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Section 43.24 (0B0D): Moving Lemma—The Stacks project
Section 43.24: Moving Lemma (cite)
43.24 Moving Lemma
The moving lemma states that given an $r$-cycle $\alpha $ and an $s$-cycle $\beta $ there exists $\alpha '$, $\alpha ' \sim _{rat} \alpha $ such that $\alpha '$ and $\beta $ intersect properly (Lemma 43.24.3). See [Samuel], [ChevalleyI], [ChevalleyII]. The key to this is Lemma 43.24.1; the reader may find this lemma in the form stated in [Example 11.4.1, F] and find a proof in [Roberts].
Lemma 43.24.1. Let $X \subset \mathbf{P}^ N$ be a nonsingular closed subvariety. Let $n = \dim (X)$ and $0 \leq d, d' < n$. Let $Z \subset X$ be a closed subvariety of dimension $d$ and $T_ i \subset X$, $i \in I$ be a finite collection of closed subvarieties of dimension $d'$. Then there exists a subvariety $C \subset \mathbf{P}^ N$ such that $C$ intersects $X$ properly and such that
\[ C \cdot X = Z + \sum \nolimits _{j \in J} m_ j Z_ j \]
where $Z_ j \subset X$ are irreducible of dimension $d$, distinct from $Z$, and
\[ \dim (Z_ j \cap T_ i) \leq \dim (Z \cap T_ i) \]
with strict inequality if $Z$ does not intersect $T_ i$ properly in $X$.
Proof. Write $\mathbf{P}^ N = \mathbf{P}(V_ N)$ so $\dim (V_ N) = N + 1$ and set $X_ N = X$. We are going to choose a sequence of projections from points
\begin{align*} & r_ N : \mathbf{P}(V_ N) \setminus \{ p_ N\} \to \mathbf{P}(V_{N - 1}), \\ & r_{N - 1} : \mathbf{P}(V_{N - 1}) \setminus \{ p_{N - 1}\} \to \mathbf{P}(V_{N - 2}), \\ & \ldots ,\\ & r_{n + 1} : \mathbf{P}(V_{n + 1}) \setminus \{ p_{n + 1}\} \to \mathbf{P}(V_ n) \end{align*}
as in Section 43.23. At each step we will choose $p_ N, p_{N - 1}, \ldots , p_{n + 1}$ in a suitable Zariski open set. Pick a closed point $x \in Z \subset X$. For every $i$ pick closed points $x_{it} \in T_ i \cap Z$, at least one in each irreducible component of $T_ i \cap Z$. Taking the composition we obtain a morphism
\[ \pi = (r_{n + 1} \circ \ldots \circ r_ N)|_ X : X \longrightarrow \mathbf{P}(V_ n) \]
which has the following properties
$\pi $ is finite,
$\pi $ is étale at $x$ and all $x_{it}$,
$\pi |_ Z : Z \to \pi (Z)$ is an isomorphism over an open neighbourhood of $\pi (x_{it})$,
$T_ i \cap \pi ^{-1}(\pi (Z)) = (T_ i \cap Z) \cup E_ i$ with $E_ i \subset T_ i$ closed and $\dim (E_ i) \leq d + d' + 1 - (n + 1) = d + d' - n$.
It follows in a straightforward manner from Lemmas 43.23.1, 43.23.2, and 43.23.3 and induction that we can do this; observe that the last projection is from $\mathbf{P}(V_{n + 1})$ and that $\dim (V_{n + 1}) = n + 2$ which explains the inequality in (4).
Let $C \subset \mathbf{P}(V_ N)$ be the scheme theoretic closure of $(r_{n + 1} \circ \ldots \circ r_ N)^{-1}(\pi (Z))$. Because $\pi $ is étale at the point $x$ of $Z$, we see that the closed subscheme $C \cap X$ contains $Z$ with multiplicity $1$ (local calculation omitted). Hence by Lemma 43.17.2 we conclude that
\[ C \cdot X = [Z] + \sum m_ j[Z_ j] \]
for some subvarieties $Z_ j \subset X$ of dimension $d$. Note that
\[ C \cap X = \pi ^{-1}(\pi (Z)) \]
set theoretically. Hence $T_ i \cap Z_ j \subset T_ i \cap \pi ^{-1}(\pi (Z)) \subset T_ i \cap Z \cup E_ i$. For any irreducible component of $T_ i \cap Z$ contained in $E_ i$ we have the desired dimension bound. Finally, let $V$ be an irreducible component of $T_ i \cap Z_ j$ which is contained in $T_ i \cap Z$. To finish the proof it suffices to show that $V$ does not contain any of the points $x_{it}$, because then $\dim (V) < \dim (Z \cap T_ i)$. To show this it suffices to show that $x_{it} \not\in Z_ j$ for all $i, t, j$.
Set $Z' = \pi (Z)$ and $Z'' = \pi ^{-1}(Z')$, scheme theoretically. By condition (3) we can find an open $U \subset \mathbf{P}(V_ n)$ containing $\pi (x_{it})$ such that $\pi ^{-1}(U) \cap Z \to U \cap Z'$ is an isomorphism. In particular, $Z \to Z'$ is a local isomorphism at $x_{it}$. On the other hand, $Z'' \to Z'$ is étale at $x_{it}$ by condition (2). Hence the closed immersion $Z \to Z''$ is étale at $x_{it}$ (Morphisms, Lemma 29.36.18). Thus $Z = Z''$ in a Zariski neighbourhood of $x_{it}$ which proves the assertion. $\square$
The actual moving is done using the following lemma.
Lemma 43.24.2. Let $C \subset \mathbf{P}^ N$ be a closed subvariety. Let $X \subset \mathbf{P}^ N$ be subvariety and let $T_ i \subset X$ be a finite collection of closed subvarieties. Assume that $C$ and $X$ intersect properly. Then there exists a closed subvariety $C' \subset \mathbf{P}^ N \times \mathbf{P}^1$ such that
$C' \to \mathbf{P}^1$ is dominant,
$C'_0 = C$ scheme theoretically,
$C'$ and $X \times \mathbf{P}^1$ intersect properly,
$C'_\infty $ properly intersects each of the given $T_ i$.
Proof. If $C \cap X = \emptyset $, then we take the constant family $C' = C \times \mathbf{P}^1$. Thus we may and do assume $C \cap X \not= \emptyset $.
Write $\mathbf{P}^ N = \mathbf{P}(V)$ so $\dim (V) = N + 1$. Let $E = \text{End}(V)$. Let $E^\vee = \mathop{\mathrm{Hom}}\nolimits (E, \mathbf{C})$. Set $\mathbf{P} = \mathbf{P}(E^\vee )$ as in Lemma 43.23.7. Choose a general line $\ell \subset \mathbf{P}$ passing through $\text{id}_ V$. Set $C' \subset \ell \times \mathbf{P}(V)$ equal to the closed subscheme having fibre $r_ g(C)$ over $[g] \in \ell $. More precisely, $C'$ is the image of
\[ \ell \times C \subset \mathbf{P} \times \mathbf{P}(V) \]
under the morphism (43.23.6.1). By Lemma 43.23.7 this makes sense, i.e., $\ell \times C \subset U(\psi )$. The morphism $\ell \times C \to C'$ is finite and $C'_{[g]} = r_ g(C)$ set theoretically for all $[g] \in \ell $. Parts (1) and (2) are clear with $0 = [\text{id}_ V] \in \ell $. Part (3) follows from the fact that $r_ g(C)$ and $X$ intersect properly for all $[g] \in \ell $. Part (4) follows from the fact that a general point $\infty = [g] \in \ell $ is a general point of $\mathbf{P}$ and for such as point $r_ g(C) \cap T$ is proper for any closed subvariety $T$ of $\mathbf{P}(V)$. Details omitted. $\square$
Lemma 43.24.3. Let $X$ be a nonsingular projective variety. Let $\alpha $ be an $r$-cycle and $\beta $ be an $s$-cycle on $X$. Then there exists an $r$-cycle $\alpha '$ such that $\alpha ' \sim _{rat} \alpha $ and such that $\alpha '$ and $\beta $ intersect properly.
Proof. Write $\beta = \sum n_ i[T_ i]$ for some subvarieties $T_ i \subset X$ of dimension $s$. By linearity we may assume that $\alpha = [Z]$ for some irreducible closed subvariety $Z \subset X$ of dimension $r$. We will prove the lemma by induction on the maximum $e$ of the integers
\[ \dim (Z \cap T_ i) \]
The base case is $e = r + s - \dim (X)$. In this case $Z$ intersects $\beta $ properly and the lemma is trivial.
Induction step. Assume that $e > r + s - \dim (X)$. Choose an embedding $X \subset \mathbf{P}^ N$ and apply Lemma 43.24.1 to find a closed subvariety $C \subset \mathbf{P}^ N$ such that $C \cdot X = [Z] + \sum m_ j[Z_ j]$ and such that the induction hypothesis applies to each $Z_ j$. Next, apply Lemma 43.24.2 to $C$, $X$, $T_ i$ to find $C' \subset \mathbf{P}^ N \times \mathbf{P}^1$. Let $\gamma = C' \cdot X \times \mathbf{P}^1$ viewed as a cycle on $X \times \mathbf{P}^1$. By Lemma 43.22.2 we have
\[ [Z] + \sum m_ j[Z_ j] = \text{pr}_{X, *}(\gamma \cdot X \times 0) \]
On the other hand the cycle $\gamma _\infty = \text{pr}_{X, *}(\gamma \cdot X \times \infty )$ is supported on $C'_\infty \cap X$ hence intersects $\beta $ transversally. Thus we see that $[Z] \sim _{rat} - \sum m_ j[Z_ j] + \gamma _\infty $ by Lemma 43.17.1. Since by induction each $[Z_ j]$ is rationally equivalent to a cycle which properly intersects $\beta $ this finishes the proof. $\square$
Comment #2758 by Ko Aoki on August 03, 2017 at 16:49
Typo in the first line: "a
s
\beta
" should be replaced by "an
s
\beta
Thanks. Fixed here
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140) Show a displacement - time graph of a wave, the amplitude and time period of a wave (141) - Physics - Propagation of Sound Waves - 11982051 | Meritnation.com
140) Show a displacement - time graph of a wave, the amplitude and time period of a wave.
(141) Draw a displacement time graph of a wave and mark on it the amplitude of wave by the letter 'a' and wavelength of the wave by the letter
\text{'}\lambda \text{'}
|
The protocol automatically determines the premium or policy fees by taking numerous aspects such as policy term/cover duration, adverse selection, sum insured, utilization ratio, total pool balance, reassurance pool, NPM provision pool, etc into account.
P_F
x : x < C \cap x > F \\ x = \frac{C_A+c}{t+(I_P*I_R)} \\ P_F = \frac{C_D}{100} + x
F
C
c
t
C_D
C_A
I_P
I_R
{P_F=(C_D/100) + (C_A+c)/(t+(I_P*I_R))}
Policy costs vary among pools and may be very expensive or cheap depending on available liquidity and demand. Higher policy costs entice more liquidity providers to take risks, which helps to reduce the policy charge.
Market vs Adverse Selection
When there is high liquidity available (supply) and relatively low proposers (demand), the policy fees are lower. Unless there is massive fear in the market, well-established projects, exchanges, and custody providers generally fall into this category. The assumption is--liquidity providers want to pool liquidity for high-quality and secure projects because there may be a relatively lesser likelihood of exploits. On the demand side, users may also feel less nervous about purchasing protection which equates to less demand.
The soft market is the seller's market and signifies strong brand trust and customer loyalty. Returns are less to the liquidity providers, but the risks are also lower.
In the hard market, the number of proposers would be high, but the cover pool would have relatively less liquidity available. Because of the high demand from proposers, the cover fees get higher. This attracts more liquidity providers to take more risks to get higher returns. The assumption is--the users may feel nervous about an unproven or relatively new project that there is a higher likelihood of attacks and exploits. They would therefore want to purchase a policy to protect against possible attacks. On the supply side, liquidity providers feel nervous about possible liquidations and therefore want higher returns.
The hard market is the buyer's market and a good opportunity for liquidity providers to earn a handsome return by taking risks. Reach out to the Neptune Mutual team to get assistance on increasing your brand awareness and attracting more liquidity providers.
Adverse selection refers to a situation when either the seller or buyer has more information than the other party. For example, a cover creator could have insider information on their project and use that information to attack or gain an unfair advantage against Neptune Mutual protocol, liquidity providers, or policyholders.
A malicious project can cleverly word their cover rules (or parameters) and purchase large protection. They can then attack their project to claim and receive a payout.
As a cover project, building the trust of the community and liquidity providers takes time and great effort. Offer reassurance support to the liquidity providers, do not create deceptive cover rules, use best security practices, and demonstrate to the community that you are trustworthy.
As a liquidity provider, carefully examine all cover parameters and reach out to the cover projects for clarification if you have doubts. Offer to pool the risk only if you can trust a project.
Definitions - Previous
Next - NPM Pools
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Finding the Area of Irregular Figures | Prealgebra | Course Hero
Combine area of regular shapes to find the area of irregular shapes.
The blue rectangle has a width of
12
4
. The red rectangle has a width of
2
, but its length is not labeled. The right side of the figure is the length of the red rectangle plus the length of the blue rectangle. Since the right side of the blue rectangle is
4
units long, the length of the red rectangle must be
6
60
The rectangle has a length of
8
units and a width of
4
Since both sides of the rectangle are
4
, the vertical side of the triangle is
3
7 - 4
8
, so the base of the triangle will be
3
8 - 4
36.5
[ohm_question]146949]
A high school track is shaped like a rectangle with a semi-circle (half a circle) on each end. The rectangle has length
105
meters and width
68
meters. Find the area enclosed by the track. Round your answer to the nearest hundredth.
105
m and a width of
68
m. The semi-circles have a diameter of
68
m, so each has a radius of
34
Prealgebra. Provided by: OpenStax. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected]
Integrated+II+Unit+IV+-+Areas+of+Irregular+Figures (1).ppt
FILM 101 • University of the Fraser Valley
Area_of_Irregular_Figures.ppt
MATHEMATICS 1008 • Rockport-Fulton High School
Activity 4 - MEASUREMENT OF AREA OF PLANE AND IRREGULAR FIGURES.docx
SEA 203 • Holy Angel University
FORMULA FOR FINDING THE AREA OF IRREGULAR SHAPES Report.docx
There are approximate methods for finding the area of a figure with an irregular boundary.docx
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5-4 Areas of Irregular figures classwork + homework (back)-day .pdf
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Lect 2 - Area of Irregular Plane Figures.docx
FINANCE 320 • Notre Dame College
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AREA OF IRREGULAR PLANE FIGURES.pdf
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Hudson_Pond_-_Area_of_irregular_figures.pdf
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MATH 101 • Roosevelt High School, Roosevelt
Finding the Area of Sector Using an Inscribed Kite in a Circle.pdf
GEO 111 000 • Sacred Heart Cathedral Preparato
Finding the Circumference and Area of Circles - Copy.pptx
MATH 101 MATH101 • Mindanao State University
5.1.9.2Finding the Area of a Sector of a Circle.pdf
MATH TRIGONOMET • American Military University
Nov 11_12_Law of Cosines & Finding the Area of a Triangle.pdf
Finding The Area by Double Integral.docx
CIVIL ENGI 360 • University of Baghdad
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LP for finding the area of parallelogram.docx
MARKEN_DILLARD_-_Finding_the_Area_of_a_Triangle.docx.pdf
LIT 211 • City Colleges of Chicago, Wilbur Wright College
Finding the Area of an Oblique Triangle Using the Sine Function.docx
MATH 315 • University of Ghana
formula for finding the area A of a circle in terms of the radius r of the circle.docx
Finding the Area of a Polygon on the Coordinate Plane.pdf
MATH 2 • Academy For Academic Excellence
Copy of Topic 9.8 - Finding the Area of a Polar Region Defined by One Curve.pdf
MATH 1120 • Auburn University
Finding the Area of Squares, Rectangles, Parallelograms and Triangles.pptx
MATHEMATIC 127 • Woodstock High School, Woodstock
I need help on finding the Circumference and Area of Circles. _ Wyzant Ask An Expert.pdf
ENGLISH 100 • Chaminade College School
Math5_Q4_Mod1_Finding_The_Area_of_A_Given_Circle_lesson12_v4.docx
MATHEMATIC 10 • Virgen Milagrosa University Foundation
Math5_Q4_Mod1_Finding_The_Area_of_A_Given_Circle_lesson12_v4.pdf
MOD 200 • Harvard University
Finding the Area of Polygons.doc
BUSINESS 101 B12 • Johnson County High School, Mountain City
|
Should I Pay off My Mortgage? | Eqtor
When you have spare money that you don't need for the remaining term on your mortgage, paying off the mortgage can seem like an attractive choice. Owning your home outright can feel very good and bring you the peace of mind that comes with being debt-free. However, the satisfaction of paying down debt has to be weighed against the drawbacks of parting with your money. With today's interest rates at a record low, you might be better off refinancing your mortgage and investing your cash instead.
How can you repay your mortgage early?
You can pay off your mortgage faster either by making a lump sum payoff, or by paying extra in your mortgage payments. Check with your bank on the conditions and restrictions for paying off the mortgage early and making extra payments.
One way to repay the mortgage early is by simply pay off the outstanding amount and any penalty imposed by the bank. This eliminates the debt amd saves interest cost, but leaves you with less cash that you could use for other opportunities. Another way is to pay an extra amount in addition to your normal monthly payment. This amount pays off a part of the debt. Over time, these extra mortgage payments can result in earlier payoff of the loan. This saves you interest cost, because for the remaining months you will be charged interest for a reduced outstanding balance.
How do you work out how much money you will save by paying off the mortgage?
You can use our mortgage savings calculator to find out the savings from paying off the loan or from paying extra on some months in the year.
How do extra mortgage payments reduce your interest cost?
When you make an extra payment, you avoid the interest cost of the extra payment amount for the remaining term of your mortgage. In addition, more of the monthly payment in subsequent months gets allocated to the repaying the loan, which can cause an earlier payoff.
Suppose you have a mortgage with 3% interest and 10 years left in the term, and your outstanding balance is $100,000. Let's compare paying off $1,000 now, with sticking to the normal payments plan. Our extra payments savings calculator, makes this comparison easy by showing you the differences in payments each month, along with the resulting savings and new payoff date.
$100,000 outstanding balance
Without extra payments:
Loan paid off after 120 months
$15,872.91 in interest payments
With $1,000 extra payment now:
Loan paid off on 119 months
Paying $1000 to save $345.05. Isn't this interesting?
How will you see this saving? The last payment you make, on month 119, would be $585.00 which pays the interest and the remaining balance. If you had not made the $1000 extra mortgage payment, you would instead pay $965.61 on month 119, and still owe $961.92 to be settled on month 120 along with $2.40 interest. Here are the differences:
Month 1 -$1000 $0
Month 119 -$586.17 -$965.61
Month 120 $0 -$965.61
When you add up the columns, you see the difference is $345.05, and $345.05 is 34.51% of your $1000 payment.
Your $1000 payment acts like like an investment that pays 34.51% in 10 years. It is the same as investing the $1000 and getting back $1345.05 in 10 years. After using this $1345.05 to cover the last two months, you are in the same state as if you had made the extra payment instead.
Are there other options where you are better off not making the extra payment to your mortgage?
What other things can you do with a $1000 instead of using it for an extra mortgage payment? You've guessed it! One option is to spend the $1000. You could also let the $1000 sit in your bank account for 10 years, but the interest rates on saving accounts are very low these days. Your $1000 would get eroded by inflation unless you earn sufficient interest.
Another option would be to invest the $1000. You will need to find an investment that has a 10-year expected return of better than 34.51%, after tax. Let's consider some investment options.
Paying off the mortgage vs. investing in Treasuries
U.S. Treasuries are backed by "the full faith and credit" of the US government, and are considered the safest investments available. We need to find out the needed annual yield for a 34.51% return in 10 years. That is easy to calculate:
\begin{alignedat}{3} (&1 + &\text{yield}&)^{10} \quad &=& \quad 1.3451 \\ &1 + &\text{yield}& \quad &=& \quad 1.3451^{1/10} \\ & &\text{yield}& &=& \quad 0.030091 \\ & && &=& \quad 3.0091\% \\ \end{alignedat}
As of writing, the 10 Year Treasury Rate is below 3.0091%. It has not been at that level since late 2018.
If the return from risk-free investments is not enough to match the savings from paying off the loan, you can also consider investments with a reasonable amount of risk, according to your personal risk tolerance level.
Paying off the mortgage vs. investing in high quality bonds
We can crank up the risk a notch and look into an investment in high quality corporate bonds. An easy way to invest in a diverse basket of bonds is via Exchange Traded Funds (ETFs). Lets consider the LQD Investment Grade Corporate Bond ETF. The investment return depends on a variety of factors such as interest rates and market demand, and there is no guarantee that you won't lose your money.
For the 10 year period until the end of 2020, an investment in this investment grade ETF would have returned roughly 78% before tax. Taxation can vary depending on your country and circumstances. Unless your tax rate is higher than 55%, you would net a higher return than with making the $1000 extra payment to your mortgage.
This investment involved some risk. For example, you would have experienced a dip of about 19% during the corona crises that took a month to recover from.
If your risk appetite allows, you could instead take on riskier investments that have a better yield. Let's compare with investing in the stock market.
Paying off the mortgage vs. investing in stocks
Like with investment grade bonds, you can invest in stocks via ETFs. This way, you buy into a diverse basket of different stocks which is less risky than investing in only one stock. Over the 10 year period until the end of 2020, the IVV S&P 500 ETF returned about 260% before tax. The worst drawdown was during the COVID-19 crises, where your investment would have lost 34% and taken around five months to recover.
Paying off the mortgage vs. the realm possibilities
There may be other, more rewording investments out there, available to you. For example, you could use your capital to buy another property, or payoff a higher-interest debt, or contribute to your pension, or maybe even start a business.
"Money in your pocket is better than money in someone else's (the bank) pocket"
Before deciding to pay off your mortgage, or make extra payments, it's important to consider whether this is the best choice given the other options available to you. You can find out your savings when paying off the mortgage, or making extra payments, and then check whether a better return is possible when investing according to your personal risk tolerance level. Consider also other options like using the money to buy another property or pursue a business opportunity.
You save in interest costs. There is no risk of loss like with investing.
You no longer have to pay the monthly payment for the mortgage, and this leaves you with a higher cash flow.
It becomes easier to sell your home when there is no mortgage on the property.
Cons of paying off debt
You lose some liquidity. If you don't have enough money left in you emergency fund, then you could run into financial difficulties in case of an unexpected event. It is usually recommended to keep an emergency reserve of at least three to six months of living costs.
You can't use the money for other, higher yield opportunities.
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Describe how entries of matrix variable X relate to decision variables - MATLAB decinfo - MathWorks United Kingdom
Describe how entries of matrix variable X relate to decision variables
decX = decinfo(lmisys,X)
decinfo(lmisys)
decinfo expresses the entries of a matrix variable X in terms of the decision variables x1, . . ., xN. Recall that the decision variables are the free scalar variables of the problem, or equivalently, the free entries of all matrix variables described in lmisys. Each entry of X is either a hard zero, some decision variable xn, or its opposite –xn.
If X is the identifier of X supplied by lmivar, the command decX = decinfo(lmisys,X) returns an integer matrix decX of the same dimensions as X whose (i, j) entry is
0 if X(i, j) is a hard zero
n if X(i, j) = xn (the n-th decision variable)
–n if X(i, j) = –xn
decX clarifies the structure of X as well as its entry-wise dependence on x1, . . ., xN. This is useful to specify matrix variables with atypical structures (see lmivar).
decinfo can also be used in interactive mode by invoking it with a single argument, as decinfo(lmisys). It then prompts the user for a matrix variable and displays in return the decision variable content of this variable.
Consider an LMI with two matrix variables X and Y with structure:
X = x I3 with x scalar
Y rectangular of size 2-by-1
If these variables are defined by
X = lmivar(1,[3 0])
Y = lmivar(2,[2 1])
the decision variables in X and Y are given by
dX = decinfo(lmis,X)
dY = decinfo(lmis,Y)
This indicates a total of three decision variables x1, x2, x3 that are related to the entries of X and Y by
X=\left(\begin{array}{ccc}{x}_{1}& 0& 0\\ 0& {x}_{1}& 0\\ 0& 0& {x}_{1}\end{array}\right),Y=\left(\begin{array}{c}{x}_{2}\\ x3\end{array}\right)
Note that the number of decision variables corresponds to the number of free entries in X and Y when taking structure into account.
Suppose that the matrix variable X is symmetric block diagonal with one 2-by-2 full block and one 2-by-2 scalar block, and is declared by
X = lmivar(1,[2 1;2 0])
The decision variable distribution in X can be visualized interactively as follows:
decinfo(lmis)
There are 4 decision variables labeled x1 to x4 in this problem.
Matrix variable Xk of interest (enter k between 1 and 1, or 0 to quit):
The decision variables involved in X1 are among {-x1,...,x4}.
Their entry-wise distribution in X1 is as follows
(0,j>0,-j<0 stand for 0,xj,-xj, respectively):
lmivar | mat2dec | dec2mat
|
Thiosemicarbazones Synthesized from Acetophenones: Tautomerism, Spectrometric Data, Reactivity and Theoretical Calculations
1Laboratory of Organic Compounds Studies (LADECOR), Department of Chemistry, Facultad de Ciencias Exactas, National University of La Plata, La Plata, Argentina
2Institute of Physics of Liquids and Biological Systems (IFLySIB, CONICET), National University of La Plata, La Plata, Argentina
3Laboratory of Research and Development of Analytical Methods (LIDMA), Department of Chemistry, Facultad de Ciencias Exactas, National University of La Plata, La Plata, Argentina
Tautomeric forms of Thiosemicarbazones have been investigated by spectrometric methods, their chemical reactivity and theoretical calculations of the relative tautomers stabilities. The mass spectral fragmentation of thiosemicarbazones synthesized from acetophenones has been studied by CG/MS. The analysis of the corresponding spectra shows not only the regular fragmentation mechanisms but homolytic ruptures from even-electron species. 1H NMR spectra exhibit signals for the most intense open thioketo tautomeric structure, although when using TFA a ring structure is observed in the corresponding tautomeric equilibrium. Density Functional Theory calculations (DFT) also provide evidence to support the experimental observations by GC-MS and 1H NMR. Methylation reactions give support to the occurrence of the thioenol tautomeric form which would be the second most abundant according to the Density Functional Theoretical calculations.
Thiosemicarbazones, Tautomerism, MS, NMR, Theoretical Calculations
{\text{CH}}_{2}^{+}
Gastaca, B., Sánchez, H.R., Menestrina, F., Caputo, M., de las Mercedes Schiavoni, M. and Furlong, J.J.P. (2019) Thiosemicarbazones Synthesized from Acetophenones: Tautomerism, Spectrometric Data, Reactivity and Theoretical Calculations. International Journal of Analytical Mass Spectrometry and Chromatography, 7, 19-34. https://doi.org/10.4236/ijamsc.2019.72003
1. Tenório, R.P., Góes A.J.S., de Lima, J.G., de Faria, A.R., Alves, A.J. and de Aquino, T.M. (2005) Tiosemicarbazonas: métodos de obtenção, aplicações sintéticas e importância biológica. Química Nova, 28, 1030-1037. https://doi.org/10.1590/S0100-40422005000600018
2. Du, X.-H., Guo, C., Hansell, E., Doyle, P.S., Caffrey, C.R., Holler, T.P., McKerrow, J.H. and Cohen, F.E. (2002) Synthesis and Structure-Activity Relationship Study of Potent Trypanocidal Thio Semicarbazone Inhibitors of the Trypanosomal Cysteine Protease Cruzain. Journal of Medicinal Chemistry, 45, 2695-2707. https://doi.org/10.1021/jm010459j
3. Yu. Ershov, A., Koshmina A.N.V., Mokeev, M.V. and Gribanov, A.V. (2003) The Isoxazolidine-1,2,4-Triazolidine-3-Thione Tautomeric System. Chemistry of Heterocyclic Compounds, 39, 1257-1258. https://doi.org/10.1023/B:COHC.0000008277.45834.c5
4. Zelenin, K.N., Kuznetsova, O.B., Alekseyev, V.V., Terentyev, P.B., Torocheshnikov, V.N. and Ovcharenko, V.V. (1993) Ring-Chain Tautomerism of N-Substituted Thiosemicarbazones. Tetrahedron, 49, 1257-1270. https://doi.org/10.1016/S0040-4020(01)85816-6
5. Singh, N.K., Singh, S.B., Shrivastav, A. and Singh, S.M. (2001) Spectral, Magnetic and Biological Studies of 1,4-Dibenzoyl-3-Thiosemicarbazide Complexes with Some First Row Transition Metal Ions. Journal of Chemical Sciences, 113, 257-273. https://doi.org/10.1007/BF02708645
6. Offiong, O.E. and Martelli, S. (1997) Stereochemistry and Antitumor Activity of Platinum Metal Complexes of 2-Acetylpyridine Thiosemicarbazones. Transition Metal Chemistry, 22, 263-269. https://doi.org/10.1023/A:1018416624951
7. Samanta, B., Chakraborty, J., Shit,S., Batten, S.R., Jensen, P., Masuda, J.D. and Mitra, S. (2007) Synthesis, Characterization and Crystal Structures of a Few Coordination Complexes of Nickel(II), Cobalt(III) and Zinc(II) with N'-[(2-pyridyl)methylene]Salicyloylhydrazone Schiff Base. Inorganica Chimica Acta, 360, 2471-2484. https://doi.org/10.1016/j.ica.2006.12.019
8. Arab, S.T. and Emran, K.M. (2008) Structure Effect of Some Thiosemicarbazone Derivatives on the Corrosion Inhibition of Fe78B13Si9 Glassy Alloy in Na2SO4 Solution. Materials Letters, 62, 1022-1032. https://doi.org/10.1016/j.matlet.2007.05.088
9. Gastaca, B., Galletti, G., Sánchez, H.R., Pis Diez, R., de las Mercedes Schiavoni, M. and Furlong, J.J.P. (2015) GC/MS Analyses of Thiosemicarbazones Synthesized from Acetophenones: Thermal Decay and Mass Spectra Features. International Journal of Analytical Mass Spectrometry and Chromatography, 3, 1-13. https://doi.org/10.4236/ijamsc.2015.31001
10. Hohenberg, P. and Kohn, W. (1964) Inhomogeneous Electron Gas. Physical Review, 136, B864-B871. https://doi.org/10.1103/PhysRev.136.B864
11. Levy, M. (1979) Universal Variational Functionals of Electron Densities, First-Order Density Matrices, and Natural Spin-Orbitals and Solution of the V-Representability Problem. Proceedings of the National Academy of Sciences of the United States of America, 76, 6062-6065. https://doi.org/10.1073/pnas.76.12.6062
12. Lieb, E.H. (1983) Density Functionals for Coulomb Systems. International journal of quantum chemistry, 24, 243-277. https://doi.org/10.1002/qua.560240302
13. Kohn,W. and Sham, L.J. (1965) Self-Consistent Equations Including Exchange and Correlation Effects. Physical Review, 140, A1133-A1138. https://doi.org/10.1103/PhysRev.140.A1133
14. Wheeler, A.S. and Bost, R.W. (1924) 4-Para-Tolylsemicarbazide and Certain Derivatives. Journal of the American Chemical Society, 46, 2813-2816. https://doi.org/10.1021/ja01677a032
15. Cheronis, N.D.I., Hodnett, E.M.A. and Entrikin, J.B.E. (1968) Semimicro Qualitative Organic Analysis: The Systematic Identification of Organic Compounds. Interscience, Olney, UK.
17. Lee, C., Yang, W.-Y. and Parr, R.G. (1988) Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Physical Review B, 37, 785-789. https://doi.org/10.1103/PhysRevB.37.785
18. Stephens, P.J., Devlin, F.J., Chabalowski, C.F. and Frisch, M. J. (1994) Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields. The Journal of Physical Chemistry, 98, 11623-11627. https://doi.org/10.1021/j100096a001
19. Goerigk, L. and Grimme, S. (2011) Efficient and Accurate Double-Hybrid-Meta-GGA Density Functionals—Evaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions. Journal of Chemical Theory and Computation, 7, 291-309. https://doi.org/10.1021/ct100466k
20. Grimme, S., Antony, J., Ehrlich, S. and Krieg, H. (2010) A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. The Journal of Chemical Physics, 132, Article ID: 154104. https://doi.org/10.1063/1.3382344
21. Grimme, S., Ehrlich, S. and Goerigk, L. (2011) Effect of the Damping Function in Dispersion Corrected Density Functional Theory. Journal of Computational Chemistry, 32, 1456-1465. https://doi.org/10.1002/jcc.21759
23. Neese, F. (2012) The ORCA Program System. Wiley Interdisciplinary Reviews: Computational Molecular Science, 2, 73-78. https://doi.org/10.1002/wcms.81
24. Furlong, J.J.P., Schiavoni, M.M., Castro, E.A. and Allegretti, P.E. (2008) Mass Spectrometry as a Tool for Studying Tautomerism. Russian Journal of Organic Chemistry, 44, 1725-1736. https://doi.org/10.1134/S1070428008120014
25. Ghogomu J.N. and Nkungli, N.K. (2016) A DFT Study of Some Structural and Spectral Properties of 4-Methoxyacetophenone Thiosemicarbazone and Its Complexes with Some Transition Metal Chlorides: Potent Antimicrobial Agents. Advances in Chemistry, 2016, Article ID: 9683630. https://doi.org/10.1155/2016/9683630
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Is Loop Quantum Gravity a Physically Correct Quantization?
Abstract: Dirac’s rule in which only special phase space variables should be promoted to operators in canonical quantization is applied to loop quantum gravity. For this theory, Dirac’s rule is violated, and as a result loop quantum gravity fails the test to be a valid quantization. Indications are included on how to create and deal with valid versions of quantum gravity.
Keywords: Quantum, Gravity, Affine Quantization, Loop Quantum Gravity
1. Dirac’s Rule for Canonical Quantization
For a single degree of freedom, a momentum p and a position q, where
-\infty <p,q<\infty
, the Poisson bracket is
\left\{q,p\right\}=1
, and the Hamiltonian function is given by
H\left(p,q\right)
. In addition, new variables1 may also be used, say,
\stackrel{¯}{p}
\stackrel{¯}{q}
\left\{\stackrel{¯}{q},\stackrel{¯}{p}\right\}=1
-\infty <\stackrel{¯}{p},\stackrel{¯}{q}<\infty
\stackrel{¯}{H}\left(\stackrel{¯}{p}\mathrm{,}\stackrel{¯}{q}\right)=H\left(p\mathrm{,}q\right)
For canonical quantization, we promote
p\to P
q\to Q
\left[Q,P\right]=i\hslash
H\left(p,q\right)\to H\left(P,Q\right)
\stackrel{¯}{p}\to \stackrel{¯}{P}
\stackrel{¯}{q}\to \stackrel{¯}{Q}
\left[\stackrel{¯}{Q},\stackrel{¯}{P}\right]=i\hslash
\stackrel{¯}{H}\left(\stackrel{¯}{p}\mathrm{,}\stackrel{¯}{q}\right)\to \stackrel{¯}{H}\left(\stackrel{¯}{P}\mathrm{,}\stackrel{¯}{Q}\right)
\stackrel{¯}{H}\left(\stackrel{¯}{P}\mathrm{,}\stackrel{¯}{Q}\right)\ne H\left(P\mathrm{,}Q\right)
. At most, only one such quan- tization can be valid while all others lead to false quantizations.
Although the classical Hamiltonians can be equal the quantum Hamiltonians are different, and the question arises which is the physically correct Hamiltonian operator. Dirac [1] asserts that the proper choice of the quantum Hamiltonian is the one which has been promoted from Cartesian coordinates as classical variables. Dirac does not prove his rule, but Dirac’s rule has recently been established [2] leading to a flat space (Fubini-Study) metric given by
\text{d}\sigma {\left(p,q\right)}^{2}=A\text{d}{p}^{2}+{A}^{-1}\text{d}{q}^{2}
A>0
is a constant. Although we have focussed on a single degree of freedom, the case of scalar fields, for example, relies on a set of degrees of freedom so that
\text{d}\sigma {\left(\pi ,\varphi \right)}^{2}=\int \left[B\left(x\right)\text{d}\pi {\left(x\right)}^{2}+B{\left(x\right)}^{-1}\text{d}\varphi {\left(x\right)}^{2}\right]\text{d}x
B\left(x\right)>0
is a fixed positive field2.
These variables enjoy
\text{d}p\wedge \text{d}q
as measures of the appropriate phase space. The same can be said about
\int \left\{\text{d}\pi \left(x\right)\wedge \text{d}\varphi \left(x\right)\right\}\text{d}x
Using canonical quantization, the case of loop quantum gravity involves two sets of fields classically denoted by
{E}_{i}^{a}\left(x\right)
{A}_{a}^{i}\left(x\right)
a,i=1,2,3
, and x denotes a 3-dimensional spatial point in space. These variables admit the phase- space measure
\int \left\{\text{d}{A}_{a}^{i}\left(x\right)\wedge \text{d}{E}_{i}^{a}\left(x\right)\right\}\text{d}x
. However, their natural metric expressions, such as
\text{d}\sigma {\left(A,E\right)}^{2}=\int \left[C\left(x\right){\left({E}_{i}^{a}\left(x\right)\text{d}{A}_{a}^{i}\left(x\right)\right)}^{2}+C{\left(x\right)}^{-1}{\left({A}_{a}^{i}\left(x\right)\text{d}{E}_{i}^{a}\left(x\right)\right)}^{2}\right]\text{d}x
0<C\left(x\right)<\infty
, fail to exhibit suitable Cartesian coordinates, and thus signal a false quantization because it does not follow Dirac’s rule.
What is affine quantization? While canonical quantization employs Q and P, with
\left[Q,P\right]=i\hslash
, as basic operators, affine quantization employs Q and
D\equiv \frac{1}{2}\left(PQ+QP\right)
, the dilation operator, with
\left[Q,D\right]=i\hslash Q
; note: the operator D can be self-adjoint even when
Q>0
is self-adjoint, but then P can not be self-adjoint.
There are some systems that canonical quantization can solve, and there are some systems that affine quantization can solve. If they solve using one system they typically fail to solve using the other system. For example, canonical quantization
can solve the Hamiltonian
H=\frac{1}{2}\left({p}^{2}+{q}^{2}\right)
-\infty <p,q<\infty
, while affine can not solve it. On the other hand, the same Hamiltonian,
H=\frac{1}{2}\left({p}^{2}+{q}^{2}\right)
-\infty <p<\infty
0<q<\infty
, can be solved with affine quantization but not with canonical quantization. This example is used to illustrate the power of affine quantization in [2], and it points the way to affine quantum gravity.
Articles [2] [3] offer an approach to resolve quantum gravity by affine quan- tization, and they lead to positive results. Although paper [3] is older, the author recommends that [2] is read first. This recommendation is because [2] employs a familiar Schrödinger representation, while [3] normally employs a less familiar current commutation representation.
The representations of the analysis in these two papers may be different, but the physics is the same: specifically, for example, the quantum gravitational metrics are not discrete, but continuous.
1e.g.,
\stackrel{¯}{p}=p/{q}^{2}
\stackrel{¯}{q}={q}^{3}/3
2In particular, in the mid-page of 114 Dirac wrote “However, if the system does have a classical analogue, its connexion with classical mechanics is specially close and one can usually assume that the Hamiltonian is the same function of the canonical coordinates and momenta in the quantum theory as in the classical theory. †” Footnote †: “This assumption is found in practice to be successful only when applied with the dynamical coordinates and momenta referring to a Cartesian system of axes and not to more general curvilinear coordinates.”
Cite this paper: Klauder, J. (2020) Is Loop Quantum Gravity a Physically Correct Quantization?. Journal of High Energy Physics, Gravitation and Cosmology, 6, 49-51. doi: 10.4236/jhepgc.2020.61006.
[1] Dirac, P.A.M. (1958) The Principles of Quantum Mechanics. Claredon Press, Oxford, 114p.
[2] Klauder, J.R. (2019) Quantum Gravity Made Easy. arXiv: 1903.11211.
[3] Klauder, J.R. (2019) Building a Genuine Quantum Gravity. arXiv:1811. 09582.
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(Redirected from Hectogon)
{\displaystyle \left(1-{\tfrac {2}{n}}\right)\pi }
{\displaystyle 180-{\tfrac {360}{n}}}
{\displaystyle {\tfrac {p}{q}}}
{\displaystyle {\tfrac {\pi (p-2q)}{p}}}
{\displaystyle {\tfrac {180(p-2q)}{p}}}
{\displaystyle (x_{0},y_{0}),(x_{1},y_{1}),\ldots ,(x_{n-1},y_{n-1})}
{\displaystyle A={\frac {1}{2}}\sum _{i=0}^{n-1}(x_{i}y_{i+1}-x_{i+1}y_{i})\quad {\text{where }}x_{n}=x_{0}{\text{ and }}y_{n}=y_{0},}
{\displaystyle 16A^{2}=\sum _{i=0}^{n-1}\sum _{j=0}^{n-1}{\begin{vmatrix}Q_{i,j}&Q_{i,j+1}\\Q_{i+1,j}&Q_{i+1,j+1}\end{vmatrix}},}
{\displaystyle Q_{i,j}}
{\displaystyle (x_{i},y_{i})}
{\displaystyle (x_{j},y_{j}).}
{\displaystyle {\begin{aligned}A={\frac {1}{2}}(a_{1}[a_{2}\sin(\theta _{1})+a_{3}\sin(\theta _{1}+\theta _{2})+\cdots +a_{n-1}\sin(\theta _{1}+\theta _{2}+\cdots +\theta _{n-2})]\\{}+a_{2}[a_{3}\sin(\theta _{2})+a_{4}\sin(\theta _{2}+\theta _{3})+\cdots +a_{n-1}\sin(\theta _{2}+\cdots +\theta _{n-2})]\\{}+\cdots +a_{n-2}[a_{n-1}\sin(\theta _{n-2})]).\end{aligned}}}
{\displaystyle p^{2}>4\pi A}
{\displaystyle A={\tfrac {1}{2}}\cdot p\cdot r.}
{\displaystyle A=R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=R^{2}\cdot n\cdot \sin {\frac {\pi }{n}}\cdot \cos {\frac {\pi }{n}}}
{\displaystyle \alpha ,}
{\displaystyle A={\frac {ns^{2}}{4}}\cot {\frac {\pi }{n}}={\frac {ns^{2}}{4}}\cot {\frac {\alpha }{n-2}}=n\cdot \sin {\frac {\alpha }{n-2}}\cdot \cos {\frac {\alpha }{n-2}}.}
{\displaystyle C_{x}={\frac {1}{6A}}\sum _{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}),}
{\displaystyle C_{y}={\frac {1}{6A}}\sum _{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}y_{i+1}-x_{i+1}y_{i}).}
{\displaystyle A}
{\displaystyle c_{x}={\frac {1}{n}}\sum _{i=0}^{n-1}x_{i},}
{\displaystyle c_{y}={\frac {1}{n}}\sum _{i=0}^{n-1}y_{i}.}
{\displaystyle P=(x_{0},y_{0})}
{\displaystyle \lim _{n\to +\infty }R^{2}\cdot {\frac {n}{2}}\cdot \sin {\frac {2\pi }{n}}=\pi \cdot R^{2}}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Polygon&oldid=1085462006#Naming"
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3 Ionic conductivity of YSZ and its degradation
See also: Zirconium dioxide § Engineering properties
{\displaystyle \leftrightarrow }
{\displaystyle \leftrightarrow }
{\displaystyle \leftrightarrow }
Thermal expansion coefficient[edit]
Ionic conductivity of YSZ and its degradation[edit]
{\displaystyle {\text{Y}}_{2}{\text{O}}_{3}\rightarrow 2{\text{Y}}_{\text{Zr}}^{'}+3{\text{O}}_{\text{O}}^{\text{x}}+{\text{V}}_{\text{O}}^{\bullet \bullet }{\text{ with }}[{\text{V}}_{\text{O}}^{\bullet \bullet }]={\frac {1}{2}}[{\text{Y}}_{\text{Zr}}^{'}],}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Yttria-stabilized_zirconia&oldid=1067083292"
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Acan u give formula for each Fast 0 6ft-cro à X File"anager GEMS portal -find current - Physics - Propagation of Sound Waves - 13484805 | Meritnation.com
Acan u give formula for each? Fast??
Acan u give formula for each? Fast?? 0.6ft-cro ö X File"anager GEMS portal -find current w if—X Ohm's L aw O 2. 3. 4. 6. x 1 of 1 REVISION + Automatic Zoom I. A body of mass 20 kg moves initially with a velocity of 50 km/hr and then after 2 minute, it changes its velocity to 60 m/s. Find the force required to change its velocity. Classify the following into scalars and vectors. Velocity, distance, pressure, buoyancy, electricity, momentum, frequency, amplitude, resistance, A longitudinal wave travels with a frequency of 500 Hz. Find its time and velocity with which it travels. A sound wave has 40 troughs and 40 crests passing a point in 4 seconds. Find its wavelength. 5. A longitudinal wave produced by a flute has a frequency of 200 Hz and has 2 cm wavelength. After some time the frequency changes producing sound of wavelength 4cm. Find the value of second frequency. If a charge of 40 C flows through a conductor in 2.5 S, find the number of charges passing through the wire.
Please post different queries in different thread.
q=ne\phantom{\rule{0ex}{0ex}}40=n×1.6×{10}^{-19}\phantom{\rule{0ex}{0ex}}n=\frac{40}{1.6×{10}^{-19}}=2.5×{10}^{20}
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How Do Loans Work? | Eqtor
Interest is a compensation you pay to the lender for the privilege of borrowing money.
An interest rate is the amount of interest due per period as a percentage (or fraction) of the borrowed amount. The period is typically one year. For example, an interest rate of 1% on $100 would amount to $1 per year.
Compounding occurs if unpaid interest for a period is added to the borrowed amount,and in the next period interest is charged on the increased amount. For a savings account, compounding occurs if you don't withdraw the interest you earn. For a loan,compounding occurs in case of negative amortization, where the payment does not fully cover the interest due for the period. If the payment covers the interest for the period, then the loan balance does not increase and no compounding occurs.
With simple interest, the interest is fully paid at each period and does not compound. On the other hand, with compound interest, interest is not fully repaid at each period and compounds.
The purchasing power of an amount of money typically decreases over time. When you borrow money, the bank considers this effect in the interest they charge. On the other hand, when you deposit money in a bank account, the value of your deposit decreases with time, unless you earn enough interest to cover inflation.
Depending on the context, can mean either: The interest rate without adjustment for compounding. This is usually what is meant in the context of loans. The interest rate without adjustment for inflation, in contrast to the "real" interest rate that accounts for inflation. Basically, the "nominal" part of the term indicates that the interest is not adjusted, either for compounding or for inflation, depending on the context.
Can you compare loans using the nominal interest rate?
You can directly compare nominal interest rate only if the loans have the same compounding period. If the loans have different compounding periods, then you need to instead compare their effective interest rates.
The effective interest rate is the annual interest rate taking into account compounding and fees.
But, does an effective interest rate mean that a loan has compound interest?
The interest on a loan or mortgage does not compound except in the case of negative amortization. The effective interest rate helps compare loans with different payment periods by converting to a common base. For an amortized loan, a lower effective interest rate would mean that you will have less total interest cost over the lifetime of the loan. The effective interest rate decreases with lower fees or more frequent payments, in case of amortization.
Amortization is when the the debt decreases with time, reducing successive interest payments. Negative amortization, on the other hand, is when the debt increases over time due to compounding.
What is a fully amortizing mortgage?
A fully amortizing mortgage is repaid with each payment such that the debt is fully paid off by the end of the term. Payments include not only interest but also a repayment portion.
How can I calculate the monthly payment for fixed-rate mortgage?
For a fixed-rate mortgage (FRM), the interest rate and the monthly payment are fixed through out the term. The required payment amount to have the debt paid off at the end of the term can be calculated using the formula:
\text{monthly payment} = P \dfrac{i \, (1 + i)^n}{(1 + i)^n - 1}
P
is the loan amount,
is the interest rate per period as a fraction, and
n
is the number of payments. For example, a 30-year mortgage of $200,000 with 3% interest and monthly payments would have
n
= 30 x 12 and
i
= 3 /(100 x 12) which gives a monthly payment of $843.21. You can also use our mortgage calculator to calculate the monthly payment and the amortization schedule for your loan.
The amortization schedule (also called amortization table, or repayment plan) shows the interest payments and the
The term "principle" refers to the outstanding debt amount. Principle payments repay a part of the debt and reduce the outstanding amount.
over the term. When payments are enough to cover the interest due for the period, the remainder pays off part of the outstanding debt, and the interest payment for the next period decreases because of the reduced outstanding debt. If, on the other hand, the payment is not sufficient to cover the interest, then negative amortization occurs, where the outstanding debt increases. You can use our mortgage calculator to calculate the amortization schedule for your loan.
Why is it useful to look at the amortization schedule?
The amortization table is useful if you want to find out the outstanding debt an any point in time. This information is particularly relevant when planing to refinance or take out a new loan.
Refinancing is when the loan agreement is modified or replaced. Refinancing can allow the borrower to reduce monthly payments, or take advantage of a new lower interest rate. However, there are typically costs associated with refinancing which need to be weighed against any benefits of the new loan conditions.
Does making extra mortgage payments reduce interest cost?
Yes. Consider the simple example where $100 are borrowed for one year at 50% interest. If you repay the $100 only at the end of the year, then you pay $50 in interest. On the other hand, if you repay half the amount after 6 months, you pay $25 for the first 6 months, and $12.5 for the second 6 months, which add up to $37.5. The $12.5 were saved because you didn't need to pay interest for the full $100 in the second 6 months, but instead you had to pay interest on $50. Likewise, when you make an extra payment to a loan, you save the cost of interest for this payment amount over the remaining lifetime of the loan.
Is it a good idea to make extra mortgage payments or repay early?
It depends. Although an earlier repayment helps save on interest cost, you have to assess whether the savings are interesting in your situation, or your money could have a better use. You can use our extra payments savings calculator to find out the savings you would get from extra payments. See our post "Should You Repay Your Mortgage Early?," for a detailed discussion of the important aspects to consider before you make extra payments.
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A Note on the Rate of Strong Convergence for Weighted Sums of Arrays of Rowwise Negatively Orthant Dependent Random Variables
Qingxia Zhang, Dingcheng Wang, "A Note on the Rate of Strong Convergence for Weighted Sums of Arrays of Rowwise Negatively Orthant Dependent Random Variables", Discrete Dynamics in Nature and Society, vol. 2014, Article ID 368702, 7 pages, 2014. https://doi.org/10.1155/2014/368702
Qingxia Zhang1 and Dingcheng Wang1
Let be an array of rowwise negatively orthant dependent (NOD) random variables. The authors discuss the rate of strong convergence for weighted sums of arrays of rowwise NOD random variables and solve an open problem posed by Huang and Wang (2012).
Firstly, let us recall the definitions of negatively associated (NA) random variables and NOD random variables as follows.
Definition 1. A finite collection of random variables is said to be NA if for every pair of disjoint subsets and of , whenever and are nondecreasing functions such that the covariance exists. An infinite collection of random variables is NA if every finite subcollection is NA.
An array of random variables is called rowwise NA random variables if for every is a sequence of NA random variables.
Definition 2. A finite collection of random variables is said to be NOD if for all . An infinite collection of random variables is said to be NOD if every finite subcollection is NOD.
An array of random variables is called rowwise NOD random variables if for every is a sequence of NOD random variables.
The concepts of NA and NOD random variables were introduced by Joag-Dev and Proschan [1]. Obviously, independent random variables are NOD, and NA implies NOD from the definition of NA and NOD, but NOD does not imply NA. So, NOD is much weaker than NA. Because of the wide applications of NOD random variables, the notion of NOD random variables has been received more and more attention recently. Many applications have been found. We can refer to Volodin [2], Asadian et al. [3], Amini et al. [4, 5], Kuczmaszewska [6], Zarei and Jabbari [7], Wu and Zhu [8], Wu [9], Sung [10], Wang et al. [11], Huang and Wang [12], and so forth. Hence, it is very significant to study limit properties of this wider NOD random variables in probability theory and practical applications.
Let be a sequence of independent and identically distributed (i.i.d.) random variables and let be an array of real constants. As Bai and Cheng [13] remarked, many useful linear statistics, for example, least-squares estimators, nonparametric regression function estimators, and jackknife estimates, are based on weighted sums of i.i.d. random variables. In this respect, the strong convergence for weighted sums has been studied by many authors (see, e.g., Bai and Cheng [13]; Cuzick [14]; Sung [15]; Tang [16]; etc.).
Cai [17] proved the following complete convergence result for weighted sums of NA random variables.
Theorem A. Let be a sequence of identically distributed NA random variables, and let be an array of real constants satisfying for some . Suppose that when . If then, for ,
Wang et al. [11] extended the above result of Cai [17] to arrays of rowwise NOD random variables as follows.
Theorem B. Let be an array of rowwise NOD random variables which is stochastically dominated by a random variable and let be an array of real constants. Assume that there exist some with and some with such that and assume further that if . If for some and such that (4), then where and .
Recently, Huang and Wang [12] partially extended the corresponding theorems of Cai [17] and Wang et al. [11] to NOD random variables under a mild moment condition.
Theorem C. Let be a sequence of NOD random variables which is stochastically dominated by a random variable and let be a triangular array of real constants such that for . Let where for some , , and . Assume that for and . Then, where .
As Huang and Wang [12] pointed out, Theorem C partially extends only the case of of Theorems A and B. They left an open problem whether the case of of Theorem C holds for NOD random variables.
The main purpose of this paper is to further study strong convergence for weighted sums of NOD random variables and to obtain the rate of strong convergence for weighted sums of arrays of rowwise NOD random variables under a suitable moment condition. We solve the above problem posed by Huang and Wang [12].
We will use the following concept in this paper.
Definition 3. An array of random variables is said to be stochastically dominated by a random variable if there exists a positive constant such that for all , and .
Now, we will present the main results of this paper; the detailed proofs will be given in the next section.
Theorem 4. Let be an array of rowwise NOD random variables which is stochastically dominated by a random variable and let be an array of real constants satisfying for some . Assume further that for and . Then, where .
Similar to the proof of Theorem 4, we can obtain the following result for NOD random variable sequences.
Corollary 5. Let be a sequence of NOD random variables which is stochastically dominated by a random variable and let be an array of real constants satisfying for some . Assume further that for and . Then, where .
Remark 6. In Theorem 4 and Corollary 5, we consider the case of for and obtain some strong convergence results for arrays of rowwise NOD random variables and NOD random variable sequences without assumption of identical distribution. The main result settles the open problem posed by Huang and Wang [12]. In addition, it is still an open problem whether holds true under the same moment condition of Theorem 4.
In order to prove our main results, the following lemmas are needed.
Lemma 7 (see Bozorgnia et al. [18]). Let be a sequence of NOD random variables, and let be a sequence of Borel functions all of which are monotone nondecreasing (or all are monotone nonincreasing). Then, is a sequence of NOD random variables.
Lemma 8 (see Asadian et al. [3]). Let and let be a sequence of NOD random variables with and for all . Then, there exists a positive constant depending only on such that, for all ,
Lemma 9. Let be a sequence of random variables which is stochastically dominated by a random variable . For any and , the following two statements hold: where and are positive constants.
Lemma 10 (see Sung [15]). Let be a random variable and let be an array of real constants satisfying for some . Let for some . Then,
Lemma 11 (see Sung [19]). Let be a random variable and let be an array of real constants satisfying or and for some . Let . If , then
Throughout this paper, let be the indicator function of the set . denotes a positive constant, which may be different in various places and stands for .
Proof of Theorem 4. Without loss of generality, suppose that and , for all . For fixed , define Denote It is easily seen that, for all , which implies that First, we will prove that Actually, for , by (14) of Lemma 9, Markov inequality, and , we have that Next, for , by , (15) of Lemmas 9 and 10, Markov inequality, and , we also have that From the above statements, we can get (22) immediately. Hence, for large enough, To prove (10), it is sufficient to show that It follows from Lemma 10 and that For fixed , it is easily seen that is still a sequence of NOD random variables with mean zero by Lemma 7. Hence, it follows from (14) of Lemmas 9 and 8 and Markov inequality (for ) that
It follows from Lemma 10, (14) of Lemma 9, and Markov inequality that From Lemma 10 and , we can obtain that
For fixed , we divide into three subsets , , and , where . Then, By Lemma 11 and again, it follows that Noting that , for and fixed , we have that Noting that and , for , we have that Finally, we will prove that Hence, by inequality, Markov inequality, Lemmas 9–11, and , we have that Therefore, the desired result (10) follows from the above statements. This completes the proof of Theorem 4.
The authors are most grateful to the referees and to the editor Professor Baodong Zheng for their valuable suggestions and some helpful comments which greatly improved the clarity and readability of this paper. This paper is partially supported by the National Nature Science Foundation of China (71271042), the Fundamental Research Funds for the Central Universities of China (ZYGX2012J119), the Nature Science Foundation of Guangxi Province (2014GXNSFBA118006, 2013GXNSFDA019001), and the Guangxi Provincial Scientific Research Projects (201204LX157, 2013YB104).
K. Joag-Dev and F. Proschan, “Negative association of random variables, with applications,” The Annals of Statistics, vol. 11, no. 1, pp. 286–295, 1983. View at: Publisher Site | Google Scholar | MathSciNet
A. Volodin, “On the Kolmogorov exponential inequality for negatively dependent random variables,” Pakistan Journal of Statistics, vol. 18, no. 2, pp. 249–254, 2002. View at: Google Scholar | MathSciNet
N. Asadian, V. Fakoor, and A. Bozorgnia, “Rosental's type inequalities for negatively orthant dependent random variables,” Journal of Iranian Statistical Society, vol. 5, no. 1-2, pp. 66–75, 2006. View at: Google Scholar
M. Amini and A. Bozorgnia, “Complete convergence for negatively dependent random variables,” Journal of Applied Mathematics and Stochastic Analysis, vol. 16, no. 2, pp. 121–126, 2003. View at: Publisher Site | Google Scholar | MathSciNet
M. Amini, H. Zarei, and A. Bozorgnia, “Some strong limit theorems of weighted sums for negatively dependent generalized Gaussian random variables,” Statistics & Probability Letters, vol. 77, no. 11, pp. 1106–1110, 2007. View at: Publisher Site | Google Scholar | MathSciNet
A. Kuczmaszewska, “On some conditions for complete convergence for arrays of rowwise negatively dependent random variables,” Stochastic Analysis and Applications, vol. 24, no. 6, pp. 1083–1095, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Zarei and H. Jabbari, “Complete convergence of weighted sums under negative dependence,” Statistical Papers, vol. 52, no. 2, pp. 413–418, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Wu and D. Zhu, “Convergence properties of partial sums for arrays of rowwise negatively orthant dependent random variables,” Journal of the Korean Statistical Society, vol. 39, no. 2, pp. 189–197, 2010. View at: Publisher Site | Google Scholar | MathSciNet
Q. Wu, “Complete convergence for negatively dependent sequences of random variables,” Journal of Inequalities and Applications, vol. 2010, Article ID 507293, 10 pages, 2010. View at: Publisher Site | Google Scholar | MathSciNet
S. H. Sung, “A note on the complete convergence for arrays of dependent random variables,” Journal of Inequalities and Applications, vol. 2011, article 76, 2011. View at: Publisher Site | Google Scholar | MathSciNet
X. Wang, S. Hu, and W. Yang, “Complete convergence for arrays of rowwise negatively orthant dependent random variables,” RACSAM, vol. 106, no. 2, pp. 235–245, 2012. View at: Publisher Site | Google Scholar | MathSciNet
H. W. Huang and D. C. Wang, “A note on the strong limit theorem for weighted sums of sequences of negatively dependent random variables,” Journal of Inequalities and Applications, vol. 2012, article 233, 2012. View at: Publisher Site | Google Scholar | MathSciNet
Z. D. Bai and P. E. Cheng, “Marcinkiewicz strong laws for linear statistics,” Statistics & Probability Letters, vol. 46, no. 2, pp. 105–112, 2000. View at: Publisher Site | Google Scholar | MathSciNet
J. Cuzick, “A strong law for weighted sums of i.i.d.\ random variables,” Journal of Theoretical Probability, vol. 8, no. 3, pp. 625–641, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. H. Sung, “On the strong convergence for weighted sums of random variables,” Statistical Papers, vol. 52, no. 2, pp. 447–454, 2011. View at: Publisher Site | Google Scholar | MathSciNet
X. Tang, “Strong convergence results for arrays of rowwise pairwise NQD random variables,” Journal of Inequalities and Applications, vol. 2013, article 102, 2013. View at: Publisher Site | Google Scholar | MathSciNet
G. Cai, “Strong laws for weighted sums of NA random variables,” Metrika, vol. 68, no. 3, pp. 323–331, 2008. View at: Publisher Site | Google Scholar | MathSciNet
A. Bozorgnia, R. F. Patterson, and R. L. Taylor, “Limit theorems for dependent random variables,” in World Congress of Nonlinear Analysts '92, Vol. I-IV (Tampa, FL, 1992), pp. 1639–1650, de Gruyter, Berlin, Germany, 1996. View at: Google Scholar | MathSciNet
S. H. Sung, “On the strong convergence for weighted sums of
{\rho }^{*}
- mixing random variables,” Statistical Papers, vol. 54, pp. 773–781, 2013. View at: Google Scholar
Copyright © 2014 Qingxia Zhang and Dingcheng Wang. 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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Cauchy surface - Wikipedia
In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as defining an "instant of time"; in the mathematics of general relativity, Cauchy surfaces are important in the formulation of the Einstein equations as an evolutionary problem.
They are named for French mathematician Augustin Louis Cauchy (1789-1857) due to their relevance for the Cauchy problem of general relativity.
2 Mathematical definition and basic properties
3 Cauchy developments
5 Cauchy horizon
Although it is usually phrased in terms of general relativity, the formal notion of a Cauchy surface can be understood in familiar terms. Suppose that humans can travel at a maximum speed of 20 miles per hour. This places constraints, for any given person, upon where they can reach by a certain time. For instance, it is impossible for a person who is in Mexico at 3 o'clock to arrive in Libya by 4 o'clock; however it is possible for a person who is in Manhattan at 1 o'clock to reach Brooklyn by 2 o'clock, since the locations are ten miles apart. So as to speak semi-formally, ignore time zones and travel difficulties, and suppose that travelers are immortal beings who have lived forever.
The system of all possible ways to fill in the four blanks in
"A person in (location 1) at (time 1) can reach (location 2) by (time 2)"
defines the notion of a causal structure. A Cauchy surface for this causal structure is a collection of pairs of locations and times such that, for any hypothetical traveler whatsoever, there is exactly one location and time pair in the collection for which the traveler was at the indicated location at the indicated time.
There are a number of uninteresting Cauchy surfaces. For instance, one Cauchy surface for this causal structure is given by considering the pairing of every location with the time of 1 o'clock (on a certain specified day), since any hypothetical traveler must have been at one specific location at this time; furthermore, no traveler can be at multiple locations at this time. By contrast, there cannot be any Cauchy surface for this causal structure that contains both the pair (Manhattan, 1 o'clock) and (Brooklyn, 2 o'clock) since there are hypothetical travelers that could have been in Manhattan at 1 o'clock and Brooklyn at 2 o'clock.
There are, also, some more interesting Cauchy surfaces which are harder to describe verbally. One could define a function τ from the collection of all locations into the collection of all times, such that the gradient of τ is everywhere less than 1/20 hours per mile. Then another example of a Cauchy surface is given by the collection of pairs
{\displaystyle {\Big \{}{\big (}p,\tau (p){\big )}:p{\text{ a location on earth}}{\Big \}}.}
The point is that, for any hypothetical traveler, there must be some location p which the traveler was at, at time τ(p); this follows from the intermediate value theorem. Furthermore, it is impossible that there are two locations p and q and that there is some traveler who is at p at time τ(p) and at q at time τ(q), since by the mean value theorem they would at some point have had to travel at speed dist(p,q)/|τ(p) − τ(q)|, which must be larger than "20 miles per hour" due to the gradient condition on τ: a contradiction.
The physical theories of special relativity and general relativity define causal structures which are schematically of the above type ("a traveler either can or cannot reach a certain spacetime point from a certain other spacetime point"), with the exception that locations and times are not cleanly separable from one another. Hence one can speak of Cauchy surfaces for these causal structures as well.
Mathematical definition and basic properties[edit]
Let (M, g) be a Lorentzian manifold. One says that a map c : (a,b) → M is an inextendible differentiable timelike curve in (M, g) if:
it is differentiable
c′(t) is timelike for each t in the interval (a, b)
c(t) does not approach a limit as t increases to b or as t decreases to a.[1]
A subset S of M is called a Cauchy surface if every inextendible differentiable timelike curve in (M, g) has exactly one point of intersection with S; if there exists such a subset, then (M, g) is called globally hyperbolic.
The following is automatically true of a Cauchy surface S:
The subset S ⊂ M is topologically closed and is an embedded continuous (and even Lipschitz) submanifold of M. The flow of any continuous timelike vector field defines a homeomorphism S × ℝ → M. By considering the restriction of the inverse to another Cauchy surface, one sees that any two Cauchy surfaces are homeomorphic.
It is hard to say more about the nature of Cauchy surfaces in general. The example of
{\displaystyle {\Big \{}(t,x,y,z):t^{2}={\frac {x^{2}+y^{2}+z^{2}}{2}}{\Big \}}}
as a Cauchy surface for Minkowski space ℝ3,1 makes clear that, even for the "simplest" Lorentzian manifolds, Cauchy surfaces may fail to be differentiable everywhere (in this case, at the origin), and that the homeomorphism S × ℝ → M may fail to be even a C1-diffeomorphism. However, the same argument as for a general Cauchy surface shows that if a Cauchy surface S is a Ck-submanifold of M, then the flow of a smooth timelike vector field defines a Ck-diffeomorphism S × ℝ → M, and that any two Cauchy surfaces which are both Ck-submanifolds of M will be Ck-diffeomorphic.
Furthermore, at the cost of not being able to consider arbitrary Cauchy surface, it is always possible to find smooth Cauchy surfaces (Bernal & Sánchez 2003):
Given any smooth Lorentzian manifold (M, g) which has a Cauchy surface, there exists a Cauchy surface S which is an embedded and spacelike smooth submanifold of M and such that S × ℝ is smoothly diffeomorphic to M.
Cauchy developments[edit]
Let (M, g) be a time-oriented Lorentzian manifold. One says that a map c : (a,b) → M is a past-inextendible differentiable causal curve in (M, g) if:
c′(t) is either future-directed timelike or future-directed null for each t in the interval (a, b)
c(t) does not approach a limit as t decreases to a
One defines a future-inextendible differentiable causal curve by the same criteria, with the phrase "as t decreases to a" replaced by "as t increases to b". Given a subset S of M, the future Cauchy development D+(S) of S is defined to consist of all points p of M such that if c : (a,b) → M is any past-inextendible differentiable causal curve such that c(t) = p for some t in (a,b), then there exists some s in (a,b) with c(s) ∈ S. One defies the past Cauchy development D−(S) by the same criteria, replacing "past-inextendible" with "future-inextendible".
The future Cauchy development of S consists of all points p such that any observer arriving at p must have passed through S; the past Cauchy development of S consists of all points p such that any observer leaving from p will have to pass through S.
The Cauchy development D(S) is the union of the future Cauchy development and the past Cauchy development.
When there are no closed timelike curves,
{\displaystyle D^{+}}
{\displaystyle D^{-}}
are two different regions. When the time dimension closes up on itself everywhere so that it makes a circle, the future and the past of
{\displaystyle {\mathcal {S}}}
are the same and both include
{\displaystyle {\mathcal {S}}}
. The Cauchy surface is defined rigorously in terms of intersections with inextensible curves in order to deal with this case of circular time. An inextensible curve is a curve with no ends: either it goes on forever, remaining timelike or null, or it closes in on itself to make a circle, a closed non-spacelike curve.
When there are closed timelike curves, or even when there are closed non-spacelike curves, a Cauchy surface still determines the future, but the future includes the surface itself. This means that the initial conditions obey a constraint, and the Cauchy surface is not of the same character as when the future and the past are disjoint.
If there are no closed timelike curves, then given
{\displaystyle {\mathcal {S}}}
a partial Cauchy surface and if
{\displaystyle D^{+}({\mathcal {S}})\cup {\mathcal {S}}\cup D^{-}({\mathcal {S}})={\mathcal {M}}}
, the entire manifold, then
{\displaystyle {\mathcal {S}}}
is a Cauchy surface. Any surface of constan{\displaystyle t}
in Minkowski space-time is a Cauchy surface.
Cauchy horizon[edit]
{\displaystyle D^{+}({\mathcal {S}})\cup {\mathcal {S}}\cup D^{-}({\mathcal {S}})\not ={\mathcal {M}}}
then there exists a Cauchy horizon between
{\displaystyle D^{\pm }({\mathcal {S}})}
and regions of the manifold not completely determined by information on
{\displaystyle {\mathcal {S}}}
. A clear physical example of a Cauchy horizon is the second horizon inside a charged or rotating black hole. The outermost horizon is an event horizon, beyond which information cannot escape, but where the future is still determined from the conditions outside. Inside the inner horizon, the Cauchy horizon, the singularity is visible and to predict the future requires additional data about what comes out of the singularity.
Since a black hole Cauchy horizon only forms in a region where the geodesics are outgoing, in radial coordinates, in a region where the central singularity is repulsive, it is hard to imagine exactly how it forms. For this reason, Kerr and others suggest that a Cauchy horizon never forms, instead that the inner horizon is in fact a spacelike or timelike singularity. The inner horizon corresponds to the instability due to mass inflation.[2]
A homogeneous space-time with a Cauchy horizon is anti-de Sitter space.
^ One is requiring that for all points p in M, there exists an open neighborhood U of p and a sequence tk which increases to b and a sequence sk decreasing to a such that c(tk) and c(sk) are not contained in U for any k. This definition makes sense even if M only has the structure of a topological space.
^ Hamilton, Andrew J.S.; Avelino, Pedro P. (2010), "The physics of the relativistic counter-streaming instability that drives mass inflation inside black holes", Physics Reports, 495 (1): 1–32, arXiv:0811.1926, doi:10.1016/j.physrep.2010.06.002, ISSN 0370-1573
Choquet-Bruhat, Yvonne; Geroch, Robert. Global aspects of the Cauchy problem in general relativity. Comm. Math. Phys. 14 (1969), 329–335.
Geroch, Robert. Domain of dependence. J. Mathematical Phys. 11 (1970), 437–449.
Bernal, Antonio N.; Sánchez, Miguel. On smooth Cauchy hypersurfaces and Geroch's splitting theorem. Comm. Math. Phys. 243 (2003), no. 3, 461–470.
Bernal, Antonio N.; Sánchez, Miguel. Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Comm. Math. Phys. 257 (2005), no. 1, 43–50.
Beem, John K.; Ehrlich, Paul E.; Easley, Kevin L. Global Lorentzian geometry. Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 202. Marcel Dekker, Inc., New York, 1996. xiv+635 pp. ISBN 0-8247-9324-2
Hawking, S.W.; Ellis, G.F.R. The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973. xi+391 pp.
O'Neill, Barrett. Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp. ISBN 0-12-526740-1
Penrose, Roger. Techniques of differential topology in relativity. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 7. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1972. viii+72 pp.
Wald, Robert M. General relativity. University of Chicago Press, Chicago, IL, 1984. xiii+491 pp. ISBN 0-226-87032-4; 0-226-87033-2
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cauchy_surface&oldid=1046071034"
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ReplaceChild - Maple Help
Home : Support : Online Help : Connectivity : Web Features : XMLTools : ReplaceChild
replace a child node of an XML tree
ReplaceChild( spec, xmlTree )
equation acceptable to subsop
The ReplaceChild(spec, xmlTree) command replaces a specific child of an XML document tree with new content.
The first argument is an equation whose left-hand side specifies which child to replace, and whose right-hand side is the replacement content. The syntax mimics that of subsop. A new XML document tree is returned with the replacement content appearing instead of the child element specified in the left-hand side of the first argument.
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{doc}≔\mathrm{XMLElement}\left("a",[],\mathrm{XMLElement}\left("b",["colour"="red"],"foo"\right),\mathrm{XMLElement}\left("c",["colour"="blue"],"bar"\right),\mathrm{XMLElement}\left("d",[],"baz"\right)\right):
\mathrm{Print}\left(\mathrm{doc}\right)
\mathrm{Print}\left(\mathrm{ReplaceChild}\left(2=\mathrm{XMLElement}\left("NEW",[],"text"\right),\mathrm{doc}\right)\right)
<NEW>text</NEW>
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Cohomology - Wikipedia
1 Singular cohomology
3 The diagonal
4 Poincaré duality
6 Eilenberg–MacLane spaces
7 Cap product
8 Brief history of singular cohomology
9 Sheaf cohomology
10 Cohomology of varieties
11 Axioms and generalized cohomology theories
12 Other cohomology theories
{\displaystyle \cdots \to C_{i+1}{\stackrel {\partial _{i+1}}{\to }}C_{i}{\stackrel {\partial _{i}}{\to }}\ C_{i-1}\to \cdots }
{\displaystyle C_{i}^{*}:=\mathrm {Hom} (C_{i},A),}
{\displaystyle \partial _{i}}
{\displaystyle d_{i-1}:C_{i-1}^{*}\to C_{i}^{*}.}
{\displaystyle \cdots \leftarrow C_{i+1}^{*}{\stackrel {d_{i}}{\leftarrow }}\ C_{i}^{*}{\stackrel {d_{i-1}}{\leftarrow }}C_{i-1}^{*}\leftarrow \cdots }
{\displaystyle C_{i}^{*}}
{\displaystyle f:X\to Y}
{\displaystyle f_{*}:H_{i}(X)\to H_{i}(Y)}
{\displaystyle f^{*}:H^{i}(Y)\to H^{i}(X)}
{\displaystyle \cdots \to H^{i}(X)\to H^{i}(U)\oplus H^{i}(V)\to H^{i}(U\cap V)\to H^{i+1}(X)\to \cdots }
{\displaystyle H^{i}(X,Y;A)}
{\displaystyle \cdots \to H^{i}(X,Y)\to H^{i}(X)\to H^{i}(Y)\to H^{i+1}(X,Y)\to \cdots }
{\displaystyle 0\to \operatorname {Ext} _{\mathbb {Z} }^{1}(\operatorname {H} _{i-1}(X,\mathbb {Z} ),A)\to H^{i}(X,A)\to \operatorname {Hom} _{\mathbb {Z} }(H_{i}(X,\mathbb {Z} ),A)\to 0.}
{\displaystyle H^{i}(X,F)}
{\displaystyle H_{i}(X,F)}
{\displaystyle H^{i}(X,A)}
{\displaystyle H^{i}(X,R)\times H^{j}(X,R)\to H^{i+j}(X,R),}
{\displaystyle H^{*}(X,R)=\bigoplus _{i}H^{i}(X,R)}
{\displaystyle uv=(-1)^{ij}vu,\qquad u\in H^{i}(X,R),v\in H^{j}(X,R).}
{\displaystyle f\colon X\to Y,}
{\displaystyle f^{*}:H^{*}(Y,R)\to H^{*}(X,R)}
{\displaystyle [S][T]=[S\cap T]\in H^{i+j}(X),}
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Z} /p}
{\displaystyle H^{i}(X)}
{\displaystyle H^{i}(X)}
{\displaystyle f^{*}([N])\in H^{i}(X)}
{\displaystyle f^{-1}(N)}
{\displaystyle f^{*}([N])}
{\displaystyle X-f^{-1}(N).}
{\displaystyle f^{*}([N])}
{\displaystyle S^{n}}
{\displaystyle (S^{1})^{n}}
{\displaystyle S^{1}}
{\displaystyle (S^{1})^{2}}
{\displaystyle H^{*}(X\times Y,R)\cong H^{*}(X,R)\otimes _{R}H^{*}(Y,R).}
{\displaystyle uv=\Delta ^{*}(u\times v)\in H^{i+j}(X,R).}
{\displaystyle u\times v=(f^{*}(u))(g^{*}(v))\in H^{i+j}(X\times Y,R).}
Main article: Poincaré duality
{\displaystyle H^{i}(X,F)\times H^{n-i}(X,F)\to H^{n}(X,F)\cong F}
Main article: Characteristic class
Main article: Eilenberg–MacLane space
{\displaystyle K(A,j)}
{\displaystyle H^{j}(K(A,j),A)}
{\displaystyle X\to K(A,j)}
{\displaystyle [X,K(A,j)]{\stackrel {\cong }{\to }}H^{j}(X,A)}
{\displaystyle [X,Y]}
{\displaystyle K(\mathbb {Z} ,1)}
{\displaystyle S^{1}}
{\displaystyle H^{1}(X,\mathbb {Z} )}
{\displaystyle S^{1}}
{\displaystyle X\to S^{1}}
{\displaystyle H^{1}(X,A)}
{\displaystyle H^{1}(X,A)}
{\displaystyle \operatorname {Hom} (\pi _{1}(X),A)}
{\displaystyle \pi _{1}(X)}
{\displaystyle H^{1}(X,\mathbb {Z} /2)}
{\displaystyle 0\in H^{1}(X,\mathbb {Z} /2)}
Main article: Cap product
{\displaystyle \cap :H^{i}(X,R)\times H_{j}(X,R)\to H_{j-i}(X,R)}
{\displaystyle H^{*}(X,R)\times H_{*}(X,R)\to H_{*}(X,R)}
{\displaystyle H^{i}(X,R)\to \operatorname {Hom} _{R}(H_{i}(X,R),R),}
{\displaystyle H^{i}(X,R){\overset {\cong }{\to }}H_{n-i}(X,R)}
{\displaystyle H_{i}(M)\times H_{j}(M)\to H_{i+j-n}(M),}
{\displaystyle 0}
{\displaystyle \mathbb {P} ^{n}}
{\displaystyle p}
{\displaystyle p}
{\displaystyle \ell }
{\displaystyle \ell \neq p}
{\displaystyle H^{k}(X;\mathbb {Q} _{\ell }):=\varprojlim H_{et}^{k}(X;\mathbb {Z} /(\ell ^{n}))\otimes _{\mathbb {Z} _{\ell }}\mathbb {Q} _{\ell }}
{\displaystyle X={\text{Proj}}\left({\frac {\mathbb {Z} \left[x_{0},\ldots ,x_{n}\right]}{\left(f_{1},\ldots ,f_{k}\right)}}\right)}
{\displaystyle X(\mathbb {C} )}
{\displaystyle \ell }
{\displaystyle X(\mathbb {F} _{q})}
{\displaystyle \geq 2}
{\displaystyle Z\subset X}
{\displaystyle {\begin{matrix}E&\longrightarrow &Bl_{Z}(X)\\\downarrow &&\downarrow \\Z&\longrightarrow &X\end{matrix}}}
{\displaystyle \cdots \to H^{n}(X)\to H^{n}(Z)\oplus H^{n}(Bl_{Z}(X))\to H^{n}(E)\to H^{n+1}(X)\to \cdots }
{\displaystyle Z}
{\displaystyle H^{n}(Bl_{Z}(X))\oplus H^{n}(Z)\cong H^{n}(X)\oplus H^{n}(E)}
{\displaystyle f:(X,A)\to (Y,B)}
{\displaystyle g:(X,A)\to (Y,B)}
{\displaystyle \cdots \to h_{i}(A){\overset {f_{*}}{\to }}h_{i}(X){\overset {g_{*}}{\to }}h_{i}(X,A){\overset {\partial }{\to }}h_{i-1}(A)\to \cdots .}
{\displaystyle h_{i}(A,A\cap B){\overset {f_{*}}{\to }}h_{i}(X,B)}
{\displaystyle \bigoplus _{\alpha }h_{i}(X_{\alpha },A_{\alpha })\to h_{i}(X,A)}
{\displaystyle \cdots \to h^{i}(X,A){\overset {g_{*}}{\to }}h^{i}(X){\overset {f_{*}}{\to }}h^{i}(A){\overset {d}{\to }}h^{i+1}(X,A)\to \cdots .}
{\displaystyle h^{i}(X,B){\overset {f_{*}}{\to }}h^{i}(A,A\cap B)}
{\displaystyle h^{i}(X,A)\to \prod _{\alpha }h^{i}(X_{\alpha },A_{\alpha })}
{\displaystyle \pi _{S}^{*}(X).}
{\displaystyle \pi _{*}^{S}(X).}
{\displaystyle MO^{*}(X)}
{\displaystyle MSO^{*}(X),}
{\displaystyle MU^{*}(X),}
{\displaystyle KO^{*}(X)}
{\displaystyle ko^{*}(X)}
{\displaystyle K^{*}(X)}
{\displaystyle ku^{*}(X)}
{\displaystyle E^{*}(X)}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Cohomology&oldid=1086341171"
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Enumeration Knowpia
An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem.
Some sets can be enumerated by means of a natural ordering (such as 1, 2, 3, 4, ... for the set of positive integers), but in other cases it may be necessary to impose a (perhaps arbitrary) ordering. In some contexts, such as enumerative combinatorics, the term enumeration is used more in the sense of counting – with emphasis on determination of the number of elements that a set contains, rather than the production of an explicit listing of those elements.
In combinatorics, enumeration means counting, i.e., determining the exact number of elements of finite sets, usually grouped into infinite families, such as the family of sets each consisting of all permutations of some finite set. There are flourishing subareas in many branches of mathematics concerned with enumerating in this sense objects of special kinds. For instance, in partition enumeration and graph enumeration the objective is to count partitions or graphs that meet certain conditions.
In set theory, the notion of enumeration has a broader sense, and does not require the set being enumerated to be finite.
ListingEdit
When an enumeration is used in an ordered list context, we impose some sort of ordering structure requirement on the index set. While we can make the requirements on the ordering quite lax in order to allow for great generality, the most natural and common prerequisite is that the index set be well-ordered. According to this characterization, an ordered enumeration is defined to be a surjection (an onto relationship) with a well-ordered domain. This definition is natural in the sense that a given well-ordering on the index set provides a unique way to list the next element given a partial enumeration.
Countable vs. uncountableEdit
The most common use of enumeration in set theory occurs in the context where infinite sets are separated into those that are countable and those that are not. In this case, an enumeration is merely an enumeration with domain ω, the ordinal of the natural numbers. This definition can also be stated as follows:
As a surjective mapping from
{\displaystyle \mathbb {N} }
(the natural numbers) to S (i.e., every element of S is the image of at least one natural number). This definition is especially suitable to questions of computability and elementary set theory.
We may also define it differently when working with finite sets. In this case an enumeration may be defined as follows:
As a bijective mapping from S to an initial segment of the natural numbers. This definition is especially suitable to combinatorial questions and finite sets; then the initial segment is {1,2,...,n} for some n which is the cardinality of S.
In the first definition it varies whether the mapping is also required to be injective (i.e., every element of S is the image of exactly one natural number), and/or allowed to be partial (i.e., the mapping is defined only for some natural numbers). In some applications (especially those concerned with computability of the set S), these differences are of little importance, because one is concerned only with the mere existence of some enumeration, and an enumeration according to a liberal definition will generally imply that enumerations satisfying stricter requirements also exist.
Enumeration of finite sets obviously requires that either non-injectivity or partiality is accepted, and in contexts where finite sets may appear one or both of these are inevitably present.
The natural numbers are enumerable by the function f(x) = x. In this case
{\displaystyle f:\mathbb {N} \to \mathbb {N} }
is simply the identity function.
{\displaystyle \mathbb {Z} }
, the set of integers is enumerable by
{\displaystyle f(x):={\begin{cases}-(x+1)/2,&{\mbox{if }}x{\mbox{ is odd}}\\x/2,&{\mbox{if }}x{\mbox{ is even}}.\end{cases}}}
{\displaystyle f:\mathbb {N} \to \mathbb {Z} }
is a bijection since every natural number corresponds to exactly one integer. The following table gives the first few values of this enumeration:
0 −1 1 −2 2 −3 3 −4 4
All (non empty) finite sets are enumerable. Let S be a finite set with n > 0 elements and let K = {1,2,...,n}. Select any element s in S and assign f(n) = s. Now set S' = S − {s} (where − denotes set difference). Select any element s' ∈ S' and assign f(n − 1) = s' . Continue this process until all elements of the set have been assigned a natural number. Then
{\displaystyle f:K\to S}
is an enumeration of S.
The real numbers have no countable enumeration as proved by Cantor's diagonal argument and Cantor's first uncountability proof.
There exists an enumeration for a set (in this sense) if and only if the set is countable.
If a set is enumerable it will have an uncountable infinity of different enumerations, except in the degenerate cases of the empty set or (depending on the precise definition) sets with one element. However, if one requires enumerations to be injective and allows only a limited form of partiality such that if f(n) is defined then f(m) must be defined for all m < n, then a finite set of N elements has exactly N! enumerations.
An enumeration e of a set S with domain
{\displaystyle \mathbb {N} }
induces a well-order ≤ on that set defined by s ≤ t if and only if
{\displaystyle \min e^{-1}(s)\leq \min e^{-1}(t)}
. Although the order may have little to do with the underlying set, it is useful when some order of the set is necessary.
In set theory, there is a more general notion of an enumeration than the characterization requiring the domain of the listing function to be an initial segment of the Natural numbers where the domain of the enumerating function can assume any ordinal. Under this definition, an enumeration of a set S is any surjection from an ordinal α onto S. The more restrictive version of enumeration mentioned before is the special case where α is a finite ordinal or the first limit ordinal ω. This more generalized version extends the aforementioned definition to encompass transfinite listings.
Under this definition, the first uncountable ordinal
{\displaystyle \omega _{1}}
can be enumerated by the identity function on
{\displaystyle \omega _{1}}
so that these two notions do not coincide. More generally, it is a theorem of ZF that any well-ordered set can be enumerated under this characterization so that it coincides up to relabeling with the generalized listing enumeration. If one also assumes the Axiom of Choice, then all sets can be enumerated so that it coincides up to relabeling with the most general form of enumerations.
Since set theorists work with infinite sets of arbitrarily large cardinalities, the default definition among this group of mathematicians of an enumeration of a set tends to be any arbitrary α-sequence exactly listing all of its elements. Indeed, in Jech's book, which is a common reference for set theorists, an enumeration is defined to be exactly this. Therefore, in order to avoid ambiguity, one may use the term finitely enumerable or denumerable to denote one of the corresponding types of distinguished countable enumerations.
Comparison of cardinalitiesEdit
Formally, the most inclusive definition of an enumeration of a set S is any surjection from an arbitrary index set I onto S. In this broad context, every set S can be trivially enumerated by the identity function from S onto itself. If one does not assume the axiom of choice or one of its variants, S need not have any well-ordering. Even if one does assume the axiom of choice, S need not have any natural well-ordering.
This general definition therefore lends itself to a counting notion where we are interested in "how many" rather than "in what order." In practice, this broad meaning of enumeration is often used to compare the relative sizes or cardinalities of different sets. If one works in Zermelo–Fraenkel set theory without the axiom of choice, one may want to impose the additional restriction that an enumeration must also be injective (without repetition) since in this theory, the existence of a surjection from I onto S need not imply the existence of an injection from S into I.
Computability and complexity theoryEdit
In computability theory one often considers countable enumerations with the added requirement that the mapping from
{\displaystyle \mathbb {N} }
(set of all natural numbers) to the enumerated set must be computable. The set being enumerated is then called recursively enumerable (or computably enumerable in more contemporary language), referring to the use of recursion theory in formalizations of what it means for the map to be computable.
In this sense, a subset of the natural numbers is computably enumerable if it is the range of a computable function. In this context, enumerable may be used to mean computably enumerable. However, these definitions characterize distinct classes since there are uncountably many subsets of the natural numbers that can be enumerated by an arbitrary function with domain ω and only countably many computable functions. A specific example of a set with an enumeration but not a computable enumeration is the complement of the halting set.
Furthermore, this characterization illustrates a place where the ordering of the listing is important. There exists a computable enumeration of the halting set, but not one that lists the elements in an increasing ordering. If there were one, then the halting set would be decidable, which is provably false. In general, being recursively enumerable is a weaker condition than being a decidable set.
The notion of enumeration has also been studied from the point of view of computational complexity theory for various tasks in the context of enumeration algorithms.
The dictionary definition of enumeration at Wiktionary
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Nonlinear Constraints with Gradients - MATLAB & Simulink - MathWorks América Latina
Objective Function and Nonlinear Constraint
Constraint Function with Gradient
Set Options to Use Derivative Information
This example shows how to solve a nonlinear problem with nonlinear constraints using derivative information.
Ordinarily, minimization routines use numerical gradients calculated by finite-difference approximation. This procedure systematically perturbs each variable in order to calculate function and constraint partial derivatives. Alternatively, you can provide a function to compute partial derivatives analytically. Typically, when you provide derivative information, solvers work more accurately and efficiently.
\underset{x}{\mathrm{min}}f\left(x\right)={e}^{{x}_{1}}\left(4{x}_{1}^{2}+2{x}_{2}^{2}+4{x}_{1}{x}_{2}+2{x}_{2}+1\right),
\begin{array}{l}{x}_{1}{x}_{2}-{x}_{1}-{x}_{2}\le -1.5\\ {x}_{1}{x}_{2}\ge -10.\end{array}
Because the fmincon solver expects the constraints to be written in the form
c\left(x\right) \le 0
, write your constraint function to return the following value:
\mathit{c}\left(\mathit{x}\right)=\left[\begin{array}{c}{\mathit{x}}_{1}{\mathit{x}}_{2}-{\mathit{x}}_{1}-{\mathit{x}}_{2}+1.5\\ -10-{\mathit{x}}_{1}{\mathit{x}}_{2}\end{array}\right]
f\left(x\right)={e}^{{x}_{1}}\left(4{x}_{1}^{2}+2{x}_{2}^{2}+4{x}_{1}{x}_{2}+2{x}_{2}+1\right)
Compute the gradient of
f\left(x\right)
{x}_{1}
{x}_{2}
\nabla \mathit{f}\left(\mathit{x}\right)=\left[\begin{array}{c}\mathit{f}\left(\mathit{x}\right)+\mathrm{exp}\left({\mathit{x}}_{1}\right)\left(8{\mathit{x}}_{1}+4{\mathit{x}}_{2}\right)\\ \mathrm{exp}\left({\mathit{x}}_{1}\right)\left(4{\mathit{x}}_{1}+4{\mathit{x}}_{2}+2\right)\end{array}\right]
The objfungrad helper function at the end of this example returns both the objective function
f\left(x\right)
and its gradient in the second output gradf. Set @objfungrad as the objective.
fun = @objfungrad;
The helper function confungrad is the nonlinear constraint function; it appears at the end of this example.
The derivative information for the inequality constraint has each column correspond to one constraint. In other words, the gradient of the constraints is in the following format:
\left[\begin{array}{cc}\frac{\partial {c}_{1}}{\partial {x}_{1}}& \frac{\partial {c}_{2}}{\partial {x}_{1}}\\ \frac{\partial {c}_{1}}{\partial {x}_{2}}& \frac{\partial {c}_{2}}{\partial {x}_{2}}\end{array}\right]=\left[\begin{array}{cc}{x}_{2}-1& -{x}_{2}\\ {x}_{1}-1& -{x}_{1}\end{array}\right].
Set @confungrad as the nonlinear constraint function.
nonlcon = @confungrad;
Indicate to the fmincon solver that the objective and constraint functions provide derivative information. To do so, use optimoptions to set the SpecifyObjectiveGradient and SpecifyConstraintGradient option values to true.
'SpecifyObjectiveGradient',true,'SpecifyConstraintGradient',true);
Set the initial point to [-1,1].
The problem has no bounds or linear constraints, so set those argument values to [].
The solution is the same as in the example Nonlinear Inequality Constraints, which solves the problem without using derivative information. The advantage of using derivatives is that solving the problem takes fewer function evaluations while gaining robustness, although this advantage is not obvious in this example. Using even more derivative information, as in fmincon Interior-Point Algorithm with Analytic Hessian, gives even more benefit, such as fewer solver iterations.
This code creates the objfungrad helper function.
function [f,gradf] = objfungrad(x)
% Gradient of the objective function:
if nargout > 1
gradf = [ f + exp(x(1)) * (8*x(1) + 4*x(2)),
exp(x(1))*(4*x(1)+4*x(2)+2)];
This code creates the confungrad helper function.
% Gradient of the constraints:
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Because array is a Maple function, it must be enclosed in single quotes to prevent evaluation.
a≔\mathrm{array}\left(1..3,[1,2,{\left(x-1\right)}^{2}]\right)
\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& {\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}\end{array}]
b≔\mathrm{array}\left(1..3,[1,2,{\left(x-1\right)}^{2}]\right)
\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& {\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)}^{\textcolor[rgb]{0,0,1}{2}}\end{array}]
c≔\mathrm{array}\left(1..3,[1,2,{x}^{2}-2x+1]\right)
\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& {\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\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}{+}\textcolor[rgb]{0,0,1}{1}\end{array}]
\mathrm{evalb}\left(a=b\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(a,b,'\mathrm{array}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left(a,c,'\mathrm{array}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(a,c,'\mathrm{array}\left(\mathrm{expand}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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ONE day very soon thereafter my servant presented me a box, which he said had been brought there by an Indian from Colonel d'Ortez, with the request that it be delivered into my own hand. And further, to beg I would make him a visit as soon as my duties would permit.
The evening being far advanced I could not go that night, so contented myself with the promise I would cross the bay on the morrow.
Later, my company being my own, I gave attention to the box, such a metal receptacle as was commonly used for articles of value. It responded easily to the key, and opened without difficulty.
The reasons for d'Ortez's fear and retirement lay bare before me, if I would but search them out. Within the box, bound together by deerskin thongs, were many writings, some on parchment, some paper, of different dates and degrees of preservation. Some were well worn from age and handling, others more recent, were in better condition. Some there were which appeared quite new and fresh; these must have been the latest to find a resting place in his keeping.
All were arranged in due and systematic order; of whatever age, each bore a careful superscription, giving in brief the contents of the paper written by his own exact hand. Beside this, each document was numbered and placed in sequence. Verily, it was most methodically done, so any child could read and understand.
It was with much misgiving I approached the task of making myself familiar with my old friend's secret. Had he committed some youthful crime which weighed heavily upon his trembling age, and had driven him to these savage shores, where, shut out from all companionship with his kind, he did a lonely penance? If so, I preferred to remain in ignorance, for his was a friendship so dear, so pure, I desired not to taint it with the odor of guilt.
He had, however, made his request in such urgent terms, even pathetic, I could not disregard it, and putting aside the reluctance I felt, I took up the paper which lay on top, directed to myself, and began its perusal. It was as follows:
My dear Placide:
The great feebleness of my worn-out frame warns me again that time for me is almost past. It may be, when you recross the seas, I shall have gone to final judgment. * * * remember my request, and carry on to the end that work which generations of cowards have left undone. * * * All is here contained in these papers, except some recent news I have of the Pasquiers from the northern colonies.
Possibly if you went to Quebec and sought out the Cure of St. Martin's (who wrote this last letter, No. 32) you may right it all, and give to my soul its eternal peace. * * * With the strong affection which my bodily infirmities have in no wise diminished, I am,
Your old friend.
Raoul Armand Xavier d'Ortez.
of Cartillon, Normandy.
Having carefully read this letter, I then proceeded to peruse the various documents in the order he had arranged them.
The first, written by the hand of the Benedictine, Laurent of Lorraine, Abbot of Vaux, told of the admission to the monastery of a child, son of Henri d'Artin, to whom the good monks gave the name Bartholomew Pasquier. This child, though designed for orders, left the monastery, cast his fortunes with the King of Navarre, and became a great officer in the household of King Henri the Fourth.
Other documents gave an account of the posterity of this child down to one Francois Rene Alois de Pasquier, who fled to America in 1674 to escape the vengeance of a certain great lord whose son he slew in a duel. This was he who was reputed to have been killed in battle, and to have left no issue. And this was he whom I afterward found to be my own good father.
There was also contained an account of the later life of Pedro d'Ortez, who, profiting not by his blood-gotten gains, threw himself, while in delirium, into the same old well whereon he had hanged his brother, Henri d'Artin.
Some further notes by the good abbot told of how Raoul, the second son of Pedro, slew his own brother, before their father's eyes, in order that he, Raoul, might be Count of Cartillon. And this same Raoul, some years later, did have the locket made and forced his own son to swear that he would restore the real sons of d'Artin, the true children of the Black Wolf's Breed, to their own again. All of these accounts are of surpassing interest, old and quaint, to a perusal of which I recommend my children.[1]
For the first time, in reading these manuscripts, did I begin clearly to associate the name d'Ortez with the name used by the madman in his story at the old Norman ruin. With this new light, link by link did the whole knotted chain untangle. Curiously enough, the tale I had heard at the ruined castle tallied in the main with the monkish documents here preserved. Indeed it supplied me with knowledge of much which otherwise I would not have comprehended so completely. The horrible reality of that weird recital was still fresh and distinct before me, undimmed by time and unforgotten through all my troubles.
I had sought refuge many times from brooding over my own affairs by turning to this for interest and occupation. Every further detail was supplied by a number of quaint documents, which Colonel d'Ortez had digested into this:
Table Showing The Male Descendants Of
HENRI d'ARTIN
Henri Francois Placide d'Artin, died Aug. 28, 1572.
Bartholomew Pasquier (son of above, died 1609.
Bartholomew Placide Pasquier killed in wars of the Fronde.
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}}
Sons of above.
Henri Louis John (brother to above, died 1654.
Francois Rene Xavier de Pasquier (ennobled), killed 1650.
Francois Rene Alois de Pasquier, fled to America. Supposed to have been killed about 1681. No known descendants. Well known to the Cure of St. Martin's, Quebec.
PEDRO ORTEZ.
Pedro d'Ortez, suicided 1604.
Charles Pedro, killed by Raoul 1602.
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}}
Raoul, died 1618.
Charles Francis Peter (son of Raoul, died without issue.
Pedro d'Ortez (brother to above, died 1663
Henry (son of above), killed in battle.
Alphonze, killed in battle.
{\displaystyle \scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}}
Felix, died in infancy.
Raoul Armand Xavier d'Ortez, born 1641 (myself.) Died —. No children.
She who was born my daughter I disowned, and she died without issue.
It appeared that the only thing to be done was to visit the good Cure of St. Martin's, and, enlisting him in the search, find whatever descendants might have been left by this Francois Rene Alois de Pasquier. The task need not be a difficult one, as many old people should still be living who might have known of the man.[2]
I now bethought me of this enterprise as a fair excuse whereby I could leave Biloxi for a space. I would, therefore, call upon my old friend, and having obtained leave, matters now being safe with the colony, make the journey to Quebec.
But, alas for the weakness of fallen humanity; my last act before putting myself out of temptation's way was to run full tilt into it.
While this came so near to causing my dishonourable death, yet it was, under Divine Providence, the direct means of spreading before me a long life of happiness and honour. After a hard battle with my weaker self I lost the fight.
Just as on the day I departed from Versailles, I determined, cost what it would, to see Agnes once again. So I wrote her a note. Such a blunt and clumsy billet as only a love-sick soldier or a country clown could have written. It craved pardon for the heat and the haste displayed by me when we parted at Sceaux; it implored one last interview before I left the colonies forever. I had not the art to conceal or veil my meaning, but told it out and plainly. Such a note as an idiotic boy might pen, or a simpering school lass be set fluttering to receive.
I bade my man deliver this to Madame de la Mora on the morrow, charging him minutely and repeatedly to see it safe in her own hands. So careful was I, I did not doubt that even so stupid a lout as Jacques understood me perfectly.
His further instructions were to meet me at the Bay when I should return in the evening from my visit to Colonel d'Ortez, and there beside its rippling waters—or so I had arranged—I was to receive her answer.
It had now turned late of the night, and I sought repose. Sleep evaded my bed. What with my own restless desires, my chiding sense of ill-doing, and the d'Ortez story I had read, I tossed and tumbled through the remaining hours of darkness. Tumbled and tossed, whilst the sins and sufferings of men long dead passed and repassed with their spectral admonitions.
Early on the morrow, while the day was yet cool, I crossed the Bay, and climbed the slope of sand before the lonely house. It looked more deserted and desolate than I had ever seen it. The stillness of solitary death clung as a pall about the place. Pachaco, the Indian servant, sat beside the gate, as motionless as the post against which he leaned.
"How is the master, Pachaco?" I inquired, passing in.
"Him die yesterday," came the stolid reply.
"What? Dead! When?"
"The shadows were at the longest," he answered, indicating by a gesture the western horizon.
I hurried into the master's room. In the same position he had occupied, when, months ago, he had beckoned me to remain, he sat there, dead in his chair. His clothing hung about him in that sharply angular fashion in which garments cling to a corpse. Long, thin locks were matted above his brow, awesomely disarranged. But the pose of his head, drooped a little forward, suggested a melancholy reverie, nothing more.
The golden locket, which he had shown me that well-remembered night, rested within his shrunken palm. I noted that the side was open which revealed the blazing bar of red. As if absorbed in that same unpleasant thought, there sat the master, dead; dead, and I alone knew his story. How vividly the old man's sorrow came back; how it oppressed me.
I bent down in tender sympathy to look again upon his wasted features, and kneeling, gazed into his wide-open eyes. The calm of promised peace upon his brow was distorted by the unsatisfied expression of one who has left his work undone.
So are the sins of the fathers visited upon their children, for I was no longer in doubt but that the murderer, Pedro Ortez, was the sinning ancestor of my old-time friend. Even in his presence my thoughts flew to Agnes; had she not spoken of her grandsire as being such a man? The stiffening body at my side was speedily forgotten in the music of this meditation.
I gained my feet again and looked down upon him, fascinated by the changeless features of the dead. It was probably natural that standing there I should revolve the whole matter over and over again, from the first I knew of it until the last. A young man's plans, though, work ever with the living; the dead he places in their tomb, covers them with earth, bids them "God-speed," and banishes the recollection. I was already busy with my contemplated search for the last d'Artin, and stood there leaning against the oaken table pondering over the question, "Where is the last d'Artin?"
My mind wandered, returning with a dogged persistence to that one thought, "Where is the last d'Artin?" "Where could I find him?" My restless eyes roamed round the cheerless room, coming always back to rest upon a long dust-covered mirror set in the wall across the way.
As wind-driven clouds gather and group themselves in fantastic shapes, so, deep in that mirror's shadowy depths, a vague figure gradually took form and character—myself.
With the vacant glance of a man whose mind is intensely preoccupied, I studied minutely the reflection, my own bearing, my dress, my weapons. I even noted a button off my coat, and tried dimly to remember where I had lost it, until—great God—this chamber of death and revelation had turned my brain.
What face was that I saw? My own, assuredly, but so like another.
Aghast, powerless to move or cry out, I stared helplessly into the glass. Every other sensation vanished now before this new-born terror which held my soul enslaved. I closed my eyes, I dared not look.
My body seemed immovable with horror, but a trembling hand arose and pointed at the mirror. Scant need there was to call attention to that dim, terrible presence; my whole soul shrank from the ghostly face reflected in the glass. For there, there was the same pallid countenance, death-distorted and drawn, which I had conjured up in many a frightened dream as that of the murdered Count—there was Henri d'Artin.
How long I stood transfixed, pointing into the mirror, I know not. As men think of trifles even in times of deadly fear, so did my lips frame over and over again the last question I had in mind before all sense forsook me, "Where is the last d'Artin? Where is the last d'Artin? Where—?"
And in answer to my question, that long, rigid finger pointed directly at me from out the dusty glass. It was as if the hand of the dead had told me who I was.
It had been no blind chance, then, which led me to the Paris house of the "Black Wolf's Head;" the girl's ring with the same device, and the grewsome narrative beneath the shadow of the Wolf at the Norman ruin—nothing less than fate had brought these lights to me.
Verily some more logical power than unreasoning accident must direct the steps of men. A God of justice perhaps had placed these tokens in my path. And soldiers call this "Fortune."
•••••••
I dispatched Pachaco to Biloxi with the news of death, and long before the afternoon our few simple arrangements for his funeral had been made.
"Bury me here, Placide, beneath this great oak," he had said to me one day. "The Infinite Mercy will consecrate the grave of penitence, wherever it may be."
He had his wish.
That long rigid finger pointed directly at me from out the dusty glass
↑ These documents have been included in an appendix to this volume.
↑ A very slight investigation showed that this last named Francois Rene Alois de Pasquier was none other than my own good father, who assumed the name de Mouret to avoid the consequences of a fatal duel in France. This I learned from the pious Cure of St. Martin's, who knew him well.
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Modify a System of LMIs - MATLAB & Simulink - MathWorks United Kingdom
Deleting an LMI
Deleting a Matrix Variable
Instantiating a Matrix Variable
Once specified, a system of LMIs can be modified in several ways with the functions dellmi, delmvar, and setmvar.
The first possibility is to remove an entire LMI from the system with dellmi. For instance, suppose that the LMI system of Specify LMI System is described in LMISYS and that we want to remove the positivity constraint on X. This is done by
NEWSYS = dellmi(LMISYS,2)
where the second argument specifies deletion of the second LMI. The resulting system of two LMIs is returned in NEWSYS.
The LMI identifiers (initial ranking of the LMI in the LMI system) are not altered by deletions. As a result, the last LMI
remains known as the third LMI even though it now ranks second in the modified system. To avoid confusion, it is safer to refer to LMIs via the identifiers returned by newlmi. If BRL, Xpos, and Slmi are the identifiers attached to the three LMIs described in Specify LMI System, Slmi keeps pointing to S > I even after deleting the second LMI by
NEWSYS = dellmi(LMISYS,Xpos)
Another way of modifying an LMI system is to delete a matrix variable, that is, to remove all variable terms involving this matrix variable. This operation is performed by delmvar. For instance, consider the LMI
ATX + XA + BW + WTBT + I < 0
with variables X = XT ∊ R4×4 and W ∊ R2×4. This LMI is defined by
X = lmivar(1,[4 1]) % X
W = lmivar(2,[2 4]) % W
lmiterm([1 1 1 W],B,1,'s')
lmiterm([1 1 1 0],1)
LMISYS = getlmis
To delete the variable W, type the command
NEWSYS = delmvar(LMISYS,W)
The resulting NEWSYS now describes the Lyapunov inequality
ATX + XA + I < 0
Note that delmvar automatically removes all LMIs that depended only on the deleted matrix variable.
The matrix variable identifiers are not affected by deletions and continue to point to the same matrix variable. For subsequent manipulations, it is therefore advisable to refer to the remaining variables through their identifier. Finally, note that deleting a matrix variable is equivalent to setting it to the zero matrix of the same dimensions with setmvar.
The function setmvar is used to set a matrix variable to some given value. As a result, this variable is removed from the problem and all terms involving it become constant terms. This is useful, for instance, to fixsetmvar some variables and optimize with respect to the remaining ones.
Consider again Specify LMI System and suppose we want to know if the peak gain of G itself is less than one, that is, if
∥G∥∞ < 1
This amounts to setting the scaling matrix D (or equivalently, S = DTD) to a multiple of the identity matrix. Keeping in mind the constraint S > I, a legitimate choice is S = 2-βψ-I. To set S to this value, enter
NEWSYS = setmvar(LMISYS,S,2)
The second argument is the variable identifier S, and the third argument is the value to which S should be set. Here the value 2 is shorthand for 2-by-I. The resulting system NEWSYS reads
\begin{array}{c}\left(\begin{array}{cc}{A}^{T}X+XA+2{C}^{TC}& XB\\ {B}^{T}X& -2I\end{array}\right)<0\\ X>0\\ 2I>I.\end{array}
Note that the last LMI is now free of variable and trivially satisfied. It could, therefore, be deleted by
NEWSYS = dellmi(NEWSYS,3)
NEWSYS = dellmi(NEWSYS,Slmi)
if Slmi is the identifier returned by newlmi.
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3 Ways to Convert Nanometers to Meters - wikiHow
1 Using a Conversion Factor
2 Counting Decimals
3 Using an Online Conversion Calculator
Nanometers and meters are units of measuring length in the SI (metric) system. A nanometer is a very small unit of measure, used to measure things like atoms and molecules.[1] X Research source The meter is the standard unit of length in the metric system. There are times when you may know the length of something in nanometers but want to convert it to meters. To do this, you can use a conversion factor fraction. You may also simply move the decimal place if you understand the metric units. Finally, you can use a conversion calculator available online.
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Know the conversion factor between nanometers and meters. You can look online or in a textbook to find the relationship between nanometers and meters. The relationship between the two units is:[2] X Research source
1 meter = 1,000,000,000 nanometers (nm)
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Write the conversion factor as a fraction. Recall that any fraction in which the numerator and denominator are equal has an overall value of 1. This is useful, because you can use the conversion fraction to change units without changing the value of the measurement. When you begin with nanometers and want to convert to meters, you will use the following conversion factor:[3] X Research source
{\displaystyle {\frac {1{\text{ meter}}}{1,000,000,000{\text{ nm}}}}}
This is the fraction to convert from nanometers into meters. If you want to begin with meters and convert to nanometers, you just need to flip the fraction upside down.
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Set up a conversion equation. To convert from one unit to another, you need to set up a simple multiplication equation. You begin with your initial measurement and multiply it by the conversion factor. The result will be your answer.[4] X Research source
Remember that you can cancel units that appear once in a numerator and once in a denominator. This is the purpose of the conversion fraction. Your original measurement begins with nanometers, and the conversion fraction has nanometers in the denominator. Thus, the units of nanometers will cancel, and what remains will be the answer in meters.
For this conversion, suppose you have measured a germ that is 2,600 nanometers wide. The conversion equation will be:
{\displaystyle 2,600{\text{ nm}}=2,600{\text{ nm}}*{\frac {1{\text{ meter}}}{1,000,000,000{\text{ nm}}}}}
Perform the conversion. If your conversion factor is correct, the final step should be a simple multiplication. Finish the conversion as follows:[5] X Research source
{\displaystyle 2,600{\text{ nm}}=2,600{\text{ nm}}*{\frac {1{\text{ meter}}}{1,000,000,000{\text{ nm}}}}=0.0000026{\text{ m}}}
Counting Decimals
Learn the relative positions of the units. The strength and simplicity of the metric system stem from the fact that the units are based on multiples of 10. Therefore, converting from one unit to another is simply a matter of moving the decimal point the correct number of spaces. To use this shortcut tool, you need to know the exponent measures of the units involved.[6] X Research source
The meter is the basic unit of length measurement in the metric system. Written as a base 10 exponent,
{\displaystyle 1{\text{ m}}=10^{0}{\text{ m}}}
The nanometer is one-billionth of a meter. Written as a base 10 exponent,
{\displaystyle 1{\text{ nm}}=10^{-9}{\text{ m}}}
Know the LARS mnemonic. For moving decimal places, "LARS" is a handy memory device. It stands for "Left-Add, Right-Subtract." This tells you what to do with the decimal point if you wish to either add or subtract to change exponents. If you want to convert your measurement from a small exponent to a larger one, you will be adding units to the exponent, and for each one addition to the exponent, you will move the decimal point one space to the left. Moving from a larger unit of measure to a smaller, you will be subtracting exponents, which will involve moving the decimal point to the right.[7] X Research source
For example, consider converting kilometers to meters. A kilometer is
{\displaystyle 10^{3}}
meters, but the single meter is
{\displaystyle 10^{0}}
. Thus, you want to subtract 3 in the exponent, so you will move the decimal to the right three places ("Left-Add, RIGHT-SUBTRACT"). Therefore, if you begin with a measure of 1.234 km, you would make the following change to meters:
1.234 km ............(original unit)
12.34 .................(move decimal one space right)
123.4..................(move decimal two spaces right)
1,234 m..................(move decimal three spaces right)
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Decide how you want to change the exponent. To convert from nanometers, which are units of 10-9 m, into meters, which are units of 100 m, you are moving from -9 to 0. This requires you to add 9. According to the LARS mnemonic, adding 9 units in the exponent means moving the decimal place 9 spaces to the left ("Left-Add").[8] X Research source
Move the decimal as needed. For this example, suppose you have the measurement of a germ at 2,600 nm. You need to move the decimal point 9 spaces to the left. Since this is a whole number with no decimal point showing, you can insert the decimal at the end of the number and then count the spaces to move it. When you run out of numbers, you will fill in all remaining spaces with zeroes.[9] X Research source
Write 2,600 nm as "2,600. nm" with a decimal point. Then move the decimal as follows:
260.0………. (one space left)
26.00……….(two spaces left)
2.600……….(three spaces left)
.2600……….(four spaces left)
.02600………(five spaces left)
.002600…….(six spaces left)
.0002600……(seven spaces left)
.00002600…..(eight spaces left)
.000002600…..(nine spaces left)
Using an Online Conversion Calculator
Search the Internet. Perform a simple Internet search for "convert nanometers to meters." You should get several results with links to a number of different conversion calculators. Look them over and choose the one that you believe will be easiest for you to use. The results should be the same, no matter which conversion calculator you choose. Select one that is easy for entering data with easy-to-read results.[10] X Research source
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Enter the number of nanometers that you wish to convert. When you find an online conversion calculator that you like, it should have a box or space for you to enter your initial measurement of nanometers. You do not need to worry about counting decimal places or knowing conversion factors. Just enter your measurement, and your answer should appear.[11] X Research source
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Perform the conversion. On some conversion calculators, after you enter your initial measurement, you need to press a "convert" button. On others, the result may be automatic.[12] X Research source
For example, at the conversion site www.Metric-conversion.net, as soon as you enter the number of nanometers in the first box, an equivalent number of meters will appear immediately in the second box. Entering the number 2,600 in the first box will cause the value of 0.0000026 m to appear in the second box.
To help remember some of the most common units of measure in the metric system, recall the sentence, “King Hector Died Monday Drinking Chocolate Milk.” The initials of these words, K-H-D-M-D-C-M, represent the common prefixes in the metric system, in decreasing decimal positions: Kilo- (103), Hecto- (102), Deka- (101), Meter/Gram/Liter (100), Deci- (10-1), Centi- (10-2), Milli (10-3). The nanometer discussed in this article is at 10-9, which is too far down the scale for this memory device.
↑ http://www.nanooze.org/how-big-is-a-nanometer/
↑ http://greenway.guhsdaz.org/UserFiles/Servers/Server_759379/File/IValenzuela/Metric%20System%20Rdg.pdf
↑ http://www.metric-conversion.net/convert_nanometers_meters.htm
Italiano:Convertire i Nanometri in Metri
Español:convertir nanómetros a metros
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Symbolic logical constant true - MATLAB symtrue - MathWorks United Kingdom
Generate Array of Symbolic Logical symtrues
Convert symtrue to Other Data Types
Symbolic logical constant true
T = symtrue(n)
T = symtrue(sz)
T = symtrue(sz1,...,szN)
symtrue is the symbolic logical constant for the true condition.
T = symtrue(n) returns an n-by-n matrix of symbolic logical symtrues.
T = symtrue(sz) returns an array of symbolic logical symtrues where the size vector, sz, defines size(T). For example, symtrue([2 3]) returns a 2-by-3 array of symbolic logical symtrues.
T = symtrue(sz1,...,szN) returns a sz1-by-...-by-szN array of symbolic logical symtrues where sz1,...,szN indicates the size of each dimension. For example, symtrue(2,3) returns a 2-by-3 array of symbolic logical symtrues.
{x}^{2}>4
4<{x}^{2}
x>2
Simplify the condition represented by the symbolic inequality eq. The simplify function returns the symbolic logical constant symtrue since the condition always holds for the assumption
x>2
T = simplify(eq)
\mathrm{symtrue}
You can also use isAlways to check if the inequality holds under the assumption being made. In this example, isAlways returns logical 1 (true).
Use symtrue to generate a 3-by-3 square matrix of symbolic logical symtrues.
T = symtrue(3)
\left(\begin{array}{ccc}\mathrm{symtrue}& \mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}& \mathrm{symtrue}\end{array}\right)
Next, use symtrue to generate a 3-by-2-by-2 array of symbolic logical symtrue's.
T = symtrue(3,2,2)
\left(\begin{array}{cc}\mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}\end{array}\right)
\left(\begin{array}{cc}\mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}\end{array}\right)
T = symtrue([3,2,2])
\left(\begin{array}{cc}\mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}\end{array}\right)
\left(\begin{array}{cc}\mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}\\ \mathrm{symtrue}& \mathrm{symtrue}\end{array}\right)
\left(\begin{array}{cc}\mathrm{symtrue}& \mathrm{symfalse}\end{array}\right)
\left(\begin{array}{c}\mathrm{symtrue}\\ \mathrm{symfalse}\end{array}\right)
\left(\begin{array}{cc}\mathrm{symtrue}& \mathrm{symfalse}\\ \mathrm{symfalse}& \mathrm{symfalse}\end{array}\right)
\mathrm{symtrue}
\mathrm{symtrue}
Convert the symbolic logical constant symtrue to a logical value.
T1 = logical(symtrue)
Convert the symbolic logical constant symtrue to numeric values in double precision and variable precision.
T2 = double(symtrue)
T3 = vpa(symtrue)
1.0
Size of square matrix, specified as an integer. n sets the output array size to n-by-n. For example, symtrue(3) returns a 3-by-3 array of symbolic logical symtrues.
Size vector, specified as a row vector of integers. For example, symtrue([2 3]) returns a 2-by-3 array of symbolic logical symtrues.
If any trailing dimensions greater than 2 have a size of 1, then the output T does not include those dimensions. For example, symtrue([2 2 1 1]) returns a 2-by-2 array and symtrue([2 2 1 2 1]) returns a 2-by-2-by-1-by-2 array.
Size inputs, specified by a comma-separated list of integers. For example, symtrue(2,3) returns a 2-by-3 array of symbolic logical symtrues.
If any trailing dimensions greater than 2 have a size of 1, then the output T does not include those dimensions. For example, symtrue([2,2,1,1]) returns a 2-by-2 array and symtrue([2,2,1,2,1]) returns a 2-by-2-by-1-by-2 array.
T — Symbolic logical constant for true condition
Symbolic logical constant for true condition, returned as a scalar, vector, matrix, or N-D array.
The command sym(true) returns a symbolic number 1, and sym(symtrue) returns symtrue.
isAlways | symfalse | and | or | not | xor
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1 Learning Basic Conversion
2 Using Powers of Ten
If you used the same unit to weigh molecules and elephants, your measurements would end up really complicated. The international system of units, called SI units, solves this problem by using prefixes. Each prefix multiplies the base unit by a certain power of ten (10, 100, 1000, and so on). This keeps the numbers you're working with manageable, while quickly identifying the relationship between very large and very small units. All SI units use the same prefixes, so once you learn them for units of length (metres), you already know how to use prefixes for units of mass (grams),
Learning Basic Conversion Download Article
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Learn the definition of "kilo." "Kilo" is one of the most common unit prefixes, so you might already recognize it from words like "kilogram" and "kilometre". This prefix means "1000" (one thousand). Kilo- is abbreviated as "k-", so a kilogram is "kg" and a kilometre is "km".[1] X Research source
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Convert kilometres to metres. Since kilo- means 1 000, one kilometre (km) is equal to 1000 metres (m). Multiply the number of kilometres by 1 000 to get the same measurement in metres.[2] X Research source
For example, 75 kilometres =
{\displaystyle 75km*1\ 000{\frac {m}{km}}}
= 75 000 metres.
To make sure you've set up the conversion correctly, try reading it in plain English:
{\displaystyle 1000{\frac {m}{km}}}
means "1 000 metres per kilometre".
Compare to other units. The base unit does not make any difference in this calculation. As long as you know the definition of the prefix, you can use the same math to convert between it and the base unit. Here are a couple examples:
75 kilograms (kg) =
{\displaystyle 75kg*1\ 000{\frac {g}{kg}}}
= 75 000 grams (g).
14 kilowatts (kW) =
{\displaystyle 14kW*1\ 000{\frac {W}{kW}}}
= 14 000 watts (W).
Learn common larger prefixes. Another common prefix is mega-, which means one million (1 000 000) and is abbreviated M. Even larger units use the prefix giga-, which means one billion (1 000 000 000) and is abbreviated G.[3] X Research source Here are a couple conversions that use these prefixes:
A 3 megawatt (MW) factory produces
{\displaystyle 3MW*1\ 000\ 000{\frac {W}{MW}}}
= 3 000 000 watts (W) of power.
A 2 gigajoule (GJ) explosion releases
{\displaystyle 2GJ*1\ 000\ 000\ 000{\frac {J}{GJ}}}
= 2 000 000 000 joules (J) of energy.
Convert from a base unit to a larger unit. To convert to a larger unit of measurement, you'll divide by the conversion factor instead of multiplying. As always, you're less likely to make a mistake if you include the units at each point in your calculation:[4] X Research source
65 300 metres is equal to
{\displaystyle 65\ 300m*{\frac {1\ km}{1\ 000m}}}
= 65.3 kilometres.
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Learn prefixes with factors less than one. What if you want to measure a length shorter than one meter, or the volume of a cup smaller than one litre? That's when you use prefixes that represent a fraction of the base unit. Here are the most common prefixes of this type:[5] X Research source
centi- (c) means "one hundredth" (0.01). 1 centimetre = 0.01 metres.
milli- (m) means "one thousandth" (0.001). 1 millimetre = 0.001 metres.
micro- (µ) means "one millionth" (0.000 001). 1 micrometre = 0.000 001 metres.
nano- (n) means "one billionth" (0.000 000 001). 1 nanometre = 0.000 000 001 metres.
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Convert between the base unit and smaller units. These conversions work exactly the same. Just make sure to set up your units so they cancel out, and to keep track of decimals:[6] X Research source
Centimetres to metres: 33 centimetres =
{\displaystyle 33cm*0.01{\frac {m}{cm}}}
= 0.33 metres.
Metres to millimetres: 2.15 metres =
{\displaystyle 2.15m*{\frac {1mm}{0.001m}}}
= 2150 millimetres.
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Check your answer. Probably the most common mistake with these conversions is to accidentally divide instead of multiply or vice versa. There are several ways to catch this mistake:
Check the units in your equation. If you set it up correctly, the original units should cancel out. For example
{\displaystyle 75km*1000{\frac {m}{km}}}
gives you an answer in terms of
{\displaystyle {\frac {km*m}{km}}}
. The km units are on top and bottom, so they cancel out and leave you with m (metres).
Compare the units logically. The smaller unit should always have the larger number next to it. Metres are smaller than kilometres, so it takes more of them to fill the same length. For example, a result of 75 000 metres = 75 kilometres makes sense, since a larger number of metres equals a smaller number of kilometres.
Using Powers of Ten Download Article
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Learn the prefixes as powers of ten. Every SI prefix represents a different power of ten: 100, 1000, 0.001, and so on. Here are the most common examples written in scientific notation (standard form):[7] X Research source
kilo- = 1 000 = 103
milli- = 0.001 = 10-3
micro- = 0.000 001 = 10-6
You can also write a negative exponent as a fraction with a positive exponent in the denominator:
{\displaystyle 10^{-2}={\frac {1}{10^{2}}}}
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Use scientific notation to convert to the base unit. Scientific notation makes it much easier to convert between units of very different size. No matter what the final unit you want is, you can start by converting the initial value to the base unit, using scientific notation.
Example: How many centimetres are in 13.78 kilometres?
The answer isn't obvious, but you do know that the kilometre is equal to 103 metres.
Therefore, 13.78 km = 13.78 * 103 metres. You'll use this in the next step.
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Find the conversion to your final unit. Now that you have a base unit value (with no prefix), you can set up an ordinary conversion to the final unit. Write the conversion in powers of ten notation.
To continue the example, you know want to convert 13.78 * 103 metres into centimetres.
The prefix centi- means 10-2, so there is
{\displaystyle {\frac {1cm}{10^{-2}m}}}
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Set up the conversion as a single expression. Set up your conversion using power of ten notation for all values. This gives you a fraction that divides one power of ten by another:
{\displaystyle 13.78*10^{3}m*{\frac {1cm}{10^{-2}m}}={\frac {13.78*10^{3}}{10^{-2}}}\ }
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Simplify the exponents. To divide one exponential expression by the other (when they both have the same base), take the top exponent and subtract the bottom exponent. The answer is the new exponent in your answer, with the same base (which is always 10 in these conversion problems).[8] X Research source
{\displaystyle {\frac {13.78*10^{3}}{10^{-2}}}}
{\displaystyle 13.78*10^{3-(-2)}=13.78*10^{5}}
It's usually helpful to write your answer in scientific notation (standard form):
{\displaystyle 1.378*10^{6}}
Simplify the formula. Once you are comfortable with this process, you can combine the whole process into a couple quick steps:
Write the initial prefix and the final prefix as powers of 10. For base units without prefixes, use
{\displaystyle 10^{0}}
Divide the initial power of 10 by the final power of 10. (To do this, subtract the final exponent from the initial exponent.)
Multiply your initial value by this answer.
Example: How many centilitres (cL) are in 85 500 millilitres (mL)?
The initial prefix is "milli-" =
{\displaystyle 10^{-3}}
and the final prefix is "centi-" =
{\displaystyle 10^{-2}}
{\displaystyle {\frac {10^{-3}}{10^{-2}}}=10^{(-3)-(-2)}=10^{-1}}
85 500 millilitres =
{\displaystyle 85\ 500*10^{-1}}
centilitres.
Optionally, write this in scientific notation:
{\displaystyle 8.55*10^{3}}
What's the speed of a car in kph if it travels 16 km in 13 minutes?
Divide 60 minutes by 13 minutes. That's 4.615. Multiply that by 16 km: (4.615)(16 km) = 73.84 km. That means that if a car travels 16 km in 13 minutes, at the same speed it would travel 73.84 km in one hour. The car is traveling at 73.84 kph.
The bold rows represent the most common prefixes.
yotta | 1024 | Y | 1,000,000,000,000,000,000,000,000 base units
zetta | 1021 | Z | 1,000,000,000,000,000,000,000
exa | 1018 | E | 1,000,000,000,000,000,000
peta | 1015 | P | 1,000,000,000,000,000
tera | 1012 | T | 1,000,000,000,000
giga | 109 | G | 1,000,000,000
mega | 106 | M | 1,000,000
kilo | 103 | k | 1,000
hecto | 102 | h | 100
deka | 101 | da | 10
(no prefix) | 100 | | 1 base unit
deci | 10-1 | d | 0.1
centi | 10-2 | c | 0.01
milli | 10-3 | m | 0.001
micro | 10-6 | µ | 0.000001
nano | 10-9 | n | 0.000000001
pico | 10-12 | p | 0.000000000001
femto | 10-15 | f | 0.000000000000001
atto | 10-18 | a | 0.000000000000000001
zepto | 10-21 | z | 0.000000000000000000001
yocto | 10-24 | y | 0.000000000000000000000001 base units
Common Units that Use Prefixes
metre (length) (m)
litre (volume) (L)
gram (mass/weight) (g)
joule (heat/energy) (J)
coulomb (charge) (C)
watt (power) (W)
volt (electrical potential difference) (V)
hertz (frequency) (Hz)
newton (force) (N)
pascal (pressure) (Pa)
seconds (time) (s)
These units are almost always seen in their base form, but they can use SI prefixes.
moles (number of discreet particles in a sample) (mol)
kelvins (temperature) (K)
degrees Celsius (temperature) (ºC)
amperes (electrical current) (A)
candela (luminous intensity) (cd)
You probably don't need to memorize prefixes larger than giga- or smaller than nano-. Those are rarely used except in specific fields of study.
Remember which "direction" you are going! Check your answers against reality, if possible.
You can only use this method if the base units are the same. Converting between kilograms and pounds, for example, requires a different formula.
↑ https://www.mathsisfun.com/definitions/kilo-.html
↑ https://www.nuffieldfoundation.org/sites/default/files/files/FSMA%20Convert%20lengths%20student.pdf
↑ http://physics.nist.gov/cuu/Units/prefixes.html
↑ https://www.nist.gov/pml/weights-and-measures/metric-si-prefixes
↑ https://sciencenotes.org/powers-ten-metric-prefixes/
↑ http://www.purplemath.com/modules/simpexpo.htm
Español:convertir unidades con prefijos
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Scaling - MATLAB & Simulink - MathWorks Deutschland
Fixed-point numbers can be encoded according to the scheme
\text{real-world value}=\left(\text{slope}×\text{integer}\right)+\text{bias}
\text{slope}=\text{slope adjustment factor }×{\text{ 2}}^{\text{fixed exponent}}
The integer is sometimes called the stored integer. This is the raw binary number, in which the binary point assumed to be at the far right of the word. In Fixed-Point Designer™ documentation, the negative of the fixed exponent is often referred to as the fraction length.
The slope and bias together represent the scaling of the fixed-point number. In a number with zero bias, only the slope affects the scaling. A fixed-point number that is only scaled by binary point position is equivalent to a number in [Slope Bias] representation that has a bias equal to zero and a slope adjustment factor equal to one. This is referred to as binary point-only scaling or power-of-two scaling:
\text{real-world value}={2}^{\text{fixed exponent}}×\text{integer}
\text{real-world value}={2}^{\text{-fraction length}}×\text{integer}
Fixed-Point Designer software supports both binary point-only scaling and [Slope Bias] scaling.
For examples of binary point-only scaling, see the Fixed-Point Designer Perform Binary-Point Scaling example.
For an example of how to compute slope and bias in MATLAB®, see Compute Slope and Bias
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Percolation test - LID SWM Planning and Design Guide
Revision as of 19:47, 25 August 2021 by Sahlla Abbasi (talk | contribs) (→Making measurements)
Schematic of percolation test setup
2.2.1 Paper basket to protect hole
A percolation test is one way of determining a design infiltration rate. If a site disturbance must be minimized, this test can be performed in a small augered well. Adding a permeameter may speed up analysis.
Percolation test results cannot be translated into field saturated hydraulic conductivity (Kfs) by simple manuipulation of the units. However, an acceptable approximation can be made by accounting for the soil texture and by making measurements under non-saturated conditions[1]. The required calculations have been written into an easy to use spreadsheet, formatted for printing:
Download Percolation test spreadsheet(.xlsx)
For reference, the relationship between measured percolation test (PT) and field saturated hydraulic conductivity (Kfs) used in the spreadsheet is:
{\displaystyle K_{fs}={\frac {\bar {C_{i}a^{2}}}{PT_{i}\left[2{\bar {H_{i}^{2}}}+{\bar {C_{i}}}a^{2}+\left({\frac {2{\bar {H_{i}}}}{a^{*}}}\right)\right]}}}
H = Mean water depth during a test
a = the radius of the test pit
α* = the representative soil sorptive number, and
C = is a shape function.
Additional details may be found in the original paper[1].
Measurement procedure[edit]
Adapted from[2].
Shovel or other excavation apparatus
Study paper bags (or similar craft paper)
Clean, fine gravel e.g. pea gravel or HPB, not 'screenings' or 'crusher run'.
Water supply (on site?)
Record sheets for field measurements
Perc test holes should be made at points and elevations selected as typical in the area of the proposed infiltration facility.
Typically, test holes are dug at each end of the area of the infiltration facility and near the centerline. Testing of the receiving area may also be necessary. Further holes could be needed, depending upon the nature of the soil, the results of the first tests and the size of the infiltration facility footprint.
Initial excavations should be made to the proposed depth of the infiltration facility (or as instructed by the designer). It is easiest to dig a larger hole part way down, then dig a 18 - 20 cm deep accurately sized test hole in the base of the larger hole.
To make the percolation test more accurate, any smeared soil should be removed from the walls of the test holes. This is best achieved by digging the hole approximately 5 cm undersized and then enlarging the hole to the accurate size as follows: using a rigid knife, insert the blade into the top side of the hole opposite you approximately 2.5 cm deep, holding the blade with its cutting edge vertical. Pull the blade away to break out a chunk of soil, repeat every ~ 2.5 cm around the hole, then repeat for another “ring” below until reaching the base. The result will be a hole with a ragged inner surface which looks like a freshly broken clod of soil.
The base of the hole should be cleaned of debris and be approximately flat, use a metal scoop or similar. It should also be picked to present a natural surface. Note that a picking action (use a pointed tool) is needed, not a scratching action (which just produces smears that are indented).
Place 5 cm of clean fine gravel in the bottom of the hole. If the sidewalls are likely to collapse, use a paper basket to support the sidewalls.
Place a piece of white plastic or similar provided with clear marks at 12.5 and 15 cm (5" and 6") from the bottom of the test hole prior to adding the gravel. For greater accuracy a float and pointer arrangement can be set up.
If the soil contains considerable amounts of silt or clay, and certainly for any soil with “clay” as part of the texture description, the test holes should be pre-soaked before proceeding with the test. Pre-soaking is accomplished by keeping the hole filled with water for 4 hours or more. The water should be added carefully and slowly to avoid disturbing the soil (including the sidewall soils). The test should be carried out immediately after pre-soaking;
Making measurements[edit]
To undertake the test, fill the test hole (the accurately sized test hole) with water. The water should be added carefully and slowly to avoid disturbing the soil (including the sidewall soils). When the water level falls anywhere below the 12.5 cm mark, refill the hole to the top. No recording of time needs be done for these 2 fillings.
After the second filling, when the water level falls below 12.5 cm, add enough water to bring the depth of water to 15 cm or slightly more. Note that these measurements are from the base of the soil bottom (using the installed marker), not the gravel layer.
Observe the water level until it drops to the 15 cm depth, at precisely 15 cm, commence timing, when the water level reaches the 12.5 cm depth, stop timing, record the time in minutes.
Repeat the procedure until the last 2 rates of fall do not vary more than 2 minutes or by more than 10% (whichever is less).
Backfill the holes with the excavated soil and flag and label their locations so you can pick them up for the plan.
If a test hole is discarded due to flow in a root channel or similar, record the information and make a replacement test. If there is a large variation (greater than or equal to 50%) between tests in the same soil layer, increase the number of tests.
Paper basket to protect hole[edit]
If sidewalls of the hole are likely to collapse, one option is to make a paper basket to protect and support the sidewalls as follows:
Cut the bottom out of a large paper bag (grocery bag) and cut the bag open along a side.
Lay bag on a soft surface. Punch holes in the bag about 5 to 7.5 cm apart using a pencil or similar.
Roll into a tube, with the short dimension being the axis of the tube, and place in the test hole.
Open the tube until the paper is in contact with the sidewalls of the test hole, then roll the top of the tube over to stiffen it.
After placing the tube in the hole, place the plastic marker and add the base gravel layer.
↑ 1.0 1.1 Reynolds, W.D., Galloway, K., and Radcliffe, D.E. (2015). "The relationship between perc time and field-saturated hydraulic conductivity for cylindrical test holes.", National Onsite Wastewater Recycling Association (NOWRA) 2015 Onsite Wastewater Mega-Conference, Virginia Beach, VA, USA, November 3-6, 2015
↑ Registered Onsite Wastewater Professional Program BC, PERCOLATION TEST PROCEDURE & FORM (2016) https://owrp.asttbc.org/wp-content/uploads/2016/11/Percolation-Test-Procedure-and-Form.pdf
Retrieved from "https://wiki.sustainabletechnologies.ca/index.php?title=Percolation_test&oldid=11894"
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XRF Analysis of Impurity Metal Elements in Iron Base Alloy ()
Bo Zhang, Jie Shi, Yunrui Jiang, Jinge Zhou, Shaoqin Li
School of Nuclear Technology & Automation Engineering, Chengdu University of Technology, Chengdu, China.
The content of Si, Mn, V, Cu and other impurity elements in tool steel samples was analyzed by wavelength dispersive X-ray fluorescence spectrometry (XRF). Using Ag target X-ray tube and the SDD detector (FWHM < 135 eV@5.9 keV). The best working conditions of the instrument are tested before measurement, including tube pressure, flow, filter used and measuring time. The repeatability of the instrument is verified to meet the standard. Finally, the detection limit of the instrument is tested. In the analysis and comparison, linear regression method is used to eliminate the matrix effect among the elements in the sample, and it is found that the multiple linear regression method has a good correction effect on the matrix effect. The results show that the average relative errors of Si, Mn, V and Cu are 3.24%, 3.05%, 0.29% and 0.59% respectively by using the optimal linear regression analysis. The method improves the control requirements of impurity elements in iron base alloy.
Iron Base Alloy, XRF, Matrix Effect, Linear Regression
Zhang, B., Shi, J., Jiang, Y.R., Zhou, J.G. and Li, S.Q. (2021) XRF Analysis of Impurity Metal Elements in Iron Base Alloy. Open Access Library Journal, 8, 1-5. doi: 10.4236/oalib.1107846.
With the rapid development of China’s iron and steel industry and national economy, the production capacity of the iron and steel industry is saturated, and the demand for high-performance steel is increasing due to its wide range of uses [1] . Adding specific elements to iron-based alloys will make them have certain characteristics, such as adding CR to iron-based alloys to increase their corrosion resistance; the addition of Mn increases its firmness and strength. Due to the continuous innovation of iron making technology, the requirements for the proportion of elements in iron-based alloys are more strict [2] [3] . It is necessary to study and control the proportion of trace elements in iron-based alloys to improve the properties of iron-based alloys [4] .
The analysis of iron-based alloys is mainly to determine the contents of various major and trace elements in various iron-based alloys, so as to control the proportion of elements in iron-based alloys. The contents of Impurity Elements Si, Mn, V and Cu in iron-based alloys were analyzed by X-ray fluorescence analysis.
According to Moseley’s law, the X-ray excites the target element in the sample to be tested to produce the characteristic X-ray of the target element, and its intensity is directly proportional to the content (Wi) of the target element in the sample to be tested. The calculation formula is as follows:
{W}_{i}=A+B\times {I}_{i}
where, Ii is the characteristic X-ray intensity of the target element in the aluminum alloy; A and B are the undetermined coefficient.
3. Selection of Experimental Instruments and Optimum Working Conditions
3.1. Preparation of Experimental Instruments
The energy dispersive X-ray fluorescence analyzer has high detection efficiency, compact spectrometer, convenient installation and use process, and is suitable for on-site or on-line analysis [5] . Malvern Panalytical is the manufacturer of energy dispersive X-ray fluorescence analyzer selected in this paper, and the instrument model is Epsilon 1. The working voltage range of the instrument is 10 kV - 50 kV and the working current range is 1 μA - 500 μA. It can be measured with or without Cu-50, Cu-300, Al-50, Al-200, Ag and Ti filters.
3.2. Repeatability Inspection of Experimental Instruments
Set the instrument parameters with voltage of 15 kV and current of 15 μA. The same stainless steel sample BX1 was measured 10 times without filter, and the repeatability of the instrument was measured and calculated. According to JJF1047-2011 calibration specification for energy dispersive X-ray fluorescence analyzer, the repeatability of the instrument shall be less than or equal to 3%, that is, the relative standard deviation (RSD) shall be less than or equal to 3%.
When measuring element Si, RSD is 2.364%; when measuring element V, RSD is 0.065%; when measuring element Mn, RSD is 0.00005%; when measuring element Cu, RSD is 0.002%. The RSD calculated in this paper meets the standard, and the repeatability of the instrument is good.
3.3. X-Light Tube Best Working Parameter Selection
The spectrum used to excite the sample is a continuous spectrum in the X-ray primary spectrum, while the working voltage of the X-ray tube and the change in operating current will affect the strength of the continuous spectrum [6] . Therefore, it is necessary to determine the optimal operating voltage and current before starting the sample. The instrument used herein is divided into four channels for measurement, namely Na-Si, K-V, Cr-Co, Ni-Mo, and four measurement channels can employ different operating conditions, and the four elements Si, V, Mn, and Cu analyzed herein are also among these four measurement channels.
The working voltages of the four measuring channels Na-Si, K-V, Cr-Co and Ni-Mo are 25 kV, 38 kV, 32 kV and 22 kV respectively, and the working currents are respectively 15 μA, 14 μA, 45 μA and 145 μA respectively. The filters are Ti, Al-50, Al-200 and Al-200 respectively, and the measurement time of each channel is 600 s.
4.1. Analysis of Impurity Elements
Select tool steel GSB03-1366 for measurement. The data analysis results of Elements Si, V, Mn and Cu are shown in the table below:
Table 1 shows that binary linear regression also has a good effect on the analysis of Si element in tool steel, in which the maximum absolute error is −0.022%, the average absolute error is 0.008%, the maximum relative error is −7.94%, and the average relative error is 3.24%.
Table 2 shows that the effect of ternary linear regression analysis on the analysis of V content in tool steel is the best, in which the maximum absolute error is −0.011%, the average absolute error is 0.005%, the maximum relative error is −4.66%, and the average relative error is 3.05%.
Table 3 shows that when considering Mn and Cu elements, the influence of other elements can be better eliminated by using quaternary linear regression analysis, with minimal error, and has a good effect in the analysis of tool steel.
Table 1. Analysis results of Si in tool steel samples.
Table 2. Analysis results of V in tool steel samples.
Table 3. Analysis results of Mn and Cu in tool steel samples.
The maximum absolute errors of Mn and Cu quaternion linear regression were −0.002% and 0.003%, the average absolute errors were 0.002% and 0.001%, the maximum relative errors were −0.50% and −1.82%, and the average relative errors were 0.29% and 0.59% respectively.
From the above analysis of Tables 1-3, it can be seen that multiple linear regression analysis has a good effect in analyzing samples with many kinds of elements, and can well eliminate the influence of other elements. Especially for iron-based alloys, the various elements contained in them are basically adjacent in the periodic table. When analyzing the content of one element, the matrix effect caused by multiple elements should be eliminated.
4.2. Calculation of Detection Limit
The detection limit is the minimum value that a certain element in the sample can be accurately detected. The detection limit includes method detection limit and instrument detection limit. The detection limit of the method reflects the sensitivity of the analytical method. The content corresponding to the standard deviation of 3 times the blank value is taken as the minimum detection limit.
Table 4. Experimental results of detection limits of each element.
Select the lowest group of elements in all samples to calculate the detection limit, and the results are shown in Table 4.
Table 4 shows that the detection limit of the instrument meets the analysis needs of Impurity Elements Si, V, Mn and Cu.
In this paper, after the best working parameters of the instrument are determined through experiments, the tool steel samples are measured. When processing the data, the best multiple linear regression analysis is used to deal with the matrix effect of each element. The measurement results meet the expectations, and it is concluded that the multiple linear regression method has a good correction effect on the matrix effect.
I would like to thank Jiang Yunrui and Li Shaoqin for their help in the process of experimental data processing, and Shijie and Zhou Jinge for improving my experimental methods.
[1] Chen, X.Q. (2010) Study on XRF Method and Technology for Analyzing the Content of Main Elements in Copper Alloys. Chengdu University of Technology, Chengdu.
[2] Ji, A., Zhuo, S.J. and Tao, G.Y. (2001) Application of Energy Dispersive X-Ray Fluorescence Spectroscopy in Iron and Steel Industry. Iron and Steel, 10, 64-68.
[3] Tian, W.H., Wang, Z.Q. and Zhang, M. (2008) Determination of Molybdenum, Lead, Iron and Copper in Molybdenum Ore by Energy Dispersive X-Ray Fluorescence Spectrometry. Rock Ore Test, 3, 236.
[4] Zhu, L.P., Li, X.X. and Sun, H.R. (2020) Discussion and Application of Detection Methods of Al, Cu, Mn, Si and P in Ferrotitanium. Scientific and Technological Innovation, 31, 22-23.
[5] Liang, X.Y. (2011) Quantitative Analysis of Standard Free X-Ray Fluorescence Spectrometer. Hangzhou University of Electronic Science and Technology, Hangzhou.
[6] Li, D. (2008) Study on the Application of EDI-Ⅲ X-Ray Tube in Energy Dispersive X-Ray Fluorescence Analysis. Chengdu University of Technology, Chengdu.
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Partial Quantum Tensors of Input and Output Connections
Partial Quantum Tensors of Input and Output Connections
Andrew Dente
I show how many connections of
\Gamma
are presently existing from
ℝ
\beta
as they are being inputted simultaneously through tensor products. I plan to address the Quantum state of this tensor connection step by step throughout the application presented. Also, I will show you how to prove that the connection is true for this tensor connection through its output method using a small bit of tensor calculus and mostly number theory.
Tensors Products, Tensor Connections, Number Theory, Tensor Calculus, Input, Output, Quantum States
Partial Quantum Tensors in summary, are network connections within the Quantum networks. These connections can be well understood using tensor calculus and even number theory. Tensor Calculus for one example, can be used to analyze the flow of electrons and to verify the movement within the electrons directions. Well, the reason why this application is called, “Partial Quantum Tensors” is because we only need to use partial methods within tensor calculus to analyze and verify the flow of Quantum input and output connections. The reason being is that Tensor Calculus can only verify the flow of particles or electrons that are perceptible through Euclidean space as this was first thought of by Neugebauer (1969) [1] . But in Quantum Networking, particles or electrons could be in two places at once. So how does that work? Well with Quantum connections we can’t just use only tensor calculus to prove my application; we will have to use a reliable mathematical method that works well with Quantum mechanics, which will be number theory in this case. Number Theory combined with Tensor Calculus should give off effective results with this application because the many methods of Number Theory are extremely useful in relation to Quantum Mechanics such as the Riemann Zeta functions expressed within the Quantum Circuits. I will first start off this application by introducing several definitions to make this application come alive in the Quantum Networks.
2. The Formulation of Quantum Networks
1.0 Definition: The tensor connection
{\Gamma }^{2\left(ℝ\right)}\beta
holds a double input connection with
{\Gamma }^{2}
clearly in conjunction with of course
ℝ
\beta
1.0 Theorem: Now, we can only locate the connections through tensor processing of all individual connections
\cup
. Only then will we be able to send input signals
\Sigma
ℝ
\beta
1.1 Definition: Let
\otimes
equals stable input connection and
\oplus
equals unstable but working input connection. These are the main mechanisms for the tensor input connections.
1.2 Definition: Out of an arbitrary instance,
\gamma
will indicate the best signal
\Sigma
1.0 Proposition: Assume that
\epsilon
\xi
are the Quantum Network areas that allow for a constant simultaneous connection for the tensor input methods.
1.3 Definition: We have to now assume that once our connection is in working order, we have to verify
\cup
’s simultaneous tensor connections of
ℝ
\beta
\Sigma
’s latency.
†
\text{‡}
\nabla
\Delta
if input connection results are precise.
1.5 Definition: Let n substitute for
\Gamma
if first connection is effective.
3. Verifying the Flow of the Quantum Connections
{\Gamma }^{2\left(ℝ\right)}\beta =\left\{\underset{\Gamma =1}{\overset{\infty }{\cup }}\frac{{\Sigma }^{\left(ℝ\otimes \beta \right)}\cdot {\Sigma }^{2}}{\gamma \left(\Sigma -\Sigma \right)}\right\}+\underset{n=1}{\overset{\infty }{\cup }}\underset{n}{\underset{︸}{{n}^{\epsilon }+2\left(\nabla \right)+\Sigma }}=\frac{2{n}^{\xi }}{ℝ+{\left(\beta \right)}_{\gamma }†}
As we can see from above in Expression (1), we used the tensor methods to find the stable connection between
ℝ
\beta
. To make things more coherent, we used
\underset{}{\underset{︸}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\text{\hspace{0.17em}}}}
to carry out our mixed functional input connection and used
\nabla
to balance both
ℝ
\beta
into the stable first wave of connections as presented above.
4. Pinpointing the Network Groups
We will be using tensor group theory to verify the result of the first wave of connections
\frac{2{n}^{\xi }}{ℝ+{\left(\beta \right)}_{\gamma }†}
ℝ
{\Gamma }^{2\left(ℝ\right)}\beta
. Lo-keng Hua, Luogeng Hua, Yuan Wang, Springer-Verlag, (1981), made great contributions to understanding this mathematical method by using Number Theory to pinpoint the operator groups of numerical analysis’s which we use today in Quantum Mechanics and such as this application [2] . M. Hassani (2004) helped us draw a path in understanding the connection made on network graphs by unique ways of using Number theory and Integer sequences combined [3] . Bombieri, Enrico (1990), helped form min/max of bounded finite and infinite sets for Mordell revised conjecture which created new mathematics for understanding the limits and bounds of Quantum group sets and operators [4] .
{\Gamma }^{2\left(ℝ\right)}\beta =\underset{n}{\sum }\frac{{n}^{2}}{\beta }+\underset{n}{\sum }\left\{\underset{n}{\sum }\text{ }\Gamma \frac{2\nabla +\Sigma }{\frac{{\xi }^{2}\cdot {\left(\gamma \otimes ℝ\right)}^{2}}{n-1}}+\underset{\Gamma }{\sum }\frac{ℝ-\xi }{{\gamma }^{2}+\Sigma }\right\}=\frac{\Sigma \cdot {\Sigma }^{\left(\beta \oplus ℝ\right)}}{\gamma +\Sigma }
\frac{\Sigma \cdot {\Sigma }^{\left(\beta \oplus ℝ\right)}}{\gamma +\Sigma }+{\Gamma }^{2\left(ℝ\right)}\beta
5. Acquiring Network Efficiency
1.1 Theorem: As seen in Expression (2) and (3), first wave of connections for the tensor product inputs yielded effective but not the best of tensor connections
\simeq
due to there not being
†
\nabla
. First wave connection results:
\frac{\Sigma \cdot {\Sigma }^{\left(\beta \oplus ℝ\right)}}{\gamma +\Sigma }=ℝ
will have to overlap with the second wave connection results
\beta
to gain a more precise input connection to verify that
ℝ
{\Gamma }^{2\left(ℝ\right)}
will indeed benefit from the convergence of the second wave input connections
\beta
. Halmos, Paul (1974) first sought out an overview of this with his findings in vector calculus based off of finite dimensions [5] . We can see this working out precisely in Expression (4).
\frac{\Sigma \cdot {\Sigma }^{\left(\beta \oplus ℝ\right)}}{\gamma +\Sigma }+{\Gamma }^{2\left(ℝ\right)}\beta \simeq \underset{n=1}{\overset{\infty }{\cup }}\underset{n}{\underset{︸}{{n}^{\epsilon }+2\left(\nabla \right)+\Sigma }}
6. Input Connections Start to Conjoin Simultaneously
1.2 Theorem: As the transition is taking place in Expression (4) and (5),
\cup
will be redundant because all futile input connections have just been eradicated from the input source of the Quantum Network areas and will only be left with solving for the exact product of
\beta
which in return will help with solving the exact connection of
ℝ
{\Gamma }^{2\left(ℝ\right)}
. Faltings, Gerd (1994), found that in algebraic geometry, algebra and geometry gave input to simultaneous expressions to explain elliptic systems that conjoined together [6] . Aguiar, M., Mahajan, S. (2010) used Monoidal Functors to understand The Schubert Statistic which in return helped us understand the inverse function of such connections that were made in his mathematical experiment and this also helped us form our input connection for the Quantum Networks [7] .
{\Gamma }^{2\left(ℝ\right)}\beta ={\left(\nabla \right)}^{2}\underset{n=1}{\overset{\infty }{\prod }}\left\{\beta \cdot \gamma \frac{n-\Sigma }{ℝ+\xi }\right\}+2\Gamma \left(\nabla \right){\left({\gamma }_{{n}^{2}}-{ℝ}^{2}\right)}^{\text{‡}}+\beta
Due to there being
\text{‡}
ℝ
\beta
in Expression (5), means we have found a precise input connection for both first wave connections and second wave connections. Also,
\gamma
\Sigma
in our tensor input product verifies that both of these connections are indeed truly accurate.
{\Gamma }^{2\left(ℝ\right)}\beta =\underset{n=1}{\overset{\infty }{\prod }}\left\{\frac{\Sigma \left(n\right)}{{ℝ}^{2}}\right\}+\Gamma \left(\beta \right)
\Gamma +\left[\beta \right]\left(\frac{\Sigma \left(n\right)}{{ℝ}^{2}}\right)\equiv {\Gamma }^{2\left(ℝ\right)}\beta
7. Both Connections Verified
1.0 Lemma: Expression (6) is unraveling to us how
ℝ
benefited from
\beta
and while Expression (7) is showing us how the effectiveness of the connections n has remained constant throughout both first and second wave connections. Also, it’s showing us that the first and second wave connection of
ℝ
\beta
do indeed have a precise verifiable simultaneous connection with the double input tensor connection
{\Gamma }^{2\left(ℝ\right)}\beta
that we have evaluated within the Quantum networks.
8. Proving That the Verified Networks Are True
To prove that the input connection is correct even further, we will trace it from the source of its output connection through:
\Gamma +\left[\beta \right]\left(\frac{\Sigma \left(n\right)}{{ℝ}^{2}}\right)
with respect to n and
\Gamma
\Sigma
for the output tensor method of integration for
{\Gamma }^{2\left(ℝ\right)}\beta
. These results based on Quantum Mechanics can be scrutinized thoroughly in a similar manner using, (2007) Grillet Pierre’s methods of abstract algebra where he helps us understand his advanced workings of group theory which then shaped the way we apply tensor calculus and number theory [8] .
n={\int }_{\Sigma }\text{ }\Gamma \left(\beta \right)2ℝ\left\{{n}^{2}\right\}\Delta \frac{n}{{ℝ}^{2}}+{\int }_{\Sigma }\left\{\frac{ℝ\Delta }{{\gamma }^{2}+\beta }+\underset{n=1}{\overset{n}{\sum }}\frac{2\gamma }{{\xi }^{2}+{\epsilon }^{\left(\beta \cdot ℝ\right)}}\right\}
=\left\{{\int }_{\Sigma }\underset{n=1}{\overset{n}{\sum }}\text{ }\Delta \left(2{\xi }_{\epsilon }-{n}^{2+\left(\beta \right)ℝ}\right)\left[\frac{2-{\gamma }^{2}}{ℝ\Delta }\right]\right\}\text{d}\Gamma
1.1 Lemma: So far, the imbalance and balance between
\frac{n}{{ℝ}^{2}}
of Expression (8) has yielded a semi stable number of connections for n with respect to
\Gamma
\Sigma
\beta
second wave of output connections seem to be providing the most fluent output signals
\gamma
with respect to the integration of
\Sigma
ℝ
is showing signs of balance
\Delta
for its first wave of output connections but it does not have the best response signal for its output connection
\gamma
as precisely shown in Expression (9). However, it’s still showing signs as an effective output source in relation to its input connection described in Expression (7) due to n being constant overall with
ℝ
and which can be seen here unraveling in Expression (10).
n=\frac{{n}^{2}}{ℝ}{\int }_{\Sigma }\Delta \frac{{\epsilon }_{2}+{\gamma }_{\xi }+\beta }{\left(n-2\Delta \right)}\text{d}\Gamma
9. Verifying the Output Source of the Input Connections
1.3 Theorem: We can clearly see that n is conjoining with
\beta
to increase the output sources speed and fluency of
ℝ
to indeed prove that:
\Gamma +\left[\beta \right]\left(\frac{\Sigma \left(n\right)}{{ℝ}^{2}}\right)
input source measures the effectiveness of it’s stable output source:
\frac{{n}^{2}}{ℝ}
and to prove its original network connection
{\Gamma }^{2\left(ℝ\right)}\beta
. This can be seen in Expression (10) and Expression (11).
\frac{{n}^{2}}{ℝ}+\left\{\frac{{\int }_{\Sigma }\Gamma \cdot {\left({ℝ}^{\epsilon }-2{\xi }_{\left(\beta +\gamma \right)}\right)}^{†}}{n{\Delta }^{2}}\right\}\simeq \frac{ℝ}{n}\left({\Gamma }_{\left(2-\gamma \right)}\cdot {\Delta }^{2}\right)+\beta
Now that we have the integration of our output connections reaching fluent and efficient stability for
ℝ
, we can see that our first output source has a precise reading
\text{‡}
with the help of the accurate output readings from β’s second output source. Koshlyakov (1964), made this methodology possible by giving a clear and thorough understanding of second order partial differential equations and now we can formulate a better understanding of how the Quantum states work [9] .
n\approx {\int }_{\Sigma }\frac{{ℝ}^{2}}{n}+\left({\Gamma }^{2}-\Delta \right)\beta \text{d}\Gamma
10. Stable Output Source of the Input Connections Has Been Achieved
We can now clearly see that after much stability balancing between both output connections
\Delta
with much respect to n,
\Gamma
has found the best stable output connection readings for
ℝ
and with β’s output source connections for:
\Gamma +\left[\beta \right]\left(\frac{\Sigma \left(n\right)}{{ℝ}^{2}}\right)
{\Gamma }^{2\left(ℝ\right)}\beta
as seen in Expression (12).
{\int }_{\Sigma }\frac{{ℝ}^{2}}{n}+\left({\Gamma }^{2}-\Delta \right)\beta \equiv \Gamma +\left[\beta \right]\left(\frac{\Sigma \left(n\right)}{{ℝ}^{2}}\right)\equiv {\Gamma }^{2\left(ℝ\right)}\beta
1.4 Theorem: Since n has shown the effectiveness of both
ℝ
\beta
, we can clearly see the accuracy between our input and output sources that are feeding off of the connections simultaneously and with the fluency of both output readings that lead back to our main tensor connection
{\Gamma }^{2\left(ℝ\right)}\beta
without any latency as seen above in Expression (13). This is our final answer.
It was Gregorio Ricci who first invented tensor calculus and Pierre de Fermat who invented number theory. But it became very interesting when Quantum Mechanics grabbed a chair to sit next to his friends, “Tensor Calculus and Number Theory”. Now we came up with new mathematical expressions and notations. It’s the very same reason why I was inspired to do this paper and especially the mathematical Expressions of (1)-(13) in which I used some tensor product applications but mostly number theory to prove the connections of both input and output scenarios. I made sure my methods of explanations are as clear and concise as possible for the reader. I would also like to mention on a side note that I’m hoping this manuscript will be possibly used in applied mathematics and applied physics in the near future. To back up my hopefulness, the reason being is because you can yield results with my formula using applied mathematics. Not just for finding the input and output sources of a Quantum network. Here’s a great example of what I’m talking about. My equation formula:
{\Gamma }^{2\left(ℝ\right)}\beta
used as an applied mathematical formula would look like this:
{7}^{2\left(3\right)}4={7}^{\left(6\right)}4=\left(117649\right)4=470596
. This is our answer.
It can also be written as:
\left({7}^{2\left(3\right)}\right)\cdot 4=470596
to make the process quicker but still yielding the same answer as the first formula that I presented. I firmly believe my formula could be used for several other mathematical applications. I take pleasure in the creation of this paper. Thank you for reading.
Dente, A. (2018) Partial Quantum Tensors of Input and Output Connections. Advances in Pure Mathematics, 8, 764-769. https://doi.org/10.4236/apm.2018.88046
1. Neugebauer, O. (1969) The Exact Sciences in Antiquity. 36-40. http://store.doverpublications.com/0486223329.html
2. Hua, L.-K. and Wang, Y. (1981) Applications of Number Theory to Numerical Analysis. Springer, Berlin, Heidelberg, 30-220.
3. Hassani, M. (2003) Derangements and Applications. Journal of Integer Sequences (JIS), 6, 123-126.
4. Bombieri, E. (1990) The Mordell Conjecture Revisited. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 17, 615-640.
5. Halmos, P.R. (1974) Finite Dimensional Vector Spaces. Springer, Berlin, 24-25. https://doi.org/10.1007/978-1-4612-6387-6
6. Faltings, G. (1994) The General Case of S. Lang’s Conjecture. In: Cristante, V. and Messing, W., Eds., Perspectives in Mathematics, Academic Press, San Diego, CA, 9-10.
7. Aguiar, M. and Mahajan, S. (2010) Monoidal Functors, Species and Hopf Algebras. In: CRM Monograph Series, Vol. 29, American Mathematical Society, Providence, 33-35. https://doi.org/10.1090/crmm/029
8. Grillet, P.A. (2007) Abstract Algebra. Springer, Berlin, 43-104.
9. Koshlyakov, N.S., Smirnov, M.M. and Gliner, E.B. (1964) Differential Equations of Mathematical Physics. 3rd Edition, John Wiley and Sons, New York, 19-80.
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Deep Learning - ProductizeML
You will learn: why deep learning is considered on of the pioneer ML types and its fundamentals.
Artificial Neural Networks, or simply Neural Networks (NN) in the ML field, are a category of machine learning algorithms whose design has been inspired by the neurophysiological workings of the human brain.
These computational models have a unique ability to extract meaning from complex data by finding patterns and detect trends that for the human mind can be hard to achieve. They are widely used nowadays in many applications: image classification tasks, voice recognition, facial recognition, character recognition, stock market prediction, among many more.
Illustration of a neuron. Credit: David Baillot/ UC San Diego.
NNs use a mathematical model of a neuron as
f(\sum_i x_i w_i + b)
, in which the input nerve impulse (
x_i
) is multiplied by a learnable matrix of weights (
w_i
), that represent the synaptic strengths of neurons.
The second parameter that the model can learn is called the bias term (
b
), which is directly added to the element-wise multiplication of previous matrices.
Mathematical model of the biological neuron.
The mathematical model of the neuron will fire the output signal (
x_iw_i
) according to an activation function (
f
), which introduces a non-linearity to the equation. Considering the multiple impulses a neuron can receive as an input, the output of the mathematical model can be expressed as previously stated
f(\sum_i x_i w_i + b)
Neural Network contains millions of neurons organized into three kinds of layers:
Input layer: neurons in this layer do not have inputs themselves, they just hold the data that will be shown to the training.
Hidden layers: connected to the input layer, act as intermediary values computed as the weighted sum of the input node values and apply an activation function before passing on the next subsequential hidden layer.
Output layer: connected to the hidden layers, these are computed as the weighted sum of the hidden node values.
Types of NNs
Common types of Neural Networks algorithms include Autoencoders (AE), Boltzmann Machines (BM), Convolutional Neural Networks (CNN), Deep Residual Networks (DRN), Generative Adversarial Networks (GAN), Long/Short Term Memory (LSTM), Recurrent Neural Network (RNN), and Support Vector Machine (SVM), among many other that are less used.
Depending on the goal that your algorithm is expected to achieve, some of these will do better than others based on the data format and performance expectations.
Convolutional Neural Networks (also known as CNNs or ConvNets) maintain a strong relationship with Artificial Neural Networks: they are also inspired by the behavior of biological systems through artificial neurons with learnable weights and biases. The layered architecture that Neural Networks performs based on matrix multiplications enables its application for image classification tasks. For this reason, ConvNets architectures assume that the inputs are images that have to be transformed into an output holding the class score predicted.
CNNs work similarly to Neural Networks: each neuron receives an input, a dot product (Hadamard product or elementwise multiplication) between each input, and its associated weight is performed, followed by a non-linearity. The most common hierarchical distribution of ConvNets layers contains:
Input layer: containing the raw pixel values from input images.
Convolutional layers: the core block of ConvNets, computes a locally dot product (2D in
the case of images) between the weights and a certain tiny region of the input volume.
Non-linear layers: most of the times using a ReLU activation function which applies an elementwise activation by thresholding at zero.
Pooling layers: that apply a spatial downsampling along the output volume.
Fully Connected layers: that compute the class scores, where each neuron is fully paired with adjacent neurons.
A CNN structure is made up of repetitive patterns (which explains the expression deep learning) of Convolutional, ReLU, and Pooling layers (considered hidden layers) and finally the fully-connected layers. The resulting volume structure is called a feature map (in the case of images, it has a two-dimension volume). The learning process (also referred to network training) where weights are optimized is achieved through backpropagation, a technique to efficiently compute gradients for its weights with respect to the loss function.
LeNet-5 CNN architecture for handwritten digits recognition.
Before describing one of the most known neural networks training algorithm called backpropagation, we should stop ourselves and quickly review to a simpler and predecessor forward propagation algorithm.
If we depict the simplest of the Neural Network cases as follows:
The way that the input data
x
is "forward propagated" through the network layer by layer (in this case, just a single hidden layer
h
) till the end
o
follows the mathematical expression:
output_{prediction} = A(A(xw_h)w_o)
x
is the input to the network,
w_h
w_o
are the hidden layer and last layer (output) weights, and
A
represents an Activation Functions such as Rectified Linear Unit, Sigmoid, Hyperbolic tangent, etc.
The backpropagation algorithm has the objective of helping the neural network to learn the parameters by adjusting each weight in proportion to how much it reduces the overall error. Following this idea, the better we can adjust the weights, the smaller the error will be and the better the model predictions will become.
The same way forwardpropagation follows a long series of nested equations, the backpropagation algorithm can be seen as the "transposed" process by applying the Chain rule to find the derivatives of the loss function with respect to any variable in the forward-pass equation
f(x) = A(B(C(x)))
A, B
C
activation functions at different layers.
If we apply the Chain rule to
f(x)
, it will look like:
f'(x) = f'(A) \cdot A'(B) \cdot B'(C) \cdot C'(x)
In simple terms, after each forward pass through a network, backpropagation performs a backward pass while adjusting the model’s parameters (weights and biases) and an optimization algorithm called Gradient Descent is used to update the parameters of our model.
The gradient descent is used as an optimization algorithm in order to minimize (typically in ML) a loss function by trying to find the directions for which the function will decrease its values the most. These directions will be given by the slope of the loss function that guarantees that the model will perform more accurately, and they are computed as the gradients of the function.
In this topographic example, our goal would be to move from the top red areas (highest loss values) to the bottom yellow areas (lowest loss values).
The step size for which the weights of a model are updated regarding gradient descent is called the learning rate. Common types of gradient descent algorithms include Stochastic Gradient Descent.
Activation functions try to simulate the natural behavior of neuronal firing when neurons communicate with each other through electrical impulses. This time, though, the activation function is found inside the neural network and is responsible for by-passing the model's feature information from one layer to the following ones. In order to replicate the neuronal firing process, activation functions are mathematical "gates" that can be represented as linear or non-linear functions.
Activation function representation by missinglink.ai
The most common activation functions used in neural networks architectures are:
Sigmoid: logistic transformation with expression
\sigma(x)=\frac{1}{1+e^{-x}}
that reduces extrem values and outliers without removing them.
Sigmoid graphic by missinglink.ai
Softmax: returns the distribution of probabilities over mutually exclusive output classes and it can be seen as a generalization of logistic regression since it can be applied to continuous data as
\sigma(x)_i = \frac{e^{x_i}}{\sum_{j}^{ }e^{x_j}}
ReLU (Rectified Linear Unit): one of the current state-of-the-art activation function since it solves gradient vanishing issues. It is represented as
f(x)=max(0,x)
that applies a threshold at zero.
ReLU graphic by missinglink.ai
Leaky ReLU: adds a small positive slope in the negative area of the ReLU so it can help the backpropagation training process.
Leaky ReLU graphic by missinglink.ai
The loss function is used to measure how well the network's predicted scores agree with the ground truth labels in the input data. As we have previously mentioned when describing the gradient descent algorithm, in order to obtain the gradients, the loss function needs to be differentiable at all times. Most used loss functions are MSE (L2) and Cross-entropy.
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Synthesis of gain-scheduled H∞ controllers - MATLAB hinfgs - MathWorks América Latina
hinfgs
Synthesis of gain-scheduled H∞ controllers
[gopt,pdK,R,S] = hinfgs(pdP,r,gmin,tol,tolred)
Given an affine parameter-dependent plant
P\left\{\begin{array}{c}\stackrel{˙}{x}=A\left(p\right)x+{B}_{1}\left(p\right)w+{B}_{2}u\\ z={C}_{1}\left(p\right)x+{D}_{11}\left(p\right)w+{D}_{12}u\\ y={C}_{2}x+{D}_{21}w+{D}_{22}u\end{array}
where the time-varying parameter vector p(t) ranges in a box and is measured in real time, hinfgs seeks an affine parameter-dependent controller
K\left\{\begin{array}{c}\stackrel{˙}{\zeta }={A}_{K}\left(p\right)\zeta +{B}_{K}\left(p\right)y\\ u={C}_{K}\left(p\right)\zeta +{D}_{K}\left(P\right)y\end{array}
scheduled by the measurements of p(t) and such that
K stabilizes the closed-loop system
for all admissible parameter trajectories p(t)
K minimizes the closed-loop quadratic H∞ performance from w to z.
The description pdP of the parameter-dependent plant P is specified with psys and the vector r gives the number of controller inputs and outputs (set r=[p2,m2] if y ∊ Rp2 and u ∊ Rm2). Note that hinfgs also accepts the polytopic model of P returned, e.g., by aff2pol.
hinfgs returns the optimal closed-loop quadratic performance gopt and a polytopic description of the gain-scheduled controller pdK. To test if a closed-loop quadratic performance γ is achievable, set the third input gmin to γ. The arguments tol and tolred control the required relative accuracy on gopt and the threshold for order reduction. Finally, hinfgs also returns solutions R, S of the characteristic LMI system.
The gain-scheduled controller pdK is parametrized by p(t) and characterized by the values KΠj of
\left(\begin{array}{cc}{A}_{K}\left(p\right)& {B}_{K}\left(p\right)\\ {C}_{K}\left(p\right)& {D}_{K}\left(p\right)\end{array}\right)
at the corners ³j of the parameter box. The command
Kj = psinfo(pdK,'sys',j)
returns the j-th vertex controller KΠj while
pv = psinfo(pdP,'par')
vertx = polydec(pv)
Pj = vertx(:,j)
gives the corresponding corner ³j of the parameter box (pv is the parameter vector description).
The controller scheduling should be performed as follows. Given the measurements p(t) of the parameters at time t,
Express p(t) as a convex combination of the ³j:
p\left(t\right)={\alpha }_{1}{{}^{3}}_{1}+\dots +{\alpha }_{N}{{}^{3}}_{N},\text{ }{\alpha }_{j}\ge 0,\sum _{i=1}^{N}{\alpha }_{j}=1
This convex decomposition is computed by polydec.
Compute the controller state-space matrices at time t as the convex combination of the vertex controllers KΠj:
\left(\begin{array}{cc}{A}_{K}\left(t\right)& {B}_{K}\left(t\right)\\ {C}_{K}\left(t\right)& {D}_{K}\left(t\right)\end{array}\right)=\sum _{i=1}^{N}{\alpha }_{j}{K}_{\Pi }{}_{{}_{\iota }}.
Use AK(t), BK(t), CK(t), DK(t) to update the controller state-space equations.
Apkarian, P., P. Gahinet, and G. Becker, “Self-Scheduled H∞ Control of Linear Parameter-Varying Systems,” Automatica, 31 (1995), pp. 1251–1261.
Becker, G., Packard, P., “Robust Performance of Linear-Parametrically Varying Systems Using Parametrically-Dependent Linear Feedback,” Systems and Control Letters, 23 (1994), pp. 205–215.
Packard, A., “Gain Scheduling via Linear Fractional Transformations,” Syst. Contr. Letters, 22 (1994), pp. 79–92.
psys | polydec
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Section 47.15 (0A7A): Dualizing complexes—The Stacks project
Section 47.15: Dualizing complexes (cite)
47.15 Dualizing complexes
In this section we define dualizing complexes for Noetherian rings.
Definition 47.15.1. Let $A$ be a Noetherian ring. A dualizing complex is a complex of $A$-modules $\omega _ A^\bullet $ such that
$\omega _ A^\bullet $ has finite injective dimension,
$H^ i(\omega _ A^\bullet )$ is a finite $A$-module for all $i$, and
$A \to R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet )$ is a quasi-isomorphism.
This definition takes some time getting used to. It is perhaps a good idea to prove some of the following lemmas yourself without reading the proofs.
Lemma 47.15.2. Let $A$ be a Noetherian ring. Let $K, L \in D_{\textit{Coh}}(A)$ and assume $L$ has finite injective dimension. Then $R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)$ is in $D_{\textit{Coh}}(A)$.
Proof. Pick an integer $n$ and consider the distinguished triangle
\[ \tau _{\leq n}K \to K \to \tau _{\geq n + 1}K \to \tau _{\leq n}K[1] \]
see Derived Categories, Remark 13.12.4. Since $L$ has finite injective dimension we see that $R\mathop{\mathrm{Hom}}\nolimits _ A(\tau _{\geq n + 1}K, L)$ has vanishing cohomology in degrees $\geq c - n$ for some constant $c$. Hence, given $i$, we see that $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, L) \to \mathop{\mathrm{Ext}}\nolimits ^ i_ A(\tau _{\leq n}K, L)$ is an isomorphism for some $n \gg - i$. By Derived Categories of Schemes, Lemma 36.11.5 applied to $\tau _{\leq n}K$ and $L$ we see conclude that $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(K, L)$ is a finite $A$-module for all $i$. Hence $R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)$ is indeed an object of $D_{\textit{Coh}}(A)$. $\square$
Lemma 47.15.3. Let $A$ be a Noetherian ring. If $\omega _ A^\bullet $ is a dualizing complex, then the functor
\[ D : K \longmapsto R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ) \]
is an anti-equivalence $D_{\textit{Coh}}(A) \to D_{\textit{Coh}}(A)$ which exchanges $D^+_{\textit{Coh}}(A)$ and $D^-_{\textit{Coh}}(A)$ and induces an anti-equivalence $D^ b_{\textit{Coh}}(A) \to D^ b_{\textit{Coh}}(A)$. Moreover $D \circ D$ is isomorphic to the identity functor.
Proof. Let $K$ be an object of $D_{\textit{Coh}}(A)$. From Lemma 47.15.2 we see $R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet )$ is an object of $D_{\textit{Coh}}(A)$. By More on Algebra, Lemma 15.98.2 and the assumptions on the dualizing complex we obtain a canonical isomorphism
\[ K = R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet ) \otimes _ A^\mathbf {L} K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ A(R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ), \omega _ A^\bullet ) \]
Thus our functor has a quasi-inverse and the proof is complete. $\square$
Let $R$ be a ring. Recall that an object $L$ of $D(R)$ is invertible if it is an invertible object for the symmetric monoidal structure on $D(R)$ given by derived tensor product. In More on Algebra, Lemma 15.126.4 we have seen this means $L$ is perfect, $L = \bigoplus H^ n(L)[-n]$, this is a finite sum, each $H^ n(L)$ is finite projective, and there is an open covering $\mathop{\mathrm{Spec}}(R) = \bigcup D(f_ i)$ such that $L \otimes _ R R_{f_ i} \cong R_{f_ i}[-n_ i]$ for some integers $n_ i$.
Lemma 47.15.4. Let $A$ be a Noetherian ring. Let $F : D^ b_{\textit{Coh}}(A) \to D^ b_{\textit{Coh}}(A)$ be an $A$-linear equivalence of categories. Then $F(A)$ is an invertible object of $D(A)$.
Proof. Let $\mathfrak m \subset A$ be a maximal ideal with residue field $\kappa $. Consider the object $F(\kappa )$. Since $\kappa = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(\kappa , \kappa )$ we find that all cohomology groups of $F(\kappa )$ are annihilated by $\mathfrak m$. We also see that
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(\kappa , \kappa ) = \text{Ext}^ i_ A(F(\kappa ), F(\kappa )) = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(F(\kappa ), F(\kappa )[i]) \]
is zero for $i < 0$. Say $H^ a(F(\kappa )) \not= 0$ and $H^ b(F(\kappa )) \not= 0$ with $a$ minimal and $b$ maximal (so in particular $a \leq b$). Then there is a nonzero map
\[ F(\kappa ) \to H^ b(F(\kappa ))[-b] \to H^ a(F(\kappa ))[-b] \to F(\kappa )[a - b] \]
in $D(A)$ (nonzero because it induces a nonzero map on cohomology). This proves that $b = a$. We conclude that $F(\kappa ) = \kappa [-a]$.
Let $G$ be a quasi-inverse to our functor $F$. Arguing as above we find an integer $b$ such that $G(\kappa ) = \kappa [-b]$. On composing we find $a + b = 0$. Let $E$ be a finite $A$-module wich is annihilated by a power of $\mathfrak m$. Arguing by induction on the length of $E$ we find that $G(E) = E'[-b]$ for some finite $A$-module $E'$ annihilated by a power of $\mathfrak m$. Then $E[-a] = F(E')$. Next, we consider the groups
\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(A, E') = \text{Ext}^ i_ A(F(A), F(E')) = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(F(A), E[-a + i]) \]
The left hand side is nonzero if and only if $i = 0$ and then we get $E'$. Applying this with $E = E' = \kappa $ and using Nakayama's lemma this implies that $H^ j(F(A))_\mathfrak m$ is zero for $j > a$ and generated by $1$ element for $j = a$. On the other hand, if $H^ j(F(A))_\mathfrak m$ is not zero for some $j < a$, then there is a map $F(A) \to E[-a + i]$ for some $i < 0$ and some $E$ (More on Algebra, Lemma 15.65.7) which is a contradiction. Thus we see that $F(A)_\mathfrak m = M[-a]$ for some $A_\mathfrak m$-module $M$ generated by $1$ element. However, since
\[ A_\mathfrak m = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(A, A)_\mathfrak m = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(F(A), F(A))_\mathfrak m = \mathop{\mathrm{Hom}}\nolimits _{A_\mathfrak m}(M, M) \]
we see that $M \cong A_\mathfrak m$. We conclude that there exists an element $f \in A$, $f \not\in \mathfrak m$ such that $F(A)_ f$ is isomorphic to $A_ f[-a]$. This finishes the proof. $\square$
Lemma 47.15.5. Let $A$ be a Noetherian ring. If $\omega _ A^\bullet $ and $(\omega '_ A)^\bullet $ are dualizing complexes, then $(\omega '_ A)^\bullet $ is quasi-isomorphic to $\omega _ A^\bullet \otimes _ A^\mathbf {L} L$ for some invertible object $L$ of $D(A)$.
Proof. By Lemmas 47.15.3 and 47.15.4 the functor $K \mapsto R\mathop{\mathrm{Hom}}\nolimits _ A(R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ), (\omega _ A')^\bullet )$ maps $A$ to an invertible object $L$. In other words, there is an isomorphism
\[ L \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , (\omega _ A')^\bullet ) \]
Since $L$ has finite tor dimension, this means that we can apply More on Algebra, Lemma 15.98.2 to see that
\[ R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , (\omega '_ A)^\bullet ) \otimes _ A^\mathbf {L} K \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ A(R\mathop{\mathrm{Hom}}\nolimits _ A(K, \omega _ A^\bullet ), (\omega _ A')^\bullet ) \]
is an isomorphism for $K$ in $D^ b_{\textit{Coh}}(A)$. In particular, setting $K = \omega _ A^\bullet $ finishes the proof. $\square$
Lemma 47.15.6. Let $A$ be a Noetherian ring. Let $B = S^{-1}A$ be a localization. If $\omega _ A^\bullet $ is a dualizing complex, then $\omega _ A^\bullet \otimes _ A B$ is a dualizing complex for $B$.
Proof. Let $\omega _ A^\bullet \to I^\bullet $ be a quasi-isomorphism with $I^\bullet $ a bounded complex of injectives. Then $S^{-1}I^\bullet $ is a bounded complex of injective $B = S^{-1}A$-modules (Lemma 47.3.8) representing $\omega _ A^\bullet \otimes _ A B$. Thus $\omega _ A^\bullet \otimes _ A B$ has finite injective dimension. Since $H^ i(\omega _ A^\bullet \otimes _ A B) = H^ i(\omega _ A^\bullet ) \otimes _ A B$ by flatness of $A \to B$ we see that $\omega _ A^\bullet \otimes _ A B$ has finite cohomology modules. Finally, the map
\[ B \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet \otimes _ A B, \omega _ A^\bullet \otimes _ A B) \]
is a quasi-isomorphism as formation of internal hom commutes with flat base change in this case, see More on Algebra, Lemma 15.99.2. $\square$
Lemma 47.15.7. Let $A$ be a Noetherian ring. Let $f_1, \ldots , f_ n \in A$ generate the unit ideal. If $\omega _ A^\bullet $ is a complex of $A$-modules such that $(\omega _ A^\bullet )_{f_ i}$ is a dualizing complex for $A_{f_ i}$ for all $i$, then $\omega _ A^\bullet $ is a dualizing complex for $A$.
Proof. Consider the double complex
\[ \prod \nolimits _{i_0} (\omega _ A^\bullet )_{f_{i_0}} \to \prod \nolimits _{i_0 < i_1} (\omega _ A^\bullet )_{f_{i_0}f_{i_1}} \to \ldots \]
The associated total complex is quasi-isomorphic to $\omega _ A^\bullet $ for example by Descent, Remark 35.3.10 or by Derived Categories of Schemes, Lemma 36.9.4. By assumption the complexes $(\omega _ A^\bullet )_{f_ i}$ have finite injective dimension as complexes of $A_{f_ i}$-modules. This implies that each of the complexes $(\omega _ A^\bullet )_{f_{i_0} \ldots f_{i_ p}}$, $p > 0$ has finite injective dimension over $A_{f_{i_0} \ldots f_{i_ p}}$, see Lemma 47.3.8. This in turn implies that each of the complexes $(\omega _ A^\bullet )_{f_{i_0} \ldots f_{i_ p}}$, $p > 0$ has finite injective dimension over $A$, see Lemma 47.3.2. Hence $\omega _ A^\bullet $ has finite injective dimension as a complex of $A$-modules (as it can be represented by a complex endowed with a finite filtration whose graded parts have finite injective dimension). Since $H^ n(\omega _ A^\bullet )_{f_ i}$ is a finite $A_{f_ i}$ module for each $i$ we see that $H^ i(\omega _ A^\bullet )$ is a finite $A$-module, see Algebra, Lemma 10.23.2. Finally, the (derived) base change of the map $A \to R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet )$ to $A_{f_ i}$ is the map $A_{f_ i} \to R\mathop{\mathrm{Hom}}\nolimits _ A((\omega _ A^\bullet )_{f_ i}, (\omega _ A^\bullet )_{f_ i})$ by More on Algebra, Lemma 15.99.2. Hence we deduce that $A \to R\mathop{\mathrm{Hom}}\nolimits _ A(\omega _ A^\bullet , \omega _ A^\bullet )$ is an isomorphism and the proof is complete. $\square$
Lemma 47.15.8. Let $A \to B$ be a finite ring map of Noetherian rings. Let $\omega _ A^\bullet $ be a dualizing complex. Then $R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet )$ is a dualizing complex for $B$.
Proof. Let $\omega _ A^\bullet \to I^\bullet $ be a quasi-isomorphism with $I^\bullet $ a bounded complex of injectives. Then $\mathop{\mathrm{Hom}}\nolimits _ A(B, I^\bullet )$ is a bounded complex of injective $B$-modules (Lemma 47.3.4) representing $R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet )$. Thus $R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet )$ has finite injective dimension. By Lemma 47.13.4 it is an object of $D_{\textit{Coh}}(B)$. Finally, we compute
\[ \mathop{\mathrm{Hom}}\nolimits _{D(B)}(R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet ), R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet )) = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet ), \omega _ A^\bullet ) = B \]
and for $n \not= 0$ we compute
\[ \mathop{\mathrm{Hom}}\nolimits _{D(B)}(R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet ), R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet )[n]) = \mathop{\mathrm{Hom}}\nolimits _{D(A)}(R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet ), \omega _ A^\bullet [n]) = 0 \]
which proves the last property of a dualizing complex. In the displayed equations, the first equality holds by Lemma 47.13.1 and the second equality holds by Lemma 47.15.3. $\square$
Lemma 47.15.9. Let $A \to B$ be a surjective homomorphism of Noetherian rings. Let $\omega _ A^\bullet $ be a dualizing complex. Then $R\mathop{\mathrm{Hom}}\nolimits (B, \omega _ A^\bullet )$ is a dualizing complex for $B$.
Lemma 47.15.10. Let $A$ be a Noetherian ring. If $\omega _ A^\bullet $ is a dualizing complex, then $\omega _ A^\bullet \otimes _ A A[x]$ is a dualizing complex for $A[x]$.
Proof. Set $B = A[x]$ and $\omega _ B^\bullet = \omega _ A^\bullet \otimes _ A B$. It follows from Lemma 47.3.10 and More on Algebra, Lemma 15.69.5 that $\omega _ B^\bullet $ has finite injective dimension. Since $H^ i(\omega _ B^\bullet ) = H^ i(\omega _ A^\bullet ) \otimes _ A B$ by flatness of $A \to B$ we see that $\omega _ A^\bullet \otimes _ A B$ has finite cohomology modules. Finally, the map
\[ B \longrightarrow R\mathop{\mathrm{Hom}}\nolimits _ B(\omega _ B^\bullet , \omega _ B^\bullet ) \]
Proposition 47.15.11. Let $A$ be a Noetherian ring which has a dualizing complex. Then any $A$-algebra essentially of finite type over $A$ has a dualizing complex.
Proof. This follows from a combination of Lemmas 47.15.6, 47.15.9, and 47.15.10. $\square$
Lemma 47.15.12. Let $A$ be a Noetherian ring. Let $\omega _ A^\bullet $ be a dualizing complex. Let $\mathfrak m \subset A$ be a maximal ideal and set $\kappa = A/\mathfrak m$. Then $R\mathop{\mathrm{Hom}}\nolimits _ A(\kappa , \omega _ A^\bullet ) \cong \kappa [n]$ for some $n \in \mathbf{Z}$.
Proof. This is true because $R\mathop{\mathrm{Hom}}\nolimits _ A(\kappa , \omega _ A^\bullet )$ is a dualizing complex over $\kappa $ (Lemma 47.15.9), because dualizing complexes over $\kappa $ are unique up to shifts (Lemma 47.15.5), and because $\kappa $ is a dualizing complex over $\kappa $. $\square$
In the proof of 0A7C, it should be "
\ge c - n
\ge n - c
p + q + n > n
n > p + q + c
In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A7A. Beware of the difference between the letter 'O' and the digit '0'.
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Gravitational energy - Simple English Wikipedia, the free encyclopedia
type of potential energy
(Redirected from Gravitational potential energy)
Gravitational energy is the potential energy held by an object because of its high position compared to a lower position. In other words, it is energy associated with gravity or gravitational force. For example, a pen being held above a table has a higher gravitational potential than a pen sitting on the table. Gravitational potential energy is mechanical energy minus kinetic energy. It has a scalar quantity measured in Joules (J).
An object gets gravitational potential energy when it moves uphill. The energy used up when moving up the hill is changed into gravitational potential energy. When the object comes back down again, the gravitational potential energy is turned back into kinetic energy (movement). That is why riding a bike up a steep hill is very hard, but when you are coming back down, you will not need to pedal at all - you are being powered by the gravitational potential energy that you stored when you rode up the hill. Another way gravitational potential energy helps us is by keeping earth and the other planets in orbit in our solar system around the Sun.
Formula and exampleEdit
The mathematical formula for the gravitational potential energy:
Gravitational potential energy =
{\displaystyle m\cdot g\cdot h}
m is the mass of the object ,
g is the gravitational acceleration of the object , and
h is the height above a chosen point.
Most scientists and students use these measurement unit:
The value m is in kilograms.
The value h is in metres.
The value g is a physical constant with the value of 9.81 metres per second squared. This is known as the gravitational constant.
The value of the gravitational potential energy that is calculated using values in the above units is called the Joule (J).
For example, an object with a mass of 1.5 kilograms that is 2.5 metres above the ground would have a gravitational potential energy value of:
{\displaystyle 1.5kg\cdot 9.81m/s^{2}\cdot 2.5m=38.8}
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Gravitational_energy&oldid=7414667"
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Purchase a Policy - Documentation
WebsiteChatBlogGithub
v1 (en-US)
Project & Repositories
Cover Reassurance
Contract Creators
Cover Fee Redemption
Purchasing Covers
USD Cover Token
NPM Pools
Recovering Cryptocurrencies
The protocol automatically determines the premium or policy fees by taking numerous aspects such as policy term/cover duration, adverse selection, sum insured, utilization ratio, total pool balance, reassurance pool, etc into account.
The policy fee is determined by the supply and demand in the cover market. A pool with a low utilization ratio charges much less than one with a high utilization ratio. A pool with a high utilization ratio costs higher fees, which makes it more enticing and lucrative for liquidity providers to contribute more cash, lowering the price gradually.
The policy fee denoted
P_F
x : x < C \cap x > F \\ x = \frac{C_A+c}{t+(I_P*I_R)} \\ P_F = \frac{C_D}{100} + x
where the value of x is always between constants
F
C
c
is the cover commitment amount,
t
is the total balance of the pool,
C_D
is the policy duration desired,
C_A
is the desired cover amount,
I_P
is incident support pool amount for the cover pool ,
I_R
is incident support pool capitalization ratio.
For simplicity, this equation can also be written as:
{P_F=(C_D/100) + (C_A+c)/(t+(I_P*I_R))}
Check the following page to calculate cover fees:
https://calculator.npm.claims
import { ChainId, policy } from '@neptunemutual/sdk'
import { info } from './info.js'
import { getProvider } from './provider.js'
import { weiAsPercent, weiAsDollars, ether } from './bn.js'
const { key } = info
const provider = getProvider()
amount: ether(100_000)
console.info('Getting %s cover for %d months', weiAsDollars(args.amount), args.duration)
console.info('--------------------------------------')
const response = await policy.getCoverFee(ChainId.Mumbai, key, args, provider)
utilizationRatio,
totalAvailableLiquidity,
coverRatio,
} = response.result
console.info('Rate: %s', weiAsPercent(rate))
console.info('Fee: %s', weiAsDollars(fee, 'xYZ'))
console.info('Utilization Ratio: %s', weiAsPercent(utilizationRatio))
console.info('Total Available Liquidity: %s', weiAsDollars(totalAvailableLiquidity))
console.info('Cover Ratio: %s', weiAsPercent(coverRatio))
[info] Getting $100,000.00 cover for 2 months
[info] Rate: 16.66%
[info] Fee: $2,776.78
[info] Utilization Ratio: 0.00%
[info] Total Available Liquidity: $299,700.00
[info] Cover Ratio: 66.73%
The amount required in DAI to purchase the policy for the specified amount.
utilizationRatio
Utilization Ratio = Total Commitment / Total Liquidity
totalAvailableLiquidity
Total Available Liquidity = Total Liquidity - Total Commitment + Amount in NPM Provision + Weighted Amount in Assurance Pool
coverRatio
Cover Ratio = Utilization Ratio + Cover Duration * (Amount to Cover / Total Liquidity)
The base policy floor rate
The base policy ceiling rate
The annualized percentage required to purchase this policy for the specified amount
import { ether } from './bn.js'
const purcahse = async () => {
amount: ether(10_000) // <-- Amount to Cover (In DAI)
// First approve the Policy contract to spend your DAI, USDC, or BUSD
await policy.approve(ChainId.Mumbai, {}, provider)
const response = await policy.purchaseCover(ChainId.Mumbai, key, args, provider)
purcahse()
gasPrice: BigNumber { _hex: '0x06fc23ac00', _isBigNumber: true },
gasLimit: BigNumber { _hex: '0x2f788c', _isBigNumber: true },
to: '0x50443BED0C90987D3A3715f0A90e60503BB548dD',
data: '0x02bbd6e570726f746f3a636f6e7472616374733a636f7665723a6366633a303100000002000000000000000000000000000000000000000000000000000000000000000200000000000000000000000000000000000000000000021e19e0c9bab2400000',
v: 160038,
r: '0xc6539f52dfb2fc026bdef71082dccdcfd886fa6cef06c1d507ba18d1bbe6ed33',
s: '0x2b2ca2256d0a9513bc503e5f8496f54def3b2fb2ef9e2a988a0f21167a3c7cbe',
from: '0x076F91C0A411197e6Fce476F37c6385CCeacd26D',
hash: '0x1e6d351385c19d13e7e08fb372d23100d8000ef916909b18fabeb40252f9eb47',
Neptune Mutual SDK - Previous
Next - Neptune Mutual SDK
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Home : Support : Online Help : Science and Engineering : Units : Known Units : area
Area has the dimension length squared. The SI composite unit of area is the meter squared.
Maple knows the units of area listed in the following table.
planck_area
planck_areas
An asterisk ( * ) indicates the default context, an at sign (@) indicates an abbreviation, and under the prefixes column, SI indicates that the unit takes all SI prefixes, IEC indicates that the unit takes IEC prefixes, and SI+ and SI- indicate that the unit takes only positive and negative SI prefixes, respectively. Refer to a unit in the Units package by indexing the name or symbol with the context, for example, are[SI] or b[SI]; or, if the context is indicated as the default, by using only the unit name or symbol, for example, are or b.
The units of area are defined as follows.
An are is defined as
100
square meters.
The hectare and centare are alternate spellings of the hectoare and centiare, respectively.
A barn is defined as
1.×{10}^{-28}
square meter. This allows physicists the phrase as big as a barn.
A shed is defined as
1.×{10}^{-52}
square meter. The origin of this unit is the same as that of the barn.
A circular mil is defined as the area of a circle with a diameter of
1
standard mil, that is,
7.853981635×{10}^{-7}
inch squared.
Each of the following units can be based on the standard or US survey mile.
A rood is defined as
\frac{1}{4}
acre.
An acre is defined as
10
square Gunter chains.
A lot is defined as
\frac{1}{10}
square mile.
A section is defined as
1
A township is defined as
36
A planck area is defined as a square planck length.
\mathrm{convert}\left({'m'}^{2},'\mathrm{dimensions}','\mathrm{base}'=\mathrm{true}\right)
{\textcolor[rgb]{0,0,1}{\mathrm{length}}}^{\textcolor[rgb]{0,0,1}{2}}
\mathrm{convert}\left(4.,'\mathrm{units}',{'\mathrm{kilometers}'}^{2},'\mathrm{sections}'\right)
\textcolor[rgb]{0,0,1}{1.544408634}
\mathrm{convert}\left(4.,'\mathrm{units}',{'\mathrm{kilometers}'}^{2},'\mathrm{sections}[\mathrm{US_survey}]'\right)
\textcolor[rgb]{0,0,1}{1.544402457}
\mathrm{convert}\left(37,'\mathrm{units}','\mathrm{barns}',{'\mathrm{nanometers}'}^{2}\right)
\frac{\textcolor[rgb]{0,0,1}{37}}{\textcolor[rgb]{0,0,1}{10000000000}}
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CFD Simulations AC 7 01 - KBwiki
CFD Simulations AC 7 01
If CFD simulations of the AC have been performed then provide a brief overview. This should cover the scope of the calculations and the main aspects of the modelling strategy e.g. equations solved, turbulence and other physical models employed). All computational domain simplifications/idealisations, and the treatment of subgrid features should also be identified (e.g. imposed symmetry plane, omission of detailed features, simplification of complex/small scale features, i.e. porous media, use of equivalent wall roughness). If important details of the geometry representation are uncertain then the impact of these uncertainties on the DOAPs should be discussed, including possible ways for managing their effect.
It is left to the discretion of each author to decide the most appropriate way for structuring and summarising the CFD results. Ideally, the data structures used should be consistent with those used for the test data.
A summary table for all CFD simulation results should be included, as shown below in Table CFD-A. Available data should be clearly identified, (e.g. UVW, k, concentration, etc). As with test data, a distinction should be drawn between detailed local data (e.g. p(x,y,z)) and data relating to DOAPs which are likely to be global/summary parameters (e.g. coefficient of lift, CL).
All available detailed data should be stored in separate electronic datafiles (according to guidelines set out by the Knowledge Base team at the University of Surrey). These should be summarised as shown below in CFD-B, with links to each of the datafiles.
CFD 1 (dense gas dispersion)
{\displaystyle 10^{5}-10^{6}}
{\displaystyle 0.2-10}
{\displaystyle 1-3}
{\displaystyle 1.22-3.00}
{\displaystyle C,\ U}
{\displaystyle {\ C/C_{0}}}
CFD 2 (passive gas releases)
{\displaystyle 10^{5}-10^{6}}
{\displaystyle C,\ U,\ V,\ W,\ k}
{\displaystyle {\ C/C_{0}}}
{\displaystyle U,\ V,\ W\ (ms^{-1})}
{\displaystyle k\ (m^{2}s^{-2})}
{\displaystyle C\ (kg/m^{3})}
CFD 1 ✔✔✔ cfd11.dat ✔ ✔ cfd13.dat cfd14.dat, cfd15.dat
CFD 2 ✔✔✔ cfd21.dat ✔ cfd22.dat ✔ cfd23.dat cfd24.dat, cfd25.dat,
cfd26.dat
Table CFD-B Summary description of all available datafiles and simulated parameters
Describe the geometry of the computational domain including the location of all boundaries. Does this domain coincide with the test geometry or does it represent a simplification? Discuss the impact of any simplification on the DOAPs. In particular justify any 2-D idealisation. Describe in detail the mesh (or meshes) used, including the total number of cells/grid points and mesh density distribution.
{\displaystyle x,\ y,\ z,\ U,\ V,\ W}
{\displaystyle x,\ y,\ z,\ C}
Retrieved from "http://kbwiki.ercoftac.org/w/index.php?title=CFD_Simulations_AC_7_01&oldid=19026"
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feature\_importance_{F} = \displaystyle\sum\limits_{trees, leafs_{F}} \left(v_{1} - avr \right)^{2} \cdot c_{1} + \left( v_{2} - avr \right)^{2} \cdot c_{2} { , }
avr = \displaystyle\frac{v_{1} \cdot c_{1} + v_{2} \cdot c_{2}}{c_{1} + c_{2}} { , where}
c_1, c_2
v_1, v_2
feature\_total\_importance_{j} = feature\_importance + \sum\limits_{i=1}^{N}average\_feature\_importance_{i} { , where}
feature\_importance_{j}
average\_feature\_importance_{i}
O(trees\_count \cdot depth \cdot 2 ^ {depth} \cdot dimension)
feature\_importance_{i} = \pm (metric E_{i}v) - metric(v))
feature\_importance_{i} = abs(metric(E_{i}v) - best\_value) - abs(metric(v) - best\_value)
E_{i}v
i
is on the path to a leaf, the new leaf value is set to the weighted average of values of leaves that have different paths by feature value. Weights represent the total weight of objects in the corresponding leaf. This weight is equal to the number of objects in each leaf if weights are not specified for the dataset.
F = (f_{1}, ..., f_{n})
E_fv = \displaystyle\left(\frac{(n - 1) v + E_{f}v}{n}\right)
v
subset\_size = min(documentCount, max(2e5, \frac{2e9}{featureCount}))
O(trees\_count \cdot (depth + sub\_samples\_count) \cdot 2 ^ {depth} +
+ Eval\_metric\_complexity(model, sub\_samples\_count) \cdot features\_count) { ,}
sub\_samples\_count = Min(samples\_count, Max(10^5, 10^9 / features\_count))
i
feature\_importance_{F} = \displaystyle\sum\limits_{trees, leafs_{F}} \left(v_{1} - avr \right)^{2} \cdot c_{1} + \left( v_{2} - avr \right)^{2} \cdot c_{2} { , }
avr = \displaystyle\frac{v_{1} \cdot c_{1} + v_{2} \cdot c_{2}}{c_{1} + c_{2}} { , where}
c_{1}, c_{2}
v_{1}, v_{2}
feature\_total\_importance_{j} = feature\_importance + \sum\limits_{i=1}^{N}average\_feature\_importance_{i} { , where}
feature\_importance_{j}
average\_feature\_importance_{i}
O(trees\_count \cdot depth \cdot 2 ^ {depth} \cdot dimension)
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Home : Support : Online Help : Connectivity : Web Features : XMLTools : GetAttribute
retrieve the value of an attribute from an XML element
GetAttribute(xmlTree, attrName)
The GetAttribute(xmlTree, attrName) command retrieves the value of the attribute attrName from the XML element xmlTree as a Maple equation. If the specified attribute is not present, then the value FAIL is used as the right-hand side of the returned equation.
Note: The presence of an attribute with a given name is tested by using HasAttribute.
\mathrm{with}\left(\mathrm{XMLTools}\right):
\mathrm{GetAttribute}\left(\mathrm{XMLElement}\left("a",["b"="c"],"some text"\right),"b"\right)
\textcolor[rgb]{0,0,1}{"b"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"c"}
\mathrm{GetAttribute}\left(\mathrm{XMLElement}\left("a",["b"="c"],"some text"\right),"d"\right)
\textcolor[rgb]{0,0,1}{"d"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{FAIL}}
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Finite 1-Regular Cayley Graphs of Valency 5
Jing Jian Li, Ben Gong Lou, Xiao Jun Zhang, "Finite 1-Regular Cayley Graphs of Valency 5", International Journal of Combinatorics, vol. 2013, Article ID 125916, 3 pages, 2013. https://doi.org/10.1155/2013/125916
Jing Jian Li,1,2 Ben Gong Lou,1 and Xiao Jun Zhang2,3
1School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650031, China
2School of Mathematics and Information Sciences, Guangxi University, Nanning 530004, China
Academic Editor: Cai Heng Li
Let and . We say is -regular Cayley graph if acts regularly on its arcs. is said to be core-free if is core-free in some . In this paper, we prove that if an -regular Cayley graph of valency is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free -regular Cayley graphs of valency .
We assume that all graphs in this paper are finite, simple, and undirected.
Let be a graph. Denote the vertex set, arc set, and full automorphism group of by , , and , respectively. A graph is called -vertex-transitive or -arc-transitive if acts transitively on or , where . is simply called vertex-transitive, arc-transitive for the case where . In particular, is called -regular if acts regularly on its arcs and then 1-regular when .
Let be a finite group with identity element . For a subset of with , the Cayley graph of (with respect to ) is defined as the graph with vertex set such that are adjacent if and only if . It is easy to see that a Cayley graph has valency , and it is connected if and only if .
Li proved in [1] that there are only finite number of core-free -transitive Cayley graphs of valency for and and that, with the exceptions and , every -transitive Cayley graph is a normal cover of a core-free one. It was proved in [2] that there are core-free -transitive cubic Cayley graphs up to isomorphism, and there are no core-free -regular cubic Cayley graphs. A natural problem arises. Characterize -transitive Cayley graphs, in particular, which graphs are -regular? Until now, the result about -regular graphs mainly focused constructing examples. For example, Frucht gave the first example of cubic -regular graph in [3]. After then, Conder and Praeger constructed two infinite families of cubic -regular graphs in [4]. Marušič [5] and Malnič et al. [6] constructed two infinite families of tetravalent -regular graphs. Classifying such graphs has aroused great interest. Motivated by above results and problem, we consider -regular Cayley graphs of valency 5 in this paper.
A graph can be viewed as a Cayley graph of a group if and only if contains a subgroup that is isomorphic to and acts regularly on the vertex set. For convenience, we denote this regular subgroup still by . If contains a normal subgroup that is regular and isomorphic to , then is called an X-normal Cayley graph of ; if is not normal in but has a subgroup which is normal in and semiregular on with exactly two orbits, then is called an X-bi-normal Cayley graph; furthermore if , is called normal or bi-normal. Some characterization of normal and bi-normal Cayley graphs has given in [1, 2].
For a Cayley graph , is said to be core-free (with respect to ) if is core-free in some ; that is, .
The main result of this paper is the following assertion.
Theorem 1. Let be an -regular Cayley graph of valency , where . Let be the number of nonisomorphic core-free -regular Cayley graph of valency with the regular subgroup equal to . Then either (i)is an -normal or -bi-normal Cayley graph or(ii) is a nontrivial normal cover of one line of Table 1.In particular, there are no core-free -regular Cayley graphs of valency .
Number 1-regular Remark
1 1 No Icosahedron
By Theorem 1, we can get the following remark immediately.
Remark 2. Let be an -regular Cayley graph of valency . Then is normal or bi-normal.
In this section we give some examples of graphs appearing in Theorem 1.
Example 3. Let be a cyclic group. Assume that is of order and . Let Suppose that where is an involution such that . Let be the Cayley graph of the dihedral group with respect to . Then is a connected -regular Cayley graph of valency . In particular, is -normal.
Proof. Let where .
Noting , we may assume that . Since the involution is not equal to , we may let for some such that . Then , and so . Thus the element is of order as . So ; that is, is connected.
Obviously, and . However, ; then is an -regular normal Cayley graph of of valency .
Example 4. Let . Set and . Then and . Let such that . It follows that . Then for , and furthermore is -regular. Obviously is not normal in . However, is semiregular and has exactly two orbits on ; then is an -regular Cayley graph of valency . In particular, is -bi-normal.
In this section, we will prove our main results. We first present some properties about normal Cayley graphs.
For a Cayley graph , we have a subgroup of :
Clearly it is a subgroup of the stabilizer in of the vertex corresponding to the identity of . Since is connected, acts faithfully on . By Godsil [7, Lemma 2.1], the normalizer . So is a normal Cayley graph if and only if .
Let be an -regular Cayley graph of valency such that . Then contains at least one involution. Let , which is the core of in .
Lemma 5. Assume that . Then or .
Proof. Let be the stabilizer in of the vertex corresponding to the identity of . Then , , and . Let be the set of right cosets of in . Consider the action of on by the right multiplication. Then we get that is a primitive permutation group of degree and is a stabilizer of . Since has valency , , and so . Then we can show or , and then or , respectively.
Lemma 6. Suppose that and . Then is the icosahedron graph. Moreover, and is not -regular.
Proof. Note that , where , , and . Since has no nontrivial normal subgroup, is not bipartite. So is the icosahedron graph. Further by Magma [8], , so is not -regular.
Lemma 7. Suppose that and . Then the graph is not -regular and there is only one isomorphism class of these graphs.
Proof. Note that , , and . Let , where . By considering the right multiplication action of on the right cosets of in , can be viewed as a stabilizer of acting on . Without lost generality, we may assume that 1 is fixed by . Take an involution . Then, by [2], and we can identify with . Note that and ; then is one of the following: , , , , , , , and . Note . Assume that ; by calculation, we have , , , , and . Then , , , . Thus the corresponding is since for each . By similar argument, for every , we can work out explicitly, which is one of the following four cases: , , , and .
Now let . We declare that . Assume that . Note that ; then . Let . Since , and for any possible . Therefore , which leads to a contradiction. So the assertion is right; that is, is not -regular.
Let and for . Set , then and . It follows that and , namely, and . Now we consider . Note that and , then . Since , . On the other hand, and , then and . By the assumption, is not the graph satisfying conditions. So far we get the result that there is only one isomorphism class of graphs when .
To finish our proof, we need to introduce some definitions and properties. Assume that is an -vertex transitive graph with being a subgroup of . Let be a normal subgroup of . Denote the set of -orbits in by . The normal quotient of induced by is defined as the graph with vertex set , and two vertices , are adjacent if there exist and such that they are adjacent in . It is easy to show that acts transitively on the vertex set of . Assume further that is -edge-transitive. Then acts transitively on the edge set of , and the valency for some positive integer . If , then is called a normal cover of .
Proof of Theorem 1. Let be an -regular Cayley graph of valency , where . Then it is trivial to see that is connected. Let be the core of in . Assume that is not trivial. Then either or . The former implies ; that is, is an -normal Cayley graph with respect to . For the case where , it is easy to verify is an -bi-normal Cayley graph. Suppose that ; namely, has at least three orbits on . Since is a prime and is -regular, is a cover of and . We have that is a Cayley graph of and is core-free with respect to . Now suppose that is trivial, then is a core-free one. According to Lemmas 5, 6, and 7, there are two core-free -regular Cayley graphs of valency (up to isomorphism) as in Table 1. As far, Theorem 1 holds.
The project was sponsored by the Foundation of Guangxi University (no. XBZ110328), the Fund of Yunnan University (no. 2012CG015), NNSF (nos. 11126343, 11226141, and 11226045), and NSF of Guangxi (no. 2012GXNSFBA053010).
C. H. Li, “Finite
s
-arc transitive Cayley graphs and flag-transitive projective planes,” Proceedings of the American Mathematical Society, vol. 133, no. 1, pp. 31–41, 2005. View at: Publisher Site | Google Scholar | MathSciNet
J. J. Li and Z. P. Lu, “Cubic
s
-arc transitive Cayley graphs,” Discrete Mathematics, vol. 309, no. 20, pp. 6014–6025, 2009. View at: Publisher Site | Google Scholar | MathSciNet
R. Frucht, “A one-regular graph of degree three,” Canadian Journal of Mathematics, vol. 4, pp. 240–247, 1952. View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. D. E. Conder and C. E. Praeger, “Remarks on path-transitivity in finite graphs,” European Journal of Combinatorics, vol. 17, no. 4, pp. 371–378, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Marušič, “A family of one-regular graphs of valency 4,” European Journal of Combinatorics, vol. 18, no. 1, pp. 59–64, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Malnič, D. Marušič, and N. Seifter, “Constructing infinite one-regular graphs,” European Journal of Combinatorics, vol. 20, no. 8, pp. 845–853, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
C. D. Godsil, “On the full automorphism group of a graph,” Combinatorica, vol. 1, no. 3, pp. 243–256, 1981. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system. I. The user language,” Journal of Symbolic Computation, vol. 24, no. 3-4, pp. 235–265, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Copyright © 2013 Jing Jian Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Autoregressive all-pole model parameters — Yule-Walker method - MATLAB aryule - MathWorks France
Parameter Estimation Using the Yule-Walker Method
Estimate Model order Using Decay of Reflection Coefficients
Autoregressive all-pole model parameters — Yule-Walker method
a = aryule(x,p) returns the normalized autoregressive (AR) parameters corresponding to a model of order p for the input array x.
[a,e,rc] = aryule(x,p) also returns the estimated variance, e, of the white noise input and the reflection coefficients, rc.
Use a vector of polynomial coefficients to generate an AR(4) process by filtering 1024 samples of white noise. Reset the random number generator for reproducible results. Use the Yule-Walker method to estimate the coefficients.
Generate 50 realizations of the process, changing each time the variance of the input noise. Compare the Yule-Walker-estimated variances to the actual values.
Use a vector of polynomial coefficients to generate an AR(2) process by filtering 1024 samples of white noise. Reset the random number generator for reproducible results.
Use the Yule-Walker method to fit an AR(10) model to the process. Output and plot the reflection coefficients. Only the first two coefficients lie outside the 95% confidence bounds, indicating that an AR(10) model significantly overestimates the time dependence in the data. See AR Order Selection with Partial Autocorrelation Sequence for more details.
The weights on the p past outputs minimize the mean squared prediction error of the autoregression. If y(n) is the current value of the output and x(n) is a zero-mean white noise input, the AR(p) model is
\sum _{k=0}^{p}a\left(k\right)y\left(n-k\right)=x\left(n\right).
The reflection coefficients are the partial autocorrelation coefficients scaled by –1.
The reflection coefficients indicate the time dependence between y(n) and y(n – k) after subtracting the prediction based on the intervening k – 1 time steps.
aryule uses the Levinson-Durbin recursion on the biased estimate of the sample autocorrelation sequence to compute the parameters.
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Parameter tuning | CatBoost
Bagging temperature
Border count
Internal dataset order
Tree growing policy
Methods for hyperparameter search
Methods for hyperparameter search by optuna
CatBoost provides a flexible interface for parameter tuning and can be configured to suit different tasks.
This section contains some tips on the possible parameter settings.
Sometimes when categorical features don't have a lot of values, one-hot encoding works well.
Usually one-hot encoding does not significantly improve the quality of the model. But if it is required, use the inbuilt parameters instead of preprocessing the dataset.
It is recommended to check that there is no obvious underfitting or overfitting before tuning any other parameters. In order to do this it is necessary to analyze the metric value on the validation dataset and select the appropriate number of iterations.
This can be done by setting the number of iterations to a large value, using the overfitting detector parameters and turning the use best model options on. In this case the resulting model contains only the first k best iterations, where k is the iteration with the best loss value on the validation dataset.
Also, the metric for choosing the best model may differ from the one used for optimizing the objective value. For example, it is possible to set the optimized function to Logloss and use the AUC function for the overfitting detector. To do so, use the evaluation metric parameter.
Command-line version parameters: -i, --iterations
Python parameters: --iterations
R parameters: --iterations
Command-line version parameters: --use-best-model
Python parameters: --use-best-model
R parameters: --use-best-model
Command-line version parameters: --eval-metric
Python parameters: --eval-metric
R parameters: --eval-metric
Command-line version parameters: Overfitting detection settings
Command-line version parameters: --od-type
Python parameters: od_type
R parameters: od_type
Command-line version parameters: --od-pval
Python parameters: od_pval
R parameters: od_pval
[10^{–10}; 10^{-2}]
Command-line version parameters: --od-wait
Python parameters: od_wait
R parameters: od_wait
This setting is used for reducing the gradient step. It affects the overall time of training: the smaller the value, the more iterations are required for training. Choose the value based on the performance expectations.
By default, the learning rate is defined automatically based on the dataset properties and the number of iterations. The automatically defined value should be close to the optimal one.
Possible ways of adjusting the learning rate depending on the overfitting results:
There is no overfitting on the last iterations of training (the training does not converge) — increase the learning rate.
Overfitting is detected — decrease the learning rate.
Command-line version parameters: -w, --learning-rate
Python parameters: learning_rate
R parameters: learning_rate
The learning rate. Used for reducing the gradient step.
In most cases, the optimal depth ranges from 4 to 10. Values in the range from 6 to 10 are recommended.
The maximum depth of the trees is limited to 8 for pairwise modes (YetiRank, PairLogitPairwise and QueryCrossEntropy) when the training is performed on GPU.
Command-line version parameters: -n, --depth
Python parameters: depth
R parameters: depth
Depth of the tree. The range of supported values depends on the processing unit type and the type of the selected loss function:
GPU — Any integer up to 8 pairwise modes (YetiRank, PairLogitPairwise and QueryCrossEntropy) and up to 16 for all other loss functions.
Try different values for the regularizer to find the best possible.
Command-line version parameters: --l2-leaf-reg
Python parameters: l2_leaf_reg
R parameters: l2_leaf_reg
Try setting different values for the random_strength parameter.
Command-line version parameters: --random-strength
Python parameters: random_strength
R parameters: random_strength
Try setting different values for the bagging_temperature parameter
Command-line version parameters: --bagging-temperature
Python parameters: bagging_temperature
R parameters: bagging_temperature
[0; \inf)
The value of this parameter significantly impacts the speed of training on GPU. The smaller the value, the faster the training is performed (refer to the Number of splits for numerical features section for details).
128 splits are enough for many datasets. However, try to set the value of this parameter to 254 when training on GPU if the best possible quality is required.
Command-line version parameters: -x, --border-count
Python parameters: border_count
R parameters: border_count
Recommended values are up to 255. Larger values slow down the training.
Use this option if the objects in your dataset are given in the required order. In this case, random permutations are not performed during the Transforming categorical features to numerical features and Choosing the tree structure stages.
Command-line version parameters: --has-time
Python parameters: --has-time
R parameters: --has-time
By default, CatBoost uses symmetric trees, which are built if the growing policy is set to SymmetricTree.
Such trees are built level by level until the specified depth is reached. On each iteration, all leaves from the last tree level are split with the same condition. The resulting tree structure is always symmetric.
Symmetric trees have a very good prediction speed (roughly 10 times faster than non-symmetric trees) and give better quality in many cases.
However, in some cases, other tree growing strategies can give better results than growing symmetric trees.
Try to analyze the results obtained with different growing trees strategies.
Specifics: Symmetric trees, that are used by default, can be applied much faster (up to 10 times faster).
Command-line version parameters: --grow-policy
Python parameters: grow_policy
R parameters: grow_policy
SymmetricTree — A tree is built level by level until the specified depth is reached. On each iteration, all leaves from the last tree level are split with the same condition. The resulting tree structure is always symmetric.
Command-line version parameters: --min-data-in-leaf
Python parameters: min_data_in_leaf
Alias:min_child_samples
R parameters: min_data_in_leaf
Command-line version parameters: --max-leaves
Python parameters: max_leaves
R parameters: max_leaves
If the dataset has a feature, which is a strong predictor of the result, the pre-quantisation of this feature may decrease the information that the model can get from it. It is recommended to use an increased number of borders (1024) for this feature.
An increased number of borders should not be set for all features. It is recommended to set it for one or two golden features.
--per-float-feature-quantization A semicolon separated list of quantization descriptions.
--per-float-feature-quantization 0:border_count=1024
--per-float-feature-quantization 0:border_count=1024;1:border_count=1024
per_float_feature_quantization The quantization description for the specified feature or list of features.
The Python package provides Grid and Randomized search methods for searching optimal parameter values for training the model with the given dataset.
CatBoost grid_search A simple grid search over specified parameter values for a model.
CatBoost randomized_search A simple randomized search on hyperparameters.
CatBoostClassifier grid_search A simple grid search over specified parameter values for a model.
CatBoostClassifier randomized_search A simple randomized search on hyperparameters.
CatBoostRegressor grid_search A simple grid search over specified parameter values for a model.
CatBoostRegressor randomized_search A simple randomized search on hyperparameters.
Optuna is a famous hyperparameter optimization framework.
Optuna enables efficient hyperparameter optimization by adopting state-of-the-art algorithms for sampling hyperparameters and pruning efficiently unpromising trials.
Catboost supports to stop unpromising trial of hyperparameter by callbacking after iteration functionality. Pull Request
The following is an optuna example that demonstrates a pruner for CatBoost. Example
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Measure active and reactive powers of voltage-current pair - Simulink
Measure active and reactive powers of voltage-current pair
powerlib_extras/Measurements
A discrete version of this block is available in the powerlib_extras/Discrete Measurements library.
A phasor version of this block is available in the powerlib_extras/Phasor Library.
The Measurements section of the Control and Measurements library contains the Power block. This is an improved version of the Active & Reactive Power block. The new block features a mechanism that eliminates duplicate continuous and discrete versions of the same block by basing the block configuration on the simulation mode. If your legacy models contain the Active & Reactive Power block, they continue to work. However, for best performance, use the Power block in your new models.
The Active & Reactive Power block measures the active power P and reactive power Q associated with a periodic voltage-current pair that can contain harmonics. P and Q are calculated by averaging the V I product with a running average window over one cycle of the fundamental frequency, so that the powers are evaluated at fundamental frequency.
\begin{array}{c}P=\frac{1}{T}\underset{t-T}{\overset{t}{\int }}V\left(\omega t\right)\cdot I\left(\omega t\right)\text{\hspace{0.17em}}dt\\ Q=\frac{1}{T}\underset{t-T}{\overset{t}{\int }}V\left(\omega t\right)\cdot I\left(\omega t-\pi /2\right)\text{\hspace{0.17em}}dt,\end{array}
where T = 1/(fundamental frequency).
A current flowing into an RL branch, for example, produces positive active and reactive powers.
As this block uses a running window, one cycle of simulation has to be completed before the output gives the correct active and reactive powers.
The discrete version of this block, available in the powerlib_extras/Discrete Measurements library, allows you to specify the initial input voltage and current (magnitude and phase). For the first cycle of simulation, the outputs are held constant using the values specified by the initial input parameters.
The fundamental frequency, in hertz, of the instantaneous voltage and current.
The first input is the instantaneous voltage, in volts.
The second input is the instantaneous current, in amperes.
The output is a vector [P Q] of the active and reactive powers, in watts and vars, respectively.
The power_transfo example simulates a three-winding distribution transformer rated at 75 kVA:14400/120/120 V.
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m
n
Matrix (or 2-dimensional Array), then it is assumed to contain
m
\mathrm{with}\left(\mathrm{SignalProcessing}\right):
\mathrm{f1}≔12.0:
\mathrm{f2}≔24.0:
\mathrm{signal}≔\mathrm{Vector}\left({2}^{10},i↦\mathrm{sin}\left(\frac{\mathrm{f1}\cdot \mathrm{\pi }\cdot i}{50}\right)+1.5\cdot \mathrm{sin}\left(\frac{\mathrm{f2}\cdot \mathrm{\pi }\cdot i}{50}\right),\mathrm{datatype}=\mathrm{float}[8]\right)
{\textcolor[rgb]{0,0,1}{\mathrm{_rtable}}}_{\textcolor[rgb]{0,0,1}{36893628696909549564}}
\mathrm{Periodogram}\left(\mathrm{signal},\mathrm{samplerate}=100\right)
\mathrm{audiofile}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/maplesim.wav"\right):
\mathrm{Periodogram}\left(\mathrm{audiofile},\mathrm{frequencyscale}="kHz"\right)
\mathrm{audiofile2}≔\mathrm{cat}\left(\mathrm{kernelopts}\left(\mathrm{datadir}\right),"/audio/stereo.wav"\right):
\mathrm{Periodogram}\left(\mathrm{audiofile2},\mathrm{compactplot}\right)
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Specify term content of LMIs - MATLAB lmiterm - MathWorks United Kingdom
lmiterm
Specify LMI Terms
Specify term content of LMIs
lmiterm(termID,A,B,flag)
lmiterm specifies the term content of an LMI one term at a time. Recall that LMI term refers to the elementary additive terms involved in the block-matrix expression of the LMI. Before using lmiterm, the LMI description must be initialized with setlmis and the matrix variables must be declared with lmivar. Each lmiterm command adds one extra term to the LMI system currently described.
LMI terms are one of the following entities:
constant terms (fixed matrices)
variable terms AXB or AXTB where X is a matrix variable and A and B are given matrices called the term coefficients.
When describing an LMI with several blocks, remember to specify only the terms in the blocks on or below the diagonal (or equivalently, only the terms in blocks on or above the diagonal). For instance, specify the blocks (1,1), (2,1), and (2,2) in a two-block LMI.
In the calling of lmiterm, termID is a four-entry vector of integers specifying the term location and the matrix variable involved.
\text{termID }\left(1\right)=\left\{\begin{array}{c}+\text{p}\\ -\text{p}\end{array}
where positive p is for terms on the left-side of the p-th LMI and negative p is for terms on the right-side of the p-th LMI.
Recall that, by convention, the left side always refers to the smaller side of the LMI. The index p is relative to the order of declaration and corresponds to the identifier returned by newlmi.
\begin{array}{l}\text{termID}\left(2:3\right)=\left\{\begin{array}{l}\left[\text{0,0}\right]\text{ for outer factors}\hfill \\ \left[i,j\right]\text{ for terms in the }\left(i,j\right)\text{-th block of the left or right inner factor}\hfill \end{array}\\ \\ \text{termID}\left(4\right)=\left\{\begin{array}{l}\text{0 for outer factors}\hfill \\ \begin{array}{l}x\text{ for variable terms }AXB\\ -x\text{ for variable terms }A{X}^{T}B\end{array}\hfill \end{array}\end{array}
where x is the identifier of the matrix variable X as returned by lmivar.
The arguments A and B contain the numerical data and are set according to:
outer factor N
matrix value of N
constant term C
matrix value of C
AXB or AXTB
matrix value of A
(1 if A is absent)
matrix value of B
(1 if B is absent)
Note that identity outer factors and zero constant terms need not be specified.
The extra argument flag is optional and concerns only conjugated expressions of the form
(AXB) + (AXB)T = AXB + BTXTAT
in diagonal blocks. Setting flag = 's' allows you to specify such expressions with a single lmiterm command. For instance,
lmiterm([1 1 1 X],A,1,'s')
adds the symmetrized expression AX + XTAT to the (1,1) block of the first LMI and summarizes the two commands
lmiterm([1 1 1 X],A,1)
lmiterm([1 1 1 -X],1,A')
Aside from being convenient, this shortcut also results in a more efficient representation of the LMI.
Consider the LMI
\left(\begin{array}{cc}2A{X}_{2}{A}^{T}-{x}_{3}E+D{D}^{T}& {B}^{T}{X}_{1}\\ {X}_{1}^{T}B& -I\end{array}\right)<{M}^{T}\left(\begin{array}{cc}C{X}_{1}{C}^{T}+C{X}_{1}^{T}{C}^{T}& 0\\ 0& -f{X}_{2}\end{array}\right)M
where X1, X2 are matrix variables of Types 2 and 1, respectively, and x3 is a scalar variable (Type 1).
After you initialize the LMI description using setlmis and declare the matrix variables using lmivar, specify the terms on the left side of this LMI.
lmiterm([1 1 1 X2],2*A,A') % 2*A*X2*A'
lmiterm([1 1 1 x3],-1,E) % -x3*E
lmiterm([1 1 1 0],D*D') % D*D'
lmiterm([1 2 1 -X1],1,B) % X1'*B
lmiterm([1 2 2 0],-1) % -I
Here X1, X2, and x3 are the variable identifiers returned by lmivar when you declare the variables.
Similarly, specify the term content of the right side.
lmiterm([-1 0 0 0],M) % outer factor M
lmiterm([-1 1 1 X1],C,C','s') % C*X1*C'+C*X1'*C'
lmiterm([-1 2 2 X2],-f,1) % -f*X2
Note that CX1CT + CX1TCT is specified by a single lmiterm command with the flag 's' to ensure proper symmetrization.
setlmis | lmivar | getlmis | lmiedit | newlmi
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Torrecillasite, Na(As,Sb)43+, a new mineral from the Torrecillas mine, Iquique Province, Chile: description and crystal structure | Mineralogical Magazine | GeoScienceWorld
A. R. Kampf, B. P. Nash, M. Dini, A. A. Molina Donoso; Torrecillasite,
Na(As,Sb)43+
, a new mineral from the Torrecillas mine, Iquique Province, Chile: description and crystal structure. Mineralogical Magazine 2014;; 78 (3): 747–755. doi: https://doi.org/10.1180/minmag.2014.078.3.20
The new mineral torrecillasite (IMA2013-112),
Na(As,Sb)43+
, was found at the Torrecillas mine, Iquique Province, Chile, where it occurs as a secondary alteration phase in association with anhydrite, cinnabar, gypsum, halite, lavendulan, magnesiokoritnigite, marcasite, quartz, pyrite, scorodite, wendwilsonite and other potentially new As-bearing minerals. Torrecillasite occurs as thin colourless prisms up to 0.4 mm long in jack-straw aggregates, as very thin fibres in puff balls and as massive intergrowths of needles. Prisms are elongated on [100] with diamond-shaped cross-section and irregular terminations. Crystals are transparent, with adamantine lustre and white streak. The Mohs hardness is 2½, tenacity is brittle and fracture is irregular. Cleavage on (001) is likely. The calculated density is 4.056 g cm−3. Optically, torrecillasite is biaxial (−) with α = 1.800(5), β = 1.96(1), γ = 2.03(calc.) (measured in white light). The measured 2V is 62.1(5)°, no dispersion or pleochroism were observed, the optical orientation is X = c, Y = b, Z = a. The mineral is very slowly soluble in H2O, slowly soluble in dilute HCl and rapidly soluble in concentrated HCl. The empirical formula, determined from electron-microprobe analyses, is (Na1.03Mg0.02)Σ1.05(As3.39Sb0.62)Σ4.01O6.07Cl0.93. Torrecillasite is orthorhombic, Pmcn, a = 5.2580(9), b = 8.0620(13), c = 18.654(3) Å, V = 790.7(2) Å3 and Z = 4. The eight strongest X-ray powder diffraction lines are [dobs Å(I)(hkl)]: 4.298(33)(111), 4.031(78)(014,020), 3.035(100)(024,122), 2.853(39)(115,123), 2.642(84)(124,200), 2.426(34)(125), 1.8963(32)(225) and 1.8026(29)(0·1·10,233). The structure, refined to R1 = 4.06% for 814 Fo >4σF reflections, contains a neutral, wavy As2O3 layer parallel to (001) consisting of As3+O3 pyramids that share O atoms to form six-membered rings. Successive layers are flipped relative to one another and successive interlayer regions contain alternately either Na or Cl atoms. Torrecillasite is isostructural with synthetic orthorhombic NaAs4O6Br.
Canutite, NaMn3[AsO4][AsO3(OH)]2, a new protonated alluaudite-group mineral from the Torrecillas mine, Iquique Province, Chile
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Bulk's modulus - Physics - Propagation of Sound Waves - 12260803 | Meritnation.com
Adarsh Dubey answered this
Bulk Modulus is termed as the proportion of volumetric stress related to the volumetric strain of a specified material, while the material deformation is within elastic limit.
Bulk modulus =\frac{Volumetric Stress}{Volumetric strain}
The relative change in the volume of a body produced by a unit compressively or tensile stress acting uniformly over its surface
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For other uses, see Shape (disambiguation).
"Geometric shape" redirects here. For the Unicode symbols, see Geometric Shapes.
A shape or figure is the graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on a plane, in contrast to solid 3D shapes. A two-dimensional shape or two-dimensional figure (also: 2D shape or 2D figure) may lie on a more general curved surface (a non-Euclidean two-dimensional space).
A children's toy used for learning various shapes
1 Classification of simple shapes
3 Equivalence of shapes
3.4 Similarity classes
4 Human perception of shapes
Classification of simple shapesEdit
In geometryEdit
Geometric shapes in 2 dimensions: parallelogram, triangle & circle
Geometric shapes in 3 dimensions: pyramid, Sphere & cube
A geometric shape consists of the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object.[1] That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape.
Sometimes, two similar or congruent objects may be regarded as having a different shape if a reflection is required to transform one into the other. For instance, the letters "b" and "d" are a reflection of each other, and hence they are congruent and similar, but in some contexts they are not regarded as having the same shape. Sometimes, only the outline or external boundary of the object is considered to determine its shape. For instance, a hollow sphere may be considered to have the same shape as a solid sphere. Procrustes analysis is used in many sciences to determine whether or not two objects have the same shape, or to measure the difference between two shapes. In advanced mathematics, quasi-isometry can be used as a criterion to state that two shapes are approximately the same.
Equivalence of shapesEdit
Congruence and similarityEdit
HomeomorphismEdit
One way of modeling non-rigid movements is by homeomorphisms. Roughly speaking, a homeomorphism is a continuous stretching and bending of an object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a donut are not. An often-repeated mathematical joke is that topologists cannot tell their coffee cup from their donut,[4] since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle.
A described shape has external lines that you can see and make up the shape. If you were putting you coordinates on and coordinate graph you could draw lines to show where you can see a shape, however not every time you put coordinates in a graph as such you can make a shape. This shape has a outline and boundary so you can see it and is not just regular dots on a regular paper.
Shape analysisEdit
The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of statistical shape analysis. In particular, Procrustes analysis is a technique used for comparing shapes of similar objects (e.g. bones of different animals), or measuring the deformation of a deformable object. Other methods are designed to work with non-rigid (bendable) objects, e.g. for posture independent shape retrieval (see for example Spectral shape analysis).
Similarity classesEdit
All similar triangles have the same shape. These shapes can be classified using complex numbers u, v, w for the vertices, in a method advanced by J.A. Lester[5] and Rafael Artzy. For example, an equilateral triangle can be expressed by the complex numbers 0, 1, (1 + i √3)/2 representing its vertices. Lester and Artzy call the ratio
{\displaystyle S(u,v,w)={\frac {u-w}{u-v}}}
For any affine transformation of the complex plane,
{\displaystyle z\mapsto az+b,\quad a\neq 0,}
a triangle is transformed but does not change its shape. Hence shape is an invariant of affine geometry. The shape p = S(u,v,w) depends on the order of the arguments of function S, but permutations lead to related values. For instance,
{\displaystyle 1-p=1-(u-w)/(u-v)=(w-v)/(u-v)=(v-w)/(v-u)=S(v,u,w).}
{\displaystyle p^{-1}=S(u,w,v).}
Combining these permutations gives
{\displaystyle S(v,w,u)=(1-p)^{-1}.}
{\displaystyle p(1-p)^{-1}=S(u,v,w)S(v,w,u)=(u-w)/(v-w)=S(w,v,u).}
{\displaystyle p=(1-q)^{-1},}
then the quadrilateral is a parallelogram.
If a parallelogram has | arg p | = | arg q |, then it is a rhombus.
{\displaystyle p=r(1-q^{-1})}
and sgn r = sgn(Im p), then the quadrilateral is a trapezoid.
A polygon
{\displaystyle (z_{1},z_{2},...z_{n})}
has a shape defined by n – 2 complex numbers
{\displaystyle S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.}
The polygon bounds a convex set when all these shape components have imaginary components of the same sign.[6]
Human perception of shapesEdit
Human vision relies on a wide range of shape representations.[7][8] Some psychologists have theorized that humans mentally break down images into simple geometric shapes (e.g., cones and spheres) called geons.[9] Others have suggested shapes are decomposed into features or dimensions that describe the way shapes tend to vary, like their segmentability, compactness and spikiness.[10] When comparing shape similarity, however, at least 22 independent dimensions are needed to account for the way natural shapes vary. [7]
There is also clear evidence that shapes guide human attention.[11][12]
^ Kendall, D.G. (1984). "Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces" (PDF). Bulletin of the London Mathematical Society. 16 (2): 81–121. doi:10.1112/blms/16.2.81.
^ Here, scale means only uniform scaling, as non-uniform scaling would change the shape of the object (e.g., it would turn a square into a rectangle).
^ J.A. Lester (1996) "Triangles I: Shapes", Aequationes Mathematicae 52:30–54
^ Rafael Artzy (1994) "Shapes of Polygons", Journal of Geometry 50(1–2):11–15
^ a b Morgenstern, Yaniv; Hartmann, Frieder; Schmidt, Filipp; Tiedemann, Henning; Prokott, Eugen; Maiello, Guido; Fleming, Roland (2021). "An image-computable model of visual shape similarity". PLOS Computational Biology. 17 (6): 34. doi:10.1371/journal.pcbi.1008981.
^ Andreopoulos, Alexander; Tsotsos, John K. (2013). "50 Years of object recognition: Directions forward". Computer Vision and Image Understanding. 117 (8): 827–891. doi:10.1016/j.cviu.2013.04.005.
^ Marr, D., & Nishihara, H. (1978). Representation and recognition of the spatial organization of three-dimensional shapes. Proceedings of the Royal Society of London, 200, 269-294.
^ Huang, Liqiang (2020). "Space of preattentive shape features". Journal of Vision. 20 (4): 10. doi:10.1167/jov.20.4.10. PMC 7405702. PMID 32315405.
^ Alexander, R. G.; Schmidt, J.; Zelinsky, G.Z. (2014). "Are summary statistics enough? Evidence for the importance of shape in guiding visual search". Visual Cognition. 22 (3–4): 595–609. doi:10.1080/13506285.2014.890989. PMC 4500174. PMID 26180505.
^ Wolfe, Jeremy M.; Horowitz, Todd S. (2017). "Five factors that guide attention in visual search". Nature Human Behaviour. 1 (3). doi:10.1038/s41562-017-0058. S2CID 2994044.
The dictionary definition of shape at Wiktionary
Retrieved from "https://en.wikipedia.org/w/index.php?title=Shape&oldid=1089489475"
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Home : Support : Online Help : Mathematics : Algebra : Expression Manipulation : Combining : range
combine(e, 'range')
Expressions involving calls to an operator F of two arguments, the second of which is of the form name = range, in which the range is either a discrete or continuous range, are combined by applying the transformations:
F\left(f\left(x\right),x=a\mathrm{..}b\right)+F\left(g\left(x\right),x=c\mathrm{..}d\right)\to F\left(f\left(x\right),x=a\mathrm{..}d\right)
whenever the operator F, first arguments
f\left(x\right)
g\left(x\right)
, and ranges
a..b
c..d
satisfy one of the following conditions.
F=\mathrm{Int}
F=\mathrm{int},c=b
f\left(x\right)=g\left(x\right)
F=\mathrm{Int}
F=\mathrm{int},c=a
f\left(x\right)=-g\left(x\right)
F=\mathrm{Int}
F=\mathrm{int},b=d
f\left(x\right)=-g\left(x\right)
F=\mathrm{Sum}
F=\mathrm{sum},c=1+b
f\left(x\right)=g\left(x\right)
F=\mathrm{Sum}
F=\mathrm{sum},c=a
b<d
f\left(x\right)=-g\left(x\right)
For the results of `combine/range` to be valid, the operator F must be a function of two arguments, the second of which has the form name = range and is additive over ranges.
Knowledge of the summation and integral operators is supported in the library. You can add to the knowledge of which operators can be manipulated by `combine/range` by assigning to one of the environment variables _EnvRangeCombinableContinuous or _EnvRangeCombinableDiscrete. Each should be a set of names of operators F. Operators in the set _EnvRangeCombinableDiscrete are combined in the same way as Sum, while those in _EnvRangeCombinableContinuous are combined like integrals. The environment variable _EnvRangeCombinable is used internally and should not be modified directly by users.
By calling the procedure combine/range directly, you can set the values of combinable operators for a single call by passing an extra (second) argument in the form of a list of two sets. The first member of the list is the set of operators that can be combined using continuous ranges (Int-like), and the second member of the list should be a set of the names of Sum-like operators whose ranges are to be treated as discrete.
The power of `combine/range` is controlled by the values of the environment variables Testzero and Normalizer. These are used to detect matching range endpoints and matching first arguments. Resetting Testzero gives you local control over the combining power of this procedure.
\mathrm{Int}\left(f\left(x\right),x=a..b\right)+\mathrm{Int}\left(-f\left(x\right),x=a..c\right)
{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{c}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}
=\mathrm{combine}\left(,'\mathrm{range}'\right)
{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{c}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\int }}_{\textcolor[rgb]{0,0,1}{c}}^{\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{ⅆ}\textcolor[rgb]{0,0,1}{x}
\left(\mathrm{Sum}\left(f\left(j\right),j=a..b\right)\right)+\left(\mathrm{Sum}\left(f\left(j\right),j=b+1..d\right)\right)
\left(\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)\right)
=\mathrm{combine}\left(,'\mathrm{range}'\right)
\left(\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)\right)\textcolor[rgb]{0,0,1}{+}\left(\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)
`combine/range` is sensitive to assumptions:
\left(\mathrm{Sum}\left(f\left(j\right),j=a..b\right)\right)+\mathrm{Sum}\left(-f\left(j\right),j=a..d\right)
\left(\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)\right)
=\mathrm{combine}\left(,'\mathrm{range}'\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}b<d
\left(\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{b}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{-}\left(\textcolor[rgb]{0.564705882352941,0.564705882352941,0.564705882352941}{\sum }_{\textcolor[rgb]{0,0,1}{j}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{d}}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{j}\right)\right)
G\left(f\left(x\right),x=a..b\right)+G\left(f\left(x\right),x=b..c\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{f}\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}{=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{b}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{f}\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}{=}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{c}\right)
=\mathrm{combine}\left(,'\mathrm{range}',[{'G'},\varnothing ]\right)
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{f}\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}{=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{b}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{f}\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}{=}\textcolor[rgb]{0,0,1}{b}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{c}\right)\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{f}\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}{=}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{..}\textcolor[rgb]{0,0,1}{c}\right)
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TemporaryEntries - Maple Help
Home : Support : Online Help : Programming : Procedures and Functions : Cache Package : TemporaryEntries
return a sequence of the temporary entries
TemporaryEntries( cache )
The TemporaryEntries command returns the temporary entries of the given cache table. The cache table can be given directly as cache, or cache can refer to a procedure that has, or can have, a cache remember table. If such a procedure is given and it has a cache remember table, the temporary entries from that table are returned. If the procedure does not have a remember table, NULL is returned.
TemporaryEntries returns the entries in same format as entries, that is a sequence of lists where the contents of each list is the value of a temporary entry from the table.
The TemporaryIndices command can be used to get the indices of the temporary entries.
\mathrm{c1}≔\mathrm{Cache}\left(\right)
\textcolor[rgb]{0,0,1}{\mathrm{c1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{Cache}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{512}\right)
\mathrm{c1}[1]≔2
{\textcolor[rgb]{0,0,1}{\mathrm{c1}}}_{\textcolor[rgb]{0,0,1}{1}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{2}
\mathrm{c1}[2]≔3
{\textcolor[rgb]{0,0,1}{\mathrm{c1}}}_{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{3}
\mathrm{Cache}:-\mathrm{TemporaryEntries}\left(\mathrm{c1}\right)
[\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}]
\mathrm{Cache}:-\mathrm{TemporaryIndices}\left(\mathrm{c1}\right)
[\textcolor[rgb]{0,0,1}{1}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{2}]
\textcolor[rgb]{0,0,1}{p}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathbf{proc}}\left(\textcolor[rgb]{0,0,1}{x}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{option}}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathrm{cache}}\textcolor[rgb]{0,0,1}{;}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{^}\textcolor[rgb]{0,0,1}{2}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{end proc}}
p\left(2\right)
\textcolor[rgb]{0,0,1}{4}
p\left(3\right)
\textcolor[rgb]{0,0,1}{9}
\mathrm{Cache}:-\mathrm{TemporaryEntries}\left(p\right)
[\textcolor[rgb]{0,0,1}{4}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{9}]
\mathrm{Cache}:-\mathrm{TemporaryIndices}\left(p\right)
[\textcolor[rgb]{0,0,1}{2}]\textcolor[rgb]{0,0,1}{,}[\textcolor[rgb]{0,0,1}{3}]
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Section 59.4 (03N4): The étale topology—The Stacks project
Section 59.4: The étale topology (cite)
59.4 The étale topology
It is very hard to simply “add” extra open sets to refine the Zariski topology. One efficient way to define a topology is to consider not only open sets, but also some schemes that lie over them. To define the étale topology, one considers all morphisms $\varphi : U \to X$ which are étale. If $X$ is a smooth projective variety over $\mathbf{C}$, then this means
$U$ is a disjoint union of smooth varieties, and
$\varphi $ is (analytically) locally an isomorphism.
The word “analytically” refers to the usual (transcendental) topology over $\mathbf{C}$. So the second condition means that the derivative of $\varphi $ has full rank everywhere (and in particular all the components of $U$ have the same dimension as $X$).
A double cover – loosely defined as a finite degree $2$ map between varieties – for example
\[ \mathop{\mathrm{Spec}}(\mathbf{C}[t]) \longrightarrow \mathop{\mathrm{Spec}}(\mathbf{C}[t]), \quad t \longmapsto t^2 \]
will not be an étale morphism if it has a fibre consisting of a single point. In the example this happens when $t = 0$. For a finite map between varieties over $\mathbf{C}$ to be étale all the fibers should have the same number of points. Removing the point $t = 0$ from the source of the map in the example will make the morphism étale. But we can remove other points from the source of the morphism also, and the morphism will still be étale. To consider the étale topology, we have to look at all such morphisms. Unlike the Zariski topology, these need not be merely open subsets of $X$, even though their images always are.
Definition 59.4.1. A family of morphisms $\{ \varphi _ i : U_ i \to X\} _{i \in I}$ is called an étale covering if each $\varphi _ i$ is an étale morphism and their images cover $X$, i.e., $X = \bigcup _{i \in I} \varphi _ i(U_ i)$.
This “defines” the étale topology. In other words, we can now say what the sheaves are. An étale sheaf $\mathcal{F}$ of sets (resp. abelian groups, vector spaces, etc) on $X$ is the data:
for each étale morphism $\varphi : U \to X$ a set (resp. abelian group, vector space, etc) $\mathcal{F}(U)$,
for each pair $U, \ U'$ of étale schemes over $X$, and each morphism $U \to U'$ over $X$ (which is automatically étale) a restriction map $\rho ^{U'}_ U : \mathcal{F}(U') \to \mathcal{F}(U)$
These data have to satisfy the condition that $\rho ^ U_ U = \text{id}$ in case of the identity morphism $U \to U$ and that $\rho ^{U'}_ U \circ \rho ^{U''}_{U'} = \rho ^{U''}_ U$ when we have morphisms $U \to U' \to U''$ of schemes étale over $X$ as well as the following sheaf axiom:
for every étale covering $\{ \varphi _ i : U_ i \to U\} _{i \in I}$, the diagram
\[ \xymatrix{ \emptyset \ar[r] & \mathcal{F} (U) \ar[r] & \Pi _{i \in I} \mathcal{F} (U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \Pi _{i, j \in I} \mathcal{F} (U_ i \times _ U U_ j) } \]
is exact in the category of sets (resp. abelian groups, vector spaces, etc).
Remark 59.4.2. In the last statement, it is essential not to forget the case where $i = j$ which is in general a highly nontrivial condition (unlike in the Zariski topology). In fact, frequently important coverings have only one element.
Since the identity is an étale morphism, we can compute the global sections of an étale sheaf, and cohomology will simply be the corresponding right-derived functors. In other words, once more theory has been developed and statements have been made precise, there will be no obstacle to defining cohomology.
Comment #1618 by Kien Nguyen on September 22, 2015 at 16:58
In (2) of the definition of an etale sheaf, should the arrow of the restriction map the other way round, to have a contravariant functor?
Yes, you are right. I also added the necessary conditions on composition of restriction mappings. See here.
minor typos In condition
(*)
: I think
U=X
. There are two instances of
U
\emptyset \to \mathcal{F}(U) \to \cdots \to \mathcal{F}(U_i \times_U U_j)
should that be
\mathcal{F}(X)
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Pauli Principle, Inflation and Simple Statistical Treatment of Free-Fermions
Angelo Plastino1,2,3, Mario Carlos Rocca1,2,4, Gustavo Ferri5
1 Departamento de Física, Universidad Nacional de La Plata, La Plata, Argentina.
2 Consejo Nacional de Investigaciones Científicas y Tecnológicas (IFLP-CCT-CONICET)-C. C. 727, La Plata, Argentina.
4 Departamento de Matemática, Universidad Nacional de La Plata, La Plata, Argentina.
5 Fac. de C. Exactas-National University La Pampa, Peru y Uruguay, Santa Rosa, La Pampa, Argentina.
Abstract: We study the dependence of the of microstates number (for free fermions-bosons) as a function of the volume-size in quantum statistics and fermions, and show then that fermions can not be accommodated in arbitrarily small volumes V. A minimum V = Vmin for that purpose is determined. Fermions can not exist for V < Vmin. This fact might have something to do with inflation. More precisely, in order to accommodate N fermions in a Slater determinant, we need a minimum radius, which is a consequence of the Pauli principle. This does not happen for bosons. As a consequence, extrapolating this statistical feature to a cosmological setting, we are able to “predict” a temperature-value for the final-stage of the inflationary period. This value agrees with current estimates.
Keywords: Microstates’s Number Ω, Fermions, Bosons
Cosmic inflation is a widely accepted phenomenon. The detailed particle physics mechanism responsible for inflation is unknown. The basic inflationary paradigm is accepted by most physicists, as a number of inflation model predictions have been confirmed by observation. The inflationary epoch lasted from 10−36 seconds after the conjectured Big Bang singularity till about 10−32 seconds after the singularity [1] [2] [3]. Here we will develop a statistical micro-canonical argument that seems to indicate that the Pauli principle might perhaps have played some kind of role in the inflation process. Our protagonist is the number of micro-states (multiplicity)
\Omega
for N particles of energy E enclosed in a volume V. We will see that, in the case of fermions, V can not be arbitrarily small. This observation motivates a hopefully attractive interpretation of cosmic inflation.
2. Multiplicities Ω
The multiplicity
\Omega \left(E,V,N\right)
of an N-particle, mono-atomic, ideal gas with energy E and volume V is the product of the number of cells of volume
\delta {s}^{3}
available in position space (that equals
V/\delta {s}^{3}
) and the number of cells available in momentum space [4]. The volume of momentum space through which the system may move is the p-volume
{\left(2\text{e}\pi /3N\right)}^{3N/2}3N{R}^{3N-1}\delta p
of a one-cell-wide shell of radius
R=\sqrt{2mE}
. As a consequence, the number of permissible momentum-space cells in this shell is (e is the basis of natural logarithms)
{V}_{\text{p-shell}}={\left(2\text{e}\pi /3N\right)}^{3N/2}\sqrt{3N}{\left(2mE/3N\right)}^{\left(3N-1\right)/2}\left(\delta p/\delta {p}^{3N}\right),
so that [4]
\Omega \left(E,V,N\right)={\left(V/\delta V\right)}^{N}{\left(2\text{e}\pi \right)}^{3N/2}\sqrt{3N}{\left(2mE/3N\right)}^{\left(3N-1\right)/2}\left(\delta p/\delta {p}^{3N}\right).
After some lengthy considerations, the author of [4] [pp. 56-57] [Eqs. (3.8)-(3.15)] rephrases the above relations as
\Omega \left(E/N,V,1\right)=V{\left(E/N\right)}^{3/2}{\left(4\text{e}\pi m/3{h}^{2}\right)}^{3/2},
an equation to be discussed below.
It has been known since at least 1925 that quantum mechanics includes, among its tenets, the following tree conditions: 1) phase space cells have a size determined by Planck’s constant h, 2) the energy, momenta, and other dynamical properties of an isolated system are quantized, and 3) for the purpose of determining multiplicity, identical particles are, indistinguishable from one another. Thus, (3) gives the multiplicity
\Omega
of an ideal gas composed of N distinguishable particles that occupy volume V and share an energy E.
\Omega \left(E,V,N\right)
is the multiplicity of an ideal gas composed of N distinguishable particles with total energy E and placed in a volume V, it follows that
\Omega \left(E/N,V,1\right)
should be the multiplicity of just a single particle of ideal gas, characterized by E/N and, and occupying volume V. This particle composes a larger system of N distinguishable particles. A single particle of an ideal gas occupies
\Omega \left(E/N,V,1\right)
equally probable microstates. Accordingly,
\Omega \left(E,V,N\right)={\left[\Omega \left(E/N,V,1\right)\right]}^{N}.
A crucial insight is here gotten: the number of cells that can be occupied by a single particle of ideal gas must be independent of whether that particle is itself distinguishable or indistinguishable from the other particles that compose the gas system and if indistinguishable whether fermion or boson. A system of N distinguishable particles of ideal gas may occupy
{\left[\Omega \left(E/N,V,1\right)\right]}^{N}
equally probable microstates [4].
Focus attention now upon an ideal Fermi gas. Let us call n the number of cells that can be occupied by a single fermion E/N. We have
n/N\ge 1
. According to (3) we have
n=V{\left(E/N\right)}^{3/2}{\left(4\text{e}\pi m/3{h}^{2}\right)}^{3/2},
V=\frac{4}{3}\pi {r}^{3}.
We face the problem of filling n cells with
N\le n
identical fermions. Thus the multiplicity becomes [4]
\Omega =\frac{n!}{N!\left(n-N\right)!}.
This multiplicity derived above is the result of the so-called average energy approximation. According to this approximation, each gas particle possesses the same energy E/N for all fermions in the system. That the average energy approximation produces the exact multiplicity when identical particles are considered distinguishable. The average energy approximation has been extensively used and produces quite reasonable results [4].
We approximate the Gamma function using the Stirling recipe
\Gamma \left(z\right)=\sqrt{2\pi }{z}^{z-\frac{1}{2}}{\text{e}}^{-z},
and find (
\Theta
is Heaviside’s step function)
\Omega =\frac{{\text{ee}}^{\left(n+1\right)\mathrm{ln}\left(n+1\right)-\Theta \left(n-N+1\right)\left(n-N+1\right)\mathrm{ln}\left(n-N+1\right)}}{\sqrt{2\pi }{\left(N+1\right)}^{\left(N+1\right)}},
and introduce the notationally simplifying definition (change of variables from r to s)
s={\left(\frac{4\pi }{3N}\right)}^{\frac{1}{3}}{\left(\frac{E}{N}\right)}^{\frac{1}{2}}{\left(\frac{4\pi em}{3{h}^{2}}\right)}^{\frac{1}{2}}r,
\Omega =\frac{{\text{ee}}^{\left(1+N{s}^{3}\right)ln\left(1+N{s}^{3}\right)}{\text{e}}^{-\Theta \left[N{s}^{3}-\left(N-1\right)\right]\left[N{s}^{3}-\left(N-1\right)\right]ln\left[N{s}^{3}-\left(N-1\right)\right]}}{\sqrt{2\pi }{\left(N+1\right)}^{\left(N+1\right)}}.
Of course, it must be
\Omega \ge 1;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}ln\left(\Omega \right)\ge 0.
From (11) one gets
\begin{array}{c}\mathrm{ln}\left(\Omega \right)=\left(1+N{s}^{3}\right)\mathrm{ln}\left(1+N{s}^{3}\right)-\Theta \left[N{s}^{3}-\left(N-1\right)\right]\left[N{s}^{3}-\left(N-1\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\times \mathrm{ln}\left[N{s}^{3}-\left(N-1\right)\right]+1-\frac{1}{2}\mathrm{ln}\left(2\pi \right)-\left(N+1\right)\mathrm{ln}\left(N+1\right),\end{array}
and we ask for a possible vanishing of
ln\left(\Omega \right)
. This is the critical novel issue that we address in this paper.
N\gg 1
we are led to answer with the relation
\frac{ln\left(\Omega \right)}{N}\simeq {s}^{3}ln\left({s}^{3}\right)-\Theta \left({s}^{3}-1\right)\left({s}^{3}-1\right)ln\left({s}^{3}-1\right)=0,
that has two possible solutions, namely,
s=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}s=1.
Figure 1 shows that the multiplicity
\Omega
would be negative for
0\le s\le 1
, which is absurd. We gather that the system is subjected to a kind of “phase transition” at
s=0
and forced to “jump” to
s=1
. One could putatively associate this jump to a sort of inflation-phenomenon motivated by the Pauli principle that would forbid N fermions to be accommodated in a too small volume V. A critical “accommodating” volume is reached at
s=1
. Emphasize that this happens for free fermions micro-canonically described. Equilibrium prevails.
The paper could well finish here. However, it is too tempting to extrapolate a bit further. We do this below, after discussing Bosons.
2.3. Bosons
The multiplicity is, for n microstates and N bosons [4],
Figure 1. FERMIONS: The logarithm of the number of microstates
ln\Omega
versus the N fermions’ size-indicator s. Note the sudden size-increase at the origin, which one might be willing to associate to inflation, that in turn would be motivated by the Pauli principle.
\Omega =\frac{\left(N+n-1\right)!}{N!\left(n-1\right)!}.
The counting performed above is structurally identical to counting the number of distinct manners of ordering a set of N identical balls and
n-1
identical white dividers. If placed in a row, these
n-1
dividers separate our N balls into n ordered (distinct) groups [4].
Thus, things are quite different for bosons, not subjected to the exclusion principle. The above
\Omega
, for N bosons of mass m contained in V with total energy E will yield [4]
\Omega =\frac{{\text{e}}^{\left(N+N{s}^{3}\right)\mathrm{ln}\left(N+N{s}^{3}\right)}{\text{e}}^{-\left(N{s}^{3}\right)\mathrm{ln}\left(N{s}^{3}\right)}}{\sqrt{2\pi }{N}^{N}},
with s given by (2).
\mathrm{ln}\left(\Omega \right)=\left(N+N{s}^{3}\right)\mathrm{ln}\left(N+N{s}^{3}\right)-\left(N{s}^{3}\right)\mathrm{ln}\left(N{s}^{3}\right)-\frac{1}{2}\mathrm{ln}\left(2\pi \right)-N\mathrm{ln}\left(N\right).
From (7) it follows that, in the spirit of last Subsection,
\frac{ln\left(\Omega \right)}{N}\simeq \left(1+{s}^{3}\right)ln\left(1+{s}^{3}\right)-\Theta \left(s\right){s}^{3}ln\left({s}^{3}\right)=0,
s=0,
2.4. A Putative Interpretation
What might have happened when fermions began to emerge at the Big-bang?
ln\Omega
versus s for bosons. N bosons can be accommodated at the origin. Remaining details are as in Figure 1.
A tiny fraction of a second after the singularity, some fermions began to emerge, out of a quark-gluon plasma [5], N of them, at a tiny region of size V, whose total fermion-energy was E (this variables determine a microcanonical ensemble). This region had to “explode” in order to accommodate them (transition from
s=0
s=1
above).
If we accept Siegel’s estimate for the Universe’s radius at the end of the inflation period of
r={r}_{0}=17\text{\hspace{0.17em}}\text{cm}
[5], then, from Equation (11), one gathers that one has for N, when
s=1
N=\frac{4\pi }{3}{\left(mc\right)}^{\frac{3}{2}}{\left(\frac{4\pi \text{e}m}{3{h}^{2}}\right)}^{\frac{3}{2}}{r}_{0}.
\frac{E}{N}=m{c}^{2}
(lower bound for E), with
m=1.6\times {10}^{-27}\text{kg}
the baryon
mass, then the fermion number becomes
N=1.15\times {10}^{48}
at the critical volume referred to above, a very small number compared to today’s estimate of 1079. Of course, radiation, neutrinos and plasma predominate at this stage [5].
2.5. Temperature at the End of the Inflationary Period
The entropy at the end of the inflationary period (
s=1
) reads
{S|}_{s=1}={{k}_{B}\mathrm{ln}\Omega |}_{s=1},
{k}_{B}
the Boltzmann constant. For the temperature T we have
{\frac{1}{T}|}_{s=1}={\frac{\partial S}{\partial E}|}_{\left(V,N,s=1\right)},
{\frac{1}{T}|}_{s=1}={\frac{\partial S}{\partial {s}^{3}}|}_{\left(V,N,s=1\right)}{\frac{\partial {s}^{3}}{\partial E}|}_{\left(V,N,s=1\right)}.
More explicitly, one has
{\frac{1}{T}|}_{s=1}=\frac{2\pi {r}_{0}^{3}{k}_{B}{m}^{2}c}{{h}^{3}}{\left(\frac{4\pi \text{e}}{3}\right)}^{\frac{3}{2}}\mathrm{ln}\left(1+N\right).
Using the appropriate values for
{k}_{B}
, h, etc., we obtain
T=3.9\times {10}^{18}
Kelvin degrees, which agrees with the value estimated in reference [6]. Note that in our case we are advancing a statistical prediction.
We have here introduced a simple, micro-canonical statistical argumentation purporting to show that, on account of Pauli’s principle, N fermions can not be accommodated in an arbitrarily small volume V, as bosons can. We statistically determined a minimum critical volume-value for N fermions. One is then tempted to extrapolate the above findings to a cosmological setting and predict a numerical value for the temperature prevailing at the end of the inflationary period. Our prediction agrees with current estimates.
Cite this paper: Plastino, A. , Rocca, M. , Ferri, G. , (2020) Pauli Principle, Inflation and Simple Statistical Treatment of Free-Fermions. Journal of High Energy Physics, Gravitation and Cosmology, 6, 443-449. doi: 10.4236/jhepgc.2020.63034.
[1] The Kavli Foundation (2014) “First Second of the Big Bang”. How the Universe Works, Discovery Science.
[2] Tyson, A., de Grasse, N. and Goldsmith, D. (2004) Origins: Fourteen Billion Years of Cosmic Evolution. W. W. Norton & Co., New York, pp. 84-85.
[3] Tsujikawa, S. (2003) Introductory Review of Cosmic Inflation. https://arxiv.org/abs/hep-ph/0304257
[4] Lemons, D.S. (2014) A Student’s Guide to Entropy. Cambridge University Press, Cambridge.
[5] Siegel, E. (2017) FORBES, Starts with a Bang.
https://www.forbes.com/sites/startswithabang/2017/03/24/how-big-was-the-universe-at-the-
moment-of-its-creation/#43e61dcf4cea
[6] Jose F. Garcia. http://lahoracero.org/fisica-de-particulas-big-bang-temperatura-energia-y-lhc/
|
Revenge - The RuneScape Wiki
Revenge is a threshold Defence ability that requires a shield to use. When activated, any attacks the user receives increases the user's own max hit by 10% of the normalised damage range, up to a maximum of 100% stacked. This ability lasts 20 seconds. Blocked and missed attacks do add Revenge stacks, as do hits blocked by Barricade. Removing the shield while Revenge is still active removes all stacks immediately. Hard typeless hits do not add Revenge stacks.
Revenge has a unique buff icon: the icon's upper half shows Revenge's remaining time active in seconds, and the lower half shows the player's current number of Revenge stacks, which are enclosed in parenthesis. Revenge increases max hit by only 5% per stack if activated with a defender.
If Revenge is used with Raids armour and a defensive reset occurs and Revenge is activated again before the duration of the first has ended, the stacks are all reset to zero.
3 Monsters that use Revenge
The following calculations assume the use of a shield and no Raids armour.
Average hit formula for abilities
Without Revenge :
{\displaystyle a={\frac {M+m}{2}}}
With Revenge :
{\displaystyle A={\frac {M\cdot (100\%+10\%\cdot {\frac {M-m}{M}}\cdot s)+m}{2}}}
Average hit increase formula:
{\displaystyle {\frac {A-a}{a}}=a_{increase}}
{\displaystyle m}
= Minimum ability damage
{\displaystyle M}
= Maximum ability damage
{\displaystyle s}
= number of revenge stacks
{\displaystyle a_{increase}}
= Average hit increase
{\displaystyle a}
= Average hit without revenge
{\displaystyle A}
= Average hit with revenge
Average hit increase for Slice (damage range 30–120%) with 10 Revenge stacks:
{\displaystyle a={\frac {120\%+30\%}{2}}=75\%}
{\displaystyle A={\frac {120\%\cdot (100\%+10\%\cdot {\frac {120\%-30\%}{120\%}}\cdot 10)+30\%}{2}}=120\%}
{\displaystyle a_{increase}={\frac {129\%-75\%}{75\%}}=60\%}
Comparison with 2 handed or dual wield weapons
The downside of revenge is that you need to wear a shield to use it, this means you ability damage will be reduced to 2/3 of the ability damage of 2 handed or dual wield. So we need a minimum average hit increase to be on par or better than 2 handed or dual wield. This minimum can be calculated as follows:
{\displaystyle {\frac {2}{3}}\cdot (a+a\cdot a_{increase}^{min})\geq a}
{\displaystyle \Rightarrow {\frac {2}{3}}\cdot (1+a_{increase}^{min})\geq 1}
{\displaystyle \Rightarrow a_{increase}^{min}\geq {\frac {3}{2}}-1=50\%}
Now we can use this to calculate the average damage increase compared to 2 handed or dual wield. We can calculate this as follows:
{\displaystyle a_{increase}^{2H}={\frac {2}{3}}\cdot (a_{increase}^{Revenge}-50\%)}
The following tables assume the use of a shield and no Raids armour.
Typically, abilities have ability damage ranges from 20% to 100%; this can be represented as a ratio 1:5. As an example, Havoc's ability damage range is 31.4–157%; this can be calculated as 20–100%, as the percentage range ratio is 1:5.
As such, Revenge increases the ability damage of abilities with a 1:5 ratio as follows:
Avg hit increase
Avg hit increase compared to 2H or dual wield
0 0% 60% 0% -33,33%
1 10% 64% 6,67% -28,89%
2 20% 68% 13,33% -24,44%
3 30% 72% 20% -20%
6 60% 84% 40% -6,67%
7 70% 88% 46,67% -2,22%
8 80% 92% 53,33% 2,22%
9 90% 96% 60% 6,67%
10 100% 100% 66,67% 11,11%
The table above applies to most abilities, with the exception of abilities with bleed effects.
For abilities with an ability damage ratio of 1:4 such as Slice (30–120%), the average hit increase of Revenge varies. The table below lists the average hit increase of these abilities:
Max hit (Normal)
Max hit (w/ 10 stacks)
Average hit (Normal)
Average hit (w/ 10 stacks)
Average hit increase
Slice 30% 120% 210% 75% 120% 60% 6,67%
Slice (Stun) 80% 146% 212% 113% 146% 29% -13,86%
Overpower 200% 400% 600% 300% 400% 33% -11,11%
Hurricane 84% 161% 238% 123% 161% 31% -12,38%
Hurricane (Adjacent) 66% 219% 372% 143% 219% 54% 2,46%
Meteor Strike 250% 350% 450% 300% 350% 17% -22,22%
Pulverise 250% 350% 450% 300% 350% 17% -22,22%
Frenzy (Hit #1) 80% 180% 280% 130% 180% 38% -7,69%
Frenzy (Hit #3) 100% 220% 340% 160% 220% 38% -8,33%
Snap Shot (Hit #1) 100% 120% 140% 110% 120% 9% -27,27%
Snap Shot (Hit #2) 100% 210% 320% 155% 210% 35% -9,68%
Snipe 125% 219% 313% 172% 219% 27% -15,12%
Shadow Tendrils (Self) 33% 250% 467% 142% 250% 77% 17,79%
Shadow Tendrils (Enemy) 66% 500% 934% 283% 500% 77% 17,79%
Omnipower 200% 400% 600% 300% 400% 33% -11,11%
Detonate 100% 350% 600% 225% 350% 56% 3,70%
Tsunami 200% 300% 400% 250% 300% 20% -20,00%
Storm Shards &
Shatter (1 stack) 75% 95% 115% 85% 95% 12% -25,49%
Onslaught (Hit #1) 33% 150% 267% 92% 150% 64% 9,29%
Onslaught (Hit #2) 44% 183% 322% 114% 183% 61% 7,49%
Onslaught (Hit #8) 110% 381% 652% 246% 381% 55% 3,46%
Onslaught (Hit #10) 132% 447% 762% 290% 447% 54% 2,94%
Monsters that use Revenge[edit | edit source]
The target info panel in PvP will no longer display the amount of stacks the Revenge ability has on every icon.
The Revenge ability has had its effect time doubled and cooldown increased to 45 seconds.
Retrieved from ‘https://runescape.wiki/w/Revenge?oldid=35832109’
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Sensors | Special Issue : Mobile Communications in 5G Networks
Mobile Communications in 5G Networks
Special Issue "Mobile Communications in 5G Networks"
Prof. Dr. Peppino Fazio
Department of Molecular Sciences and Nanosystems, Ca' Foscari University of Venice, Via Torino 155, 30172 Mestre, Italy
Interests: mobile communications; channel modeling; vehicular protocols; predictive algorithms; mobility analysis; wireless communications
Special Issue in Electronics: Unmanned Aerial Vehicles: Platforms, Applications, Security and Services
Prof. Dr. Miroslav Voznak
Faculty of Electrical Engineering and Computer Science VŠB-Technical University of Ostrava, 708 00 Ostrava, Czech Republic
Interests: non-intrusive speech quality evaluation methods; quality of service (QoS); quality of experience (QoE) and security of real-time applications in networks; Traffic modeling; wireless communication; internet of things; energy harvesting; network security; big data analytics in networks
Special Issue in Journal of Sensor and Actuator Networks: Energy Management in Distributed Wireless Networks
Special Issue in Sensors: Internet of Things and Machine-to-Machine Communication
Prof. Dr. Miralem Mehić
Department of Telecommunications, Faculty of Electrical Engineering, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina
Interests: quantum cryptography; network security; network simulations; routing protocols; quality of service; steganography; authentication and VoIP
In the last few decades, the paradigm of mobile communications redefined the way users take advantage of technology, both from a comfort and performance point of view. The deployment of the new protocols and technologies for mobile scenarios, such as 5G, HetNets, UAVs, and Autonomous Vehicles, and the integration of different heterogeneous research fields with them, has led to the enhancement of life quality, and the way human actions are conceived. In fact, in 5G environments, the pervasive nature of IP, e.g., IoT, gives us the possibility to put under observation several aspects of our life (environment, health, energy consumption, etc.), in a completely transparent way, especially if mobility is taken into consideration, and having the possibility to exploit the main benefits of new generation networks.
For these reasons, it is important to contribute to the scientific community, with solid ideas regarding the exploiting of new mobile TLC/IT technologies, and their integration with the last available technologies (such as 5G), along with several devices and/or sensors, generally used in heterogeneous research fields. So, the main aim of this SI is to collect solid works related to the aforementioned concepts; in particular, the possible research topics of interest include (but are not limited to) the following:
Integration of different and recent mobile communications technologies with 5G standards;
Protocols and algorithms for environmental sensing suitable for 5G networks;
LPWAN technologies for Wireless Sensor Networks and IoT;
Applications of mobile and/or wearable sensing for healthcare based on 5G connectivity;
Ultra-low latency and real time applications in 5G networks;
Vehicular protocols and their integration with other technologies (VANETs, 4G, 5G, mobile IP, etc.);
Network planning and network slicing in the context of 5G networks;
Applications and Protocols related to sustainable computing in the context of 5G;
Beyond 5G: 6G enabling technologies.
Ales Povalac
The last decade has transformed wireless access technologies and crystallized a new direction for the internet of things (IoT). The modern low-power wide-area network (LPWAN) technologies have been introduced to deliver connectivity for billions of devices while keeping the costs and consumption low, [...] Read more.
The last decade has transformed wireless access technologies and crystallized a new direction for the internet of things (IoT). The modern low-power wide-area network (LPWAN) technologies have been introduced to deliver connectivity for billions of devices while keeping the costs and consumption low, and the range of communication high. While the 5G (fifth generation mobile network) LPWAN-like radio technologies, namely NB-IoT (narrowband internet of things) and LTE-M (long-term evolution machine type communication) are emerging, the long-range wide-area network (LoRaWAN) remains extremely popular. One unique feature of this technology, which distinguishes it from the competitors, is the possibility of supporting both public and private network deployments. In this paper we focus on this aspect and deliver original results comparing the performance of the private and public LoRAWAN deployment options; these results should help understand the LoRaWAN technology and give a clear overview of the advantages and disadvantages of the private versus public approaches. Notably, we carry the comparison along the three dimensions: the communication performance, the security, and the cost analysis. The presented results illustratively demonstrate the differences of the two deployment approaches, and thus can support selection of the most efficient deployment option for a target application. Full article
(This article belongs to the Special Issue Mobile Communications in 5G Networks)
Moment-Based Parameter Estimation for the Γ-Parameterized TWDP Model
Two-wave with diffuse power (TWDP) is one of the most promising models for description of a small-scale fading effects in the emerging wireless networks. However, its conventional parameterization based on parameters K and
\Delta
is not in line with model’s underlying physical mechanisms. [...] Read more.
\Delta
is not in line with model’s underlying physical mechanisms. Accordingly, in this paper, we first identified anomalies related to usage of conventional TWDP parameterization in moment-based estimation, showing that the existing
\Delta
-based estimators are unable to provide meaningful estimates in some channel conditions. Then, we derived moment-based estimators of recently introduced physically justified TWDP parameters K and
\Gamma
and analyzed their performance through asymptotic variance (AsV) and Cramer–Rao bound (CRB) metrics. Performed analysis has shown that
\Gamma
-based estimators managed to overcome all anomalies observed for
\Delta
-based estimators, simultaneously improving the overall moment-based estimation accuracy. Full article
In this work, toward an intelligent radio environment for 5G/6G, design methodologies of active split-ring resonators (SRRs) for more efficient dynamic control of metasurfaces are investigated. The relationship between the excitation of circulating-current eigenmode and the asymmetric structure of SRRs is numerically analyzed, [...] Read more.
In this work, toward an intelligent radio environment for 5G/6G, design methodologies of active split-ring resonators (SRRs) for more efficient dynamic control of metasurfaces are investigated. The relationship between the excitation of circulating-current eigenmode and the asymmetric structure of SRRs is numerically analyzed, and it is clarified that the excitation of the circulating-current mode is difficult when the level of asymmetry of the current path is decreased by the addition of large capacitance such as from semiconductor-based devices. To avoid change in the asymmetry, we incorporated an additional gap (slit) in the SRRs, which enabled us to excite the circulating-current mode even when a large capacitance was implemented. Prototype devices were fabricated according to this design methodology, and by the control of the intensity/phase distribution, the variable focal-length and beamsteering capabilities of the transmitted waves were demonstrated, indicating the high effectiveness of the design. The presented design methodology can be applied not only to the demonstrated case of discrete varactors, but also to various other active metamaterials, such as semiconductor-integrated types for operating in the millimeter and submillimeter frequency bands as potential candidates for future 6G systems. Full article
Trong-Dinh Huy
Ahmed Younes Shdefat
Maryam Altalhi
Reaching a flat network is the main target of future evolved packet core for the 5G mobile networks. The current 4th generation core network is centralized architecture, including Serving Gateway and Packet-data-network Gateway; both act as mobility and IP anchors. However, this architecture [...] Read more.
Reaching a flat network is the main target of future evolved packet core for the 5G mobile networks. The current 4th generation core network is centralized architecture, including Serving Gateway and Packet-data-network Gateway; both act as mobility and IP anchors. However, this architecture suffers from non-optimal routing and intolerable latency due to many control messages. To overcome these challenges, we propose a partially distributed architecture for 5th generation networks, such that the control plane and data plane are fully decoupled. The proposed architecture is based on including a node Multi-session Gateway to merge the mobility and IP anchor gateway functionality. This work presented a control entity with the full implementation of the control plane to achieve an optimal flat network architecture. The impact of the proposed evolved packet Core structure in attachment, data delivery, and mobility procedures is validated through simulation. Several experiments were carried out by using NS-3 simulation to validate the results of the proposed architecture. The Numerical analysis is evaluated in terms of total transmission delay, inter and intra handover delay, queuing delay, and total attachment time. Simulation results show that the proposed architecture performance-enhanced end-to-end latency over the legacy architecture. Full article
Performance Analyses of Energy Detection Based on Square-Law Combining in MIMO-OFDM Cognitive Radio Networks
Cognitive radio (CR) technology has the potential to detect and share the unutilized spectrum by enabling dynamic spectrum access. To detect the primary users’ (PUs) activity, energy detection (ED) is widely exploited due to its applicability when it comes to sensing a large [...] Read more.
Cognitive radio (CR) technology has the potential to detect and share the unutilized spectrum by enabling dynamic spectrum access. To detect the primary users’ (PUs) activity, energy detection (ED) is widely exploited due to its applicability when it comes to sensing a large range of PU signals, low computation complexity, and implementation costs. As orthogonal frequency-division multiplexing (OFDM) transmission has been proven to have a high resistance to interference, the ED of OFDM signals has become an important local spectrum-sensing (SS) concept in cognitive radio networks (CRNs). In combination with multiple-input multiple-output (MIMO) transmissions, MIMO-OFDM-based transmissions have started to become a widely accepted air interface, which ensures a significant improvement in spectral efficiency. Taking into account the future massive implementation of MIMO-OFDM systems in the fifth and sixth generation of mobile networks, this work introduces a mathematical formulation of expressions that enable the analysis of ED performance based on the square-law combining (SLC) method in MIMO-OFDM systems. The analysis of the ED performance was done through simulations performed using the developed algorithms that enable the performance analysis of the ED process based on the SLC in the MIMO-OFDM systems having a different number of transmit (Tx) and receive (Rx) communication branches. The impact of the distinct factors including the PU Tx power, the false alarm probability, the number of Tx and Rx MIMO branches, the number of samples in the ED process, and the different modulation techniques on the ED performance in environments with different levels of signal-to-noise ratios are presented. A comprehensive analysis of the obtained results indicated how the appropriate selection of the analyzed factors can be used to enhance the ED performance of MIMO-OFDM-based CRNs. Full article
An Alternative Statistical Characterization of TWDP Fading Model
Two-wave with diffuse power (TWDP) is one of the most promising models for the description of small-scale fading effects in 5G networks, which employs mmWave band, and in wireless sensor networks deployed in different cavity environments. However, its current statistical characterization has several [...] Read more.
Two-wave with diffuse power (TWDP) is one of the most promising models for the description of small-scale fading effects in 5G networks, which employs mmWave band, and in wireless sensor networks deployed in different cavity environments. However, its current statistical characterization has several fundamental issues. Primarily, conventional TWDP parameterization is not in accordance with the model’s underlying physical mechanisms. In addition, available TWDP expressions for PDF, CDF, and MGF are given either in integral or approximate forms, or as mathematically untractable closed-form expressions. Consequently, the existing TWDP statistical characterization does not allow accurate evaluation of system performance in all fading conditions for most modulation and diversity techniques. In this regard, physically justified TWDP parameterization is proposed and used for further calculations. Additionally, exact infinite-series PDF and CDF are introduced. Based on these expressions, the exact MGF of the SNR is derived in a form suitable for mathematical manipulations. The applicability of the proposed MGF for derivation of the exact average symbol error probability (ASEP) is demonstrated with the example of M-ary PSK modulation. The derived M-ary PSK ASEP expression is further simplified for large SNR values in order to obtain a closed-form asymptotic ASEP, which is shown to be applicable for SNR > 20 dB. All proposed expressions are verified by Monte Carlo simulation in a variety of TWDP fading conditions. Full article
PHY, MAC, and RLC Layer Based Estimation of Optimal Cyclic Prefix Length
This work is motivated by growing evidence that the standard Cyclic Prefix (CP) length, adopted in the Long Term Evolution (LTE) physical layer (PHY) specifications, is oversized in propagation environments ranging from indoor to typical urban. Although this ostensibly seems to be addressed [...] Read more.
This work is motivated by growing evidence that the standard Cyclic Prefix (CP) length, adopted in the Long Term Evolution (LTE) physical layer (PHY) specifications, is oversized in propagation environments ranging from indoor to typical urban. Although this ostensibly seems to be addressed by 5G New Radio (NR) numerology, its scalable CP length reduction is proportionally tracked by the OFDM symbol length, which preserves the relative CP overhead of LTE. Furthermore, some simple means to optimize fixed or introduce adaptive CP length arose from either simulations or models taking into account only the bit-oriented PHY transmission performance. On the contrary, in the novel crosslayer analytical model proposed here, the closed-form expression for the optimal CP length is derived such as to minimize the effective average codeblock length, by also considering the error recovery retransmissions through the layers above PHY—the Medium Access Control (MAC) and the Radio Link Control (RLC), in particular. It turns out that, for given protective coding, the optimal CP length is determined by the appropriate rms delay spread of the channel power delay profile part remaining outside the CP span. The optimal CP length values are found to be significantly lower than the corresponding industry-standard ones, which unveils the potential for improving the net throughput. Full article
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Canutite, NaMn3[AsO4][AsO3(OH)]2, a new protonated alluaudite-group mineral from the Torrecillas mine, Iquique Province, Chile | Mineralogical Magazine | GeoScienceWorld
GPO Box 666, Melbourne 3001, Victoria
F. Hatert;
F. Hatert
Université de Liège, Laboratoire de Minéralogie
B18, B-4000 Liège
A. R. Kampf, S. J. Mills, F. Hatert, B. P. Nash, M. Dini, A. A. Molina Donoso; Canutite, NaMn3[AsO4][AsO3(OH)]2, a new protonated alluaudite-group mineral from the Torrecillas mine, Iquique Province, Chile. Mineralogical Magazine 2014;; 78 (4): 787–795. doi: https://doi.org/10.1180/minmag.2014.078.4.02
The new mineral canutite (IMA2013-070), NaMn3[AsO4][AsO3(OH)]2, was found at two different locations at the Torrecillas mine, Salar Grande, Iquique Province, Chile, where it occurs as a secondary alteration phase in association with anhydrite, halite, lavendulan, magnesiokoritnigite, pyrite, quartz and scorodite. Canutite is reddish brown in colour. It forms as prisms elongated on [201İ] and exhibiting the forms {010}, {100}, {102}, {201} and {102İ}, or as tablets flattened on {102} and exhibiting the forms {102} and {110}. Crystals are transparent with a vitreous lustre. The mineral has a pale tan streak, Mohs hardness of 2½, brittle tenacity, splintery fracture and two perfect cleavages, on {010} and {101}. The calculated density is 4.112 g cm−3. Optically, canutite is biaxial (+) with α = 1.712(3), β = 1.725(3) and γ = 1.756(3) (measured in white light). The measured 2V is 65.6(4)°, the dispersion is r < v (slight), the optical orientation is Z = b; X ^ a = 18° in obtuse β and pleochroism is imperceptible. The mineral is slowly soluble in cold, dilute HCl. The empirical formula (for tabular crystals from near the mine shaft), determined from electron-microprobe analyses, is (Na1.05Mn2.64Mg0.34Cu0.14Co0.03)Σ4.20As3O12H1.62. Canutite is monoclinic, C2/c, a = 12.3282(4), b = 12.6039(5), c = 6.8814(5) Å, β = 113.480(8)°, V = 980.72(10) Å3 and Z = 4. The eight strongest X-ray powder diffraction lines are [dobs Å(I)(hkl)]: 6.33(34)(020), 4.12(26)(2İ21), 3.608(29)(310,1İ31), 3.296(57)(1İ12), 3.150(28)(002,131), 2.819(42)(400,041,330), 2.740(100)(240,4İ02,112) and 1.5364(31)(multiple). The structure, refined to R1 = 2.33% for 1089 Fo > 4σF reflections, shows canutite to be isostructural with protonated members of the alluaudite group.
Na(As,Sb)43+
ITSIITE, Ba2Ca(BSi2O7)2, A NEW MINERAL SPECIES FROM YUKON, CANADA: DESCRIPTION AND CRYSTAL STRUCTURE
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Detached Eddy Simulation of Film Cooling Performance on the Trailing Edge Cutback of Gas Turbine Airfoils | J. Turbomach. | ASME Digital Collection
P. Martini,
Lehrstuhl und Institut für Thermische Strömungsmaschinen,
, Kaiserstr. 12, 76128 Karlsruhe, Germany
H. -J. Bauer,
H. -J. Bauer
C. F. Whitney
ALSTROM Power Technology Centre
, Cambridge Road, Whetstone, Leicester LE8 6LH, UK
Martini, P., Schulz, A., Bauer, H. -., and Whitney, C. F. (February 1, 2005). "Detached Eddy Simulation of Film Cooling Performance on the Trailing Edge Cutback of Gas Turbine Airfoils." ASME. J. Turbomach. April 2006; 128(2): 292–299. https://doi.org/10.1115/1.2137739
The present study deals with the unsteady flow simulation of trailing edge film cooling on the pressure side cut back of gas turbine airfoils. Before being ejected tangentially on the inclined cut-back surface, the coolant air passes a partly converging passage that is equipped with turbulators such as pin fins and ribs. The film mixing process on the cut back is complicated. In the near slot region, due to the turbulators and the blunt pressure side lip, turbulence is expected to be anisotropic. Furthermore, unsteady flow phenomena like vortex shedding from the pressure side lip might influence the mixing process (i.e., the film cooling effectiveness on the cut-back surface). In the current study, three different internal cooling designs are numerically investigated starting from the steady
RaNS
solution, and ending with unsteady detached eddy simulations (DES). Blowing ratios
M=0.5
; 0.8; 1.1 are considered. To obtain both, film cooling effectiveness as well as heat transfer coefficients on the cut-back surface, the simulations are performed using adiabatic and diabatic wall boundary conditions. The DES simulations give a detailed insight into the unsteady film mixing process on the trailing edge cut back, which is indeed influenced by vortex shedding from the pressure side lip. Furthermore, the time averaged DES results show very good agreement with the experimental data in terms of film cooling effectiveness and heat transfer coefficients.
flow simulation, gas turbines, aerodynamics, flow instability, mixing, cooling, design engineering, vortices
Coolants, Cooling, Eddies (Fluid dynamics), Film cooling, Gas turbines, Pressure, Reynolds-averaged Navier–Stokes equations, Simulation, Turbulence, Heat transfer coefficients, Vortex shedding, Boundary-value problems, Flow (Dynamics), Airfoils, Temperature, Computation
,” AIAA J.
Pressure Side Bleed Film Cooling: Part 1—Steady Framework for Experimental and Computational Results
Pressure Side Bleed Film Cooling: Part 2—Unsteady Framework for Experimental and Computational Results
Calculation of Steady and Unsteady Flows in a Film-Cooling Arrangement Using a Two-Layer Algebraic Stress Model
1040-7782, Part A,
,” 1st AFOSR Int. Conf. on DNS/LES, 4–8, Aug. 1997, Ruston, LA,
Advances in DNS/LES
,” AIAA Paper.
S. -E.
DES of Turbulent Flow Over an Airfoil at High Incidence
Detached-Eddy Simulation of the Separated Flow Around a Forebody Cross-Section
Direct and Large Eddy Simulation—IV
), ISBN-1-4020-0177-0, Kluwer, Dordrecht, pp.
, December,
Recent Enhancements to USM3D Unstructured Flow Solver for Unsteady Flows
First Hybrid Turbulence Modeling for Turbine Blade Cooling
Film Cooling Effectiveness and Heat Transfer on the Trailing Edge Cut-back of Gas Turbine Airfoils With Various Internal Cooling Designs
FLUENT 6.1 User’s Guide, 2003,
On the Application of the Eddy Viscosity Concept in the Inertial Subrange of Turbulence
,” NCAR Manuscript 123.
,” NASA/CR-2001-211032, pp.
ALESSIA Report, 2002, “
EP 28189: Application of Large Eddy Simulation to the Solution of Industrial Problems, Best Practice Guide: LES and Acoustics
,” Issue 1, Internal version 1.4.
Detached Eddy Simulation of Film Cooling Performance on the Trailing Edge Cut-Back of Gas Turbine Airfoils
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Notional AMM - Notional
This document offers a detailed technical description of Notional's liquidity curve
Special Requirements for Trading fCash:
Dynamic Curve Sensitivity: The optimal liquidity curve sensitivity (how much slippage a given trade size incurs) will vary vastly as a function of time to maturity. A static curve sensitivity will only be appropriate for a narrow window of time within a fCash token’s lifespan and will make trading uneconomical for the rest of its lifespan.
Interest Rate Continuity: The same exchange-rate produces a different interest rate depending on a fCash token’s time to maturity. The interest rate is the more relevant measure of a fCash token’s “price”, and the AMM must keep this rate constant between trades, not the exchange-rate. Failing to do so will result in a fCash token’s prevailing interest rate constantly drifting off-market with time. The effect would be minimal throughout most of a fCash token’s lifespan but would blow up exponentially as it approached maturity. The result would be increasingly volatile and unpredictable trading close to maturity and a negative outcome for all involved.
Notional AMM Basics:
First, we need a curve with the right shape. The curve needs to be relatively flat most of the time so that normal trading conditions produce reasonable levels of slippage. But the curve also needs to be able to accommodate market repricings - if the curve is too flat, or flat throughout, it won’t be able to respond effectively to large changes in the equilibrium interest rate. The logit curve has the right general characteristics. Here’s what it looks like.
This curve maps a 0 to 1 x-range onto an exchange-rate. In order to use this curve, we need a measure of the balance between a given pool’s currency to fCash that sits on a 0 to 1 scale. We can use the following:
Proportion = totalFCash / (totalCurrency + totalFCash)
This gives us the following formula for the currency / fCash exchange-rate and the associated implied interest rate:
Exchange Rate = (1 / scalar) * ln(proportion / (1 - proportion)) + anchor \\ ~ \\ Interest Rate = (Exchange Rate - 1) * periodSize / timeToMaturity
Trading on the Notional AMM:
When a user trades on the Notional AMM, the traded exchange rate is calculated in two steps according to the following formulas:
tradeProportion = (totalFCash ± tradeSize) / (totalFCash + totalCash) \\ ~ \\ Trade Exchange Rate = (1 / scalar) * ln(tradeProportion / (1 - tradeProportion)) \\+ anchor ± liquidityFee
Unlike Uniswap, Notional’s AMM does not allow us to easily solve for an exchange rate such that the traded exchange rate equals the prevailing exchange rate after the trade has occurred. The traded exchange rate we use is an approximation of what the exchange rate will be post-trade. Because of this, we need to ensure that this approximation method is suitable for our purposes. The relevant discussion and proofs are included in this document’s appendix.
When a user places a trade, the Notional AMM first updates the mid rate to account for the user’s trade (this is always against the user’s favor) and then adds a liquidity fee on top of that updated mid rate. For example, a given trade might move the prevailing mid interest rate from 5% to 5.25% and execute at 5.5% (a liquidity fee of .25%). This user therefore traded at .5% from the prevailing mid at the time of their trade - .25% from the difference in the mids and .25% from the liquidity fee. So a user’s all-in fee - and a liquidity provider’s all-in charge - can be thought of as a combination of the explicit liquidity fee and the difference between the old mid and the new mid (the slippage). Noting this distinction helps inform the discussion of parameter choice later on in this document.
Trading on the Notional AMM (Example):
Consider a Notional liquidity pool between Dai and 1M Dai (fDai maturing in one month’s time) with the following values:
Anchor: 1.01
Liquidity Fee (absolute terms): .00025
Liquidity Fee (annualized interest rate terms): .30%
Total Currency: 100,000 Dai
Total fCash: 100,000 1M Dai
Proportion: .5
Prevailing Exchange Rate = (1 / 100) * ln(1) + 1.01 = 1.01
Prevailing Interest Rate (annualized) = 12%
A lender comes to Notional to buy 1,000 1M Dai. Notional calculates the user’s trade exchange rate and updates the pool balances accordingly.
tradeProportion = (100,000 - 1,000) / (100,000 + 100,000) = .495
Trade Exchange Rate = (1/100) * ln(.495/.505) + 1.01 - .00025 = 1.00955
Trade Interest Rate (annualized) = 11.46%
Dai sold in exchange for 1,000 1M Dai = 990.5403 Dai
All-In Trading Fee (Dai) = 990.5403 - (1000 / 1.01) = .4403 Dai
All-In Trading Fee (annualized interest rate terms) = .54%
Updated Pool Details:
Total fCash: 99,000 1M Dai
Proportion: .49502
Prevailing Exchange Rate = 1.0098
Prevailing Interest Rate = 11.76%
Notional AMM Parameters:
The Notional AMM is parameterized by three variables - the scalar, anchor and liquidity fee. The scalar and anchor allow us to vary the steepness of the curve and its position in the xy-plane, respectively. Here is the same logit curve with different scalar and anchor values.
Increasing the scalar value flattens the curve, and decreasing the value steepens it. This translates to a less sensitive curve, and less slippage on a given trade, with a greater scalar value. Conversely, a smaller scalar value results in a more sensitive curve, and more slippage on a given trade. Varying the anchor value shifts the curve up and down in the xy-plane.
Moving from a Static Curve to a Dynamic Curve:
The problem of static sensitivity is relevant not only to the liquidity curve itself, but also to the liquidity fee. The same reasoning applies - a constant fee in exchange rate terms will grow exponentially more punitive to end users as fCash tokens approach maturity. To solve this problem, we convert the scalar and liquidity fee into functions of time to maturity, each parameterized by a root value. Making the scalar a function of time to maturity means that the shape of the liquidity curve changes as we approach maturity:
In the Notional system, we normalize rates associated with a given fCash market to periodSize - the amount of time between the inception of a fCash token and its maturity.
timeToMaturity(t) = (maturity - t) \\ ~ \\ scalar(t) = scalarRoot * periodSize / timeToMaturity(t) \\ ~ \\ liquidityFee(t) = liquidityFeeRoot * timeToMaturity(t) / periodSize
Expressing the scalar and liquidity fee in this form maintains a consistent sensitivity and fee throughout the lifecycle of a fCash token. It may seem intuitive that this is true, but it’s not totally obvious - we include a more detailed exposition and proof in the appendix.
Substituting in the changes to scalar and liquidityFee, we have the final exchange rate equations for the Notional AMM.
Exchange Rate = (1 / scalar(t)) * ln(proportion / (1 - proportion)) + anchor \\ ~ \\ Trade Exchange Rate = (1/scalar(t)) * ln(tradeProportion / (1-tradeProportion)) \\+ anchor ± liquidityFee(t)
Preserving Interest Rate Continuity:
As fCash tokens approach maturity, the exchange rate will drift in the absence of any trading. This is problematic because it means that the prevailing interest rate of a pool will also drift over time in the absence of trading. This drift will increase exponentially as fCash tokens approach maturity. Interest rate drift presents clever traders the opportunity to gradually siphon value from liquidity providers over time.
The Notional AMM prevents this by updating the anchor upon each trade such that the pre-trade interest rate equals the interest rate immediately after the last trade occurred. This mechanism preserves consistent interest rates over time in the absence of trading. Implementation is relatively straightforward, details are included in the appendix.
Parameter Choice Implications - Economics:
The values chosen for the scalar and the liquidity fee have economic implications for the system. Both of these variables have the effect of shifting the economic balance between users and liquidity providers. The liquidity fee has that effect explicitly. But the scalar changes that balance as well, though somewhat less directly. Recall that the all-in fee a user pays can be decomposed into the liquidity fee + the slippage. Changing the scalar changes the slippage. It’s important to pick appropriate scalar and liquidity fee values that balance the interests of users and liquidity providers. Liquidity provider profitability is critical, but charging users too much will impede the system’s growth and success.
Parameter Choice Implications - Interest Rate Boundaries:
Choosing the anchor value decides where the flatter part of the liquidity curve sits in interest rate terms. Given the shape of the logit curve, trading conditions between proportion values of .1 and .9 can broadly be described as “normal”. The exponential curvature really starts to kick in past those points. The anchor value chosen upon the curve’s instantiation determines the range of interest rates that can be traded “normally”. The scalar value determines the absolute distance in interest rate terms between the interest rate at a proportion of .9 and the interest rate at a proportion of .1. For example, a scalar value of 100 implies a distance of 52.73% between the interest rate at .1 and the interest rate at .9 in a one-month maturity. The anchor value determines what interest rate sits in the middle of that range.
Traded Exchange Rates:
We need to ensure that the traded exchange rate is always worse than (from the user’s perspective) or equal to the pool’s exchange rate after the trade has occurred. If this is not true, the mechanism is vulnerable to arbitrage and manipulation. Recall the formulas for determining traded exchange rates:
tradeProportion = (totalFCash ± tradeSize) / (totalFCash + totalCash) \\ ~ \\ Trade Exchange Rate = (1 / scalar) * ln(tradeProportion / (1 - tradeProportion)) \\+ anchor ± liquidityFee
For the purposes of this proof we are going to allow tradeSize to be a negative number (this is easier to deal with mathematically than the above equation for tradeProportion). This gives the below equation.
tradeProportion = (totalFCash + tradeSize) / (totalFCash + totalCash)
A positive value of tradeSize means that a user has sold fCash and increased the supply of fCash within the pool. A negative value of X means that a user has bought fCash and decreased the supply of fCash within the pool. Trades and exchange rates are always specified in terms of fCash, never in terms of current cash. Thus, somewhat counterintuitively, a user would always prefer to sell her fCash at a lower exchange rate (a lower exchange rate implies that fCash is worth more in current cash terms).
Here is the proof that tradeExchangeRate is always worse than endExchangeRate (the exchange rate after the trade has occurred). Note - we rely on the fact that we do not allow exchange rates below 1 (i.e. negative interest rates) in this proof. Notional will revert upon a trade if it produces a negative interest rate.
X = tradeSize \\ Y = cashAmountTraded \\ TFC = totalFCash \\ TC = totalCash \\ ~ \\ If X > 0 \\ ~ \\tradeProportion = TFC + X / (TFC + TC) \\ endProportion = TFC+X/(TFC+X+TC-Y)=TFC+X/((TFC+TC)+(X-Y)) \\ exchangeRate > 1 -> X > Y \\ -> (X - Y) > 0 \\ -> ((TFC + TC) + (X - Y)) > (TFC + TC) \\ -> tradeProportion > endProportion \\ -> tradeExchangeRate > endExchangeRate \\ ~ \\ If X < 0 \\ ~\\tradeProportion = TFC + X / (TFC + TC) \\ endProportion = TFC+X/(TFC+X+TC-Y)=TFC+X/((TFC+TC)+(X-Y)) \\ exchangeRate > 1 -> X < Y \\ -> (X - Y) < 0 \\ -> ((TFC + TC) + (X - Y)) < (TFC + TC) \\ -> tradeProportion < endProportion \\ -> tradeExchangeRate < endExchangeRate
Consistent Curve Sensitivity:
We can represent the sensitivity of the liquidity curve as the following derivative.
d \text{ }interestRate / d \text{ }proportion
It’s straightforward to show that this derivative changes as a function of time to maturity.
interestRate=(ln(proportion / (1-proportion)) / scalar+anchor-1)\\*periodSize / timeToMaturity \\~\\= ln(p / (1-p)) / scalar *periodSize / tTM + (anchor-1) * periodSize / tTM \\ ~ \\ ~\\ d \text{ } interestRate / d \text{ }proportion = (p / (1-p))' * ((1-p) / p) * (periodSize / (scalar*tTM)) \\ ~ \\ = 1 / (1-p)2 * ((1-p) / p) * (periodSize / (scalar*tTM)) \\~\\ = 1 / p(1-p)* (periodSize / (scalar*tTM)) \\~\\(d\text{ }interestRate / d \text{ }proportion) / d\text{ } tTM = -periodSize / scalar * tTM2 / p(1-p)
We want the sensitivity of the liquidity curve to be constant through time - in effect we want the derivative of the sensitivity with respect to time to maturity to equal 0. Varying scalar with time to maturity achieves this goal.
scalar =scalarRoot * periodSize / timeToMaturity \\ ~ \\ d \text{ } interestRate / d\text{ } proportion =1 / p(1-p)* (periodSize / (scalar*tTM)) \\ ~ \\ =1 / p(1-p)* (1 / scalarRoot) \\~\\ (d \text{ }interestRate / d\text{ } proportion) / d\text{ } timeToMaturity = 0
Interest Rate Continuity:
To counteract implied interest rate drift, we use the anchor variable to keep interest rates consistent over time in the absence of any trading. After each trade we save the implied interest rate. Upon the next trade we check to see if the current implied interest rate == the saved implied interest rate. If it does not, we update the anchor such that it does prior to executing the trade. The anchor is just a constant in Notional’s exchange rate formula, so we don’t need to worry that this has any unintended effects. Here is how we solve for a new anchor value.
interestRateDifference =currentInterestRate - savedInterestRate \\ ~ \\newAnchor =anchor -interestRateDifference * (tTM / periodSize)
newExchangeRate = (1 / scalar(t)) * ln(proportion / (1 - proportion)) + newAnchor \\~\\newExchangeRate = (1 / scalar(t)) * ln(proportion / (1 - proportion)) \\ + anchor -interestRateDifference * (tTM / periodSize) \\~\\newExchangeRate = currentExchangeRate - interestRateDifference * (tTm / periodSize)\\~\\newInterestRate = (currentExchangeRate -interestRateDifference *(tTm / periodSize)-1) \\* periodSize/tTM\\~\\= (currentExchangeRate - 1) * periodSize/tTM - interestRateDifference\\~\\= currentInterestRate - interestRateDifference\\~\\= savedInterestRate
Risk and Collateralization - Previous
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Simple Singular Perturbation Problems with Turning Points
Simple Singular Perturbation Problems with Turning Points ()
Department of Applied Mathematics, Shanghai Institute of Technology, Shanghai, China.
The paper considers the asymptotic solution of two-point boundary value problems εy” + A(x)y’ = 0, 0 ≤ x ≤ 1, when 0 < ε ≪ 1, A(x) is smooth with isolated zeros, y(0) = 0 and y(1) = 1. By using perturbation method, the limit asymptotic solutions of various cases are obtained. We provide a reliable and direct method for solving similar problems. The limiting solutions are constants in this paper, except in narrow boundary and interior layers of nonuniform convergence. These provide simple examples of boundary layer resonance.
Singular Perturbations, Asymptotic Methods, Boundary Value Problems, Turning Points, Boundary and Interior Layers, Boundary Layer Resonance
Wang, N. (2019) Simple Singular Perturbation Problems with Turning Points. Journal of Applied Mathematics and Physics, 7, 2979-2989. doi: 10.4236/jamp.2019.712208.
A typical turning point problem consists of the linear differential equation
\epsilon {y}^{″}-x{y}^{\prime }+ny=0,
for a nonnegative integer n on
-1\le x\le 1
with prescribed boundary values
y\left(\pm 1\right)
and a small positive parameter
\epsilon
0<\epsilon \ll 1
. Limiting solutions, away from narrow so-called boundary and interior shock layers of rapid change, take the form
{Y}_{0}\left(x\right)={x}^{n}C
\epsilon \to 0
for constants C, so satisfy the limiting reduced equation
x{{Y}^{\prime }}_{0}=n{Y}_{0}.
Determining constants C and the location of layers is a nontrivial task, the subject of boundary layer resonance [1]. It involves detailed asymptotic analysis and often uses special functions. The classical techniques of matched asymptotic expansions [2] [3] and the boundary function method of Vasil’eva et al. [4] may break down, though the newer composite asymptotic expansions [5] seem to apply. Many experts have studied such problems over the last fifty years [6] [7] [8] for surveys. An important application to stochastic differential equations in described in the 2017 SIAM von Neumann lecture by Matkowsky [9]. Computing solutions to such problems remains a challenge, although Trefethen et al. [10] succeed for some examples using the program Chebfun.
A simple, but still rich, the related problem concerns the asymptotic solution of the two-point problem [11]
\epsilon {y}^{″}+A\left(x\right){y}^{\prime }=0,0\le x\le 1,
with the special boundary values
y\left(0\right)=0\text{and}y\left(1\right)=1
and a smooth coefficient
A\left(x\right)
. Its unique exact solution is
y\left(x,\epsilon \right)=\frac{{\int }_{0}^{x}\text{ }\text{ }{\text{e}}^{-\frac{1}{\epsilon }{\int }_{0}^{s}\text{ }\text{ }A\left(t\right)\text{d}t}\text{d}s}{{\int }_{0}^{1}\text{ }\text{ }{\text{e}}^{-\frac{1}{\epsilon }{\int }_{0}^{s}\text{ }\text{ }A\left(t\right)\text{d}t}\text{d}s}.
{y}^{\prime }\left(x,\epsilon \right)=\frac{{\text{e}}^{-\frac{1}{\epsilon }{\int }_{0}^{x}\text{ }\text{ }A\left(t\right)\text{d}t}}{{\int }_{0}^{1}\text{ }\text{ }{\text{e}}^{-\frac{1}{\epsilon }{\int }_{0}^{s}\text{ }\text{ }A\left(t\right)\text{d}t}\text{d}s}>0,
the solution y will increase monotonically with x. The asymptotic value of
{\int }_{0}^{x}\text{ }\text{ }I\left(s,\epsilon \right)\text{d}s
in (6) is the area under the curve
I\left(s,\epsilon \right)\equiv {\text{e}}^{-\frac{1}{\epsilon }{\int }_{0}^{s}\text{ }\text{ }A\left(t\right)\text{d}t}
\epsilon \to 0
. Sophisticated techniques to obtain the asymptotic evaluation of integrals can be found in Olver [12], Wong [13] and elsewhere. Simple arguments often provide the limiting ratio (6), often after rescaling I.
The variety of limiting behaviors to singularly perturbed linear two-point boundary value problems with turning points has not been clearly described. The first papers by Pearson in 1968 stressed a numerical approach. In the intervening fifty years, software has improved tremendously, though finding the limiting solution is extremely ill-conditioned as Trefethen recently observed. Due to the serious instability of direct numerical methods, the examples found in scattered literature are usually less detailed. Inspired by this, in this paper we consider the asymptotic solution of two-point boundary value problems (4)-(5). In our examples, we’ll find the constant “outer” limits 0, 0.5, and 1.
A\left(x\right)>0
I\left(s,\epsilon \right)
\epsilon \to 0
, so for any fixed
x>0
, the numerator and denominator of (6) are both
O\left(\epsilon \right)
and the ratio (6) is asymptotically one. Since
y\left(0,\epsilon \right)=0
, there is an initial boundary layer region of
O\left(\epsilon \right)
thickness involving nonuniform convergence of y. Here, we’re using the big O order symbol.
As an example, take
A\left(x\right)\equiv 1
\epsilon =0.1,0.01
and 0.001, and plot the solution (6). One gets
y\left(x,\epsilon \right)=\frac{\frac{1}{\epsilon }{\int }_{0}^{x}\text{ }\text{ }{\text{e}}^{-\frac{s}{\epsilon }}\text{d}s}{\frac{1}{\epsilon }{\int }_{0}^{1}\text{ }\text{ }{\text{e}}^{-\frac{s}{\epsilon }}\text{d}s}=\frac{1-{\text{e}}^{-\frac{x}{\epsilon }}}{1-{\text{e}}^{-\frac{1}{\epsilon }}}.
The constant limiting solution
{Y}_{0}\left(x\right)=1
x>0
\epsilon \to 0
, satisfies the reduced equation
{{Y}^{\prime }}_{0}=0
x=0
. We plot the solution for three small
\epsilon
values in Figure 1.
y\left(x,\epsilon \right)
x=0
, signaling nonuniform convergence.
A\left(x\right)<0
I\left(x,\epsilon \right)
grows exponentially large as
\epsilon \to 0
. This causes y to be asymptotically zero for any
x<1
and a terminal boundary layer of nonuniform convergence to occur near
x=1
A\left(x\right)\equiv -1
and plot
y\left(x,\epsilon \right)=\frac{\frac{{\text{e}}^{-\frac{1}{\epsilon }}}{\epsilon }{\int }_{0}^{x}\text{ }\text{ }{\text{e}}^{\frac{s}{\epsilon }}\text{d}s}{\frac{{\text{e}}^{-\frac{1}{\epsilon }}}{\epsilon }{\int }_{0}^{1}\text{ }\text{ }{\text{e}}^{\frac{s}{\epsilon }}\text{d}s}=\frac{{\text{e}}^{\frac{x-1}{\epsilon }}-{\text{e}}^{-\frac{1}{\epsilon }}}{1-{\text{e}}^{-\frac{1}{\epsilon }}}.
x<1
, the limiting solution
{\text{e}}^{\frac{x-1}{\epsilon }}
\epsilon \to 0
is trivial. The limiting terminal layer will have
O\left(\epsilon \right)
y\left(x,\epsilon \right)
A\left(x\right)=x-0.5
Now there’s a simple turning point at
x=0.5
I\left(s,\epsilon \right)={\text{e}}^{-\frac{1}{2\epsilon }\left[{\left(s-\frac{1}{2}\right)}^{2}-\frac{1}{4}\right]}={\text{e}}^{\frac{1}{8\epsilon }}{\text{e}}^{-\frac{1}{2\epsilon }{\left(s-\frac{1}{2}\right)}^{2}}.
We write the ratio (6) as
y\left(x,\epsilon \right)=\frac{{\int }_{0}^{x}\text{ }\text{ }I\left(s,\epsilon \right)\text{d}s}{{\int }_{0}^{1}\text{ }\text{ }I\left(s,\epsilon \right)\text{d}s}.
y\left(x,\epsilon \right)=\frac{1}{2}\left(1+\frac{erf\left(\frac{x-\frac{1}{2}}{\sqrt{2\epsilon }}\right)}{erf\left(\frac{1}{2\sqrt{2\epsilon }}\right)}\right),
erf\left(z\right)=\frac{2}{\sqrt{\pi }}{\int }_{0}^{z}\text{ }\text{ }{\text{e}}^{-{t}^{2}}\text{d}t
is the error function [14]. It satisfies
{z}^{″}+x{z}^{\prime }=0
, it is odd, it increases monotonically, and it tends to ±1 as
x\to \pm \infty
Since the integrands of (11) peak at the turning point and are asymptotically negligible elsewhere, we will have
y\left(x,\epsilon \right)\sim \left(\begin{array}{l}0,\text{for}\text{\hspace{0.17em}}\text{ }x<\frac{1}{2}\\ 1,\text{ }\text{for}\text{\hspace{0.17em}}\text{ }x>\frac{1}{2}.\end{array}
The numerator and denominator of (11) are both
O\left(\sqrt{\epsilon }\right)
y\left(\frac{1}{2},\epsilon \right)=\frac{1}{2}
, there’s antisymmetry about
s=\frac{1}{2}
, and an
O\left(\sqrt{\epsilon }\right)
thick region of nonuniform convergence about the midpoint.
Plotting the solution (12) for
\epsilon ={10}^{-3}
, we get Figure 2.
A\left(x\right)=x-\alpha ,0<\alpha <1
We rescale I to get
{\text{e}}^{-\frac{{\alpha }^{2}}{2\epsilon }}I\left(s,\epsilon \right)={\text{e}}^{-\frac{1}{2\epsilon }{\left(s-\alpha \right)}^{2}},
a function that peaks in an
O\left(\sqrt{\epsilon }\right)
interval about
s=\alpha
and is asymptotically negligible elsewhere. This implies that a shock layer of nonuniform convergence occurs about the turning point. The exact solution (6) is
y\left(x,\epsilon \right)=\frac{erf\left(\frac{x-\alpha }{\sqrt{2\epsilon }}\right)+erf\left(\frac{\alpha }{\sqrt{2\epsilon }}\right)}{erf\left(\frac{1-\alpha }{\sqrt{2\epsilon }}\right)+erf\left(\frac{\alpha }{\sqrt{2\epsilon }}\right)}.
y\left(x{,10}^{-3}\right)
from (12).
\alpha =\frac{1}{3}
\epsilon ={10}^{-3}
Not surprisingly, the asymptotic solution is essentially a translation of that for
\alpha =0.5
\alpha =0
y\left(x,\epsilon \right)=\frac{{\int }_{0}^{x}\text{ }\text{ }{\text{e}}^{-\frac{{s}^{2}}{2\epsilon }}\text{d}s}{{\int }_{0}^{1}\text{ }\text{ }{\text{e}}^{-\frac{{s}^{2}}{2\epsilon }}\text{d}s},
so we again get an
O\left(\sqrt{\epsilon }\right)
initial layer (see Figure 4(a) and Figure 4(b)).
\alpha =1
, there is an analogous terminal layer. For
\alpha <0
\alpha >1
, the boundary layer is
O\left(\epsilon \right)
, i.e. thinner.
A\left(x\right)={\left(x-\alpha \right)}^{3},0<\alpha <1
We have a third order turning point at
x=\alpha
. Again, the rescaled integral
I\left(s,\epsilon \right)
peaks at
s=\alpha
, causing y to jump there. The shock layer is now
O\left({\epsilon }^{\frac{1}{4}}\right)
thick. The exact solution is
y\left(x,\epsilon \right)=\frac{\Gamma \left(\frac{1}{4},\frac{{\left(x-\alpha \right)}^{4}}{4\epsilon }\right)-\Gamma \left(\frac{1}{4},\frac{{\alpha }^{4}}{4\epsilon }\right)}{\Gamma \left(\frac{1}{4},\frac{{\left(1-\alpha \right)}^{4}}{4\epsilon }\right)-\Gamma \left(\frac{1}{4},\frac{{\alpha }^{4}}{4\epsilon }\right)},
\Gamma \left(a,z\right)={\int }_{z}^{+\infty }\text{ }\text{ }{u}^{a-1}{\text{e}}^{-u}\text{d}u
is the incomplete gamma function.
y\left(x,\epsilon \right)
\alpha =\frac{1}{3}
\epsilon ={10}^{-3}
To steepen the shock layer, we must take
\epsilon
much smaller.
We change the sign of A for the next three examples.
A\left(x\right)=\frac{1}{2}-x
y\left(x,\epsilon \right)=\frac{{\int }_{0}^{x}\text{ }\text{ }{\text{e}}^{\frac{s}{2\epsilon }\left(s-1\right)}\text{d}s}{{\int }_{0}^{1}\text{ }\text{ }{\text{e}}^{\frac{s}{2\epsilon }\left(s-1\right)}\text{d}s},
y\left(x{,10}^{-3}\right)
from (14) with
\alpha =\frac{1}{3}
y\left(x{,10}^{-3}\right)
from (15) in the initial layer; (b)
y\left(x{,10}^{-3}\right)
y\left(x{,10}^{-3}\right)
from (16) for
\alpha =\frac{1}{3}
y\left(x,\epsilon \right)=\frac{1}{2}\left(1+\frac{{\int }_{0}^{\frac{x-\frac{1}{2}}{\sqrt{2\epsilon }}}\text{ }\text{ }{\text{e}}^{{t}^{2}}\text{d}t}{{\int }_{0}^{\frac{1}{2\sqrt{2\epsilon }}}\text{ }\text{ }{\text{e}}^{{t}^{2}}\text{d}t}\right).
We note that the solution could be expressed in terms of Dawson’s integral
F\left(z\right)={\text{e}}^{-{z}^{2}}{\int }_{0}^{z}\text{ }\text{ }{\text{e}}^{{t}^{2}}\text{d}t
. The integrands in (17) peak symmetrically at
s=0
and 1,
being asymptotically negligible elsewhere. Moreover,
y\left(\frac{1}{2}\right)=\frac{1}{2}
y\left(x\right)\sim \frac{1}{2}
0<x<1
, and twin
O\left(\epsilon \right)
boundary layers occur near both endpoints. For
\epsilon ={10}^{-3}
, we have Figure 6.
A\left(x\right)=\alpha -x,0<\alpha <\frac{1}{2}
y\left(x,\epsilon \right)=\frac{{\int }_{0}^{x}\text{ }\text{ }{\text{e}}^{\frac{s}{2\epsilon }\left(s-2\alpha \right)}\text{d}s}{{\int }_{0}^{1}\text{ }\text{ }{\text{e}}^{\frac{s}{2\epsilon }\left(s-2\alpha \right)}\text{d}s}.
y\left(x,\epsilon \right)=\frac{{\int }_{0}^{\frac{x-\alpha }{\sqrt{2\epsilon }}}\text{ }\text{ }{\text{e}}^{{t}^{2}}\text{d}t+{\int }_{0}^{\frac{\alpha }{\sqrt{2\epsilon }}}\text{ }\text{ }{\text{e}}^{{t}^{2}}\text{d}t}{{\int }_{0}^{\frac{1-\alpha }{\sqrt{2\epsilon }}}\text{ }\text{ }{\text{e}}^{{t}^{2}}\text{d}t+{\int }_{0}^{\frac{\alpha }{\sqrt{2\epsilon }}}\text{ }\text{ }{\text{e}}^{{t}^{2}}\text{d}t}.
The integrand of (19) is asymptotically negligible for
s<2\alpha
, but asymptotically large for
s>2\alpha
y\left(x\right)\sim 0\text{for}\text{\hspace{0.17em}}\text{ }x<1,
so there is as
O\left(\epsilon \right)
-thick terminal layer.
{10}^{-3}{y}^{″}+\left(\frac{1}{3}-x\right){y}^{\prime }=0.
We have Figure 7 for picture of
y\left(x{,10}^{-3}\right)
y\left(x{,10}^{-3}\right)
y\left(x{,10}^{-3}\right)
\alpha =\frac{1}{3}
A\left(x\right)=\alpha -x,\text{\hspace{0.17em}}\alpha >\frac{1}{2}
, the integrand in (19) decays for
0<s<2\alpha
y\left(x\right)\sim 1\text{for}\text{\hspace{0.17em}}\text{ }x>0
and there is an
O\left(\epsilon \right)
thick initial layer (see Figure 8).
\alpha \sim \frac{1}{2}+o\left(1\right)
, we’d expect that the shock layer moves across the interval. We’re now using the little o order symbol, which admittedly isn’t very explicit.
A\left(\left(x\right)=\left(x-\frac{1}{4}\right)\left(x-\frac{3}{4}\right)
, we have simple turning points at
\frac{1}{4}
\frac{3}{4}
I\left(x,\epsilon \right)={\text{e}}^{-\frac{x}{3\epsilon }{\left(x-\frac{3}{4}\right)}^{2}}
x=0
\frac{3}{4}
and is asymptotically negligible elsewhere. The sizes of the contributions to the integral differ, however. The area under I near
x=0
O\left(\epsilon \right)
, but that near
x=\frac{3}{4}
O\left(\sqrt{\epsilon }\right)
, i.e., larger. Thus, the ratio (6) is
O\left(\sqrt{\epsilon }\right)
0<x<\frac{3}{4}
O\left(1\right)
\frac{3}{4}<x<1
Computing for
\epsilon ={10}^{-3}
This relies on the following figures. We’ve increased
\epsilon
in Figure 10 to show the relative contributions. Normalizing to get
y\left(1\right)=1
, we get the solution in Figure 9. And Figure 11 shows the picture of integral for
I\left(x,\epsilon \right)
\epsilon ={10}^{-3}
y\left(x{,10}^{-3}\right)
\alpha =\frac{3}{4}
y\left(x{,10}^{-3}\right)
I\left(x,0.002\right)
{\int }_{0}^{x}\text{ }I\left(s{,10}^{-3}\right)\text{d}s
We have not been exhaustive, but we have certainly demonstrated a wide variety of asymptotic solutions to turning point problems of the form (4) - (5). They mimic the asymptotics of the more general boundary layer resonance problem. When the problem of turning points becomes complicated, numerical methods will become unreliable. Finding the limiting solution is extremely ill conditioned as Trefethen recently observed. Due to the serious instability of direct numerical methods, the examples found in scattered literature are usually less detailed. In this paper, we only give asymptotic solutions for a class of singularly perturbed with a turning point. Indeed, the techniques developed here might be expected to apply to that problem. Readers are encouraged to study other limiting possibilities for (4) - (5).
This research is supported by the Natural Science Foundation of Shanghai Institute of Technology, Research Fund nos. ZQ2018-22, 391100190016027.
[1] Ackerberg, R.C. and O’Malley Jr., R.E. (1970) Boundary Layer Problems Exhibiting Resonance. Studies in Applied Mathematics, 49, 277-295.
[2] Fraenkel, L.E. (1969) On the Method of Matched Asymptotic Expansions. Mathematical Proceedings of the Cambridge Philosophical Society, 65, 263-284.
[3] Skinner, L.A. (2011) Singular Perturbation Theory. Springer, New York.
[4] Vasil’eva, A.B., Butuzov, V.F. and Kalachev, L.V. (1995) The Boundary Function Method for Singular Perturbation Problems. SIAM, Philadelphia.
[5] Fruchard, A. and Schäfke, R. (2013) Composite Asymptotic Expansions. Lecture Notes in Mathematics. Vol. 2066, Springer, Heidelberg.
[6] O’Malley, R.E. (2008) Singularly Perturbed Linear Two-Point Boundary Value Problems. SIAM Review, 50, 459-482.
[7] O’Malley, R.E. (2014) Historical Developments in Singular Perturbations. Springer, Cham.
[8] Sharma, K.K., Rai, P. and Patidar, K.C. (2013) A Review on Singularly Perturbed Differential Equations with Turning Points and Interior Layers. Applied Mathematics and Computation, 219, 10575-10609.
[9] Matkowsky, B.J. (2018) Singular Perturbations in Noisy Dynamical Systems. European Journal of Applied Mathematics, 29, 570-593.
[10] Trefethen, L.N., Birkisson, A. and Driscoll, T.A. (2018) Exploring ODEs. SIAM, Philadelphia.
[11] Wasow, W. (1985) Linear Turning Point Theory. Springer, New York.
[12] Olver, F.W.J. (1974) Asymptotics and Special Functions. Academic Press, New York.
[13] Wong, R. (1989) Asymptotic Approximations of Integrals. Academic Press, New York.
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A Binary Search Tree is a binary tree where each node contains a key and an optional associated value. It allows particularly fast lookup, addition, and removal of items.
The nodes are arranged in a binary search tree according to the following properties:
The left subtree of a particular node will always contain nodes with keys less than that node’s key.
The right subtree of a particular node will always contain nodes with keys greater than that node’s key.
The left and the right subtree of a particular node will also, in turn, be binary search trees.
In average cases, the above mentioned properties enable the insert, search and deletion operations in
O(log n)
time where n is the number of nodes in the tree. However, the time complexity for these operations is
O(n)
in the worst case when the tree becomes unbalanced.
The space complexity of a binary search tree is
O(n)
in both the average and the worst cases.
Types of Traversals
The Binary Search Tree can be traversed in the following ways:
The pre-order traversal will visit nodes in Parent-LeftChild-RightChild order.
The in-order traversal will visit nodes in LeftChild-Parent-RightChild order. In this way, the tree is traversed in an ascending order of keys.
The post-order traversal will visit nodes in LeftChild-RightChild-Parent order.
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Unsupervised Learning - ProductizeML
You will learn: most-used unsupervised learning methods such as clustering and dimensionality reduction algorithms.
A really important requirement we asked our data to fulfill when talking about Supervised Learning techniques is this to contain associated labels.
This is, in fact, a really time-expensive demand given the large volumes of data some Machine Learning algorithms require generalizing well during training. Not only time but also the economic cost associated with the labelling process can be expensive when the domain knowledge belongs to a specific field like for instance medical data.
https://prooffreaderswhimsy.blogspot.com/2014/11/machine-learning.html
Despite this, it is common to find scenarios where large volumes of unlabelled and unstructured data are available, and we still want to understand it better. These are the cases where unsupervised learning techniques can help to find hidden patterns in this unlabelled data.
A simple way to find insights in unlabelled data is via clustering techniques in such a way those data points that share similar properties will belong to the same groupings, and dissimilar points will form other groupings. These algorithms can clearly help with classification tasks, as well as statistical data analyses.
Overview of clustering algorithms supported in scikit-learn.
In this fast-paced course, we will only review one of the most well-known clustering algorithms, K-Means, but we should be aware of many more such as Hierarchical clustering, Gaussian mixtures, Mean-shift, and others.
K-means is one of the simplest unsupervised learning algorithms that works as a clustering technique by selecting k central points or means (hence K-Means). Once these means are computed, they are used as the centroid of the cluster and an association among neighbour points is done in a way that any point that is closest to a given mean is assigned to that mean’s cluster.
The algorithm works iteratively until it converges to a local optimum as follows:
Select K initial cluster means,
\{ \mu_1, ..., \mu_k \}
Iterate until the stopping criterion is satisfied:
Assign each data sample to the closest cluster
x \in C_i , d(x, \mu_i) \leq d (x, \mu_j),\forall i\neq j
Update K means from assigned samples
\mu_i = E\{x\}, x \in X_i, 1 \leq i \leq K
Once all the points have been assigned to a cluster and the stopping criterion has been reached, the algorithm is said to have converged.
The concept behind dimensionality reduction is quite simple: reduce the dimensionality of your feature set. Particularly when computing features using Deep Learning techniques, these are usually of high magnitude values and therefore challenging to manage and take insights from the raw data.
Working with a dimensional feature subspace may help when analyzing the data in the form of visual graphics, or even come up with the best dimensionality for your feature set to be fed to the training algorithms. This last observation is hard to answer since some algorithms might do well on large feature vectors, and others might struggle to converge. This is called the "Curse of Dimensionality", and it can be mitigated via the dimensionality reduction techniques that follow.
Popularly used for dimensionality reduction in continuous data, PCA is used to decompose a multivariate dataset in a set of successive orthogonal components that explain a maximum amount of the variance.
The iteration includes rotating and projecting the data along the direction of increasing variance, in a way that the features with the maximum variance become the principal components.
Visualization of the Principal Component Analysis (PCA) algorithm by Shervine Amidi
t-Distributed Stochastic Neighbor Embedding (t-SNE) is a technique for dimensionality reduction that is particularly well suited for the visualization of high-dimensional datasets. The reduction consists in converting similarities between data points to joint probabilities and tries to minimize the Kullback-Leibler divergence between the joint probabilities of the low-dimensional embedding and the high-dimensional data. This way, we can convert feature maps of thousands of dimensions to a minimal two-dimensional map easy to interpret.
You can find some examples applied to real-world datasets in Laurens van der Maaten's blog, t-SNE visualizations of the MNIST dataset in Colah's blog, and how to use t-SNE effectively since its flexibility makes it hard to interpret sometimes.
Visualizations by t-SNE of 6,000 handwritten digits from the MNIST data set by Maaten & Hinton. Visualizing High-Dimensional Data using t-SNE.
Another popular dimensionality reduction method that gives successful results is autoencoders, a type of artificial neural network that aims to mimic the inputs to the outputs. These networks reduce the input data into a latent-space representation and then reconstructs the output from this representation.
An autoencoder is composed of two parts :
Encoder: compresses the input into a latent-space representation.
Decoder: reconstruct the input from the latent space representation.
Illustration of autoencoder model architecture by Lilian Weng
https://towardsdatascience.com/dimensionality-reduction-for-machine-learning-80a46c2ebb7e
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Multivariate normal probability density function - MATLAB mvnpdf - MathWorks Deutschland
Standard Multivariate Normal pdf
Multivariate Normal pdfs Evaluated at Different Points
Multivariate Normal pdf
Multivariate Normal pdfs Evaluated at Same Point
y = mvnpdf(X)
y = mvnpdf(X,mu)
y = mvnpdf(X,mu,Sigma)
y = mvnpdf(X) returns an n-by-1 vector y containing the probability density function (pdf) values for the d-dimensional multivariate normal distribution with zero mean and identity covariance matrix, evaluated at each row of the n-by-d matrix X. For more information, see Multivariate Normal Distribution.
y = mvnpdf(X,mu) returns pdf values of points in X, where mu determines the mean of each associated multivariate normal distribution.
y = mvnpdf(X,mu,Sigma) returns pdf values of points in X, where Sigma determines the covariance of each associated multivariate normal distribution.
Evaluate the pdf of a standard five-dimensional normal distribution at a set of random points.
Randomly sample eight points from the standard five-dimensional normal distribution.
mu = zeros(1,5);
X = mvnrnd(mu,Sigma,8)
Evaluate the pdf of the distribution at the points in X.
Find the point in X with the greatest pdf value.
[maxpdf,idx] = max(y)
maxpdf = 0.0054
maxPoint = X(idx,:)
maxPoint = 1×5
The fifth point in X has a greater pdf value than any of the other randomly selected points.
Create six three-dimensional normal distributions, each with a distinct mean. Evaluate the pdf of each distribution at a different random point.
Specify the means mu and covariances Sigma of the distributions. Each distribution has the same covariance matrix—the identity matrix.
Randomly sample once from each of the six distributions.
X = mvnrnd(mu,Sigma)
Evaluate the pdfs of the distributions at the points in X. The pdf of the first distribution is evaluated at the point X(1,:), the pdf of the second distribution is evaluated at the point X(2,:), and so on.
Evaluate the pdf of a two-dimensional normal distribution at a set of given points.
Specify the mean mu and covariance Sigma of the distribution.
Sigma = [0.9 0.4; 0.4 0.3];
Randomly sample from the distribution 100 times. Specify X as the matrix of sampled points.
X = mvnrnd(mu,Sigma,100);
Plot the probability density values.
scatter3(X(:,1),X(:,2),y)
Create ten different five-dimensional normal distributions, and compare the values of their pdfs at a specified point.
Set the dimensions n and d equal to 10 and 5, respectively.
Specify the means mu and the covariances Sigma of the multivariate normal distributions. Let all the distributions have the same mean vector, but vary the covariance matrices.
mu = ones(1,d)
mat = eye(d);
nMat = repmat(mat,1,1,n);
var = reshape(1:n,1,1,n);
Sigma = nMat.*var;
Display the first two covariance matrices in Sigma.
Sigma(:,:,1:2)
Set x to be a random point in five-dimensional space.
x = normrnd(0,1,1,5)
Evaluate the pdf at x for each of the ten distributions.
scatter(1:n,y,'filled')
xlabel('Distribution Index')
ylabel('Probability Density at x')
Evaluation points, specified as a 1-by-d numeric vector or an n-by-d numeric matrix, where n is a positive scalar integer and d is the dimension of a single multivariate normal distribution. The rows of X correspond to observations (or points), and the columns correspond to variables (or coordinates).
If X is a vector, then mvnpdf replicates it to match the leading dimension of mu or the trailing dimension of Sigma.
vector of zeros (default) | numeric vector | numeric matrix
Means of multivariate normal distributions, specified as a 1-by-d numeric vector or an n-by-d numeric matrix.
If mu is a vector, then mvnpdf replicates the vector to match the trailing dimension of Sigma.
identity matrix (default) | symmetric, positive definite matrix | numeric array
Covariances of multivariate normal distributions, specified as a d-by-d symmetric, positive definite matrix or a d-by-d-by-n numeric array.
If Sigma is a matrix, then mvnpdf replicates the matrix to match the number of rows in mu.
If Sigma is an array, then each page of Sigma, Sigma(:,:,i), is the covariance matrix of a single multivariate normal distribution and, therefore, is a symmetric, positive definite matrix.
If the covariance matrices are diagonal, containing variances along the diagonal and zero covariances off it, then you can also specify Sigma as a 1-by-d vector or a 1-by-d-by-n array containing just the diagonal entries.
pdf values, returned as an n-by-1 numeric vector, where n is one of the following:
Number of rows in X if X is a matrix
Number of times X is replicated if X is a vector
If X is a matrix, mu is a matrix, and Sigma is an array, then mvnpdf computes y(i) using X(i,:), mu(i,:), and Sigma(:,:,i).
y\text{ = }f\left(x,\mu ,\Sigma \right)\text{ = }\frac{\text{1}}{\sqrt{|\Sigma |{\text{(2}\pi \text{)}}^{d}}}\mathrm{exp}\left(-\frac{\text{1}}{\text{2}}\text{(}x\text{-}\mu \text{) }{\Sigma }^{\text{-1}}\text{(}x\text{-}\mu \text{)'}\right)
\mathrm{Pr}\left\{v\left(1\right)\le x\left(1\right),v\left(2\right)\le x\left(2\right),...,v\left(d\right)\le x\left(d\right)\right\}.
In the one-dimensional case, Sigma is the variance, not the standard deviation. For example, mvnpdf(1,0,4) is the same as normpdf(1,0,2), where 4 is the variance and 2 is the standard deviation.
mvncdf | mvnrnd | normpdf
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Specifically, its base power is equal to
{\displaystyle {\bigg \lfloor }{\frac {150\times HP_{current}}{HP_{max}}}{\bigg \rfloor }}
. However, if this formula produces a value less than 1, the move's base power becomes 1 instead.
SwSh Converting its life-force into power, the user attacks opposing Pokémon. The lower the user's HP, the lower the move's power.
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Solving Applied Problems Involving Ellipses | College Algebra | Course Hero
Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper.
Example 7: Locating the Foci of a Whispering Chamber
a. What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a horizontal ellipse, and let the center of the room be the point
\left(0,0\right)
b. If two senators standing at the foci of this room can hear each other whisper, how far apart are the senators? Round to the nearest foot.
a. We are assuming a horizontal ellipse with center
\left(0,0\right)
, so we need to find an equation of the form
\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1
a>b
. We know that the length of the major axis,
2a
, is longer than the length of the minor axis,
2b
. So the length of the room, 96, is represented by the major axis, and the width of the room, 46, is represented by the minor axis.
a
2a=96
a=48
{a}^{2}=2304
b
2b=46
b=23
{b}^{2}=529
Therefore, the equation of the ellipse is
\frac{{x}^{2}}{2304}+\frac{{y}^{2}}{529}=1
b. To find the distance between the senators, we must find the distance between the foci,
\left(\pm c,0\right)
{c}^{2}={a}^{2}-{b}^{2}
c
\begin{array}{ll}{c}^{2}={a}^{2}-{b}^{2} & \\ {c}^{2}=2304 - 529 & \begin{array}{cccc}& & & \end{array}\text{Substitute using the values found in part (a)}. \\ c=\pm \sqrt{2304 - 529} & \begin{array}{cccc}& & & \end{array}\text{Take the square root of both sides}. \\ c=\pm \sqrt{1775} & \begin{array}{cccc}& & & \end{array}\text{Subtract}. \\ c\approx \pm 42 & \begin{array}{cccc}& & & \end{array}\text{Round to the nearest foot}. \end{array}
\left(\pm 42,0\right)
represent the foci. Thus, the distance between the senators is
2\left(42\right)=84
a. What is the standard form of the equation of the ellipse representing the room? Hint: assume a horizontal ellipse, and let the center of the room be the point
\left(0,0\right)
b. If two people are standing at the foci of this room and can hear each other whisper, how far apart are the people? Round to the nearest foot.
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Estimating and Solving Applied Problems Using Decimals
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MATHEMATICS TRIGONOMET • Calamba Doctors' College - Calamba City
Week 6. Solving Problems involving parabola.pdf
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Transition from Low to High-Fidelity UAV Models in Three Stages - MATLAB & Simulink - MathWorks í•œêµ
This example shows how to continuously evolve your UAV plant model to keep in sync with the latest information available. 
Towards the end of the design cycle, there is enough information to develop a high-fidelity plant. To accurately model the UAV, a high-fidelity model incorporates modeling all forces and moments, wind and environmental effects and sensors in detail. However, this level of information may be unavailable to a designer early in the design process. To build such a complex model, it can take several flight and wind tunnel tests to create enough detailed aerodynamic coefficients to compute all forces and moments that affect the UAV. These factors can potentially block guidance algorithm design until the end of the design process, when a more realistic estimate of UAV dynamics is obtained.   
To concurrently design a guidance algorithm sooner, a UAV algorithm designer can start with a low-fidelity model and evolve their plant model as and when additional data becomes available. 
Designing a guidance algorithm using only a low-fidelity model can also pose a risk. Without controller or aerodynamic constraints, an optimistic guidance technique can fail for a real UAV with slower aircraft dynamics. 
This example highlights an alternative approach. You progress from the low-fidelity Guidance Block to a medium and then high-fidelity model by progressively adding layers of control and dynamics to the simulation. In this process, the medium-fidelity model becomes a useful tool for leveraging limited information about a plant model to tune and test guidance algorithms.
The medium-fidelity model is thus used to test a given path following an algorithm. Since the high-fidelity model is unavailable until the end of the design process, the high-fidelity model is only used later to validate our modelling approach by comparing step response and path following behavior. 
Assume your UAV has the following design specifications shown in the table below. The low-fidelity variant provided in this model is tuned to achieve the desired response, but you can tune these gains for your specific requirements. The low-fidelity plant uses the UAV Guidance Block which is a reduced order model for a UAV. To run the low-fidelity variant, click the Simulate Plant shortcut under the Low Fidelity group of the project toolstrip.
Open the UAV Fixed Wing Guidance Model block in the FidelityStepResponse/FixedWingModel/LowFidelity subsystem. In the Configuration tab, inspect the gains set for height, airspeed, and roll response. This guidance block integrates the controller with the dynamics of the aircraft. The low-fidelity variant gives a first estimate of how fast the UAV can realistically respond to help tune high-level planners.
To design a medium-fidelity model, the model needs only preliminary aerodynamic coefficients, thrust curves, and response time specifications. To model a medium-fidelity UAV, you can use the Fixed-Wing Point Mass Block. The block only requires lift, drag and thrust force inputs, which are much easier to approximate at an early design stage than detailed forces and moments of an aircraft. To set up the medium-fidelity variant, click the Setup Plant shortcut under the Medium Fidelity group of the project toolstrip.
\mathrm{Î}={\mathrm{γ}}_{\mathit{a}}
\mathrm{α}
\mathrm{Î}
{\mathrm{γ}}_{\mathit{a}}
\mathrm{α}
Unlike the low-fidelity model, the medium-fidelity model splits the autopilot from the plant dynamics. The medium-fidelity plant needs an outer-loop controller for height-pitch and airspeed-throttle control to be added. The predefined controllers provided are using standard PID-tuning loops to reach satisfactory response without overshoot. To inspect the outer-loop controller, open the Outer_Loop_Autopilot Simulink model.
The low-fidelity plant was tuned in the previous step by assuming that all response time specifications are met by the UAV. To test this assumption, use the medium-fidelity plant. The study of the step response of the improved plant is used to contrast the performance of the low-fidelity and medium-fidelity variant. To simulate the medium-fidelity step response, click the Simulate Plant shortcut under the Medium Fidelity group of the project toolstrip. The step response plots appear as figures.
To simulate and visualize the medium-fidelity UAV path following the model, click the Simulate Path Follower shortcut under the Medium Fidelity group of the project toolstrip.
The high-fidelity plant inputs all forces and moments to a 6-DOF block, adds on-board sensors, and models actuator dynamics for the UAV. Unlike the mid-fidelity plant, the high-fidelity version does not take attitude inputs directly. Instead, an inner loop controller is added to control attitude. Additionally, a yaw compensation loop balances the non-zero sideslip. The model reuses the outer-loop controller designed for the medium-fidelity model. To validate that the medium-fidelity model provided useful intermediate information, use the response of the higher fidelity model.
To simulate and visualize the high-fidelity step response, click the Simulate Plant shortcut under the High-Fidelity group of the project toolstrip. Notice that despite added complexity, the trajectory matches well with the medium-fidelity model. Also, notice the design specifications are relatively the same for the high-fidelity stage. This similarity shows that the medium-fidelity plant modelled UAV dynamics accurately.
Towards the end of the design cycle, the high-fidelity model finally becomes available. To get the final UAV path following characteristics, you can now test the guidance algorithm developed in previous steps on the high-fidelity plant. Click the Simulate Path Follower shortcut under the High-Fidelity group of the project toolstrip.
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Given particular instance of decision variables, evaluate all variable terms in system of LMIs - MATLAB evallmi - MathWorks United Kingdom
evallmi
Given particular instance of decision variables, evaluate all variable terms in system of LMIs
evalsys = evallmi(lmisys,decvars)
evallmi evaluates all LMI constraints for a particular instance decvars of the vector of decision variables. Recall that decvars fully determines the values of the matrix variables X1, . . ., XK. The “evaluation” consists of replacing all terms involving X1, . . ., XK by their matrix value. The output evalsys is an LMI system containing only constant terms.
The function evallmi is useful for validation of the LMI solvers' output. The vector returned by these solvers can be fed directly to evallmi to evaluate all variable terms. The matrix values of the left and right sides of each LMI are then returned by showlmi.
evallmi is meant to operate on the output of the LMI solvers. To evaluate all LMIs for particular instances of the matrix variables X1, . . ., XK, first form the corresponding decision vector x with mat2dec and then call evallmi with x as input.
Consider the feasibility problem of finding X > 0 such that
ATXA – X + I < 0
A=\left(\begin{array}{cc}0.5& -0.2\\ 0.1& -0.7\end{array}\right).
This LMI system is defined by:
X = lmivar(1,[2 1]) % full symmetric X
lmiterm([1 1 1 X],A',A) % LMI #1: A'*X*A
lmiterm([1 1 1 X],-1,1) % LMI #1: -X
lmiterm([1 1 1 0],1) % LMI #1: I
lmiterm([-2 1 1 X],1,1) % LMI #2: X
To compute a solution xfeas, call feasp by
[tmin,xfeas] = feasp(lmis)
xfeas' =
1.1029e+02 -1.1519e+01 1.1942e+02
The LMI constraints are therefore feasible since tmin < 0. The solution X corresponding to the feasible decision vector xfeas would be given by X = dec2mat(lmis,xfeas,X).
To check that xfeas is indeed feasible, evaluate all LMI constraints by typing
evals = evallmi(lmis,xfeas)
The left and right sides of the first and second LMIs are then given by
[lhs1,rhs1] = showlmi(evals,1)
eig(lhs1-rhs1)
confirms that the first LMI constraint is satisfied by xfeas.
showlmi | setmvar | dec2mat | mat2dec
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Kinetics (physics) — Wikipedia Republished // WIKI 2
Subfield of physics
Not to be confused with Kinematics.
See also: Analytical dynamics
{\displaystyle {\textbf {F}}={\frac {d}{dt}}(m{\textbf {v}})}
In physics and engineering, kinetics is the branch of classical mechanics that is concerned with the relationship between motion and its causes, specifically, forces and torques.[1][2][3] Since the mid-20th century, the term "dynamics" (or "analytical dynamics") has largely superseded "kinetics" in physics textbooks,[4] though the term is still used in engineering.
In plasma physics, kinetics refers to the study of continua in velocity space. This is usually in the context of non-thermal (non-Maxwellian) velocity distributions, or processes that perturb thermal distributions. These "kinetic plasmas" cannot be adequately described with fluid equations.
The term kinetics is also used to refer to chemical kinetics, particularly in chemical physics and physical chemistry.[5][6][7][8][9] In such uses, a qualifier is often used or implied, for example: "physical kinetics", "crystal growth kinetics", and so on.
Kinetics 1 Intro
Plug flow reactor with second order kinetics (design equation)
^ kinetics. Encyclopædia Britannica Online
^ Louis Adolphe Martin (1907). Text-book of Mechanics. Wiley. p. Section X, pp. 69ff.
^ Kinetics must not be confused with kinematics, the study of motion without consideration of the physical circumstances causing it (see, e.g., Edmund Taylor Whittaker (1988). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Reprint of the Fourth Edition of 1936 with a foreword by William McCrea ed.). Cambridge University Press. p. Chapter 1. ISBN 0-521-35883-3. ), which is a separate branch of classical mechanics.
^ See this discussion.
^ Lifshitz, E. M.; Pitaevskii, L. P.; Sykes, J. B.; Franklin, R.N. (1981). Physical Kinetics. Butterworth-Heinemann. ISBN 0-7506-2635-6.
^ Alexeev, Boris V.; Alexeev (2004). Generalized Boltzmann Physical Kinetics. Elsevier. ISBN 0-444-51582-8.
^ Gorelik, G. E.; N. V. Pavlyukevish; V. V. Levdansky; V. G. Leitsina; G. I. Rudin (1995). Physical Kinetics and Transfer Processes in Phase Transitions. Begell House. ISBN 1-56700-044-4.
^ Krainov, Vladimir P.; Kevin Hendzel (1992). Qualitative Methods in Physical Kinetics and Hydrodynamics. Springer. ISBN 0-88318-953-4.
^ American Chemical Society, Division of Physical Chemistry (1976). Evolution of Kinetics: A Centennial Symposium of the Division of Physical Chemistry, American Chemical Society. American Chemical Society.
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Abdullah Aljouiee, Samir Kumar Bhowmik, "Windowed Fourier Frames to Approximate Two-Point Boundary Value Problems", Abstract and Applied Analysis, vol. 2015, Article ID 153010, 7 pages, 2015. https://doi.org/10.1155/2015/153010
Abdullah Aljouiee 1 and Samir Kumar Bhowmik1
1Department of Mathematics, College of Science, Al Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia
Academic Editor: Maria A. Ragusa
Numerical approximation of various ordinary and partial differential equations is of ongoing interest [1–5]. There are several popular schemes to approximate such models. Schemes based on special functions are increasingly popular [6].
The windowed Fourier transform (Gabor transform) has been a widely used tool in signal processing. This technique uses a single window function to Fourier-transform a signal locally. This process is repeated while shifting the window through the real line. This single window shifting and modulation mechanism of the Gabor transform produces some undesirable effects [5]. A set of frame functions, the windowed Fourier frames (WFFs), have been used to serve such purpose as well [5].
In recent time, WFFs have been popularly used for solving partial differential equations (PDE) [7]. In [7], the authors consider an elliptic PDE and develop an efficient solver using a combination of the symbol of the operator and WFFs. They discuss window functions, discretisation, and implementations in detail. They also study the efficiency of using such functions.
The author develops a general recipe for higher order BVPs in [2]. He considers Tchebychev polynomials to approximate BVPs by reducing the order. In fact, the higher order BVPs problems have been converted to first-order BVPs to approximate the problem using global polynomials efficiently. The author exhibits some numerical results to demonstrate the efficiency of the proposed scheme.
Here, in this paper, we focus on approximating the solutions of two-point boundary value problems using windowed Fourier frames. We motivate ourselves to develop a scheme based on windowed frame functions to approximate various operators in a spare way for one-dimensional academic problems (with an aim to approximate higher dimensional operators using WFFs in the near future). One needs a single window function to generate a family of windowed Fourier frame functions. Thus presentation of the operator becomes neat and simple. The advantage of using windowed frame functions is that they have a flexibility to use for various purpose; the windowed Fourier transformation operator generates a spare differential operator which is easy to store; as a result computations become simple (compared to the spectral collocation/global polynomial approximations for the differential operator). The superiority of the technique has been well discussed in [5, 8]. In this paper we use tight frames to approximate a function, its derivatives, as well as various inner products. Then we apply the frame representations to approximate the solutions of the BVPs.
This paper is organized as follows.(i)We start by discussing WFFs with some properties, followed by an approximation of a function using WFFs in Section 2.(ii)We discuss representation of various operators using WFFs in Section 3.(iii)In Section 4 we approximate some two-point BVPs using WFFs.(iv)We finish with a conclusion in Section 5.
2. A Short Review of Frames
In this section we review in short frames, windowed Fourier frame functions, and the windowed Fourier frame transformation (WFFT) (to approximate any function ). We start by discussing frames and windowed Fourier frames. Then we discuss construction of an efficient window function briefly and use this function to construct windowed Fourier frames.
A frame is a family of vectors that characterizes any function from its inner product . It is possible to recover a vector in a Hilbert space from its inner products with a family of vectors . The index set might be finite or infinite and one can define a frame operator so that
Theorem 1. The sequence is a frame of if there exist two constants , such that for any If this condition is satisfied then is called a frame operator. When the frame is said to be tight [5, 9].
It is well established that it is possible to reconstruct a signal from its frame transformation using the concept of pseudo inverse which is a bounded operator expressed with a dual frame [5]. Note that a pseudo inverse is denoted by and satisfies where is the adjoint of . If is a frame operator with frame bounds and then . The pseudo inverse of a frame operator is related to a dual fame family, which is expressed by the following result.
Theorem 2 (see [5]). Let be a frame with bounds and . The dual frame defined by satisfies Then the frame is tight (i.e., ). Here is adjoint of and is the pseudo inverse of .
We discuss windowed Fourier frame and its transformation next. For , the translation can be defined by and modulation operator can be defined as . Operators of the form or are called time-frequency shifts. Given a nonzero window function and lattice parameters , the set of time-frequency shifts is called a Gabor system. If is a frame for , it is called Gabor frame, Weyl-Heisenberg frame, or windowed Fourier frame [8]. From now we will be denoting such frames by the windowed Fourier frames. Figure 1 shows a sample window function and its Fourier transform. Gabor frame can be constructed using A detailed construction process of window functions can be found in [7].
The figure shows window function , its Fourier transform , and .
Theorem 3 (see [5]). Let be a window whose support is included in , . If for all , then is a tight frame with a frame bound equal to .
Thus construction of a window function is important. So we aim to present some ideas to design window functions. Let us first give a brief explanation of forming a window function. Consider an interval , and . We want to form a function so that if and there exists such that we have the following. Property 1 is increasing monotonically in . Property 2 in . Property 3 is decreasing monotonically in . Property 4 for all .
2.1. Computation of a Function and Sparsity
Now we can define as where are the Fourier windowed frames and are frame coefficients given by . Here many of the frame coefficients are very close to (see an example in Figure 2) and thus the coefficient matrix is sparse if we use frames. To define windowed Fourier frames we consider the following steps. We fix(i)the number of windows ,(ii)the number of points on each window ,(iii),(iv)the window function , where .We plot a cartoon of an original function and recovered () from its frame coefficients with sparsity set up in Figure 3. From this computation we notice at the approximation is accuracy as , . We notice that a large number of windowed Fourier frame coefficients are numerically zero or very small. This result motivates us to perform further study on the computation of operators using WFF functions.
This figure shows frame coefficients (sorted) for where . Here we have considered window functions with grid points on each window.
This figure shows the original function and recovered function from the frame coefficients () and the error we made with such an approximation.
3. Computation of Operators Using Windowed Fourier Frames
Here we intend to define and compute the solution of a differential equation using the frame transformation of the initial function over a bounded interval . Actually we intend to present some operators over a bounded interval to facilitate the approximation of several BVPs with boundary conditions at and . Let us approximate functions and by where are frame coefficients and where are frame coefficients, respectively. We define where is an operator. It is our goal to find the matrix that maps to and to find a way to compute the matrices. Here and thus the frame coefficients to approximate can be defined as and the first approach of is simpler since we need to compute the integral over the support of the window function only. In practice, we are interested in the following operators: (1)multiplication by a function,(2)differentiation operator (derivative as an operator),which are used to formulate PDEs and to present solutions.
3.1. as Multiplication:
We define as multiplication operator , where is an arbitrary function. Then the matrix elements of (11) can be written as where is a function which depends on . Thus we can think of the following three cases.(1)When , both the windows have identical support, and so the integral can be written as (2)When , then the windows do not have overlapping support, and so (3)When , then two successive windows overlap and we can compute the elements by Thus the computation of the elements becomes the frame transformation of the functions , , .
3.2. as Differential Operators and
Let us start with computing using windowed Fourier frames. Consider and we find frame coefficients where is defined by (9). Then we can define Thus the computation that needs to be focused on is . Now where the integral is exactly the same as the integral (12) we discussed above. is also of the same form and can be computed similarly to once we know . Here we have to define two matrices, one with a derivative () and the other integral that needs to be multiplied by frequencies ().
Now where and for each and , computation of can be performed as of if the window and its derivative have the same support.
Next we consider where the frame coefficients are defined by and is defined by (9). Thus can be presented using frames following similar steps as of . Figure 4 shows a window function and its first and second derivatives computed using central difference formula.
We show , , and where .
3.3. Inner Product of Two Derivatives
There are some cases when the second derivative is replaced by its variational form. Now with appropriate boundary conditions the variational form can be written as where is a test function. We replace by and by , respectively. Thus we are interested in computing for all or , where
4. Numerical Solution of Two-Point BVPs
In the previous section we discuss the frame representation of various functions and operators. Here we aim to use the representations to approximate two-point boundary value problems. We consider for all with some boundary conditions . Let be a test function; then the variational form of the solutions can be written as and we consider . Now considering and replacing by Thus we need to find of (29).
Now we aim to display some computational results obtained with the scheme discussed in this paper. Here we solve some one-dimensional BVPs of the form (26) to demonstrate the scheme.
Example 4. Let , , , and where . The variational form (29) is then which can be written as a system of linear equations with unknowns as We present the approximate solution in Figure 5. We compare the result with that of the exact solution.
Solution of the BVP: using windowed Fourier frames.
Example 5. Consider the BVP with , if , and , if . Using similar steps as of the previous discussion we present the approximate solution in Figure 6.
Example 6. Consider the BVP We present the approximate solution in Figure 7. We compare the result with that of an approximation using a standard finite difference scheme.
Solution of the BVP: using windowed Fourier frames and using a standard finite difference scheme.
From Figures 5, 6, and 7 we see that the approximate solutions agree with the exact solutions, which show the efficiency of the numerical scheme presented in this paper.
In the paper windowed Fourier frames have been used to approximate two-point BVPs. From the approximation of functions we notice that the results agree with the exact solutions. We also note that a small amount of WFF coefficients is needed to reconstruct a function, and huge storage costs can be minimized. The scheme also does not require a lot of knowledge concerning the behavior of the solutions. The illustrative examples have been included to demonstrate the validity and applicability of the technique. These examples also exhibit the efficiency of the present method. There are some drawbacks: design of an efficient window function is very important. For one-dimensional problems the advantage of using the method is not very highly visible compared to the other existing numerical schemes, but we have a conjecture (our computational experiences) that this scheme can be used for higher dimensional problem where storage is a real problem for numerical computations. A multidimensional approximation using WFF would be of interest which is left as an open problem.
In this study we apply WFFs to some second-order linear BVPs. WFFs are not limited to these problems only. There are many linear higher order BVPs and nonlinear BVPs modeling scientific and engineering problems where WFFs can be applied to approximate the solutions which are left as open research problems.
Samir Kumar Bhowmik is on leave from the Department of Mathematics, University of Dhaka, Dhaka, Bangladesh.
Samir Kumar Bhowmik would like to thank Chris C. Stolk of University of Amsterdam for his kind and cordial help.
A. Ashyralyev and O. Yildirim, “On stability of a third order of accuracy difference scheme for hyperbolic nonlocal BVP with self-adjoint operator,” Abstract and Applied Analysis, vol. 2013, Article ID 959216, 15 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
S. K. Bhowmik, “Tchebychev polynomial approximations for
m
th order boundary value problems,” International Journal of Pure and Applied Mathematics, vol. 98, no. 1, pp. 45–63, 2015. View at: Google Scholar
S. K. Bhowmik, F. M. Al Faqih, and N. Islam, “A note on some numerical approaches to solve a
\stackrel{˙}{\theta }
neuron networks model,” Abstract and Applied Analysis, vol. 2014, Article ID 863842, 7 pages, 2014. View at: Publisher Site | Google Scholar
S. Kumar, S. Dhawan, and S. Kapoor, “Numerical method for advection diffusion equation using fem and b-splines,” Journal of Computational Science, vol. 3, no. 5, pp. 429–437, 2012. View at: Publisher Site | Google Scholar
S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Amsterdam, The Netherlands, 3rd edition, 2009. View at: MathSciNet
C. C. Stolk, “A fast method for linear waves based on geometrical optics,” SIAM Journal on Numerical Analysis, vol. 47, no. 2, pp. 1168–1194, 2009. View at: Publisher Site | Google Scholar | MathSciNet
S. K. Bhowmik and C. C. Stolk, “Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations,” Journal of Pseudo-Differential Operators and Applications, vol. 2, no. 3, pp. 317–342, 2011. View at: Publisher Site | Google Scholar | MathSciNet
K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, Mass, USA, 2000. View at: Publisher Site | MathSciNet
O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, 2004.
Copyright © 2015 Abdullah Aljouiee and Samir Kumar Bhowmik. 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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Design System – Hillarys Boat Harbour
Hillarys Boat Harbour - Design System 🎨
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Applied Colors .c-icon .icon--currentColor
Applied Colors .c-icon .icon--currentColor-stroke
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This one has no set colours. It's just really small.
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A more complicated combination of components
Food & Drink See & Do Shopping Nightlife Family Friendly
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This relies on the SEO Press Pro plugin
These are just padding sizes with breakpoints.
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This type is generally used for full width sections.
And there’s lots to love about our exciting new website – coming soon!
In the meantime, stay up to date with what’s happening at the Harbour by subscribing to hear about our latest giveaways, upcoming events and heaps more!
This one is used for half width sections.
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This one is a horizontal rule that you can apply any background colour.
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The .o-readable class is to be used as a wrapper to make the text more legible.
This might include things like links having good hover styles.
This really applies a lot when you have no control over the content that is being inserted.
HTML5 Kitchen Sink
Jump to: Headings | Sections | Phrasing | Palpable Content | Embeds | Forms | Tables
This section seves as the header.
Elements h1, h2, h3, h4, h5, h6 make up the heading content category.
See the Heading Content spec.
Note: these two paragraphs are contained in a footer element.
Elements article, aside, nav, section make up the sectioning content category.
These links are contained in a nav element.
This paragraph is nested inside an article element. It contains many different, sometimes useful, HTML5 elements. Of course there are classics like emphasis, strong, and small but there are many others as well. Hover the following text for abbreviation element: abbr. You can define deleted text which often gets replaced with inserted text.
You can also use keyboard text, which sometimes is styled similarly to the <code> or samp elements. Even more specifically, there is an element just for variables. Not to be mistaken with block quotes below, the quote element lets you denote something as quoted text . Lastly don't forget the sub (H2O) and sup (E = MC2) elements.
This paragraph is contained in a section element of its parent article element.
↓ The following paragraph has the hidden attribute and should not be displayed. ↓
→ You should not see this paragraph. ←
↑ The previous paragraph should not be displayed. ↑
This is contained in an aside element.
See the Sectioning Content spec.
# Phrasing
Elements abbr, b, bdi, bdo, br, cite, code, data, del, dfn, em, i, ins, kbd, mark, meter, progress, q, s, samp, small, span, strong, sub, sup, time, u, var, wbr, and others make up the phrasing content category.
abbr: Some vehicles meet the SULEV standard.
br was used to make this sentence start on a new line.
bdi: Some languages read right to left, مرحبا. bdo: The normal direction has been overridden.
em is used for emphasis and usually renders as italics, contrast that with i which is used for alternate voice or to offset from the normal (such as a phrase from a different language or taxonomic designation): E. coli can be bad. strong is used for importance or urgency and usually renders as bold, contrast that with b which is used to draw attention without the semantic meaning of importance.
cite: In the words of Charles Bukowski — An intellectual says a simple thing in a hard way. An artist says a hard thing in a simple way.
data can be used to specify 5 A.M. that is machine-readable, but time is a better choice for specifying 5 A.M. in a machine-readable format.
del can be varily used to mark deletions. ins marks insertions. s: similar to del, but used to mark content that is no longer relevant. Windows XP version available. u: a holdover with no real meaning that should be removed. mark: the HTML equivalent of the yellow highlighter. span: a generic element with no meaning by itself.
dfn: Foreign phrases add a certain je ne sais quoi to one's prose.
q: The W3C page About W3C says the W3C’s mission is To lead the World Wide Web to its full potential by developing protocols and guidelines that ensure long-term growth for the Web .
kbd and samp: I did this:
c:\>format c: /yes
Is that bad? Press Ctrl+F5 for a hard reload.
var: To log in, type ssh user@example.com, where user is your user ID.
meter and progress: Storage space usage: 6 blocks used (out of 8 total) Progress:
sub is used for subscripts: H2O. sup is used for superscripts: E = MC2. small is used for side comments: I wrote this whole document. [Editor's note: no he did not] wbr: used to specify where a word may break and it is supercalifragilisticexpialidocious.
Donec id elit non mi porta gravida at eget metus. Cras mattis consectetur purus sit amet fermentum. Donec sed odio dui. Nullam quis risus eget urna mollis ornare vel eu leo. Nullam quis risus eget urna mollis ornare vel eu leo. Praesent commodo cursus magna, vel scelerisque nisl consectetur et. Nulla vitae elit libero, a pharetra augue.
Duis mollis, est non commodo luctus, nisi erat porttitor ligula, eget lacinia odio sem nec elit. Maecenas faucibus mollis interdum. Aenean lacinia bibendum nulla sed consectetur. Praesent commodo cursus magna, vel scelerisque nisl consectetur et.
Fusce dapibus, tellus ac cursus commodo, tortor mauris condimentum nibh, ut fermentum massa justo sit amet risus. Donec id elit non mi porta gravida at eget metus. Nullam id dolor id nibh ultricies vehicula ut id elit. Vestibulum id ligula porta felis euismod semper. Curabitur blandit tempus porttitor. Donec sed odio dui. Praesent commodo cursus magna, vel scelerisque nisl consectetur et.
See the Phrasing Content spec.
# Palpable Content
Elements a, address, blockquote, button, details, dl, fieldset, figure, form, input, label, map, ol, output, pre, select, table, textarea, ul, and others make up the palpable content category.
a: Example.
I quickly explained that many big jobs involve few hazards
This is a mult-line blockquote with a cite reference. People think focus means saying yes to the thing you’ve got to focus on. But that’s not what it means at all. It means saying no to the hundred other good ideas that there are. You have to pick carefully. I’m actually as proud of the things we haven’tdone as the things I have done. Innovation is saying no to 1,000 things.
SD (6-1-6)
Figure 1: A picture of Bill Murray from fillmurray.com
ul and ol:
In the following table, characteristics are given in the second column, with the negative side in the left column and the positive side in the right column.
Complex table with a thead, multiple tbody elements, and a tfoot.
See the Palpable Content spec.
Elements audio, canvas, embed, iframe, img, math, object, picture, svg, video make up the embedded content category.
audio: By Cqdx [CC BY-SA 3.0 ], from Wikimedia Commons.
\begin{array}{cc}\text{Quadratic Equation}& x=\frac{-b±\sqrt{{b}_{}^{2}-4ac}}{2a}\end{array}
See the Embedded Content spec.
Find this document on GitHub.
Seven is just a beautiful number here.
If you're not seeing a component here there may be no content. Might need to add some dummy content.
.c-harbour-dropdown
Lorem ipsum dolor sit amet consectetur adipisicing elit. Repellendus et eaque eum eos a fugiat, deserunt nisi incidunt ad sequi tempora voluptate expedita atque. Vero sed nesciunt cumque fuga sequi!
Sidecar .c-sidecar || Sidecar Mirrored .c-sidecar .c-sidecar--mirrored
|
In the hallway of Jefferson Middle School, there is a
30
-foot distance between classroom doors. From her classroom door, Rebecca walked down the hall
17
feet, then turned around (
180°
) and walked back
6
feet to use the drinking fountain.
How far is she from her starting point? When she turns again (
180°
) to continue to the next classroom, how far does she have to walk? Draw a diagram to show Rebecca's movement and clearly explain your thinking.
This problem is very similar to problem 1-33 on page
21
of your textbook.
Use the solution to that problem as a guide if you need help. Also, remember
to draw a diagram to illustrate your thinking and Rebecca's movement.
Use the above diagram to help you solve the problem and create your own diagram.
|
Generative Adversarial Networks | Machine Learning Medium
Image Source: https://user-images.githubusercontent.com/8406802/45344464-683fb800-b569-11e8-90fe-a228401ffadd.png
The basic adversarial framework of the GAN architecture can be broken down into the following two players:
A generative model
G
, that tries to capture the latent data distribution.
A discriminative model
D
, that estimates the probability that a sample came from training data rather than
G
The framework is adversarial in the sense that the training procedure for
G
tries to maximize the probability of
D
making a mistake. The framework thus corresponds to a minimax two-player game.
Related Generative Models
Generative Stochastic Network
Easiest to implement GANs when the models are multilayer perceptrons for both generator and discriminator.
p_g
is the generator’s distribution over data
x
p_z(z)
is an input noise function and
G(z; \theta_g)
is the mapping to data space.
G
is differentiable function represented by a paramter
\theta_g
D
is another differentiable function that outputs a scalar.
D(x)
represents the probability of assigning the correct label to both training examples and samples from
G
G
is simultaneously trained to minimize
log(1-D(G(z)))
The training framework between
D
G
can be represented by a two player minimax game in value function
V(G,D)
G
D
are trained iteratively one after the other
D
is not optimized to completion as it would lead to overfitting
Alternate between
k
steps of optimizing
D
and one step of
G
D
near its optimal, so long as
G
changes slowly.
Early in learning when
G
is poor
D
can reject samples with high confidence which causes
log(1-D(G(z)))
to saturate
Instead of minimizing
log(1-D(G(z)))
, maximize
log(D(G(z)))
for stronger gradients early in the learning.
G
, the optimal discriminator can be found by differentiating the objective function w.r.t.
D(x)
. The objective function is of the form,
Differentiating w.r.t $y$ gives,
Since we are maximising this, the maximum can be found by estimating the point of 0 derivative, i.e,
So the optimal discriminator for a fixed
G
For this maximized
D
, the optimization objective can be rewritten as,
We can show that this expression is minimized for
p_g=p_{data}
D_G^*(x)
1/2
p_g=p_{data}
C(G) = -log\,4
To see that this is the minimu possible value, consider the following modification to the
C(G)
expression above,
The last term is the Jensen-Shannon divergence between two distributions which is always non-negative and zero only when the two distributions are equal. So
C^* = -log\,4
is the global minimum of
C(G)
p_g=p_{data}
, i.e. generative model perfectly replicating the data distribution.
Complexity Comparison of Generative Models
There is no explicit representation of
p_g(x)
G
must be synchronized well with
D
during training. There are possibilities of
D
being too strong leading to zero gradient for
G
D
being too weak which causes
G
to collapse to many values of
z
to the same value of
x
which would not have enough diversity to model
p_{data}
RBMs and DBMs
A fast learning algorithm for deep belief nets by Hinton et al.
Deep boltzman machines by Salakhutdinov et al.
Information processing in dynamical systems: Foundations of harmony theory by Smolensky
Better mixing via deep representations by Bengio et al.
Deep generative stochastic networks trainable by backprop by Bengio et al.
What is the best multi-stage architecture for object recognition? by Jarett et al.
Generalized denoising auto-encoders as generative models by Bengio et al.
Deep sparse rectifier neural networks by Glorot et al.
Maxout networks by Goodfellow et al.
Auto-encoding variational bayes by Kingma et al.
Stochastic backpropagation and approximate inference in deep generative models by Rezende et al.
Learning deep architectures for AI by Bengio Y.
Generative Adversarial Nets by Goodfellow et al.
Privacy Preserving Predictive Modeling GANs Deep Learning without Poor Local Minima
|
Section 8.4: Moments of Inertia (Second Moments)
The connection between second moments and moments of inertia for was developed in Section 6.6 for plane regions. The present section deals with moments of inertia and the associated radii of gyration for three-dimensional regions.
m
be the total mass of a three-dimensional region
R
having density
\mathrm{δ}\left(x,y,z\right)
\mathrm{δ}\left(r,\mathrm{θ},z\right)
in Cartesian or cylindrical coordinates, respectively. Table 8.4.1 lists expressions for the moments of inertia (second moments)
{I}_{x},{I}_{y},{I}_{z}
. For Cartesian coordinates,
\mathrm{dv}
is one of the six permutations of the differentials
\mathrm{dx},\mathrm{dy},\mathrm{dz}
. For cylindrical coordinates,
\mathrm{dv}\prime
\mathrm{dz},\mathrm{dr},d\mathrm{θ}
Second Moments - Cartesian
Second Moments - Cylindrical
{I}_{x}=∫∫{∫}_{R}\mathrm{δ} \left({y}^{2}+{z}^{2}\right)\mathrm{dv}
{I}_{x}=∫∫∫\mathrm{δ} \left({r}^{2}{\mathrm{sin}}^{2}\left(\mathrm{θ}\right)+{z}^{2}\right) r \mathrm{dv}\prime
{k}_{x}=\sqrt{{I}_{x}/m}
{I}_{y}=∫∫{∫}_{R}\mathrm{\delta } \left({x}^{2}+{z}^{2}\right)\mathrm{dv}
{I}_{y}=∫∫∫\mathrm{\delta } \left({r}^{2}{\mathrm{cos}}^{2}\left(\mathrm{\theta }\right)+{z}^{2}\right) r \mathrm{dv}\prime
{k}_{y}=\sqrt{{I}_{y}/m}
{I}_{z}=∫∫{∫}_{R}\mathrm{\delta } \left({x}^{2}+{y}^{2}\right)\mathrm{dv}
{I}_{z}=∫∫∫\mathrm{\delta } \left({r}^{2}\right) r \mathrm{dv}\prime
{k}_{z}=\sqrt{{I}_{z}/m}
Table 8.4.1 Moments of inertia and radii of gyration
{I}_{x}
, the expressions
{y}^{2}+{z}^{2}
{r}^{2}{\mathrm{sin}}^{2}\left(\mathrm{\theta }\right)+{z}^{2}
represent, for a point in
R
, the square of the distance from the
x
{I}_{y}
{x}^{2}+{z}^{2}
{r}^{2}{\mathrm{cos}}^{2}\left(\mathrm{\theta }\right)+{z}^{2}
R
y
{I}_{z}
{x}^{2}+{y}^{2}
{r}^{2}
R
z
Because Maple uses
I
for the imaginary unit
\sqrt{-1}
, it is troublesome to assign to a symbol such as
{I}_{x}
. Hence, whenever such an assignment is needed in the accompanying examples, a symbol such as
\mathrm{Ix}
will be used instead.
The radii of gyration represent distances from the Cartesian coordinate-axes where, if all the mass
m
R
were concentrated, the rotational properties of the region would be preserved.
In each of the following examples, the region
R
and the density
\mathrm{δ}
is taken from the corresponding example in Section 8.3.3. For each example, obtain the second moments
{I}_{x},{I}_{y},{I}_{z}
, and the radii of gyration
{k}_{x},{k}_{y},{k}_{z}
R
\mathrm{\delta }\left(r,\mathrm{\theta },z\right)=z {r}^{2}\mathrm{sin}\left(\mathrm{\theta }/6\right)
R
\mathrm{\delta }\left(r,\mathrm{\theta },z\right)=r {z}^{2}\mathrm{cos}\left(\mathrm{\theta }/3\right)
R
\mathrm{\delta }\left(x,y,z\right)=3+x+y+z
R
\mathrm{\delta }\left(r,\mathrm{\theta },z\right)=\left({r}^{2}+z\right) \mathrm{sin}\left(\mathrm{\theta }/4\right)
R
\mathrm{δ}\left(r,\mathrm{θ},z\right)=z
R
\mathrm{\delta }\left(r ,\mathrm{\theta },z\right)=r z \mathrm{cos}\left(\mathrm{\theta }/6\right)
R
\mathrm{\delta }\left(x,y,z\right)=2 {x}^{2}+3 {y}^{2}+4 {z}^{2}
R
\mathrm{\delta }\left(r,\mathrm{\theta },z\right)={z}^{2}r \mathrm{sin}\left(\mathrm{\theta }/3\right)
R
\mathrm{\delta }\left(x,y,z\right)=x {y}^{2}{z}^{3}
|
V/Line A class - Wikipedia
V/Line liveried A66 in Inglewood on 707 Operations' 'The Grainlander' - March 2021
EMD AAT22C-2R
1,000 imp gal (4,540 l)
137 imp gal (625 l)
EMD AR10-A4-D18
(Replaced with CA5)
(Replaced with D78)
74 VDC, 10 kW
Starting 66,094 lbf (294 kN)
Continuous 47,659 lbf (212 kN)
2 preserved, 1 in service, 2 stored, 6 scrapped
The A class are a class of diesel locomotives rebuilt from Victorian Railways B class locomotives by Clyde Engineering, Rosewater in South Australia for V/Line in 1984–1985.
Freight Australia liveried A85 with Pacific National decals in Melbourne in February 2007
V/Line liveried A70 at Flinders Street in August 2006
The class were rebuilt from B class locomotives originally constructed in the 1950s, as part of the New Deal reforms of passenger rail operations in Victoria. The rebuild contract was let in January 1983 to Clyde Engineering, Rosewater with the first locomotive entering service in May 1984, but the project was abandoned in mid 1985 after rising costs due to structural fatigue, with the 11th rebuild delivered in August 1985.[1] It was decided to instead built more of the N class locomotives, mechanically similar to the A class. The major difference was the addition of head end power generators, as it was believed this was a more efficient way of supplying power for air-conditioning and lighting than power vans or individual generator sets under carriages.[1]
Four locomotives were named after Australian rules football players in September 1984,[2] while A60 was named after former railway commissioner Harold Clapp. In July 1986, A85 was regeared for 160 km/h (99 mph) operation, and tested between Glenorchy and Lubeck, to test an H type carriage set fitted with high speed bogies,[3][4] but was returned to the standard 133 km/h (83 mph) gearing soon after.[2] In preparation for the privatisation of V/Line, four were allocated to passenger services and seven to freight services.
When the engines were initially converted from B class units they were intended for light, high speed passenger traffic, but in Freight Australia service this had limited value. The engine frames could not be easily reinforced, but by 2002 the D57 traction motors were swapped for D77, which raised the current axle from 825A to 1050A. To better make use of the increased power the gear ratios in each unit were changed from 59:18 to 61:16, improving adhesion and overall providing an increase in tractive effort of 35%. The lower-power traction motors were cascaded to the Y Class fleet.[5]
The V/Line units were not upgraded in this way, and continued to operate in regular service supplementing the N Class locomotives on mainline services. Another fixed roster was to Stony Point, as the MTH carriages allocated to that service were fitted with on-board head-end-power generators. V/Line withdrew their fleet with A60 operating the final service, the 16:15 Southern Cross to Bacchus Marsh on 24 April 2013.[6][7] V/Line subsequently returned A66 and A70 to service, with the both since withdrawn. Pacific National withdrew its last examples in May 2014.[8]
V/Line continued to operate A66 in revenue service until August 2018, where it used to operate on the interurban service to Bacchus Marsh on a regular basis operating in peak times. In August 2018, timetable alterations had seen the loco no longer required on the Bacchus Marsh services and A66 was subsequently sent to Bendigo Roundhouse and later Melbourne where it resides currently in dry storage as an emergency locomotive.[9]
During the time period of 7 and 18 January 2019, Pacific National units A73, A77, A81 and A85 were scrapped at South Dynon broad gauge turntable.[10] In May 2019, Pacific National scrapped A71 and A79 at Melbourne Freight Terminal to further clear out redundant assets at Dynon Workshops. Currently, A78 is the last remaining ex-Freight Australia / Pacific National A class locomotive remaining which has been stored in North Bendigo Workshops with an uncertain future.[9]
As of June 2020, The two V/Line A class locomotives that used to be located at South Dynon Locomotive Depot have since been allocated to heritage preservation, they are locomotives A60 and A62 due to their significance and contributions to the Victorian Railways which marked many historical milestones. A60 has been allocated to Seymour Railway Heritage Centre while A62 has been allocated to 707 Operations.The respective transfers of the two locomotives took place on 6 June 2020.
·A60 (formerly B60) being the first the Victorian Railways mainline diesel-electric locomotive, which also has been named after the Victorian Railway commissioner Sir Harold Clapp.
·A62 (formerly B62) being the first diesel-electric locomotive to hit a million miles within the Victorian Railway system. It too has been named in honor of the railway employees of the Victorian Railways.
The class were delivered in the V/Line orange and tangerine scheme. In February 1988, A66 which was painted in a green and gold livery to celebrate the Australian Bicentenary,[11] later appearing in a second special livery to advertise the Melbourne bid for the 1996 Olympic Games.[2] Today[timeframe?] the class appears in either the 1995 red and blue or 2008 red and yellow V/Line Passenger liveries, or the green and yellow Freight Australia livery with Pacific National logos.[9]
In Service Preserved Stored or Withdrawn Scrapped
Locomotive Pre-conversion Named Entered service Withdrawn Scrapped Livery Owners Status Gauge Notes
A60 B60 Sir Harold Clapp 7 September 1984 1 August 2015 V/Line Mk2 V/Line (Rebuilt), V/Line Passenger (1995), VicTrack Heritage allocated to Seymour Railway Heritage Centre (2020) Stored 1,600 mm (5 ft 3 in) broad gauge This loco was reportedly used on the 1954 Royal Tour when it was a B class locomotive[citation needed]
A62 B62 17 July 1984 1 August 2015 V/Line Mk2 V/Line (Rebuilt), V/Line Passenger (1995), VicTrack Heritage allocated to 707 Operations (2020) Stored 1,600 mm (5 ft 3 in) broad gauge
A66 B66 22 March 1985 V/Line Orange and Grey V/Line (Rebuilt), V/Line Passenger (1995) Out of Traffic, Repaint 1,600 mm (5 ft 3 in) broad gauge Reactivated
A70 B70 10 May 1985 1 August 2015 V/Line Mk2 V/Line (Rebuilt), V/Line Passenger (1995) Stored South Dynon 1,600 mm (5 ft 3 in) broad gauge
A71 B71 Dick Reynolds 20 March 1984 Unknown 1 June 2019 Freight Australia Green and Yellow with Pacific National logos V/Line (Rebuilt), V/Line Freight (1995), Pacific National (????) Scrapped 1,600 mm (5 ft 3 in) broad gauge
A73 B73 Bob Skilton 20 March 1984 Unknown 14 January 2019 Freight Australia Green and Yellow with Pacific National logos V/Line (Rebuilt), V/Line Freight (1995), Pacific National (????) Scrapped 1,600 mm (5 ft 3 in) broad gauge
A77 B77 Ian Stewart 17 May 1984 Unknown 17 January 2019 Freight Australia Green and Yellow with Pacific National logos V/Line (Rebuilt), V/Line Freight (1995), Pacific National (????) Scrapped 1,600 mm (5 ft 3 in) broad gauge
A78 B78 21 December 1984 Unknown Freight Australia Green and Yellow with Pacific National logos V/Line (Rebuilt), V/Line Freight (1995), Pacific National (????) Stored - North Bendigo 1,600 mm (5 ft 3 in) broad gauge
A79 B79 22 June 1985 Unknown 17 May 2019 Freight Australia Green and Yellow with Pacific National logos V/Line (Rebuilt), V/Line Freight (1995), Pacific National (????) Scrapped 1,600 mm (5 ft 3 in) broad gauge
A81 B81 15 August 1985 Unknown 10 January 2019 Freight Australia Green and Yellow with Pacific National logos V/Line (Rebuilt), V/Line Freight (1995), Pacific National (????) Scrapped 1,600 mm (5 ft 3 in) broad gauge
A85 B85 Haydn Bunton 5 June 1984 Unknown 14 January 2019 Freight Australia Green and Yellow with Pacific National logos V/Line (Rebuilt), V/Line Freight (1995), Pacific National (????) Scrapped 1,600 mm (5 ft 3 in) broad gauge
^ a b Scott Martin & Chris Banger (October 2006). "'New Deal' for County Passengers - 25 years on". Newsrail. Australian Railway Historical Society (Victorian Division): 319.
^ a b c Railmac Publications (1992). Australian Fleetbooks: V/Line locomotives. Kitchner Press. ISBN 0-949817-76-7.
^ "News". Newsrail. Australian Railway Heritage Society: 24–27. September 1986.
^ "General News". Newsrail. Australian Railway Historical Society. September 1986. p. 275.
^ Hewison, P, Kimpton, R. (2002) Freight Australia's Rollingstock Initiatives, Conference on Railway Engineering, Wollongong, 10–13 November 2002, pp. 294–295
^ Railway Digest June 2013
^ A Class (A60-A85) Railpage
^ "A Class Swansong" Motive Power issue 94 July 2014 pages 66, 68
^ a b c A Class Vicsig
^ Pacific National Scrapping A Class Locomotives Wongm's Rail Gallery
^ "Rollingstock". Newsrail. Australian Railway Historical Society. May 1988. p. 152.
Media related to V/Line A class at Wikimedia Commons
{\displaystyle \mathrm {D_{E}^{D}} }
Retrieved from "https://en.wikipedia.org/w/index.php?title=V/Line_A_class&oldid=1089667145"
Clyde Engineering locomotives
Pacific National diesel locomotives
Victorian Railways diesel locomotives
Broad gauge locomotives in Australia
|
permute_elements - Maple Help
Home : Support : Online Help : Programming : Logic : Boolean : verify : permute_elements
verify equality up to the permutation of the given elements
verify(expr1, expr2, permute_elements(elems))
verify(expr1, expr2, permute_elements(elems, ver))
either a set or a list
verification for the permute_elements operands
The verify(expr1, expr2, permute_elements(elems)) and verify(expr1, expr2, permute_elements(elems, ver)) calling sequences return true if, after the permutations of elems is substituted into expr2, the test for equality or the verification ver is true.
This function is useful for comparing objects which may contain session dependent permutations of arbitrary constants.
The verification permute_elements(elems) is always symmetric. The verification permute_elements(elems, ver) is symmetric if and only if the verification ver is symmetric.
\mathrm{evalb}\left(\mathrm{_C1}\mathrm{exp}\left(x\right)+\mathrm{_C2}\mathrm{exp}\left(-x\right)=\mathrm{_C2}\mathrm{exp}\left(x\right)+\mathrm{_C1}\mathrm{exp}\left(-x\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left(\mathrm{_C1}\mathrm{exp}\left(x\right)+\mathrm{_C2}\mathrm{exp}\left(-x\right),\mathrm{_C2}\mathrm{exp}\left(x\right)+\mathrm{_C1}\mathrm{exp}\left(-x\right),'\mathrm{permute_elements}'\left({\mathrm{_C1},\mathrm{_C2}}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
|
Normal distribution - Wikiversity
(Redirected from Bell curve)
A normal distribution can be described by four moments: mean, standard deviation, skewness and kurtosis. Statistical properties of normal distributions are important for parametric statistical tests which rely on assumptions of normality.
4 Normal probability plots
6 Testing for normality
6.2 Descriptive indicators of normality
6.3 Inferential tests of normality
7 Dealing with non-normality
Probability density function[edit | edit source]
The probability density function of the standard normal distribution (with the standard deviation and area under the curve standardized to 1 and the mean and skewness standardized to 0) is given by
{\displaystyle f(x)={\frac {e^{-x^{2}/2}}{\sqrt {2\pi }}}}
This function has no elementary antiderivative, and thus normal distribution problems are mainly limited to using numerical integration to find a probability. However, using multivariable calculus, the value of the Gaussian integral can be determined to be exactly
{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}dx={\sqrt {\pi }}}
, and through the change of variables
{\displaystyle u=x{\sqrt {2}},du=dx{\sqrt {2}}}
, we can be assured that the total area beneath the curve is 1, and thus it is normalized correctly.
A similar approach can be used to prove that its standard deviation is also 1. By the definition of the standard deviation,
{\displaystyle \sigma ^{2}=\Sigma {(x_{k}-\mu )^{2}}}
, so we form the Riemann sum
{\displaystyle \Sigma (x-0)^{2}{\frac {e^{-x^{2}/2}}{\sqrt {2\pi }}}\Delta x}
. Taking the limit of the sum leads to the integral
{\displaystyle \int _{-\infty }^{\infty }x^{2}{\frac {e^{-x^{2}/2}}{\sqrt {2\pi }}}dx}
. Using integration by parts, we can determine that this is in fact equal to the area under the normal curve, and thus the standard deviation is 1. Using differentiation of the probability density function, we find that the inflection points of the normal distribution curve are each exactly one standard deviation away from the mean.
Any other normal distribution can be standardized through a change of variables (such as if the mean is not 0).
Learn about the general properties of a Probability Distribution and check why does the normal distribution fullfil these properities.
Some normal distributions with various parameters.
Normal probability plots[edit | edit source]
A normality probability plot - data falling around this straight indicates the degree of departure from normality. For more information, see * Q-Q plot
Population mean = μ (Mu)
Population variance = σ2 = (Sigma squared)
Testing for normality[edit | edit source]
No single indicator of normality should be overly relied upon. Graphical, descriptive, and inferential can be used, each with strengths and limitations. The most important result is to actually describe and show the distribution. Simply listing statistical properties does not demonstrate understanding.
Graphical analysis[edit | edit source]
Whether it is reported as a Figure or not, responses to interval or continuous variables should be visualised as a:
Histogram, with normal curve imposed
This is the single most basic and important way of examining the central tendency and shape of distribution for participants' responses.
It may also be helpful to examine a:
Descriptive indicators of normality[edit | edit source]
A rule of thumb for assessing normality for the purposes of assumption testing for inferential statistical tests such as ANOVA is that if skewness and kurtosis are between -1 and +1 and there is a reasonable sample size (e.g., at least 20 per cell), then you are unlikely to run into issues related to violations of the assumption of normality.
Some authors suggest that variables with skewness and kurtosis values between -2 to +2 or even -3 to +3 can be treated as being drawn from a normally distributed population.
The larger the sample size, the more robust inferential tests are to departure from normality.
The skewness of a Normal Distribution is always 0; +ve scores indicate a tail to the right; -ve scores indicate a tail to the left.
The kurtosis of a Normal Distribution is always 0; +ve scores indicate a peaked distribution; -ve scores indicate a relatively flat distribution.
If concerned about non-normality, then consider recoding data to a lower level of measurement.
For more information, see: Judging severity of skewness and kurtosis
Inferential tests of normality[edit | edit source]
Significance tests of (non-)normality become overly sensitive when the sample size is large. Thus, do not rely on significance tests of normality alone in making an assessment (e.g., for assumption-testing purposes):
These tests are overly sensitive to minor departures from normality, particularly with large sample samples (e.g., > 200). This doesn't mean that it should be discounted as an indicator, just that a sig. (p < .05) test value does not necessarily indicate a notable or problematic departure from normality. Also check normality using other indicators.
Take-home message[edit | edit source]
It is recommended that both graphical indicators and descriptive indicators be used for testing the assumption that a sample is derived from a normally distributed population. Inferential normality tests may also be useful.
Dealing with non-normality[edit | edit source]
Standardise data in SPSS
Non-parametric statistics[edit | edit source]
Testing for normality using SPSS (Laerd Statistics]]
Retrieved from "https://en.wikiversity.org/w/index.php?title=Normal_distribution&oldid=1877576"
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7+3·4+2
What movements does this represent for Cecil walking on his tightrope? Draw a diagram to show his movements and the length of his walk.
Which of the lengths does Cecil repeat? How many times does he repeat it?
If you were to write the expression using only addition, it would look like this:
7+4+4+4+2
How many different answers can you get by grouping differently? Add parentheses to the expression
7+3·4+2
to create new expressions with as many different values as possible.
Add parentheses to any possible place in the expression. Remember to evaluate each expression for its value.
One possible solution to this problem is
7+3(4+2)=25
. In total, there are four answers possible for adding parentheses to this expression.
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Constrained Tibial Vibration in Mice: A Method for Studying the Effects of Vibrational Loading of Bone | J. Biomech Eng. | ASME Digital Collection
Department of Orthopaedic Surgery, and Department of Biomedical Engineering,
, Campus Box 8233, St. Louis, MO 63110
e-mail: bchrist1@bidmc.harvard.edu
Christiansen, B. A., Bayly, P. V., and Silva, M. J. (May 16, 2008). "Constrained Tibial Vibration in Mice: A Method for Studying the Effects of Vibrational Loading of Bone." ASME. J Biomech Eng. August 2008; 130(4): 044502. https://doi.org/10.1115/1.2917435
Vibrational loading can stimulate the formation of new trabecular bone or maintain bone mass. Studies investigating vibrational loading have often used whole-body vibration (WBV) as their loading method. However, WBV has limitations in small animal studies because transmissibility of vibration is dependent on posture. In this study, we propose constrained tibial vibration (CTV) as an experimental method for vibrational loading of mice under controlled conditions. In CTV, the lower leg of an anesthetized mouse is subjected to vertical vibrational loading while supporting a mass. The setup approximates a one degree-of-freedom vibrational system. Accelerometers were used to measure transmissibility of vibration through the lower leg in CTV at frequencies from
20Hzto150Hz
. First, the frequency response of transmissibility was quantified in vivo, and dissections were performed to remove one component of the mouse leg (the knee joint, foot, or soft tissue) to investigate the contribution of each component to the frequency response of the intact leg. Next, a finite element (FE) model of a mouse tibia-fibula was used to estimate the deformation of the bone during CTV. Finally, strain gages were used to determine the dependence of bone strain on loading frequency. The in vivo mouse leg in the CTV system had a resonant frequency of
60Hz
±0.5G
vibration (
1.0G
peak to peak). Removing the foot caused the natural frequency of the system to shift from
60Hzto70Hz
, removing the soft tissue caused no change in natural frequency, and removing the knee changed the natural frequency from
60Hzto90Hz
. By using the FE model, maximum tensile and compressive strains during CTV were estimated to be on the cranial-medial and caudolateral surfaces of the tibia, respectively, and the peak transmissibility and peak cortical strain occurred at the same frequency. Strain gage data confirmed the relationship between peak transmissibility and peak bone strain indicated by the FE model, and showed that the maximum cyclic tibial strain during CTV of the intact leg was
330±82με
and occurred at
60–70Hz
. This study presents a comprehensive mechanical analysis of CTV, a loading method for studying vibrational loading under controlled conditions. This model will be used in future in vivo studies and will potentially become an important tool for understanding the response of bone to vibrational loading.
biomechanics, bone, deformation, finite element analysis, vibrations, vibration, osteoporosis, transmissibility, WBV
Bone, Vibration, Strain gages, Deformation, Frequency response, Finite element analysis, Knee, Finite element model
Genetic Predisposition to Low Bone Mass is Paralleled by an Enhanced Sensitivity to Signals Anabolic to the Skeleton
Anabolism. Low Mechanical Signals Strengthen Long Bones
Mechanical Strain, Induced Noninvasively in the High-Frequency Domain, Is Anabolic to Cancellous Bone, But Not Cortical Bone
Low Magnitude Mechanical Loading is Osteogenic in Children With Disabling Conditions
Low-Magnitude Mechanical Signals That Stimulate Bone Formation in the Ovariectomized Rat are Dependent on the Applied Frequency But Not on the Strain Magnitude
Quantity and Quality of Trabecular Bone in the Femur are Enhanced by a Strongly Anabolic, Noninvasive Mechanical Intervention
Rest Insertion Combined With High-Frequency Loading Enhances Osteogenesis
Low-Level Accelerations Applied in the Absence of Weight Bearing Can Enhance Trabecular Bone Formation
Age-Related Bone Loss. A Hypothesis and Initial Assessment in Mice
Whole-Body Vibration in the Skeleton: Development of a Resonance-Based Testing Device
Finite Element Analysis of the Mouse Tibia: Estimating Endocortical Strain During Three-Point Bending in SAMP6 Osteoporotic Mice
0003-276X A Discov. Molec., Cell., Evol. Biol.,
Nanoindentation and Whole-Bone Bending Estimates of Material Properties in Bones From the Senescence Accelerated Mouse SAMP6
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40 CFR Appendix A to Subpart S of Part 51 - Calibrations, Adjustments and Quality Control | CFR | US Law | LII / Legal Information Institute
Appendix A to Subpart S of Part 51 - Calibrations, Adjustments and Quality Control
40 CFR Appendix A to Subpart S of Part 51 - Calibrations, Adjustments and Quality Control
(I) Steady-State Test Equipment
States may opt to use transient emission test equipment for steady-state tests and follow the quality control requirements in paragraph (II) of this appendix instead of the following requirements.
(a) Equipment shall be calibrated in accordance with the manufacturers' instructions.
(b) Prior to each test -
(1) Hydrocarbon hang-up check. Immediately prior to each test the analyzer shall automatically perform a hydrocarbon hang-up check. If the HC reading, when the probe is sampling ambient air, exceeds 20 ppm, the system shall be purged with clean air or zero gas. The analyzer shall be inhibited from continuing the test until HC levels drop below 20 ppm.
(2) Automatic zero and span. The analyzer shall conduct an automatic zero and span check prior to each test. The span check shall include the HC, CO, and CO2 channels, and the NO and O2 channels, if present. If zero and/or span drift cause the signal levels to move beyond the adjustment range of the analyzer, it shall lock out from testing.
(3) Low flow. The system shall lock out from testing if sample flow is below the acceptable level as defined in paragraph (I)(b)(6) of appendix D to this subpart.
(c) Leak check. A system leak check shall be performed within twenty-four hours before the test in low volume stations (those performing less than the 4,000 inspections per year) and within four hours in high-volume stations (4,000 or more inspections per year) and may be performed in conjunction with the gas calibration described in paragraph (I)(d)(1) of this appendix. If a leak check is not performed within the preceding twenty-four hours in low volume stations and within four hours in high-volume stations or if the analyzer fails the leak check, the analyzer shall lock out from testing. The leak check shall be a procedure demonstrated to effectively check the sample hose and probe for leaks and shall be performed in accordance with good engineering practices. An error of more than ±2% of the reading using low range span gas shall cause the analyzer to lock out from testing and shall require repair of leaks.
(d) Gas calibration.
(1) On each operating day in high-volume stations, analyzers shall automatically require and successfully pass a two-point gas calibration for HC, CO, and CO2 and shall continually compensate for changes in barometric pressure. Calibration shall be checked within four hours before the test and the analyzer adjusted if the reading is more than 2% different from the span gas value. In low-volume stations, analyzers shall undergo a two-point calibration within seventy-two hours before each test, unless changes in barometric pressure are compensated for automatically and statistical process control demonstrates equal or better quality control using different frequencies. Gas calibration shall be accomplished by introducing span gas that meets the requirements of paragraph (I)(d)(3) of this appendix into the analyzer through the calibration port. If the analyzer reads the span gas within the allowable tolerance range (i.e., the square root of sum of the squares of the span gas tolerance described in paragraph (I)(d)(3) of this appendix and the calibration tolerance, which shall be equal to 2%), no adjustment of the analyzer is necessary. The gas calibration procedure shall correct readings that exceed the allowable tolerance range to the center of the allowable tolerance range. The pressure in the sample cell shall be the same with the calibration gas flowing during calibration as with the sample gas flowing during sampling. If the system is not calibrated, or the system fails the calibration check, the analyzer shall lock out from testing.
(2) Span points. A two point gas calibration procedure shall be followed. The span shall be accomplished at one of the following pairs of span points:
(A) 300 - ppm propane (HC)
1.0 - % carbon monoxide (CO)
6.0 - % carbon dioxide (CO2)
1000 - ppm nitric oxide (if equipped with NO)
1200 - ppm propane (HC)
12.0 - % carbon dioxide (CO2)
(B) - ppm propane
0.0 - % carbon monoxide
0.0 - % carbon dioxide
0 - ppm nitric oxide (if equipped with NO)
600 - ppm propane (HC)
(3) Span gases. The span gases used for the gas calibration shall be traceable to National Institute of Standards and Technology (NIST) standards ±2%, and shall be within two percent of the span points specified in paragraph (d)(2) of this appendix. Zero gases shall conform to the specifications given in § 86.114-79(a)(5) of this chapter.
(e) Dynamometer checks -
(1) Monthly check. Within one month preceding each loaded test, the accuracy of the roll speed indicator shall be verified and the dynamometer shall be checked for proper power absorber settings.
(2) Semi-annual check. Within six months preceding each loaded test, the road-load response of the variable-curve dynamometer or the frictional power absorption of the dynamometer shall be checked by a coast down procedure similar to that described in § 86.118-78 of this chapter. The check shall be done at 30 mph, and a power absorption load setting to generate a total horsepower (hp) of 4.1 hp. The actual coast down time from 45 mph to 15 mph shall be within ±1 second of the time calculated by the following equation:
\text{Coast Down Time}=\frac{0.0508×W}{\mathrm{HP}}
where W is the total inertia weight as represented by the weight of the rollers (excluding free rollers), and any inertia flywheels used, measured in pounds. If the coast down time is not within the specified tolerance the dynamometer shall be taken out of service and corrective action shall be taken.
(f) Other checks. In addition to the above periodic checks, these shall also be used to verify system performance under the following special circumstances.
(1) Gas Calibration.
(A) Each time the analyzer electronic or optical systems are repaired or replaced, a gas calibration shall be performed prior to returning the unit to service.
(B) In high-volume stations, monthly multi-point calibrations shall be performed. Low-volume stations shall perform multi-point calibrations every six months. The calibration curve shall be checked at 20%, 40%, 60%, and 80% of full scale and adjusted or repaired if the specifications in appendix D(I)(b)(1) to this subpart are not met.
(2) Leak checks. Each time the sample line integrity is broken, a leak check shall be performed prior to testing.
(II) Transient Test Equipment
(a) Dynamometer. Once per week, the calibration of each dynamometer and each fly wheel shall be checked by a dynamometer coast-down procedure comparable to that in § 86.118-78 of this chapter between the speeds of 55 to 45 mph, and between 30 to 20 mph. All rotating dynamometer components shall be included in the coast-down check for the inertia weight selected. For dynamometers with uncoupled rolls, the uncoupled rollers may undergo a separate coast-down check. If a vehicle is used to motor the dynamometer to the beginning coast-down speed, the vehicle shall be lifted off the dynamometer rolls before the coast-down test begins. If the difference between the measured coast-down time and the theoretical coast-down time is greater than + 1 second, the system shall lock out, until corrective action brings the dynamometer into calibration.
(b) Constant volume sampler.
(1) The constant volume sampler (CVS) flow calibration shall be checked daily by a procedure that identifies deviations in flow from the true value. Deviations greater than ±4% shall be corrected.
(2) The sample probe shall be cleaned and checked at least once per month. The main CVS venturi shall be cleaned and checked at least once per year.
(3) Verification that flow through the sample probe is adequate for the design shall be done daily. Deviations greater than the design tolerances shall be corrected.
(c) Analyzer system -
(1) Calibration checks.
(A) Upon initial operation, calibration curves shall be generated for each analyzer. The calibration curve shall consider the entire range of the analyzer as one curve. At least 6 calibration points plus zero shall be used in the lower portion of the range corresponding to an average concentration of approximately 2 gpm for HC, 30 gpm for CO, 3 gpm for NOX, and 400 gpm for CO2. For the case where a low and a high range analyzer is used, the high range analyzer shall use at least 6 calibration points plus zero in the lower portion of the high range scale corresponding to approximately 100% of the full-scale value of the low range analyzer. For all analyzers, at least 6 calibration points shall also be used to define the calibration curve in the region above the 6 lower calibration points. Gas dividers may be used to obtain the intermediate points for the general range classifications specified. The calibration curves generated shall be a polynomial of no greater order than 4th order, and shall fit the date within 0.5% at each calibration point.
(B) For all calibration curves, curve checks, span adjustments, and span checks, the zero gas shall be considered a down-scale reference gas, and the analyzer zero shall be set at the trace concentration value of the specific zero gas used.
(2) The basic curve shall be checked monthly by the same procedure used to generate the curve, and to the same tolerances.
(3) On a daily basis prior to vehicle testing -
(A) The curve for each analyzer shall be checked by adjusting the analyzer to correctly read a zero gas and an up-scale span gas, and then by correctly reading a mid-scale span gas within 2% of point. If the analyzer does not read the mid-scale span point within 2% of point, the system shall lock out. The up-scale span gas concentration for each analyzer shall correspond to approximately 80 percent of full scale, and the mid-point concentration shall correspond to approximately 15 percent of full scale; and
(B) After the up-scale span check, each analyzer in a given facility shall analyze a sample of a random concentration corresponding to approximately 0.5 to 3 times the cut point (in gpm) for the constituent. The value of the random sample may be determined by a gas blender. The deviation in analysis from the sample concentration for each analyzer shall be recorded and compared to the historical mean and standard deviation for the analyzers at the facility and at all facilities. Any reading exceeding 3 sigma shall cause the analyzer to lock out.
(4) Flame ionization detector check. Upon initial operation, and after maintenance to the detector, each Flame Ionization Detector (FID) shall be checked, and adjusted if necessary, for proper peaking and characterization. Procedures described in SAE Paper No. 770141 are recommended for this purpose. A copy of this paper may be obtained from the Society of Automotive Engineers, Inc. (SAE), 400 Commonwealth Drive, Warrendale, Pennsylvania, 15096-0001. Additionally, every month the response of each FID to a methane concentration of approximately 50 ppm CH4 shall be checked. If the response is outside of the range of 1.10 to 1.20, corrective action shall be taken to bring the FID response within this range. The response shall be computed by the following formula:
\text{Ratio of Methane Response}=\frac{\text{FID response in ppmC}}{\text{ppm methane in cylinder}}
(5) Spanning frequency. The zero and up-scale span point shall be checked, and adjusted if necessary, at 2 hour intervals following the daily mid-scale curve check. If the zero or the up-scale span point drifts by more than 2% for the previous check (except for the first check of the day), the system shall lock out, and corrective action shall be taken to bring the system into compliance.
(6) Spanning limit checks. The tolerance on the adjustment of the up-scale span point is 0.4% of point. A software algorithm to perform the span adjustment and subsequent calibration curve adjustment shall be used. However, software up-scale span adjustments greater than ±10% shall cause the system to lock out, requiring system maintenance.
(7) Integrator checks. Upon initial operation, and every three months thereafter, emissions from a randomly selected vehicle with official test value greater than 60% of the standard (determined retrospectively) shall be simultaneously sampled by the normal integration method and by the bag method in each lane. The data from each method shall be put into a historical data base for determining normal and deviant performance for each test lane, facility, and all facilities combined. Specific deviations exceeding ±5% shall require corrective action.
(8) Interference. CO and CO2 analyzers shall be checked prior to initial service, and on a yearly basis thereafter, for water interference. The specifications and procedures used shall generally comply with either § 86.122-78 or § 86.321-79 of this chapter.
(9) NOXconverter check. The converter efficiency of the NO2 to NO converter shall be checked on a weekly basis. The check shall generally conform to § 86.123-78 of this chapter, or EPA MVEL Form 305-01. Equivalent methods may be approved by the Administrator.
(10) NO/NOXflow balance. The flow balance between the NO and NOX test modes shall be checked weekly. The check may be combined with the NOX convertor check as illustrated in EPA MVEL Form 305-01.
(11) Additional checks. Additional checks shall be performed on the HC, CO, CO2, and NOX analyzers according to best engineering practices for the measurement technology used to ensure that measurements meet specified accuracy requirements.
(12) System artifacts (hang-up). Prior to each test a comparison shall be made between the background HC reading, the HC reading measured through the sample probe (if different), and the zero gas. Deviations from the zero gas greater than 10 parts per million carbon (ppmC) shall cause the analyzer to lock out.
(13) Ambient background. The average of the pre-test and post-test ambient background levels shall be compared to the permissible levels of 10 ppmC HC, 20 ppm CO, and 1 ppm NOX. If the permissible levels are exceeded, the test shall be voided and corrective action taken to lower the ambient background concentrations.
(14) Analytical gases. Zero gases shall meet the requirements of § 86.114-79(a)(5) of this chapter. NOX calibration gas shall be a single blend using nitrogen as the diluent. Calibration gas for the flame ionization detector shall be a single blend of propane with a diluent of air. Calibration gases for CO and CO2 shall be single blends using nitrogen or air as a diluent. Multiple blends of HC, CO, and CO2 in air may be used if shown to be stable and accurate.
(III) Purge Analysis System
On a daily basis each purge flow meter shall be checked with a simulated purge flow against a reference flow measuring device with performance specifications equal to or better than those specified for the purge meter. The check shall include a mid-scale rate check, and a total flow check between 10 and 20 liters. Deviations greater than ±5% shall be corrected. On a monthly basis, the calibration of purge meters shall be checked for proper rate and total flow with three equally spaced points across the flow rate and the totalized flow range. Deviations exceeding the specified accuracy shall be corrected. The dynamometer quality assurance checks required under paragraph (II) of this appendix shall also apply to the dynamometer used for purge tests.
(IV) Evaporative System Integrity Test Equipment
(a) On a weekly basis pressure measurement devices shall be checked against a reference device with performance specifications equal to or better than those specified for the measurement device. Deviations exceeding the performance specifications shall be corrected. Flow measurement devices, if any, shall be checked according to paragraph III of this appendix.
(b) Systems that monitor evaporative system leaks shall be checked for integrity on a daily basis by sealing and pressurizing.
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V/Line Sprinter - Wikipedia
Refurbished car 7002 at Jacana
A Goninan & Co
1 (7019; accident damage)
V/Line, Metro Trains Melbourne
Ballarat (up to Bacchus Marsh)
Metro Trains Melbourne: Stony Point
4 plug doors
2 × Deutz BF8L513C
Voith T211RZ hydraulic
Davies and Metcalfe EBC/5 EP pneumatic disc
1,600 mm (5 ft 3 in) broad gauge
The Sprinter is a diesel railcar built by A Goninan & Co in Broadmeadow for V/Line between 1993 and 1995.
1 Design origins
2 Manufacturing & testing
3 Passenger experience
6 Accidents and other problems
Design origins[edit]
7004 at Wallan in November 2007 in original livery
Three Sprinters in Geelong in November 2007 in original livery
A refurbished two car set at Dandenong, 2013
The Sprinter concept dates back to 1989, when the Public Transport Corporation, having seen a substantial increase in patronage and reduction in costs following the introduction of faster, more frequent services as part of the New Deal for Country Passengers program of the 1980s, required additional train capacity to meet demand.[1] Initial talks suggested an order for 24 new vehicles,[2] though the tenders for the construction of the 22 railcars closed in November 1989.[3]
At the time they were designed to supplement locomotive-hauled H type carriage sets on shorter runs (such as on the outer suburban Melton and Sunbury lines, as well as the interurban Geelong and Seymour lines) and thus provide faster and more frequent service to Melbourne's fringe areas, and indirectly (primarily by freeing up other rollingstock) to more distant regions. Their introduction also enabled the retirement of the four DRC railcars from service.
In keeping with their intended operation, they feature high-capacity single-class seating and a single-car railmotor design over a multiple unit design, allowing a large number of passengers to be carried with greatly reduced operating costs and increased flexibility.
Manufacturing & testing[edit]
On 11 October 1991, the Federal Government announced the purchase of 22 Sprinters,[4] built at a total cost of $65 million, of which the Federal Government provided $24 million through its Better Cities program.[1]
Twenty-two single-car stainless-steel-body railcars were ordered from A Goninan & Co, Broadmeadow, for introduction to service between 1993 and 1995. Construction commenced in March 1993 with the final unit outshopped in January 1995. The first two units were fully fitted out at Goninan, with the other 20 units internally fitted out at the PTC North Bendigo workshops.[3][5][6] All were transferred to Melbourne by rail on standard gauge transfer bogies.
Passenger experience[edit]
Sprinters feature a mix of 3x2 and 2x2 economy seats arranged so that half of them face the direction of travel at any one time. Reflecting the nature of the sets and their intended use, these seats are slightly smaller than the seats found in H and N sets and VLocity DMUs. They are finished in blue patterned cloth.
The cars are also fitted with a toilet and drinking fountains. Provision is made at one end of each carriage for one wheelchair and occupant. When this space is not needed able-bodied passengers may make use of the three wall-mounted fold-down seats. The toilet has a wide door and grab bars for use by disabled passengers.
While the capacity exists for several units to be coupled in service, passengers may not under normal circumstances move between coupled cars. A door is however provided, along with a detraining ladder, at the end of each car in case emergency egress is required. Conductors may move between cars during travel.
Passenger luggage can be carried in the overhead racks, between the backs of seats, or in the luggage/bicycle storage area usually found at the Melbourne end of the carriage. Ordinary access to the car is via four automatic plug doors, one on each side of the car at each end. These doors are opened by a push button mounted beside them and closed by the driver, and are wide enough to permit the access of a standard wheelchair.
Each car is powered by two air-cooled Deutz turbocharged V8 diesel engines. Power is transferred via a Voith T211RZ hydraulic transmission. Sprinters use a Davies and Metcalfe EBC/5 EP anti-slide pneumatic disc brake system.
To facilitate use in multiple-unit formations, they are fitted with Scharfenberg couplers. This allows them to be coupled to other Sprinters to form a train as long as eight carriages.
Sprinter operation commenced in December 1993, with an official launch on the Ballarat corridor on 16 December 1993, with a special service running to Ballarat station and back, with the then Transport Minister, Alan Brown and other guests on board. A maximum speed of 143 kilometres per hour (89 mph) was achieved during this trip.[3] They were launched on the Bendigo corridor on 17 March 1994, with a special service running to Bendigo station and back. The Sprinters were launched on the Geelong line on 1 September 1994, with a special service running to Geelong station, launching on the Seymour line on 14 December 1994, with a special service running to Seymour station, and finally launching on the Traralgon line on 16 June 1995, to coincide with the newly built Traralgon station.[3]
For operation on the longer routes a portable buffet cart was trialled - the first such service since 1961, on the Horsham and Warrnambool lines. The cart was delivered to V/Line Passenger at Spencer Street Station on 12 August 1996, and its first use was on the 10:30am Melbourne to Bendigo and 1:20pm return service on the 15th of that month. By September 1996 the cart was in regular use between Seymour and Wodonga.[7]
Over time, the Sprinters also operated to interurban destinations such as Warragul and Kyneton, along with outer suburban destinations like Craigieburn and Sunbury.
Sprinters were the first Victorian passenger trains to run at 130 kilometres per hour (81 mph), and enabled acceleration of some services by up to 15 minutes. Furthermore, their relatively low operating costs allowed for an increase in service frequency on the Ballarat and Bendigo lines. These improved services contributed to an increase in V/Line patronage from 6.5 million passenger journeys in 1993/94 to 7.0 million in 1995/96.[1]
In addition, they performed some longer trips to destinations such as Albury and Echuca, although this was not specially catered for in their designs and thus led to some concerns over amenities such as lack of catering.
Following the introduction of VLocity stock, Sprinters have been returned to short-haul duties. Since 2008, a pair have been hired to Metro Trains Melbourne to operate services on the Stony Point line with units periodically rotated.
Sprinters are used on lines including:
North-East line to Seymour
Stony Point line between Frankston and Stony Point, leased to and operated by Metro Trains Melbourne
Until 31 January 2021 the trains were still in regular use along the Gippsland line to Traralgon and the Geelong line to Geelong, however with the timetable change on that day they are no longer scheduled to run on these lines. In emergencies, they can as required.
The Sprinters were specifically designed to also run on the South Gippsland line to Leongatha and still includes the town on its list of V/Line routes.[8] Although they never ran this service, these trains were operated along the line from Dandenong to Cranbourne for a short period of time from when the V/Line rail service to Leongatha was withdrawn on 24 July 1993 until 24 March 1995 when the line to Cranbourne was electrified, in order to avoid having no rail service to Cranbourne.
When operating on the Stony Point line, the toilet and drinking fountain are locked out of use.
Accidents and other problems[edit]
The Sprinter fleet had a number of teething problems, including failures to trigger level crossings, which led to their temporary removal from service on 9 January 1996 on all lines except the Bendigo line.[3] They were also noted to have a high fault incidence mainly due to unreliable componentry and electrical circuitry.[9]
The first Sprinter to derail on a public service occurred on 1 May 1994 at Bacchus Marsh. The leading bogie on a unit derailed after rocks were placed on both rails.[3]
Only three major accidents involving a Sprinter have been recorded – the first occurred on 20 November 1996 at the Spencer Street Rail Motor Depot when at around 17:30 Sprinters 7010+7008 shunted out from the platforms at low speed after disembarking passengers from an up service. The cars were to run into the sidings at the Rail Motor Depot, but when the driver got up to adjust the offside rear-vision mirror he was locked out of the cab by the self-closing door. Unable to regain control of the vehicle, the train proceeded to collide with stabled Sprinters 7019+7016.[citation needed]
Despite being the aggressor, 7010’s damage was mostly superficial, consisting of broken windows and bent side panelling. 7019 came off much worse; with a buckled frame, the No. 2 cab bent downwards nearly 30 degrees at the saloon doors. The coupler was never recovered. Both trailing Sprinters received little damage.[citation needed]
After the accident investigation was concluded in April 1997, both Sprinters were hauled to Goninan’s Bendigo Workshops for assessment. 7010 returned to service in 1998, but it was decided that repairing 7019 would be too costly and so the car was written off. Its interior was gutted before the car was hauled back to Melbourne on 12 July 1998. After being stored in the East Block of Newport Workshops it was later scrapped.[10]
The second occurred on 15 November 2003. Sprinters 7003, 7004 & 7005 were on the 15:49 service to Ballarat[11] when 7003 being the lead unit struck a vehicle stuck on the tracks between Ballan and Gordon. Over 60 people were injured with 7003 rolling on to its side and finishing in a ditch while 7004 & 7005 derailed. 7003 sustained significant damage to its driver compartment and side, 7005 to its driver compartment after striking 7003 as it jackknifed and 7004 sustained only minor damage. The Ballarat line was closed for three days for the clean up.[12]
The third occurred on 2 December 2016 when 7012 caught fire at Seymour railway station while preparing to haul a passenger train to Melbourne at around 12:30pm. The cause of the fire is currently[when?] under investigation.[13]
V/Line sprinter 7011 arrives into Seymour.
A refurbishment program for the Sprinters was announced in 2007 by Transport Minister Lynne Kosky. The works included reupholstery of the interiors and repainting of the exterior.[14]
In September 2018 Sprinter railcar 7012 re-entered service following a repaint into PTV livery and interior refurbishment.[15]
All but the last 3 Sprinters have been named after prominent Victorian sportspersons.[16]
7001 24 Mar 1993 20 Dec 1993 Sir Hubert Opperman
7002 16 Apr 1993 20 Dec 1993 Steve Moneghetti
7003 27 May 1993 8 Jan 1994 James Tomkins
7004 1 Jul 1993 8 Jan 1994 Nick Green
7005 6 Aug 1993 8 Feb 1994 Michael McKay
7006 3 Sep 1993 1 Mar 1994 Andrew Cooper
7007 29 Sep 1993 28 Mar 1994 Faith Leech
7008 27 Oct 1993 20 Jun 1994 Gary Ablett
7009 24 Nov 1993 15 Jul 1994 Bob Davis
7010 14 Jan 1994 19 Jul 1994 Andrew Gaze
7011 11 Feb 1994 24 Aug 1994 Raelene Boyle
7012 9 Mar 1994 1 Sep 1994 Roy Higgins 1st to have PTV Livery
7013 20 Apr 1994 26 Oct 1994 Lionel Rose
7014 25 May 1994 24 Oct 1994 Kirstie Marshall
7015 6 Jul 1994 14 Nov 1994 Louise Dobson
7016 27 Jul 1994 28 Nov 1994 Michael Tuck
7017 17 Aug 1994 28 Dec 1994 Debbie Flintoff-King
7018 14 Sep 1994 7 Feb 1995 Bill Roycroft
7019 7 Oct 1994 1 Mar 1995 Danni Roche Scrapped
7020 4 Nov 1994 5 May 1995
7021 16 Dec 1994 8 May 1995
On Track Models is producing HO Scale V/Line Sprinters. Models are sold in single or two-packs, with all liveries aside from PTV represented.
As of November 2021, no ready-to-run N scale models of the V/Line Sprinter exist. Brimbank Models is currently selling an N Scale body designed to fit over a readily available Tomix mechanism.[17]
^ a b c Lee, Robert (2007). The Railways of Victoria 1854-2004. Melbourne University Publishing. pp. 255, 257. ISBN 978-0-522-85134-2.
^ a b c d e f Chris Banger (November 1997). "Sprinters". Newsrail. Australian Railway Historical Society. p. 338.
^ "General News". Newsrail. Australian Railway Historical Society. December 1991. p. 396.
^ Sprinter Railpage
^ Sprinter Vicsig
^ Newsrail February 1997 p.87
^ 'High speed with Sprinter', Tracks Magazine (Railways of Australia Network)): pages 14-17. January, February, March 1992
^ "Public transport reforms - Moving from a system to a service - Part 8: Efficiency of public transport". Victorian Auditor-General's Office (Australia). 14 May 1998.
^ "VICSIG". vicsig.net. Retrieved 25 October 2021.
^ Ballan Vicsig
^ Berry, Jamie; La Canna, Xavier (17 November 2003). "Two men charged over train accident". The Age. Melbourne.
^ "V/Line train catches fire at train station". www.heraldsun.com.au. 2 December 2016. Retrieved 1 February 2020.
^ "New Look V/Line Fleet Takes to the". From the Minister for Public Transport. Government of Victoria. 12 September 2007. Archived from the original on 6 May 2008. Retrieved 5 May 2018.
^ "VICSIG - Rollingstock". vicsig.net. Retrieved 10 January 2020.
^ Sprinter naming summary V/Line Cars
^ "V/Line Sprinter". Brimbank Models. 1 January 2019. Retrieved 25 October 2021.
Wikimedia Commons has media related to Sprinter (Victorian train).
{\displaystyle \mathrm {D_{E}^{D}} }
Retrieved from "https://en.wikipedia.org/w/index.php?title=V/Line_Sprinter&oldid=1083641168"
Diesel multiple units of Victoria (Australia)
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Invention master cape - The RuneScape Wiki
The cape worn by the true masters of Invention.
Item JSON: {"edible":"no","members":"yes","stackable":"no","stacksinbank":"yes","death":"reclaimable","name":"Invention master cape","bankable":"yes","gemw":false,"equipable":"yes","disassembly":"no","release_date":"25 January 2016","id":"36356","release_update_post":"New Skill - Invention","lendable":"no","destroy":"Drop","highalch":48000,"weight":0.453,"tradeable":"no","examine":"The cape worn by the true masters of Invention.","noteable":"no"}
The Invention master cape is the cape awarded for achieving True skill mastery by reaching level 120 Invention. Its appearance is different from regular skillcapes, and has a separate, unique emote from their level 99 skillcape counterpart. The cape is sold by Doc or in the Max Guild by Elen Anterth for 120,000 coins.
It can be stored in a cape rack of a player-owned house.
The Invention master cape emote
2.1.1 Charge drain
2.1.2 Disassembly chances
The Invention master cape inherits the perk of the base cape: it provides a 2% charge drain reduction, which stacks multiplicatively with other reductions.
The Invention master cape gains a new perk that increases the chance of receiving uncommon and rare components when disassembling by 20%.
Charge drain[edit | edit source]
The charge drain reduction is simply a 0.98 multiplier on top of any and all other reductions currently in effect - see charge pack for more details.
Disassembly chances[edit | edit source]
Reason: need a large amount of disassembly data to fully confirm this - it may adjust weights without scaling back the total weight - see Disassemble#Invention master cape
The disassembly chance increase applies to rare and uncommon materials, at the expense of the chance of getting common materials. The after-junk chance of getting a rare/uncommon material is increased by 20%, and the increase is removed proportionally from the common materials. The cape does not change junk chance. This is better understood with some worked examples, as well as understanding how materials are rolled when disassembling (see Disassemble#Material distribution).
The chance increase only applies when disassembling items by hand with the Disassemble action or the Breakdown perk - the mk. I and mk. II machines do not benefit from this.
Example 1 - logs
The basic disassembly information for logs is:
Simple parts - 99%
Living components - 1%
1 material per log
Junk chance:
Maple logs: 50.5% base, 40.5% with maximum reduction
Elder logs: 0%
While wearing a master cape, the chance of the uncommon material, living components, is multiplied by 1.2 to get 1.2%. The additional chance is removed from the common material, simple parts, giving
{\displaystyle 99-0.2=98.8\%}
Thus when disassembling elder logs you should expect to get 9880 simple parts and 120 living components instead of 9900 and 100 without the cape, per 10,000 logs. Junk applies before material distribution, so when disassembling maple logs, you should expect 4050 junk and 5950 non-junk per 10,000 logs, which then is distributed into 5878.6 simple parts and 71.4 living components with the cape, and 5890.5 and 59.5 without the cape.
Example 2 - jewellery
The basic disassembly information for jewellery is:
Delicate parts - 35%
Connector parts - 30%
Smooth parts - 30%
Precious components - 3%
Enhancing components - 2%
5 materials per item
Ring of slaying: 4.2% base, 3.4% with maximum reduction
The two uncommon materials are multiplied by 1.2 to give them a 3.6% and 2.4% chance respectively - the sum of uncommon chances is increased from 5% to 6%. The common material chances are then reduced by multiplying each by a scaling factor -
{\displaystyle (100-sum\ of\ new\ uncommon\ chances)/(sum\ of\ original\ common\ chances)}
; which in this case is
{\displaystyle (100-6)/(35+30+30)=94/95}
. This then modifies the common chances into 34.632% and 29.684%.
Thus, for 10,000 rings of slaying disassembled, you should expect:
Delicate 16,905 16,727.1
Connector/Smooth 14,490 14,337.5
Precious 1,449 1,738.8
Enhancing 966 1159.2
Invention skillcape (shop) Invention Guild 120,000 Coins ∞
Players will no longer disconnect when trying to disassemble Doc's Spare Parts while wearing the Invention master cape.
Invention master cape gives an increased chance of receiving uncommon and rare components when disassembling.
Altered the colour of the particles on the Invention master cape.
The cape was formerly blue with a yellow trim. The colours were inverted in an update on 1 February 2016. The particle trail however remained the same until the update on 22 February 2016.
The Invention master emote is the only Cape of Accomplishment not to feature sound to the player who uses it.
Retrieved from ‘https://runescape.wiki/w/Invention_master_cape?oldid=35834094’
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FactorDerivative - Maple Help
Home : Support : Online Help : Mathematics : Differential Equations : Differential Algebra : Tools : FactorDerivative
extracts the derivation operator of a derivative
FactorDerivative(v, R, opts)
notation = jet, tjet, diff or Diff. Specifies the notation used for the result of the function call. If not specified, the notation of v is used.
The function call FactorDerivative(v,R) returns a sequence
\mathrm{\theta }
u
\mathrm{\theta }
is the derivation operator, and, u is the dependent variable, associated to
u
(see DifferentialAlgebra). The argument v must be a derivative of R, or of its embedding ring if R is an ideal.
This command is part of the DifferentialAlgebra:-Tools package. It can be called using the form FactorDerivative(...) after executing the command with(DifferentialAlgebra:-Tools). It can also be directly called using the form DifferentialAlgebra[Tools][FactorDerivative](...).
\mathrm{with}\left(\mathrm{DifferentialAlgebra}\right):
\mathrm{with}\left(\mathrm{Tools}\right):
R≔\mathrm{DifferentialRing}\left(\mathrm{derivations}=[x,y],\mathrm{blocks}=[[v,u],p],\mathrm{parameters}=[p]\right)
\textcolor[rgb]{0,0,1}{R}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{differential_ring}}
\mathrm{\theta },\mathrm{indep}≔\mathrm{FactorDerivative}\left(u[x,y],R\right)
\textcolor[rgb]{0,0,1}{\mathrm{\theta }}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{indep}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}
\mathrm{Differentiate}\left(\mathrm{indep},\mathrm{\theta },R\right)
{\textcolor[rgb]{0,0,1}{u}}_{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}}
\mathrm{FactorDerivative}\left(u,R\right)
\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}
\mathrm{FactorDerivative}\left(\mathrm{diff}\left(u\left(x,y\right),x\right),R,\mathrm{notation}=\mathrm{diff}\right)
\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{u}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{y}\right)
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Thermodynamic potential - Simple English Wikipedia, the free encyclopedia
In thermodynamics, thermodynamic potentials are parameters associated with a thermodynamic system and have the dimensions of energy. They are called "potentials" because in a sense, they describe the amount of potential energy in a thermodynamic system when it is subjected to certain constraints. The different potentials correspond to different constraints to which the system may be subjected. The four most common thermodynamic potentials are:
{\displaystyle U\,}
{\displaystyle ~~~~~S,V,\{N_{i}\}\,}
{\displaystyle A=U-TS\,}
{\displaystyle ~~~~~T,V,\{N_{i}\}\,}
{\displaystyle H=U+pV\,}
{\displaystyle ~~~~~S,p,\{N_{i}\}\,}
{\displaystyle G=U+pV-TS\,}
{\displaystyle ~~~~~T,p,\{N_{i}\}\,}
where T = temperature, S = entropy, p = pressure, V = volume. The Helmholtz free energy is often denoted by the symbol F, but the use of A is preferred by IUPAC (See Alberty, 2001).
{\displaystyle N_{i}}
is the number of particles of type i in the system. For the sake of completeness, the set of all
{\displaystyle N_{i}}
are also included as natural variables, although they are sometimes ignored.
Thermodynamic potentials are very useful when calculating the equilibrium results of a chemical reaction, or when measuring the properties of materials in a chemical reaction. The chemical reactions usually take place under some simple constraints such as constant pressure and temperature, or constant entropy and volume, and when this is true, there is a corresponding thermodynamic potential which comes into play. Just as in mechanics, the system will tend towards lower values of potential and at equilibrium, under these constraints, the potential will take on an unchanging minimum value. The thermodynamic potentials can also be used to estimate the total amount of energy available from a thermodynamic system under the appropriate constraint.
Alberty, R. A. (2001). "Use of Legendre transforms in chemical thermodynamics" (PDF). Pure Appl. Chem. 73 (8): 1349–1380. doi:10.1351/pac200173081349. S2CID 98264934.
Callen, Herbert B. (1985). Thermodynamics and an Introduction to Themostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8.
Thermodynamic Potentials - Georgia State University
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Thermodynamic_potential&oldid=8050224"
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Electronics/Resistors - Wikiversity
6 Resistor color code
This is a lesson in the course Electronics, which is a part of the School:Electronics
A resistor (
{\displaystyle R}
) is an electronic component that resists, restricts, or opposes the flow of electricalcurrent.
The volume of water flow following a constriction in a water pipe is reduced
It can be visualized as constriction or narrowing in a pipe, where the constricted area is the resistance (resistor), and the flow of water is current. The volume of water flow following a constriction in a water pipe is reduced.The resistive property of this type of component can be attributed to a material which has much lower electrical conductivity than regular conductive materials such as metals.
The electrical resistance of a given object depends primarily on two factors:
What material is it made?
therefore, can be computed as:
{\displaystyle R=\rho {\frac {l}{A}}}
{\displaystyle l}
is the length of the conductor, measured in metres [m], A is the cross-section area of the conductor measured in square metres [m²] and ρ (rho) is the electrical resistivity (also called specific electrical resistance) of the material, measured in ohm-metres (Ω·m).
In This formula
{\displaystyle \rho ,{l},{A}}
are constant,therefore
{\displaystyle R}
already is constant.
The inverse resistance R is conductance G
The inverse resistance
{\displaystyle R}
is conductance
{\displaystyle G}
, the ease at which an electric current passes.therefore, can be computed as:
{\displaystyle G={\frac {1}{R}}={\frac {1}{\rho }}\cdot {\frac {A}{l}}}
{\displaystyle {\frac {1}{\rho }}=\sigma }
{\displaystyle \sigma }
([[Sigma (sigma) is the electrical conductivity measured in siemens per meter(S·m−1)
{\displaystyle G={\frac {1}{R}}=\sigma \cdot {\frac {A}{l}}}
{\displaystyle \sigma ,{A},{l},}
{\displaystyle G}
Electrical current (
{\displaystyle I}
) results when a voltage (
{\displaystyle V}
) (or electromagnetic force) causes movement of electrons.
{\displaystyle I\propto V}
then: ↑
{\displaystyle V}
→ ↑
{\displaystyle I}
and ↓
{\displaystyle V}
→ ↓
{\displaystyle I}n the electric circuit, if temperature and all other conditions remain constant, R and G are constant therefore relationship between I and V is linearity then:
{\displaystyle I=G\cdot V\;\;\;\;\;\;\;and\;\;\;\;\;\;\;G={\frac {1}{R}}}
{\displaystyle I={\frac {1}{R}}\cdot V\;\;\;\;\;\;\;or\;\;\;\;\;\;\;I={\frac {V}{R}}}
Ohm's law is linearity
R=1,G=1
R>1,G<1
R<1,G>1
Ohm's law can be used to calculate the resistance present in a DC circuit if voltage and current are known.
Every resistor has a resistance calculated as:
{\displaystyle I={\frac {V}{R}}}
{\displaystyle R={\frac {V}{I}}}
R is resistance (ohms, Ω),
V is voltage (volts, V),
I is current (amperes, A).
Conductance (G) is defined as the inverse of resistance, calculated as:
{\displaystyle G={\frac {1}{R}}}
G is conductance (siemens, S).
Voltage is calculated as:
{\displaystyle I={\frac {V}{R}}}
{\displaystyle V=I\cdot R}
Current is calculated as:
{\displaystyle I={\frac {V}{R}}}
Power is calculated as:
{\displaystyle P=I\cdot V\;\;\;\;\;\;\ and\;\;\;\;\;\;\ I={\frac {V}{R}}\;\;\;\;\;\;\ and\;\;\;\;\;\;\ V=R\cdot I}
{\displaystyle P=I^{2}R\;\;\;\;\;\;\ P={\frac {V^{2}}{R}}}
Resistor color code[edit | edit source]
Any temperature coefficent not assigned its own letter shall be marked "Z", and the coefficient found in other documentation.
Yellow and Gray are used in high-voltage resistors to avoid metal particles in the lacquer.
Resistors found in electronic kits and hobby projects have up to four bars indicating the color code. The first two bars represent the first and second significant digit, the third represents the exponent or decimal multiplier, and the fourth represents the tolerance. The chart to the right shows information about the color coding on resistors.
Wikipedia: Resistor
Retrieved from "https://en.wikiversity.org/w/index.php?title=Electronics/Resistors&oldid=2259852"
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In draft mode, the Pub editor automatically saves text while you’re working on it. Your work is saved when the saving label is green and says “Saved.” The editor can work offline and save your work when you go back online.
If the editor fails to save work, the saving label will show red say “Disconnected.” It will prompt you to copy any changes and refresh the page before you leave the page.
To save a public version or private snapshot version of your Pub, click the Publish button.
Release notes: If entered, these notes will be publicly available on the Pub Dashboard overview.
You can embed LaTeX equations into your Pubs. We use KaTeX to render equations in real-time. For a full list of supported equations, see KaTeX’s support table and supported functions. Separately, we support importing full Pubs from LaTex. See importing for more details.
sum_ix^i
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AVL, named after inventors Adelson-Velsky and Landis, is a binary tree that self-balances by keeping a check on the balance factor of every node.
The balance factor of a node is the difference in the heights of the left and right subtrees. The balance factor of every node in the AVL tree should be either +1, 0 or -1.
After each insertion or deletion, if the balance factor of any node does not follow the AVL balance property, the AVL tree balances itself through the following rotation techniques:
Due to the balancing property, the insertion, deletion and search operations take
O(log n)
in both the average and the worst cases. Therefore, AVL trees give us an edge over Binary Search Trees which have an
O(n)
time complexity in the worst case scenario.
The space complexity of an AVL tree is
O(n)
in both the average and the worst case.
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1 N.N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia.
2 Faculty of Biomedicine, N.I. Pirogov Russian National Research Medical University, Moscow, Russia.
Abstract: A conventionally synthesized thio- and cyano-modified single-stranded poly(dNTP) sequences of different molecular sizes (20n - 200n) and the same lengths routine poly(dNTP) and poly(NTP) species were obtained through the good services provided by the Russian Federal Bioorganic Products Group and by the ThermoFischer, Inc., and then tested for their impact on catalytic activities of β-like DNA polymerases from chromatin of HL-60, WERI-1A and Y-79 cells as well as for the affinity patterns in DNApolβ-poly(dNTP)/ (NTP) pairs, respectively. An essential link between the lengths of ultrashort (50n - 100n) single-stranded poly(dNTP) sequences of different structures and their inhibitory effects towards the cancer-specific DNA polymerases β has been found. A possible significance of this phenomenon for both DNA repair suppression in tumors and a consequent anti-cancer activity of the DNA repair related short poly(dNTP) fragments has been for the first time emphasized with a respect to their pharmacophore revealing potential. Thus, this work presents an experimental attempt to upgrade a contemporary attitude towards the DNA derived products applied for anti-cancer agenda, particularly, for acute myeloid leukemia and retinoblastoma cell DNA repair machinery breakdown. In this study, tumor specific DNA polymerases β were found of being the targets for attack promoted with the primer-like single-stranded DNA fragments followed by consequent cytostatic phenomena. A novel concept of the DNA related anti-cancer medicines is under discussion.
Keywords: Magnetic Isotope Effects (MIE), DNA Repair, DNA Polymerases, DNA-Enzyme Binding
\text{R}+\text{L}\underset{\beta }{\overset{\alpha }{⇄}}\text{RL}
\alpha
\beta
{R}_{0}
{R}_{0}-x
{L}_{0}
{f}_{1}=\alpha {L}_{0}\left({R}_{0}-x\right)
{f}_{-1}=\beta x
\begin{array}{c}\frac{\partial M\left(\theta ,t\right)}{\partial t}={L}_{0}{R}_{0}\alpha \left({\text{e}}^{\theta }-1\right)M\left(\theta ,t\right)-{L}_{0}\alpha \left({\text{e}}^{\theta }-1\right)\frac{\partial M\left(\theta ,t\right)}{\partial \theta }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\beta \left({\text{e}}^{-\theta }-1\right)\frac{\partial M\left(\theta ,t\right)}{\partial \theta }\end{array}
\frac{\text{d}{k}_{1}\left(t\right)}{\text{d}t}={L}_{0}{R}_{0}\alpha -{L}_{0}\alpha {k}_{1}\left(t\right)-\beta {k}_{1}\left(t\right),
\frac{\text{d}{k}_{2}\left(t\right)}{\text{d}t}={L}_{0}{R}_{0}\alpha -{L}_{0}\alpha {k}_{1}\left(t\right)-2{L}_{0}\alpha {k}_{2}\left(t\right)-\beta {k}_{1}\left(t\right)+2\beta {k}_{2}\left(t\right),
{k}_{1}\left(t\right)=m\left(t\right)=\frac{\beta lr}{\beta l+\alpha }\left(1-\mathrm{exp}\text{}\left[-\left(\beta l+\alpha \right)t\right]\right),
\begin{array}{l}{k}_{2}\left(t\right)={\sigma }^{2}\left(t\right)=\frac{\alpha \beta lr}{{\left(\beta l+\alpha \right)}^{2}}\left(1-\mathrm{exp}\left[-\left(\beta l+\alpha \right)t\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{{\beta }^{2}{l}^{2}r}{{\left(\beta l+\alpha \right)}^{2}}\mathrm{exp}\left[-\left(\beta l+\alpha \right)t\right]\left(1-\mathrm{exp}\left[-\left(\beta l+\alpha \right)t\right]\right]\right)\end{array}
{A}_{c}=\left[{A}_{254}/\left({A}_{280}-{A}_{254}\right)\right]/{A}_{210}
\mathrm{ln}{K}_{d}\cong \mathrm{ln}{K}_{0}-E/R{T}_{0}+E\Delta T/R{T}_{0}^{2}
{K}_{d}={K}_{0}\mathrm{exp}\left(-E/RT\right)
\mathrm{ln}{K}_{d}\cong \mathrm{ln}{K}_{0}-E/\left(R{T}_{0}\right)+E\Delta T/R{T}_{0}^{2}
\Delta T
{K}_{d}=f\left(T\right)
{U}_{d}~1/{r}^{6}
\epsilon ~RT
{\tau }_{20}/{\tau }_{50}=\mathrm{exp}\left(-\Delta n\epsilon /RT\right)\cong {10}^{-7}
\Delta =30
{\tau }_{20}
Cite this paper: Stovbun, S. , Ermakov, K. , Bukhvostov, A. , Vedenkin, A. and Kuznetsov, D. (2019) Primer-Like Inhibitors for DNA Repair Enzymes of the AML-HL60 and WERI-1A/Y79 Malignant Cells. Advances in Enzyme Research, 7, 27-44. doi: 10.4236/aer.2019.73003.
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On graph paper, graph the shape that has coordinates
\left(−2, −1\right)
\left(1, 2\right)
\left(−2, 3\right)
Predict the coordinates of the shape after it is translated three units to the right and one unit down. Check your prediction on the graph.
Dilate the original shape by multiplying both the
x
y
2
Reflect the original shape over the
y
-axis. What are the new coordinates?
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Partial function — Wikipedia Republished // WIKI 2
Not to be confused with the partial application of a function of several variables, by fixing some of them.
More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to exactly one element of the second set.
A partial function is often used when its exact domain of definition is not known or difficult to specify. This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function. In computability theory, a general recursive function is a partial function from the integers to the integers; for many of them no algorithm can exist for deciding whether they are in fact total.
When arrow notation is used for functions, a partial function
{\displaystyle f}rom
{\displaystyle X}
{\displaystyle Y}
{\displaystyle f:X\rightharpoonup Y,}
f: X ⇸ Y,
{\displaystyle f:X\nrightarrow Y,}
{\displaystyle f:X\hookrightarrow Y.}
However, there is no general convention, and the latter notation is more commonly used for injective functions.[citation needed]
Specifically, for a partial function
{\displaystyle f:X\rightharpoonup Y,}
{\displaystyle x\in X,}
one has either:
{\displaystyle f(x)=y\in Y}
(it is a single element in Y), or
{\displaystyle f(x)}
For example, i{\displaystyle f}
is the square root function restricted to the integers
{\displaystyle f:\mathbb {Z} \to \mathbb {N} ,}
{\displaystyle f(n)=m}
{\displaystyle m^{2}=n,}
{\displaystyle m\in \mathbb {N} ,n\in \mathbb {Z} ,}
{\displaystyle f(n)}
is only defined if
{\displaystyle n}
is a perfect square (that is,
{\displaystyle 0,1,4,9,16,\ldots }
{\displaystyle f(25)=5}
{\displaystyle f(26)}
Partial Fraction Decomposion
Partial Fractions in 5 minutes
total vs. partial functions [episode 12]
Partial Fraction - Lecture 1 | Unacademy Foundation - Mathematics | Surabhi Ma'am
2 Function spaces
3 Discussion and examples
3.1 Natural logarithm
3.2 Subtraction of natural numbers
3.3 Bottom element
3.4 In category theory
3.5 In abstract algebra
3.6 Charts and atlases for manifolds and fiber bundles
An example of a partial function that is injective.
An example of a function that is not injective.
A partial function arises from the consideration of maps between two sets X and Y that may not be defined on the entire set X. A common example is the square root operation on the real numbers
{\displaystyle \mathbb {R} }
: because negative real numbers do not have real square roots, the operation can be viewed as a partial function from
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {R} .}
The domain of definition of a partial function is the subset S of X on which the partial function is defined; in this case, the partial function may also be viewed as a function from S to Y. In the example of the square root operation, the set S consists of the nonnegative real numbers
{\displaystyle [0,+\infty ).}
The notion of partial function is particularly convenient when the exact domain of definition is unknown or even unknowable. For a computer-science example of the latter, see Halting problem.
Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective.[1]
The notion of transformation can be generalized to partial functions as well. A partial transformation is a function
{\displaystyle f:A\rightharpoonup B,}
{\displaystyle A}
{\displaystyle B}
are subsets of some set
{\displaystyle X.}
The set of all partial functions
{\displaystyle f:X\rightharpoonup Y}
from a set
{\displaystyle X}
to a set
{\displaystyle Y,}
{\displaystyle [X\rightharpoonup Y],}
is the union of all functions defined on subsets of
{\displaystyle X}
with same codomain
{\displaystyle Y}
{\displaystyle [X\rightharpoonup Y]=\bigcup _{D\subseteq {X}}[D\to Y],}
the latter also written as
{\textstyle \bigcup _{D\subseteq {X}}Y^{D}.}
In finite case, its cardinality is
{\displaystyle |[X\rightharpoonup Y]|=(|Y|+1)^{|X|},}
because any partial function can be extended to a function by any fixed value
{\displaystyle c}
{\displaystyle Y,}
so that the codomain is
{\displaystyle Y\cup \{c\},}
an operation which is injective (unique and invertible by restriction).
Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore, the natural logarithm function is not a function when viewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a function.
Subtraction of natural numbers (non-negative integers) can be viewed as a partial function:
{\displaystyle f:\mathbb {N} \times \mathbb {N} \rightharpoonup \mathbb {N} }
{\displaystyle f(x,y)=x-y.}
It is defined only when
{\displaystyle x\geq y.}
In denotational semantics a partial function is considered as returning the bottom element when it is undefined.
In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEE floating point standard defines a not-a-number value which is returned when a floating point operation is undefined and exceptions are suppressed, e.g. when the square root of a negative number is requested.
In a programming language where function parameters are statically typed, a function may be defined as a partial function because the language's type system cannot express the exact domain of the function, so the programmer instead gives it the smallest domain which is expressible as a type and contains the domain of definition of the function.
In category theory, when considering the operation of morphism composition in concrete categories, the composition operation
{\displaystyle \circ \;:\;\hom(C)\times \hom(C)\to \hom(C)}
is a function if and only if
{\displaystyle \operatorname {ob} (C)}
has one element. The reason for this is that two morphisms
{\displaystyle f:X\to Y}
{\displaystyle g:U\to V}
can only be composed as
{\displaystyle g\circ f}
{\displaystyle Y=U,}
that is, the codomain o{\displaystyle f}
must equal the domain of
{\displaystyle g.}
The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps.[2] One textbook notes that "This formal completion of sets and partial maps by adding “improper,” “infinite” elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science."[3]
The category of sets and partial bijections is equivalent to its dual.[4] It is the prototypical inverse category.[5]
Partial algebra generalizes the notion of universal algebra to partial operations. An example would be a field, in which the multiplicative inversion is the only proper partial operation (because division by zero is not defined).[6]
The set of all partial functions (partial transformations) on a given base set,
{\displaystyle X,}
forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on
{\displaystyle X}
), typically denoted by
{\displaystyle {\mathcal {PT}}_{X}.}
[7][8][9] The set of all partial bijections on
{\displaystyle X}
forms the symmetric inverse semigroup.[7][8]
Charts in the atlases which specify the structure of manifolds and fiber bundles are partial functions. In the case of manifolds, the domain is the point set of the manifold. In the case of fiber bundles, the domain is the space of the fiber bundle. In these applications, the most important construction is the transition map, which is the composite of one chart with the inverse of another. The initial classification of manifolds and fiber bundles is largely expressed in terms of constraints on these transition maps.
{\displaystyle X}
{\displaystyle \mathbb {B} }
{\displaystyle \mathbb {B} }
{\displaystyle X}
{\displaystyle \mathbb {B} ^{n}}
{\displaystyle X}
{\displaystyle X}
{\displaystyle \mathbb {Z} }
{\displaystyle \mathbb {Z} }
{\displaystyle X}
{\displaystyle X}
{\displaystyle \mathbb {R} }
{\displaystyle \mathbb {R} }
{\displaystyle X}
{\displaystyle \mathbb {R} ^{n}}
{\displaystyle X}
{\displaystyle X}
{\displaystyle \mathbb {C} }
{\displaystyle \mathbb {C} }
{\displaystyle X}
{\displaystyle \mathbb {C} ^{n}}
{\displaystyle X}
Analytic continuation – Extension of the domain of an analytic function (mathematics)
Multivalued function – Generalization of a function that may produce several outputs for each input
Densely defined operator – Function that is defined almost everywhere (mathematics)
^ a b Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.
^ Lutz Schröder (2001). "Categories: a free tour". In Jürgen Koslowski and Austin Melton (ed.). Categorical Perspectives. Springer Science & Business Media. p. 10. ISBN 978-0-8176-4186-3.
^ Neal Koblitz; B. Zilber; Yu. I. Manin (2009). A Course in Mathematical Logic for Mathematicians. Springer Science & Business Media. p. 290. ISBN 978-1-4419-0615-1.
^ Marco Grandis (2012). Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups. World Scientific. p. 55. ISBN 978-981-4407-06-9.
^ Peter Burmeister (1993). "Partial algebras – an introductory survey". In Ivo G. Rosenberg; Gert Sabidussi (eds.). Algebras and Orders. Springer Science & Business Media. ISBN 978-0-7923-2143-9.
^ a b Alfred Hoblitzelle Clifford; G. B. Preston (1967). The Algebraic Theory of Semigroups. Volume II. American Mathematical Soc. p. xii. ISBN 978-0-8218-0272-4.
^ a b Peter M. Higgins (1992). Techniques of semigroup theory. Oxford University Press, Incorporated. p. 4. ISBN 978-0-19-853577-5.
^ Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. pp. 16 and 24. ISBN 978-1-84800-281-4.
Wikimedia Commons has media related to Partial mappings.
Martin Davis (1958), Computability and Unsolvability, McGraw–Hill Book Company, Inc, New York. Republished by Dover in 1982. ISBN 0-486-61471-9.
Stephen Kleene (1952), Introduction to Meta-Mathematics, North-Holland Publishing Company, Amsterdam, Netherlands, 10th printing with corrections added on 7th printing (1974). ISBN 0-7204-2103-9.
Harold S. Stone (1972), Introduction to Computer Organization and Data Structures, McGraw–Hill Book Company, New York.
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Tesseractic honeycomb - Wikipedia
Perspective projection of a 3x3x3x3 red-blue chessboard.
Type Regular 4-space honeycomb
Family Hypercubic honeycomb
Schläfli symbols {4,3,3,4}
t0,4{4,3,3,4}
{4,3,4}x{∞}
{4,4}x{∞}2
{∞}4
4-face type {4,3,3}
Face type {4}
Edge figure {3,4}
Vertex figure {3,3,4}
{\displaystyle {\tilde {C}}_{4}}
, [4,3,3,4]
{\displaystyle {\tilde {B}}_{4}}
, [4,3,31,1]
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive, 4-face-transitive
In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets.
Its vertex figure is a 16-cell. Two tesseracts meet at each cubic cell, four meet at each square face, eight meet on each edge, and sixteen meet at each vertex.
It is an analog of the square tiling, {4,4}, of the plane and the cubic honeycomb, {4,3,4}, of 3-space. These are all part of the hypercubic honeycomb family of tessellations of the form {4,3,...,3,4}. Tessellations in this family are Self-dual.
4 Related polytopes and tessellations
4.1 Birectified tesseractic honeycomb
Vertices of this honeycomb can be positioned in 4-space in all integer coordinates (i,j,k,l).
Sphere packing[edit]
Like all regular hypercubic honeycombs, the tesseractic honeycomb corresponds to a sphere packing of edge-length-diameter spheres centered on each vertex, or (dually) inscribed in each cell instead. In the hypercubic honeycomb of 4 dimensions, vertex-centered 3-spheres and cell-inscribed 3-spheres will both fit at once, forming the unique regular body-centered cubic lattice of equal-sized spheres (in any number of dimensions). Since the tesseract is radially equilateral, there is exactly enough space in the hole between the 16 vertex-centered 3-spheres for another edge-length-diameter 3-sphere. (This 4-dimensional body centered cubic lattice is actually the union of two tesseractic honeycombs, in dual positions.)
This is the same densest known regular 3-sphere packing, with kissing number 24, that is also seen in the other two regular tessellations of 4-space, the 16-cell honeycomb and the 24-cell-honeycomb. Each tesseract-inscribed 3-sphere kisses a surrounding shell of 24 3-spheres, 16 at the vertices of the tesseract and 8 inscribed in the adjacent tesseracts. These 24 kissing points are the vertices of a 24-cell of radius (and edge length) 1/2.
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3,3,4}. Another form has two alternating tesseract facets (like a checkerboard) with Schläfli symbol {4,3,31,1}. The lowest symmetry Wythoff construction has 16 types of facets around each vertex and a prismatic product Schläfli symbol {∞}4. One can be made by stericating another.
Related polytopes and tessellations[edit]
(1), (2), (13), 18
(6), 19, 20
The [4,3,31,1], , Coxeter group generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the 16-cell honeycomb and snub 24-cell honeycomb respectively.
[4,3,31,1]:
<[4,3,31,1]>:
↔[4,3,3,4]
(10), 15, 16, (13), 17, 18, 19
[3[1+,4,3,31,1]]
↔ [3[3,31,1,1]]
[(3,3)[1+,4,3,31,1]]
The 24-cell honeycomb is similar, but in addition to the vertices at integers (i,j,k,l), it has vertices at half integers (i+1/2,j+1/2,k+1/2,l+1/2) of odd integers only. It is a half-filled body centered cubic (a checkerboard in which the red 4-cubes have a central vertex but the black 4-cubes do not).
The tesseract can make a regular tessellation of the 4-sphere, with three tesseracts per face, with Schläfli symbol {4,3,3,3}, called an order-3 tesseractic honeycomb. It is topologically equivalent to the regular polytope penteract in 5-space.
The tesseract can make a regular tessellation of 4-dimensional hyperbolic space, with 5 tesseracts around each face, with Schläfli symbol {4,3,3,5}, called an order-5 tesseractic honeycomb.
Birectified tesseractic honeycomb[edit]
A birectified tesseractic honeycomb, , contains all rectified 16-cell (24-cell) facets and is the Voronoi tessellation of the D4* lattice. Facets can be identically colored from a doubled
{\displaystyle {\tilde {C}}_{4}}
×2, [[4,3,3,4]] symmetry, alternately colored from
{\displaystyle {\tilde {C}}_{4}}
, [4,3,3,4] symmetry, three colors from
{\displaystyle {\tilde {B}}_{4}}
, [4,3,31,1] symmetry, and 4 colors from
{\displaystyle {\tilde {D}}_{4}}
, [31,1,1,1] symmetry.
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) - Model 1
Klitzing, Richard. "4D Euclidean tesselations". x∞o x∞o x∞o x∞o, x∞x x∞o x∞o x∞o, x∞x x∞x x∞o x∞o, x∞x x∞x x∞x x∞o,x∞x x∞x x∞x x∞x, x∞o x∞o x4o4o, x∞o x∞o o4x4o, x∞x x∞o x4o4o, x∞x x∞o o4x4o, x∞o x∞o x4o4x, x∞x x∞x x4o4o, x∞x x∞x o4x4o, x∞x x∞o x4o4x, x∞x x∞x x4o4x, x4o4x x4o4x, x4o4x o4x4o, x4o4x x4o4o, o4x4o o4x4o, x4o4o o4x4o, x4o4o x4o4o, x∞x o3o3o *d4x, x∞o o3o3o *d4x, x∞x x4o3o4x, x∞o x4o3o4x, x∞x x4o3o4o, x∞o x4o3o4o, o3o3o *b3o4x, x4o3o3o4x, x4o3o3o4o - test - O1
{\displaystyle {\tilde {A}}_{n-1}}
{\displaystyle {\tilde {C}}_{n-1}}
{\displaystyle {\tilde {B}}_{n-1}}
{\displaystyle {\tilde {D}}_{n-1}}
{\displaystyle {\tilde {G}}_{2}}
{\displaystyle {\tilde {F}}_{4}}
{\displaystyle {\tilde {E}}_{n-1}}
Retrieved from "https://en.wikipedia.org/w/index.php?title=Tesseractic_honeycomb&oldid=1047612280"
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superset - Maple Help
Home : Support : Online Help : Programming : Logic : Boolean : verify : superset
verify that the first set is a superset of the second
verify(
\mathrm{expr1}
\mathrm{expr2}
, superset)
\mathrm{expr1}
\mathrm{expr2}
, superset(
\mathrm{ver}
The verify(expr1, expr2, superset) and verify(expr1, expr2, superset(ver)) calling sequences return true if, for every operand in the second set, expr2, it can be determined that there is an operand in the first set, expr1, that satisfies the relationship ver. If ver is omitted, then equality is used.
If true is returned, then it has been determined that each operand of the second set satisfies the relationship with at least one element of the first set. If false is returned, then there is at least one operand in the second set that does not satisfy the relationship (a result of type verify(false)) with each operand in the first set. If neither case can be determined, FAIL is returned.
\mathrm{verify}\left({a,b,c,d,e},{a,b,c},'\mathrm{superset}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left({a,b,c,d,e},{a,b,c,f},'\mathrm{superset}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left({a,b,{x}^{2}-x},{a,b,x\left(x-1\right)},'\mathrm{superset}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left({a,b,{x}^{2}-x},{a,b,x\left(x-1\right)},'\mathrm{superset}\left(\mathrm{expand}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left({a,b,c,{x}^{2}-x},{a,b,x\left(x-1\right)},'\mathrm{superset}\left(\mathrm{expand}\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{verify}\left({0.102221,0.2334},{0.10222,0.2333},'\mathrm{superset}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{verify}\left({0.102221,0.2334},{0.10222,0.2333},'\mathrm{superset}\left(\mathrm{float}\left({10}^{6}\right)\right)'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Engineering Acoustics/Sonar - Wikibooks, open books for an open world
Engineering Acoustics/Sonar
The use of navigation instruments was always of great importance to those who depend on the sea. Methods to find the position on earth according to the stars were available since long ago, but the first apparatus that track what is underwater are relatively recent. One of these instruments, which improved the safety of navigation is the fathometer. It has the simple concept of measuring how much time a sound wave generated at the ship takes to reach the bottom and return. If one knows the speed of sound in the medium, the depth could be easily determined. Another mechanism consists of underwater bells on lightships or lighthouses and hydrophones on ships that are to find the distance between them. These could be considered the forerunners of the SONAR (SOund Navigation And Ranging).
There are a lot of animals that also take advantage of underwater sound propagation to communicate.
Speed of SoundEdit
In 1841, Jean-Daniel Colladon [1] was able to measure the speed of sound in water for the first time.
The equation for speed of sound(m/s) in water developed by Del Grosso,[1] applicable in Neptunian[2] waters, depends on the Temperature(T) in Celsius, Salinity(S) in ppt(part per thousand) and gauge Pressure(P) in atmospheres.:
{\displaystyle c(T,S,P)=1449.08+4.57Te^{-(T/86.9+(T/360)^{2})}+1.33(S-35)e^{-T/120}+0.1522Pe^{T/1200+(S-35)/400}+1.4610^{-5}P^{2}e^{-T/20+(S-35)/10}}
where the pressure is a function of depth[Km] and the Latitude, given by:
{\displaystyle P=99.5(1-0.00263cos2\phi )Z+0.239Z^{2}}
Figure 1: Sea speed of sound profile at low latitudes. Salinity gradient was not taken into account.
The speed of sound is very sensible to the temperature, which changes considerably in the thermocline [2]. After 1000m meters deep, the pressure governs the equation, increasing slowly the speed with depth. The salinity has very low effect on the equation unless in very specific situations, such as heavy rain or the encounter between a river and the sea.
↑ Del Grosso (1974), J. Acoust. Soc. Am.,56,1084(1974)., p. 56,1084
↑ Leroy (1969), J. Acoust. Soc. Am.,56,1084(1974)., p. 46,216
Retrieved from "https://en.wikibooks.org/w/index.php?title=Engineering_Acoustics/Sonar&oldid=3254476"
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Pub Editor · PubPub Help
PubPub Help
Reference for editing and formatting your Pubs.
This Release (#5) was created on Jan 11, 2019 ( 3 years ago )
The latest Release (#7) was created on Nov 23, 2020 ( 2 years ago ).
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sum_ix^i
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https://en.wikipedia.org/wiki/Science#/media/File:CMB_Timeline300_no_WMAP.jpg
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Radio hiss from Enceladus, recorded by NASA’s Cassini probe. NASA/JPL/University of Iowa
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Pirri J, Rayes D, Alkema M
https://commons.wikimedia.org/wiki/File:A-Change-in-the-Ion-Selectivity-of-Ligand-Gated-Ion-Channels-Provides-a-Mechanism-to-Switch-Behavior-pbio.1002238.s008.ogv
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Miss Goodall and the Wild Chimpanzees (1965) - Wildlife Documentary - A National Geographic Film
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The crystal chemistry of ‘wheatsheaf’ tourmaline from Mogok, Myanmar | Mineralogical Magazine | GeoScienceWorld
A. J. Lussier, F. C. Hawthorne, Y. Abdu, S. Herwig, V. K. Michaelis, P. M. Aguiar, S. Kroeker; The crystal chemistry of ‘wheatsheaf’ tourmaline from Mogok, Myanmar. Mineralogical Magazine 2011;; 75 (1): 65–86. doi: https://doi.org/10.1180/minmag.2011.075.1.65
Tourmalines of unusual (mushroom) habit are common in granitic pegmatites of Momeik, northeast of Mogok, Myanmar. Here, we examine a sample of elbaite of significantly different habit, consisting of a series of diverging crystals, resembling a sheaf of wheat and ranging in colour from light purplish-red at the base to dark purplish-red at the tip with a thin green cap at the termination. The crystal structures of eight crystals are refined to R1-indices of ~2.5% using graphite-monochromated Mo-Kα X-radiation; the same crystals were analysed by electron microprobe. 11B and 27Al magic angle spinning nuclear magnetic resonance (MAS NMR) spectra were collected on four regions of the wheatsheaf crystal, and show ~0.3 a.p.f.u. [4]B and <0.1 a.p.f.u. [4]Al in the structure. 57Fe Mössbauer spectroscopy was done on the dark green rim at the termination of the crystal, showing all Fe in this region (~0.6 a.p.f.u.) to be Fe2+. Detailed electron-microprobe traverses show that the principal compositional variation involves the substitutions [4]B + YAl → Si + YFe*, where transition metals are present, and [4]B2 + YAl → Si2 + YLi, where transition metals are not present, although several other minor substitutions also affect crystal composition. Successive microscopic bifurcation of crystallites causes divergence of growth directions along the c axis, imparting the overall ‘wheatsheaf’ shape to the crystal aggregate. We suggest that such bifurcation is common in pegmatitic elbaite crystals, resulting in their common divergent habit.
Mn22+
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Solving Boundary Value Problems - MATLAB & Simulink - MathWorks España
Initial Guess of Solution
Finding Unknown Parameters
BVP Solver Selection
BVP Examples and Files
In a boundary value problem (BVP), the goal is to find a solution to an ordinary differential equation (ODE) that also satisfies certain specified boundary conditions. The boundary conditions specify a relationship between the values of the solution at two or more locations in the interval of integration. In the simplest case, the boundary conditions apply at the beginning and end (or boundaries) of the interval.
The MATLAB® BVP solvers bvp4c and bvp5c are designed to handle systems of ODEs of the form
y\text{'}=f\left(x,y\right)
y′ represents the derivative of y with respect to x, also written as dy/dx
In the simplest case of a two-point BVP, the solution to the ODE is sought on an interval [a, b], and must satisfy the boundary conditions
g\left(y\left(a\right),y\left(b\right)\right)=0\text{\hspace{0.17em}}.
To specify the boundary conditions for a given BVP, you must:
Write a function of the form res = bcfun(ya,yb), or use the form res = bcfun(ya,yb,p) if there are unknown parameters involved. You supply this function to the solver as the second input argument. The function returns res, which is the residual value of the solution at the boundary point. For example, if y(a) = 1 and y(b) = 0, then the boundary condition function is
In the initial guess for the solution, the first and last points in the mesh specify the points at which the boundary conditions are enforced. For the above boundary conditions, you can specify bvpinit(linspace(a,b,5),yinit) to enforce the boundary conditions at a and b.
The BVP solvers in MATLAB also can accommodate other types of problems that have:
Unknown parameters p
Singularities in the solutions
Multipoint conditions (internal boundaries that separate the integration interval into several regions)
In the case of multipoint boundary conditions, the boundary conditions apply at more than two points in the interval of integration. For example, the solution might be required to be zero at the beginning, middle, and end of the interval. See bvpinit for details on how to specify multiple boundary conditions.
Unlike initial value problems, a boundary value problem can have:
An important part of the process of solving a BVP is providing a guess for the required solution. The quality of this guess can be critical for the solver performance and even for a successful computation.
Use the bvpinit function to create a structure for the initial guess of the solution. The solvers bvp4c and bvp5c accept this structure as the third input argument.
Creating a good initial guess for the solution is more an art than a science. However, some general guidelines include:
Have the initial guess satisfy the boundary conditions, since the solution is required to satisfy them as well. If the problem contains unknown parameters, then the initial guess for the parameters also should satisfy the boundary conditions.
Try to incorporate as much information about the physical problem or expected solution into the initial guess as possible. For example, if the solution is supposed to oscillate or have a certain number of sign changes, then the initial guess should as well.
Consider the placement of the mesh points (the x-coordinates of the initial guess of the solution). The BVP solvers adapt these points during the solution process, so you do not need to specify too many mesh points. Best practice is to specify a few mesh points placed near where the solution changes rapidly.
If there is a known, simpler solution on a smaller interval, then use it as an initial guess on a larger interval. Often you can solve a problem as a series of relatively simpler problems, a practice called continuation. With continuation, a series of simple problems are connected by using the solution of one problem as the initial guess to solve the next problem.
Often BVPs involve unknown parameters p that have to be determined as part of solving the problem. The ODE and boundary conditions become
\begin{array}{l}y\text{'}=f\left(x,y,p\right)\\ g\left(y\left(a\right),y\left(b\right),p\right)=0\end{array}
In this case, the boundary conditions must suffice to determine the values of the parameters p.
You must provide the solver with an initial guess for any unknown parameters. When you call bvpinit to create the structure solinit, specify the initial guess as a vector in the third input argument parameters.
solinit = bvpinit(x,v,parameters)
Additionally, the functions odefun and bcfun that encode the ODE equations and boundary conditions must each have a third argument.
dydx = odefun(x,y,parameters)
res = bcfun(ya,yb,parameters)
While solving the differential equations, the solver adjusts the value of the unknown parameters to satisfy the boundary conditions. The solver returns the final values of these unknown parameters in sol.parameters.
bvp4c and bvp5c can solve a class of singular BVPs of the form
\begin{array}{c}{y}^{\prime }=\frac{1}{x}Sy+f\left(x,y\right),\\ 0=g\left(y\left(0\right),y\left(b\right)\right).\end{array}
The solvers can also accommodate unknown parameters for problems of the form
\begin{array}{c}{y}^{\prime }=\frac{1}{x}Sy+f\left(x,y,p\right),\\ 0=g\left(y\left(0\right),y\left(b\right),p\right).\end{array}
Singular problems must be posed on an interval [0,b] with b > 0. Use bvpset to pass the constant matrix S to the solver as the value of the 'SingularTerm' option. Boundary conditions at x = 0 must be consistent with the necessary condition for a smooth solution, Sy(0) = 0. The initial guess of the solution also should satisfy this condition.
When you solve a singular BVP, the solver requires that your function odefun(x,y) return only the value of the f(x, y) term in the equation. The term involving S is handled by the solver separately using the 'SingularTerm' option.
MATLAB includes the solvers bvp4c and bvp5c to solve BVPs. In most cases you can use the solvers interchangeably. The main difference between the solvers is that bvp4c implements a fourth-order formula, while bvp5c implements a fifth-order formula.
The bvp5c function is used exactly like bvp4c, with the exception of the meaning of error tolerances between the two solvers. If S(x) approximates the solution y(x), bvp4c controls the residual |S′(x) – f(x,S(x))|. This approach indirectly controls the true error |y(x) – S(x)|. Use bvp5c to control the true error directly.
bvp4c is a finite difference code that implements the 3-stage Lobatto IIIa formula. This is a collocation formula, and the collocation polynomial provides a C1-continuous solution that is fourth-order accurate uniformly in the interval of integration. Mesh selection and error control are based on the residual of the continuous solution.
The collocation technique uses a mesh of points to divide the interval of integration into subintervals. The solver determines a numerical solution by solving a global system of algebraic equations resulting from the boundary conditions and the collocation conditions imposed on all the subintervals. The solver then estimates the error of the numerical solution on each subinterval. If the solution does not satisfy the tolerance criteria, then the solver adapts the mesh and repeats the process. You must provide the points of the initial mesh, as well as an initial approximation of the solution at the mesh points.
bvp5c is a finite difference code that implements the four-stage Lobatto IIIa formula. This is a collocation formula and the collocation polynomial provides a C1-continuous solution that is fifth-order accurate uniformly in [a,b]. The formula is implemented as an implicit Runge-Kutta formula. bvp5c solves the algebraic equations directly, whereas bvp4c uses analytical condensation. bvp4c handles unknown parameters directly, while bvp5c augments the system with trivial differential equations for unknown parameters.
The collocation methods implemented in bvp4c and bvp5c produce C1-continuous solutions over the interval of integration [a,b]. You can evaluate the approximate solution, S(x), at any point in [a,b] using the helper function deval and the structure sol returned by the solver. For example, to evaluate the solution sol at the mesh of points xint, use the command
Sxint = deval(sol,xint)
The deval function is vectorized. For a vector xint, the ith column of Sxint approximates the solution y(xint(i)).
Several available example files serve as excellent starting points for most common BVP problems. To easily explore and run examples, simply use the Differential Equations Examples app. To run this app, type
To open an individual example file for editing, type
This table contains a list of the available BVP example files, as well as the solvers and the options they use.
emdenbvp
bvp4c or bvp5c
'SingularTerm'
Emden's equation, a singular BVP
fsbvp
Falkner-Skan BVP on an infinite interval
mat4bvp
Fourth eigenfunction of Mathieu's equation
rcbvp
bvp4c and bvp5c
'FJacobian'
Example comparing the errors controlled by bvp4c and bvp5c
Compare bvp4c and bvp5c Solvers (bvp4c)
shockbvp
'BCJacobian'
Solution with a shock layer near x = 0
twobvp
BVP with exactly two solutions
threebvp
Three-point boundary value problem
[1] Ascher, U., R. Mattheij, and R. Russell. “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations.” Philadelphia, PA: SIAM, 1995, p. 372.
bvp4c | bvp5c | bvpinit | bvpset | pdepe | ode45
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Model Evaluation - ProductizeML
You will learn: how to assess to evaluation of ML models.
Accuracy = \frac{number \space correct \space predictions}{number \space of \space samples}
Sensitivity = \frac{number \space of \space true \space positives}{number \space of \space true \space positives + number \space of \space false \space negatives}
Precision: computed as the fraction of retrieved instances that are relevant.
Precision = \frac{number \space of \space true \space positives}{number \space of \space true \space positives + number \space of \space false \space positives}
Specificity = \frac{number \space of \space true \space negatives}{number \space of \space true \space negatives + number \space of \space false \space positives}
🚧 This section is still under construction!
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Option set for tfest - MATLAB - MathWorks América Latina
{F}_{cutoff}=\frac{\text{FilterTimeConstant}}{{T}_{s}}
1/|G\left(\omega \right)|
as the weighting filter, where G(ω) is the complex frequency-response data. Use this option for capturing relatively low amplitude dynamics in data, or for fitting data with high modal density. This option also makes it easier to specify channel-dependent weighting filters for MIMO frequency-response data.
1/\sqrt{|G\left(\omega \right)|}
\mathrm{det}\left(E\text{'}*E/N\right)
\frac{‖{y}_{p,z}-{y}_{meas}‖}{‖{y}_{p,e}-{y}_{meas}‖}>\text{AutoInitThreshold}
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Home : Support : Online Help : Connectivity : Database Package : Connection : GetOptions
get the current value of Connection module options
connection:-GetOptions( opts )
(optional) name of the option to get, one of readonly, autocommit or isolation
GetOptions gets the current values of connection's options.
GetOptions can query all the options that are set by SetOptions.
readonly= true or false
The readonly argument informs the database that it can enable optimizations associated with a read-only connection. The effect that this has depends on the database. Setting readonly=true may not prevent the execution of updates. By default, connections are not opened read-only.
The autocommit optional argument controls if updates are automatically committed on execution. If autocommit is true, then updates are automatically committed to the database. Otherwise, transactions will be used and updates will need to be committed before they are reflected in the database (see Commit and Rollback for more information). By default, autocommit is false.
isolation= uncommittedread, committedread, repeatableread, or serializable
The isolation argument determines how isolated transactions using this connection are from transactions occurring at the same time.
- At committedread isolation, only dirty reads are prevented.
\mathrm{driver}≔\mathrm{Database}[\mathrm{LoadDriver}]\left(\right):
\mathrm{conn}≔\mathrm{driver}:-\mathrm{OpenConnection}\left(\mathrm{url},\mathrm{name},\mathrm{pass}\right):
\mathrm{conn}:-\mathrm{GetOptions}\left(\right)
[\textcolor[rgb]{0,0,1}{\mathrm{readonly}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{isolation}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{committedread}}]
\mathrm{conn}:-\mathrm{SetOptions}\left('\mathrm{readonly}'=\mathrm{true}\right);
\mathrm{conn}:-\mathrm{GetOptions}\left('\mathrm{readonly}'\right)
[\textcolor[rgb]{0,0,1}{\mathrm{readonly}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{true}}]
\mathrm{conn}:-\mathrm{SetOptions}\left('\mathrm{isolation}'='\mathrm{serializable}'\right);
\mathrm{conn}:-\mathrm{GetOptions}\left('\mathrm{readonly}','\mathrm{isolation}'\right)
[\textcolor[rgb]{0,0,1}{\mathrm{readonly}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{true}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{isolation}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{serializable}}]
\mathrm{conn}:-\mathrm{SetOptions}\left('\mathrm{isolation}'='\mathrm{committedread}','\mathrm{readonly}'=\mathrm{false}\right);
\mathrm{conn}:-\mathrm{GetOptions}\left('\mathrm{readonly}','\mathrm{isolation}'\right)
[\textcolor[rgb]{0,0,1}{\mathrm{readonly}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{false}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{isolation}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{committedread}}]
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How to Convert Meters to Yards: 10 Steps (with Pictures) - wikiHow
1 Quickly Converting
2 Estimating Without a Calculator
Converting meters to yards is as simple as plugging a number of meters into a simple multiplication problem. This works only if you know that the conversion rate is 1 meter to 1.0936 yards. Even if you don't have a calculator, however, this is a particularly easy conversion to estimate.
Quickly Converting Download Article
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Note the number of meters you want to convert. A conversion from meters to yards is nothing but a simple multiplication operation. All you need to do is start with a number of meters.
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Multiply the number of meters by 1.0936 to get the number of yards. There are 1.0936 yards in every meter. So if 1 meter is 1.0963 yards, then 2 meters is 2.1872 yards, and so on. You could keep adding 1.0936 yards for each additional meter, but the easiest way to do it is by multiplication:[1] X Research source
For example, convert 12 meters into yards.
{\displaystyle 12m*1.0936{\frac {yards}{meters}}=13.1234}
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Provide your "units" at the end to complete the problem. In the example, you determined the answer was "13.1234." But on a test or a project you need to be sure to answer the simple question, "13.1234 of what?" The answer, of course, is 13.1234 yards, but don't forget to add these units to get full credit.
So to convert 12 meters into yards, multiply 12 meters by 1.0936 yards per meter.
{\displaystyle 12m*1.0936{\frac {yards}{meters}}=13.1234}
{\displaystyle 12meters=13.1234yards}
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Use more or fewer decimal places to get a more precise or less precise answer. The formula "1 meter = 1.0936 yards" is just an approximation. The most accurate conversion factor is an infinite string of numbers. The more decimal places you use, the more precise the conversion will be. However, it would be a rare case where four decimal places wouldn't be enough. For example, you could also use the following:
Rough Calculation: 1 meter = 1.1 yards
Fairly Precise Calculation: 1 meter = 1.0936 yards
Extremely Precise Calculation: 1 meter = 1.0936133 yards[2] X Research source
Estimating Without a Calculator Download Article
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Get used to the idea that for estimation purposes 1 meter = 1.1 yards. As noted above, the actual conversion is closer to 1.0936 yards, but you can round this number up slightly to make it easier to calculate by hand. While 1.1 slightly exceeds the actual conversion, it is only by .0064 yard -- not enough to throw off a basic estimation.
Though this method is close, remember that your estimation will be slightly larger than the actual conversion. [3] X Research source
Write down the number of meters followed by a plus sign. Meters and yards are almost the exact same length. The difference is in the .1, which is easy to calculate. Since anything multiplied by 1 is itself (
{\displaystyle 1*15=15}
), all you need to do is find the .1 portion and add it to the number of meters.
For example, estimate the number of yards in 15 meters.
15 meters = 15 yards + ?
Multiply the number of meters by .1. Do this by moving the decimal point one place to the left. Simply take the number of meters and shift the decimal point one place to the left -- this is the exact same thing as "multiplying by .1" So in this example:
Estimate the number of yards in 15 meters.
{\displaystyle 15meters=15yards+}
{\displaystyle 15meters*.1=1.5}
Add the new number to the original number of meters to get your final answer. The only math you have to do at this point is some quick addition. Once you've moved the decimal point over, add this new number to the original number of meters. For example:
{\displaystyle 15meters=15yards+}
{\displaystyle 15meters*.1=1.5}
{\displaystyle 15meters=15yards+1.5}
{\displaystyle 15meters=16.5yards}
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Consider subtracting your overestimation for an even more precise estimate. The actual conversion of 15 meters is 16.4042 yards, meaning the estimation was only about .1 yard off. Note that the overestimation will get bigger if you're working with bigger numbers. 150 meters would be estimated at 165 yards, and that's 1.6 yards too high. A good way to eliminate this over-estimation is to subtract .1 yard for every ten yards:
In the first example, you have 16.5 yards. To correct it, you can subtract .1, because there is only one "10" in 16. Your final estimation would be 16.4 yards.
For the bigger example, you'd subtract 1.6 yards, because there are sixteen "10s" in 165. The estimate this time would be 163.4 yards.[4] X Research source
Which one is greater: a yard or meter?
A meter is slightly longer than a yard.
How do you calculate how many yards is in 220 meters?
1 meter is equivalent to 1.0936 yards. To find how many yards are in 220 meters you would multiply 220 x 1.0936. The answer is 240.592 yards.
If someone ran 9.5 seconds for 100 yards, what would the seconds be for 91.44 meters?
As you are probably already aware, 100 yards is the same as 91.44 meters.
How do I convert kilometers to miles?
Multiply the number of kilometers by 0.62137.
You can check your work by using division. Simply divide the number of yards in the answer by 1.0936. You should get back to the original number of meters. If not, you've made a mistake either in the original conversion or the later check.
↑ http://www.metric-conversions.org/length/meters-to-yards-table.htm
↑ http://www.asknumbers.com/MetersToYardsConversion.aspx
↑ http://www.mathgoodies.com/lessons/decimals_part2/estimate_products.html
Español:convertir metros a yardas
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Please solve the Q 28 Velocity of sound in steel is 5000 m s-1 A sound wave of - Physics - Propagation of Sound Waves - 12297275 | Meritnation.com
Q.28. Velocity of sound in steel is 5000
m {s}^{-1}
. A sound wave of frequency 1000 Hz is produced in a steel rail of length 20 m. Calculate total number of waves produced in the rail.
[14 Waves]
Here in this case , n = 1000 Hz and V = 5000 m/s
So the wavelength is ,
\lambda =\frac{V}{n}=\frac{5000}{1000}=5 m
So the number of waves produced in 20 m length of the steel rail is = 20 /5 = 4 nos.
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Electric field - Simple English Wikipedia, the free encyclopedia
An electric field is a vector field that shows the direction that a positively charged particle will move when placed in the field. More precisely, if a particle has an electric charge
{\displaystyle q}
and is in an electric field
{\displaystyle {\vec {E}}}
, the electric force the charge will feel is
{\displaystyle {\vec {F}}=q\cdot {\vec {E}}}
.Electric fields are produced around objects that have electrical charge, or by a magnetic field that changes with time. Electric field lines are used to represent the influence of electric field. [1] The idea of an electric field was first made by Michael Faraday.[2]
Electric fields are caused by electric charges, described by Gauss's law,[3] or varying magnetic fields, described by Faraday's law of induction.[4] The equations of both fields are coupled and together form Maxwell's equations that describe both fields as a function of charges and currents.[5]
↑ "Michael Faraday". Retrieved 2015-06-30.
↑ Purcell, p 25: "Gauss's Law: the flux of the electric field E through any closed surface... equals 1/e times the total charge enclosed by the surface."
↑ Purcell, p 356: "Faraday's Law of Induction."
↑ Purcell, Edward & Morin, David 2013. Electricity and magnetism. 3rd ed, Cambridge University Press. New York. ISBN 978-1-107-01402-2
Retrieved from "https://simple.wikipedia.org/w/index.php?title=Electric_field&oldid=7197832"
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Wind_speed Knowpia
An anemometer is commonly used to measure wind speed.
Wind speed affects weather forecasting, aviation and maritime operations, construction projects, growth and metabolism rate of many plant species, and has countless other implications.[1] Note that wind direction is usually almost parallel to isobars (and not perpendicular, as one might expect), due to Earth's rotation.
Meters per second (m/s) is the SI unit for velocity and the unit recommended by the World Meteorological Organization for reporting wind speeds, and is amongst others used in weather forecasts in the Nordic countries.[2] Since 2010 the International Civil Aviation Organization (ICAO) also recommends meters per second for reporting wind speed when approaching runways, replacing their former recommendation of using kilometers per hour (km/h).[3] For historical reasons, other units such as miles per hour (mph), knots (kn)[4] or feet per second (ft/s) are also sometimes used to measure wind speeds. Historically, wind speeds have also been classified using the Beaufort scale, which is based on visual observations of specifically defined wind effects at sea or on land.
Factors affecting wind speedEdit
Wind speed is affected by a number of factors and situations, operating on varying scales (from micro to macro scales). These include the pressure gradient, Rossby waves and jet streams, and local weather conditions. There are also links to be found between wind speed and wind direction, notably with the pressure gradient and terrain conditions.
Pressure gradient is a term to describe the difference in air pressure between two points in the atmosphere or on the surface of the Earth. It is vital to wind speed, because the greater the difference in pressure, the faster the wind flows (from the high to low pressure) to balance out the variation. The pressure gradient, when combined with the Coriolis effect and friction, also influences wind direction.
Rossby waves are strong winds in the upper troposphere. These operate on a global scale and move from West to East (hence being known as Westerlies). The Rossby waves are themselves a different wind speed from what we experience in the lower troposphere.
Local weather conditions play a key role in influencing wind speed, as the formation of hurricanes, monsoons and cyclones as freak weather conditions can drastically affect the flow velocity of the wind.[citation needed]
Highest speedEdit
The original anemometer that measured The Big Wind in 1934 at Mount Washington Observatory
The fastest wind speed not related to tornadoes ever recorded was during the passage of Tropical Cyclone Olivia on 10 April 1996: an automatic weather station on Barrow Island, Australia, registered a maximum wind gust of 113.3 m/s (408 km/h; 253 mph; 220.2 kn; 372 ft/s)[5][6] The wind gust was evaluated by the WMO Evaluation Panel who found that the anemometer was mechanically sound and the gust was within statistical probability and ratified the measurement in 2010. The anemometer was mounted 10 m above ground level (and thus 64 m above sea level). During the cyclone, several extreme gusts of greater than 83 m/s (300 km/h; 190 mph; 161 kn; 270 ft/s) were recorded, with a maximum 5-minute mean speed of 49 m/s (180 km/h; 110 mph; 95 kn; 160 ft/s) the extreme gust factor was in the order of 2.27–2.75 times the mean wind speed. The pattern and scales of the gusts suggest that a mesovortex was embedded in the already strong eyewall of the cyclone.[5]
Currently, the second-highest surface wind speed ever officially recorded is 103.266 m/s (371.76 km/h; 231.00 mph; 200.733 kn; 338.80 ft/s) at the Mount Washington (New Hampshire) Observatory 1,917 m (6,288 ft) above sea level in the US on 12 April 1934, using a hot-wire anemometer. The anemometer, specifically designed for use on Mount Washington was later tested by the US National Weather Bureau and confirmed to be accurate.[7]
Wind speeds within certain atmospheric phenomena (such as tornadoes) may greatly exceed these values but have never been accurately measured. Directly measuring these tornadic winds is rarely done as the violent wind would destroy the instruments. A method of estimating speed is to use Doppler on Wheels to sense the wind speeds remotely,[8] and, using this method, the figure of 135 m/s (490 km/h; 300 mph; 262 kn; 440 ft/s) during the 1999 Bridge Creek–Moore tornado in Oklahoma on 3 May 1999 is often quoted as the highest-recorded surface wind speed,[9] although another figure of 142 m/s (510 km/h; 320 mph; 276 kn; 470 ft/s) has also been quoted for the same tornado.[10] Yet another number used by the Center for Severe Weather Research for that measurement is 135 ± 9 m/s (486 ± 32 km/h; 302 ± 20 mph; 262 ± 17 kn; 443 ± 30 ft/s).[11] However, speeds measured by Doppler radar are not considered official records.[10]
The fastest wind speed observed on an exoplanet was observed on HD 189733b by scientists at the University of Warwick in 2015, and was measured at 5,400 mph, or 2,414 metres per second. In a press release, the University announced that the methods used from measuring HD 189733b's wind speeds could be used to measure wind speeds on Earth-like exoplanets.[12]
Modern day anemometer used to capture wind speed.
FT742-DM acoustic resonance wind sensor, one of the instruments now used to measure wind speed at Mount Washington Observatory
An anemometer is one of the tools used to measure wind speed.[13] A device consisting of a vertical pillar and three or four concave cups, the anemometer captures the horizontal movement of air particles (wind speed).
Unlike traditional cup and vane anemometers, ultrasonic wind sensors have no moving parts and are therefore used to measure wind speed in applications that require maintenance-free performance, such as on the top of wind turbines. As the name suggests, ultrasonic wind sensors measure the wind speed using high-frequency sound. An ultrasonic anemometer has two or three pairs of sound transmitters and receivers. Stand it in the wind and each transmitter constantly beams high-frequency sound to its respective receiver. Electronic circuits inside measure the time it takes for the sound to make its journey from each transmitter to the corresponding receiver. Depending on how the wind blows, it will affect some of the sound beams more than the others, slowing it down or speeding it up very slightly. The circuits measure the difference in speeds of the beams and use that to calculate how fast the wind is blowing.[14]
Acoustic resonance wind sensors are a variant of the ultrasonic sensor. Instead of using time of flight measurement, acoustic resonance sensors use resonating acoustic waves within a small purpose-built cavity in order to perform their wind speed measurement. Built into the cavity is an array of ultrasonic transducers, which are used to create the separate standing-wave patterns at ultrasonic frequencies. As wind passes through the cavity, a change in the wave's property occurs (phase shift). By measuring the amount of phase shift in the received signals by each transducer, and then by mathematically processing the data, the sensor is able to provide an accurate horizontal measurement of wind speed and direction.[15]
Another tool used to measure wind velocity includes a GPS combined with pitot tube.[citation needed] A fluid flow velocity tool, the Pitot tube is primarily used to determine the air velocity of an aircraft.
Design of structuresEdit
Anemometer on an outdoor stage set, to measure wind speed
Wind speed is a common factor in the design of structures and buildings around the world. It is often the governing factor in the required lateral strength of a structure's design.
In the United States, the wind speed used in design is often referred to as a "3-second gust" which is the highest sustained gust over a 3-second period having a probability of being exceeded per year of 1 in 50 (ASCE 7-05, updated to ASCE 7-16).[16] This design wind speed is accepted by most building codes in the United States and often governs the lateral design of buildings and structures.
In Canada, reference wind pressures are used in design and are based on the "mean hourly" wind speed having a probability of being exceeded per year of 1 in 50. The reference wind pressure
{\displaystyle q}
is calculated using the following equation:
{\displaystyle q={\frac {1}{2}}pv^{2}}
{\displaystyle p}
is the air density and
{\displaystyle v}
is wind speed.[17]
Historically, wind speeds have been reported with a variety of averaging times (such as fastest mile, 3-second gust, 1-minute and mean hourly) which designers may have to take into account. To convert wind speeds from one averaging time to another, the Durst Curve was developed which defines the relation between probable maximum wind speed averaged over t seconds, Vt, and mean wind speed over one hour V3600.[18]
American Society of Civil Engineers (promulgator of ASCE 7-05, current version is ASCE 7-16)
Fujita scale and Enhanced Fujita Scale
International Building Code (promulgator of NBC 2005)
ICAO recommendations – International System of Units
^ Hogan, C. Michael (2010). "Abiotic factor". In Emily Monosson; C. Cleveland (eds.). Encyclopedia of Earth. Washington D.C.: National Council for Science and the Environment. Archived from the original on 2013-06-08.
^ Windspeed | Icelandic Meteorological office "The Icelandic Meteorological Office now uses the SI (Systeme Internationale d'Unites) measurement metres per second (m/s) [..] other Nordic meteorological institutes have used this system for years with satisfactory results"
^ International Civil Aviation Organization - International Standards and Recommended Practices - Units of Measurement to be Used in Air and Ground Operations - Annex 5 to the Convention on International Civil Aviation
^ Measuring Wind Speed in Knots "The reason why sea winds are measured in knots at all has to do with maritime tradition"
^ a b "Documentation and verification of the world extreme wind gust record: 113.3 m s–1 on Barrow Island, Australia, during passage of tropical cyclone Olivia" (PDF). Australian Meteorological and Oceanographic Journal.
^ "World record wind gust". World Meteorological Association. Retrieved 12 February 2017.
^ "The story of the world record wind". Mount Washington Observatory. Retrieved 26 January 2010.
^ "Massive Okla. tornado had windspeed up to 200 mph". CBS News. 20 May 2013. Retrieved 17 May 2014.
^ "Historical Tornadoes". National Weather Service.
^ a b "Highest surface wind speed-Tropical Cyclone Olivia sets world record". World Record Academy. 26 January 2010. Retrieved 17 May 2014.
^ Wurman, Joshua (2007). "Doppler On Wheels". Center for Severe Weather Research. Archived from the original on 2011-07-19.
^ "5400mph winds discovered hurtling around planet outside solar system". warwick.ac.uk. Retrieved 2020-08-08.
^ Koen, Joshua. "Make and Use an Anemometer to measure Wind Speed". www.ciese.org. Retrieved 2018-04-18.
^ Chris Woodford. Ultrasonic anemometers. https://www.explainthatstuff.com/anemometers.html
^ "Wind and Structures". Korea Science (in Korean). Retrieved 2018-04-18.
^ NBC 2005 Structural Commentaries - Part 4 of Div. B, Comm. I
^ ASCE 7-05 commentary Figure C6-4, ASCE 7-10 C26.5-1
Media related to Wind speed at Wikimedia Commons
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It seems to me that we can reproduce the argument in Tag 092C instead of referring to it, since it is even clearer in this context, without passing to stalks:
We need to check that the functor
\mathcal G\mapsto\mathcal G\otimes\mathcal F
is exact. The point is that we can take the "external tensor product" (this is essentially the argument in Tag 092C, but it seems even clearer when we write it in the language of algebraic geometry): let
p_1,p_2\colon X\times_YX\to X
be two projections of which a common section is the diagonal
j\colon X\to X\times_YX
\mathcal G\otimes\mathcal F\cong j^*(p_1^*\mathcal G\otimes p_2^*\mathcal F)
. Note that the pushforward of
p_1^*\mathcal G\otimes p_2^*\mathcal F
Y
\mathcal G
\mathcal F
\mathcal O_Y
, the result follows from the assumed flatness.
OK, yes, but I am going to leave as is for now.
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Glossary - Integral SIZE
A pair (or pool) in SIZE is a smart contract that holds the ERC20 tokens where actual events (swap/deposit/withdraw) happen. Currently SIZE only supports 2-token swapping (does not support multi-token swapping), so each pool contains only 2 tokens. Each trading pair (such as WBTC-WETH, USDC-WETH) has its own pool.
Due to the delay mechanism, only the delay contract can directly interact with the pair contract, i.e. all the swap/deposit/withdraw actions have to be invoked by the delay contract, and cannot be triggered by anyone else.
Pair creation is a governance-controlled process instead of a templated system such as in Uniswap. This is because pair creation in SIZE involves many other parameters including oracle choice, fee rate, etc, that need to be decided by governance.
Oracle in the SIZE is a smart contract that holds the reference to the external price oracle and contains the math that swaps involve (given tokenIn calculate tokenOut etc). Currently SIZE only supports Uniswap V2 oracle.
The curve is a mathematical function that calculates the swap amount. In Uniswap, the curve is a constant product formula x*y=k, which implies
(x+\Delta x)(y+\Delta y)=k
thus given
\Delta x
, one can calculate
\Delta y
In SIZE, the curve is a constant swap function, meaning that it always fills the order at the oracle TWAP price, regardless of order size. Therefore, there is no price impact.
Delay in the SIZE is a smart contract that serves as an intermediary layer between users and the SIZE pool contracts. All user interactions with the SIZE (swap/deposit/withdraw, etc) will be done through the delay contract. In particular, every user action (swap/deposit/withdraw) will experience a delay.
The key parameter (governance controlled) in the delay contract is the delay parameter. It dictates the time period that has to elapse before any particular user action can start to be processed. The delay parameter can range from minutes to a week. This delay parameter is also the time duration of the TWAP window.
Order is an instruction struct that a user sends to the delay contract to be processed. An order can be swap, deposit, or withdraw. Once an order is successfully sent, it will be enqueued in the system. After the delay time has passed, the order will become eligible and will be processed in the order it is received. A swap order will be filled at the TWAP price of the oracle over the window between the submit time and the execution time.
In order for orders to be executed smoothly, a bot is set up to run continuously to look for eligible orders to execute. Once the bot sees eligible orders, it will initiate an Ethereum transaction to execute the order. Currently, the bot is set by admin, i.e. not everyone can become a bot.
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