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Section 15.12: Henselization of pairs
Lemma 15.12.5. Let $(A, I) = \mathop{\mathrm{colim}}\nolimits (A_ i, I_ i)$ be a filtered colimit of pairs. The functor of Lemma 15.12.1 gives $A^ h = \mathop{\mathrm{colim}}\nolimits A_ i^ h$ and $I^ h = \mathop{\mathrm{colim}}\nolimits I_ i^ h$.
Proof. By Categories, Lemma 4.24.5 we see that $(A^ h, I^ h)$ is the colimit of the system $(A_ i^ h, I_ i^ h)$ in the category of henselian pairs. Thus for a henselian pair $(B, J)$ we have
\[ \mathop{\mathrm{Mor}}\nolimits ((A^ h, I^ h), (B, J)) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Mor}}\nolimits ((A_ i^ h, I_ i^ h), (B, J)) = \mathop{\mathrm{Mor}}\nolimits (\mathop{\mathrm{colim}}\nolimits (A_ i^ h, I_ i^ h), (B, J)) \]
Here the colimit is in the category of pairs. Since the colimit is filtered we obtain $\mathop{\mathrm{colim}}\nolimits (A_ i^ h, I_ i^ h) = (\mathop{\mathrm{colim}}\nolimits A_ i^ h, \mathop{\mathrm{colim}}\nolimits I_ i^ h)$ in the category of pairs; details omitted. Again using the colimit is filtered, this is a henselian pair (Lemma 15.11.13). Hence by the Yoneda lemma we find $(A^ h, I^ h) = (\mathop{\mathrm{colim}}\nolimits A_ i^ h, \mathop{\mathrm{colim}}\nolimits I_ i^ h)$. $\square$
Lemma 0038 says that
(A,I)^h
is the colimit of the system
(A_i^h,I_i^h)
in the category of henselian pairs. This colimit is (looking at the universal properties) the henselization of
(\operatorname{colim} A_i^h, \operatorname{colim} I_i^h)
. The latter is henselian if the colimit is filtered, but in general this is not clear to me.
Follow-up to Comment #4896: consider for instance, for an odd prime
p
, the coproduct of
(\mathbb{Z}_p,(p))
with itself: in the category of pairs, this is
(A,(p))
A:=\mathbb{Z}_p \otimes \mathbb{Z}_p
; however
A/(p)=\mathbb{F}_p
A
is not connected, so
(A,I)
is not even a Zariski pair. (To see that
A
is not connected, let
t:=\sqrt{1+p}\in 1+p\mathbb{Z}_p
\mathbb{Z}_p
is faithfully flat over
A_0:=\mathbb{Z}[t]\left[\frac{1}{t+1}\right]
A
A_0\otimes_\mathbb{Z} A_0
which is not connected.)
Good catch! I will fix this later today or tomorrow. Thanks very much!
OK, this is now done. See changes here.
Comment #5019 by 羽山籍真 on April 09, 2020 at 12:33
In the comment 4897, if the prime
p=3
t=2
A_0=\mathbb{Z}[t]\left[\frac{1}{1+t} \right]=\mathbb{Z}[2]\left[\frac{1}{3}\right]=\mathbb{Z}\left[\frac{1}{3}\right]
A_0\rightarrow \mathbb{Z}_3
\mathbb{Z}\left[\frac{1}{3}\right]\rightarrow \mathbb{Z}_3
, which is problematic because the image
x
\frac{1}{3}
3\cdot x=1
, but such
x
does not exist in
\mathbb{Z}_3
@ 羽山籍真: No,
t
1+p
\equiv1
p
, so in this case it is
-2
2
A_0=\mathbb{Z}
@Laurent Moret-Bailly Thanks. But in this case,
A_0\rightarrow \mathbb{Z}_p
\mathbb{Z}\rightarrow \mathbb{Z}_3
, which is not faithfully flat.
The example by Moret-Bailly is now Example 15.11.14.
Comment #5035 by slogan_bot on April 16, 2020 at 03:30
Suggested slogan: "Henselization of pairs commutes with filtered colimits"
|
{\displaystyle n}
{\displaystyle n}
{\displaystyle n}
{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle 13}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle (n-1)^{\textrm {th}}}
case. In such situations, strong induction assumes that the conjecture is true for ALL cases of lower value tha{\displaystyle n}
{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i+n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}+n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}+{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
|
Significant Figures and Order of Magnitude | Boundless Physics | Course Hero
In scientific notation all numbers are written in the form of
\text{a}\cdot 10^{\text{b}}
(a times ten raised to the power of b).
exponent: The power to which a number, symbol or expression is to be raised. For example, the 3 in $x^3$.
Scientific notation is a way of writing numbers that are too big or too small in a convenient and standard form. Scientific notation has a number of useful properties and is commonly used in calculators and by scientists, mathematicians and engineers. In scientific notation all numbers are written in the form of
\text{a}\cdot 10^{\text{b}}
\text{a}
multiplied by ten raised to the power of
\text{b}
), where the exponent
\text{b}
is an integer, and the coefficient
\text{a}
Most of the interesting phenomena in our universe are not on the human scale. It would take about 1,000,000,000,000,000,000,000 bacteria to equal the mass of a human body. Thomas Young's discovery that light was a wave preceded the use of scientific notation, and he was obliged to write that the time required for one vibration of the wave was "
\displaystyle \frac{1}{500}
of a millionth of a millionth of a second"; an inconvenient way of expressing the point. Scientific notation is a less awkward and wordy way to write very large and very small numbers such as these.
32 = 3.2\cdot 10^{1}
320 = 3.2\cdot 10^{2}
3200 = 3.2\cdot 10^{3}
, and so forth...
Each number is ten times bigger than the previous one. Since
10^{1}
is ten times smaller than
10^{2}
, it makes sense to use the notation
10^{0}
to stand for one, the number that is in turn ten times smaller than
10^{1}
. Continuing on, we can write
10^{-1}
to stand for 0.1, the number ten times smaller than
10^{0}
. Negative exponents are used for small numbers:
3.2=3.2\cdot 10^{0}
0.32=3.2\cdot 10^{-1}
0.032=3.2\cdot 10^{-2}
Scientific notation displayed calculators can take other shortened forms that mean the same thing. For example,
3.2\cdot 10^{6}
(written notation) is the same as
3.2\text{E+6}
(notation on some calculators) and
3.2^{6}
(notation on some other calculators).
Calculations rarely lead to whole numbers. As such, values are expressed in the form of a decimal with infinite digits. The more digits that are used, the more accurate the calculations will be upon completion. Using a slew of digits in multiple calculations, however, is often unfeasible if calculating by hand and can lead to much more human error when keeping track of so many digits. To make calculations much easier, the results are often 'rounded off' to the nearest few decimal places.
For example, the equation for finding the area of a circle is
\text{A} = \pi \text{r}^2
\pi
(pi) has infinitely many digits, but can be truncated to a rounded representation of as 3.14159265359. However, for the convenience of performing calculations by hand, this number is typically rounded even further, to the nearest two decimal places, giving just 3.14. Though this technically decreases the accuracy of the calculations, the value derived is typically 'close enough' for most estimation purposes.
\sqrt{4.58^{2} + 3.28^{2}} = \sqrt{21.0+10.8}=5.64
Order of Magnitude: The class of scale or magnitude of any amount, where each class contains values of a fixed ratio (most often 10) to the class preceding it. For example, something that is 2 orders of magnitude larger is 100 times larger; something that is 3 orders of magnitude larger is 1000 times larger; and something that is 6 orders of magnitude larger is one million times larger, because 102 = 100, 103 = 1000, and 106 = one million
An order of magnitude is the class of scale of any amount in which each class contains values of a fixed ratio to the class preceding it. In its most common usage, the amount scaled is 10, and the scale is the exponent applied to this amount (therefore, to be an order of magnitude greater is to be 10 times, or 10 to the power of 1, greater). Such differences in order of magnitude can be measured on the logarithmic scale in "decades," or factors of ten. It is common among scientists and technologists to say that a parameter whose value is not accurately known or is known only within a range is "on the order of" some value. The order of magnitude of a physical quantity is its magnitude in powers of ten when the physical quantity is expressed in powers of ten with one digit to the left of the decimal.
Orders of magnitude are generally used to make very approximate comparisons and reflect very large differences. If two numbers differ by one order of magnitude, one is about ten times larger than the other. If they differ by two orders of magnitude, they differ by a factor of about 100. Two numbers of the same order of magnitude have roughly the same scale -- the larger value is less than ten times the smaller value.
An Example Estimation
Incorrect solution: Let's say the trucker needs to make a profit on the trip. Taking into account her benifits, the cost of gas, and maintenance and payments on the truck, let's say the total cost is more like 2000. You might guess about 5000 tomatoes would fit in the back of the truck, so the extra cost per tomato is 40 cents. That means the cost of transporting one tomato is comparable to the cost of the tomato itself.
The problem here is that the human brain is not very good at estimating area or volume -- it turns out the estimate of 5000 tomatoes fitting in the truck is way off. (This is why people have a hard time in volume-estimation contests, such as the one shown below.) When estimating area or volume, you are much better off estimating linear dimensions and computing the volume from there.
So, here's a better solution: As before, let's say the cost of the trip is $2000. The dimensions of the bin are probably 4m by 2m by 1m, for a volume of
8\text{m}^{3}
. Since our goal is just an order-of-magnitude estimate, let's round that volume off to the nearest power of ten:
10\text{m}^{3}
. The shape of a tomato doesn't follow linear dimensions, but since this is just an estimate, let's pretend that a tomato is an 0.1m by 0.1m by 0.1m cube, with a volume of
1\cdot10^{-3} \text{m}^{3}
. We can find the total number of tomatoes by dividing the volume of the bin by the volume of one tomato:
\displaystyle \frac{10^{3}\text{m}^{3}}{10^{-3}\text{m}^{3}} = 10^{6}
tomatoes. The transportation cost per tomato is
\displaystyle \frac{\$2000}{10^{6}\text{tomatoes}} = \$0.002
per tomato. That means that transportation really doesn't contribute very much to the cost of a tomato. Approximating the shape of a tomato as a cube is an example of another general strategy for making order-of-magnitude estimates.
Scientific notation. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Scientific%20notation. License: CC BY-SA: Attribution-ShareAlike
Provided by: http://www.thechembook.com/chemistry/index.php/Scientific_notation. Located at: http://www.thechembook.com/chemistry/index.php/Scientific_notation. License: CC BY: Attribution
Round-off error. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Round-off_error. License: CC BY-SA: Attribution-ShareAlike
approximation. Provided by: Wiktionary. Located at: http://en.wiktionary.org/wiki/approximation. License: CC BY-SA: Attribution-ShareAlike
Order of magnitude. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Order_of_magnitude. License: CC BY-SA: Attribution-ShareAlike
Order of Magnitude. Provided by: Wikipedia. Located at: http://en.wikipedia.org/wiki/Order%20of%20Magnitude. License: CC BY-SA: Attribution-ShareAlike
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|
Fall time - Wikipedia
This article is about the electronics concept. For the film, see Fall Time.
In electronics, fall time (pulse decay time)
{\displaystyle t_{f}}
is the time taken for the amplitude of a pulse to decrease (fall) from a specified value (usually 90% of the peak value exclusive of overshoot or undershoot) to another specified value (usually 10% of the maximum value exclusive of overshoot or undershoot).
Limits on undershoot and oscillation (also known as ringing and hunting) are sometimes additionally stated when specifying fall time limits.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Fall_time&oldid=1028805958"
|
The mascot of High Tech High is Wendell the Wizard. For the homecoming football game, the cheerleaders have made seven large cards that together spell out “WIZARDS.” Each card has one letter on it, and each cheerleader is supposed to hold up one card. At the end of the first quarter, they realize that someone has mixed up the cards.
How many ways are there to arrange the cards?
Try using a factorial to represent the different arrangements.
If they had not noticed the mix up, what would be the probability that the cards would have correctly spelled out “WIZARDS”?
\frac{1}{5040}
|
Lemma 36.31.1 (0BDI)—The Stacks project
Lemma 36.31.1. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be pseudo-coherent (for example perfect). For any $i \in \mathbf{Z}$ consider the function
\[ \beta _ i : X \longrightarrow \{ 0, 1, 2, \ldots \} ,\quad x \longmapsto \dim _{\kappa (x)} H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \kappa (x)) \]
the level sets of $\beta _ i$ are locally constructible in $X$.
Proof. Consider a morphism of schemes $f : Y \to X$ and a point $y \in Y$. Let $x$ be the image of $y$ and consider the commutative diagram
\[ \xymatrix{ y \ar[r]_ j \ar[d]_ g & Y \ar[d]^ f \\ x \ar[r]^ i & X } \]
Then we see that $Lg^* \circ Li^* = Lj^* \circ Lf^*$. This implies that the function $\beta '_ i$ associated to the pseudo-coherent complex $Lf^*E$ is the pullback of the function $\beta _ i$, in a formula: $\beta '_ i = \beta _ i \circ f$. This is the meaning of (1).
Fix $i$ and let $x \in X$. It is enough to prove (2) and (3) holds in an open neighbourhood of $x$, hence we may assume $X$ affine. Then we can represent $E$ by a bounded above complex $\mathcal{F}^\bullet $ of finite free modules (Lemma 36.13.3). Then $P = \sigma _{\geq i - 1}\mathcal{F}^\bullet $ is a perfect object and $P \to E$ induces an isomorphism
\[ H^ i(P \otimes _{\mathcal{O}_ X}^\mathbf {L} \kappa (x')) \to H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \kappa (x')) \]
for all $x' \in X$. Thus we may assume $E$ is perfect. In this case by More on Algebra, Lemma 15.75.6 there exists an affine open neighbourhood $U$ of $x$ and $a \leq b$ such that $E|_ U$ is represented by a complex
\[ \ldots \to 0 \to \mathcal{O}_ U^{\oplus \beta _ a(x)} \to \mathcal{O}_ U^{\oplus \beta _{a + 1}(x)} \to \ldots \to \mathcal{O}_ U^{\oplus \beta _{b - 1}(x)} \to \mathcal{O}_ U^{\oplus \beta _ b(x)} \to 0 \to \ldots \]
(This also uses earlier results to turn the problem into algebra, for example Lemmas 36.3.5 and 36.10.7.) It follows immediately that $\beta _ i(x') \leq \beta _ i(x)$ for all $x' \in U$. This proves that $\beta _ i$ is upper semi-continuous.
To prove (3) we may assume that $X$ is affine and $E$ is given by a complex of finite free $\mathcal{O}_ X$-modules (for example by arguing as in the previous paragraph, or by using Cohomology, Lemma 20.46.3). Thus we have to show that given a complex
\[ \mathcal{O}_ X^{\oplus a} \to \mathcal{O}_ X^{\oplus b} \to \mathcal{O}_ X^{\oplus c} \]
the function associated to a point $x \in X$ the dimension of the cohomology of $\kappa _ x^{\oplus a} \to \kappa _ x^{\oplus b} \to \kappa _ x^{\oplus c}$ in the middle has constructible level sets. Let $A \in \text{Mat}(a \times b, \Gamma (X, \mathcal{O}_ X))$ be the matrix of the first arrow. The rank of the image of $A$ in $\text{Mat}(a \times b, \kappa (x))$ is equal to $r$ if all $(r + 1) \times (r + 1)$-minors of $A$ vanish at $x$ and there is some $r \times r$-minor of $A$ which does not vanish at $x$. Thus the set of points where the rank is $r$ is a constructible locally closed set. Arguing similarly for the second arrow and putting everything together we obtain the desired result. $\square$
Comment #5399 by Davis Lazowski on July 17, 2020 at 13:31
I think whenever you write
K
, you mean
E
. You write
K
in three places:
Lf^\star K
, at the start of the proof of 1);
K|_U
, in the proof of 2);
K
at the start of the proof of 3)
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 0BDI. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 0BDI, in case you are confused.
|
Comment #5985 by Manolis Tsakiris on March 19, 2021 at 03:19
Hi, i would just like to point out that Proposition 00KJ does not explicitly say that an Artinian ring is the product of its localizations at its maximal ideals. It would be actually nice to include that statement in the proposition. Eric Wofsey has given a nice scheme-theoretic proof here: https://math.stackexchange.com/questions/2384868/any-artinian-ring-is-direct-product-of-its-localizations-at-the-maximal-ideals
@#5985: Do you mean there is something wrong with the proof? Because the proof of this lemma doesn't use the fact that an Artinian local ring is a product of the localizations at its maximal ideals. I'm asking because if you are commenting on Proposition 00KJ, why are you leaving a comment on the page for this lemma?
Dear Johan, my point is the following. We have
S_m \cong k
for every maximal ideal of
S
S
is Artinian. Now, the last sentence of the proof says that "This implies the result because
S
is the product of the localizations at its maximal ideals by Lemma 00J6 and Proposition 00KJ again." So it seems to me that the proof uses the fact that "an Artinian ring is the product of its localizations at its maximal ideals". On the other hand, this is not mentioned explicitly in Proposition 00KJ. So i suppose my comment is a joint comment for the present tag and Proposition 00KJ.
@#5996. Yes, you are totally right. Sorry! OK, I will fix this the next time I go through all the comments.
Okay, this is now fixed. I decided to work around the issue you raise and I tried to "simplify" the proof of the lemma. See changes.
There does not seem to be a way of including in the statement of a result every possible way this result will be used in the future as the discussion above shows with regards to Proposition 10.60.7. In fact, I think the tendency to write very long lemmas with many equivalent formulations of the same thing actually often backfires I think. An example is Lemma 10.153.3 characterizing henselian local rings. Another one is Nakayama's lemma 10.20.1.
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Simulate regression coefficients and disturbance variance of Bayesian linear regression model - MATLAB simulate - MathWorks Deutschland
{\text{GNPR}}_{t}={\beta }_{0}+{\beta }_{1}{\text{IPI}}_{t}+{\beta }_{2}{\text{E}}_{t}+{\beta }_{3}{\text{WR}}_{t}+{\epsilon }_{t}.
t
{\epsilon }_{t}
{\sigma }^{2}
\beta |{\sigma }^{2}\sim {N}_{4}\left(M,{\sigma }^{2}V\right)
M
V
{\sigma }^{2}\sim IG\left(A,B\right)
A
B
Although the marginal and conditional posterior distributions of
\beta
{\sigma }^{2}
are analytically tractable, this example focuses on how to implement the Gibbs sampler to reproduce known results.
Estimate the model again, but use a Gibbs sampler. Alternate between sampling from the conditional posterior distributions of the parameters. Sample 10,000 times and create variables for preallocation. Start the sampler by drawing from the conditional posterior of
\beta
{\sigma }^{2}=2
\mathit{k}
{\beta }_{k}|{\sigma }^{2},{\gamma }_{k}={\gamma }_{k}\sigma \sqrt{{V}_{k1}}{Z}_{1}+\left(1-{\gamma }_{k}\right)\sigma \sqrt{{V}_{k2}}{Z}_{2}
{\mathit{Z}}_{1}
{\mathit{Z}}_{2}
{\sigma }^{2}\sim IG\left(A,B\right)
A
B
{\gamma }_{\mathit{k}}\in \left\{0,1\right\}
\beta
{\sigma }^{2}\text{\hspace{0.17em}}
Consider a Bayesian linear regression model containing one predictor, and a t distributed disturbance variance with a profiled degrees of freedom parameter
\nu
{\lambda }_{j}\sim IG\left(\nu /2,2/\nu \right)
{\epsilon }_{j}|{\lambda }_{j}\sim N\left(0,{\lambda }_{j}{\sigma }^{2}\right)
f\left(\beta ,{\sigma }^{2}\right)\propto \frac{1}{{\sigma }^{2}}
{\epsilon }_{j}\sim t\left(0,{\sigma }^{2},\nu \right)
{\lambda }_{j}|{\epsilon }_{j}\sim IG\left(\frac{\nu +1}{2},\frac{2}{\nu +{\epsilon }_{j}^{2}/{\sigma }^{2}}\right)
\lambda
is a vector of latent scale parameters that attributes low precision to observations far from the regression line.
\nu
\lambda
For this problem, the Gibbs sampler is well suited to estimate the coefficients because you can simulate the parameters of a Bayesian linear regression model conditioned on
\lambda
\lambda
n=100
responses from
{y}_{t}=1+2{x}_{t}+{e}_{t},
x\in \left[0,2\right]
{e}_{t}\sim N\left(0,0.{5}^{2}\right)
Introduce outlying responses by inflating all responses below
x=0.25
\beta ,{\sigma }^{2}|y,x,\lambda
\lambda
, create a diffuse prior model with two regression coefficients, and draw a set of parameters from the posterior. The first regression coefficient corresponds to the intercept, so specify that bayeslm not include an intercept.
\lambda
Run the Gibbs sampler for 20,000 iterations and apply a burn-in period of 5,000. Specify
\nu =1
, preallocate for the posterior draws, and initialize
\lambda
\beta
{\sigma }^{2}
\beta
from the posterior model, and extract the posterior covariance of
\beta
estBetaMean is a 4-by-1 vector representing the mean of the marginal posterior of
\beta
. EstBetaCov is a 4-by-4 matrix representing the covariance matrix of the posterior of
\beta
Because the posterior of
\beta
is symmetric and unimodal, the posterior mean and MAP should be the same. Compare the MAP estimate of
\beta
with its posterior mean.
Estimate the analytical mode of the posterior of
{\sigma }^{2}
. Compare it to the estimated MAP of
{\sigma }^{2}
\ell \left(\beta ,{\sigma }^{2}|y,x\right)=\prod _{t=1}^{T}\varphi \left({y}_{t};{x}_{t}\beta ,{\sigma }^{2}\right).
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evenfunc - Maple Help
Home : Support : Online Help : Programming : Data Types : Type Checking : Types : evenfunc
check for an even function
check for an odd function
type(f, evenfunc(x))
type(f, oddfunc(x))
expression, regarded as a function of x
These procedures test the parity of the expression f, that is, whether it is even or odd, with respect to x.
More precisely,
\mathrm{type}\left(f\left(x\right),\mathrm{evenfunc}\left(x\right)\right)
returns true if Maple can determine that
f\left(x\right)-f\left(-x\right)
is zero and false otherwise. Maple performs this test by calling Testzero on this difference. If false is returned, that is not a guarantee that f is not even: it may be zero, but if so, Testzero is not strong enough to recognize that the result is zero. Type
\mathrm{oddfunc}\left(x\right)
is determined in a similar way but using the expression
f\left(x\right)+f\left(-x\right)
\mathrm{type}\left({x}^{2},\mathrm{evenfunc}\left(x\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left({x}^{2}{y}^{3}+2,\mathrm{evenfunc}\left(x\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\mathrm{sin}\left(x\right),\mathrm{oddfunc}\left(x\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\frac{{x}^{2}}{x-1},\mathrm{oddfunc}\left(x\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
f
is even, but the default zero testing algorithm does not recognize this.
f≔\mathrm{piecewise}\left(x=0,1,\frac{1}{{x}^{2}}\right)
\textcolor[rgb]{0,0,1}{f}\textcolor[rgb]{0,0,1}{≔}{\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\\ \frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}}& \textcolor[rgb]{0,0,1}{\mathrm{otherwise}}\end{array}
\mathrm{type}\left(f,\mathrm{evenfunc}\left(x\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
By changing the zero testing algorithm, we can make this return true. By default, Testzero calls Normalizer. In this case, we strengthen Normalizer.
\mathrm{Normalizer}≔\mathrm{simplify}
\textcolor[rgb]{0,0,1}{\mathrm{Normalizer}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{\mathrm{simplify}}
\mathrm{type}\left(f,\mathrm{evenfunc}\left(x\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
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Sums of the divisors, in Cuisenaire rods, of the first six highly abundant numbers
In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.
Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on the subject was done by Alaoglu and Erdős (1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N.
Formal definition and examples
Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n,
{\displaystyle \sigma (n)>\sigma (m)}
where σ denotes the sum-of-divisors function. The first few highly abundant numbers are
1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... (sequence A002093 in the OEIS).
For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ.
The only odd highly abundant numbers are 1 and 3.[1]
Relations with other sets of numbers
Although the first eight factorials are highly abundant, not all factorials are highly abundant. For example,
but there is a smaller number with larger sum of divisors,
so 9! is not highly abundant.
Alaoglu and Erdős noted that all superabundant numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by Jean-Louis Nicolas (1969).
Despite the terminology, not all highly abundant numbers are abundant numbers. In particular, none of the first seven highly abundant numbers (1, 2, 3, 4, 6, 8, and 10) is abundant. Along with 16, the ninth highly abundant number, these are the only highly abundant numbers that are not abundant.
7200 is the largest powerful number that is also highly abundant: all larger highly abundant numbers have a prime factor that divides them only once. Therefore, 7200 is also the largest highly abundant number with an odd sum of divisors.[2]
^ See Alaoglu & Erdős (1944), p. 466. Alaoglu and Erdős claim more strongly that all highly abundant numbers greater than 210 are divisible by 4, but this is not true: 630 is highly abundant, and is not divisible by 4. (In fact, 630 is the only counterexample; all larger highly abundant numbers are divisible by 12.)
^ Alaoglu & Erdős (1944), pp. 464–466.
Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers" (PDF). Transactions of the American Mathematical Society. 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR 0011087.
Nicolas, Jean-Louis (1969). "Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers"". Bull. Soc. Math. France. 97: 129–191. MR 0254130.
Pillai, S. S. (1943). "Highly abundant numbers". Bull. Calcutta Math. Soc. 35: 141–156. MR 0010560.
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In the past, many states had license plates composed of three letters followed by three digits (
0\text{ to }9
). Recently, many states have responded to the increased number of cars by adding one digit (
1\text{ to }9
) ahead of the three letters. How many more license plates of the second type are possible? What is the probability of being randomly assigned a license plate containing
\text{ALG }2
Since repetition is allowed, the total number of license plates of the first type is
26\cdot26\cdot26\cdot10\cdot10\cdot10 = 17,576,000
The total number of license plates of the second type is
9\cdot26\cdot26\cdot26\cdot10\cdot10\cdot10 = 158,184,000
Find the number of license plates containing
\text{ALG }2
9\cdot1\cdot1\cdot1\cdot1\cdot10\cdot10 =
P(\text{ALG }2)=\frac{900}{158,184,000}=?
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12
toppings other than cheese. How many different pizzas can they create with five or fewer toppings? List all subproblems and calculate the solution.
\begin{array}{l} \qquad \; \; \; \text{five toppings:}\quad {\color{blue}{\mathbf{_{12}C_5}}}\\ \qquad \; \; \text{four toppings:}\quad {\color{blue}{\mathbf{_{12}C_5}}}\\ \qquad \text{three toppings:}\quad {\color{blue}{\mathbf{_{12}C_5}}}\\ \qquad \; \; \text{two toppings:}\quad {\color{blue}{\mathbf{_{12}C_5}}}\\ \qquad \; \; \text{one toppings:}\quad \; \; \; {\color{blue}{\mathbf{?}}}\\ \underline{+\qquad \; \text{no toppings:}\quad \; \; \; {\color{blue}{\mathbf{?}}}\qquad\qquad \qquad \quad \; }\\ \; \end{array}
\begin{array}{l} \qquad \; \; \; \text{five toppings:}\quad {\color{blue}{\mathbf{_{12}C_5}}} {\color{red}{\boldsymbol{=792}}}\\ \qquad \; \; \text{four toppings:}\quad {\color{blue}{\mathbf{_{12}C_5}}} {\color{red}{\boldsymbol{=495}}}\\ \qquad \text{three toppings:}\quad {\color{blue}{\mathbf{_{12}C_5}}} {\color{red}{\boldsymbol{=220}}}\\ \qquad \; \; \text{two toppings:}\quad {\color{blue}{\mathbf{_{12}C_5}}} {\color{red}{\boldsymbol{=?}}}\\ \qquad \; \; \text{one toppings:}\quad \; \; \; {\color{blue}{\mathbf{?}}} \quad {\color{red}{\boldsymbol{=?}}}\\ \underline{+\qquad \; \text{no toppings:}\quad \; \; \; {\color{blue}{\mathbf{?}}} \quad {\color{red}{\boldsymbol{=?}}}\qquad \qquad \qquad }\\ \qquad \qquad \qquad \qquad \qquad \quad \; \; \; {\color{red}{\boldsymbol{=1586}\textbf{ pizzas}}} \end{array}
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Missing data - Wikipedia
Missing data can occur because of nonresponse: no information is provided for one or more items or for a whole unit ("subject"). Some items are more likely to generate a nonresponse than others: for example items about private subjects such as income. Attrition is a type of missingness that can occur in longitudinal studies—for instance studying development where a measurement is repeated after a certain period of time. Missingness occurs when participants drop out before the test ends and one or more measurements are missing.
Data often are missing in research in economics, sociology, and political science because governments or private entities choose not to, or fail to, report critical statistics,[1] or because the information is not available. Sometimes missing values are caused by the researcher—for example, when data collection is done improperly or mistakes are made in data entry.[2]
These forms of missingness take different types, with different impacts on the validity of conclusions from research: Missing completely at random, missing at random, and missing not at random. Missing data can be handled similarly as censored data.
1.1 Missing completely at random
1.2 Missing at random
1.3 Missing not at random
2 Techniques of dealing with missing data
2.2 Partial deletion
2.3 Full analysis
3 Model-based techniques
Understanding the reasons why data are missing is important for handling the remaining data correctly. If values are missing completely at random, the data sample is likely still representative of the population. But if the values are missing systematically, analysis may be biased. For example, in a study of the relation between IQ and income, if participants with an above-average IQ tend to skip the question ‘What is your salary?’, analyses that do not take into account this missing at random (MAR pattern (see below)) may falsely fail to find a positive association between IQ and salary. Because of these problems, methodologists routinely advise researchers to design studies to minimize the occurrence of missing values.[2] Graphical models can be used to describe the missing data mechanism in detail.[3][4]
Missing completely at random[edit]
Missing at random[edit]
Missing at random (MAR) occurs when the missingness is not random, but where missingness can be fully accounted for by variables where there is complete information.[7] Since MAR is an assumption that is impossible to verify statistically, we must rely on its substantive reasonableness.[8] An example is that males are less likely to fill in a depression survey but this has nothing to do with their level of depression, after accounting for maleness. Depending on the analysis method, these data can still induce parameter bias in analyses due to the contingent emptiness of cells (male, very high depression may have zero entries). However, if the parameter is estimated with Full Information Maximum Likelihood, MAR will provide asymptotically unbiased estimates.[citation needed]
Missing not at random[edit]
Samuelson and Spirer (1992) discussed how missing and/or distorted data about demographics, law enforcement, and health could be indicators of patterns of human rights violations. They gave several fairly well documented examples. [Samuelson, D A, and Spirer, H F, "Use of Missing and Distorted Deta in Inferences About Human Rights Violations," in "Jabine, T., and Claude, R., eds., "Human Rights and Statistics: Getting the Record Straight," U of Pennsylvania Press, 1992.]
Techniques of dealing with missing data[edit]
Missing data reduces the representativeness of the sample and can therefore distort inferences about the population. Generally speaking, there are three main approaches to handle missing data: (1) Imputation—where values are filled in the place of missing data, (2) omission—where samples with invalid data are discarded from further analysis and (3) analysis—by directly applying methods unaffected by the missing values. One systematic review addressing the prevention and handling of missing data for patient-centered outcomes research identified 10 standards as necessary for the prevention and handling of missing data. These include standards for study design, study conduct, analysis, and reporting.[9]
In some practical application, the experimenters can control the level of missingness, and prevent missing values before gathering the data. For example, in computer questionnaires, it is often not possible to skip a question. A question has to be answered, otherwise one cannot continue to the next. So missing values due to the participant are eliminated by this type of questionnaire, though this method may not be permitted by an ethics board overseeing the research. In survey research, it is common to make multiple efforts to contact each individual in the sample, often sending letters to attempt to persuade those who have decided not to participate to change their minds.[10]: 161–187 However, such techniques can either help or hurt in terms of reducing the negative inferential effects of missing data, because the kind of people who are willing to be persuaded to participate after initially refusing or not being home are likely to be significantly different from the kinds of people who will still refuse or remain unreachable after additional effort.[10]: 188–198
In situations where missing values are likely to occur, the researcher is often advised on planning to use methods of data analysis methods that are robust to missingness. An analysis is robust when we are confident that mild to moderate violations of the technique's key assumptions will produce little or no bias, or distortion in the conclusions drawn about the population.
Imputation[edit]
Some data analysis techniques are not robust to missingness, and require to "fill in", or impute the missing data. Rubin (1987) argued that repeating imputation even a few times (5 or less) enormously improves the quality of estimation.[2] For many practical purposes, 2 or 3 imputations capture most of the relative efficiency that could be captured with a larger number of imputations. However, a too-small number of imputations can lead to a substantial loss of statistical power, and some scholars now recommend 20 to 100 or more.[11] Any multiply-imputed data analysis must be repeated for each of the imputed data sets and, in some cases, the relevant statistics must be combined in a relatively complicated way.[2]
In the comparison of two paired samples with missing data, a test statistic that uses all available data without the need for imputation is the partially overlapping samples t-test.[12] This is valid under normality and assuming MCAR
Partial deletion[edit]
Full analysis[edit]
Discriminative approaches:
Max-margin classification of data with absent features [13][14]
Partial identification methods may also be used.[15]
Model-based techniques[edit]
For any three variables X,Y, and Z where Z is fully observed and X and Y partially observed, the data should satisfy:
{\displaystyle X\perp \!\!\!\perp R_{y}|(R_{x},Z)}
{\displaystyle {\begin{aligned}P(X,Y)&=P(X|Y)P(Y)\\&=P(X|Y,R_{x}=0,R_{y}=0)P(Y|R_{y}=0)\end{aligned}}}
{\displaystyle R_{x}=0}
{\displaystyle R_{y}=0}
denote the observed portions of their respective variables.
Different model structures may yield different estimands and different procedures of estimation whenever consistent estimation is possible. The preceding estimand calls for first estimating
{\displaystyle P(X|Y)}
from complete data and multiplying it by
{\displaystyle P(Y)}
estimated from cases in which Y is observed regardless of the status of X. Moreover, in order to obtain a consistent estimate it is crucial that the first term be
{\displaystyle P(X|Y)}
{\displaystyle P(Y|X)}
In many cases model based techniques permit the model structure to undergo refutation tests.[19] Any model which implies the independence between a partially observed variable X and the missingness indicator of another variable Y (i.e.
{\displaystyle R_{y}}
), conditional on
{\displaystyle R_{x}}
can be submitted to the following refutation test:
{\displaystyle X\perp \!\!\!\perp R_{y}|R_{x}=0}
A special class of problems appears when the probability of the missingness depends on time. For example, in the trauma databases the probability to lose data about the trauma outcome depends on the day after trauma. In these cases various non-stationary Markov chain models are applied. [21]
^ Messner SF (1992). "Exploring the Consequences of Erratic Data Reporting for Cross-National Research on Homicide". Journal of Quantitative Criminology. 8 (2): 155–173. doi:10.1007/bf01066742. S2CID 133325281.
^ a b c d Hand, David J.; Adèr, Herman J.; Mellenbergh, Gideon J. (2008). Advising on Research Methods: A Consultant's Companion. Huizen, Netherlands: Johannes van Kessel. pp. 305–332. ISBN 978-90-79418-01-5.
^ a b Mohan, Karthika; Pearl, Judea; Tian, Jin (2013). "Graphical Models for Inference with Missing Data". Advances in Neural Information Processing Systems 26. pp. 1277–1285.
^ Karvanen, Juha (2015). "Study design in causal models". Scandinavian Journal of Statistics. 42 (2): 361–377. arXiv:1211.2958. doi:10.1111/sjos.12110. S2CID 53642701.
^ a b Polit DF Beck CT (2012). Nursing Research: Generating and Assessing Evidence for Nursing Practice, 9th ed. Philadelphia, USA: Wolters Klower Health, Lippincott Williams & Wilkins.
^ Deng (2012-10-05). "On Biostatistics and Clinical Trials". Archived from the original on 15 March 2016. Retrieved 13 May 2016.
^ Little, Roderick J. A.; Rubin, Donald B. (2002), Statistical Analysis with Missing Data (2nd ed.), Wiley .
^ Li, Tianjing; Hutfless, Susan; Scharfstein, Daniel O.; Daniels, Michael J.; Hogan, Joseph W.; Little, Roderick J.A.; Roy, Jason A.; Law, Andrew H.; Dickersin, Kay (2014). "Standards should be applied in the prevention and handling of missing data for patient-centered outcomes research: a systematic review and expert consensus". Journal of Clinical Epidemiology. 67 (1): 15–32. doi:10.1016/j.jclinepi.2013.08.013. PMC 4631258. PMID 24262770.
^ a b Stoop, I.; Billiet, J.; Koch, A.; Fitzgerald, R. (2010). Reducing Survey Nonresponse: Lessons Learned from the European Social Survey. Oxford: Wiley-Blackwell. ISBN 978-0-470-51669-0.
^ Graham J.W.; Olchowski A.E.; Gilreath T.D. (2007). "How Many Imputations Are Really Needed? Some Practical Clarifications of Multiple Imputation Theory". Preventative Science. 8 (3): 208–213. CiteSeerX 10.1.1.595.7125. doi:10.1007/s11121-007-0070-9. PMID 17549635. S2CID 24566076.
^ Derrick, B; Russ, B; Toher, D; White, P (2017). "Test Statistics for the Comparison of Means for Two Samples That Include Both Paired and Independent Observations". Journal of Modern Applied Statistical Methods. 16 (1): 137–157. doi:10.22237/jmasm/1493597280.
^ Chechik, Gal; Heitz, Geremy; Elidan, Gal; Abbeel, Pieter; Koller, Daphne (2008-06-01). "Max-margin Classification of incomplete data" (PDF). Neural Information Processing Systems: 233–240.
^ Chechik, Gal; Heitz, Geremy; Elidan, Gal; Abbeel, Pieter; Koller, Daphne (2008-06-01). "Max-margin Classification of Data with Absent Features". The Journal of Machine Learning Research. 9: 1–21. ISSN 1532-4435.
^ Tamer, Elie (2010). "Partial Identification in Econometrics". Annual Review of Economics. 2 (1): 167–195. doi:10.1146/annurev.economics.050708.143401.
^ Mohan, Karthika; Pearl, Judea (2014). "On the testability of models with missing data". Proceedings of AISTAT-2014, Forthcoming.
^ Darwiche, Adnan (2009). Modeling and Reasoning with Bayesian Networks. Cambridge University Press.
^ Potthoff, R.F.; Tudor, G.E.; Pieper, K.S.; Hasselblad, V. (2006). "Can one assess whether missing data are missing at random in medical studies?". Statistical Methods in Medical Research. 15 (3): 213–234. doi:10.1191/0962280206sm448oa. PMID 16768297. S2CID 12882831.
^ a b Pearl, Judea; Mohan, Karthika (2013). Recoverability and Testability of Missing data: Introduction and Summary of Results (PDF) (Technical report). UCLA Computer Science Department, R-417.
^ Mohan, K.; Van den Broeck, G.; Choi, A.; Pearl, J. (2014). "An Efficient Method for Bayesian Network Parameter Learning from Incomplete Data". Presented at Causal Modeling and Machine Learning Workshop, ICML-2014.
^ Mirkes, E.M.; Coats, T.J.; Levesley, J.; Gorban, A.N. (2016). "Handling missing data in large healthcare dataset: A case study of unknown trauma outcomes". Computers in Biology and Medicine. 75: 203–216. arXiv:1604.00627. Bibcode:2016arXiv160400627M. doi:10.1016/j.compbiomed.2016.06.004. PMID 27318570. S2CID 5874067. Archived from the original on 2016-08-05.
Acock AC (2005), "Working with missing values", Journal of Marriage and Family, 67 (4): 1012–28, doi:10.1111/j.1741-3737.2005.00191.x, archived from the original on 2013-01-05
Allison, Paul D. (2001), Missing Data, SAGE Publishing
Bouza-Herrera, Carlos N. (2013), Handling Missing Data in Ranked Set Sampling, Springer
Enders, Craig K. (2010), Applied Missing Data Analysis, Guilford Press
Graham, John W. (2012), Missing Data, Springer
Molenberghs, Geert; Fitzmaurice, Garrett; Kenward, Michael G.; Tsiatis, Anastasios; Verbeke, Geert, eds. (2015), Handbook of Missing Data Methodology, Chapman & Hall
Raghunathan, Trivellore (2016), Missing Data Analysis in Practice, Chapman & Hall
Little, Roderick J. A.; Rubin, Donald B. (2002), Statistical Analysis with Missing Data (2nd ed.), Wiley
Tsiatis, Anastasios A. (2006), Semiparametric Theory and Missing Data, Springer
Van den Broeck J, Cunningham SA, Eeckels R, Herbst K (2005), "Data cleaning: detecting, diagnosing, and editing data abnormalities", PLOS Medicine, 2 (10): e267, doi:10.1371/journal.pmed.0020267, PMC 1198040, PMID 16138788, S2CID 5667073
Zarate LE, Nogueira BM, Santos TR, Song MA (2006). "Techniques for Missing Value Recovering in Imbalanced Databases: Application in a marketing database with massive missing data". IEEE International Conference on Systems, Man and Cybernetics, 2006. SMC '06. Vol. 3. pp. 2658–2664. doi:10.1109/ICSMC.2006.385265.
Missing Data, Department of Medical Statistics, London School of Hygiene & Tropical Medicine
Spatial and temporal Trend Analysis of Long Term rainfall records in data-poor catchments with missing data, a case study of Lower Shire floodplain in Malawi for the period 1953–2010.
R-miss-tastic, A unified platform for missing values methods and workflows.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Missing_data&oldid=1088779457"
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Difference between revisions of "'''Solution for the Problems of Chapter 8-Polynomials 10 STD'''" - Karnataka Open Educational Resources
Difference between revisions of "'''Solution for the Problems of Chapter 8-Polynomials 10 STD'''"
m (removed Category:Algebraic expressions using HotCat)
m (added Category:Algebraic expressions using HotCat)
#Is A=<math>s^2</math> is a polynomial function where A is the Area and s is the side of Square .Can we plot this on a graph?
#Plot p=4s where p is the perimeter and s is the side length of a square.Find the Zeroes of both the graphs
[[Category:Algebraic expressions]]
{\displaystyle x^{2}}
{\displaystyle x^{2}}
{\displaystyle s^{2}}
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(Created page with "== Formal Definition == We say <math style="vertical-align: 0px">L</math> is the limit of <math style="vertical-align: 0px">f</math> at <math style="vertical-align: 0px">c</m...")
We say <math style="vertical-align: 0px">L</math> is the limit of <math style="vertical-align: 0px">f</math> at <math style="vertical-align: 0px">c</math> if, for any <math style="vertical-align: 0px">\epsilon>0</math>,
We say <math style="vertical-align: 0px">L</math> is the limit of <math style="vertical-align: -5px">f</math> at <math style="vertical-align: 0px">c</math> if, for any <math style="vertical-align: 0px">\epsilon>0</math>,
there exists a <math style="vertical-align: 0px">\delta>0</math> such that whenever <math style="vertical-align: 0px">0<|x-c|<\delta</math>,
there exists a <math style="vertical-align: 0px">\delta>0</math>  such that whenever <math style="vertical-align: -5px">0<|x-c|<\delta</math>,
::<math>\left|f(x)-L\right|\,<\,\epsilon.</math>
{\displaystyle L}
is the limit o{\displaystyle f}
{\displaystyle c}
{\displaystyle \epsilon >0}
{\displaystyle \delta >0}
{\displaystyle 0<|x-c|<\delta }
{\displaystyle \left|f(x)-L\right|\,<\,\epsilon .}
{\displaystyle |f(x)-L|<\epsilon }
{\displaystyle -\epsilon \,<\,f(x)-L\,<\,\epsilon .}
{\displaystyle L}
{\displaystyle L-\epsilon \,<\,f(x)\,<\,L+\epsilon .}
{\displaystyle f(x)}
{\displaystyle \epsilon }
{\displaystyle L}
{\displaystyle (L-\epsilon ,L+\epsilon )}
{\displaystyle |x-c|<\delta }
{\displaystyle c-\delta \,<\,x\,<\,c+\delta .}
{\displaystyle x}
{\displaystyle \delta }
{\displaystyle c,}
which is the interval
{\displaystyle (c-\delta ,c+\delta )}
{\displaystyle 0<|x-c|?}
This means we ignore what happens at
{\displaystyle c}
. This neighborhood - the interval
{\displaystyle (c-\delta ,c+\delta )}
{\displaystyle c}
{\displaystyle \delta }
{\displaystyle \epsilon }
{\displaystyle \delta <\epsilon }
{\displaystyle \delta <\epsilon /3}
{\displaystyle \delta <\min\{1,\epsilon /3\}}
. But in each case, we usually build our proof ``backwards in what we can refer to as scratchwork.
{\displaystyle |f(x)-L|\,<\,\epsilon .}
{\displaystyle |f(x)-L|}
{\displaystyle |x-c|}
{\displaystyle |x-c|\,<\,{\textrm {something}},}
{\displaystyle \delta }
that will work. We then just ``reverse the chain of equalities in our scratchwork to construct the proof. It's easier to see in a few examples.
{\displaystyle {\displaystyle \lim _{x\rightarrow 1}10=10}}
{\displaystyle f(x)=10,\ L=10}
{\displaystyle c=1}
{\displaystyle x}
{\displaystyle f(x)=10}
{\displaystyle x,}
{\displaystyle |f(x)-L|\,=\,|10-10|\,=\,0\,<\epsilon }
{\displaystyle \epsilon }
{\displaystyle \delta }
{\displaystyle \epsilon >0}
{\displaystyle \delta =1}
{\displaystyle |x-1|<\delta =1}
{\displaystyle |f(x)-L|\ =\ |10-10|\ =\ 0\ <\ \epsilon .}
A few quick notes about these.
Every one will begin with ``let
{\displaystyle \epsilon >0}
be given. That's because most of the time, our
{\displaystyle \delta }
{\displaystyle \epsilon }
{\displaystyle {\displaystyle \lim _{x\rightarrow 3}2x-4=2}}
{\displaystyle f(x)=2x-4,\ L=2}
{\displaystyle c=3}
{\displaystyle |f(x)-L|<\epsilon }
{\displaystyle |x-c|=|x-3|}
{\displaystyle {\begin{array}{ccccrcl}&&&&\left|(2x-4)-2\right|&<&\epsilon \\\\\Rightarrow &&&&|2x-6|&<&\epsilon \\\\\Rightarrow &&&&|2(x-3)|&<&\epsilon \\\\\Rightarrow &&&&2|x-3|&<&\epsilon \\\\\Rightarrow &&&&|x-3|&<&{\displaystyle {\frac {\epsilon }{2}}.}\end{array}}}
{\displaystyle \delta }
{\displaystyle \delta <\epsilon /2}
{\displaystyle \epsilon >0}
{\displaystyle \delta <\epsilon /2}
{\displaystyle |x-c|=|x-3|<\delta <\epsilon /2}
{\displaystyle {\begin{array}{ccccrcl}&&&&|x-3|&<&{\displaystyle {\frac {\epsilon }{2}}}\\\\\Rightarrow &&&&|2x-6|&<&\epsilon \\\\\Rightarrow &&&&|2(x-3)|&<&\epsilon \\\\\Rightarrow &&&&|2x-6|&<&\epsilon \\\\\Rightarrow &&&&|(2x-4)-2|&<&{\displaystyle \epsilon }\\\\\Rightarrow &&&&|f(x)-L|&<&\epsilon ,\end{array}}}
{\displaystyle f(x)=mx+b}
{\displaystyle m\neq 0}
{\displaystyle \delta <\epsilon /|m|}
{\displaystyle {\displaystyle \lim _{x\rightarrow 1}x^{2}=1}}
{\displaystyle f(x)=x^{2},\ L=1}
{\displaystyle c=1}
{\displaystyle |f(x)-L|<\epsilon }
{\displaystyle |x-c|=|x-1|}
{\displaystyle {\begin{array}{ccccrcl}&&&&\left|x^{2}-1\right|&<&\epsilon \\\\\Rightarrow &&&&|(x-1)(x+1)|&<&\epsilon \\\\\Rightarrow &&&&|x+1|\cdot |x-1|&<&\epsilon \\\\\Rightarrow &&&&|x-1|&<&{\displaystyle {\frac {\epsilon }{|x+1|}}.}\end{array}}}
{\displaystyle |x-1|}
{\displaystyle \epsilon }
{\displaystyle x}
on the right hand side! This requires us to pick an ``initial </math>\delta</math>. Let's choose
{\displaystyle \delta =1}
{\displaystyle |x-c|=|x-1|<\delta =1}
{\displaystyle -\delta \,=\,-1\,<\,x-1\,<\,1\,=\,\delta ,}
in the manner explained in An Explanation'. More importantly, by adding two to the inequality we have
{\displaystyle 1\,<\,x+1\,<\,3.\qquad \qquad \qquad (\dagger )}
{\displaystyle \epsilon }
{\displaystyle {\frac {\epsilon }{3}}\,<\,{\frac {\epsilon }{x+1}}\,=\,{\frac {\epsilon }{|x+1|}}\,<\,{\frac {\epsilon }{1}}\,=\,\epsilon .}
{\displaystyle x}
{\displaystyle |x-1|<1}
{\displaystyle \epsilon /3<\epsilon /|x+1|}
{\displaystyle \delta <\epsilon /3}
{\displaystyle \delta =1}
. The way around this is to use the minimum function.
{\displaystyle \min\{a,b\}}
, it means to take whichever is the
least of both
{\displaystyle a}nd
{\displaystyle b}
{\displaystyle \epsilon >0}
{\displaystyle \delta =\min\{1,\epsilon /3\}}
{\displaystyle |x-c|=|x-1|<\delta }
{\displaystyle {\begin{array}{ccrcrclccccc}&&&&|x-1|&<&{\displaystyle {\frac {\epsilon }{3}}}\\\\\Rightarrow &&&&3|x-1|&<&\epsilon \\\\\Rightarrow &&|x+1||x-1|&<&3|x-1|&<&\epsilon &&&&&{\textrm {(using}}\dagger )\\\\\Rightarrow &&&&|x^{2}-1|&<&\epsilon \\\\\Rightarrow &&&&|f(x)-L|&<&\epsilon ,\end{array}}}
{\displaystyle {\displaystyle \lim _{x\rightarrow 2}3x^{2}=12}}
{\displaystyle f(x)=3x^{2},\ L=12}
{\displaystyle c=2}
{\displaystyle |f(x)-L|<\epsilon }
{\displaystyle {\begin{array}{ccccrcl}&&&&\left|3x^{2}-12\right|&<&\epsilon \\\\\Rightarrow &&&&3|x^{2}-4|&<&\epsilon \\\\\Rightarrow &&&&3|(x+2|\cdot |x-2)|&<&\epsilon \\\\\Rightarrow &&&&|x-2|&<&{\displaystyle {\frac {\epsilon }{3|x+2|}}.}\end{array}}}
{\displaystyle \delta =1}
{\displaystyle -\delta \,=\,-1\,<\,x-2\,<\,1\,=\,\delta .}
{\displaystyle 3\,<\,x+2\,<\,5.\qquad \qquad \qquad (\dagger \dagger )}
{\displaystyle \epsilon }
{\displaystyle {\frac {\epsilon }{5}}\,<\,{\frac {\epsilon }{|x+2|}}\,<\,{\frac {\epsilon }{3}}.}
{\displaystyle \delta =\min \left\{1,{\frac {1}{3}}\cdot {\frac {\epsilon }{3}}\right\}=\min \left\{1,{\frac {\epsilon }{9}}\right\}.}
{\displaystyle \epsilon >0}
{\displaystyle \delta =\min \left\{1,{\frac {\epsilon }{9}}\right\}}
{\displaystyle |x-c|=|x-2|<\delta }
{\displaystyle {\begin{array}{ccrcrclccccc}&&&&|x-2|&<&{\displaystyle {\frac {\epsilon }{9}}}\\\\\Rightarrow &&&&9|x-2|&<&\epsilon \\\\\Rightarrow &&3|x+2||x-2|&<&9|x-2|&<&\epsilon &&&&&{\textrm {(using}}\dagger \dagger )\\\\\Rightarrow &&&&|3x^{2}-12|&<&\epsilon \\\\\Rightarrow &&&&|f(x)-L|&<&\epsilon ,\end{array}}}
{\displaystyle {\displaystyle \lim _{x\rightarrow 2}x^{2}+3x+1=11}}
{\displaystyle f(x)=x^{2}+3x+1,\ L=11}
{\displaystyle c=2}
{\displaystyle |f(x)-L|<\epsilon }
{\displaystyle {\begin{array}{ccccrcl}&&&&\left|x^{2}+3x+1-11\right|&<&\epsilon \\\\\Rightarrow &&&&|x^{2}+3x-10|&<&\epsilon \\\\\Rightarrow &&&&|(x+5)(x-2)|&<&\epsilon \\\\\Rightarrow &&&&|(x+5|\cdot |x-2)|&<&\epsilon \\\\\Rightarrow &&&&|x-2|&<&{\displaystyle {\frac {\epsilon }{|x+5|}}.}\end{array}}}
{\displaystyle \delta =1}
{\displaystyle -\delta \,=\,-1\,<\,x-2\,<\,1\,=\,\delta ,\qquad \qquad \qquad (\natural )}
{\displaystyle -\delta \,=\,6\,<\,x+5\,<\,1\,=\,7.}
{\displaystyle \epsilon }
{\displaystyle {\frac {\epsilon }{7}}\,<\,{\frac {\epsilon }{|x+5|}}\,<\,{\frac {\epsilon }{5}}}
{\displaystyle x}
{\displaystyle |x-2|<1.}
{\displaystyle \epsilon >0}
{\displaystyle \delta =\min \left\{1,{\frac {\epsilon }{7}}\right\}}
{\displaystyle |x-c|=|x-2|<\delta ,}
{\displaystyle {\begin{array}{ccrcrclccccl}&&&&|x-2|&<&{\displaystyle {\frac {\epsilon }{7}}}\\\\\Rightarrow &&&&7|x-2|&<&\epsilon \\\\\Rightarrow &&|x+5||x-2|&<&7|x-2|&<&\epsilon &&&&&{\textrm {(using}}\natural )\\\\\Rightarrow &&&&|x^{2}+3x-10|&<&\epsilon \\\\\Rightarrow &&&&|x^{2}+3x+1-11&<&\epsilon \\\\\Rightarrow &&&&|f(x)-L|&<&\epsilon ,\end{array}}}
{\displaystyle {\displaystyle \lim _{x\rightarrow 1}{\sqrt {x}}=1}}
{\displaystyle f(x)={\sqrt {x}},\ L=1}
{\displaystyle c=1}
{\displaystyle |f(x)-L|<\epsilon }
{\displaystyle {\begin{array}{cccrcl}&&&|{\sqrt {x}}-1|&<&{\displaystyle \epsilon }\\\\\Rightarrow &&&|{\sqrt {x}}+1|\cdot |{\sqrt {x}}-1&<&|{\sqrt {x}}+1|\cdot \epsilon \\\\\Rightarrow &&&|x-1|&<&|{\sqrt {x}}+1|\cdot \epsilon .\end{array}}}
{\displaystyle \delta =1.}
{\displaystyle -\delta \,=\,-1\,<\,x-1\,<\,1\,=\,\delta ,}
{\displaystyle 0\,<\,x\,<\,2.}
{\displaystyle 1\,<\,{\sqrt {x}}+1\,<\,{\sqrt {2}}+1.}
{\displaystyle \epsilon }
{\displaystyle \epsilon \,<\,|{\sqrt {x}}+1|\,<\,\left({\sqrt {2}}+1\right)\epsilon \qquad {\mbox{while}}\qquad {\frac {\epsilon }{{\sqrt {2}}+1}}\,<\,{\frac {\epsilon }{|{\sqrt {x}}+1|}}\,<\,{\frac {\epsilon }{1}}.}
{\displaystyle x}
{\displaystyle |x-1|<1=\delta }
{\displaystyle \delta }
{\displaystyle \epsilon >0}
{\displaystyle \delta =\min \left\{1,\epsilon \right\}}
{\displaystyle |x-c|=|x-1|<\delta }
{\displaystyle {\begin{array}{cccrclcc}&&&|x-1|&<&{\displaystyle \epsilon }\\\\\Rightarrow &&&|({\sqrt {x}}+1)({\sqrt {x}}-1)|&<&\epsilon \\\\\Rightarrow &&&\left|{\sqrt {x}}+1\right|\cdot \left|{\sqrt {x}}-1\right|&<&\epsilon \\\\\Rightarrow &&&\left|{\sqrt {x}}-1\right|&<&{\frac {\epsilon }{\left|{\sqrt {x}}+1\right|}}&<&\epsilon ,\end{array}}}
as required. Notice that we used both of the inequalities involving </math>\epsilon</math> and
{\displaystyle \left|{\sqrt {x}}+1\right|}
There are many more difficult examples, but these are meant as an introduction. \end{document}
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{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle 13}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle (n+1)^{\textrm {th}}}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1,}}
{\displaystyle 1}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1,}}
{\displaystyle n=1.}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
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This should extend to henselian pairs, right? This would involve extending theorem 07QY (approximation) to henselian pairs
(R,I)
R
is a noetherian G-ring, and extending (part of) Lemma 07QR to show that the henselization of a noetherian G-ring along any ideal is a G-ring. Am I missing something?
Yes, the statement you suggest sounds correct. Fun! I guess the proof as you sketched it should work too. The proof will use Lemmas 15.50.14 and 15.50.15 and (as you say) the extension of Theorem 15.50.8 to henselian pairs, namely Lemma 16.14.1. I will add this the next time I go through all the comments. Thanks very much!
OK, I added this result but it doesn't have a tag yet. It will in a few days appear as the lemma following this one.
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Patterning PLA Packaging Films for Implantable Medical Devices | J. Med. Devices | ASME Digital Collection
Radheshyam Tewari,
Tewari, R., and Friedrich, C. (June 13, 2011). "Patterning PLA Packaging Films for Implantable Medical Devices." ASME. J. Med. Devices. June 2011; 5(2): 027522. https://doi.org/10.1115/1.3590627
biomedical materials, cochlear implants, embossing, etching, packaging, polymer films
Medical devices, Packaging, Embossing, Etching, Biomaterials, Cochlear implants, Polymer films
Microhot-embossing is a less complex and inexpensive alternative over standard photolithography for patterning poly(lactic acid) (PLA) films. However, direct patterning of discrete or through-thickness microstructures by conventional microhot-embossing is not possible due to embossing-caused residual film. Use of complex modifications in the embossing process can further prohibit its integration with other standard semiconductor fabrication processes. Plasma-based reactive ion etching (RIE) of embossing-caused PLA residual film can be a viable option potentially allowing integration of the conventional hot-embossing process with standard semiconductor fabrication processes. RIE etch-rates of PLA packaging films, hot-embossed with parylene-based thin-film cochlear implant-shaped stiffener structures, were characterized for oxygen
(O2)
, nitrogen
(N2)
, and argon (Ar) plasmas under two different process conditions. The etch-rates of PLA films for
O2
N2
, and Ar plasmas were 0.29–0.72
μm
/min, 0.09–0.14
μm
/min, and 0.11–0.15
μm
/min, respectively. Complete removal of embossing-caused residual film has been demonstrated utilizing the etching results for
O2
plasma. Also, the effect of RIE etching on resultant PLA film surface roughness has been quantified for the three plasmas.
A Packaging Solution for Pressure Sensor MEMS
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Yet another Wordle solver
about • blog • software • rss
Tags: probability programming data
Like everyone and their dog, I've been playing Wordle. And like everyone and their dog, I've written a Wordle solver. Details below.
But first, a quick introduction to the game. If you haven't come across it yet, Wordle is a language puzzle game. You have 6 guesses to find a secret 5-letter word. Every time you make a guess, you are told which letters are in the correct position, which letters are correct but in the wrong position, and which letters are incorrect.
In the example below, our first guess is WORDY. The letters 'R' and 'D' are highlighted in yellow, which means that they're in the secret word somewhere, but not in those positions. The remaining letters ('W', 'O' and 'Y'), are greyed out, which means that they're not in the secret word.
We use that information about the secret word to form our next guess, DRAPE, which tells us that there's an 'E' in the secret word, in addition to the 'R' and 'D'. Finally, we guess ELDER, which turns out to be the correct word. All the letters are highlighted in green, which means they're all in the correct position.
My Wordle solver (let's call it YAWS for short - Yet Another Wordle Solver) is simple. It keeps a list of possible secret words, starting with all 2315 words in Wordle's dictionary. It removes words from the list after each guess, according to the hints it receives. If the solver knows that the secret word begins with 'e', for example, it can remove all words that don't begin with 'e'.
How does YAWS pick the next word to guess? It picks the word that, on average, rules out as many secret words as possible. If we expect there to be an average of 61 possible secret words left after guessing 'raise', then it's a better guess than 'album', which leaves an average of 254 possible secret words, and so YAWS picks 'raise' over 'album'.
Very Slightly More Formally, let's say
S
is the set of possible secret words,
G
is the set of allowed guesses (of which there are around 12 thousand),
X
is the actual secret word (which is random), and
r(g, X, S)
is the new set of possible secret words after we guess the word
g
r
X
S
g
, since it depends both on what possibilities are left and on the actual secret word.
The next word picked by YAWS is given by
\text{argmin}_{g \in G} E[\lvert r(g, X, S) \rvert],
which is just the original explanation in (crappy) maths notation. We minimise (using argmin) the expected size (E[|...|]) of the new set of secret words.
The expected size can be expressed as
E[\lvert r(g,X,S) \rvert] = \sum_{x \in S} P(X=x) \,\lvert r(g,x,S) \rvert,
where each secret is equally probable, i.e.
P(X=x)=\frac{1}{\lvert S \rvert}
The final piece of the puzzle is to compute the new secret set
r(g, x, S)
. If we have a hint function
h
that accepts a guess and a secret word, and returns a string representing the hint, such as "GYBBG" (green, yellow, black, black, green), then the new secret set is
r(g,x,S) = \{y \in S : h(g,x)=h(g,y)\}.
In other words, we only keep secret words that return the same hint as the true secret word.
This is great! We've fully specified how YAWS greedily picks the best next guess. The only problem is complexity. Let's say that the size of
G
and the size of
S
are both proportional to a number
n
. YAWS has to loop over all possible guesses, which is
\mathcal{O}(n)
complexity. For each guess, it has to check all possible secrets, which is also
\mathcal{O}(n)
complexity. And for each secret, it has to loop over all the other secrets to figure out which ones will be in the new set of secrets, which is - you guessed it -
\mathcal{O}(n)
complexity. Overall, that makes it an
\mathcal{O}(n^3)
algorithm, which is verrrrryyyy slow for Wordle's
n \approx 10000
A better approach is to group the secrets based on the hint they will return for a given guess
g
. Then we don't need to loop over the secrets a third time, because we already know which secrets give the same hint. This makes the complexity
O(n^2)
. Here's the improved algorithm in Python pseudocode.
def expected_size(g, S):
buckets = collections.defaultdict(list)
buckets[h(g,x)].append(x)
return sum(1/len(S) * len(buckets[h(g,x)]) for x in S)
If that handwavey explanation didn't lose you completely, here are the results! This is the number of guesses made by YAWS when it's tested on all possible secret words, with its first guess being RAISE. I've compared it to my first 28 Wordle scores.
YAWS gets a score of 3 much more consistently than I do, and it never makes more than 5 guesses. It makes an average (mean) of 3.495 guesses, which is pretty close to the best possible average (3.4212). Not bad for a greedy algorithm. For a description of the optimal Wordle algorithm, I would recommend checking out this write-up, which is where I nicked some of the maths notation that I used above.
One last thing. Here's a graph of letter frequency among the Wordle secret words. ETAOIN SHRDLU is often used as a mnemonic to remember the most commonly used letters in the English language, but for Wordle it looks like that should be EAROT LISNCU.
I'd be happy to hear from you at galligankevinp@gmail.com.
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Find the area of a regular polygon with
100
sides and with a perimeter of
100
A triangle, with top vertex labeled, 3.6 degrees. Find the measure of a central angle.
\text{Central angle}=\frac{360^{\large\circ}}{100}=3.6^{\large\circ}
Determine the length of the sides, which will be the length of the base of the triangles.
P=100
\frac{100 \text{ units}}{100 \text{ sides}}=1
unit per side Label, of 1, added to bottom side of triangle.
Added: A dashed line, labeled h, from the top vertex, perpendicular to the bottom side. Use the tan trig ratio to determine the height of the triangle. Use this and the length of the base to find the area of the triangle.
\frac{3.6^{\large\circ}}{2}=1.8^{\large\circ}
\tan1.8^{\large\circ} =\frac{\text{opp}}{\text{adj}}=\frac{0.5}{h}
h=\frac{0.5}{\text{tan}1.8^{\large\circ}}\approx 15.9103
A=\frac{1}{2}(1)(15.9103)\approx 7.9551 \text{ units}^{2}
for each triangle.
Use this area to determine the area of the entire polygon.
\text{Area}≈ (100)(7.9551)
≈ 795.51\text{ square units}
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Home : Support : Online Help : Programming : Data Types : Type Checking : Types : package
check for a Maple package
type(expr, 'package')
The call type(expr, 'package') checks if expr is a Maple package. A Maple package is a module that has option package. For information on Maple packages, see module,package.
Not all modules are packages. Package semantics differ from module semantics in two ways. First, (module-based) package exports are automatically protected. Second, packages can be used as the first argument to with.
For historical reasons, type package also recognizes Maple tables and procedures with option package as packages. (The use of tables and procedures to implement packages in Maple is deprecated.)
\mathrm{type}\left(\mathrm{foo},'\mathrm{package}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left(\mathrm{plots},'\mathrm{package}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\mathrm{Statistics},'\mathrm{package}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\mathrm{LinearAlgebra},'\mathrm{package}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
Q≔\mathbf{module}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{_export}\left(e,f\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}e≔x→2*x-1;\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}f≔x→\mathrm{sin}\left(x\right)/\mathrm{cos}\left(x-1\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end module}:
\mathrm{type}\left(Q,'\mathrm{package}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
P≔\mathbf{module}\left(\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{option}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{package};\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathrm{_export}\left(e,f\right);\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}e≔x→2*x-1;\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}f≔x→\mathrm{sin}\left(x\right)/\mathrm{cos}\left(x-1\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{end module}:
\mathrm{type}\left(P,'\mathrm{package}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(\mathrm{Record}\left(a=2,b=3\right),'\mathrm{package}'\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
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Numbers with special prime factorization
Demonstration, with Cuisenaire rods, of the number 72 being powerful
An Achilles number is a number that is powerful but not a perfect power.[1] A positive integer n is a powerful number if, for every prime factor p of n, p2 is also a divisor. In other words, every prime factor appears at least squared in the factorization. All Achilles numbers are powerful. However, not all powerful numbers are Achilles numbers: only those that cannot be represented as mk, where m and k are positive integers greater than 1.
Achilles numbers were named by Henry Bottomley after Achilles, a hero of the Trojan war, who was also powerful but imperfect. Strong Achilles numbers are Achilles numbers whose Euler totients are also Achilles numbers.[2]
Sequence of Achilles numbers
A number n = p1a1p2a2…pkak is powerful if min(a1, a2, …, ak) ≥ 2. If in addition gcd(a1, a2, …, ak) = 1 the number is an Achilles number.
The Achilles numbers up to 5000 are:
72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000 (sequence A052486 in the OEIS).
The smallest pair of consecutive Achilles numbers is:[3]
108 is a powerful number. Its prime factorization is 22 · 33, and thus its prime factors are 2 and 3. Both 22 = 4 and 32 = 9 are divisors of 108. However, 108 cannot be represented as mk, where m and k are positive integers greater than 1, so 108 is an Achilles number.
Finally, 784 is not an Achilles number. It is a powerful number, because not only are 2 and 7 its only prime factors, but also 22 = 4 and 72 = 49 are divisors of it. Nonetheless, it is a perfect power:
{\displaystyle 784=2^{4}\cdot 7^{2}=(2^{2})^{2}\cdot 7^{2}=(2^{2}\cdot 7)^{2}=28^{2}.\,}
So it is not an Achilles number.
500 = 22 × 53 is a strong Achilles number as its Euler totient of 200 = 23 × 52 is also an Achilles number.
^ Weisstein, Eric W. "Achilles Number". MathWorld.
^ "Problem 302 - Project Euler". projecteuler.net.
^ Carlos Rivera, The Prime Puzzles and Problem Connection, Problem 53
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How to calculate parallel resistance
Other uses of the parallel resistor calculator
The parallel resistor calculator is a tool for determining the equivalent resistance of a circuit with up to five resistors in parallel. Read on or jump to the series resistor calculator.
Prefer watching over reading? Learn all you need in 90 seconds with this video we made for you:
A parallel circuit is characterized by a common potential difference (voltage) across the ends of all resistors. The equivalent resistance for this kind of circuit is calculated according to the following formula:
1/R = 1/R_1 + 1/R_2 + ... + 1/R_n
R
is the equivalent parallel resistance
R_1, R₂, ..., R_n
are the resistances of individual resistors numbered
1, 2, ..., n
The units of all values are Ohms (symbol: Ω). 1 Ohm is defined as electrical resistance between two points that, when applied with a potential difference of 1 volt, produces a current of 1 ampere. Hence, 1 Ω = 1 V / 1 A or, in SI base units, Ω = kg·m²/(s³·A²).
The formula for resistors in parallel is similar to the formula for inductors in parallel.
The parallel resistor calculator has two different modes. The first mode allows you to calculate the total resistance equivalent to a group of individual resistors in parallel. In contrast, the second mode allows you to set the desired total resistance of the bunch and calculate the one missing resistor value, given the rest.
To keep it simple, we only show you a few rows to input numbers, but new fields will magically appear as you need them. You can input up to 10 resistors in total.
Let's look at an example for the second, slightly more complicated, mode:
Select Calculate missing resistor under Mode.
Now input the total resistance you want your circuit/collection of resistors to have.
Start by introducing the values of the resistors you already know (new fields will appear as needed).
The calculator automatically gives you the required missing resistor after each input.
The principle is the same as when determining capacitance in series or induction in parallel – you can use it for these calculations too. Just remember that the units are not the same!
If you would like to find out the value of power dissipated in the resistor, try the Ohm's law calculator or Resistor wattage calculator.
How do you calculate two resistors in parallel?
Take their reciprocal values, add the two together and take the reciprocal again. For example, if one resistor is 2 Ω and the other is 4 Ω, then the calculation to find the equivalent resistance is 1 / (1/2 + 1/4) = 1 / (3/4) = 4/3 = 1.33.
Is the voltage the same in a parallel circuit?
Yes, the voltage across all the components is the same in a parallel circuit.
Why does resistance decrease in parallel?
This phenomenon happens because the current has many more paths that it could take. Imagine a shop opens up several new check-out tills. The overall resistance to people going through the check-out will decrease as the workload is shared in parallel.
How do you find an unknown resistor in a parallel circuit?
Rearrange the parallel resistor formula 1/R = 1/R₁ + 1/R₂ + ... + 1/Rn in terms of Rn, given that you know the desired overall resistance. That gives you Rn = (1/R - 1/R₁ + 1/R₂ + ...)-1.
For example, if you have R1 = 4, R2 = 2 and want R = 1, then R4 = 1 / (1 - 1/4 - 1/2) = 4 Ω.
Calculate equivalent resistance
You can add up to 10 resistors, fields will appear as you need them.
The Stokes' law calculator will help you determine either the viscosity of a fluid or the terminal velocity of a particle in a falling ball viscometer.
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An Experimental Investigation of the Effects of the Environmental Conditions and the Channel Depth for an Air-Breathing Polymer Electrolyte Membrane Fuel Cell | J. Electrochem. En. Conv. Stor | ASME Digital Collection
An Experimental Investigation of the Effects of the Environmental Conditions and the Channel Depth for an Air-Breathing Polymer Electrolyte Membrane Fuel Cell
Yong Hun Park,
Yong Hun Park
, 762 Peach Creek Cut Off Road, College Station, TX 77845
E3 (Engines, Emissions, Energy) Research Laboratory, Department of Mechanical Engineering,
Park, Y. H., and Caton, J. A. (September 11, 2008). "An Experimental Investigation of the Effects of the Environmental Conditions and the Channel Depth for an Air-Breathing Polymer Electrolyte Membrane Fuel Cell." ASME. J. Fuel Cell Sci. Technol. November 2008; 5(4): 041016. https://doi.org/10.1115/1.2971196
The effects of the environmental conditions and the channel depth for an air-breathing polymer electrolyte membrane fuel cell were investigated experimentally. The fuel cell used in this work included a membrane and electrode assembly, which possessed an active area of
25 cm2
with Nafion® 117 membrane. Triple serpentine designs for the flow fields with two different flow depths were used in this research. The experimental results indicated that the relative humidity and temperature play an important role with respect to fuel cell performance. The fuel cell needs to be operated at least 20 min to obtain stable performance. When the shallow flow field was used, the performance increased dramatically for low humidity and slightly for high humidity. The current density was obtained around only
120 mA/cm2
30°C
with an 80% relative humidity, which was nearly double the performance for the deep flow field. The minimum operating temperature for an air-breathing fuel cell would be
20°C
. When it was
10°C
at 60% relative humidity, the open circuit voltage dropped to around 0.65 V. The fuel cell performance improved with increasing relative humidity from 80% to 100% at high current density.
current density, electrochemical electrodes, humidity, proton exchange membrane fuel cells
Current density, Flow (Dynamics), Fuel cells, Proton exchange membrane fuel cells, Temperature, Circuits, Electrodes, Power density
Fuel Cells: Theory and Practice
Annual Feed Air Breathing Fuel Cell Stack
,” U.S. Patent Nos. 5,514,486 and 5,595,834.
Performance of Polymer Electrolyte Membrane Fuel Cell (PEMFC) Stacks: Part I: Evaluation and Simulation of an Air-Breathing PEMFC Stack
Development of a PEM Fuel Cell Powered Portable Field Generator for the Dismounted Soldier
The Performance of PEM Fuel Cells Fed With Oxygen Through the Free-Convection Mode
Three-Dimensional Modeling and Experimental Investigation for an Air-Breathing Polymer Electrolyte Membrane Fuel Cell (PEMFC)
Three-Dimensional Analysis for Effect of Channel Configuration on the Performance of a Small Air-Breathing Proton Exchange Membrane Fuel Cell (PEMFC)
Lundlad
Adhesive Copper Films for an Air-Breathing Polymer Electrolyte Fuel Cell
Fabrication of Novel MEMS-Based Polymer Electrolyte Fuel Cell Architectures With Catalytic Electrodes Supported on Porous SiO2
Mass Transport in Polymer Electrolyte Membrane Fuel Cells Using Natural Convection for Air Supply
,” Ph.D. thesis, Helsinki University of Technology, Espoo, Finland.
Arbin Spring, 2006,
, College Station, TX, Newsletter available at http://www.arbin.com/Download/newsletter/1Q-2006-Newsletter.pdfhttp://www.arbin.com/Download/newsletter/1Q-2006-Newsletter.pdf
, 2006, Arbin-006 MITS Pro-FCTS 5.0–FCTS User Manual,
, available at http://www.fuelcelltestingsystem.com/Download/products/MITSPro_FCT.pdfhttp://www.fuelcelltestingsystem.com/Download/products/MITSPro_FCT.pdf
Experimental Investigations of the Anode Flow Field of a Micro Direct Methanol Fuel Cell
Effect of Gas Channel Depth on Current Density Distribution of Polymer Electrolyte Fuel Cell by Numerical Analysis Including Gas Flow through Gas Diffusion Layer
Effect of Flow Pattern of Gas and Cooling Water on Relative Humidity Distribution in Polymer Electrolyte Fuel Eell
Analysis of Single PEM Fuel Cell Performances Based on Current Density Distribution Measurement
Direct Measurement of the Local Current Density Under Land-Channel Areas With Partially-Catalyzed MEAs
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Micro-Mechanical Deformation Analysis of Surface Laminar Circuit in Organic Flip-Chip Package: An Experimental Study | J. Electron. Packag. | ASME Digital Collection
B. Han, Mem. ASME,
B. Han, Mem. ASME
Department of Mechanical Engineering, University of Maryland, College Park, MD 20742
P. Kunthong
Mechanical Engineering Department, Clemson University, Clemson, SC 29634
Contributed by the Electrical and Electronic Packaging Division and presented at the 11th Symposium on Mechanics of Surface Mount Assembly, ASME International Mechanical Engineering Congress and Exposition, Nashville, TN, November 14–19, 1999. Revised manuscript received May 15, 2000. Associate Technical Editor: B. Courtois.
Han, B., and Kunthong, P. (May 15, 2000). "Micro-Mechanical Deformation Analysis of Surface Laminar Circuit in Organic Flip-Chip Package: An Experimental Study ." ASME. J. Electron. Packag. December 2000; 122(4): 294–300. https://doi.org/10.1115/1.1290000
Thermo-mechanical deformations of microstructures in a surface laminar circuit (SLC) substrate are quantified by microscopic moire´ interferometry. Two specimens are analyzed; a bare SLC substrate and a flip chip package assembly. The specimens are subjected to a uniform thermal loading of
ΔT=−70°C
and the microscopic displacement fields are documented at the identical region of interest. The nano-scale displacement sensitivity and the microscopic spatial resolution obtained from the experiments provide a faithful account of the complex deformation of the surface laminar layer and the embedded microstructures. The displacement fields are analyzed to produce the deformed configuration of the surface laminar layer and the strain distributions in the microstructures. The high modulus of underfill produces a strong coupling between the chip and the surface laminar layer, which produces a DNP-dependent shear deformation of the layer. The effect of the underfill on the deformation of the microstructures is investigated and its implications on the package reliability are discussed. [S1043-7398(00)01304-9]
plastic packaging, flip-chip devices, micromechanics, circuit reliability, shear deformation, thermal stresses, light interferometry, Surface Laminar Circuit, Flip Chip, Underfill, Microscopic Moire´ Interferometry, Photosensitive Dielectric Layer, Solder Resist Mask, Metal Via
Circuits, Deformation, Displacement, Flip-chip devices, Flip-chip packages, Interferometry, Flip-chip assemblies, Metals, Shear deformation, Reliability, Flip-chip, Solder masks, Manufacturing, Shear (Mechanics)
Darveaux, R., and Banerji, K., 1991, “Fatigue Analysis of Flip Chip Assemblies Using Thermal Stress Simulations and a Coffin-Manson Relation,” Proceedings of the 41st Electronic Components and Technology Conference, pp. 797–805.
Tsukada, Y., Mashimoto, Y., Nishio, T., and Mii, N., 1992, “Reliability and Stress of Encapsulated Flip Chip Joint on Epoxy Base Printed Circuit Board,” Proceedings of the Advances in Electronic Packaging, ASME, pp. 827–835.
Powell, D., and Trivedi, A., 1993, “Flip-Chip on FR4 Integrated Circuit Package,” Proceedings of the 43rd Electronic Components and Technology Conference, pp. 182–186.
Semiconductor Industry Association, 1997, The National Technology Roadmap for Semiconductors, SIA.
Tsukada, Y., 1994, “Chapter 9: Solder Bumped Flip Chip Attach on SLC Board and Multichip Module,” Chip on Board, Lau, J. H., ed., Van Nostrand Reinhold, pp. 410–443.
High Sensitivity Moire´ Interferometry for Micromechanics Studies
Post, D., Han, B., and Ifju, P., 1994, High Sensitivity Moire´: Experimental Analysis for Mechanics and Materials, Springer-Verlag, NY.
Immersion Interferometer for Microscopic Moire´ Interferometry
Han, B., Guo, Y., and Caletka, D., 1995, “On the Effect of Moire´ Specimen Preparation on Solder Ball Strains of Ball Grid Array Package Assembly,” Proceedings of the 1995 SEM Spring Conference on Experimental Mechanics, Grand Rapid, MI.
Verma, K., Kunthong, P., and Han, B., 1998, “Thermo-Mechanical Strains of Flip-Chip Solder Bumps on Organic PCB Substrate,” Proceedings of the International Mechanical Engineering Congress and Exposition, ASME Winter Meeting, Anaheim, CA.
Solder Ball Connect (SBC) Assemblies Under Thermal Loading: I. Deformation Measurement via Moire´ Interferometry, and its Interpretation
Deformation Mechanism of Two-Phase Solder Column Interconnections Under Highly Accelerated Thermal Cycling Condition: An Experimental Study
Hsisa
T. P., J. M.
Enhancement of Flip-Chip Fatigue Life by Encapsulation
Encapsultants Used in Flip-Chip Packages
Encapsulant for Fatigue Life Enhancement of Controlled-Collapse Chip Connection (C4)
Zheng, W. C., Harren, S. V., and Skipor, A. F., 1994, “Thermo-Mechanical Analysis of Flip-Chip on Board Electronic Packaging Assembly,” Presented at the 1994 International Mechanical Engineering Congress & Exhibition, ASME, Chicago, IL, Nov. 6–11.
Han, B., Chopra, M., Park, S.-B., Li, L., and Verma, K., 1996, “Effect of Substrate CTE on Solder Ball Reliability of Flip Chip Ball Grid Array Package Assembly,” Proceedings of the 1996 Surface Mount International Conference, SMT, San Jose, CA, pp. 43–52.
Hecht, E., 1987, Optics, Addison-Wesley, Reading, MA.
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Phasor approach to fluorescence lifetime and spectral imaging – Violet-Wall INFO
Phasor approach refers to a method which is used for vectorial representation of sinusoidal waves like alternative currents and voltages or electromagnetic waves. The amplitude and the phase of the waveform is transformed into a vector where the phase is translated to the angle between the phasor vector and X axis and the amplitude is translated to vector length or magnitude. In this concept the representation and the analysis becomes very simple and the addition of two wave forms is realized by their vectorial summation.
Sinusoidal wave with phase of φ.
Vectorial representation of waves and their superposition.
In Fluorescence lifetime and spectral imaging, phasor can be used to visualize the spectra and decay curves.[1][2] In this method the Fourier transformation of the spectrum or decay curve is calculated and the resulted complex number is plotted on a 2D plot where the X axis represents the Real component and the Y axis represents the Imaginary component. This facilitate the analysis since each spectrum and decay is transformed into a unique position on the phasor plot which depends on its spectral width or emission maximum or to its average lifetime. The most important feature of this analysis is that it is fast and it provides a graphical representation of the measured curve.
. . . Phasor approach to fluorescence lifetime and spectral imaging . . .
If we have decay curve which is represented by an exponential function with lifetime of τ:
{displaystyle d(t)={d_{0}{e}^{-t/tau }}}
Temporal phasor for decay curves with different lifetimes.
Then the Fourier transformation at frequency ω of
{displaystyle d(t)}
(normalized to have area under the curve 1) is represented by the Lorentz function:
{displaystyle D(omega )={frac {1}{1+jomega tau }}={frac {1}{1+jomega tau }}{frac {1-jomega tau }{1-jomega tau }}={frac {1-jomega tau }{1+(omega tau )^{2}}}={frac {1}{1+(omega tau )^{2}}}-j{frac {omega tau }{1+(omega tau )^{2}}}}
This is a complex function and drawing the Imaginary versus real part of this function for all possible lifetimes will be a semicircle where the zero lifetime is located at (1,0) and the infinite lifetime located at (0,0). By changing the lifetime from zero to infinity the phasor point moves along a semicircle from (1,0) to (0,0). This suggest that by taking the Fourier transformation of a measured decay curve and mapping the result on the phasor plot the lifetime can be estimated from the position of the phasor on the semicircle.
Explicitly, the lifetime can be measured from the magnitude of the phasor as follow:
{displaystyle tau ={frac {1}{omega }}{frac {operatorname {Im} D(omega )}{operatorname {Re} D(omega )}}}
This is a very fast approach compared to methods where they use fitting to estimate the lifetime.
The intensity,phasor and lifetime image of cells transfected with Alexa 488 and Alexa 555.
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1 School of Economy and Management, China University of Geosciences, Wuhan, China.
2 School of Business, Central South University, Changsha, China.
Abstract: Influenza is a kind of infectious disease, which spreads quickly and widely. The outbreak of influenza has brought huge losses to society. In this paper, four major categories of flu keywords, “prevention phase”, “symptom phase”, “treatment phase”, and “commonly-used phrase” were set. Python web crawler was used to obtain relevant influenza data from the National Influenza Center’s influenza surveillance weekly report and Baidu Index. The establishment of support vector regression (SVR), least absolute shrinkage and selection operator (LASSO), convolutional neural networks (CNN) prediction models through machine learning, took into account the seasonal characteristics of the influenza, also established the time series model (ARMA). The results show that, it is feasible to predict influenza based on web search data. Machine learning shows a certain forecast effect in the prediction of influenza based on web search data. In the future, it will have certain reference value in influenza prediction. The ARMA(3,0) model predicts better results and has greater generalization. Finally, the lack of research in this paper and future research directions are given.
Keywords: Data Mining, Web Search, Machine Learning, Baidu Index, Influenza Prediction
D=\left({x}^{\left(1\right)},{x}^{\left(2\right)},\cdots ,{x}^{\left(m\right)}\right)
{n}^{\prime }
{D}^{\prime }
{x}^{\left(i\right)}={x}^{\left(i\right)}-\frac{1}{m}{\sum }_{j=1}^{m}{x}^{\left(j\right)}
{n}^{\prime }
\left({w}_{1},{w}_{2},\cdots ,{w}_{{n}^{\prime }}\right)
{x}^{\left(i\right)}
{z}^{\left(i\right)}={W}^{\text{T}}{x}^{\left(i\right)}
{D}^{\prime }=\left({z}^{\left(1\right)},{z}^{\left(2\right)},\cdots ,{z}^{\left(m\right)}\right)
\left({x}_{i},{y}_{i}\right)
\left(i=1,2,\cdots ,l\right)
f\left(x,\omega \right)=\left(\omega \cdot x\right)+b
\epsilon
\underset{\alpha }{\mathrm{min}}\Phi \left(w\right)=\frac{1}{2}{‖w‖}^{2}
\text{s}\text{.t}\text{.}\left\{\begin{array}{l}\left(w\cdot {x}_{i}+b\right)-{y}_{i}\le \epsilon ,i=1,2,\cdots ,l\\ {y}_{i}-\left(w\cdot {x}_{i}+b\right)\le \epsilon ,i=1,2,\cdots ,l\end{array}
{\xi }_{i}
{\xi }_{i}^{\text{*}}
\underset{a,{\xi }_{i},{\xi }_{i}^{*},b}{\mathrm{min}}\frac{1}{2}{‖w‖}^{2}+C\ast \frac{1}{l}{\sum }_{i=1}^{l}\left({\xi }_{i}+{\xi }_{i}^{*}\right)
\text{s}\text{.t}\text{.}\left\{\begin{array}{l}\left(w\cdot {x}_{i}+b\right)-{y}_{i}\le \epsilon +{\xi }_{i},i=1,2,\cdots ,l\\ {y}_{i}-\left(w\cdot {x}_{i}+b\right)\le \epsilon +{\xi }_{i}^{\text{*}},i=1,2,\cdots ,l\\ {\xi }_{i}\ge 0,{\xi }_{i}^{\text{*}}\ge 0,i=1,2,\cdots ,l\end{array}
w={\sum }_{i=1}^{l}\left({\alpha }_{i}^{\text{*}}-{\alpha }_{i}\right){x}_{i}
f\left(x\right)={\sum }_{i=1}^{l}\left({\alpha }_{i}^{\text{*}}-{\alpha }_{i}\right)\left({x}_{i}\cdot x\right)+b
\left({x}_{i}\cdot x\right)
{x}_{i}
J\left(w\right)=\underset{m}{\mathrm{min}}\left\{\frac{1}{2N}{‖{X}^{\text{T}}w-y‖}_{2}^{2}+\alpha {‖w‖}_{1}\right\}
{Y}_{t}=F\left({Y}_{t-1},{Y}_{t-2},\cdots ,{u}_{t}\right)
Cite this paper: Bu, Y. , Bai, J. , Chen, Z. , Guo, M. and Yang, F. (2018) The Study on China’s Flu Prediction Model Based on Web Search Data. Journal of Data Analysis and Information Processing, 6, 79-92. doi: 10.4236/jdaip.2018.63006.
[1] Huang, L.R., Yuan, L., Li, R.C., et al. (2011) Research on Safety and Immunogenicity of Domestic Influenza Virus Split Vaccine. Fifth National Symposium on Immunodiagnosis and Vaccine, Yinchuan, 1 August 2011, 298-301. (In Chinese)
[2] WHO (2014) Seasonal Influenza.
[3] Brady, R.C. (2010) Influenza. Adolescent Medicine State of the Art Reviews, 21, 236-250.
[4] Shimao, T. (2009) Spanish Flu Related Data. Kekkaku, 84, 685-689.
[5] Centers for Disease Control and Prevention (CDC) (2010) Influenza Activity-United States and Worldwide, June 13-September 25, 2010. Morbidity and Mortality Weekly Report, 59, 1270-1273.
[6] Dijk, A.V., Aramini, J., Edge, G. and Moore, K.M. (2009) Real-Time Surveillance for Respiratory Disease Outbreaks, Ontario, Canada. Emerging Infectious Diseases, 15, 799-801.
[7] Li, X.T., Liu, F., Dong, J.C.H., et al. (2013) Chinese Influenza Surveillance Based on Internet Search Data. Systems Engineering Theory and Practice, 33, 3028-3034. (In Chinese)
[8] FOXS (2016) Online Health Search 2006.
http://www.Pewinternet.Org/2006/10/29/online-health-search-2006/
[9] Polgreen, P., Chen, Y.D. and Nelson, F. (2008) Using Internet Searches for Influenza Surveillance. Clinical Infectious Diseases, 47, 1443-1448.
[10] Ginsberg, J., Mohebbi, M.H., Patel, R.S., Brammer, L., Smolinski, M.S. and Brilliant, L. (2009) Detecting Influenza Epidemics Using Search Engine Query Data. Nature, 457, 1012.
[11] Wang, R.J. (2016) Comparison and Optimization of Influenza Alerting Models Based on Internet Search Data. Doctoral Dissertation, Nankai University, Tianjing. (In Chinese)
[12] Yuan, Q., Nsoesie, E.O., Lv, B., et al. (2013) Monitoring Influenza Epidemics in China with Search Query from Baidu. PLoS ONE, 8, e64323.
[13] Lu, L., Zou, Y.Q., Peng, Y.S., et al. (2016) Comparative Analysis of Baidu Index and Micro Index in Chinese Influenza Surveillance. Computer Applied Research, 33, 392-395. (In Chinese)
[14] Collier, N. (2011) Omg u Got Flu? Analysis of Shared Health Messages for Bio-Surveillance. Journal of Biomedical Semantics, 2, S9.
[15] Lampos, V., Bie, T.D. and Cristianini, N. (2010) Flu Detector-Tracking Epidemics on Twitter. In: European Conference on Machine Learning and Knowledge Discovery in Databases, Vol. 6323, Springer-Verlag, Berlin, 599-602.
[16] Xie, Y., Chen, Z., Cheng, Y., Zhang, K., Agrawal, A., Liao, W.K., et al. (2013) Detecting and Tracking Disease Outbreaks by Mining Social Media Data. Proceedings of the 23rd International Joint Conference on Artificial Intelligence, Beijing, 3-9 August 2013, 2958-2960.
[17] Huang, J., Zhao, H. and Zhang, J. (2013) Detecting Flu Transmission by Social Sensor in China. IEEE International Conference on Green Computing and Communications and IEEE Internet of Things and IEEE Cyber, Physical and Social Computing, Beijing, 20-23 August 2013, 1242-1247.
[18] Lu, L. (2015) Prediction of Chinese Flu Trends Based on Internet Data. Doctoral Dissertation, Hunan University, Changsha. (In Chinese)
[19] Chinese Center for Disease Control and Prevention (2005) National Implementation Plan for Influenza/Human Bird Flu Surveillance. (In Chinese)
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The Effect of Film Thickness on Initial Friction of Elastic-Plastically Rough Surface With a Soft Thin Metallic Film | J. Tribol. | ASME Digital Collection
Department of Mechanical and Chemical Engineering, Heriot-Watt University, Edinburgh EH14 4AS, UK
Contributed by the Tribology Division for publication in the ASME JOURNAL OF TRIBOLOGY. Manuscript received by the Tribology Division September 14, 2001; revised manuscript received July 19, 2001. Associate Editor: T. C. Ovaert.
Liu , Z., Neville , A., and Reuben, R. L. (May 31, 2002). "The Effect of Film Thickness on Initial Friction of Elastic-Plastically Rough Surface With a Soft Thin Metallic Film ." ASME. J. Tribol. July 2002; 124(3): 627–636. https://doi.org/10.1115/1.1454103
The effect of a deposited soft thin metallic film on friction properties of a hardened steel substrate has been investigated experimentally and theoretically. The dependency of the film thickness and contact load on the static friction coefficient is presented. The experimental observations show that deformation of the film in contact was plastic, thereby confirming the assumption of the theoretical calculation. The effect of the film thickness on the contact area has been analyzed. A model for calculating the static friction coefficient of contacting rough surfaces in the presence of a soft thin film has been used. Results from the numerical calculations have been compared with the present static friction measurements performed on a pin on plate reciprocating apparatus. The rise in friction that occurs with increasing thickness for very thin films is discussed in detail. The calculated results, which predict the correct trend of the friction behavior from the present experiment, cover an extremely large range of
F/AnE
10−12
10−2,
where three different dependencies of
F/AnE
on the static friction coefficient can be identified. An investigation into the discrepancy between the calculated and experimental values for the static friction coefficient μ suggests that an accurate prediction of the magnitude of μ depends to a great extent on the level of accuracy in measuring the value of the constant ζ, the effective hardness of the film.
rough surfaces, mechanical contact, friction, elastoplasticity, metallic thin films
Film thickness, Friction, Stiction, Stress, Surface roughness, Thin films, Metallic films, Deformation
Bowden, F. P., and Tabor, D., 1954, The Friction and Lubrication of Solids, Oxford University Press, Oxford.
The Friction of Very Thin Solid Film Lubrication on Surface of Finite Roughness
A Contribution to the Theory of Mechanical Wear
El-Sherbiney
The Hertizian Contact of Surfaces Covered With Metallic Films
Friction and Wear of Ion-Plated Soft Metallic Films
An Experimental Study on the Hertzian Contact of Surfaces Covered by Soft Metal Films
Predicting the Friction and Durability of MoS2 Coatings Using a Numerical Contact Model
Initial Wear Rates of Soft Metallic Films
Friction Properties of a Surface Covered with a Soft Metal Film—Part 2: Analysis of Friction Between a Single Protuberance and a Surface
Finite Element Analysis of an Elastic-Plastic Two Layer Half-Space: Normal Contact
On Two Methods of Calculation of the Force of Sticking of an Elastic Sphere to a Rigid Plane
Contact Adhesion Between Solids in Vacuum: II. Deformation and Interfacial Energy
Lozunskli, M. G., 1961, High Temperature Metallography, Pergamon, New York.
Adhesion of Contacting Rough Surfaces in the Presence of Sub-Boundary Lubrication
Liu, Z., Neville, A. and Reuben, R. L., 2001, “Static Friction Modeling in the Presence of Soft Thin Metallic Films,” ASME Paper No. 462(2000)P/gga.
Relating Profile Instrument Measurements of the Functional Performance of Rough Surfaces
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For each of the geometric relationships represented below, write and solve an equation for the given variable. For parts (a) and (b), assume that
C
is the center of the circle. Show all work.
Since both are radii,
5m + 1 = 3m + 9
m = 4
The measure of the central angle is twice the measure of the inscribed angle. So
2(x + 4) = 3x - 9
How are the sides of a right triangle related?
p = 10
What is the total sum of the angles of this polygon?
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In number theory, a branch of mathematics, a Hilbert number is a positive integer of the form 4n + 1 (Flannery & Flannery (2000, p. 35)). The Hilbert numbers were named after David Hilbert. The sequence of Hilbert numbers begins 1, 5, 9, 13, 17, ... (sequence A016813 in the OEIS))
The Hilbert number sequence is the arithmetic sequence with
{\displaystyle a_{1}=1,d=4}
, meaning the Hilbert numbers follow the recurrence relation
{\displaystyle a_{n}=a_{n-1}+4}
The sum of a Hilbert number amount of Hilbert numbers (1 number, 5 numbers, 9 numbers, etc.) is also a Hilbert number.
Hilbert primes
A Hilbert prime is a Hilbert number that is not divisible by a smaller Hilbert number (other than 1). The sequence of Hilbert primes begins
5, 9, 13, 17, 21, 29, 33, 37, 41, 49, ... (sequence A057948 in the OEIS).
A Hilbert prime is not necessarily a prime number; for example, 21 is a composite number since 21 = 3 ⋅ 7. However, 21 is a Hilbert prime since neither 3 nor 7 (the only factors of 21 other than 1 and itself) are Hilbert numbers. It follows from multiplication modulo 4 that a Hilbert prime is either a prime number of the form 4n + 1 (called a Pythagorean prime), or a semiprime of the form (4a + 3) ⋅ (4b + 3).
Flannery, S.; Flannery, D. (2000), In Code: A Mathematical Journey, Profile Books
Weisstein, Eric W. "Hilbert Number". MathWorld.
OEIS sequence A057949 (Numbers with more than one factorization into Hilbert primes)
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Golden rock - The RuneScape Wiki
For the item used to create the Dahmaroc statue, see strange rock.
For other uses, see gold rock (disambiguation).
A golden rock is an item required to build the Statue of Rhiannon in the Shattered Heart Distraction and Diversion. To start receiving golden rocks, one must first toggle the ability by having the required levels and talking to Auron Ithell who is standing by a golden plinth in the Tower of Voices in Prifddinas. Though both rocks in a pair are identical, they do not stack in the bank or inventory.
The golden rocks can be added to the statue collection bag in the same way strange rocks can be added to their respective bag.
Players require a skill level of at least 75 in all the skills listed below before they can start adding pairs of rocks to the plinth. Boosts do not work. Speaking to Auron will give a message in chat saying which skill levels have not been met.
Although they are not restricted to only being found in Prifddinas, it is more likely to find one while skilling in a district of the city that has the Voice of Seren active. The chance to get the second rock for a skill is reduced, however obtaining a golden rock in a different skill will remove this reduction.[1]
The rocks are awarded at random per action, so consider training a skill in a way that many quick actions are performed.
The motherlode maw in Prifddinas can also give a random golden rock. The metamorphic geode guarantees a strange or golden rock that the player does not have yet.
A player can identify what skill a golden rock is by examining it. Also, the different skills' rocks have different appearances. Skilling in a clan citadel does yield golden rocks, along with using the appropriate portable skilling station.
Agility A rock found while practicing agility. Completion of the following courses:
Completing obstacles on the Anachronia Agility Course
Using either of the Surefooted Aura will help with rock collection by eliminating obstacle failure.
Cycling on a Manual Auto-cycle in the Empty Throne Room is the fastest way to obtain golden rocks.
Alternatively, completing the Hefin Agility if the Voice of Seren is active over Hefin will offer good chances at getting rocks.
Construction A rock found while constructing furniture. Working on furniture or any construction activity that yields experience in the Construction skill, including in building mode and creating flatpacks on a workbench.
Protean planks + Flatpack depacker makes this option significantly more AFK
Limestone fireplaces are recommended as limestone bricks can be acquired for 21 coins each from the stonemason in Keldagrim and sawmill operator in Prifddinas, with the latter being in close proximity to a bank chest and the Prifddinas house portal.
Wield the crystal saw made with harmonic dust to double the chance of getting a rock.
Crafting A rock found while crafting. Making gold jewellery (best is to make gold bracelets); making items from leather or dragonhide; making molten glass in a furnace (as well as blowing it into glass objects); throwing clay on a potter's wheel (but not firing clay); making slayer rings; grinding sandstone; making cloth on the loom at a clan citadel; spinning items on a spinning wheel; crafting any type of battlestaff; blowing potion flasks; playing harps in Prifddinas. Crafting silver items will not yield golden rocks.
Creating crystal flasks or playing the harps in Ithell district during Voice of Seren will yield higher golden rock chances.
Divination A rock found while divining. Gathering wisps and converting memories from any wisp colony. Gathering and depositing memory jars in Hall of Memories. Converting Shadow Cores at the Light Rift in Amlodd Clan District.
It is recommended to use the Divine Conversion relic as the chance for a rock is given per memory deposited by the relic.
Protean Memories do work.
Dungeoneering A rock found while spelunking. Completing dungeons in Daemonheim.
Rocks are awarded at the winterface. The amount of experience rewarded does not affect the chance of obtaining a rock, so it's advisable to complete small, complexity 4 floors with a friend for quick rocks (complexity 1 has more key doors, making it more maze-like and more guard doors, both of which take longer than skill doors found on complexity 2-4).
Completing Sinkholes or Elite Dungeons (Temple of Aminishi, Dragonkin Laboratory and Shadow Reef). Golden rocks can be obtained from any monster in elite dungeons, even on story mode.
Farming A rock found while harvesting produce. Harvesting/planting vegetables, hops, herbs, flowers, bushes, fruit trees, and calquat trees; planting seeds or saplings. Raking weeds, mucking out Manure, and turning the Manure mound, as well as checking player owned farm animals.
The fastest way to receive these rocks is to do a standard herb/allotment run while using supercompost on every patch. Have the magic secateurs in the inventory or tool belt when harvesting. If players have completed the hard Falador achievements, it is also advisable to plant white lilies at all flower patches to protect allotment patches.
Harvesting crops from the fruit tree in Meilyr district or the herb or bush patch in Crwys district during the respective Voice of Seren will have higher chances for golden rocks.
Herblore A rock found when making potions. Combining unfinished potions with secondary ingredients, including barbarian mixes; making unfinished potions; or by cleaning herbs
It is highly recommended to either clean cheap herbs, such as tarromin or herbs that yield a profit when cleaned, while having the statue collection bag in inventory. The Herblore rock can be one of the fastest rocks to obtain by using this method.
Magic A rock found while fighting magic users. Killing any monster that uses magic attacks and drops summoning charms.
Melee A rock found while fighting melee warriors. Killing any monster that uses melee attacks and drops summoning charms.
Cows in Burthorpe are recommended. Cows only have 25 life points and respawn extremely quickly.
Mining A rock found while mining.
Mining ore, coal, granite, clay, stone, sandstone, gems, or rune essence
Mining in the Living Rock Caverns (coal and gold)
Mining in Miscellania
Mining in the Lava Flow Mine
Mining anything in the Trahaearn district during the Voice of Seren is extremely fast.
It does not matter what is mined. There is a chance to get a rock at every animation.
Prayer A rock found while performing last rites.
Burying bones or scattering demonic ashes
Burying bones with the bonecrusher
Offering bones or ashes to a player-owned house altar
Offering bones or ashes to the altar at the chaos temple (in the Wilderness)
Teleporting completed urns
Worshipping the ectofuntus
Since bones buried by the bonecrusher count, prayer rocks can be gained at the same time as the combat and slayer rocks. Consuming bones while wearing bone necklaces also works.
The following methods do not yield any golden rocks:
Using cleansing crystals at the Corrupted Seren Stone in the Hefin Cathedral
Ranged A rock found while fighting rangers. Killing any monster that uses ranged attacks and drops summoning charms.
Slayer A rock found while on assignment. Killing monsters on a slayer task. The highest slayer master available is recommended so more monsters are given per assignment.
Slayer contracts from Markus will yield slayer rocks without needing a task. Recommended repeating crawling hands contract and can double up slayer rocks and melee rocks. Using a bonecrusher triples efficiency by including prayer rocks.
Smithing A rock found while smithing. Smelting normal ores, including gold ore and silver ore; smithing items from normal bars; making Cannonballs out of Steel bars; working at the Artisans' Workshop. Smelting corrupted ore, especially during the Voice of Seren, is particularly fast.
It does not matter what is smithed, so if a player is on a budget, it is advisable to smith with cheap materials such as bronze bars or gold ore.
Activities that do not yield rocks: Superheating and Superheat Form curse, smelting ores in the Blast Furnace and protean bar smithing.
Summoning A rock found while crafting familiar pouches. Creating summoning pouches. Unlike other rocks, the chance of a rock is given per pouch and not per action.
For best chances of obtaining the rock per pouch made, it is best to create pouches at the Amlodd obelisk during Voice of Seren. Making spirit terrorbird pouches is recommended with buying raw bird meat packs from Chargurr in Oo'glog.
Woodcutting A rock found while cutting trees. Cutting any wood, ivy, Jadinko Lair roots, overgrown idol, or jungle near Tai Bwo Wannai as well as training at the sawmill.
Chopping trees or ivy in Crwys during Voice of Seren will yield rocks the fastest.
Otherwise, it does not matter which woodcutting option is selected. There is a chance to get a rock every animation.
Golden rocks tab of the statue collection bag interface.
No special equipment is required to collect golden rocks; however the player can talk to Barnabus Hurma (or any of the other archaeologists within Varrock Museum), or Auron Ithell by the golden plinth, to receive a statue collection bag. This bag, if equipped or held in the inventory, will automatically store any statue pieces a player gains from a skill activity (giving the message " You find a golden rock and add it to your bag."). Otherwise, it can be right-clicked to fill with any golden rocks that are in the inventory.
Wearing a skillcape while training its corresponding skill provides a boost to obtaining strange and/or golden rocks. Using crystal tools will double chances of obtaining rocks.
When a player has two corresponding statue pieces in the bag, they may empty the bag and use the rocks on the statue base, or continue to collect further rocks until the bag is full, or anything in between. The experience will be given at any time, until the last rock pair is ready. Thus, if a player needs that experience immediately to level, it can be received immediately. The player may also 'Look-in' the bag to check how many rocks have currently been collected.
If the player loses or destroys the statue collection bag, the contents will also be lost, requiring the player to complete skill tasks to regain the rocks that were lost.
Similar to the Dahmaroc statue, adding pairs of golden rocks gives experience in the related skill. Experience in combat skills may be allocated to Defence or Constitution instead.
{\displaystyle Experience=1.2x^{2}-2.4x+120}
, where x is the skill level. This is 20% more than the experience from Dahmaroc's statue.
form = shf
result = shr
Subsequently, the Stones throw away achievement is no longer time gated.
Strange and Golden Rocks can now be added to their respective statues directly from the Statue collection bag or the Backpack.
Added feedback messaging upon trying to add Rocks without currently possessing any.
Strange and Golden Rocks will now be sent to the Bank if you don't have enough Backpack space.
If your Bank is full, the Rock will either replace the last item received or be lost entirely, depending on the skilling method.
Info-boxes will now be displayed when receiving Strange or Golden Rocks.
The Statue Collection Bag can now be filled via the Bank while it's in the Backpack.
Placeholders for golden rocks are no longer removed when building the statue of Rhiannon.
Metamorphic geodes now give strange/golden rocks for each one opened if all are not owned or the statue is not built.
The skill name has been added to Strange rocks and Golden rocks.
The following items can no longer have alchemy cast on them: Golden rocks.
In the tower, you can activate and add pieces to the golden Shattered Heart statue.
^ ImRubic. TL;DW 453 - Mythbusters. Reddit. 5 October 2019. (Archived from the original on 5 November 2021.) ImRubic: "You are 50% less likely to get the same type of strange rock, however this goes away if you get another strange rock of a different skill."
Retrieved from ‘https://runescape.wiki/w/Golden_rock?oldid=35821966’
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(→Higher-Order Partial Derivatives)
{\displaystyle z=f(x,y)}
{\displaystyle x}
{\displaystyle y}
{\displaystyle {\frac {\partial z}{\partial x}}}
{\displaystyle {\frac {\partial z}{\partial x}}}
{\displaystyle {\frac {\partial z}{\partial x}}=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x,y)-f(x,y)}{\Delta x}}}
{\displaystyle {\frac {\partial z}{\partial y}}=\lim _{\Delta y\to 0}{\frac {f(x,y+\Delta y)-f(x,y)}{\Delta y}}}
{\displaystyle {\frac {\partial z}{\partial x}}}
{\displaystyle f_{x}(x,y)}
{\displaystyle {\frac {\partial z}{\partial y}}}
{\displaystyle f_{y}(x,y)}
{\displaystyle {\frac {\partial z}{\partial x}}}
{\displaystyle {\frac {\partial z}{\partial y}}}
{\displaystyle z=f(x,y)=2x^{2}-4xy}
{\displaystyle {\frac {\partial z}{\partial x}}=4x^{2}-4y}
{\displaystyle {\frac {\partial z}{\partial y}}=-4x}
{\displaystyle z=f(x,y)=x^{2}y^{3}}
{\displaystyle {\frac {\partial z}{\partial x}}=2xy^{3}}
{\displaystyle {\frac {\partial z}{\partial y}}=3x^{2}y^{2}}
{\displaystyle z=f(x,y)=x^{2}e^{x^{2}y}}
{\displaystyle {\frac {\partial z}{\partial x}}=2xe^{x^{2}y}+x^{2}e^{x^{2}y}2xy}
{\displaystyle {\frac {\partial z}{\partial y}}=x^{2}e^{x^{2}y}(x^{2})=x^{4}e^{x^{2}y}}
{\displaystyle {\frac {\partial }{\partial x}}({\frac {\partial f}{\partial x}})={\frac {\partial ^{2}f}{\partial x^{2}}}=f_{xx}}
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Development and Characterization of Ductile Mg∕Y2O3 Nanocomposites | J. Eng. Mater. Technol. | ASME Digital Collection
Mg∕Y2O3
S. F. Hassan,
Hassan, S. F., and Gupta, M. (January 11, 2007). "Development and Characterization of Ductile
Mg∕Y2O3
Nanocomposites." ASME. J. Eng. Mater. Technol. July 2007; 129(3): 462–467. https://doi.org/10.1115/1.2744418
Y2O3
particulates containing ductile magnesium nanocomposites were synthesized using blend-press-sinter powder metallurgy technique followed by hot extrusion. Microstructural characterization of the nanocomposite samples showed fairly uniform reinforcement distribution, good reinforcement-matrix interfacial integrity, significant grain refinement of magnesium matrix with increasing presence of reinforcement, and the presence of minimal porosity. Mechanical properties characterization revealed that the presence of nano-
Y2O3
reinforcement leads to marginal increases in hardness, 0.2% yield strength and ultimate tensile strength, but a significant increase in ductility and work of fracture of magnesium. The fracture mode was changed from brittle for pure Mg to mix ductile and intergranular in the case of nanocomposites.
magnesium, yttrium compounds, particle reinforced composites, nanocomposites, ductility, pressing, powder metallurgy, sintering, extrusion, grain refinement, porosity, hardness, yield strength, fracture, magnesium, yttria, microstructure, tensile properties, ductility, powder metallurgy
Ductility, Magnesium (Metal), Nanocomposites, Powder metallurgy, Porosity, Particulate matter, Fracture (Materials), Tensile strength, Mechanical properties, Extruding, Yield strength
Die-Casting Magnesium, Magnesium-Alloys and Technologies
, Weinham, Germany.
Global Outlook on the Use of Magnesium Die-Casting in Automotive Applications
Proceeding of Conference on Magnesium Alloys and Their Applications
Werkstoff-Informationsge-Sellschaft
, Wolfsburg, Germany, pp.
Fabrication and Characterization of Pure Magnesium-30 vol.% SiCP Particle Composite
Synthesis, Microstructure and Properties Characterization of Disintegrated Melt Deposited Mg∕SiC Composites
Microstructure and Mechanical Properties of Magnesium Containing High Volume Fractions of Yttria Dispersoids
Development of High Strength Magnesium Based Composites Using Elemental Nickel Particulates as Reinforcement
Effect of Volume Fraction and Particle Size on the Microstructure and Plastic Deformation of Mg-Y2O3 Composites
Development of a Ductile Magnesium Composite Materials Using Titanium as Reinforcement
Dispersion-Strengthening of Magnesium by Nanoscaled Ceramic Powder
,” Conference on Magnesium Alloys and Their Applications, April 28–30, Wolfsburg, Germany, pp.
Quasicrystal Strengthened Mg-Zn-Y Alloys by Extrusion
Development of High-Performance Magnesium Nano-Composites Using Solidification Processing Route
Effect of Different Types of Nano-size Oxide Particulates on Microstructural and Mechanical Properties of Elemental Mg
Wettability at High Temperatures
Powder Metal Matrix Composites: Selection and Processing
Effects of Particle Size on Inhibited Grain Growth
Dispersion Hardening of Metals by Nanoscaled Ceramic Powder
ASM Metal Reference Book
, West Latayette IN.
Microstructure and Grain Growth Behavior of an Aluminum Alloy Metal Matrix Composite Processed by Disintegrated Melt Deposition
The Effects of Processing on the Microstructures and Properties of Gr∕Mg Composites
The Influence of Porosity on the Characteristics of Sintered Materials
Mesoplasticity and its Applications
Mechanical Properties of a Mg-10 (vol.%)Ti Composite
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On Paranorm Zweier -Convergent Sequence Spaces
Vakeel A. Khan, Khalid Ebadullah, Ayhan Esi, Nazneen Khan, Mohd Shafiq, "On Paranorm Zweier -Convergent Sequence Spaces", Journal of Mathematics, vol. 2013, Article ID 613501, 6 pages, 2013. https://doi.org/10.1155/2013/613501
Vakeel A. Khan,1 Khalid Ebadullah,1 Ayhan Esi ,2 Nazneen Khan,1 and Mohd Shafiq1
2Department of Mathematics, University of Adiyaman, Altinsehir, 02040 Adiyaman, Turkey
In this paper, we introduce the paranorm Zweier -convergent sequence spaces , , and , a sequence of positive real numbers. We study some topological properties, prove the decomposition theorem, and study some inclusion relations on these spaces.
Let , and be the sets of all natural, real, and complex numbers, respectively. We write the space of all real or complex sequences.
Let , , and denote the Banach spaces of bounded, convergent, and null sequences, respectively, normed by .
The following subspaces of were first introduced and discussed by Maddox [1]: , , , , where is a sequence of strictly positive real numbers.
After that Lascarides [2, 3] defined the following sequence spaces: where , for all .
Each linear subspace of , for example, , is called a sequence space.
A sequence space with linear topology is called a -space provided each map defined by is continuous for all .
A -space is called an -space provided is a complete linear metric space.
An FK-space whose topology is normable is called a BK-space.
Let and be two sequence spaces and an infinite matrix of real or complex numbers , where . Then we say that defines a matrix mapping from to , and we denote it by writing .
If for every sequence the sequence , the transform of is in , where By , we denote the class of matrices such that .
Thus, if and only if series on the right side of (3) converges for each and every .
The approach of constructing the new sequence spaces by means of the matrix domain of a particular limitation method has been recently employed by Altay et al. [4], Başar and Altay [5], Malkowsky [6], Ng and Lee [7], and Wang [8].
Şengönül[9] defined the sequence which is frequently used as the transform of the sequence , that is, where , and denotes the matrix defined by Following Başar and Altay [5], Şengönül[9] introduced the Zweier sequence spaces and as follows: Here we quote below some of the results due to Şengönül [9] which we will need in order to establish the results of this paper.
Theorem 1 (see [9, Theorem 2.1]). The sets and are the linear spaces with the coordinate wise addition and scalar multiplication which are the BK-spaces with the norm
Theorem 2 (see [9, Theorem 2.2]). The sequence spaces and are linearly isomorphic to the spaces and , respectively, that is, and .
Theorem 3 (see [9, Theorem 2.3]). The inclusions strictly hold for .
Theorem 4 (see [9, Theorem 2.6]). is solid.
Theorem 5 (see [9, Theorem 3.6]). is not a solid sequence space.
The concept of statistical convergence was first introduced by Fast [10] and also independently by Buck [11] and Schoenberg [12] for real and complex sequences. Further this concept was studied by Connor [13, 14], Connor et al. [15], and many others. Statistical convergence is a generalization of the usual notion of convergence that parallels the usual theory of convergence. A sequence is said to be statistically convergent to if for a given as The notion of -convergence is a generalization of the statistical convergence. At the initial stage, it was studied by Kostyrko et al. [16]. Later on, it was studied by Šalát et al. [17, 18], Demirci [19], Tripathy and Hazarika [20, 21], and Khan et al. [22–24].
Here we give some preliminaries about the notion of -convergence.
Let X be a nonempty set. Then a family of sets (denoting the power set of ) is said to be an ideal if is additive, that is, , and hereditary, that is, , .
A nonempty family of sets is said to be a filter on if and only if , for we have and for each and implies .
An ideal is called nontrivial if .
A non-trivial ideal is called admissible if .
A non-trivial ideal is maximal if there cannot exist any non-trivial ideal containing as a subset.
For each ideal , there is a filter () corresponding to I. that is, () = , where .
Definition 6. A sequence is said to be -convergent to a number if for every . In this case we write .
The space of all -convergent sequences converging to is given by
Definition 7. A sequence is said to be -null if . In this case we write .
Definition 8. A sequence is said to be -Cauchy if for every there exists a number such that for all .
Definition 9. A sequence is said to be -bounded if there exists such that .
Definition 10. Let be two sequences. We say that for almost all k relative to (a.a.k.r.I), if .
The following lemma will be used for establishing some results of this paper.
Lemma 11. If and . If , then (see [20, 21]) cf. ([17, 18, 20–24]).
Recently Khan and Ebadullah [25] introduced the following classes of sequence spaces: We also denote by
In this paper we introduce the following classes of sequence spaces: We also denote by where , is a sequence of positive real numbers.
Throughout the paper, for the sake of convenience now we will denote by for all .
Theorem 12. The classes of sequences and are linear spaces.
Proof. We shall prove the result for the space .
The proof for the other spaces will follow similarly.
Let , and let be scalars. Then for a given : we have where Let be such that .
Then Thus . Hence . Therefore is a linear space. The rest of the result follows similarly.
Theorem 13. Let . Then and are paranormed spaces, paranormed by where .
Proof. Let .(1)Clearly, if and only if .(2) is obvious.(3)Since and , using Minkowski’s inequality, we have (4)Now for any complex , we have such that , .
Let such that .
Therefore, , where .
Hence as .
Hence is a paranormed space.
The rest of the result follows similarly.
Theorem 14. is a closed subspace of .
Proof. Let be a Cauchy sequence in such that .
Since , then there exists such that We need to show that (1) converges to ,(2)if , then .
(1) Since is a Cauchy sequence in then for a given , there exists such that For a given , we have Then .
We choose , then for each , we have Then is a Cauchy sequence of scalars in , so there exists a scalar such that , as .
(2) Let be given. Then we show that if , then .
Since , then there exists such that which implies that .
The number can be so chosen that together with (23), we have such that .
Since . Then we have a subset of such that , where Let , where .
Therefore for each , we have Then the result follows.
Since the inclusions and are strict, so in view of Theorem 14 we have the following result.
Theorem 15. The spaces and are nowhere dense subsets of .
Theorem 16. The spaces and are not separable.
Let be an infinite subset of of increasing natural numbers such that .
Let Let .
Clearly is uncountable.
Consider the class of open balls .
Let be an open cover of containing .
Since is uncountable, so cannot be reduced to a countable subcover for .
Thus is not separable.
Theorem 17. Let and an admissible ideal. Then the following is equivalent.(a); (b)there exists such that , for a.a.k.r.I;(c)there exists and such that for all and ;(d)there exists a subset of such that and .
Proof. (a) implies (b).
Let . Then there exists such that Let be an increasing sequence with such that Define a sequence as For . Then and form the following inclusion: we get , for a.a.k.r.I.
(b) implies (c).
For . Then there exists such that , for a.a.k.r.I.
Define a sequence as Then and .
(c) implies (d).
Suppose (c) holds.
Let be given.
Let and Then we have .
(d) implies (a).
Then for any , and Lemma 11, we have Thus .
Theorem 18. Let and . Then the following results are equivalent.(a) and .(b) .
Proof. Suppose that and , then the inequalities hold for any and for all .
Therefore the equivalence of (a) and (b) is obvious.
Theorem 19. Let and be two sequences of positive real numbers. Then if and only if , where such that .
Proof. Let and . Then there exists such that , for all sufficiently large .
Since for a given , we have Let Then .
Then for all sufficiently large , Therefore .
The converse part of the result follows obviously.
Proof. The proof follows similarly as the proof of Theorem 19.
Theorem 21. Let and be two sequences of positive real numbers. Then if and only if , and , where such that .
Proof. By combining Theorems 19 and 20, we get the required result.
The authors would like to record their gratitude to the reviewer for his careful reading and making some useful corrections which improved the presentation of this paper.
I. J. Maddox, “Some properties of paranormed sequence spaces,” Journal of the London Mathematical Society, vol. 1, pp. 316–322, 1969. View at: Publisher Site | Google Scholar
C. G. Lascarides, “A study of certain sequence spaces of Maddox and a generalization of a theorem of Iyer,” Pacific Journal of Mathematics, vol. 38, pp. 487–500, 1971. View at: Google Scholar | Zentralblatt MATH
C. G. Lascarides, “On the equivalence of certain sets of sequences,” Indian Journal of Mathematics, vol. 25, no. 1, pp. 41–52, 1983. View at: Google Scholar | Zentralblatt MATH
B. Altay, F. Başar, and M. Mursaleen, “On the Euler sequence spaces which include the spaces
{l}_{p}
{l}_{\infty }
. I,” Information Sciences, vol. 176, no. 10, pp. 1450–1462, 2006. View at: Publisher Site | Google Scholar
p
-bounded variation and related matrix mappings,” Ukrainian Mathematical Journal, vol. 55, no. 1, pp. 136–1147, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
E. Malkowsky, “Recent results in the theory of matrix transformations in sequence spaces,” Matematichki Vesnik, vol. 49, no. 3-4, pp. 187–196, 1997. View at: Google Scholar
P. N. Ng and P. Y. Lee, “Cesàro sequence spaces of non-absolute type,” Commentationes Mathematicae (Prace Matematyczne), vol. 20, no. 2, pp. 429–433, 1978. View at: Google Scholar | Zentralblatt MATH
C. S. Wang, “On Nörlund sequence spaces,” Tamkang Journal of Mathematics, vol. 9, no. 2, pp. 269–274, 1978. View at: Google Scholar
M. Şengönül, “On the Zweier sequence space,” Demonstratio Mathematica, vol. 40, no. 1, pp. 181–196, 2007. View at: Google Scholar | Zentralblatt MATH
H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951. View at: Google Scholar | Zentralblatt MATH
R. C. Buck, “Generalized asymptotic density,” American Journal of Mathematics, vol. 75, pp. 335–346, 1953. View at: Publisher Site | Google Scholar | Zentralblatt MATH
I. J. Schoenberg, “The integrability of certain functions and related summability methods,” The American Mathematical Monthly, vol. 66, pp. 361–375, 1959. View at: Publisher Site | Google Scholar | Zentralblatt MATH
P
-Cesàro convergence of sequences,” Analysis, vol. 8, no. 1-2, pp. 47–63, 1988. View at: Google Scholar | Zentralblatt MATH
J. S. Connor, “On strong matrix summability with respect to a modulus and statistical convergence,” Canadian Mathematical Bulletin, vol. 32, no. 2, pp. 194–198, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH
J. Connor, J. Fridy, and J. Kline, “Statistically pre-Cauchy sequences,” Analysis, vol. 14, no. 4, pp. 311–317, 1994. View at: Google Scholar | Zentralblatt MATH
I
-convergence,” Real Analysis Exchange, vol. 26, no. 2, pp. 669–686, 2000. View at: Google Scholar
T. Šalát, B. C. Tripathy, and M. Ziman, “On some properties of
I
-convergence,” Tatra Mountains Mathematical Publications, vol. 28, no. 2, pp. 279–286, 2004. View at: Google Scholar | Zentralblatt MATH
T. Šalát, B. C. Tripathy, and M. Ziman, “On
I
-convergence field,” Italian Journal of Pure and Applied Mathematics, no. 17, pp. 45–54, 2005. View at: Google Scholar | Zentralblatt MATH
K. Demirci, “
I
-limit superior and limit inferior,” Mathematical Communications, vol. 6, no. 2, pp. 165–172, 2001. View at: Google Scholar | Zentralblatt MATH
B. C. Tripathy and B. Hazarika, “Paranorm
I
-Convergent sequence spaces,” Mathematica Slovaca, vol. 59, no. 4, pp. 485–494, 2009. View at: Publisher Site | Google Scholar
B. C. Tripathy and B. Hazarika, “Some
I
-Convergent sequence spaces defined by Orlicz functions,” Acta Mathematicae Applicatae Sinica, vol. 27, no. 1, pp. 149–154, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
V. A. Khan and K. Ebadullah, “On some I-Convergent sequence spaces defined by a modulus function,” Theory and Applications of Mathematics and Computer Science, vol. 1, no. 2, pp. 22–30, 2011. View at: Google Scholar
V. A. Khan, K. Ebadullah, and A. Ahmad, “I-pre-Cauchy sequences and Orlicz functions,” Journal of Mathematical Analysis, vol. 3, no. 1, pp. 21–26, 2012. View at: Google Scholar
V. A. Khan and K. Ebadullah, “I-Convergent difference sequence spaces defined by a sequence of moduli,” Journal of Mathematical and Computational Science, vol. 2, no. 2, pp. 265–273, 2012. View at: Google Scholar
V. A. Khan and K. Ebadullah, “Zweier I-Convergent sequence spaces,” submitted to Acta Mathematica Scientia. View at: Google Scholar
Copyright © 2013 Vakeel A. Khan 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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Florian Cajori - Wikiquote
Florian Cajori (1859 – 1930) was a Swiss-American professor of mathematics and physics. He was one of the most celebrated historians of mathematics in his day. Cajori's A History of Mathematics (1894) was the first popular presentation of the history of mathematics in the United States and his 1928 –1929 History of Mathematical Notations has been described as "unsurpassed."
See also: A History of Mathematics
2 Quotes about Cajori
Professor Sylvester's first high class at the new university Johns Hopkins consisted of only one student, G. B. Halsted, who had persisted in urging Sylvester to lecture on the modern algebra. The attempt to lecture on this subject led him into new investigations in quantics.
F. Cajori's Teaching and History of Mathematics in the U. S. (Washington, 1890), p. 265; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book, (1914) p. 171; Persons and anecdotes.
The opinion is widely prevalent that even if the subjects are totally forgotten, a valuable mental discipline is acquired by the efforts made to master them. While the Conference admits that, considered in itself this discipline has a certain value, it feels that such a discipline is greatly inferior to that which may be gained by a different class of exercises, and bears the same relation to a really improving discipline that lifting exercises in an ill-ventilated room bear to games in the open air. The movements of a race horse afford a better model of improving exercise than those of the ox in a tread-mill.
Simon Newcomb, Henry Burchard Fine, Florian Cajori et al. Report of the Committee [of Ten on Secondary School Studies Appointed at the Meeting of the National Educational Association July 9, 1892: With the Reports of the Conferences Arranged by this Committee and Held December 28-30, 1892]. p. 108
Explanatory Appendix, Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World (1934) Tr. Andrew Motte, p. 674
The history of mathematics may be instructive as well as agreeable ; it may not only remind us of what we have, but may also teach us to increase our store. Says De Morgan, "The early history of the mind of men with regards to mathematics leads us to point out our own errors; and in this respect it is well to pay attention to the history of mathematics." It warns us against hasty conclusions; it points out the importance of a good notation upon the progress of the science; it discourages excessive specialization on the part of the investigator, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure; it teaches that fortifications can be taken by other ways than by direct attack, that when repulsed from a direct assault it is well to reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken.
pp. 1-2; Cited in: Robert Edouard Moritz. Memorabilia mathematica; or, The philomath's quotation-book, (1914) p. 90; Study and research in mathematics
p. 4; Cited in: Moritz (1914, 90); Study and research in mathematics
p. 30 Reported in Memorabilia mathematica or, The philomath's quotation-book by Robert Edouard Moritz. Published 1914.
Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him.
p. 30, Reported in Moritz (1914)
The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard.
p. 37. Reported in Moritz (1914)
The miraculous powers of modern calculation are due to three inventions : the Arabic Notation, Decimal Fractions and Logarithms.
p. 161; Cited in: Moritz (1914, 263); Arithmetics
{\displaystyle 2^{2^{n}}+1=}
{\displaystyle 2^{2^{5}}+1=}
p. 180; also cited in Moritz, Memorabilia Mathematica; Or, The Philomath's Quotation-book (1914) pp. 156-157.
In 1735 the solving of an astronomical problem, proposed by the Academy, for which several eminent mathematicians had demanded several months' time, was achieved in three days by Euler with aid of improved methods of his own... With still superior methods this same problem was solved by the illustrious Gauss in one hour.
p. 248; As cited in: Moritz (1914, 155); Persons and anecdotes.
Most of his [Euler's] memoirs are contained in the transactions of the Academy of Sciences at St. Petersburg, and in those of the Academy at Berlin. From 1728 to 1783 a large portion of the Petropolitan transactions were filled by his writings. He had engaged to furnish the Petersburg Academy with memoirs in sufficient number to enrich its acts for twenty years a promise more than fulfilled, for down to 1818 [Euler died in 1793] the volumes usually contained one or more papers of his. It has been said that an edition of Euler's complete works would fill 16,000 quarto pages.
pp. 253-254; Cited in: Moritz (1914, 155); Persons and anecdotes.
J. J. Sylvester was an enthusiastic supporter of reform [in the teaching of geometry]. The difference in attitude on this question between the two foremost British mathematicians, J. J. Sylvester, the algebraist, and Arthur Cayley, the algebraist and geometer, was grotesque. Sylvester wished to bury Euclid "deeper than e'er plummet sounded" out of the schoolboy's reach; Cayley, an ardent admirer of Euclid, desired the retention of Simson's Euclid. When reminded that this treatise was a mixture of Euclid and Simson, Cayley suggested striking out Simson's additions and keeping strictly to the original treatise.
p. 285; Cited in: Moritz (1914, 148); Persons and anecdotes
Quotes about Cajori[edit]
What is perhaps the greatest blow that has ever come to the student body of Colorado College came last Friday when it was announced that Dean Florian Cajori, for about thirty years the best-known and best-liked professor in the College, had resigned and will not be back with us next year. It was not only on account of the value of his service as an instructor that the students felt such a sense of loss at the announcement, but more on account of the friendship and intimate relationship which he has shown to us. "Caj"… has been closer to this student body than any other one man. It was usually "Caj" who made the speech at the Barbecue, it was "Caj" who talked upcoming events in chapel, it was "Caj" who was always out there at the picnic or the Festival or the ball game. No form of student activity has seemed entirely complete unless our "Caj" has been there or has had something to do with it.
The Tiger student newspaper, May 14, 1918 editorial, Colorado College
They [the students] looked upon "Caj" as one of their best friends in all the College, and thought of him as the one responsible for their later success. He has had a personal appeal to a great many students... It was the appeal of the one with a human interest in what somebody else is doing, the appeal of the true friend and the hearty well-wisher. It was the appeal of "Caj".
As a mathematician Dean Cajori has achieved a name which very few in this world can equal, a name which is respected all over the globe. His text books and his writings have been published all over the world. We are proud of all the achievements of our "Caj", of course, but we are especially proud of what he has done for us here, and it is for this reason that we shall always hold him in our memory. As a friend and as an instructor he has been more to us than we can ever measure, and we shall always look back upon the days when we had "Caj".
Professor Florian Cajori died August 14, 1934. In May of the following year I was invited by the University of California Press to edit this work. ...this is a revision of Motte's translation of the Principia. From many conversations with Professor Cajori, I know that he had long cherished the idea of revising Newton's immortal work by rendering certain parts into modern phraseology (e.g., to change the reading of "reciprocally in the subduplicate ratio of " to "inversely as the square root of") and to append historical and critical notes which would provide instruction to some readers and interest to all. This is his last work; one of the most fitting to crown a life devoted to investigation and to the history of the sciences in his chosen field.
R. T. Crawford, Editor's Note, Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World (1934) Isaac Newton, Florian Cajori.
Author:Florian Cajori
A History of Mathematics (1906, original-1893)
A History of Elementary Mathematics with Hints of Methods of Teaching (1896)
A History of Physics in its Elementary Branches:Including the Evolution of Physical Laboratories (1899)
''An Introduction to the Modern Theory of Equations (1904)
History of the Logarithmic Slide Rule and Allied Instruments (1909)
A History of the Conceptions of Limits and Fluxions in Great Britain: from Newton to Woodhouse (1911)
William Oughtred, a great seventeenth-century Teacher of Mathematics (1916)
On the History of Gunter's Scale and the Slide Rule During the Seventeenth Century (1920)
A History Of Mathematical Notations Vol I (1928)
Retrieved from "https://en.wikiquote.org/w/index.php?title=Florian_Cajori&oldid=2937025"
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Analysis of Influencing Factors of Silt Solidified Soil in Flowing State
Every year, a large amount of silt is generated by river dredging. For the characteristics of dredging silt, such as high moisture content, low strength and high compressibility, the traditional solidification method can no longer better solve this kind of silt problem. This paper mainly studies the fluidized solidification treatment of high water content sludge, not only makes the silt soil achieve a good solidification effect, but also the project cost is lower, the construction method is more environmentally friendly and green. The influencing factors of the solidified soil are mainly investigated by the unconfined compression test and the fluidity test. The experiment result shows 1)When the cement to sludge mass ratio (RCS) is 0.09 - 0.16 and the fly ash to cement mass ratio (RFC) is 0.35 - 0.80, as the amount of RFC increases, the flow rate of the solidified soil gradually decreases. With the increase of time, the liquidity is significantly reduced, and the influence of cement on the fluidity is greater than that of fly ash. 2) When RCS = 0.09 - 0.16, the strength of the sludge solidified soil at 28 d age increased by 4.5 - 6 times. 3) When RCS = 0.09 - 0.16 and RFC = 35% - 80%, the intensity of 14 d increased by 1.23 times than that of 7 d, and the intensity of 28 days increased by 1.29 times than that of 14 d. This experiment can provide the mix ratio design of solidified materials for different needs of the project, which can better provide a basis for engineering application and strength prediction.
Flowing Solidified Sludge, Influencing Factors, Fluidity, Unconfined Compressive Strength
Ding, J. , Feng, Z. , Sun, D. , You, K. , Yan, M. and Xun, Y. (2019) Analysis of Influencing Factors of Silt Solidified Soil in Flowing State. World Journal of Engineering and Technology, 7, 455-464. doi: 10.4236/wjet.2019.73033.
China is a country with a large water area, and thousands of lakes. The large amount of silt produced every year is a major problem that we face. According to statistics, the annual dredging volume of large and medium-sized dredging enterprises nationwide reached 1.3 billion cubic meters. The dredged soil has high initial moisture content, high compressibility, low strength, long self-weight consolidation time, and is difficult to directly use and occupies a large amount of yardland [1] . The flow curing treatment method is more suitable for this high water content dredged sludge. Compared with the previous principle of water cut rate, the fluidization treatment is to increase the moisture content of the silt and add a curing agent to make it flow, which greatly reduces the engineering cost and improves the engineering quality. No need for pre-pressing treatment, it can take advantage of its fluidity and self-hardening characteristics, without compression molding, and can directly adopt the pumping construction method [2] . This method can reduce noise and environmental pollution and meet the requirements of green construction.
Wang Dongxing [3] used MgO as a solidification material for sludge to analyze the appearance quality stress-strain relationship and unconfined compressive strength of samples under different immersion time. It was found that MgO can significantly change the water stability of solidified sludge. Wang Yuhua [4] eliminated the influence of organic matter by adding alkaline oxidant to the solidified sludge. It was found that the 7 d unconfined compressive strength of the solidified sludge reached 1.536 MPa, and the strength was 121.6% higher than that of cement-solidified sludge. Zhang Rongjun [5] and others used cement as the solidified material of sea mud, and found that the curing temperature has a significant impact on the strength development. The high curing temperature not only significantly increases the early strength of the solidified sludge, but also significantly increases the late strength of the solidified sludge. Boutoui et al. [6] also proposed various strength prediction formulas. More scholars have studied the solidified materials. Shao Wei [7] conducted an experimental study on cement-pulverized ash-reinforced organic soil. Miura, N. [8] and others have shown through experiments that the clay-water-cement ratio is the main parameter for analyzing the strength and deformation behavior of cement soil induced by high water content. The cement bond strength increases as the clay water to cement ratio decreases. Yong Yong [9] added cement lime and other solidified materials to the sludge to form a physical and chemical reaction with the sludge and pore water to increase the strength of the sludge.
Fluidity is the main indicator for describing flow-solidified sludge [10] . Gu Huanda [11] and others summarized the liquidity measurement method and liquidity index. Ding Jianwen, Zhang Shuai et al. [12] and other experiments through the high-water content dredging sludge flow curing treatment, found that the initial water content, the amount of curing agent and the time after preparation have a certain impact on the flowability of the flowing soil. And after the addition of gypsum in the curing agent, the curing strength far exceeds the strength of the cement only. The fluidity test is first seen in the determination of the workability of the concrete to evaluate the pumpability of the concrete mixture. This article is aimed at high moisture content sludge, starting from the concept of “disposing waste with waste” [2] . In this paper, composite cement and fly ash were used as curing agent materials to study and analyze sludge solidified soil and the factors affecting fluidity and strength were analyzed.
2. Experimental Material and Material Content Ratio
The sludge used in this test was taken from the Baiyangdian marshland, and the sampling position was 5 - 15 cm below the surface layer. The initial moisture content was 52%, and water was added to increase the moisture content to 160% of the slurry. The liquid limit is 38.2% and the plastic limit is 26.2%. It belongs to the clay soil. The solidified material is made of cement and fly ash. The cement is made of P·O42.5 ordinary Portland cement. Fly ash is used as the primary auxiliary material. The industrial waste used in the experiment was a silica-alumina low-calcium fly ash.
2.2. Material Content Ratio
The experiment uses 1 L of mud as a unit amount, Cement content: 110 g - 200 g, i.e. RCS = 0.09 - 0.16; Cement fly ash: cement quality 35% - 80%, i.e. RFC = 0.35 - 0.8; Cement phosphogypsum: 10 g. The specific experimental material dosage scheme is shown in Table 1.
Table 1. Material content ratio scheme.
Figure 1 shows the relationship between the fluidity of the sludge and the amount of curing agent. The effect of cement content and fly ash content on the fluidity of sludge solidified soil was studied. As can be seen from the figure, the amount of the curing agent has a significant influence on the fluidity, and the fluidity decreases as the amount of the curing agent increases. The fluidity index of the solidified soil is between 27 and 36 cm. Since the curing agent does not completely react with the water in the slurry at the beginning, the initial fluidity is large.
As can be seen from Figure 2, as time increases, the liquidity decreases significantly,
Figure 1. Relationship between curing agent dosage and fluidity.
Figure 2. The relationship between time and liquidity.
The liquidity decreases slowly during the period of 0 - 40 min, and the variation range is between 32.5 - 27 cm. The fluidity decreased significantly during the 40 - 60 min period, and the variation ranged from 30 to 23.5 cm. The flow rate decreases slowly during the period of 60 - 180 min, and the variation range is between 27 - 18 cm. After being placed for 3 hours, the fluidity is as low as 18 cm, which meets the requirements of pumping and construction of the flowing soil.
4. Analysis of Factors Affecting Strength
4.1. Impact of Environmental Conditions on the Strength of Curing
Adjust the temperature of the Constant temperature and humidity box to 20˚C, 30˚C and outdoor natural conditions, the outdoor time is 14 d, and the time period is from early May to mid-May. The highest average daytime temperature in the outdoor is 23˚C, and the lowest average temperature in the night is 6˚C. The average temperature difference between day and night is 12˚C, and there is rainfall for 2 and half days. Figure 3(a) shows the relationship between unconfined compressive strength and RCS in solidified soil when RFC = 0.65. It can be seen from Figure 3(a) that the strength of the solidified soil under outdoor natural conditions is higher in three different environments. And as the amount of cement and fly ash is increased, the highest strength of solidified sludge at 30˚C. The lowest strength is the solidified sludge at 20˚C, the sludge at 30˚C is 1.23 times the strength of the sludge at 20˚C.
It can be seen from Figure 3(b) that when the amount of cement is certain (RCS = 0.14), the strength of the solidified soil at 30˚C is not much different from the strength under outdoor natural conditions, but both are stronger than the strength of the solidified soil at 20˚C. When the amount of fly ash is RFC = 0.5, the strength of the solidified soil under outdoor conditions is 1.36 times the strength of 20˚C
4.2. Effect of Curing Age and Dosage on Strength
As can be seen from Figures 4-6, When the cement content is: RCS = 0.09 - 0.16, the strength of the sludge solidified soil increases with the increase of cement content, and the strength can be increased by 4.5 - 6 times, indicating that the cement content plays an important role in strength. When the cement content is RCS = 0.09 - 0.14, the amount of fly ash is increased, and the strength of the solidified soil is not obvious. However, when the amount of cement is RCS = 0.16 and the proportion of fly ash RFC increases from 35% to 65%, the strength can be increased by 1.72 times. However, the amount of fly ash continued to increase, and the strength of the solidified soil decreased. This also shows that when the cement content is 0.16, the optimum dosage of fly ash is 65%, so the excessive addition of fly ash has little effect on the strength increase. When the cement content is low, too much fly ash is added, which will reduce the strength of the solidified soil.
As can be seen from Figures 4-6, the strength of the sludge solidified soil and
Figure 3. Variation of unconfined compressive strength h at 14 d at different temperatures. (a) Cement content RCS (RFC = 0.65); (b) Fly ash content RFc (RCS = 0.14).
the curing time are in an increasing relationship. When the ratio of cement is RCS = 0.11 - 0.16 and the amount of fly ash is RFC = 65% - 80%, the strength at 28d curing age is 1.25 - 1.6 times the intensity of 14 d. However, when the amount of cement and fly ash is relatively small, the effect of curing time on strength is not obvious. When the curing time is short, the effect of cement on the strength of the solidified soil is greater than that of the fly ash. This also means that the solidifying agent of fly ash must function after reaching a certain value at the age.
According to the fitting results of Figure 7 & Figure 8, Equations (1)-(2) give
Figure 4. qu-7d/kPa.
Figure 5. qu-14d/kPa.
Figure 7. Relationship between f7d - f14d.
Figure 8. Relationship between f14d - f28.
the relationship between curing time and strength. It can better judge the strength and provide valuable reference for the project. It can be seen that the intensity of 14 d is 1.23 times higher than that of 7 d, and the intensity of 28 days is 1.29 times higher than that of 14 d.
{f}_{\text{14d}}=1.23{f}_{\text{7d}}
{f}_{\text{28d}}=1.23{f}_{\text{14d}}
1) When the cement content (RCS = 0.09 - 0.16) and the fly ash content (RFC = 35% - 80%), the fluidity of the sludge solidified soil decreases linearly with time, and the fluidity decreases with the amount of solidified material. When RCS = 0.09 - 0.16 and RFC = 0.60, the flow value changes in each time period are different, and the fluidity decreases slowly in the 0 - 40 min time period
2) When RCS = 0.14, RFC = 0.65, and the curing time is 14 d, the sludge at 30˚C is 1.23 times the strength of the sludge at 20˚C. When the amount of cement is RCS = 0.14 and the amount of fly ash is RFC = 0.5, the strength of the solidified soil under outdoor conditions is 1.36 times the strength of 20˚C.
3) When the cement content is: RCS = 0.14 - 0.16 and the curing time is 28 d, the strength of the sludge solidified soil increases with the cement content, and the strength of the solidified soil can increase by 4.5 - 6 times. When the cement content is RCS = 0.09 - 0.14, the amount of fly ash increased, and the strength of the solidified soil not significantly increased. However, when the amount of cement is RCS = 0.16, the proportion of fly ash RFC increases from 35% to 65%, and the strength can increase by 1.72 times.
4) When RCS = 0.09 - 0.16 and RFC = 35% - 80%, the intensity of 14 d increased by 1.23 times than that of 7 d, and the intensity of 28 days increased by 1.29 times than that of 14 d.
[1] Zhou, Y., Pu, Y. and Li, Z. (2019) Horizontal Drainage Board—Vacuum Preloading Combined Treatment of High Water Content Dredging Sludge Model Test. Journal of Rock Mechanics and Engineering, 35, 3246-3251.
[2] Ding, J., Liu, T. and Cao, Y. (2013) Pressure Test and Strength Prediction of High Moisture Content Dredged Sludge Solidified Soil. Chinese Journal of Geotechnical Engineering, 35, 55-60.
[3] Wang, D., Wang, H. and Xiao, J. (2018) Experimental Study on Water Stability Characteristics of Activated MgO Solidified Sludge. Journal of Geotechnical Engineering, 52, 719-726.
[4] Wang, Y., Xiang, W., Wu, X. and Cui, D. (2019) Study on the Effect of Alkaline Oxidant on the Strength of Cement Solidified Sludge. Chinese Journal of Geotechnical Engineering, 41, 693-699.
[5] Zhang, R., Zheng, J., Cheng, Y. and Dong, R. (2016) Experimental Study on Influence of Curing Temperature on Cement Solidified Sludge Strength. Rock and Soil Mechanics, 37, 3463-3471.
[6] Boutouil, M. and Lcvacher, D. (2005) Effect of High Initial Water Content on Cement-Based Fludge Solidification. Ground Improvement, 9, 169-174.
https://doi.org/10.1680/grim.2005.9.4.169
[7] Shao, Y., Liu, S., Du, G., et al. (2008) Experimental Study on Cemented Fly Ash to Reinforce Organic Soil. Journal of Engineering Geology, 16, 408-413.
[8] Miura, N., Horpibulsuk, S. and Nagaraj, T.S. (2001) Engineering Behavior of Cement Stabilized Clay at High Water Content. Soils and Foundations, 41, 33-45.
[9] Yong, Y. (2000) Experimental Study on Soft Soil Reinforced by Cement-Based Curing Agent Containing Industrial Waste. Chinese Journal of Geotechnical Engineering, 22, 210-213.
[10] Ding, J., Hong, Z. and Liu, S. (2011) Experimental Study on Flow Curing Treatment and Fluidity of Dredged Sludge. Rock and Soil Mechanics, 32, 280-284.
[11] Gu, H. and Chen, S. (2002) Experimental Study on Fluidization Treatment of River Silt and Its Engineering Properties.. Chinese Journal of Geotechnical Engineering, 24, 108-111.
[12] Ding, J., Zhang, S., et al. (2009) Experimental Study on Flow Solidification Treatment of High Water Content Dredged Sludge. Water Transport Engineering, No. 6, 30-34.
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Womersley_number Knowpia
The Womersley number (
{\displaystyle \alpha }
{\displaystyle {\text{Wo}}}
) is a dimensionless number in biofluid mechanics and biofluid dynamics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscous effects. It is named after John R. Womersley (1907–1958) for his work with blood flow in arteries.[1] The Womersley number is important in keeping dynamic similarity when scaling an experiment. An example of this is scaling up the vascular system for experimental study. The Womersley number is also important in determining the thickness of the boundary layer to see if entrance effects can be ignored.
This square root of this number is also referred to as Stokes number,
{\displaystyle {\text{Stk}}={\sqrt {\text{Wo}}}}
, due to the pioneering work done by Sir George Stokes on the Stokes second problem.
The Womersley number, usually denoted
{\displaystyle \alpha }
, is defined by the relation
{\displaystyle \alpha ^{2}={\frac {\text{transient inertial force}}{\text{viscous force}}}={\frac {\rho \omega U}{\mu UL^{-2}}}={\frac {\omega L^{2}}{\mu \rho ^{-1}}}={\frac {\omega L^{2}}{\nu }}\,,}
{\displaystyle L}
is an appropriate length scale (for example the radius of a pipe),
{\displaystyle \omega }
is the angular frequency of the oscillations, and
{\displaystyle \nu }
{\displaystyle \rho }
{\displaystyle \mu }
are the kinematic viscosity, density, and dynamic viscosity of the fluid, respectively.[2] The Womersley number is normally written in the powerless form
{\displaystyle \alpha =L\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}\,.}
In the cardiovascular system, the pulsation frequency, density, and dynamic viscosity are constant, however the Characteristic length, which in the case of blood flow is the vessel diameter, changes by three orders of magnitudes (OoM) between the aorta and fine capillaries. The Womersley number thus changes due to the variations in vessel size across the vasculature system. The Womersley number of human blood flow can be estimated as follows:
{\displaystyle \alpha =L\left({\frac {\omega \rho }{\mu }}\right)^{\frac {1}{2}}\,.}
Below is a list of estimated Womersley numbers in different human blood vessels:
{\displaystyle \alpha }
Aorta 0.025 13.83
Artery 0.004 2.21
Arteriole 3×10−5 0.0166
Capillary 8×10−6 4.43×10−3
Venule 2×10−5 0.011
Veins 0.005 2.77
Vena cava 0.03 16.6
It can also be written in terms of the dimensionless Reynolds number (Re) and Strouhal number (St):
{\displaystyle \alpha =\left(2\pi \,\mathrm {Re} \,\mathrm {St} \right)^{1/2}\,.}
The Womersley number arises in the solution of the linearized Navier–Stokes equations for oscillatory flow (presumed to be laminar and incompressible) in a tube. It expresses the ratio of the transient or oscillatory inertia force to the shear force. When
{\displaystyle \alpha }
is small (1 or less), it means the frequency of pulsations is sufficiently low that a parabolic velocity profile has time to develop during each cycle, and the flow will be very nearly in phase with the pressure gradient, and will be given to a good approximation by Poiseuille's law, using the instantaneous pressure gradient. When
{\displaystyle \alpha }
is large (10 or more), it means the frequency of pulsations is sufficiently large that the velocity profile is relatively flat or plug-like, and the mean flow lags the pressure gradient by about 90 degrees. Along with the Reynolds number, the Womersley number governs dynamic similarity.[3]
The boundary layer thickness
{\displaystyle \delta }
that is associated with the transient acceleration is inversely related to the Womersley number. This can be seen by recognizing the Womersley number as the square root of the Stokes number.[4]
{\displaystyle \delta =\left(L/\alpha \right)=\left({\frac {L}{\sqrt {\mathrm {Stk} }}}\right),}
{\displaystyle L}
is a characteristic length.
In a flow distribution network that progresses from a large tube to many small tubes (e.g. a blood vessel network), the frequency, density, and dynamic viscosity are (usually) the same throughout the network, but the tube radii change. Therefore, the Womersley number is large in large vessels and small in small vessels. As the vessel diameter decreases with each division the Womersley number soon becomes quite small. The Womersley numbers tend to 1 at the level of the terminal arteries. In the arterioles, capillaries, and venules the Womersley numbers are less than one. In these regions the inertia force becomes less important and the flow is determined by the balance of viscous stresses and the pressure gradient. This is called microcirculation.[4]
Some typical values for the Womersley number in the cardiovascular system for a canine at a heart rate of 2 Hz are:[4]
Ascending aorta – 13.2
Descending aorta – 11.5
Abdominal aorta – 8
Femoral artery – 3.5
Carotid artery – 4.4
Arterioles – 0.04
Capillaries – 0.005
Venules – 0.035
Inferior vena cava – 8.8
Main pulmonary artery – 15
It has been argued that universal biological scaling laws (power-law relationships that describe variation of quantities such as metabolic rate, lifespan, length, etc., with body mass) are a consequence of the need for energy minimization, the fractal nature of vascular networks, and the crossover from high to low Womersley number flow as one progresses from large to small vessels.[5]
^ Womersley, J.R. (March 1955). "Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known". J. Physiol. 127 (3): 553–563. doi:10.1113/jphysiol.1955.sp005276. PMC 1365740. PMID 14368548.
^ Fung, Y. C. (1990). Biomechanics – Motion, flow, stress and growth. New York (USA): Springer-Verlag. p. 569. ISBN 978-0-387-97124-7.
^ Nichols, W. W.; O'Rourke, M. F. (2005). McDonald's Blood Flow in Arteries (5th ed.). London (England): Hodder-Arnold. ISBN 978-0-340-80941-9.
^ a b c Fung, Y.C. (1996). Biomechanics Circulation. Springer Verlag. p. 571. ISBN 978-0-387-94384-8.
^ West GB, Brown JH, Enquist BJ (4 April 1997). "A general model for the origin of allometric scaling laws in biology". Science. 276 (5309): 122–6. doi:10.1126/science.276.5309.122. PMID 9082983.
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Paper published: A sensorimotor paradigm for Bayesian model selection – Tim Genewein
Paper published: A sensorimotor paradigm for Bayesian model selection
Genewein T, Braun DA (2012) A sensorimotor paradigm for Bayesian model selection. Frontiers in Human Neuroscience 6:291. doi: 10.3389/fnhum.2012.00291
Our paper on a sensorimotor paradigm for Bayesian model selection got published in Frontiers in Human Neuroscience.
In the paper we present a sensorimotor paradigm that allows for testing whether human choice behavior is quantiatively consistent with Bayesian model selection. In Bayesian model selection, models
M_1
M_2
are compared by taking their posterior probability ratio (given the data
D
\theta
M_1
M_2
. If the strength of preference is ignored, the previous rule leads to a deterministic model selection mechanism. Additionally, the magnitude of the posterior odds ratio is an indicator of the strength of the preference which can be used to derive stochastic model selection mechanisms (for instance by combination with a softmax selection rule). In case of equal prior probabilities for both models, the quantity that governs model selection is the Bayes factor, which is the ratio of marginal likelihoods.
In the paper, we designed a virtual-reality experiment that required participants to report their belief about the model
M_i
after observing some data
D
. Importantly, the experiment was designed such that we did not ask participants to report their belief over the parameter
\theta
but rather “integrate out” the parameter dimension . To do so, we set up an experiment with a separate model- and parameter-dimension and presented an ambiguous stimulus that was compatible with a whole range of
\theta
-values. This required participants to “integrate out” the parameters compatible with the stimulus, similar to the marginalization when computing the model evidence.
We found participants’ behavior to be quantitatively consistent with the theoretical model of Bayesian model selection combined with a softmax model selection mechanism.
The document under the download button is protected by copyright and was first published by Frontiers. All rights reserved. It is reproduced with permission.
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Isoperimetric inequality - Wikipedia
Geometric inequality which sets a lower bound on the surface area of a set given its volume
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. I{\displaystyle n}
{\displaystyle \mathbb {R} ^{n}}
the inequality lower bounds the surface area or perimeter
{\displaystyle \operatorname {per} (S)}
{\displaystyle S\subset \mathbb {R} ^{n}}
by its volume
{\displaystyle \operatorname {vol} (S)}
{\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{\frac {n-1}{n}}\,\operatorname {vol} (B_{1})^{\frac {1}{n}}}
{\displaystyle B_{1}\subset \mathbb {R} ^{n}}
is a unit sphere. The equality holds only when
{\displaystyle S}
{\displaystyle \mathbb {R} ^{n}}
On a plane, i.e. when
{\displaystyle n=2}
, the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. Isoperimetric literally means "having the same perimeter". Specifically in
{\displaystyle \mathbb {R} ^{2}}
, the isoperimetric inequality states, for the length L of a closed curve and the area A of the planar region that it encloses, that
{\displaystyle L^{2}\geq 4\pi A,}
1 The isoperimetric problem in the plane
3 On a sphere
4 In '"`UNIQ--postMath-00000019-QINU`"'
5 In Hadamard manifolds
6 In a metric measure space
7 For graphs
7.1 Example: Isoperimetric inequalities for hypercubes
7.1.1 Edge isoperimetric inequality
7.1.2 Vertex isoperimetric inequality
8 Isoperimetric inequality for triangles
The isoperimetric problem in the plane[edit]
The classical isoperimetric problem dates back to antiquity.[2] The problem can be stated as follows: Among all closed curves in the plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region? This question can be shown to be equivalent to the following problem: Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimizes the perimeter?
On a plane[edit]
{\displaystyle 4\pi A\leq L^{2},}
and that the equality holds if and only if the curve is a circle. The area of a disk of radius R is πR2 and the circumference of the circle is 2πR, so both sides of the inequality are equal to 4π2R2 in this case.
{\displaystyle Q={\frac {4\pi A}{L^{2}}}}
{\displaystyle Q_{n}={\frac {\pi }{n\tan {\tfrac {\pi }{n}}}}.}
{\displaystyle C}
be a smooth regular convex closed curve. Then the improved isoperimetric inequality states the following
{\displaystyle L^{2}\geqslant 4\pi A+8\pi \left|{\widetilde {A}}_{0.5}\right|,}
{\displaystyle L,A,{\widetilde {A}}_{0.5}}
{\displaystyle C}
, the area of the region bounded by
{\displaystyle C}
and the oriented area of the Wigner caustic of
{\displaystyle C}
, respectively, and the equality holds if and only if
{\displaystyle C}
is a curve of constant width.[4]
On a sphere[edit]
{\displaystyle L^{2}\geq A(4\pi -A),}
This inequality was discovered by Paul Lévy (1919) who also extended it to higher dimensions and general surfaces.[5]
{\displaystyle L^{2}\geq 4\pi A-{\frac {A^{2}}{R^{2}}}.}
{\displaystyle \mathbb {R} ^{n}}
The isoperimetric inequality states that a sphere has the smallest surface area per given volume. Given a bounded set
{\displaystyle S\subset \mathbb {R} ^{n}}
with surface area
{\displaystyle \operatorname {per} (S)}
{\displaystyle \operatorname {vol} (S)}
, the isoperimetric inequality states
{\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{\frac {n-1}{n}}\,\operatorname {vol} (B_{1})^{\frac {1}{n}}}
{\displaystyle B_{1}\subset \mathbb {R} ^{n}}
is a unit ball. The equality holds when
{\displaystyle S}
is a ball in
{\displaystyle \mathbb {R} ^{n}}
. Under additional restrictions on the set (such as convexity, regularity, smooth boundary), the equality holds for a ball only. But in full generality the situation is more complicated. The relevant result of Schmidt (1949, Sect. 20.7) (for a simpler proof see Baebler (1957)) is clarified in Hadwiger (1957, Sect. 5.2.5) as follows. An extremal set consists of a ball and a "corona" that contributes neither to the volume nor to the surface area. That is, the equality holds for a compact set
{\displaystyle S}
{\displaystyle S}
contains a closed ball
{\displaystyle B}
{\displaystyle \operatorname {vol} (B)=\operatorname {vol} (S)}
{\displaystyle \operatorname {per} (B)=\operatorname {per} (S).}
For example, the "corona" may be a curve.
The proof of the inequality follows directly from Brunn–Minkowski inequality between a set
{\displaystyle S}
and a ball with radius
{\displaystyle \epsilon }
{\displaystyle B_{\epsilon }=\epsilon B_{1}}
. By taking Brunn–Minkowski inequality to the power
{\displaystyle n}
, subtracting
{\displaystyle \operatorname {vol} (S)}
from both sides, dividing them by
{\displaystyle \epsilon }
, and taking the limit as
{\displaystyle \epsilon \to 0.}
(Osserman (1978); Federer (1969, §3.2.43)).
In full generality (Federer 1969, §3.2.43), the isoperimetric inequality states that for any set
{\displaystyle S\subset \mathbb {R} ^{n}}
whose closure has finite Lebesgue measure
{\displaystyle n\,\omega _{n}^{\frac {1}{n}}L^{n}({\bar {S}})^{\frac {n-1}{n}}\leq M_{*}^{n-1}(\partial S)}
{\displaystyle M_{*}^{n-1}}
is the (n-1)-dimensional Minkowski content, Ln is the n-dimensional Lebesgue measure, and ωn is the volume of the unit ball in
{\displaystyle \mathbb {R} ^{n}}
. If the boundary of S is rectifiable, then the Minkowski content is the (n-1)-dimensional Hausdorff measure.
The n-dimensional isoperimetric inequality is equivalent (for sufficiently smooth domains) to the Sobolev inequality on
{\displaystyle \mathbb {R} ^{n}}
with optimal constant:
{\displaystyle \left(\int _{\mathbb {R} ^{n}}|u|^{\frac {n}{n-1}}\right)^{\frac {n-1}{n}}\leq n^{-1}\omega _{n}^{-{\frac {1}{n}}}\int _{\mathbb {R} ^{n}}|\nabla u|}
{\displaystyle u\in W^{1,1}(\mathbb {R} ^{n})}
In Hadamard manifolds[edit]
Hadamard manifolds are complete simply connected manifolds with nonpositive curvature. Thus they generalize the Euclidean space
{\displaystyle \mathbb {R} ^{n}}
, which is a Hadamard manifold with curvature zero. In 1970's and early 80's, Thierry Aubin, Misha Gromov, Yuri Burago, and Viktor Zalgaller conjectured that the Euclidean isoperimetric inequality
{\displaystyle \operatorname {per} (S)\geq n\operatorname {vol} (S)^{\frac {n-1}{n}}\operatorname {vol} (B_{1})^{\frac {1}{n}}}
holds for bounded sets
{\displaystyle S}
in Hadamard manifolds, which has become known as the Cartan–Hadamard conjecture. In dimension 2 this had already been established in 1926 by André Weil, who was a student of Hadamard at the time. In dimensions 3 and 4 the conjecture was proved by Bruce Kleiner in 1992, and Chris Croke in 1984 respectively.
In a metric measure space[edit]
Most of the work on isoperimetric problem has been done in the context of smooth regions in Euclidean spaces, or more generally, in Riemannian manifolds. However, the isoperimetric problem can be formulated in much greater generality, using the notion of Minkowski content. Let
{\displaystyle (X,\mu ,d)}
be a metric measure space: X is a metric space with metric d, and μ is a Borel measure on X. The boundary measure, or Minkowski content, of a measurable subset A of X is defined as the lim inf
{\displaystyle \mu ^{+}(A)=\liminf _{\varepsilon \to 0+}{\frac {\mu (A_{\varepsilon })-\mu (A)}{\varepsilon }},}
{\displaystyle A_{\varepsilon }=\{x\in X|d(x,A)\leq \varepsilon \}}
The isoperimetric problem in X asks how small can
{\displaystyle \mu ^{+}(A)}
be for a given μ(A). If X is the Euclidean plane with the usual distance and the Lebesgue measure then this question generalizes the classical isoperimetric problem to planar regions whose boundary is not necessarily smooth, although the answer turns out to be the same.
{\displaystyle I(a)=\inf\{\mu ^{+}(A)|\mu (A)=a\}}
is called the isoperimetric profile of the metric measure space
{\displaystyle (X,\mu ,d)}
. Isoperimetric profiles have been studied for Cayley graphs of discrete groups and for special classes of Riemannian manifolds (where usually only regions A with regular boundary are considered).
For graphs[edit]
Isoperimetric inequalities for graphs relate the size of vertex subsets to the size of their boundary, which is usually measured by the number of edges leaving the subset (edge expansion) or by the number of neighbouring vertices (vertex expansion). For a graph
{\displaystyle G}
{\displaystyle k}
, the following are two standard isoperimetric parameters for graphs.[8]
{\displaystyle \Phi _{E}(G,k)=\min _{S\subseteq V}\left\{|E(S,{\overline {S}})|:|S|=k\right\}}
{\displaystyle \Phi _{V}(G,k)=\min _{S\subseteq V}\left\{|\Gamma (S)\setminus S|:|S|=k\right\}}
{\displaystyle E(S,{\overline {S}})}
denotes the set of edges leaving
{\displaystyle S}
{\displaystyle \Gamma (S)}
denotes the set of vertices that have a neighbour in
{\displaystyle S}
. The isoperimetric problem consists of understanding how the parameters
{\displaystyle \Phi _{E}}
{\displaystyle \Phi _{V}}
behave for natural families of graphs.
Example: Isoperimetric inequalities for hypercubes[edit]
{\displaystyle d}
-dimensional hypercube
{\displaystyle Q_{d}}
is the graph whose vertices are all Boolean vectors of length
{\displaystyle d}
{\displaystyle \{0,1\}^{d}}
. Two such vectors are connected by an edge in
{\displaystyle Q_{d}}
if they are equal up to a single bit flip, that is, their Hamming distance is exactly one. The following are the isoperimetric inequalities for the Boolean hypercube.[9]
Edge isoperimetric inequality[edit]
The edge isoperimetric inequality of the hypercube is
{\displaystyle \Phi _{E}(Q_{d},k)\geq k(d-\log _{2}k)}
. This bound is tight, as is witnessed by each set
{\displaystyle S}
that is the set of vertices of any subcube of
{\displaystyle Q_{d}}
Vertex isoperimetric inequality[edit]
Harper's theorem[10] says that Hamming balls have the smallest vertex boundary among all sets of a given size. Hamming balls are sets that contain all points of Hamming weight at most
{\displaystyle r}
and no points of Hamming weight larger than
{\displaystyle r+1}
for some intege{\displaystyle r}
. This theorem implies that any set
{\displaystyle S\subseteq V}
{\displaystyle |S|\geq \sum _{i=0}^{r}{d \choose i}}
{\displaystyle |S\cup \Gamma (S)|\geq \sum _{i=0}^{r+1}{d \choose i}.}
As a special case, consider set sizes
{\displaystyle k=|S|}
{\displaystyle k={d \choose 0}+{d \choose 1}+\dots +{d \choose r}}
for some intege{\displaystyle r}
. Then the above implies that the exact vertex isoperimetric parameter is
{\displaystyle \Phi _{V}(Q_{d},k)={d \choose r+1}.}
Isoperimetric inequality for triangles[edit]
The isoperimetric inequality for triangles in terms of perimeter p and area T states that[13][14]
{\displaystyle p^{2}\geq 12{\sqrt {3}}\cdot T,}
with equality for the equilateral triangle. This is implied, via the AM–GM inequality, by a stronger inequality which has also been called the isoperimetric inequality for triangles:[15]
{\displaystyle T\leq {\frac {\sqrt {3}}{4}}(abc)^{\frac {2}{3}}.}
^ Blåsjö, Viktor (2005). "The Evolution of the Isoperimetric Problem". Amer. Math. Monthly. 112 (6): 526–566. doi:10.2307/30037526. JSTOR 30037526.
^ Olmo, Carlos Beltrán, Irene (4 January 2021). "Sobre mates y mitos". EL PAÍS (in Spanish). Retrieved 14 January 2021.
^ Zwierzyński, Michał (2016). "The improved isoperimetric inequality and the Wigner caustic of planar ovals". J. Math. Anal. Appl. 442 (2): 726–739. arXiv:1512.06684. doi:10.1016/j.jmaa.2016.05.016. S2CID 119708226.
^ Gromov, Mikhail; Pansu, Pierre (2006). "Appendix C. Paul Levy's Isoperimetric Inequality". Metric Structures for Riemannian and Non-Riemannian Spaces. Modern Birkhäuser Classics. Dordrecht: Springer. p. 519. ISBN 9780817645830.
^ Osserman, Robert. "The Isoperimetric Inequality." Bulletin of the American Mathematical Society. 84.6 (1978) http://www.ams.org/journals/bull/1978-84-06/S0002-9904-1978-14553-4/S0002-9904-1978-14553-4.pdf
^ Hoory, Linial & Widgerson (2006)
^ Definitions 4.2 and 4.3 of Hoory, Linial & Widgerson (2006)
^ See Bollobás (1986) and Section 4 in Hoory, Linial & Widgerson (2006)
^ Cf. Calabro (2004) or Bollobás (1986)
^ cf. Leader (1991)
^ Also stated in Hoory, Linial & Widgerson (2006)
^ "The isoperimetric inequality for triangles".
^ Dragutin Svrtan and Darko Veljan, "Non-Euclidean Versions of Some Classical Triangle Inequalities", Forum Geometricorum 12, 2012, 197–209. http://forumgeom.fau.edu/FG2012volume12/FG201217.pdf
Bollobás, Béla (1986). Combinatorics: set systems, hypergraphs, families of vectors, and combinatorial probability. Cambridge University Press. ISBN 978-0-521-33703-8.
Burago (2001) [1994], "Isoperimetric inequality", Encyclopedia of Mathematics, EMS Press
Capogna, Luca; Donatella Danielli; Scott Pauls; Jeremy Tyson (2007). An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Birkhäuser Verlag. ISBN 978-3-7643-8132-5.
Fenchel, Werner; Bonnesen, Tommy (1934). Theorie der konvexen Körper. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 3. Berlin: 1. Verlag von Julius Springer.
Hadwiger, Hugo (1957). Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag. .
Hoory, Shlomo; Linial, Nathan; Widgerson, Avi (2006). "Expander graphs and their applications" (PDF). Bulletin of the American Mathematical Society. New Series. 43 (4): 439–561. doi:10.1090/S0273-0979-06-01126-8.
Leader, Imre (1991). "Discrete isoperimetric inequalities". Proceedings of Symposia in Applied Mathematics. Vol. 44. pp. 57–80.
Zwierzyński, Michał (2016). "The improved isoperimetric inequality and the Wigner caustic of planar ovals". J. Math. Anal. Appl. 442 (2): 726–739. arXiv:1512.06684. doi:10.1016/j.jmaa.2016.05.016. S2CID 119708226.
Schmidt, Erhard (1949). "Die Brunn-Minkowskische Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Hugel in der euklidischen und nichteuklidischen Geometrie. II". Math. Nachr. 2 (3–4): 171–244. doi:10.1002/mana.19490020308. .
Baebler, F. (1957). "Zum isoperimetrischen Problem". Arch. Math. (Basel). 8: 52–65. doi:10.1007/BF01898439. S2CID 123704157. .
Wikimedia Commons has media related to Isoperimetric inequality.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Isoperimetric_inequality&oldid=1068546891"
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Bayesian Coin Flips—The bcf Package – Tom Shafer
Bayesian Coin Flips—The bcf Package
In an experiment with Approximate Bayesian Computation and R packages, I uploaded a new R package of my own to GitHub a few days ago named bcf for Bayesian Coin Flip. It simulates
N
-person games of skill, approximating these games as multiple players flipping coins with different “fairness parameters”
\theta_i \sim \mathrm{Beta}(\alpha_i, \beta_i)
. The first player to obtain a “Heads” result first wins, dealing with ties in a sensible way.1
The ABC concept is well explained in a pair of articles. First, Rasmus Bååth introduces ABC through an exercise involving mismatched socks in the laundry (thanks for pointing me to this, Kenny). And Darren Wilkinson also does a nice job explaining how ABC works.
As far as bcf is concerned, the probability of a coin coming up Heads is picked from a distribution assigned to each player,
\theta_i^* \sim \mathrm{Beta}(\alpha_i, \beta_i)
. The package then simulates a set of games
d^*
using these parameters and provides samples from the joint distribution
(d^*, \theta^*)
. Finally, by keeping only the data that match an observed result, we end up with a distribution proportional to the posterior
p(\theta|d)
I built the package to better understand ABC and to humorously model our office’s dart-playing abilities. To keep things simple, the bcf package only provides a few functions: We can initialize a player, run a game, and use the results of the game to update a player’s statistics.
A basic game might go like so: We assume a pretty weak prior (
\theta \sim \mathrm{Beta}(1.2, 1.2)
) for each player before running a few games. After each game, we update the involved players.
In practice, I think it’s pretty easy to use. First, instantiate a few players:
library(bcf)
tom <- new_player("Tom", alpha = 1.2, beta = 1.2)
david <- new_player("David", alpha = 1.2, beta = 1.2)
kevin <- new_player("Kevin", alpha = 1.2, beta = 1.2)
## UUID: d8cfe17e-c81d-11e7-9f88-f45c899c4b7b
## Name: Tom
## Games: 0
## Wins: 0
## Losses: 0
## Est. Distribution: Beta(1.200, 1.200)
## MAP Win Percentage: 50.000
Then simulate three games for which we already have results:
# Tom wins, David places second, Kevin finishes third
game_1 <- abc_coin_flip_game(
players = list(tom, david, kevin),
result = c(1, 2, 3),
iters = 5000, cores = 5L)
## No. players: 3
## Assign result: 1, 2, 3
## Iters: 5000
## CPU cores: 5
## Workloads: 1000, 1000, 1000, 1000, 1000
tom <- update_player(tom, game_1)
david <- update_player(david, game_1)
kevin <- update_player(kevin, game_1)
# Tom wins, Kevin places second, David finishes third
# Tom finishes second, David wins, Kevin finishes third
bcf then provides methods for examining both players and games:
print(game_3)
## Tom David Kevin outcome n pct
## <dbl> <dbl> <dbl> <chr> <int> <dbl>
## 1 1 2 3 1627 32.5
## 3 2 1 3 *** 439 8.8
## 4 2 3 1 590 11.8
## 5 3 1 2 222 4.4
plot(game_3)
plot(tom)
This has been a pretty fun experiment in ABC and in R packaging. I’ll update this post if I ever return to the project.
If mote than one player obtains the same result on a given flip, these players play one or more sub-games to break the tie. ↩︎
One gotcha—for now—is that bcf imposes a beta distribution for each player’s win probability. After working out the likelihood on paper, I don’t think the posterior is actually a beta…just almost a beta. The possibility of ties and additional rounds adds complication. ↩︎
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Explore Single-Period Asset Arbitrage - MATLAB & Simulink Example
Define Parameters of the Portfolio
Solve for Risk-Neutral Probabilities
Test for No-Arbitrage Violations
Plot Call Price as a Surface
This example explores basic arbitrage concepts in a single-period, two-state asset portfolio. The portfolio consists of a bond, a long stock, and a long call option on the stock.
It uses these Symbolic Math Toolbox™ functions:
equationsToMatrix to convert a linear system of equations to a matrix.
linsolve to solve the system.
Symbolic equivalents of standard MATLAB® functions, such as diag.
This example symbolically derives the risk-neutral probabilities and call price for a single-period, two-state scenario.
Create the symbolic variable r representing the risk-free rate over the period. Set the assumption that r is a positive value.
syms r positive
Define the parameters for the beginning of a single period, time = 0. Here S0 is the stock price, and C0 is the call option price with strike, K.
syms S0 C0 K positive
Now, define the parameters for the end of a period, time = 1. Label the two possible states at the end of the period as U (the stock price over this period goes up) and D (the stock price over this period goes down). Thus, SU and SD are the stock prices at states U and D, and CU is the value of the call at state U. Note that
SD<=K<=SU
syms SU SD CU positive
The bond price at time = 0 is 1. Note that this example ignores friction costs.
Collect the prices at time = 0 into a column vector.
prices = [1 S0 C0]'
\left(\begin{array}{c}1\\ {S}_{0}\\ {C}_{0}\end{array}\right)
Collect the payoffs of the portfolio at time = 1 into the payoff matrix. The columns of payoff correspond to payoffs for states D and U. The rows correspond to payoffs for bond, stock, and call. The payoff for the bond is 1 + r. The payoff for the call in state D is zero since it is not exercised (because
SD<=K
payoff = [(1 + r), (1 + r); SD, SU; 0, CU]
\left(\begin{array}{cc}r+1& r+1\\ \mathrm{SD}& \mathrm{SU}\\ 0& \mathrm{CU}\end{array}\right)
CU is worth SU - K in state U. Substitute this value in payoff.
payoff = subs(payoff, CU, SU - K)
\left(\begin{array}{cc}r+1& r+1\\ \mathrm{SD}& \mathrm{SU}\\ 0& \mathrm{SU}-K\end{array}\right)
Define the probabilities of reaching states U and D.
syms pU pD real
Under no-arbitrage, eqns == 0 must always hold true with positive pU and pD.
eqns = payoff*[pD; pU] - prices
\left(\begin{array}{c}\mathrm{pD} \left(r+1\right)+\mathrm{pU} \left(r+1\right)-1\\ \mathrm{SD} \mathrm{pD}-{S}_{0}+\mathrm{SU} \mathrm{pU}\\ -{C}_{0}-\mathrm{pU} \left(K-\mathrm{SU}\right)\end{array}\right)
Transform equations to use risk-neutral probabilities.
syms pDrn pUrn real;
eqns = subs(eqns, [pD; pU], [pDrn; pUrn]/(1 + r))
\left(\begin{array}{c}\mathrm{pDrn}+\mathrm{pUrn}-1\\ \frac{\mathrm{SD} \mathrm{pDrn}}{r+1}-{S}_{0}+\frac{\mathrm{SU} \mathrm{pUrn}}{r+1}\\ -{C}_{0}-\frac{\mathrm{pUrn} \left(K-\mathrm{SU}\right)}{r+1}\end{array}\right)
The unknown variables are pDrn, pUrn, and C0. Transform the linear system to a matrix form using these unknown variables.
[A, b] = equationsToMatrix(eqns, [pDrn, pUrn, C0]')
\left(\begin{array}{ccc}1& 1& 0\\ \frac{\mathrm{SD}}{r+1}& \frac{\mathrm{SU}}{r+1}& 0\\ 0& -\frac{K-\mathrm{SU}}{r+1}& -1\end{array}\right)
\left(\begin{array}{c}1\\ {S}_{0}\\ 0\end{array}\right)
Using linsolve, find the solution for the risk-neutral probabilities and call price.
x = linsolve(A, b)
\left(\begin{array}{c}\frac{{S}_{0}-\mathrm{SU}+{S}_{0} r}{\mathrm{SD}-\mathrm{SU}}\\ -\frac{{S}_{0}-\mathrm{SD}+{S}_{0} r}{\mathrm{SD}-\mathrm{SU}}\\ \frac{\left(K-\mathrm{SU}\right) \left({S}_{0}-\mathrm{SD}+{S}_{0} r\right)}{\left(\mathrm{SD}-\mathrm{SU}\right) \left(r+1\right)}\end{array}\right)
Verify that under risk-neutral probabilities, x(1:2), the expected rate of return for the portfolio, E_return equals the risk-free rate, r.
E_return = diag(prices)\(payoff - [prices,prices])*x(1:2);
E_return = simplify(subs(E_return, C0, x(3)))
E_return =
\left(\begin{array}{c}r\\ r\\ r\end{array}\right)
As an example of testing no-arbitrage violations, use the following values: r = 5%, S0 = 100, and K = 100. For SU < 105, the no-arbitrage condition is violated because pDrn = xSol(1) is negative (SU >= SD). Further, for any call price other than xSol(3), there is arbitrage.
xSol = simplify(subs(x, [r,S0,K], [0.05,100,100]))
\left(\begin{array}{c}-\frac{\mathrm{SU}-105}{\mathrm{SD}-\mathrm{SU}}\\ \frac{\mathrm{SD}-105}{\mathrm{SD}-\mathrm{SU}}\\ \frac{20 \left(\mathrm{SD}-105\right) \left(\mathrm{SU}-100\right)}{21 \left(\mathrm{SD}-\mathrm{SU}\right)}\end{array}\right)
Plot the call price, C0 = xSol(3), for 50 <= SD <= 100 and 105 <= SU <= 150. Note that the call is worth more when the "variance" of the underlying stock price is higher for example, SD = 50, SU = 150.
fsurf(xSol(3), [50,100,105,150])
xlabel SD
ylabel SU
title 'Call Price'
Advanced Derivatives, Pricing and Risk Management: Theory, Tools and Programming Applications by Albanese, C., Campolieti, G.
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When Erica and Ken explored a cave, they each found a gold nugget. Erica’s nugget is similar to Ken’s nugget. They measured the length of two matching parts of the nuggets and found that Erica’s nugget is five times as long as Ken’s. When they took their nuggets to the metallurgist to be analyzed, they learned that it would cost
\$30
to have the surface area and weight of the smaller nugget calculated, and
\$150
to have the same analysis done on the larger nugget.
“I won’t have that kind of money until I sell my nugget, and then I won’t need it analyzed!” Erica says.
“Wait, Erica. Don’t worry. I’m pretty sure we can get all the information we need for only
\mathit{\$30}
Explain how they can get all the information they need for
\$30
Since the two nuggets are similar, if they spend
\$30
to get Ken's nugget analyzed, they can calculate the surface area and weight of Erica's nuggets using the ratios of similarity.
If Ken’s nugget has a surface area of
20 \; \text{cm}^2
, what is the surface area of Erica’s nugget?
We know the linear scale factor is
5
, so the ratio of the areas is
5^2
25
If Ken’s nugget weighs
5.6
g (about
0.2
oz), what is the weight of Erica’s nugget?
Since the nuggets are solid gold and have the same density throughout, calculate the weight using the ratio of the volumes.
700 \; \text{g} \approx 25 \; \text{ounces}
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Compressible flow using the Noble-Abel Stiffened Gas model
The Noble-Abel Stiffened Gas model
e\left(p,v\right)=\frac{p+\gamma {p}_{\infty }}{\gamma -1}\left(v-b\right)+q,
is a simple extension of the perfect gas model to treat compressible flows in dense media, liquid and solids and obtain closed form analytical solutions.
What started as an example in a gas dynamics course taught in Fall 2019, is now a complete description of the gasdynamics in closed form, now published in Physics of Fluids.
Find here your favourite analytical expressions for Riemann variables, expansion and shock jump conditions, isentropes, detonations and deflagrations and the solution to the Riemann problem, as an example.
Phys. Fluids 32, 056101 (2020); doi: 10.1063/1.5143428
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Loss given default - Wikipedia
Loss given default or LGD is the share of an asset that is lost if a borrower defaults.
It is a common parameter in risk models and also a parameter used in the calculation of economic capital, expected loss or regulatory capital under Basel II for a banking institution. This is an attribute of any exposure on bank's client. Exposure is the amount that one may lose in an investment.
The LGD is closely linked to the expected loss, which is defined as the product of the LGD, the probability of default (PD) and the exposure at default (EAD).
2 How to calculate LGD
2.1 Calculating LGD under the foundation approach (for corporate, sovereign and bank exposure)
2.1.1 Exposure without collateral
2.1.2 Exposure with collateral
2.2 Calculating LGD under the advanced approach (and for the retail-portfolio under the foundation approach)
3 Downturn LGD
4 Correcting for different default definitions
5 Country-specific LGD
LGD is the share of an asset that is lost when a borrower defaults. The recovery rate is defined as 1 minus the LGD, the share of an asset that is recovered when a borrower defaults.[1]
Loss given default is facility-specific because such losses are generally understood to be influenced by key transaction characteristics such as the presence of collateral and the degree of subordination.
How to calculate LGD[edit]
The LGD calculation is easily understood with the help of an example: If the client defaults with an outstanding debt of $200,000 and the bank or insurance is able to sell the security (e.g. a condo) for a net price of $160,000 (including costs related to the repurchase), then the LGD is 20% (= $40,000 / $200,000).
Theoretically, LGD is calculated in different ways, but the most popular is 'gross' LGD, where total losses are divided by exposure at default (EAD). Another method is to divide losses by the unsecured portion of a credit line (where security covers a portion of EAD). This is known as 'Blanco' LGD.[a] If collateral value is zero in the last case then Blanco LGD is equivalent to gross LGD. Different types of statistical methods can be used to do this.
Gross LGD is most popular amongst academics because of its simplicity and because academics only have access to bond market data, where collateral values often are unknown, uncalculated or irrelevant. Blanco LGD is popular amongst some practitioners (banks) because banks often have many secured facilities, and banks would like to decompose their losses between losses on unsecured portions and losses on secured portions due to depreciation of collateral quality. The latter calculation is also a subtle requirement of Basel II, but most banks are not sophisticated enough at this time to make those types of calculations.[citation needed]
Calculating LGD under the foundation approach (for corporate, sovereign and bank exposure)[edit]
To determine required capital for a bank or financial institution under Basel II, the institution has to calculate risk-weighted assets. This requires estimating the LGD for each corporate, sovereign and bank exposure. There are two approaches for deriving this estimate: a foundation approach and an advanced approach.
Exposure without collateral[edit]
Under the foundation approach, BIS prescribes fixed LGD ratios for certain classes of unsecured exposures:
Senior claims on corporates, sovereigns and banks not secured by recognized collateral attract a 45% LGD.
All subordinated claims on corporates, sovereigns and banks attract a 75% LGD.
Exposure with collateral[edit]
Simple LGD example: If the client defaults, with an outstanding debt of 200,000 (EAD) and the bank is able to sell the security for a net price of 160,000 (including costs related to the repurchase), then 40,000, or 20%, of the EAD are lost - the LGD is 20%.
The effective loss given default (
{\displaystyle L_{GD}^{*}}
) applicable to a collateralized transaction can be expressed as
{\displaystyle L_{GD}^{*}=L_{GD}\cdot {\frac {E^{*}}{E}}}
Haircut appropriate for currency mismatch between the collateral and exposure (The standard supervisory haircut for currency risk where exposure and collateral are denominated in different currencies is 8%)
The *He and *Hc has to be derived from the following table of standard supervisory haircuts:
However, under certain special circumstances the supervisors, i.e. the local central banks may choose not to apply the haircuts specified under the comprehensive approach, but instead to apply a zero H.
Calculating LGD under the advanced approach (and for the retail-portfolio under the foundation approach)[edit]
Under the A-IRB approach and for the retail-portfolio under the F-IRB approach, the bank itself determines the appropriate loss given default to be applied to each exposure, on the basis of robust data and analysis. The analysis must be capable of being validated both internally and by supervisors. Thus, a bank using internal loss given default estimates for capital purposes might be able to differentiate loss given default values on the basis of a wider set of transaction characteristics (e.g. product type, wider range of collateral types) as well as borrower characteristics. These values would be expected to represent a conservative view of long-run averages. A bank wishing to use its own estimates of LGD will need to demonstrate to its supervisor that it can meet additional minimum requirements pertinent to the integrity and reliability of these estimates.
An LGD model assesses the value and/or the quality of a security the bank holds for providing the loan – securities can be either machinery like cars, trucks or construction machines. It can be mortgages or it can be a custody account or a commodity. The higher the value of the security the lower the LGD and thus the potential loss the bank or insurance faces in the case of a default. Banks using the A-IRB approach have to determine LGD values, whereas banks within the F-IRB do only have to do so for the retail-portfolio. For example, as of 2013, there were nine companies in the United Kingdom with their own mortgage LGD models. In Switzerland there were two banks as of 2013. In Germany many thrifts – especially the market leader Bausparkasse Schwäbisch Hall – have their own mortgage LGD models. In the corporate asset class many German banks still only use the values given by the regulator under the F-IRB approach.
Repurchase value estimators (RVEs) have proven to be the best kind of tools for LGD estimates. The repurchase value ratio provides the percentage of the value of the house/apartment (mortgages) or machinery at a given time compared to its purchase price.
Downturn LGD[edit]
Under Basel II, banks and other financial institutions are recommended to calculate 'downturn LGD' (downturn loss given default), which reflects the losses occurring during a 'downturn' in a business cycle for regulatory purposes. Downturn LGD is interpreted in many ways, and most financial institutions that are applying for IRB approval under BIS II often have differing definitions of what Downturn conditions are. One definition is at least two consecutive quarters of negative growth in real GDP. Often, negative growth is also accompanied by a negative output gap in an economy (where potential production exceeds actual demand).
The calculation of LGD (or downturn LGD) poses significant challenges to modelers and practitioners. Final resolutions of defaults can take many years and final losses, and hence final LGD, cannot be calculated until all of this information is ripe. Furthermore, practitioners are of want of data since BIS II implementation is rather new and financial institutions may have only just started collecting the information necessary for calculating the individual elements that LGD is composed of: EAD, direct and indirect Losses, security values and potential, expected future recoveries. Another challenge, and maybe the most significant, is the fact that the default definitions between institutions vary. This often results in a so-called differing cure-rates or percentage of defaults without losses. Calculation of LGD (average) is often composed of defaults with losses and defaults without. Naturally, when more defaults without losses are added to a sample pool of observations LGD becomes lower. This is often the case when default definitions become more 'sensitive' to credit deterioration or 'early' signs of defaults. When institutions use different definitions, LGD parameters therefore become non-comparable.
Many institutions are scrambling to produce estimates of downturn LGD, but often resort to 'mapping' since downturn data is often lacking. Mapping is the process of guesstimating losses under a downturn by taking existing LGD and adding a supplement or buffer, which is supposed to represent a potential increase in LGD when a downturn occurs. LGD often decreases for some segments during a downturn since there is a relatively larger increase of defaults that result in higher cure-rates, often the result of temporary credit deterioration that disappears after the downturn period is over. Furthermore, LGD values decrease for defaulting financial institutions under economic downturns because governments and central banks often rescue these institutions in order to maintain financial stability.
In 2010 researchers at Moody’s Analytics quantify an LGD in line with the target probability event intended to be captured under Basel. They illustrate that the Basel downturn LGD guidelines may not be sufficiently conservative.[2] Their results are based on a structural model that incorporates systematic risk in recovery.[3]
Correcting for different default definitions[edit]
One problem facing practitioners is the comparison of LGD estimates (usually averages) arising from different time periods where differing default definitions have been in place. The following formula can be used to compare LGD estimates from one time period (say x) with another time period (say y):
LGDy=LGDx*(1-Cure Ratey)/(1-Cure Ratex)
Country-specific LGD[edit]
In Australia, the prudential regulator APRA has set an interim minimum downturn LGD of 20 per cent on residential mortgages for all applicants for the advanced Basel II approaches. The 20 per cent floor is not risk sensitive and is designed to encourage authorised deposit-taking institutions (ADIs) to undertake further work, which APRA believes would be closer to the 20 per cent on average than ADIs’ original estimates.
LGD warrants more attention than it has been given in the past decade, where credit risk models often assumed that LGD was time-invariant. Movements in LGD often result in proportional movements in required economic capital. According to BIS (2006) institutions implementing Advanced-IRB instead of Foundation-IRB will experience larger decreases in Tier 1 capital, and the internal calculation of LGD is a factor separating the two Methods.[citation needed]
^ Named after academic Roberto Blanco
^ Altman, Edward; Resti, Andrea; Sironi, Andrea (July 2004). "Default Recovery Rates in Credit Risk Modelling: A Review of the Literature and Empirical Evidence". Economic Notes. 33 (2): 183–208. CiteSeerX 10.1.1.194.4041. doi:10.1111/j.0391-5026.2004.00129.x. S2CID 8312724.
^ Levy, Amnon; Meng, Qiang; Kaplin, Andrew; Wang, Yashan; Hu, Zhenya (2010). "Implications of PD-LGD Correlation in a Portfolio Setting" (PDF). Moody's Analytics Whitepaper.
^ Levy, Amnon; Hu, Zhenya (2007). "Incorporating Systematic Risk in Recovery: Theory and Evidence" (PDF). Moody's Analytics Whitepaper.
Basel II: Revised international capital framework (BCBS)
Basel II: International Convergence of Capital Measurement and Capital Standards: a Revised Framework (BCBS)
Basel II: International Convergence of Capital Measurement and Capital Standards: a Revised Framework (BCBS) (November 2005 Revision)
Basel II: International Convergence of Capital Measurement and Capital Standards: a Revised Framework, Comprehensive Version (BCBS) (June 2006 Revision)
Validating Risk Rating Systems under the IRB Approaches, HKMA
Basel II: Benefiting from improved risk management and governance
Loss Given Default Information and Basel II Mini Exam Updates (broken link)
What Do We Know About Loss Given Default at The Wharton Financial Institutions Center
Dynamic Prediction of LGD at Moody's K M V (broken link)
Practical articles, on BIS2 and risk modelling, submitted by professionals to help create an industry standard.
Downturn LGDs for Basel II, RMA Capital Working Group.
Business Cycle and Downturn LGD in Italian banks: a case study.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Loss_given_default&oldid=1044198075"
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Agent Simulator Loop
The figure above shows AI2-THOR's agent-simulator interaction loop. Our backend scenes, actions, and observations are stored within Unity, a powerful real-time game engine. From the Python side, we sequentially interact with Unity by taking actions using AI2-THOR's Python
, 3) stored in the
, except image channels are in BGR ordering. This is often useful with Python's OpenCV module (i.e.,
), which expects images with BGR ordering.
) stored in the
format. Each unique pixel color corresponds to a different object, which are indexable with
s visible in the frame and each value is the boolean
of size (
The keys are object IDs and the values are [Upper Left
x
, Upper Left
y
, Lower Right
x
y
], where each element is the number of pixels it is from the top left corner of the image.
The name of the action passed into the
This object provides all coordinates that are within bounds of the scene. This can be used in tandem with actions like
to make sure the coordinate used is not out of bounds. This returns a
object that includes an 8x3 matrix of xyz coordinates that represent the 8 corners of the box encompassing the entire scene, an xyz dictionary for the coordinates of the center of that box, and an xyz dictionary for the size (extents) of that box. Example:
Within the metadata dictionary, the
key contains the pose of the agent after the action has executed.
cameraHorizon: {...},
isStanding: {...},
position: {...},
rotation: {...},
values correspond to the agent looking up, whereas positive
if the agent is currently in a standing position, otherwise
. This bool can be changed if the agent uses the
agent is currently the only agent with the ability to stand.
The global position of the agent, with keys for
x
y
z
y
coordinate corresponds to upwards in 3D space.
The local rotation of the agent's body, with keys for
x
(pitch),
y
(yaw), and
z
(roll). Since the
agent's body can only change its yaw rotation, both
x
z
will always be approximately
0
Each object has a plethora of information exposed about it in each event.
Beyond what is shown here, the object metadata also provides information for each object state change action, which is documented on the Object State Changes page.
event.metadata["objects"][i]
{...Object State Changes...}
The global position of the object, with keys for
x
y
z
y
corresponds to the upward coordinate in 3D space.
The local rotation of the object, with keys for
x
y
z
(roll).
Environment queries are actions that query the environment to extract additional metadata regarding the current state. Since there is a performance cost that comes from calculating each environment query, they are not automatically provided in the metadata after each action.
Query actions do not alter the state of the environment. Thus, they are often substantially faster than non-query actions (e.g., MoveAhead, RotateRight), since image frames and object metadata can be reused from the previous Event.
Get Object in Frame
GetObjectInFrame
queries the current view of the agent for the object that appears at specified
(x, y)
coordinate, relative to its current view. If there is an object that appears at the coordinate, its objectId is provided in
query.metadata["actionReturn"]
. Alternatively, if no object is at the provided coordinate,
bool(query)
This action can be used in tandem with object interaction actions, where
is first called to extract an
, and if the objectId is extracted.
query = controller.step(
action="GetObjectInFrame",
checkVisible=False
object_id = query.metadata["actionReturn"]
Get Coordinate from Raycast Parameters
x
coordinate from the current image frame, corresponding to the relative distance from the left of the frame. Valid values are in
[0:1]
y
coordinate from the current image frame, corresponding to the relative distance from the top of the frame. Valid values are in
[0:1]
checkVisible
, the returned object will only be provided if it is within a distance of the initialized
(default: 1.5 meters) from the agent. This is set to
by default so that the agent can only interact with objects in-front of it, rather than objects far across the room.
Get Coordinate from Raycast
GetCoordinateFromRaycast
sends a raycast out from the camera in the direction of the
(x, y)
screen coordinate, relative to the agent's current view. The world
(x, y, z)
coordinate of the first point of collision that is hit on an object by the raycast is returned in
action="GetCoordinateFromRaycast",
coordinate = query.metadata["actionReturn"]
x
[0:1]
y
[0:1]
finds all the positions that the agent can reach in a scene. It does an optimized BFS over a grid spaced out by the initialized
. The valid positions are then added and returned in a list. The action can be used in tandem with Teleport, to actually travel to a given position.
(x, y, z)
positions that the agent can reach in the scene.
Get Interactable Poses
GetInteractablePoses
returns all the agent poses where an object is
to the agent. A pose assigns every degree of freedom on the agent to a specific value. A pose can then be passed into TeleportFull to teleport the agent to a given pose.
In order for an object to be
at a certain pose, the object has to both be within
of the agent's camera and it must be in the agent's field of view. Thus, if the agent is too far away from an object, but it appears in the agent's field of view, the object will return
The action also provides the ability to restrict which poses should be searched, for each degree of freedom. For instance, one can restrict the poses they want to poses where
. Such restrictions can often make the action execute faster, if the search space is more constrained.
This action is expected to solely be used in tandem with TeleportFull. Hence, in future releases, if the agent is given extra degrees of freedom, those degrees of freedom will be included in each pose, and thus change the returned poses. Therefore, we recommend not indexing into any pose (i.e.,
pose["horizon"]
), and instead passing the entire pose to TeleportFull using Python's
feature (i.e.,
**pose
action="GetInteractablePoses",
objectId="Apple|-1.0|+1.0|+1.5",
positions=[dict(x=0, y=0.9, z=0)],
rotations=range(0, 360, 10),
horizons=np.linspace(-30, 60, 30),
standings=[True, False]
poses = event.metadata["actionReturn"]
x=(...),
y=(...),
z=(...),
horizon=(...),
rotation=(...),
standing=(...)
TeleportFull to a Pose
pose = random.choice(poses)
controller.step("TeleportFull", **pose)
Get Interactable Poses Attributes
The objectId of the object with which the interactable poses will be queried.
: Optional[list[dict[str, float]]] = None
Restricts which positions should be searched. If not specified, all positions from
: Optional[list[float]] = None
Restricts which rotation values should appear in the returned response. For instance, if
is passed in, only such values may appear as the rotation for each returned pose. By default, the rotation values are
\text{range}(A_r\,\%\, D,\;\; D,\;\; 360 + A_r\,\%\, D),
A_r
is the current rotation of the agent and
D
is the initialized
rotateStepDegrees
A_r = 10^\circ
D = 90
, then the default rotations will be
An exception is thrown if 360 mod the initialized
does not equal 0 and rotations has not been provided. Here, the agent cannot rotate in a circular manner, and hence, an infinite number of rotations would be possible.
Restricts which horizons should be searched. For instance, if
is passed in, only such values may appear as the horizon for each returned pose. Defaults to using
[-30, 0, 30, 60]
Each horizon must be in
[-30:60]
: Optional[list[bool]] = None
Restricts which
poses should be added to the search. For instance, if
is passed in, only values of
may appear in the response. Defaults to
Add Third Party Camera
AddThirdPartyCamera
adds an invisible camera to the scene, with images available for each successive event, until reset has been called. When reset is called, the camera is removed from the scene.
action="AddThirdPartyCamera",
position=dict(x=-1.25, y=1, z=-1),
rotation=dict(x=90, y=0, z=0),
event.third_party_camera_frames
Add Third Party Camera Parameters
The global
(x, y, z)
position of where the camera will be placed.
(x, y, z)
rotation of where the camera.
: dict[str, float] = 90
Changes the camera's optical field of view, in degrees. Valid values are in the domain
(0:180)
Image frames are then accessible with the
third_party_camera_frames
attribute on successive events:
.third_party_camera_frames
Third Party Camera Response
: list[numpy.ndarray]
A list of RGB frames from rendered from each third party camera. Images are stored as numpy.uint8 ndarrays of size (height, width, 3), where the height and width are the same as the agent's egocentric image height and width.
Camera frames in the list appear in the order that they were added to the scene.
Update Third Party Camera
UpdateThirdPartyCamera
updates the state of a previously added third party camera that is currently in the scene. Any values that are unspecified will remain the same.
action="UpdateThirdPartyCamera",
thirdPartyCameraId=0,
Update Third Party Camera Parameters
thirdPartyCameraId
Targets which third party camera to modify. Third party camera ids are based on the order that they were added to the scene.
Valid values are in
[0:\text{len}(\text{third party cameras}) - 1]
(x, y, z)
(x, y, z)
rotation of how the camera will be orientated.
(0:180)
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{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle i=13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle (n+1)^{\mathrm {th} }}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1,}}
{\displaystyle 1}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1,}}
{\displaystyle n=1.}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{4}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
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Abstract: We claim that the linking of a shrinking prior universe to our own via a wormhole bridge solution of about ten to the minus forty four power seconds permits the formation of a short-term quintessence scalar field. Symmetries allow for creating high-frequency gravitational waves at the onset of inflation, which has consequences in our present cosmological era. This instantaneous energy transfer between prior to present universes permits relic graviton production which we claim is a viable candidate for future propulsion technologies in space craft design. The Big Bang started as the passage of thermal energy from an existing universe into ours resulting in another Big Bang, and helps us understand how a graviton burst can occur in the first place.
Keywords: Wormhole, High-Frequency Gravitational Waves (HGW), Symmetry, Causal Discontinuity
w=-1
z=1100
w=0
{r}_{0}\left(t\right)
{l}_{P}
\text{d}{S}^{2}=-F\left(r\right)\cdot \text{d}{t}^{2}+\frac{\text{d}{r}^{2}}{F\left(r\right)}+\text{d}{\Omega }^{2}
F\left(r\right)=1-\frac{2M}{r}+\frac{{Q}^{2}}{{r}^{2}}-\frac{\Lambda }{3}\cdot {r}^{2}\underset{T\to {10}^{32}\text{Kelvin}~\infty }{\to }-\frac{\Lambda }{3}\cdot {\left(r={l}_{P}\right)}^{2}
\frac{\partial F}{\partial r}~-2\cdot \frac{\Lambda }{3}\cdot \left(r\approx {l}_{P}\right)\equiv \eta \left(T\right)\cdot \left(r\approx {l}_{P}\right)
\Psi \left(T\right)\propto -A\cdot \left\{{\eta }^{2}\cdot {C}_{1}\right\}+A\cdot \eta \cdot {\omega }^{2}\cdot {C}_{2}
{C}_{1}={C}_{1}\left(\omega ,t,r\right)
{C}_{2}={C}_{2}\left(\omega ,t,r\right)
{C}_{1}={C}_{1}\left(\omega ,t,r\right)\ne {C}_{2}\left(\omega ,t,r\right)
|{\Lambda }_{5-\mathrm{dim}}|\approx {c}_{1}\cdot \left(1/{T}^{\alpha }\right)
{\Lambda }_{4-\mathrm{dim}}\approx {c}_{2}\cdot {T}^{\beta }
{m}_{P}\approx 2.17645\times {10}^{-8}
{\Lambda }_{4-\mathrm{dim}}\propto {c}_{2}\cdot T\underset{\text{graviton-production}}{\to }360\cdot {m}_{P}^{2}\ll {c}_{2}\cdot \left[T\approx {10}^{32}\text{K}\right]
t\approx {\delta }^{1}\cdot {t}_{P}
0<{\delta }^{1}\le 1
T\approx {10}^{12}\text{Kelvin}
{t}_{P}~{10}^{-44}
\frac{{\Lambda }_{4-\mathrm{dim}}}{|{\Lambda }_{5-\mathrm{dim}}|}-1\approx \frac{1}{n}
\left(-\Lambda /3\right)
{t}_{P}
\Psi \left(r,t,T\right)
\Psi \left(r,t,T\right)=\Psi \left(r,-t,T\right)
|t|<{t}_{P}
\Psi \left(r,t,T\right)
{t}_{P}
{\left(\stackrel{˙}{a}/a\right)}^{2}=\frac{8\text{π}G}{3}\cdot \left[{\rho }_{\text{rel}}+{\rho }_{\text{matter}}\right]+\frac{\Lambda }{3}
\prec
x\prec y
y\prec z
x\prec z
x\prec y
y\prec x
x=y
x,y\in C
\left\{y|x\prec y\prec z\right\}
a\left({t}^{\ast }\right)<{l}_{P}
{t}^{\ast }<{t}_{P}
{a}_{0}\equiv {l}_{P}
{a}_{0}/a\left({t}^{\ast }\right)\equiv {10}^{\alpha }
\alpha \gg 0
a\left(t\right)
\left[\frac{a\left({t}^{\ast }+\delta t\right)}{a\left({t}^{\ast }\right)}\right]-1<\frac{\left(\delta t\cdot {l}_{P}\right)}{\sqrt{\Lambda /3}}\cdot {\left[1+\frac{8\text{π}}{\Lambda }\cdot \left[{\left({\rho }_{rel}\right)}_{0}\cdot {10}^{4\alpha }+{\left({\rho }_{m}\right)}_{0}\cdot {10}^{3\alpha }\right]\right]}^{1/2}\underset{\Lambda \to \infty }{\to }0
\left[\frac{a\left({t}^{\ast }+\delta t\right)}{a\left({t}^{\ast }\right)}\right]<1
t~{10}^{-44}
\left[#\text{operations}/\mathrm{sec}\right]\le 2E/\text{π}\hslash
\left[#\text{operations}\right]\le S\left(entropy\right)/\left({k}_{B}\cdot \mathrm{ln}2\right)
{c}^{3}\cdot {t}^{3}
E\left(\text{energy}\right)~\rho \cdot {c}^{2}
\rho
\rho \cdot {c}^{2}
\left[\#\text{operations}/\mathrm{sec}\right]\le \rho \cdot {c}^{2}\times {c}^{3}\cdot {t}^{3}
\rho ~{10}^{-27}\text{kil}/{\text{meter}}^{3}
t~{10}^{10}\text{years}
\left[#\text{operations}\right]\approx \rho \cdot {c}^{5}\cdot {t}^{4}\le 10{}^{120}
#\text{operations}=\frac{4E}{\hslash }\cdot \left({t}_{1}-\sqrt{{t}_{1}{t}_{0}}\right)\approx \left({t}_{\text{Final}}/{t}_{P}\right)\le {10}^{120}
{t}_{0}={t}_{P}~10{}^{-43}
{N}^{+}\ne {\epsilon }^{+}\ll 1
0<{N}^{+}<1
\begin{array}{l}E=\left({V}_{4-\text{Dim}}\right)\cdot \left[{\rho }_{Vac}=\frac{\Lambda }{8\text{π}G}\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}}~{N}^{+}\cdot \left[{\rho }_{\text{graviton}}\cdot {V}_{4-\text{vol}}\approx \hslash \cdot {\omega }_{\text{graviton}}\right]\end{array}
T\approx {10}^{32}-{10}^{29}
t=1/H
\approx c/H
\approx {\rho }_{C}\cdot {c}^{3}/\left(H{}^{4}\cdot \hslash \right)\approx 1/\left({t}_{P}^{2}\cdot H\right)
H=\sqrt{8\text{π}G\cdot \left[{\rho }_{crit}\right]/3\cdot {c}^{2}}
{\rho }_{crit}~{\rho }_{\text{graviton}}~\hslash \cdot {\omega }_{\text{graviton}}/{V}_{4-\text{Vol}}
\begin{array}{c}#\text{operations}\approx 1/\left[{t}_{P}^{2}\cdot H\right]\approx \sqrt{{V}_{4-\text{Vol}}}\cdot {t}_{P}^{-2}/\sqrt{\left[8\text{π}G\hslash {\omega }_{\text{graviton}}/3{c}^{2}\right]}\\ \approx {\left[3\mathrm{ln}2/4\right]}^{4/3}\cdot {\left[{S}_{\text{Entrophy}}/{k}_{B}\mathrm{ln}2\right]}^{4/3}\end{array}
V\left(t\right)\equiv V\left(\phi \right)~\frac{3{H}^{2}}{8\text{π}G}\cdot \left(1+\frac{\stackrel{˙}{H}}{3{H}^{2}}\right)
\phi \left(t\right)~\int \text{d}t\cdot \sqrt{\frac{-\stackrel{˙}{H}}{4\text{π}G}}
H=\stackrel{˙}{a}/a
a\left({t}_{\text{INITIAL}}\right)~{\text{e}}^{H\cdot t}
H=\stackrel{˙}{a}/a
a\left({t}_{\text{later}}\right)~{\text{e}}^{\left(\Lambda \left[\text{present-day}\right]\cdot t\right)}
H=\stackrel{˙}{a}/a
\varphi =0
\varphi \ne 0
\varphi \left(t\right)
H=\stackrel{˙}{a}/a
a\left(t\left(\text{initial}\right)\right)~\mathrm{exp}\left(H\cdot t\left(\text{initial}\right)\right)
a\left(t\left(\text{later}\right)\right)~\mathrm{exp}\left(\Lambda \left(\text{today}\right)\cdot t\left(\text{later}\right)\right)
H=\stackrel{˙}{a}/a
\varphi \left(t\right)
\varphi \left(t\right)
{\omega }_{\text{graviton}}
{t}_{P}^{-2}~{10}^{86}{\mathrm{sec}}^{-2}
T\approx {10}^{32}\text{Kelvin}
z\approx {10}^{25}
z\approx {10}^{25}
Cite this paper: Beckwith, A. (2018) Symmetries in Evolving Space-Time and Their Connection to High-Frequency Gravitational Wave Production. Journal of High Energy Physics, Gravitation and Cosmology, 4, 492-503. doi: 10.4236/jhepgc.2018.43027.
[1] Ryden, B. (2017) Introduction to Cosmology. 2nd Edition, Cambridge University Press, Cambridge.
[2] Jerzy Plebasnki, J. and Krasinki, A. (2007) An Introduction to General Relativity and Cosmology. Cambridge University Press, Cambridge.
[3] Lloyd, S. (2002) Computational Capacity of the Universe. Physical Review Letters 88, Article ID: 237901.
[4] Visser, M. (1995) Lorentzian Wormholes, from Einstein to Hawking. AIP Press, Woodbury, New York.
[5] Crowell, L. (2005) Quantum Fluctuations of Space Time. World Scientific Series in Contemporary Chemical Physics, Volume 25. World Scientific Publishing Company, Singapore.
[6] Hamber, H. and Williams, R. (2011) Discrete Wheeler-DeWitt Equation. Physical Review D, 84, Article ID: 104033.
[8] Beckwith, A. (2007) Symmetries in Evolving Space Time from Present to Prior Universes. arXiv:math-ph/0501028
[9] Barvinsky, A., Kamenschick, A. and Yu, A. (2006) Thermodynamics from Nothing: Limiting the Cosmological Constant Landscape. Physical Review D, 74, Article ID: 121502.
[12] Simovici, D. and Djeraba, C. (2008) Partially Ordered Sets. Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics. Springer, Berlin.
[13] Schröder, B.S.W. (2003) Ordered Sets: An Introduction. Birkhäuser, Boston.
[15] Soto-Andrade, J., Jaramillo, S., Gutierrez, C. and Letelier, J.-C. (2011) Ouroboros Avatars: A Mathematical Exploration of Self-reference and Metabolic Closure. MIT Press, Paris.
[16] Padmanabhan, T. (2006) An Invitation to Astro Physics. World Scientific Publishing Co. Pte. Ltd., Singapore.
[17] Tsujikawa, S. (2013) Quintessence: A Review. Classical and Quantum Gravity, 30, Article ID: 214003.
[18] Smoot, G. (2007) Nobel Lecture: Cosmic Microwave Background Radiation Anisotropies: Their Discovery and Utilization. Reviews of Modern Physics, 79, 1-31.
https://cdn.journals.aps.org/files/RevModPhys.79.1349.pdf
[19] Volovik, G. (2003) The Universe in a Helium Droplet. International Series of Monographs on Physics 117, Clarendon Press, Oxford.
[20] Koplik, J. and Levine, H. (1993) Vortex Reconnection in Superfluid Helium. Physical Review Letters, 71, 1375-1378.
[21] Li, F., Baker Jr., R.M.L., Fang, Z., Stephenson, G.V. and Chen, Z. (2008) Perturbative Photon Fluxes Generated by High-Frequency Gravitational Waves and Their Physical Effects. The European Physical Journal C, 56, 407-423.
[26] Barcelo, C. and Visser, M. (1999) Transversable Wormholes from Massless Conformally Coupled Scalar Fields. Physics Letters B, 66, 127-134.
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Rules of Probability | Codecademy
Pro Pricing
Probability: Course OverviewCourse Overview
Reset Progress
The union of two sets encompasses any element that exists in either one or both of them. We can represent this visually as a venn diagram as shown. Union is often represented as:
(A\ or\ B)
Probability is a way to quantify uncertainty. When we flip a fair coin, we say that there is a 50 percent chance (probability = 0.5) of it coming up tails. This means that if we flip INFINITELY man…
Let’s dive into some key concepts we will use throughout this lesson: union, intersection, and complement. #### Union The union of two sets encompasses any element that exists in either …
Imagine that we flip a fair coin 5 times and get 5 heads in a row. Does this affect the probability of getting heads on the next flip? Even though we may feel like it’s time to see “tails”, it is i…
Two events are considered mutually exclusive if they cannot occur at the same time. For example, consider a single coin flip: the events “tails” and “heads” are mutually exclusive because we cann…
Now, it’s time to apply these concepts to calculate probabilities. Let’s go back to one of our first examples: event A is rolling an odd number on a six-sided die and event B is rolling a num…
If we want to calculate the probability that a pair of dependent events both occur, we need to define conditional probability. Using a bag of marbles as an example, let’s remind ourselves of the de…
We have looked at the addition rule, which describes the probability one event OR another event (or both) occurs. What if we want to calculate the probability that two events happen simultaneously?…
We have introduced conditional probability as a part of the multiplication rule for dependent events. However, let’s go a bit more in-depth with it as it is a powerful probability tool that has rea…
Imagine that you are a patient who has recently tested positive for strep throat. You may want to know the probability that you HAVE strep throat, given that you tested positive: P(ST \mid +) T…
Congratulations, we have finished our exploration into the rules of probability! To recap, we have covered: * The union and intersection of two events * The complement of an event * Independent an…
We roll a 6-sided die. Let’s say event A represents rolling an odd number and event B represents rolling a number greater than one. Which of the following represents the intersection of these two events?
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Change number of summation terms for calculating periodic Green's function - MATLAB - MathWorks í•œêµ
numSummationTerms
Change Number of Summation Terms in Infinite Array
Change number of summation terms for calculating periodic Green's function
numSummationTerms(array,num)
numSummationTerms(array,num) changes the number of summation terms used to calculate periodic Green's function of the infinite array. This method calculates
2*num+1
of the periodic Green's function. The summation is carried out from –num to +num. A higher number of terms results in better accuracy but increases the overall computation time.
array — Infinite array
Infinite array, specified as a scalar.
num — Number to calculate summation terms
Number to calculate summation terms, specified as a scalar. The summation is carried out from –num to +num.
Create an infinite array with the scan elevation at 45 degrees. Calculate the scan impedance. By default, the number of summation terms used is 21.
h = infiniteArray('ScanElevation',45);
s = impedance(h,1e9)
s = 84.7925 + 70.6722i
Change the number of summation terms to 51. Calculate the scan impedance again.
numSummationTerms(h,25)
Change the number of terms to 101. Increasing the number of summation terms results in a more accurate scan impedance. However, the time required to calculate the scan impedance increases.
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(Created page with "right|350px <span class="exam"> '''Use calculus to set up and solve the word problem:''' Find the length and width of a rectangle that has a perimet...")
|As with all word problems, start with a picture. Using the variables <math style="vertical-align: 0%">x</math> and <math style="vertical-align: 0%">y</math> as shown in the image, we need to remember the equations for perimeter:
::<math>P\,=\,2x+2y,</math>
|and that for area:
|and for area:
::<math>A\,=\,xy.</math>
|Since we want to maximize area, we will have to rewrite it as a function of a single variable, and then find when the first derivative is zero. From this, we will find the dimensions which provide the maximum area.
::<math>A'(x)\,=\,24-2x\,=\,2(12-x).</math>
|This derivative is zero precisely when <math style="vertical-align: -20%">x=y=12</math>, and these are the values that will maximize area.
|This derivative is zero precisely when <math style="vertical-align: -20%">x=y=12</math>, and these are the values that will maximize area. Also, don't forget the units - meters!
{\displaystyle x}
{\displaystyle y}
{\displaystyle P\,=\,2x+2y,}
{\displaystyle A\,=\,xy.}
{\displaystyle P\,=\,2x+2y}
{\displaystyle y}
{\displaystyle x}
{\displaystyle 48\,=\,P\,=\,2x+2y,}
{\displaystyle y=24-x.}
{\displaystyle A(x)\,=\,xy\,=\,x(24-x)\,=\,24x-x^{2}.}
{\displaystyle A(x)=24x-x^{2}}
{\displaystyle A'(x)\,=\,24-2x\,=\,2(12-x).}
{\displaystyle x=y=12}
|
High‐Quality Revision of the Israeli Seismic Bulletin | Seismological Research Letters | GeoScienceWorld
High‐Quality Revision of the Israeli Seismic Bulletin
Lewis Schardong;
Lewis Schardong *
Department of Geophysics, Tel Aviv University, Tel Aviv, Israel
Corresponding author: lschardong@tauex.tau.ac.il
Yochai Ben Horin;
Yochai Ben Horin
National Data Center, Soreq Nuclear Research Center, Yavne, Israel
Geophysical Monitoring Programs, Lawrence Livermore National Laboratory, Livermore, California, U.S.A.
Hillel Wust‐Bloch;
Hillel Wust‐Bloch
Yael Radzyner
Lewis Schardong, Yochai Ben Horin, Alon Ziv, Stephen C. Myers, Hillel Wust‐Bloch, Yael Radzyner; High‐Quality Revision of the Israeli Seismic Bulletin. Seismological Research Letters 2021;; 92 (4): 2668–2678. doi: https://doi.org/10.1785/0220200422
Seismic bulletins, with trustworthy phase picks, origin times, and source locations are key for regional seismic studies, such as travel‐time (TT) tomography, attenuation tomography, and anisotropy studies. To lay the groundwork for such studies in Israel, we revised the seismic bulletin of Israel and the surrounding area and obtained a trustworthy TT data set. From the earthquake and explosion bulletins of the Geophysical Institute of Israel, we compiled a starting data set of about 123,000 earthquakes and explosions that occurred during the past 40 yr. After screening out the poorly recorded events, we were left with a data set of
∼38,000
well‐recorded events. We then revised the remaining data set in two consecutive steps. In the first, we reviewed and updated station metadata, including changes in station metadata parameters over time. In the second step, we jointly relocated a list of selected seismic events, using the Bayesian hierarchical location software package (BayesLoc) of Myers et al. (2007) that performs joint relocation of multiple events. We observed striking dissimilarities between the spatial distributions of the newly relocated catalog and the initial locations. Although the depth distribution of the starting catalog is trimodal with peaks at 0, 5, and 10 km, the distribution in this study is unimodal, with a broad peak between 7.5 and 12.5 km. By differencing the observed arrival times and the origin times obtained through relocation with BayesLoc, we obtained a revised TT database that consists of 261,336 Pg, 132,876 Pn, 114,816 Sg, and 60,394 Sn arrivals, from a set of 30,458 jointly relocated seismic sources. We compared prerevision and postrevision TTs as a function of epicentral distance and concluded that the revised data set contains far fewer outliers and inconsistencies than the original data set. The revised TT data set may be used for seismic studies, such as TT tomography, attenuation tomography, and anisotropy studies.
Sn-waves
BayesLoc
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Validate quality of compact credit scorecard model - MATLAB validatemodel - MathWorks América Latina
Validate a Compact Credit Scorecard Model
Validate a Compact Credit Scorecard Model with Weights
Validate a Compact Credit Score Card Model When Using the 'BinMissingData' Option
Use validatemodel with Weights
Validate quality of compact credit scorecard model
Stats = validatemodel(csc,data)
[Stats,T] = validatemodel(___,Name,Value)
[Stats,T,hf] = validatemodel(___,Name,Value)
Stats = validatemodel(csc,data) validates the quality of the compactCreditScorecard model for the data set specified using the argument data.
[Stats,T] = validatemodel(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax and returns the outputs Stats and T.
[Stats,T,hf] = validatemodel(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax and returns the outputs Stats and T and the figure handle hf to the CAP, ROC, and KS plots.
Compute model validation statistics for a compact credit scorecard model.
To create a compactCreditScorecard object, you must first develop a credit scorecard model using a creditscorecard object.
Convert the creditscorecard object into a compactCreditScorecard object. A compactCreditScorecard object is a lightweight version of a creditscorecard object that is used for deployment purposes.
Validate the compact credit scorecard model by generating the CAP, ROC, and KS plots. This example uses the training data. However, you can use any validation data, as long as:
The data has the same predictor names and predictor types as the data used to create the initial creditscorecard object.
The data has a response column with the same name as the 'ResponseVar' property in the initial creditscorecard object.
The data has a weights column (if weights were used to train the model) with the same name as 'WeightsVar' property in the initial creditscorecard object.
[Stats,T] = validatemodel(csc,data,'Plot',{'CAP','ROC','KS'});
Compute model validation statistics for a compact credit scorecard model with weights.
Create a creditscorecard object using the optional name-value pair argument 'WeightsVar'.
Perform automatic binning. By default, autobinning uses the Monotone algorithm.
sc = formatpoints(sc,'PointsOddsAndPDO',[500,2,50]);
Validate the compact credit scorecard model by generating the CAP, ROC, and KS plots. When you use the optional name-value pair argument 'WeightsVar' to specify observation (sample) weights in the original creditscorecard object, the T table for validatemodel uses statistics, sums, and cumulative sums that are weighted counts.
This example uses the training data (dataWeights). However, you can use any validation data, as long as:
The data has a weights column (if weights were used to train the model) with the same name as the 'WeightsVar' property in the initial creditscorecard object.
[Stats,T] = validatemodel(csc,dataWeights,'Plot',{'CAP','ROC','KS'});
Compute model validation statistics and assign points for missing data when using the 'BinMissingData' option.
Predictors in a creditscorecard object that have missing data in the training set have an explicit bin for <missing> with corresponding points in the final scorecard. These points are computed from the Weight-of-Evidence (WOE) value for the <missing> bin and the logistic model coefficients. For scoring purposes, these points are assigned to missing values and to out-of-range values, and after you convert the creditscorecard object to a compactCreditScorecard object, you can use the final score to compute model validation statistics with validatemodel.
Predictors in a creditscorecard object with no missing data in the training set have no <missing> bin, so no WOE can be estimated from the training data. By default, the points for missing and out-of-range values are set to NaN resulting in a score of NaN when running score. For predictors in a creditscorecard object that have no explicit <missing> bin, use the name-value argument 'Missing' in formatpoints to specify how the function treats missing data for scoring purposes. After converting the creditscorecard object to a compactCreditScorecard object, you can use the final score to compute model validation statistics with validatemodel.
Create a creditscorecard object using the CreditCardData.mat file to load dataMissing, a table that contains missing values.
To make any negative age or income information invalid or "out of range," set a minimum value of zero for 'CustAge' and 'CustIncome'. For scoring and probability-of-default computations, out-of-range values are given the same points as missing values.
For the 'CustAge' and 'ResStatus' predictors, the training data contains missing data (NaNs and <undefined> values. For missing data in these predictors, the binning process estimates WOE values of -0.15787 and 0.026469, respectively.
Because the training data contains no missing values for the 'EmpStatus' and 'CustIncome' predictors, neither predictor has an explicit bin for missing values.
Use fitmodel to fit a logistic regression model using Weight of Evidence (WOE) data. fitmodel internally transforms all the predictor variables into WOE values by using the bins found in the automatic binning process. fitmodel then fits a logistic regression model using a stepwise method (by default). For predictors that have missing data, there is an explicit <missing> bin, with a corresponding WOE value computed from the data. When you use fitmodel, the function applies the corresponding WOE value for the <missing> bin when performing the WOE transformation.
Notice that points for the <missing> bin for 'CustAge' and 'ResStatus' are explicitly shown (as 64.9836 and 74.1250, respectively). The function computes these points from the WOE value for the <missing> bin and the logistic model coefficients.
For predictors that have no missing data in the training set, there is no explicit <missing> bin during the training of the model. By default, displaypoints reports the points as NaN for missing data resulting in a score of NaN when you use score. For these predictors, use the name-value pair argument 'Missing' in formatpoints to indicate how missing data should be treated for scoring purposes.
Use compactCreditScorecard to convert the creditscorecard object into a compactCreditScorecard object. A compactCreditScorecard object is a lightweight version of a creditscorecard object that is used for deployment purposes.
For the purpose of illustration, take a few rows from the original data as test data and introduce some missing data. Also introduce some invalid, or out-of-range, values. For numeric data, values below the minimum (or above the maximum) are considered invalid, such as a negative value for age (recall that in a previous step, you set 'MinValue' to 0 for 'CustAge' and 'CustIncome'). For categorical data, invalid values are categories not explicitly included in the scorecard, for example, a residential status not previously mapped to scorecard categories, such as "House", or a meaningless string such as "abc123."
This example uses a very small validation data set only to illustrate the scoring of rows with missing and out-of-range values and the relationship between scoring and model validation.
tdata = dataMissing(11:200,mdl.PredictorNames); % Keep only the predictors retained in the model
tdata.status = dataMissing.status(11:200); % Copy the response variable value, needed for validation purposes
disp(tdata(1:10,:))
NaN Home Owner Employed 51000 11 Yes 519.46 1
52 Other Unknown 42000 12 Yes 1269.2 0
Use validatemodel for a compactCreditScorecard object with the validation data set (tdata).
[ValStats,ValTable] = validatemodel(csc,tdata,'Plot',{'CAP','ROC','KS'});
disp(ValTable(1:10,:))
597.33 NaN 0 1 135 54 0 0.0073529 0.0052632
598.54 NaN 0 2 134 54 0 0.014706 0.010526
601.18 NaN 1 2 134 53 0.018519 0.014706 0.015789
637.3 NaN 1 3 133 53 0.018519 0.022059 0.021053
NaN 0.69421 2 3 133 52 0.037037 0.022059 0.026316
390.86 0.58964 4 5 131 50 0.074074 0.036765 0.047368
404.09 0.57902 6 5 131 48 0.11111 0.036765 0.057895
Validation data, specified as a MATLAB® table, where each table row corresponds to individual observations. The data must contain columns for each of the predictors in the credit scorecard model. The columns of data can be any one of the following data types:
In addition, the table must contain a binary response variable and the name of this column must match the name of the ResponseVar property in the compactCreditScorecard object. (The ResponseVar property in the compactCreditScorecard is copied from the ResponseVar property of the original creditscorecard object.)
If a different validation data set is provided using the optional data input, observation weights for the validation data must be included in a column whose name matches WeightsVar from the original creditscorecard object, otherwise unit weights are used for the validation data. For more information, see Using validatemodel with Weights.
Example: csc = validatemodel(csc,data,'Plot','CAP')
'CAP' — Cumulative Accuracy Profile. Plots the fraction of borrowers up to score “s” against the fraction of defaulters up to score “s” ('PctObs' against 'Sensitivity' columns of T optional output argument). For details, see Cumulative Accuracy Profile (CAP).
'ROC' — Receiver Operating Characteristic. Plots the fraction of non-defaulters up to score “s” against the fraction of defaulters up to score “s” ('FalseAlarm' against 'Sensitivity' columns of T optional output argument). For details, see Receiver Operating Characteristic (ROC).
'KS' — Kolmogorov-Smirnov. Plots each score “s” against the fraction of defaulters up to score “s,” and also against the fraction of nondefaulters up to score “s” ('Scores' against both 'Sensitivity' and 'FalseAlarm' columns of the optional output argument T). For details, see Kolmogorov-Smirnov statistic (KS).
For the Kolmogorov-Smirnov statistic option, you can enter either 'KS' or 'K-S'.
Validation statistics data, returned as an N-by-9 table of validation statistics data, sorted by score from riskiest to safest. N is equal to the total number of unique scores, that is, scores without duplicates.
'Scores' — Scores sorted from riskiest to safest. The data in this row corresponds to all observations up to and including the score in this row.
'TrueBads' — Cumulative number of “bads” up to and including the corresponding score.
'FalseBads' — Cumulative number of “goods” up to and including the corresponding score.
'FalseAlarm' — Fraction of nondefaulters (or the cumulative number of “goods” divided by total number of “goods”). This is the distribution of “goods” up to and including the corresponding score.
The scores of given observations are sorted from riskiest to safest. For a given fraction M (0% to 100%) of the total borrowers, the height of the CAP curve is the fraction of defaulters whose scores are less than or equal to the maximum score of the fraction M. This fraction of defaulters is also known as the “Sensitivity.”.
AR=\frac{{A}_{R}}{{A}_{P}}
where AR is the area between the CAP curve and the diagonal, and AP is the area between the perfect model and the diagonal. This represents a “random” model, where scores are assigned randomly and therefore the proportion of defaulters and nondefaulters is independent of the score. The perfect model is the model for which all defaulters are assigned the lowest scores, and therefore perfectly discriminates between defaulters and nondefaulters. Thus, the closer to unity AR is, the better the scoring model.
This proportion is known as the true positive rate (TPR). Also, the proportion of nondefaulters up to score “s,“ or “False Alarm Rate,” is also computed. This proportion is also known as the false positive rate (FPR). The ROC curve is the plot of the “Sensitivity” vs. the “False Alarm Rate.” Computing the ROC curve is similar to computing the equivalent of a confusion matrix at each score level.
AR=2\left(AUROC\right)-1
The Kolmogorov-Smirnov (KS) plot, also known as the fish-eye graph, is a common statistic for measuring the predictive power of scorecards.
The KS plot shows the distribution of defaulters and the distribution of nondefaulters on the same plot. For the distribution of defaulters, each score “s” is plotted against the proportion of defaulters up to “s," or “Sensitivity." For the distribution of non-defaulters, each score “s” is plotted against the proportion of nondefaulters up to "s," or "False Alarm." The statistic of interest is called the KS statistic and is the maximum difference between these two distributions (“Sensitivity” minus “False Alarm”). The score at which this maximum is attained is also of interest.
If you provide observation weights, the validatemodel function incorporates the observation weights when calculating model validation statistics.
If you do not provide weights, the validation statistics are based on how many good and bad observations fall below a particular score. If you do provide weights, the weight (not the count) is accumulated for the good and the bad observations that fall below a particular score.
When you define observation weights using the optional WeightsVar name-value pair argument when creating a creditscorecard object, the weights stored in the WeightsVar column are used when validating the model on the training data. When a different validation data set is provided using the optional data input, observation weights for the validation data must be included in a column whose name matches WeightsVar. Otherwise, the unit weights are used for the validation data set.
The observation weights of the training data affect not only the validation statistics but also the credit scorecard scores themselves. For more information, see Using fitmodel with Weights and Credit Scorecard Modeling Using Observation Weights.
[3] Loeffler, G. and P. N. Posch. Credit Risk Modeling Using Excel and VBA. Wiley Finance, 2007.
compactCreditScorecard | probdefault | displaypoints | score
|
{\displaystyle n}
{\displaystyle n}
{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle 13}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle (n-1)^{\textrm {th}}}
case. In such situations, strong induction assumes that the conjecture is true for ALL cases of lower value tha{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
|
{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle 13}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle (n+1)^{\textrm {th}}}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1,}}
{\displaystyle 1}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1,}}
{\displaystyle n=1.}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{4}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
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Paper published: Occam's Razor in sensorimotor learning – Tim Genewein
Paper published: Occam's Razor in sensorimotor learning
Genewein T, Braun DA (2104) Occam's Razor in sensorimotor learning. Proceedings of the Royal Societey B 281:20132952. doi: 10.1098/rspb.2013.2952
Experimental design - marginal likelihood of a Gaussian processes
Our paper on sensorimotor regression and the preference of simpler models got published in Proceedings B of the Royal Society.
In the paper we present a sensorimotor regression paradigm that allows for testing whether humans act in line with Occam’s razor and prefer the simpler explanation if two models explain the data equally well.
We modeled human choice behavior with Bayesian model selection. In Bayesian model selection, models
M_1
M_2
are compared by taking their posterior probability ratio (given the data $D$)
where the model evidence or marginal likelihood is given by marginalizing over the model parameters
\theta
If the ratio of posterior odds is larger than one, model
M_1
should be preferred - if the posterior odds ratio is smaller than one, the data is in favor of
M_2
. If the strength of preference is ignored, the previous rule leads to a deterministic model selection mechanism. Additionally, the magnitude of the posterior odds ratio is an indicator of the strength of the preference which can be used to derive stochastic model selection mechanisms (for instance by combination with a softmax selection rule). In case of equal prior probabilities for both models, the quantity that governs model selection is the Bayes factor (sometimes called Occam factor), which is the ratio of marginal likelihoods.
It is known that Bayesian model selection embodies Occam’s razor, that is if two models explain the data equally well, prefer the simpler model. Below are two intuitions why this happens - for a more rigorous discussion see Chapter 28 in David MacKay’s Information Theory, Inference, and Learning Algorithms (Cambridge University Press, 2003).
The marginal likelihood
P(D\vert M_i)
is obtained by considering the likelihood of the parameters given the data
P(D\vert \theta,M_i)
under all possible parameter-settings (by taking the integral, see above). A very complex model is highly likely to have a particular parameter-setting that explains the data very well and leads to a large value of the likelihood function (think of polynomial regression with polynomials of a high degree). However, because of the marginalization, all parameters have to be taken into account and a very complex model will also contain many parameter-settings that yield a very bad likelihood. In contrast, an overly simple model will on average neither explain the data very badly but also not very well, leading to a low marginal likelihood as well. Only models that are complex enough but not overly complex will get a large marginal likelihood score. Thus Bayesian model selection is intrinsically regularized.
A complex model can explain many data-sets and must thus spread its probability mass over a quite large range (or volume). In contrast, a simple model can only explain few data-sets and has its probability mass more concentrated. If observed data happens to fall in a region where both models can explain the data, the simpler model is very likely to have more probability mass in that region and will thus be favored.
In our experiment we test whether humans follow Occam’s razor and prefer the simpler model if two models explained the data equally well. We translated the classical regression problem of finding a curve underlying noisy observations into a sensorimotor task: in our experiment participants would see some dots that represent noisy observations of an underlying curve. Their task was to draw their best guess of the underlying curve. In training trials, after participants had drawn their curve, they were shown the true underlying curve that generated the noisy observations. Importantly, the underlying curve could only be generated by one of two models - a simple model that leads to smooth curves and a complex model that leads to more wiggly curves. In training trials, participants were informed at the start of the trial about the generative model with a color cue. In test trials we showed participants the same stimulus (noisy observations) but with a neutral color cue that did not indicate the underlying model. We asked participants to draw their best guess of the underlying function, informing them that the generative model could only be one of the two models experienced previously. From their drawing we could infer their model choice and could thus test human behavior against theoretical choice behavior modeled with Bayesian model selection.
The two underlying models that generated the curves were Gaussian processes (GPs) with a squared-exponential kernel with either a long or a short length-scale (corresponding to a simple or complex model respectively). We found Gaussian processes very suitable for generating natural trajectories (which is much more difficult with polynomials for instance) but perhaps most importantly we could use the Gaussian processes to generate trials where the noisy observations can be fitted with both models equally well. Inspecting the (closed-form) analytical expression for the marginal log likelihood of the Gaussian process reveals that it is composed of a data-fit-error term and a data-independent complexity term
y
denotes (noisy) observations at locations
X
M_{\lambda}
indicates the models with different length-scales
\lambda
K_{\lambda}
is the covariance matrix of the GP obtained by evaluating the kernel function for all input-pairs
k_{\lambda}(x_i, x_j)
(which depends on the length-scale
\lambda
N
Importantly, the model complexity term is independent of the observations
y
(in our case the number of observations $N$ was fixed throughout the whole experiment) and is governed by the length-scale parameter of the different models. The complexity term consists of the log determinant of the covariance matrix and a factor that depends on the number of observations. One explanation why this term corresponds to a complexity term is that it is the entropy of the (posterior) GP (since the GP is essentially a multivariate Gaussian) - a large entropy would thus imply a large complexity and vice versa. See the publication for more discussion on the complexity term and also some simulations that confirm that the shorter-lengthscale model also incurs a larger complexity-term.
Given the analytical expression of the data-fit-error term that is independent of the model complexity allowed us to use the GP framework to create stimuli where the noisy observations lead to the same data-fit-error term under both models. We used these trials to test whether humans would prefer the simpler model as an explanation in case both models would fit the data equally well. Additionally, we could also generate trials where the data-fit-error term was different under both models and use the marginal likelihood ratio to predict theoretical choice probabilities according to Bayesian model selection (paired with a softmax selection rule).
In equal-error trials where the data-fit-error term was the same under both models, we found that humans followed Occam’s razor and strongly preferred the simpler model. In general trials with arbitrary data-fit-error terms we found participants’ behavior to be quantitatively consistent with the theoretical model of Bayesian model selection combined with a softmax model selection mechanism.
Additionally we performed a control experiment to rule out that the preference of the simpler model was simply due to a preference for lower physical effort (as smooth trajectories require less effort to draw) by having participants indicate their model choice with a mouse-click an then drawing a trajectory automatically according to their model choice. We found that the physical effort of drawing had on impact no model choice in our task.
We also performed a control experiment to rule out that the preference was not due to model complexity but simply due to smoothness of trajectories. We designed the two GP models in a way such that the wiggly model became the simple model. To achieve this we used a wiggly mean-function in the short length-scale model but then tuned the parameters of the GP such that the trajectories generated by the model are very similar to the mean and have very little variation. This lead to a wiggly generative model that produced functions of high spatial frequency but with very little variation (simple model, because it only explains a very narrow range of observations) and a smooth generative model that produced a large variety of functions (complex model, even though the generated functions have low spatial frequency). See the publication for more details.
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Shallow Structure and Geomorphology along the Offshore Northern San Andreas Fault, Tomales Point to Fort Ross, CaliforniaShallow Structure and Geomorphology along the Offshore Northern San Andreas Fault | Bulletin of the Seismological Society of America | GeoScienceWorld
U.S. Geological Survey, Pacific Coastal and Marine Science Center, 2885 Mission Street, Santa Cruz, California 95060 U.S.A., sjohnson@usgs.gov
Fugro USA Marine, Inc., P.O. Box 740010, Houston, Texas 77274 U.S.A., j.beeson@fugro.com
Samuel Y. Johnson, Jeffrey W. Beeson; Shallow Structure and Geomorphology along the Offshore Northern San Andreas Fault, Tomales Point to Fort Ross, California. Bulletin of the Seismological Society of America 2019;; 109 (3): 833–854. doi: https://doi.org/10.1785/0120180158
We mapped a poorly documented 35‐km‐long section of the northern San Andreas fault (NSAF) zone between Tomales Point and Fort Ross, California. Mapping is largely based on high‐resolution seismic‐reflection profiles (38 fault crossings), multibeam bathymetry, and onshore geology. NSAF strike in this section is nearly parallel to plate motion, characterized by a slight (
∼2°
) northerly (transtensional) bend in the south between Tomales Bay and the Bodega isthmus, and a northwesterly (transpressional)
∼5°
bend in the north between the Bodega isthmus and Fort Ross. The southern transtensional bend is the northern part of the now‐submerged, linear,
∼50‐km‐long
and 1–2‐km‐wide, Tomales–Bodega valley. The valley floor is cut by a complex zone of subparallel, variably continuous fault strands, and the deformed valley fill is an inferred mix of late Quaternary marine and nonmarine strata. In the northern part of this elongate valley, Holocene fault offset occurred on two fault strands about 740 m apart. The northern transpressional bend is characterized by narrow, elongate, asymmetric basins containing as much as 56 m of inferred latest Pleistocene to Holocene sediment.
Between Bodega Head and Fort Ross, the gently dipping (
∼0.8°
) shelf includes two large (4.8 and
5.9 km2
) zones of sediment failure that we speculatively correlate with the 1906 San Francisco NSAF earthquake. Similar sediment‐failure zones should be common along offshore reaches of the NSAF and other nearshore fault zones but have apparent limited preservation potential. Onland geomorphic impacts of the mainly offshore NSAF include: (1) northward upwarping of uplifted marine terraces in the transpressional zone north of Bodega Bay; and (2) blocking of littoral sediment transport by uplifts on the west flank of the NSAF at Bodega Head and Tomales Point, resulting in rapidly accreting beaches and large coastal sand dune complexes.
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Jeremy Siek: Big-step, diverging or stuck?
In a naive big-step semantics, one can't tell the difference between the following two programs, even though their behavior is quite different according to single-step reduction rules. The first program diverges (runs forever) whereas the second program is stuck, that is, it results in a run-time type error because you can't perform function application with a number in the function position. However, a naive big-step semantics is undefined for both programs. That is, you can't build a derivation of the evaluation relation for either program.
The naive big-step semantics that I'm referring to is the following. The
\rho
maps variables to values (parameters to their arguments), and is called the environment.
Trouble formulating type safety
This inability to distinguish between divergence and stuck makes it difficult to formulate type safety using a naive big-step semantics. The following statement is true but not very strong. It captures the notion of type preservation, but not progress. That is, the above does not rule out the possibility of type errors because the premise
\emptyset \vdash e \Downarrow v
is false in such cases, making the entire statement vacuously true. On the other hand, the following statement is so strong that it is false. One counter-example to the above statement is the diverging program above. The program is well typed, but it does not terminate and return a value.
Several textbooks stop at this point and just say that the above is a problem with big-step semantics. Not true! There are several solutions to this problem in the literature. I've included papers that address the divergence/stuck issue at the bottom of this blog. If you know of others, please let me know! The solutions I've seen fall into three categories:
Introduce a separate co-inductive definition of divergence for the language in question.
Develop a notion of partial derivations.
Add a time counter that causes a time-out error when it gets to zero. That is, make the semantics step-indexed.
Of the three kinds of solutions, the counter-based solution is the simplest, and it gets the job done, so I won't discuss the other two. I first learned of this approach from the paper A Virtual Class Calculus by Ernst, Ostermann, and Cook.
Big-step Semantics with Counters
The first part of the solution is to augment the big-step semantics with a counter k, making sure to decrement the counter in the premises of each rule. In addition, all of the above rules have a side condition requiring
k > 0
. Next, we add a rule to handle the
k = 0
case. Finally, we add the usual rules for propagating errors such as this time out.
With the addition of the counter, we can always build a derivation for a diverging program, with the result being a time out, no matter how high you set the counter to begin with. However, for a program that gets stuck, there will be some starting counter that is high enough so that you can't build a derivation anymore (you might have been able to get time outs with lower values).
The type safety theorem can now be stated in a way that is both strong enough and true! Let r range over both values and the time out. (We let the time out take on any type.) At first glance this statement might seem weak. If the counter k is 0, then its trivial to build a derivation of
\emptyset; k \vdash e \Downarrow r
. However, note that the statement says for any k. Thus, if the program get's stuck, there will be some value for k for which you can't build a derivation, making the above statement false. So this indeed is a good formulation of type safety. If the above holds, then no well-typed program will get stuck!
One parting comment. The addition of propagation rules was rather annoying. I'll discuss how to avoid propagation rules in big-step semantics in an upcoming blog post, summarizing a talk I gave at Scottish Programming Languages Seminar in 2010.
C. A. Gunter and D. Rémy. A Proof-Theoretic Assesment of Runtime Type Errors. AT&T Bell Laboratories Technical Memo 11261-921230-43TM, 1993.
A. Stoughton. An operational semantics framework supporting the incremental construction of derivation trees. Electr. Notes Theor. Comput. Sci., 10, 1997.
H. Ibraheem and D. A. Schmidt. Adapting big-step semantics to small-step style: Coinductive interpretations and “higher-order” derivations. In Second Workshop on Higher-Order Techniques in Operational Semantics (HOOTS2). Elsevier, 1997.
Kathleen Fisher and John Reppy. Statically typed traits. University of Chicago, technical report TR-2003-13, December 2003.
Erik Ernst, Klaus Ostermann, and William R. Cook. A Virtual Class Calculus. In POPL 2006.
Xavier Leroy and Herve Grall. Coinductive big-step operational semantics. Journal of Information and Computation. Vol. 207, Issue 2, February 2009.
Big-step Operational Semantics Revisited, by Jarosław Dominik, Mateusz Kuśmierek, and Viviana Bono. In Fundamenta Informaticae Volume 103, Issue 1-4, January 2010.
I did a double take on the statement 42 3 because it looked to me like 423. You might want to increase the space to \: or even \;.
Typo: JarosłAw should be Jarosław.
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{\displaystyle n}
{\displaystyle n}
{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle 13}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle (n-1)^{\textrm {th}}}
case. In such situations, strong induction assumes that the conjecture is true for ALL cases of lower value tha{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1.}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
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Plot antenna or transducer element directivity and pattern versus azimuth - MATLAB patternAzimuth - MathWorks Deutschland
Plot antenna or transducer element directivity and pattern versus azimuth
patternAzimuth(element,FREQ)
patternAzimuth(element,FREQ,EL)
patternAzimuth(element,FREQ,EL,Name,Value)
patternAzimuth(element,FREQ) plots the 2-D element directivity pattern versus azimuth (in dBi) for the element element at zero-degrees elevation angle. The argument FREQ specifies the operating frequency.
patternAzimuth(element,FREQ,EL), in addition, plots the 2-D element directivity pattern versus azimuth (in dBi) at the elevation angle specified by EL. When EL is a vector, multiple overlaid plots are created.
patternAzimuth(element,FREQ,EL,Name,Value) plots the element pattern with additional options specified by one or more Name,Value pair arguments.
Example: Azimuth,[-90:90],Type,'directivity'
D=4\pi \frac{{U}_{\text{rad}}\left(\theta ,\phi \right)}{{P}_{\text{total}}}
directivity | pattern | patternElevation | beamwidth
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Topic: Systems/Cybernetics
You are looking at all articles with the topic "Systems/Cybernetics". We found 4 matches.
{\displaystyle \Delta E}
{\displaystyle \Delta t={\frac {\pi \hbar }{2\Delta E}}.}
{\displaystyle \Delta t={\frac {\pi \hbar }{2E}}}
Systems Robotics Systems/Cybernetics
The sonar guidance system was developed for Mod I and improved for Mod II. It used two ultrasonic transducers to determine distance, location within the halls, and obstructions in its path. This provided "The Beast" with bat-like guidance. At this point, it could detect obstructions in the hallway, such as people in the hallway. Once an obstruction was detected, the Beast would slow down and then decide whether to stop or divert around the obstruction. It could also ultrasonically recognize the stairway and doorways to take appropriate action.
"Johns Hopkins Beast" | 2022-03-30 | 198 Upvotes 48 Comments
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February - Volume 100, Number 1 April - Volume 100, Number 2 June - Volume 100, Number 3 August - Volume 100, Number 4 October - Volume 100, Number 5A November - Volume 100, Number 5B December - Volume 100, Number 6
Earthquake Monitoring in Southern California for Seventy-Seven Years (1932–2008)
Kate Hutton; Jochen Woessner; Egill Hauksson
Petra Adamová; Jan Šílený
Fabian Walter; Douglas S. Dreger; John F. Clinton; Nicholas Deichmann; Martin Funk
Slip-Length Scaling Law for Strike-Slip Multiple Segment Earthquakes Based on Dynamic Rupture Simulations
Physics-Based Earthquake Source Characterization and Modeling with Geostatistics
Seok Goo Song; Paul Somerville
On Possible Plume-Guided Seismic Waves
Bruce R. Julian; John R. Evans
Precursory Rise of P-Wave Attenuation before the 2004 Parkfield Earthquake
K.-Y. Chun; Q.-Y. Yuan; G. A. Henderson
Triggering Effect of M 4–5 Earthquakes on the Earthquake Cycle of Repeating Events at Parkfield, California
Kate Huihsuan Chen; Roland Bürgmann; Robert M. Nadeau
Localized Surface Disruptions Observed by InSAR during Strong Earthquakes in Java and Hawai‘i
Five Short Historical Earthquake Surface Ruptures near the Silk Road, Gansu Province, China
Xiwei Xu; Robert S. Yeats; Guihua Yu
A Catalog of Felt Intensity Data for 570 Earthquakes in India from 1636 to 2009
Stacey Martin; Walter Szeliga
Walter Szeliga; Susan Hough; Stacey Martin; Roger Bilham
Chin-Jen Lin; Han-Pang Huang; Chun-Chi Liu; Hung-Chie Chiu
An Optical Seismometer without Force Feedback
Mark Zumberge; Jonathan Berger; Jose Otero; Erhard Wielandt
On the Composition of Earth’s Short-Period Seismic Noise Field
Keith D. Koper; Kevin Seats; Harley Benz
J. Díaz; A. Villaseñor; J. Morales; A. Pazos; D. Córdoba; J. Pulgar; J. L. García-Lobón; M. Harnafi; R. Carbonell; J. Gallart; TopoIberia Seismic Working Group
Detection of Systematic Errors in Travel-Time Data Using a Minimum 1D Model: Application to Costa Rica Seismic Tomography
V. Maurer; E. Kissling; S. Husen; R. Quintero
Stephen C. Myers; Michael L. Begnaud; Sanford Ballard; Michael E. Pasyanos; W. Scott Phillips; Abelardo L. Ramirez; Michael S. Antolik; Kevin D. Hutchenson; John J. Dwyer; Charlotte A. Rowe; Gregory S. Wagner
P-Wave Back-Azimuth and Slowness Anomalies Observed by an IMS Seismic Array LZDM
Chunyue Hao; Zhong Zheng
Polarization Analysis in the Discrete Wavelet Domain: An Application to Volcano Seismology
L. D’Auria; F. Giudicepietro; M. Martini; M. Orazi; R. Peluso; G. Scarpato
Tpd, a Damped Predominant Period Function with Improvements for Magnitude Estimation
Mark William Hildyard; Andreas Rietbrock
John X. Zhao
2D Analysis of Earthquake Ground Motion in Haifa Bay, Israel
Zohar Gvirtzman; John N. Louie
Ground-Motion Prediction Equations for Hawaii from a Referenced Empirical Approach
Hidenori Mogi; Santa Man Shrestha; Hideji Kawakami; Shinya Okamura
Statistical Validity Control on SPAC Microtremor Observations Recorded with a Restricted Number of Sensors
M. Claprood; M. W. Asten
Site-Specific and Spatially Distributed Ground-Motion Prediction of Acceleration Spectrum Intensity
Giovanna Laurenzano; Enrico Priolo; Maria Rosaria Gallipoli; Marco Mucciarelli; Felice C. Ponzo
Moderate Earthquake Ground-Motion Validation in the San Francisco Bay Area
Ahyi Kim; Douglas S. Dreger; Shawn Larsen
Nonlinear Behavior of Strong Surface Waves Trapped in Sedimentary Basins
Scattering and Intrinsic Attenuation of Short-Period S Waves in the Gyeongsang Basin, South Korea, Revealed from S-Wave Seismogram Envelopes Based on the Radiative Transfer Theory
Won Sang Lee; Sukyoung Yun; Ji-Young Do
Segmentally Iterative Ray Tracing in Complex 2D and 3D Heterogeneous Block Models
Tao Xu; Zhongjie Zhang; Ergen Gao; Guoming Xu; Lian Sun
Lg Attenuation in a Region with Both Continental and Oceanic Environments
Earthquake Source Scaling: m1 versus Other Magnitude Scales
Shengzao Chen
Assessment of Regional-Distance Location Calibration Using a Multiple-Event Location Algorithm
Megan L. Anderson; Stephen C. Myers
Yoshiki Tsukakoshi
The Vallo di Diano Fault System: New Evidence for an Active Range-Bounding Fault in Southern Italy Using Shallow, High-Resolution Seismic Profiling
Pier Paolo Bruno; Luigi Improta; Antonio Castiello; Fabio Villani; Paola Montone
An Investigation into the Seismic Potential of the Irrawaddy Region, Northern Sunda Arc
Bhaskar Kundu; V. K. Gahalaut
Comment on “Is There a Basis for Preferring Characteristic Earthquakes over a Gutenberg–Richter Distribution in Probabilistic Earthquake Forecasting” by Tom Parsons and Eric L. Geist
Reply to “Comment on ‘Is There a Basis for Preferring Characteristic Earthquakes over a Gutenberg–Richter Distribution in Probabilistic Earthquake Forecasting?’ by Tom Parsons and Eric L....
Tom Parsons; Eric L. Geist
Mw
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Quick Help - Detailed Information - Maple Help
Home : Support : Online Help : Getting Started : Quick Help - Detailed Information
Math Editor Shortcuts
Evaluate and Display on New Line
Toggle Math/Text
Toggle Executable Math
Exit Fraction/Superscript/Subscript
Navigate Placeholders
Keyboard shortcuts are available for all operations, in addition to palette and mouse operations.
Common Keyboard Shortcuts (See full list)
Symbol/Formats
Toggle between math and text mode
{}^{1}
Example using fraction:
\frac{1}{4}
(Math mode) versus
1/4 (Text mode)
Esc, Mac and Windows
Command + Shift + Space, Mac
{}^{2}
\frac{1}{4}
{}^{2}
{x}^{2}
{}^{2}
Ctrl + _ (Command + _, Mac)
{x}_{a}
{}^{2}
\mathrm{x__max}
\^ (caret)
u^v
sqrt+command/symbol completion keys
\sqrt{25}
Navigating the expressions
\frac{1}{x+\frac{1}{2}}
{}^{1}
F5 is a three way toggle between math, text, and nonexecutable math modes. See Toggle Math/Text.
{}^{2}
use right arrow key to leave denominator, superscript, or subscript region
Use Enter to evaluate your mathematical expression or Maple command. Maple calculates and displays the result on a new line.
\int {x}^{3}+x ⅆx
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{4}}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}
From the Expression palette, select
∫\textcolor[rgb]{0,0.627450980392157,0.313725490196078}{f}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}ⅆ\textcolor[rgb]{0.784313725490196,0,0.784313725490196}{x}
x^3 + x Tab x and press Enter.
factor(x^3+x);
\textcolor[rgb]{0,0,1}{x}\left({\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}\right)
Enter text as it appears and press Enter.
To display the result beside the original expression, see Evaluate and Display Inline.
In Document mode, use Alt+Enter (Alt+Return, for Mac) to evaluate your mathematical expression or Maple command and display inline.
1+1
\textcolor[rgb]{0,0,1}{2}
1 + 1 Alt+Enter (Alt+Return, for Mac)
\mathrm{sin}\left(\frac{\mathrm{\pi }}{3}\right)
\frac{\sqrt{\textcolor[rgb]{0,0,1}{3}}}{\textcolor[rgb]{0,0,1}{2}}
From the Common Symbols palette, select
\mathrm{\pi }
Alt + Enter (Alt + Return, for Mac)
To display the Maple palettes, see Arranging Palettes in Your Worksheet.
Clickable Math via the Context Panel
To easily manipulate an expression in Maple, select the expression to display a context-sensitive list of applicable options in the Context Panel, located on the right side if the Maple window. When you choose an item in the context panel, the result is displayed inline.
{x}^{2}
View the Context Panel for this expression. The options include: differentiate, evaluate at a point, and more.
{x}^{2}
\stackrel{\text{evaluate at point}}{\to }
\textcolor[rgb]{0,0,1}{0.25}
For example, select Evaluate at a point, and then x=0.5
Context-sensitive options are available for Maple input and output.
\mathrm{diff}\left(\mathrm{sin}\left(x\right)\cdot \mathrm{cos}\left(x\right),x\right)
{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}
\mathrm{diff}\left(\mathrm{sin}\left(x\right)\cdot \mathrm{cos}\left(x\right),x\right)
{\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}{\textcolor[rgb]{0,0,1}{\mathrm{sin}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{x}\right)}^{\textcolor[rgb]{0,0,1}{2}}
\stackrel{\text{combine}}{=}
\textcolor[rgb]{0,0,1}{\mathrm{cos}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{x}\right)
Select from the Context Panel Combine > trig.
In Document mode, toggling between Text and Math modes switches between entering text and entering 2-D mathematical expressions
\left(\frac{{x}^{2}}{y}\right)
Use the F5 key to toggle between three states: executable math, text entry, and nonexecutable math. Math mode is characterized by a slanted, italic prompt (/) whereas the Text mode is characterized by a regular prompt (|). Executable math is math you want to evaluate, whereas nonexecutable math is for display purposes only.
In Document Mode
Here, the first example uses nonexecutable math, and the second example uses executable math, and then evaluates and displays output.
A parabola has the form
y={a}^{2}\cdot x+b\cdot x+c
A parabola has the form F5 y=a*x^2+b*x+c
A simple problem: 1+3 = 4
F5 A simple problem: F5 F5 1 + 3 Alt + Enter
(Alt + Return, for Mac)
Note: For more information see Select Entry Mode.
In Worksheet Mode
In Worksheet mode, toggling between Math and Text modes switches between 2-D and 1-D Math commands.
∫{x}^{3}+{x}^{2} ⅆx
∫\textcolor[rgb]{0,0.627450980392157,0.313725490196078}{f}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}ⅆ\textcolor[rgb]{0.784313725490196,0,0.784313725490196}{x}
x^3 + x^2 Tab x
int(x^3+x^2, x);
Enter text as it appears.
To prevent execution of a math expression, make the math nonexecutable. Click the expression and use the shortcut Shift + F5.
For details, see Executable and Nonexecutable Math.
To leave the superscript, subscript, or denominator of a fraction, use the right-arrow key.
{x}^{2}
Right-arrow to leave the superscript region
\mathrm{x__a}
For a literal subscript (subscripted variable name), use two underscores:
Right-arrow to leave the subscript region
\frac{1}{x+\frac{2}{y}+3}
Right-arrow to leave the denominator of
\frac{2}{y}
Right-arrow to leave the denominator
Note: In Maple, subscripts can be indexed or literal. For details, see 2-D Math Shortcut Keys and Hints.
Use := (colon equals) to assign a value to a variable.
Note: To make the assignment, you must execute the statement. You can terminate the statement with a colon (:) if you do not want to see the output. In document mode, you can alternatively Toggle Input/Output Display to prevent display of the output.
x≔5
F5 Let F5 a:= 5 Enter
(F5 is the toggle between Text mode and Math mode.)
To define a function, use the arrow notation.
f:=x→{x}^{2}:
f\left(3\right)
\textcolor[rgb]{0,0,1}{9}
\textcolor[rgb]{0,0.627450980392157,0.313725490196078}{f}:=\textcolor[rgb]{0.784313725490196,0,0.784313725490196}{a}→\textcolor[rgb]{0.784313725490196,0,0.784313725490196}{y}
x^2: Enter
f(3) Alt + Enter (Alt + Return, for Mac)
G := (x,y,z) -> x^2+y^2+z^2;
\textcolor[rgb]{0,0,1}{G}\textcolor[rgb]{0,0,1}{≔}\left(\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}{z}\right)\textcolor[rgb]{0,0,1}{↦}{\textcolor[rgb]{0,0,1}{x}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{z}}^{\textcolor[rgb]{0,0,1}{2}}
G(-1,0,1);
\textcolor[rgb]{0,0,1}{2}
Enter text as it appears. Press enter to evaluate.
Equations are represented using the = sign.
y=m\cdot x+b
To assign an equation to a variable, use the assignment operator.
\mathrm{eqn}≔ y=m\cdot x+b
After assigning the equation to a variable name, you can use the name to manipulate the equation.
\mathrm{rhs}\left(\mathrm{eqn}\right)
\textcolor[rgb]{0,0,1}{m}\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}
When you type a symbol name in Math mode, Esc (see command/symbol completion keys) automatically converts the name into a properly displayed symbol. When you type the first few letters of a symbol name, Esc displays a popup list of matching symbols. Use Tab to select the proper completion.
\sqrt{x}
sqrt then Esc then Tab x
In Math and Text modes, the command/symbol completion keys are used to complete a Maple command or a user-defined variable name. All possible completions are displayed in a popup list. If there is a unique choice, the command is completed automatically.
Stud Esc [L Esc [Gr
The Expression palette and task templates use fill-in-the-blank placeholders. To fill the placeholders, complete the first entry and then Tab to the next.
The Tab key is also used for indenting in Text mode. You can control whether the Tab key is for moving between placeholders or indenting by using Format > Tab Navigation. See Using the Tab Key.
To display the Maple Help Navigator: Select Help > Maple Help or press F1.
To display the Quick Reference Card: Select Help > Quick Reference or press Ctrl + F2.
To display help on a particular topic: Enter ?
\mathrm{topic}
or place the cursor on the topic name and press F2. For example ?
\mathrm{factor}
displays the help page for the factor command.
Interactive Assistants, Tutors, Task Templates, and Demonstrations
Access an extensive list of interactive assistants, tutors, task templates, and demonstrations through the Tools menu.
Tools > Assistants provides access to easy-to-use assistants that allow you to accomplish various tasks through point-and-click interfaces. Assistants include code translation, creating plots, and converting units.
Tools > Math Apps provides access to interactive demonstrations for exploring concepts in mathematics and science.
Tools > Tutors provides access to interactive tutors covering topics in precalculus, calculus, vector calculus, multivariate calculus, linear algebra, numerical analysis, differential equations, and statistics.
Tools > Tasks > Browse is the entry point to an extensive list of task templates, which show you the steps needed to solve a particular problem and provide you with a fill-in-the-blank template of the corresponding Maple command.
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An agent is a capsule shaped entity that can navigate within scenes and interact with objects.
A scene within AI2-THOR represents a virtual room that an agent can navigate and interact in. There are 4 scene Categories, each with 30 unique scenes within them: Kitchen, Living Room, Bedroom, Bathroom.
An action is a discrete command for the agent to perform within a scene (e.g. MoveAhead, RotateRight, PickupObject).
Sim Object
A sim object is an object that can be interacted with by the agent. Objects have a range of interactions based on their assigned
An object is considered
when it satisfies two conditions:
In frame. The object appears in the camera's frame (i.e., it is within the camera's field of view). Even if an object is partially blocked by another object, as long as some of it is in the frame, that is sufficient for this condition.
Proximity. The object is within a distance of visibilityDistance from the agent's central vertical axis. By default, visibilityDistance is set to
1.5
meters, but it can be overwritten upon initialization.
Obstruction. The object is not obstructed. Here, a ray emitted from the camera must hit the object without first hitting another object.
An object rendered in an image will not always be visible to the agent. For example, an object outside the
1.5
meter threshold could be seen in an image, but will be reported as not-visible to the agent.
Each object's visibility is reported in the metadata.
Object Interactability
An object is said to be interactable if it is both
and if it is unobstructed by any other objects. For instance, if a
, then it will report
, but the agent will be unable to interact with it due to the obstruction.
If an object is interactable, then that implies the object is visible. But, if an object is visible, that does not imply it is interactable.
is a type of object that may be the parent of another object. Examples include
. The full set of
object types can be filtered on the Object Types.
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Joint_probability_distribution Knowpia
{\displaystyle X}
{\displaystyle Y}
{\displaystyle p(X)}
{\displaystyle p(Y)}
Draws from an urnEdit
Suppose each of two urns contains twice as many red balls as blue balls, and no others, and suppose one ball is randomly selected from each urn, with the two draws independent of each other. Let
{\displaystyle A}
{\displaystyle B}
be discrete random variables associated with the outcomes of the draw from the first urn and second urn respectively. The probability of drawing a red ball from either of the urns is 2/3, and the probability of drawing a blue ball is 1/3. The joint probability distribution is presented in the following table:
Moreover, the final row and the final column give the marginal probability distribution for A and the marginal probability distribution for B respectively. For example, for A the first of these cells gives the sum of the probabilities for A being red, regardless of which possibility for B in the column above the cell occurs, as 2/3. Thus the marginal probability distribution for
{\displaystyle A}
{\displaystyle A}
's probabilities unconditional on
{\displaystyle B}
, in a margin of the table.
Coin flipsEdit
Consider the flip of two fair coins; let
{\displaystyle A}
{\displaystyle B}
be discrete random variables associated with the outcomes of the first and second coin flips respectively. Each coin flip is a Bernoulli trial and has a Bernoulli distribution. If a coin displays "heads" then the associated random variable takes the value 1, and it takes the value 0 otherwise. The probability of each of these outcomes is 1/2, so the marginal (unconditional) density functions are
{\displaystyle P(A)=1/2\quad {\text{for}}\quad A\in \{0,1\};}
{\displaystyle P(B)=1/2\quad {\text{for}}\quad B\in \{0,1\}.}
The joint probability mass function of
{\displaystyle A}
{\displaystyle B}
defines probabilities for each pair of outcomes. All possible outcomes are
{\displaystyle (A=0,B=0),(A=0,B=1),(A=1,B=0),(A=1,B=1).}
{\displaystyle P(A,B)=1/4\quad {\text{for}}\quad A,B\in \{0,1\}.}
{\displaystyle P(A,B)=P(A)P(B)\quad {\text{for}}\quad A,B\in \{0,1\}.}
Rolling a dieEdit
Consider the roll of a fair dice and let
{\displaystyle A=1}
if the number is even (i.e. 2, 4, or 6) and
{\displaystyle A=0}
otherwise. Furthermore, let
{\displaystyle B=1}
if the number is prime (i.e. 2, 3, or 5) and
{\displaystyle B=0}
Then, the joint distribution of
{\displaystyle A}
{\displaystyle B}
, expressed as a probability mass function, is
{\displaystyle \mathrm {P} (A=0,B=0)=P\{1\}={\frac {1}{6}},\quad \quad \mathrm {P} (A=1,B=0)=P\{4,6\}={\frac {2}{6}},}
{\displaystyle \mathrm {P} (A=0,B=1)=P\{3,5\}={\frac {2}{6}},\quad \quad \mathrm {P} (A=1,B=1)=P\{2\}={\frac {1}{6}}.}
These probabilities necessarily sum to 1, since the probability of some combination of
{\displaystyle A}
{\displaystyle B}
occurring is 1.
Marginal probability distributionEdit
If the joint probability density function of random variable X and Y is
{\displaystyle f_{X,Y}(x,y)}
, the marginal probability density function of X and Y, which defines the Marginal distribution, is given by:
{\displaystyle f_{X}(x)=\int f_{X,Y}(x,y)\;dy}
{\displaystyle f_{Y}(y)=\int f_{X,Y}(x,y)\;dx}
Joint cumulative distribution functionEdit
For a pair of random variables
{\displaystyle X,Y}
, the joint cumulative distribution function (CDF)
{\displaystyle F_{XY}}
is given by[3]: p. 89
{\displaystyle F_{X,Y}(x,y)=\operatorname {P} (X\leq x,Y\leq y)}
where the right-hand side represents the probability that the random variable
{\displaystyle X}
takes on a value less than or equal to
{\displaystyle x}
{\displaystyle Y}
{\displaystyle y}
{\displaystyle N}
{\displaystyle X_{1},\ldots ,X_{N}}
, the joint CDF
{\displaystyle F_{X_{1},\ldots ,X_{N}}}
{\displaystyle F_{X_{1},\ldots ,X_{N}}(x_{1},\ldots ,x_{N})=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})}
Interpreting the
{\displaystyle N}
random variables as a random vector
{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{N})^{T}}
yields a shorter notation:
{\displaystyle F_{\mathbf {X} }(\mathbf {x} )=\operatorname {P} (X_{1}\leq x_{1},\ldots ,X_{N}\leq x_{N})}
Joint density function or mass functionEdit
Discrete caseEdit
The joint probability mass function of two discrete random variables
{\displaystyle X,Y}
{\displaystyle p_{X,Y}(x,y)=\mathrm {P} (X=x\ \mathrm {and} \ Y=y)}
{\displaystyle p_{X,Y}(x,y)=\mathrm {P} (Y=y\mid X=x)\cdot \mathrm {P} (X=x)=\mathrm {P} (X=x\mid Y=y)\cdot \mathrm {P} (Y=y)}
{\displaystyle \mathrm {P} (Y=y\mid X=x)}
{\displaystyle Y=y}
{\displaystyle X=x}
The generalization of the preceding two-variable case is the joint probability distribution of
{\displaystyle n\,}
{\displaystyle X_{1},X_{2},\dots ,X_{n}}
{\displaystyle p_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})=\mathrm {P} (X_{1}=x_{1}{\text{ and }}\dots {\text{ and }}X_{n}=x_{n})}
{\displaystyle {\begin{aligned}p_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})&=\mathrm {P} (X_{1}=x_{1})\cdot \mathrm {P} (X_{2}=x_{2}\mid X_{1}=x_{1})\\&\cdot \mathrm {P} (X_{3}=x_{3}\mid X_{1}=x_{1},X_{2}=x_{2})\\&\dots \\&\cdot P(X_{n}=x_{n}\mid X_{1}=x_{1},X_{2}=x_{2},\dots ,X_{n-1}=x_{n-1}).\end{aligned}}}
{\displaystyle \sum _{i}\sum _{j}\mathrm {P} (X=x_{i}\ \mathrm {and} \ Y=y_{j})=1,\,}
which generalizes for
{\displaystyle n\,}
{\displaystyle X_{1},X_{2},\dots ,X_{n}}
{\displaystyle \sum _{i}\sum _{j}\dots \sum _{k}\mathrm {P} (X_{1}=x_{1i},X_{2}=x_{2j},\dots ,X_{n}=x_{nk})=1.\;}
Continuous caseEdit
The joint probability density function
{\displaystyle f_{X,Y}(x,y)}
for two continuous random variables is defined as the derivative of the joint cumulative distribution function (see Eq.1):
{\displaystyle f_{X,Y}(x,y)={\frac {\partial ^{2}F_{X,Y}(x,y)}{\partial x\partial y}}}
{\displaystyle f_{X,Y}(x,y)=f_{Y\mid X}(y\mid x)f_{X}(x)=f_{X\mid Y}(x\mid y)f_{Y}(y)}
{\displaystyle f_{Y\mid X}(y\mid x)}
{\displaystyle f_{X\mid Y}(x\mid y)}
are the conditional distributions of
{\displaystyle Y}
{\displaystyle X=x}
{\displaystyle X}
{\displaystyle Y=y}
{\displaystyle f_{X}(x)}
{\displaystyle f_{Y}(y)}
are the marginal distributions for
{\displaystyle X}
{\displaystyle Y}
{\displaystyle f_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})={\frac {\partial ^{n}F_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})}{\partial x_{1}\ldots \partial x_{n}}}}
{\displaystyle \int _{x}\int _{y}f_{X,Y}(x,y)\;dy\;dx=1}
{\displaystyle \int _{x_{1}}\ldots \int _{x_{n}}f_{X_{1},\ldots ,X_{n}}(x_{1},\ldots ,x_{n})\;dx_{n}\ldots \;dx_{1}=1}
Mixed caseEdit
{\displaystyle {\begin{aligned}f_{X,Y}(x,y)=f_{X\mid Y}(x\mid y)\mathrm {P} (Y=y)=\mathrm {P} (Y=y\mid X=x)f_{X}(x).\end{aligned}}}
One example of a situation in which one may wish to find the cumulative distribution of one random variable which is continuous and another random variable which is discrete arises when one wishes to use a logistic regression in predicting the probability of a binary outcome Y conditional on the value of a continuously distributed outcome
{\displaystyle X}
. One must use the "mixed" joint density when finding the cumulative distribution of this binary outcome because the input variables
{\displaystyle (X,Y)}
were initially defined in such a way that one could not collectively assign it either a probability density function or a probability mass function. Formally,
{\displaystyle f_{X,Y}(x,y)}
{\displaystyle (X,Y)}
with respect to the product measure on the respective supports of
{\displaystyle X}
{\displaystyle Y}
. Either of these two decompositions can then be used to recover the joint cumulative distribution function:
{\displaystyle {\begin{aligned}F_{X,Y}(x,y)&=\sum \limits _{t\leq y}\int _{s=-\infty }^{x}f_{X,Y}(s,t)\;ds.\end{aligned}}}
Joint distribution for independent variablesEdit
In general two random variables
{\displaystyle X}
{\displaystyle Y}
are independent if and only if the joint cumulative distribution function satisfies
{\displaystyle F_{X,Y}(x,y)=F_{X}(x)\cdot F_{Y}(y)}
Two discrete random variables
{\displaystyle X}
{\displaystyle Y}
are independent if and only if the joint probability mass function satisfies
{\displaystyle P(X=x\ {\mbox{and}}\ Y=y)=P(X=x)\cdot P(Y=y)}
{\displaystyle x}
{\displaystyle y}
{\displaystyle f_{X,Y}(x,y)=f_{X}(x)\cdot f_{Y}(y)}
{\displaystyle x}
{\displaystyle y}
. This means that acquiring any information about the value of one or more of the random variables leads to a conditional distribution of any other variable that is identical to its unconditional (marginal) distribution; thus no variable provides any information about any other variable.
Joint distribution for conditionally dependent variablesEdit
If a subset
{\displaystyle A}
{\displaystyle X_{1},\cdots ,X_{n}}
is conditionally dependent given another subset
{\displaystyle B}
of these variables, then the probability mass function of the joint distribution is
{\displaystyle \mathrm {P} (X_{1},\ldots ,X_{n})}
{\displaystyle \mathrm {P} (X_{1},\ldots ,X_{n})}
{\displaystyle P(B)\cdot P(A\mid B)}
. Therefore, it can be efficiently represented by the lower-dimensional probability distributions
{\displaystyle P(B)}
{\displaystyle P(A\mid B)}
. Such conditional independence relations can be represented with a Bayesian network or copula functions.
{\displaystyle \sigma _{XY}=E[(X-\mu _{x})(Y-\mu _{y})]=E(XY)-\mu _{x}\mu _{y}}
{\displaystyle \rho _{XY}={\frac {cov(X,Y)}{\sqrt {V(X)V(Y)}}}={\frac {\sigma _{XY}}{\sigma _{X}\sigma _{Y}}}}
Important named distributionsEdit
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Rewrite the following using a single trigonometric function. You may wish to review your trigonometric identities in Chapter 1.
10\operatorname{ sin}(3x)\operatorname{ cos}(3x)
Double Angle identity (factor first).
\operatorname{sin }x\operatorname{ cos }3x −\operatorname{sin }3x\operatorname{ cos }x
Sum and Difference (Angle Sum) identity.
\operatorname{cos}^4 x −\operatorname{ sin}^4 x
\operatorname{tan }x +\operatorname{cot }x
Start by rewriting
\operatorname{tan}(x)
\operatorname{cot }(x)
as fractions.
You will use more than one identity.
\frac{\text{sin}x}{\text{cos}x}+\frac{\text{cos}x}{\text{sin}x}=\frac{\text{sin}^{2}x+\text{cos}^{2}x}{(\text{cos}x)(\text{sin}x)}=
\frac{1}{\frac{1}{2}\text{sin}(2x)}=2\text{csc}(2x)
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Transform IIR lowpass filter to IIR complex multiband filter - MATLAB iirlp2mbc - MathWorks í•œêµ
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{â1}+\cdots +{b}_{n}{z}^{ân}}{{a}_{0}+{a}_{1}{z}^{â1}+\cdots +{a}_{n}{z}^{ân}},
b=\left[\begin{array}{ccccc}{b}_{01}& {b}_{11}& {b}_{21}& ...& {b}_{Q1}\\ {b}_{02}& {b}_{12}& {b}_{22}& ...& {b}_{Q2}\\ ⋮& ⋮& ⋮& â±& ⋮\\ {b}_{0P}& {b}_{1P}& {b}_{2P}& \cdots & {b}_{QP}\end{array}\right]
H\left(z\right)=\underset{k=1}{\overset{P}{â}}{H}_{k}\left(z\right)=\underset{k=1}{\overset{P}{â}}\frac{{b}_{0k}+{b}_{1k}{z}^{â1}+{b}_{2k}{z}^{â2}+\cdots +{b}_{Qk}{z}^{âQ}}{{a}_{0k}+{a}_{1k}{z}^{â1}+{a}_{2k}{z}^{â2}+\cdots +{a}_{Qk}{z}^{âQ}},
H\left(z\right)=\frac{B\left(z\right)}{A\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{â1}+\cdots +{b}_{n}{z}^{ân}}{{a}_{0}+{a}_{1}{z}^{â1}+\cdots +{a}_{n}{z}^{ân}},
a=\left[\begin{array}{ccccc}{a}_{01}& {a}_{11}& {a}_{21}& \cdots & {a}_{Q1}\\ {a}_{02}& {a}_{12}& {a}_{22}& \cdots & {a}_{Q2}\\ ⋮& ⋮& ⋮& â±& ⋮\\ {a}_{0P}& {a}_{1P}& {a}_{2P}& \cdots & {a}_{QP}\end{array}\right]
H\left(z\right)=\underset{k=1}{\overset{P}{â}}{H}_{k}\left(z\right)=\underset{k=1}{\overset{P}{â}}\frac{{b}_{0k}+{b}_{1k}{z}^{â1}+{b}_{2k}{z}^{â2}+\cdots +{b}_{Qk}{z}^{âQ}}{{a}_{0k}+{a}_{1k}{z}^{â1}+{a}_{2k}{z}^{â2}+\cdots +{a}_{Qk}{z}^{âQ}},
IIR lowpass filter to IIR complex multiband filter transformation effectively places one feature of the original filter, located at frequency wo, at the required target frequency locations, wt1, … ,wtM.
[1] Krukowski, A., and I. Kale. “High-Order Complex Frequency Transformations,†Internal report No. 27/2001. Applied DSP and VLSI Research Group, University of Westminster.
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Paper by F. Leibfried: Bounded rational decision-making in feedforward neural networks – Tim Genewein
Paper by F. Leibfried: Bounded rational decision-making in feedforward neural networks
Felix Leibfried just published a very interesting application of the information-theoretic principle for bounded rationality to feedforward neural networks (including convolutional neural networks). This work will be presented in a plenary talk at UAI 2016.
In a nutshell, the information-theoretic optimality principle (similar to the rate-distortion principle, see here) was used to derive a gradient-based on-line update rule to learn the parameters of a neural network. In the principle, gains in utility must be traded off against the computational cost that these gains incur
where a parametric model
p_\theta(a\vert w)
is used to describe the stochastic mapping from an input
w
to an output
a
. In the paper, Felix shows how to derive a gradient-ascent rule for finding a (locally) optimal
\theta
. Then, the paper shows how to apply the same principle and derive an on-line gradient-based update rule when using a feedforward neural network for the parametric model. Interestingly, the result is an update-rule very similar to the (well-known) error-backpropagation with an additional regularization-term that results from the mutual information constraint (that formalizes limited computational capacity).
This result adds an interesting angle to the notion of limited computational resources: an alternative way to interpret computational limitations is to view them as a regularizer, that (sort of) artificially imposes computational limitations in order to be robust. This regularizer reflects the computational limitation that results from having small sample sizes - with an infinitely large sample size, corresponding to no computational limitation, the regularizer is not needed. However, under limited sample size (limited information to update the parameters), a regularizer “emulates” limited computational resources that reflect the lack of rich parameter-update information.
The paper concludes by showing simulations that apply the learning rule to a regular multi-layer perceptron as well as a convolutional neural network on the MNIST data set.
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assignable - Maple Help
Home : Support : Online Help : Programming : Data Types : Type Checking : Types : assignable
check if an object can be assigned to
type(expr, assignable)
type(expr, assignable(t))
The function type(expr, 'assignable') returns true if the expression expr can be assigned to; otherwise, it returns false.
If the parameter t is included, it checks that expr is also of that type.
An expression is assignable if one of the following is true:
- It is a symbol and it is not protected.
- It is of type indexed and its zeroth operand is either assignable or a table or an rtable.
- It is of type function and its zeroth operand is assignable.
\mathrm{type}\left(x[0]\left(4\right),\mathrm{assignable}\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
x[0]\left(4\right)≔7
{\textcolor[rgb]{0,0,1}{x}}_{\textcolor[rgb]{0,0,1}{0}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\right)\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{7}
\mathrm{type}\left(x[0],\mathrm{assignable}\left(\mathrm{name}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
\mathrm{type}\left(f\left(x\right),\mathrm{assignable}\left(\mathrm{name}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
\mathrm{type}\left(\mathrm{true},\mathrm{assignable}\left(\mathrm{name}\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
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Model Reduction Techniques - MATLAB & Simulink - MathWorks 日本
{âGâGredâ}_{\infty }
{â{G}^{â1}\left(GâGred\right)â}_{\infty }
These theoretical bounds are based on the “tails†of the Hankel singular values of the model, which are given as follows.
{âGâGredâ}_{\infty }â¤2\underset{k+1}{\overset{n}{â}}{\mathrm{Ï}}_{i}
Here, σi are denoted the ith Hankel singular value of the original system G.
{â{G}^{â1}\left(GâGred\right)â}_{\infty }â¤\underset{k+1}{\overset{n}{â}}\left(1+2{\mathrm{Ï}}_{i}\left(\sqrt{1+{\mathrm{Ï}}_{i}^{2}}+{\mathrm{Ï}}_{i}\right)\right)â1
Here, σi are denoted the ith Hankel singular value of the phase matrix of the model G (see the bstmr reference page).
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Hyperbola - Wikipedia @ WordDisk
Hyperbolas as plane sections of quadrics
{\displaystyle y(x)=1/x}
{\displaystyle y(x)=1/x}
This article uses material from the Wikipedia article Hyperbola, and is written by contributors. Text is available under a CC BY-SA 4.0 International License; additional terms may apply. Images, videos and audio are available under their respective licenses.
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6 Regulatory agencies
{\displaystyle V{dC \over dt}=\sum ({d^{b}m \over dt^{b}})}
{\displaystyle totalmass=mass_{a}+mass_{b}+mass_{c}...mass_{n}}
{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}+b(x)=0,}
{\displaystyle C=C_{0}/(1+\tau *k)}
Water supply and treatmentEdit
Environmental impact assessment and mitigationEdit
Environmental Protection AgencyEdit
National Academies of Sciences, Engineering, and Medicine (2019). Environmental Engineering for the 21st Century: Addressing Grand Challenges (Report). Washington, DC: The National Academies Press. doi:10.17226/25121. {{cite report}}: CS1 maint: multiple names: authors list (link)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Environmental_engineering&oldid=1089748385"
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1st Senators from Puerto Rico both Democrats | Metaculus
1st Senators from Puerto Rico both Democrats
Will Puerto Rico become a US state prior to 2035?
If Puerto Rico becomes a US state by 2035, will their first 2 elected Senators both be Republicans?
Puerto Rico was aquired as a territory of the USA in 1898. Since then, there has been ongoing discussion to admit Puerto Rico as a US state, though there has been much disagreement among Puerto Ricans among factions who favor statehood, favor national independence, or who favor the status quo.
In a related debate on statehood for the District of Columbia, the Republican Party is opposed to statehood, predicting that Democrats would gain an advantage in the Senate:
Expecting DC to elect 2 Democratic Senators upon statehood is almost certain, but the outcome of a Puerto Rican statehood is less so. The Republican Party's official platform from 2008 to 2020 stated:
The Democratic Party has also expressed support of PR statehood, on the condition that it is will of PR's citizens in a fair referendum.
Several referendums have been held on PR's future political status; in 2020, 52% of voters favored statehood.
If Puerto Rico becomes a US state by 2035, will their first 2 elected Senators both be Democrats?
This question will resolve positively if the first 2 elected Senators of Puerto Rico are both members of the Democratic Party
^{†}
, as of their date of election. It will resolve negatively if they are a member of any other party
^{†}
, including if they are independents who caucus with Democrats.
If Puerto Rico is not a state at any time prior to January 1, 2035, or if Puerto Rico will not elect at least 2 senators by that time, this question will resolve ambiguously.
Senators must be elected by the general populace. If Senators are appointed for PR, this question will wait to resolve on the first Senators who are elected. This question will resolve for the first 2 elected senators, regardless of whether those senators are elected in the same year or in the same election.
^{†}
If both elected senators are members of a Democratic Party Affiliate (for example, the Minnesota Democratic-Farmer-Labor Party), they will be considered Democrats for this question, assuming the Democratic Party does not endorse or support competing candidates in Puerto Rico (on or immediately prior to the general election day).
Senators will be "elected prior to 2035-01-01" if their election day is prior to 2035-01-01, regardless of when they are projected by election media, or when they take office.
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The RoboTHOR challenge at CVPR 2020 involves deploying navigation models onto the LoCoBot and running it through the physical RoboTHOR environment. This physical environment is located at the Allen Institute for AI (AI2) offices in Seattle. Due to the COVID-19 situation, all employees of AI2 are currently working from home and will be doing so for the foreseeable future. This prevents us from running experiments in the physical robot. As a result, the RoboTHOR challenge at CVPR 2020 will only involve simulated scenes. The final winners will be chosen based on performance in the simulated scenes in the Test-Challenge set. The component of the RoboTHOR challenge involving the physical robot will be held at a later date, to be determined as we get more clarity on the COVID-19 stay at home order. We wish you all the best of health, through this challenging time.
The RoboTHOR challenge is an embodied AI challenge focused on the problem of simulation-to-real transfer. The goals of the challenge are to encourage researchers to work on this important problem, to create a unified benchmark and track progress over time.
The RoboTHOR Challenge 2020 deals with the task of Visual Semantic Navigation. An agent starts from a random location in an apartment and is expected to navigate towards an object that is specified by its name. The 2020 challenge will restrict teams to using an ego-centric RGB camera mounted on the robot. Participants will train their models in simulation and these models will be evaluated by the challenge organizers using a real robot in physical apartments.
Training and evaluation is performed within the RoboTHOR framework. RoboTHOR is composed of 4 parts: 60 simulated apartments in Train, 15 simulated apartments in Val, 4 simulated apartments with real counterparts in Test-Dev and 10 simulated apartments with real counterparts in Test-Challenge. The RoboTHOR challenge consists of 3 phases:
Phase 1: Training (in Simulation)
Participants are provided with 60 simulated apartments (Train set) to train their models and 15 simulated apartments (Val set) to validate their performance in simulation. Each scene comes with a set of data points (a data point is a starting location along with a target object category name). The data points in train are provided for convenience. However, participants may generate a different (and potentially larger) set of Train data points using metadata provided in the RoboTHOR environment.
Phase 2: Training (in Simulation and Reality)
The goal of this phase is to enable teams to iterate on their models and improve their generalization to the real world. This is similar to a practitioner improving their model, while having access to a development set of data.
Participants will upload their models via EvalAI. These models will be evaluated on the Test-Dev-Simulation apartments. In this phase, the top 5 entries each week as per the simulation leaderboard will be evaluated on the Test-Dev-Real environment, that consists of a LocoBot navigating in RoboTHOR apartments built at the AI2 office in Seattle. Teams will gain access to the results of their model as well as metadata such as images observed, actions taken by the agent and a video feed, which should help them create improved models.
Phase 3: Test-Challenge (in Reality)
The top 3 entries as per the Test-Dev-Real leaderboard will be evaluated on the Test-Challenge-Real apartments. These apartments will have different layouts, object instances and object configurations compared to the Train, Val and Test-Dev apartments. Final results will be posted on this website.
Simulation Training and Validation Data Released
Simulation Test-Dev evaluation opens on Eval AI
Test-Challenge Phase
The evaluation metric used is Success weighted by (normalized inverse) Path Length (SPL), defined in Anderson et al. as
\mathrm{SPL} = \frac{1}{\mathit N}\sum_{\mathit i = 1}^{\mathit N}\mathit {S_i}\cdot\frac{\mathit{\ell_i}}{\max(p_i, \ell_i)},
\mathit i \in \{1, 2, 3\ldots\mathit N\}
\mathit{S_i}
is the binary indicator variable denoting if the episode was successful (as defined below),
\mathit{\ell_i}
\mathit{p_i}
is the path length (in meters) that the agent took. A navigation episode is considered successful if both of the following criteria are met:
Yes, you can use any external data (in addition to the provided training apartments) for training your models.
Can we access the real robot and apartments to evaluate our models?
During the Test-Dev Phase, we will evaluate your model in the real apartments and provide you with results and other metadata. During the Test-Challenge phase we will evaluate your model in the real apartments and only provide you with the final metrics.
What set do I use to report numbers in a paper?
To report an ablation analysis in simulation, we recommend using the Val set. To report final numbers in real, we recommend using the Real-Dev set. If your entry is selected for a final evaluation on Test-Challenge, we also recommend that you report these numbers.
We would like to thank Rishabh Jain for the help for setting up the EvalAI leaderboard.
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on which is more convenient, or frequently which is more appealing to the teacher grading the work. We will use this style for the proofs on this page.
{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle 13}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle (n+1)^{\textrm {th}}}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1,}}
{\displaystyle 1}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1,}}
{\displaystyle n=1.}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{4}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
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Assign properties of material for electromagnetic model - MATLAB electromagneticProperties - MathWorks India
electromagneticProperties
Specify Relative Permittivity
Specify Relative Permeability
Specify Material Properties for Harmonic Analysis
Specify Relative Permittivity for Each Face
Specify Nonconstant Relative Permittivity
Assign properties of material for electromagnetic model
electromagneticProperties(emagmodel,"RelativePermittivity",epsilon)
electromagneticProperties(emagmodel,"RelativePermeability",mu)
electromagneticProperties(emagmodel,"RelativePermittivity",epsilon,"RelativePermeability",mu,"Conductivity",sigma)
electromagneticProperties(___,RegionType,RegionID)
mtl = electromagneticProperties(___)
electromagneticProperties(emagmodel,"RelativePermittivity",epsilon) assigns the relative permittivity epsilon to the entire geometry. Specify the permittivity of vacuum using the electromagnetic model properties. The solver uses a relative permittivity for electrostatic and harmonic analyses.
For a nonconstant material, specify epsilon as a function handle.
electromagneticProperties(emagmodel,"RelativePermeability",mu) assigns the relative permeability to the entire geometry. Specify the permeability of vacuum using the electromagnetic model properties. The solver uses a relative permeability for magnetostatic and harmonic analyses.
For a nonconstant material, specify mu as a function handle.
electromagneticProperties(emagmodel,"RelativePermittivity",epsilon,"RelativePermeability",mu,"Conductivity",sigma) assigns the relative permittivity, relative permeability, and conductivity to the entire geometry. Specify the permittivity and permeability of vacuum using the electromagnetic model properties. The solver requires all three parameters for a harmonic analysis.
For a nonconstant material, specify epsilon, mu, and sigma as function handles.
electromagneticProperties(___,RegionType,RegionID) assigns the material properties to specified faces of a 2-D geometry or cells of a 3-D geometry. Use this syntax with any of the input argument combinations in the previous syntaxes.
mtl = electromagneticProperties(___) returns the material properties object.
Specify relative permittivity for an electrostatic analysis.
Import and plot a geometry of a plate with a hole in its center.
mtl = electromagneticProperties(emagmodel,"RelativePermittivity",2.25)
mtl =
ElectromagneticMaterialAssignment with properties:
RelativePermittivity: 2.2500
RelativePermeability: []
Conductivity: []
Specify relative permeability for a magnetostatic analysis.
pdegplot(gm,"EdgeLabels","on","FaceLabels","on")
mtl = electromagneticProperties(emagmodel,"RelativePermeability",5000)
RelativePermittivity: []
RelativePermeability: 5000
Specify relative permittivity, relative permeability, and conductivity of a material for harmonic analysis.
Specify the vacuum permittivity and permeability values in the SI system of units.
electromagneticProperties(emagmodel,"RelativePermittivity",2.25, ...
"RelativePermeability",5000, ...
"Conductivity",1)
Specify relative permittivity for individual faces in an electrostatic model.
Create a 2-D geometry with two faces. First, import and plot a 2-D geometry representing a plate with a hole.
Then, fill the hole by adding a face and plot the resulting geometry.
gm = addFace(gm,5);
pdegplot(gm,"FaceLabels","on")
Specify relative permittivities separately for faces 1 and 2.
RelativePermittivity: 1
Use a function handle to specify a relative permittivity that depends on the spatial coordinates.
Create a square geometry and include it in the model.
Specify the relative permittivity of the material as a function of the x-coordinate,
\epsilon =\sqrt{1+{\mathit{x}}^{2}}
perm = @(location,~)sqrt(1 + location.x.^2);
electromagneticProperties(emagmodel,"RelativePermittivity",perm)
RelativePermittivity: @(location,~)sqrt(1+location.x.^2)
Electromagnetic model, specified as an ElectromagneticModel object. The model contains a geometry, a mesh, the electromagnetic properties of the material, the electromagnetic sources, and the boundary conditions.
positive number | complex number | function handle
Relative permittivity, specified as a positive or complex number or a function handle.
Use a positive number to specify a relative permittivity for an electrostatic analysis.
Use a complex number to specify a relative permittivity for a harmonic electromagnetic analysis.
Use a function handle to specify a relative permittivity that depends on the coordinates or, for a harmonic analysis, on the frequency.
For details, see More About.
mu — Relative permeability
Relative permeability, specified as a positive or complex number or a function handle.
Use a positive number to specify a relative permeability for a magnetostatic analysis.
Use a complex number to specify a relative permeability for а harmonic electromagnetic analysis.
Use a function handle to specify a relative permeability that depends on the coordinates or, for a harmonic analysis, on the frequency.
nonnegative number | function handle
Conductivity, specified as a nonnegative number or a function handle. Use a function handle to specify a conductivity that depends on the coordinates or, for a harmonic analysis, on the frequency. For details, see More About.
"Face" for a 2-D model | "Cell" for a 3-D model
Geometric region type, specified as "Face" for a 2-D geometry or "Cell" for a 3-D geometry.
Region ID, specified as a vector of positive integers. Find the face or cell IDs by using pdegplot with the "FaceLabels" or "CellLabels" name-value argument set to "on".
Example: electromagneticProperties(emagmodel,"RelativePermeability",5000,"Face",1:3)
ElectromagneticMaterialAssignment object
Handle to material properties, returned as an ElectromagneticMaterialAssignment object. For more information, see ElectromagneticMaterialAssignment Properties.
mtl associates material properties with the geometric faces.
ElectromagneticModel | ElectromagneticMaterialAssignment Properties | createpde | electromagneticSource | electromagneticBC | solve | assembleFEMatrices
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Lemma 30.8.1 (01XT)—The Stacks project
Section 30.8: Cohomology of projective space
[III Proposition 2.1.12, EGA]
Reference: EGA, Chapitre III, "Étude cohomologique des faisceaux cohérents", Proposition (2.1.12)
Comment #3244 by Aravind Asok on April 02, 2018 at 21:07
NEG(\vec(e)) = \{0,\ldots,n\}
, I think the expression
\frac{1}{T^{-e_0}\cdots T^{-e_n}}
\frac{1}{T_0^{-e_0}\cdots T_n^{-e_n}}
(twice). In the case
NEG(\vec(e)) = \emptyset
(near the end),
R \cdot T^{e_0}\cdots T^{e_n}
R \cdot T_0^{e_0} \cdots T_n^{e_n}
Actually, I think we should perhaps replace this whole proof by thinking about the Koszul complex discussed in Section 15.28. But for now I have corrected the typos you pointed out, see here. Thanks!
I think there are some little problem for the discussion of the last case. If
NEG(e)
has only one integral then its zero cech complex is not zero.(But undoubtly the differential is injective in this case).
Thanks! This is now fixed and I added the computation that this homotopy works. See this commit on github.
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{\displaystyle {\displaystyle \sum _{n=1}^{\infty }}\,(-1)^{n}\sin {\frac {\pi }{n}}.}
{\displaystyle {\displaystyle \sum _{n=1}^{\infty }}\,(-1)^{n}\cos {\frac {\pi }{n}}.}
{\displaystyle n\geq 2}
{\displaystyle {\frac {\pi }{n}}}
{\displaystyle \sum _{k=1}^{\infty }a_{k}}
{\displaystyle \lim _{k\rightarrow 0}|a_{k}|=0,}
{\displaystyle {\displaystyle \lim _{k\rightarrow \infty }a_{k}\neq 0,}}
{\displaystyle \sum _{k=0}^{\infty }a_{k}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \lim _{n\rightarrow \infty }\left|(-1)^{n}\sin \left({\frac {\pi }{n}}\right)\right|}&=&\left|\sin \left({\displaystyle \lim _{n\rightarrow \infty }{\frac {\pi }{n}}}\right)\right|\\\\&=&0,\end{array}}}
as we can pass the limit through absolute value and sine because both are continuous functions. Since the series alternates for
{\displaystyle n\geq 2}
, the series is convergent by the alternating series test.
In this case, we find
{\displaystyle {\begin{array}{rcl}{\displaystyle \lim _{n\rightarrow \infty }\left|(-1)^{n}\cos \left({\frac {\pi }{n}}\right)\right|}&=&\left|\cos \left({\displaystyle \lim _{n\rightarrow \infty }{\frac {\pi }{n}}}\right)\right|\\\\&=&\cos(0)\\\\&=&1,\end{array}}}
where we can again pass the limit through absolute value and cosine because both are continuous. Since the terms do not converge to zero absolutely, they do not converge to zero. Thus, by the divergence test, the series is divergent.
The series for (a) is convergent, while the series (b) is divergent.
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On -Derivations in BCI-Algebras
G. Muhiuddin, Abdullah M. Al-Roqi, "On -Derivations in BCI-Algebras", Discrete Dynamics in Nature and Society, vol. 2012, Article ID 403209, 11 pages, 2012. https://doi.org/10.1155/2012/403209
G. Muhiuddin1 and Abdullah M. Al-Roqi1
1Department of Mathematics, University of Tabuk, P. O. Box 741, Tabuk 71491, Saudi Arabia
The notion of (regular) -derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. The condition for a -derivation to be regular is provided. The concepts of a -invariant -derivation and -ideal are introduced, and their relations are discussed. Finally, some results on regular -derivations are obtained.
BCK-algebras and BCI-algebras are two classes of nonclassical logic algebras which were introduced by Imai and Iséki in 1966 [1, 2]. They are algebraic formulation of BCK-system and BCI-system in combinatory logic. However, these algebras were not studied any further until 1980. Iséki published a series of notes in 1980 and presented a beautiful exposition of BCI-algebras in these notes (see [3–5]). The notion of a BCI-algebra generalizes the notion of a BCK-algebra in the sense that every BCK-algebra is a BCI-algebra but not vice versa (see [6]). Later on, the notion of BCI-algebras has been extensively investigated by many researchers (see [7–9] and references therein).
Throughout our discussion, X will denote a BCI-algebra unless otherwise mentioned. In the year 2004, Jun and Xin [10] applied the notion of derivation in ring and near-ring theory to BCI-algebras, and as a result they introduced a new concept, called a (regular) derivation, in BCI-algebras. Using this concept as defined, they investigated some of its properties. Using the notion of a regular derivation, they also established characterizations of a -semisimple BCI-algebra. For a self map of a BCI-algebra, they defined a -invariant ideal and gave conditions for an ideal to be -invariant. According to Jun and Xin, a self-map is called a left-right derivation (briefly -derivation) of if holds for all . Similarly, a self-map is called a right-left derivation (briefly -derivation) of if holds for all . Moreover, if is both - and -derivation, it is a derivation on . After the work of Jun and Xin [10], many research articles have been appeared on the derivations of BCI-algebras and a greater interest has been devoted to the study of derivation in BCI-algebras on various aspects (see [11–15]).
Several authors [16–19] have studied derivations in rings and near-rings. Inspired by the notions of -derivation [20], left derivation [21] and generalized derivation [19, 22] in rings and near rings theory, many authors have applied these notions in a similar way to the theory of BCI-algebras (see [11, 14, 15]). For instant, in 2005 [15], Zhan and Liu have given the notion of -derivation of BCI-algebras as follows: a self-map is said to be a left-right -derivation or --derivation of if it satisfies the identity for all . Similarly, a self map is said to be a right-left -derivation or --derivation of if it satisfies the identity for all . Moreover, if is both and --derivation, it is said that is an -derivation where is an endomorphism. In the year 2007, Abujabal and Al-Shehri [11] defined and studied the notion of left derivation of BCI-algebras as follows: a self-map is said to be a left derivation of if satisfying for all . Furthermore, in 2009 [14], Öztürk et al. have introduced the notion of generalized derivation in BCI-algebras. A self map is called a generalized -derivation if there exists an -derivation such that for all . If there exists an -derivation such that for all , the mapping is called generalized -derivation. Moreover, if is both a generalized - -derivation, is a generalized derivation on .
In fact, the notion of derivation in ring theory is quite old and playsa significant role in analysis, algebraic geometry, and algebra. In his famous book “Structures of Rings” Jacobson [23] introduced the notion of -derivation which was later more commonly known as or -derivation. After that a number of research articles have been appeared on or -derivations in the theory of rings (see [16, 24, 25] and references therein).
Motivated by the notion of or -derivation in the theory of rings, in the present paper, we introduce the notion of -derivation in a BCI-algebra and investigate related properties. We provide a condition for a -derivation to be regular. We also introduce the concepts of a -invariant -derivation and -ideal, and then we investigate their relations. Furthermore, we obtain some results on regular -derivations.
We begin with the following definitions and properties that will be needed in the sequel.
A nonempty set with a constant and a binary operation is called a -algebra if for all the following conditions hold:(I), (II), (III), (IV) and imply .
Define a binary relation on by letting if and only if . Then is a partially ordered set. A BCI-algebra satisfying for all , is called BCK-algebra.
A -algebra has the following properties: for all (a1),(a2), (a3) implies and , (a4), (a5), (a6), (a7) implies .
For a BCI-algebra , denote by (resp. ) the -part (resp. the BCI-G part) of , that is, is the set of all such that (resp. ). Note that (see [26]). If , then is called a -semisimple BCI-algebra. In a -semisimple BCI-algebra , the following hold:(a8), (a9) for all , (a10), (a11) implies , (a12) implies , (a13) implies , (a14).
Let be a -semisimple BCI-algebra. We define addition “+” as for all . Then is an abelian group with identity and . Conversely let be an abelian group with identity and let . Then is a -semisimple -algebra and for all (see [9]).
For a BCI-algebra we denote , in particular , and , for all . We call the elements of the -atoms of . For any , let , which is called the branch of with respect to . It follows that whenever and for all and all . Note that , which is the -semisimple part of , and is a -semisimple -algebra if and only if (see [27, Proposition 3.2]). Note also that , that is, , which implies that for all . It is clear that , and and for all and all . A BCI-algebra is said to be torsion free if for all [14]. For more details, refer to [7–10, 26, 27].
3. -Derivations in BCI-Algebras
In what follows, and are endomorphisms of a -algebra unless otherwise specified.
Definition 3.1. Let be a -algebra. Then a self map is called a -derivation of if it satisfies:
Example 3.2. Consider a -algebra with the following Cayley table: (1)Define a map and define two endomorphisms It is routine to verify that is a -derivation of .(2)Define a map and define two endomorphisms It is routine to verify that is a -derivation of .
Lemma 3.3 (see [8]). Let be a BCI-algebra. For any , if , then and are contained in the same branch of .
Lemma 3.4 3.4 (see [8]). Let be a BCI-algebra. For any , if and are contained in the same branch of , then .
Proposition 3.5. Let be a commutative BCI-algebra. Then every -derivation of satisfies the following assertion: that is, every -derivation of is isotone.
Proof. Let be such that . Since is commutative, we have . Hence Since every endomorphism of is isotone, we have . It follows from Lemma 3.3 that and so that there exists such that . Hence (3.8) implies that . Using (a3), (a2), and (III), we have and so , that is, by (a7).
Example 3.6. In Example 3.2 (1), the -derivation does not satisfy the inequality (3.7).
Proposition 3.7. Every -derivation of a -algebra satisfies the following assertion:
Proof. Let be an -derivation of . Using (a2) and (a4), we have Obviously by (II). Therefore, the equality (3.10) is valid.
Theorem 3.8. Let be a -derivation on a BCI-algebra . Then (1)for all ,(2)for all ,(3)for all .
Proof. (1) For any , we have for all . Thus .
(2) For any and , it follows from (1) that
(3) The proof follows directly from .
Definition 3.9. Let be a BCI-algebra and , be two self maps of , we define by for all .
Theorem 3.10. Let be a -semisimple BCI-algebra. If and are two -derivations on such that . Then is a -derivation on .
Proof. For any , it follows from (a14) that This completes the proof.
Theorem 3.11. Let , be two endomorphisms and be a self map on a p-semisimple BCI-algebra such that for all . Then is a -derivation on .
Proof. Let us take for all . Since . Using (a14), we have This completes the proof.
Definition 3.12. A -derivation of a -algebra is said to be regular if .
Example 3.13. (1) The -derivation of in Example 3.2 (1) is not regular.
(2) The -derivation of in Example 3.2 (2) is regular.
We provide conditions for a -derivation to be regular.
Theorem 3.14. Let be a -derivation of a -algebra . If there exists such that for all , then is regular.
Proof. Assume that there exists such that for all . Then and so . Hence is regular.
Definition 3.15. For a -derivation of a BCI-algebra , we say that an ideal of is a -ideal (resp. -ideal) if (resp. ).
Definition 3.16. For a -derivation of a BCI-algebra , we say that an ideal of is -invariant if .
Example 3.17. (1) Let be a -derivation of which is described in Example 3.2 (1). We know that is both a -ideal and a -ideal of . But is an ideal of which is not -invariant.
(2) Let be a -derivation of which is described in Example 3.2 (2). We know that is both a -ideal and a -invariant ideal of . But is not a -ideal of .
Next, we prove some results on regular -derivations in a -algebra.
Theorem 3.18. Let be a regular -derivation of a -algebra . Then (1)for all ,(2)for all ,(3)for all ,(4).
Proof. (1) Let be a regular -derivation, that is, . Then the proof follows directly form Proposition 3.7.
(2) Let . Then , and so . Thus . Similarly, .
(3) Let . Using (2), (a1) and (a14), we have
(4) Let . Then . Using (3), we have This completes the proof.
Theorem 3.19. Let be a torsion free BCI-algebra and be a regular -derivation on such that . If on , then on .
Proof. Let us suppose on . If , then and so by using Theorem 3.18 (3) and (4), we have Since is a torsion free. Therefore, for all implying thereby . This completes the proof.
Theorem 3.20. Let be a torsion free BCI-algebra and , be two regular -derivations on such that . If on , then on .
Proof. Let us suppose on . If , then and so by using Theorem 3.18 (1) and (2), we have Since is a torsion free. Therefore for all and so . This completes the proof.
Proposition 3.21. Let be a regular -derivation of a -algebra . If on , then for all .
Proof. Assume that on . If , then and so by using Theorem 3.18 (3) and (4), we have Hence for all .
Proposition 3.22. Let and be two regular -derivations of a -algebra . If on , then for all .
Proof. Let . Then , and so by Theorem 3.18 (1). It follows from Theorem 3.18 (3) and (4) that so that for all .
The authors would like to express their sincere thanks to the anonymous referees. This research is supported by the Deanship of Scientific Research, University of Tabuk, Tabuk, Kingdom of Saudi Arabia.
Y. Imai and K. Iséki, “On axiom systems of propositional calculi. XIV,” Proceedings of the Japan Academy, vol. 42, pp. 19–22, 1966. View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Iséki, “An algebra related with a propositional calculus,” Proceedings of the Japan Academy, vol. 42, pp. 26–29, 1966. View at: Publisher Site | Google Scholar | Zentralblatt MATH
K. Iséki, “On BCI-algebras,” Kobe University. Mathematics Seminar Notes, vol. 8, no. 1, pp. 125–130, 1980. View at: Google Scholar | Zentralblatt MATH
K. Iséki, “A variety of BCI-algebras,” Kobe University. Mathematics Seminar Notes, vol. 8, no. 1, pp. 225–226, 1980. View at: Google Scholar | Zentralblatt MATH
K. Iséki, “Some examples of BCI-algebras,” Kobe University. Mathematics Seminar Notes, vol. 8, no. 1, pp. 237–240, 1980. View at: Google Scholar | Zentralblatt MATH
Q. P. Hu and K. Iséki, “On some classes of BCI-algebras,” Mathematica Japonica, vol. 29, no. 2, pp. 251–253, 1984. View at: Google Scholar | Zentralblatt MATH
M. Aslam and A. B. Thaheem, “A note on
p
-semisimple BCI-algebras,” Mathematica Japonica, vol. 36, no. 1, pp. 39–45, 1991. View at: Google Scholar | Zentralblatt MATH
S. A. Bhatti, M. A. Chaudhry, and B. Ahmad, “On classification of BCI-algebras,” Mathematica Japonica, vol. 34, no. 6, pp. 865–876, 1989. View at: Google Scholar | Zentralblatt MATH
M. Daoji, “BCI-algebras and abelian groups,” Mathematica Japonica, vol. 32, no. 5, pp. 693–696, 1987. View at: Google Scholar | Zentralblatt MATH
Y. B. Jun and X. L. Xin, “On derivations of BCI-algebras,” Information Sciences, vol. 159, no. 3-4, pp. 167–176, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. A. S. Abujabal and N. O. Al-Shehri, “On left derivations of BCI-algebras,” Soochow Journal of Mathematics, vol. 33, no. 3, pp. 435–444, 2007. View at: Google Scholar | Zentralblatt MATH
S. Ilbira, A. Firat, and Y. B. Jun, “On symmetric bi-derivations of BCI-algebras,” Applied Mathematical Sciences, vol. 5, no. 57–60, pp. 2957–2966, 2011. View at: Google Scholar | Zentralblatt MATH
G. Mudiuddin and A. M. Al-roqi, “On t-derivations of BCI-algebras,” Abstract and Applied Analysis, vol. 2012, Article ID 872784, 12 pages, 2012. View at: Google Scholar
M. A. Öztürk, Y. Çeven, and Y. B. Jun, “Generalized derivations of BCI-algebras,” Honam Mathematical Journal, vol. 31, no. 4, pp. 601–609, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
f
-derivations of BCI-algebras,” International Journal of Mathematics and Mathematical Sciences, no. 11, pp. 1675–1684, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
\left(\sigma ,\tau \right)
-derivation,” Doga Turkish Journal of Mathematics, vol. 16, no. 3, pp. 169–176, 1992. View at: Google Scholar
H. E. Bell and L.-C. Kappe, “Rings in which derivations satisfy certain algebraic conditions,” Acta Mathematica Hungarica, vol. 53, no. 3-4, pp. 339–346, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH
H. E. Bell and G. Mason, “On derivations in near-rings, near-rings and Near-fields,” North-Holland Mathematics Studies, vol. 137, pp. 31–35, 1987. View at: Publisher Site | Google Scholar
B. Hvala, “Generalized derivations in rings,” Communications in Algebra, vol. 26, no. 4, pp. 1147–1166, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. A. M. Kamal, “
\sigma
-derivations on prime near-rings,” Tamkang Journal of Mathematics, vol. 32, no. 2, pp. 89–93, 2001. View at: Google Scholar | Zentralblatt MATH
M. Brešar and J. Vukman, “On left derivations and related mappings,” Proceedings of the American Mathematical Society, vol. 110, no. 1, pp. 7–16, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH
M. Brešar, “On the distance of the composition of two derivations to the generalized derivations,” Glasgow Mathematical Journal, vol. 33, no. 1, pp. 89–93, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH
N. Jacobson, Structures of Rings, vol. 37, American Mathematical Society, Colloquium Publications, Providence, RI, USA.
O. Golbasi and N. Aydin, “Results on prime near-ring with
\left(\sigma ,\tau \right)
-derivation,” Mathematical Journal of Okayama University, vol. 46, pp. 1–7, 2004. View at: Google Scholar
K. Kaya, “On
\left(\sigma ,\tau \right)
-derivations of prime rings,” Doga Turkish Journal of Mathematics, vol. 12, no. 2, pp. 42–45, 1988. View at: Google Scholar
Y. B. Jun and E. H. Roh, “On the BCI-G part of BCI-algebras,” Mathematica Japonica, vol. 38, no. 4, pp. 697–702, 1993. View at: Google Scholar | Zentralblatt MATH
Y. B. Jun, X. Xin, and E. H. Roh, “The role of atoms in BCI-algebras,” Soochow Journal of Mathematics, vol. 30, no. 4, pp. 491–506, 2004. View at: Google Scholar | Zentralblatt MATH
Copyright © 2012 G. Muhiuddin and Abdullah M. Al-Roqi. 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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Empirical Bayes: UMBC Over Virginia – Tom Shafer
Empirical Bayes: UMBC Over Virginia
This year’s NCAA tournament finally brought the mother of all upsets: 16-seeded UMBC (the Retrievers!) beat the Virginia Cavaliers by 21 points!1 Just how unlikely was UMBC’s upset?
To get a more precise without delving deeply into basketball metrics we can use Empirical Bayes as a framework for estimating the likelihood of a 16-over-1 upset. Our NCAA problem turns out to be especially suited for this approach because we’re asking a question about probabilities (this involves a Beta distribution) using win-loss data (“Bernoulli trials”). Under this setup, the probability of a 16-over-1 upset is described by a probability distribution
\mathrm{Beta}(\alpha, \beta)
\mathrm{Beta}
gives us us the “probability of a probability”: it tells us what the probability of a 16-over-1 upset could be. Maybe it’s 1%, maybe 10%, etc.
We need two pieces of information to apply Empirical Bayes: an initial guess at the probability of an upset (a “prior” belief) and some hard data. The data is easy to gather; in the 33 years since the NCAA tournament format expanded to include 64 teams, 16 seeds had won zero games in
33 \cdot 4 = 132
attempts.
Our “initial guess” is subjective, but it won’t turn out to matter much. For fun, let’s pick two extreme guesses and see how they affect our estimates.
Prior 1: We have no idea what the probability of an upset is; it’s just as likely for a 16 to beat a 1 as for the reverse outcome. This is clearly not a realistic position, but it (in a sense) keeps us from imposing our point of view on the problem. This belief corresponds to a
\mathrm{Beta}(1,1)
Prior 2: We’re pretty sure 1’s will beat 16’s; maybe call it 20-to-1 odds against an upset. This is a much stronger initial position and corresponds to an initial
\mathrm{Beta}(1, 20)
distribution.2
Our two guesses look like this:
The first guess is completely flat; before seeing hard data, we do not claim to know if 16 seeds win 0% of the time, 55%, or 99%, etc. The second guess, though, is much more opinionated: we’re pretty sure that the 16 seed is going to lose.
Flat Prior 0 1.000
Opinionated Prior 0 0.197
With the “flat” prior, we don’t claim to know the probability of an upset: The 99% credible interval for the upset probability ranges from 0% to 100%. On the other hand, our 20-to-1 guess really does make an impact: The 99% CI ranges between 0% and 20%. That seems more reasonable.
Now, let’s add some data and answer the question: Going into this year, how unlikely was it that any individual 16 seed would beat a 1 seed?3 Since 1985 we’d observed 132 failures and zero successes, so we can update our initial guesses very easily:
\alpha_{2018} = \alpha_\mathrm{prior} + 0
\beta_{2018}=\beta_\mathrm{prior} + 132
. This is possible because of the Beta distribution/Bernoulli trial problem type—they are conjugate priors:
\mathrm{Beta}(1, 1) \to \mathrm{Beta}(1, 133)
\mathrm{Beta}(1, 20) \to \mathrm{Beta}(1, 152)
The resulting probability distributions are below—we have to really zoom in to even see where the distribution is nonzero.
These are very similar distributions.
The probability of a 16-over-1 upset is very small (we are zoomed in 20 times on the
x
Just how small is that probability?
Flat Prior 0 0 0.0340
Opinionated Prior 0 0 0.0298
Even for the flat prior, our 99% CI only ranges up to a 3.4% chance of upset. But, as promised, the opinionated prior hasn’t made much of a difference: incorporating our 20-to-1 guess only shrinks the probability to a 3% chance. In both cases, the most likely probability of an upset (the mode) is 0%!
So—we could not have ruled out an upset being impossible, given these results. That’s good, given what happened next! But, also happily, the estimated probability of an upset is very small and matches our intuition. Pretty cool!
So…what about after this year? To find out, we just follow the same procedure as before, adding three non-upsets and our one shiny new upset. The posterior distributions are now:
\to \mathrm{Beta}(2, 136)
Opinionated Prior
\to \mathrm{Beta}(2, 155)
And as a result, our updated probabilities are:
Flat Prior 0.0074 1e-04 0.0474
Opinionated Prior 0.0065 1e-04 0.0417
Still small, with the 99% CI ranging from about 0.01% to 4.7% and the most likely probability somewhere around 0.7%. But notice—now that we have finally observed an upset—that our posterior distribution no longer allows for a 0% chance of upset. As it shouldn’t!
Even worse: Virginia was the overall number one seed, the highest-ranked team in the nation. ↩︎
This is the only constraint given for a two-parameter problem; it’s only enough to guarantee
\alpha
\beta
are related:
20\alpha = \beta
. In keeping with the flat prior, we set
\alpha=1
\beta=20
The probability that any of the four 16 seeds would win is slightly different:
p_\mathrm{any} = 1 - (1 - p_\mathrm{one})^4
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A Tight-Binding for Calculating the Raman Spectra and the Hartree–Fock Method Treatment for C13 NMR Spectra of Fullerenes C60Br6 | J. Nanotechnol. Eng. Med. | ASME Digital Collection
A Tight-Binding for Calculating the Raman Spectra and the Hartree–Fock Method Treatment for
C13
NMR Spectra of Fullerenes
C60Br6
Rashid Nizam,
, Aligarh, India
e-mail: rashid.nizam@gmail.com
S. Mahdi A. Rizvi,
e-mail: mahdirizvi@yahoo.com
e-mail: azam222@rediffmail.com
Nizam, R., Rizvi, S. M. A., and Azam, A. (February 16, 2011). "A Tight-Binding for Calculating the Raman Spectra and the Hartree–Fock Method Treatment for
C13
C60Br6
." ASME. J. Nanotechnol. Eng. Med. February 2011; 2(1): 011014. https://doi.org/10.1115/1.4003258
The structural Raman spectra and NMR of the fullerene derivative
(C60Br6)
were simulated at room temperature from the tight-binding method and the first-principles calculations, respectively. The calculated Raman spectra results are compared with available experimental fullerene derivative
(C60Br6)
data. The simulated Raman spectrum of
C60Br6
obtained almost all frequencies that are observed in experimental data; besides, some other frequencies are also obtained, which are not observed in the experimental data. In addition to these frequencies, further simulation to give P-polarization and U-polarization more accuracy and the polarization of experimental sample acquired become a complicated task. Besides the spectroscopic characterization of fullerene derivative
(C60Br6)
, it also includes the prediction of
C13
NMR fullerene derivative
(C60Br6)
ab initio calculations, density functional theory, fullerene compounds, HF calculations, nuclear magnetic resonance, Raman spectra, tight-binding calculations
Fullerenes, Nuclear magnetic resonance, Raman spectra, Tight-binding model, Spectra (Spectroscopy), Tensors, Polarization (Light)
Abdul-Sada
Isolation, Separation and Characterisation of the Fullerenes C60 and C70: The Third Form of Carbon
2D Nuclear Magnetic Resonance Study of the Structure of the Fullerene C70
NMR Characterization of Isomers of C78, C82 and C84 Fullerenes
C13 NMR Spectroscopy of C76, C78, C84 and Mixtures of C86–C102; Anomalous Chromatographic Behaviour of C82, and Evidence for C70H12
G. -W.
C36, a New Carbon Solid
Ground State of C84: Two Almost Isoenergetic Isomers
Structures of Large Fullerenes: C60 to C94 Structures of Large Fullerenes: C60 to C94
An Ab Initio Study of the C78 Fullerene Isomers
Bakowies
Quantum-Chemical Study of C84 Fullerene Isomers
Computational Evidence for a New C84 Isomer
A Theoretical Study on Vibrational Spectra of C84 Fullerenes: Results for C2, D2, and D2d Isomers
J. Mol. Struct.: THEOCHEM
Pentagon Adjacency as a Determinant of Fullerene Stability
Porezag
Construction of Tight-Binding-Like Potentials on the Basis of Density-Functional Theory: Application to Carbon
Calculations of Molecules, Clusters, and Solids With a Simplified LCAO-DFT-LDA Scheme
Geometric and Electronic Structure of Clusters
Z. Phys. D: At., Mol. Clusters
C36, a Hexavalent Building Block for Fullerene Compounds and Solids
The Chemistry of Fullerenes
The Theory of the Electric and Magnetic Properties of Molecules
Magnetic Shielding of Nuclei in Molecules
, 1966, Adv. Magn. Reson., 2, p. 137.
Circularly Polarized Raman Spectroscopy: Direct Determination of Antisymmetric Scattering in the Resonance Raman Spectrum of Ferrocytrochrome c
Nanoparticles of Cadmium Nitrate and Cobalt Nitrate Complexes Bearing Phosphoramide Ligands Designed for Application in Dye Sensitized Solar Cells
|
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)
|
Lemma 29.15.6. Let $f : X \to S$ be a morphism. If $S$ is (locally) Noetherian and $f$ (locally) of finite type then $X$ is (locally) Noetherian.
Proof. This follows immediately from the fact that a ring of finite type over a Noetherian ring is Noetherian, see Algebra, Lemma 10.31.1. (Also: use the fact that the source of a quasi-compact morphism with quasi-compact target is quasi-compact.) $\square$
Comment #5043 by Taro konno on April 20, 2020 at 05:51
I would like to propose that the statement of this lemma be changed to the following : Let
S
be (locally) Noetherian scheme and
X
be arbitrary scheme. If there exists a morphism (locally) of finite
f :X \longrightarrow S
X
is (locally) Noetherian.
@#5043: What would be the reason for that?
2 comment(s) on Section 29.15: Morphisms of finite type
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{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
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{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
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{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle i=13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
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{\displaystyle 1}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1,}}
{\displaystyle n=1.}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{4}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
|
This is the basis for weak, or simple induction; we must first prove our conjecture is true for the lowest value (such as <math style="vertical-align: -1px">1</math>), and then
It is also equivalent to prove that whenever the conjecture is true for <math style="vertical-align: -4px">n-1, </math> it's true for <math style="vertical-align: 0px">n. </math> Which approach you choose can depend
Although we won't show examples here, there are induction proofs that require '''strong induction'''. This occurs when proving it for the <math style="vertical-align: 0px">n^{\mathrm{th}} </math> case requires assuming more than just the <math style="vertical-align: -5px">(n-1)^{\textrm{th}} </math>
case. In such situations, strong induction assumes that the conjecture is true for ALL cases of lower value than <math style="vertical-align: 0px">n </math> down to our base case.
{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle i=13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle (n+1)^{\mathrm {th} }}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1,}}
{\displaystyle 1}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1,}}
{\displaystyle n=1.}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{4}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
|
Enter the vector function and the values of the point in the gradient calculator. Click calculate to solve.
for( x, y )
for( x, y, z )
Enter Function f(x, y) f(x, y, z)
This gradient calculator finds the partial derivatives of functions. You can enter the values of a vector line passing from 2 points and 3 points. For detailed calculation, click “show steps”.
The gradient is similar to the slope. It is represented by ∇(nabla symbol). A gradient in calculus and algebra can be defined as:
“A differential operator applied to a vector-valued function to yield a vector whose components are the partial derivatives of the function with respect to its variables.”
\nabla f (x, y, z) = \Huge [ \small \begin{array}{ccc} \dfrac{df}{dx}\ \ \dfrac{df}{dy}\ \ \dfrac{df}{dz} \end{array} \Huge]
A gradient is calculated by finding the partial derivative of the function with respect to the variable.
A function is {(X^2) + (Y^2)}. Find its gradient for point (2,2).
Step 1: Find the differentiate.
\nabla f = \Big( \dfrac{df}{dx}, \dfrac{df}{dy} \Big)=ddx[d(x2+ y2) , = ddy[d(x2+ y2)
=\dfrac{d}{dx}[d(x^2+ y^2) \enspace ,=\dfrac{d}{dy}[d(x^2+ y^2)
=d(\dfrac{d}{dx}[x^2]+\dfrac{d}{dx}[y^2]) \enspace, = d(\dfrac{d}{dy}[x^2]+\dfrac{d}{dy}[y^2])
= d (2x + 0) \enspace , = d (0 + 2y)
= 2x , = 2y
Step 2: Plug in the point.
\nabla f (x,y) = [(2)(x) , (2)(y)]
\nabla f (2,2) = [(2)(2) , (2)(2)]
Step 3: Write equation.
\nabla(x^2+y^2)(x,y)=(2x , 2y)
\nabla(x^{2} + y^{3})|_{(x,y)=(2,2)} = (4,4)
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Lemma 37.58.12 (0G2E)—The Stacks project
Comment #6652 by Brian Shin on October 23, 2021 at 11:01
Very small typo:
i_*\mathcal{O}_X
i_*\mathcal{O}_Z
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 0G2E. Beware of the difference between the letter 'O' and the digit '0'.
The tag you filled in for the captcha is wrong. You need to write 0G2E, in case you are confused.
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1 Department of Physics, Imo State University, Owerri, Nigeria.
2 Department of Physics, Federal College of Education (Technical), Omoku, Nigeria.
3 Department of Physics and Astronomy, University of Nigeria, Nsukka, Nigeria.
Abstract: In contributing to the improvement of Ferrite Magnetic nanoparticles, the effects of Poly (Vinyl Pyrrolidone) (PVP) and annealing on the structural and magnetic properties of Zinc ferrite nanoparticles (ZFNPs) synthesis were investigated in this work. The effects were evaluated using the X-ray diffraction (XRD) spectroscopy, Scanning electron microscopy (SEM), Fourier transform infrared spectroscopy (FTIR) and Vi-brating sample magnetometer (VSM). The XRD analysis confirms a good formation of the inverse spinel crystal structure with an average particle size of 1.3 nm to 15.2 nm and from 1.6 nm to 21.1 nm for the ZFNPs as-prepared and PVP mediated ZFNPs for the un-annealed and annealed samples, respectively. The SEM image reveals an increase in the particle size for both the as-prepared and PVP mediated samples after annealing at 500°C. The FTIR also reveals the inverse spinel structure for the as-prepared and annealed samples, which witnesses a vibrational red shift towards a higher wave number for the annealed samples. The VSM analysis indicates the superparamagnetic behavior of PVP mediated and annealed sample with zero remanence magnetization (Mr) and Coercivity (Hc). The saturation magnetization (Ms) increases from 1.31 emu/g, for the as-prepared samples, to 4.31 emu/g after the annealing and from 1.18 emu/g, for the PVP mediated, to 6.38 emu/g after annealing. These effects have been attributed to the cationic re-arrangement on the lattice site after the annealing. This presents a superior material for various applications in nanotechnology.
Keywords: ZFNPs, Poly (Vinyl Pyrrolidone), Annealing, Super Paramagnetic, Properties
{\text{Zn}}^{\text{2+}}+{\text{Fe}}^{2+}+{\text{H}}^{\text{+}}\to {\text{ZnFe}}_{\text{2}}{\text{O}}_{\text{4}}+{\text{H}}_{\text{2}}\text{O}
D=\frac{0.9\lambda }{\beta \mathrm{cos}\mathrm{cos}\theta }
Cite this paper: Okoroh, D. , Ozuomba, J. , Aisida, S. and Asogwa, P. (2019) Properties of Zinc Ferrite Nanoparticles Due to PVP Mediation and Annealing at 500°C. Advances in Nanoparticles, 8, 36-45. doi: 10.4236/anp.2019.82003.
[1] Naidu, K.C.B. and Madhuri, W. (2017) Microwave Processed Bulk and Nano NiMg Ferrites: A Comparative Study on X-Band Electromagnetic Interference Shielding Properties. Materials Chemistry and Physics, 187, 164-176.
[2] Samavati, A. and Ismail, A.F. (2017) Antibacterial Properties of Copper-Substituted Cobalt Ferrite Nanoparticles Synthesized by Co-Precipitation Method. Particuology, 30, 158-163.
[3] Sharma, R., Thakur, P., Kumar, M., Thakur, N., Negi, N.S., Sharma, P. and Sharma, V. (2016) Improvement in Magnetic Behaviour of Cobalt Doped Magnesium Zinc Nano-Ferrites via Co-Precipitation Route. Journal of Alloys and Compounds, 684, 569-581.
[4] Tartaj, P., del Puerto Morales, M., Veintemillas-Verdaguer, S., González-Carreno, T. and Serna, C.J. (2003) The Preparation of Magnetic Nanoparticles for Applications in Biomedicine. Journal of Physics D: Applied Physics, 36, R182.
[5] Dantas, J., Leal, E., Mapossa, A.B., Cornejo, D.R. and Costa, A.C.F.M. (2017) Magnetic Nanocatalysts of Ni0. 5Zn0. 5Fe2O4 Doped with Cu and Performance Evaluation in Transesterification Reaction for Biodiesel Production. Fuel, 191, 463-471.
[6] Naseri, M.G., Saion, E.B., Hashim, M., Shaari, A.H. and Ahangar, H.A. (2011) Synthesis and Characterization of Zinc Ferrite Nanoparticles by a Thermal Treatment Method. Solid State Communications, 151, 1031-1035.
[7] Yeary, L.W., Moon, J.W., Rawn, C.J., Love, L.J., Rondinone, A.J., Thompson, J.R., Chakoumakos, B.C. and Phelps, T.J. (2011) Magnetic Properties of Bio-Synthesized Zinc Ferrite Nanoparticles. Journal of Magnetism and Magnetic Materials, 323, 3043-3048.
[8] Zaharieva, K., Rives, V., Tsvetkov, M., Cherkezova-Zheleva, Z., Kunev, B., Trujillano, R., Mitov, I. and Milanova, M. (2015) Preparation, Characterization and Application of Nanosized Copper Ferrite Photocatalysts for Dye Degradation under UV Irradiation. Materials Chemistry and Physics, 160, 271-278.
[9] Grasset, F., Labhsetwar, N., Li, D., Park, D.C., Saito, N., Haneda, H., Cador, O., Roisnel, T., Mornet, S., Duguet, E., Etourneau, J. and Portier, J. (2002) Synthesis and Magnetic Characterization of Zinc Ferrite Nanoparticles with Different Environments: Powder, Colloidal Solution, and Zinc Ferrite-Silica Core-Shell Nanoparticles. Langmuir, 18, 8209-8216.
[10] Naseri, M.G., Halimah, M.K., Dehzangi, A., Kamalianfar, A., Saion, E.B. and Majlis, B.Y. (2014) A Comprehensive Overview on the Structure and Comparison of Magnetic Properties of Nanocrystalline Synthesized by a Thermal Treatment Method. Journal of Physics and Chemistry of Solids, 75, 315-327.
[11] Wu, W., He, Q. and Jiang, C. (2008) Magnetic Iron Oxide Nano-particles: Synthesis and Surface Functionalization Strategies. Nanoscale Research Letters, 3, 397.
[12] Laurent, S., Forge, D., Port, M., Roch, A., Robic, C., Vander Elst, L. and Muller, R.N. (2008) Magnetic Iron Oxide Nanoparticles: Synthesis, Stabilization, Vectorization, Physicochemical Characterizations, and Biological Applications. Chemical Reviews, 108, 2064-2110.
[13] Berry, C.C. and Curtis, A.S. (2003) Functionalisation of Magnetic Nanoparticles for Applications in Biomedicine. Journal of physics D: Applied physics, 36, R198.
[14] Afzal, S., Khan, R., Zeb, T., ur Rahman, M., Ali, S., Khan, G., ur Rahman, Z., Hussain, A., et al. (2018) Structural, Optical, Dielectric and Magnetic Properties of PVP Coated Magnetite (Fe3O4) Nanoparticles. Journal of Materials Science: Materials in Electronics, 29, 20040-20050.
[15] Kombaiah, K., Vijaya, J.J., Kennedy, L.J., Bououdina, M., Kaviyarasu, K., Ramalingam, R.J., Munusamy, M.A. and AlArfaj, A. (2017) Effect of Cd2+ Concentration on ZnFe2O4 Nanoparticles on the Structural, Optical and Magnetic Properties. Optik, 135, 190-199.
[16] Hcini, S., Selmi, A., Rahmouni, H., Omri, A. and Bouazizi, M.L. (2017) Structural, Dielectric and Complex Impedance Properties of T0. 6Co0. 4Fe2O4 (T= Ni, Mg) Ferrite Nanoparticles Prepared by Sol Gel Method. Ceramics International, 43, 2529-2536.
[17] Satalkar, M., Kane, S.N., Kumaresavanji, M. and Araujo, J.P. (2017) On the Role of Cationic Distribution in Determining Magnetic Properties of Zn0.7-xNixMg0.2Cu0.1Fe2O4 Nano Ferrite. Materials Research Bulletin, 91, 14-21.
[18] Smit, J. (1971) Magnetic Properties of Materials. McGraw-Hill, New York.
[19] Snelling, E.C. (1988) Soft Ferrites, Properties and Applications. Butter Worth and Co. Ltd., London.
[20] Petrova, E., Kotsikau, D. and Pankov, V. (2015) Structural Characterization and Magnetic Properties of Sol-Gel Derived ZnxFe3–xO4 Nanoparticles. Journal of Magnetism and Magnetic Materials, 378, 429-435.
[21] Sharma, R., Thakur, P., Sharma, P. and Sharma, V. (2017) Ferrimagnetic Ni2+ Doped Mg-Zn Spinel Ferrite Nanoparticles for High Density Information Storage. Journal of Alloys and Compounds, 704, 7-17.
[22] Abbas, M., Rao, B.P. and Kim, C. (2014) Shape and Size-Controlled Synthesis of Ni Zn Ferrite Nanoparticles by Two Different Routes. Materials Chemistry and Physics, 147, 443-451.
[23] Karimi, Z., Karimi, L. and Shokrollahi, H. (2013) Nano-Magnetic Particles Used in Biomedicine: Core and Coating Materials. Materials Science and Engineering: C, 33, 2465-2475.
[24] Hankare, P.P., Sankpal, U.B., Patil, R.P., Jadhav, A.V., Garadkar, K.M. and Chougule, B.K. (2011) Magnetic and Dielectric Studies of Nanocrystalline Zinc Substituted Cu-Mn Ferrites. Journal of Magnetism and Magnetic Materials, 323, 389-393.
[25] Makovec, D. and Drofenik, M. (2008) Non-Stoichiometric Zinc-Ferrite Spinel Nanoparticles. Journal of Nanoparticle Research, 10, 131-141.
[26] Kamari, H., Naseri, M. and Saion, E. (2014) A Novel Research on Behavior of Zinc Ferrite Nanoparticles in Different Concentration of Poly (vinyl pyrrolidone) (PVP). Metals, 4, 118-129.
[27] Gordon, T., Perlstein, B., Houbara, O., Felner, I., Banin, E. and Margel, S. (2011) Synthesis and Characterization of Zinc/Iron Oxide Composite Nanoparticles and Their Antibacterial Properties. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 374, 1-8.
[28] Cullity, B.D. (1978) Elements of X-Ray Diffraction. Addison-Wesley, Boston, MA, 359.
[29] Bushroa, A.R., Rahbari, R.G., Masjuki, H.H. and Muhamad, M.R. (2012) Approximation of Crystallite Size and Microstrain via XRD Line Broadening Analysis in TiSiN Thin Films. Vacuum, 86, 1107-1112.
[30] Jaberolansar, E., Kameli, P., Ahmadvand, H. and Salamati, H. (2016) Synthesis and Characterization of PVP-Coated Co0. 3Zn0. 7Fe2O4 Ferrite Nanoparticles. Journal of Magnetism and Magnetic Materials, 404, 21-28.
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Crystals | Free Full-Text | Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification
The Influence of Silicateins on the Shape and Crystalline Habit of Silica Carbonate Biomorphs of Alkaline Earth Metals (Ca, Ba, Sr)
Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal
Crystal Plasticity Simulation of Magnesium and Its Alloys: A Review of Recent Advances
Effect of Argon Flow on Oxygen and Carbon Coupled Transport in an Industrial Directional Solidification Furnace for Crystalline Silicon Ingots
Podmaniczky, F.
Gránásy, L.
Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest, Hungary
BCAST, Brunel University, Uxbridge, Middlesex UB8 3PH, UK
Academic Editors: Wolfram Miller and Koichi Kakimoto
(This article belongs to the Special Issue Modeling of Crystal Growth)
Two-step nucleation and subsequent growth processes were investigated in the framework of the single mode phase-field crystal model combined with diffusive dynamics (corresponding to colloid suspensions) and hydrodynamical density relaxation (simple liquids). It is found that independently of dynamics, nucleation starts with the formation of solid precursor clusters that consist of domains with noncrystalline ordering (ringlike projections are seen from certain angles), and regions that have amorphous structure. Using the average bond order parameter
{\overline{q}}_{6}
, we distinguished amorphous, medium range crystallike order (MRCO), and crystalline local orders. We show that crystallization to the stable body-centered cubic phase is preceded by the formation of a mixture of amorphous and MRCO structures. We have determined the time dependence of the phase composition of the forming solid state. We also investigated the time/size dependence of the growth rate for solidification. The bond order analysis indicates similar structural transitions during solidification in the case of diffusive and hydrodynamic density relaxation. View Full-Text
Keywords: classical density functional theory; molecular modelling; two-step nucleation; growth kinetics; hydrodynamic theory of freezing classical density functional theory; molecular modelling; two-step nucleation; growth kinetics; hydrodynamic theory of freezing
Podmaniczky, F.; Gránásy, L. Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification. Crystals 2021, 11, 437. https://doi.org/10.3390/cryst11040437
Podmaniczky F, Gránásy L. Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification. Crystals. 2021; 11(4):437. https://doi.org/10.3390/cryst11040437
Podmaniczky, Frigyes, and László Gránásy. 2021. "Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification" Crystals 11, no. 4: 437. https://doi.org/10.3390/cryst11040437
Frigyes Podmaniczky is a PhD student at the Budapest University of Technology and Economics and works at the Wigner Research Centre for Physics, Budapest, HUngary. His research areas include the investigation of complex solidification problems within phase-field crystal models, such as homogeneous, heterogeneous, and nonclassic nucleation processes, dendritic solidification, polycrystalline growth, and the determination of the anisotropy of the crystal–liquid interface.
Prof. László Gránásy (66) received his PhD degree in solid-state physics at the Eötvös University and CSc and DSc degrees at the Hungarian Academy of Sciences (HAS) in 1989 and 2004, respectively. He has been a corresponding member of HAS since 2018 and member of the Academia Europaea (London) since 2014. He is the leader of the Computational Materials Science Group at the Wigner Research Centre for Physics, Budapest, and is a honorary Professor at Brunel University, Uxbridge, UK. His recent field of interest is mathematical and numerical modeling of complex solidification microstructures, using continuum models (including phase-field and classic density functional methods), with special emphasis on nucleation and polycrystalline freezing.
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Serre's criterion for affineness.
[Serre-criterion], [II, Theorem 5.2.1 (d') and IV (1.7.17), EGA]
Comment #1332 by Paul Lessard on March 03, 2015 at 18:28
Affinenes in the slogan should be changed to affineness (missing the second s).
A reference is EGA II, Theorem 5.2.1 (d') and EGA IV_1, 1.7.17
Comment #7115 by Liu on March 15, 2022 at 07:37
The assumption in EGA II (5.2.1) & EGA IV (1.7.17) says that the scheme X should be qcqs. Here we only assume quasi-compact, does it harm the argument given here?
@#7115: Do you think so?
Comment #7158 by DatPham on March 30, 2022 at 08:23
@#7115: That
X
is qcqs is in fact a consequence of the fact that
X
has a finite cover of affine open subsets of the form
X_f
X_f\cap X_g=D_{X_f}(g|_{X_f})
is affine (hence quasi-compact) as soon as
X_f,X_g
are affine.
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On coproducts in varieties, quasivarieties and prevarieties
KEYWORDS: coproduct of algebras in a variety or quasivariety or prevariety, free algebra on $n$ generators containing a subalgebra free on more than $n$ generators, amalgamation property, number of algebras needed to generate a quasivariety or prevariety, symmetric group on an infinite set, 08B25, 08B26, 08C15, 03C05, 08A60, 08B20, 20M30
If the free algebra
F
on one generator in a variety
V
of algebras (in the sense of universal algebra) has a subalgebra free on two generators, must it also have a subalgebra free on three generators? In general, no; but yes if
F
generates the variety
V
Generalizing the argument, it is shown that if we are given an algebra and subalgebras,
{A}_{0}\supseteq \cdots \supseteq {A}_{n}
, in a prevariety (
\mathbb{S}ℙ
-closed class of algebras)
P
{A}_{n}
P
, and also subalgebras
{B}_{i}\subseteq {A}_{i-1}
\left(0<i\le n\right)
i>0
the subalgebra of
{A}_{i-1}
{A}_{i}
{B}_{i}
is their coproduct in
P
, then the subalgebra of
A
{B}_{1},\dots ,{B}_{n}
is the coproduct in
P
of these algebras.
Some further results on coproducts are noted:
P
satisfies the amalgamation property, then one has the stronger “transitivity” statement, that if
A
has a finite family of subalgebras
{\left({B}_{i}\right)}_{i\in I}
such that the subalgebra of
A
{B}_{i}
is their coproduct, and each
{B}_{i}
{\left({C}_{ij}\right)}_{j\in {J}_{i}}
with the same property, then the subalgebra of
A
generated by all the
{C}_{ij}
is their coproduct.
P
a residually small prevariety or an arbitrary quasivariety, relationships are proved between the least number of algebras needed to generate
P
as a prevariety or quasivariety, and behavior of the coproduct operation in
P
It is shown by example that for
B
a subgroup of the group
S=Sym\left(\Omega \right)
of all permutations of an infinite set
\Omega
S
need not have a subgroup isomorphic over
B
to the coproduct with amalgamation
S\phantom{\rule{0.3em}{0ex}}{\coprod }_{B}S
. But under various additional hypotheses on
B
, the question remains open.
Exponential sums nondegenerate relative to a lattice
KEYWORDS: exponential sums, $p$-adic cohomology, 11L07, 11T23, 14F30
Our previous theorems on exponential sums often did not apply or did not give sharp results when certain powers of a variable appearing in the polynomial were divisible by
p
. We remedy that defect in this paper by systematically applying
p
-power reduction, making it possible to strengthen and extend our earlier results.
F-adjunction
KEYWORDS: F-pure, F-split, test ideal, log canonical, center of log canonicity, subadjunction, adjunction conjecture, different, 14B05, 13A35
In this paper we study singularities defined by the action of Frobenius in characteristic
p>0
. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if
X
is a Gorenstein normal variety then to every normal center of sharp
F
-purity
W\subseteq X
X
F
-pure at the generic point of
W
, there exists a canonically defined
ℚ
-divisor
{\Delta }_{W}
W
\left({K}_{X}\right){|}_{W}{\sim }_{ℚ}{K}_{W}+{\Delta }_{W}
. Furthermore, the singularities of
X
W
are “the same” as the singularities of
\left(W,{\Delta }_{W}\right)
. As an application, we show that there are finitely many subschemes of a quasiprojective variety that are compatibly split by a given Frobenius splitting. We also reinterpret Fedder’s criterion in this context, which has some surprising implications.
KEYWORDS: minimal models, Mori fibre spaces, 14E30
d
d
4
Centers of graded fusion categories
Shlomo Gelaki, Deepak Naidu, Dmitri Nikshych
KEYWORDS: fusion categories, braided categories, graded tensor categories, 16W30, 18D10
\mathsc{C}
be a fusion category faithfully graded by a finite group
G
\mathsc{D}
be the trivial component of this grading. The center
\mathsc{Z}\left(\mathsc{C}\right)
\mathsc{C}
is shown to be canonically equivalent to a
G
-equivariantization of the relative center
{\mathsc{Z}}_{\mathsc{D}}\left(\mathsc{C}\right)
. We use this result to obtain a criterion for
\mathsc{C}
to be group-theoretical and apply it to Tambara–Yamagami fusion categories. We also find several new series of modular categories by analyzing the centers of Tambara–Yamagami categories. Finally, we prove a general result about the existence of zeroes in
S
-matrices of weakly integral modular categories.
|
Convolutional sparse coding for noise attenuation in seismic dataCSC denoising | Geophysics | GeoScienceWorld
Convolutional sparse coding for noise attenuation in seismic data
Zhaolun Liu;
, Department of Earth Science and Engineering, Thuwal 23955-6900, Saudi Arabia; presently
, Department of Geosciences, Princeton, New Jersey 08544, USA. E-mail: zhaolunl@princeton.edu (corresponding author).
, Department of Earth Science and Engineering, Thuwal 23955-6900, Saudi Arabia. E-mail: Kai.lu@kaust.edu.sa.
Zhaolun Liu, Kai Lu; Convolutional sparse coding for noise attenuation in seismic data. Geophysics 2021;; 86 (1): V23–V30. doi: https://doi.org/10.1190/geo2019-0746.1
We have developed convolutional sparse coding (CSC) to attenuate noise in seismic data. CSC gives a data-driven set of basis functions whose coefficients form a sparse distribution. The noise attenuation method by CSC can be divided into the training and denoising phases. Seismic data with a relatively high signal-to-noise ratio are chosen for training to get the learned basis functions. Then, we use all (or a subset) of the basis functions to attenuate the random or coherent noise in the seismic data. Numerical experiments on synthetic data show that CSC can learn a set of shifted invariant filters, which can reduce the redundancy of learned filters in the traditional sparse-coding denoising method. CSC achieves good denoising performance when training with the noisy data and better performance when training on a similar but noiseless data set. The numerical results from the field data test indicate that CSC can effectively suppress seismic noise in complex field data. By excluding filters with coherent noise features, our method can further attenuate coherent noise and separate ground roll.
Q
Effective Q-compensated reverse time migration using new decoupled fractional Laplacian viscoacoustic wave equation
Q
filtering via synchrosqueezed wavelet transform
A stable and efficient approach of Q reverse time migration
|
Fool's mate - Wikipedia
Fastest checkmate in the game of chess
For other uses, see Fool's mate (disambiguation).
Fool's mate: White is checkmated.
1. (f3, f4 or g4)
1... (e6 or e5)
2. (g4 if 1. f3 or 1. f4,
or either f3 or f4 if 1. g4)
Gioachino Greco (c. 1620),
via Francis Beale (1656)
Barnes Opening,
Bird Opening,
or Grob's Attack
In chess, the fool's mate, also known as the two-move checkmate, is the checkmate delivered after the fewest possible moves from the game's starting position.[1] It can be achieved only by Black, giving checkmate on the second move with the queen. The fool's mate received its name because it can only occur if White commits an extraordinary blunder. Even among rank beginners, this checkmate rarely occurs in practice.
The mate is an illustration of the kingside weakness shared by both players along the f- and g-files during the opening phase of the game. Black can be mated in a complementary situation, although this requires an additional move. A player may also suffer an early checkmate if the f- and g-pawns are advanced prematurely and the kingside is not properly defended, as shown in historical miniature games recorded in chess literature.
2.1 White to mate in three moves
2.2 Teed vs. Delmar
2.3 Greco vs. NN
The fool's mate was named and described in The Royall Game of Chesse-Play, a 1656 text by Francis Beale that adapted the work of the early chess writer Gioachino Greco.[2]
Prior to the mid-19th century there was not a prevailing convention as to whether White or Black moved first; according to Beale, the matter was to be decided in some prior contest or decision of the players' choice.[3] In Beale's example Black was the player to move first, with each player making two moves to various squares or "houses", after which White achieved checkmate.
Black Kings Biſhops pawne one houſe.
Black kings knights pawne two houſes
White Queen gives Mate at the contrary kings Rookes fourth houſe
— Beale, The Royall Game of Chesse-Play[4]
Beale's example can be paraphrased in modern terms (shown above), where White always moves first, algebraic notation is used, and Black delivers the fastest possible mate after each player makes two moves:
There are eight distinct ways in which the fool's mate can be reached.[1] In the sequence there are three independent points where one of two options may be chosen, resulting in
{\displaystyle 2^{3}=8}
possibilities. White may alternate the order of movement of the f- and g-pawns, Black may play either e6 or e5, allowing the queen to move, and White may move their f-pawn to f3 or f4. Hooper and Whyld expressed this generalized sequence as 1. g4 e5 (or e6) 2. f3 (or f4) Qh4#.[1]
Similar mating patterns can occur early in the game. White can mate Black using an extra turn, and historical games illustrate the early weakness alongside the kingside f- and g-files, exploited in the fool's mate.
White to mate in three moves[edit]
A problem with White to mate instead, given by Fischer and Polgár.
It is possible for White to achieve a very similar checkmate. When the roles are reversed, however, White requires an extra third turn or half-move, known in computer chess as a ply. In both cases, the principle is the same: a player advances their f- and g-pawns such that the opponent's queen can mate along the unblocked diagonal. A board position illustrating White's version of the fool's mate—with White to mate—was given as a problem in Bobby Fischer Teaches Chess, and also as an early example in a compendium of problems by László Polgár.[5] The solution in Fischer's book bore the comment "Black foolishly weakened his King's defenses. This game took three moves!!"[6] One possible sequence leading to the position is 1. e4 g5 2. d4 f6?? 3. Qh5#.
A possibly apocryphal variant of the fool's mate has been reported by several sources. The 1959 game 1. e4 g5 2. Nc3 f5?? 3. Qh5# has been attributed to Masefield and Trinka, although the players' names have also been reported as Mayfield, Mansfield, Trinks, or Trent.[7][8][9][10][11] Further, a similar mate can occur in From's Gambit: 1. f4 e5 2. g3? exf4 3. gxf4?? Qh4#.
There is another possible three move mate for white, 1. e4 e5 2. Qh5 Ke7?? 3. Qxe5#.
Teed vs. Delmar[edit]
Teed vs. Delmar, 1896
After 6...Rh6?? White mates in two moves.
A well-known trap in the Dutch Defence occurred in the game Frank Melville Teed–Eugene Delmar, 1896:[12][13]
1. d4 f5 2. Bg5 h6 3. Bh4 g5 4. Bg3 f4
Probably better is 6.Be2, but the move played sets a trap.
Greco vs. NN[edit]
Final position after 8.Bg6#
3. Bd3 f5?
4. exf5 Bxg2?
5. Qh5+ g6
6. fxg6 Nf6??
Opening up a flight square for the king at f8 with 6...Bg7 would have prolonged the game. White still wins with 7.Qf5! Nf6 8.Bh6 Bxh6 9.gxh7 Bxh1 (9...e6 opens another flight square at e7; then White checks with 10.Qg6+ Ke7) 10.Qg6+ Kf8 11.Qxh6+ Kf7 12.Nh3, but much slower than in the game.[14]
^ a b c Hooper, David; Whyld, Kenneth (1992). The Oxford Companion to Chess (2nd ed.). Oxford University Press. p. 143. ISBN 9780198661641.
^ Beale, Francis (29 August 2021). The Royall Game of Chesse-Play. babel.hathitrust.org. p. 17, .pdf p. 49.
^ Beale 1656, p. 10 (.pdf p. 42).
^ Polgár, László (1994). Chess: 5334 Problems, Combinations, and Games. Tess Press. p. 57. ISBN 9781579121303. Problem No. 14.
^ Fischer, Bobby; Margulies, Stuart; Mosenfelder, Donn (1972). Bobby Fischer Teaches Chess. Bantam. pp. 95–96. ISBN 9780553263152. Problem No. 73.
^ Mike Fox and Richard James (1993). The Even More Complete Chess Addict. Faber and Faber. p. 177.
^ Winter, Edward (2005). Chess Facts and Fables. McFarland & Co. pp. 253–254. ISBN 978-0-7864-2310-1.
^ Edward G. Winter (August 2006). "Chess Notes 4493. Short game".
^ Edward G. Winter (August 2006). "Chess Notes 4506. Short game (C.N. 4493)".
^ Averbakh, Yuri Lvovich; Beilin, Mikhail Abramovich (1972). Путешествие в шахматное королевство (in Russian). Fizkultura i sport. p. 227.
^ "Teed vs. Delmar". Retrieved December 16, 2020.
^ Edward G. Winter (September 3, 2006). "Chess Notes 4561. 1 d4 f5 2 Bg5".
^ Lev Alburt (2011). Chess Openings for White, Explained. Chess Information Research Center. p. 509.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Fool%27s_mate&oldid=1087066343"
|
Unitary group - Wikipedia
"U(2)" redirects here. For other topics, see U2 (disambiguation).
3.1 2-out-of-3 property
3.2 Special unitary and projective unitary groups
4 G-structure: almost Hermitian
5.1 Indefinite forms
5.3 Degree-2 separable algebras
5.4 Algebraic groups
5.4.1 Unitary group of a quadratic module
7 Classifying space
{\displaystyle \det \colon \operatorname {U} (n)\to \operatorname {U} (1).}
{\displaystyle 1\to \operatorname {SU} (n)\to \operatorname {U} (n)\to \operatorname {U} (1)\to 1.}
{\displaystyle A=S\,\operatorname {diag} \left(e^{i\theta _{1}},\dots ,e^{i\theta _{n}}\right)\,S^{-1}.}
{\displaystyle t\mapsto S\,\operatorname {diag} \left(e^{it\theta _{1}},\dots ,e^{it\theta _{n}}\right)\,S^{-1}.}
{\displaystyle \pi _{1}(\operatorname {U} (n))\cong \mathbf {Z} .}
{\displaystyle \pi _{1}(\operatorname {U} (n))\cong \pi _{1}(\operatorname {SU} (n))\times \pi _{1}(\operatorname {U} (1)).}
Now the first unitary group U(1) is topologically a circle, which is well known to have a fundamental group isomorphic to Z, whereas
{\displaystyle \operatorname {SU} (n)}
is simply connected.[2]
{\displaystyle \operatorname {diag} \left(e^{i\theta _{1}},\dots ,e^{i\theta _{n}}\right)\mapsto \operatorname {diag} \left(e^{i\theta _{\sigma (1)}},\dots ,e^{i\theta _{\sigma (n)}}\right)}
2-out-of-3 propertyEdit
{\displaystyle \operatorname {U} (n)=\operatorname {O} (2n)\cap \operatorname {GL} (n,\mathbf {C} )\cap \operatorname {Sp} (2n,\mathbf {R} ).}
{\displaystyle {\begin{array}{r|r}{\text{Symplectic}}&A^{\mathsf {T}}JA=J\\\hline {\text{Complex}}&A^{-1}JA=J\\\hline {\text{Orthogonal}}&A^{\mathsf {T}}=A^{-1}\end{array}}}
Special unitary and projective unitary groupsEdit
The above is for the classical unitary group (over the complex numbers) – for unitary groups over finite fields, one similarly obtains special unitary and projective unitary groups, but in general
{\displaystyle \operatorname {PSU} \left(n,q^{2}\right)\neq \operatorname {PU} \left(n,q^{2}\right)}
G-structure: almost HermitianEdit
From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group
{\displaystyle {}^{2}\!A_{n}}
, which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group (reversing the Dynkin diagram An, which corresponds to transpose inverse) and the field automorphism of the extension C/R (namely complex conjugation). Both these automorphisms are automorphisms of the algebraic group, have order 2, and commute, and the unitary group is the fixed points of the product automorphism, as an algebraic group. The classical unitary group is a real form of this group, corresponding to the standard Hermitian form Ψ, which is positive definite.
generalizing to other diagrams yields other groups of Lie type, namely the other Steinberg groups
{\displaystyle {}^{2}\!D_{n},{}^{2}\!E_{6},{}^{3}\!D_{4},}
(in addition to
{\displaystyle {}^{2}\!A_{n}}
) and Suzuki-Ree groups
{\displaystyle {}^{2}\!B_{2}\left(2^{2n+1}\right),{}^{2}\!F_{4}\left(2^{2n+1}\right),{}^{2}\!G_{2}\left(3^{2n+1}\right);}
Indefinite formsEdit
{\displaystyle \lVert z\rVert _{\Psi }^{2}=\lVert z_{1}\rVert ^{2}+\dots +\lVert z_{p}\rVert ^{2}-\lVert z_{p+1}\rVert ^{2}-\dots -\lVert z_{n}\rVert ^{2}}
{\displaystyle \Psi (w,z)={\bar {w}}_{1}z_{1}+\cdots +{\bar {w}}_{p}z_{p}-{\bar {w}}_{p+1}z_{p+1}-\cdots -{\bar {w}}_{n}z_{n}.}
Finite fieldsEdit
Over the finite field with q = pr elements, Fq, there is a unique quadratic extension field, Fq2, with order 2 automorphism
{\displaystyle \alpha \colon x\mapsto x^{q}}
(the rth power of the Frobenius automorphism). This allows one to define a Hermitian form on an Fq2 vector space V, as an Fq-bilinear map
{\displaystyle \Psi \colon V\times V\to K}
{\displaystyle \Psi (w,v)=\alpha \left(\Psi (v,w)\right)}
{\displaystyle \Psi (w,cv)=c\Psi (w,v)}
for c ∈ Fq2.[clarification needed] Further, all non-degenerate Hermitian forms on a vector space over a finite field are unitarily congruent to the standard one, represented by the identity matrix; that is, any Hermitian form is unitarily equivalent to
{\displaystyle \Psi (w,v)=w^{\alpha }\cdot v=\sum _{i=1}^{n}w_{i}^{q}v_{i}}
{\displaystyle w_{i},v_{i}}
represent the coordinates of w, v ∈ V in some particular Fq2-basis of the n-dimensional space V (Grove 2002, Thm. 10.3).
Thus one can define a (unique) unitary group of dimension n for the extension Fq2/Fq, denoted either as U(n, q) or U(n, q2) depending on the author. The subgroup of the unitary group consisting of matrices of determinant 1 is called the special unitary group and denoted SU(n, q) or SU(n, q2). For convenience, this article will use the U(n, q2) convention. The center of U(n, q2) has order q + 1 and consists of the scalar matrices that are unitary, that is those matrices cIV with
{\displaystyle c^{q+1}=1}
. The center of the special unitary group has order gcd(n, q + 1) and consists of those unitary scalars which also have order dividing n. The quotient of the unitary group by its center is called the projective unitary group, PU(n, q2), and the quotient of the special unitary group by its center is the projective special unitary group PSU(n, q2). In most cases (n > 1 and (n, q2) ∉ {(2, 22), (2, 32), (3, 22)}), SU(n, q2) is a perfect group and PSU(n, q2) is a finite simple group, (Grove 2002, Thm. 11.22 and 11.26).
Degree-2 separable algebrasEdit
First, there is a unique k-automorphism of K
{\displaystyle a\mapsto {\bar {a}}}
which is an involution and fixes exactly k (
{\displaystyle a={\bar {a}}}
if and only if a ∈ k).[5] This generalizes complex conjugation and the conjugation of degree 2 finite field extensions, and allows one to define Hermitian forms and unitary groups as above.
Algebraic groupsEdit
The equations defining a unitary group are polynomial equations over k (but not over K): for the standard form Φ = I, the equations are given in matrices as A∗A = I, where
{\displaystyle A^{*}={\bar {A}}^{\mathsf {T}}}
is the conjugate transpose. Given a different form, they are A∗ΦA = Φ. The unitary group is thus an algebraic group, whose points over a k-algebra R are given by:
{\displaystyle \operatorname {U} (n,K/k,\Phi )(R):=\left\{A\in \operatorname {GL} (n,K\otimes _{k}R):A^{*}\Phi A=\Phi \right\}.}
{\displaystyle {\begin{aligned}\operatorname {U} (n,\mathbf {C} /\mathbf {R} )(\mathbf {R} )&=\operatorname {U} (n)\\\operatorname {U} (n,\mathbf {C} /\mathbf {R} )(\mathbf {C} )&=\operatorname {GL} (n,\mathbf {C} ).\end{aligned}}}
Unitary group of a quadratic moduleEdit
Let R be a ring with anti-automorphism J,
{\displaystyle \varepsilon \in R^{\times }}
{\displaystyle r^{J^{2}}=\varepsilon r\varepsilon ^{-1}}
for all r in R and
{\displaystyle \varepsilon ^{J}=\varepsilon ^{-1}}
{\displaystyle {\begin{aligned}\Lambda _{\text{min}}&:=\left\{r\in R\ :\ r-r^{J}\varepsilon \right\},\\\Lambda _{\text{max}}&:=\left\{r\in R\ :\ r^{J}\varepsilon =-r\right\}.\end{aligned}}}
Let Λ ⊆ R be an additive subgroup of R, then Λ is called form parameter if
{\displaystyle \Lambda _{\text{min}}\subseteq \Lambda \subseteq \Lambda _{\text{max}}}
{\displaystyle r^{J}\Lambda r\subseteq \Lambda }
. A pair (R, Λ) such that R is a ring and Λ a form parameter is called form ring.
Let M be an R-module and f a J-sesquilinear form on M (i.e.,
{\displaystyle f(xr,ys)=r^{J}f(x,y)s}
{\displaystyle x,y\in M}
{\displaystyle r,s\in R}
). Define
{\displaystyle h(x,y):=f(x,y)+f(y,x)^{J}\varepsilon \in R}
{\displaystyle q(x):=f(x,x)\in R/\Lambda }
, then f is said to define the Λ-quadratic form (h, q) on M. A quadratic module over (R, Λ) is a triple (M, h, q) such that M is an R-module and (h, q) is a Λ-quadratic form.
{\displaystyle U(M):=\{\sigma \in GL(M)\ :\ \forall x,y\in M,h(\sigma x,\sigma y)=h(x,y){\text{ and }}q(\sigma x)=q(x)\}.}
The special case where Λ = Λmax, with J any non-trivial involution (i.e.,
{\displaystyle J\neq id_{R},J^{2}=id_{R}}
and ε = −1 gives back the "classical" unitary group (as an algebraic group).
Polynomial invariantsEdit
{\displaystyle {\begin{aligned}C_{1}&=\left(u^{2}+v^{2}\right)+\left(w^{2}+x^{2}\right)+\left(y^{2}+z^{2}\right)+\ldots \\C_{2}&=\left(uv-vu\right)+\left(wx-xw\right)+\left(yz-zy\right)+\ldots \end{aligned}}}
These are easily seen to be the real and imaginary parts of the complex form
{\displaystyle Z{\overline {Z}}}
. The two invariants separately are invariants of O(2n) and Sp(2n). Combined they make the invariants of U(n) which is a subgroup of both these groups. The variables must be non-commutative in these invariants otherwise the second polynomial is identically zero.
Classifying spaceEdit
^ Arnold, V.I. (1989). Mathematical Methods of Classical Mechanics (Second ed.). Springer. p. 225.
^ Baez, John. "Symplectic, Quaternionic, Fermionic". Retrieved 1 February 2012.
^ Milne, Algebraic Groups and Arithmetic Groups, p. 103
^ Bak, Anthony (1969), "On modules with quadratic forms", Algebraic K-Theory and its Geometric Applications (editors—Moss R. M. F., Thomas C. B.) Lecture Notes in Mathematics, Vol. 108, pp. 55-66, Springer. doi:10.1007/BFb0059990
Retrieved from "https://en.wikipedia.org/w/index.php?title=Unitary_group&oldid=1078063336"
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Heat Transfer Characteristics of Oscillating Heat Pipe With Water and Ethanol as Working Fluids | J. Heat Transfer | ASME Digital Collection
Haizhen Xian,
Haizhen Xian
School of Energy and Power Engineering, Beijing Key Laboratory of Energy Safety and Clean Utilization,
e-mail: yyp@ncepu.edu.cn
Dengying Liu,
Xian, H., Yang, Y., Liu, D., and Du, X. (September 21, 2010). "Heat Transfer Characteristics of Oscillating Heat Pipe With Water and Ethanol as Working Fluids." ASME. J. Heat Transfer. December 2010; 132(12): 121501. https://doi.org/10.1115/1.4002366
In this paper, experiments were conducted to achieve a better understanding of the oscillating heat pipe (OHP) operating behavior with water and ethanol as working fluid. The experimental results showed that there existed a necessary temperature difference between the evaporator and the condenser section to keep the heat pipe working. The maximum effective conductivity of the water OHP reached up to
259 kW/m K
, while that of the ethanol OHP is of
111 kW/m K
. Not all the OHPs are operated in the horizontal operation mode. The heat transfer performance of the ethanol OHP was obviously affected by the filling ratio and the inclination angle but the influence law is irregular. The effect of the filling ratio and the inclination angle of the water OHP were smaller than that of the ethanol one. The heat transfer performance of the OHP was improved with increase of operating temperature. The startup characteristics of the OHP depended on the establishment of the integral oscillating process, which was determined by the operating factors. The startup temperature of the ethanol OHP varied from
40°C
50°C
and that of the water, OHP varied from
40°C
60°C
without considering the horizontal operating mode. The water OHP showed a better performance and more stable heat transfer characteristics than the ethanol OHP, which had no obvious advantages of the startup capability as well.
heat pipes, heat transfer, water, oscillating heat pipe, heat transfer characteristics, water, ethanol
Condensers (steam plant), Ethanol, Fluids, Heat pipes, Heat transfer, Operating temperature, Temperature, Water
Heat Transfer Characteristics of Looped Capillary Heat Pipe
Proceedings of the Fifth International Heat Pipe Symposium
Closed Loop Pulsating Heat Pipes, Part A: Parametric Experimental Investigations
Closed Loop Pulsating Heat Pipes, Part B: Visualization and Semi-Empirical Modeling
High Speed Flow Visualization of a Closed Loop Pulsating Heat Pipe
Study on Heat Transfer Enhancement of Oscillating-Flow Heat Pipe for Drying
Study on the Heat Transfer Enhancement of Oscillating-Flow Heat Pipe by Pulse Heating
CPU Cooling of Notebook PC by Oscillating Heat Pipe
Dangeton
Soponronnarit
Close-Ended Oscillating Heat-Pipe (CEOHP) Air-Preheater for Energy Thrift in a Dryer
Wannapakne
Experimental Study of the Performance of a Solar Collector by Closed-End Oscillating Heat Pipe (CEOHP)
Investigations on the Heat Transport Capability of a Cryogenic Oscillating Heat Pipe and Its Application in Achieving Ultra-Fast Cooling Rates for Cell Vitrification Cryopreservation
|
If one takes Pascal's triangle with
{\displaystyle 2^{n}}
rows and colors the even numbers white, and the odd numbers black, the result is an approximation to the Sierpinski triangle. More precisely, the limit as
{\displaystyle n}
approaches infinity of this parity-colored
{\displaystyle 2^{n}}
-row Pascal triangle is the Sierpinski triangle.[12]
The Towers of Hanoi puzzle involves moving disks of different sizes between three pegs, maintaining the property that no disk is ever placed on top of a smaller disk. The states of a{\displaystyle n}
-disk puzzle, and the allowable moves from one state to another, form an undirected graph, the Hanoi graph, that can be represented geometrically as the intersection graph of the set of triangles remaining after the
{\displaystyle n}
th step in the construction of the Sierpinski triangle. Thus, in the limit as
{\displaystyle n}
goes to infinity, this sequence of graphs can be interpreted as a discrete analogue of the Sierpinski triangle.[13]
For integer number of dimensions
{\displaystyle d}
, when doubling a side of an object,
{\displaystyle 2^{d}}
copies of it are created, i.e. 2 copies for 1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For the Sierpinski triangle, doubling its side creates 3 copies of itself. Thus the Sierpinski triangle has Hausdorff dimension
{\displaystyle {\tfrac {\log 3}{\log 2}}\approx 1.585}
, which follows from solving
{\displaystyle 2^{d}=3}
{\displaystyle d}
The area of a Sierpinski triangle is zero (in Lebesgue measure). The area remaining after each iteration is
{\displaystyle {\tfrac {3}{4}}}
of the area from the previous iteration, and an infinite number of iterations results in an area approaching zero.[15]
The points of a Sierpinski triangle have a simple characterization in barycentric coordinates.[16] If a point has barycentric coordinates
{\displaystyle (0.u_{1}u_{2}u_{3}\dots ,0.v_{1}v_{2}v_{3}\dots ,0.w_{1}w_{2}w_{3}\dots )}
, expressed as binary numerals, then the point is in Sierpinski's triangle if and only if
{\displaystyle u_{i}+v_{i}+w_{i}=1}
{\displaystyle i}
A generalization of the Sierpinski triangle can also be generated using Pascal's triangle if a different modulus
{\displaystyle P}
is used. Iteratio{\displaystyle n}
can be generated by taking a Pascal's triangle with
{\displaystyle P^{n}}
rows and coloring numbers by their value modulo
{\displaystyle P}
{\displaystyle n}
approaches infinity, a fractal is generated.
The same fractal can be achieved by dividing a triangle into a tessellation of
{\displaystyle P^{2}}
similar triangles and removing the triangles that are upside-down from the original, then iterating this step with each smaller triangle.
Conversely, the fractal can also be generated by beginning with a triangle and duplicating it and arranging
{\displaystyle {\tfrac {n(n+1)}{2}}}
of the new figures in the same orientation into a larger similar triangle with the vertices of the previous figures touching, then iterating that step.[17]
A tetrix constructed from an initial tetrahedron of side-length
{\displaystyle L}
has the property that the total surface area remains constant with each iteration. The initial surface area of the (iteration-0) tetrahedron of side-length
{\displaystyle L}
{\displaystyle L^{2}{\sqrt {3}}}
. The next iteration consists of four copies with side length
{\displaystyle {\tfrac {L}{2}}}
, so the total area is
{\textstyle 4{\bigl (}{\tfrac {L}{2}}{\bigr )}^{2}{\sqrt {3}}=L^{2}{\sqrt {3}}}
again. Subsequent iterations again quadruple the number of copies and halve the side length, preserving the overall area. Meanwhile, the volume of the construction is halved at every step and therefore approaches zero. The limit of this process has neither volume nor surface but, like the Sierpinski gasket, is an intricately connected curve. Its Hausdorff dimension is
{\textstyle {\tfrac {\log 4}{\log 2}}=2}
; here "log" denotes the natural logarithm, the numerator is the logarithm of the number of copies of the shape formed from each copy of the previous iteration, and the denominator is the logarithm of the factor by which these copies are scaled down from the previous iteration. If all points are projected onto a plane that is parallel to two of the outer edges, they exactly fill a square of side length
{\displaystyle {\tfrac {L}{\sqrt {2}}}}
without overlap.[18]
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{\displaystyle 1+2+3+4+5+6+7+8+9+10+11+12+13,}
{\displaystyle 1+2+\cdots +13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}\,i,}}
{\displaystyle \left(\Sigma \right)}
{\displaystyle i}
{\displaystyle i}
{\displaystyle \Sigma }
{\displaystyle i}
{\displaystyle 1}
{\displaystyle 13}
{\displaystyle i=1,}
{\displaystyle i=2,}
{\displaystyle i=3,}
{\displaystyle i=13.}
{\displaystyle {\displaystyle \sum _{i=1}^{13}}\,i\,=\,1+2+3+4+5+6+7+8+9+10+11+12+13.}
{\displaystyle {\displaystyle \sum _{i=1}^{5}}\,i^{2}\,=\,1^{2}+2^{2}+3^{2}+4^{2}+5^{2}.}
{\displaystyle {\displaystyle \sum _{i=n}^{2n}}\,i\,=\,n+(n+1)+\cdots +(2n-1)+2n,}
{\displaystyle {\displaystyle \sum _{i=1}^{n}}\,i^{3}\,=\,1^{3}+2^{3}+3^{3}+\cdots +n^{3}.}
{\displaystyle \mathbb {N} }
{\displaystyle {\text{The Natural Numbers}}\,=\,\mathbb {N} \,=\,\{1,2,3,\ldots \}\,=\,\{1,1+1,1+1+1,1+1+1+1,\ldots \}.}
{\displaystyle 1}
{\displaystyle n,}
{\displaystyle n+1}
{\displaystyle n-1,}
{\displaystyle n.}
{\displaystyle (n+1)^{\mathrm {th} }}
{\displaystyle n^{\mathrm {th} }}
{\displaystyle n}
{\displaystyle n}atural numbers is
{\displaystyle {\displaystyle \sum _{i=1}^{n}i\,=\,1+2+\cdots +n\,=\,{\frac {n(n+1)}{2}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)}{2}}\,=\,{\frac {1(1+1)}{2}}\,=\,1,}}
{\displaystyle 1}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)}{2}}\,=\,{\frac {(n-1)n}{2}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i}&=&{\displaystyle \sum _{i=1}^{n-1}i\,+\,n}\\\\&=&{\displaystyle {\frac {(n-1)n}{2}}\,+\,n\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{2}-n}{2}}\,+\,{\frac {2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}-n+2n}{2}}}\\\\&=&{\displaystyle {\frac {n^{2}+n}{2}}}\\\\&=&{\displaystyle {\frac {n(n+1)}{2}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{2}\,=\,1^{2}+2^{2}+\cdots +n^{2}\,=\,{\frac {n(n+1)(2n+1)}{6}}.}}
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n(n+1)(2n+1)}{6}}\,=\,{\frac {1(1+1)(2+1)}{6}}\,=\,1,}}
{\displaystyle n=1.}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i\,=\,{\frac {(n-1)\left(\left(n-1\right)+1\right)\left(2\left(n-1\right)+1\right)}{6}}\,=\,{\frac {(n-1)n(2n-1)}{6}}\,=\,{\frac {2n^{3}-3n^{2}+n}{6}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{2}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{2}+n^{2}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+n^{2}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {2n^{3}-3n^{2}+n}{6}}+{\frac {6n^{2}}{6}}}\\\\&=&{\displaystyle {\frac {2n^{3}+3n^{2}+n}{6}}}\\\\&=&{\displaystyle {\frac {n(2n^{2}+3n+1)}{6}}}\\\\&=&{\displaystyle {\frac {n(n+1)(2n+1)}{6}}},\end{array}}}
{\displaystyle \square }
{\displaystyle n}
{\displaystyle {\displaystyle \sum _{i=1}^{n}i^{3}\,=\,1^{3}+2^{3}+\cdots +n^{3}\,=\,{\frac {n^{2}(n+1)^{2}}{4}}.}}
{\displaystyle n}atural numbers.
{\displaystyle n=1,}
{\displaystyle {\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}\,=\,{\frac {1^{2}(1+1)^{2}}{4}}\,=\,1,}}
{\displaystyle n-1,}
{\displaystyle {\displaystyle \sum _{i=1}^{n-1}i^{3}\,=\,{\frac {(n-1)^{2}\left(\left(n-1\right)+1\right)^{2}}{4}}\,=\,{\frac {(n-1)^{2}n^{2}}{4}}.}}
{\displaystyle {\begin{array}{rcl}{\displaystyle \sum _{i=1}^{n}i^{3}}&=&{\displaystyle \sum _{i=1}^{n-1}i^{3}+n^{3}}\\\\&=&{\displaystyle {\frac {(n-1)^{2}n^{2}}{4}}+n^{3}\qquad \qquad {\mbox{(by the induction assumption)}}}\\\\&=&{\displaystyle {\frac {n^{4}-2n^{3}+n^{2}}{4}}+{\frac {4n^{3}}{4}}}\\\\&=&{\displaystyle {\frac {n^{4}+2n^{3}+n^{2}}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n^{2}+2n+1)}{4}}}\\\\&=&{\displaystyle {\frac {n^{2}(n+1)^{2}}{4}}},\end{array}}}
{\displaystyle \square }
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Bagnold_number Knowpia
The Bagnold number (Ba) is the ratio of grain collision stresses to viscous fluid stresses in a granular flow with interstitial Newtonian fluid, first identified by Ralph Alger Bagnold.[1]
The Bagnold number is defined by
{\displaystyle \mathrm {Ba} ={\frac {\rho d^{2}\lambda ^{1/2}{\dot {\gamma }}}{\mu }}}
{\displaystyle \rho }
is the particle density,
{\displaystyle d}
is the grain diameter,
{\displaystyle {\dot {\gamma }}}
is the shear rate and
{\displaystyle \mu }
is the dynamic viscosity of the interstitial fluid. The parameter
{\displaystyle \lambda }
is known as the linear concentration, and is given by
{\displaystyle \lambda ={\frac {1}{\left(\phi _{0}/\phi \right)^{\frac {1}{3}}-1}}}
{\displaystyle \phi }
is the solids fraction and
{\displaystyle \phi _{0}}
is the maximum possible concentration (see random close packing).
In flows with small Bagnold numbers (Ba < 40), viscous fluid stresses dominate grain collision stresses, and the flow is said to be in the "macro-viscous" regime. Grain collision stresses dominate at large Bagnold number (Ba > 450), which is known as the "grain-inertia" regime. A transitional regime falls between these two values.
^ Bagnold, R. A. (1954). "Experiments on a Gravity-Free Dispersion of Large Solid Spheres in a Newtonian Fluid under Shear". Proc. R. Soc. Lond. A. 225 (1160): 49–63. Bibcode:1954RSPSA.225...49B. doi:10.1098/rspa.1954.0186.
^ Hunt, M. L.; Zenit, R.; Campbell, C. S.; Brennen, C.E. (2002). "Revisiting the 1954 suspension experiments of R. A. Bagnold". Journal of Fluid Mechanics. 452: 1–24. CiteSeerX 10.1.1.564.7792. doi:10.1017/S0022112001006577.
Granular Material Flows at N.A.S.A
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Grandi's series - Wikipedia
The infinite sum of alternating 1 and -1 terms
In mathematics, the infinite series 1 − 1 + 1 − 1 + ⋯, also written
{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}}
is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703. It is a divergent series, meaning that it lacks a sum in the usual sense. On the other hand, its Cesàro sum is 1/2.
1 Unrigorous methods
2 Relation to the geometric series
5.1 Cognitive impact
Unrigorous methods[edit]
One obvious method to attack the series
1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + ...
is to treat it like a telescoping series and perform the subtractions in place:
(1 − 1) + (1 − 1) + (1 − 1) + ... = 0 + 0 + 0 + ... = 0.
On the other hand, a similar bracketing procedure leads to the apparently contradictory result
1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + ... = 1 + 0 + 0 + 0 + ... = 1.
Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value". (Variations of this idea, called the Eilenberg–Mazur swindle, are sometimes used in knot theory and algebra.)
Treating Grandi's series as a divergent geometric series and using the same algebraic methods that evaluate convergent geometric series to obtain a third value:
S = 1 − 1 + 1 − 1 + ..., so
1 − S = 1 − (1 − 1 + 1 − 1 + ...) = 1 − 1 + 1 − 1 + ... = S
1 − S = S
1 = 2S,
resulting in S = 1/2. The same conclusion results from calculating −S, subtracting the result from S, and solving 2S = 1.[1]
The above manipulations do not consider what the sum of a series actually means and how said algebraic methods can be applied to divergent geometric series. Still, to the extent that it is important to be able to bracket series at will, and that it is more important to be able to perform arithmetic with them, one can arrive at two conclusions:
The series 1 − 1 + 1 − 1 + ... has no sum.[1][2]
...but its sum should be 1/2.[2]
In fact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts that arose in the 19th century. After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians.[3][4]
Relation to the geometric series[edit]
For any numbe{\displaystyle r}
{\displaystyle (-1,1)}
, the sum to infinity of a geometric series can be evaluated via
{\displaystyle \lim _{N\to \infty }\sum _{n=0}^{N}r^{n}=\sum _{n=0}^{\infty }r^{n}={\frac {1}{1-r}}.}
{\displaystyle \varepsilon \in (0,2)}
, one thus finds
{\displaystyle \sum _{n=0}^{\infty }(-1+\varepsilon )^{n}={\frac {1}{1-(-1+\varepsilon )}}={\frac {1}{2-\varepsilon }},}
and so the limit
{\displaystyle \varepsilon \to 0}
of series evaluations is
{\displaystyle \lim _{\varepsilon \to 0}\lim _{N\to \infty }\sum _{n=0}^{N}(-1+\varepsilon )^{n}={\frac {1}{2}}.}
However, as mentioned, the series obtained by switching the limits,
{\displaystyle \lim _{N\to \infty }\lim _{\varepsilon \to 0}\sum _{n=0}^{N}(-1+\varepsilon )^{n}=\sum _{n=0}^{\infty }(-1)^{n}}
In the terms of complex analysis,
{\displaystyle {\tfrac {1}{2}}}
is thus seen to be the value at
{\displaystyle z=-1}
of the analytic continuation of the series
{\displaystyle \sum _{n=0}^{N}z^{n}}
, which is only defined on the complex unit disk,
{\displaystyle |z|<1}
Main article: History of Grandi's series
In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. The sequence of partial sums of Grandi's series is 1, 0, 1, 0, ..., which clearly does not approach any number (although it does have two accumulation points at 0 and 1). Therefore, Grandi's series is divergent.
It can be shown that it is not valid to perform many seemingly innocuous operations on a series, such as reordering individual terms, unless the series is absolutely convergent. Otherwise these operations can alter the result of summation.[5] Further, the terms of Grandi's series can be rearranged to have its accumulation points at any interval of two or more consecutive integer numbers, not only 0 or 1. For instance, the series
{\displaystyle 1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1-\cdots }
(in which, after five initial +1 terms, the terms alternate in pairs of +1 and −1 terms) is a permutation of Grandi's series in which each value in the rearranged series corresponds to a value that is at most four positions away from it in the original series; its accumulation points are 3, 4, and 5.
Cognitive impact[edit]
Around 1987, Anna Sierpińska introduced Grandi's series to a group of 17-year-old precalculus students at a Warsaw lyceum. She focused on humanities students with the expectation that their mathematical experience would be less significant than that of their peers studying mathematics and physics, so the epistemological obstacles they exhibit would be more representative of the obstacles that may still be present in lyceum students.
Sierpińska initially expected the students to balk at assigning a value to Grandi's series, at which point she could shock them by claiming that 1 − 1 + 1 − 1 + · · · = 1⁄2 as a result of the geometric series formula. Ideally, by searching for the error in reasoning and by investigating the formula for various common ratios, the students would "notice that there are two kinds of series and an implicit conception of convergence will be born." However, the students showed no shock at being told that 1 − 1 + 1 − 1 + · · · = 1⁄2 or even that 1 + 2 + 4 + 8 + · · · = −1. Sierpińska remarks that a priori, the students' reaction shouldn't be too surprising given that Leibniz and Grandi thought 1⁄2 to be a plausible result;
"A posteriori, however, the explanation of this lack of shock on the part of the students may be somewhat different. They accepted calmly the absurdity because, after all, 'mathematics is completely abstract and far from reality', and 'with those mathematical transformations you can prove all kinds of nonsense', as one of the boys later said."
The students were ultimately not immune to the question of convergence; Sierpińska succeeded in engaging them in the issue by linking it to decimal expansions the following day. As soon as 0.999... = 1 caught the students by surprise, the rest of her material "went past their ears".[6]
Preconceptions[edit]
In another study conducted in Treviso, Italy around the year 2000, third-year and fourth-year Liceo Scientifico pupils (between 16 and 18 years old) were given cards asking the following:
"In 1703, the mathematician Guido Grandi studied the addition: 1 – 1 + 1 – 1 + ... (addends, infinitely many, are always +1 and –1). What is your opinion about it?"
The students had been introduced to the idea of an infinite set, but they had no prior experience with infinite series. They were given ten minutes without books or calculators. The 88 responses were categorized as follows:
(26) the result is 0
(18) the result can be either 0 or 1
(5) the result does not exist
(4) the result is 1⁄2
(3) the result is 1
(2) the result is infinite
(30) no answer
The researcher, Giorgio Bagni, interviewed several of the students to determine their reasoning. Some 16 of them justified an answer of 0 using logic similar to that of Grandi and Riccati. Others justified 1⁄2 as being the average of 0 and 1. Bagni notes that their reasoning, while similar to Leibniz's, lacks the probabilistic basis that was so important to 18th-century mathematics. He concludes that the responses are consistent with a link between historical development and individual development, although the cultural context is different.[7]
Joel Lehmann describes the process of distinguishing between different sum concepts as building a bridge over a conceptual crevasse: the confusion over divergence that dogged 18th-century mathematics.
"Since series are generally presented without history and separate from applications, the student must wonder not only "What are these things?" but also "Why are we doing this?" The preoccupation with determining convergence but not the sum makes the whole process seem artificial and pointless to many students—and instructors as well."
As a result, many students develop an attitude similar to Euler's:
"...problems that arise naturally (i.e., from nature) do have solutions, so the assumption that things will work out eventually is justified experimentally without the need for existence sorts of proof. Assume everything is okay, and if the arrived-at solution works, you were probably right, or at least right enough. ...so why bother with the details that only show up in homework problems?"
Lehmann recommends meeting this objection with the same example that was advanced against Euler's treatment of Grandi's series by Callet.[clarification needed]
Main article: Summation of Grandi's series
Main article: Occurrences of Grandi's series
The series 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + .... (up to infinity) is also divergent, but some methods may be used to sum it to 1⁄4. This is the square of the value most summation methods assign to Grandi's series, which is reasonable as it can be viewed as the Cauchy product of two copies of Grandi's series.
^ a b Devlin p.77
^ a b Davis p.152
^ Kline 1983 p.307
^ Knopp p.457
^ Protter, Murray H.; Morrey, Charles B., Jr. (1991), A First Course in Real Analysis, Undergraduate Texts in Mathematics, Springer, p. 249, ISBN 9780387974378 .
^ Sierpińska pp. 371–378
^ Bagni pp. 6–8
Bagni, Giorgio T. (2005-06-30). "Infinite Series from History to Mathematics Education" (PDF). International Journal for Mathematics Teaching and Learning. Archived from the original (PDF) on 2006-12-29.
Davis, Harry F. (May 1989). Fourier Series and Orthogonal Functions. Dover. ISBN 978-0-486-65973-2.
Devlin, Keith (1994). Mathematics, the science of patterns: the search for order in life, mind, and the universe. Scientific American Library. ISBN 978-0-7167-6022-1.
Kline, Morris (November 1983). "Euler and Infinite Series". Mathematics Magazine. 56 (5): 307–314. CiteSeerX 10.1.1.639.6923. doi:10.2307/2690371. JSTOR 2690371.
Knopp, Konrad (1990) [1922]. Theory and Application of Infinite Series. Dover. ISBN 978-0-486-66165-0.
Hobson, E. W. (1907). The theory of functions of a real variable and the theory of Fourier's series. The University of Michigan Historical Mathematics Collection: Cambridge University Press. section 331. {{cite book}}: External link in |location= (help)
Lehmann, Joel (1995). "Converging Concepts of Series: Learning from History". Learn from the Masters. ISBN 0-88385-703-0.
Sierpińska, Anna (November 1987). "Humanities students and epistemological obstacles related to limits". Educational Studies in Mathematics. 18 (4): 371–396. doi:10.1007/BF00240986. JSTOR 3482354.
Whittaker, E. T.; Watson, G. N. (1962). A Course of Modern Analysis (4th, reprinted ed.). Cambridge University Press. § 2.1.
One minus one plus one minus one – Numberphile, Grandi's series
Retrieved from "https://en.wikipedia.org/w/index.php?title=Grandi%27s_series&oldid=1071485568"
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One-class kernel subspace ensemble for medical image classification | SpringerLink
{\mathbb{H}}_{\kappa }
\mathbb{X}\subset {\Re }^{n}
{\mathbb{H}}_{\kappa }
x,y\in \mathbb{X}.\phantom{\rule{0.3em}{0ex}}\phi \left(·\right)
\phi \left(x\right):\mathbb{X}\to {\mathbb{H}}_{\kappa }
\left\{{x}_{1},{x}_{2},\dots ,{x}_{N}\right\}\in \mathbb{X}
{\mathbb{H}}_{\kappa }
\mathit{\text{HKH}}=\mathrm{U\Lambda }{U}^{\prime }
H=I-\frac{1}{N}\mathbf{\text{1}}{\mathbf{\text{1}}}^{\prime }
\stackrel{̄}{\phi }=\frac{1}{N}\sum _{j=1}^{N}\phi \left({x}_{j}\right)
\stackrel{~}{\phi }\left({x}_{i}\right)
\stackrel{~}{\phi }\left({x}_{i}\right)=\phi \left({x}_{i}\right)-\stackrel{̄}{\phi }.
\stackrel{~}{\phi }\left({x}_{i}\right)
{V}_{k}=\sum _{i=1}^{N}{\alpha }_{\mathit{\text{ki}}}\stackrel{~}{\phi }\left({x}_{i}\right)=\stackrel{~}{\phi }{\mathbit{\alpha }}_{k},
\stackrel{~}{\phi }=\left[\stackrel{~}{\phi }\left({x}_{1}\right),\stackrel{~}{\phi }\left({x}_{2}\right),\mathrm{...},\stackrel{~}{\phi }\left({x}_{N}\right)\right]
\begin{array}{ll}{\beta }_{k}& =\stackrel{~}{\phi }{\left(x\right)}^{\prime }{V}_{k}=\sum _{i=1}^{N}{\alpha }_{\mathit{\text{ki}}}\stackrel{~}{\phi }{\left(x\right)}^{\prime }\stackrel{~}{\phi }\left({x}_{i}\right)\phantom{\rule{2em}{0ex}}\\ =\sum _{i=1}^{N}{\alpha }_{\mathit{\text{ki}}}\stackrel{~}{\kappa }\left(x,{x}_{i}\right),\phantom{\rule{2em}{0ex}}\end{array}
\begin{array}{ll}\stackrel{~}{\kappa }\left(x,y\right)& =\stackrel{~}{\phi }{\left(x\right)}^{\prime }\stackrel{~}{\phi }\left(y\right)\\ ={\left(\phi \left(x\right)-\stackrel{̄}{\phi }\right)}^{\prime }\left(\phi \left(y\right)-\stackrel{̄}{\phi }\right)\\ =\kappa \left(x,y\right)-\frac{1}{N}{\mathbf{1}}^{\prime }{\mathbf{k}}_{x}-\frac{1}{N}{\mathbf{1}}^{\prime }{\mathbf{k}}_{y}+\frac{1}{{N}^{2}}{\mathbf{1}}^{\prime }\mathbf{\text{K}}\mathbf{1}\phantom{\rule{2em}{0ex}}\end{array}
\begin{array}{ll}{\stackrel{~}{\kappa }}_{x}& ={\left[\stackrel{~}{\kappa }\left(x,{x}_{1}\right),\dots ,\stackrel{~}{\kappa }\left(x,{x}_{N}\right)\right]}^{\prime }\\ ={\mathbf{k}}_{x}-\frac{1}{N}{\mathbf{11}}^{\prime }{\mathbf{k}}_{x}-\frac{1}{N}\mathbf{K}\mathbf{1}+\frac{1}{{N}^{2}}{\mathbf{11}}^{\prime }\mathbf{K}\mathbf{1}\\ =\mathbf{H}\left({\mathbf{k}}_{x}-\frac{1}{N}\mathbf{K}\mathbf{1}\right),\phantom{\rule{2em}{0ex}}\end{array}
{\beta }_{k}={\mathbit{\alpha }}_{k}^{{}^{\prime }}{\stackrel{~}{\kappa }}_{x}
\begin{array}{ll}P\left(\phi \left(x\right)\right)& =\sum _{k=1}^{M}{\beta }_{k}{V}_{k}+\stackrel{̄}{\phi }=\sum _{k=1}^{M}\left({\mathbit{\alpha }}_{k}^{\prime }{\stackrel{~}{\kappa }}_{x}\right)\left(\stackrel{~}{\phi }{\mathbit{\alpha }}_{k}\right)+\stackrel{̄}{\phi }\\ =\stackrel{~}{\phi }\mathbf{L}{\stackrel{~}{\kappa }}_{x}+\stackrel{̄}{\phi },\phantom{\rule{2em}{0ex}}\end{array}
\mathbf{L}=\sum _{k=1}^{M}{\mathbit{\alpha }}_{k}{\mathbit{\alpha }}_{k}^{\prime }
{\mathbb{H}}_{\kappa }
{x}_{1}^{{}^{\prime }},\dots ,{x}_{m}^{{}^{\prime }}
{x}_{i}^{{}^{\prime }}
\stackrel{̂}{x}
\phi \left(\stackrel{̂}{x}\right)=P\left(\phi \left(x\right)\right).
\stackrel{̂}{x}
\phantom{\rule{2.77626pt}{0ex}}\stackrel{~}{d}\left(\phi \left({x}_{i}\right),\phi \left({x}_{j}\right)\right)
{\stackrel{~}{d}}_{\mathit{\text{ij}}}^{\phantom{\rule{0.3em}{0ex}}2}={\mathbf{K}}_{\mathit{\text{ii}}}+{\mathbf{K}}_{\mathit{\text{jj}}}-2\kappa \left({d}_{\mathit{\text{ij}}}^{2}\right).
\kappa \left({d}_{\mathit{\text{ij}}}^{2}\right)=\frac{1}{2}\left({\mathbf{K}}_{\mathit{\text{ii}}}+{\mathbf{K}}_{\mathit{\text{jj}}}-\phantom{\rule{0.3em}{0ex}}{\stackrel{~}{d}}_{\mathit{\text{ij}}}^{\phantom{\rule{0.3em}{0ex}}2}\right).
\phantom{\rule{2.77626pt}{0ex}}{\stackrel{~}{d}}_{\mathit{\text{ij}}}^{\phantom{\rule{0.3em}{0ex}}2}
\phantom{\rule{2.77626pt}{0ex}}\stackrel{~}{d}\left(P\left(\phi \left(x\right)\right),\phi \left({x}_{i}\right)\right)
\begin{array}{ll}{\stackrel{~}{d}}^{\phantom{\rule{0.3em}{0ex}}2}\left(P\left(\phi \left(x\right)\right),\phi \left(x\right)\right)=\parallel P\left(\phi \left(x\right)\right){\parallel }^{2}& +\parallel \phi \left({x}_{i}\right){\parallel }^{2}\\ -2P{\left(\phi \left(x\right)\right)}^{\prime }\phi \left({x}_{i}\right).\end{array}
{\mathbit{d}}^{2}=\left[{d}_{1}^{2},{d}_{2}^{2},\dots ,{d}_{n}^{2}\right].
\stackrel{̂}{x}
{d}^{2}\left(\stackrel{̂}{x},{x}_{i}\right)
{d}^{2}\left(\stackrel{̂}{x},{x}_{i}\right)\simeq {d}_{i}^{2},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}i=1,\dots ,\mathrm{n.}
\stackrel{̂}{x}
{\text{KPCA}}_{i}^{j}
{\text{KPCA}}_{i}^{j}
{\text{KPCA}}_{i}^{j}
{\text{KPCA}}_{i}^{1},i=1,\dots ,m
{f}_{j}^{\prime }=\left\{{f}_{j}^{\prime 1},{f}_{j}^{\prime 2},\dots ,{f}_{j}^{\mathrm{\prime m}}\right\}
{D}_{j}=\left[{d}_{j}^{1},{d}_{j}^{2},\dots ,{d}_{j}^{m}\right],
{d}_{i}^{j}=\parallel {f}_{j}-{f}_{j}^{\mathrm{\prime i}}{\parallel }^{2},i=1,\dots ,m
D=\left[\begin{array}{c}{D}_{1}\\ {D}_{2}\\ ⋮\\ {D}_{n}\end{array}\right]=\left(\begin{array}{cccc}{d}_{1}^{1}& {d}_{1}^{2}& \cdots & {d}_{1}^{m}\\ {d}_{2}^{1}& {d}_{2}^{2}& \cdots & {d}_{2}^{m}\\ ⋮& ⋮& \cdots & ⋮\\ {d}_{n}^{1}& {d}_{n}^{2}& \cdots & {d}_{n}^{m}\end{array}\right).
{\stackrel{~}{d}}_{i}^{\phantom{\rule{0.3em}{0ex}}j}=\text{exp}\left(-{d}_{i}^{\phantom{\rule{0.3em}{0ex}}j}/s\right),
{d}_{i}^{j}
\stackrel{~}{D}
\stackrel{~}{D}
\stackrel{~}{D}
{\mathit{\text{cs}}}_{k}\left(x\right)=\frac{\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{T}}\right)}{\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{T}}\right)+\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{O}}\right)},
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{T}}\right)
\stackrel{~}{D}
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{T}}\right)=\prod _{j=1\dots n}\phantom{\rule{2.77626pt}{0ex}}{\stackrel{~}{d}}_{j}^{\phantom{\rule{0.3em}{0ex}}k}.
\prod {P}_{k}\left(x|{w}_{\mathrm{O}}\right)
\stackrel{~}{D}
\phantom{\rule{-10.0pt}{0ex}}\prod {P}_{k}\left(x|{w}_{\mathrm{O}}\right)=\prod \phantom{\rule{0.3em}{0ex}}{\stackrel{~}{d}}_{j}^{\phantom{\rule{0.3em}{0ex}}i},\phantom{\rule{0.3em}{0ex}}j=1\dots n,i=1\dots m\phantom{\rule{2.77626pt}{0ex}}\text{and}\phantom{\rule{2.77626pt}{0ex}}i\ne \mathrm{k.}
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{O}}\right)
\prod {P}_{k}\left(x|{w}_{\mathrm{T}}\right)
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{O}}\right)
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{O}}\right)
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{O}}\right)
{\mathit{\text{cs}}}_{k}\left(x\right)=\frac{\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{T}}\right)}{\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{T}}\right)+\prod _{j}{P}_{k}^{\phantom{\rule{2.77626pt}{0ex}}j}\left(x|{w}_{\mathrm{O}}\right)},
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{T}}\right)
{P}_{k}^{\phantom{\rule{2.77626pt}{0ex}}j}\left(x|{w}_{\mathrm{O}}\right)
\prod _{j}{P}_{k}^{\phantom{\rule{2.77626pt}{0ex}}j}\left(x|{w}_{\mathrm{O}}\right)
{P}_{k}^{\phantom{\rule{2.77626pt}{0ex}}j}\left(x|{w}_{\mathrm{O}}\right)=max\left\{\phantom{\rule{2.77626pt}{0ex}}{\stackrel{~}{d}}_{j}^{\phantom{\rule{0.3em}{0ex}}i}\right\},\phantom{\rule{1em}{0ex}}i=1\dots m\text{and}\phantom{\rule{1em}{0ex}}i\ne k.
\stackrel{~}{D}
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{T}}\right)
\prod _{j}{P}_{k}^{\phantom{\rule{2.77626pt}{0ex}}j}\left(x|{w}_{\mathrm{O}}\right)
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{O}}\right)
\prod {P}_{k}\left(x|{w}_{\mathrm{T}}\right)
\stackrel{~}{D}
\prod _{k}{P}_{k}\left(x|{w}_{\mathrm{T}}\right)
\text{TPr}=\frac{\text{True positive}}{\text{True positive + False negative}}
\text{FPr}=\frac{\text{False positive}}{\text{False positive + True negative}}
|
On the Endurance Limit of Fiberglass Pipes Using Acoustic Emission | J. Pressure Vessel Technol. | ASME Digital Collection
Michael D. Engelhardt,
Michael D. Engelhardt Professor
Timothy J. Fowler Adjunct Professor (retired)
Ramirez, G., Engelhardt, M. D., and Fowler, T. J. (November 18, 2005). "On the Endurance Limit of Fiberglass Pipes Using Acoustic Emission." ASME. J. Pressure Vessel Technol. August 2006; 128(3): 454–461. https://doi.org/10.1115/1.2218351
This paper proposes a method for determining endurance limits to leakage for fiberglass pipes under internal pressure using acoustic emission signals and static pressure tests. Results are presented from a series of cyclic internal pressure tests performed on fiberglass pipes fabricated according to ASME RTP-1 (1989) recommendations. The specimens were
8in.
(20.32cm)
internal diameter and
5ft
(1.525m)
long. The construction layup consisted of an internal corrosion barrier followed by several layers of continuous filament wound glass. A state of pure hoop stress was imposed via an internal pressure system that allowed free axial movement of the pipe. The purpose of the tests was to evaluate current design procedures and to assess the capabilities of acoustic emission (AE) monitoring to detect damage and to predict the leakage pressure for the pipes. Twenty-four specimens were tested to failure with AE monitoring. The results of these tests are presented, including data on the measured cyclic life of the specimens, comparisons to the static leakage capacity, and the effectiveness of AE in determining damage induced by cyclic loading.
leak detection, flaw detection, acoustic emission testing, glass, design engineering, corrosion testing, pipes, fiber composites, cyclic, internal pressure, fiberglass, fatigue endurance
Acoustic emissions, Endurance limit, Failure, Glass reinforced plastics, Leakage, Pipes, Pressure, Corrosion, Knee, Signals, Damage, Glass, Cycles
, 2004, Boiler and Pressure Vessel Code, Section 10, ASME, New York.
Reinforced Thermoset Plastic Corrosion Resistant Equipment
, 1989, The American Society of Mechanical Engineers, ASME-ANSI-RTP-1989.
”Design of Fiberglass Reinforced Plastic Chemical Storage Tanks
21st Annual Meeting of the Reinforced Plastics Division
, Chicago IL, The Society of the Plastics Industry, Inc.
Acoustic Emission of Fiber Reinforced Plastics
,” J. Technical Councils of ASCE, pp.
External Pressure Testing of a Large Scale Composite Pipe
Proceedings of the 12th International Conference in Composite Materials
, ICCM-12,
Detection of Deterioration and Damage of Corrosion Resistance of Thermoplastic and FRP Barrier
,” Business Report, Anderson Consultants, 535 Oak Drive, Lake Jackson, TX 77566.
Use of Acoustic Emission in the Design of Glass and Carbon Fiber Reinforced Plastic Pipe
Proceedings 2nd International Symposium on Acoustic Emission from Reinforced Composites
New Technology for Wave Based Acoustic Emission and Acousto-Ultrasonics
,” AMD-Vol.
, Wave Propagation and Emerging Technologies, ASME, New York.
The Composites Institute of the Society of Plastics Industry
Recommended Practice for Acoustic Emission Testing of Fiberglass Reinforced Plastic Resin (RP) Tanks/Vessels
,” published by the Committee f Acoustic Emission from Reinforced Plastics (CARP).
Fiber Reinforced Polymer Vessel Design With a Damage Approach
J. Compos. Constr.
,” ASME RTP-1, ASME, New York.
(ASTM) E1067, “
Standard Practice for Acoustic Emission Examination of Fiberglass Reinforced Plastic Resin (FRP) Tanks and Vessels
,” ASTM, Philadelphia, PA.
, 1998, Technical Data, January.
British Standard for Design and Construction of Vessels and Tanks in Reinforced Plastics
, BS4994–1987.
Code de Construction des Appareils Chaudronnes en Plastique Arme
, 1976, Commission Chaudronnerie, Genie Chimique du Syndicat General de L’Industrie du Plastique Arme, France, December.
Acoustic Emission From Depressurization to Detect/Evaluate Significance of Impact Damage to Graphite/Epoxy Pressure Vessels
,” Technical Product Information, March.
Modal AE: A New Understanding of Acoustic Emission
0730-0050, Technical Note.
Long Duration AE Events in Filament Wound Graphite/Epoxy in the 100-300KHz Band Pass Region
,” First International Symposium on Acoustic Emission From Reinforced Composites, The Society of the Plastics Industry, Inc.
Correlation of Residual Strength With Acoustic Emission From Impact-Damaged Composite Structures Under Constant Biaxial Load
|
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