity in physics - describes how warm or cold a system is. [e]
Retrieved from "https://citizendium.org/wiki/index.php?title=Heat_capacity/Related_Articles&oldid=680561"
|
Ask Answer - Linear Inequalities - Expert Answered Questions for School Students
Find ?Hf
for acetic acid, HC2H3O2, using the following thermochemical data.
HC2H3O2 (l) + 2 O2 (g) ? 2 CO2 (g) + 2 H2O (l) ?H = -875. kJ/mole
C (s, graphite) + O2 (g) ? CO2 (g) ?H = -394.51 kJ/mole
H2 (g) + 1/2 O2 (g) ? H2O (l) ?H = -285.8 kJ/moleFind ?Hf
H2 (g) + 1/2 O2 (g) ? H2O (l) ?H = -285.8 kJ/mole
x + 2y \le 10 , x + y \ge 1 , x - y \le 0, x , y \ge 0
Sree Harine Govindaraj
How had the expression become equal to 1? (Please view the attatched photo).
20. Solve the inequality
\left|\frac{2 - 3x}{4}\right| < 5.
A robot is designed to move in a peculiar way and it can be set in motion by a microprocessor program. The program can be initiated by assigning a positive rational value to its variable n. The program directs the robot to move in the following way. As soon as the program is started, the robot starts from the point O, moves 2n metres northward and changes its direction by n° to the right. It then moves 2n metres forward and again changes its direction by n° to the right and continues in this manner till it reaches the starting point O, or till it covers a total distance of 1000 m, whichever happens first, and then it stops.
a. I assigned a value for n and started the program. If the robot finally came back to O and stopped, what is the total distance that it has covered?
2. 360 m
4. Cannot be determined.
b. For how many values of n in the intervals [1, 60] does the robot cover less than 1000 m, before it stops?
4. Infinte
Please solve question number 17
17. In the first four papers each of 100 marks Devansh got 95, 72, 73, 83 marks. If he wants an average of greater than or equal to 75 marks and less than 80 marks, find the range of marks he should score in the fifth paper.
what is the difference between isupper and toupper function in c++.
please whoever knows pls help asp
how much is 10000 more than 2025?
Is the ans 7975??
\mathrm{Solve} \mathrm{for} \mathrm{x}\phantom{\rule{0ex}{0ex}}\left(\mathrm{a}\right) \left|\mathrm{x}-1\right|+\left|\mathrm{x}-3\right|=5\phantom{\rule{0ex}{0ex}}\left(\mathrm{b}\right) \left|\mathrm{x}\right|-\left|\mathrm{x}-2\right|=2
\mathrm{Q}1 \mathrm{If} \frac{3}{\mathrm{x}}<2,\mathrm{then} \mathrm{x}\in \phantom{\rule{0ex}{0ex}}\mathrm{Choose} \mathrm{one}:\phantom{\rule{0ex}{0ex}}\circ \left(-\infty ,0\right)\cup \left(\frac{3}{2},\infty \right)\phantom{\rule{0ex}{0ex}}\circ \left(0,\frac{3}{2}\right)\phantom{\rule{0ex}{0ex}}\circ \left(-\infty , \frac{3}{2}\right)\phantom{\rule{0ex}{0ex}}\circ \left(\frac{3}{2},\infty \right)\phantom{\rule{0ex}{0ex}}
Q. 2 If \frac{3x-2}{5x-3}\ge 3, then x\in \phantom{\rule{0ex}{0ex}}\circ \left[\frac{7}{12},\frac{3}{5}\right]\phantom{\rule{0ex}{0ex}}\circ R- \left[\frac{7}{12},\frac{3}{5}\right]\phantom{\rule{0ex}{0ex}}\circ \left[\frac{7}{12},\frac{3}{5}\right)\phantom{\rule{0ex}{0ex}}\circ \left(\frac{7}{12},\frac{3}{5}\right]
\mathrm{Q}. \mathrm{If} \frac{{\mathrm{x}}^{4}}{{\left(\mathrm{x}-2\right)}^{2}}>0, \mathrm{then} \mathrm{x}\in \phantom{\rule{0ex}{0ex}}\mathrm{Choose} \mathrm{one}\phantom{\rule{0ex}{0ex}}\circ \mathrm{R}\phantom{\rule{0ex}{0ex}}\circ \mathrm{R}-\left\{2\right\}\phantom{\rule{0ex}{0ex}}\circ \left(0,2\right)\phantom{\rule{0ex}{0ex}}\circ \mathrm{R}-\left\{0,2\right\}
Q.4. If | x | < 4, then
x\in
\infty
\infty
|x|\ge 5, then x\in
[5,
\infty
R - [ -5, 5]
R - (- 5, 5)
Q.6. For
1\le x\le 5
, then value of | 2x - 7 | is
|
Athletes in the Middle Plains School District regularly receive personal advising on their nutrition. Coaches wondered if the nutritional advising was having an impact, so they divided athletes into two groups. One group received advice and one group did not. After six months, they collected the following data:
Received Nutrition Advice
Regularly Ate a
46
39
Often Did Not Eat a
89
73
What were the two groups in the study?
Which variable, if changed, had an effect on the outcome of the study?
Receiving nutrition advice.
What is the total number of athletes in each row?
What percentage of each row received nutrition advice?
What percentage of each row did not?
Is there an association between receiving the nutritional advice and regularly eating a balanced breakfast?
There does not appear to be an association, because about the same percentage of athletes who did receive advice and who did not receive advice ate a balanced breakfast
|
Tree_diagram_(probability_theory) Knowpia
Tree diagram (probability theory)
In probability theory, a tree diagram may be used to represent a probability space.
Tree diagram for events
{\displaystyle A}
{\displaystyle B}
Tree diagrams may represent a series of independent events (such as a set of coin flips) or conditional probabilities (such as drawing cards from a deck, without replacing the cards).[1] Each node on the diagram represents an event and is associated with the probability of that event. The root node represents the certain event and therefore has probability 1. Each set of sibling nodes represents an exclusive and exhaustive partition of the parent event.
The probability associated with a node is the chance of that event occurring after the parent event occurs. The probability that the series of events leading to a particular node will occur is equal to the product of that node and its parents' probabilities.
^ "Tree Diagrams". BBC GCSE Bitesize. BBC. p. 1,3. Retrieved 25 October 2013.
Charles Henry Brase, Corrinne Pellillo Brase: Understanding Basic Statistics. Cengage Learning, 2012, ISBN 9781133713890, pp. 205–208 (online copy at Google)
tree diagrams - examples and applications
Wikimedia Commons has media related to Probability trees.
|
{\mathrm{Zn}}^{2+}\left(\mathrm{aq}\right)+2{\mathrm{e}}^{-}⇌\mathrm{Zn}\left(\mathrm{s}\right)
\mathrm{At}\mathrm{the}\mathrm{zinc}\mathrm{rod}:\mathrm{Zn}\left(\mathrm{s}\right)\to {\mathrm{Zn}}^{2+}\left(\mathrm{aq}\right)+2{\mathrm{e}}^{-}\mathrm{At}\mathrm{the}\mathrm{copper}\mathrm{rod}:{\mathrm{Cu}}^{2+}\left(\mathrm{aq}\right)+ 2{\mathrm{e}}^{-}\to \mathrm{Cu}\left(\mathrm{s}\right)
{\mathrm{Zn}}^{2+}\left(\mathrm{aq}\right)+2{\mathrm{e}}^{-}\to \mathrm{Zn}\left(\mathrm{s}\right)\mathrm{E}°=-0.76\mathrm{V}
Buffers Learn
pH and pKa Learn
Reaction Quotient Learn
|
Your Oxalis is O. Valdiviensis, Gay. of Chili—1
I am glad you are getting on with your movement of Cotyledon’s researches—2
My Address for Royal is nowhere I have not thought of a word for it, & every time I try it makes my head & heart ache— One’s last address ought to be good.3 I have this last
\frac{1}{2}
hour (moved thereto by your letter) maundered over the matter & written to De La Rue for some information relative to Electric discharges apropos of Spottiswoodes researches.4 Hitherto I have not, (like my predecessors) sponged on my Fellows for matter for my addresses Now I must, if, as I am advised, I am to give a resumè of some of the advances in Physical & Biological Sciences that have rendered the Societys labors noteworthy during my Presidentship.5 Would Frank give me some crude data in reference to your & his labors? & as to what they point to? I would work them up.6 Pray do not allude to it to him if you think better not. I should like to give a short analysis of the question of biogenesis—& so forth, but it makes me giddy to think of it. I shall consult the Godlike Huxley on this.7 I must keep off controversial questions.
I am very busy at Grays & my joint paper on the Botany of Colorado in relation to the rest of America & the Universe I suppose. It has I find curious relations with Altai which I hope to shew are not shared by the Floras of either Eastern or Western America. but these comparisons are very laborious.8
Balls & my Marocco Journals are nearly out, they await a brief Essay from me on the comparison of the Floras of Maroccoo & the Canaries— the differences are marvellous & quite unexpected.9 There are no Islands in the world so near the main land with such a difference in their vegetation— they beat the Galapagos in certain respects, but then the separate Islands do not differ much.
I must clear the American & Marocco works off before I begin my Address: happily the matter of these is in my head— then I must go to Paris on the 18th. to be present at the Prize Giving of the Exhibition—which is to be my only duty as a Royal Commissioner!10 I have shirked every other without exception & cannot have the impudence to decline this—though I do hate it.—
I am still looking out for a country cottage within easy distance of Kew to retire to on Sundays & perhaps in the end for weeks—months—years of Sundays;11 for between you & me I am getting giddy with Science in all shapes—& with the worry of Social, Scientific & official life, & I long for rest & nothing but the Library & Herbarium to busy myself with—
This is the best & most sensible growl you have had from me for a long time—
Ever yrs affec | J. D. Hooker.
See letter to J. D. Hooker, 3 October [1878] and n. 2. Oxalis valdiviensis is Chilean yellow-sorrel. Throughout Movement in plants, CD referred to the species as Oxalis valdiviana.
See letter to J. D. Hooker, 3 October [1878] and n. 3. CD began to study movement in cotyledons or seed-leaves in the late summer of 1877 (see Correspondence vol. 25, Appendix II).
See letter to J. D. Hooker, 3 October [1878] and n. 4. Hooker was preparing his final presidential address to the Royal Society of London; he was president of the society from 1873 to 1878 (ODNB).
In his presidential address, Hooker referred to the experimental work of Warren De la Rue, Hugo Müller, and William Spottiswoode on electrical discharge in a gas (Hooker 1878b, p. 49–50).
See Hooker 1878b, pp. 48–63.
In his presidential address, Hooker mentioned Francis Darwin’s work on the protoplasmic filaments in teasel and nutrition in Drosera (sundew) (Hooker 1878b, p. 58; see also F. Darwin 1877b and 1878a). CD’s work on botanical topics is summarised in Hooker 1878b, pp. 58–9.
No correspondence between Hooker and Thomas Henry Huxley on the subject of spontaneous generation has been identified. Hooker wrote that the discovery of bacteria afforded a morphological argument against the doctrine of spontaneous generation (Hooker 1878b, p. 61).
Hooker and Asa Gray’s paper, ‘The vegetation of the Rocky Mountain region and a comparison with that of other parts of the world’, was published in 1880 (Hooker and Gray 1880). The Altai is a mountain range in central Asia, largely in Russia and Kazakhstan, extending into Mongolia.
Hooker’s essay comparing the floras of Marocco (Morocco) and the Canary Islands is Appendix E in Journal of a tour in Marocco and the Great Atlas (Hooker and Ball 1878). The book was based on the journals kept by Hooker and John Ball during their 1871 trip (ibid., p. vi).
Hooker was a Royal Commissioner of the British section of the Paris Universal International Exhibition of 1878 (Paris Exhibition 1878, British section 1: vi). The exhibition was held from 20 May until 1 November 1878. A partial list of prizes awarded appeared in The Times, 19 October 1878, p. 10.
Hooker bought land in Sunningdale, Berkshire, in 1881; he built a house, later known as ‘The Camp’, in 1882 (Allan 1967, p. 238).
Paris Exhibition 1878, British section: Paris Universal International Exhibition, 1878. Official catalogue of the British section. 2d edition. London: HMSO. 1878.
|
Observer-Based H∞ Feedback Control for Arbitrarily Time-Varying Discrete-Time Systems With Intermittent Measurements and Input Constraints | J. Dyn. Sys., Meas., Control. | ASME Digital Collection
Observer-Based
H∞
Feedback Control for Arbitrarily Time-Varying Discrete-Time Systems With Intermittent Measurements and Input Constraints
Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 15, 2010; final manuscript received November 17, 2011; published online September 13, 2012. Assoc. Editor: Marcio de Queiroz.
Zhang, H., and Shi, Y. (September 13, 2012). "Observer-Based
H∞
Feedback Control for Arbitrarily Time-Varying Discrete-Time Systems With Intermittent Measurements and Input Constraints." ASME. J. Dyn. Sys., Meas., Control. November 2012; 134(6): 061008. https://doi.org/10.1115/1.4006070
In this paper, we investigate the observer-based
H∞
feedback control problem for discrete-time systems subject intermittent measurements and constrained control inputs. To characterize the practical scenario of the intermittent measurement phenomenon, we model it using a stochastic Bernoulli approach. We assume that the control action is constrained to be below a prescribed level. Sufficient conditions are obtained for the observer-based
H∞
feedback control problem. The estimator and the controller are derived by solving a linear matrix inequality (LMI)-based optimization problem. Moreover, the proposed method is extended to systems with arbitrarily time-varying parameters within a polytope with unknown vertices. Three examples are given to illustrate the effectiveness and efficacy of the proposed method.
arbitrarily time-varying systems, intermittent measurements, input constraints, observer-based H∞ performance
Control equipment, Design, Discrete time systems, Feedback, Pendulums, Time-varying systems, Probability, Theorems (Mathematics), Closed loop systems, Sensors
Robust Stabilization of Nonlinear Systems by
H∞
.10.1016/0167-6911(94)00032-Q
Parameterization of Nonlinear
H∞
State-Feedback Controllers
H∞
State Feedback Control for Linear Systems With State Delay and Parameter Uncertainty
H∞
Control for Uncertain Discrete Stochastic Time-Delay Systems
A State-Feedback Approach to Event-Based Control
H∞
Control in Multiple Channel Networked Control Systems With Random Packet Dropouts
Robust Stabilization for Composite Observer-Based Control of Discrete Systems
H∞
Controller Design for State Delayed Linear Systems
Observer-Based Stabilization of Switching Linear Systems
Robust Stabilization Conditions and Observer-Based Controllers for Fuzzy Systems With Input Delay
(A)), pp.
Fuzzy Observer Design for Near Space Vehicle With Application to Sensor Fault Estimation
ICIC Express Lett.
H∞
Control for Networked Systems With Random Communication Delays
Observer-Based Networked Control for Continuous-Time Systems With Random Sensor Delays
Robust Optimal Stabilizing Observer-Based Control Design of Decentralized Stochastic Singularly-Perturbed Computer Controlled Systems With Multiple Time-Varying Delays
H∞
Control for Systems With Repeated Scalar Nonlinearities and Multiple Packet Losses
A Parameter-Dependent Approach to Robust
H∞
Filtering for Time-Delay Systems
Parameter-Dependent Robust
H∞
Filtering for Uncertain Discrete-Time Systems
Optimal Recursive Estimation With Uncertain Observation
.10.1109/TIT.1969.1054329
Robust Finite-Horizon Filtering for Stochastic Systems With Missing Measurements
.10.1109/LSP.2005.847890
Robust Filtering With Stochastic Nonlinearities and Multiple Missing Measurements
H∞
Fuzzy Filtering of Nonlinear Systems With Intermittent Measurements
H∞
Fuzzy Output-Feedback Control With Multiple Probabilistic Delays and Multiple Missing Measurements
.10.1109/TFUZZ.2010.2047648
Robust Energy-to-Peak Filtering for Networked Systems With Time-Varying Delays and Randomly Missing Data
Robust Weighted
H∞
Filtering for Networked Systems With Intermittent Measurements of Multiple Sensors
Robust Minimum Variance Linear State Estimators for Multiple Sensors With Different Failure Rates
Fault Detection for Fuzzy Systems With Intermittent Measurements
Kalman Filter Based Adaptive Control for Networked Systems With Unknown Parameters and Randomly Missing Outputs
.10.1002/rnc.v19:18
H∞
Control for a Class of Nonlinear Discrete Time-Delay Stochastic Systems With Missing Measurements
H∞
Filtering for Stochastic Time-Delay Systems With Missing Measurements
Robust Non-Fragile Dynamic Vibration Absorbers With Uncertain Factors
AitRami
Nachidi
Static Output-Feedback for Takagicsugeno Systems With Delays
.10.1002/acs.v25.4
ASME J. Dyn. Sys., Meas., Control
Improved Robust Energy-to-Peak Filtering for Uncertain Linear Systems
Robust FIR Equalization for Time-Varying Communication Channels With Intermittent Observations via an LMI approach
Robust Mixed
H2/H∞
Control of Networked Control Systems With Random Time Delays in Both Forward and Backward Communication Links
|
Vector space - Citizendium
A vector space, also known as a linear space, is an abstract mathematical construct with many important applications in the natural sciences, in particular in physics and numerous areas of mathematics. Some vector spaces make sense somewhat intuitively, such as the space of 2D vectors in standard Euclidean plane, and the language that we use when talking about these intuitive spaces has been taken to describe the more abstract notion as well. For example, we know how to add vectors and multiply them by real numbers (scalars) in
{\displaystyle \mathbb {R} ^{3}}
, and these notions of vector addition and scalar multiplication are defined in a more general sense (as we will see below).
Vector spaces are important because many different mathematical objects that at first glance seem unrelated in fact share a common structure. By defining this structure and proving things about it in general, we are then able to apply these results to each specific case without having to re-prove them each time. Besides vectors in
{\displaystyle \mathbb {R} ^{3}}
that are relatively easy to visualize, we can make a vector space out of
{\displaystyle \mathbb {R} ^{n}}
for any natural number n; or the complex plane or powers of it,
{\displaystyle \mathbb {C} ^{n}}
; or polynomials of degree n.
Analyzing the structure of vector spaces in abstraction is also important for understanding which properties of a particular space follow solely from it having the structure of a vector space, and which require imposing additional structure on top of the vector space structure. For instance, vectors in every vector space can always be uniquely identified by assigning them a set of coordinates. However, the useful notion of the angle between vectors in
{\displaystyle \mathbb {R} ^{3}}
cannot be defined solely in terms of the vector space structure; it requires imposing the additional structure given an inner product on the space. Compartmentalizing mathematical information in this way can greatly aid mathematical intuition.
No matter what vector space you have to work with though, it is often useful to keep a picture of either 2D or 3D space in mind. This helps when thinking of things such as orthogonal polynomials or matrices.
2 Axioms of a vector space
3 Some important theorems
4 Examples of vector spaces
4.1.2 Negative vector
4.1.4 Multiplication by real number
5 Applications of vector spaces
{\displaystyle V}
{\displaystyle F}
is a set that satisfies certain axioms (see below) and which is equipped with two operations, vector addition and scalar multiplication. Vector addition is defined as a map
{\displaystyle +:\quad V\times V\to V}
that takes the ordered pair
{\displaystyle ({\vec {u}},{\vec {v}})\in V\times V}
to the vector
{\displaystyle {\vec {u}}+{\vec {v}}}
{\displaystyle \times }
represents the Cartesian product between sets. Scalar multiplication is defined in a similar way, as a map
{\displaystyle \cdot :\quad F\times V\to V}
{\displaystyle (a,{\vec {u}})\in F\times V}
{\displaystyle a\cdot {\vec {u}}}
. Note that frequently the dot representing scalar multiplication is omitted, the result being written simply as
{\displaystyle a{\vec {u}}}
instead. This is especially common when an inner product will also be defined on the vector space, with the dot then representing the inner product between two vectors. It is important to keep in mind the distinction between scalar multiplication, which multiplies one vector by a scalar, and an inner or scalar product, that combined two vectors to yield a scalar.
{\displaystyle V}
be a set,
{\displaystyle {\vec {u}}}
{\displaystyle {\vec {v}}}
{\displaystyle {\vec {w}}}
elements of that set, and
{\displaystyle a}nd
{\displaystyle b}
scalar elements of a field,
{\displaystyle F}
{\displaystyle V}
is a vector space if the following axioms hold true for all choices of
{\displaystyle {\vec {u}},\ {\vec {v}},\ a,\ b}
{\displaystyle V}
is closed under addition
{\displaystyle {\vec {u}}+{\vec {v}}}
{\displaystyle V}
. This is automatically satisfied when the addition operation is defined as being injective as it was above. Care must be taken however if
{\displaystyle V}
is a subset of some larger set
{\displaystyle W}
{\displaystyle +:\,\,V\times V\to W}
, as is often the case when looking at subspaces.
2. Addition is commutative
The order in which two vectors are added does not affect the result,
{\displaystyle {\vec {u}}+{\vec {v}}={\vec {v}}+{\vec {u}}}
3. Addition is associative
{\displaystyle {\vec {u}}+({\vec {v}}+{\vec {w}})=({\vec {u}}+{\vec {v}})+{\vec {w}}}
. This means that even though addition is strictly defined as a binary operation, the object
{\displaystyle {\vec {u}}+{\vec {v}}+{\vec {w}}}
4. An additive identity exists in
{\displaystyle V}
{\displaystyle {\vec {0}}}
, the additive identity or zero vector satisfies
{\displaystyle {\vec {0}}+{\vec {u}}={\vec {u}}+{\vec {0}}={\vec {u}}}
5. The additive inverse exists in
{\displaystyle V}
{\displaystyle -{\vec {u}}}
can be found such that
{\displaystyle -{\vec {u}}+{\vec {u}}={\vec {u}}+(-{\vec {u}})={\vec {0}}}
{\displaystyle V}
is closed under scalar multiplication
{\displaystyle a{\vec {u}}}
is itself an element of
{\displaystyle V}
7. Scalar multiplication is distributive over addition in
{\displaystyle F}
{\displaystyle (a+b){\vec {u}}=a{\vec {u}}+b{\vec {u}}}
. It is important to note that the addition occurring on the left-hand side of this equality is a 'different operation' from the addition on the right-hand side. While the latter is vector addition as defined above, the former is the addition operation defined on the field
{\displaystyle F}
8. Vector addition is distributive over scalar multiplication
{\displaystyle a({\vec {u}}+{\vec {v}})=a{\vec {u}}+a{\vec {v}}}
. In this case vector addition takes place on both sides of the equality.
9. Scalar multiplication is associative
{\displaystyle a(b{\vec {u}})=(ab){\vec {u}}}
. This means that the algebraic structure of the underlying field
{\displaystyle F}
is preserved. Note that the left-hand side of this equality contains two subsequent applications of the scalar multiplication defined above, while the right-hand side contains one scalar multiplication as defined in
{\displaystyle F}
(that of
{\displaystyle ab}
), followed by scalar multiplication with the vector
{\displaystyle {\vec {u}}}
10. The multiplicative identity of
{\displaystyle F}
provides a scalar multiplicative identity
{\displaystyle 1{\vec {u}}={\vec {u}}}
{\displaystyle 1}
is the multiplicative identity of the field
{\displaystyle F}
Properties 1 - 5 state that a vector space is an Abelian group with addition as group operation.
These axioms can be expressed concisely in mathematical notation as follows:
{\displaystyle \forall {\vec {u}},{\vec {v}},{\vec {w}}\in V,\ \forall a,b\in F,}
{\displaystyle {\vec {u}}+{\vec {v}}\in V}
{\displaystyle {\vec {u}}+{\vec {v}}={\vec {v}}+{\vec {u}}}
{\displaystyle {\vec {u}}+({\vec {v}}+{\vec {w}})=({\vec {u}}+{\vec {v}})+{\vec {w}}}
{\displaystyle \exists {\vec {0}}\in V:{\vec {0}}+{\vec {u}}={\vec {u}}+{\vec {0}}={\vec {u}}}
{\displaystyle \exists \ -\!{\vec {u}}\in V:-{\vec {u}}+{\vec {u}}={\vec {u}}+(-{\vec {u}})={\vec {0}}}
{\displaystyle a{\vec {u}}\in V}
{\displaystyle (a+b){\vec {u}}=a{\vec {u}}+b{\vec {u}}}
{\displaystyle a({\vec {u}}+{\vec {v}})=a{\vec {u}}+a{\vec {v}}}
{\displaystyle a(b{\vec {u}})=(ab){\vec {u}}}
{\displaystyle 1{\vec {u}}={\vec {u}}}
For more information, see: Linear independence.
A system of p ( ≥ 1 ) vectors
{\displaystyle {\vec {u}}_{1},\dots ,{\vec {u}}_{p}}
of a vector space V is called linearly dependent if there exist coefficients (elements in F ) a1, ..., ap not all zero, such that the linear combination is the zero vector in V,
{\displaystyle \sum _{\nu }\,a_{\nu }\,{\vec {u}}_{\nu }={\vec {0}}\in V.}
Otherwise, the vectors
{\displaystyle {\vec {u}}_{1},\dots ,{\vec {u}}_{p}}
are called linearly independent. A single vector not equal to the zero vector is obviously linearly independent.
If all a1, ..., ap are zero (in F ) then
{\displaystyle \sum _{\nu }\,a_{\nu }\,{\vec {u}}_{\nu }={\vec {0}}.}
{\displaystyle {\vec {u}}_{1},\dots ,{\vec {u}}_{p}}
is linearly independent then the relation
{\displaystyle \sum _{\nu }\,a_{\nu }\,{\vec {u}}_{\nu }={\vec {0}}}
implies that all a1, ..., ap are zero. Hence a set of p vectors in V is linearly independent if
{\displaystyle \sum _{\nu }\,a_{\nu }\,{\vec {u}}_{\nu }={\vec {0}}\quad \Longleftrightarrow \quad a_{\nu }=0,\quad {\hbox{for}}\quad \nu =1,\ldots ,p.}
Every set of vectors containing the zero vector is linearly dependent.
A system of linearly independent vectors
{\displaystyle {\vec {u}}_{1},\dots ,{\vec {u}}_{p}}
remains linearly independent if some vectors are omitted from the system. For, let a subset of the first q vectors
{\displaystyle {\vec {u}}_{1},\dots ,{\vec {u}}_{q}}
, with q < p, be linearly dependent then one or more coefficients not equal to zero can be found while the following is true
{\displaystyle a_{1}{\vec {u}}_{1}+\cdots +a_{q}{\vec {u}}_{q}={\vec {0}}.}
Add to the left- and right-hand side of this expression
{\displaystyle {\vec {0}}=0\cdot {\vec {u}}_{q+1}+\cdots +0\cdot {\vec {u}}_{p}}
and we get a contradiction.
In general there are infinitely many linearly independent vectors in a vector space. When the maximum number of linearly independent vectors is finite, say n, the vector space is called of finite dimension n. Otherwise the space is called infinite-dimensional. If V′ is a linear subspace of the n-dimensional space V (all elements of V′ belong simultaneously to V ), and V′ contains a set B of m linearly independent vectors then m < n, because B belongs to the n-dimensional space V. It follows that m is finite and that all subspaces of finite-dimensional spaces are finite-dimensional. If m is the maximum number of linearly independent vectors in V′ then this subspace is of dimension m < n. For finite n it can be shown that V′ coincides with V (is an "improper" subspace) if and only if n = m.
The set of all sequences {x1, x2, …, xn} of n elements of a field, in particular, the real numbers. Except for the Euclidean plane, the best known vector space is the space
{\displaystyle \mathbb {R} ^{n}}
. For integral finite n > 0 this space can be represented as columns (stacks) of n real numbers. In order to make the discussion concrete we consider the case n = 4. It will be clear how the rules apply to general finite n.
{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\\end{pmatrix}}+{\begin{pmatrix}y_{1}\\y_{2}\\y_{3}\\y_{4}\\\end{pmatrix}}={\begin{pmatrix}x_{1}+y_{1}\\x_{2}+y_{2}\\x_{3}+y_{3}\\x_{4}+y_{4}\\\end{pmatrix}}}
Because xk and yk are real numbers, xk+yk is a well-defined real number.
{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\\end{pmatrix}}+{\begin{pmatrix}-x_{1}\\-x_{2}\\-x_{3}\\-x_{4}\\\end{pmatrix}}={\begin{pmatrix}x_{1}-x_{1}\\x_{2}-x_{2}\\x_{3}-x_{3}\\x_{4}-x_{4}\\\end{pmatrix}}={\begin{pmatrix}0\\0\\0\\0\\\end{pmatrix}}}
{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\\end{pmatrix}}+{\begin{pmatrix}0\\0\\0\\0\\\end{pmatrix}}={\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\\end{pmatrix}}}
Multiplication by real number
{\displaystyle a{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\\end{pmatrix}}={\begin{pmatrix}a\,x_{1}\\a\,x_{2}\\a\,x_{3}\\a\,x_{4}\\\end{pmatrix}}}
Because a and xk are real numbers, a xk is well-defined and real.
The reader may easily convince him/herself, using the known properties of real numbers, that these columns of real numbers satisfy the postulates of a vector space. Its dimension is at least 4, because the following 4 vectors are linearly independent,
{\displaystyle {\begin{pmatrix}1\\0\\0\\0\\\end{pmatrix}},\quad {\begin{pmatrix}0\\1\\0\\0\\\end{pmatrix}},\quad {\begin{pmatrix}0\\0\\1\\0\\\end{pmatrix}},\quad {\begin{pmatrix}0\\0\\0\\1\\\end{pmatrix}}.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1)}
Indeed, assume that one or more of the coefficients (real numbers) ak is not equal to zero, then the equation
{\displaystyle a_{1}{\begin{pmatrix}1\\0\\0\\0\\\end{pmatrix}}+a_{2}{\begin{pmatrix}0\\1\\0\\0\\\end{pmatrix}}+a_{3}{\begin{pmatrix}0\\0\\1\\0\\\end{pmatrix}}+a_{4}{\begin{pmatrix}0\\0\\0\\1\\\end{pmatrix}}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\\a_{4}\\\end{pmatrix}}={\begin{pmatrix}0\\0\\0\\0\\\end{pmatrix}}.}
leads to all four a′s are zero (two vectors are equal if and only if their corresponding elements are equal). This is in contradiction to the assumption that one or more of the coefficients ak is not equal to zero.
The set (1) is maximally linearly independent because any non-zero vector can be expressed in the four vectors,
{\displaystyle {\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\\end{pmatrix}}=x_{1}{\begin{pmatrix}1\\0\\0\\0\\\end{pmatrix}}+x_{2}{\begin{pmatrix}0\\1\\0\\0\\\end{pmatrix}}+x_{3}{\begin{pmatrix}0\\0\\1\\0\\\end{pmatrix}}+x_{4}{\begin{pmatrix}0\\0\\0\\1\\\end{pmatrix}}\quad \Longrightarrow \quad -1{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\\x_{4}\\\end{pmatrix}}+x_{1}{\begin{pmatrix}1\\0\\0\\0\\\end{pmatrix}}+x_{2}{\begin{pmatrix}0\\1\\0\\0\\\end{pmatrix}}+x_{3}{\begin{pmatrix}0\\0\\1\\0\\\end{pmatrix}}+x_{4}{\begin{pmatrix}0\\0\\0\\1\\\end{pmatrix}}={\begin{pmatrix}0\\0\\0\\0\\\end{pmatrix}}.}
The equation on the right is a valid equation between five vectors that do not have a prefactor zero and yet give the zero vector. Hence it is not possible to find a fifth vector linearly independent of the vectors (1): any five vectors form a linearly dependent set. In other words, the four vectors in Eq. (1) form a basis of the vector space
{\displaystyle \mathbb {R} ^{n}}
The set of all polynomials of n variables {xi} with various coefficients {αi} from a field. Given polynomials like:
{\displaystyle f=\sum _{i=1}^{n}\ \alpha _{i}x_{i}^{n}\ ,}
{\displaystyle g=\sum _{i=1}^{n}\ \beta _{i}x_{i}^{n}\ ,}
it is clear that the various operations above are directly represented by a mapping:
{\displaystyle f\ \rightarrow \ \{\alpha _{1},\ \ldots \ ,\ \alpha _{n}\};\ g\ \rightarrow \ \{\beta _{1},\ \ldots \ ,\ \beta _{n}\}\ ,}
with the various powers of {xi} serving as place markers, so all the operations surveyed above for sequences apply equally here.
Consider the field ℝ of real numbers and I an interval in ℝ. The set C(I) of all real valued continuous functions on I, the set D(I) of all real differentiable functions and the set A(I) of all real analytic function on I are linear spaces contained in the linear space of all real valued functions defined on I.
Retrieved from "https://citizendium.org/wiki/index.php?title=Vector_space&oldid=26647"
|
사용자:Bin/원본 - 위키백과, 우리 모두의 백과사전
사용자:Bin/원본
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.[1]
Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear ones.
{\displaystyle T:V\to W}
{\displaystyle T(u+v)=T(u)+T(v),\quad T(av)=aT(v)}
{\displaystyle \quad T(au+bv)=T(au)+T(bv)=aT(u)+bT(v)}
{\displaystyle a_{1}v_{1}+a_{2}v_{2}+\cdots +a_{k}v_{k},}
{\displaystyle \dim(U_{1}+U_{2})=\dim U_{1}+\dim U_{2}-\dim(U_{1}\cap U_{2})}
{\displaystyle a_{1}v_{1}+a_{2}v_{2}+\cdots +a_{n}v_{n}.\,}
{\displaystyle Tv-\lambda v=(T-\lambda \,{\text{I}})v=0,}
{\displaystyle T(v)=T(a_{1}v_{1})+\cdots +T(a_{n}v_{n})=a_{1}T(v_{1})+\cdots +a_{n}T(v_{n})=a_{1}\lambda _{1}v_{1}+\cdots +a_{n}\lambda _{n}v_{n}.}
{\displaystyle \langle \cdot ,\cdot \rangle :V\times V\rightarrow \mathbf {F} }
{\displaystyle \langle u,v\rangle ={\overline {\langle v,u\rangle }}.}
{\displaystyle \langle au,v\rangle =a\langle u,v\rangle .}
{\displaystyle \langle u+v,w\rangle =\langle u,w\rangle +\langle v,w\rangle .}
{\displaystyle \langle v,v\rangle \geq 0}
{\displaystyle \|v\|^{2}=\langle v,v\rangle ,}
{\displaystyle |\langle u,v\rangle |\leq \|u\|\cdot \|v\|.}
{\displaystyle {\frac {|\langle u,v\rangle |}{\|u\|\cdot \|v\|}}\leq 1,}
{\displaystyle \langle u,v\rangle =0}
{\displaystyle a_{i}=\langle v,v_{i}\rangle }
{\displaystyle \langle Tu,v\rangle =\langle u,T^{*}v\rangle .}
{\displaystyle {\begin{alignedat}{7}2x&&\;+\;&&y&&\;-\;&&z&&\;=\;&&8&\qquad (L_{1})\\-3x&&\;-\;&&y&&\;+\;&&2z&&\;=\;&&-11&\qquad (L_{2})\\-2x&&\;+\;&&y&&\;+\;&&2z&&\;=\;&&-3&\qquad (L_{3})\end{alignedat}}}
{\displaystyle L_{2}+{\tfrac {3}{2}}L_{1}\rightarrow L_{2}}
{\displaystyle L_{3}+L_{1}\rightarrow L_{3}}
{\displaystyle {\begin{alignedat}{7}2x&&\;+&&y&&\;-&&\;z&&\;=\;&&8&\\&&&&{\frac {1}{2}}y&&\;+&&\;{\frac {1}{2}}z&&\;=\;&&1&\\&&&&2y&&\;+&&\;z&&\;=\;&&5&\end{alignedat}}}
{\displaystyle L_{3}+-4L_{2}\rightarrow L_{3}}
{\displaystyle {\begin{alignedat}{7}2x&&\;+&&y\;&&-&&\;z\;&&=\;&&8&\\&&&&{\frac {1}{2}}y\;&&+&&\;{\frac {1}{2}}z\;&&=\;&&1&\\&&&&&&&&\;-z\;&&\;=\;&&1&\end{alignedat}}}
{\displaystyle z=-1\quad (L_{3})}
{\displaystyle y=3\quad (L_{2})}
{\displaystyle x=2\quad (L_{1})}
{\displaystyle Ax=b.}
{\displaystyle x_{0}+N=\{x_{0}+n:n\in N\}}
{\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\,[a_{n}\cos(nx)+b_{n}\sin(nx)].}
{\displaystyle \langle f,g\rangle ={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)g(x)\,dx.}
{\displaystyle \langle f,h_{k}\rangle ={\frac {a_{0}}{2}}\langle h_{0},h_{k}\rangle +\sum _{n=1}^{\infty }\,[a_{n}\langle h_{n},h_{k}\rangle +b_{n}\langle \ g_{n},h_{k}\rangle ],}
{\displaystyle \langle f,h_{k}\rangle =a_{k}}
{\displaystyle a_{k}={\frac {1}{\pi }}\int _{-\pi }^{\pi }f(x)\cos(kx)\,dx.}
{\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }|\phi |^{2}dxdydz}
{\displaystyle H=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(x,y,z)}
Bin/원본
원본 주소 "https://ko.wikipedia.org/w/index.php?title=사용자:Bin/원본&oldid=11994784"
|
Differences from IEEE ASTM SI 10-1997 - Maple Help
Home : Support : Online Help : Science and Engineering : Units : Differences from IEEE ASTM SI 10-1997
Differences between Maple and the IEEE/ASTM SI 10-1997 Standard
Modifier Use for Plane Angles, Solid Angles, and Human Health Units
For plane and solid angles, radians and steradians are dimensionalized as modified ratios of lengths, namely length/length(radius) and length^2/length(radius)^2, respectively. The symbols in parentheses are called Unit annotations. For more information see the Units,angle and Units,solid_angle help pages.
Similarly, SI derived units related to human health, for example, units of absorbed dose, dose equivalent, and exposure, use annotations.
In the Simple Units environment, these differences are minimized: this is still how these units work behind the scenes, but whenever units are combined, the decision whether this succeeds or not is made while ignoring any annotations. For example, it is possible to add a unit-free quantity to an angle.
Unless all units are associated with symbols or abbreviations, an expression is displayed using unit names. For example, joules per kilogram is displayed as
\frac{J}{\mathrm{kg}}
, whereas joules per clove is displayed as
\frac{\mathrm{joule}}{\mathrm{clove}}
. This does not conform to the IEEE/ASTM standards, in which the latter is written as the statement joules per clove.
In Maple output, there is no separation between groups of three digits. Furthermore, in documentation, the comma is used as a separator for groups of three digits when four or more digits appear on either side of the decimal marker. The IEEE standard is to use a space separator.
In table A.1 of the IEEE/ASTM standard, the floating-point approximations for converting degrees, grad, and mil (angle) to radians, kilometers per hour to meters per second and lamberts to candela per square meter are labeled as exact conversions. In the Units package, the exact, rational conversion rates are used.
The table indicates that an acre equals 43560 square US survey feet, a chain equals 66 US survey feet, and a rod equals 16.5 US survey feet. Because these units are not intrinsically associated with US survey units, these conversions are used for US survey acres (acre[US_survey]), US survey chains (chain[US_survey]), and US survey rods (rod[US_survey]).
In all other cases, the Units package uses either the exact conversion rate from or a floating-point approximation at least as accurate as that listed in table A.1 of the IEEE/ASTM standards.
|
CGS - Maple Help
Home : Support : Online Help : Science and Engineering : Units : Systems : CGS
The Centimeter-Gram-Second (CGS) System of Units
The CGS system of units has the centimeter, gram, and second as its base units. Although the default CGS system does not include units of electricity or magnetism, the electrostatic and electromagnetic systems, which are also based in the centimeter, gram, and second, do.
When used in conjunction with the UseSystem command, the CGS system sets the following units for the given dimensions.
\mathrm{unit}
\mathrm{with}\left(\mathrm{Units}[\mathrm{Standard}]\right):
\mathrm{Units}[\mathrm{UseSystem}]\left('\mathrm{CGS}'\right)
5\mathrm{Unit}\left(\frac{'\mathrm{kg}''m'}{'s'}\right)
\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{\mathrm{kg}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{s}}⟧
\mathrm{combine}\left(,'\mathrm{units}'\right)
\textcolor[rgb]{0,0,1}{500000}\textcolor[rgb]{0,0,1}{}⟦\frac{\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{cm}}}{\textcolor[rgb]{0,0,1}{s}}⟧
\frac{\mathrm{Unit}\left('s'\right)}{}
\textcolor[rgb]{0,0,1}{500000}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{dyn}}⟧
\frac{{\mathrm{Unit}\left('\mathrm{cm}'\right)}^{2}}{}
\textcolor[rgb]{0,0,1}{500000}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{barye}}⟧
Units[EMU]
Units[ESU]
|
Science In Everyday Life for Class 7 Science Chapter 1 - Nutrition In Plants
Science In Everyday Life Solutions for Class 7 Science Chapter 1 Nutrition In Plants are provided here with simple step-by-step explanations. These solutions for Nutrition In Plants are extremely popular among Class 7 students for Science Nutrition In Plants Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Science In Everyday Life Book of Class 7 Science Chapter 1 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Science In Everyday Life Solutions. All Science In Everyday Life Solutions for class Class 7 Science are prepared by experts and are 100% accurate.
1. Modes of nutrition .................... ....................
2. Parasitic plants .................... ....................
3. Saprophytic plants .................... ....................
4. Insectivorous Plants .................... ....................
5. Symbiotic association .................... ....................
Autotrophic, Heterotrophic
Cuscuta, Mistletoe
Indian Pipe, Coral root
Drosera, Venus flytrap
Lichens, Pea plant
1. The substance that is broken down in the body to obtain energy ....................
2. The green pigment present in leaves ....................
3. Structures in leaves that contain the green pigment ....................
4. Stacks of thylakoids ....................
5. Structures that carry water and minerals from the roots to the leaves of a plant ....................
6. Structures that carry stratch to various parts of a plant ....................
Which of these is/are required for photosynthesis?
(b) Sunlight and water
During photosynthesis sunlight is used to convert carbon dioxide and water into oxygen and carbohydrates in presence of chlorophyll.
Plants obtain carbon dioxide from the atmosphere through these structures.
Plants obtain carbon dioxide in atmosphere through structures called stomata present on the underside of leaves.
Plant structures that carry food from leaces to the roots are
(d) both xylem and phloem
The plant structures, phloem carry the starch produced during photosynthesis from leaves to roots and other parts of the plants.
Which of these obtain nutrition from dead and decaying matter?
(a) Parasitic plants
(b) Saprophytic plants
(c) Insectivorous plants
(d) Symbiotic plants
Saprophytic plants obtain their nutrition from dead and decayed animal and plant matter.
Pea plants form a symbiotic association with a/an
Pea plant form a symbiotic association with a bacteria called Rhizobium, which helps to fix atmospheric nitrogen.
Autotrophs Water
Stroma Guard cells
Phloem Green plants
Stomata Chloroplast
Xylem Starch
Autotrophs Green Plants
Stroma Chloroplasts
Phloem Starch
Stomata Guard cells
Describe the process of photosynthesis in green plants with the help of a labelled diagram.
Photosynthesis is the process of converting water and carbon dioxide into carbohydrates or starch and oxygen in the presence of sunlight. Plants obtain water from the soil and carbon dioxide from air. Water reacts with carbon dioxide, to form starch and oxygen in the presence of sunlight and the green pigment in leaves called chlorophyll.
The reaction of photosynthesis can be represented as:
Carbon dioxide + Water
\underset{\mathit{a}\mathit{n}\mathit{d}\mathit{ }\mathit{c}\mathit{h}\mathit{l}\mathit{o}\mathit{r}\mathit{o}\mathit{p}\mathit{h}\mathit{y}\mathit{l}\mathit{l}}{\overset{\mathit{I}\mathit{n}\mathit{ }\mathit{p}\mathit{r}\mathit{e}\mathit{s}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathit{ }\mathit{o}\mathit{f}\mathit{ }\mathit{s}\mathit{u}\mathit{n}\mathit{l}\mathit{i}\mathit{g}\mathit{h}\mathit{t}}{\mathit{\to }}}
Starch + Oxygen
Describe how green plants obtain the things that are necessary for photosynthesis.
The things required for photosynthesis are chlorophyll, sunlight, water and carbon dioxide. Chlorophyll, which absorbs light energy, is present in chloroplasts in the leaves. Plants take atmospheric carbon dioxide through their openings called stomata on the underside of the leaves. Each stoma has two guard cells that swell and move away to open the stoma. The root system of the plants absorbs water from the soil, which reaches the leaves through xylem.
Describe the structure of a chloroplast with the help of a labelled diagram.
Chloroplasts are structures in the leaves that contain chlorophyll, which is required for photosynthesis. The chloroplast has an outer membrane and an inner membrane. Inside every chloroplast, stacks of discs are present. Each disc is called thylakoid. These contain the pigment, chlorophyll and are helpful in absorbing sunlight. The discs of thylakoid are present in stacks and are called grana. The grana are arranged in a fluid called stroma.
Define heterotrophic nutrition. Discuss the different types of heterotrophic nutrition in plants. Give two examples of each type.
Heterotrophic nutrition is a mode of nutrition in which an organism is unable to prepare its own food and depends on other organisms for food. The types of heterotrophic plants include:
Parasitic plants partially or completely depend on another host plant for their nutrition. E.g. Cuscuta and Mistletoe.
Saprophytic plants obtain their nutrition from dead and decaying animal or plant matter. E.g. - Indian Pipe and Coral root.
Insectivorous plants mostly obtain their nutrition by trapping and consuming animals, particularly insects. E.g. - Drosera, Venus flytrap and sun dew plant.
Symbiotic plants live in association with other organisms to share food and other resources. E.g. - Lichens and leguminous plants.
With the help of three examples, discuss how leaves of insectivorous plants are modified to trap insects.
Insectivorous plants trap insects and digest them for nutrition. Insectivorous plants have modifications to their leaves to help them trap insects:
The leaves of bladderwort are slender and have many small, pear shaped bladders that trap insects by sucking them in. The insects are then digested in the bladders.
The leaves of sundew plant have tentacles, which are long, thin structures having drops of sticky substance called mucilage at their ends. When an insect touches them, it sticks in the mucilage and is then digested.
The Venus flytrap has leaves with short, stiff hair on their inner surface. On touching the hair, the insects get snap shut in the leaves, where the trapped insect is eventually digested.
What do you understand by the term nutrition? Name the two main modes of nutrition.
Nutrition is the process by which an organism takes in food and uses it to obtain energy to live and grow.
The two main modes of nutrition are autotrophic and heterotrophic nutrition.
What is photosynthesis? Write the reaction that takes place during photosynthesis.
Photosynthesis is the process of converting water and carbon dioxide into carbohydrates or starch and oxygen in the presence of sunlight. The reaction of photosynthesis is:
\underset{\mathrm{and} \mathrm{chlorophyll}}{\overset{\mathrm{In} \mathrm{the} \mathrm{presence} \mathrm{of} \mathrm{light}}{\to }}
What are the general conditions necessary for photosynthesis?
The general conditions necessary for photosynthesis include the presence of chlorophyll or the green pigment, presence of light and the availability of water and carbon dioxide.
Explain how plants take in carbon dioxide through stomata.
Stomata are small openings present on the underside of the leaves. Each stoma is bounded by two guard cells that swell, when there is light and water and then move away from each other to open the stoma. This opening in stoma helps to obtain carbon dioxide from the atmosphere.
Differentiate between xylem and phloem.
Xylem transports water and soluble nutrients from the roots. The phloem transport starch produced by photosynthesis in the leaves.
The xylem transports raw materials. The phloem transports starch.
Differentiate between parasitic and saprophytic plants.
Parasitic Plants Saprophytic Plants
A parasitic plant depends upon another plant (the host) partially or entirely for nutrition. A saprophytic plant gets nutrients from dead and decaying organisms.
Parasitic plants have special roots that penetrate the host's stem or roots to directly absorb nutrients. Saprophytic plants have fungi in their roots to help break down dead and decaying plant or animal matter.
Parasitic plants may have leaves and are usually green. Saprophytic plants are generally white and do not have leaves.
Parasitic plants grow on other plants, which serve as the host. Saprophytic plants generally grow in deep shade in tropical forests.
Examples: Mistletoe and Cuscuta. Examples: Indian pipe and coral root are common saprophytic plants.
Differentiate between insectivorous and symbiotic plants.
Insectivorous plants Symbiotic plants
Insectivorous plants trap and consume insects for nutrition. Symbiotic plants live in association with other plants/organisms and both benefit from the association.
Examples: Venus fly trap and pitcher plant. Examples: Rhizobium and lichens.
|
NCERT Solutions for Class 12 Science Math Chapter 5 - Continuity And Differentiability
NCERT Solutions for Class 12 Science Math Chapter 5 Continuity And Differentiability are provided here with simple step-by-step explanations. These solutions for Continuity And Differentiability are extremely popular among Class 12 Science students for Math Continuity And Differentiability Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the NCERT Book of Class 12 Science Math Chapter 5 are provided here for you for free. You will also love the ad-free experience on Meritnation’s NCERT Solutions. All NCERT Solutions for class Class 12 Science Math are prepared by experts and are 100% accurate.
\mathrm{The} \mathrm{given} \mathrm{function} \mathrm{is} f\left(x\right)=5x-3\phantom{\rule{0ex}{0ex}}\mathrm{At} x=0, f\left(0\right)=5×0-3=-3\phantom{\rule{0ex}{0ex}}\underset{x\to 0}{\mathrm{lim}}f\left(x\right)=\underset{x\to 0}{\mathrm{lim}}\left(5x-3\right)=5×0-3=-3\phantom{\rule{0ex}{0ex}}\therefore \underset{x\to 0}{\mathrm{lim}}f\left(x\right)=f\left(0\right)
(a) The given function is
Case I: c < 5
Case II : c = 5
(i) c < 2
(ii) c > 2
(iii) c = 2
Case (i) c < 2
Case (ii) c > 2
Case (iii) c = 2
Is the function defined by continuous at x =
\mathrm{\pi }
It is evident that f is defined at x =
\mathrm{\pi }
(a) f (x) = sin x + cos x
(b) f (x) = sin x − cos x
(c) f (x) = sin x × cos x
(a) f (x) = g (x) + h (x) = sin x + cos x is a continuous function
(b) f (x) = g (x) − h (x) = sin x − cos x is a continuous function
(c) f (x) = g (x) × h (x) = sin x × cos x is a continuous function
⇒-{x}^{2}\le {x}^{2}\mathrm{sin}\frac{1}{x}\le {x}^{2}
\mathrm{Let} f\left(x\right)=\mathrm{sin}\left({x}^{2}+5\right), u\left(x\right)={x}^{2}+5, \mathrm{and} v\left(t\right)=\mathrm{sin}t
\mathrm{Then}, \left(vou\right)=v\left(u\left(x\right)\right)=v\left({x}^{2}+5\right)=\mathrm{tan}\left({x}^{2}+5\right)=f\left(x\right)
\mathrm{We} \mathrm{have},\phantom{\rule{0ex}{0ex}}y = {\mathrm{sin}}^{-1}\left[\frac{2x}{1 + {x}^{2}}\right]\phantom{\rule{0ex}{0ex}}\mathrm{put} x = \mathrm{tan} \theta ⇒ \theta = {\mathrm{tan}}^{-1}x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Now},\phantom{\rule{0ex}{0ex}} y = {\mathrm{sin}}^{-1}\left[\frac{2 \mathrm{tan} \theta }{1 + {\mathrm{tan}}^{2}\theta }\right]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒y = {\mathrm{sin}}^{-1}\left(\mathrm{sin} 2\theta \right), \left(\mathrm{as} \mathrm{sin} 2\theta =\frac{2 \mathrm{tan} \theta }{1 + {\mathrm{tan}}^{2}\theta }\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒y = 2\theta , \left(\mathrm{as} {\mathrm{sin}}^{-1}\left(\mathrm{sin} x\right)=x\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒y = 2 {\mathrm{tan}}^{-1}x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dx} = 2 × \frac{1}{1 + {x}^{2}}, \left\{\mathrm{because} \frac{d\left({\mathrm{tan}}^{-1}x\right)}{dx}=\frac{1}{1 + {x}^{2}}\right\}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dx} = \frac{2}{1 + {x}^{2}}
{\mathrm{sec}}^{2}\left(\frac{\mathrm{y}}{2}\right).\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{y}}{2}\right)=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}\right)
⇒{\mathrm{sec}}^{2}\frac{y}{2}×\frac{1}{2}\frac{\mathrm{dy}}{\mathrm{dx}}=1
⇒\frac{dy}{dx}=\frac{2}{se{c}^{2}\frac{y}{2}}
⇒\frac{dy}{dx}=\frac{2}{1+{\mathrm{tan}}^{2}\frac{y}{2}}
\frac{dy}{dx}=\frac{2}{1+{x}^{2}}
Differentiate in three ways mentioned below
(i) By using product rule.
(ii) By expanding the product to obtain a single polynomial.
(iii By logarithmic differentiation.
Do they all give the same answer?
From the above three observations, it can be concluded that all the results of are same.
(a) f is continuous on [a, b]
(b) f is differentiable on (a, b)
(c) f (a) = f (b)
(a) f is continuous on [−5, 5].
(b) f is differentiable on (−5, 5).
⇒\mathrm{sin}\left(a+y-y\right)\frac{dy}{dx}={\mathrm{cos}}^{2}\left(a+y\right)\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dx}=\frac{{\mathrm{cos}}^{2}\left(a+y\right)}{\mathrm{sin}a}
y=\left\{\begin{array}{l}\left|x\right| -\infty <x\le 1 \\ 2-x 1\le x\le \infty \end{array}\right\
|
Log-stretch DMO correction - SEG Wiki
The frequency-wavenumber DMO correction [1][2] described in this section is computationally intensive. Specifically, for each output frequency ω0, one has to apply the phase-shift exp(−iω0tnA), scale by (2A2 − 1)/A3, and sum the resulting output over input time tn as described by equation ( 14a ). A computationally more efficient DMO correction can be formulated in the logarithmic time domain [3][4] [5] [6][7]. The log-stretch time variable enables linearization of the coordinate transform equation ( 12b ), and as a result, the DMO correction is achieved by a simple multiplication of the input data with a phase-shift operator in the Fourier transform domain.
{\displaystyle P_{0}\left(k_{y},\omega _{0};h\right)=\int {\frac {2A^{2}-1}{A^{3}}}\times P_{n}\left(k_{y},t_{n};h\right){\rm {exp}}\left(-i\omega _{0}t_{n}A\right)dt_{n}.}
{\displaystyle \tau _{0}={\frac {t_{n}}{A}}}
Define the following logarithmic variables that correspond to the time variables τ0 and tn of equation ( 12b ):
{\displaystyle T_{0}={\rm {ln}}\ \tau _{0},}
{\displaystyle T_{n}={\rm {ln}}\ t_{n},}
where, for convenience, a constant scalar with its unit in time is omitted. Hence, the inverse relationships are given by
{\displaystyle \tau _{0}=e^{T_{0}},}
{\displaystyle t_{n}=e^{T_{n}}.}
Our goal is to derive equations for DMO correction in the log-stretch coordinates (y0, T0). The transform relation between the input log-stretch time variable Tn and the output log-stretch time variable T0 is given by
{\displaystyle T_{0}=T_{n}-{\rm {ln}}\ A_{e},}
and the expression for the midpoint variable y0 in the log-stretch domain is given by
{\displaystyle y_{0}=y_{n}-{\frac {h^{2}k_{y}}{A_{e}\Omega _{0}}},}
{\displaystyle A_{e}={\sqrt {1+{\frac {h^{2}k_{y}^{2}}{\Omega _{0}^{2}}}}}.}
The variable Ω0 is the Fourier transform dual of the variable T0 in the log-stretch domain. Equations ( 17a , 17b ) and ( 18 ) correspond to equations ( 12a , 12b ) and ( 13 ) in the log-stretch domain. Mathematical details of the derivation of equations ( 17a , 17b ) are left to Section E.3.
{\displaystyle y_{0}=y_{n}-{\frac {h^{2}k_{y}}{t_{n}A\omega _{0}}},}
{\displaystyle A={\sqrt {1+{\frac {h^{2}k_{y}^{2}}{t_{n}^{2}\omega _{0}^{2}}}}}.}
The log-stretch dip-moveout correction process is achieved by the following relationship (Section E.3):
{\displaystyle P_{0}\left(k_{y},\Omega _{0};h\right)={\rm {exp}}\left(-i{\frac {h^{2}k_{y}^{2}}{A_{e}\Omega _{0}}}+i\Omega _{0}{\rm {ln}}\ A_{e}\right)P_{n}\left(k_{y},\Omega _{0};h\right).}
Note that the relationship of input Pn(ky, Ω0; h) to output P0(ky, Ω0; h) given by equation ( 19 ) computationally is much simpler than that of equation ( 14a ). The log-stretch domain implementation of DMO correction involves application of a phase-shift given by the exponential in equation ( 19 ) to the input data; whereas, the frequency-wavenumber implementation involves an integral transform given by equation ( 14a ).
Figure 5.1-11 Impulse response of a log-stretch dip-moveout operator with source-receiver (S-G) offsets (a) 0 m, (b) 1000 m, (c) 2000 m, and (d) 3000 m.
To circumvent the logarithmic computation, a variation of the phase-shift term in equation ( 19 ) is given by Notfors and Godfrey [5]. As in most log-stretch formulations of DMO correction, this reference assumes that under DMO correction the midpoint variable is invariant; hence, by way of equation ( 17b ), the first term in the exponential of equation ( 19 ) drops out. A further approximation, In Ae = Ae − 1, and use of the definition for Ae given by equation ( 18 ) then lead to the following expression for DMO correction:
{\displaystyle P_{0}\left(k_{y},\Omega _{0};h\right)={\rm {exp}}\left[i\Omega _{0}\left({\sqrt {1+{\frac {h^{2}k_{y}^{2}}{\Omega _{0}^{2}}}}}-1\right)\right]P_{n}\left(k_{y},\Omega _{0};h\right).}
We now outline the steps in dip-moveout correction in the log-stretch domain:
Apply the logarithmic stretch in the time direction based on equation ( 15b ) so as to map each common-offset section Pn(yn, tn; h) in yn − tn coordinates to Pn(yn, Tn; h) in yn − Tn coordinates.
Perform 2-D Fourier transform of each common-offset section in the log-stretch domain.
Apply the phase-shift given by the exponential in equation ( 20 ) to each common-offset section Pn(ky, Ω0; h), and obtain the dip-moveout-corrected data P0(ky, Ω0; h) in the log-stretch Fourier transform domain.
Perform 2-D inverse Fourier transform to obtain the dip-moveout corrected common-offset section P0(y0, T0; h) in the log-stretch domain.
Undo the logarithmic stretch as in step (c) in the time direction based on equation ( 16a ) so as to obtain the dip-moveout-corrected data P(y0, τ0; h).
Figure 5.1-11 shows the impulse responses of a log-stretch DMO operator based on equation ( 20 ) for 1000-m, 2000-m and 3000-m offsets. The impulse responses greatly resemble those of the frequency-wave-number DMO correction described earlier (Figure 5.1-6b). Field data examples of DMO correction presented in this chapter mostly have been created using a log-stretch algorithm.
↑ Hale (1984), Hale, D., 1984, Dip moveout by Fourier transform: Geophysics, 49, 741–757.
↑ Black et al. (1993), Black, J., Schleicher, K. L. and Zhang, L., 1993, True-amplitude imaging and dip moveout: Geophysics, 58, 47–66.
↑ Bolondi et al., 1982, Bolondi, G., Loinger, E. and Rocca, F., 1982, Offset continuation of seismic sections: Geophys. Prosp., 30, 813–828.
↑ Bale and Jacubowicz, 1987, Bale, R. and Jacubowitz, H., 1987, Poststack prestack migration: 57th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 714–717.
↑ 5.0 5.1 Notfors and Godfrey, 1987, Marcoux, M. O., Godfrey, R. J., and Notfors, C. D., 1987, Migration for optimum velocity evaluation and stacking: Presented at the 49th Ann. Mtg. European Asn. Expl. Geophys.
↑ Liner, 1990, Liner, C. L., 1990, General theory and comparative anatomy of dip-moveout: Geophysics, 55, 595–607.
↑ Zhou et al. (1996), Zhou, B, Mason, I. M., and Greenalgh, S.A., 1996, Accurate and efficient shot-gather dip-moveout processing in the log-stretch domain: Geophys. Prosp., 43, 963–978.
Retrieved from "https://wiki.seg.org/index.php?title=Log-stretch_DMO_correction&oldid=19783"
|
Carbon - Marspedia
Abundance: 32% (atmosphere
as CO2 and CO)
Carbon, periodic table C, is a nonmetallic element. carbon on Mars is abundant in the the form of carbon dioxide. Elemental carbon can be produced via the bosch reaction. One of the important uses of carbon in a colony would be in the production of plastics and hydrocarbons.
Carbon makes up about 0.39%[1] of the matter in the solar system.
The most common isotope is carbon 12, carbon 14 is used for radiocarbon dating.
2 Role of carbon-formation in the future of Mars
4 In situ production
Massive stars (more than about half a solar mass) are capable of burning helium in the so-called triple-alpha process:
{\displaystyle {}^{4}He+{}^{4}He\rightleftharpoons {}^{8}Be}
{\displaystyle {}^{4}He+{}^{8}Be\rightleftharpoons {}^{12}C^{\star }}
{\displaystyle {}^{12}C^{\star }\rightarrow {}^{12}C+2\gamma }
{\displaystyle {}^{12}C^{\star }\rightarrow {}^{12}C+e^{+}+e^{-}}
In brief, two helium-4 nuclei are fused to create highly unstable beryllium-8 nuclei. While most of these nuclei simply decay back to helium, a small fraction will fuse with another helium-4 nucleus to form yet another unstable nucleus, an excited state of carbon-12 (here denoted
{\displaystyle {}^{12}C^{\star }}
). While once again most of these nuclei will simply decay back to helium-4 and beryllium-8, a tiny fraction will instead randomly decay to the ground state of carbon-12, where they will remain. Over time, this process produces a lot of energy and carbon.[1]
Once C12 has been created in a star, it can speed the fusion process using the CNO cycle (where C12 absorbs successive protons, becoming first Nitrogen then Oxygen, which then absorbs another proton and splits off a Helium nuclei and returns to C12). The CNO cycle acts as a catalyst, allowing stars to burn Hydrogen into Helium faster, increasing their temperature.
Role of carbon-formation in the future of Mars
Our sun, while massive enough to fuse helium, has not yet begun this process. When the helium core ignites in the distant future, the core will become very hot and dense, causing the outer layers of the sun to expand and cool[1]. This "red giant" phase of the sun's life will almost certainly destroy all life on Earth[2][3], quite possibly evaporating the planet, but Mars is likely to survive[2]. Whether conditions on Mars would be tolerable for any Earth-origin lifeforms at that time depends on less accurately known aspects of the process, mainly how much mass the sun loses and how much drag the planet experiences, but it seems likely[4] that the planet will be reasonably tolerable for hundreds of millions, if not billions of years.
Because there are no stable atomic nuclei of mass numbers 5 and 8, the triple-alpha process is the only way in which stars can create elements beyond helium on a large scale[1]. As a result, carbon is the fourth most common element in the universe. (It lies after oxygen because the same stars that create carbon-12 mostly convert it into oxygen-16 by the addition of another helium-4 nucleus[1].)
Carbon is readily available on Mars in the form of CO2 from the atmosphere, and from carbonate deposits in the Martian regolith. Elemental Carbon may be produced via the bosch reaction. The availability of Elemental Carbon as diamonds or graphite will need to be evaluated by exploration. Elemental carbon as graphene, carbon nanotubes or Buckyballs may be useful for certain applications.
Carbon is an essential element in organic molecules. Atmospheric CO2 and CO should be adequate as sources for all hydrocarbons and organic chemistry requirements for a settlement.
↑ 1.0 1.1 1.2 1.3 1.4 A.C. Phillips - The physics of stars 2nd ed. 1999. Wiley. ISBN 0-471-98798-0. pp. 127-135.
↑ 2.0 2.1 K.R. Rybicki & C. Denis - On the Final Destiny of the Earth and the Solar System 2001. Icarus, Vol. 151(1) pp. 130–137. Abstract available here.
↑ K.-P. Schröder & R.C. Smith - Distant future of the Sun and Earth revisited 2008. Monthly Notices of the Royal Astronomical Society, Vol. 386(1) pp. 155–163. Abstract available here.
↑ B. Lopez, J. Schneider & W.C. Danchi - Can Life Develop in the Expanded Habitable Zones around Red Giant Stars? 2005. The Astrophysical Journal, Vol. 627(2). Full text here.
Retrieved from "https://marspedia.org/index.php?title=Carbon&oldid=137965"
|
On Simple Graphs Arising from Exponential Congruences
2012 On Simple Graphs Arising from Exponential Congruences
M. Aslam Malik, M. Khalid Mahmood
We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integers
and
b
G\left(n\right)
denote the graph for which
V=\left\{0,1,\dots ,n-1\right\}
is the set of vertices and there is an edge between
and
b
{a}^{x}\equiv b \left(\text{mod\hspace{0.17em}}n\right)
is solvable. Let
n={p}_{1}^{{k}_{1}}{p}_{2}^{{k}_{2}}\cdots {p}_{r}^{{k}_{r}}
be the prime power factorization of an integer
n
{p}_{1}<{p}_{2}<\cdots <{p}_{r}
are distinct primes. The number of nontrivial self-loops of the graph
G\left(n\right)
has been determined and shown to be equal to
{\prod }_{i=1}^{r}\left(\varphi \left({p}_{i}^{{k}_{i}}\right)+1\right)
. It is shown that the graph
G\left(n\right)
{2}^{r}
components. Further, it is proved that the component
{\mathrm{\Gamma }}_{p}
of the simple graph
G\left({p}^{2}\right)
is a tree with root at zero, and if
n
is a Fermat's prime, then the component
{\mathrm{\Gamma }}_{\varphi \left(n\right)}
G\left(n\right)
M. Aslam Malik. M. Khalid Mahmood. "On Simple Graphs Arising from Exponential Congruences." J. Appl. Math. 2012 1 - 10, 2012. https://doi.org/10.1155/2012/292895
M. Aslam Malik, M. Khalid Mahmood "On Simple Graphs Arising from Exponential Congruences," Journal of Applied Mathematics, J. Appl. Math. 2012(none), 1-10, (2012)
|
Pile of Crabs | Toph
Pile of Crabs
There is a pile of crabs standing on one top of another.
There are total
N crabs numbered
1 to
N. The crab numbered
1 is at the bottom of the pile. Crab numbered
2 is on top of crab
1 and so on. The crab numbered
N is at the top of the pile.
Each crab has an initial health value represented with an integer number.
You have to observe
Q events.
For each event, a crab numbered
X will start climbing from his position to the top of the pile. Every crab he will cross on his way, he will decrease their health value by
1. If one’s health is equal to 0, the crab is dead and his value will not get decreased further. If the crab numbered
X is already on the top of the pile, he will not move.
If the crab numbered
X is already dead, he will not climb the pile. You have to report the crab is dead.
After observing all the events, you’ve to print the health value of each crab.
The first line on input consists of two integers
N and
Q
(3 ≤ N, Q ≤ 5 × 10^5)
(3≤N,Q≤5×105) represents the number of crabs and the number of events respectively.
N lines consists of an integer value
H_i
(1 ≤ H_i ≤ 10^9)
(1≤Hi≤109) represents the health of
i^{th}
ithcrab for each
i from
1 to
N.
Q lines consists of an integer
X represents the crab that will climb up the pile.
Q events, if the crab numbered
X is already dead, print
Dead in separate lines. Otherwise, do not print anything.
After observing all the events, print a single line with
N integers
H_i
Hifor each
i from
1 to
Nseperated by space.
H_i
Hi represent the health of
i^{th}
ith crab after all the events.
After the first event, the health of crab 1, 2, 3, and 4 is equal to 2, 3, 0, 3 respectively.
In the second event, we can see crab 3 is already dead. So, it won’t move.
After the third event, the health of crabs 1, 2, 3, and 4 is equal to 2, 2, 0, 2 respectively.
towfiq379Earliest, Apr '21
pathanLightest, 11 MB
This problem can be solved using SegmentSegmentSegment TreeTreeTree. First, take an array sized N+Q...
|
Teimpléad:Wikidata - Vicipéid
Teimpléad:Wikidata
For the module that should be used in infobox templates instead of this teimpléad, see Module:WikidataIB.
This teimpléad is intended to fetch data from Wikidata with or without a link to the connected Wikipedia article and with many other features.
The general structure of a call to this teimpléad is as follows. Note that the basic structure consists of positional commands, flags and arguments, which all have a fixed position.
Teimpléad:Tnull
Use different [[#Commands|Teimpléad:Background color]] to get different kinds of values from Wikidata. At least one command must be given and multiple commands can be combined into one call as shown above (in any order, more than two is also possible), but this only applies to commands from the claim class; calls containing a command from the general class cannot contain any other command. Each command can be followed by any number of [[#Command flags|Teimpléad:Background color]], which are optional and can be used to tweak the output generated by that command.
The commands and their flags may be followed by any number of [[#Configuration flags|Teimpléad:Background color]], which are also optional and affect the selection of data and the teimpléad's behaviour in general. The call is closed with the [[#Positional arguments|Teimpléad:Background color]], which may be required depending on the given command(s). Some named arguments (i.e. name-value pairs) also exist, as well as a set of named flags for advanced usage that can be used to change the way the fetched values are merged together into the output.
This teimpléad was designed to provide the basic needs for fetching data from Wikidata, but a lot can be achieved through different combinations of calls. For convenience, such combinations could be wrapped into new templates that serve a specific need. See also the section on common use cases below for some examples of useful "building blocks". Likewise, the functionality of this teimpléad can be extended by creating wrapper templates that use the main command provided by Module:Wd which is being used by this template (just like {{WikidataOI}} does).
Common use cases[athraigh foinse]
Teimpléad:Tnull Returns the Q-identifier of the Wikidata item connected to the current page (e.g. "Q55").
{{#if:Teimpléad:Tnull|...}} Performs a check to determine if the current page has a Wikidata item.
Commands[athraigh foinse]
The commands (Teimpléad:Background color, Teimpléad:Background color, ...) determine what kind of values are returned. One call can only contain commands from a single class.
Claim class[athraigh foinse]
Combine multiple commands into one call to this teimpléad, instead of making multiple calls to this teimpléad with one command each, to be sure that all the returned pieces of information belong to each other (see also the examples below).
first matchTeimpléad:Efn Teimpléad:Tnull Returns the requested property – or list of properties – from the current item-entity or from a given entity.
all matches Teimpléad:Tnull
first matchTeimpléad:Efn Teimpléad:Tnull Returns the requested qualifier – or list of qualifiers – from the given property of the current item-entity or of a given entity.
first matchTeimpléad:Efn Teimpléad:Tnull Returns a reference – or list of references – from the given property of the current item-entity or of a given entity.Teimpléad:Efn
General class[athraigh foinse]
Teimpléad:Tnull Returns the label of the current item-entity or of a given entity if present.
Teimpléad:Tnull Returns the title of the page connected to the current item-entity or to a given item-entity if such page exists.
Teimpléad:Tnull Returns the description of the current item-entity or of a given entity if present.
first matchTeimpléad:Efn Teimpléad:Tnull Returns an alias – or list of aliases – of the current item-entity or of a given entity if present.
first matchTeimpléad:Efn Teimpléad:Tnull Returns a badge – or list of badges – for the page connected to the current item-entity or to a given item-entity if such page exists.
Flags[athraigh foinse]
The following (optional) flags are available which can be used to alter this teimpléad's behaviour. They must be given after the (first) Teimpléad:Background color and before the Teimpléad:Background color. For convenience, empty flags (i.e. ||) are allowed and will simply be ignored.
Command flags[athraigh foinse]
These flags (Teimpléad:Background color, Teimpléad:Background color, ...) apply to the command that precedes them directly.
[EXPENSIVE] Returns the Teimpléad:Wpl of any entity returned if they have one attached. If that is not the case, then the default behaviour of returning the entity's label will occur.
Configuration flags[athraigh foinse]
These flags (Teimpléad:Background color) are general configuration flags and can be given anywhere after the first Teimpléad:Background color (but before the Teimpléad:Background color).
Sets a time constraint for the selected claim(s). Uses the claims' qualifiers of Teimpléad:Wpl and Teimpléad:Wpl to determine if the claim is valid for the selected time period(s).
Arguments[athraigh foinse]
Positional arguments[athraigh foinse]
The following table shows the available positional arguments (Teimpléad:Background color) in their fixed order. For each command, the applicable set of arguments is marked. If multiple commands are given, then the applicable set is the union of the individual sets. For instance, if the commands properties and qualifiers have been given, then at least both the arguments property_id and qualifier_id should be given as well.
Named arguments[athraigh foinse]
This argument can be used to set a particular date (e.g. |date=1731-02-11) relative to which claim matching using the future, current and former flags is done, instead of relative to today. It overrides the default of these flags to current so that by default only claims that were valid at the given date are returned (based on the claims' qualifiers of Teimpléad:Wpl and Teimpléad:Wpl).
Property aliases[athraigh foinse]
Advanced usage[athraigh foinse]
To use two opening square brackets that directly follow each other (i.e. Teimpléad:!((), use {{!((}}.
%p[ <span style="font-size:85\%">(%q)</span>][%s][%r]
if the property/properties command was given and the qualifier/qualifiers command was given
Teimpléad:Dfn default The fixed separator between each pair of claims, aliases or badges.
Teimpléad:Dfn if only the reference/references command was given without the raw flag
Teimpléad:Dfn default The separator between each pair of qualifiers of a single claim. These are the value separators for the %q1, %q2, %q3, ... parameters.
Teimpléad:Dfn if exactly one qualifier/qualifiers command was given The separator between each set of qualifiers of a single claim. This is the value separator for the %q parameter.
Teimpléad:Dfn if more than one qualifier/qualifiers command was given
Teimpléad:Dfn default The separator between each pair of references of a single claim. This is the value separator for the %r parameter.
Teimpléad:Dfn if the raw flag was given for the reference/references command
Teimpléad:Dfn default A punctuation mark placed at the end of the output. This will be placed on the %s parameter applied to the last claim (or alias or badge) in the list.
[[[:Teimpléad:Smallcaps]]]
If the teimpléad is transcluded on the Netherlands page (which is linked to Q55), then the Q55 can be omitted.
[[[:Teimpléad:Smallcaps]]], [[[:Teimpléad:Smallcaps]]]
[[[:Teimpléad:Smallcaps]]], [[[:Teimpléad:Smallcaps]]], [[[:Teimpléad:Smallcaps]]]
A total of Teimpléad:Tnull people live in the Netherlands.
The Netherlands has a population of Teimpléad:Tnull
<ul>Teimpléad:Tnull</ul>
0.922,[6] 0.787, 0.799, 0.829,[6][7] 0.861,[7] 0.877,[6] 0.891,[7] 0.909,[6] 0.919,[6] 0.920,[6] 0.920,[6] 0.834,[7] 0.835,[7] 0.839,[7] 0.864,[7] 0.866,[7] 0.865,[7] 0.867,[7] 0.870,[7] 0.876,[7] 0.879,[7] 0.878,[7] 0.883,[7] 0.886,[7] 0.897,[7] 0.904,[7] 0.906,[7] 0.906,[7] 0.910,[7] 0.921,[7] 0.921,[7] 0.923,[7] 0.924,[7] 0.926,[7] 0.928,[7] 0.931[7]
0.922,[6] 0.829,[6][7] 0.861,[7] 0.877,[6] 0.891,[7] 0.909,[6] 0.919,[6] 0.920,[6] 0.920,[6] 0.834,[7] 0.835,[7] 0.839,[7] 0.864,[7] 0.866,[7] 0.865,[7] 0.867,[7] 0.870,[7] 0.876,[7] 0.879,[7] 0.878,[7] 0.883,[7] 0.886,[7] 0.897,[7] 0.904,[7] 0.906,[7] 0.906,[7] 0.910,[7] 0.921,[7] 0.921,[7] 0.923,[7] 0.924,[7] 0.926,[7] 0.928,[7] 0.931[7]
{\displaystyle {\frac {n^{2}-1}{n^{2}+2}}={\frac {4\pi }{3}}N\alpha }
{\displaystyle \alpha }
{\displaystyle \pi }
If the teimpléad is transcluded on the Utrecht (province) page (which is linked to Q776), then the Q776 can be omitted.
If the teimpléad is transcluded on the Earth page (which is linked to Q2), then the Q2 can be omitted.
Q28865 = "Python", P548 = "version type", P348 = "software version identifier",
Get Python's latest stable release version with its references. You may want to use P548=Q2122918 to get the latest preview release version.
Example references[athraigh foinse]
↑ 5.0 5.1 https://data.worldbank.org/indicator/SP.POP.TOTL; World Bank Open Data; retrieved: 8 April 2019.
TemplateData[athraigh foinse]
This template fetches data from the centralized knowledge base Wikidata. To edit the data, click on "Wikidata item" in the left sidebar.
Module:Wd, the source module that is called by this template
{{Wikidata editnotice}}, an editnotice for articles that make extensive use of this template
an thuas doiciméadú Is é transcluded ó Teimpléad:Wikidata/doc. (in eagar | stair)
Aisghafa ó "https://ga.wikipedia.org/w/index.php?title=Teimpléad:Wikidata&oldid=971347"
|
Augmented Seren godbow (ice) - The RuneScape Wiki
For this item's standard variant, see seren godbow (ice).
Augmented Seren godbow (ice)Augmented Seren godbow (ice)(uncharged)? (edit)? (edit) ? (edit)chargeduncharged26 February 2018 (Update)? (edit)YesAn intricate bow of pure crystal.NoNo? (edit)falsefalseYesYestrueNoinfobox-cell-shown? (edit)falsefalseNo? (edit)falsefalseNo? (edit)falsefalseYestrueNoNo? (edit)falsefalseYou must augment another one using an augmentor. In addition, you will lose all the item's XP and gizmos.? (edit)Wield, Check, Disassembletrue? (edit)Check? (edit)? (edit)? (edit)falsefalseAugmented Seren godbow (ice)Augmented Seren godbow (ice)(uncharged)0? (edit)Not soldNo data to display-? (edit)-12500001,250,000 coinsreclaimable1830697432922398736895948Reclaimable
Sacrifice: 36,895,948--true? (edit)0.6750000500000750,000 coins500,000 coins? (edit)2.267 kg2.267? (edit)surface? (edit)4219242193? (edit)4219242193MRIDMRID • recipeMRID • recipe? (edit){"edible":"no","members":"yes","stackable":"no","stacksinbank":"no","death":"reclaimable","name":"Augmented Seren godbow (ice)","bankable":"yes","gemw":false,"equipable":"yes","disassembly":"yes","release_date":"26 February 2018","release_update_post":"Clue Scroll Overhaul - Lightning Weapons","lendable":"no","destroy":"You must augment another one using an augmentor. In addition, you will lose all the item's XP and gizmos.","highalch":500000,"weight":2.267,"tradeable":"no","examine":"An intricate bow of pure crystal.","noteable":"no"}{"edible":"no","members":"yes","noteable":"no","stackable":"no","stacksinbank":"no","death":"reclaimable","name":"Augmented Seren godbow (ice)","bankable":"yes","gemw":false,"equipable":"yes","disassembly":"yes","release_date":"26 February 2018","id":"42192","release_update_post":"Clue Scroll Overhaul - Lightning Weapons","lendable":"no","destroy":"You must augment another one using an augmentor. In addition, you will lose all the item's XP and gizmos.","highalch":500000,"weight":2.267,"tradeable":"no","examine":"An intricate bow of pure crystal.","version":"charged"}{"edible":"no","members":"yes","noteable":"no","stackable":"no","stacksinbank":"no","death":"reclaimable","name":"Augmented Seren godbow (ice)(uncharged)","bankable":"yes","gemw":false,"equipable":"yes","disassembly":"yes","release_date":"26 February 2018","id":"42193","release_update_post":"Clue Scroll Overhaul - Lightning Weapons","lendable":"no","destroy":"You must augment another one using an augmentor. In addition, you will lose all the item's XP and gizmos.","highalch":500000,"weight":2.267,"tradeable":"no","examine":"An intricate bow of pure crystal.","version":"uncharged"}Versions: 2
Item JSON: {"edible":"no","members":"yes","noteable":"no","stackable":"no","stacksinbank":"no","death":"reclaimable","name":"Augmented Seren godbow (ice)","bankable":"yes","gemw":false,"equipable":"yes","disassembly":"yes","release_date":"26 February 2018","id":"42192","release_update_post":"Clue Scroll Overhaul - Lightning Weapons","lendable":"no","destroy":"You must augment another one using an augmentor. In addition, you will lose all the item's XP and gizmos.","highalch":500000,"weight":2.267,"tradeable":"no","examine":"An intricate bow of pure crystal.","version":"charged"}Item JSON: {"edible":"no","members":"yes","noteable":"no","stackable":"no","stacksinbank":"no","death":"reclaimable","name":"Augmented Seren godbow (ice)(uncharged)","bankable":"yes","gemw":false,"equipable":"yes","disassembly":"yes","release_date":"26 February 2018","id":"42193","release_update_post":"Clue Scroll Overhaul - Lightning Weapons","lendable":"no","destroy":"You must augment another one using an augmentor. In addition, you will lose all the item's XP and gizmos.","highalch":500000,"weight":2.267,"tradeable":"no","examine":"An intricate bow of pure crystal.","version":"uncharged"}
SMW Subobject for unchargedHigh Alchemy value: 750000Examine: An intricate bow of pure crystal.Stacksinbank: falseIs variant of: Augmented Seren godbow (ice)Low Alchemy value: 500000Kept on death: reclaimableWeight: 2.267Value: 1250000Bankable: trueLendable: falseDisassembleable: trueIs members only: trueLocation restriction: surfaceTradeable: falseStackable: falseItem name: Augmented Seren godbow (ice)(uncharged)Release date: 26 February 2018Version anchor: unchargedNoteable: falseItem ID: 42193
SMW Subobject for chargedHigh Alchemy value: 750000Examine: An intricate bow of pure crystal.Stacksinbank: falseIs variant of: Augmented Seren godbow (ice)Low Alchemy value: 500000Kept on death: reclaimableWeight: 2.267Value: 1250000Bankable: trueLendable: falseDisassembleable: trueIs members only: trueLocation restriction: surfaceTradeable: falseStackable: falseItem name: Augmented Seren godbow (ice)Release date: 26 February 2018Version anchor: chargedNoteable: falseItem ID: 42192
The ice-dyed augmented seren godbow is a level 90 Ranged two-handed weapon, created by using an augmentor on a Seren godbow (Ice), or by using ice dye on an Augmented Seren godbow.
Augmented dyed equipment do not feature the cogs associated with most augmented items (they look identical to non-augmented versions). Disassembling augmented dyed items does not return the dye. Dying in unsafe player-versus-player (PvP) combat with this item drops a broken Seren godbow - the dye, augmentor, and gizmos are lost.
Weapon gizmos charged with perks can be used to enhance the item's abilities. As a two-handed slot item, the Augmented Seren godbow (ice) can hold 2 gizmo's, allowing up to 4 perks (2 perks each).
The augmented Seren godbow (ice) uses charges stored in the universal charge pack. When the charge pack runs out of charges, the item loses stats and gizmo effects, and can also no longer gain any equipment experience. Function will resume when the pack is recharged with divine charges.
SMW Subobject for chargedEquipment armour: 0.0Attack range: 9Is variant of: Augmented Seren godbow (ice)Weapon accuracy: 2577Ranged bonus: 0Prayer bonus: 0Equipment tier: 92Weapon attack speed: averageEquipment JSON: {"ranged":0,"class":"ranged","attack_range":9,"lp":0,"tier":92,"speed":"average","damage":2056.2,"type":"Chargebow","armour":"0.0","slot":"2h weapon","strength":0,"style":"arrows","invention":92,"magic":0,"prayer":0,"accuracy":2577}Magic bonus: 0Invention tier: 92Combat class: rangedEquipment type: ChargebowAttack style: arrowsEquipment life points: 0Weapon damage: 2056.2Strength bonus: 0PvP damage reduction: 0Is cosmetic recolour: trueEquipment slot: 2h weaponPvM damage reduction: 0
{\displaystyle d_{i}^{\text{fixed}}}
{\displaystyle i}
{\displaystyle i}
{\displaystyle d_{i}^{\text{random}}}
{\displaystyle i}
{\displaystyle i}
{\displaystyle d_{i}^{\text{fixed}}+d_{i}^{\text{random}}}
{\displaystyle d_{i}^{\text{fixed}}+r\cdot d_{i}^{\text{random}}}
{\displaystyle r}
{\displaystyle d_{1}^{\text{fixed}}=80\%}
{\displaystyle d_{1}^{\text{random}}=120\%}
{\displaystyle d_{i}^{\text{fixed}}=d_{1}^{\text{fixed}}}
{\displaystyle d_{i}^{\text{random}}=\left\lfloor {\frac {{\text{max}}(0,8-i)}{3}}\cdot d_{1}^{\text{fixed}}\right\rfloor }
{\displaystyle d_{1}^{\text{random}}}
The augmented Seren godbow (ice) can be levelled up by using it. The equipment experience required for each level and the effect of disassembly is as follows:
Retrieved from ‘https://runescape.wiki/w/Augmented_Seren_godbow_(ice)?oldid=35691667’
|
Is It A Square? | Toph
By fsshakkhor · Limits 1.5s, 512 MB
Shikamaru is the most brilliant student of his class. He is very good at Mathematics. Mr. Asuma is their math teacher. Today he taught the class about square numbers. A square number is a non-negative integer which is the product of some integer with itself. For example, 0, 1, 4, 9, 16, 25 are square numbers, but 2, 3, 5, 6 are not. Mr. Asuma wanted to test whether his students can identify square numbers or not. So he gave them a homework.
He gave them an array A[ consisting of n integers and asked q questions. In each question, he gave them two integers l and r which denotes two positions of the array. The students have to tell whether the product of all the integers between the two positions (l and r inclusive) is a square number or not. In other words, they have to compute the value of P represented as the following:
P = \prod\limits_{i=l}^r A_i
P=i=l∏rAi
And check whether it is a square or not. The array follows 1-based indexing.
After returning home, Shikamaru started doing his homework. He was able to answer some of the questions correctly, but later some of the products P became so huge and he could not identify whether they are square or not. Can you help Shikamaru doing his homework?
The first line contains two integers n (1 ≤ n ≤ 105) and q (1 ≤ q ≤ 105). The second line contains n integers A1, A2, ... , An ( -105 ≤ Ai ≤ 105 ). Each of the next q lines contains two integers l and r (1 ≤ l ≤ r ≤ n).
For each question print "Yes" if the product is a square, otherwise print "No" in a separate line.
For the first input, the product is 4. It is a square number.
For the second input, the product is 1×2 = 2. It is not a square number.
For the third input, the product is 0. It is a square number.
For the fourth input, the product is (-3)×(-3) = 9. It is a square number.
Hashing, MOSAlgorithm, NumberTheory uDebug
NirjhorEarliest, Jun '17
nitaibanikLightest, 1.8 MB
BRUR CSE Fest Programming Contest 2017 Mock
|
Solve the equations below by first changing each equation to a simpler, equivalent equation. Check your solution(s). Are any values of the variable not allowed?
50x^2+200x=-150
\frac{a}{9}+\frac{1}{a}=\frac{2}{3}
1.2m-0.2=3.8+m
\frac{3x}{4}+\frac{x}{3}-\frac{1}{2}=0
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.