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Fermat's Last Theorem/Appendix - Wikibooks, open books for an open world
This appendix collects some proofs and studies in depth some mathematical concepts that can be interesting in order to examine in depth some aspects of the book. It seeks to maintain a simple approach but the proofs being correct, in some cases however it was necessary to make recourse to some non immediate concepts and passages that have been specified and clarified where possible.
2 Fundamental theory of arithmetic
3 Principle of Induction
4.1 Properties of congruence
4.2 Invariance with respect to arithmetical operations
Pythagoras’ theorem[edit | edit source]
Being one of the most noted theorems in the history of mathematics, many proofs exist, the work of astronomers, stockbrokers, and even one by Leonardo da Vinci. Probably, together with quadratic reciprocity, it contends for the prize for the theorem with the most absolute proofs. It will be proved in a graphical manner utilising only the concepts of elementary geometry. Pythagoras’ proof will not be quoted being very complex and not immediate.
The proof is very simple, as one sees from the graphic one constructs first a square formed by four triangles and by two squares, one of side a and the second of side b. The area of a square is calculated multiplying the side by itself or in modern notation the area is the side raised to the power two. The area of the large square is therefore the sum of the areas of the squares values a2 and b2 plus the sum of the four triangles. In the second square we have again four triangles and a square of area c2. The four triangles being present both to the left of the equality and to the right can be eliminated. Therefore remaining in the equation:
As one wished to prove. It is to be noted that the proof is generic and effectively covers every possible right angled triangle given that numbers are not utilised in the proof but only generic segments length a, b and c. The proof depends solely on the fact that the triangles are right angled and on the fact that one is utilising Euclidean geometry. This is a small gallery of some geometrical proofs of the theorem even if as was said above there are very many proofs and in fact there are also purely algebraic proofs, proofs that utilise complex numbers and even proofs written in the form of sonnets.
Fundamental theory of arithmetic[edit | edit source]
The fundamental theory of arithmetic states that every natural number that is not 1 admits one and only one factorisation in prime numbers not taking account of the order of the factors. (The exclusion of 1 is due to the fact that it does not have prime factors.) This theorem was the basis of the proofs of Gabriel Lamé and Augustin Luis Cauchy and as was said in general is not valid for complex numbers therefore it cannot be utilised for Fermat’s theorem but given its importance it was decided however to include the proof in the appendix. The statement is easily verifiable for “small” natural numbers: it is easy to discover that 70 is equal to 2 × 5 × 7 while 100 equates to 2 × 2 × 5 × 5 or 22 × 52, and furthermore it is easy to verify that for these numbers other decompositions into prime factors cannot exist. Conversely the general proof is rather longer: Here is an extract of it. It deals with a proof for absurd: that is it starts from the hypothesis contrary to that of the statement in order to be able to prove it unfounded.
One supposes that there exist some number reducible into prime factors in more than one way, and one calls the smallest m. Firstly one proves that, given two factorisations of m, the prime numbers that present themselves in the first factorisation are all distinct from those in the second factorisation. They are in fact two different factorisations:
{\displaystyle m=p_{1}p_{2}p_{3}\dots p_{s}}
{\displaystyle m=q_{1}q_{2}q_{3}\dots q_{t}}
where pi and qj are prime. (Note: within a single factorisation there can be repeated factors, naturally: for example, 100 = 2 × 2 × 5 × 5). Were there to be an identical factor ph = qk, we could divide m by such a factor and obtain a number m' , less than m, that would also itself have two distinct factorisations. At this point we know that p1 is different from q1; without loss of generality we can suppose that p1 < q1. We put then
{\displaystyle n=(q_{1}-p_{1})q_{2}q_{3}\dots q_{t}}
Evidently, n < m, given that [3] can be written as:
{\displaystyle n=q_{1}q_{2}q_{3}\dots q_{t}-p_{1}q_{2}q_{3}\dots q_{t}=m-p_{1}q_{2}q_{3}\dots q_{t}\;\!}
We now demonstrate that n admits at least two distinct factorisations. We begin by considering if the first factor of n, q1 - p1, can or cannot be prime; in the case that it was not, we suppose factorising it. The factorisation of n thus obtained does not admit p1 among its factors. In fact, from the first part of the proof, we know that p1 is different from q2, q3, ..., qt; but it cannot none the less appear in the eventual factorisation of q1 - p1. If in fact it happened that would mean that
{\displaystyle q_{1}-p_{1}=p1\cdot b\Rightarrow q_{1}=p_{1}(1+b)}
and therefore q1 would be divisible by p1, which is not possible in as much as q1 is a prime number. Taking now the final equality [4] and substituting for m the value from [1], we obtain:
{\displaystyle n=p_{1}p_{2}p_{3}\dots p_{s}-p_{1}q_{2}q_{3}\dots q_{t}\rightarrow n=p_{1}(p_{2}p_{3}\dots p_{s}-q_{2}q_{3}\dots q_{t})}
In whatever way the second factor in [5], may be factorised, we will have obtained a factorisation of n that contains p1, and that therefore is different from that in [3], contrary to the hypothesis that m was the smallest number that admitted more than one factorisation. The theorem is therefore proved.
Principle of Induction[edit | edit source]
The Principle of induction states that
{\displaystyle U}
is a subset of the set
{\displaystyle \mathbb {N} }
of natural numbers that satisfy the following two properties:
{\displaystyle U}
{\displaystyle 0}
every time that
{\displaystyle U}
contains a number
{\displaystyle n}
{\displaystyle U}
contains also the successive number
{\displaystyle n+1}
{\displaystyle U}
coincides with all of the set of natural numbers
{\displaystyle \mathbb {N} }
This is found in general within the axioms of Peano and furnishes a potent instrument for proofs in all sectors of mathematics. In the book it is recalled at times given that many mathematicians have sought to prove a base case of Fermat's theorem and then of generalising the solution in order to be able include therein the successive cases by means of this principle.
Proof by induction[edit | edit source]
In order to prove that a certain assertion
{\displaystyle P(n)}
in which appears a natural number
{\displaystyle n}
valid for whatver
{\displaystyle n\in \mathbb {N} }
one must apply the principle of induction in the following way:
One puts
{\displaystyle U=\{n\in \mathbb {N} :{\mbox{vale }}P(n)\}}
one proves
{\displaystyle P(0)}
as valid, that is that
{\displaystyle 0}
is in the set of natural numbers
{\displaystyle U}
{\displaystyle P(n)}
is valid;
one assumes as a hypothesis that the assertion
{\displaystyle P(n)}
may be valid for a generic
{\displaystyle n}
and from such an assumption one also proves that
{\displaystyle P(n+1)}
(that is that
{\displaystyle n\in U\Rightarrow n+1\in U}
and therefore one concludes that the set
{\displaystyle U}
of numbers for which
{\displaystyle P(n)}
is valid is valid coincides with all the set of natural numbers. Point 1 is generally called base of the induction, point 2 the inductive step.
The following is an intuitive mode in which one can look at this type of proof: if we have available a proof of the base
{\displaystyle P(0)}
of the inductive step
{\displaystyle P(n)\Rightarrow P(n+1)}
then clearly we can make use of these proofs in order to prove
{\displaystyle P(1)}
using the logical rule modus ponens on
{\displaystyle P(0)}
(the base) and
{\displaystyle P(0)\Rightarrow P(1)}
(which is a particular case of the inductive step for
{\displaystyle n=0}
), then we can prove
{\displaystyle P(2)}
since now we are using the modus ponens on
{\displaystyle P(1)}
{\displaystyle P(1)\Rightarrow P(2)}
, thus for
{\displaystyle P(3)}
{\displaystyle P(4)}
, etcetera... it is clear at this point that we can produce a proof in a finite number of steps (eventually very lengthy) of
{\displaystyle P(n)}
for whatever natural number
{\displaystyle n}
{\displaystyle P(n)}
{\displaystyle n\in \mathbb {N} }
We will prove that the following formula is valid:
{\displaystyle n\in \mathbb {N} }
{\displaystyle 0+1+2+3+4+...+n={\frac {n(n+1)}{2}}}
In this case we have
{\displaystyle P(n)\quad \equiv \quad 0+1+2+3+4+...+n={\frac {n(n+1)}{2}}}
Base base of the induction: we have to prove the assertion
{\displaystyle P(n)}
{\displaystyle n=0}
, and so, substituting, that
{\displaystyle 0={\frac {0\cdot 1}{2}}}
, and in effect there is very little work, one is talking of an elementary calculation;
Inductive step: we must show that for every n the implication
{\displaystyle P(n)\Rightarrow P(n+1)}
is valid, that is, substituting:
{\displaystyle 0+1+2+3+4+...+n={\frac {n(n+1)}{2}}\quad \Rightarrow \quad 0+1+2+3+4+...+n+(n+1)={\frac {(n+1)((n+1)+1)}{2}}}
Therefore we must assume as true
{\displaystyle P(n)\quad \equiv \quad 0+1+2+3+4+...+n={\frac {n(n+1)}{2}}}
working on this equality and concluding with the analogous equality for
{\displaystyle n+1}
: We could for example add
{\displaystyle n+1}
to both sides of the equality P(n):
{\displaystyle 0+1+2+3+4+...+n+(n+1)={\frac {n(n+1)}{2}}+(n+1)}
then we make some simple algebraic steps:
{\displaystyle 0+1+2+3+4+...+n+(n+1)={\frac {n(n+1)}{2}}+{\frac {2(n+1)}{2}}}
{\displaystyle 0+1+2+3+4+...+n+(n+1)={\frac {(n+1)(n+2)}{2}}}
{\displaystyle 0+1+2+3+4+...+n+(n+1)={\frac {(n+1)((n+1)+1)}{2}}}
and this final equality is exactly
{\displaystyle P(n+1)}
. This concludes the proof of the inductive step.
Having therefore proved the base of the induction and the inductive step we can conclude (from the principle of induction) that
{\displaystyle P(n)}
must be true for every
{\displaystyle n\in \mathbb {N} }
{\displaystyle \square }
Modular Arithmetic[edit | edit source]
Modular arithmetic (sometimes called clock arithmetic since the calculation of the hours in cycles of 12 or 24 is based on such principle) represents an important branch of mathematics. It finds applications in cryptography, in the theory of numbers (in particular in the research of prime numbers), and it is the basis of many of the most common arithmetical and algebraic operations. It deals with a system of integer arithmetic, in which the numbers "wrap round on themselves " every time that they reach multiples of a specific number n, called the modulo. Modular arithmetic was formally introduced by Carl Friedrich Gauss in his treatise Disquisitiones Arithmeticae, published in 1801. Modular arithmetic is based on the concept of congruence modulo n . Given three whole numbers a, b, n, with n≠0, we say that a and b are congruent modulo n if their difference (a−b) is a multiple of n. In this case we write
{\displaystyle a\equiv b{\pmod {n}}}
and we say that a is congruent to b modulo n. For example, we can write
{\displaystyle 38\equiv 14{\pmod {12}}}
since 38 − 14 = 24, which is a multiple of 12. In the case that both numbers are positive, one can also say that a and b are congruent modulo n if they have the same remainder from division by n. Therefore we can also say that 38 is congruent 14 modulo 12 since both 38 and 14 have remainder 2 after division by 12.
Properties of congruence[edit | edit source]
Congruence is a relationship of equivalence between integer numbers as one sees from the following properties:
Reflexive property: every number is congruent to itself modulo n, for every n different from fixed 0.
{\displaystyle a\equiv a{\pmod {n}}\qquad \forall a\in \mathbb {N} ,\forall n\in \mathbb {N} _{0}}
Proof: one has a - a = 0. But as is noted, every non-zero integer is a divisor of 0. Therefore n divides (a - a)
Symmetrical property: if a is congruent to b modulo n then b is congruent to a modulo n
{\displaystyle a\equiv b{\pmod {n}}\Rightarrow b\equiv a{\pmod {n}}\qquad \forall a,b\in \mathbb {N} ,\forall n\in \mathbb {N} _{0}}
Proof: if n divides (a - b), then n also divides the opposite (b - a) = - (a - b
Transitive property: if a is congruent to b modulo n and b is congruent to c modulo n then also a is congruent to c modulo n.
{\displaystyle a\equiv b{\pmod {n}}\quad \land \quad b\equiv c{\pmod {n}}\Rightarrow a\equiv c{\pmod {n}}\qquad \forall a,b,c\in \mathbb {N} ,\forall n\in \mathbb {N} _{0}}
Proof: if n divides (a - b) and n divides (a - c) then, because of the distributive property of the division with respect to the subtraction, n also divides [(a - c) - (a - b)]=[a - c - a + b] = (b - c).
Invariance with respect to arithmetical operations[edit | edit source]
Another important characteristic of the relationship of congruence is the fact of being preserved by the usual arithmetical operations between integers:
Invariance by addition: increasing or reducing two numbers congruent modulo n by the same quantity, the new numbers obtained are still congruent between themselves modulo n. More synthetically
{\displaystyle a\equiv b{\pmod {n}}\Leftrightarrow (a+c)\equiv (b+c){\pmod {n}}\qquad \forall a,b,c\in \mathbb {N} ,\forall n\in \mathbb {N} _{0}}
Proof: we write (a - b) = (a - b + c - c) = (a + c) - (b + c)
Invariance by multiplication: multiplying two numbers congruent modulo n by the same quantity, the new numbers obtained are still congruent between themselves modulo n.
{\displaystyle a\equiv b{\pmod {n}}\Rightarrow a\cdot c\equiv b\cdot c{\pmod {n}}\qquad \forall a,b,c\in \mathbb {N} ,\forall n\in \mathbb {N} _{0}}
Proof: if n divides (a - b) then n divides (a - b)×c
Note: One can only invert this property if n does not divide c, that is if c is not congruent to 0 (modulo n).
Invariance with respect to raising to a power: raising two numbers congruent modulo n to the same power k, the numbers obtained are still congruent between themselves modulo n.
{\displaystyle a\equiv b{\pmod {n}}\Rightarrow a^{k}\equiv b^{k}{\pmod {n}}\qquad \forall a,b,c\in \mathbb {N} ,\forall k\in \mathbb {N} ,\forall n\in \mathbb {N} _{0}}
Proof: if a ≡ b ≡ 0 (mod n) the proposition is banal. If a ≡ b (mod n) is non null, we suppose to know that
{\displaystyle a^{k-1}\equiv b^{k-1}{\pmod {n}}}
. Multiplying both the terms by a thanks to invariance by multiplication, we will have
{\displaystyle a^{k}\equiv b^{k-1}\cdot a{\pmod {n}}}
. We start now from the congruence a ≡ b (mod n) and we multiply both the members by
{\displaystyle b^{k-1}{\pmod {n}}}
, thanks always to invariance by multiplication. We obtain:
{\displaystyle a\cdot b^{k-1}\equiv b^{k}{\pmod {n}}}
. Comparing the two expressions, and utilising the symmetrical and transitive properties, one deduces that
{\displaystyle a^{k}\equiv b^{k}{\pmod {n}}}
. Since the proposition is true for k = 1 and the fact that it may be true for k-1 implies that it is true for k, by the principle of induction the proposition is true for every k.
Note: This property can be inverted only if k is not 0.
Retrieved from "https://en.wikibooks.org/w/index.php?title=Fermat%27s_Last_Theorem/Appendix&oldid=3233355"
|
m
n
Matrix (or 2-dimensional Array), then it is assumed to contain
m
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{audiofile}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/stereo.wav"\right):
\mathrm{Spectrogram}\left(\mathrm{audiofile},\mathrm{compactplot}\right)
\mathrm{Spectrogram}\left(\mathrm{audiofile},\mathrm{channel}=1,\mathrm{includesignal}=[\mathrm{color}="Navy"],\mathrm{includepowerspectrum},\mathrm{colorscheme}=["Orange","SteelBlue","Navy"]\right)
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Inline Plot - Maple Help
Home : Support : Online Help : Programming : Document Tools : Layout : Inline Plot
generate XML for an InlinedPlot element
InlinePlot( p, opts )
(optional) ; content for the InlinedPlot element, such as the result of a plotting command
gridlinevisibility truefalse:=true ; Specifies whether gridlines are visible, if present in the plot content.
height : {posint,identical(NoUserValue)}:=NoUserValue ; The height in pixels. If height is NoUserValue and p is a 2D plot constructed with the size option specifying a positive pixel value for height then that value is used for height. Otherwise the default value is 500.
legendvisibility {truefalse,identical(NoUserValue)}:=NoUserValue ; Specifies whether a legend may be visible. The default behaviour is for the legend to be visible if any legend is present for any curve or other constituent of p. The option values true and false override the default.
scale : realcons:=1.0 ; The scaling factor for 3D plots, a positive real value.
width : {posint,identical(NoUserValue)}:=NoUserValue ; The width in pixels. If width is NoUserValue and p is a 2D plot constructed with the size option specifying a positive pixel value for width then that value is used for width. Otherwise the default value is 500.
xtrans : realcons:=0.0 ; The horizontal translation for 3D plots, in units of 10 pixels.
ytrans : realcons:=0.0 ; The vertical translation for 3D plots, in units of 10 pixels.
The InlinePlot command in the Layout Constructors package returns an XML function call which represents a InlinedPlot element of a worksheet.
An InlinedPlot element is used to display a plot inlined within a Document or Worksheet.
For construction of a PlotComponent see the Plot Component command.
\mathrm{with}\left(\mathrm{DocumentTools}\right):
\mathrm{with}\left(\mathrm{DocumentTools}:-\mathrm{Layout}\right):
Executing the InlinePlot command produces a function call, in which the supplied plot structure p has been encoded.
p≔\mathrm{plot}\left(\mathrm{sin}\left(x\right),x=0..\mathrm{\pi },\mathrm{adaptive}=\mathrm{false},\mathrm{numpoints}=5\right):
P≔\mathrm{InlinePlot}\left(p\right)
\textcolor[rgb]{0,0,1}{P}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_XML_Plot}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"gridlinevisibility"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"0"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"height"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"500"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"legendvisibility"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"false"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"plot-scale"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"1.0"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"type"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"two-dimensional"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"width"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"500"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"plot-xtrans"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"0."}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"plot-ytrans"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"0."}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"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"}\right)
\mathrm{xml}≔\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(P\right)\right)\right)\right):
\mathrm{InsertContent}\left(\mathrm{xml}\right):
The height and width options can be used to control the displayed dimensions of both 2D and 3D plots.
p≔\mathrm{plot}\left(\mathrm{sin}\left(x\right){x}^{2},x=-2\mathrm{\pi }..2\mathrm{\pi }\right):
P≔\mathrm{InlinePlot}\left(p,\mathrm{height}=200,\mathrm{width}=500\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(P\right)\right)\right)\right)\right):
p≔\mathrm{plot3d}\left(\mathrm{sin}\left(x\right){y}^{2},x=-2\mathrm{\pi }..2\mathrm{\pi },y=-1..1\right):
P≔\mathrm{InlinePlot}\left(p,\mathrm{height}=200,\mathrm{width}=300\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(P\right)\right)\right)\right)\right):
The size option used in constructing a 2D plot, when specifying width and height as positive integer pixel values, is respected by default.
p≔\mathrm{plot}\left(\mathrm{sin}\left(x\right){x}^{2},x=-2\mathrm{\pi }..2\mathrm{\pi },\mathrm{size}=[500,300]\right):
P≔\mathrm{InlinePlot}\left(p\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(P\right)\right)\right)\right)\right):
A legend in a 2D plot is made visible by default.
p≔\mathrm{plot}\left(\mathrm{sin}\left(x\right){x}^{2},x=-2\mathrm{\pi }..2\mathrm{\pi },\mathrm{legend}=\mathrm{typeset}\left(\mathrm{sin}\left(x\right){x}^{2}\right),\mathrm{size}=[200,200]\right):
\mathrm{P1}≔\mathrm{InlinePlot}\left(p\right):
\mathrm{P2}≔\mathrm{InlinePlot}\left(p,\mathrm{legendvisibility}=\mathrm{false}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Table}\left(\mathrm{Row}\left(\mathrm{P1},\mathrm{P2}\right)\right)\right)\right):
The initial pan and scale of a 3D plot can be controlled by options.
p≔\mathrm{plot3d}\left({x}^{2}+{y}^{2},x=-1..1,y=-1..1\right):
\mathrm{P1}≔\mathrm{InlinePlot}\left(p\right):
\mathrm{P2}≔\mathrm{InlinePlot}\left(p,\mathrm{xtrans}=10.0,\mathrm{ytrans}=5.0\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(\mathrm{P1},\mathrm{P2}\right)\right)\right)\right)\right):
p≔\mathrm{plot3d}\left({x}^{2}+{y}^{2},x=-1..1,y=-1..1\right):
\mathrm{P1}≔\mathrm{InlinePlot}\left(p\right):
\mathrm{P2}≔\mathrm{InlinePlot}\left(p,\mathrm{scale}=1.5\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(\mathrm{P1},\mathrm{P2}\right)\right)\right)\right)\right):
The gridlinevisibility option only controls whether gridlines present in a plot will be visible. Supplying the gridlinevisibility option will not by itself cause gridlines to be added to a plot result.
\mathrm{pg}≔\mathrm{plot}\left(\mathrm{sin},\mathrm{gridlines}=\mathrm{true}\right):
\mathrm{pn}≔\mathrm{plot}\left(\mathrm{sin},\mathrm{gridlines}=\mathrm{false}\right):
T≔\mathrm{Table}\left(\mathrm{widthmode}=\mathrm{pixels},\mathrm{width}=300,\mathrm{alignment}=\mathrm{center},\mathrm{Row}\left(\mathrm{InlinePlot}\left(\mathrm{pn},\mathrm{gridlinevisibility}=\mathrm{true}\right),\mathrm{InlinePlot}\left(\mathrm{pn},\mathrm{gridlinevisibility}=\mathrm{false}\right)\right),\mathrm{Row}\left(\mathrm{InlinePlot}\left(\mathrm{pg},\mathrm{gridlinevisibility}=\mathrm{true}\right),\mathrm{InlinePlot}\left(\mathrm{pg},\mathrm{gridlinevisibility}=\mathrm{false}\right)\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
P≔\mathrm{InlinePlot}\left(\mathrm{plot3d}\left({x}^{2}\mathrm{sin}\left(y\right),x=-1..1,y=-2\mathrm{\pi }..2\mathrm{\pi },\mathrm{axis}=[\mathrm{color}=\mathrm{cyan}],\mathrm{labels}=["","",""]\right)\right):
C≔\mathrm{Cell}\left(\mathrm{Textfield}\left(P\right),\mathrm{fillcolor}="black"\right):
T≔\mathrm{Table}\left(\mathrm{Column}\left(\right),\mathrm{Row}\left(C\right),\mathrm{widthmode}=\mathrm{pixels},\mathrm{width}=300,\mathrm{alignment}=\mathrm{center}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
The DocumentTools:-Layout:-InlinePlot command was introduced in Maple 2015.
The and options were updated in Maple 2020.
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Molecules | Free Full-Text | Columnar Aggregates of Azobenzene Stars: Exploring Intermolecular Interactions, Structure, and Stability in Atomistic Simulations
In Silico Investigation of the Biological Implications of Complex DNA Damage with Emphasis in Cancer Radiotherapy through a Systems Biology Approach
New Unsymmetrically Substituted Benzothiadiazole-Based Luminophores: Synthesis, Optical, Electrochemical Studies, Charge Transport, and Electroluminescent Characteristics
Morphologically Diverse Micro- and Macrostructures Created via Solvent Evaporation-Induced Assembly of Fluorescent Spherical Particles in the Presence of Polyethylene Glycol Derivatives
We present a simulation study of supramolecular aggregates formed by three-arm azobenzene (Azo) stars with a benzene-1,3,5-tricarboxamide (BTA) core in water. Previous experimental works by other research groups demonstrate that such Azo stars assemble into needle-like structures with light-responsive properties. Disregarding the response to light, we intend to characterize the equilibrium state of this system on the molecular scale. In particular, we aim to develop a thorough understanding of the binding mechanism between the molecules and analyze the structural properties of columnar stacks of Azo stars. Our study employs fully atomistic molecular dynamics (MD) simulations to model pre-assembled aggregates with various sizes and arrangements in water. In our detailed approach, we decompose the binding energies of the aggregates into the contributions due to the different types of non-covalent interactions and the contributions of the functional groups in the Azo stars. Initially, we investigate the origin and strength of the non-covalent interactions within a stacked dimer. Based on these findings, three arrangements of longer columnar stacks are prepared and equilibrated. We confirm that the binding energies of the stacks are mainly composed of
\pi
\pi
interactions between the conjugated parts of the molecules and hydrogen bonds formed between the stacked BTA cores. Our study quantifies the strength of these interactions and shows that the
\pi
\pi
interactions, especially between the Azo moieties, dominate the binding energies. We clarify that hydrogen bonds, which are predominant in BTA stacks, have only secondary energetic contributions in stacks of Azo stars but remain necessary stabilizers. Both types of interactions,
\pi
\pi
stacking and H-bonds, are required to maintain the columnar arrangement of the aggregates. View Full-Text
Keywords: azobenzenes; supramolecular assembly; hydrogen bonding; molecular dynamics; computer simulations azobenzenes; supramolecular assembly; hydrogen bonding; molecular dynamics; computer simulations
Koch, M.; Saphiannikova, M.; Guskova, O. Columnar Aggregates of Azobenzene Stars: Exploring Intermolecular Interactions, Structure, and Stability in Atomistic Simulations. Molecules 2021, 26, 7598. https://doi.org/10.3390/molecules26247598
Koch M, Saphiannikova M, Guskova O. Columnar Aggregates of Azobenzene Stars: Exploring Intermolecular Interactions, Structure, and Stability in Atomistic Simulations. Molecules. 2021; 26(24):7598. https://doi.org/10.3390/molecules26247598
Koch, Markus, Marina Saphiannikova, and Olga Guskova. 2021. "Columnar Aggregates of Azobenzene Stars: Exploring Intermolecular Interactions, Structure, and Stability in Atomistic Simulations" Molecules 26, no. 24: 7598. https://doi.org/10.3390/molecules26247598
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Upper bounds for the number of spanning trees of graphs | Journal of Inequalities and Applications | Full Text
Upper bounds for the number of spanning trees of graphs
Ş Burcu Bozkurt1
In this paper, we present some upper bounds for the number of spanning trees of graphs in terms of the number of vertices, the number of edges and the vertex degrees.
Let G be a simple graph with n vertices and e edges. Let
V\left(G\right)=\left\{{v}_{1},{v}_{2},\dots ,{v}_{n}\right\}
be the vertex set of G. If two vertices
{v}_{i}
{v}_{j}
are adjacent, then we use the notation
{v}_{i}\sim {v}_{j}
{v}_{i}\in V\left(G\right)
, the degree of the vertex
{v}_{i}
{d}_{i}
, is the number of vertices adjacent to
{v}_{i}
. Throughout this paper, we assume that the vertex degrees are ordered by
{d}_{1}\ge {d}_{2}\ge \cdots \ge {d}_{n}
The complete graph, the complete bipartite graph and the star of order n are denoted by
{K}_{n}
{K}_{p,q}
p+q=n
{S}_{n}
G-m
be the graph obtained by deleting any edge m from the graph G and let
\overline{G}
be the complement of G. Let
G\cup H
be the vertex-disjoint union of the graphs G and H and let
G\vee H
be the graph obtained from
G\cup H
by adding all possible edges from vertices of G to vertices of H, i.e.,
G\vee H=\overline{\overline{G}\cup \overline{H}}
L\left(G\right)=D\left(G\right)-A\left(G\right)
be the Laplacian matrix of the graph G, where
A\left(G\right)
D\left(G\right)
are the adjacency matrix and the diagonal matrix of the vertex degrees of G, respectively. The normalized Laplacian matrix of G is defined as
L=D{\left(G\right)}^{-\frac{1}{2}}L\left(G\right)D{\left(G\right)}^{-\frac{1}{2}}
D{\left(G\right)}^{-\frac{1}{2}}
is the matrix which is obtained by taking
\left(-\frac{1}{2}\right)
-power of each entry of
D\left(G\right)
. The Laplacian eigenvalues and the normalized Laplacian eigenvalues of G are the eigenvalues of
L\left(G\right)
and L, respectively. Let
{\mu }_{1}\ge {\mu }_{2}\ge \cdots \ge {\mu }_{n}
be the Laplacian eigenvalues and
{\lambda }_{1}\ge {\lambda }_{2}\ge \cdots \ge {\lambda }_{n}
be the normalized Laplacian eigenvalues of G. It is well known that
{\mu }_{n}=0
{\lambda }_{n}=0
and the multiplicities of these zero eigenvalues are equal to the number of connected components of G; see [2, 3].
The number of spanning trees (also known as complexity),
t\left(G\right)
, of G is given by the following formula in terms of the Laplacian eigenvalues (see [1], p.39):
t\left(G\right)=\frac{1}{n}\prod _{i=1}^{n-1}{\mu }_{i}.
It is known that the number of spanning trees of G is also expressed by the normalized Laplacian eigenvalues as follows (see [1], p.49):
t\left(G\right)=\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right)\prod _{i=1}^{n-1}{\lambda }_{i}.
Now we list some known upper bounds for
t\left(G\right)
Grimmett [4]:
t\left(G\right)\le \frac{1}{n}{\left(\frac{2e}{n-1}\right)}^{n-1}.
Grone and Merris [5]:
t\left(G\right)\le {\left(\frac{n}{n-1}\right)}^{n-1}\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right).
Nosal [6]: For r-regular graphs,
t\left(G\right)\le {n}^{n-2}{\left(\frac{r}{n-1}\right)}^{n-1}.
Kelmanns ([1], p.222):
t\left(G\right)\le {n}^{n-2}{\left(1-\frac{2}{n}\right)}^{\overline{e}},
wher\overline{e}
is the number of edges of
\overline{G}
Das [7]:
t\left(G\right)\le {\left(\frac{2e-{d}_{1}-1}{n-2}\right)}^{n-2}.
Zhang [8]:
t\left(G\right)\le \left(1+\left(n-2\right)a\right){\left(1-a\right)}^{n-2}\frac{1}{n}{\left(\frac{2e}{n-1}\right)}^{n-1},
a={\left(\frac{n\left(n-1\right)-2e}{2en\left(n-2\right)}\right)}^{1/2}
Feng et al. [9]:
t\left(G\right)\le \left(\frac{{d}_{1}+1}{n}\right){\left(\frac{2e-{d}_{1}-1}{n-2}\right)}^{n-2}
t\left(G\right)\le {\left(\frac{{\sum }_{i=1}^{n}{d}_{i}^{2}+2e-{\left({d}_{1}+1\right)}^{2}}{n-2}\right)}^{\frac{n-2}{2}}.
Li et al. [10]:
t\left(G\right)\le {d}_{n}{\left(\frac{2e-{d}_{1}-1-{d}_{n}}{n-3}\right)}^{n-3}.
In [4] Grimmett observed that (3) is the generalization of (5). Grone and Merris [5] stated that by the application of arithmetic-geometric mean inequality, (4) leads to (3). In [7] Das indicated that (7) is sharp for
{S}_{n}
{K}_{n}
, but (3), (4), (5) and (6) are sharp only for
{K}_{n}
. Li et al. [10] pointed out that (11) is sharp for
{S}_{n}
{K}_{n}
G\cong {K}_{1}\vee \left({K}_{1}\cup {K}_{n-2}\right)
{K}_{n}-m
, but (3) is sharp only for
{K}_{n}
, (7) and (9) are sharp for
{S}_{n}
{K}_{n}
. In [8, 9] the authors showed that (8) is always better than (3), and (9) is always better than (7) and (10).
This paper is organized as follows. In Section 2, we give some useful lemmas. In Section 3, we obtain some upper bounds for the number of spanning trees of graphs in terms of the number of vertices, the number of edges and the vertex degrees of graphs. We also show that one of these upper bounds is always better than the upper bound (4).
In this section, we give some lemmas which will be used later. Firstly, we introduce an auxiliary quantity of a graph G on the vertex set
V\left(G\right)=\left\{{v}_{1},{v}_{2},\dots ,{v}_{n}\right\}
\phantom{\rule{0.25em}{0ex}}P=1+\sqrt{\frac{2}{n\left(n-1\right)}\sum _{{v}_{i}\sim {v}_{j}}\frac{1}{{d}_{i}{d}_{j}}},
{d}_{i}
is the degree of the vertex
{v}_{i}
Let G be a graph with n vertices and normalized Laplacian matrix L without isolated vertices. Then
\sum _{i=1}^{n}{\lambda }_{i}=tr\left(L\right)=n
\sum _{i=1}^{n}{\lambda }_{i}^{2}=tr\left({L}^{2}\right)=n+2\sum _{{v}_{i}\sim {v}_{j}}\frac{1}{{d}_{i}{d}_{j}}.
Let G be a graph with n vertices and normalized Laplacian eigenvalues
{\lambda }_{1}\ge {\lambda }_{2}\ge \cdots \ge {\lambda }_{n}=0
0\le {\lambda }_{i}\le 2.
{\lambda }_{1}=2
if and only if a connected component of G is bipartite and nontrivial.
{\lambda }_{1}\ge {\lambda }_{2}\ge \cdots \ge {\lambda }_{n}=0
{\lambda }_{1}\ge \frac{n}{n-1}.
Moreover, the equality holds in (12) if and only if G is a complete graph
{K}_{n}
{\lambda }_{1}\ge {\lambda }_{2}\ge \cdots \ge {\lambda }_{n}=0
{\lambda }_{1}\ge P.
{K}_{n}
The lower bound (13) is always better than the lower bound (12).
Let G be a connected graph with
n>2
vertices. Then
{\lambda }_{2}={\lambda }_{3}=\cdots ={\lambda }_{n-1}
G\cong {K}_{n}
G\cong {K}_{p,q}
Let G be a graph with n vertices and without isolated vertices. Suppose G has the maximum vertex degree equal to
{d}_{1}
\sum _{{v}_{i}\sim {v}_{j}}\frac{1}{{d}_{i}{d}_{j}}\ge \frac{n}{2{d}_{1}}.
Moreover, the equality holds in (14) if and only if G is a regular graph.
{x}_{i}>-1
1\le i\le n
{\sum }_{i=1}^{n}{x}_{i}=0
{\sum }_{i=1}^{n}{x}_{i}^{2}\ge {c}^{2}\left(1-{n}^{-1}\right)
\sum _{i=1}^{n}ln\left(1+{x}_{i}\right)\le ln\left(1+c-c{n}^{-1}\right)+\left(n-1\right)ln\left(1-c{n}^{-1}\right).
Now we present the main results of this paper following the ideas in [8] and [9]. Note that P was defined earlier in the previous section.
Theorem 1 Let G be a graph with n vertices and without isolated vertices. Then
t\left(G\right)\le \left(1+\left(n-2\right)b\right){\left(1-b\right)}^{n-2}{\left(\frac{n}{n-1}\right)}^{n-1}\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right),
b={\left(\frac{n-1-{d}_{1}}{n\left(n-2\right){d}_{1}}\right)}^{1/2}
Proof If G is disconnected, then
t\left(G\right)=0
and (15) follows. Now we assume that G is connected. From (2), we have
0<t\left(G\right)=\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right){\lambda }_{1}\cdots {\lambda }_{n-1}
{\lambda }_{n-1}>0
q=\frac{n}{n-1}
{x}_{i}=\frac{{\lambda }_{i}}{q}-1
1\le i\le n-1
{x}_{i}>-1
. Moreover, by Lemma 1 and Lemma 7, we get
\sum _{i=1}^{n-1}{x}_{i}=\sum _{i=1}^{n-1}\left(\frac{{\lambda }_{i}}{q}-1\right)=0
\begin{array}{rcl}\sum _{i=1}^{n-1}{x}_{i}^{2}& =& \sum _{i=1}^{n-1}{\left(\frac{{\lambda }_{i}}{q}-1\right)}^{2}\\ =& \left(n-1\right)-\frac{2{\sum }_{i=1}^{n-1}{\lambda }_{i}}{q}+\frac{{\sum }_{i=1}^{n-1}{\lambda }_{i}^{2}}{{q}^{2}}\\ \ge & \left(n-1\right)-2\left(n-1\right)+{\left(\frac{n-1}{n}\right)}^{2}\left(n+\frac{n}{{d}_{1}}\right)\\ =& \frac{{\left(n-1\right)}^{2}}{n{d}_{1}}-\left(\frac{n-1}{n}\right)\\ =& \frac{{\left(n-1\right)}^{2}\left(n-1-{d}_{1}\right)}{n\left(n-2\right){d}_{1}}\left(1-\frac{1}{n-1}\right)\\ =& {\left(\left(n-1\right)b\right)}^{2}\left(1-\frac{1}{n-1}\right).\end{array}
Then by Lemma 8, we obtain
\prod _{i=1}^{n-1}\left(1+{x}_{i}\right)\le \left(1+\left(n-1\right)b-\frac{\left(n-1\right)b}{n-1}\right){\left(1-b\right)}^{n-2}.
\prod _{i=1}^{n-1}{\lambda }_{i}\le \left(1+\left(n-2\right)b\right){\left(1-b\right)}^{n-2}{\left(\frac{n}{n-1}\right)}^{n-1}
t\left(G\right)\le \left(1+\left(n-2\right)b\right){\left(1-b\right)}^{n-2}{\left(\frac{n}{n-1}\right)}^{n-1}\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right).
Hence, the result holds. □
f\left(b\right)=\left(1+\left(n-2\right)b\right){\left(1-b\right)}^{n-2}
{f}^{\mathrm{\prime }}\left(b\right)=-\left(n-2\right)\left(n-1\right)b{\left(1-b\right)}^{n-3}\le 0
0\le b\le 1
f\left(b\right)\le f\left(0\right)=1
; see [8]. Hence, we conclude that the upper bound (15) is always better than the upper bound (4). Moreover, if G is the complete graph
{K}_{n}
, then the equality holds in (15).
Theorem 2 Let G be a connected graph with
n>2
t\left(G\right)\le P{\left(\frac{n-P}{n-2}\right)}^{n-2}\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right).
Moreover, the equality holds in (16) if and only if G is the complete graph
{K}_{n}
Proof From (2) and Lemma 1, we get
\begin{array}{rcl}t\left(G\right)& =& \left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right)\prod _{i=1}^{n-1}{\lambda }_{i}=\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right){\lambda }_{1}\prod _{i=2}^{n-1}{\lambda }_{i}\\ \le & \left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right){\lambda }_{1}{\left(\frac{{\sum }_{i=2}^{n-1}{\lambda }_{i}}{n-2}\right)}^{n-2}\\ =& \left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right){\lambda }_{1}{\left(\frac{{\sum }_{i=1}^{n-1}{\lambda }_{i}-{\lambda }_{1}}{n-2}\right)}^{n-2}=\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right){\lambda }_{1}{\left(\frac{n-{\lambda }_{1}}{n-2}\right)}^{n-2}.\end{array}
P\le x\le 2
f\left(x\right)=x{\left(n-x\right)}^{n-2}.
By Lemma 4 and Lemma 5, we have that
{\lambda }_{1}\ge P\ge \frac{n}{n-1}
{f}^{\mathrm{\prime }}\left(x\right)=f\left(x\right)\frac{n-\left(n-1\right)x}{x\left(n-x\right)}\le 0
P\le x\le 2
f\left(x\right)
x=P
and (16) follows.
If the equality holds in (16), then all inequalities in the above argument must be equalities. Hence, we have
{\lambda }_{1}=P\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{\lambda }_{2}=\cdots ={\lambda }_{n-1}.
Then by Lemma 4 and Lemma 6, we conclude that G is the complete graph
{K}_{n}
Conversely, we can easily see that the equality holds in (16) for the complete graph
{K}_{n}
Now we consider the bipartite graph case of the above theorem.
Theorem 3 Let G be a connected bipartite graph with
n>2
t\left(G\right)\le \frac{{\prod }_{i=1}^{n}{d}_{i}}{e}.
Moreover, the equality holds in (17) if and only if
G\cong {K}_{p,q}
Proof Since G is a connected bipartite graph, by Lemma 2, we have
{\lambda }_{1}=2
. Considering this, (2) and Lemma 1, we obtain
\begin{array}{rcl}t\left(G\right)& =& \left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right)\prod _{i=1}^{n-1}{\lambda }_{i}=\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{2e}\right){\lambda }_{1}\prod _{i=2}^{n-1}{\lambda }_{i}\\ \le & \left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{e}\right){\left(\frac{{\sum }_{i=2}^{n-1}{\lambda }_{i}}{n-2}\right)}^{n-2}=\left(\frac{{\prod }_{i=1}^{n}{d}_{i}}{e}\right){\left(\frac{n-{\lambda }_{1}}{n-2}\right)}^{n-2}=\frac{{\prod }_{i=1}^{n}{d}_{i}}{e}.\end{array}
{\lambda }_{2}=\cdots ={\lambda }_{n-1}
, by Lemma 6, i.e., if and only if
G\cong {K}_{p,q}
Fiedler M: Algebraic connectivity of graphs. Czechoslov. Math. J. 1973, 23: 298–305.
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Zumstein, P: Comparison of spectral methods through the adjacency matrix and the Laplacian of a graph. Diploma Thesis, ETH Zürich (2005)
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The author thanks the referees for their helpful comments and suggestions concerning the presentation of this paper. The author is also thankful to TUBITAK and the Office of Selcuk University Scientific Research Project (BAP). This study is based on a part of the author’s PhD thesis.
Department of Mathematics, Science Faculty, Selçuk University, Campus, Konya, 42075, Turkey
Ş Burcu Bozkurt
Correspondence to Ş Burcu Bozkurt.
Bozkurt, Ş.B. Upper bounds for the number of spanning trees of graphs. J Inequal Appl 2012, 269 (2012). https://doi.org/10.1186/1029-242X-2012-269
normalized Laplacian eigenvalues
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Iterative process for a strictly pseudo-contractive mapping in uniformly convex Banach spaces | Journal of Inequalities and Applications | Full Text
Iterative process for a strictly pseudo-contractive mapping in uniformly convex Banach spaces
Peiyuan Wang1
This paper is concerned with a new method to prove the weak convergence of a strictly pseudo-contractive mapping in a p-uniformly convex Banach space with more relaxed restrictions on the parameters. Our results extend and improve the corresponding earlier results.
MSC:41A65, 47H17, 47J20.
In 1967, Browder and Petryshyn [1] gave the classical definition for strictly pseudo-contractive mappings in Hilbert spaces for the first time.
Definition 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H.
T:C\to H
is called a Browder-Petryshyn-type k-strictly pseudo-contractive mapping. Then there exists
k\in \left[0,1\right)
x,y\in C
〈Tx-Ty,j\left(x-y\right)〉\le {\parallel x-y\parallel }^{2}-k{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{2}.
In 2010, Zhou [2] gave a new definition for k-strictly pseudo-contractive mappings in q-uniformly smooth Banach spaces.
Definition 1.2 Let C be a nonempty closed convex subset of a q-uniformly smooth Banach space X.
T:C\to C
is called a Zhou-type k-strictly pseudo-contractive mapping, if there exists
k\in \left[0,1\right)
x,y\in C
〈Tx-Ty,{j}_{q}\left(x-y\right)〉\le {\parallel x-y\parallel }^{q}-\frac{1-k}{2}{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{q}.
In 2009, Hu and Wang [3] gave another definition for k-strictly pseudo-contractive mappings in p-uniformly convex Banach spaces.
Definition 1.3 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X.
T:C\to C
is called a Hu-type k-strictly pseudo-contractive mapping, if there exists
k\in \left[0,1\right)
x,y\in C
{\parallel Tx-Ty\parallel }^{p}\le {\parallel x-y\parallel }^{p}+k{\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{p}.
Remark 1.1 The mappings defined by (1.1) and (1.2) are pseudo-contractive mappings, but the mapping defined by (1.3) may not be pseudo-contractive in general Banach spaces.
Remark 1.2 If and only if
q=2
, the mappings defined by (1.1) and (1.2) are equivalent.
p=q=2
, the mappings defined by (1.1), (1.2), and (1.3) are equivalent in Hilbert space.
In 1979, Reich [4] established a weak convergence theorem via a Mann-type iterative process for nonexpansive mapping in a uniformly convex Banach space with Fréchet differentiable norm.
Theorem R Let C be a closed convex subset of a uniformly convex Banach space X with a Fréchet differentiable norm and
T:C\to C
F\left(T\right)\ne \mathrm{\varnothing }
{x}_{1}\in C
\left\{{x}_{n}\right\}
{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n}
, where the real sequence
\left\{{\alpha }_{n}\right\}\subset \left[0,1\right]
{\sum }_{n=1}^{\mathrm{\infty }}\left(1-{\alpha }_{n}\right){\alpha }_{n}=\mathrm{\infty }
\left\{{x}_{n}\right\}
In 2007, Marino and Xu [5] improved Reich’s [4] result and gave several weak convergence theorems via the normal Mann iterative algorithm for strictly pseudo-contractive mappings in Hilbert spaces. Further, they proposed an open problem: Do the main results of [5]still hold true in the framework of Banach spaces which are uniformly convex and have a Fréchet differentiable norm?
In 2009, Hu and Wang [3] considered above problem in a p-uniformly convex Banach space and established the following theorem.
Theorem H Let C be a closed convex subset of a p-uniformly convex Banach space X with a Fréchet differentiable norm and
T:C\to C
be a k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients
p,k<min\left\{1,{2}^{-\left(p-2\right)}{c}_{p}\right\}
F\left(T\right)\ne \mathrm{\varnothing }
{x}_{1}\in C
n>1
\left\{{x}_{n}\right\}
{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n}
\left\{{\alpha }_{n}\right\}\subset \left[0,1\right]
0<\epsilon \le {\alpha }_{n}\le 1-\epsilon <1-\frac{{2}^{p-2}k}{{c}_{p}}
\left\{{x}_{n}\right\}
Question Can one relax the restriction on the parameters
{\alpha }_{n}
in Theorem H and simplify its proof?
The purpose of this paper is to solve the question mentioned above. To prove our results, we need the following lemmas.
Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X and
T:C\to C
be a Hu-type strictly pseudo-contractive mapping in the light of (1.3). For
\alpha \in \left(0,1\right)
{T}_{\alpha }:C\to C
{T}_{\alpha }=\left(1-\alpha \right)x+\alpha Tx
x\in C
\alpha \in \left(0,1-\left(k{2}^{p-2}\right)/{c}_{p}\right)
{T}_{\alpha }
F\left({T}_{\alpha }\right)=F\left(T\right)
T:C\to C
\mu \in \left(0,1\right)
{T}_{\mu }:C\to C
{T}_{\mu }=\left(1-\mu \right)x+\mu Tx
x\in C
{\parallel {T}_{\mu }x-{T}_{\mu }y\parallel }^{p}\le {\parallel x-y\parallel }^{p}-\left({W}_{p}\left(\mu \right){c}_{p}-\mu \lambda \right){\parallel \left(I-T\right)x-\left(I-T\right)y\parallel }^{p},\phantom{\rule{1em}{0ex}}\mathrm{\forall }x,y\in C,
{W}_{p}\left(\mu \right)={\mu }^{p}\left(1-\mu \right)+\mu {\left(1-\mu \right)}^{p}
T:C\to C
be a nonexpansive mapping, then
I-T
Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X which satisfies the Opial condition and
T:C\to C
be a quasi-nonexpansive mapping with
F\left(T\right)\ne \mathrm{\varnothing }
I-T
is demiclosed at zero, then for any
{x}_{0}\in C
, the normal Mann iteration
\left\{{x}_{n}\right\}
{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0,
converges weakly to a fixed point of T, where
\left\{{\alpha }_{n}\right\}\subset \left[0,1\right]
{\sum }_{n=0}^{\mathrm{\infty }}min\left\{{\alpha }_{n},\left(1-{\alpha }_{n}\right)\right\}=\mathrm{\infty }
Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X whose dual space
{X}^{\ast }
satisfies Kadec-Klee property and
T:C\to C
F\left(T\right)\ne \mathrm{\varnothing }
{x}_{0}\in C
\left\{{x}_{n}\right\}
{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 0,
\left\{{\alpha }_{n}\right\}\subset \left[0,1\right]
{\sum }_{n=0}^{\mathrm{\infty }}min\left\{{\alpha }_{n},\left(1-{\alpha }_{n}\right)\right\}=\mathrm{\infty }
Now we are in a position to state and prove the main results in this paper.
Theorem 2.1 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X with Fréchet differential norm. Let
T:C\to C
be a Hu-type k-strictly pseudo-contractive mapping in the light of (1.3) with coefficients
p,k<min\left\{1,{2}^{-\left(p-2\right)}{c}_{n}\right\}
F\left(T\right)\ne \mathrm{\varnothing }
. Assume that a real sequence
\left\{{\alpha }_{n}\right\}
\left[0,1\right]
0\le {\alpha }_{n}\le \alpha =1-\left(k{2}^{p-2}/{c}_{p}\right)
n\ge 0
{\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}\left[\left(1-{\alpha }_{n}\right){2}^{2-p}{c}_{p}-k\right]=\mathrm{\infty }
{x}_{0}\in C
, the normal Mann iterative sequence
\left\{{x}_{n}\right\}
{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.
\left\{{x}_{n}\right\}
defined by (2.1) converges weakly to a fixed point of T.
{T}_{\alpha }
be given as in Lemma 1.1. Then
{T}_{\alpha }:C\to C
F\left({T}_{\alpha }\right)=F\left(T\right)
{\beta }_{n}=\frac{\alpha -{\alpha }_{n}}{\alpha }
{x}_{n+1}={\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){T}_{\alpha }{x}_{n}
\begin{array}{rl}\sum _{n=0}^{\mathrm{\infty }}{\beta }_{n}\left(1-{\beta }_{n}\right)& =\frac{1}{{\alpha }^{2}}\sum _{n=0}^{\mathrm{\infty }}{\alpha }_{n}\left(\alpha -{\alpha }_{n}\right)\\ =\frac{1}{{\alpha }^{2}}\sum _{n=0}^{\mathrm{\infty }}{\alpha }_{n}\left(1-{\alpha }_{n}-\frac{k{2}^{p-2}}{{c}_{p}}\right)\\ =\frac{{2}^{p-2}}{{\alpha }^{2}{c}_{p}}\sum _{n=0}^{\mathrm{\infty }}{\alpha }_{n}\left[\left(1-{\alpha }_{n}\right){2}^{p-2}{c}_{p}-k\right]\\ =\mathrm{\infty }.\end{array}
By using Theorem R, we conclude that
\left\{{x}_{n}\right\}
{T}_{\alpha }
, and of T. The proof is complete. □
Remark 2.2 Theorem 2.1 relaxes the iterative parameters in Theorem H and our proof method is also quite concise.
Theorem 2.3 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X which satisfies the Opial condition. Let
T:C\to C
p,k<min\left\{1,{2}^{-\left(p-2\right)}{c}_{p}\right\}
F\left(T\right)\ne \mathrm{\varnothing }
. Assume that the real sequence
\left\{{\alpha }_{n}\right\}
\left[0,1\right]
0\le {\alpha }_{n}\le \alpha =1-\left(k{2}^{p-2}/{c}_{p}\right)
n\ge 0
{\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}\left[\left(1-{\alpha }_{n}\right){2}^{2-p}{c}_{p}-k\right]=\mathrm{\infty }
{x}_{0}\in C
\left\{{x}_{n}\right\}
{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.
\left\{{x}_{n}\right\}
defined by (2.2) converges weakly to the fixed point of T.
{T}_{\alpha }
{T}_{\alpha }:C\to C
F\left({T}_{\alpha }\right)=F\left(T\right)
{\beta }_{n}=\frac{\alpha -{\alpha }_{n}}{\alpha }
{x}_{n+1}={\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){T}_{\alpha }{x}_{n}
. As shown in Theorem 2.1,
{\sum }_{n=0}^{\mathrm{\infty }}{\beta }_{n}\left(1-{\beta }_{n}\right)=\mathrm{\infty }
I-{T}_{\alpha }
is demiclosed at zero. By Lemma 1.4, we conclude that
\left\{{x}_{n}\right\}
{T}_{\alpha }
Theorem 2.4 Let C be a nonempty closed convex subset of a p-uniformly convex Banach space X with the dual space
{X}^{\ast }
satisfying the Kadec-Klee property. Let
T:C\to C
p,k<min\left\{1,{2}^{-\left(p-2\right)}{c}_{p}\right\}
F\left(T\right)\ne \mathrm{\varnothing }
\left\{{\alpha }_{n}\right\}
\left[0,1\right]
0\le {\alpha }_{n}\le \alpha =1-\left(k{2}^{p-2}/{c}_{p}\right)
n\ge 0
{\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_{n}\left[\left(1-{\alpha }_{n}\right){2}^{2-p}{c}_{p}-k\right]=\mathrm{\infty }
{x}_{0}\in C
\left\{{x}_{n}\right\}
{x}_{n+1}=\left(1-{\alpha }_{n}\right){x}_{n}+{\alpha }_{n}T{x}_{n},\phantom{\rule{1em}{0ex}}n\ge 0.
\left\{{x}_{n}\right\}
{T}_{\alpha }
{T}_{\alpha }:C\to C
F\left({T}_{\alpha }\right)=F\left(T\right)
{\beta }_{n}=\frac{\alpha -{\alpha }_{n}}{\alpha }
{x}_{n+1}={\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){T}_{\alpha }{x}_{n}
{\sum }_{n=0}^{\mathrm{\infty }}{\beta }_{n}\left(1-{\beta }_{n}\right)=\mathrm{\infty }
. By using Lemma 1.5,
\left\{{x}_{n}\right\}
defined by (2.3) converges weakly to a fixed point of
{T}_{\alpha }
Browder FE, Petryshyn WV: Contraction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 82-90.
Zhou HY: Convergence theorem for strict pseudo-contractions in uniformly smooth Banach spaces. Acta Math. Sin. Engl. Ser. 2010,26(4):743-758. 10.1007/s10114-010-7341-2
Hu LG, Wang JP: Mann iteration of weak convergence theorem in Banach space. Acta Math. Sin. Engl. Ser. 2009,25(2):217-224. 10.1007/s10255-007-7054-1
Reich S: Weak convergence theorem for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 1979, 67: 274-276. 10.1016/0022-247X(79)90024-6
Marino G, Xu HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2003, 279: 336-349.
Browder FE: Semicontractive and semiaccretive nonlinear mappings in Banach spaces. Bull. Am. Math. Soc. 1968, 74: 660-665. 10.1090/S0002-9904-1968-11983-4
Agarwal RP, O’Regan D, Sahu DR: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, Berlin; 2008:299-302.
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang, 050003, China
Yu Zhou, Haiyun Zhou & Peiyuan Wang
Department of Mathematics and Information, Hebei Normal University, Shijiazhuang, 050016, China
Zhou, Y., Zhou, H. & Wang, P. Iterative process for a strictly pseudo-contractive mapping in uniformly convex Banach spaces. J Inequal Appl 2014, 377 (2014). https://doi.org/10.1186/1029-242X-2014-377
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Methylmalonyl-CoA carboxytransferase - Wikipedia
Methylmalonyl-CoA carboxytransferase homohexamer, Propionibacterium
In enzymology, a methylmalonyl-CoA carboxytransferase (EC 2.1.3.1) is an enzyme that catalyzes the chemical reaction
(S)-methylmalonyl-CoA + pyruvate
{\displaystyle \rightleftharpoons }
propanoyl-CoA + oxaloacetate
Thus, the two substrates of this enzyme are (S)-methylmalonyl-CoA and pyruvate, whereas its two products are propanoyl-CoA and oxaloacetate.
This enzyme belongs to the family of transferases that transfer one-carbon groups, specifically the carboxy- and carbamoyltransferases. The systematic name of this enzyme class is (S)-methylmalonyl-CoA:pyruvate carboxytransferase. Other names in common use include transcarboxylase, methylmalonyl coenzyme A carboxyltransferase, methylmalonyl-CoA transcarboxylase, oxalacetic transcarboxylase, methylmalonyl-CoA carboxyltransferase, methylmalonyl-CoA carboxyltransferase, (S)-2-methyl-3-oxopropanoyl-CoA:pyruvate carboxyltransferase, (S)-2-methyl-3-oxopropanoyl-CoA:pyruvate carboxytransferase, and carboxytransferase [incorrect]. This enzyme participates in propanoate metabolism. It has 3 cofactors: zinc, Biotin, and Cobalt.
As of late 2007, 12 structures have been solved for this class of enzymes, with PDB accession codes 1DCZ, 1DD2, 1ON3, 1ON9, 1RQB, 1RQE, 1RQH, 1RR2, 1S3H, 1U5J, 2D5D, and 2EVB.
Hoffmann A, Hilpert W, Dimroth P (1989). "The carboxyltransferase activity of the sodium-ion-translocating methylmalonyl-CoA decarboxylase of Veillonella alcalescens". Eur. J. Biochem. 179 (3): 645–50. doi:10.1111/j.1432-1033.1989.tb14596.x. PMID 2920730.
Swick RW; Wood HG (1960). "The role of transcarboxylation in propionic acid fermentation". Proc. Natl. Acad. Sci. USA. 46 (1): 28–41. Bibcode:1960PNAS...46...28S. doi:10.1073/pnas.46.1.28. PMC 285006. PMID 16590594.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Methylmalonyl-CoA_carboxytransferase&oldid=1043456021"
Cobalt enzymes
|
Quiz | Chemical Equilibrium
\frac{\left[{\mathrm{N}}_{2}\right]\left[{\mathrm{H}}_{2}\right]}{\left[{\mathrm{NH}}_{3}\right]}
\frac{\left[{\mathrm{NH}}_{3}\right]}{\left[{\mathrm{N}}_{2}\right]\left[{\mathrm{H}}_{2}\right]}
\frac{\left[{\mathrm{N}}_{2}\right]{\left[{\mathrm{H}}_{2}\right]}^{3}}{{\left[{\mathrm{NH}}_{3}\right]}^{2}}
\frac{{\left[{\mathrm{NH}}_{3}\right]}^{2}}{\left[{\mathrm{N}}_{2}\right]{\left[{\mathrm{H}}_{2}\right]}^{3}}
Equilibrium constant KC: ratio of product concentrations to reactant concentrations at the equilibrium
The concentrations are raised to a power equal to the stoichiometric coefficient
What is the equilibrium constant expression KC of the reaction: 2 ICl (s)
⇄
I2 (s) + Cl2 (g)?
\frac{\left[{\mathrm{I}}_{2}\right]}{{\left[\mathrm{ICl}\right]}^{2}}
\frac{\left[{\mathrm{Cl}}_{2}\right]\left[{\mathrm{I}}_{2}\right]}{2 \left[\mathrm{ICl}\right]}
\frac{\left[{\mathrm{Cl}}_{2}\right]\left[{\mathrm{I}}_{2}\right]}{{\left[\mathrm{ICl}\right]}^{2}}
Pure solids reactants and products do not appear in the equilibrium-constant expression (= 1 by convention)
How can we increase the total amount of produced Cl2 during the reaction: 2 ICl (s)
⇄
By adding more ICl
By removing the Cl2 as it is formed
By decreasing the volume of the container
According to Le Chatelier’ principle, the position of equilibrium will move in such a way as to counteract the change: by removing the Cl2 as it is formed, the equilibrium will move to the right to produce more Cl2
Additionally, the right is the side with most moles of gas. The equilibrium will move to the right if the concentration of a reactant decreases OR the volume increases OR the pressure decreases
What happens when the reaction quotient QC is less than the equilibrium constant K?
The reaction will move to the left
The reaction will move to the right
It depends on whether the reaction is endothermic or exothermic
It depends on the reaction conditions
Reaction quotient QC: ratio of product concentrations to reactant concentrations at a given time
If QC < K, the reaction will move to the right ⇒ net formation of product
|
This is particularly useful for documenting isotopes, relative atomic and relative molecular mass, empirical and molecular formulae, balanced equations (full and ionic), equilibrium reactions and themodynamics.
Note that mhchem is not currently installed in this documentation wiki. Therefore, the chemical display of chemical formulae and equations is simulated and may not exactly correspond to rendering using mhchem, which is usually nicer!.
{\displaystyle \mathrm {H_{2}O} }
{\displaystyle \mathrm {Sb_{2}O_{3}} }
{\displaystyle \mathrm {H^{+}} }
{\displaystyle \mathrm {CrO_{4}^{2-}} }
{\displaystyle \mathrm {AgCl_{2}^{-}} }
{\displaystyle \mathrm {[AgCl_{2}]^{-}} }
{\displaystyle \mathrm {Y^{99+}} }
{\displaystyle \mathrm {H_{2}(aq)} }
{\displaystyle \mathrm {NO_{3}^{-}} }
{\displaystyle \mathrm {(NH_{4})_{2}S} }
{\displaystyle \mathrm {2H_{2}O} }
{\displaystyle \mathrm {{\frac {\scriptstyle {1}}{\scriptstyle {2}}}H_{2}O} }
{\displaystyle \mathrm {{}_{90}^{227}Th^{+}} }
{\displaystyle {\mathit {V}}\mathrm {_{H_{2}O}} }
{\displaystyle \mathrm {H_{2}O} }
{\displaystyle \mathrm {H_{2}O} }
{\displaystyle \mathrm {Ce^{IV}} }
{\displaystyle \mathrm {KCr(SO_{4})_{2}\cdot 12H_{2}O} }
{\displaystyle \mathrm {KCr(SO_{4})_{2}\cdot 12H_{2}O} }
{\displaystyle \mathrm {[Cd\{SC(NH_{2})_{2}\}_{2}]\cdot [Cr(SCN)_{4}(NH_{3})_{2}]_{2}} }
{\displaystyle \mathrm {RNO_{2}^{-.}} }
{\displaystyle \mathrm {RNO_{2}^{-.}} }
{\displaystyle \mu \!-\!\mathrm {Cl} }
{\displaystyle \mathrm {C_{6}H_{5}\!-\!CHO} }
{\displaystyle \mathrm {X\!=\!Y\!\equiv \!Z} }
{\displaystyle \mathrm {A\!-\!B\!=\!C\!\equiv \!D} }
{\displaystyle \mathrm {A\!-\!B\!=\!C\!\equiv \!D} }
{\displaystyle \mathrm {A\!\sim \!B\!\simeq \!C} }
{\displaystyle \mathrm {A\!\cong \!B\!\cong \!C\!\cong \!D} }
{\displaystyle \mathrm {A\!\cdots \!B\!\cdot \cdots \!C} }
{\displaystyle \mathrm {A\!\rightarrow \!B\!\leftarrow \!C} }
{\displaystyle \mathrm {Fe(CN)_{\frac {6}{2}}} }
{\displaystyle \mathrm {{\mathit {x}}Na(NH_{4})HPO_{4}{\overset {\Delta }{\rightarrow }}(NaPO_{3})_{\mathit {x}}+{\mathit {x}}NH_{3}\uparrow +{\mathit {x}}H_{2}O} }
{\displaystyle \mathrm {CH_{4}(g)+2O_{2}(g)\rightarrow CO_{2}(g)+2H_{2}O(l)} \quad \Delta H_{\mathrm {c} }^{\ominus }=-890.3\;\mathrm {kJ} \;\mathrm {mol} ^{-1}}
{\displaystyle \mathrm {CO_{2}+C\rightarrow 2CO} }
{\displaystyle \mathrm {CO_{2}+C\leftarrow 2CO} }
{\displaystyle \mathrm {CO_{2}+C\rightleftharpoons 2CO} }
{\displaystyle \mathrm {H^{+}+OH^{-}{\overset {-\!-\!-\!\rightharpoonup }{\quad \scriptstyle {\leftharpoondown }\quad }}H_{2}O} }
{\displaystyle A\leftrightarrow A'}
{\displaystyle \mathrm {CO_{2}+C\;{\overset {\alpha }{\rightarrow }}\;2CO} }
{\displaystyle \mathrm {CO_{2}+C\;{\underset {\beta }{\overset {\alpha }{\rightarrow }}}\;2CO} }
{\displaystyle \mathrm {CO_{2}+C\;{\overset {above}{-\!\!\!\longrightarrow }}\;2CO} }
{\displaystyle \mathrm {CO_{2}+C\;{\underset {below}{\overset {above}{-\!\!\!\longrightarrow }}}\;2CO} }
{\displaystyle \mathrm {CO_{2}+C\;{\underset {below}{\overset {above}{-\!\!\!\longrightarrow }}}\;2CO} }
{\displaystyle A\;\mathrm {\overset {H_{2}O}{\longrightarrow }} \;B}
{\displaystyle A\;\mathrm {\overset {H_{2}O}{\longrightarrow }} \;B}
{\displaystyle \mathrm {SO_{4}^{2-}+Ba^{2+}\rightarrow BaSO_{4}\downarrow } }
{\displaystyle A\mathrm {\overset {\text{Enclose spaces!}}{\longleftarrow \!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!\longrightarrow }} A'}
{\displaystyle \mathrm {Zn^{2+}{\underset {+2H^{+}}{\overset {+2OH^{-}}{\rightleftharpoons }}}{\underset {\text{amphoteric hydroxide}}{Zn(OH)_{2}\downarrow }}{\underset {+2H^{+}}{\overset {+2OH^{-}}{\rightleftharpoons }}}{\underset {\text{tetrahydroxozincate}}{[Zn(OH)_{4}]^{2-}}}} }
{\displaystyle K=\mathrm {\frac {[Hg^{2+}][Hg]}{[Hg_{2}^{2+}]}} }
{\displaystyle \mathrm {Hg^{2+}\;{\overset {I^{-}}{\rightarrow }}\;{\underset {\text{red}}{HgI_{2}}}\;{\overset {I^{-}}{\rightarrow }}\;{\underset {\text{red}}{[Hg^{II}I_{4}]^{2-}}}} }
Retrieved from "https://docs.moodle.org/310/en/index.php?title=Chemistry_notation_using_mhchem&oldid=139588"
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Heart refining in the world of mortals 2021-09-15 04:18:40
suggestions vue code readability comments
This article has participated in the review and gift drawing activity , Nuggets Badge *2, Comment on relevant content to participate in the lucky draw Check the details
Recently, I went into the pit Vue project , It feels like falling into the ancestral shit mountain , Poor readability , Not to mention maintainability . So I'd like to take this column to mention a few points about Vue Suggestions for code readability , If you think it's useful, give it a compliment , If you think the suggestion is unreasonable, comment and criticize , There are better suggestions. Welcome to comment and add .
One 、 Make good use of components to make the code more organized
Never put the implementation code of a page in one .vue In file , Unless this page is very simple , Otherwise, this .vue The code in the file will be long and smelly .
Vue The purpose of providing components is not just to reuse , It can also be used to split code , Even making good use of components can optimize the rendering and updating speed of pages . This is because Vue When the page is rendered and updated, the components in the page will not be updated , Unless the component props perhaps slot The referenced data changes .
You can use the following steps to put a Vue The page is divided into components to make the code more organized
1.1、 extract UI Components
How to define UI What about the components ？ It is recommended to distinguish by whether the server data is processed or not UI Components and business components . For example, load pop-up window 、 The second confirmation pop-up window 、 Message prompt box and so on belong to UI Interactive components .
take UI After the component is extracted , You can put UI The interactive code is separated from the business interactive code . Remember not to UI Write business code in component , such UI Components will not be reusable .
Take a counterexample , Add the business code to be processed after secondary confirmation in the secondary confirmation pop-up window , Lead to UI Components will not be reusable . We can imitate ElementUI The secondary confirmation pop-up window is called to implement a secondary confirmation pop-up window component .
this.$confirm(message, title, options)
.then(res =>{})
.catch(err =>{})
In this way, the business code can be written in then In the callback function of , The core implementation code of the component is as follows :
//confirm.vue
<div @click="ok"></div>
<div @click="cancel"></div>
import options from './confirm.vue';
const Confirm = Vue.extend(options);
let confirm = undefined;
const ConfirmInit = (options = {}) => {
options.resolve = resolve;
options.reject = reject;
confirm = new Confirm({
document.body.appendChild(confirm.$el);
if(confirm) confirm.show = true;
Vue.prototype.$confirm = ConfirmInit;
import 'components/confirm/index.js';// Global registration secondary confirmation pop-up window confirm Components
1.2、 Extract business components by module
A page can be divided into multiple areas , Like the head 、 Bottom 、 Sidebar 、 List of goods 、 Member list, etc , Each region can be used as a module to extract business components .
1.3、 Extract functional components by function
Extract business components by module , At this time, the business component may still be very large , Therefore, the functional components should be further extracted according to the function .
The functions are big and small , We should pay attention to several principles ：
Too simple functions are not extracted
For example, a collection function , Just request an interface , Don't extract functions like this . Only the functions of logical operations with a certain complexity can be extracted .
Single function , A functional component handles only one business .
For example, a file reader component , There is a need , It is required to automatically collect the file after it is opened , So where to write the collection logic code ？
Maybe you write the collection logic code in the method of listening for the successful opening of files in the component without thinking about it , After a while , The requirement should be added to the reading record first, and then click the collect button to collect , When you modify the code in the component, you find that another page also references this component , Therefore, an additional parameter should be added to the component to distinguish business scenarios , Business scenarios are superimposed as requirements change , Various judgment logic will be added to the code of the component , Over time, it becomes long and smelly , Obviously, this practice is not allowed to .
The correct approach is to customize an event on the component label on-fileOpen-success, use handleFileOpenSuccess Function to listen for this event .
<fileReader @on-fileOpen-success="handleFileOpenSuccess"></fileReader>
Execute in the method of listening for the successful opening of the file in the component this.$emit('on-fileOpen-success',data) Trigger this event , among data You can pass file information out , stay handleFileOpenSuccess Function to handle business interactions such as collecting or adding history records and then collecting . This approach makes the file reader component single .
Functional components should contain as few as possible UI part ,UI Partly with slot Slot incoming , This makes the component more pure , More reusable .
For example, the upload icon of the upload component , It's impossible to follow UI When the design changes, add an upload icon to it , You can use slot The slot passes the upload icon into .
//upload.vue
// Upload icon
Two 、 utilize v-bind Make component properties more readable
If you want to treat all the attributes of an object as prop Incoming components componentA, You can use... Without parameters v-bind. for example , For a given object params：
<componentA :id=params.id :name="params.name"></componentA>
<componentA v-bind="params"></componentA>
3、 ... and 、 utilize
attrs And
listeners To encapsulate third-party components
In encapsulating third-party components , There is always a problem , How to use the properties and events of third-party components through encapsulated components .
Like encapsulating a elementUi In the component Input Input box components myInput, When an error is entered, an error prompt is displayed below the input box .
myInput The component code is as follows ：
<el-input v-model="input"></el-input>
<div>{{errorTip}}</div>
errorTip:{
this.$emit('input',val)
This call myInput Components , among errorTip Enter an error prompt for the input box .
<myInput v-model="input" :errorTip="errorTip"></myInput>
If you want to in myInput Add a... To the component disabled Property to disable the input box , How to achieve it ？ Most students do this
<el-input v-model="input" :disabled="disabled"></el-input>
After a period of time, I will be in myInput Add... To the component el-input Other properties of the component ,el-input The total number of components is 27 More than a , What should I do , One by one prop Pass in , This is not only poor readability, but also cumbersome , It can be used $attrs One step in place , Let's take a look first attrs The definition of .
$attrs: Contains inaction in the parent scope prop Be identified ( And get ) Of attribute binding (class and style With the exception of ). When a component does not declare anything prop when , This will include all bindings for the parent scope (class and style With the exception of ), And through v-bind="$attrs" Incoming internal components
It is not enough , You have to inheritAttrs Option set to false, Why? , Take a look at inheritAttrs The definition of options is clear .
By default, the parent scope is not recognized as props Of attribute binding (attribute bindings) will “ Back off ” And as ordinary HTML attribute Apply to the root element of the subcomponent . When composing a component that wraps a target element or another component , This may not always be in line with expected behavior . By setting inheritAttrs by false, These default behaviors will be removed . And by $attrs You can make these attribute take effect , And through v-bind Explicitly bound to non root elements . Be careful ： This option does not affect class and style binding .
Simply speaking , hold inheritAttrs Set to false,v-bind="$attrs" To take effect .
So you can clearly el-input The properties and properties of the component myinput The properties of components are distinguished , Component's props The readability of options is greatly improved .
So how to achieve in myIpput Use... On components el-input What about the custom events on the component , Maybe your first reaction was this.$emit.
<myInput v-model="input" :errorTip="errorTip" @blur="handleBlur"></myInput>
el-input The components are 4 A custom event , Not much , If you encounter custom events, more third-party components , do , Can you add them one by one , Not only will it add a bunch of unnecessary methods, And the readability is poor, it is easy to communicate with myInput Self methods Mixed in with . You can actually use $listeners One step in place , Let's take a look first $listeners The definition of .
$listeners： Contains... In the parent scope ( Not included .native Decorator's ) v-on Event listener . It can go through v-on="$listeners" Incoming internal components .
stay myInput In the component, as long as el-input Add... To the component v-on="$listeners", You can go to myInput Use... On components el-input Component custom events .
The way of drawing prizes
By the end of 9 month 10 Japan , If the comment area exceeds 10 Human interaction （ Not including the author himself ）, The author can draw in his own name to give out the Nuggets badge 2 gold （ Nuggets officials bear ）
Commentators can't be trumpets , such as LV0, No historical developments , Or just like an author's number ;
Comments cannot appear 「 Trample 」「 Support 」「 Welcome back 」 Etc. or similar comments ;
The way of drawing prizes ：
All qualified commentators , Random sampling 2 name .
Lottery tools ：
choujiang.wiicha.com/
2021 year 9 month 9 Japan 21 spot
本文为[Heart refining in the world of mortals]所创，转载请带上原文链接，感谢
https://qdmana.com/2021/09/20210909111329225x.html
Copyright © 2019-2020 前端知识 All Rights Reserved.
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How many Solution line represented by 3x+10=0 has Also find the point where 3x+10=0 meet Y axis Pls - Maths - Linear Equations in Two Variables - 8870479 | Meritnation.com
How many Solution line represented by 3x+10=0 has? Also find the point where 3x+10=0 meet Y axis. Pls Solve this Quickly.Revision exam is on Tuesday.
we have equation 3x + 10 = 0 , So
-\frac{10}{3}
, So that equation have only one solution
And This equation is passing through x =
-\frac{10}{3}
and parallel to y axis , So
That equation never intersect at y - axis .
Neil Lasrado answered this
there are infinite solutions for the line represented by 3x +10 = 0 .
the line never meets the y axis as the line is always parallel to the y axis
|
A brief, whirlwind tour of data science
A whirlwind tour of Data Science
manycupsofcoffee.com
BA, MS, PhD (almost) Computer Science
10 years building banking and healthcare software
5 years leading "skunkworks" R&D team
Poor speller (relevant)
What's this talk?
Broad, shallow tour of data science tasks and methods
Intuition, not math*
Biased to a computer science pov
Sorry statisticians, physicists, signal processing folk
* ok it turns out there is a little math
What's a Data Science?
No pedantic definitions here, but themes:
Tools to extract meaning from lots of data
Explore structure and relationships in data
Multi-disciplinary, yay!
Multi-disciplinary, boo!
Different names for same thing
Math notation, conventions
Obligatory Graphics
Single points of data
Task Families
It's an iterative process
Problem definition - who cares about this?
Data preparation - easy systematic access
Data exploration - signal vs noise, patterns
Modelling - noisy inputs -> something useful
Evaluation - what is good?
Deployment - from science to end users
Discrete: Categorical: Red, Blue,...
Continuous: Numerical: [0, 1], x > 0
Exploring single attributes
Skew, Kurtosis
Exploring Single Values (cont)
Median, quantiles, box-plots
Exploring Pairs of Values
Co-variance - how two attributes change together
Pearson correlation coefficient - how two attributes change together vs how change individually
t/z-test, ANOVA
Exploring structure of data aka dimensionality reduction
what directions explain the most variance
Kernel tricks + linear methods
Assume there is some lower dimensional structure
Neural Networks trained on the identity function
Model? Explain/predict some output by some inputs
Why build models at all?
Incomplete noisy data
Discover some latent, hidden process
Describe phenomena in more compact form
Bias vs Variance: two sources of error
Bias - how much does this model differ from true answer on average
Variance - if I build a lot of models using the same process how much will they vary from one another
Want low+low, but often they're antagonistic
Intuition: predicting election results
Only poll people from phone book, that model is biased towards home-phone owning folks -- doesn't matter how many people you poll
Only poll 30 people from phone book and you do it multiple times--each time the results might vary. If you increase the number of people, variance will go down
Generally the challenge of model fitting: do not want to over-fit or under-fit
In machine learning, we use a methodology of cross-validation
train vs dev vs test
n-fold validation
Variance via model complexity
h_0(x) = b
h_1(x) = a x + b
\theta
A few ways to say the same thing?
Is there a hidden process that can be described by finite, fixed parameters and can explain the observed data?
Can the data or process be described by a shape that has convenient math properties?
Parametric statistical tests assume distributions
Non-parametric make fewer assumptions but are often harder to interpret and less powerful
Philosophical difference over interpretation of probability - we'll skip that
How it matters to the everyday data scientist?
Quick probability review:
% chance of clouds at any moment
% chance of rain, given its cloudy
P( \text{Rain} | \text{Clouds} ) = \frac{P( \text{Clouds} | \text{Rain} ) P( \text{Rain} )}{P( \text{Clouds} )}
P( \text{Rain} | \text{Clouds} ) = \frac{P( \text{Clouds} | \text{Rain} ) P( \text{Rain} )}{P( \text{Clouds} )}
P( \text{Clouds} )
P( \text{Clouds} )
P( \text{Rain} | \text{Clouds} )
P( \text{Rain} | \text{Clouds} )
\text{Posterior} \propto \text{Likelihood} \times \text{Prior}
\text{Posterior} \propto \text{Likelihood} \times \text{Prior}
P( \text{Stroke} | \text{Headache} ) = \frac{P( \text{Headache} | \text{Stroke} ) P( \text{Stroke} )}{P( \text{Headache} )}
P( \text{Stroke} | \text{Headache} ) = \frac{P( \text{Headache} | \text{Stroke} ) P( \text{Stroke} )}{P( \text{Headache} )}
Our models have parameters, θ, which we set via machine learning based on training data, D
Allows us to engineer some expert knowledge about the parameters to combat problems with data sparsity, noise, etc.
Learning process doesn't find a set of specific parameter values, it finds a distribution over all possible parameter values
Google for maximum a posteriori estimation (MAP) vs maximum likelihood estimation (MLE) if you're interested in more
P( \theta | D ) = \frac{P( D | \theta ) P( \theta )}{P( D )}
P( \theta | D ) = \frac{P( D | \theta ) P( \theta )}{P( D )}
Task & Method Families
Stochastic/Evolutionary methods
Predict a continuous output value given some input
E.g. How many inches of water will fall for cloudy may day?
Predict a categorical output value given some input
E.g. Is this credit card transaction fraudulent?
Given this sequence of inputs, predict a sequence of outputs
E.g. Assign Parts-of-speech tags to each word
Group data points in some way that provides meaning
E.g. discovering customers that have similar purchasing habits
Object extraction, identification in images
no exponents in terms
each input variable has a parameter representing the weight of how much that input contributes to the output
multiply weight * input and add 'em up to get output
Regularization - try and encourage the parameter values to stay within a given range
Lot's of math tricks that work with this constraint
y(x, \theta) = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \cdots + \theta_n x_n
y(x, \theta) = \theta_0 + \theta_1 x_1 + \theta_2 x_2 + \cdots + \theta_n x_n
Discriminative: model the desired target variable directly
Generative: model a conditional process that generates values then use bayes rule
Simplest, generative classifier
Powerful discriminative method of classification
Weights the "log odds" of each input
P(C_1|x) = \frac{P(x | C_1) P(C_1)}{P(x | C_1) P(C_1) + P(x | C_2) P(C_2)}
P(C_1|x) = \frac{P(x | C_1) P(C_1)}{P(x | C_1) P(C_1) + P(x | C_2) P(C_2)}
E.g. bayes nets, markov random fields, factor graphs
Framework for encoding dependence relationships
Hidden Markov Model (generative)
Conditional Random Fields (discriminative)
C4.5, C5 popular; especially good at categorical data
Typically used for classification, but CART does regression in the leaves
Each node in the tree represents the best way to split the data so each sub-tree is more homogenous than the parent
At test time, follow tree from root to leaf
Build lots of trees from resampled training data
Average or vote the results
Example of the technique: bagging
Find an optimally separating plane
Math tricks (finding support vectors, kernel trick)
Excellent discriminative method, commonly limited to binary classification
Feed forward, recurrent, hopfield, ARTMap, boltzmann machines, oh my!
Deep learning: stacking networks + clever training
Stochastic/Evolutionary
When all else fails, search forever
Particle Swarm (PSO) et al
Many methods deliberately assume distributions or constrain the function such that it is convex
Calculus tricks (gradients, line search, LBFGS)
Large graphical models with lots of dependence
Incomplete training data
Expectation constraints, posterior regularization
Data Science is hard
Which methods, which constraints on methods
Mathematical details of model leads to intuition
Mind boggling amount of information, growing quickly
Multi-disciplinary history -> overlapping concepts, words, notation -> confusion
Tools are maturing quickly but there is still large gap between day to day exploration and modelling and production end-user value
Unicorns are hard to find
Easy for ignorant charlatans to convince otherwise unsuspecting non-technical people that they know how to data science
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2022 Differential invariance of the multiplicity of real and complex analytic sets
José Edson Sampaio1
1Departamento de Matemática, Universidade Federal do Ceará, Rua Campus do Pici, s/n, Bloco 914, Pici 60440-900, Fortaleza-CE, Brazil
This paper is devoted to proving the differential invariance of the multiplicity of real and complex analytic sets. In particular, we prove the real version of the Gau–Lipman theorem, i.e., it is proved that the multiplicity mod
2
of real analytic sets is a differential invariant. We also prove a generalization of the Gau–Lipman theorem.
The author was partially supported by CNPq-Brazil grant 303811/2018-8.
José Edson Sampaio. "Differential invariance of the multiplicity of real and complex analytic sets." Publ. Mat. 66 (1) 355 - 368, 2022. https://doi.org/10.5565/PUBLMAT6612214
Primary: 14B05 , 14Pxx , 32S50
Keywords: analytic sets , multiplicity , Zariski’s multiplicity conjecture
José Edson Sampaio "Differential invariance of the multiplicity of real and complex analytic sets," Publicacions Matemàtiques, Publ. Mat. 66(1), 355-368, (2022)
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Mr. Benesh is in charge of facilities at Walt Clark Middle School. He is organizing a project to paint all
36
classrooms during the school’s summer break. He estimates that it will take one person five hours to paint each classroom.
How many total hours would it take for one person to paint all of the classrooms?
If it takes five hours for one person to paint one classroom, how many hours would it take to paint
36
classrooms?
180
Mr. Benesh has a team of four workers he is planning to assign to the job. Assuming they all paint at the same rate of five hours per classroom, how many hours would it take the team to do the painting?
With a team of
4
, it would take five hours to finish painting
4
classrooms. If they wanted to paint all 36 classrooms, how many sets of classrooms would they have to paint, or how may would each person have to paint?
36 ÷ 4 = 9 \text{ sets}
Mr. Benesh realized that he needs the painting to be finished in nine hours so that a different team can come in to wax the floors before school starts. How many people will he need to assign to do the painting in order to do this?
Based on part (a), all
36
classrooms would take a total of
180
20
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Hua Dong, Xianghua Zhao, "Numerical Method for a Markov-Modulated Risk Model with Two-Sided Jumps", Abstract and Applied Analysis, vol. 2012, Article ID 401562, 9 pages, 2012. https://doi.org/10.1155/2012/401562
Hua Dong1 and Xianghua Zhao1
This paper considers a perturbed Markov-modulated risk model with two-sided jumps, where both the upward and downward jumps follow arbitrary distribution. We first derive a system of differential equations for the Gerber-Shiu function. Furthermore, a numerical result is given based on Chebyshev polynomial approximation. Finally, an example is provided to illustrate the method.
The risk model with two-sided jumps was first proposed by Boucherie et al. [1] and has been further investigated by many authors during the last few years. For example, Kou and Wang [2] studied the Laplace transform of the first passage time and the overshoot for a perturbed compound Poisson model with double exponential jumps. Xing et al. [3] extended the results of Kou and Wang [2] to the case that the surplus process with phase-type downward and arbitrary upward jumps. Zhang et al. [4] assumed that the downward jumps follow arbitrary distribution and the upward jumps have a rational Laplace transform. They derived the Laplace transform of the Gerber-Shiu function by using the roots of the generalized Lundberg equation. Under the assumption that the upward jumps follow Laplace distribution and arbitrary downward jumps, Chi [5] obtained a closed-form expression for the Gerber-Shiu function by applying Wiener-Hopf factorization technique. The applications of the model in finance were also discussed. Jacobsen [6] studied a perturbed renewal risk model with phase-type interclaim times and two-sided jumps, where both the jumps have rational Laplace transforms. Based on the roots of the Cramér-Lundberg equation, the joint Laplace transform on the time to ruin, and the undershoot at ruin were given. However, in all the aforemetioned papers, the topic that the jumps in both directions are arbitrary distributions is still not discussed. The Markov-modulated risk model (Markovian regime switching model) was first proposed by Asmussen [7] to extend the classical risk model. Since then, it has received remarkable attention in actuarial mathematics, see, for example, Zhu and Yang [8, 9], Zhang et al. [4], Ng and Yang [10], Li and Lu [11], Lu and Tsai [12], and references therein. Motivated by the papers mentioned above, in this paper, we will study the Markov-modulated risk model with two-sided jumps.
Let be a homogenous, irreducible, and recurrent Markov process with finite state space . Denote the intensity matrix of by with and for . Let be a sequence of independent random variables representing the jumps, and be a standard Brownian motion with . Here we assume that the premium rates, claim interarrival times, the distributions of the jumps, and the diffusion parameter are all influenced by the environment process . When , the premium rate is , jumps arrive according to a Poisson process with intensity , the diffusion parameter is , and the size of the jumps which arrives at time follows the distribution with density and finite mean . Then the Markov-modulated diffusion risk model is defined by where is the initial surplus. If we denote the stationary distribution of by , then the positive security loading condition is given by
In this paper, we further assume that the jumps in (1.1) are two-sided. The upward jumps can be explained as the random income (premium or investment), while the downward jumps are interpreted as the random loss. In this case, the density function is given by where , , is the indicator function, and are two arbitrary functions on .
Let ( otherwise) be the time to ruin. For , let be the Gerber-Shiu function at ruin given that the initial state is , where is a nonnegative penalty function, is the surplus immediately prior to ruin, and is the deficit at ruin. Without loss of generality, we assume that . Thus for . When , (1.4) reduces to the Laplace transform of the time to ruin when and , (1.4) reduces to the probability of ruin
The purpose of this paper is to present some numerical results on the Gerber-Shiu function for the Markov-modulated diffusion risk model with arbitrary upward and downward jumps. In Section 2 we derive a system of integrodifferential equations and approximate solutions for . Numerical example is given in the last section.
2. Integrodifferential Equations and Approximate Solution
Theorem 2.1. For , satisfies the following integrodifferential equation where with boundary conditions
Proof. Similar to Ng and Yang [10].
Remark 2.2. When , (2.1) is identical to in Zhang et al. [4].
Clearly, (2.1) is a system of second order linear integrodifferential equations of Fredholm-Volterra type. As is well known, it is very difficult to find analytical solution of this system. Motivated by Akyüz-Dascioglu [13], we will study an alternative system defined on by Chebyshev collocation method. First, we transform the interval to . Following Diko and Usábel [14], we set , that is, . Furthermore, we assume that is an arbitrary strictly monotone, twice continuously differentiable function throughout the paper.
Theorem 2.3. Let be a monotone increase function and for . Then satisfies the following integrodifferential equation where with boundary conditions
Proof. By the definitions of function and , we have
Substituting (2.7) and into (2.1) and simplifying lead to (2.4). The boundary conditions are direct result of the boundary conditions in Theorem 2.1. This completes the proof.
Remark 2.4. The existence of the solution for the system of integrodifferential equations (2.4) can be found in Fariborzi and Behzadi [15].
According to Akyüz-Dascioglu [13], and its derivatives have truncated Chebyshev series expression where , are shifted Chebyshev polynomials of the first kind and are the unknown coefficients to be determined.
Let , , . Then (2.8) can be written in the matrix form where for odd , and for even .
Similarly, the kernel functions and can be expanded to univariate Chebyshev series where with and are Chebyshev coefficients determined by Clenshaw and Curtis [16].
Theorem 2.5. For , an approximate expression for is given by where the column vector can be determined by the following systems where matrix with elements matrix with elements and are collocations.
Proof. Using (2.8) and (2.12), one obtains Substituting (2.18) into (2.4), we have which is identical to (2.15) in form. Substituting the collocations into (2.19) leads to (2.15). and can be obtained by (2.6).
Example 2.6. To illustration our method, we use the example of Zhang et al. [4]. Let , , , , , , , the downward jumps are exponentially distributed with parameter , and the upward jump density is given by . We set and the collocation points are .
Figure 1 shows that the approximate solution is very near to the exact solution for any initial surplus . We remark that the horizontal axis in Figure 1 is and .
Laplace transform of the time to ruin.
From Table 1 we can see that the errors between the approximate solutions and the exact solutions decrease when increases. The initial surplus can also influence the approximate solution: the bigger need a bigger to decrease the error.
2 0.36228 0.36702 0.36918 0.37053 0.37149 0.3720
8 0.082052 0.11061 0.12152 0.12482 0.129 0.1430
This work is supported by the Natural Science Foundation of Shandong (no. ZR2010AQ015), the Tianyuan fund for Mathematics (no. 11226251), and the Natural Science Foundation of Qufu Normal University (no. 2012ZRB01473).
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Copyright © 2012 Hua Dong and Xianghua Zhao. 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.
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Classical Kuiper belt object - WikiMili, The Best Wikipedia Reader
A classical Kuiper belt object, also called a cubewano ( /ˌkjuːbiːˈwʌnoʊ/ "QB1-o"), [lower-alpha 1] is a low-eccentricity Kuiper belt object (KBO) that orbits beyond Neptune and is not controlled by an orbital resonance with Neptune. Cubewanos have orbits with semi-major axes in the 40–50 AU range and, unlike Pluto, do not cross Neptune's orbit. That is, they have low-eccentricity and sometimes low-inclination orbits like the classical planets.
Orbits: 'hot' and 'cold' populations
Cold and hot populations: physical characteristics
DES classification
SSBN07 classification
The name "cubewano" derives from the first trans-Neptunian object (TNO) found after Pluto and Charon, 15760 Albion, which until January 2018 had only the provisional designation (15760) 1992 QB1 . [2] Similar objects found later were often called "QB1-o's", or "cubewanos", after this object, though the term "classical" is much more frequently used in the scientific literature.
15760 Albion [3] (aka 1992 QB1 and gave rise to term 'Cubewano')
136472 Makemake, the largest known cubewano and a dwarf planet [3]
50000 Quaoar and 20000 Varuna, each considered the largest TNO at the time of discovery [3]
(33001) 1997 CU29 , (55636) 2002 TX300 , (55565) 2002 AW197 , (55637) 2002 UX25
136108 Haumea was provisionally listed as a cubewano by the Minor Planet Center in 2006, [4] but was later found to be in a resonant orbit. [3]
The majority of classical objects, the so-called cold population, have low inclinations (< 5°) and near-circular orbits, lying between 42 and 47 AU. A smaller population (the hot population) is characterised by highly inclined, more eccentric orbits. [5] The terms 'hot' and 'cold' has nothing to do with surface or internal temperatures. Instead, the terms 'hot and 'cold' refer to the orbits of the objects, by analogy to particles in a gas, which increase their relative velocity as they become heated up. [6]
The Deep Ecliptic Survey reports the distributions of the two populations; one with the inclination centered at 4.6° (named Core) and another with inclinations extending beyond 30° (Halo). [7]
When the orbital eccentricities of cubewanos and plutinos are compared, it can be seen that the cubewanos form a clear 'belt' outside Neptune's orbit, whereas the plutinos approach, or even cross Neptune's orbit. When orbital inclinations are compared, 'hot' cubewanos can be easily distinguished by their higher inclinations, as the plutinos typically keep orbits below 20°. (No clear explanation currently exists for the inclinations of 'hot' cubewanos. [8] )
The difference in colour between the red cold population, such as 486958 Arrokoth, and more heterogeneous hot population was observed as early as in 2002. [9] Recent studies, based on a larger data set, indicate the cut-off inclination of 12° (instead of 5°) between the cold and hot populations and confirm the distinction between the homogenous red cold population and the bluish hot population. [10]
Another difference between the low-inclination (cold) and high-inclination (hot) classical objects is the observed number of binary objects. Binaries are quite common on low-inclination orbits and are typically similar-brightness systems. Binaries are less common on high-inclination orbits and their components typically differ in brightness. This correlation, together with the differences in colour, support further the suggestion that the currently observed classical objects belong to at least two different overlapping populations, with different physical properties and orbital history. [11]
There is no official definition of 'cubewano' or 'classical KBO'. However, the terms are normally used to refer to objects free from significant perturbation from Neptune, thereby excluding KBOs in orbital resonance with Neptune (resonant trans-Neptunian objects). The Minor Planet Center (MPC) and the Deep Ecliptic Survey (DES) do not list cubewanos (classical objects) using the same criteria. Many TNOs classified as cubewanos by the MPC are classified as ScatNear (possibly scattered by Neptune) by the DES. Dwarf planet Makemake is such a borderline classical cubewano/scatnear object. (119951) 2002 KX14 may be an inner cubewano near the plutinos. Furthermore, there is evidence that the Kuiper belt has an 'edge', in that an apparent lack of low-inclination objects beyond 47–49 AU was suspected as early as 1998 and shown with more data in 2001. [12] Consequently, the traditional usage of the terms is based on the orbit's semi-major axis, and includes objects situated between the 2:3 and 1:2 resonances, that is between 39.4 and 47.8 AU (with exclusion of these resonances and the minor ones in-between). [5]
These definitions lack precision: in particular the boundary between the classical objects and the scattered disk remains blurred. As of 2020 [update] , there are 634 objects with perihelion (q) > 40 AU and aphelion (Q) < 47 AU. [13]
Introduced by the report from the Deep Ecliptic Survey by J. L. Elliott et al. in 2005 uses formal criteria based on the mean orbital parameters. [7] Put informally, the definition includes the objects that have never crossed the orbit of Neptune. According to this definition, an object qualifies as a classical KBO if:
An alternative classification, introduced by B. Gladman, B. Marsden and C. van Laerhoven in 2007, uses a 10-million-year orbit integration instead of the Tisserand's parameter. Classical objects are defined as not resonant and not being currently scattered by Neptune. [14]
have their eccentricity
{\displaystyle e<0.240}
(to exclude detached objects)
Unlike other schemes, this definition includes the objects with major semi-axis less than 39.4 AU (2:3 resonance)—termed inner classical belt, or more than 48.7 (1:2 resonance) – termed outer classical belt, and reserves the term main classical belt for the orbits between these two resonances. [14]
The first known collisional family in the classical Kuiper belt—a group of objects thought to be remnants from the breakup of a single body—is the Haumea family. [15] It includes Haumea, its moons, 2002 TX300 and seven smaller bodies.† The objects not only follow similar orbits but also share similar physical characteristics. Unlike many other KBO their surface contains large amounts of ice (H2O) and no or very little tholins. [16] The surface composition is inferred from their neutral (as opposed to red) colour and deep absorption at 1.5 and 2. μm in infrared spectrum. [17] Several other collisional families might reside in the classical Kuiper belt. [18] [19]
As of January 2019, only one classical Kuiper belt object has been observed up close by spacecraft. Both Voyager spacecraft have passed through the region before the discovery of the Kuiper belt. [20] New Horizons was the first mission to visit a classical KBO. After its successful exploration of the Pluto system in 2015, the NASA spacecraft has visited the small KBO 486958 Arrokoth at a distance of 3,500 kilometres (2,200 mi) on 1 January 2019. [21]
Here is a very generic list of classical Kuiper belt objects. As of October 2020 [update] , there are about 779 objects with q > 40 AU and Q < 48 AU. [22]
↑ Somewhat old-fashioned, but “cubewano” is still used by the Minor Planet Center for their list of Distant Minor Planets. [1]
Makemake is a dwarf planet and perhaps the second-largest Kuiper belt object in the classical population, with a diameter approximately two-thirds that of Pluto. It has one known satellite. Its extremely low average temperature, about 40 K (−230 °C), means its surface is covered with methane, ethane, and possibly nitrogen ices.
The Haumea or Haumean family is the only identified trans-Neptunian collisional family; that is, the only group of trans-Neptunian objects (TNOs) with similar orbital parameters and spectra that suggest they originated in the disruptive impact of a progenitor body. Calculations indicate that it is probably the only trans-Neptunian collisional family. Members are known as Haumeids.
(416400) 2003 UZ117 is a trans-Neptunian object and suspected member of the Haumea family, located in the Kuiper belt in the outermost region of the Solar System. It was discovered on 24 October 2003, by astronomers of the Spacewatch survey project at Kitt Peak Observatory, Arizona. The object may also be a non-resonant cubewano.
(505448) 2013 SA100, provisional designation 2013 SA100 and also known as o3l79, is a trans-Neptunian object from the classical Kuiper belt in the outermost region of the Solar System. It was discovered on 5 August 2013, by astronomer with the Outer Solar System Origins Survey at the Mauna Kea Observatories, Hawaii, in the United States. The classical Kuiper belt object belongs to the hot population and is a weak dwarf planet candidate, approximately 260 kilometers (160 miles) in diameter.
(516977) 2012 HZ84, provisional designation 2012 HZ84, is a small trans-Neptunian object from the Kuiper belt located in the outermost region of the Solar System, approximately 74 kilometers (46 miles) in diameter. It was discovered on 17 April 2012, by a team of astronomers using one of the Magellan Telescopes in Chile during the New Horizons KBO Search in order to find a potential flyby target for the New Horizons spacecraft. In December 2017, this classical Kuiper belt object was imaged by the spacecraft from afar at a record distance from Earth.
↑ "Distant Minor Planets".
↑ Jewitt, David. "Classical Kuiper Belt Objects". UCLA. Retrieved 1 July 2013.
1 2 3 4 Brian G. Marsden (30 January 2010). "MPEC 2010-B62: Distant Minor Planets (2010 FEB. 13.0 TT)". IAU Minor Planet Center. Harvard-Smithsonian Center for Astrophysics. Archived from the original on 4 September 2012. Retrieved 26 July 2010.
↑ "MPEC 2006-X45: Distant Minor Planets". IAU Minor Planet Center & Tamkin Foundation Computer Network. 12 December 2006. Retrieved 3 October 2008.
1 2 Jewitt, D.; Delsanti, A. (2006). "The Solar System Beyond The Planets" (PDF). Solar System Update : Topical and Timely Reviews in Solar System Sciences (PDF). Springer-Praxis. ISBN 978-3-540-26056-1. Archived from the original (PDF) on 29 January 2007. Retrieved 2 March 2006. )
↑ Levison, Harold F.; Morbidelli, Alessandro (2003). "The formation of the Kuiper belt by the outward transport of bodies during Neptune's migration". Nature . 426 (6965): 419–421. Bibcode:2003Natur.426..419L. doi:10.1038/nature02120. PMID 14647375. S2CID 4395099.
1 2 J. L. Elliot; et al. (2006). "The Deep Ecliptic Survey: A Search for Kuiper Belt Objects and Centaurs. II. Dynamical Classification, the Kuiper Belt Plane, and the Core Population". Astronomical Journal . 129 (2): 1117–1162. Bibcode:2005AJ....129.1117E. doi: 10.1086/427395 . ( "Preprint" (PDF). Archived from the original (PDF) on 23 August 2006. )
↑ Jewitt, D. (2004). "Plutino". Archived from the original on 19 April 2007.
↑ A. Doressoundiram; N. Peixinho; C. de Bergh; S. Fornasier; P. Thebault; M. A. Barucci; C. Veillet (October 2002). "The Color Distribution in the Edgeworth-Kuiper Belt". The Astronomical Journal. 124 (4): 2279. arXiv: astro-ph/0206468 . Bibcode:2002AJ....124.2279D. doi:10.1086/342447. S2CID 30565926.
↑ Peixinho, Nuno; Lacerda, Pedro; Jewitt, David (August 2008). "Color-inclination relation of the classical Kuiper belt objects". The Astronomical Journal. 136 (5): 1837. arXiv: 0808.3025 . Bibcode:2008AJ....136.1837P. doi:10.1088/0004-6256/136/5/1837. S2CID 16473299.
↑ K. Noll; W. Grundy; D. Stephens; H. Levison; S. Kern (April 2008). "Evidence for two populations of classical transneptunian objects: The strong inclination dependence of classical binaries". Icarus. 194 (2): 758. arXiv: 0711.1545 . Bibcode:2008Icar..194..758N. doi:10.1016/j.icarus.2007.10.022. S2CID 336950.
↑ Trujillo, Chadwick A.; Brown, Michael E. (2001). "The Radial Distribution of the Kuiper Belt" (PDF). The Astrophysical Journal. 554 (1): L95–L98. Bibcode:2001ApJ...554L..95T. doi:10.1086/320917. Archived from the original (PDF) on 19 September 2006.
↑ "JPL Small-Body Database Search Engine". JPL Solar System Dynamics. Retrieved 26 July 2010.
1 2 Gladman, B. J.; Marsden, B.; van Laerhoven, C. (2008). "Nomenclature in the Outer Solar System" (PDF). In Barucci, M. A.; et al. (eds.). The Solar System Beyond Neptune. Tucson: University of Arizona Press. ISBN 978-0-8165-2755-7.
↑ Brown, Michael E.; Barkume, Kristina M.; Ragozzine, Darin; Schaller, Emily L. (2007). "A collisional family of icy objects in the Kuiper belt" (PDF). Nature. 446 (7133): 294–6. Bibcode:2007Natur.446..294B. doi:10.1038/nature05619. PMID 17361177. S2CID 4430027.
↑ Pinilla-Alonso, N.; Brunetto, R.; Licandro, J.; Gil-Hutton, R.; Roush, T. L.; Strazzulla, G. (2009). "The surface of (136108) Haumea (2003 EL61), the largest carbon-depleted object in the trans-Neptunian belt". Astronomy and Astrophysics. 496 (2): 547. arXiv: 0803.1080 . Bibcode:2009A&A...496..547P. doi:10.1051/0004-6361/200809733. S2CID 15139257.
↑ Pinilla-Alonso, N.; Licandro, J.; Gil-Hutton, R.; Brunetto, R. (2007). "The water ice rich surface of (145453) 2005 RR43: a case for a carbon-depleted population of TNOs?". Astronomy and Astrophysics. 468 (1): L25–L28. arXiv: astro-ph/0703098 . Bibcode:2007A&A...468L..25P. doi:10.1051/0004-6361:20077294. S2CID 18546361.
↑ Chiang, E.-I. (July 2002). "A Collisional Family in the Classical Kuiper Belt". The Astrophysical Journal . 573 (1): L65–L68. arXiv: astro-ph/0205275 . Bibcode:2002ApJ...573L..65C. doi:10.1086/342089. S2CID 18671789.
↑ de la Fuente Marcos, Carlos; de la Fuente Marcos, Raúl (11 February 2018). "Dynamically correlated minor bodies in the outer Solar system". Monthly Notices of the Royal Astronomical Society . 474 (1): 838–846. arXiv: 1710.07610 . Bibcode:2018MNRAS.474..838D. doi:10.1093/mnras/stx2765. S2CID 73588205.
↑ Stern, Alan (28 February 2018). "The PI's Perspective: Why Didn't Voyager Explore the Kuiper Belt?" . Retrieved 13 March 2018.
↑ Lakdawalla, Emily (24 January 2018). "New Horizons prepares for encounter with 2014 MU69". Planetary Society. Retrieved 13 March 2018.
↑ "q > 40 AU and Q < 48 AU". IAU Minor Planet Center. minorplanetcenter.net. Harvard-Smithsonian Center for Astrophysics.
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My.SUPA has been setup to allow maths to be written quickly using LaTeX notation. This can be included anywhere you see a text box in your course area -- including news or social forums, web pages and wikis. The format for entering LaTeX in My.SUPA is to wrap the code between two pairs of dollar signs. $$ a=b+c $$
If you are looking at this for the first time, please read the entries under 01 Getting Started for an overview. The list of entries may be viewed by categories or alphabetically.
01 Getting Started | 02 Arithmetic expressions | 03 Font Styles | 04 Delimiters
05 Spaces | 06 Symbols | 07 Relations | 09 Structures | 10 Feynman Diagrams
11 Other LaTeX Software
\_ (where _ is blank)
Ordinary whitespace to be used after a dot not denoting the end of a sentence
After commands without parameters use \~ (tilde) instead in order to avoid browser specific problems
\, inserts the smallest predefined space in a formula
Equivalent: \hspace{2}
Ex.: $$a\,b$$ gives
a\,b
Ex.: $$a~\hspace{2}~b$$ gives also
a~\hspace{2}~b
\; (backslash semicolon) inserts the third smallest predefined space in a formula
Ex.: $$a\;b$$ gives
a\;b
a~\hspace{6}~b
\: inserts the second smallest predefined space in a formula
Ex.: $$a\:b$$ gives
a\:b
a~\hspace{4}~b
\/ (backslash slash) avoids ligatures
Ex.: $$V\/A$$ gives
V\/A
in contrast to $$VA$$ which gives
VA
In order to prevent some browser specific problems with whitespaces, it is advisable to use ~ (tilde) as the whitespace instead of the normal blank key (in places where whitespaces are mandatory, e.g. after commands).
Ex.: $$\frac~xy$$ to produce
\frac~xy
Ex.: $$\sqrt~n$$ to produce
\sqrt~n
\hspace{n}
inserts a space of n pixels
Ex.: $$f(x)\hspace{6}=\hspace{6}0$$ gives
f(x)\hspace{6}=\hspace{6}0
can be combined with the preceding command \unitlength{m}(default: m=1px) , which defines the applied unit
Ex.: $$\unitlength{20}a\hspace{2}b$$ gives
\unitlength{20}a\hspace{2}b
, i.e. a space of 20x2=40px
\LARGE (all capital letters)
Everthing following the \LARGE command will be output in the largest predefined font size until the system encounters another font size command.
Note: This command is case sensitive, since large, Large and LARGE are different sizes!
Ex.: $$\LARGE~3x$$ gives
\LARGE~3x
\Large (L capital letter)
Everthing following the \Large command will be output in the second largest font size until the system encounters another font size command.
\Large~3x
\large (all lower case letters)
Everthing following the \large command will be output in the large font size until the system encounters another font size command.
\large~3x
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PolySCIP is a solver for multi-criteria integer programming and multi-criteria linear programming. In other words, it aims at solving optimization problems of the form:
\text{min/max}\phantom{\rule{0.2em}{0ex}}\left({c}_{1}\cdot x,\dots ,{c}_{k}\cdot x\right)\phantom{\rule{0.2em}{0ex}}\text{s.t.}\phantom{\rule{0.2em}{0ex}}Ax\le b,x\in {\mathrm{\mathbb{Z}}}^{n}\mathmr{and} {\mathrm{\mathbb{Q}}}^{n}
Bi-criteria integer program.
Image of a bi-criteria integer program (where objectives are maximized) with non-dominated points in blue.
Bi-criteria linear program.
Image of a bi-criteria linear program (where objectives are maximized) with non-dominated points in blue.
The name PolySCIP is composed of Poly (from the Greek πολύς meaning "many") and SCIP. The current version of PolySCIP is able to compute supported non-dominated vertices for problems with an arbitrary number of objectives and the entire set of non-dominated points for bi-criteria and tri-criteria integer programs. The file format of PolySCIP is based on the MPS file format.
13/Nov/2015 Website launched.
29/Feb/2016 SCIP Version 3.2.1 with PolySCIP version 1.0 released.
09/Mar/2017 SCIP Version 4.0 with PolySCIP version 2.0 released.
27/May/2017 Visualisation tool PolyNondom available.
The current version of PolySCIP is 2.0 (the development is inactive at the moment). As part of SCIP its source code resides in the directory 'applications/PolySCIP'. You can download the source code via the SCIP website.
SCIP Optimization Suite with PolySCIP 2.0
A description of features and improvements of PolySCIP 2.0 can be found in section 7.2 of this technical report.
(See also the corresponding INSTALL file in the PolySCIP directory.)
Build SCIP: see the corresponding INSTALL file in the SCIP directory
Build PolySCIP: change into the PolySCIP directory and execute make on the command line
if SCIP was built with make [options], then run make [options] with the same options in the PolySCIP directory
Build PolySCIP documentation:
Run make doc to build doxygen documentation in 'doc/html'
Run cd doc; pdflatex userguide.tex to compile the user guide
Please include a reference if you use PolySCIP for your work:
R. Borndörfer, S. Schenker, M. Skutella, T. Strunk: PolySCIP.
Mathematical Software - Proceedings of ICMS 2016, G.-M. Greuel, T. Koch, P. Paule, A. Sommese (Eds.),
Lecture Notes in Computer Science Vol. 9725, ISBN: 978-3-319-42431-6
For more details about the usage, file format of PolySCIP and an easy way to generate .mop problem files containing (your) mathematical programs see the user guide.
Sebastian Schenker Timo Strunk
PolySCIP is part of SCIP and distributed under the ZIB Academic License. You are allowed to retrieve (Poly)SCIP as a member of a non-commercial and academic institution. If you want to use PolySCIP, but you do not comply with the above criteria, please contact me.
If you find any bugs, please send a description.
MOPLIB (short for Multi-Objective Problem LIBrary) is a collection of multi-objective optimization problems. PolySCIP supports the following problem classes: molp, mobp, moip, (momip)
If you develop a solver for multi-criteria optimization problems, please let me know.
Bensolve - a vector linear program solver
inner - a multi-objective linear program solver
MOPS - a solver for non-linear multiobjective optimization problems
The development of PolySCIP started in the project A5 Multicriteria Optimisation within the Collaborative Research Center 1026 Sustainable Manufacturing - Shaping Global Value Creation.
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Obtaining the Fluorescent Chitosan for Investigations in the Analytical Ultracentrifuge
Namangan State University, Namangan, Uzbekistan
1) In order to achieve the visibility of the chitosan macromolecule for the UV optical system of the analytical ultracentrifuge on investigation of the molecular characteristics and polymers interactions, the labeling of chitosan by a new fluorophore of fluorescein-5-isothiocyanat was carried out. 2) Samples of fluorescent chitosan with two different degrees of fluorophore substitution and various degrees of acetylation were obtained. 3) The labeled chitosans with the fluorescein-5-isothiocyanat allowed estimating the sedimentation coefficient and the molecular characteristic in the analytical ultracentrifuge. 4) The sensitivity of the UV-optical system of the analytical ultracentrifuge for the obtained fluorescent samples of chitosan relatively to the fixation of the meniscus and the influence of the wavelength and rotation speed were estimated.
Polysaccharide, Chitosan, Analytical Ultracentrifuge, Fluorescein-5-Isothiocyanat, Fluorescent Chitosan, Labeling, Sedimentation, Degree of Substitution, UV Absorption
Due to the features of the chemical structure and organization of the macromo lecules, chitosan possesses many properties, that enable to use in many spheres [1] - [6] . Chitosan is polysaccharide consisting of 2-acetamido-2-deoxy-β-D-glucopyranose and 2-amino-2-deoxy-β-D-glucopyranose units obtained with partial deacetylation of chitin. Such kind of chemical composition conditions some problem in completely quantitative analysis and identification of chitosan. That’s why the chemical structure of chitosan lacks the fluorophore group with necessary UV extinction, and as a result the molecule becomes transparent to the UV spectrum. Such invisibility of the chitosan also limits investigations of molecular characteristics and other advantaged [7] [8] applications of analytical ultracentrifuge using of UV optical system.
As is known, one possible way of overcoming this invisibility is to label the polysaccharide with a chromophore containing compounds. At present time a labeling procedure effectively used in the field of polymers and on the strength of practical needs this method which develops with searching of the new systems of fluoroagents and polymers.
For example, a non-invasive analytical tool was developed to assess the use of in situ biomaterials for surgical implants or scaffolds in tissue engineering and on the basis of polymeric methods of treatment. In this study, a method for fluorescence monitoring of the chitosan membrane framework degradation was established for in vitro use in bioreactors and, ultimately, in vivo. The basis of this tracking system is the fluorescence-radiating biomaterial obtained by covalent binding of tetramethylboramine isothiocyanate (TRITC) fluorophore based on chitosan [9] . In the work of Coelfen et al. (1996), incorporation of the fluorophore 9-anthraldehyde onto chitosans has been considered and the effect of increase in degree of substitution of the fluorophore has been investigated on 2 chitosans of differing degrees of acetylation by analytical ultracentrifugation. Four chitosans with chemical composition ranging from a fraction of N-acetylated units (FA) of 0.01 to 0.61 were used to prepare fluorescence labeled chitosans with 9-anthraldehyde. The efficiency of the labeling of chitosans was determined by UV and 1H NMR spectroscopy and the influence of substituted fluorescent amounts to conformational characteristic of labeled chitosans in solution was investigated [10] .
At present work the samples with high extinction of UV-adsorption fluorescent chitosan modified with fluorescein-5-isothiocyanat (FITC) were obtained and possibilities of using this fluorescence labeled chitosans for characterization in the analytical ultracentrifuge were investigated. Labeled with FITC chitosans intended to use especially for identification of polymers interactions by synthetic boundary methods in AUS [8] .
Fluorescein-5-isothiocyanat. As fluoroagent is used fluorescein-5-isothiocyanat (FITC, the product of ALDRICH, F22502-1G), M = 389.4 g/mol.
Chitosan samples with different viscosities (η) and fractions of N-acetylated units (FA), were provided by SIGMA. The chitosan-1, η = 400 mPa, FA = 0.15; The chitosan-2, η = 200 mPa, FA = 0.18.
Chitosans were labelled with the fluorescein-5-isothiocyanat by method described in work (Coelfen [10] ) and efficiency of labelling process was observed with UV spectroscopy and analytical ultracentrifuge by the technique described in [7] [8] . This was done for each chitosan for two degrees of substitution: −0.5% and −1%, with the amount of label incorporated assayed by its absorption at a wavelength of 240 nm, where the labelled chitosan has a strong absorption maximum.
So, chitosan is hygroscopic polymer therefore for correlation density and solutions concentration the thermogravimetric analysis has performed the dates which are given in Table 1. The values of density and partial specific volume of the used chitosans samples also are presented in this table. Thermo-Mikrowaage TG 209 F1 (NETZSCH, Germany) and Densitymeter DMA 5000 (Anton Paar) equipments were used for the thermogravimetric and density analysis correspondingly.
The chitosans dissolved in acetate buffer which made up with the following composition (Dawson et al., 1986) [11] : 0.4MCH3COONa/0.4M CH3COOH/0.2M NaCl. To keep stable pH 4.5 and ionic strength 0.1 of solution during all experiments the dialyzed with polymer solution the buffer solution was used.
2.3.1. Analytical Ultracentrifugation
An Optima XL-A analytical ultracentrifuge (Beckman, Palo Alto, CA, USA) was used for all the experiments. It included two integrated detection systems, scanning UV:vis and Rayleigh interference optics systems. For the sedimentation analysis the rotor speed of 50,000 rpm, temperature of 20˚C and scanning wavelengths of 210 nm - 270 nm were employed. Sedimentation coefficient distributions were calculated using of SEDFITBETA2 data evaluation program.
2.3.2. UV-Spectroscopy
UV/VIS Spectrometer Lambda 2 (Perkin Elmer) was used for determining of efficiency of the fluorescence labelling reaction of chitosans and to calculate the extinction coefficient of chitosan solutions.
Table 1. Loss on drying (LOD), Degree of acetylation (DA), the solvent density (ρ), partial-specific volume (υ) and molecular weight (MW) of used Chitosansamples.
2.3.3. Dialyzing and Freeze Drying
All dialyzing equilibrium of polymer solutions was performed with Spectra Por Dialysis Membrane, MWCO 3500. Vacuum drying of samples was performed with equipment CHRIST LOC-1M, The freezing conditions was performed in vacuum 0.431 mbar and temperature 20˚C.
2.3.4. The Coupling Reaction
The modification of chitosan with FITC realized by method described in works Coelfen [10] and Tømmeraasa [12] . It is expected that, coupling of FITC will occur through amination of isothiocyanate by amino group of chitosan [12] [13] .
Preliminary dissolved the FITC in DMSO is added to solution of chitosan in acetate buffer
0.4{\text{MCH}}_{3}\text{COOH}/0.4{\text{MCH}}_{3}\text{COONa}/0.2\text{MNaCl}
. The amount of the reacting compounds correlated as for the first case to 100 structured units of chitosan corresponded one molecule of FITC (DS 1.0%) and in the second case to 200 units corresponded one molecule of FITC (DS 0.5%). The reactionary mixture was stirred 24 h. Excesses amount of FITC removed with dialyzing in system Acetic buffer/DMSO taken in 12:1 ratio. With evaporation of the solution under vacuum and low temperature, the sample of fluorescent chitosan is received.
The fluorescein 5-isotiocyanatewas introduced to the different chitosans (Table 1). Theconditions of these reactions kept as it is described in works [11] [14] for similar reaction system. The condensation reaction of a primary amino group (nucleophile) of chitosan (I) is the nucleophile condensing with the carbonyl group of fluorescein 5-isotiocyanate (II), resulting in the secondary amine of the fluorescent chitosan (III) as shown in Scheme 1.
Scheme 1. Reductive amination by chitosan (I) of fluorescein 5-isotiocyanate (II) to obtain the fluorescent chitosan (III).
The molar ratios of compounds in the coupling reaction as mentioned before were 100:1, 200:1 in order to obtain the chitosans with a chromophore which can conveniently be visualized and quantified by UV or fluorescence spectroscopy, without altering the conformation of the chitosan.
The qualitative estimation of productivity to reactions was performed with UV spectroscopy measurements of modified chitosans solutions with different concentrations in acetate buffer. The maximum emission wavelength of 240 nm with excitation ε = 9.934 was found in specters. Fluorescent chitosans with DS 0.5% show the same maximum emission wavelength of 240 nm but with decreased excitation. No significant difference was observed between the maximum emission wavelengths (εmax) of different MW chitosans.
The quantity estimation of degree substitution of the fluoroagent is definite with measuring of compactness charge in macromolecule with the help of devise ParticleCarge Detector PCD 3. According to this measurement for Chitosan-1 with DS 1% it was 322.556 k/g and for Chitosan-1 with DS 0.5% was 162.52 k/g.
The fluorescent chitosans were characterized and some compared analysis of the sedimentation coefficients as a function of the degree of substitution of the chromophore and UV extinction for the two chitosans by treatment of UV absorption scans of the XL-A optic system of the analytical ultracentrifuge was carried out. The data of measurements show that for the chitosan-1 with DA 85%, MW = 100,000 and substituted with fluoroagent 0.5% in macrochain has average sedimentation coefficient value 2.3 Svb and extinction coefficient at 240 nm wavelength of UV corresponded to 5.582. For this chitosan sample with contents of fluoreagent 1% the sedimentation coefficient is negligibly increased, at extinction coefficient increased on 9.934 as it was expected. For another chitosan sample with DA 82%, MW = 50,000 also observed the same sequence of dependence of the sedimentation coefficient (S = 1.49 Svb) with DS and extinction coefficient.
Sedimentation behavior of fluorescein modified chitosans determines that the chromophor group does not render the essential influence upon molecular and hydrodynamic parameters of chitosan. In the following discussion, we have chosen the modified chitosan-1 (see Table 1) with DS 1% as an arbitrary example.
As mentioned before, the fluorescein modified chitosan intended to use to obtain the membrane by synthetic boundary method in the analytical ultracentrifuge through interpolymer interactions [8] . In this connection, evaluation of UV absorption properties in different conditions is necessary. Absorption scan exhibit a meniscus peak. This would probably influence the detection of the initial membrane growth at the meniscus interface. Therefore, the occurrence of such a meniscus peak is dependent on the solution concentration range of 1% - 2% (Figure 1), the wavelength range of 210 - 270 (Figure 2) and the run velocity range of 3000 - 10,000 (Figure 3) was studied. As an example, for the detection of chitosan for the membrane formation in synthetic boundary method in the analytical ultracentrifuge in optimum was assigned to a velocity range 3000 - 5000 rpm and wavelength of 240 nm. These scans follow that the obtained fluorescein modified chitosan might be identified and analyzed by UV optics of AUC during membrane formation.
Figure 1. Meniscus detection as function of chitosan concentration: 2.0% (──); 1.5% (∙∙∙∙∙); 1.0% ( ̵ ̵ ̵ ). The velocity 3000 rpm; the wavelength 240 nm.
Figure 2. Meniscus detection as function of wavelength: 210 nm (──); 240 nm (∙∙∙∙∙); 270 nm ( ̵ ̵ ̵ ). Chitosan concentration 2%; the velocity 3000 rpm.
Figure 3. Meniscus detection as function of run velocity: 3000 rpm (──); 5000 rpm (∙∙∙∙∙); 8000 rpm ( ̵ ̵ ̵ ); 12,000 rpm (-× -×). Chitosan concentration 2%; wavelength 240 nm.
The fluorescent chitosan without significant depolymerisation of the polysaccharide with high extinction was prepared. Like chitosan, fluorophore 5-isotioceanat chitosan was water soluble at acidic pH values.
The increased incorporation of the fluorophore 5-isotioceanat at least up to a degree of substitution of 1% has no deleterious effects on the molar mass of two chitosans of differing degrees of acetylation. The inclusion of the fluorophore allows the use of the absorption optical system of the analytical ultracentrifuge to evaluate the sedimentation velocity and membrane formation process in synthetic boundary methods, for which these samples of chitosan will be used.
I express my gratitude to Professor of University of Konstanz, Germany Helmut Coelfen who provided laboratories and help on carrying out this work, also I thank the professor of Institute of chemistry and physics of polymers of the Academy sciences of Uzbekistan Sayora Rashidova for her assistance.
Kodirkhonov, M.R. (2019) Obtaining the Fluorescent Chitosan for Investigations in the Analytical Ultracentrifuge. Advances in Biological Chemistry, 9, 23-30. https://doi.org/10.4236/abc.2019.91002
1. Roberts, G.A.F. (1992) Chitin Chemistry. Macmillan Press Ltd., Hong Kong, 368 p. https://doi.org/10.1007/978-1-349-11545-7
2. Kean, T., Roth, S. and Thanou, M. (2005) Trimethylated Chitosans as Non-Viral Gene Delivery Vectors: Cytotoxicity and Transfection Efficiency. Journal of Controlled Release, 103, 643-653. https://doi.org/10.1016/j.jconrel.2005.01.001
3. Roberts, G.A.F. (2008) Thirty Years of Progress in Chitin and Chitosan. In: Jaworska, M.M., Ed., Progress on Chemistry and Application of Chitin and Its Derivatives, Polish Chitin Soc., Lodz, Vol. 13, 7-15.
4. USDA NOP and EPA Rule on Chitosan (2007) Federal Register No. 236, Vol. 72.
5. Chitin and Chitosan Final Registration Review Decision. Document ID: EPA-HQ-OPP-2007-0566-0019, 10-15.
6. Rashidova, S.Sh. and Milusheva, R.Yu. (2009) Chitin and Chitosan of Bombyx Mori: Synthesis, Properties and Applications. Fan, Tashkent, 246 p. (In Russian)
7. Wandrey, C. and Bartkowiak, A. (2001) Membrane Formation at Interfaces Examined by Analytical Ultracentrifugation Techniques. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 180, 141-153. https://doi.org/10.1016/S0927-7757(00)00767-6
8. Bourdillon, L. and Wandrey, C. (2004) On-Line Study of Polyelectrolyte Network Formation by Interfacial Reaction. Colloid and Polymer Science, 282, 1247-1257. https://doi.org/10.1007/s00396-004-1076-5
9. Cunha-Reis, C., El Haj, A.J., Yang, X.B. and Yang, Y. (2013) Fluorescent Labeling of Chitosan for Use in Non-Invasive Monitoring of Degradation in Tissue Engineering. Tissue Engineering and Regenerative Medicine, 7, 39-50. https://doi.org/10.1002/term.494
10. Coelfen, H., Harding, S.E. and Varum, K.-M. (1996) Investigation, Using Analytical Ultracentrifugation, of the Effect of the Incorporation of the Fluorophore 9-Anthraldehyde on Two Chitosans of Differing Degrees of Acetylation. Carbohydrate Polymers, 30, 55-60. https://doi.org/10.1016/S0144-8617(96)00042-2
11. Dawson, R.M.C., Elliott, D.C., Elliott, W.H. and Jones, K.M. (1986) Data for Biochemical Research. 3rd Edition, Oxford University Press, Oxford, 429 p.
12. Tømmeraasa, K., Stranda, S.P., Tianb, W., Kenneband, L. and Vårum, K.M. (2001) Preparation and Characterisation of Fluorescent Chitosans Using 9-Anthraldehyde as Fluorophore. Carbohydrate Research, 336, 291-296.
13. Yalpani, M. and Hall, L.D. (1981) Synthesis of Fluorescent Probe—Carbohydrate Conjugates. Canadian Journal of Chemistry, 59, 2934-2939. https://doi.org/10.1139/v81-425
14. Borch, R.F., Bernstein, M.D. and Dupont Durst, H. (1971) Cyanohydridoborate Anion as a Selective Reducing Agent. Journal of the American Chemical Society, 93, 2897-2904. https://doi.org/10.1021/ja00741a013
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1 Factors that affect infiltration
1.3 Soil moisture content
1.4 Organic materials in soils
4 Infiltration in wastewater collection
5 Infiltration calculation methods
5.1 General hydrologic budget
5.2 Richards' equation (1931)
5.3 Finite water-content vadose zone flow method
5.4 Green and Ampt
5.5 Horton's equation
5.6 Kostiakov equation
Factors that affect infiltration[edit]
Soil characteristics[edit]
Soil moisture content[edit]
Organic materials in soils[edit]
Infiltration in wastewater collection[edit]
Infiltration calculation methods[edit]
General hydrologic budget[edit]
{\displaystyle F=B_{I}+P-E-T-ET-S-I_{A}-R-B_{O}}
{\displaystyle B_{I}}
is the boundary input, which is essentially the output watershed from adjacent, directly connected impervious areas;
{\displaystyle B_{O}}
is the boundary output, which is also related to surface runoff, R, depending on where one chooses to define the exit point or points for the boundary output;
{\displaystyle I_{A}}
is the initial abstraction, which is the short term surface storage such as puddles or even possibly detention ponds depending on size;
Richards' equation (1931)[edit]
Finite water-content vadose zone flow method[edit]
Green and Ampt[edit]
{\displaystyle \int _{0}^{F(t)}{F \over F+\psi \,\Delta \theta }\,dF=\int _{0}^{t}K\,dt}
{\displaystyle {\psi }}
is wetting front soil suction head (L);
{\displaystyle \theta }
is water content (-);
{\displaystyle K}
is hydraulic conductivity (L/T);
{\displaystyle F(t)}
is the cumulative depth of infiltration (L).
{\displaystyle F(t)=Kt+\psi \,\Delta \theta \ln \left[1+{F(t) \over \psi \,\Delta \theta }\right].}
Using this model one can find the volume easily by solving for
{\displaystyle F(t)}
. However the variable being solved for is in the equation itself so when solving for this one must set the variable in question to converge on zero, or another appropriate constant. A good first guess for
{\displaystyle F}
is the larger value of either
{\displaystyle Kt}
{\displaystyle {\sqrt {2\psi \,\Delta \theta Kt}}}
. These values can be obtained by solving the model with log replaced with its Taylor-Expansion around one, of the zeroth and second order respectively. The only note on using this formula is that one must assume that
{\displaystyle h_{0}}
, the water head or the depth of ponded water above the surface, is negligible. Using the infiltration volume from this equation one may then substitute
{\displaystyle F}
into the corresponding infiltration rate equation below to find the instantaneous infiltration rate at the time,
{\displaystyle t}
{\displaystyle F}
was measured.
{\displaystyle f(t)=K\left[{\psi \,\Delta \theta \over F(t)}+1\right].}
Horton's equation[edit]
Named after the same Robert E. Horton mentioned above, Horton's equation[14] is another viable option when measuring ground infiltration rates or volumes. It is an empirical formula that says that infiltration starts at a constant rate,
{\displaystyle f_{0}}
, and is decreasing exponentially with time,
{\displaystyle t}
. After some time when the soil saturation level reaches a certain value, the rate of infiltration will level off to the rate
{\displaystyle f_{c}}
{\displaystyle f_{t}=f_{c}+(f_{0}-f_{c})e^{-kt}}
{\displaystyle f_{t}}
is the infiltration rate at time t;
{\displaystyle f_{0}}
is the initial infiltration rate or maximum infiltration rate;
{\displaystyle f_{c}}
is the constant or equilibrium infiltration rate after the soil has been saturated or minimum infiltration rate;
{\displaystyle k}
is the decay constant specific to the soil.
{\displaystyle F_{t}=f_{c}t+{(f_{0}-f_{c}) \over k}(1-e^{-kt})}
Kostiakov equation[edit]
{\displaystyle f(t)=akt^{a-1}\!}
{\displaystyle a}nd
{\displaystyle k}
are empirical parameters.
{\displaystyle f(t)=akt^{a-1}+f_{0}\!}
{\displaystyle F(t)=kt^{a}+f_{0}t\!}
{\displaystyle f_{0}}
approximates, but does not necessarily equate to the final infiltration rate of the soil.
Darcy's law[edit]
This method used for infiltration is using a simplified version of Darcy's law.[14] Many would argue that this method is too simple and should not be used. Compare it with the Green and Ampt (1911) solution mentioned previously. This method is similar to Green and Ampt, but missing the cumulative infiltration depth and is therefore incomplete because it assumes that the infiltration gradient occurs over some arbitrary length
{\displaystyle L}
. In this model the ponded water is assumed to be equal to
{\displaystyle h_{0}}
and the head of dry soil that exists below the depth of the wetting front soil suction head is assumed to be equal to
{\displaystyle -\psi -L}
{\displaystyle f=K\left[{h_{0}-(-\psi -L) \over L}\right]}
{\displaystyle {\psi }}
is wetting front soil suction head
{\displaystyle h_{0}}
is the depth of ponded water above the ground surface;
{\displaystyle K}
is the hydraulic conductivity;
{\displaystyle L}
is the vague total depth of subsurface ground in question. This vague definition explains why this method should be avoided.
{\displaystyle f=K\left[{L+S_{f}+h_{0} \over L}\right]}
{\displaystyle {f}}
Infiltration rate f (mm hour−1))
{\displaystyle K}
is the hydraulic conductivity (mm hour−1));
{\displaystyle L}
is the vague total depth of subsurface ground in question (mm). This vague definition explains why this method should be avoided.
{\displaystyle {S_{f}}}
is wetting front soil suction head (
{\displaystyle {-\psi }}
{\displaystyle {-\psi _{f}}}
) (mm)
{\displaystyle h_{0}}
is the depth of ponded water above the ground surface (mm);
Retrieved from "https://en.wikipedia.org/w/index.php?title=Infiltration_(hydrology)&oldid=1077891391"
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Accessing - Maple Help
Home : Support : Online Help : Programming : Modules : Module Exports : Accessing
retrieve the exported locals of a module
exports(m)
exports(m, options)
The procedure exports returns an expression sequence containing the names (symbols) of the exported members of a module m.
In addition to the module argument, exports accepts several optional arguments.
By default, the global instances of the exported member names are returned. The instances of the names local to the module can be requested by specifying the option instance as an optional argument.
By default, only the name portion (first operand) of an exported member that has been declared as an expression of type :: is returned. The entire structure, including the type, can be retrieved by passing the optional argument typed.
The string option causes exports to return the exported names as strings instead of names. In cases where it is necessary to know only the names of the exports and not their values (for example, for reporting purposes), this avoids any danger of accidental unintended evaluation.
The typed option can be used in conjunction with instance or string. The instance and string options cannot be used together.
Scope Selection Options
A module can contain both per-instance and static exports, the latter of which are shared by all instances of a module. By default, the exports function returns only the per-instance exports. Specifying the static option causes it to return only the static exports instead.
The all option causes exports to return both the per-instance and static exports. All of the per-instance exports will appear in the result before the static exports.
The static and all options cannot be used together.
Type Specification Options
The option type=T, where T is a valid Maple type specification, causes exports to return only those exports whose current value is of that type.
The method option selects per-instance and static exports whose value is of type callable, returning the global names of those exports. It is equivalent to the sequence of options, all,type=callable.
The type and method options cannot be used together.
The exports command is thread-safe as of Maple 15.
m≔\mathbf{module}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{_export}\left(\mathrm{e1},\mathrm{e2}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end module}:
e≔\mathrm{exports}\left(m\right)
\textcolor[rgb]{0,0,1}{e}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e2}}
\mathrm{evalb}\left(e[1]=\mathrm{e1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
e≔\mathrm{exports}\left(m,'\mathrm{instance}'\right)
\textcolor[rgb]{0,0,1}{e}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e2}}
\mathrm{evalb}\left(e[1]=\mathrm{e1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
m≔\mathbf{module}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{_export}\left(\mathrm{e1}::\mathrm{integer}≔3,\mathrm{e2}::\mathrm{`module`}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end module}:
\mathrm{exports}\left(m\right)
\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e2}}
\mathrm{exports}\left(m,'\mathrm{typed}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{::}\textcolor[rgb]{0,0,1}{\mathrm{Typesetting}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{_Hold}}\textcolor[rgb]{0,0,1}{}\left(\left[\textcolor[rgb]{0,0,1}{'}\textcolor[rgb]{0,0,1}{\mathrm{integer}}\textcolor[rgb]{0,0,1}{'}\right]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{::}\textcolor[rgb]{0,0,1}{\mathrm{module}}
\mathrm{exports}\left(m,'\mathrm{instance}','\mathrm{typed}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{::}\textcolor[rgb]{0,0,1}{\mathrm{Typesetting}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{_Hold}}\textcolor[rgb]{0,0,1}{}\left(\left[\textcolor[rgb]{0,0,1}{'}\textcolor[rgb]{0,0,1}{\mathrm{integer}}\textcolor[rgb]{0,0,1}{'}\right]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{e2}}\textcolor[rgb]{0,0,1}{::}\textcolor[rgb]{0,0,1}{\mathrm{module}}
\mathrm{exports}\left(m,'\mathrm{string}','\mathrm{typed}'\right)
\textcolor[rgb]{0,0,1}{"e1"}\textcolor[rgb]{0,0,1}{::}\textcolor[rgb]{0,0,1}{\mathrm{Typesetting}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{_Hold}}\textcolor[rgb]{0,0,1}{}\left(\left[\textcolor[rgb]{0,0,1}{'}\textcolor[rgb]{0,0,1}{\mathrm{integer}}\textcolor[rgb]{0,0,1}{'}\right]\right)\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"e2"}\textcolor[rgb]{0,0,1}{::}\textcolor[rgb]{0,0,1}{\mathrm{module}}
\mathrm{exports}\left(m,'\mathrm{type}'='\mathrm{posint}','\mathrm{typed}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{e1}}\textcolor[rgb]{0,0,1}{::}\textcolor[rgb]{0,0,1}{\mathrm{Typesetting}}\textcolor[rgb]{0,0,1}{:-}\textcolor[rgb]{0,0,1}{\mathrm{_Hold}}\textcolor[rgb]{0,0,1}{}\left(\left[\textcolor[rgb]{0,0,1}{'}\textcolor[rgb]{0,0,1}{\mathrm{integer}}\textcolor[rgb]{0,0,1}{'}\right]\right)
The exports command was updated in Maple 2021.
The all, type and method options were introduced in Maple 2021.
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Inflation Schedule | Solana Docs
Subject to change. Follow most recent economic discussions in the Solana forums: https://forums.solana.com
Validator-clients have two functional roles in the Solana network:
Validate (vote) the current global state of their observed PoH.
Be elected as ‘leader’ on a stake-weighted round-robin schedule during which time they are responsible for collecting outstanding transactions and incorporating them into their observed PoH, thus updating the global state of the network and providing chain continuity.
Validator-client rewards for these services are to be distributed at the end of each Solana epoch. As previously discussed, compensation for validator-clients is provided via a commission charged on the protocol-based annual inflation rate dispersed in proportion to the stake-weight of each validator-node (see below) along with leader-claimed transaction fees available during each leader rotation. I.e. during the time a given validator-client is elected as leader, it has the opportunity to keep a portion of each transaction fee, less a protocol-specified amount that is destroyed (see Validation-client State Transaction Fees).
The effective protocol-based annual staking yield (%) per epoch received by validation-clients is to be a function of:
the current global inflation rate, derived from the pre-determined dis-inflationary issuance schedule (see Validation-client Economics)
the fraction of staked SOLs out of the current total circulating supply,
the commission charged by the validation service,
the up-time/participation [% of available slots that validator had opportunity to vote on] of a given validator over the previous epoch.
The first factor is a function of protocol parameters only (i.e. independent of validator behavior in a given epoch) and results in an inflation schedule designed to incentivize early participation, provide clear monetary stability and provide optimal security in the network.
As a first step to understanding the impact of the Inflation Schedule on the Solana economy, we’ve simulated the upper and lower ranges of what token issuance over time might look like given the current ranges of Inflation Schedule parameters under study.
Initial Inflation Rate: 7-9%
Dis-inflation Rate: -14-16%
Long-term Inflation Rate: 1-2%
Using these ranges to simulate a number of possible Inflation Schedules, we can explore inflation over time:
In the above graph, the average values of the range are identified to illustrate the contribution of each parameter. From these simulated Inflation Schedules, we can also project ranges for token issuance over time.
Finally we can estimate the Staked Yield on staked SOL, if we introduce an additional parameter, previously discussed, % of Staked SOL:
\%~\text{SOL Staked} = \frac{\text{Total SOL Staked}}{\text{Total Current Supply}}
In this case, because % of Staked SOL is a parameter that must be estimated (unlike the Inflation Schedule parameters), it is easier to use specific Inflation Schedule parameters and explore a range of % of Staked SOL. For the below example, we’ve chosen the middle of the parameter ranges explored above:
Dis-inflation Rate: -15%
The values of % of Staked SOL range from 60% - 90%, which we feel covers the likely range we expect to observe, based on feedback from the investor and validator communities as well as what is observed on comparable Proof-of-Stake protocols.
Again, the above shows an example Staked Yield that a staker might expect over time on the Solana network with the Inflation Schedule as specified. This is an idealized Staked Yield as it neglects validator uptime impact on rewards, validator commissions, potential yield throttling and potential slashing incidents. It additionally ignores that % of Staked SOL is dynamic by design - the economic incentives set up by this Inflation Schedule.
Adjusted Staking Yield#
A complete appraisal of earning potential from staking tokens should take into account staked Token Dilution and its impact on staking yield. For this, we define adjusted staking yield as the change in fractional token supply ownership of staked tokens due to the distribution of inflation issuance. I.e. the positive dilutive effects of inflation.
We can examine the adjusted staking yield as a function of the inflation rate and the percent of staked tokens on the network. We can see this plotted for various staking fractions here:
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Formaldehyde Exposure, Health Symptoms and Risk Assessment among Hospital Workers in Malaysia
Sharifah Mazrah Sayed Mohamed Zain* , Wan Nurul Farah Wan Azmi, Yuvaneswary Veloo, Rafiza Shaharudin
Environmental Health Research Centre, Institute for Medical Research, Ministry of Health Malaysia, Shah Alam, Malaysia
Formaldehyde is a chemical commonly used in hospitals as a tissue preservative; histopathology laboratory personnel are therefore among the workers most heavily exposed to formaldehyde. This study measured the formaldehyde exposure through ambient and personal air sampling, assessed the symptoms of poor health as well as estimating the health risk among hospital workers. We conducted a comparative cross-sectional study of both histopathology laboratory (exposed) and administration (nonexposed) workers in four hospitals in the Klang Valley, Selangor, Malaysia. Ambient and personal exposure to formaldehyde was measured using the OSHA 52 and NIOSH 2541 methods, respectively. The 8-hr time-weighted-average formaldehyde concentration was higher in exposed areas (0.25 ± 0.11 ppm) than nonexposed areas (0.08 ± 0.02 ppm). Histopathology workers were exposed to between 140% and 480% higher concentrations of formaldehyde than administration workers. Personal exposure was highest during grossing tasks (0.797 ± 0.436 ppm). A total of 67% of the exposed workers exhibited the same ten health symptoms related to formaldehyde exposure, and 57% of the nonexposed workers reported similar symptoms at their current workplace. Notably, symptoms of eye irritation, headache, drowsiness, and chest tightness were significantly more prevalent (p < 0.05; chi square and Fisher’s exact tests) among the exposed workers than the nonexposed workers. Among those with symptoms, 37% of the exposed workers, and 16% of the nonexposed workers believed that the symptoms were related to their current working environment. The noncancer effect of formaldehyde from air inhalation poses a potential risk of eye irritation among exposed workers. The cancer risk was not significant in both groups. Formaldehyde levels and symptoms of poor health were significantly higher among the exposed group. Exposure and risk could be minimised by strengthening control measures to improve indoor air quality in the workplace.
Healthcare Workers, Histopathology Laboratory, Health Risk Estimation, Occupational Exposure
\text{EC}=\frac{\text{C}\times \text{ET}\times \text{EF}\times \text{ED}}{\text{AT}}
\text{HQ}=\frac{\text{EC}}{\text{RfC}\times 1000}
\text{ELCR}=\text{EC}\times \text{IUR}
Zain, S.M.S.M., Azmi, W.N.F.W., Veloo, Y. and Shaharudin, R. (2019) Formaldehyde Exposure, Health Symptoms and Risk Assessment among Hospital Workers in Malaysia. Journal of Environmental Protection, 10, 861-879. https://doi.org/10.4236/jep.2019.106051
1. Bono, R., Romanazzi, V., Pirro, V., Degan, R., Pignata, C., Suppo, E., et al. (2012) Formaldehyde and Tobacco as Alkylating Agents: The Formation of N-Methylenvaline in Pathologists and in Plastic Laminate Workers. Science of the Total Environment, 414, 701-707. https://doi.org/10.1016/j.scitotenv.2011.10.047
2. Xu, Y. and Zhang, Y.-P. (2003) An Improved Mass Transfer Based Model for Analyzing VOC Emission from Building Materials. Atmospheric Environment, 37, 2497-2250. https://doi.org/10.1016/S1352-2310(03)00160-2
3. World Health Organization, Regional Office for Europe (2010) WHO Guidelines for Indoor Air Quality: Selected Pollutants. World Health Organization. Regional Office for Europe. http://www.who.int/iris/handle/10665/260127
4. Jerusalem, J.G., and Galarpe, V.R.K.R. (2015) Determination of Formaldehyde in Air in Selected Hospital Histopathology-Laboratories in Cagayan de Oro, Philippines. Journal of Chemical Health & Safety, 22, 10-14. https://doi.org/10.1016/j.jchas.2014.07.012
5. Norback, D., Hashim, J.H., Hashim, Z. and Ali, F. (2017) Volatile Organic Compound (VOC), Formaldehyde and Nitrogen Dioxide (NO2) in Schools in Johor Bahru, Malaysia: Association with Rhinitis, Ocular, Throat and Dermal Symptoms, Headache and Fatigue. Science of the Total Environment, 592, 153-160. https://doi.org/10.1016/j.scitotenv.2017.02.215
6. Ya’acob, S.H., Suis, A.J., Awang, N. and Sahani, M. (2013) Exposure Assessment of Formaldehyde and Its Symptoms among Anatomy Laboratory Workers and Medical Students. Asian Journal of Applied Sciences, 6, 50-55. https://doi.org/10.3923/ajaps.2013.50.55
7. Horvath, E.P., Anderson, J., Pierce, W.E., Hanrahan, L. and Wendlick, J.D. (1988) Effects of Formaldehyde on the Mucous Membranes and Lungs: A Study of an Industrial Population. Journal of the American Medical Association, 259, 701-707. https://doi.org/10.1001/jama.1988.03720050037020
8. World Health Organization (WHO) (1989) Environmental Health Criteria for Formaldehyde: Vol. 89. Geneva, Switzerland.
9. Agency for Toxic Substances and Disease Registry (ATSDR) (1997) Toxicological Profile for Formaldehyde (Draft). Public Health Services, United States Department of Health and Human Services, Atlanta, GA.
10. Feron, V.J., Arts, J.H., Kuper, C.F., Slootweg, P.J. and Woutersen, R.A. (2001) Health Risks Associated with Inhaled Nasal Toxicants. Critical Reviews in Toxicology, 31, 313-347. https://doi.org/10.1080/20014091111712
11. Kriebel, D., Sama, S.R. and Cocanour, B. (1993) Reversible Pulmonary Responses to Formaldehyde. A Study of Clinical Anatomy Students. American Review of Respiratory Disease, 148, 1509-1515. https://doi.org/10.1164/ajrccm/148.6_Pt_1.1509
12. Takigawa, T., Usami, M., Yamasaki, Y., Wang, B., Sakano, N., Horike, T., et al. (2005) Reduction of Indoor Formaldehyde Concentrations and Subjective Symptoms in a Gross Anatomy Laboratory. Bulletin of Environmental Contamination and Toxicology, 74, 1027-1033. https://doi.org/10.1007/s00128-005-0683-2
13. Takahashi, S., Tsuji, K., Fujii, K., Okazaki, F., Takigawa, T., Ohtsuka, A., et al. (2007) Prospective Study of Clinical Symptoms and Skin Test Reactions in Medical Students Exposed to Formaldehyde Gas. The Journal of Dermatology, 34, 283-289. https://doi.org/10.1111/j.1346-8138.2007.00274.x
14. Ghasmenkhani, M., Jahanpeyma, F. and Azam, K. (2005) Formaldehyde Exposure in Some Educational Hospitals of Tehran. Industrial Health, 43, 703-707.
15. Orsière, T., Sari-Minodier, I., Iarmarcovai, G. and Botta, A. (2006) Genotoxic Risk Assessment of Pathology and Anatomy Laboratory Workers Exposed to Formaldehyde by the Use of Personal Air Sampling and Analysis of DNA Damage in Peripheral Lymphocytes. Mutation Research/Genetic Toxicology and Environmental Mutagenesis, 605, 30-41. https://doi.org/10.1016/j.mrgentox.2006.01.006
16. Doty, R.L., Cometto-Muniz, J.E., Jalowayski, A.A., Dalton, P., Kendel-Reed, M. and Hodqson, M. (2004) Assessment of Upper Respiratory Tract and Ocular Irritative Effects of Volatile Chemicals in Humans. Critical Reviews in Toxicology, 34, 85-142. https://doi.org/10.1080/10408440490269586
17. Agency for Toxic Substances and Disease Registry (ATSDR) (1987) Toxicological Profile for Formaldehyde (Draft). Public Health Services, United States Department of Health and Human Services, Atlanta, GA.
18. IARC Working Group on the Evaluation of Carcinogenic Risk to Humans (2006) Formaldehyde, 2-Butoxyethanol and 0-Tert-Butoxypropan-2-Ol. International Agency for Research on Cancer, Lyon, France.
19. NIOSH Bulletin (1981) Current Intelligence Bulletin 34. U.S. Department of Health and Human Services, Public Health Service, Center for Disease Control.
20. Walrath, J. and Fraumeni, J.F. (1984) Cancer and Other Causes of Death among Embalmers. Cancer Research, 44, 4638-4641.
21. Harrington, J.M. and Shannon, H. (1975) Mortality Study of Pathologists and Medical Laboratory Technicians. British Medical Journal, 4, 329-332. https://doi.org/10.1136/bmj.4.5992.329
22. Stellman, S.D., Demers, P.A., Colin, D. and Boffetta, P. (1988) Cancer Mortality and Wood Dust Exposure among Participants in the American Cancer Society Cancer Prevention Study-II (CPS-II). American Journal of Industrial Medicine, 34, 229-237. https://doi.org/10.1002/(SICI)1097-0274(199809)34:3<229::AID-AJIM4>3.0.CO;2-Q
23. Beane-Freeman, L.E., Blair, A., Lubin, J.H., Stewart, P.A., Hayes, R.B., Hoover, R.N., et al. (2009) Mortality from Lymphohematopoietic Malignancies among Workers in Formaldehyde Industries: The National Cancer Institute Cohort. Journal of the National Cancer Institute, 101, 751-761. https://doi.org/10.1093/jnci/djp096
24. Gardner, M.J., Pannett, B., Winter, P.D. and Cruddes, A.M. (1993) A Cohort Study of Workers Exposed to Formaldehyde in the British Chemical Industry: An Update. British Journal of Industrial Medicine, 50, 827-834. https://doi.org/10.1136/oem.50.9.827
25. Coggon, D., Harris, E.C., Poole, J. and Palmer, K.T. (2003) Extended Follow-Up of a Cohort of British Chemical Workers Exposed to Formaldehyde. Journal of the National Cancer Institute, 95, 1608-1615. https://doi.org/10.1093/jnci/djg046
26. United States Occupational Safety and Health Administration (1993) Occupational Safety and Health Standards, Toxic and Hazardous Substances, Standard Number 1910-1048 (c). United States Department of Labor, Washington DC.
27. Malaysia Occupational Safety and Health Act and Regulations Act 514 (1994) Use and Standards of Exposure of Chemicals Hazardous to Health Regulations 2000. MDC Publishers Sdn Bhd, Kuala Lumpur, Malaysia.
28. Wilhelmsson, B. and Holmstrom, M (1992) Possible Mechanisms of Formaldehyde-Induced Discomfort in the Upper Airways. Scandinavian Journal of Work, Environment & Health, 18, 403-407. https://doi.org/10.5271/sjweh.1556
29. Emory University (2012) Formaldehyde Questionnaire. Environmental Health and Safety Office, Georgia.
30. United States Occupational Safety and Health Administration (1993) Non-Mandatory Medical Disease Questionnaire, Occupational Safety and Health Standards, Toxic and Hazardous Substances, Standard Number 1910.1048 App D. United States Department of Labor, Washington DC.
31. Industry Code of Practice on Indoor Air (2010) Questionnaire for Building Occupants. Quality Ministry of Human Resources, Department of Occupational Safety and Health, Malaysia.
32. United States Environmental Protection Agency (USEPA) (1989) Risk Assessment Guidance for Superfund Volume I: Human Health Evaluation Manual (Part A). Washington DC.
33. United States Environmental Protection Agency (USEPA) (2009) Risk Assessment Guidance for Superfund Volume I: Human Health Evaluation Manual (Part F, Supplemental Guidance for Inhalation Risk Assessment). Washington DC.
34. United States Environmental Protection Agency (USEPA) (2010) IRIS Toxicological Review of Formaldehyde (Inhalation) (External Review Draft 2010). Washington DC, EPA/635/R-10/002A.
35. Liu, K.S., Huang, F.Y., Hayward, S.B., Wesolowski, J. and Sexton, K. (1991) Irritant Effects of Formaldehyde Exposure in Mobile Homes. Environmental Health Perspectives, 94, 91-94. https://doi.org/10.1289/ehp.94-1567965
36. National Research Council (2011) Review of the Environmental Protection Agency’s Draft IRIS Assessment of Formaldehyde. Reference Concentrations for Non-Cancer Effects and Unit Risks for Cancers. The National Academies Press, Washington DC.
37. United States Environmental Protection Agency (USEPA) (1986) Guidelines for Carcinogen Risk Assessment, Federal Regulation. United States Environmental Protection Agency, Washington DC.
38. World Health Organization (WHO) (2000) World Health Organization Guidelines for Air Quality. Geneva, Switzerland.
39. Woodruff, T.J., Caldwell, J., Cogliano, V.J. and Axelrad, D.A. (1990) Estimating Cancer Risk from Outdoor Concentrations of Hazardous Air Pollutants in 1990. Environmental Research, 82, 194-206. https://doi.org/10.1006/enrs.1999.4021
40. Morello-Frosch, R.A., Woodruff, T.J., Axelrad, D.A. and Caldwell, J.C. (2005) Air Toxic and Health Risk in California: The Public Health Implications of Outdoor Concentrations. Risk Analysis, 20, 273-292. https://doi.org/10.1111/0272-4332.202026
41. United States Environmental Protection Agency (USEPA) (2005) Guidelines for Carcinogen Risk Assessment, Risk Assessment Forum. United States Environmental Protection Agency, Washington DC.
42. Industry Code of Practice on Indoor Air Quality (2010) Table I: List of Indoor Air Contaminants and the Maximum Limits. Ministry of Human Resources, Department of Occupational Safety and Health, Malaysia.
43. Salthammer, T., Mentese, S. and Marutzky, R. (2010) Formaldehyde in the Indoor Environment. Chemical Reviews, 110, 2536-2572. https://doi.org/10.1021/cr800399g
44. Du, Z.-J., Mo, J.-H., Zhang, Y.-P. and Xu, Q.-J. (2014) Benzene, Toluene and Xylenes in Newly Renovated Homes and Associated Health Risk in Guangzhou, China. Building Environment, 72, 75-81. https://doi.org/10.1016/j.buildenv.2013.10.013
45. Ladeira, C., Viegas, S., Carolino, E., Prsista, J., Gomes, M.C. and Brito, M. (2011) Genotoxicity Biomarker in Occupational Exposure to Formaldehyde—The Case of Histopathology Laboratories. Mutation Research/Genetic Toxicology and Environmental Mutagenesis, 721, 15-20. https://doi.org/10.1016/j.mrgentox.2010.11.015
46. Ahmed, H.O. (2011) Preliminary Study: Formaldehyde Exposure in Laboratories of Sharjah University in UAE. Indian Journal of Occupational & Environmental Medicine, 15, 33-37. https://doi.org/10.4103/0019-5278.82997
47. Xu, W. and Stewart, E.J. (2016) A Comparison of Engineering Controls for Formaldehyde Exposure during Grossing Activities in Health Care Anatomic Pathology Laboratories. Journal of Occupational and Environmental Hygiene, 13, 529-537. https://doi.org/10.1080/15459624.2016.1149182
48. Ogawa, M., Kabe, I., Terauchi, Y. and Tanaka, S. (2018) A Strategy for the Reduction of Formaldehyde Concentration in a Hospital Pathology Laboratory. Journal of Occupational Health, 61, 135-142. https://doi.org/10.1002/1348-9585.12018
49. Roland, O.W., Douglas, K.E. and Akaranta, O. (2019) Comparative Assessment of Formaldehyde Concentrations in Public and Private Mortuaries in Rivers State, Nigeria. Journal of Scientific Research & Reports, 23, 1-11. https://doi.org/10.9734/jsrr/2019/v23i230115
50. Azari, M.R., Asadi, P., Jafari, M.J., Soori, H. and Hosseini, V. (2012) Occupational Exposure of a Medical School Staff to Formaldehyde in Tehran. Tanaffos, 11, 36-41.
51. Amoore, J.E. and Hautala, E. (1983) Odor as an Ald to Chemical Safety: Odor Thresholds Compared with Threshold Limit Values and Volatilities for 214 Industrial Chemicals in Air and Water Dilution. Journal of Applied Toxicology, 3, 272-290. https://doi.org/10.1002/jat.2550030603
52. Harvey, C., Linda, D.D., Boffetta, P., Gallagher, A.E., Crawford, L., Lees, P.J.L., et al. (2015) Formaldehyde Exposure and Mortality Risks from Acute Myeloid Leukemia and Other Lymphohematopoietic Malignancies in the Us National Cancer Institute Cohort Study of Workers in Formaldehyde Industries. Journal of Occupational and Environmental Medicine, 57, 785-794. https://doi.org/10.1097/JOM.0000000000000466
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This problem is a checkpoint for division of fractions and decimals. It will be referred to as Checkpoint 8B.
\frac { 3 } { 8 } \div \frac { 1 } { 2 }
\frac { 1 } { 3 }÷ 4
1 \frac { 1 } { 2 } \div \frac { 1 } { 6 }
\frac { 7 } { 8 } \div 1 \frac { 1 } { 4 }
27.42 ÷ 1.2
19.5 ÷ 0.025
Answers and extra practice for the Checkpoint problems are located in the back of your printed textbook or in the Reference Tab of your eBook. If you have an eBook for Core Connections, Course 1, login and then click the following link: Checkpoint 8B: Division of Fractions and Decimals
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Home : Support : Online Help : Mathematics : Linear Algebra : LinearAlgebra Package : VectorSpaces : ColumnSpace
The RowSpace(A) (ColumnSpace(A)) function returns a list of row (column) Vectors that form a basis for the Vector space spanned by the rows (columns) of Matrix A. The Vectors are returned in canonical form with leading entries 1.
This function is part of the LinearAlgebra package, and so it can be used in the form RowSpace(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[RowSpace](..).
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
A≔〈〈1,2,0〉|〈0,2,6〉|〈0,0,4〉|〈0,0,0〉〉
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{6}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{0}\end{array}]
\mathrm{RowSpace}\left(A\right)
[[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{0}\end{array}]]
\mathrm{ColumnSpace}\left(A\right)
[[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}\end{array}]]
\mathrm{RowSpace}\left(〈〈0,0〉|〈0,0〉〉\right)
[]
B≔〈〈x,0〉|〈y,1〉〉
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{x}& \textcolor[rgb]{0,0,1}{y}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{ColumnSpace}\left(B\right)
[[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{0}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{1}\end{array}]]
C≔〈〈\frac{15}{4},-\frac{3}{4}\mathrm{sqrt}\left(10\right),\frac{1}{4}\mathrm{sqrt}\left(165\right)〉|〈-\frac{3}{4}\mathrm{sqrt}\left(10\right),\frac{3}{2},-\frac{1}{4}\mathrm{sqrt}\left(66\right)〉|〈\frac{1}{4}\mathrm{sqrt}\left(165\right),-\frac{1}{4}\mathrm{sqrt}\left(66\right),\frac{11}{4}〉〉
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\frac{\textcolor[rgb]{0,0,1}{15}}{\textcolor[rgb]{0,0,1}{4}}& \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{10}}}{\textcolor[rgb]{0,0,1}{4}}& \frac{\sqrt{\textcolor[rgb]{0,0,1}{165}}}{\textcolor[rgb]{0,0,1}{4}}\\ \textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}\sqrt{\textcolor[rgb]{0,0,1}{10}}}{\textcolor[rgb]{0,0,1}{4}}& \frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}& \textcolor[rgb]{0,0,1}{-}\frac{\sqrt{\textcolor[rgb]{0,0,1}{66}}}{\textcolor[rgb]{0,0,1}{4}}\\ \frac{\sqrt{\textcolor[rgb]{0,0,1}{165}}}{\textcolor[rgb]{0,0,1}{4}}& \textcolor[rgb]{0,0,1}{-}\frac{\sqrt{\textcolor[rgb]{0,0,1}{66}}}{\textcolor[rgb]{0,0,1}{4}}& \frac{\textcolor[rgb]{0,0,1}{11}}{\textcolor[rgb]{0,0,1}{4}}\end{array}]
\mathrm{Normalizer}≔\mathrm{radnormal}
\textcolor[rgb]{0,0,1}{\mathrm{Normalizer}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{radnormal}}
\mathrm{ColumnSpace}\left(C\right)
[[\begin{array}{c}\textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{-}\frac{\sqrt{\textcolor[rgb]{0,0,1}{10}}}{\textcolor[rgb]{0,0,1}{5}}\\ \frac{\sqrt{\textcolor[rgb]{0,0,1}{165}}}{\textcolor[rgb]{0,0,1}{15}}\end{array}]]
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27.3: Examples of Problematic Model Fit - Statistics LibreTexts
https://stats.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fstats.libretexts.org%2FBookshelves%2FIntroductory_Statistics%2FBook%253A_Statistical_Thinking_for_the_21st_Century_(Poldrack)%2F27%253A_The_General_Linear_Model_in_R%2F27.03%253A_Examples_of_Problematic_Model_Fit
## (Intercept) 10.978 0.117 93.65 <2e-16 ***
## x 0.270 0.119 2.27 0.025 *
## F-statistic: 5.17 on 1 and 98 DF, p-value: 0.0252
## (Intercept) 10.5547 0.0844 125.07 <2e-16 ***
## x -0.0419 0.0854 -0.49 0.62
Now we see that there is no significant linear relationship between
X
and Y/ But if we look at the residuals the problem with the model becomes clear:
## x -0.0118 0.0600 -0.2 0.84
## x_squared 0.4557 0.0451 10.1 <2e-16 ***
Now we see that the effect of
X
is significant, and if we look at the residual plot we should see that things look much better:
27.3: Examples of Problematic Model Fit is shared under a not declared license and was authored, remixed, and/or curated by Russell A. Poldrack via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.
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Products of Interest. Computer Music Journal 2020; 44 (2-3): 167–187. doi: https://doi.org/10.1162/comj_r_00566
Grid is a set of portable, modular MIDI controllers from Hungarian company Intech Studio (see Figure 1). There are four controllers currently available with plans to expand the offerings in the future. The PO16 is a 4
×
4 grid of 16 potentiometers. The BU16 module features a 4
×
4 grid of tactile buttons, which can be set to toggle or momentary action. The EN16A is a 4
×
4 grid of 16 push-button encoders. The PBF4 controller is a mixing module with four 30-mm faders, four potentiometers, and four buttons.
Each of the GRID controllers measure 106.6
×
×
32 mm and weigh 250 g. They have an aluminum front panel with programmable RGB LEDs for all controls. Each module has a USB-C connector for power and data. They also have magnetic connectors on all four sides of the module that are used to connect them...
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(Redirected from BLAST)
Bioinformatics search algorithm
This article is about the bioinformatics software tool. For other uses, see Blast (disambiguation).
Stephen Altschul, Warren Gish, Webb Miller, Eugene Myers, and David Lipman
2.12.0+ / 28 June 2021; 10 months ago (2021-06-28)
C and C++[1]
UNIX, Linux, Mac, MS-Windows
3.1 Parallel BLAST
4.2 Accelerated versions
5 Alternatives to BLAST
5.1 Comparing BLAST and the Smith-Waterman Process
6 BLAST output visualization
7 Uses of BLAST
BLAST is available on the web on the NCBI website. Different types of BLASTs are available according to the query sequences and the target databases. Alternative implementations include AB-BLAST (formerly known as WU-BLAST), FSA-BLAST (last updated in 2006), and ScalaBLAST.[8][9]
The original paper by Altschul, et al.[7] was the most highly cited paper published in the 1990s.[10]
Fig. 1 The method to establish the k-letter query word list.[13]
Fig. 2 The process to extend the exact match. Adapted from Biological Sequence Analysis I, Current Topics in Genome Analysis [2].
Fig. 3 The positions of the exact matches.
{\displaystyle p\left(S\geq x\right)=1-\exp \left(-e^{-\lambda \left(x-\mu \right)}\right)}
{\displaystyle \mu ={\frac {\log \left(Km'n'\right)}{\lambda }}\;}
The statistical parameters
{\displaystyle \lambda }
{\displaystyle \mathrm {K} }
are estimated by fitting the distribution of the un-gapped local alignment scores, of the query sequence and a lot of shuffled versions (Global or local shuffling) of a database sequence, to the Gumbel extreme value distribution. Note that
{\displaystyle \lambda }
{\displaystyle \mathrm {K} }
depend upon the substitution matrix, gap penalties, and sequence composition (the letter frequencies).
{\displaystyle m'}
{\displaystyle n'}
are the effective lengths of the query and database sequences, respectively. The original sequence length is shortened to the effective length to compensate for the edge effect (an alignment start near the end of one of the query or database sequence is likely not to have enough sequence to build an optimal alignment). They can be calculated as
{\displaystyle m'\approx m-{\frac {\ln Kmn}{H}}\;}
{\displaystyle n'\approx n-{\frac {\ln Kmn}{H}}\;}
{\displaystyle \mathrm {H} }
is the average expected score per aligned pair of residues in an alignment of two random sequences. Altschul and Gish gave the typical values,
{\displaystyle \lambda =0.318}
{\displaystyle \mathrm {K} =0.13}
{\displaystyle \mathrm {H} =0.40}
, for un-gapped local alignment using BLOSUM62 as the substitution matrix. Using the typical values for assessing the significance is called the lookup table method; it is not accurate. The expect score E of a database match is the number of times that an unrelated database sequence would obtain a score S higher than x by chance. The expectation E obtained in a search for a database of D sequences is given by
{\displaystyle E\approx 1-e^{-p\left(s>x\right)D}}
{\displaystyle p<0.1}
, E could be approximated by the Poisson distribution as
{\displaystyle E\approx pD}
Parallel BLAST[edit]
This program, given a DNA query, returns the most similar DNA sequences from the DNA database that the user specifies.
Protein-protein BLAST (blastp)
This program, given a protein query, returns the most similar protein sequences from the protein database that the user specifies.
Position-Specific Iterative BLAST (PSI-BLAST) (blastpgp)
This program is used to find distant relatives of a protein. First, a list of all closely related proteins is created. These proteins are combined into a general "profile" sequence, which summarises significant features present in these sequences. A query against the protein database is then run using this profile, and a larger group of proteins is found. This larger group is used to construct another profile, and the process is repeated.
This program compares the six-frame conceptual translation products of a nucleotide query sequence (both strands) against a protein sequence database to find a protein-coding gene in a genomic sequence or to see if the cDNA corresponds to a known protein.
Nucleotide 6-frame translation-nucleotide 6-frame translation (tblastx)
This program is the slowest of the BLAST family. It translates the query nucleotide sequence in all six possible frames and compares it against the six-frame translations of a nucleotide sequence database. The purpose of tblastx is to find very distant relationships between nucleotide sequences.
This program compares a protein query against the all six reading frames of a nucleotide sequence database. It may be used to map a protein to genomic DNA.
Large numbers of query sequences (megablast)
When comparing large numbers of input sequences via the command-line BLAST, "megablast" is much faster than running BLAST multiple times. It concatenates many input sequences together to form a large sequence before searching the BLAST database, then post-analyzes the search results to glean individual alignments and statistical values.
Alternatives to BLAST[edit]
Comparing BLAST and the Smith-Waterman Process[edit]
BLAST output visualization[edit]
Fig. 4 Circos-style visualisation of BLAST results generated using SequenceServer software.
Fig. 5 Length distribution of BLAST hits generated using SequenceServer software showing that the query (a predicted gene product) is longer compared to similar database sequences.
Uses of BLAST[edit]
With the use of BLAST, you can possibly correctly identify a species or find homologous species. This can be useful, for example, when you are working with a DNA sequence from an unknown species.
Using the results received through BLAST you can create a phylogenetic tree using the BLAST web-page. Phylogenies based on BLAST alone are less reliable than other purpose-built computational phylogenetic methods, so should only be relied upon for "first pass" phylogenetic analyses.
When working with a known species, and looking to sequence a gene at an unknown location, BLAST can compare the chromosomal position of the sequence of interest, to relevant sequences in the database(s). NCBI has a "Magic-BLAST" tool built around BLAST for this purpose.[31]
^ "BLAST Developer Information". blast.ncbi.nlm.nih.gov.
^ a b c Douglas Martin (21 February 2008). "Samuel Karlin, Versatile Mathematician, Dies at 83". The New York Times.
^ R. M. Casey (2005). "BLAST Sequences Aid in Genomics and Proteomics". Business Intelligence Network.
^ Lipman, DJ; Pearson, WR (1985). "Rapid and sensitive protein similarity searches". Science. 227 (4693): 1435–41. Bibcode:1985Sci...227.1435L. doi:10.1126/science.2983426. PMID 2983426.
^ "BLAST topics".
^ Dan Stober (January 16, 2008). "Sam Karlin, mathematician who improved DNA analysis, dead at 83". Stanford.edu.
^ a b Stephen Altschul; Warren Gish; Webb Miller; Eugene Myers; David J. Lipman (1990). "Basic local alignment search tool". Journal of Molecular Biology. 215 (3): 403–410. doi:10.1016/S0022-2836(05)80360-2. PMID 2231712.
^ Oehmen, C.; Nieplocha, J. (2006). "ScalaBLAST: A Scalable Implementation of BLAST for High-Performance Data-Intensive Bioinformatics Analysis". IEEE Transactions on Parallel and Distributed Systems. 17 (8): 740. doi:10.1109/TPDS.2006.112. S2CID 11122366.
^ Oehmen, C. S.; Baxter, D. J. (2013). "ScalaBLAST 2.0: Rapid and robust BLAST calculations on multiprocessor systems". Bioinformatics. 29 (6): 797–798. doi:10.1093/bioinformatics/btt013. PMC 3597145. PMID 23361326.
^ "Sense from Sequences: Stephen F. Altschul on Bettering BLAST". ScienceWatch. July–August 2000. Archived from the original on 7 October 2007.
^ Steven Henikoff; Jorja Henikoff (1992). "Amino Acid Substitution Matrices from Protein Blocks". PNAS. 89 (22): 10915–10919. Bibcode:1992PNAS...8910915H. doi:10.1073/pnas.89.22.10915. PMC 50453. PMID 1438297.
^ Mount, D. W. (2004). Bioinformatics: Sequence and Genome Analysis (2nd ed.). Cold Spring Harbor Press. ISBN 978-0-87969-712-9.
^ Adapted from Biological Sequence Analysis I, Current Topics in Genome Analysis [1].
^ Yim, WC; Cushman, JC (2017). "Divide and Conquer (DC) BLAST: fast and easy BLAST execution within HPC environments". PeerJ. 5: e3486. doi:10.7717/peerj.3486. PMC 5483034. PMID 28652936.
^ "Program Selection Tables of the Blast NCBI web site".
^ "Which BLAST program should I use?". resources.qiagenbioinformatics.com. Retrieved 18 January 2022.
^ Camacho, C.; Coulouris, G.; Avagyan, V.; Ma, N.; Papadopoulos, J.; Bealer, K.; Madden, T. L. (2009). "BLAST+: Architecture and applications". BMC Bioinformatics. 10: 421. doi:10.1186/1471-2105-10-421. PMC 2803857. PMID 20003500.
^ Vouzis, P. D.; Sahinidis, N. V. (2010). "GPU-BLAST: using graphics processors to accelerate protein sequence alignment". Bioinformatics. 27 (2): 182–8. doi:10.1093/bioinformatics/btq644. PMC 3018811. PMID 21088027.
^ Liu W, Schmidt B, Müller-Wittig W (2011). "CUDA-BLASTP: accelerating BLASTP on CUDA-enabled graphics hardware". IEEE/ACM Trans Comput Biol Bioinform. 8 (6): 1678–84. doi:10.1109/TCBB.2011.33. PMID 21339531. S2CID 18221547.
^ Zhao K, Chu X (May 2014). "G-BLASTN: accelerating nucleotide alignment by graphics processors". Bioinformatics. 30 (10): 1384–91. doi:10.1093/bioinformatics/btu047. PMID 24463183.
^ Loh PR, Baym M, Berger B (July 2012). "Compressive genomics". Nat. Biotechnol. 30 (7): 627–30. doi:10.1038/nbt.2241. PMID 22781691.
^ Madden, Tom; Boratyn, Greg (2017). "QuickBLASTP: Faster Protein Alignments" (PDF). Proceedings of NIH Research Festival. Retrieved 16 May 2019. Abstract page
^ Kent, W. James (2002-04-01). "BLAT—The BLAST-Like Alignment Tool". Genome Research. 12 (4): 656–664. doi:10.1101/gr.229202. ISSN 1088-9051. PMC 187518. PMID 11932250.
^ Lavenier, D.; Lavenier, Dominique (2009). "PLAST: parallel local alignment search tool for database comparison". BMC Bioinformatics. 10: 329. doi:10.1186/1471-2105-10-329. PMC 2770072. PMID 19821978.
^ Lavenier, D. (2009). "Ordered index seed algorithm for intensive DNA sequence comparison" (PDF). 2008 IEEE International Symposium on Parallel and Distributed Processing (PDF). pp. 1–8. CiteSeerX 10.1.1.155.3633. doi:10.1109/IPDPS.2008.4536172. ISBN 978-1-4244-1693-6. S2CID 10804289.
^ Buchfink, Xie and Huson (2015). "Fast and sensitive protein alignment using DIAMOND". Nature Methods. 12 (1): 59–60. doi:10.1038/nmeth.3176. PMID 25402007. S2CID 5346781.
^ Steinegger, Martin; Soeding, Johannes (2017-10-16). "MMseqs2 enables sensitive protein sequence searching for the analysis of massive data sets". Nature Biotechnology. 35 (11): 1026–1028. doi:10.1038/nbt.3988. hdl:11858/00-001M-0000-002E-1967-3. PMID 29035372. S2CID 402352.
^ "Bioinformatics Explained: BLAST versus Smith-Waterman" (PDF). 4 July 2007.
^ Neumann, Kumar and Shalchian-Tabrizi (2014). "BLAST output visualization in the new sequencing era". Briefings in Bioinformatics. 15 (4): 484–503. doi:10.1093/bib/bbt009. PMID 23603091.
^ "NCBI Magic-BLAST". ncbi.github.io. Retrieved 16 May 2019.
Retrieved from "https://en.wikipedia.org/w/index.php?title=BLAST_(biotechnology)&oldid=1086970095"
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Fast Tool for Evaluation of Iliac Crest Tissue Elastic Properties Using the Reduced-Basis Methods | J. Biomech Eng. | ASME Digital Collection
, Block E1, No. 08-03, 9 Engineering Drive 1, 117576, Singapore
e-mail: bielt@nus.edu.sg
Revanth Reddy Garlapati,
Revanth Reddy Garlapati
, Block E3A, No. 07-15, 7 Engineering Drive 1, 117574, Singapore
e-mail: biegrr@nus.edu.sg
Kathy Lam,
e-mail: ktlam85@hotmail.com
Peter Vee Sin Lee,
Department of Mechanical Engineering, Melbourne School of Engineering,
e-mail: pvlee@unimelb.edu.au
Yoon-Sok Chung,
Department of Endocrinology and Metabolism, School of Medicine,
, Suwon 443-749, Korea
e-mail: yschung@ajou.ac.kr
Jae Bong Choi,
Jae Bong Choi
, 389 samsoon-dong 2-ga, Seongbuk-gu, Seoul, Korea
e-mail: jbchoi@hansung.ac.kr
Tan Beng Chye Vincent,
Tan Beng Chye Vincent
e-mail: mpetanbc@nus.edu.sg
Department of Orthopaedic Surgery, Yong Loo Lin School of Medicine,
e-mail: dosdasde@nus.edu.sg
Lee, T., Garlapati, R. R., Lam, K., Lee, P. V. S., Chung, Y., Choi, J. B., Vincent, T. B. C., and Das De, S. (November 9, 2010). "Fast Tool for Evaluation of Iliac Crest Tissue Elastic Properties Using the Reduced-Basis Methods." ASME. J Biomech Eng. December 2010; 132(12): 121009. https://doi.org/10.1115/1.4001254
Computationally expensive finite element (FE) methods are generally used for indirect evaluation of tissue mechanical properties of trabecular specimens, which is vital for fracture risk prediction in the elderly. This work presents the application of reduced-basis (RB) methods for rapid evaluation of simulation results. Three cylindrical transiliac crest specimens (diameter: 7.5 mm, length: 10–12 mm) were obtained from healthy subjects (20 year-old, 22 year-old, and 24 year-old females) and scanned using microcomputed tomography imaging. Cubic samples of dimensions
5×5×5 mm3
were extracted from the core of the cylindrical specimens for FE analysis. Subsequently, a FE solution library (test space) was constructed for each of the specimens by varying the material property parameters: tissue elastic modulus and Poisson’s ratio, to develop RB algorithms. The computational speed gain obtained by the RB methods and their accuracy relative to the FE analysis were evaluated. Speed gains greater than 4000 times, were obtained for all three specimens for a loss in accuracy of less than 1% in the maxima of von-Mises stress with respect to the FE-based value. The computational time decreased from more than 6 h to less than 18 s. RB algorithms can be successfully utilized for real-time reliable evaluation of trabecular bone elastic properties.
biological tissues, biomechanics, computerised tomography, elasticity, finite element analysis, fracture, iliac crest trabeculae, reduced-basis method, elastic property, computational speed gain, finite element methods
Biological tissues, Bone, Elastic moduli, Elasticity, Finite element analysis, Stress, Errors, Materials properties, Poisson ratio, Approximation, Fracture (Materials), Dimensions, Boundary-value problems
Heterogeneous Linear Elastic Trabecular Bone Modelling Using Micro-CT Attenuation Data and Experimentally Measured Heterogeneous Tissue Properties
Specimen-Specific Beam Models for Fast and Accurate Prediction Of Human Trabecular Bone Mechanical Properties
Regional Variations in the Apparent and Tissue-Level Mechanical Parameters of Vertebral Trabecular Bone With Aging Using Micro-Finite Element Analysis
Reduced-Basis Approximation and A Posteriori Error Bounds for Non-Affine and Nonlinear Partial Differential Equations: Application to Inverse Analysis
,” Ph.D. thesis, Singapore-MIT Alliance, National University of Singapore, Singapore.
Certified Real-Time Solution of the Parametrized Steady Incompressible Navier-Stokes Equations: Rigorous Reduced-Basis A Posteriori Error Bounds
Investigation of Bovine Bone by Resonant Ultrasound Spectroscopy and Transmission Ultrasound
Rapid Identification of Elastic Modulus of the Interface Tissue on Dental Implants Surfaces Using Reduced-Basis Method and a Neural Network
Certified Real-Time Solution of Parametrized Partial Differential Equations
Reduced Basis Approximation and A Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive Partial Differential Equations Application to Transport and Continuum Mechanics
Novel Approach of Predicting Fracture Load in the Human Proximal Femur Using Non-Invasive QCT Imaging Technique
Predicting Failure Load of the Femur With Simulated Osteolytic Defects Using Noninvasive Imaging Technique in a Simplified Load Case
Shape Design by Optimal Flow Control and Reduced Basis Techniques: Applications to Bypass Configurations in Haemodynamics
,” Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland.
The Quality of Trabecular Bone Evaluated With Micro-Computed Tomography, FEA and Mechanical Testing
Anisotropy of the Fatigue Behavior of Cancellous Bone
|
Quiz - Early Quantum Theory | Early Quantum Theory
What is the frequency of radiation which has an energy of 1.52 x 106 J per mole of photons?
n = number of photons = 1 mol = 6.022 x 1023 photons
Which statement below is incorrect regarding early quantum theory?
Bohr’s model of the atom accounts for the absorption and emission spectrum of hydrogen atom
Bohr’s model of the atom accounts for the Rydberg equation
Energy states in the hydrogen atom are quantized
Atomic Line Spectra The Line Spectrum of Hydrogen
What is the energy of one electron in the second orbital of a hydrogen atom?
-1.83 x 1018 J
\frac{- 2.1799 × {10}^{-18}}{{\mathrm{n}}^{2}}
En = energy of the orbital n
n = number of the orbital = 2
According to the Bohr model for the hydrogen atom, how is the energy required to excite an electron from n=3 to n=4 compared to the energy required to excite an electron from n=2 to n=3?
Either equal or greater
\frac{1}{{3}^{2}}
\frac{1}{{4}^{2}}
<
\frac{1}{{2}^{2}}
\frac{1}{{3}^{2}}
The Line Spectrum of Hydrogen
Wave model of light - 1
Photons & Photoelectric effect - 1
Wave-particle duality - 1
Alabama Requirements for Passing High School Chemistry
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Energy and All of Its Potentials
Predicting the Relative Properties of Elements
What are Patterns of Electrons?
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LaTeX/索引 - 维基教科书，自由的教学读本
LaTeX/索引
A useful feature of many books, index is an alphabetical list of words and expressions with the pages of the book upon which they are to be found. LaTeX supports the creation of indices with its package makeidx, and its support program makeindex, called on some systems makeidx.
1 使用 makeidx
1.1 Compiling Indexes
1.1.1 MakeIndex Settings in WinEdt
1.2 Sophisticated Indexing
1.2.1 Subentries
1.2.2 Controlling Sorting
1.2.3 Changing Page Number Style
3 Multiple indexes
4 Adding Index to Table Of Contents
5 International indexes
5.1 Generating index
5.1.1 xindy in kile
使用 makeidx[编辑]
To enable the indexing feature of LaTeX, the makeidx package must be loaded in the preamble with: \documentclass{artical} \begin{document} Hallo! \end{document}
and the special indexing commands must be enabled by putting the \documentclass{artical} \begin{document} Hallo! \end{document}
command into the input file preamble. This should be done within the preamble, since it tells LaTeX to create the files needed for indexing. To tell LaTeX what to index, use \documentclass{artical} \begin{document} Hallo! \end{document}
where key is the index entry and does not appear in the final layout. You enter the index commands at the points in the text that you want to be referenced in the index, likely near the reason for the key. For example, the text \documentclass{artical} \begin{document} Hallo! \end{document} can be re-written as \documentclass{artical} \begin{document} Hallo! \end{document} to create an entry called 'Fourier Series' with a reference to the target page. Multiple uses of \index with the same key on different pages will add those target pages to the same index entry.
To show the index within the document, merely use the command \documentclass{artical} \begin{document} Hallo! \end{document} It is common to place it at the end of the document. The default index format is two columns.
The showidx package that comes with LaTeX prints out all index entries in the left margin of the text. This is quite useful for proofreading a document and verifying the index.
Compiling Indexes[编辑]
When the input file is processed with LaTeX, each ± command writes an appropriate index entry, together with the current page number, to a special file. The file has the same name as the LaTeX input file, but a different extension (.idx). This .idx file can then be processed with the makeindex program. Type in the command line:
Note that filename is without extension: the program will look for filename.idx and use that. You can optionally pass filename.idx directly to the program as an argument. The makeindex program generates a sorted index with the same base file name, but this time with the extension .ind. If now the LaTeX input file is processed again, this sorted index gets included into the document at the point where LaTeX finds ±.
The index created by latex with the default options may not look as nice or as suitable as you would like it. To improve the looks of the index makeindex comes with a set of style files, usually located somewhere in the tex directory structure, usually below the makeindex subdirectory. To tell makeindex to use a specific style file, run it with the command line option:
makeindex -s <style file> filename
If you use a GUI for compiling latex and index files, you may have to set this in the options. Here are some configuration tips for typical tools:
MakeIndex Settings in WinEdt[编辑]
Say you want to add an index style file named simpleidx.ist
Texify/PDFTexify: Options→Execution Modes→Accessories→PDFTeXify, add to the Switches: --mkidx-option="-s simpleidx.ist"
MakeIndex alone: Options->Execution Modes→Accessories→MakeIndex, add to command line: -s simpleidx.ist
Sophisticated Indexing[编辑]
Below are examples of ± entries:
±} hello, 1 Plain entry
±} Peter, 3 Subentry under 'hello'
±}} Sam, 2 Formatted entry
±}} Lin, 7 Same as above
±} Jenny, 3 Formatted page number
±} Joe, 5 Same as above
±} école, 4 Handling of accents
±}} Peter, see hello Cross-references
±}} Jen, see also Jenny Same as above
Subentries[编辑]
If some entry has subsections, these can be marked off with !. For example, \documentclass{artical} \begin{document} Hallo! \end{document} would an index with 'cp850' categorized under 'input' (which itself is categorized into 'encodings'). These are called subsubentries and subentries in makeidx terminology.
Controlling Sorting[编辑]
In order to determine how an index key is sorted, place a value to sort by before the key with the @ as a separator. This is useful if there is any formatting or math mode, so one example may be \documentclass{artical} \begin{document} Hallo! \end{document} so that the entry in the index will show as '
{\displaystyle {\vec {F}}}
' but be sorted as 'F'.
Changing Page Number Style[编辑]
To change the formatting of a page number, append a | and the name of some command which does the formatting. This command should only accept one argument.
For example, if on page 3 of a book you introduce bulldogs and include the command \documentclass{artical} \begin{document} Hallo! \end{document} and on page 10 of the same book you wish to show the main section on bulldogs with a bold page number, use \documentclass{artical} \begin{document} Hallo! \end{document} This will appear in the index as bulldog, 3, 10
If you use texindy in place of makeindex, the classified entries will be sorted too, such that all the bolded entries will be placed before all others by default.
Multiple Pages[编辑]
To perform multi-page indexing, add a |( and |) to the end of the ± command, as in \documentclass{artical} \begin{document} Hallo! \end{document} The entry in the index for the subentry 'History' will be the range of pages between the two ± commands.
Using special characters[编辑]
In order to place values with !, @, or | in the ± command, one must quote these characters by using a double quotation mark (") and can only show " by quoting it (i.e., a key for " would be ±}).
This rule does not hold for \", so to put ä in the index, one may still use ±}}.
Note that the ± command can affect your layout if not used carefully. Here is an example:
θ 斜体文字
Abbreviation list[编辑]
You can make a list of abbreviations with the package nomencl [1]. You may also be interested in using the glossaries package described in the Glossary chapter.
To enable the Nomenclature feature of LaTeX, the nomencl package must be loaded in the preamble with: \documentclass{artical} \begin{document} Hallo! \end{document}
Issue the ± command for each symbol you want to have included in the nomenclature list. The best place for this command is immediately after you introduce the symbol for the first time. Put ± at the place you want to have your nomenclature list.
Run LaTeX 2 times then
makeindex filename.nlo -s nomencl.ist -o filename.nls
followed by running LaTeX once again.
To add the abbreviation list to the table of content, \left.\begin{array}{r}
-\frac{1}{m}=\frac{\sum_{i=1}^n{X_iY_i}-\frac{1}{n}\sum_{i=1}^n{X_i\sum_{i=1}^n{Y_i}}}{\sum_{i=1}^n{X_{i}^{2}-\frac{1}{n}\left(\sum_{i=1}^n{X_i}\right)^2}}\\
\frac{1}{m}\lg C=\frac{1}{n}\left(\sum_{i=1}^n{Y_i+\frac{1}{m}\sum_{i=1}^n{X_i}}\right)\\
\end{array} \right\} option can be used when declare the nomencl package, i.e. \documentclass{artical} \begin{document} Hallo! \end{document} instead of using the code in Adding Index to Table Of Contents section.
The title of the list can be changed using the following command: \documentclass{artical} \begin{document} Hallo! \end{document}
Multiple indexes[编辑]
If you need multiple indexes you can use the package multind [2].
This package provides the same commands as makeidx, but now you also have to pass a name as the first argument to every command. \documentclass{artical} \begin{document} Hallo! \end{document}
Adding Index to Table Of Contents[编辑]
By default, Index won't show in Table Of Contents, you have to add it manually.
To add index as a chapter, use this commands: \documentclass{artical} \begin{document} Hallo! \end{document}
If you use book class, you may want to start it on odd page, for this, use ±.
International indexes[编辑]
If you want to sort entries that have international characters (such as ő, ą, ó, ç, etc.) you may find that the sorting "is not quite right". In most cases the characters are treated as special characters and end up in the same group as @, ¶ or µ. In most languages that use Latin alphabet it's not correct.
Generating index[编辑]
Unfortunately, current version of xindy and hyperref are incompatible. When you use \left.\begin{array}{r}
\end{array} \right\} or \left.\begin{array}{r}
\end{array} \right\} modifiers, texindy will print error message:unknown cross-reference-class `hyperindexformat'! (ignored) and won't add those pages to index. Work-around for this bug is described on the talk page.
To generate international index file you have to use texindy instead of makeindex.
xindy is a much more extensible and robust indexing system than the makeindex system.
For example, one does not need to write: \documentclass{artical} \begin{document} Hallo! \end{document} to get the Lin entry after LAN and before LZA, instead, it's enough to write \documentclass{artical} \begin{document} Hallo! \end{document}
But what is much more important, it can properly sort index files in many languages, not only english.
Unfortunately, generating indexes ready to use by LaTeX using xindy is a bit more complicated than with makeindex.
First, we need to know in what encoding the .tex project file is saved. In most cases it will be UTF-8 or ISO-8859-1, though if you live, for example in Poland it may be ISO-8859-2 or CP-1250. Check the parameter to the inputenc package.
Second, we need to know which language is prominently used in our document. xindy can natively sort indexes in albanian, dutch, hebrew, latin, norwegian, slovak, belarusian, english, georgian, hungarian, latvian, polish, slovenian, vietnamese, bulgarian, esperanto, german, icelandic, lithuanian, portuguese, spanish, croatian, estonian, greek, italian, romanian, sorbian, swedish, czech, finnish, gypsy, klingon, macedonian, russian, turkish, danish, french, hausa, kurdish, mongolian, serbian and ukrainian.
I don't know if other languages have similar problems, but with polish, if your .tex is saved using UTF-8, the .ind produced by texindy will be encoded in ISO-8859-2 if you use only -L polish. While it's not a problem for entries containing polish letters, as LaTeX internally encodes all letters to plain ASCII, it is for accented letters at beginning of words, they create new index entry groups, if you have, for example an "średnia" entry, you'll get a "Ś" encoded in ISO-8859-2 .ind file. LaTeX doesn't like if part of the file is in UTF-8 and part is in IS-8859-2. The obvious solution (adding -C utf8) doesn't work, texindy stops with
error. The fix this, you have to load the definiton style for the headings using -M switch:
In the end we have to run such command:
texindy -L polish -M lang/polish/utf8 filename.idx
xindy in kile[编辑]
To use texindy instead of makeindex in kile, you have to either redefine the MakeIndex tool in Settings → Configure Kile... → Tools → Build, or define new tool and redefine other tools to use it.
The xindy definition should look similar to this:
Previous: Labels and Cross-referencing Index Next: Glossary
取自“https://zh.wikibooks.org/w/index.php?title=LaTeX/索引&oldid=72138”
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BySeries - Maple Help
Home : Support : Online Help : Education : Student Packages : ODEs : Computation : Solve : BySeries
Find a series solution for a linear homogeneous ODE with polynomial coefficients
BySeries(ODE, y(x))
The BySeries(ODE, y(x)) command finds a particular series solution of a linear homogeneous ODE with polynomial coefficients.
Note that the series solution may not represent the complete solution of the given ODE.
\mathrm{with}\left(\mathrm{Student}[\mathrm{ODEs}][\mathrm{Solve}]\right):
\mathrm{ode1}≔\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right),x\right),x\right)+x\mathrm{diff}\left(y\left(x\right),x\right)+y\left(x\right)=0
\textcolor[rgb]{0,0,1}{\mathrm{ode1}}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}
\mathrm{BySeries}\left(\mathrm{ode1},y\left(x\right)\right)
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}}}{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}]
\mathrm{ode3}≔{x}^{2}\mathrm{diff}\left(y\left(x\right),x,x\right)+{x}^{2}\mathrm{diff}\left(y\left(x\right),x\right)+\left({x}^{3}-6\right)y\left(x\right)=0
\textcolor[rgb]{0,0,1}{\mathrm{ode3}}\textcolor[rgb]{0,0,1}{≔}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{+}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{6}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}
\mathrm{BySeries}\left(\mathrm{ode3},y\left(x\right)\right)
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{5}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{2}}}{\left(\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}\right)\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\right)}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{0}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{=}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{0}}}{\textcolor[rgb]{0,0,1}{7}}]
\mathrm{ode4}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+\mathrm{diff}\left(y\left(x\right),x\right)+{x}^{2}y\left(x\right)=0
\textcolor[rgb]{0,0,1}{\mathrm{ode4}}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}
\mathrm{BySeries}\left(\mathrm{ode4},y\left(x\right)\right)
[\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}}^{\textcolor[rgb]{0,0,1}{\mathrm{\infty }}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{3}}}{{\textcolor[rgb]{0,0,1}{k}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{k}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{12}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{1}}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{,}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{=}\frac{{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{1}}}{\textcolor[rgb]{0,0,1}{6}}]
\mathrm{ode5}≔\mathrm{diff}\left(\left(-{x}^{2}+1\right)\mathrm{diff}\left(y\left(x\right),x\right),x\right)+12y\left(x\right)=0
\textcolor[rgb]{0,0,1}{\mathrm{ode5}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{ⅆ}}{\textcolor[rgb]{0,0,1}{ⅆ}\textcolor[rgb]{0,0,1}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)\textcolor[rgb]{0,0,1}{}\left(\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{12}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}
\mathrm{BySeries}\left(\mathrm{ode5},y\left(x\right)\right)
\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{a}}_{\textcolor[rgb]{0,0,1}{0}}\textcolor[rgb]{0,0,1}{}\left(\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{5}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{3}}\right)
\mathrm{ode6}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=\mathrm{sin}\left(x\right)y\left(x\right)
\textcolor[rgb]{0,0,1}{\mathrm{ode6}}\textcolor[rgb]{0,0,1}{≔}\frac{{\textcolor[rgb]{0,0,1}{ⅆ}}^{\textcolor[rgb]{0,0,1}{2}}}{\textcolor[rgb]{0,0,1}{ⅆ}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)
\mathrm{BySeries}\left(\mathrm{ode6},y\left(x\right)\right)
Error, (in Student:-ODEs:-SeriesSolve) series solutions are only available for linear homogeneous ODEs with polynomial coefficients
The Student[ODEs][Solve][BySeries] command was introduced in Maple 2021.
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On the spectral radius of bipartite graphs which are nearly complete | Journal of Inequalities and Applications | Full Text
On the spectral radius of bipartite graphs which are nearly complete
Ismail Naci Cangul2,
Ayse Dilek Maden3 &
Ahmet Sinan Cevik3
p,q,r,s,t\in {\mathbb{Z}}^{+}
rt\le p
st\le q
G=G\left(p,q;r,s;t\right)
be the bipartite graph with partite sets
U=\left\{{u}_{1},\dots ,{u}_{p}\right\}
V=\left\{{v}_{1},\dots ,{v}_{q}\right\}
such that any two edges
{u}_{i}
{v}_{j}
are not adjacent if and only if there exists a positive integer k with
1\le k\le t
\left(k-1\right)r+1\le i\le kr
\left(k-1\right)s+1\le j\le ks
. Under these circumstances, Chen et al. (Linear Algebra Appl. 432:606-614, 2010) presented the following conjecture:
p\le q
k<p
|U|=p
|V|=q
|E\left(G\right)|=pq-k
. Then whether it is true that
{\lambda }_{1}\left(G\right)\le {\lambda }_{1}\left(G\left(p,q;k,1;1\right)\right)=\sqrt{\frac{pq-k+\sqrt{{p}^{2}{q}^{2}-6pqk+4pk+4q{k}^{2}-3{k}^{2}}}{2}}.
In this paper, we prove this conjecture for the range
{min}_{{v}_{h}\in V}\left\{deg{v}_{h}\right\}\le ⌊\frac{p-1}{2}⌋
Let G be a (simple) graph with the vertex and edge sets given by
V\left(G\right)=\left\{{v}_{1},{v}_{2},\dots ,{v}_{n}\right\}
E\left(G\right)=\left\{{v}_{i}{v}_{j}\mid {v}_{i}\text{ and }{v}_{j}\text{ are adjacent}\right\}
, respectively. The adjacency matrix of G on n vertices is an
n×n
A\left(G\right)
whose entries
{a}_{ij}
{a}_{ij}=\left\{\begin{array}{cc}1;\hfill & \text{if }{v}_{i}{v}_{j}\in E\left(G\right),\hfill \\ 0;\hfill & \text{otherwise}.\hfill \end{array}
A\left(G\right)
is symmetric, all the eigenvalues of
A\left(G\right)
are real. In fact, the eigenvalues of
A\left(G\right)
are called eigenvalues of the graph G. We can list the eigenvalues of the graph G in a non-increasing order as follows:
{\lambda }_{1}\left(G\right)\ge {\lambda }_{2}\left(G\right)\ge \cdots \ge {\lambda }_{n-1}\left(G\right)\ge {\lambda }_{n}\left(G\right).
The largest eigenvalue
{\lambda }_{1}\left(G\right)
is often called the spectral radius of G.
Throughout this paper, we will consider only finite, simple, undirected, bipartite graphs. So, let us suppose that
G=\left(U\cup V,E\right)
is such a bipartite graph, where
U=\left\{{u}_{1},{u}_{2},\dots ,{u}_{p}\right\}
V=\left\{{v}_{1},{v}_{2},\dots ,{v}_{q}\right\}
are two sets of vertices and E is the set of edges defined as a subset of
U×V
. As a usual notation, the degrees of vertices
{u}_{i}\in U
{v}_{j}\in V
deg{u}_{i}
deg{v}_{j}
, respectively. For the integers
p,q,r,s,t\in {\mathbb{Z}}^{+}
rt\le p
st\le q
, let us denote the bipartite graph G by
G\left(p,q;r,s;t\right)
with the above partite sets U and V such that
{u}_{i}\in U
{v}_{j}\in V
are not adjacent if and only if there exists a
k\in {\mathbb{Z}}^{+}
1\le k\le t
\left(k-1\right)r+1\le i\le kr
\left(k-1\right)s+1\le j\le ks
In the literature, upper bounds for the spectral radius in terms of various parameters for unweighted and weighted graphs have been widely investigated [1–10]. As a special case, in [3], Chen et al. studied the spectral radius of bipartite graphs which are close to a complete bipartite graph. For partite sets U and V having
|U|=p
|V|=q
p\le q
, in the same reference, the authors also gave an affirmative answer to the conjecture [[11], Conjecture 1.2] by taking
|E\left(G\right)|=pq-2
into account of a bipartite graph. Furthermore, refining the same conjecture for the number of edges is at least
pq-p+1
, there still exists the following conjecture.
Conjecture 1 [3]
For positive integers p, q and k satisfying
p\le q
k<p
, let G be a bipartite graph with partite sets U and V having
|U|=p
|V|=q
|E\left(G\right)|=pq-k
\lambda \left(G\right)\le \lambda \left(G\left(p,q;k,1;1\right)\right)=\sqrt{\frac{pq-k+\sqrt{{p}^{2}{q}^{2}-6pqk+4pk+4q{k}^{2}-3{k}^{2}}}{2}}.
We note that similar conjectures in this topic have been resolved by the first author in the papers [12–16]. In here, as the main goal, we present the proof of Conjecture 1 for the range
{min}_{{v}_{h}\in V}\left\{deg{v}_{h}\right\}\le ⌊\frac{p-1}{2}⌋
The following lemma will be needed for the proof of our main result.
{\lambda }_{1}
be the spectral radius of the bipartite graph
G\left(p,q;k,1;1\right)
{\lambda }_{1}=\sqrt{\frac{pq-k+\sqrt{{p}^{2}{q}^{2}-6pqk+4pk+4q{k}^{2}-3{k}^{2}}}{2}}.
We now present an upper bound on the spectral radius of the bipartite graph G.
Theorem 1 For positive integers p, q and k satisfying
p\le q
k<p
|U|=p
|V|=q
|E\left(G\right)|=pq-k
{min}_{{v}_{h}\in V}\left\{deg{v}_{h}\right\}\le ⌊\frac{p-1}{2}⌋
{\lambda }_{1}\left(G\right)\le \sqrt{\frac{pq-k+\sqrt{{p}^{2}{q}^{2}-6pqk+4pk+4q{k}^{2}-3{k}^{2}}}{2}}
G\cong G\left(p,q;k,1;1\right)
\mathbf{Z}={\left({x}_{1},{x}_{2},\dots ,{x}_{p},{y}_{1},{y}_{2},\dots ,{y}_{q}\right)}^{T}
A\left(G\right)
corresponding to an eigenvalue
{\lambda }_{1}\left(G\right)
. For the sets U and V, let
{x}_{i}={max}_{1\le h\le p}{x}_{h}
{y}_{j}={max}_{1\le h\le q}{y}_{h}
, respectively. Also, let us suppose that
{v}_{1}
is the vertex having minimum degree in V. Then we have
⌊\frac{p-1}{2}⌋\ge \underset{{v}_{h}\in V}{min}\left\{deg{v}_{h}\right\}=deg{v}_{1}={d}_{1}\phantom{\rule{1em}{0ex}}\text{(say).}
A\left(G\right)\mathbf{Z}={\lambda }_{1}\left(G\right)\mathbf{Z}.
Considering (2), we get
{\lambda }_{1}\left(G\right){x}_{i}\le \left(q-1\right){y}_{j}+{y}_{1}\phantom{\rule{1em}{0ex}}\text{for }{u}_{i}\in U
{\lambda }_{1}\left(G\right){y}_{1}\le {d}_{1}{x}_{i}\phantom{\rule{1em}{0ex}}\text{for }{v}_{1}\in V.
However, from (3) and (4), we clearly obtain
{\lambda }_{1}^{2}\left(G\right){y}_{1}\le {d}_{1}\left[\left(q-1\right){y}_{j}+{y}_{1}\right],
which can be written shortly as
\left({\lambda }_{1}^{2}\left(G\right)-{d}_{1}\right){y}_{1}\le \left(q-1\right){d}_{1}{y}_{j}.
{v}_{1}
is the vertex with the minimum degree
{d}_{1}
in V and the total number of edges in bipartite graph G is
pq-k
\sum _{h=1}^{p}{\lambda }_{1}\left(G\right){x}_{h}\le \left(pq-k-{d}_{1}\right){y}_{j}+{d}_{1}{y}_{1}.
{v}_{j}\in V
, from (2) we get
{\lambda }_{1}\left(G\right){y}_{j}=\sum _{{u}_{h}:{u}_{h}{v}_{j}\in E}{x}_{h}.
In other words, by (6),
{\lambda }_{1}^{2}\left(G\right){y}_{j}=\sum _{{u}_{h}:{u}_{h}{v}_{j}\in E}{\lambda }_{1}\left(G\right){x}_{h}\le \sum _{h=1}^{p}{\lambda }_{1}\left(G\right){x}_{h}\le \left(pq-k-{d}_{1}\right){y}_{j}+{d}_{1}{y}_{1},
\left({\lambda }_{1}^{2}\left(G\right)-pq+k+{d}_{1}\right){y}_{j}\le {d}_{1}{y}_{1}.
{\lambda }_{1}^{4}\left(G\right)-\left(pq-k\right){\lambda }_{1}^{2}\left(G\right)+{d}_{1}\left(pq-k-q{d}_{1}\right)\le 0,
{\lambda }_{1}\left(G\right)\le \sqrt{\frac{pq-k+\sqrt{{p}^{2}{q}^{2}-2pqk+{k}^{2}-4pq{d}_{1}+4k{d}_{1}+4q{d}_{1}^{2}}}{2}}.
f\left(x\right)=4q{x}^{2}+4kx-4pqx,\phantom{\rule{1em}{0ex}}\text{where }x\le ⌊\frac{p-1}{2}⌋.
{f}^{\mathrm{\prime }}\left(x\right)=-4q\left(p-\frac{k}{q}-2x\right)<0,\phantom{\rule{1em}{0ex}}\text{as }x\le ⌊\frac{p-1}{2}⌋\text{ and }k<p\le q.
f\left(x\right)
1\le x\le ⌊\frac{p-1}{2}⌋
p-k\le {d}_{1}\le ⌊\frac{p-1}{2}⌋
, from (8), we get the required result (1).
Suppose now that equality holds in (1). Then all inequalities in the above argument must become equalities. Thus we have
{d}_{1}=p-k
. From the equality in (3), we get
From the equality in (4), we get
G\cong G\left(p,q;k,1;1\right)
Conversely, by Lemma 1, one can easily see that the equality holds in (1) for the graph
G\left(p,q;k,1;1\right)
Remark 1 In Theorem 1, we proved Conjecture 1 for the range
{min}_{{v}_{h}\in V}\left\{deg{v}_{h}\right\}\le ⌊\frac{p-1}{2}⌋
. However, this conjecture is still open for the range
⌊\frac{p-1}{2}⌋<{min}_{{v}_{h}\in V}\left\{deg{v}_{h}\right\}<p
Berman A, Zhang XD: On the spectral radius of graphs with cut vertices. J. Comb. Theory, Ser. B 2001, 83: 233–240. 10.1006/jctb.2001.2052
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Das KC, Kumar P: Bounds on the greatest eigenvalue of graphs. Indian J. Pure Appl. Math. 2003, 34(6):917–925.
Das KC, Kumar P: Some new bounds on the spectral radius of graphs. Discrete Math. 2004, 281: 149–161. 10.1016/j.disc.2003.08.005
Das KC, Bapat RB: A sharp upper bound on the spectral radius of weighted graphs. Discrete Math. 2008, 308: 3180–3186. 10.1016/j.disc.2007.06.020
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Bhattacharya A, Friedland S, Peled UN: On the first eigenvalue of bipartite graphs. Electron. J. Comb. 2008., 15: Article ID #R144
Das KC: On conjectures involving second largest signless Laplacian eigenvalue of graphs. Linear Algebra Appl. 2010, 432: 3018–3029. 10.1016/j.laa.2010.01.005
Das KC: Conjectures on index and algebraic connectivity of graphs. Linear Algebra Appl. 2010, 433: 1666–1673. 10.1016/j.laa.2010.06.012
Das KC: Proofs of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs. Linear Algebra Appl. 2011, 435: 2420–2424. 10.1016/j.laa.2010.12.018
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The first author is supported by BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea, and the other authors are partially supported by Research Project Offices of Uludag (2012-15 and 2012-19) and Selcuk Universities.
Department of Mathematics, Faculty of Science, Selcuk University, Campus, Konya, 42075, Turkey
Ayse Dilek Maden & Ahmet Sinan Cevik
Ayse Dilek Maden
All authors completed the paper together. Moreover, all authors read and approved the final manuscript.
Das, K.C., Cangul, I.N., Maden, A.D. et al. On the spectral radius of bipartite graphs which are nearly complete. J Inequal Appl 2013, 121 (2013). https://doi.org/10.1186/1029-242X-2013-121
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Symbolic Objects to Represent Mathematical Objects - MATLAB & Simulink - MathWorks España
Symbolic Scalar Variable, Function, and Expression
Symbolic Vector and Matrix
Symbolic Matrix Function
Comparison of Symbolic Objects
To solve mathematical problems with Symbolic Math Toolbox™, you can define symbolic objects to represent various mathematical objects. This example discusses the usage of these symbolic objects in the Command Window:
Defining a number as a symbolic number instructs MATLAB® to treat the number as an exact form instead of using a numeric approximation. For example, use a symbolic number to represent the argument of an inverse trigonometric function
\theta ={\mathrm{sin}}^{-1}\left(1/\sqrt{2}\right)
Create the symbolic number
\text{1/}\sqrt{2}
using sym, and assign it to a.
a = sym(1/sqrt(2))
Find the inverse sine of a. The result is the symbolic number pi/4.
thetaSym = asin(a)
thetaSym =
You can convert a symbolic number to variable-precision arithmetic by using vpa. The result is a decimal number with 32 significant digits.
thetaVpa = vpa(thetaSym)
thetaVpa =
To convert the symbolic number to a double-precision number, use double. For more information about whether to use numeric or symbolic arithmetic, see Choose Numeric or Symbolic Arithmetic.
thetaDouble = double(thetaSym)
thetaDouble =
Defining variables, functions, and expressions as symbolic objects enables you to perform algebraic operations with those symbolic objects, including simplifying formulas and solving equations. For example, use a symbolic scalar variable, function, and expression to represent the quadratic function
f\left(x\right)={x}^{2}+x-2
. For brevity, a symbolic scalar variable is also called a symbolic variable.
Create a symbolic scalar variable x using syms. You can also use sym to create a symbolic scalar variable. For more information about whether to use syms or sym, see Choose syms or sym Function.
Define a symbolic expression x^2 + x - 2 to represent the right side of the quadratic equation and assign it to f(x). The identifier f(x) now refers to a symbolic function that represents the quadratic function. A symbolic function accepts scalars as input arguments.
f(x) = x^2 + x - 2
You can then evaluate the quadratic function by providing its input argument inside the parentheses. For example, evaluate f(2).
fVal = f(2)
You can also solve the quadratic equation
f\left(x\right)=0
. Use solve to find the roots of the quadratic equation. solve returns the two solutions as a vector of two symbolic numbers.
sols = solve(f)
Defining a mathematical equation as a symbolic equation enables you to find the solution of the equation. For example, use a symbolic equation to solve the trigonometric problem
2\mathrm{sin}\left(t\right)\mathrm{cos}\left(t\right)=1
Create a symbolic function g(t) using syms. Assign the symbolic expression 2*sin(t)*cos(t) to g(t).
syms g(t)
g(t) = 2*sin(t)*cos(t)
To define the equation, use the == operator and assign the mathematical relation g(t) == 1 to eqn. The identifier eqn is a symbolic equation that represents the trigonometric problem.
eqn = g(t) == 1
2*cos(t)*sin(t) == 1
Use solve to find the solution of the trigonometric problem.
Use a symbolic vector and matrix to represent and solve a system of linear equations.
\begin{array}{c}x+2y=u\\ 4x+5y=v\end{array}
You can represent the system of equations as a vector of two symbolic equations. You can also represent the system of equations as a matrix problem involving a matrix of symbolic numbers and a vector of symbolic variables. For brevity, any vector of symbolic objects is called a symbolic vector and any matrix of symbolic objects is called a symbolic matrix.
Create two symbolic equations eq1 and eq2. Combine the two equations into a symbolic vector.
eq1 = x + 2*y == u;
eq2 = 4*x + 5*y == v;
eqns = [eq1, eq2]
[x + 2*y == u, 4*x + 5*y == v]
Use solve to find the solutions of the system of equations represented by eqns. solve returns a structure S with fields named after each of the variables in the equations. You can access the solutions using dot notation, as S.x and S.y.
(2*v)/3 - (5*u)/3
(4*u)/3 - v/3
Another way to solve the system of linear equations is to convert it to matrix form. Use equationsToMatrix to convert the system of equations to matrix form and assign the output to A and b. Here, A is a symbolic matrix and b is a symbolic vector. Solve the matrix problem by using the matrix division \ operator.
[A,b] = equationsToMatrix(eqns,x,y)
sols = A\b
Use symbolic matrix variables to evaluate differentials with respect to vectors.
\begin{array}{l}\alpha ={y}^{\text{T}}Ax\\ \frac{\partial \alpha }{\partial x}={y}^{\text{T}}A\\ \frac{\partial \alpha }{\partial y}={x}^{\text{T}}{A}^{\text{T}}\end{array}
Symbolic matrix variables represent matrices, vectors, and scalars in compact matrix notation. Symbolic matrix variables offer a concise display in typeset and show mathematical formulas with more clarity. You can enter vector- and matrix-based expressions as symbolic matrix variables in Symbolic Math Toolbox.
Create three symbolic matrix variables x, y, and A using the syms command with the matrix syntax. Nonscalar symbolic matrix variables are displayed as bold characters in the Command Window and in the Live Editor.
Define alpha. Find the differential of alpha with respect to the vectors x and y that are represented by the symbolic matrix variables x and y.
y.'*A*x
y.'*A
x.'*A.'
Substitute y with [1; 2; 3] in Dx and substitute x with [-1; 2; 0; 1] in Dy using subs. When evaluating a symbolic expression, you must substitute values that have the same size as the defined symbolic matrix variables.
Dx = subs(Dx,y,[1; 2; 3])
symmatrix([1;2;3]).'*A
Dy = subs(Dy,x,[-1; 2; 0; 1])
symmatrix([-1;2;0;1]).'*A.'
Use a symbolic matrix function to evaluate a matrix polynomial.
f\left(A\right)={A}^{2}-3A+{I}_{2}
A symbolic matrix function represents a parameter-dependent function that accepts matrices, vectors, and scalars as input arguments. Symbolic matrix function operates on matrices in compact matrix notation, offering a concise display in typeset and showing mathematical formulas with more clarity. You can enter vector- and matrix-based formulas as symbolic matrix functions in Symbolic Math Toolbox.
Create a 2-by-2 symbolic matrix variable A using the syms command with the matrix syntax. Create a symbolic matrix function f(A) that accepts A as an input argument using the syms command with the matrix keepargs syntax to keep the previous definition of A.
Assign the polynomial expression to the symbolic matrix function.
f(A) = A^2 - 3*A + 2*eye(2)
2*symmatrix(eye(2)) - 3*A + A^2
Evaluate the function for the matrix value A = [1 2; -2 -1]. When evaluating a symbolic matrix function, you must substitute values that have the same size as the defined input arguments.
- 3*symmatrix([1,2;-2,-1]) + symmatrix([1,2;-2,-1])^2 + 2*symmatrix(eye(2))
This table compares the symbolic objects that are available in Symbolic Math Toolbox.
Examples of MATLAB Command
Size of Symbolic Object
a = 1/sqrt(sym(2))
theta = asin(a)
1-by-1 sym
Symbolic scalar variable
syms x y u v
syms g(t) [1 3]
[g1(t), g2(t), g3(t)]
Size of unevaluated function, such as size(g), is 1-by-1.
Size of evaluated function, such as size(g(t)), is m-by-n, where m is the row size and n is the column size.
Data type of unevaluated function, such as class(g), is symfun.
Data type of evaluated function, such as class(g(t)), is sym.
expr = x^2 + x - 2
expr2 = 2*sin(x)*cos(x)
eq1 = x + 2*y == u
eq2 = 4*x + 5*y == v
x + 2*y == u
4*x + 5*y == v
b = [u v]
1-by-n or m-by-1, where m is the row size and n is the column size sym
A = [x y; x*y y^2]
[ x, y]
[x*y, y^2]
m-by-n, where m is the row size and n is the column size sym
Symbolic multidimensional array
syms A [2 1 2]
sz1-by-sz2-...-szn, where szn is the size of the nth dimension sym
(since R2021a)
m-by-n, where m is the row size and n is the column size symmatrix
syms f(X,Y) [2 2] matrix keepargs
f(X,Y) = X*Y - Y*X
X*Y - Y*X
Size of unevaluated matrix function, such as size(f), is 1-by-1.
Size of evaluated function, such as size(f(X,Y)), is m-by-n, where m is the row size and n is the column size.
Data type of unevaluated matrix function, such as class(f), is symfunmatrix.
Data type of evaluated function, such as class(f(X,Y)), is symmatrix.
syms | sym | symfun | symmatrix | symfunmatrix | symfunmatrix2symfun | symmatrix2sym | str2sym
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Some Properties of Furuta Type Inequalities and Applications
Jiangtao Yuan, Caihong Wang, "Some Properties of Furuta Type Inequalities and Applications", Abstract and Applied Analysis, vol. 2014, Article ID 457367, 7 pages, 2014. https://doi.org/10.1155/2014/457367
Jiangtao Yuan 1 and Caihong Wang1
This work is to consider Furuta type inequalities and their applications. Firstly, some Furuta type inequalities under are obtained via Loewner-Heinz inequality; as an application, a proof of Furuta inequality is given without using the invertibility of operators. Secondly, we show a unified satellite theorem of grand Furuta inequality which is an extension of the results by Fujii et al. At the end, a kind of Riccati type operator equation is discussed via Furuta type inequalities.
Throughout this paper, an operator means a bounded linear operator on a Hilbert space. and mean a positive operator and an invertible positive operator, respectively, (see [1, page 103]). The classical Loewner-Heinz inequality (L-H) is stated below (see [2, page 127]).
Theorem 1 (Loewner-Heinz inequality (L-H)). Let ; then ensures
In general, (L-H) is not true for . As a celebrated development of (L-H), Furuta provided a kind of order preserving operator inequality [2, page 129], the so-called Furuta inequality (FI).
Theorem 2 (Furuta inequality (FI), [3]). Let , ; then ensures
Tanahashi proved that the outer exponent above is optimal; see [3] for related topics. In order to establish the order structure on Aluthge transform of nonnormal operators, the complete form of Furuta inequality was showed in [4].
Theorem 3 (Complete form [4]). Let , , and . Then and such that ensures
We call the theorem above the complete form of Furuta inequality because the case of it implies the essential part () of Furuta inequality by the Loewner-Heinz inequality for . For convenience, we call Furuta inequality (Theorem 2) the original form of Furuta inequality.
It is known that there are many applications of Furuta type inequalities; we cite [5–7].
Based on Ito et al. [8] which is a continuation of [9], the equivalent relations between two operator inequalities are useful. For , means the projection .
Theorem 4 (see [8]). Let , , and .(1)If , then, for each , , and , the following inequalities are equivalent to each other: In particular, (4) implies (5) without condition .(2)For each , , and , the following inequalities are equivalent to each other:
It should be pointed out that (5) ensures (4) is not true without the condition [8, Remark 1]. Moreover, the proof of Theorem 4 is independent of (L-H).
In Section 2, some Furuta type inequalities under are proved via Loewner-Heinz inequality; as applications, we show alternate proofs of some well-known Furuta type inequalities (proofs of Theorems 10 and 2).
In 1995, Furuta [10] proved the so-called grand Furuta inequality which is also an extension of Theorem 2.
Theorem 5 (grand Furuta inequality [10]). Let , , and . If with ; then
Fujii et al. proved some satellite theorems of grand Furuta inequality.
Theorem 6 (see [11]). Let , , and . If with ; then
Theorems 6 and 7 are extensions of Theorem 5.
In Section 3, we will show a unified satellite theorem which is an extension of Theorems 6 and 7 via the complete forms of Furuta inequality with negative powers.
Lastly, it is known that Riccati type operator equations relate to control theory closely and have been studied extensively [13]. Pedersen and Takesaki [14] developed the special kind of Riccati equation as a useful tool for the noncommutative Radon-Nikodym theorem.
Yuan and Gao [15] discussed the Riccati type equation:
In Section 4, as a continuation of [15, 16], we will consider the Riccati type equation: via Furuta type inequalities.
2. Furuta Type Inequalities under the Order
Reference [17] proved a kind of equivalent relations which can be regarded as a parallel result to Theorem 4.
Theorem 8 (see [17]). Let , , and . If , then, for each , and , the following inequalities are equivalent to each other: In particular, (12) implies (13) without condition .
The proof of Theorem 8 is different from Theorem 4 and independent of (L-H).
In this section, we consider some Furuta type inequalities under the order . As applications, alternate proofs of some Furuta type inequalities are given (proofs of Theorems 10 and 2). Especially, we prove (FI) without using the invertibility of operators.
Theorem 9. Let , . (1)For each and with , the following inequalities hold and they are equivalent to each other: (2)For each and with , the following inequalities hold: (3)If , then
Proof. (1) Since , follows. By and (L-H) for and , we have Hence, (14) holds. Since , follows. So, the equivalency follows by Theorem 8.
(2) Similar to the proof of (14), we have Hence, (16) holds. Since (12) implies (13) without kernel condition, (17) follows by (16).
(3) By (15), there exists the function defined on satisfying [18, Lemma 2.6(1)]. Hence, case of [18, Lemma 2.6(2)] implies So (18) holds by and (L-H) for . It is easy to prove (19) in a similar way.
As prompt applications, we show alternate proofs of some Furuta type inequalities.
Theorem 10 (see [19]). Let , , , . For such that , if then where . Moreover, for each , the function is decreasing (resp., increasing) for .
Proof. It is enough to prove the case because the case can be proved in a similar manner. Denote (23) by ; that is, For , , by (19) of Theorem 9, we have By putting , the inequality above becomes This implies that (24) holds for . Denote and ; repeating this process, (24) holds for .
For each , , by (24) and (L-H), where . This together with Theorem 8 and (L-H) deduce that where . So, the monotonicity of the function holds.
It should be pointed out that, if and , the assertion that (23) ensures (24) is not true [15, Theorem 2.8].
Theorem 11 (see [15]). Given any positive numbers , , , and with , there exist invertible positive operators and such that where is an arbitrary positive number.
Alternate Proof of Theorem 2. The case and of Theorem 2 follows by (L-H) directly. Theorem 9(3) means the case and of Theorem 2; this together with Theorem 10 implies the case and of Theorem 2. So, the proof is complete.
The proof above says that the original form of Furuta inequality (Theorem 2) is a composition of (L-H), Theorems 9 and 10. The proof here is independent of the invertibility of the operators and .
3. A Unified Satellite Theorem of Grand Furuta Inequalities
Denote , where .
Theorem 12. Let , , , with .(1)If , , and , then (2)If and , then
The case of Theorem 12(2) is just Theorem 7. The special case of Theorem 12(1) implies the result below.
Corollary 13. Let , , , with . If and , then
It is obvious that the special case of Corollary 13 is a unified result of Theorems 6 and 7; that is, it is an extension of Theorems 6 and 7. So, we call Theorem 12 a unified satellite theorem of grand Furuta inequality (Theorem 5).
In order to give a proof, we prepare some results in advance.
Lemma 14 (see [18]). Let , and . Then with ensures that the function is decreasing for . In particular,
Lemma 15 (see [18]). Let , , , and . Then with ensures
Lemma 16. Let , and . Then the following assertion (1) implies (2).(1)There exists an increasing function such that, for each , if , then (2)The function in (1) satisfies that, for each , if , then
Lemma 16 is a complement to [18, Lemma 2.6].
Proof. It is sufficient to prove the case for the case can be proved in a similar manner. For each and , if , then (2) follows by (1) immediately. Suppose that for some positive integer and . By , for , we have Noting that and , these together with (L-H) deduce that Therefore, the function in satisfies .
Lemma 17. Let , and ; then with ensures
Proof. Firstly, we prove the case of Lemma 17. By [10, Lemma 1], (42) is equivalent to On the other hand, holds by Loewner-Heinz inequality for . So (42) holds for .
Now, it is proved that (42) holds when . Meanwhile, it is easy to see that the increasing function satisfies of Lemma 16, so (42) holds for .
Proof of Theorem 12. By the case of Lemma 15 and (L-H) for , (1)For , Theorem 3 and (L-H) deduce that Meanwhile, for and , Lemma 17 and (L-H) imply Hence, follows by the case of (44), (45), and (46).(2)By (L-H), (44), , Theorem 3 and Lemma 17 ensure The above is the same as the function in Lemma 14.
4. Riccati Type Operator Equations
Theorem 18 (see [15]). Let , and assume that .(1)The following statements are equivalent for each , and .(a) for some .(b)There exists a unique operator that satisfies and (48). If in additional is invertible, (1) holds for .(2)If there exists satisfying (48) for fixed , and , then, for and , there exists satisfying
One of the applications of Riccati equation (48) is to show that the inclusion relations among class operators are strict [15, Theorem 3.1]. Recently, there are some developments on operator equations including the following equation (see [16, 20]):
Obviously, the special case of (50) is just (48).
Theorem 19 (see [16]). Let , and assume that . The following statements are equivalent for each , , and .(1) for some .(2)There exists a unique operator which satisfies and (50).
If in additional is invertible, the condition can be replaced with where means the set of all real numbers, and if and are both invertible, the conditions and can be replaced with and .
The case of Theorem 19 is a generalization of Theorem 18(1). In this section, we give a generalization of Theorem 18(2).
Lemma 20. Let , , , , , . For and such that and , if then, for , where .
The case , and of Lemma 20 implies Theorem 10. The case , and of Lemma 20 implies Yanagida's result [21, Proposition 4].
Proof. It is enough to prove the case because the case can be proved in a similar manner. Denote (51) by ; that is, For , , by (FI) (Theorem 2), we have By putting , the inequality above becomes Denote and ; then and , so that (52) holds by the inequality above.
Theorem 21. Let , and assume that . For each , and , if there exists satisfying the equation where , and . Then, for and , there exists satisfying If is invertible, the condition can be replaced with .
Proof. By the assumption, (1) of Theorem 19 holds for some ; that is, So, the following holds by Lemma 20: where ; that is, where . Hence, (57) is solvable by Theorem 19.
The result below is the case and of Theorem 21.
Corollary 22. Let , and assume that . For each , and , if there exists satisfying the equation where and . Then, for , there exists satisfying If is invertible, the condition can be replaced with .
It is obvious that Corollary 22 is a generalization of the case of Theorem 18(2).
This work was supported in part by the National Natural Science Foundation of China (11301155) and Project of Education Department of Henan Province of China (2012GGJS-061), and Project of Science and Technology Department of Henan Province of China (142300410143).
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Tezos Amendment Process - Nomadic Labs knowledge center
The aim of this document is to offer a detailed view of the Tezos on-chain governance model by going through the process used to consider protocol amendments.
Baking: The creation of new blocks on the Tezos blockchain by its validators (also known as bakers or delegates), who receive compensation for each block produced.
Endorsement: Each baked block is validated by a set of bakers who have not baked the block, but are selected to endorse it. They are known as endorsers of the block, and receive compensation for each endorsement realized.
Delegation: All Tez holders can delegate their baking and voting rights to a “delegate”, while still maintaining control over their funds.
Stake: To have the right to be recognized as a baker in a network, a Tez holder must possess at least 6,000
in their staking balance (own balance + delegated balance). Voting rights are also indexed to the staking balance.
Cycle: The time required for 8,192 blocks to be created on the Tezos blockchain. It lasts around 2 days, 20 hours, and 16 minutes (or 30 seconds. per block, if all bakers cooperate effectively)
Proposal: A request for addition, adjustment, or removal of features of the protocol.
On Tezos, any baker can submit a proposal to amend the protocol.
The proposal/voting process then takes place entirely on-chain and is known as the amendment process.
As each period lasts 5 cycles (approx. 14 days), a complete amendment process requires 25 cycles, or around 2 months and 10 days to be adopted and merged with the main network.
First period: Proposal
First period - Proposal (Cycles 1-5)
During this period, users can submit up to 20 amendment proposals which are then subjected, over subsequent periods, to voting and testing.
Proposals can relate to a variety of features, for example:
The threshold to register as a baker
The gas consumption process
Aspects/features of the smart contract language
Bakers across the network then proceed to vote (in the form of up-voting). The most highly voted proposal then moves on to the Exploration period.
Second period - Exploration
Second period - Exploration Vote (Cycles 6-10)
During this period, all voters must decide whether or not the proposal brought forward will proceed to the next period: the cooldown period. Unlike in the previous period, Tezos uses the concepts of Quorum and supermajority vote – or qualified majority vote – to decide whether or not to send it to the cooldown period.
What we mean by these terms :
Quorum: the representation weight required for a vote to actually take place.
A supermajority vote or qualified majority vote – establishes a minimum percentage of positive expressed votes needed (80% for Tezos) for a decision to be taken.
In order to move to the next period, a proposal must have a greater Voter Turnout than the Quorum and a greater percentage of positive votes than the supermajority.
Voting during the Exploration period (Cycle 6-10)
Calculating the Quorum:
When the Tezos Mainnet was launched, the Quorum was set at 80% and updated at the end of each vote which was successfully approved, based on the Voter Turnout.
The Babylon amendment introduced two major changes to the calculation of the Quorum:
The calculation now takes into account the exponential moving average (EMA) of the Voter Turnout
The Quorum is now bounded between 30% and 70%
The following formula is used to calculate the Quorum:
Quorum = 0.3 + MME_t ∗ (0.7 − 0.3)
The following formula is then used to update the moving average for the next vote:
MME_{t+1} = 0.8 * MME_t + 0.2 * Participation_t
Voting system :
There are 3 possible ways to vote :
Yay (Y): For
Nay (N): Against
Pass (A): Neutral
To vote, each active baker on the network must have at least 6,000
. Each holder of the tez cryptocurrency can delegate their “associated voting right” to a baker, while retaining control over their funds.
The weight of a baker’s vote is determined on a pro-rata basis by the size of their staking balance.
The more tez a baker has, the greater the weight of his vote.
Voting example 1/2
To illustrate this process, let us assume a total of 100 active voting rigths managed by bakers and a Voter Turnout EMA of 75%, and then 90 votes (Yay, Nay, and Pass) during the Exploration period.
1. Quorum = 0.3 + 75% ∗ (0.7 − 0.3) = 60%
2. Update of the Exponential Moving Average
3. Positive voter turnout: = 88%
1. Quorum = 0.3 + 75% * (0.7 - 0.3) = 60%
3. Proposal rejected: Although the Yays have reached the number required for a supermajority, the proposal is rejected as the Quorum has not been reached. We must therefore go back to the initial proposals stage.
Third period - Cooldown
Third period - Cooldown (Cycles 11-15)
If a proposal is accepted by a supermajority during the Exploration period, the Cooldown period starts. This period replaces the previous Testing period that was underused since most of the testing happens on a dedicated testnest.
Fourth period - Promotion
Fourth period - Promotion (Cycles 16-20
At the end of the Cooldown period, the network decides whether or not to adopt the amendment proposal, based on off-chain discussions and its behavior on the test network.
The same concepts of both the Quorum and the supermajority are applied here. After the vote, the Voter Turnout moving average is updated again.
The version of the protocol integrating the amendment is then disseminated by the network and transmitted to the nodes which are automatically and openly updated.
Fifth period – Adoption
Fifth period – Adoption (Cycles 21-25)
During the adoption period, bakers update their infrastructure. At the end of this period, the new protocol will be adopted. This period is dedicated to helping validators to migrate on the new Mainnet version.
At the end of the Adoption period, a new Proposal period starts.
The person whose proposal is ultimately adopted as a protocol amendment has the merit of having evolved the Tezos blockchain, and is compensated in tez by an amount determined beforehand in the source code of their proposal.
May 2019, Athens amendment :
Reduced roll size for bakers (from 10,000 to 8,000 tez)
Increased gas limit per operation and per block
October 2019, Babylon amendment:
Adjusted “emmy+” consensus algorithm
New features in a low-level language (Michelson) for smart contracts, adjusted cost of gas
Adjusted formula for updating Quorum (30% ≤ Quorum ≤ 70%)
March 2020, Carthage amendment:
Improved formula for calculating compensation for baking
November 2020, Delphi amendment:
Reduced storage costs by 4
General recomputation of the gas costs
February 2021, Edo amendment:
Inclusion of Sapling protocol
New Michelson data structure: tickets
New Michelson functions, especially hash functions SHA3 and Keccak
Updating of the amendment protocol :
A fifth period to allow bakers to update their infrastructure: The adoption period
Reduction of the overall time of each period, from 8 to 5 cycles
May 2021, Florence amendment:
Increase maximum operation data size
Depth-First execution order
Updating of the amendment protocol: testing period is now replaced by a cooldown period
August 2021, Granada amendment:
Emmy *, new consensus algorithm:
smaller block times
Liquidity Baking, a small amount of tez from each block issued to provide liquidity to the tez/tzBTC pair
December 2021, Hangzhou amendment:
Cache Context storage flattening
April 2022, Ithaca2 amendment:
Tenderbake, a new consensus algorithm
Backend storage improvements
Reduction of the amount of Tez required to become a baker (from 8,000 to 6,000 Tez + no more notion of rolls)
On-chain governance reduces the risks of hard forks
The development of Bitcoin has shown that hard forks are often necessary in order to make changes to a protocol.
There are risks involved here, as two chains can co-exist if a consensus cannot be reached within the community, which can then lead to operational risks for some blockchain projects.
However, whenever a critical bug is found, the protocol must be updated urgently – for Tezos just as for Bitcoin. However, on the Tezos blockchain, this type of situation has never led to a chain splitting into two, as this type of change is always driven by consensus.
It is possible to vote by using the Tezos client :
To know the current period :
$ tezos−client show voting period
To vote for the proposal whose hash is <proposal > with the address <delegate >:
$ tezos−client submit ballot for <delegate> <proposal > <yay | nay | pass>
Summary diagram of the amendment process
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Existence of Nontrivial Solutions of p-Laplacian Equation with Sign-Changing Weight Functions
Ghanmi Abdeljabbar, "Existence of Nontrivial Solutions of p-Laplacian Equation with Sign-Changing Weight Functions", International Scholarly Research Notices, vol. 2014, Article ID 461965, 7 pages, 2014. https://doi.org/10.1155/2014/461965
Ghanmi Abdeljabbar1
Academic Editor: L. Gasinski
This paper shows the existence and multiplicity of nontrivial solutions of the p-Laplacian problem for with zero Dirichlet boundary conditions, where is a bounded open set in , if , if ), , is a smooth function which may change sign in , and . The method is based on Nehari results on three submanifolds of the space .
In this paper, we are concerned with the multiplicity of nontrivial nonnegative solutions of the following elliptic equation: where is a bounded domain of , if , if , , is positively homogeneous of degree ; that is, holds for all and the sign-changing weight function satisfies the following condition:
(A) with , , and .
In recent years, several authors have used the Nehari manifold and fibering maps (i.e., maps of the form , where is the Euler function associated with the equation) to solve semilinear and quasilinear problems. For instance, we cite papers [1–9] and references therein. More precisely, Brown and Zhang [10] studied the following subcritical semilinear elliptic equation with sign-changing weight function: where . Also, the authors in [10] by the same arguments considered the following semilinear elliptic problem: where . Exploiting the relationship between the Nehari manifold and fibering maps, they gave an interesting explanation of the well-known bifurcation result. In fact, the nature of the Nehari manifold changes as the parameter crosses the bifurcation value.
Inspired by the work of Brown and Zhang [10], Nyamouradi [11] treated the following problem: where is positively homogeneous of degree .
In this work, motivated by the above works, we give a very simple variational method to prove the existence of at least two nontrivial solutions of problem (1). In fact, we use the decomposition of the Nehari manifold as vary to prove our main result.
Before stating our main result, we need the following assumptions:(H1) is a function such that (H2), , and for all .We remark that using assumption (H1), for all , , we have the so-called Euler identity: Our main result is the following.
Theorem 1. Under the assumptions (A), (H1), and (H2), there exists such that for all , problem (1) has at least two nontrivial nonnegative solutions.
This paper is organized as follows. In Section 2, we give some notations and preliminaries and we present some technical lemmas which are crucial in the proof of Theorem 1. Theorem 1 is proved in Section 3.
2. Some Notations and Preliminaries
Throughout this paper, we denote by the best Sobolev constant for the operators , given by where . In particular, we have with the standard norm Problem (1) is posed in the framework of the Sobolev space . Moreover, a function in is said to be a weak solution of problem (1) if Thus, by (6) the corresponding energy functional of problem (1) is defined in by In order to verify , we need the following lemmas.
Lemma 2. Assume that is positively homogeneous of degree ; then is positively homogeneous of degree .
Proof. The proof is the same as that in Chu and Tang [4].
In addition, by Lemma 2, we get the existence of positive constant such that
Lemma 3 (see [12], Theorem A.2). Let and such that Then for every , one has ; moreover the operator defined by is continuous.
Lemma 4 (See Proposition 1 in [13]). Suppose that verifies condition (12). Then, the functional belongs to , and where denotes the usual duality between and (the dual space of the sobolev space ).
As the energy functional is not bounded below in , it is useful to consider the functional on the Nehari manifold: Thus, if and only if Note that contains every nonzero solution of problem (1). Moreover, one has the following result.
Lemma 5. The energy functional is coercive and bounded below on .
Proof. If , then by (16) and condition (A) we obtain So, it follows from (8) that Thus, is coercive and bounded below on .
Define Then, by (16) it is easy to see that for , Now, we split into three parts
Lemma 6. Assume that is a local minimizer for on and that . Then, in (the dual space of the Sobolev space E).
Proof. Our proof is the same as that in Brown-Zhang [10, Theorem 2.3].
Lemma 7. One has the following:(i)if , then ;(ii)if , then and ;(iii)if , then .
Proof. The proof is immediate from (21), (22), and (23).
From now on, we denote by the constant defined by then we have the following.
Proof. Suppose otherwise, that such that . Then for , we have From the Hölder inequality, (6) and (8), it follows that Hence, it follows from (27) that then, On the other hand, from condition (A), (8) and (26) we have So, Combining (30) and (32), we obtain , which is a contradiction.
By Lemma 8, for , we write and define Then, we have the following.
Lemma 9. If , then for some depending on , and .
Proof. Let . Then, from (23) we have So Thus, from the definition of and , we can deduce that .
Now, let . Then, using (6) and (8) we obtain this implies that In addition, by (18) and (38) Thus, since , we conclude that for some . This completes the proof.
For with , set Then, the following lemma holds.
Lemma 10. For each with , one has the following:(i)if , then there exists unique such that and (ii)if , then there are unique such that and
Proof. We fix with and we let Then, it is easy to check that achieves its maximum at . Moreover,
(i) We suppose that . Since as , for and for . There is a unique such that .
Now, it follows from (14) and (27) that Hence, . On the other hand, it is easy to see that for all Thus, .
(ii) We suppose that . Then, by (A), (8) and the fact that we obtain Then, there are unique and such that , , and . We have , and Thus, This completes the proof.
For each with , set Then we have the following.
Lemma 11. For each with , one has the following:(i)if , then there exists a unique such that and (ii)if , then there are unique such that and
Proof. For with , we can take and similar to the argument in Lemma 9, we obtain the results of Lemma 10.
Proposition 12. (i) There exist minimizing sequences in such that
(ii) There exist minimizing sequences in such that
Proof. The proof is almost the same as that in Wu [14, Proposition 9] and is omitted here.
3. Proof of Our Result
Throughout this section, the norm is denoted by for and the parameter satisfies .
Theorem 13. If , then, problem (1) has a positive solution in such that
Proof. By Proposition 12(i), there exists a minimizing sequence for on such that Then by Lemma 5, there exists a subsequence and in such that This implies that as .
Next, we will show that By Lemma 3, we have where . On the other hand, it follows from the Hölder inequality that Hence, as .
By (57) and (58) it is easy to prove that is a weak solution of (1).
Since then by (57) and Lemma 9, we have as . Letting , we obtain Now, we aim to prove that strongly in and .
Using the fact that and by Fatou's lemma, we get This implies that Let ; then by Brézis-Lieb Lemma [3] we obtain Therefore, strongly in .
Moreover, we have . In fact, if then, there exist such that and . In particular we have . Since there exists such that . By Lemma 10, we have which is a contradiction.
Finally, by (63) we may assume that is a nontrivial nonnegative solution of problem (1).
Proof. By Proposition 12(ii), there exists a minimizing sequence for on such that Moreover, by (23) we obtain So, by (38) and (72) there exists a positive constant such that This implies that By (70) and (71), we obtain clearly that is a weak solution of (1).
Now, we aim to prove that strongly in . Supposing otherwise, then By Lemma 9, there is a unique such that . Since , for all , we have which is a contradiction. Hence strongly in .
This imply that By Lemma 5 and (74) we may assume that is a nontrivial solution of problem (1).
Now, we begin to show the proof of Theorem 1: by Theorem 13, we obtain that for all , problem (1) has a nontrivial solution . On the other hand, from Theorem 14, we get the second solution . Since , then and are distinct.
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p
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Copyright © 2014 Ghanmi Abdeljabbar. 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.
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Cheat sheet/ko - Kerbal Space Program Wiki
< Cheat sheet
경고: 평서체로 번역할 경우, 바로 삭제 신청이 들어갈 것입니다. Ysjbserver (talk)
Kerbal Space Program rocket scientist's cheat sheet: Delta-v maps, equations and more for your reference so you can get from here to there and back again.
1.1 추력비(Thrust to Weight Ratio; TWR)
1.2 합산비추력(Combined Specific Impulse; Isp)
1.3 델타V (Delta-v; Δv)
1.3.1 Basic calculation
1.3.2 True Δv of a stage that crosses from atmosphere to vacuum
1.3.4 Maximum Δv Chart
2.3 Δv
2.4 Maximum Δv
2.5 True Δv
추력비(Thrust to Weight Ratio; TWR)
→ 참고하기: Thrust-to-weight ratio
이건 제 2 뉴턴 운동 법칙를 따릅니다. 만약 이 값이 1보다 작다면, 땅을 떠나지 못하겠죠. 발사하는 곳의 표면 기준의 중력가속도 값이 필요하다는 것을 잘 알아두세요!
{\displaystyle {\text{TWR}}={\frac {F_{T}}{m\cdot g}}>1}
{\displaystyle F_{T}}
엔진의 출력
{\displaystyle m}
전체 중량
{\displaystyle g}
중력 가속도 값
합산비추력(Combined Specific Impulse; Isp)
→ 참고하기: Specific impulse
If the Isp is the same for all engines in a stage, then the Isp is equal to a single engine. If the Isp is different for engines in a single stage, then use the following equation:
{\displaystyle I_{sp}={\frac {(F_{1}+F_{2}+\dots )}{{\frac {F_{1}}{I_{sp1}}}+{\frac {F_{2}}{I_{sp2}}}+\dots }}}
델타V (Delta-v; Δv)
→ 참고하기: Tutorial:Advanced Rocket Design
Basic calculation of a rocket's Δv. Use the atmospheric and vacuum thrust values for atmospheric and vacuum Δv, respectively.
{\displaystyle \Delta {v}=ln\left({\frac {M_{start}}{M_{end}}}\right)\cdot I_{sp}\cdot 9.81{\frac {m}{s^{2}}}}
{\displaystyle \Delta {v}}
is the velocity change possible in m/s
{\displaystyle M_{start}}
is the starting mass in the same unit as
{\displaystyle M_{end}}
{\displaystyle M_{end}}
is the end mass in the same unit as
{\displaystyle M_{start}}
{\displaystyle I_{sp}}
is the specific impulse of the engine in seconds
True Δv of a stage that crosses from atmosphere to vacuum
Kerbin 1000 m/s
other bodies' data missing
Calculation of a rocket stage's Δv, taking into account transitioning from atmosphere to vacuum. Δvout is the amount of Δv required to leave a body's atmosphere, not reach orbit. This equation is useful to figure out the actual Δv of a stage that transitions from atmosphere to vacuum.
{\displaystyle \Delta {v}_{T}={\frac {\Delta {v}_{atm}-\Delta {v}_{out}}{\Delta {v}_{atm}}}\cdot \Delta {v}_{vac}+\Delta {v}_{out}}
Various fan-made maps showing the Δv required to travel to a certain body.
Subway style Δv map:
Total Δv values
http://www.skyrender.net/lp/ksp/system_map.png
Δv change values
http://i.imgur.com/duY2S.png
Δv nomogram
http://ubuntuone.com/1kD39BCoV38WP1QeG6MtO6
Δv with Phase Angles
http://i.imgur.com/dXT6r7s.png
Precise Total Δv values
http://i.imgur.com/UUU8yCk.png
Maximum Δv Chart
This chart is a quick guide to what engine to use for a single stage interplanetary ship. No matter how much fuel you add you will never reach these ΔV without staging to shed mass or using the slingshot maneuver.
Max Δv (m/s)
290 6257 LV-1
320 6905 Mark-55
330 7120 Mainsail
350 7552 Skipper
360 7768 KS-25X4
370 7983 LV-T30
LV-T45
380 8199 KR-2L
390 8415 Poodle
800 17261 LV-N
TWR = F / (m * g) > 1
When Isp is the same for all engines in a stage, then the Isp is equal to a single engine. So six 200 Isp engines still yields only 200 Isp.
When Isp is different for engines in a single stage, then use the following equation:
{\displaystyle I_{sp}={\frac {(F_{1}+F_{2}+\dots )}{{\frac {F_{1}}{I_{sp1}}}+{\frac {F_{2}}{I_{sp2}}}+\dots }}}
Isp = ( F1 + F2 + ... ) / ( ( F1 / Isp1 ) + ( F2 / Isp2 ) + ... )
Isp = ( Force of Thrust of 1st Engine + Force of Thrust of 2nd Engine...and so on... ) / ( ( Force of Thrust of 1st Engine / Isp of 1st Engine ) + ( Force of Thrust of 2nd Engine / Isp of 2nd Engine ) + ...and so on... )
Two engines, one rated 200 newtons and 120 seconds Isp ; another engine rated 50 newtons and 200 seconds Isp.
Isp = (200 newtons + 50 newtons) / ( ( 200 newtons / 120 ) + ( 50 newtons / 200 ) = 130.89 seconds Isp
For atmospheric Δv value, use atmospheric thrust values.
For vacuum Δv value, use vacuum thrust values.
Use this equation to figure out the Δv per stage:
{\displaystyle \Delta {v}=ln\left({\frac {M_{start}}{M_{dry}}}\right)\cdot I_{sp}\cdot 9.81{\frac {m}{s^{2}}}}
Δv = ln ( Mstart / Mdry ) * Isp * g
Δv = ln ( Starting Mass / Dry Mass ) X Isp X 9.81
Single Stage Rocket that weighs 23 tons when full, 15 tons when fuel is emptied, and engine that outputs 120 seconds Isp.
Δv = ln ( 23 Tons / 15 Tons ) × 120 seconds Isp × 9.81m/s² = Total Δv of 503.2 m/s
Maximum Δv
Simplified version of the Δv calculation to find the maximum Δv a craft with the given ISP could hope to achieve. This is done by using a magic 0 mass engine and not having a payload.
{\displaystyle \Delta {v}=21.576745349086\cdot I_{sp}}
Δv =21.576745349086 * Isp
Explained / Examples:
This calculation only uses the mass of the fuel tanks and so the ln ( Mstart / Mdry ) part of the Δv equation has been replaced by a constant as Mstart / Mdry is always 9 (or worse with some fuel tanks) regardless of how many fuel tanks you use.
The following example will use a single stage and fuel tanks in the T-100 to Jumbo 64 range with an engine that outputs 380 seconds Isp.
Δv = ln ( 18 Tons / 2 Tons ) × 380 seconds Isp × 9.81m/s² = Maximum Δv of 8199.1632327878 m/s
Δv = 2.1972245773 × 380 seconds Isp × 9.82m/s² = Maximum Δv of 8199.1632327878 m/s (Replaced the log of mass with a constant as the ratio of total mass to dry mass is constant regardless of the number of tanks used as there is no other mass involved)
Δv = 21.576745349086 × 380 seconds Isp = Maximum Δv of 8199.1632327878 m/s (Reduced to its most simple form by combining all the constants)
True Δv
How to calculate the Δv of a rocket stage that transitions from Kerbin atmosphere to vacuum.
Assumption: It takes approximately 1000 m/s of Δv to escape Kerbin's atmosphere before vacuum Δv values take over for the stage powering the transition.
Note: This equation is an guess, approximation, and is not 100% accurate. Per forum user stupid_chris who came up with the equation: "The results will vary a bit depending on your TWR and such, but it should usually be pretty darn accurate."
Equation for Kerbin Atmospheric Escape:
{\displaystyle \Delta {v}_{T}={\frac {\Delta {v}_{atm}-\Delta {v}_{out}}{\Delta {v}_{atm}}}\cdot \Delta {v}_{vac}+\Delta {v}_{out}}
True Δv = ( ( Δv atm - 1000 ) / Δv atm ) * Δv vac + 1000
True Δv = ( ( Total Δv in atmosphere - 1000 m/s) / Total Δv in atmosphere ) X Total Δv in vacuum + 1000
Single Stage with total atmospheric Δv of 5000 m/s, and rated 6000 Δv in vacuum.
Transitional Δv = ( ( 5000 Δv atm - 1000 Δv Required to escape Kerbin atmosphere ) / 5000 Δv atm ) X 6000 Δv vac + 1000 Δv Required to escape Kerbin atmosphere = Total Δv of 5800 m/s
The Drawing Board: A library of tutorials and other useful information
Retrieved from "https://wiki.kerbalspaceprogram.com/index.php?title=Cheat_sheet/ko&oldid=100781"
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Product (category theory) - Wikipedia
Generalized object in category theory
Not to be confused with Product category.
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.
1.1 Product of two objects
1.2 Product of an arbitrary family
1.3 Equational definition
1.4 As a limit
Product of two objects[edit]
Fix a category
{\displaystyle C.}
{\displaystyle X_{1}}
{\displaystyle X_{2}}
be objects of
{\displaystyle C.}
{\displaystyle X_{1}}
{\displaystyle X_{2}}
{\displaystyle X,}
{\displaystyle X_{1}\times X_{2},}
equipped with a pair of morphisms
{\displaystyle \pi _{1}:X\to X_{1},}
{\displaystyle \pi _{2}:X\to X_{2}}
satisfying the following universal property:
For every object
{\displaystyle Y}
and every pair of morphisms
{\displaystyle f_{1}:Y\to X_{1},}
{\displaystyle f_{2}:Y\to X_{2},}
{\displaystyle f:Y\to X_{1}\times X_{2}}
Whether a product exists may depend on
{\displaystyle C}
{\displaystyle X_{1}}
{\displaystyle X_{2}.}
If it does exist, it is unique up to canonical isomorphism, because of the universal property, so one may speak of the product. This has the following meaning: let
{\displaystyle X',\pi _{1}',\pi _{2}'}
be another cartesian product, there exists a unique isomorphism
{\displaystyle h:X'\to X_{1}\times X_{2}}
{\displaystyle \pi _{1}'=\pi _{1}\circ h}
{\displaystyle \pi _{2}'=\pi _{2}\circ h}
{\displaystyle \pi _{1}}
{\displaystyle \pi _{2}}
are called the canonical projections or projection morphisms. Given
{\displaystyle Y}
{\displaystyle f_{1},}
{\displaystyle f_{2},}
the unique morphism
{\displaystyle f}
is called the product of morphisms
{\displaystyle f_{1}}
{\displaystyle f_{2}}
{\displaystyle \langle f_{1},f_{2}\rangle .}
Product of an arbitrary family[edit]
Instead of two objects, we can start with an arbitrary family of objects indexed by a set
{\displaystyle I.}
Given a family
{\displaystyle \left(X_{i}\right)_{i\in I}}
of objects, a product of the family is an object
{\displaystyle X}
equipped with morphisms
{\displaystyle \pi _{i}:X\to X_{i},}
{\displaystyle Y}
{\displaystyle I}
-indexed family of morphisms
{\displaystyle f_{i}:Y\to X_{i},}
{\displaystyle f:Y\to X}
such that the following diagrams commute for all
{\displaystyle i\in I:}
The product is denoted
{\displaystyle \prod _{i\in I}X_{i}.}
{\displaystyle I=\{1,\ldots ,n\},}
then it is denoted
{\displaystyle X_{1}\times \cdots \times X_{n}}
and the product of morphisms is denoted
{\displaystyle \langle f_{1},\ldots ,f_{n}\rangle .}
Equational definition[edit]
Alternatively, the product may be defined through equations. So, for example, for the binary product:
Existence o{\displaystyle f}
is guaranteed by existence of the operation
{\displaystyle \langle \cdot ,\cdot \rangle .}
Commutativity of the diagrams above is guaranteed by the equality: for all
{\displaystyle f_{1},f_{2}}
{\displaystyle i\in \{1,2\},}
{\displaystyle \pi _{i}\circ \left\langle f_{1},f_{2}\right\rangle =f_{i}}
Uniqueness o{\displaystyle f}
is guaranteed by the equality: for all
{\displaystyle g:Y\to X_{1}\times X_{2},}
{\displaystyle \left\langle \pi _{1}\circ g,\pi _{2}\circ g\right\rangle =g.}
As a limit[edit]
The product is a special case of a limit. This may be seen by using a discrete category (a family of objects without any morphisms, other than their identity morphisms) as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set
{\displaystyle I}
considered as a discrete category. The definition of the product then coincides with the definition of the limit,
{\displaystyle \{f\}_{i}}
being a cone and projections being the limit (limiting cone).
Just as the limit is a special case of the universal construction, so is the product. Starting with the definition given for the universal property of limits, take
{\displaystyle \mathbf {J} }
as the discrete category with two objects, so that
{\displaystyle \mathbf {C} ^{\mathbf {J} }}
is simply the product category
{\displaystyle \mathbf {C} \times \mathbf {C} .}
The diagonal functor
{\displaystyle \Delta :\mathbf {C} \to \mathbf {C} \times \mathbf {C} }
assigns to each object
{\displaystyle X}
{\displaystyle (X,X)}
and to each morphism
{\displaystyle f}
{\displaystyle (f,f).}
{\displaystyle X_{1}\times X_{2}}
{\displaystyle C}
is given by a universal morphism from the functor
{\displaystyle \Delta }
to the object
{\displaystyle \left(X_{1},X_{2}\right)}
{\displaystyle \mathbf {C} \times \mathbf {C} .}
This universal morphism consists of an object
{\displaystyle X}
{\displaystyle C}
{\displaystyle (X,X)\to \left(X_{1},X_{2}\right)}
which contains projections.
In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets
{\displaystyle X_{i}}
the product is defined as
{\displaystyle \prod _{i\in I}X_{i}:=\left\{\left(x_{i}\right)_{i\in I}:x_{i}\in X_{i}{\text{ for all }}i\in I\right\}}
with the canonical projections
{\displaystyle \pi _{j}:\prod _{i\in I}X_{i}\to X_{j},\quad \pi _{j}\left(\left(x_{i}\right)_{i\in I}\right):=x_{j}.}
{\displaystyle Y}
with a family of functions
{\displaystyle f_{i}:Y\to X_{i},}
the universal arrow
{\displaystyle f:Y\to \prod _{i\in I}X_{i}}
{\displaystyle f(y):=\left(f_{i}(y)\right)_{i\in I}.}
In the category of topological spaces, the product is the space whose underlying set is the Cartesian product and which carries the product topology. The product topology is the coarsest topology for which all the projections are continuous.
In the category of modules over some ring
{\displaystyle R,}
the product is the Cartesian product with addition defined componentwise and distributive multiplication.
In the category of groups, the product is the direct product of groups given by the Cartesian product with multiplication defined componentwise.
In the category of graphs, the product is the tensor product of graphs.
In the category of relations, the product is given by the disjoint union. (This may come as a bit of a surprise given that the category of sets is a subcategory of the category of relations.)
In the category of algebraic varieties, the product is given by the Segre embedding.
In the category of semi-abelian monoids, the product is given by the history monoid.
A partially ordered set can be treated as a category, using the order relation as the morphisms. In this case the products and coproducts correspond to greatest lower bounds (meets) and least upper bounds (joins).
An example in which the product does not exist: In the category of fields, the product
{\displaystyle \mathbb {Q} \times F_{p}}
does not exist, since there is no field with homomorphisms to both
{\displaystyle \mathbb {Q} }
{\displaystyle F_{p}.}
Another example: An empty product (that is,
{\displaystyle I}
is the empty set) is the same as a terminal object, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group
{\displaystyle G}
there are infinitely many morphisms
{\displaystyle \mathbb {Z} \to G,}
{\displaystyle G}
cannot be terminal.
{\displaystyle I}
is a set such that all products for families indexed with
{\displaystyle I}
exist, then one can treat each product as a functor
{\displaystyle \mathbf {C} ^{I}\to \mathbf {C} .}
[2] How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the binary product functor, which is a bifunctor. For
{\displaystyle f_{1}:X_{1}\to Y_{1},f_{2}:X_{2}\to Y_{2}}
we should find a morphism
{\displaystyle X_{1}\times X_{2}\to Y_{1}\times Y_{2}.}
{\displaystyle \left\langle f_{1}\circ \pi _{1},f_{2}\circ \pi _{2}\right\rangle .}
This operation on morphisms is called Cartesian product of morphisms.[3] Second, consider the general product functor. For families
{\displaystyle \left\{X\right\}_{i},\left\{Y\right\}_{i},f_{i}:X_{i}\to Y_{i}}
{\displaystyle \prod _{i\in I}X_{i}\to \prod _{i\in I}Y_{i}.}
We choose the product of morphisms
{\displaystyle \left\{f_{i}\circ \pi _{i}\right\}_{i}.}
A category where every finite set of objects has a product is sometimes called a Cartesian category[3] (although some authors use this phrase to mean "a category with all finite limits").
The product is associative. Suppose
{\displaystyle C}
is a Cartesian category, product functors have been chosen as above, and
{\displaystyle 1}
denotes a terminal object of
{\displaystyle C.}
We then have natural isomorphisms
{\displaystyle X\times (Y\times Z)\simeq (X\times Y)\times Z\simeq X\times Y\times Z,}
{\displaystyle X\times 1\simeq 1\times X\simeq X,}
{\displaystyle X\times Y\simeq Y\times X.}
These properties are formally similar to those of a commutative monoid; a Cartesian category with its finite products is an example of a symmetric monoidal category.
Main article: Distributive category
For any objects
{\displaystyle X,Y,{\text{ and }}Z}
of a category with finite products and coproducts, there is a canonical morphism
{\displaystyle X\times Y+X\times Z\to X\times (Y+Z),}
where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct
{\displaystyle X\times Y+X\times Z}
guarantees the existence of unique arrows filling out the following diagram (the induced arrows are dashed):
The universal property of the product
{\displaystyle X\times (Y+Z)}
then guarantees a unique morphism
{\displaystyle X\times Y+X\times Z\to X\times (Y+Z)}
induced by the dashed arrows in the above diagram. A distributive category is one in which this morphism is actually an isomorphism. Thus in a distributive category, there is the canonical isomorphism
{\displaystyle X\times (Y+Z)\simeq (X\times Y)+(X\times Z).}
Coproduct – the dual of the product
Diagonal functor – the left adjoint of the product functor.
Limit and colimits – Mathematical concept
Equalizer – Set of arguments where two or more functions have the same value
Inverse limit – Construction in category theory
Cartesian closed category – Type of category in category theory
Categorical pullback
^ Lambek J., Scott P. J. (1988). Introduction to Higher-Order Categorical Logic. Cambridge University Press. p. 304.
^ Lane, S. Mac (1988). Categories for the working mathematician (1st ed.). New York: Springer-Verlag. p. 37. ISBN 0-387-90035-7.
^ a b Michael Barr, Charles Wells (1999). Category Theory – Lecture Notes for ESSLLI. p. 62. Archived from the original on 2011-04-13.
Adámek, Jiří; Horst Herrlich; George E. Strecker (1990). Abstract and Concrete Categories (PDF). John Wiley & Sons. ISBN 0-471-60922-6.
Barr, Michael; Charles Wells (1999). Category Theory for Computing Science (PDF). Les Publications CRM Montreal (publication PM023). Archived from the original (PDF) on 2016-03-04. Retrieved 2016-03-21. Chapter 5.
Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5 (2nd ed.). Springer. ISBN 0-387-98403-8.
Definition 2.1.1 in Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50–51, 53 [i.e. 52]. Vol. 1. Cambridge University Press. p. 39. ISBN 0-521-44178-1.
Interactive Web page which generates examples of products in the category of finite sets. Written by Jocelyn Paine.
Product in nLab
Retrieved from "https://en.wikipedia.org/w/index.php?title=Product_(category_theory)&oldid=1089785982"
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Account : How do I get a SUPA account?
A My.SUPA account can be automatically issued if you are a member of staff or PhD student in Physics or a related area and have a university email address from a SUPA partner
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Quick self-service My.SUPA account setup for staff and PhD students
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For other account requests please use the form below.
Use the 'other account requests' if you are not a staff member or PhD student in one of the SUPA universities.
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Example: Type the following without the spaces between the dollar signs:
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PowerPoint - I'm not a Windows user. How can I view PowerPoint files?
Struggling to view your lecturer's slides? Try the TonicPoint viewer. http://tonicsystems.com/products/viewer/
Keyword(s): PowerPointplatform
Rooms. How to book one or more SUPA VC Rooms
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To book one or more SUPA VC room, please contact the local manager of the room to check availability and make the booking online @ http://www.supa.ac.uk/room_booking with your request.
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Rooms. Where are the Videoconferencing Rooms?
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6 |x|> 18
x < −3 \text{ or } x > 3
| 3 x - 2 |≤ 2
Begin by changing the inequality to an equation and solve for the boundary points.
\left|3x-2\right|=2
\left|±2\right| = 2
, this can be rewritten as two equations. Solve the two equations for
x
and plot the boundary points on a number line.
3x − 2 = 2 \quad \quad 3x − 2 = −2
Test points on the number line in the original inequality to find the regions that make it true. Shade them and describe them algebraically.
(4x − 2)^2 = 100
4x − 2 = 10 \quad \quad 4x − 2 = −10
(x−1)^3=8
Look inside. The cube root of
8
2
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High Capacity Implantable Data Recorders: System Design and Experience in Canines and Denning Black Bears | J. Biomech Eng. | ASME Digital Collection
Timothy G. Laske,
Timothy G. Laske
Departments of Surgery and Physiology,
, Minneapolis, MN 55432 and
Henry J. Harlow,
Department of Zoology and Physiology,
Jon C. Werder,
Jon C. Werder
Mark T. Marshall,
Laske, T. G., Harlow, H. J., Werder, J. C., Marshall, M. T., and Iaizzo, P. A. (July 29, 2005). "High Capacity Implantable Data Recorders: System Design and Experience in Canines and Denning Black Bears." ASME. J Biomech Eng. November 2005; 127(6): 964–971. https://doi.org/10.1115/1.2049340
Background: Implantable medical devices have increasingly large capacities for storing patient data as a diagnostic aid and to allow patient monitoring. Although these devices can store a significant amount of data, an increased ability for data storage was required for chronic monitoring in recent physiological studies. Method of Approach: Novel high capacity implantable data recorders were designed for use in advanced physiological studies of canines and free-ranging black bears. These hermitically sealed titanium encased recorders were chronically implanted and programmed to record intrabody broadband electrical activity to monitor electrocardiograms and electromyograms, and single-axis acceleration to document relative activities. Results: Changes in cardiac T-wave morphology were characterized in the canines over a
6month
period, providing new physiological data for the design of algorithms and filtering schemes that could be employed to avoid inappropriate implantable defibrillator shocks. Unique characteristics of bear hibernation physiology were successfully identified in the black bears, including: heart rate, respiratory rate, gross body movement, and shiver. An unanticipated high rejection rate of these devices occurred in the bears, with five of six being externalized during the overwintering period, including two devices implanted in the peritoneal cavity. Conclusions: High capacity implantable data recorders were designed and utilized for the collection of long-term physiological data in both laboratory and extreme field environments. The devices described were programmable to accommodate the diverse research protocols. Additionally, we have described substantial differences in the response of two species to a common device. Variations in the foreign body response of different mammals must be identified and taken into consideration when choosing tissue-contacting materials in the application of biomedical technology to physiologic research.
defibrillators, prosthetics, patient monitoring, dentistry, patient diagnosis, electrocardiography, electromyography, skin, cardiology, Chronic, Titanium Canisters, Programmable, Subcutaneous, Intraperitoneal, Lithium Batteries, Flash Memory Card, Foreign Body Response
Cables, Computer programming, Design, Electromyography, Patient monitoring, Physiology, Signals, Testing, Titanium, Waves, Data collection, Biomedicine, Cavities, Data storage systems, Biological tissues, Lithium, Electronics, Algorithms
Compass-HF Study. Retrieved March 30, 2005 from http://wwwp.medtronic.com/Newsroom/NewsReleaseDetails.do?itemId=1110237750252&lang=en_UShttp://wwwp.medtronic.com/Newsroom/NewsReleaseDetails.do?itemId=1110237750252&lang=en_US.
Reveal Plus: Information for health professionals. Retrieved March 30, 2005 from http://www.medtronic.com/reveal/revealplus.htmlhttp://www.medtronic.com/reveal/revealplus.html.
The Handbook of Cardiac Anatomy, Physiology and Devices
Techniques and Devices for Extraction of Pacemaker and Implantable Cardioverter-Defibrillator Leads
ICD System T-Wave Changes Impact of Lead and Time
Respiratory Sinus Arrhythmia in Humans: How Breathing Pattern Modulates Heart Rate
Evaluation of Subcutaneous Implants for Monitoring American Black Bear Cub Survival
Muscle Strength in Overwintering Bears
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Implement first-order lead-lag filter - Simulink - MathWorks Nordic
Implement first-order lead-lag filter
The Lead-Lag Filter block implements the following transfer function:
H\left(s\right)=\frac{1+{T}_{1}s}{1+{T}_{2}s}
\begin{array}{c}s=\text{Laplace operator}\\ {T}_{1},{T}_{2}=\text{time constants}\end{array}
This type of filter is used mainly for implementing lead-lag compensation in control systems. The key characteristics of the Lead-Lag Filter block are:
Input accepts a vectorized input of N signals, thus implementing N filters. This feature is particularly useful for designing controllers in three-phase systems (N=3).
The same block is used for continuous or discrete model. Changing the sample time Ts from 0 to a positive value automatically discretizes the filter, and vice versa.
Filter states can be initialized for specified DC inputs and outputs.
Time constant T1 (s)
Specify the filter time constant(s) T1 in seconds. Default is 5e-3.
Specify the filter time constant(s) T2 in seconds. Default is 20e-3.
DC initial input and output
Specify the DC initial value of input and output signals. If the input signal is vectorized, specify a 1-by-N vector, where each value corresponds to a particular input. Default is 0.
States One state per filter
The power_LeadLagFilter example shows two uses of a vectorized Lead-Lag Filter.
The model sample time is parameterized with variable Ts (default value Ts = 50e-6). To simulate continuous filters, specify Ts = 0 in the MATLAB® Command Window before simulating the model.
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How to Read a Logarithmic Scale
1 Reading the Axes of the Graph
2 Plotting Points on a Logarithmic Scale
Most people are familiar with reading numbers on a number line or reading data from a graph. However, under certain circumstances, a standard scale may not be useful. If the data grows or decreases exponentially, then you will need to use what is called a logarithmic scale. For example, a graph of the number of McDonald’s hamburgers sold over time would start at 1 million in 1955; then 5 million just a year later; then 400 million, 1 billion (in less than 10 years), and up to 80 billion by 1990.[1] X Research source This data would be too much for a standard graph, but it is easily displayed on a logarithmic scale. You need to understand that a logarithmic scale has a different system of displaying the numbers, which are not evenly spaced as on a standard scale. By knowing how to read a logarithmic scale you can more effectively read and represent data in graphic form.
Reading the Axes of the Graph
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Determine whether you are reading a semi-log or log-log graph. Graphs that represent rapidly growing data can use one-log scales or two-log scales. The difference is in whether both the x-axis and y-axis use logarithmic scales, or only one.[2] X Research source The choice depends on the amount of detail that you wish to display with your graph. If numbers on one axis or the other grow or decrease exponentially, you may wish to use a logarithmic scale for that axis.
A logarithmic (or just "log") scale has unevenly spaced grid lines. A standard scale has evenly spaced grid lines. Some data needs to be graphed on standard paper only, some on semi-log graphs, and some on log-log graphs.
{\displaystyle y={\sqrt {x}}}
(or any similar function with a radical term) can be graphed on a purely standard graph, a semi-log graph, or log-log graph. On a standard graph, the function appears as a sideways parabola, but the detail for very small numbers is difficult to see. On the log-log graph, the same function appears as a straight line, and the values are more spread out for better detail.[3] X Research source
If both variables in a study include great ranges of data, you would probably use a log-log graph. Studies of evolutionary effects, for example, may be measured in thousands or millions of years and might choose a logarithmic scale for the x-axis. Depending on the item being measured, a log-log scale may be necessary.
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Read the scale of the main divisions. On a logarithmic scale graph, the evenly spaced marks represent the powers of whatever base you are working with. The standard logarithms use either base 10 or the natural logarithm which uses the bas{\displaystyle e}
{\displaystyle e}
is a mathematical constant that is useful in working with compound interest and other advanced calculations. It is approximately equal to 2.718.[4] X Research source This article will focus on the base-10 logarithms, but the reading the natural logarithm scale operates in the same way.
Standard logarithms use base 10. Instead of counting 1, 2, 3, 4… or 10, 20, 30, 40… or some other evenly spaced scale, a logarithm scale counts by powers of 10. The main axis points are, therefore,
{\displaystyle 10^{1},10^{2},10^{3},10^{4}}
and so on.[5] X Research source
Each of the main divisions, usually noted on log paper with a darker line, is called a "cycle." When specifically using based 10, you can use the term "decade" because it refers to a new power of 10.
Notice that the minor intervals are not evenly spaced. If you are using printed logarithmic graph paper, you will notice that the intervals between the main units are not evenly spaced. That is, for example, the mark for 20 would actually be placed about 1/3 of the way between 10 and 100.[6] X Research source
The minor interval marks are based on the logarithm of each number. Therefore, if 10 is represented as the first major mark on the scale, and 100 is the second, the other numbers fall in between as follows:
{\displaystyle log(10)=1}
{\displaystyle log(20)=1.3}
{\displaystyle log(30)=1.48}
{\displaystyle log(40)=1.60}
{\displaystyle log(50)=1.70}
{\displaystyle log(60)=1.78}
{\displaystyle log(70)=1.85}
{\displaystyle log(80)=1.90}
{\displaystyle log(90)=1.95}
{\displaystyle log(100)=2.00}
At higher powers of 10, the minor intervals are spaced in the same ratios. Thus, the spacing between 10, 20, 30… looks like the spacing between 100, 200, 300… or 1000, 2000, 3000….
Plotting Points on a Logarithmic Scale
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Determine the type of scale you wish to use. For the explanation given below, the focus will be on a semi-log graph, using a standard scale for the x-axis and a log scale for the y-axis. However, you may wish to reverse these, depending on how you want the data to appear. Reversing the axes has the effect of shifting the graph by ninety degrees and may make the data more easily interpreted in one direction or the other. Additionally, you may wish to use a log scale to spread out certain data values and make their details more visible.[7] X Research source
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Mark the x-axis scale. The x-axis is the independent variable. The independent variable is the one that you generally control in a measurement or experiment. The independent variable is not affected by the other variable in the study. Some examples of independent variables may be such things as:[8] X Research source
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Determine that you need a logarithmic scale for the y-axis. You will use a logarithmic scale to graph data that changes extremely quickly. A standard graph is useful for data that grows or decreases at a linear rate. A logarithmic graph is for data that changes at an exponential rate. Samples of such data might be:
Product consumption rates
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Label the logarithmic scale. Review your data and decide how to mark the y-axis. If your data measures numbers only within, for example, the millions and billions, you probably do not need to have your graph begin at 0. You could label the lowest cycle on the graph as
{\displaystyle 10^{6}}
. Subsequent cycles would be
{\displaystyle 10^{7},10^{8},10^{9}}
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Find the position on the x-axis for a data point. To graph the first (or any) data point, you begin by finding its position along the x-axis. This may be an incremental scale, such as a regular number line that counts 1, 2, 3, and so on. It may be a scale of labels that you assign, such as dates or months of the year when you take certain measurements.
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Locate the position along the logarithmic scale y-axis. You need to find the corresponding position along the y-axis for the data that you wish to plot. Recall that, since you are working with a logarithmic scale, the major markings are powers of 10, and the minor scale markings in between represent the subdivisions. For example, between
{\displaystyle 10^{6}}
(one million) and
{\displaystyle 10^{7}}
(ten million), the lines represent divisions of 1,000,000’s.[9] X Research source
For example, the number 4,000,000 would be graphed at the fourth minor scale mark above
{\displaystyle 10^{6}}
. Even though, on a standard linear scale, 4,000,000 is less than halfway between 1,000,000 and 10,000,000, because of the logarithmic scale, it actually appears slightly more than halfway.
You should note that the higher intervals, closer to the upper limit, become squeezed together. This is due to the mathematical nature of the logarithmic scale.
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Continue with all the data. Continue repeating the previous steps for all the data that you need to graph. For each data point, first locate its position along the x-axis, and then find its corresponding location along the logarithmic scale of the y-axis.
Is log scale also calculus?
Log scales and similar applications of logarithms are not part of calculus, but are a significant topic in pre-calc. Historically, logarithms came first; Newton had ready access to log scale slide rules when he was inventing calculus. However, the natural logarithm and properties of Euler's constant e are topics for calculus, and don't make a lot of sense without it.
When reading data off a logarithmic scale, be sure you know what base is used for the logarithm. Data measured in base 10 will be very different than data measured on a natural log scale with base e.
↑ https://www3.physics.uoguelph.ca/tutorials/GLP/
When reading a logarithmic scale, the evenly spaced marks represent the powers of whatever base you are working with. Standard logarithms use base 10, so a logarithm scale counts by powers of 10. Each of the main divisions, noted on log paper with a darker line, is called a cycle or decade. The minor intervals are not evenly spaced since their value is based on the logarithm for each number. To learn how to plot points on a logarithmic scale, keep reading!
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I want the solution ∴1 620=x27 and 1 620=2 4yx=27×1 620 and y=2 4×201 6x = 2 16 kg - Maths - Direct Proportion and Inverse Proportion - 12457115 | Meritnation.com
I want the solution?
\therefore \frac{1.6}{20}=\frac{x}{27} and \frac{1.6}{20}=\frac{2.4}{y}\phantom{\rule{0ex}{0ex}}x=\frac{27×1.6}{20} and y=\frac{2.4×20}{1.6}\phantom{\rule{0ex}{0ex}}x = 2.16 kg and y = 30 people\phantom{\rule{0ex}{0ex}} x = 2 kg 160 g and y = 30 people
It is already solved so please mention specifically about the doubt so that we may help you appropriately.
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J. Fuel Cell Sci. Technol. August 2009, 6(3): 031001. doi: https://doi.org/10.1115/1.3005385
Topics: Anodes , Floods , Fuel cells , Hydrogen , Membranes , Oxygen , Temperature , Temperature distribution , Proton exchange membrane fuel cells , Water
Topics: Current density , Electrolytes , Flow (Dynamics) , Fuels , Solid oxide fuel cells , Fuel consumption
Topics: Catalysts , Design , Electrolysis , Fuel cells , Gas diffusion layers , Hydrogen , Oxygen , Water , Membranes , Electrodes
Topics: Electrospinning , Nanofibers , Proton exchange membrane fuel cells , Polymers , Fibers , Electrodes , Ethanol
Topics: Composite materials , Elastomers , Electrical resistivity , Fillers (Materials) , Graphite , Proton exchange membrane fuel cells , Silicones , Graphite fibers , Plates (structures) , Polymers
Matthew A. Howe, David N. Rocheleau
Topics: Control equipment , Proton exchange membrane fuel cells , Feedforward control , Fuel cells , Compressors , Oxygen , Predictive control
Topics: Electrolytes , Flow (Dynamics) , Fuel cells , Gas diffusion layers , Polymers , Two-phase flow , Water , Design , Channel flow , Floods
Mario L. Ferrari, Matteo Pascenti, Roberto Bertone, Loredana Magistri
Topics: Compressors , Flow (Dynamics) , Fuel cells , Machinery , Micro gas turbines , Pipes , Pressure , Solid oxide fuel cells , Stress , Valves
Topics: Cold ironing , Exhaust systems , Fuel cells , Fuels , Heat , Heat recovery steam generators , Solid oxide fuel cells , Steam , Water , Emissions
Cathode Properties of
SmxSr1−x(Co,Fe,Ni)O3−δ/Sm0.2Ce0.8O1.9
Composite Material for Intermediate Temperature-Operating Solid Oxide Fuel Cell
Seung-Wook Baek, Changbo Lee, Joongmyeon Bae
Topics: Composite materials , Solid oxide fuel cells , Temperature , Thermal expansion , Electrolytes , Electrodes
Abhijit Mukherjee, Anthony Bourassa
Topics: Drops , Flow (Dynamics) , Proton exchange membrane fuel cells , Slug flows , Water
Jong-Hee Kim, Rak-Hyun Song, Dong-Ryul Shin
Topics: Anodes , Brazing , Electromagnetic induction , Solid oxide fuel cells , Fillers (Materials) , Electrical conductivity , Joining , Permeability
Oxidation Behavior of Various Metallic Alloys for Solid Oxide Fuel Cell Interconnect
Chun-Lin Chu, Jian-Yih Wang, Ruey-Yi Lee, Tien-Hsi Lee, Shyong Lee
Topics: Alloys , Oxidation , Solid oxide fuel cells , Iron alloys
Topics: Creep , Durability , Failure , Leakage , Membranes , Pressure , Proton exchange membranes , Rupture , Stress , Testing
System Architectures for Solid Oxide Fuel Cell-Based Auxiliary Power Units in Future Commercial Aircraft Applications
R. J. Braun, M. Gummalla, J. Yamanis
Topics: Aircraft , Fuels , Solid oxide fuel cells , System architecture
Yong-Song Chen, Huei Peng
Topics: Anodes , Flow (Dynamics) , Fuel cells , Gas diffusion layers , Neutron radiography , Sensors , Water , Proton exchange membrane fuel cells
Innovative Design of an Air-Breathing Proton Exchange Membrane Fuel Cell With a Piezoelectric Device
Hsiao-Kang Ma, Shih-Han Huang
Topics: Proton exchange membrane fuel cells , Design
Topics: Design , Flow (Dynamics) , Fuel cells , Fuels , Hydrogen , Micro fuel cells , Polarization (Electricity) , Polarization (Light) , Polarization (Waves) , Proton exchange membrane fuel cells
Polarization Resistances of
(Ln1−xSrx)CoO3
Ln=Pr
, Nd, Sm, and Gd
x=0
, 0.3, 0.5, 0.7, and 1) as Cathode Materials for Intermediate Temperature-operating Solid Oxide Fuel Cells
Topics: Solid oxide fuel cells , Temperature , Polarization (Electricity) , Polarization (Light) , Polarization (Waves) , Thermoelectric coolers , Thermal expansion , Electrochemical impedance spectroscopy
A Solar-Hydrogen Fuel-Cell Home and Research Platform
David J. Palmer, Gregory D. Sachs, William J. Sembler
Topics: Fuel cells , Hydrogen , Solar energy
Topics: Frequency response , Gas turbines , Solid oxide fuel cells , Transfer functions , Fuel cells , Valves , Fuels , Temperature , Turbines , Flow (Dynamics)
Hydrocarbon Condensation Heating of Natural Gas by an Activated Carbon Desulfurizer
Topics: Activated carbon , Condensation , Heating , Latent heat , Natural gas , Water vapor
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Lesson 6.1 Physical Science
A car races by. A light is switched on. Sound filters out from speakers. These are all forms of energy-motion, light, and sound. Energy is the ability to make things happen. Every action is connected to energy in one form or another. Objects can have energy due to their movement or their position.
Scientists measure energy in joules (J). It takes about one joule for a person to lift an apple one meter off the ground. Eating the apple provides the human body with about 250,000 J. Every form of energy, including movement, stored energy, heat, and light, can be measured in joules.
Kinetic energy is the energy of an object’s or particle’s motion. The amount of kinetic energy depends on two things: the object’s mass and how fast the object is moving. The amount of kinetic energy (KE) in joules that an object has is determined by the equation
KE = \frac{1}{2}m v^2 w
, here m equals the mass of the object in kilograms and v equals its velocity in meters per second. Energy can be expressed in units of joules, where 1 joule is 1 (kg•m2)/2.
If two objects have equal mass, the object that is moving faster has more kinetic energy. The diagram at the top of the next page shows calculations for the kinetic energy for three vehicles. Notice that Car Band Car C have equal mass, but Car C has more kinetic energy than Car B because it is moving faster. If two objects are moving at the same speed, the object with more mass has more kinetic energy than the object with less mass. As shown in the diagram, Truck A and Car B are traveling at the same speed. However, Truck A has more kinetic energy than Car B because Truck A has more mass. Notice that a change in speed affects energy more than a change in mass. If a car doubles its speed, its kinetic energy increases by a factor of four. A truck with four times the mass of a car has four times as much energy as the car when they travel at the same speed.
Potential energy is energy an object has due to its position. Potential energy does not involve motion, it is dependent on the interaction between two objects and the forces involved. This is considered stored energy.
Consider a book on a table. Together, the book and Earth have potential energy. Gravitational potential energy is the energy resulting from the gravitational forces between two objects. Raising an object above the ground increases the gravitational potential energy because work has been done on the object against the force of gravity. Gravity is a force that is described as a field, meaning there is a region in space that has this force at every point.
Gravitational potential energy (GPE) is related to the mass and height of the object, and acceleration due to the gravitational field. This can be expressed as GPE = mgh, where m is the mass of the object in kilograms, g is the acceleration due to gravity in meters per second squared (9.80 m/s2 near Earth’s surface), and his the height the object is raised in meters.
If you lift a 2.00 kilogram book to a shelf 1.20 meters above the floor, what is the change in potential energy in joules? The gravitational field of the book is not considered because the field of an object only affects other objects.
The gravitational potential energy will increase by 23.5 J.
Potential energy can also be affected by other field forces between objects. For example, a magnetic field exerts a force on a paper clip that pulls the paper clip toward a magnet. The field forces around charged particles are called electrical fields. When a charged particle or object is moved a distance against the force of this field, its electric potential increases.
A leaf falls from a tree to the ground. At what point is the gravitational potential energy the greatest? A. while the leaf is still attached to the branch B. after the leaf has fallen a short distance C. when the leaf is about half way to the ground D. when the leaf hits the ground
You constantly use energy in your daily activities. When you turn on the lights or heat food in a microwave, you know you are using some type of energy. In other cases, your interactions with energy are less obvious. When you are sleeping, your body is using energy to maintain your internal temperature, breathe, digest food, and repair injured cells. There are many types of energy that constantly do work and cause changes around you.
The mechanical energy of an object is the sum of its kinetic energy and its potential energy. As shown below, when the roller coaster is at the top of the hill, all of its energy is stored as gravitational potential energy. When the cars travel down the hill, their kinetic energy increases and the gravitational potential energy decreases by an equal amount. Not counting friction, the mechanical energy remains the same throughout the entire ride as the cars move up and down the hills of the roller coaster.
If you throw a ball in the air, the kinetic energy of its upward motion will decrease as gravitational potential energy increases. When the ball has reached its highest height, it has no kinetic energy at all. As the ball falls back to the ground, kinetic energy again increases and gravitational potential energy decreases by an equal amount. Overall the total amount of mechanical energy does not change as the ball moves from one position to another.
While it may not be obvious, every piece of matter around you is full of energy. The atoms and molecules that make up matter are always interacting. Liquid or gas particles flow from place to place, and even the molecules of a solid constantly vibrate. Thermal energy is the sum of the kinetic energy and the potential energy of the particles that make up matter.
Thermal energy can be detected when it flows from one object to another as heat. Faster-moving particles have more kinetic energy than slower moving particles. When the particles collide, energy transfers from the faster particles to the slower particles. When you touch a hot pan, some of the energy of the rapidly vibrating metal atoms is transferred to your hand, in which atoms are moving more slowly. Wind is generated when heat is transferred from areas of the atmosphere with greater thermal energy to other regions with less thermal energy.
When a match is struck, it emits light, sound, and thermal energy. All this energy had been stored in the match in the form of chemical energy. Chemical energy is the potential energy stored in the bonds between the atoms of a substance. The sources of this stored energy are the electromagnetic force fields of the charged particles that make up the atoms. Interactions among these fields provide energy that can be released during a chemical reaction.
Chemical energy is the source of most of the energy humans need to function. Plants store chemical energy in carbohydrates formed during photosynthesis. Humans release this chemical energy during digestion and use it to power systems inside the body.
The nuclei of atoms contain a tremendous amount of potential energy. Energy stored in the nucleus of an atom is called nuclear energy. Nuclear energy holds the particles of the nucleus together. It can be released when nuclei are combined, which occurs in reactions in the Sun. Nuclear energy can also be released when nuclei are split apart, which occurs in nuclear reactors on Earth. One kilogram of uranium used as fuel in a nuclear power plant produces the same amount of energy as 14,000 kilograms of coal burned in a coal-fired power plant.
Radiant energy is emitted from a source as waves. These waves carry energy from the Sun through the vacuum of space to Earth. Radiant energy is a form of kinetic energy. In addition to the light we can see, radiant energy includes radio waves, microwaves, infrared radiation, ultraviolet radiation, gamma rays, and X-rays.
Directions: Fill in the blank.
Turning on a flashlight releases [ blank ] energy.
Thermal energy is the sum of the [ blank ] and potential energy of the particles in an object.
Energy stored in the nucleus of an atom is [ blank ] energy.
Chemical energy is a type of potential energy stored in [ blank ].
Some objects can affect other objects from a distance due to a force field that exists around the them. A force field is a push or pull exerted in a region around the object producing it. Electrical energy and magnetic energy are both the result of fields. These forms of energy are related to one another.
You may have experienced a shock after walking across a carpet and then touching a metal object, such as a doorknob. The shock comes from a transfer of electric charge. There are two types of electric charge: positive and negative. Two charges that are alike repel one another, and two charges that are different attract one another. Electrons are negatively charged atomic particles that naturally repel each other through the interaction of the electric fields that surround each electron. Electrical potential energy is the result of the positions of the charged particles within the electric fields. The friction of shuffling feet on a carpet rubs electrons from the carpet onto the feet. This buildup of charge generates electrical potential energy. Potential energy becomes kinetic energy when a static shock carries the charges toward positive charges located on the doorknob. It would be reasonable to expect, or anticipate, a static shock when you shuffle your feet on a carpet.
Electrons in a circuit have both kinetic and potential energy. When the electrons travel through a closed path, or electric circuit, some of its kinetic and potential energy can be changed to other forms of energy, such as light, thermal energy, or sound. Electrical energy powers many appliances and machines at home and at work.
A magnet produces a force that can attract or repel other magnets and can attract certain other substances. You can feel this by holding two magnets near each other. Depending on how you hold the magnets, you can feel them push or pull on each other. This push or pull is due to the force of a magnetic field. The field is produced by moving electrons as in atoms. The magnetic field is exerted in a region surrounding the magnet, and it is strongest close to the magnet. The magnetic field stretches between two magnetic poles, which are regions where the magnetic field exerted by a magnet is the strongest. The north and south poles are at opposite ends of a bar magnet.
When two magnets are brought close together, their magnetic fields interact with each other. As shown here, the north pole of one magnet will repel the north pole of another magnet. South poles also repel each other. The north pole of one magnet and the south pole of another magnet, however, attract each other and stick together.
Earth has a magnetic field that extends from its North and South Magnetic Poles. When a compass is allowed to line up with Earth’s magnetic field, the end labeled with an N points toward the magnetic North Pole. Based on this observation, if Earth’s North Pole were a labeled magnet, should it be labeled as N or as S?
People often talk about energy as if it were used up or lost during an activity. However, the energy still exists, just in a form that may not be obvious. The law of conservation of energy states that energy can be changed in form but it cannot be created or destroyed.
It can take some detective work to follow the path of energy as it changes forms. For example, it takes a lot of energy to run a race. After the race, the runner’s body has less energy than it had before. The chemical energy used to power muscles has been converted into kinetic energy and thermal energy. The total amount of energy in the universe is the same after the race as it was before.
We observe changes in energy all the time. For example, when an object moves against gravity, some of its kinetic energy is transformed to potential energy. An electric circuit that includes a light bulb transforms electrical energy into radiant energy and thermal energy. The change of one form of energy to another form of energy is called energy transformation.
Energy can change forms in many different ways. Energy conversions occur continually in living things. The human body provides many examples of energy transformations. The body takes in chemical potential energy in the form of food. The food is transformed into other chemicals in the digestive system. Sugars provide chemical energy for bodily functions, and fats store potential energy for future use. The heart and other muscles convert chemical energy to kinetic energy as blood circulates and the body moves. Some of the body’s energy is transformed to sound. The body releases thermal energy in the form of heat. Nerves use electrical energy to communicate within the body.
Plants transform radiant energy into chemical energy. Electric eels transform chemical energy into electrical energy. Running deer transform chemical energy into kinetic energy. The table includes more examples of energy transformations.
Chemical Electrical Battery discharge
Chemical Kinetic Muscle movement
Chemical Radiant and thermal Combustion
Electrical Kinetic Electric motor
Electrical Magnetic Electromagnet
Radiant Electrical Solar cell
Radiant Thermal Absorption of sunlight
What energy transformation occurs in a toaster? A. kinetic energy to electrical energy B. electrical energy to thermal energy C. magnetic energy to electrical energy D. potential energy to kinetic energy
Different types of energy on Earth come from a variety of sources. Learn how some sources are replaced as quickly as they are consumed while other energy sources take millions of years to replace.
There are many different types of potential and kinetic energy. Every action is connected to energy in one form or another. Learn that energy in the physical world can never be created or destroyed, but it can change from one type of energy to another.
The random motion of particles in a substance is a type of energy. This energy is transferred from one object to another in the form of heat. Learn how heat can transfer from one substance to another by three different methods: conduction, convection, and radiation.
Some forms of energy such as sound and light, can travel in waves. In this lesson you will learn about the wave theory, the different types of waves, and their properties.
Work, Motion, and Forces
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Rewrite each expression in simple radical form.
\sqrt { 18 }
\sqrt{9}\sqrt{2}
2\sqrt{27}
2\sqrt{9}\sqrt{3}
\sqrt { 32 b ^ { 3 } }
\sqrt{16}\sqrt{2}\sqrt{b^2}\sqrt{b}
Your answer will need absolute value. A variable can usually represent any number. In this case, when a square root is taken, the answer is always positive. For example:
\sqrt{(-5)^2}=5\text{ (not }-5)
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Solubility and Precipitation Reactions | General Chemistry 3
Solubility and precipitation reactions are studied in this chapter: solubility-product constants, common ion effect, formation of complexes, precipitation criteria and the amphoteric hydroxides
Solubility (generally in g.L-1): quantity of a substance that is dissolved in a saturated solution
The solid does not appear in the Ksp expression because it is a pure solid so equal to 1
AgBr (s)
⇌
Ag+ (aq) + Br- (aq)
Ksp = [Ag+] [Br-]
Ag2CO3 (s)
⇌
2 Ag+ (aq) + CO32- (aq)
Ksp = [Ag+]2 [CO32-]
The solubility s of an ionic solid can be determined using Ksp
⇌
According to the reaction stoichiometry: [Ag+] = [Br-]
Solubility s of AgBr (s) in mol.L-1 = [Ag+] = [Br-]
⇒ Ksp = [Ag+] [Br-] = s2
⇒ s =
\sqrt{{\mathrm{K}}_{\mathrm{sp}}}
Solubility S of AgBr (s) in g.L-1 = s x MAgBr =
\sqrt{{\mathrm{K}}_{\mathrm{sp}}}
x MAgBr
Common ion: ion already contained in the solution
Complex ion: metal ion with small molecules or ions attached to it
Complexation reaction: reaction between a metal ion and a molecular or ionic entity called ligand that forms a complex
Formation constant Kf: equilibrium-constant of the formation of a complex ion
Formation of the complex ion [Ag(NH3)2]+:
Ag+ (aq) + 2 NH3 (aq)
⇌
[Ag(NH3)2]+ (aq)
Kf =
\frac{\left[\mathrm{Ag}{{\left({\mathrm{NH}}_{3}\right)}_{2}}^{+}\right]}{\left[{\mathrm{Ag}}^{+}\right] {\left[{\mathrm{NH}}_{3}\right]}^{2}}
Solubility-product concentration quotient Qsp: same form as Ksp but is expressed in terms of arbitrary concentrations
Qsp and Ksp can be used to predict whether an ionic solid can precipitate:
Qsp > Ksp: precipitate forms
Qsp < Ksp: no precipitate forms
At the equilibrium: Qsp = Ksp
When equilibrium is disturbed:
Qsp > Ksp: more precipitate forms until Qsp = Ksp
Qsp < Ksp: precipitate dissolves (until Qsp = Ksp or until the complete dissolution of precipitate)
Amphoteric metal hydroxides: metal hydroxides insoluble in neutral aqueous solutions but soluble in both acidic and basic solutions
In neutral aqueous solutions: no reactions occur
In acidic solutions: reaction similar to an acid-base neutralization reaction
In basic solution: formation of a soluble hydroxy complex ion
Acidic solution: Al(OH)3 (s) + 3 H3O+ (aq)
⇌
Al3+ (aq) + 6 H2O (l) [acid-base neutralization]
Basic solution: Al(OH)3 (s) + HO- (aq)
⇌
[Al(OH)4]- (aq) [formation of a soluble hydroxy complex ion]
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Solve this system for y and z:
\left. \begin{array} { r } { \frac { z + y } { 2 } + \frac { z - y } { 4 } = 3 } \\ { \frac { 4 z - y } { 2 } + \frac { 5 z + 2 y } { 11 } = 3 } \end{array} \right.
Simplify the equations before solving.
4
4\left( \frac{z+y}{2}+\frac{z-y}{4} \right) =4\cdot3
2(z+y)+z-y =12
2z+2y+z-y =12
3z+y=12
2
nd equation. But multiply both sides by
22
. When both equations are simplified, solve for
z
y
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Home ⁄ StudentZone ⁄ Low-pass and high-pass filters
5th January 2022 3rd March 2022 by Kiera Sowery
After the introduction of the SMU ADALM1000 lets continue with the tenth part of our series with some small, basic measurements.
Written by Doug Mercer and Antoniu Miclaus, Analog Devices
The objective of this lab activity is to study the characteristics of passive filters by obtaining the frequency response of a low-pass RC filter and high-pass RL filter.
Passive filters consist of passive components, such as resistors, capacitors, and inductors, that have no amplifying elements, such as op amps, and transistors. The output level of the passive filters is always less than the input since they have no signal gain.
The impedance of capacitors and inductors are frequency dependent. The impedance of an inductor is proportional to frequency and the impedance of a capacitor is inversely proportional to frequency. These characteristics can be used to select or reject certain frequencies of an input signal. This selection and rejection of frequencies is called filtering, and a circuit that does this is called a filter.
If a filter passes high frequencies and rejects low frequencies, then it is a high-pass filter. Conversely, if it passes low frequencies and rejects high ones, it is a low-pass filter. Filters, like most things, aren’t perfect. They don’t absolutely pass some frequencies and absolutely reject others. A frequency is considered passed if its magnitude (voltage amplitude) is within 70% or 1/√2 of the maximum amplitude passed and rejected otherwise. The 70% frequency is called cut-off frequency, roll-off frequency, or half-power frequency.
At low frequencies, the impedance of the capacitor will be very large compared to the resistive value of the resistor, R. This means that the voltage potential, Vo, across the capacitor will be much larger than the voltage drop across the resistor. Therefore, at high frequencies the reverse is true, with Vo being small and VR1 being large due to the change in the capacitor impedance value.
The cut-off frequencies for an RC filter:
{\mathrm{f}}_{\mathrm{c}}=\frac{1}{\left(2\mathrm{\pi RC}\right)}
At low frequencies, the impedance of the inductor will be very small compared to the resistive value of the resistor, R. This means that the voltage potential, Vo, across the inductor will be much smaller than the voltage drop across the resistor. Therefore, at high frequencies the reverse is true, with Vo being large and VR1 being small due to the change in the inductor impedance value.
The cut-off frequencies for an RL filter:
{\mathrm{f}}_{\mathrm{c}}=\frac{\mathrm{R}}{\left(2 \mathrm{x} \mathrm{\pi } \mathrm{x} \mathrm{L}}
Frequency response: A graph of the magnitude of the output voltage of the filter as a function of the frequency. It is generally used to characterise the range of frequencies that the filter is designed to operate within.
Resistors (1 kΩ) X Capacitor (1 µF) X Inductor (20 mH)
Set up the RC circuit as shown in Figure 1 on your solderless bread- board, with the component values R1 = 1kΩ, C1 = 1µF.
Set the Channel A AWG min value to 0.5V and max value to 4.5V to apply a 4V p-p sine wave centred on 5V as the input voltage to the circuit. From the AWG A Mode drop-down menu, select SVMI mode. From the AWG A Shape drop-down menu, select Sine. From the AWG B Mode drop-down menu, select the Hi-Z mode.
From the ALICE Curves drop-down menu, select CA-V and CB-V for From the Trigger drop-down menu, select CA-V and Auto Level. Set the Hold Off to 2 (ms). Adjust the time base until you have approximately two cycles of the sine wave on the display grid. From the Meas CA drop-down menu, select P-P under CA-V and do the same for CB. Also, from the Meas CA menu, select A-B Phase.
Start with a low frequency, 50Hz, and measure output voltage CB-V peak to peak from the scope screen. It should be the same as the channel A output. Increase the frequency of Channel A in small increments until the peak-to-peak voltage of Channel B is roughly 0.7 times the peak-to-peak voltage for Channel A. Compute 70% of V p-p and obtain the frequency at which this happens on the oscilloscope. This gives the cut-off (roll-off) frequency for the constructed low-pass RC.
High-pass RL filter:
Set up the RL circuit as shown in Figure 2 on your solderless bread- board, with the component values R1 = 1 kΩ, L = 20
Repeat Steps 2 and 3, as in part A, to obtain the oscilloscope.
Start with a high frequency, 20 kHz, and measure output voltage CB-V peak to peak from the scope screen. It should be the same as the Channel A Lower the frequency of Channel A in small increments until the peak-to-peak voltage of Channel B is roughly 0.7 times the peak-to-peak voltage for Channel A. Compute 70% of V p-p and obtain the frequency at which this happens on the oscilloscope. This gives the cut-off (roll-off) frequency for the constructed high-pass RL filter.
Calculate the cut-off frequencies for the RC low-pass and RL high-pass filter using Equation 1 and Equation 2. Compare the computed theoretical values to the ones obtained from the experimental measurements and provide a suitable explanation for any differences.
Using other component values
It is possible to substitute other component values in cases where the specified values are not readily available. The reactance of a component (XC or XL) scales with frequency. For example, if 4.7mHz inductors are available rather than the 47mHz called for, all that is needed is to increase the test frequency from 250 Hz to 2.5 kHz. The same would be true when substituting a 1.0 µF capacitor for the 10.0 µF capacitor specified.
Using the RLC impedance meter tool
ALICE Desktop includes an impedance analyser/RLC meter that can be used to measure the series resistance (R) and reactance (X). As part of this lab activity, it might be informative to use this tool to measure the components R, L, and C used to confirm your test results.
As in all the ALM labs, we use the following terminology when referring to the connections to the ALM1000 connector and configuring the hardware. The green shaded rectangles indicate connections to the ADALM1000 analogue I/O connector. The analogue I/O channel pins are referred to as CA and CB. When configured to force voltage/measure current, –V is added (as in CA-V) or when configured to force current/measure voltage, –I is added (as in CA-I). When a channel is configured in the high impedance mode to only measure voltage, –H is added (as in CA-H).
We are using the ALICE Rev 1.1 software for those examples here. File: alice-desktop-1.1-setup.zip. Please download here.
A 2-channel oscilloscope for time domain display and analysis of volt- age and current
The X and Y display for plotting captured voltage and current voltage and current data, as well as voltage waveform histograms.
The 2-channel spectrum analyser for frequency domain display and analysis of voltage
The Bode plotter and network analyser with built-in sweep generator.
An impedance analyser for analysing complex RLC networks and as an RLC meter and vector
A dc ohmmeter measures unknown resistance with respect to known external resistor or known internal 50Ω.
Board self-calibration using the AD584 precision 2.5V reference from the ADALP2000 analogue parts kit.
How to encourage young people into engineering
NMITE bolsters its board with five new trustees
What is phase and why do we care?
21st February 2020 3rd March 2022
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2022 Integrality and cuspidality of pullbacks of nearly holomorphic Siegel Eisenstein series
Ameya Pitale, Abhishek Saha, Ralf Schmidt
Ameya Pitale,1 Abhishek Saha,2 Ralf Schmidt3
1Department of Mathematics, University of Oklahoma, Norman, OK 73019 USA
2School of Mathematical Sciences, Queen Mary University of London, London E14NS UK
3Department of Mathematics, University of North Texas, Denton, TX 76203 USA
We study nearly holomorphic Siegel Eisenstein series of general levels and characters on
{ℍ}_{2n}
, the Siegel upper half space of degree
2n
. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of
{ℚ}_{p}
for all sufficiently large primes
p
. We also prove that the pullbacks of these Eisenstein series to
{ℍ}_{n}×{ℍ}_{n}
are cuspidal under certain assumptions.
Ameya Pitale. Abhishek Saha. Ralf Schmidt. "Integrality and cuspidality of pullbacks of nearly holomorphic Siegel Eisenstein series." Publ. Mat. 66 (1) 405 - 434, 2022. https://doi.org/10.5565/PUBLMAT6612216
Received: 17 June 2020; Accepted: 2 March 2021; Published: 2022
Keywords: nearly holomorphic , pullback formula , Siegel Eisenstein series , Siegel modular forms
Ameya Pitale, Abhishek Saha, Ralf Schmidt "Integrality and cuspidality of pullbacks of nearly holomorphic Siegel Eisenstein series," Publicacions Matemàtiques, Publ. Mat. 66(1), 405-434, (2022)
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Reinforcement Learning & pricing: a complicated love story | Tryolabs
Reinforcement Learning (RL) is a recurrent topic here at Tryolabs, either internally while designing solutions for our clients or working with them. Particularly when evaluating options for Price Optimization problems, we've considered and studied its feasibility many times, under different scenarios.
I can identify at least two important reasons for that topic arising so often. First, this is an exciting field of AI, so we all want to learn more about it. And added to that, it's quite straightforward to map the pricing problem to the Reinforcement Learning framework: set reward = profit and try to maximize that (we'll explain this better below). And it works! You can play this simple game we developed to do precisely that, in case you're curious.
But in real-world scenarios, in price-optimization projects for our clients, we learned that classical demand forecasting approaches provide an excellent place to begin working. In further steps, RL models can be built on top of an existing machine learning system, as will be shown with a couple of examples.
In the following lines, I'll explain the main reasons that led us to prefer other options over RL for these cases. But since we're very interested on it and have no intention to be labeled as its public enemy, I'll cover other potentially groundbreaking use cases for it. These cases are still related to pricing systems.
What's the story with Reinforcement Learning?
If you've been snooping around the Machine Learning world lately, you may know that RL has been on the bleeding edge of research and development. Advancements in this area are astonishing, as we all have heard.
One of RL's most interesting aspects is that it can learn without the need for expert domain knowledge other than the system rules, i.e., some representation of the state of the environment, with available actions in each state, and finally the rewards/penalties received when moving from a state to the next one. That's what happens when computer scientists meet Pavlov.
The link between 'the way we believe we think' and this field of AI is clear. Behaviorism (the natural-intelligence counterpart of RL theory) is not considered the ultimate way to explain how the mind works, but it accounts for a large part of the advancements in psychological research.
That similarity in the way people and animals learn from their environment seems to bias our common sense and make us believe that we have finally solved the problem of intelligence and that RL is now the only path to follow.
But should we all follow this particular path to solve all our Machine Learning problems? Well, intense research pushes this field forward, and from that point of view, we are in a very sweet spot where computer science meets psychology (among many other fields, you can just search for 'behavioral economics' as an example). Discoveries on one field inspire research on the other, and from that synergy, we all get excited. Computer scientists think that they should be awarded psychologist certifications. Psychologists and economists learn to program in Python and R. All that motivation is truly powerful, which is awesome, but it doesn't mean that we should all throw away our previous tools.
The great power of many of our analytical tools comes from the fact that they enable different ways to look at the problems. Sometimes these tools support, and other times, they completely override our intuition. Combining tools, processing different visualizations, and leveraging the best-performing machinery in each specific domain, brings us to the best results if we do it right.
We should ask ourselves, what different visualizations, components, and sub-problems will help us optimize a pricing policy?
Focusing on solving a price optimization problem may seem shallow compared to the deep mysteries of how the mind works. But consider this if you're starting to feel bored: prices are probably the most important signals nowadays, driving this giant worldwide network of human exchange and collaboration that makes up the global economy.
So, if you feel more interested in understanding how pricing works, let's get into it.
What's the story with pricing?
Resuming our reasoning about the various ways to visualize problems, the power of combining tools and solving sub-problems, we get to one of the central points we want to argue here:
A great pricing system is first a good forecasting system.
Depending on where you come from, you might find this statement either obvious or unacceptable. In some cases, forecasting may be so inaccurate that it's not feasible for a pricing system, and thus a different approach is needed. I would argue that this was the reason to add the "great" qualification to the previous statement. The point is: when available, the best pricing system also provides forecasting, and it's based on it.
Let's explain where this sort of if and only if condition comes from and see if we can agree on a few things.
Good forecasting yields pricing strategies
Let's start with the less controversial part of our statement and explain why forecasting enables pricing. It's because if you can predict how many sales you'll have for each product, as a function of the price, and you make it with good accuracy, then you are almost at a high-school-level calculation to get the optimum price for this product:
\mathrm{profit}(\mathrm{price}) = \mathrm{predicted\_sales}(\mathrm{price}) \times (\mathrm{price} - \mathrm{cost})
Go through the entire price range, get the predicted sales for each price point, calculate the profit, and then just find the point where this profit function has the maximum value.
As you can see, we're assuming that we can forecast sales for any price we end up setting, among other variables that might have an influence in sales. So hold on, as there are many subtle things implicit behind the innocent "good forecasting" requirement—hiding and lying in wait to hit us, naive developers, with a shot of reality that could ruin our whole system.
Supply chain management: What if we run out of stock in the middle of our predicted sales?
Outliers prediction: Black Friday yay! Not so funny if you're trying to predict sales and didn't think about it.
Product cannibalization: You have 2k products to price and can't check them one by one to find out that you're trying to sell a package of two -deodorants, soap, cookies, use your imagination- more expensively than the two separate units. You'll sell near zero, very likely.
Exploration/exploitation tradeoff: What if we're always selecting prices near the historically set prices? Maybe the system gets high forecasting accuracy for the prices that it sets live. Still, we never discover that the rest of the predicted demand curve was wrong, underestimating sales for some prices that were never previously tested. If some new prices are never explored, there's no way to tell that the selected prices are the best. This dilemma is very well known in RL theory, but we'll have to deal with it here even if we don't use RL.
Each of the above is an entire research area, and this combination is particular to sales forecasting. As such, these issues won't get resolved out of the box when using a generic forecasting solution.
So, "good forecasting," uh? There's a lot more to it than meets the eye.
Great pricing requires good forecasting first
A perfect Reinforcement Learning scenario
After all, considering the important challenges in forecasting listed above, it looks like we're over-complicating things, trying to forecast sales first only to calculate the optimum price later.
That's probably the main concern for those that come from RL and probably found our previous statement "unacceptable" (about needing forecasting to solve the pricing problem). Because they will say:
"I just want to maximize revenue or profit, and I will set that as my reward function. That's all I need for RL. I may even solve this using a simple multi-armed bandit." - Hypothetical advocate for RL
Well, that might be a lot to infer about what someone would say, but I'm sure that the first argument would be around that. And it's a valid argument. There's no fallacy in that statement, except for the last part, because if you read the challenges above, you can't think of a simple multi-armed bandit for anything else than a baseline or a warm-up to start thinking about the problem.
Many papers about using RL for pricing set directly revenue or profit as the RL algorithm's reward function. Some were even validated in real e-commerce, like in this very interesting paper from Alibaba researchers, among other papers. They don't try to predict sales. They try to maximize revenue/profit.
Where RL falls short
The problem is that in most real scenarios, it's not good enough to have a black-box like oracle that only provides a numeric output (the price you need to set for each product), then sit to wait for the results and blindly trust that you're doing the best you can.
Here are some important things that a real business would need in order to actually take advantage of a pricing system:
An estimate of future sales so that the stock replenishment team can keep up. Or at least, a way to tell our system that it can't sell above a certain stock limit in a given period. This is the supply chain management mentioned before.
A clear metric to check that the system is working reasonably well and it's not just a sometimes-lucky random number generator.
A way to diagnose and explain results, to find room for improvements or serious problems. Is there cannibalization between similar products? Out of stocks? Is the system underestimating sales for some reason? Any other systematic error in the predictions?
If you only had a magic box that told you the price that you should set, in order to get maximum revenue, you might choose to believe it. But you won't know how well it's actually doing, compared to the prices you didn't test.
On the other hand, when your system also provides some kind of forecast (aside of the optimal price), you can easily test your predictions against reality.
Note that you solve all of the above issues when your pricing actions are derived from a sales forecasting model, as we'll explain on each of the following sections.
Stock management: a big deal
For the case study in the paper mentioned above from Alibaba (and others like this one), the main focus was to optimize price for a fixed stock amount. The opposite alternative is also usually found in research papers: assuming infinite stock, stock replenishment issues are not considered. In the paper, they mention that for their case (online marketplace with multiple sellers for each product), predicting sales is not possible due to very unstable market conditions. They probably tried to forecast something at first and failed to obtain anything meaningful in their scenario, so that could be a good reason to explore a different approach, after confirming that situation.
It seems like the majority of the proposed RL models rely either on fixed stock or the assumption of infinite stock, which is the opposite extreme.
But in general, real cases aren't in either extreme. Stores can usually adapt their stock policy to a certain extent for the sake of profitability. But finding a good stock replenishment policy requires some type of forecasting.
Not adapting the replenishment policy can cause either costly overstock or worse: out of stocks. In the case of an online store with multiple sellers, this might not be a problem. However, in stores with a constant flow of goods, if you don't have the product on the shelf, you won't sell it. That is certainly inefficient because clients are not getting what they want. When added up for many products, unexpected out of stocks are a very important issue in our experience for the performance of a pricing system.
Price optimization can lead to significant increases in profitability. The stock replenishment policy (whichever it is) must be considered part of the problem, sooner or later. The whole problem can be viewed as finding the price that produces the highest profit, according to the replenishment capacity or order arrival rate, as described in the 'Market Microstructure Theory' book by Maureen O'hara.
The effort required to adapt a working system to different scenarios, or just for continuous improvement, is directly related to the so-called "explainability" of the model. How easy is it to map the different components of the model to understandable concepts or features that are observable in the real world?
How will you know if profit or sales were underestimated if they were not predicted?
A system that's not doing any forecasting and only outputs the best price you should set, is quite hard to understand and evaluate.
There are many different approaches to use RL for pricing, and some of them actually have an underlying sales prediction, maybe in an implicit way. But now, if sales are predicted somehow in an RL system (e.g., related to the Q function value), then this is a forecasting system after all. And for those cases, the question is then: if your system is in some way a forecasting model, is it the best one?
And that's where the RL option doesn't seem to be the one to go for now, because so far, the best performing forecasting systems are not based on any RL model but rather on combined methods. The usual approach is to start with something based on gradient boosted machines, namely XGBoost or LightGBM, among others.
But that situation might change, and we need to stay tuned! That means, for example, that we need to keep an eye on the best performing models from competitions like M5 forecasting accuracy.
The reconcilement
We're proposing the idea that Reinforcement Learning isn't the best approach to do forecasting yet, and ideally, a pricing system is based on forecasting. For that part of the pricing system, we can move on and hang out with some boosted tree regressor that satisfies all our forecasting needs. They are straightforward to implement and do a great job.
But don't get me wrong: we are (and I am) very interested in Reinforcement Learning. Again, I'll suggest that you check the pricing game we did just for fun to reinforce this point.
In industry, it must be considered when it fits as an option because it could potentially be a breakthrough. We did consider it as an option for pricing. It's just that we preferred other alternatives for the very specific reasons explained before.
At this point, you might be wondering: in which cases do you think it could be a good option, then?
Pricing challenges where RL seems like a good fit
Following our reconciliatory mood, I'll provide some examples that are part of a pricing system where Reinforcement Learning seems like a great option to consider.
Going back to the challenges mentioned before in this post, solving the pricing problem successfully also requires:
Ensuring that there's enough stock available, coherent with our predictions.
Detecting similar products to be aware of/prevent potential cannibalization.
Handling price exploration/exploitation and ensuring that the data being gathered is helping us improve the results.
Could we use Reinforcement Learning to tackle these problems?
If you're curious about it, we're providing some ideas. 💡
This post’s underlying idea is to explain why we consider Reinforcement Learning as a second step in developing a pricing system and why we've so far preferred forecasting models adapted to the characteristics of sales predictions and used that to find the optimal prices.
Presenting also some imaginary use cases as an optional downloadable, the idea is to help the reader ground some of the most important RL concepts, and build an intuition on how it works.
Stay tuned, and hopefully, we'll have more posts about this fascinating topic and what we can do to help in your actual use case!
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Garching bei München, European Southern Observatory, see ESO Media Advisory (15:00 CEST) - Live streaming at ESO Website and ESO YouTube Channel Mexico City, CONACyT, see CONACyT Media Advisory (08:00 CDT) - Live streaming at CONACyT YouTube Channel Santiago de Chile, Joint ALMA Observatory, see ALMA Media Advisory (09:00 CLT) Shanghai, Shanghai Astronomical Observatory, see Shanghai Astronomical Observatory Media Advisory (21:00 CST) Taipei, Academia Sinica Institute for Astronomy and Astrophysics (21:00 CST), see YouTube Live Streaming. Tokyo, National Astronomical Observatory of Japan (22:00 JST), see YouTube Live Streaming. Washington D.C., National Press Club, see National Science Foundation Media Advisory (09:00 EDT) - Live streaming at NSF Webpage and NSF Facebook Madrid (15:00 CEST, see CSIC YouTube streaming) South Korea (22:00 KST, see YouTube Live Streaming
In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density
{\displaystyle \rho _{m}}
and isotropic pressure p.
{\displaystyle T^{\mu \nu }=\left(\rho _{m}+{\frac {p}{c^{2}}}\right)\,U^{\mu }U^{\nu }+p\,\eta ^{\mu \nu }\,}
where U is the 4-velocity vector field of the fluid and where
{\displaystyle \eta _{\mu \nu }=\operatorname {diag} (-1,1,1,1)}
is the metric tensor of Minkowski spacetime.
{\displaystyle T^{\mu \nu }=\left(\rho _{\text{m}}+{\frac {p}{c^{2}}}\right)\,U^{\mu }U^{\nu }-p\,\eta ^{\mu \nu }\,}
where U is the 4-velocity of the fluid and where
{\displaystyle \eta _{\mu \nu }=\operatorname {diag} (1,-1,-1,-1)}
{\displaystyle \left[{\begin{matrix}\rho _{e}&0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right]}
{\displaystyle \rho _{\text{e}}=\rho _{\text{m}}c^{2}}
{\displaystyle p}
is the pressure of the fluid.
{\displaystyle T^{\mu \nu }=\left(\rho _{m}+{\frac {p}{c^{2}}}\right)\,U^{\mu }U^{\nu }+p\,g^{\mu \nu }\,}
{\displaystyle g_{\mu \nu }}
is the metric, written with a space-positive signature.
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Ms. Nguyen wants to describe how much most middle school students read during summer vacation. She has data showing how many pages each of her
35
students read last summer, but she does not know how she should use the data.
Write a note explaining to her what the mean, median, and mode are and what each of these measures of central tendency might tell her about the reading habits of middle school students.
To refresh your memory on mean, median, and mode, refer to Lesson 1.3.2 in your textbook.
Would any of this information not be helpful? Sarah thinks that the mean and the median will help the teacher, but that the mode will not, since it does not really matter if many students read the exact same number of pages. Think about Sarah's ideas and use them to help you with the rest of the problem.
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Diode - 2D PCM Schematics - 3D Model
Diode (12052 views - Electronics & PCB)
In electronics, a diode is a two-terminal electronic component that conducts primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance to the current in one direction, and high (ideally infinite) resistance in the other. A semiconductor diode, the most common type today, is a crystalline piece of semiconductor material with a p–n junction connected to two electrical terminals. A vacuum tube diode has two electrodes, a plate (anode) and a heated cathode. Semiconductor diodes were the first semiconductor electronic devices. The discovery of crystals' rectifying abilities was made by German physicist Ferdinand Braun in 1874. The first semiconductor diodes, called cat's whisker diodes, developed around 1906, were made of mineral crystals such as galena. Today, most diodes are made of silicon, but other semiconductors such as selenium and germanium are sometimes used.
3D CAD Models - Diode
Licensed under Creative Commons Attribution-Share Alike 2.5 (The original uploader was Morcheeba at English Wikipedia).
In electronics, a diode is a two-terminal electronic component that conducts primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance to the current in one direction, and high (ideally infinite) resistance in the other. A semiconductor diode, the most common type today, is a crystalline piece of semiconductor material with a p–n junction connected to two electrical terminals.[5] A vacuum tube diode has two electrodes, a plate (anode) and a heated cathode. Semiconductor diodes were the first semiconductor electronic devices. The discovery of crystals' rectifying abilities was made by German physicist Ferdinand Braun in 1874. The first semiconductor diodes, called cat's whisker diodes, developed around 1906, were made of mineral crystals such as galena. Today, most diodes are made of silicon, but other semiconductors such as selenium and germanium are sometimes used.[6]
2.1 Vacuum tube diodes
2.4 World's smallest diode
3 Thermionic diodes
4.1 Electronic symbols
6 Numbering and coding schemes
The most common function of a diode is to allow an electric current to pass in one direction (called the diode's forward direction), while blocking current in the opposite direction (the reverse direction). Thus, the diode can be viewed as an electronic version of a check valve. This unidirectional behavior is called rectification, and is used to convert alternating current (AC) to direct current (DC), including extraction of modulation from radio signals in radio receivers—these diodes are forms of rectifiers.
Thermionic (vacuum tube) diodes and solid state (semiconductor) diodes were developed separately, at approximately the same time, in the early 1900s, as radio receiver detectors.[7] Until the 1950s vacuum tube diodes were used more frequently in radios because the early point-contact type semiconductor diodes were less stable. In addition, most receiving sets had vacuum tubes for amplification that could easily have the thermionic diodes included in the tube (for example the 12SQ7 double diode triode), and vacuum tube rectifiers and gas-filled rectifiers were capable of handling some high voltage/high current rectification tasks better than the semiconductor diodes (such as selenium rectifiers) which were available at that time.
In 1873, Frederick Guthrie discovered the basic principle of operation of thermionic diodes.[8][9] Guthrie discovered that a positively charged electroscope could be discharged by bringing a grounded piece of white-hot metal close to it (but not actually touching it). The same did not apply to a negatively charged electroscope, indicating that the current flow was only possible in one direction.
Thomas Edison independently rediscovered the principle on February 13, 1880.[10] At the time, Edison was investigating why the filaments of his carbon-filament light bulbs nearly always burned out at the positive-connected end. He had a special bulb made with a metal plate sealed into the glass envelope. Using this device, he confirmed that an invisible current flowed from the glowing filament through the vacuum to the metal plate, but only when the plate was connected to the positive supply.
Edison devised a circuit where his modified light bulb effectively replaced the resistor in a DC voltmeter. Edison was awarded a patent for this invention in 1884.[11] Since there was no apparent practical use for such a device at the time, the patent application was most likely simply a precaution in case someone else did find a use for the so-called Edison effect.
About 20 years later, John Ambrose Fleming (scientific adviser to the Marconi Company and former Edison employee) realized that the Edison effect could be used as a precision radio detector. Fleming patented the first true thermionic diode, the Fleming valve, in Britain on November 16, 1904[12] (followed by U.S. Patent 803,684 in November 1905).
In 1874 German scientist Karl Ferdinand Braun discovered the "unilateral conduction" of crystals.[13][14] Braun patented the crystal rectifier in 1899.[15] Copper oxide and selenium rectifiers were developed for power applications in the 1930s.
Indian scientist Jagadish Chandra Bose was the first to use a crystal for detecting radio waves in 1894.[16] The crystal detector was developed into a practical device for wireless telegraphy by Greenleaf Whittier Pickard, who invented a silicon crystal detector in 1903 and received a patent for it on November 20, 1906.[17] Other experimenters tried a variety of other substances, of which the most widely used was the mineral galena (lead sulfide). Other substances offered slightly better performance, but galena was most widely used because it had the advantage of being cheap and easy to obtain. The crystal detector in these early crystal radio sets consisted of an adjustable wire point-contact, often made of gold or platinum because of their incorrodible nature (the so-called "cat's whisker"), which could be manually moved over the face of the crystal in search of a portion of that mineral with rectifying qualties. This troublesome device was superseded by thermionic diodes (vacuum tubes) by the 1920s, but after high purity semiconductor materials became available, the crystal detector returned to dominant use with the advent, in the 1950s, of inexpensive fixed-germanium diodes. Bell Labs also developed a germanium diode for microwave reception, and AT&T used these in their microwave towers that criss-crossed the nation[which?] starting in the late 1940s, carrying telephone and network television signals. Bell Labs did not develop a satisfactory thermionic diode for microwave reception.
At the time of their invention, such devices were known as rectifiers. In 1919, the year tetrodes were invented, William Henry Eccles coined the term diode from the Greek roots di (from δί), meaning 'two', and ode (from ὁδός), meaning 'path'. (However, the word diode itself, as well as triode, tetrode, pentode, hexode, were already in use as terms of multiplex telegraphy; see, for example, The telegraphic journal and electrical review, September 10, 1886, p. 252).
Although all diodes rectify, the term 'rectifier' is normally reserved for higher currents and voltages than would normally be found in the rectification of lower power signals; examples include:
Power supply rectifiers (half-wave, full-wave, bridge)
A thermionic diode is a thermionic-valve device (also known as a vacuum tube, tube, or valve), consisting of a sealed evacuated glass envelope containing two electrodes: a cathode heated by a filament, and a plate (anode). Early examples were fairly similar in appearance to incandescent light bulbs.
A point-contact diode works the same as the junction diodes described below, but its construction is simpler. A pointed metal wire is placed in contact with an n-type semiconductor. Some metal migrates into the semiconductor to make a small p-type region around the contact. The 1N34 germanium version is still used in radio receivers as a detector and occasionally in specialized analog electronics.[citation needed]
Current–voltage characteristic
However, if the polarity of the external voltage opposes the built-in potential, recombination can once again proceed, resulting in a substantial electric current through the p–n junction (i.e. substantial numbers of electrons and holes recombine at the junction). For silicon diodes, the built-in potential is approximately 0.7 V (0.3 V for germanium and 0.2 V for Schottky). Thus, if an external voltage greater than and opposite to the built-in voltage is applied, a current will flow and the diode is said to be "turned on" as it has been given an external forward bias. The diode is commonly said to have a forward "threshold" voltage, above which it conducts and below which conduction stops. However, this is only an approximation as the forward characteristic is according to the Shockley equation absolutely smooth (see graph below).[clarification needed]
{\displaystyle I=I_{\mathrm {S} }\left(e^{\frac {V_{\text{D}}}{nV_{\text{T}}}}-1\right)}
{\displaystyle V_{\text{T}}={\frac {kT}{q}}\,,}
{\displaystyle I=I_{\text{S}}e^{\frac {V_{\text{D}}}{nV_{\text{T}}}}}
Small-signal behavior
Reverse-recovery effect
There are several types of p–n junction diodes, which emphasize either a different physical aspect of a diode often by geometric scaling, doping level, choosing the right electrodes, are just an application of a diode in a special circuit, or are really different devices like the Gunn and laser diode and the MOSFET:
Numbering and coding schemes
Alps ElectricSamsung Galaxy S6Raspberry PiLED DisplayCapacitorVaricapConstant-current diodeLight-emitting diodeCompact fluorescent lampTwo-port network
This article uses material from the Wikipedia article "Diode", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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Definition of power in VLSI design
Home ⁄ Learning ⁄ Year 3 ⁄ VLSI Design ⁄ Power in VLSI design
This post tells about the definition of power in VLSI design.
There is several terms of power we must consider: instantaneous power and average power. The term of power leads us to the term of energy.
Instantaneous power of the certain circuit element is the product of the current and voltage through this element and described by the formula
P\left(t\right)=I\left(t\right)V\left(t\right)
. The instantaneous power is always time dependent, measured with Watts.
Average power through the circuit element can be found through the formula
{P}_{a}=\frac{1}{T}{\int }_{0}^{T}P\left(t\right)dt
, measured with Watts.
Knowing instantaneous power we can find energy through the circuit element
E={\int }_{0}^{T}P\left(t\right)dt
, measured with Joules.
Let’s consider the power components of CMOS circuits. Power dissipation of CMOS circuits happens due to charging and discharging of the load capacitances when the gate switches and due to short-circuited current when all transistors of the circuits are ON at the same time. Those are the reasons of dynamic power dissipation. Dynamic power dissipation can be described by formula
{P}_{dynamic}={P}_{switching}+{P}_{short–circuit current}
The reasons of static power dissipation are gate leakage through the gate dielectric; junction leakage via source or drain diffusions; subthreshold leakage through the transistors in OFF state; contention current. Static power can be described by the formula
{P}_{static }={V}_{DD}{I}_{gate}+{V}_{DD}{I}_{diffusion}+{V}_{DD}{I}_{contention}+{V}_{DD}{I}_{subthreshold}
Both dynamic and static power creates total power through the circuit element
{P}_{total}={P}_{dynamic}+{P}_{static}
Another classification of power is standby, active and sleep mode power. Active power is consumed during active circuit element operation. Standby power is characterising the circuit element in the standby mode. And sleep power is supplied or consumed by the circuit elements during sleep mode.
Let’s consider power characteristics of the most common circuit elements : resistor, voltage source, capacitor that can be used in future.
Instantaneous power dissipated on the resistor is
{P}_{R}\left(t\right)={{I}_{R}}^{2}\left(t\right)R
Instantaneous power supplied by the voltage source is
{P}_{VDD}\left(t\right)={I}_{DD}\left(t\right){V}_{DD}
Energy stored by capacitor is
{E}_{C}={\int }_{0}^{\infty }I\left(t\right)V\left(t\right)dt=\frac{1}{2}C{{V}_{C}}^{2}
Let’s consider dynamic power in a detail. The biggest part of dynamic power is switching power. Considering the circuit we must consider every node of the circuit separately. Switching power depends on the capacitance of each node, and can consist of gate, diffusion and wire capacitance. The effective capacitance of the node is described by the node capacitance and activity factor. Activity factor describes possibility of the circuit to reduce power. Activity factor and dynamic power goes to zero of circuit turns off entirely. For this purpose clock gating technique is used.
More educational content can be found at Reddit community r/ElectronicsEasy.
Way to verify that circuits are executing correctly
Slimmer silicon wafers for cheaper solar panels
IC reliability and failure mechanisms
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On Approximating the Gradient of the Value Function
1Department of Economics, University of Auckland, Auckland, New Zealand
2Centre for Applied Macroeconomic Analysis (CAMA), Australian National University, Canberra, Australia
\left\{{c}_{it}\right\}
\left\{{i}_{it}\right\}
\underset{\left\{{c}_{it},{i}_{it}\right\}}{\mathrm{max}}{E}_{0}\underset{i=1}{\overset{I}{\sum }}\text{ }{\lambda }_{i}\underset{t=0}{\overset{\infty }{\sum }}\text{ }{\beta }^{t}u\left({c}_{it},{h}_{it}\right)
\underset{i=1}{\overset{I}{\sum }}\text{ }{c}_{it}+\underset{i=1}{\overset{I}{\sum }}\text{ }{i}_{it}=\underset{i=1}{\overset{I}{\sum }}\text{ }f\left({k}_{it},{\theta }_{it}\right),
i=1,\cdots ,I
{E}_{t}\underset{j=0}{\overset{\infty }{\sum }}\text{ }{\beta }^{j}u\left({c}_{it+j},{h}_{it+j}\right)\ge {V}_{i}^{a}\left({k}_{it},{h}_{it},{\theta }_{it}\right),
{k}_{it+1}=\left(1-\delta \right){k}_{it}+{i}_{it},
{h}_{it+1}={h}_{it}+\lambda \left({c}_{it}-{h}_{it}\right),
{c}_{it},{i}_{it}\ge 0
{\theta }_{t}
{k}_{i0},{h}_{i0},{\theta }_{i0}
{\lambda }_{i}
\beta \in \left(0,1\right)
\delta \in \left(0,1\right)
\lambda \in \left(0,1\right)
{V}_{i}^{a}\left({k}_{it},{h}_{it},{\theta }_{it}\right)
\underset{{\left\{{c}_{it},{i}_{it}\right\}}_{t=0}^{\infty }}{\mathrm{max}}{E}_{0}\underset{t=0}{\overset{\infty }{\sum }}\text{ }{\beta }^{t}u\left({c}_{it},{h}_{it}\right)
{c}_{it}+{i}_{it}=f\left({k}_{it},{\theta }_{it}\right),
{k}_{it+1}=f\left({k}_{it},{\theta }_{it}\right)-{c}_{it}+\left(1-\delta \right){k}_{it},
{h}_{it+1}={h}_{it}+\lambda \left({c}_{it}-{h}_{it}\right),
{k}_{it},{h}_{it},{\theta }_{it}
i,s=1,\cdots ,I
\frac{{\Lambda }_{i,t}}{{\Lambda }_{s,t}}=\frac{{\xi }_{st}}{{\xi }_{it}},
\begin{array}{c}{\Lambda }_{i,t}={u}_{c}\left(i,t\right)+\lambda \beta {E}_{t}\underset{j=0}{\overset{\infty }{\sum }}\text{ }{\beta }^{j}{\left(1-\lambda \right)}^{j}\left[\frac{{\xi }_{it+1+j}}{{\xi }_{it}}{u}_{h}\left(i,t+1+j\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{\mu }_{it+1+j}}{{\xi }_{it}}\frac{\partial {V}_{i}^{a}}{\partial {h}_{it+1+j}}\left(i,t+1+j\right)\right],\end{array}
{\Lambda }_{i,t}=\beta {E}_{t}\left[\frac{{\xi }_{it+1+j}}{{\xi }_{it}}{\Lambda }_{i,t+1}\left({f}_{k}\left({k}_{it+1},{\theta }_{it+1}\right)+1-\delta \right)-\frac{{\mu }_{it+1}}{{\xi }_{it}}\frac{\partial {V}_{i}^{a}}{\partial {k}_{it+1}}\left(i,t+1\right)\right],
{\mu }_{it}\left[{E}_{t}\underset{j=0}{\overset{\infty }{\sum }}\text{ }{\beta }^{j}u\left({c}_{it+j},{h}_{it+j}\right)-{V}_{i}^{a}\left({k}_{it},{h}_{it},{\theta }_{it}\right)\right]=0,
{M}_{it+1}={M}_{it}+{\mu }_{it}
{\xi }_{it}={\lambda }_{i}+{M}_{it+1}
{\mu }_{it}\ge 0
{M}_{i0}=0
{u}_{c}\left(i,t\right)
{V}_{i}^{a}
\frac{\partial {V}_{i}^{a}}{\partial {k}_{it}}\left({k}_{it},{h}_{it},{\theta }_{it}\right)
\frac{\partial {V}_{i}^{a}}{\partial {h}_{it}}\left({k}_{it},{h}_{it},{\theta }_{it}\right)
\stackrel{¯}{x}
\stackrel{¯}{s}
\left(\stackrel{¯}{x},\stackrel{¯}{s}\right)
V\left(\stackrel{¯}{x},\stackrel{¯}{s}\right)=\underset{\left\{{a}_{t}\right\}}{\mathrm{max}}{E}_{0}{\displaystyle \underset{t=0}{\overset{\infty }{\sum }}}\text{\hspace{0.05em}}{\beta }^{t}r\left({x}_{t},{a}_{t},{s}_{t}\right)
{x}_{t+1}=l\left({x}_{t},{a}_{t},{s}_{t}\right),\text{\hspace{0.17em}}{a}_{t}\in A\left({x}_{t},{s}_{t}\right),
{x}_{0}=\stackrel{¯}{x},{s}_{0}=\stackrel{¯}{s},
\beta \in \left(0,1\right)
\left\{{s}_{t}\right\}
{x}_{t}
{a}_{t}
{a}_{t}=f\left({x}_{t},{s}_{t}\right)
\frac{\partial }{\partial {x}_{i}}V\left(\stackrel{¯}{x},\stackrel{¯}{s}\right)
{a}_{t}=\stackrel{^}{f}\left(\omega ;{x}_{t},{s}_{t}\right)
\omega
{\left\{{s}_{t}^{n}\right\}}_{t=1}^{T}
{s}_{0}^{n}=\stackrel{¯}{s}
n=1,\cdots ,N
{\left\{{s}_{t}^{n}\right\}}_{t=0}^{T}
{\left\{{x}_{t}^{n},{a}_{t}^{n}\right\}}_{t=0}^{T}
\stackrel{^}{f}
{x}_{0}^{n}=\stackrel{¯}{x}
V\left(\stackrel{¯}{x},\stackrel{¯}{s}\right)\simeq \frac{1}{N}{\displaystyle \underset{n=1}{\overset{N}{\sum }}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\displaystyle \underset{t=0}{\overset{T}{\sum }}}\text{\hspace{0.05em}}{\beta }^{t}r\left({x}_{t}^{n}\mathrm{,}{a}_{t}^{n}\mathrm{,}{s}_{t}^{n}\right)\mathrm{.}
V\left(\stackrel{¯}{x}+ϵ{\iota }_{i}\mathrm{,}\stackrel{¯}{s}\right)
V\left(\stackrel{¯}{x}-ϵ{\iota }_{i}\mathrm{,}\stackrel{¯}{s}\right)
{\iota }_{i}
ϵ
\frac{\partial }{\partial {x}_{i}}V\left(\stackrel{¯}{x},\stackrel{¯}{s}\right)\simeq \frac{V\left(\stackrel{¯}{x}+ϵ{\iota }_{i}\mathrm{,}\stackrel{¯}{s}\right)-V\left(\stackrel{¯}{x}-ϵ{\iota }_{i}\mathrm{,}\stackrel{¯}{s}\right)}{2ϵ}\mathrm{.}
ϵ
V\left(k,h,\theta \right)=\underset{\left(c,i\right)\in A\left(k,\theta \right)}{\mathrm{max}}\left\{u\left(c,h\right)+\beta E\left[V\left({k}^{\prime },{h}^{\prime },{\theta }^{\prime }\right)|\left(k,h,\theta \right)\right]\right\}
{h}^{\prime }=h+\lambda \left(c-h\right),
{k}^{\prime }=\left(1-\delta \right)k+i,
A\left(k,\theta \right)=\left\{\left(c,i\right)\in {ℝ}_{+}^{2}:c+i=f\left(k,\theta \right)\right\}.
{V}_{h}\left(\cdot \right)
{V}_{k}\left(\cdot \right)
\left(\stackrel{¯}{k}\mathrm{,}\stackrel{¯}{h}\mathrm{,}\stackrel{¯}{\theta }\right)
V\left(\stackrel{¯}{k}+\epsilon \mathrm{,}\stackrel{¯}{h}\mathrm{,}\stackrel{¯}{\theta }\right)
V\left(\stackrel{¯}{k}-\epsilon \mathrm{,}\stackrel{¯}{h}\mathrm{,}\stackrel{¯}{\theta }\right)
{\Lambda }_{t}=\beta {E}_{t}\left[{\Lambda }_{t+1}\left({f}_{k}\left({k}_{t+1},{\theta }_{t+1}\right)+1-\delta \right)\right],
{\Lambda }_{t}={u}_{c}\left({c}_{t},{h}_{t}\right)+\beta \lambda {E}_{t}\left[\underset{j=0}{\overset{\infty }{\sum }}\text{ }{\beta }^{j}{\left(1-\lambda \right)}^{j}{u}_{h}\left({c}_{t+j+1},{h}_{t+j+1}\right)\right].
\lambda =1
u\left({c}_{t},{h}_{t}\right)=\frac{{\left({c}_{t}-b{h}_{t}\right)}^{1-\sigma }}{1-\sigma }
b\in \left(0,1\right)
\sigma >0
f\left({k}_{t},{\theta }_{t}\right)={\theta }_{t}{k}_{t}^{\alpha }
\mathrm{log}{\theta }_{t}=\rho \mathrm{log}{\theta }_{t-1}+{\epsilon }_{t}
\left\{{\epsilon }_{t}\right\}
{\sigma }_{\epsilon }^{2}
{\left\{{c}_{t},{h}_{t+1},{k}_{t+1}\right\}}_{t=0}^{\infty }
\begin{array}{l}{\left({c}_{t}-b{h}_{t}\right)}^{-\sigma }\\ =\beta {E}_{t}\left[b{\left({c}_{t+1}-b{h}_{t+1}\right)}^{-\sigma }\times \left(1+\left(\alpha {\theta }_{t+1}{k}_{t+1}^{\alpha -1}+1-\delta \right)\left(\frac{1}{b}-\beta {\left(\frac{{c}_{t+2}-b{h}_{t+2}}{{c}_{t+1}-b{h}_{t+1}}\right)}^{-\sigma }\right)\right)\right],\end{array}
{k}_{t+1}={\theta }_{t}{k}_{t}^{\alpha }+\left(1-\delta \right){k}_{t}-{c}_{t},
{h}_{t+1}={c}_{t}.
{\left\{{\theta }_{t}\right\}}_{t=1}^{T}
T=50000
{k}_{t},{h}_{t},{\theta }_{t}
\omega
{\left({c}_{t}\left(\omega \right)-b{h}_{t}\left(\omega \right)\right)}^{-\sigma }=\beta \psi \left(\omega ;{k}_{t}\left(\omega \right),{h}_{t}\left(\omega \right),{\theta }_{t}\right),
\psi \left(\omega ;{k}_{t}\left(\omega \right),{h}_{t}\left(\omega \right),{\theta }_{t}\right)=\mathrm{exp}\left({P}_{n}\left(\omega ;\mathrm{log}{k}_{t}\left(\omega \right),\mathrm{log}{h}_{t}\left(\omega \right),\mathrm{log}{\theta }_{t}\right)\right)
{P}_{n}
{c}_{t}\left(\omega \right)
{\left\{{\theta }_{t}\right\}}_{t=0}^{T}
{\left\{{c}_{t}\left(\omega \right),{k}_{t+1}\left(\omega \right),{h}_{t+1}\left(\omega \right)\right\}}_{t=0}^{T}
\omega
{Y}_{t}\left(\omega \right)=\mathrm{exp}\left({P}_{n}\left(\xi ;\mathrm{log}{k}_{t}\left(\omega \right),\mathrm{log}{h}_{t}\left(\omega \right),\mathrm{log}{\theta }_{t}\right)\right)+{\eta }_{t},
{Y}_{t}\left(\omega \right)
S\left(\omega \right)
{\omega }_{f}=S\left({\omega }_{f}\right)
{\left\{{c}_{t}\left({\omega }_{f}\right),{k}_{t+1}\left({\omega }_{f}\right),{h}_{t+1}\left({\omega }_{f}\right)\right\}}_{t=0}^{T}
{\left\{{\theta }_{t}\right\}}_{t=1}^{T}
{c}_{t}\left({k}_{t},{h}_{t},{\theta }_{t}\right)=b{h}_{t}+{\left[\beta \psi \left({\omega }_{f};{k}_{t},{h}_{t},{\theta }_{t}\right)\right]}^{-\frac{1}{\sigma }},
{k}_{t+1}\left({k}_{t},{h}_{t},{\theta }_{t}\right)={\theta }_{t}{k}_{t}^{\alpha }+\left(1-\delta \right){k}_{t}-b{h}_{t}-{\left[\beta \psi \left({\omega }_{f};{k}_{t},{h}_{t},{\theta }_{t}\right)\right]}_{t}^{-\frac{1}{\sigma }},
{h}_{t+1}\left({k}_{t},{h}_{t},{\theta }_{t}\right)=b{h}_{t}+{\left[\beta \psi \left({\omega }_{f};{k}_{t},{h}_{t},{\theta }_{t}\right)\right]}^{-\frac{1}{\sigma }}.
\left(\stackrel{¯}{k}\mathrm{,}\stackrel{¯}{h}\mathrm{,}\stackrel{¯}{\theta }\right)
{\left\{{\theta }^{n}\right\}}_{t=0}^{\stackrel{¯}{T}}
\stackrel{¯}{T}
{\theta }_{0}^{n}=\stackrel{¯}{\theta }
n=1,\cdots ,N
{\left\{{\theta }^{n}\right\}}_{t=0}^{T}
{\left\{{k}_{t}^{n},{h}_{t}^{n},{c}_{t}^{n}\right\}}_{t=0}^{\stackrel{¯}{T}}
{k}_{0}^{n}=\stackrel{¯}{k},{h}_{0}^{n}=\stackrel{¯}{h}
V\left(\stackrel{¯}{k}\mathrm{,}\stackrel{¯}{h}\mathrm{,}\stackrel{¯}{\theta }\right)\simeq \frac{1}{N}{\displaystyle \underset{n=1}{\overset{N}{\sum }}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}{\displaystyle \underset{t=0}{\overset{\stackrel{¯}{T}}{\sum }}}\text{\hspace{0.05em}}{\beta }^{t}\frac{{\left({c}_{t}^{n}-b{h}_{t}^{n}\right)}^{1-\sigma }}{1-\sigma }\mathrm{.}
{V}_{k}\left(\stackrel{¯}{k}\mathrm{,}\stackrel{¯}{h}\mathrm{,}\stackrel{¯}{\theta }\right)
V\left(k+ϵ,h,\theta \right)
V\left(k-ϵ,h,\theta \right)
ϵ
\frac{\partial V\left(\stackrel{¯}{k}\mathrm{,}\stackrel{¯}{h}\mathrm{,}\stackrel{¯}{\theta }\right)}{\partial k}\simeq \frac{V\left(\stackrel{¯}{k}+ϵ\mathrm{,}\stackrel{¯}{h}\mathrm{,}\stackrel{¯}{\theta }\right)-V\left(\stackrel{¯}{k}-ϵ\mathrm{,}\stackrel{¯}{h}\mathrm{,}\stackrel{¯}{\theta }\right)}{2ϵ}\mathrm{.}
\stackrel{¯}{T}
ϵ
{u}_{c}\left({c}_{t},{h}_{t}\right)+\beta \lambda {E}_{t}\left[{V}_{h}\left({k}_{t+1},{h}_{t+1},{\theta }_{t+1}\right)\right]=\beta {E}_{t}\left[{V}_{k}\left({k}_{t+1},{h}_{t+1},{\theta }_{t+1}\right)\right],
{V}_{k}\left({k}_{t},{h}_{t},{\theta }_{t}\right)=\beta {E}_{t}\left[{V}_{k}\left({k}_{t+1},{h}_{t+1},{\theta }_{t+1}\right)\right]\left({f}_{k}\left({k}_{t},{\theta }_{t}\right)+1-\delta \right),
{V}_{h}\left({k}_{t},{h}_{t},{\theta }_{t}\right)={u}_{h}\left({c}_{t},{h}_{t}\right)+\beta \left(1-\lambda \right){E}_{t}\left[{V}_{h}\left({k}_{t+1},{h}_{t+1},{\theta }_{t+1}\right)\right].
{V}_{h}\left({k}_{t},{h}_{t},{\theta }_{t}\right)={u}_{h}\left({c}_{t},{h}_{t}\right)+{E}_{t}\left[\underset{j=1}{\overset{\infty }{\sum }}\text{ }{\beta }^{j}{\left(1-\lambda \right)}^{j}{u}_{h}\left({c}_{t+j},{h}_{t+j}\right)\right].
\left({k}_{t},{h}_{t},{\theta }_{t}\right)
{V}_{h}\left({k}_{t},{h}_{t},{\theta }_{t}\right)
{k}_{t}
{k}_{t}
{V}_{h}\left({k}_{t},{h}_{t},{\theta }_{t}\right)
\lambda =0
, it reduces to the Brock-Mirman stochastic growth model. In this case, the analytical form of the one-period return function r, which maps the graph A of the feasibility correspondence into the real numbers is known. The correspondence describing the feasibility constraints is given by
\Gamma \left({k}_{t},{\theta }_{t}\right)=\left[f\left(1-\delta \right){k}_{t},f\left({k}_{t},{\theta }_{t}\right)+\left(1-\delta \right){k}_{t}\right],
r:A\to ℝ\text{\hspace{0.17em}}\text{given}\text{\hspace{0.17em}}\text{by}\text{\hspace{0.17em}}r\left({k}_{t},{k}_{t+1},{\theta }_{t}\right)=u\left(f\left({k}_{t},{\theta }_{t}\right)+\left(1-\delta \right){k}_{t}-{k}_{t+1}\right),
A=\left\{\left({k}_{t},{k}_{t+1},{\theta }_{t}\right)\in {ℝ}_{+}^{3}:{k}_{t+1}\in \Gamma \left({k}_{t},{\theta }_{t}\right)\right\}
{V}_{k}\left({k}_{t},{\theta }_{t}\right)={u}^{\prime }\left(f\left({k}_{t},{\theta }_{t}\right)+\left(1-\delta \right){k}_{t}-g\left({k}_{t},{\theta }_{t}\right)\right)\left[{f}_{k}\left({k}_{t},{\theta }_{t}\right)+1-\delta \right],
f\left({k}_{t},{\theta }_{t}\right)={\theta }_{t}{k}_{t}^{\alpha }
u\left({c}_{t}\right)=\mathrm{log}{c}_{t}
\delta =1
{k}_{t+1}=\alpha \beta {\theta }_{t}{k}_{t}^{\alpha }
{V}_{k}\left({k}_{t},{\theta }_{t}\right)=\frac{\alpha {\theta }_{t}{k}_{t}^{\alpha -1}}{\left(1-\alpha \beta \right){\theta }_{t}{k}_{t}^{\alpha }}=\frac{\alpha }{1-\alpha \beta }\frac{1}{{k}_{t}}.
{k}_{t}
{k}_{t}
Dmitriev, A. (2019) On Approximating the Gradient of the Value Function. Theoretical Economics Letters, 9, 126-138. https://doi.org/10.4236/tel.2019.91011
1. Cole, H. and Kubler, F. (2012) Recursive Contracts, Lotteries and Weakly Concave Pareto Sets. Review of Economic Dynamics, 15, 479-500.
2. Golosov, M., Tsyvinski, A. and Werquin, N. (2016) Recursive Contracts and Endogenously Incomplete Markets. In: Handbook of Macroeconomics, Vol. 2, Elsevier, Amsterdam, 725-841. https://doi.org/10.1016/bs.hesmac.2016.03.007
3. Marcet, A. and Marimon, R. (1992) Communication, Commitment, and Growth. Journal of Economic Theory, 58, 219-249. https://doi.org/10.1016/0022-0531(92)90054-L
4. Benveniste, L.M. and Scheinkman, J.A. (1979) On the Differentiability of the Value Function in Dynamic Models of Economics. Econometrica, 47, 727-732. https://doi.org/10.2307/1910417
5. Backus, D.K., Kehoe, P.J. and Kydland, F.E. (1992) International Real Business Cycles. Journal of Political Economy, 100, 745-775. https://doi.org/10.1086/261838
6. Kehoe, P.J. and Perri, F. (2002) International Business Cycles with Endogenous Incomplete Markets. Econometrica, 70, 907-928. https://doi.org/10.1111/1468-0262.00314
7. Dmitriev, A. and Roberts, I. (2012) International Business Cycles with Complete Markets. Journal of Economic Dynamics and Control, 36, 862-875. https://doi.org/10.1016/j.jedc.2011.12.006
8. Dmitriev, A. and Krznar, I. (2012) Habit Persistence and International Comovements. Macroeconomic Dynamics, 16, 312-330. https://doi.org/10.1017/S1365100510000957
9. Chen, X. and Ludvigson, S.C. (2009) Land of Addicts? An Empirical Investigation of Habit-Based Asset Pricing Models. Journal of Applied Econometrics, 24, 1057-1093.
10. Judd, K.L. (1998) Numerical Methods in Economics. The MIT Press, Cambridge, MA.
11. Mathews, J.H. and Fink, K.K. (2004) Numerical Methods Using Matlab. 4th Edition, Prentice-Hall, Upper Saddle River, NJ.
12. Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992) Numerical Recipes in C: The Art of Scientific Computing. 2nd Edition, Cambridge University Press, Cambridge.
13. den Haan, W. and Marcet, A. (1990) Solving the Stochastic Growth Model by Parameterizing Expectations. Journal of Business & Economic Statistics, 8, 31-34.
14. Bodenstein, M. (2008) International Asset Markets and Real Exchange Rate Volatility. Review of Economic Dynamics, 11, 688-705. https://doi.org/10.1016/j.red.2007.12.003
15. Creel, M. (2008) Using Parallelization to Solve a Macroeconomic Model: A Parallel Parameterized Expectations Algorithm. Computational Economics, 32, 343-352. https://doi.org/10.1007/s10614-008-9142-6
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\frac { 1 } { 5 } x + \frac { 1 } { 3 } x = 2
x+0.15x=\$2
\frac { x + 2 } { 3 } = \frac { x - 2 } { 7 }
\left. \begin{array} [t]{ l } { y = \frac { 2 } { 3 } x + 8 } \\ { y = \frac { 1 } { 2 } x + 10 } \end{array} \right.
Answers and extra practice for the Checkpoint problems are located in the back of your printed textbook or in the Reference Tab of your eBook. If you have an eBook for Core Connections, Course 3, login and then click the following link: Checkpoint 7: Solving Equations with Fractions and Decimals(Fraction Busters)
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Generate real or complex sinusoidal signals - Simulink - MathWorks Italia
Phase adder parameters
NCO Characterization
Quarter wave sine lookup table size
Theoretical spurious free dynamic range
The NCO block generates a multichannel real or complex sinusoidal signal, with independent frequency and phase in each output channel. The amplitude of the created signal is always 1. The NCO block supports real inputs only. All outputs are real except for the output signal in Complex exponential mode. For more information on how the block computes the output, see Algorithms.
To produce a multichannel output, specify a vector quantity for the Phase increment and Phase offset parameters. Both parameters must have the same length, which defines the number of output channels. Each element of each vector is applied to a different output channel.
Phase increment signal, specified as a real-valued scalar or vector. The input must have an integer data type, or a fixed-point data type with zero fraction length. The dimensions of the phase increment signal depend on how you choose to specify the Phase offset parameter:
When you specify the Phase offset on the block dialog box, the Phase increment must be a scalar or a vector with the same length as the Phase offset value. The block applies each element of the vector to a different channel, and therefore the vector length defines the number of output channels.
When you specify the Phase offset via the an input port, the offset port treats each column of the input as an independent channel. The Phase increment length must equal the number of columns in the input to the offset port.
To enable this port, set Phase increment source to Input port.
Phase offset signal, specified as a real-valued scalar, vector, or a full matrix. The input must have an integer data type, or a fixed-point data type with zero fraction length. The block treats each column of the input to the offset port as an independent channel. The number of channels in the phase offset must match the number of channels in the data input. For each frame of the input, the block can apply different phase offsets to each sample and channel.
To enable this port, set Phase offset source to Input port.
sin — Sine output
Sinusoidal output signal specified as a scalar, vector, or matrix. You can specify the data type of the signal using the Output data type parameter.
To enable this port, set Output signal to Sine or Sine and cosine.
cos — Cosine output
Cosinusoidal output signal specified as a scalar, vector, or matrix. You can specify the data type of the signal using the Output Data Type parameter.
To enable this port, set Output signal to Cosine or Sine and cosine.
exp — Complex exponential
Complex exponential output, specified as a scalar, vector, or matrix. You can specify the data type of the signal using the Output Data Type parameter.
To enable this port, set Output signal to Complex exponential.
Qerr — Phase quantization error
Phase quantization error specified as a scalar, vector, or matrix.
To enable this port, select the Show phase quantization error port check box.
Phase increment source — Source of phase increment value
Specify via dialog | Input port
Choose how you specify the phase increment. The phase increment can come from an input port or from the dialog box parameter.
If you select Input port, the inc input port appears on the block icon.
If you select Specify via dialog, the Phase increment parameter appears.
When the block comes from the Signal Operations library, the default value of the Phase increment Source parameter is Input port.
When the block comes from the Sources library, the default value of the Phase increment Source parameter is Specify via dialog.
Phase increment — Phase increment value
Specify the phase increment as an integer-valued scalar or vector. Only integer data types, including fixed-point data types with zero fraction length, are allowed. The dimensions of the phase increment depend on those of the phase offset:
When you specify the phase offset on the block dialog box, the phase increment must be a scalar or a vector with the same length as the phase offset. The block applies each element of the vector to a different channel, and therefore the vector length defines the number of output channels.
When you specify the phase offset via an input port, the offset port treats each column of the input as an independent channel. The phase increment length must equal the number of columns in the input to the offset port.
To enable this parameter, set Phase increment source to Specify via dialog.
Choose how you specify the phase offset. The phase offset can come from an input port or from the dialog box.
If you select Input port, the offset port appears on the block icon.
If you select Specify via dialog, the Phase offset parameter appears.
Phase offset — Phase offset value
Specify the phase offset as an integer-valued scalar or vector. Only integer data types, including fixed-point data types with zero fraction length, are allowed. When you specify the phase offset using this dialog box parameter, it must be a scalar or vector with the same length as the phase increment. Scalars are expanded to a vector with the same length as the phase increment. Each element of the phase offset vector is applied to a different channel of the input, and therefore the vector length defines the number of output channels.
To enable this parameter, set Phase offset source to Specify via dialog.
Add internal dither — Add internal dithering
Select to add internal dithering to the NCO algorithm. Dithering is added using the PN Sequence Generator (Communications Toolbox) from the Communications Toolbox™ product.
Number of dither bits — Number of dither bits
To enable this port, select the Add internal dither check box.
Quantize phase — Enable quantization of accumulated phase
To enable quantization of the accumulated phase, select this check box.
Number of quantized accumulator bits — Number of quantized accumulator bits
12 (default) | integer scalar
Specify the number of quantized accumulator bits as a scalar integer greater than one, and less than the accumulator word length. This value determines the number of entries in the lookup table.
To enable this port, select the Quantize phase check box.
Show phase quantization error port — Output quantization error
Select to output the phase quantization error. When you select this check box, the Qerr port appears on the block icon.
To enable this parameter, select the Quantize phase check box.
Output signal — Output signal
Choose whether the block outputs a Sine, Cosine, Complex exponential, or both Sine and cosine signals. If you select Sine and cosine, the two signals output on different ports.
When the block is acting as a source, specify the sample time in seconds as a positive scalar.
To enable this parameter, both Phase increment source and Phase offset source must be set to Specify via dialog. When either the phase increment or phase offset come in via a block input port, the sample time is inherited and this parameter is not visible.
Specify the number of samples per frame as a positive integer. When the value is greater than one, the phase increment and phase offset can vary from channel to channel and from frame to frame, but they are constant along each channel in a given frame.
When the phase offset input port exists, it has the same frame status as any output port present. When the phase increment input port exists, it does not support frames.
To enable this parameter set Phase increment source and/or Phase offset source to Specify via dialog.
When the input is fixed point, the NCO block always uses the rounding mode Floor.
Overflow mode — Overflow method
When the input is fixed point, the NCO block always uses the overflow mode Wrap.
Specify a Word length for the Accumulator as a positive integer from 2 to 128. The Data Type is always Binary point scaling, and the Fraction length is always 0.
double (default) | single | Binary point scaling
Specify a Data Type for the block Output.
Choose double or single for a floating-point implementation.
The lookup table for this block is constructed from double-precision floating-point values. Thus, the maximum amount of precision you can achieve in your output is 53 bits. Setting the word length of the Output data type to values greater than 53 bits does not improve the precision of your output.
Specify a Word length for the block Output as a positive integer from 2 to 128.
To enable this parameter, set the Data Type for the Output to Binary point scaling.
Specify a Fraction length for the block Output as a scalar integer.
The NCO Characterization pane provides you with read-only details on the NCO signal currently being implemented by the block:
Number of data points for lookup table — Number of points in lookup table
The lookup table is implemented as a quarter-wave sine table. The number of lookup table data points is defined by
{2}^{\text{number of quantized accumulator bits}-2}+1
Quarter wave sine lookup table size — Size of quarter wave sine lookup table
The quarter wave sine lookup table size is defined by
\frac{\left(\text{number of data points for lookup table}\right)\cdot \left(\text{output word length}\right)}{8}\text{ bytes}
Example: 2050 bytes
Theoretical spurious free dynamic range — Spurious free dynamic range
The spurious free dynamic range (SFDR) is calculated as follows for a lookup table with
{2}^{P}
entries:
\begin{array}{l}SFDR=\left(6P\right)\text{ dB without dither}\\ SFDR=\left(6P+12\right)\text{ dB with dither}\end{array}
Example: 84 dBc
Frequency resolution — Frequency resolution
The frequency resolution is the smallest possible incremental change in frequency and is defined by:
\Delta f=\frac{1}{{T}_{s}\cdot {2}^{N}}\text{Hz}
Example: 15.2588 uHz
The frequency resolution only appears when you set Phase increment source and Phase offset source to Specify via dialog.
This model example shows how to design an NCO source block from predetermined specifications.
The block implements the algorithm as shown in the following diagram:
The implementation of a numerically controlled oscillator (NCO) has two distinct parts. First, a phase accumulator accumulates the phase increment and adds in the phase offset. In this stage, an optional internal dither signal can also be added. The NCO output is then calculated by quantizing the results of the phase accumulator section and using them to select values from a lookup table. Since the lookup table contains a finite set of entries, in its normal mode of operation, the NCO block allows the adder’s numeric values to overflow and wrap around. The Fixed-Point infrastructure then causes overflow warnings to appear on the command line. This overflow is of no consequence.
Given a desired output frequency F0, calculate the value of the Phase increment block parameter with
phaseincrement=\left(\frac{{F}_{0}\cdot {2}^{N}}{{F}_{s}}\right)
where N is the accumulator word length and
{F}_{s}=\frac{1}{{T}_{s}}=\frac{1}{sampletime}
The frequency resolution of an NCO is defined by
\Delta f=\frac{1}{{T}_{s}\cdot {2}^{N}}\text{Hz}
Given a desired phase offset (in radians), calculate the Phase offset block parameter with
phaseoffset=\frac{{2}^{N}\cdot desiredphaseoffset}{2\pi }
The spurious free dynamic range (SFDR) is estimated as follows for a lookup table with
{2}^{P}
entries, where P is the number of quantized accumulator bits:
\begin{array}{l}SFDR=\left(6P\right)\text{ dB without dither}\\ SFDR=\left(6P+12\right)\text{ dB with dither}\end{array}
The NCO block uses a quarter-wave lookup table technique that stores table values from 0 to π/2. The block calculates other values on demand using the accumulator data type, then casts them into the output data type. This can lead to quantization effects at the range limits of a given data type. For example, consider a case where you would expect the value of the sine wave to be –1 at π. Because the lookup table value at that point must be calculated, the block might not yield exactly –1, depending on the precision of the accumulator and output data types.
HDL support for the NCO block will be removed in a future release. Use the NCO (DSP HDL Toolbox) block instead.
The following diagram shows the data types used within the NCO block.
You can set the accumulator and output data types in the block dialog box as discussed in Data Types.
The phase increment and phase offset inputs must be integers or fixed-point data types with zero fraction length.
You specify the number of quantized accumulator bits in the Number of quantized accumulator bits parameter.
The phase quantization error word length is equal to the accumulator word length minus the number of quantized accumulator bits, and the fraction length is zero.
PN Sequence Generator (Communications Toolbox) | Sine Wave | Digital Down-Converter | Digital Up-Converter
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Fill gaps using autoregressive modeling - MATLAB fillgaps - MathWorks Benelux
Fill Gaps in Audio File
Fill Gaps in Two-Dimensional Data
Fill Gaps in Function
Fill Gaps in Chirp
Fill gaps using autoregressive modeling
y = fillgaps(x)
y = fillgaps(x,maxlen)
y = fillgaps(x,maxlen,order)
fillgaps(___)
y = fillgaps(x) replaces any NaNs present in a signal x with estimates extrapolated from forward and reverse autoregressive fits of the remaining samples. If x is a matrix, then the function treats each column as an independent channel.
y = fillgaps(x,maxlen) specifies the maximum number of samples to use in the estimation. Use this argument when your signal is not well characterized throughout its range by a single autoregressive process.
y = fillgaps(x,maxlen,order) specifies the order of the autoregressive model used to reconstruct the gaps.
fillgaps(___) with no output arguments plots the original samples and the reconstructed signal. This syntax accepts any input arguments from previous syntaxes.
{F}_{s}=7418\phantom{\rule{0.2777777777777778em}{0ex}}Hz
. The file contains a recording of a female voice saying the word "MATLAB®." Play the sound.
Simulate a situation in which a noisy transmission channel corrupts parts of the signal irretrievably. Introduce gaps of random length roughly every 500 samples. Reset the random number generator for reproducible results.
Plot the original and corrupted signals. Offset the corrupted signal for ease of display. Play the signal with the gaps.
plot([mtlb mt+4])
% To hear, type soundsc(mt,Fs)
Reconstruct the signal using an autoregressive process. Use fillgaps with the default settings. Plot the original and reconstructed signals, again using an offset. Play the reconstructed signal.
lb = fillgaps(mt);
plot([mtlb lb+4])
% To hear, type soundsc(lb,Fs)
Load a file that contains depth measurements of a mold used to mint a United States penny. The data, taken at the National Institute of Standards and Technology, are sampled on a 128-by-128 grid.
Draw a contour plot with 25 copper-colored contour lines.
contour(P,nc)
axis ij square
Introduce four 10-by-10 gaps into the data. Draw a contour plot of the corrupted signal.
P(50:60,80:90) = NaN;
P(100:110,20:30) = NaN;
P(100:110,100:110) = NaN;
P(20:30,110:120) = NaN;
Use fillgaps to reconstruct the data, treating each column as an independent channel. Specify an 8th-order autoregressive model extrapolated from 30 samples at each end. Draw a contour plot of the reconstruction.
q = fillgaps(P,30,8);
contour(q,nc)
Generate a function that consists of the sum of two sinusoids and a Lorentzian curve. The function is sampled at 200 Hz for 2 seconds. Plot the result.
f = 1./(1+10*x.^2)+sin(2*pi*3*x)/10+cos(25*pi*x)/10;
Insert gaps at intervals (-0.8,-0.6), (-0.2,0.1), and (0.4,0.7).
h(x>-0.8 & x<-0.6) = NaN;
h(x>-0.2 & x< 0.1) = NaN;
h(x> 0.4 & x< 0.7) = NaN;
Fill the gaps using the default settings of fillgaps. Plot the original and reconstructed functions.
y = fillgaps(h);
plot(x,f,'.',x,y)
Repeat the computation, but now specify a maximum prediction-sequence length of 3 samples and a model order of 1. Plot the original and reconstructed functions. At its simplest, fillgaps performs a linear fit.
y = fillgaps(h,3,1);
Specify a maximum prediction-sequence length of 80 samples and a model order of 40. Plot the original and reconstructed functions.
y = fillgaps(h,80,40);
Change the model order to 70. Plot the original and reconstructed functions.
The reconstruction is imperfect because very high model orders often have problems with finite precision.
Generate a multichannel signal consisting of two instances of a chirp sampled at 1 kHz for 1 second. The frequency of the chirp is zero at 0.3 seconds and increases linearly to reach a final value of 40 Hz. Each instance has a different DC value.
r = chirp(t-0.3,0,0.7,40);
q = [r-f;r+f]';
Introduce gaps to the signal. One of the gaps covers the low-frequency region, and the other covers the high-frequency region.
gap = (460:720);
q(gap-300,1) = NaN;
q(gap+200,2) = NaN;
Fill the gaps using the default parameters. Plot the reconstructed signals.
y = fillgaps(q);
Fill the gaps by fitting 14th-order autoregressive models to the signal. Limit the models to incorporate 15 samples on the end of each gap. Use the functionality of fillgaps to plot the reconstructions.
fillgaps(q,15,14)
Increase the number of samples to use in the estimation to 150. Increase the model order to 140.
fillgaps(q,150,140)
Input signal, specified as a vector or matrix. If x is a matrix, then its columns are treated as independent channels. x contains NaNs to represent missing samples.
Example: cos(pi/4*(0:159))+reshape(ones(32,1)*[0 NaN 0 NaN 0],1,160) is a single-channel row-vector signal missing 40% of its samples.
Example: cos(pi./[4;2]*(0:159))'+reshape(ones(64,1)*[0 NaN 0 NaN 0],160,2) is a two-channel signal with large gaps.
maxlen — Maximum length of prediction sequences
Maximum length of prediction sequences, specified as a positive integer. If you leave maxlen unspecified, then fillgaps iteratively fits autoregressive models using all previous points for forward estimation and all future points for backward estimation.
order — Autoregressive model order
'aic' (default) | positive integer
Autoregressive model order, specified as 'aic' or a positive integer. The order is truncated when order is infinite or when there are not enough available samples. If you specify order as 'aic', or leave it unspecified, then fillgaps selects the order that minimizes the Akaike information criterion.
Reconstructed signal, returned as a vector or matrix.
[1] Akaike, Hirotugu. "Fitting Autoregressive Models for Prediction." Annals of the Institute of Statistical Mathematics. Vol. 21, 1969, pp. 243–247.
The size of order must be a compile-time constant.
arburg | resample
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Three types of supramolecular synthons, (A), (B) and (C), were discussed in this paper. (A) & (B) represent homo and heterosynthons,
{\mathit{R}}_{2}^{2}\left(8\right)
, respectively whereas (C) represents a DADA array,
{\mathit{R}}_{2}^{3}\left(8\right)
{\mathit{R}}_{2}^{2}\left(8\right)
{\mathit{R}}_{2}^{3}\left(8\right)
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Daily Challenge #203 - Pascal's Triangle - DEV Community
Daily Challenge #203 - Pascal's Triangle
In the drawing below we have a part of the Pascal's triangle, lines are numbered from zero (top).
We want to calculate the sum of the squares of the binomial coefficients on a given line with a function called easyline (or easyLine or easy-line).
Can you write a program which calculates easyline(n) where n is the line number?
The function will take n (with: n >= 0) as parameter and will return the sum of the squares of the binomial coefficients on line n.
easyline(0) => 1
easyline(4) => 70
easyline(50) => 100891344545564193334812497256
choose :: (Integral a, Num b) => a -> a -> b
n `choose` k = fromIntegral $ factorial n `div` (factorial k * factorial (n - k))
easyLine :: (Integral a, Num b) => a -> b
easyLine n = (2 * n) `choose` n
The sum of the squares of the elements of row n equals the middle element of row 2n...In general form:
\sum_{k = 0}^{n} {n \choose k}^2 = {2n \choose n}
p.s. I got to use the new latex feature for that!
Python one-liner using recursion
easyLine = lambda n : 1 if n==0 else int((2*(2*n-1)/n)*easyLine(n-1))
easyLine(n) = 2*(2*n-1)/n * easyLine(n-1) when n > 0
= 1 when n=0
// calculate nCr in O(r) time
long long binCoef(int n, int r){
ans *= (n-i);
ans /= i+1;
// sum will be 2nCn
long long easyline(int n){
return binCoef(2*n, n);
InsoGamer
I am coder, who thinks alomost all kind of problems can be solved with coding :)
Little lazy solution using 11 in python3
def easyLine(n):
print (sum( [ int(i)**2 for i in str(11**n)] ))
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Chlorobenzene reacts with Mg in dry ether to give a compound (A) which further reacts with ethanol to yield 1. phenol 2. benzene 3. ethyl benzene 4. phenyl ether NEET Practice Questions, MCQs, Past Year Questions (PYQs), NCERT Questions, Question Bank, Class 11 and Class 12 Questions, and PDF solved with answers
Chlorobenzene reacts with Mg in dry ether to give a compound (A) which further reacts
with ethanol to yield
4. phenyl ether
Ethyl alcohol is denatured by:
1. Methanol and formic acid
3. CH3OH and C6H6
4. CH3OH and pyridine
The compound B formed in the following sequence of reactions,
\stackrel{{\mathrm{PCl}}_{5}}{\to }
\stackrel{\mathrm{Alc}.\mathrm{NaOH}}{\to }
B will be:
The boiling point of ethanol is higher than that of dimethyl ether due to presence of-
1. H-bonding in ethanol
2. H-bonding in dimethyl ether
3. -CH3 group in ethanol
4. -CH3 group in dimethyl ether
The correct order of the acidic strength among for the above compounds is -
1. I>II>III
2. III>I>II
3. II>III>I
4. I>III>II
Subtopic: Phenols: Preparation & Properties |
Tautomerism is not exhibited by-
The most acidic compound among the following is-
{\mathrm{ClCH}}_{2} - {\mathrm{CH}}_{2}\mathrm{OH}
p-nitrophenol is stronger acid than phenol because nitro group is:
1. Electron withdrawing
2. Electron donating
The -OH group of an alcohol or the -COOH group of a carboxylic acid on can be replaced
by -Cl using
1. phosphorus pentachloride
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How to factor polynomials using grouping | StudyPug
x^2 + bx + c
ax^2 + bx + c
x^2 + bx + c
ax^2 + bx + c
Before we get into the details of factoring polynomials by grouping, let's do a quick review of the general process of factoring itself.
First, we need to know what exactly a "factor" is. The understanding of what factors are is crucial to all of mathematics, and it is a term you will hear again and again as you progress with your studies.
When we're looking at factoring polynomials, in particular, the meaning of a factor isn't all that different. The factor of a polynomial is just a value of the independent value (usually x) that makes an entire polynomial equation to zero. Not too complicated after all!
Check out our videos covering how to find the greatest common factor of polynomials, factoring polynomials with common factor, as well as factoring trinomials with leading coefficient not 1.
Now that we have a good understanding of what it means to factor in its most general terms, let's look at factoring by grouping. Factoring polynomials by grouping is just another technique we can use, similar to others you've likely seen in the past. What makes factoring by grouping so powerful, however, is its ability to help us to factor higher degree polynomials like cubics with relative ease.
It is important to understand what we mean when we say "grouping". When we are factoring by grouping, all we are really doing is breaking up our polynomial into easier-to-factor groups or "families" so that we can better approach the problem. Once we break it up into groups, we can factor using the methods we've learned from factoring quadratic and simpler polynomials.
The process is the same for any degree of polynomial, whether we be factoring quadratics by grouping, cubics by grouping, and beyond. Remember, the degree of a polynomial just related to the highest power on the independent variable x.
Lastly, for a video explanation of all of this, see our video on how to factor by grouping.
The best way to learn this technique is to do some factoring by grouping examples!
Factor the following polynomial by grouping:
x^3-7x^2+2x-14
Step 1: Divide Polynomial Into Groups
This is the trickiest part of solving these kinds of problems. Choosing what groups to make varies from problem to problem, but, in most cases, we are usually going to group the 2 highest powers together and then the lowest 2 or 1 powers together. You will see later that this doesn't necessarily matter, but it is the easiest way to do it.
In this case, the groups we will make are:
(x^3-7x^2)
(+2x-14)
Step 2: Factor Individual Groups
Now that we have done our grouping step, next we need to factor each of these groups using skills we've developed in the past.
In the group that is
(x^3-7x^2)
, we can take out the common factor
x^2
of this family. After taking
x^2
out, we end up with:
x^2(x-7)
Going on to the next group which is
(+2x-14)
, the common factor here is 2. So we take 2 out of the family and we end up with:
2(x-7)
Step 3: Factor the Entire Polynomial
After factoring each group individually, we now need to put the groups together. This gives us a more complicated looking polynomial that is:
x^2(x-7)+2(x-7)
What is important now is to consider each of our "groups" from before as their own terms in the polynomial, with
x^2(x-7)
being the first term, and
+2(x-7)
being the second term. With that in mind, we can factor this entire polynomial by recognizing there is a common factor between the two terms:
(x-7)
. So we factor it out and that leaves us with our final answer:
(x-7)(x^2+2)
Now, you may ask is it possible to group different terms into a family and still get the same final answer. In fact, it is possible!
(x^3+2x)
(-7x^2-14)
Now that we have done our grouping step, next we need to factor each of these groups using skills we've developed in the past, just like in the first method to solving this problem.
(x^3+2x)
x
is the factor. Giving us a factored group of:
x(x^2+2)
(-7x^2-14)
, -7 is the factor. Giving us a factored group of:
-7(x^2+2)
After factoring each group individually, again, we now need to put the groups together. This gives us a more complicated looking polynomial that is:
x(x^2+2)-7(x^2+2)
Once again, what is important now is to consider each of our "groups" from before as their own terms in the polynomial with
x(x^2+2)
-7(x^2+2)
being the second term.
Now, look for the common factor between the two terms, and hopefully you'll notice
(x^2+2)
as the common factor. So, we factor it out, and arrive at the same final answer of:
(x^2+2)(x-7)
And that's all there is to it! For more practice, take a look at this factoring by grouping worksheet, which has many more examples to try. You can find it here. Lastly, for some related work, see our videos on completing the square and the conversion of standard form to vertex form quadratic equations.
x^2 - y^2
This is a more advanced lesson on how to factor polynomials by grouping. Tips: 1) Look for the common factors and then 2) group them together.
Basic Concepts: Common factors of polynomials, Factoring polynomials by grouping, Factoring polynomials:
x^2 + bx + c
ax^2 + bx + c
When can we factor by grouping?
{x^3} - 7{x^2} + 2x - 14
15{y^3} + 25{y^2} + 3{y} + 5
x^2 - y^2
x^2 - y^2
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Wuhan Institute for Neuroscience and Neuroengineering, South-Central University for Nationalities, Wuhan, China.
DOI: 10.4236/ns.2017.99029 PDF HTML XML 906 Downloads 1,991 Views Citations
A long-term goal of theoretical physics is to develop a single simple theory or model that would unify the four known fundamental forces (or interactions) and give explanations for the origin and the evolution of the Universe. Here a “spiral wave law” has been proposed based on the previous studies that a consistent universe field presents various forms of spiral (helical) wave motions at the speed of light (c), and therefore, a mathematical equation for the relationship between the radius of a spiral wave motion (r) and its wave length (λ) is derived including a simplified formula ( or ), which could provide a novel explanation for the origin and the evolution of the Universe, and the space-time relationships. This model may give a new way for the unification of four fundamental forces and determine the moving properties of galaxies and basic particles, and the propagation characteristics of electromagnetic waves at the large or small scale.
Spiral Wave Law, Fundamental Forces, Origin and the Evolution of the Universe
Dai, J. (2017) Spiral Wave Law Is Involved in the Unification of Four Fundamental Forces and the Origin and the Evolution of the Universe. Natural Science, 9, 306-311. doi: 10.4236/ns.2017.99029.
A long-term goal of theoretical physics is to develop a single simple theory or model [ 1 , 2 ] that would unify the four known fundamental forces (or interactions) and give explanations for the origin and the evolution of the Universe, and other physical questions or problems, such as the origin of the gravity and mass [ 3 ], the nature of dark matter and dark energy [ 4 , 5 ], the horizon problem [ 6 ], the flatness/oldness problem [ 7 ], the strong CP problem [ 8 ], the neutrino oscillations [ 9 , 10 ], and the baryon asymmetry [ 11 - 13 ], etc. However, such a theory still does not appear.
2. The Derivation of Spiral Wave Law
It can be assumed that “Ether” or “Aether” that has been debated for more than five centuries in physics is existed based on the increasing research evidence [ 14 ], called the “universe field”, which was proposed before [ 15 ], and here I want to derive a spiral wave law according to this report. There have three distinct characters for the proposed universe field: 1) The universe field presents a spiral (helical) wavelike motion at the speed of light at the small scale [ 15 ] (Figure 1); 2) A microscopic matter is just the particular energy form of a spiral wavelike motion companying with a spin movement in the universe field, named as “spin wave packet”, which forms macroscopic objects; and 3) There exist the transformation between the universe field energy (Eu) and the matter energy (Em) in the Universe, which is mediated via electromagnetic waves.
We cannot directly assess Eu, however, we could define it when considering of energy transformation from Em to the part of Eu (pEu). Therefore, according to the energy conservation law and the mass energy formula, we can write an equation as follows:
p{E}_{u}={E}_{m}=m{c}^{2}
where “c” is the speed of light and “m” is the quality of a matter. It may be more understandable to evaluate the Em by taking a photon as an example rather than a macroscopic object. If the quality of a photon is “m” (actually, m is considered as a wave packet in order to avoid arguments on whether a photon has the quality or not) and its whole momentum is Pw that includes the spiral (helical) angular momentum (Ph) and the spin momentum (Ps) (Figure 2), then
{P}_{w}={P}_{h}+{P}_{s}
Because a photon presents a spiral wave motion at the small scale according to the assumption mentioned above, and therefore, if the linear velocity of spiral (helical) wave movement of a photon is “v”; then
v=s/t
where “s” is a spiral circumference, and “t” is the time for a spiral wave motion of a photon (Figure 2); then
v=2\text{π}r/t
where “r” is the radius of helix
The time for a spiral wave motion of a photon can also be determined by
t=\lambda /c
where “λ” is the wavelength of spiral wave motion; then
v=2\text{π}cr/\lambda
Figure 1. Schematic drawing of the spiral (helical) wave motion of the universe field at the small scale in a four-dimensional space (x, y, z and t).
Figure 2. Schematic drawing of the spiral (helical) wave motion of a photon in the universe field for deriving a unifying mathematical equation. c, r, λ and t is the speed of light, the radius of helix, the wavelength of spiral wave motion and the time for a spiral wave motion, respectively. The blue circular spots show the wave packets of a matter such as a photon, presenting a spiral wave motion companying with the spin movement (or spin momentum).
According to the angular momentum law, then
{P}_{h}=r\left(mv\right)=2\text{π}{r}^{2}mc/\lambda
{P}_{w}={E}_{m}/c=m{c}^{2}/c=mc
for a photon, then
2\text{π}{r}^{2}mc/\lambda +{P}_{s}=mc
\lambda =2\text{π}{r}^{2}mc/\left(mc-{P}_{s}\right)
This is a unifying equation for the movement of a matter and its relationship to the universe field, and is defined as a “spiral wave law” (SWL). Such a mathematical equation can be used to describe the movement rules of the universe field and the matters such as the galaxies and planets, the basic particles and the electromagnetic waves both at the macroscopic and microscopic levels. In addition, the wave-particle duality of mater is unified in this mathematical equation.
For example, if Ps = 0, meaning that the spin movement or momentum of a particle such as a photon is disappeared, and consequently the particle is completely transferred into the universe energy (the universe field), then
\lambda =2\text{π}{r}^{2}
\lambda /{r}^{2}=2\text{π}=\text{constant}
or if λ = ct, then
r=\sqrt{ct/2\text{π}}
These are three mathematical equations for the wave features of the university field, and the cosmic physical connotations contain: 1) The appearance of the Universe presents a shuttle shaped sphere; 2) If “r” → 0, then “λ” → 0, indicating that a singularity occurs, which may be the origin of the Universe; 3) If “c” is variable, for example “c” → 0, then “r” → 0, indicating that a black hole may be appeared under the condition of universe field collapse, in which the formed matter may just have spin movement or spin momentum, and the spiral wave movement of the universe field is disappeared there. Because the “c” value is limited (for example,
c\ne \infty
), then “r” is also limited, indicating that the Universe is bounded; and 4) The Equation (10c) describes the space-time relationship (or law). If “t” is increasing, then the space is expanding, but it could not do so forever, indicating that the accelerated expansion of the Universe may just be a local process of the cyclic motion of the Universe [ 16 ] because there exist three possibilities including the accelerating expansion (red shift), accelerating shrink (blue shift) and unaltered condition (invariant spectral feature) of the Universe based on the observation of the spectral information between the stars (Figure 3).
The macroscopic objects such as the sun and earth, etc, also present a spiral wavelike motion, and their movement characteristics including the wavelength and radius of spiral wave like motion can be determined by the mathematical Equation (9). We take the sun’s planets as examples:
The spin angular momentum of a planet can be described by as implified equation as follows:
{P}_{s}=I\omega
where “I” and “ω” are the moment of inertia and the average spin angular velocity of a planet, respectively, then
\lambda =2\text{π}{r}^{2}mc/\left(mc-I\omega \right)
Therefore, if the average radius of the revolution of a planet is considered as its radius of the spiral wave movement, then the λ of the sun’s planets can be calculated according to the other measured parameters (m, I and ω) of these planets.
It could be proposed that the wavelengths of spiral wavelike motion of the sun’s planets are identical to their wavelengths of gravitational waves, indicating that the gravity originates from the gravitational waves. In addition, the trajectory of spiral wavelike motion of the sun’s planets is curved, which may cause the precession of the sun’s planets.
Figure 3. Schematic drawing of the three possibilities of movement relationships between the stars in the Universe. Arrows indicate the movement directions of four stars (A-D). If the star B moves away the star A, then observations in A would result in redshirt effects; if the star C moves close to the star A, then would result in blue shift effects, and if the star D and A present motion synchronization, then would result in invariant spectral features. Such relationships indicate that the accelerating expansion, accelerating shrink and unaltered condition of the Universe may be existed based on the observations of spectral information.
The microscopic particles and the electromagnetic waves also present spiral wave like motions, and according to the quantum mechanics [ 17 ], the spin angular momentum of a particle can be determined by an equation as follows:
{P}_{s}=h\sqrt{j\left(j+1\right)}
where “j” is the spin quantum number,
j=0,\frac{1}{2},1,\frac{3}{2}
, etc., and “h” is the Planck constant, then
\lambda =2\text{π}{r}^{2}mc/\left[mc-h\sqrt{j\left(j+1\right)}\right]
This is a mathematical equation for the spiral wave movement of particles.
In this study, a unified theory has been proposed, which would unify the four known fundamental forces (or interactions) and give explanations for the origin and the evolution of the Universe. Therefore, it could be concluded that the nature of the universal gravitation is due to the interaction between the matters via the gravitational waves at the large scale, and similarly, the electromagnetic force, the strong and weak interactions hold same mechanism as gravity at the small scale. In addition, this unified theory may provide novel explanations for other physical questions or problems, such as the origin of the gravity and mass, the nature of dark matter and dark energy, the horizon problem, the flatness/oldness problem and the neutrino oscillations.
It should be particularly emphasized that the spiral wavelike motion of the sun (the origin of sun’s gravitational wave) and its action on the earth may be the causes of nature occurred climate phenomena on earth such as typhoons and hurricanes, etc., which usually present spiral wavelike motions and are related to the specific seasonal periods when the sun’s gravitational wave may generate the maximum effects on the earth. In addition, the lens effect of the propagation of light in the Universe is involved in the gravitational waves of the stars such as the sun.
This work was supported by the research team fund of South-Central University for Nationalities (XTZ15014).
[1] 'tHooft, G., et al. (2005) A Theory of Everything? Nature, 433, 257.
[2] Waldrop, M.M. (2011) Unification + 150. Nature, 471, 286.
[3] Abbott, B.P., et al. (2016) GW150914: The Advanced LIGO Detectors in the Era of First Discoveries. Physical Review Letters, 116, Article ID: 061102.
[4] Peebles, P.J. (2015) Dark Matter. Proceedings of the National Academy of Sciences USA, 112, Article ID: 12246.
[5] Wang, B., Abdalla, E., Atrio-Barandela, F. and Pavón, D. (2016) Dark Matter and Dark Energy Interactions: Theoretical Challenges, Cosmological Implications and Observational Signatures. Reports on Progress in Physics, 79, Article ID: 096901.
[6] Chung, D.J.H. and Freese, K. (2000) Cosmological Challenges in Theories with Extra Dimensions and Remarks on the Horizon Problem. Physical Review D, 61, 1.
[7] Lightman, A. (1993) Ancient Light: Our Changing View of the Universe. Harvard University Press, Cambridge.
[8] Mainini, R. and Bonometto, S.A. (2004) Dark Matter and Dark Energy from the Solution of the Strong CP Problem. Physical Review Letters, 93, Article ID: 121301.
[9] Hasert, F.J., et al. (1973) Observation of Neutrino-Like Interactions without Muon or Electron in the Gargamelle Neutrino Experiment. Physics Letters B, 46, 138.
[10] Vogel, P., Wen, L.J. and Zhang, C. (2015) Neutrino Oscillation Studies with Reactors. Nature Communications, 6, Article ID: 6935.
[11] Glashow, S.L. (1961) Partial-Symmetries of Weak Interactions. Nuclear Physics, 22, 579.
[12] Farrar, G.R. and Zaharijas, G. (2006) Dark Matter and the Baryon Asymmetry of the Universe. Physical Review Letters, 96, Article ID: 041302.
[13] Cho, A. (2002) Particle Physics. Hints of Greater Matter-Antimatter Asymmetry Challenge Theorists. Science, 328, 1087.
[14] Ranzan, C. (2016) The History of the Aether Theory. A Compendious Summary and Chronology of the Aether Theories. Cellular Universe Website.
[15] Dai, J. (2012) Universe Collapse Model and Its Roles in the Unification of Four Fundamental Forces and the Origin and the Evolution of the Universe. Natural Science, 4, 199.
[16] Steinhardt, P.J. and Turok, N. (2002) A Cyclic Model of the Universe. Science, 296, 1436.
[17] Griffiths, J.D. (2005) Introduction to Quantum Mechanics. 2nd Edition, Pearson Education Limited, London, 183-184.
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Explicit modeling of isoprene chemical processing in polluted air masses...
Zhang, Kun; Huang, Ling; Li, Qing; Huo, Juntao; Duan, Yusen; Wang, Yuhang; Yaluk, Elly; Wang, Yangjun; Fu, Qingyan; Li, Li
In recent years, ozone pollution has become one of the most severe environmental problems in China. Evidence from observations have showed increased frequency of high Oinline-formula3 levels in suburban areas of the Yangtze River Delta (YRD) region. To better understand the formation mechanism of local Oinline-formula3 pollution and investigate the potential role of isoprene chemistry in the budgets of inline-formulaROxinline-formula
M4inlinescrollmathmlchem\left(\mathrm{normal OH}+{\mathrm{normal HO}}_{normal 2}+{\mathrm{normal RO}}_{normal 2}\right)
91pt13ptsvg-formulamathimg4ae8019fe45adcf21a580be62ff43ac2 acp-21-5905-2021-ie00001.svg91pt13ptacp-21-5905-2021-ie00001.png radicals, synchronous observations of volatile organic compounds (VOCs), formaldehyde (HCHO), and meteorological parameters were conducted at a suburban site of the YRD region in 2018. Five episodes with elevated Oinline-formula3 concentrations under stagnant meteorological conditions were identified; an observation-based model (OBM) with the Master Chemical Mechanism was applied to analyze the photochemical processes during these high Oinline-formula3 episodes. The high levels of Oinline-formula3, nitrogen oxides (NOinline-formulax), and VOCs facilitated strong production and recycling of ROinline-formulax radicals with the photolysis of oxygenated VOCs (OVOCs) being the primary source. Our results suggest that local biogenic isoprene is important in suburban photochemical processes. Removing isoprene could drastically slow down the efficiency of ROinline-formulax recycling and reduce the concentrations of ROinline-formulax. In addition, the absence of isoprene chemistry could further lead to a decrease in the daily average concentrations of Oinline-formula3 and HCHO by 34 % and 36 %, respectively. Therefore, this study emphasizes the importance of isoprene chemistry in the suburban atmosphere, particularly with the participation of anthropogenic NOinline-formulax. Moreover, our results provide insights into the radical chemistry that essentially drives the formation of secondary pollutants (e.g., Oinline-formula3 and HCHO) in suburban areas of the YRD region.
Zhang, Kun / Huang, Ling / Li, Qing / et al: Explicit modeling of isoprene chemical processing in polluted air masses in suburban areas of the Yangtze River Delta region: radical cycling and formation of ozone and formaldehyde. 2021. Copernicus Publications.
Rechteinhaber: Kun Zhang et al.
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Return on Equity (ROE) Calculation
Return on Equity (ROE) Is Imperfect
Return on Equity (ROE) Example
Return on Equity (ROE) and Intangibles
Investing in companies that generate profits more efficiently than their rivals can be very profitable for portfolios. Return on equity (ROE) can help investors distinguish between companies that are profit creators and those that are profit burners.
On the other hand, ROE might not necessarily tell the whole story about a company and must be used carefully. Here, we dig deeper into return on equity, what it means and how it is used in practice.
Return on equity (ROE) is calculated by dividing a company's net income by its shareholders' equity, thereby arriving at a measure of how efficient a company is in generating profits.
ROE can be distorted by a variety of factors, such as a company taking a large write-down or instituting a program of share buybacks.
Another drawback of using ROE to evaluate a stock is that it excludes a company's intangible assets—such as intellectual property and brand recognition—from the calculation.
While ROE can help investors identify a potentially profitable stock, it has its drawbacks and is not the only metric an investor should review when evaluating a stock.
By measuring the earnings a company can generate from assets, ROE offers a gauge of profit-generating efficiency. ROE helps investors determine whether a company is a lean, profit machine or an inefficient operator.
Firms that do a good job of milking profit from their operations typically have a competitive advantage—a feature that normally translates into superior returns for investors. The relationship between the company's profit and the investor's return makes ROE a particularly valuable metric to examine.
To find companies with a competitive advantage, investors can use five-year averages of the ROE of companies within the same industry.
ROE is calculated by dividing a company's net income by its shareholders' equity, or book value. The formula is:
\textit{Return on equity = }\dfrac{\textit{Net income}}{\textit{Shareholders' equity}}
Return on equity = Shareholders’ equityNet income
You can find net income on the income statement, but you can also take the sum of the last four quarters worth of earnings. Shareholders' equity, meanwhile, is located on the balance sheet and is simply the difference between total assets and total liabilities. Shareholders' equity represents the tangible assets that have been produced by the business.
Both net income and shareholders' equity should cover the same period of time.
ROE offers a useful signal of financial success since it might indicate whether the company is earning profits without pouring new equity capital into the business. A steadily increasing ROE is a hint that management is giving shareholders more for their money, which is represented by shareholders' equity. Simply put, ROE indicates how well management is using investors' capital.
It turns out, however, that a company cannot grow earnings faster than its current ROE without raising additional cash. That is, a firm that now has a 15% ROE cannot increase its earnings faster than 15% annually without borrowing funds or selling more shares. However, raising funds comes at a cost. Servicing additional debt cuts into net income, and selling more shares shrinks earnings per share (EPS) by increasing the total number of shares outstanding.
So ROE is, in effect, a speed limit on a firm's growth rate, which is why money managers rely on it to gauge growth potential. In fact, many specify 15% as their minimum acceptable ROE when evaluating investment candidates.
ROE is not an absolute indicator of investment value. After all, the ratio gets a big boost whenever the value of shareholders' equity, the denominator, goes down.
If, for instance, a company takes a large write-down, the reduction in income (ROE's numerator) occurs only in the year that the expense is charged. That write-down, therefore, makes a more significant dent in shareholders' equity (the denominator) in the following years, causing an overall rise in the ROE without any improvement in the company's operations.
Having a similar effect as write-downs, share buybacks also normally depress shareholders' equity proportionately far more than they depress earnings. As a result, buybacks also give an artificial boost to ROE.
Investors looking for a profitable stock should also review other key metrics, such as return on invested capital (ROIC), earnings per share (EPS), and return on total assets (ROTA).
Moreover, a high ROE doesn't tell you if a company has excessive debt and is raising more of its funds through borrowing rather than issuing shares. Remember, shareholders' equity is assets less liabilities, which represent what the firm owes, including its long- and short-term debt. So, the more debt a company has, the less equity it has. And the less equity a company has, the higher its ROE ratio will be.
Suppose that two firms have the same amount of assets ($1,000) and the same net income ($120) but different levels of debt.
Firm A has $500 in debt and therefore $500 in shareholders' equity ($1,000 - $500), while Firm B has $200 in debt and $800 in shareholders' equity ($1,000 - $200). Firm A shows a ROE of 24% ($120/$500) while Firm B, with less debt, shows an ROE of 15% ($120/$800). As ROE equals net income divided by the equity figure, Firm A, the higher-debt firm, shows the highest return on equity.
Firm A looks as though it has higher profitability when it really just has more demanding obligations to its creditors. Its higher ROE may, therefore, be simply a mask of future problems. For a more transparent view that helps you see through this mask, make sure you also examine the company's return on invested capital (ROIC), which reveals the extent to which debt drives returns.
Another pitfall of ROE concerns the way in which intangible assets are excluded from shareholders' equity. For the sake of being conservative, the accounting profession generally omits a company's possession of things such as trademarks, brand names, and patents from asset and equity-based calculations. As a result, shareholders' equity often gets understated in relation to its value, and, in turn, ROE calculations can be misleading.
A company with no assets other than a trademark is an extreme example of a situation in which accounting's exclusion of intangibles would distort ROE. After adjusting for intangibles, the company would be left with no assets and probably no shareholder equity base. ROE measured this way would be astronomical but would offer little guidance for investors looking to gauge earnings efficiency.
Let's face it—no single metric can provide a perfect tool for examining fundamentals. But contrasting the five-year average ROEs within a specific industrial sector does highlight companies with a competitive advantage and knack for delivering shareholder value.
Think of ROE as a handy tool for identifying industry leaders. A high ROE can signal unrecognized value potential, so long as you know where the ratio's numbers are coming from.
Jensen Investment Management. "Return on Equity: A Compelling Case for Investors," Pages 8-10, 14. Accessed Feb. 9, 2022.
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1Right Triangle Trigonometry
1.1Use sine ratio to calculate angles and sides (Sin =
\frac{o}{h}
1.2Use cosine ratio to calculate angles and sides (Cos =
\frac{a}{h}
1.3Use tangent ratio to calculate angles and sides (Tan =
\frac{o}{a}
1.4Combination of SohCahToa questions (free lessons)
1.5Solving expressions using 45-45-90 special right triangles (free lessons)
1.6Solving expressions using 30-60-90 special right triangles
1.7Word problems relating ladder in trigonometry
1.8Word problems relating guy wire in trigonometry
1.9Other word problems relating angles in trigonometry
2Trigonometric Ratios and Angle Measure
2.1Angle in standard position
2.2Coterminal angles (free lessons)
2.3Reference angle
2.4Find the exact value of trigonometric ratios
2.5ASTC rule in trigonometry (All Students Take Calculus) (free lessons)
2.6Unit circle (free lessons)
2.7Converting between degrees and radians
2.8Trigonometric ratios of angles in radians
2.9Radian measure and arc length
2.10Law of sines
2.11Law of cosines
2.12Applications of the sine law and cosine law
3.1Introduction to bearings
3.2Bearings and direction word problems
3.3Angle of elevation and depression
4Graphing Trigonometric Functions
4.1Sine graph: y = sin x
4.2Cosine graph: y = cos x
4.3Tangent graph: y = tan x
4.4Cotangent graph: y = cot x
4.5Secant graph: y = sec x
4.6Cosecant graph: y = csc x
4.7Graphing transformations of trigonometric functions
4.8Determining trigonometric functions given their graphs
5Applications of Trigonometric Functions
5.1Ferris wheel trig problems
5.2Tides and water depth trig problems
5.3Spring (simple harmonic motion) trig problems
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6.1Quotient identities and reciprocal identities (free lessons)
6.2Pythagorean identities
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7Solving Trigonometric Equations
7.1Solving first degree trigonometric equations (free lessons)
7.2Determining non-permissible values for trig expressions
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7.5Solving trigonometric equations using pythagorean identities
7.6Solving trigonometric equations using sum and difference identities
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8.1Finding inverse trigonometric function from its graph
8.2Evaluating inverse trigonometric functions
8.3Finding inverse reciprocal trigonometric function from its graph
8.4Finding exact value of inverse reciprocal trig functions
9Imaginary and Complex Numbers
9.1Introduction to imaginary numbers (free lessons)
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9.3Adding and subtracting complex numbers
9.4Complex conjugates
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9.9Operations on complex numbers in polar form
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Trigonometry, Campbell High, Manchester, NH
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N-bit DAC based on R-2R weighted resistor architecture - Simulink - MathWorks América Latina
Reference (V)
Show ready port
Enable linearity impairments
Specify switch timing using
Settling time tolerance (LSB)
N-bit DAC based on R-2R weighted resistor architecture
Mixed-Signal Blockset / DAC / Architectures
The R-2R DAC is one of the most common types of Binary-Weighted DACs. It consists of a parallel binary-weighted resistor bank. Each digital level is converted to an equivalent analog signal by the resistor bank.
The input/output transfer curve of the binary weighted DAC can be nonmonotonic, which means that the transfer curve can reverse its direction.
The R-2R DAC architecture is low resolution and consumes more power due to the large number of resistors required to implement the architecture.
digital — Digital input signal to DAC
Digital input signal to DAC, specified as an integer.
If the Input polarity parameter is set to Bipolar, the allowed range of the signal is [−2NBits-1, 2NBits-1].
If the Input polarity parameter is set to Unipolar, the allowed range of the signal is [0, 2NBits-1].
External clock to start conversion, specified as a scalar. The digital-to-analog conversion process starts at the rising edge of the signal at the start port.
To enable this port, select Use external start clock in the General tab.
analog — Converted analog output signal
Converted analog output signal, returned as a scalar.
ready — Indicates whether digital-to-analog conversion is complete
Indicates whether the digital-to-analog conversion is complete, returned as a scalar.
To enable this port, select Show ready port in the General tab.
Number of bits — Number of bits in input word
Number of bits in the input word, specified as a unitless positive real integer. Number of bits determines the resolution of the DAC.
Values: positive real integer
Input polarity — Polarity of input signal to DAC
Bipolar (default) | Unipolar
Polarity of the input signal to the DAC.
Block parameter: Polarity
Values: Bipolar|Unipolar
Default: Bipolar
Select to connect to an external start conversion clock. By default, this option is selected. If you deselect this option, a Sampling Clock Source block inside the Segmented DAC is used to generate the start conversion clock
Frequency of the internal start conversion clock, specified as a real scalar in Hz. The Conversion start frequency parameter determines the conversion rate at the start of conversion.
To enable this parameter, deselect Use external start clock.
Block parameter: StartFreq
Reference (V) — Reference voltage
Reference voltage of the DAC, specified as a real scalar in volts. Reference (V) helps determine the output from the input digital code, Number of bits, and Bias (V) using the equation:
\text{DAC output = }\left(\left(\frac{\text{Digital input code}}{{2}^{\text{Number of bits}}}\right)\text{Reference}\right)+\text{Bias}
Block parameter: Ref
Bias (V) — Bias voltage added to output
Bias voltage added to the output of the DAC, specified as a real scalar in volts. Bias (V) helps determine the output from the input digital code, Number of bits, and Reference (V) using the equation:
\text{DAC output = }\left(\left(\frac{\text{Digital input code}}{{2}^{\text{Number of bits}}}\right)\text{Reference}\right)+\text{Bias}
Show ready port — Enable ready port on block
Select to enable the ready port on the block. This option is deselected by default.
Enable linearity impairments — Enable offset and gain errors in DAC simulation
Select to enable impairments such as offset error and gain error in DAC simulation. This parameter is selected. by default.
Shifts quantization steps by a specific value, specified as a scalar in %FS (percentage full scale), FS (full scale), or LSB (least significant bit).
Offset error is applied before Reference (V) and Bias (V).
To enable this parameter, select Enable linearity impairments in the Impairments tab.
Block parameter: OffsetError
Default: 0 LSB
Gain error — Error in slope of DAC transfer curve
Error in the slope of the straight line interpolating the DAC transfer curve, specified as a real scalar in %FS (percentage full scale), FS (full scale), or LSB (least significant bit).
Gain error is applied before Reference (V) and Bias (V).
Block parameter: GainError
Enable timing impairments — Enable timing impairments in DAC simulation
Select to enable timing impairments such as settling time or slew rate in DAC simulation. This parameter is selected. by default.
Specify switch timing using — Specify how DAC calculates switch timing
Settling time (default) | Slew rate
Specify whether the Binary Weighted DAC calculates switch timing using the settling time parameters or the slew rate parameters.
Settling time (s) — Time required for output to settle
The time required for the output of the DAC to settle to within some fraction of its final value, specified as a nonnegative real scalar in seconds.
To enable this parameter, select Enable timing impairments and set Specify switch timing using to Settling time in the Impairments tab.
Settling time tolerance (LSB) — Tolerance for calculating settling time
The tolerance allowed for calculating settling time, specified as a positive real scalar in LSB. The output of the DAC must settle within the Settling time tolerance (LSB) by Settling time (s).
Block parameter: SettlingTimeTolerance
Rising slew rate — Switch rising slew rate for DAC
5015625 (default) | positive real scalar | positive real vector
Switch the rising slew rate for the DAC, specified as a positive real scalar or vector. If Rising slew rate is scalar, it specifies the same slew rate for all the switches. If Rising slew rate is a vector of length Nbits, it specifies the slew rate for each individual switch.
To enable this parameter, select Enable timing impairments and set Specify switch timing using to Slew rate in the Impairments tab.
Block parameter: RisingSlewRate
Values: positive real scalar | positive real vector
Falling slew rate — Switch falling slew rate for DAC
-5015625 (default) | negative real scalar | negative real vector
Switch the falling slew rate for the DAC, specified as a positive real scalar or vector. If Falling slew rate is scalar, it specifies the same slew rate for all the switches. If Falling slew rate is a vector of length Nbits, it specifies the slew rate for each individual switch.
Block parameter: FallingSlewRate
Values: negative real scalar | negative real vector
Default: -5015625
Measure Offset and Gain Error of Binary Weighted DAC
Find the offset and gain errors of a binary weighted DAC block.
Measure AC Performance Metrics of Binary Weighted DAC
Find the AC performance metrics such as SNR, SINAD, SFDR, ENOB, noise floor and settling time of a binary weighted DAC block.
DAC Testbench | DAC DC measurement | DAC AC measurement | inldnl
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Bottomness Knowpia
In physics, bottomness (symbol B′ using a prime as plain B is used already for baryon number) or beauty is a flavour quantum number reflecting the difference between the number of bottom antiquarks (n
) and the number of bottom quarks (n
{\displaystyle B^{\prime }=-(n_{b}-n_{\bar {b}})}
Bottom quarks have (by convention) a bottomness of −1 while bottom antiquarks have a bottomness of +1. The convention is that the flavour quantum number sign for the quark is the same as the sign of the electric charge (symbol Q) of that quark (in this case, Q = −1⁄3).
As with other flavour-related quantum numbers, bottomness is preserved under strong and electromagnetic interactions, but not under weak interactions. For first-order weak reactions, it holds that
{\displaystyle \Delta B^{\prime }=\pm 1}
This term is rarely used. Most physicists simply refer to "the number of bottom quarks" and "the number of bottom antiquarks".
Anchordoqui, L.; Halzen, F. (2009). "Lessons in Particle Physics". arXiv:0906.1271 [physics.ed-ph].
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Electron Knowpia
{\displaystyle {\ce {^{0}_{-1}e}}}
Interactions involving electrons with other subatomic particles are of interest in fields such as chemistry and nuclear physics. The Coulomb force interaction between the positive protons within atomic nuclei and the negative electrons without, allows the composition of the two known as atoms. Ionization or differences in the proportions of negative electrons versus positive nuclei changes the binding energy of an atomic system. The exchange or sharing of the electrons between two or more atoms is the main cause of chemical bonding.[13] In 1838, British natural philosopher Richard Laming first hypothesized the concept of an indivisible quantity of electric charge to explain the chemical properties of atoms.[3] Irish physicist George Johnstone Stoney named this charge 'electron' in 1891, and J. J. Thomson and his team of British physicists identified it as a particle in 1897 during the cathode-ray tube experiment.[5] Electrons can also participate in nuclear reactions, such as nucleosynthesis in stars, where they are known as beta particles. Electrons can be created through beta decay of radioactive isotopes and in high-energy collisions, for instance when cosmic rays enter the atmosphere. The antiparticle of the electron is called the positron; it is identical to the electron except that it carries electrical charge of the opposite sign. When an electron collides with a positron, both particles can be annihilated, producing gamma ray photons.
Discovery of effect of electric forceEdit
Discovery of two kinds of chargesEdit
Discovery of free electrons outside matterEdit
During the 1870s, the English chemist and physicist Sir William Crookes developed the first cathode-ray tube to have a high vacuum inside.[30] He then showed in 1874 that the cathode rays can turn a small paddle wheel when placed in their path. Therefore, he concluded that the rays carried momentum. Furthermore, by applying a magnetic field, he was able to deflect the rays, thereby demonstrating that the beam behaved as though it were negatively charged.[28] In 1879, he proposed that these properties could be explained by regarding cathode rays as composed of negatively charged gaseous molecules in a fourth state of matter in which the mean free path of the particles is so long that collisions may be ignored.[29]: 394–395
Particle acceleratorsEdit
Confinement of individual electronsEdit
Fundamental propertiesEdit
The electron has an intrinsic angular momentum or spin of 1/2.[71] This property is usually stated by referring to the electron as a spin-1/2 particle.[70] For such particles the spin magnitude is ħ/2,[75][e] while the result of the measurement of a projection of the spin on any axis can only be ±ħ/2. In addition to spin, the electron has an intrinsic magnetic moment along its spin axis.[71] It is approximately equal to one Bohr magneton,[76][f] which is a physical constant equal to 9.27400915(23)×10−24 joules per tesla.[71] The orientation of the spin with respect to the momentum of the electron defines the property of elementary particles known as helicity.[77]
Quantum propertiesEdit
Virtual particlesEdit
The relative strength of the electromagnetic interaction between two charged particles, such as an electron and a proton, is given by the fine-structure constant. This value is a dimensionless quantity formed by the ratio of two energies: the electrostatic energy of attraction (or repulsion) at a separation of one Compton wavelength, and the rest energy of the charge. It is given by α ≈ 7.297353×10−3, which is approximately equal to 1/137.[71]
Atoms and moleculesEdit
Motion and energyEdit
{\displaystyle \scriptstyle \gamma =1/{\sqrt {1-{v^{2}}/{c^{2}}}}}
{\displaystyle \displaystyle K_{\mathrm {e} }=(\gamma -1)m_{\mathrm {e} }c^{2},}
Plasma applicationsEdit
Particle beamsEdit
Electrons are important in cathode-ray tubes, which have been extensively used as display devices in laboratory instruments, computer monitors and television sets.[185] In a photomultiplier tube, every photon striking the photocathode initiates an avalanche of electrons that produces a detectable current pulse.[186] Vacuum tubes use the flow of electrons to manipulate electrical signals, and they played a critical role in the development of electronics technology. However, they have been largely supplanted by solid-state devices such as the transistor.[187]
{\displaystyle {\begin{alignedat}{2}S&={\sqrt {s(s+1)}}\cdot {\frac {h}{2\pi }}\\&={\frac {\sqrt {3}}{2}}\hbar \\\end{alignedat}}}
{\displaystyle \textstyle \mu _{\mathrm {B} }={\frac {e\hbar }{2m_{\mathrm {e} }}}.}
{\displaystyle E_{\mathrm {p} }={\frac {e^{2}}{8\pi \varepsilon _{0}r}},}
{\displaystyle \textstyle E_{\mathrm {p} }=m_{0}c^{2},}
{\displaystyle \textstyle \Delta \lambda ={\frac {h}{m_{\mathrm {e} }c}}(1-\cos \theta ),}
^ Abraham Pais (1997). "The discovery of the electron – 100 years of elementary particles" (PDF). Beam Line. 1: 4–16. Archived (PDF) from the original on 2021-09-14. Retrieved 2021-09-04.
Wikiquote has quotations related to Electron.
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Home ⁄ StudentZone ⁄ Band-stop filters
29th March 2022 12th April 2022 by Kiera Sowery
After the introduction of the SMU ADALM1000 lets continue with the thirteenth part in the series with some small, basic measurements.
Construct a band-stop filter by combining a low-pass filter and a high-pass filter. A series LC circuit will be used.
Obtain the frequency response of the filter by using the Bode plotter software tool.
Figure 1. A schematic of the ADALM1000
A band-stop filter, also called a notch or band reject filter, prevents a specific range of frequencies from passing to the output, while allowing lower and higher frequencies to pass with little attenuation. It removes or notches out frequencies between the two cutoff frequencies while passing frequencies outside the cutoff frequencies.
One typical application of a band-stop filter is in audio signal processing, for removing a specific range of undesirable frequencies of sound like noise or hum, while not attenuating the rest. Another application is in the rejection of a specific signal from a range of signals in communication systems.
A band reject filter may be constructed by combining a high-pass RL filter with a roll-off frequency fL and a low-pass RC filter with a roll-off frequency fH, such that:
{f}_{L}<{f}_{H}
{f}_{L}=\frac{R}{\left(2\mathrm{\pi RC}\right)}
{f}_{H}\frac{1}{\left(2\mathrm{\pi RC}\right)}
The bandwidth of frequencies rejected is given by:
{f}_{BW}={f}_{H}–{f}_{L}
All the frequencies below fL and above fH are allowed to pass and the frequencies between are attenuated by the filter. The series combination of an L and C as shown in Figure 2 is such a filter.
Figure 2. Band reject filter circuit
From the previous article on parallel LC resonance, we can also use the formula for the LC resonance to calculate the centre frequency of the band-pass filter, the resonant frequency ωo is given by:
{\omega }_{o}\frac{1}{\sqrt{LC}} \left[rad/s\right]
{f}_{o}=\frac{1}{\left(2\mathrm{\pi }\sqrt{\mathrm{LC}}\right)} \left[Hertz\right]
To show how a circuit responds to a range of frequencies, a plot of the magnitude (amplitude) of the output voltage of the filter as a function of the frequency can be drawn. It is generally used to characterise the range of frequencies in which the filter is designed to operate. Figure 2 shows a typical frequency response of a band-pass filter.
Figure 3. Band reject filter frequency response
Resistor R1 1.0kΩ
Capacitor C1 0.1 µF (marked 104)
Inductor L1 (one 20mH or two 10mH in series)
Figure 4. Band reject filter breadboard connections
Setup the filter circuit as shown in Figure 2 on your solderless breadboard, with the component values R1 = 1kΩ, C1 = 0.1µF, L1 = 20mH.
Set the Channel A AWG Min value to 0.5 and Max value to 4.5V to apply a 4V p-p sine wave centered on 2.5V as the input voltage to the circuit. From the AWG A Mode drop-down menu, select the SVMI mode. From the AWG A Shape drop-down menu, select Sine. From the AWG B Mode drop-down menu, select Hi-Z mode.
From the ALICE Curves drop-down menu, select CA-V and CB-V for display. From the Trigger drop-down menu, select CA-V and Auto Level. Set the Hold Off to 2(ms). Adjust the time base until you have at approximately two cycles of the sine wave on the display grid. From the Meas CA drop-down menu, select P-P under CA-V and do the same for CB. Also, from the Meas CA menu, select A-B Phase.
Start with a low frequency (100Hz) and measure the output voltage CB-V peak to peak from the scope screen. It should be about the same as the Channel A output. Increase the frequency of Channel A in small increments until the peak-to-peak voltage of Channel B is roughly 0.7 times the peak-to-peak voltage for Channel A. Compute 70% of V p-p and obtain the frequency at which this happens on the oscilloscope. This gives the cutoff (roll-off) frequency for the constructed RL time constant of the filter.
Continue increasing the frequency of Channel A until the peak-to-peak voltage of Channel B drops to its minimum value. Measure the frequency at which this happens on the oscilloscope. This gives the centre frequency for the constructed series LC resonator section of the filter. Note that this 70% amplitude point occurs twice on the band reject filter, at the lower cutoff and upper cutoff frequencies.
Frequency response plots with ALICE bode plotter
Use the band-pass circuit in Figure 2, with R1 = 0kΩ, C1 = 0.1μF, and L1 = 20mH so that we can sweep the input frequency from 500Hz to 12,000Hz and plot the signal amplitude of both Channel A and Channel B and the relative phase angle between Channel B and Channel A.
With the circuit connected to the ALM1000, as in Figure 2, start the ALICE Desktop
Open the Bode Plotting Under the Curves menu, select CA-dBV, CB-dBV, and Phase B-A.
Under the Options menu, change the setting for zero-stuffing to 2.
Set the AWG Channel A Min value to 086 and the Max value to 3.914. This will be a 1V rms (0 dBV) amplitude centred on the 2.5V middle of the analogue input range. Set AWG A mode to SVMI and Shape to Sine. Set AWG Channel B to Hi-Z mode. Be sure the Sync AWG check box is selected.
Use the Start Frequency entry to set the frequency sweep to start at 100Hz and use the Stop Frequency entry to the sweep to stop at 20,000 Under Sweep Gen, select CHA as the channel to sweep. Also use the Sweep Steps entry to enter the number of frequency steps, using 200 as the number.
You should now be able to press the green Run button and run the frequency sweep. After the sweep is completed (this could take a few seconds for 200 points), you should see something like the screenshot in Figure 5. You may want to use the LVL and dB/div buttons to optimize the plots to best fit the screen grid.
Record the results and save the Bode plot as a screenshot.
Figure 5. Bode analyser settings band-stop filter
Compute the cutoff frequencies for each band reject filter constructed using the formula in Equation 1 and 2. Compare these theoretical values to the ones obtained from the experiment and provide suitable explanation for any differences.
As in all the ALM labs, we use the following terminology when referring to the connections to the ALM1000 connector and configuring the hardware. The green shaded rectangles indicate connections to the ADALM1000 analogue I/O connector. The analogue I/O channel pins are referred to as CA and CB. When configured to force voltage/measure current, -V is added (as in CA-V) or when configured to force current/measure voltage, -I is added (as in CA-I). When a channel is configured in the high impedance mode to only measure voltage, -H is added (as in CA-H).
One 47 Ω resistor
X A 2-channel oscilloscope for time domain display and analysis of voltage and current
Board self-calibration using the AD584 precision 2.5 V reference from the ADALP2000 analogue parts kit.
Figure 6. ALICE desktop 1.1 menu
Antoniu Miclaus is a system applications engineer at Analog Devices, where he works on ADI academic programs, as well as embedded software for Circuits from the Lab and QA process management. He started working at Analog Devices in February 2017 in Cluj-Napoca, Romania.
He is currently an M.Sc. student in the software engineering master’s program at Babes-Bolyai University and he has a B.Eng. in electronics and telecommunications from Technical University of Cluj-Napoca.
Coventry University’s collaboration with Aston Martin yields degree apprentice graduates
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Superstring theory - Wikipedia
(Redirected from Superstring)
Theory of strings with supersymmetry
"Superstring" redirects here. For the converse relation of "substring", see Superstring (formal languages). For the bundle of firecrackers, see Superstring (fireworks). For the album by Ron Carter, see Super Strings.
Find sources: "Superstring theory" – news · newspapers · books · scholar · JSTOR (November 2012) (Learn how and when to remove this template message)
3 Lack of experimental evidence
6 Integrating general relativity and quantum mechanics
7.1 D-branes
7.2 Why five superstring theories?
8 Beyond superstring theory
8.1 Compactification
8.2 Kac–Moody algebras
Investigating how a string theory may include fermions in its spectrum led to the invention of supersymmetry (in the West[clarification needed])[2] in 1971,[3] a mathematical transformation between bosons and fermions. String theories that include fermionic vibrations are now known as "superstring theories".
Lack of experimental evidence[edit]
Superstring theory is based on supersymmetry. No supersymmetric particles have been discovered and recent research at the Large Hadron Collider (LHC) and Tevatron has excluded some of the ranges.[4][self-published source?][5][6][7] For instance, the mass constraint of the Minimal Supersymmetric Standard Model squarks has been up to 1.1 TeV, and gluinos up to 500 GeV.[8] No report on suggesting large extra dimensions has been delivered from LHC. There have been no principles so far to limit the number of vacua in the concept of a landscape of vacua.[9]
If the extra dimensions are compactified, then the extra 6 dimensions must be in the form of a Calabi–Yau manifold. Within the more complete framework of M-theory, they would have to take form of a G2 manifold. A particular exact symmetry of string/M-theory called T-duality (which exchanges momentum modes for winding number and sends compact dimensions of radius R to radius 1/R),[12] has led to the discovery of equivalences between different Calabi–Yau manifolds called mirror symmetry.
Number of superstring theories[edit]
Integrating general relativity and quantum mechanics[edit]
D-branes[edit]
{\displaystyle \partial _{z}\rightarrow \partial _{z}+iA_{z}(z,{\overline {z}})}
Why five superstring theories?[edit]
Heterotic
{\displaystyle \partial _{z}X^{\mu }-i{\overline {\theta _{L}}}\Gamma ^{\mu }\partial _{z}\theta _{L}}
{\displaystyle \partial _{z}X^{\mu }-i{\overline {\theta _{L}}}\Gamma ^{\mu }\partial _{z}\theta _{L}-i{\overline {\theta _{R}}}\Gamma ^{\mu }\partial _{z}\theta _{R}}
{\displaystyle \partial _{z}X^{\mu }-i{\overline {\theta _{L}^{1}}}\Gamma ^{\mu }\partial _{z}\theta _{L}^{1}-i{\overline {\theta _{L}^{2}}}\Gamma ^{\mu }\partial _{z}\theta _{L}^{2}}
Beyond superstring theory[edit]
{\displaystyle \int _{-\infty }^{\infty }\exp({ax^{4}+bx^{3}+cx^{2}+dx+f})\,dx=e^{f}\sum _{n,m,p=0}^{\infty }{\frac {b^{4n}}{(4n)!}}{\frac {c^{2m}}{(2m)!}}{\frac {d^{4p}}{(4p)!}}{\frac {\Gamma (3n+m+p+{\frac {1}{4}})}{a^{3n+m+p+{\frac {1}{4}}}}}}
Compactification[edit]
Investigating theories of higher dimensions often involves looking at the 10 dimensional superstring theory and interpreting some of the more obscure results in terms of compactified dimensions. For example, D-branes are seen as compactified membranes from 11D M-theory. Theories of higher dimensions such as 12D F-theory and beyond produce other effects, such as gauge terms higher than U(1). The components of the extra vector fields (A) in the D-brane actions can be thought of as extra coordinates (X) in disguise. However, the known symmetries including supersymmetry currently restrict the spinors to 32-components—which limits the number of dimensions to 11 (or 12 if you include two time dimensions.) Some physicists (e.g., John Baez et al.) have speculated that the exceptional Lie groups E6, E7 and E8 having maximum orthogonal subgroups SO(10), SO(12) and SO(16) may be related to theories in 10, 12 and 16 dimensions; 10 dimensions corresponding to string theory and the 12 and 16 dimensional theories being yet undiscovered but would be theories based on 3-branes and 7-branes respectively. However, this is a minority view within the string community. Since E7 is in some sense F4 quaternified and E8 is F4 octonified, the 12 and 16 dimensional theories, if they did exist, may involve the noncommutative geometry based on the quaternions and octonions respectively. From the above discussion, it can be seen that physicists have many ideas for extending superstring theory beyond the current 10 dimensional theory, but so far all have been unsuccessful.
Kac–Moody algebras[edit]
^ Polchinski, Joseph. String Theory: Volume I. Cambridge University Press, p. 4.
^ Rickles, Dean (2014). A Brief History of String Theory: From Dual Models to M-Theory. Springer, p. 104. ISBN 978-3-642-45128-7
^ J. L. Gervais and B. Sakita worked on the two-dimensional case in which they use the concept of "supergauge," taken from Ramond, Neveu, and Schwarz's work on dual models: Gervais, J.-L.; Sakita, B. (1971). "Field theory interpretation of supergauges in dual models". Nuclear Physics B. 34 (2): 632–639. Bibcode:1971NuPhB..34..632G. doi:10.1016/0550-3213(71)90351-8.
^ Woit, Peter (February 22, 2011). "Implications of Initial LHC Searches for Supersymmetry".
^ Cassel, S.; Ghilencea, D. M.; Kraml, S.; Lessa, A.; Ross, G. G. (2011). "Fine-tuning implications for complementary dark matter and LHC SUSY searches". Journal of High Energy Physics. 2011 (5): 120. arXiv:1101.4664. Bibcode:2011JHEP...05..120C. doi:10.1007/JHEP05(2011)120. S2CID 53467362.
^ Falkowski, Adam (Jester) (February 16, 2011). "What LHC tells about SUSY". resonaances.blogspot.com. Archived from the original on March 22, 2014. Retrieved March 22, 2014.
^ Tapper, Alex (24 March 2010). "Early SUSY searches at the LHC" (PDF). Imperial College London.
^ CMS Collaboration (2011). "Search for Supersymmetry at the LHC in Events with Jets and Missing Transverse Energy". Physical Review Letters. 107 (22): 221804. arXiv:1109.2352. Bibcode:2011PhRvL.107v1804C. doi:10.1103/PhysRevLett.107.221804. PMID 22182023.
^ Shifman, M. (2012). "Frontiers Beyond the Standard Model: Reflections and Impressionistic Portrait of the Conference". Modern Physics Letters A. 27 (40): 1230043. Bibcode:2012MPLA...2730043S. doi:10.1142/S0217732312300431.
^ a b Jha, Alok (August 6, 2013). "One year on from the Higgs boson find, has physics hit the buffers?". The Guardian. photograph: Harold Cunningham/Getty Images. London: GMG. ISSN 0261-3077. OCLC 60623878. Archived from the original on March 22, 2014. Retrieved March 22, 2014.
^ The D = 10 critical dimension was originally discovered by John H. Schwarz in Schwarz, J. H. (1972). "Physical states and pomeron poles in the dual pion model". Nuclear Physics, B46(1), 61–74.
^ Polchinski, Joseph. String Theory: Volume I. Cambridge University Press, p. 247.
^ Polchinski, Joseph. String Theory: Volume II. Cambridge University Press, p. 198.
^ Foot, R.; Joshi, G. C. (1990). "Nonstandard signature of spacetime, superstrings, and the split composition algebras". Letters in Mathematical Physics. 19: 65–71. Bibcode:1990LMaPh..19...65F. doi:10.1007/BF00402262. S2CID 120143992.
Polchinski, Joseph (1998). String Theory Vol. 1: An Introduction to the Bosonic String. Cambridge University Press. ISBN 978-0-521-63303-1.
Polchinski, Joseph (1998). String Theory Vol. 2: Superstring Theory and Beyond. Cambridge University Press. ISBN 978-0-521-63304-8.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Superstring_theory&oldid=1068907007"
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RandomDigraph - Maple Help
Home : Support : Online Help : Mathematics : Discrete Mathematics : Graph Theory : GraphTheory Package : RandomGraphs : RandomDigraph
generate random digraph
RandomDigraph(V, p, options)
RandomDigraph(V, m, options)
RandomDigraph(n, p, options)
RandomDigraph(n, m, options)
numerical value in the closed range [0.0,1.0]
weights : range or function
m\le n
are integers, the graph returned is a weighted graph with edge weights chosen from [m,n] uniformly at random. The weight matrix W in the graph has datatype=integer, and if the edge from vertex i to j is not in the graph then W[i,j] = 0.
x\le y
are floating-point numbers is specified, the graph returned is a weighted graph with numerical edge weights chosen from [x,y] uniformly at random. The weight matrix W in the graph has datatype=float[8], that is, double precision floats (16 decimal digits), and if the edge from vertex i to j is not in the graph then W[i,j] = 0.0.
RandomDigraph(n,m) creates a directed unweighted graph on n vertices and m edges, where the m edges are chosen uniformly at random.
RandomDigraph(n,p) creates a directed unweighted graph on n vertices where each possible edge is present with probability p.
The random number generator used can be seeded with the seed option or by using the randomize function.
\mathrm{with}\left(\mathrm{GraphTheory}\right):
\mathrm{with}\left(\mathrm{RandomGraphs}\right):
G≔\mathrm{RandomDigraph}\left(10,0.5\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 1: a directed unweighted graph with 10 vertices and 39 arc\left(s\right)}}
\mathrm{IsDirected}\left(G\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
H≔\mathrm{RandomDigraph}\left(10,20\right)
\textcolor[rgb]{0,0,1}{H}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 2: a directed unweighted graph with 10 vertices, 19 arc\left(s\right), and 1 self-loop\left(s\right)}}
J≔\mathrm{RandomDigraph}\left(4,6,\mathrm{weights}=1..4\right)
\textcolor[rgb]{0,0,1}{J}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Graph 3: a directed weighted graph with 4 vertices, 3 arc\left(s\right), and 3 self-loop\left(s\right)}}
\mathrm{WeightMatrix}\left(J\right)
[\begin{array}{cccc}\textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{4}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{0}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}\end{array}]
GraphTheory:-IsDirected
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Cardinality - Simple English Wikipedia, the free encyclopedia
measure of the “number of elements of the set”, either as a cardinal number or as the equivalence class of sets admitting bijections to this set
In mathematics, the cardinality of a set means the number of its elements. For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. The cardinality of a set A can also be represented as
{\displaystyle |A|}
Two sets have the same (or equal) cardinality if and only if they have the same number of elements, which is the another way of saying that there is a 1-to-1 correspondence between the two sets.[3] The cardinality of the set A is less than or equal to the cardinality of set B if and only if there is an injective function from A to B. The cardinality of the set B is greater than or equal to the cardinality of set A if and only if there is an injective function from A to B.
The cardinality of a set is only one way of giving a number to the size of a set. The concept of measure is yet another way.
||A|| is the cardinality of the set A. Single vertical bars can also be placed around a set to mean cardinality, such as |A|.[1][2]
Finite setsEdit
The cardinality of a finite set is a natural number. The smallest cardinality is 0. The empty set has cardinality 0. If the cardinality of the set A is n, then there is a "next larger" set with cardinality n+1 (for example, the set A ∪ {A}). If ||A|| ≤ ||B|| ≤ ||A ∪ {A}||, then either ||B|| = ||A|| or ||B|| = ||A ∪ {A}||.) There is no largest finite cardinality.
If the cardinality of a set is not finite, then the cardinality is infinite.[4]
An infinite set is considered countable if they can be listed without missing any (that is, if there is a one-to-one correspondence between it and the set of natural numbers
{\displaystyle \mathbb {N} }
).[3] Examples include the rational numbers, integers, and natural numbers. Such sets have a cardinality that we call
{\displaystyle \aleph _{0}}
(pronounced "aleph null", "aleph naught" or "aleph zero"). Sets such as the real numbers are not countable, since given any finite or infinite list of real numbers, it's always possible to find a number that's not on that list. The real numbers have a cardinality of
{\displaystyle {\mathfrak {c}}}
—the cardinality of the continuum.[1]
↑ 1.0 1.1 1.2 "Comprehensive List of Set Theory Symbols". Math Vault. 2020-04-11. Retrieved 2020-08-23.
↑ 2.0 2.1 "Cardinality | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-23.
↑ In order to simplify, we will assume the Axiom of choice
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Cardinality&oldid=7414397"
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How to use the negative exponent rule? | StudyPug
Don't be scared by the negative sign! Just flip the number over to get rid of the negative sign. In other words, a number with a negative exponent should be put to the denominator, and vice versa.
(a^x)(a^y)=a^{(x+y)}
{a^x \over a^y}=a^{(x-y)}
(a^x)^y = a^{(x\cdot y)}
{a^{-n}} = \frac{1}{a^n} , a \neq 0
and \frac{1}{a^{-n}} = {a^n} , a \neq 0
{2^{-2}}
-{2^{-2}}
-(-2)^{-2}
\frac{3^{-2}}{4^{-3}}
-4( {x^3}{y^{-2}}{z^{-4}}{)^{-3}}
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Assessment of a Second-Moment Closure Model for Strongly Heated Internal Gas Flows | J. Heat Transfer | ASME Digital Collection
, Logan, UT 84322-4130
Eugen Nisipeanu,
Eugen Nisipeanu
, 1007 Church St., Evanston, IL 60201
e-mail: en@evanston.fluent.com
, 5009 Centennial Boulevard, Colorado Springs, CO 80919
e-mail: Adamh_richards@hotmail.com
Spall, R. E., Nisipeanu, E., and Richards, A. (April 10, 2007). "Assessment of a Second-Moment Closure Model for Strongly Heated Internal Gas Flows." ASME. J. Heat Transfer. December 2007; 129(12): 1719–1722. https://doi.org/10.1115/1.2768098
Both low- and high-Reynolds-number versions of the stress-
ω
model of Wilcox (Turbulence Modeling for CFD, 2nd ed., DCW Industries, Inc.) were used to predict velocity and heat transfer data in a high-heat-flux cylindrical tube for which fluid properties varied strongly with temperature. The results indicate that for accurate heat transfer calculations under the conditions considered in this study, inclusion of low-Reynolds-number viscous corrections to the model are essential. The failure of the high-Reynolds-number model to accurately predict the wall temperature was attributed to an overprediction of the near-wall velocity.
computational fluid dynamics, confined flow, heat transfer, computational fluid dynamics, forced convection, turbulence models
Computational fluid dynamics, Fluids, Gas flow, Heat transfer, Stress, Temperature, Turbulence, Wall temperature, Boundary-value problems, Heat, Reynolds number, Heat flux, Failure, Forced convection
Thermal and Hydraulic Tests of Standard Fuel Rod of HTTR With HENDEL
J. At. Energy Soc. Jpn.
Turbulence Structure in Gas Flows Laminarizing by Heating
,” Ph.D. dissertation, University of Arizona.
Mean Turbulence Structure in Strongly Heated Air Flows
Turbulence Structure in the Viscous Layer of Strongly Heated Gas Flows
,” Tech. Report No. INEL-95/0223, Idaho National Engineering Laboratory.
Mean Turbulence Structure in the Viscous Layer of Strongly-Heated Internal Gas Flows: Measurements
Temperature, Velocity and Mean Turbulence Structure in Strongly Heated Gas Flows, Comparison of Numerical Predictions With Data
Application of Energy-Dissipation Model of Turbulence to the Calculation of Flow Near a Spinning Disk
An Assessment of k−ω and v2−f Turbulence Models for Strongly Heated Internal Gas Flows
Fluent, Inc., Lebanon, NH.
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Unified Law of Subsequent Albums - Uncyclopedia, the content-free encyclopedia
During the late 1990s, the band U2 fell on hard times.
“"Noel Coward Does Liberace II" outsold all my previous recordings, which surely proves this law to be a scientific fact.”
~ Noel Coward on Unified Law of Subsequent Albums
The Unified Law of Subsequent Albums relates the pressure of popularity to the phenomenon of album release and resultant relative quality.
"Every subsequent album sucks, unless it rules."
We see examples floating on the airwaves and hear them coming from the TV daily. Hacks with talent and hard-working bands that couldn't make money on a street corner if they didn't have a thug beating ingrates just around the corner. Masters of the arcane art of sadomasochism paid pocket change by guys in faux fur coats and rainbow socks.
Without 40 year old boys that wished they were 42, the industry would collapse in upon itself and all the corporate whores members of quality bands would have to find real jobs reality. They are the primary driving factor in the industry, before internet downloads and random iPod jackings. To tell whether or not a band is good you only need to get an airboat and seek those bands which are bobbing on the tops of the airwaves. Avoiding those that are floundering on the bottom like halibut, they're no good. The fact their lead singer looks like a halibut does not affect the flavor of the music, but it does mean that you should avoid their groupers.
The theory was distilled to a few simple equations by Dr. Dre in 1872:
{\displaystyle c={\frac {\sqrt {n}}{a^{y}}}}
{\displaystyle \qquad s=c^{y}}
{\displaystyle \qquad x={\sqrt {y^{2}+z^{2}}}}
{\displaystyle \qquad e=mc^{2}}
{\displaystyle \qquad d=2\pi \times r}
{\displaystyle \qquad a=\pi \times r^{2}}
{\displaystyle \qquad b={\sqrt {t \over {f^{w}}}}}
1 Theories on the Phenomenon
2 Application of the Theory
3 Bands with Subsequent Albums that Suck
4 Bands with Subsequent Albums that Rule
5 Bands that Defy Categorization
6 Bands that Always Sucked
Theories on the Phenomenon[edit]
Sales figures after the internet.
Sir Splarka has postulated that the phenomenon may be linked with the rate at which albums are released, specifically, the size of the disk in relation to the available material. If the material is under a certain size, the album must rule.
He has also put forward the theory of bands that roughly follow a sine wave release pattern to keep the fans guessing, however he cannot pinpoint the purpose.
Dr. Dawg of the Uncyclopedia Institute believes that in modern punk music it is based on the amount of angst that has been amassed prior to album production. The theory extends to other forms of music based on the infungible material such artists use to produce fungible waves of meaningful meaning. Therefore, if a band has run out of whatever makes their music good they cannot make good music except through practice or something new to make noise about.
Application of the Theory[edit]
Applying the equations by Dr. Dre shows that more bands will suck over time. It also indicates that they will make more money if they did better on the charts at the beginning because people will be duped into buying their mindless drivel again. The good doctor decided to avoid the roller-coaster ride and just started a record company because that was where most of the money seemed to be going. Besides, nobody remembers which record company printed good or bad albums, since Americans can't read. After signing a number of guys with absolutely no talent and eating lima beans for almost a year, he diversified his portfolio with a chain of fast food stores, tuna factories, and sweatshops in Queens.
Bands with Subsequent Albums that Suck[edit]
Graph showing quality over time.
What did you expect from a band that produced something as great as "God Save the Emperor of Andorra"? You just can't top that.
They started out strong, then people noticed that they couldn't come up with a fresh tune. Temporarily gained in the charts due to a steamy public romance between the lead singer's husband and a French prostitute.
The quintessential two-hit-wonder. Everyone in the world loved his song King of the World. The first person in Nashville to meet Bubba quipped, "Woooeee, you're a long drink of water!"
Bands with Subsequent Albums that Rule[edit]
Unusual pattern seen in certain bands.
Their first album sucked so bad that they only sold it after releasing their third album, which critics consider their best. What did anyone expect from a band named after a suburb of Boston?
Very unusually named band that went from zero to awesome and has remained there ever since. A clear reflection on the intellignce of humanity.
Bands that Defy Categorization[edit]
Just when you thought they were dead, they came back to life. Just when you thought they were awesome, they ran out of steam. The world's most bipolar band.
Each album is something completely different. Considered a sine-wave band, much like the rest in this group.
Originally we thought this band wouldn't make money unless the band members made little girls swoon and corporate radio pushed them constantly. We were right! Then they tried another genre, and they weren't half bad. Surprising.
Bands that Always Sucked[edit]
The Village Hippies
No minds left after all that pot.
C.W. McNugget
Someone named after an inedible food should have tipped you off to their suckiness.
Retrieved from "https://uncyclopedia.com/w/index.php?title=Unified_Law_of_Subsequent_Albums&oldid=6121211"
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Operations with integers word problems | StudyPug
In this section, we will use all four integer operations namely, addition, subtraction, multiplication, and division to solve questions. We will also apply the integer operations to tackle related word problems.
Basic Concepts: Understanding integer multiplication, Multiplying integers, Understanding integer division, Dividing integers
Related Concepts: Using exponents to describe numbers, Solving linear equations using multiplication and division, Solving two-step linear equations:
ax + b = c
{x \over a} + b = c
a(x + b) = c
A dime = $0.1
A nickel = $0.05
Solving Problems Involving Multiple Integer Operations
(-45)÷(+15)-(-8)×(-2)
(+3)-(+8)÷(+4)×(-9)
(+6)×[(+20)-(-3)]+(+1)
(-6)+(-60)÷(-4)-(+12)×(-3)
Applications of Integer Operations – Word Problems
Paul went to a furniture store with his friend and had $1000 in cash in his wallet. When paying for 6 pieces of furniture, he did not have enough money and borrowed $410 from his friend. What is the average price of the furniture he bought?
A car manufacture makes 80 cars per week. They sell 35 cars to buyers from Europe, 20 cars to those from Asia, and 25 cars to local buyers.
How many cars are sold to other countries in 6 weeks by this manufacture?
How many more cars are sold overseas than locally in 10 weeks?
The manufacture now wants to expand their business and make 4940 cars each year. How many more cars do they need to make each week?
Vivian owed $482 on her credit card. She made a $136 payment and then bought $42 worth of groceries. What is the overall balanced on her credit card now?
Maggie's dad promised that he would double the amount of money she has in her piggybank. If Maggie counted 18 quarters, 6 dimes and 13 nickels in her piggybank, how much does her dad have to pay her?
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Copy the trapezoid at right on your paper. Then find its area and perimeter. Keep your work organized so that you can later explain how you solved it. (Note: The diagram is not drawn to scale.)
\text{Area}=\frac{1}{2}\left(b_1+b_2\right)
Look at the right triangle on the left side of the trapezoid.
Using trigonometry find the length of the hypotenuse (longest side).
Using the same right triangle on the left side of the trapezoid, determine the length of the side adjacent to the
45^\circ
Use trigonometry or Pythagorean Theorem.
When finding total perimeter you need to create another right triangle on the right side of the trapezoid.
\text{Perimeter}\approx47.66
\text{Area}=74
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Droop Control | Building DC Energy Systems
# Droop Control
If the DC grid voltage is decoupled from all power sources and sinks, the locally measured voltage of each grid participant can be used to control the power flows within the grid.
This chapter describes the control mode of the grid port for most important grid participants.
The voltage setpoints can be set arbitrarily. Here we use setpoints currently implemented in a 48V grid pilot developed by Libre Solar for demonstration.
As a convention, current/power flow towards the grid (export) has a positive sign and current consumed from the grid (import) has a negative sign.
A renewable energy source (e.g. solar panel) connected to the grid via a DC/DC converter will only export power to the grid.
The operating range of the converter at the grid side is shown in Fig. 1.
Figure 1. Droop control: Renewable energy source.
If no power is consumed from the grid, the grid controller will ramp up the grid voltage until it reaches is maximum grid voltage setpoint
v_{max} = 55\,\mathrm{V}
. As soon as power is consumed, the controller tries to maintain that voltage until it reaches its maximum input power limit
P_{max}
. For a solar panel, this is the maximum power point (MPP), which mainly depends on the solar irradiance. If the power consumed in the grid increases further, the converter cannot maintain the voltage anymore. It will drop until it reaches the maximum current limit of the device and finally completely break down if the demand in the grid is not reduced.
# Energy storage device
An energy storage device like a Lithium-ion battery can import and export power, as shown in Fig. 2.
Figure 2. Droop control: Energy storage device.
It will import at high grid voltages and export at lower voltages to support the grid.
Between importing and exporting mode, the battery needs a voltage hysteresis to prevent charge transfer between batteries.
In contrast to the solar panel, the operating curve of an energy storage device has a slope, which is called the droop curve. This droop makes the system react like a voltage source with a series resistor. If the power increases, the voltage drops, indicating that the load in the system is high. This behavior is used to control the grid, as explained further down below.
If the state of charge (SOC) of the battery is high, the operating curve can be slightly moved upwards, which causes full batteries to start exporting their energy before batteries with low state of charge.
# Smart load
Loads can be connected directly to the grid and don't have to be smart, i.e. measure the grid voltage and change their operation depending on it.
However, in order to prioritize different loads depending on the available power or energy, it makes sense to implement at least a threshold below which a load would shut off itself.
If possible, a load should also implement a droop curve behavior, as shown in Fig. 3.
Figure 3. Droop control: Smart load (pure consumer).
Loads with low priority operate only at high grid voltages.
The line indicated with "low prio" would switch the load on only after all batteries have been fully charged (i.e. don't pull the grid voltage down anymore). So it would only use abundant renewable energy. This method could be used for heating or other wasteful uses of electricity.
The high priority load with dashed lines would only get switched off when there is no energy left in the grid at all.
# Grid connection
As a demonstration how the different participants interact with each other via the droop control mechanism, Fig. 4 shows a very simple grid with only one solar panel (left side) and one battery (right side).
The red line indicates the equilibrium which is automatically reached when those two grid interfaces are connected. The solar converter operates in its maximum power point and the battery consumes exactly the same amount of current as provided by the solar panel towards the grid (neglecting any losses).
If the solar panel gets shaded, the maximum power will decrease and the
P_{max}
curve will move to the left. The battery controller will immediately react and reduce the imported current. The resulting grid voltage will be lower than before.
← Grid Architecture
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How do you find the circumference of a circle? | StudyPug
What exactly is the circumference of a circle? It's actually defined as the length of the edge surrounding a circle, that is, the perimeter of a circle. We need to use
\pi
to help us find the circumference. The formula for finding the circumference includes the diameter, and looks like this:
\pi
To make this easier, we can also find the circumference if we know the radius of a circle. We know that the diameter is equalled to 2r (2 times the radius), so in other words, the formula for a circle's circumference is:
\pi
Either of these circumference formulas can be used to help you solve problems.
We'll do a three examples to help you learn how to find the circumference of a circle using the formulas we just learned.
Find the circumference of the following circle:
Circles, radius, and circumference
\pi
\pi
In this example, we are given a circle, with only one of its characteristics given to us. The 7cm is the measurement of the line from the center of the circle to its edge, which is, in other words, the radius of the circle. Luckily, we have a circumference formula to help us out when we know the radius: C = 2
\pi
r. Simply by substituting in r with 7, we're able to find that the circumference is 43.98cm.
What is the circumference of the circle given the radius
\pi
\pi
(8½)
Once again in this example, we're given the radius of the circle. Although it's not a clean number like our previous example, but we can still simply plug the number directly into the formula like what we did above. Be aware of the units that this circle's radius is given in and remember to give your final answer in the same unit. In this question, we find that the circumference is equalled to 53.41m.
Circles, diameter, and circumference
\pi
\pi
In this example, we aren't given the radius. We're given the distance across a circle through its center, which is also called the diameter of a circle. Again, referring back to the two equations we can use to calculate a circle's circumference, we find that one of them simply uses C =
\pi
d. When we substitute "d" with 17, we find that we'll get the answer of 53.41m.
An interesting point to note is that you can still use the other formula for finding the circumference that uses the radius. All we have to do is first change the diameter into a radius. We know that the diameter is 2 times the radius, so therefore, we can divide 17 by 2 to find the radius of 8.5 You can see that this number is actually the same one as the radius given in the previous circle, and therefore, we get the same answer when we use the C = 2
\pi
r formula.
Generally, it's easier to use whichever formula corresponds with the characteristics of the circle you are given. However, if you're unable to remember both of the formulas, you can always manipulate the info you're given so that it fits into the formula you do remember.
Feel free to play around with this online circle calculator to see how the circumference changes as the diameter and radius changes.
Find the circumferences of the following circles.
Find the radius (r).
\overline {CD}
\overline {CE}
are radii of circle C.
If CD = 15 cm, what is CE?
If DF is the diameter of circle C, find its length.
Find the circumference of circle C.
The wheels of David's bicycle have a radius of 9 inches. How far has David traveled if the wheels have made 120 revolutions?
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Error, invalid input: f expects its 1st argument, x, to be of type evaln(integer), but received b := 4.5 - Maple Help
Home : Support : Online Help : Error, invalid input: f expects its 1st argument, x, to be of type evaln(integer), but received b := 4.5
\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]
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Home : Support : Online Help : Programming : Document Tools : Layout : Font
generate XML for a Font element
Font( str, opts )
(optional) ; a string of text
(optional) ; one or more keyword options as described later
background : {list(nonnegint),symbol,string}:=[255,255,255] ; The background color for text. The passed value can be either a named color or a list of three integers each between 0 and 255. A list of non-negative integers is interpreted as RGB values in a 24bit 3-channel color space. The default value is [255,255,255] which corresponds to white.
bold : truefalse:=false ; Whether text will be shown in bold.
color : {list(nonnegint),symbol,string}:=[0,0,0] ; The foreground color for text. The passed value can be either a named color or a list of three integers each between 0 and 255. A list of nonnegative integers is interpreted as RGB values in a 24bit 3-channel color space. The default value is [0,0,0] which corresponds to black.
encoding : identical("UTF-8"):=NULL ; Indicates that UTF-8 encoding shall be used. The default is NULL.
family : identical("Times New Roman","Courier","DejaVu Sans","DejaVu Serif","Helvetica","Lucida Sans"):="Times New Roman" ; The font family to be used for displayed text.
italic : truefalse:=false ; Whether text will be shown in italics.
opaque : truefalse ; Whether the background color will be opaque, in which case any underlying color such as the fillcolor of a Table Cell will show. The default value is false when background is white, and true otherwise.
size : posint:=12 ; The size of the text font.
style : identical(Text,Hyperlink):=Text ; A worksheet style for text.
superscript : truefalse:=false ; Whether the text will be displayed in the superscript position.
underline : truefalse:=false ; Whether text will be shown underlined.
The Font command provides a modifier for a string of text appearing in a Textfield.
An Font element is returned as an XML function call.
\mathrm{with}\left(\mathrm{DocumentTools}\right):
\mathrm{with}\left(\mathrm{DocumentTools}:-\mathrm{Layout}\right):
Executing the Font command produces a function call.
F≔\mathrm{Font}\left("Some text"\right)
\textcolor[rgb]{0,0,1}{F}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{_XML_Font}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{"size"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"12"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"mathsize"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"12"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"mathvariant"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"normal"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"style"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Text"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"background"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"\left[255,255,255\right]"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"mathbackground"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"\left[255,255,255\right]"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"foreground"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"\left[0,0,0\right]"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"mathcolor"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"\left[0,0,0\right]"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"family"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Times New Roman"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Some text"}\right)
\mathrm{xml}≔\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(F\right)\right)\right)\right):
\mathrm{InsertContent}\left(\mathrm{xml}\right):
Size and color can be specified as options.
F≔\mathrm{Font}\left("Some text",\mathrm{size}=20,\mathrm{color}=[0,150,0],\mathrm{background}=[255,255,120]\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(F\right)\right)\right)\right)\right):
F≔\mathrm{Font}\left("Some text",\mathrm{size}=20,\mathrm{color}="blue",\mathrm{background}=\mathrm{Orange}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(F\right)\right)\right)\right)\right):
Each of the bold, italic, and underline options can be used independently.
F≔\mathrm{Font}\left("Some text",\mathrm{size}=20,\mathrm{bold}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(F\right)\right)\right)\right)\right):
F≔\mathrm{seq}\left([\mathrm{Font}\left("Text",\mathrm{size}=20,\mathrm{op}\left(f\right)\right),""][],f\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{in}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{combinat}:-\mathrm{powerset}\left([\mathrm{underline},\mathrm{bold},\mathrm{italic}]\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(F\right)\right)\right)\right)\right):
The font family may be specified as an option.
F≔\mathrm{Font}\left("Some text",\mathrm{size}=20,\mathrm{color}=\mathrm{DarkCyan},\mathrm{family}="DejaVu Sans"\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(F\right)\right)\right)\right)\right):
Text may be placed in a superscript position.
\mathrm{FM}≔\mathrm{Font}\left("M",\mathrm{size}=20\right):
\mathrm{FE}≔\mathrm{Font}\left("e",\mathrm{size}=20,\mathrm{superscript}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(\mathrm{FM},\mathrm{FE}\right)\right)\right)\right)\right):
The worksheet's hyperlink style can be respected.
F≔\mathrm{Font}\left("Some text",\mathrm{size}=16,\mathrm{color}=\mathrm{blue},\mathrm{style}=:-\mathrm{Hyperlink}\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(\mathrm{Group}\left(\mathrm{Input}\left(\mathrm{Textfield}\left(F\right)\right)\right)\right)\right):
By default a background other than white is opaque with respect to an underlying shade such as the fillcolor of a Table Cell. And by default a white background is not opaque. In either case the opaque option can be used to force the behavior.
s≔\mathrm{size}=16,"The quick brown fox jumps over the lazy dog":
\mathrm{cf}≔\mathrm{fillcolor}=[0,120,150]:
\mathrm{F1}≔\mathrm{Font}\left(s,\mathrm{background}=\mathrm{Orange}\right):
\mathrm{F2}≔\mathrm{Font}\left(s,\mathrm{background}=\mathrm{Orange},\mathrm{opaque}=\mathrm{false}\right):
\mathrm{F3}≔\mathrm{Font}\left(s,\mathrm{background}=\mathrm{Orange},\mathrm{opaque}=\mathrm{true}\right):
\mathrm{F4}≔\mathrm{Font}\left(s\right):
\mathrm{F5}≔\mathrm{Font}\left(s,\mathrm{opaque}=\mathrm{false}\right):
\mathrm{F6}≔\mathrm{Font}\left(s,\mathrm{opaque}=\mathrm{true}\right):
T≔\mathrm{Table}\left(\mathrm{alignment}=\mathrm{center},\mathrm{width}=50,\mathrm{seq}\left(\mathrm{Row}\left(\mathrm{Cell}\left(\mathrm{cf},F‖i\right)\right),i=1..6\right)\right):
\mathrm{InsertContent}\left(\mathrm{Worksheet}\left(T\right)\right):
The DocumentTools:-Layout:-Font command was introduced in Maple 2015.
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Stability versions of Erdős–Ko–Rado type theorems via isoperimetry | EMS Press
Stability versions of Erdős–Ko–Rado type theorems via isoperimetry
Erdős–Ko–Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family.
In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of exact EKR-type results into strong stability versions of these results, without going into the proofs of the original results.
We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede–Khachatrian theorem on
t
-intersecting families of
k
\{1,\dots,n\}
k < n/(t+1)
), and to show that, somewhat surprisingly, all these results hold when the ‘intersection’ requirement is replaced by a much weaker requirement.
Other examples include stability versions of Frankl’s recent result on the Erdős matching conjecture, the Ellis–Filmus–Friedgut proof of the Simonovits–Sós conjecture, and various EKR-type results on
r
-wise (cross-)
t
-intersecting families.
David Ellis, Nathan Keller, Noam Lifshitz, Stability versions of Erdős–Ko–Rado type theorems via isoperimetry. J. Eur. Math. Soc. 21 (2019), no. 12, pp. 3857–3902
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How to factor using the greatest common factor | StudyPug
x^2 + bx + c
ax^2 + bx + c
x^2 + bx + c
ax^2 + bx + c
What happens when you are asked to factor a polynomial? It's actually quite similar to what you do when you factor numbers. You'll have to find numbers that you can divide out from the polynomials evenly. Unlike working with just numbers, in polynomials you have to divide numbers out of terms rather than just a single number.
Finding factors of polynomials
So how do you go about finding what the factors are in a polynomial? You'll have to learn to spot what can be factored out of every term. This is the common factor.
In the past, when you're asked to simplify expressions, you'd have to distribute numbers into terms in parentheses. For example, in 4(x+2), you'd evaluate it as 4x+8. When we're factoring out terms, you'll have to do the opposite of that! Other than just learning to look at a set of polynomials and identifying the factor, you can also use the method of finding the greatest common factor. Let's learn this first step to factoring polynomials.
In order to find the greatest common factor (GCF), you'll have to find the prime factors for each of the numbers you are working with. Then you'll multiply the factors that all the numbers have in common.
So for example, if you had the numbers 10 and 5 and you had to find their common factor, you'd first tackle the 10, and then the 5 as follows:
\footnotesize2\frac{)10}{5}
\footnotesize5\frac{)5}{1}
In both of these, you see that you have 5. Since there's only 1 factor in common, you don't have to multiply the 5 by anything else. Therefore, you've found that 5 is the biggest numbers that divides evenly into 10 and 5. If you don't find any common factors, then your GCF will be 1.
This is the method you'll employ in order to find the factors of polynomials. Follow along as we solve the upcoming example questions to see how the GCF is used when factoring expressions.
12p^{7} - 18p^{2} - 30
Look for the greatest common factor of this polynomial using long division. You can do each of the terms separately or you can do them altogether like what we did here:
GCF of multiple numbers
The GCF of this polynomial is found through multiplying the common factors from all three of the numbers together. This means you'll get:
We then factor out 6 from each term of the polynomial, and we'll get the final answer of:
6[2p^{7} - 3p^{2} - 5]
10z(x + 2y) - 6(x + 2y)
Firstly, let us look for the common factors of the polynomial. When you first look at the numbers, you'll likely spot out that the common factor of 10 and 6 is 2.
Another common factor is (x+2y). So, we factor out both and we'll get the final answer of:
(2)(x + 2y)[5z - 3]
If you ever need to double check your answer when trying to find the common factors of polynomials, try out this online GCF calculator. It will help you be sure of your answers when you factor more complex polynomials. As always, do remember that the calculator should only be used to check your answers rather than doing the questions for you!
Ready to move on? Learn how to complete the square in quadratic functions, convert quadratic functions from general to vertex form, and solve quadratic equations by factoring or by completing the square.
x^2 - y^2
To factor means to take out a common factor out from the expressions. In this lesson, we will try to do it by determining the greatest common factor among the terms of the expressions, and factor it out from each term.
Basic Concepts: Common factors of polynomials, Factoring polynomials:
x^2 + bx + c
, Factoring polynomials:
ax^2 + bx + c
review "factoring"
12{p^7} - 18{p^2} - 30
10z\left( {x + 2y} \right) - 6\left( {x + 2y} \right)
x^2 - y^2
x^2 - y^2
|
All functions are locally $s$-harmonic up to a small error | EMS Press
All functions are locally
s
-harmonic up to a small error
University of Melbourne, Australia, University of Milano, Italy, and WIAS, Berlin, Germany
We show that we can approximate every function
f\in C^{k}(\overline{B_1})
s
-harmonic function in
B_1
that vanishes outside a compact set.
s
-harmonic functions are dense in
C^{k}_{\mathrm {loc}}
.This result is clearly in contrast with the rigidity of harmonic functions in the classical case and can be viewed as a purely nonlocal feature.
Serena Dipierro, Ovidiu Savin, Enrico Valdinoci, All functions are locally
s
-harmonic up to a small error. J. Eur. Math. Soc. 19 (2017), no. 4, pp. 957–966
|
The Gregorian calendar is the calendar used in most of the world.[1][a] It was introduced in October 1582 by Pope Gregory XIII as a modification of, and replacement for, the Julian calendar. The principal change was to space leap years differently so as to make the average calendar year 365.2425 days long, more closely approximating the 365.2422-day 'tropical' or 'solar' year that is determined by the Earth's revolution around the Sun.
The rule for leap years is:
Calendar cycles repeat completely every 400 years, which equals 146,097 days.[c][d] Of these 400 years, 303 are regular years of 365 days and 97 are leap years of 366 days. A mean calendar year is 365+97/400 days = 365.2425 days, or 365 days, 5 hours, 49 minutes and 12 seconds.[e] During intervals that do not contain any century common years (such as 1900), the calendar repeats every 28 years, during which February 29 will fall on each of the seven days of the week once and only once. All other dates of the year fall on each day exactly four times, each day of the week having gaps of 6 years, 5 years, 6 years, and 11 years, in that order.
On 29 September 1582, Philip II of Spain decreed the change from the Julian to the Gregorian calendar.[20] This affected much of Roman Catholic Europe, as Philip was at the time ruler over Spain and Portugal as well as much of Italy. In these territories, as well as in the Polish–Lithuanian Commonwealth[21] (ruled by Anna Jagiellon) and in the Papal States, the new calendar was implemented on the date specified by the bull, with Julian Thursday, 4 October 1582, being followed by Gregorian Friday, 15 October. The Spanish and Portuguese colonies followed somewhat later de facto because of delay in communication.[22][dead link] The other major Catholic power of Western Europe, France, adopted the change a few months later: 9 December was followed by 20 December.[23]
{\displaystyle D=\left\lfloor {Y/100}\right\rfloor -\left\lfloor {Y/400}\right\rfloor -2}
{\displaystyle D}
is the secular difference and
{\displaystyle Y}
is the year using astronomical year numbering, that is, use (year BC) − 1 for BC years.
{\displaystyle \left\lfloor {x}\right\rfloor }
means that if the result of the division is not an integer it is rounded down to the nearest integer. Thus during the 1900s, 1900/400 = 4, while during the −500s, −500/400 = −2.
^ "Introduction to Calendars". United States Naval Observatory. n.d. Retrieved 9 May 2022.
^ "The Calendar FAQ: The Gregorian Calendar". Tondering.dk. Retrieved 3 May 2022.
Inter gravissimas in English
Retrieved from "https://en.wikipedia.org/w/index.php?title=Gregorian_calendar&oldid=1089167366"
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Introducing SZO token - Guide to the ShuttleOne.Network
Liquidity Rewards, Mining, Governance and Utility
The SZO token is a utility token that regulates incentivies and secures the network for token holders and suppliers of liquidity in the mining pool.
If you hold the ShuttleOne token (SZO), you get access to higher interest rates that are generated by real world assets collaterals. Businesses that are in need of working capital or growth funds collateralize their real-world assets (such as commodities, purchase orders, cargos, etc.) and get a loan from ShuttleOne.
Token Supply - 230,000,000 SZO
Token Type - ERC20
Inflation - 5% a year or 11.5mil SZO per year.
How can I get SZO token?
1. Provide liquidity into the ShuttleOne.Network with your stablecoin (DAI, USDC, USDT) and earn SZO as a reward for supporting the protocol.
2. Buy SZO off exchanges that are listing the token.
Usage of ShuttleOne.Network
The primary use case of the SZO token is access payment to utilize the products and services offered by the ShuttleOne.Network to our users.
Merchants who taps on the ShuttleOne.Network of products, for services such as onchain risk management, credit for trade financing, kyb and aml regulatory compliance and remittance pays a fee set by the network in SZO.
Token holders are incentivised by holding onto the SZO token as more SZO is burnt whenever there is a transaction done via the network.
Estimated Burn vs Circulation According to Network Usage
Rewards Mining
Liquidity suppliers to the ShuttleOne.Network enjoys a rate of return per second in SZO that is shared to liquidty providers paid out per block. The rate per 30 days is as follows:
SZOperblock30 = 0.000000385802469136
Should liquidity suppliers wish to support the network for longer terms the rewards grows exponentially of the above rewards to the power of 1.2.
Governance & Vaults
ShuttleOne will be releasing announcements of Governance & Vaults related details in Q4 2020. We plan to deliver these before Q3 2021.
|
Defaults & Liquidation - Guide to the ShuttleOne.Network
Standard Operating Procedures on how ShuttleOne.Network handles borrower defaults
Tokenizing real world assets to bridge the digital realm on the blockchain requires not only technical abilities on blockchain technologies but also process orientated procedures on how to recover defaults to protect Liquidity Providers and the SZO Token Holders.
A default is defined in Time as:
D = T+259201
In simpler, ShuttleOne codes in a estimated 3 day grace for Fiat On Ramps to happen when the borrower repays back in fiat in their local currency before the smart contracts get repaid into the liquidity pools.
Technically, the Risk Assesement Tokens (NFT) that represents the underwriting collateral moves into an auction contract while we commence the Ecosystem Processes below.
Once a loan is deemed to be in default, ShuttleOne activates a 4 step procedure to commence to recover the debt.
Reminders of Financing Penalties (4 weeks)
Suspension of Trade Ecosystem Accounts (1 day)
Confiscation of Asset for Liquidation (3-6 weeks)
Legal Process in Permited Jurisdictions
Reminders of Financing Penalties
For 4 weeks after the default has been confirmed, ShuttleOne through our ecosystem partners will try to communicate with the borrower to recover the principle with interests in arrears.
Email Communications with Defaulter
Other than the usual Emails, SalesOps also follows up with calls and visits to the borrower's operation facilities. Should defaulter choose to ignore these steps, we proceed to Step 2.
Suspension of Trade Ecosystem Accounts
ShuttleOne partners Ecosystem partners such as ecommerce platforms, institutions and various other different types of digital partners to facilitate trade financing. The merchants comes through these distribution channels.
We then proceed to put up a request to our ecosystem partners to suspend their trading accounts for a period until the arrears are recovered. With this move, the borrower is put under more pressure that their business is at risk.
Should this method not work, we will proceed to Step 3.
Confiscation of Assets for Liquidation
Every borrowing that ShuttleOne.Network facilitates, it is backed up by the receivables of the assets in the invoices. It can also be backed up by the cargo assets in trade financing.
We will therefore proceed to request for ownership of these assets to be liquidated in our Liquidators' Network to recover the arrears.
Different type of assets require different types of Liquidators. Currently within our network there represents Liquidators in:
Hospitals, Government Health Ministries
Energy & Chemical Traders
These liquidators allow ShuttleOne to facilitate the recovery of asset value that underwrites the borrowing back to the SpacePods.
Should all else fail, and/or the liquidated asset value above doesn't cover the priciple of the loan, we proceed to step 4.
ShuttleOne alongside with a network of legal partners will commence proceedings against the default to recover the value of borrowing.
|
Finding the square and square root of a number | StudyPug
To explain square roots, let's take a step back and remember what it means to square a number. To square is to raise the number to the second power. Square roots are the opposite of that, and is actually the inverse operation of squaring. To square root is to find the two identical factors of a number.
For numbers that are perfect squares, you can find whole numbers as answers. However, for numbers that aren't perfect squares, you'll have to use a method that involves estimation (or you can use a table of square and square roots).
Finding square root of perfect square numbers
Let's first take a look at this question here:
What is the square root of 64? If you have a calculator, you can always just punch in it and get the answer. But do you know how to find the square root of a number without a calculator?
Now, if you do remember your perfect squared numbers, the root of 64 is just eight. Eight times eight gives you 64. But let's say you can't freely recall perfect numbers. How would do we do this from scratch?
First, you will have to find all the prime factors of 64. So, let's go ahead and do that:
prime factors of root 64
Imagine that the question now becomes 2x2x2x2x2x2— 2 is multiplied 6 times here. So we've just determined that 64 is just a square root of six 2s, all multiplied together.
Before we move on, we must remember that the radical sign actually means "the square root". The square root symbol should really be written with a tiny little two here:
Since it's a square root, you can pick a pair of identical numbers to work with and bring them out from under the radical. In this case, we'll take out a 2 from the first pair of 2s, another 2 from the second pair, and another 2 from the last pair. It should look something like this:
taking out pairs
Now if you multiply the 2s with one another, what do you get? You'll find that you get 8, which is exactly the same as what you would have remember if you knew your perfect squares. However, this is the correct way to find the square root of a number without memorization.
Finding square root of numbers that aren't perfect squares
The basic method to find the square root of a number that is not a perfect square is as follows:
Estimate: Pick a number that if you square comes close to, but is less than, the square root of the number you're trying to find.
Divide: Divide the number that you are finding the squared root for with the number you picked in step 1
Average: Take the average of the number you got in step 2 and the square root
Repeat: Repeat steps 2 and 3 until the number is accurate enough for you
Now you've learned how to find the square root for numbers that both are and are not perfect squares. Continue on with our lessons to learn how to deal with different radical numbers examples.
To square is to raise the number to the second power. In other words, to square is to multiply the number by itself. Square root is the inverse operation of squaring. To square root is to find the two identical factors of a number.
Basic Concepts: Squares and square roots, Estimating square roots, Prime factorization
Related Concepts: Conversions involve squares and cubic, Operations with radicals, Conversion between entire radicals and mixed radicals
To square:
Raise the number to the second power
{5^2}
5\times 5 = 25
{8^2}
8\times 8 = 64
To square root:
Finding the two identical factors
\sqrt{16}
\sqrt{4\times 4}
\sqrt{49}
\sqrt{7\times 7}
Perfect squares numbers:
{0^2}
{1^2}
{2^2}
{3^2}
{4^2}
{5^2}
{6^2}
{7^2}
{8^2}
{9^2}
& so on... {100, 121, 144, 169, 196...}
Understanding the negative square roots of the following
\sqrt{225}
-\sqrt{225}
\sqrt{-225}
Find the square roots
\sqrt{64}
-\sqrt{676}
\sqrt{-81}
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MVRV - Market Value To Realized Value | Santiment Academy
MVRV - Market Value To Realized Value
Ivan, Irina Pranovich
MVRV shows the average profit/loss of all the coins currently in circulation given the current price.
We need to define two terms:
MV as in Market Value refers to the well-known capitalization. The second part is the
RV as in Realized Value. It is an alternative to the Market Value where instead of the current price, every coin/token is multiplied by its acquisition price.
The definition of MVRV is:
MVRV = \frac{MV}{RV}
MVRV value of 2 means that if all holders sell their coins/tokens at the current price, they will generate an x2 profit on average. In this sense, MVRV shows the ratio between the current price and the average price of every coin/token acquired. The more the ratio increases, the more people will be willing to sell as the potential profits increase.
The value of MVRV gives an idea of how much overvalued or undervalued an asset is.
If the MVRV value is between 0 and 1, then the market is "undervalued" on average, meaning most people will be realizing losses if they all sell their holdings at the current price.
Keep in mind that this is in the ideal case and does not account for the addresses with private keys or graveyard addresses. The way to adjust for this is to look at the historical values for the MVRV values. As the value is approaching historical maximums or minimums, then the possibility of a highly overvalued or undervalued market is much higher.
Another way to deal with lost private keys and graveyard addresses is to compute the MVRV value only taking into account the subset of tokens that have been active at least once in the last several years.
Timebound Metrics available.
The timebound metrics can help exclude the inactive addresses. These metrics are computed the same way as the MVRV metric, with the only difference that they take into account only the coins/tokens that have moved in the desired time range. Examples: mvrv_usd_365d is computed on the coins/tokens that moved at least once in the past 365 days. mvrv_usd_60d is computed by taking only the coins/tokens that moved at least once in the past 60 days.
Comparing timebound MVRV values of different time ranges can clarify how much profit/loss long-term and short-term holders can realize.
MVRV Long/Short Difference
MVRV Long/Short Difference is defined as mvrv_usd_365d - mvrv_usd_60d
Negative values mean that short-term holders will realize higher profits than long-term holders if they sell at a price at this moment. Positive values show the opposite.
During strong and long bull runs, this metric tends to grow, and during bear markets it is decreasing. The rationale is that during strong bull runs, the long term holders are determining when the bull run will end when they start selling, while during bear markets, the long term holders are at a loss on average and the short term holders manage to realize profits
All available assets have Daily Intervals A subset of the available assets that consists of some of the bigger assets have Five-Minute Intervals
The daily interval MVRV metrics are available for these assets
The 5-minute MVRV metrics are available for these assets
The daily metrics are available under the mvrv_usd name under the mvrv_usd_<interval> name for the timebound metrics. The 5-minute interval metrics are available under the mvrv_usd_intraday name under the mvrv_usd_intraday_<interval> name for the timebound metrics.
Example of query for mvrv_usd:
getMetric(metric: "mvrv_usd") {
Example of query for mvrv_usd_intraday:
getMetric(metric: "mvrv_usd_intraday") {
slug: "bitcoin"
from: "utc_now-90d"
to: "utc_now-30d"
Example of query for timebound MVRV:
getMetric(metric: "mvrv_usd_7d") {
Example of query for MVRV long-short difference:
getMetric(metric: "mvrv_long_short_diff_usd") {
|
A new characterization of chord-arc domains | EMS Press
\Omega \subset \mathbb{R}^{n+1}
n\geq 1
, is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of
\Omega
implies the existence of exterior corkscrew points at all scales, so that in fact,
\Omega
is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior corkscrew conditions, and an interior Harnack chain condition. We discuss some implications of this result for theorems of F. and M. Riesz type, and for certain free boundary problems.
Jonas Azzam, Steve Hofmann, José María Martell, Kaj Nyström, Tatiana Toro, A new characterization of chord-arc domains. J. Eur. Math. Soc. 19 (2017), no. 4, pp. 967–981
|
Modular generalized Springer correspondence II: classical groups | EMS Press
Modular generalized Springer correspondence II: classical groups
Pramod N. Achar
We construct a modular generalized Springer correspondence for any classical group, by generalizing to the modular setting various results of Lusztig in the case of characteristic-0 coefficients.We determine the cuspidal pairs in all classical types, and compute the correspondence explicitly for SL(
n
) with coefficients of arbitrary characteristic and for SO(
n
) and Sp(2
n
) with characteristic-2 coefficients.
Pramod N. Achar, Anthony Henderson, Daniel Juteau, Simon Riche, Modular generalized Springer correspondence II: classical groups. J. Eur. Math. Soc. 19 (2017), no. 4, pp. 1013–1070
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Ask Answer - Prepositions - Expert Answered Questions for School Students
Tell me why the Ali is going to post office
Please answered fast
Fill in the blank with the correct preposition given in the bracket:\phantom{\rule{0ex}{0ex}}1\right) Go straight down Kingsway Street and you will find the shop _________ Your right .\left(in , on\right)\phantom{\rule{0ex}{0ex}}2\right) The jeweler\text{'}s store is right next _______ the bus stop \left(for , to\right)\phantom{\rule{0ex}{0ex}}3\right) The florist\text{'}s stall is _______ Kamna\text{'}s house \left(Below , under\right).\phantom{\rule{0ex}{0ex}}4\right) You can find her house ________ the corner of the main street . \left( around , from\right)
Q. What is the meaning of following word.
Most of the people are annoyed __ passwords.
Thief escaped ___(through/by) a small hole
Please answer this question from anne frank's novel
Q.1. What thing interrupted the dinner of the Franks taken with Miep and Henk?
Experts and dear friends kindly tell me what I should study in order to achieve good marks in writting portion of English.
The peddler delete that the whole world is a rat trap. How did he himself get caught in the same ? ( RATTRAP)
Why the letters ABCD on keyboard not written in a straight line
Nothing holding devices for counting as developer line around the neck and Chin experts warn this new problem has been up technique and could affecting anyone who using a modern gadgets the problem was identifying after a side of neck related enquiry non surgical treatment for me too, lines around the neck area.--> please do the editing
Hlo experts ... i just wanna know that would i be able to get answers for F.I.T. subject its in my school and its a bit difficult to find its answers... hope i will get a positive response ... Thanks ^0^
Plz slove this question last question value based question only please slove at last question
Jharinath
in the lesson mrs.packledide's tiger. why were the boys posted day and night at the end of local jungle? pls answer detailedly
|
A quasiconformal composition problem for the $Q$-spaces | EMS Press
A quasiconformal composition problem for the
Q
Yi Ru-Ya Zhang
Beijing University of Aeronautics and Astronautics, China and University of Jyväskylä, Finland
Given a quasiconformal mapping
f:{\mathbb R}^n\to{\mathbb R}^n
n\ge2
, we show that (un-)boundedness of the composition operator
{\mathbf C}_f
on the spaces
Q_{\alpha}({\mathbb R}^n)
depends on the index
\alpha
and the degeneracy set of the Jacobian
J_f
. We establish sharp results in terms of the index
\alpha
and the local/global self-similar Minkowski dimension of the degeneracy set of
J_f
. This gives a solution to [3, Problem 8.4] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel–Lizorkin and Besov spaces. Consequently, Tukia–Väisälä's quasiconformal extension
f:{\mathbb R}^n\to{\mathbb R}^n
of an arbitrary quasisymmetric mapping
g:{\mathbb R}^{n-p}\to {\mathbb R}^{n-p}
is shown to preserve
Q_{\alpha} ({\mathbb R}^n)
(\alpha,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\}
Q_{\alpha}({\mathbb R}^n)
is shown to be invariant under inversions for all
0<\alpha<1
Pekka Koskela, Jie Xiao, Yi Ru-Ya Zhang, Yuan Zhou, A quasiconformal composition problem for the
Q
-spaces. J. Eur. Math. Soc. 19 (2017), no. 4, pp. 1159–1187
|
Commensurating endomorphisms of acylindrically hyperbolic groups and applications | EMS Press
We prove that the outer automorphism group Out
(G)
is residually finite when the group
G
is virtually compact special (in the sense of Haglund and Wise) or when
G
is isomorphic to the fundamental group of some compact 3-manifold.
To prove these results we characterize commensurating endomorphisms of acylindrically hyperbolic groups. An endomorphism
\phi
G
is said to be commensurating, if for every
g \in G
some non-zero power of
\phi(g)
is conjugate to a non-zero power of
g
. Given an acylindrically hyperbolic group
G
, we show that any commensurating endomorphism of
G
is inner modulo a small perturbation. This generalizes a theorem of Minasyan and Osin, which provided a similar statement in the case when
G
is relatively hyperbolic. We then use this result to study pointwise inner and normal endomorphisms of acylindrically hyperbolic groups.
Yago Antolín, Ashot Minasyan, Alessandro Sisto, Commensurating endomorphisms of acylindrically hyperbolic groups and applications. Groups Geom. Dyn. 10 (2016), no. 4, pp. 1149–1210
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Wind Turbines | Building DC Energy Systems
Harvesting wind energy is one of the oldest forms of energy production in human history. Even without an electric generator it can be used to drive pumps or mills directly. This chapter will explain the core principles of wind energy and highlight the advantages as well as the disadvantages.
# Turbine Types
The two main types of turbines are distinct by their working principles. Drag type turbines use the fact that certain bodies or geometries have different drag when moving against the wind and are comonly reffered to as "Savonius turbines". Lift type turbines use the lift effect on airfoils in moving liquids, much like an airplane.
According to Betz's Law, the maximum amount of energy that can be extracted from the wind is around
\frac{16}{27}
\approx60\%
of the total kinetic energy of the moving air [3]. The most important detail of turbines is therefore the power coefficient
c_p
and how close it is to the maximum posted by Betz's Law. It is defined by the ratio of turbine energy output to kinetic wind energy [3]:
c_p=\frac{P_{turbine}}{P_{wind}}
# Drag Runner
Drag type turbines are always Vertikal Axis Wind Turbines (VAWT), where the rotational axis is vertical to the wind.
Figure 1. Two-scooped savonius turbine [1].
The mathematical equation to describe the force on the scoops is given by
F = \frac{c_w}{2} \cdot \varrho_{air} \cdot A \cdot v_{air}^2
A
is the cross-sectional area of the scoop,
v_{air}
is the windspeed,
\varrho_{air}
is air density and
c_w
is the drag coeeficient specific for the geometry used. The coefficient is derived via measurements and can be found in corresponding tables. The force driving the rotation is the difference between the coefficients
c_{w,1}
c_{w,2}
for the flat and curved sides of the scoop, respectively:
\Delta F = \frac{c_{w,1}}{2} \cdot \varrho_{air} \cdot A \cdot v_{air}^2 - \frac{c_{w,2}}{2} \cdot \varrho_{air} \cdot A \cdot v_{air}^2 = \frac{c_{w,1} -c_{w,2}}{2} \cdot \varrho_{air} \cdot A \cdot v_{air}^2
A Savonius rotor's rotation speed can not exceed the wind velocity, so the Tip Speed Ratio (TSR) defined as
\lambda = \frac{\omega r}{v_{air}}
\omega
as angular frequency and
r
as radius, is always
\lambda\le1
. Due to their low power coeefficient (
c_p \lt 25\%
) and low rotatioal speed, Savonius turbines are not commonly used for power generation. There are however cases were a high starting torque, simple design and low maintenance requirements are beneficial.
# Lift Runner
Lift running turbines use airfoils as blade geometry and are mostly divided into VAWT and Horizontal Axis Wind Turbines (HAWT). There are other hybrids and experimental types of turbines, but those are not commonly used. Most lift types have a much higher TSR (
\lambda
=2...12) [4] than drag types thus the wind velocity at the blade is much higher, resulting in better power coeeficients. All lift types have a relativly low starting torque, resulting in a high cut-in windspeed.
Figure 2. Basic principle of lift on an airfoil [2].
Horizontal turbines are most commonly seen in commercial wind energy production due to there high power efficiency (
c_p \approx50\%
) and scalability. To control rotational speed, large systems use active pitch controller where each blade can be individually pitched to adapt generated torque. Small turbines with fixed blades need either a passive stall design or a yaw mechanism to effectivly lower the wind-facing area of the turbine. Most systems will have mechanical or hydraulic brake systems for storm protection. Since the relative windspeed attacking the blade changes along the radius, the angle of attack also changes which leads different optimal blade geometries along the blade [4].
High Power coefficient Yaw system to follow wind needed
No pulsating torque on drive train High noise generation due to high tip speed
Well understood and developed Complex blade design and production
# VAWT
Vertical axis wind turbines have a power coefficient of
c_p \approx 30\%-40\%
and are not often found in commercial setups. They have however some advantages that makes them suitable for a range of applications. Namely their low noise profile and unusal appearance give them high acceptance in urban areas. They also do not require a yaw system and are better suited for turbulent areas. Most common designs are the Darriues turbine, the helical bladed turbine and the straight-bladed Darrieus turbines (see Wikipedia (opens new window) for more details)
No yaw mechanism needed Medium Power coefficient
Low noise profile High normal forces on axis
Simple blade geometry Oscillating torque on generator
Electric generators can be classified into radial flux machines and axial flux machines. The main difference is the flow direction of magnetic flux, where it is either parallel or radial to the rotaional axis. The common synchronous and asynchronous (induction) machines belong to the radial flux machines and are mainly used in large turbines. One of the main challenges for larger wind turbines is the connection to the grid. While DC-Sources can be connected to the grid when their voltage is equal to the grid voltage, AC-Sources have to be in synch with the grid's voltage frequency and amplitude [4]. For small scale, especially off-grid turbines, axial flux generators are more common as they can be fairly easily constructed without high level machinery, especially for low-rpm usecases.
The distinctive property of induction generators is the slip between rotor and stator fields. In generator operation, the turbine moves the rotor above the synchronous speed. Because of the two magnetic fields running against each other, this type of generator can introduce high reactive power loads in the system.
The synchronous speed
n_s
in rpm is dependend on the pole count
p
(as pairs, four poles -> two pairs) of the stator and the excitation frequenzy
f_e
n=\frac{f_e\cdot60}{p}
The synchronous speed for a four pole generator on grid frequenzy of
f=50Hz
1500rpm
. To achieve these speeds, a gearbox is installed. Alternatively the pole pairs could be increased, leading to higher overall mass. To run a turbine at
n_s=120rpm
f=e=50Hz
p=25
pole pairs (!) are needed which is simply not feasible for this type of machine. To manipulate the synchronous speed to allow variable turbine speed, several approaches are possible. The quasi-standard are double-fed induction generators (DFIG) (always as wound-rotor) where the field windings are fed with an adjustable frequency AC-power [4] while the armature windings are directly connected to the grid or other equipment for conversion. With this setup, the rotational speed of the turbine can vary with windspeed while the active and reactive power of the generator can be controlled.
Figure 3. Schematics for a double fed induction generator [5].
# Field Coil Excitation
Synchronous generators can be build with a much higher pole count than asynchronous generators and are used for gearless setups which reduces the maintance requirements significantly. If they are directly coupled to the grid, they need to be synchronized first and cannot run with different speeds. In fact, when the force on the turbine increases and the generator comes out of sync with the grid, the system can be damage due to balancing currents. Most turbines have a rectifier and a power inverter to connect to the grid.
# Permanent Magnet Excitation
Exitation with permanent magnets reduces the complexity and therefor maintance. For this reason, PM generators are used mostly on offshore wind farms. There is also no energy needed for exitation so they have slightly better efficieny. Considering the large amount of rare earths needed for permanent magnets though, this effect is negligible [6].
The axial flux or axial gap generators are a special type of synchronous PM generators not widely found in larger applications. It can however be found frequently for small scale DIY turbines. Their flat structure and easy-to-wind coils makes them very suited for small workshops. They provide a fairly good efficiency and common materials such as wood, steel and epoxy can be used. Since the rotor becomes wider with increasing pole count, rotational inertia and centrifugal forces are increased as well, though higher pole count mostly means lower rpm needed.
Depending on the generator used and to what load (machine, grid, battery, ...) the sytem will be connected, different electronic components are necessary. Since this OER is aimed at makers and for off-grid (or local tiny-grid) use cases, high power AC/DC-DC/AC setups, frequency synchronizing elements for direct coupled wind turbines or storage units like pumped hydroelectric energy storage are not discussed. Most common standalone systems consist of a source, a sink and a storage unit.
As a storage unit, mostly batteries are used. While it is possible to connect a turbine directly to some types of batteries using a rectifier and a charge controller like Hugh Piggott suggested, it has some serious drawbacks.
Figure 4. Simplified layout for direct connection to the battery after Piggott's design.
Whenever there is an increase in voltage or the battery is full and no load is connected, current will be diverted to the dump load and dissipated. If the voltage of the turbine is much higher than the battery's, it's not suitable to charge the battery. The same happens when the turbine voltage is below charging voltage. To mitigate these flaws, a rectifier and a DC/DC-Converter can be used to either step-down or step-up the voltage needed to efficiently charge the battery. This is mostly done inside a charge controller which can take several parameters into account like optimal current at given generator voltage. Modern batteries like Lithium-Ion require a proper Battery Management System to work safely.
Figure 5. Simplified layout for controlled connection to the battery.
The disadvantage of this design is that all components need to be able to take the full maximum power taken by the load or given by the turbine.
One of the most famous DIY-projects is Hugh Piggott's (opens new window) 2F-Windturbine. His books (opens new window) provide many details as well as basic principles on how to design, build and run small off-grid turbines.
Another well-documented project with good video (opens new window) material is James Biggar's Reaper Turbine (opens new window)
Another seriously low-tech wind turbine is the design provided as open source by Daniel Connel (opens new window). It is a VAWT as a combination of lift and drag type and can be constructed for under 100€ in material cost.
[1] !Original: RottweilerVector: Cmglee, CC BY-SA 3.0, Link (opens new window)
[2] By Vector by רונאלדיניו המלך, Original by J Doug McLean, CC BY-SA 4.0, Link (opens new window)
[3] E. Hau, Windkraftanlagen: Grundlagen, Technik, Einsatz, Wirtschaftlichkeit. Springer-Verlag Berlin Heidelberg, 2008
[4] S. Heier, Windkraftanlagen: Systemauslegung, Netzintegration und Regelung. 6. Auflage, Springer Vieweg Verlag Wiesbaden, 2018, DOI: https://doi.org/10.1007/978-3-8348-2104-1
[5] By Funkjoker23 - Own work, CC0, https://commons.wikimedia.org/w/index.php?curid=17634829
[6] Wikipedia contributors. (2020, October 19). Rare-earth element. In Wikipedia, The Free Encyclopedia. Retrieved 07:59, November 6, 2020, Linkhttps://en.wikipedia.org/w/index.php?title=Rare-earth_element&oldid=984305200)
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You are given a bag that you are told contains eight marbles. You draw out a marble, record its color, and put it back.
If you repeat this eight times and you do not record any red marbles, can you conclude that there are not any red marbles in the bag? Explain.
Possible scenario: There are
7
blue marbles and
1
red marble in the bag.
You pick a marble eight times and every time it is blue.
If you repeat this
100
times and you do not record any red marbles, can you conclude that there are not any red marbles in the bag? Explain.
How many times do you have to draw marbles (putting them back each time) to be absolutely certain that there are no red marbles in the bag?
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