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Home : Support : Online Help : Graphics : Packages : Plot Tools : arrow
generate 2-D or 3-D plot object for an arrow
arrow(base, dir, wb, wh, hh, sh, fr, brd, options)
arrow(base, dir, pv, wb, wh, hh, sh, fr, brd, options)
base of the arrow, given as 2-D (or 3-D) point or 2-D (or 3-D) Vector
2-D (or 3-D) point or 2-D (or 3-D) direction Vector
vector indicating the plane containing the arrow
width of the body of the arrow
width of the head of the arrow
height of the head of the arrow as a ratio of the length of the body
(optional) shape of the arrow, either harpoon, arrow, double_arrow, or cylindrical_arrow
(optional) add a "fringe" to the arrowhead, only works if shape of the arrow is cylindrical_arrow
(optional) add a border around the arrow, only works for 2-D arrows having shape double_arrow
(optional) equations of the form option=value. For a complete list, see plot/options and plot3d/option.
The arrow command creates plot data objects, which when displayed create an arrow whose base is located at base.
Note: Multiple plot data objects may be generated, so if the result is intended to be part of animation, it may be necessary to enclose these in a PLOT or PLOT3D function.
If dir is a point (a list of two or three real numbers), the arrow is drawn from the point base to the point dir.
If dir is a Vector (a Vector of dimension two or three), the output is the specified vector dir with the tail at the point base.
In the 3-D case, the plane vector
\mathrm{pv}=[a,b,c]
is used to specify the plane in which the vector lies. It lies in the plane containing the arrow and cross product of the arrow and pv. If the direction of the arrow and the plane vector are collinear, the argument pv is ignored.
The wb parameter indicates the width of the body of the arrow
The wh parameter indicates the width of the head of the arrow
The hh parameter is the ratio of the height of the head of the arrow to the length of the body of the arrow.
The optional argument sh indicates how the arrow should be drawn, either harpoon or arrow for a one-dimensional arrow, double_arrow for a two-dimensional arrow, or cylindrical_arrow for a three-dimensional arrow.
The optional argument fr adds a fringe to the arrowhead. It is available only for cylindrical_arrow. The inputs must be in the form fringe = color where color can be any color specification as described in plot/color.
The optional argument brd adds a border around the arrow. It is available only for the 2-D double_arrow. The input must be in the form border = t where t can be true, false or a list of plot options as described in plot/options. The default value is true. If a list of options is given, then any that are applicable to 2-D curves (such as color or thickness) are applied to the border.
A call to arrow produces a plot data object that can be used in a PLOT or PLOT3D data structure, or displayed using the plots[display] command.
The remaining arguments are interpreted as options, which are specified as equations of the form option = value. For more information, see plottools, plot/options and plot3d/option.
In some cases, the result is an expression sequence of plot objects, and therefore in order to group these objects, place a PLOT or PLOT3D command around the result.
\mathrm{with}\left(\mathrm{plots}\right):
\mathrm{with}\left(\mathrm{plottools}\right):
\mathrm{l1}≔\mathrm{arrow}\left([0,0],[10,10],0.2,0.4,0.1,\mathrm{color}="Green"\right):
\mathrm{l2}≔\mathrm{arrow}\left([10,10],\mathrm{Vector}\left([0,10]\right),0.2,0.4,0.1,\mathrm{color}="Red"\right):
\mathrm{display}\left(\mathrm{l1},\mathrm{l2},\mathrm{axes}=\mathrm{frame}\right)
\mathrm{ll}≔\mathrm{arrow}\left([0,0,0],[2,2,2],0.2,0.4,0.1\right):
\mathrm{display}\left(\mathrm{ll},\mathrm{axes}=\mathrm{frame},\mathrm{color}="Red",\mathrm{orientation}=[-50,60]\right)
\mathrm{ll}≔\mathrm{arrow}\left(\mathrm{Vector}\left([1,0,0]\right),\mathrm{Vector}\left([2,2,2]\right),0.2,0.4,0.1,\mathrm{cylindrical_arrow},\mathrm{fringe}=\mathrm{blue},\mathrm{color}="Green"\right):
\mathrm{display}\left(\mathrm{ll},\mathrm{color}="Red",\mathrm{orientation}=[100,160]\right)
\mathrm{display}\left(\mathrm{seq}\left(\mathrm{INTERFACE_PLOT}\left(\mathrm{arrow}\left(\left[0,0\right],\left[i,10\right],0.2,0.4,0.1,\mathrm{`=`}\left(\mathrm{color},"Green"\right)\right)\right),i=1..10\right),\mathrm{insequence}=\mathrm{true}\right)
\mathrm{display}\left(\mathrm{arrow}\left([0,0],[10,10],0.3,0.5,0.1,\mathrm{color}="Green",\mathrm{border}=\mathrm{false}\right)\right)
\mathrm{display}\left(\mathrm{arrow}\left([0,0],[10,10],0.3,0.5,0.1,\mathrm{color}="Green",\mathrm{border}=[\mathrm{color}="Blue",\mathrm{thickness}=4]\right)\right)
The plottools[arrow] command was updated in Maple 2018.
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Burnside problem - Wikipedia
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The Burnside problem asks whether a finitely generated group in which every element has finite order must necessarily be a finite group. It was posed by William Burnside in 1902, making it one of the oldest questions in group theory and was influential in the development of combinatorial group theory. It is known to have a negative answer in general, as Evgeny Golod and Igor Shafarevich provided a counter-example in 1964. The problem has many refinements and variants (see bounded and restricted below) that differ in the additional conditions imposed on the orders of the group elements, some of which are still open questions.
2 General Burnside problem
3 Bounded Burnside problem
4 Restricted Burnside problem
Initial work pointed towards the affirmative answer. For example, if a group G is finitely generated and the order of each element of G is a divisor of 4, then G is finite. Moreover, A. I. Kostrikin was able to prove in 1958 that among the finite groups with a given number of generators and a given prime exponent, there exists a largest one. This provides a solution for the restricted Burnside problem for the case of prime exponent. (Later, in 1989, Efim Zelmanov was able to solve the restricted Burnside problem for an arbitrary exponent.) Issai Schur had shown in 1911 that any finitely generated periodic group that was a subgroup of the group of invertible n × n complex matrices was finite; he used this theorem to prove the Jordan–Schur theorem.[1]
Nevertheless, the general answer to the Burnside problem turned out to be negative. In 1964, Golod and Shafarevich constructed an infinite group of Burnside type without assuming that all elements have uniformly bounded order. In 1968, Pyotr Novikov and Sergei Adian supplied a negative solution to the bounded exponent problem for all odd exponents larger than 4381. In 1982, A. Yu. Ol'shanskii found some striking counterexamples for sufficiently large odd exponents (greater than 1010), and supplied a considerably simpler proof based on geometric ideas.
The case of even exponents turned out to be much harder to settle. In 1992, S. V. Ivanov announced the negative solution for sufficiently large even exponents divisible by a large power of 2 (detailed proofs were published in 1994 and occupied some 300 pages). Later joint work of Ol'shanskii and Ivanov established a negative solution to an analogue of Burnside problem for hyperbolic groups, provided the exponent is sufficiently large. By contrast, when the exponent is small and different from 2, 3, 4 and 6, very little is known.
General Burnside problem[edit]
A group G is called periodic if every element has finite order; in other words, for each g in G, there exists some positive integer n such that gn = 1. Clearly, every finite group is periodic. There exist easily defined groups such as the p∞-group which are infinite periodic groups; but the latter group cannot be finitely generated.
General Burnside problem. If G is a finitely generated, periodic group, then is G necessarily finite?
This question was answered in the negative in 1964 by Evgeny Golod and Igor Shafarevich, who gave an example of an infinite p-group that is finitely generated (see Golod–Shafarevich theorem). However, the orders of the elements of this group are not a priori bounded by a single constant.
Bounded Burnside problem[edit]
The Cayley graph of the 27-element free Burnside group of rank 2 and exponent 3.
Part of the difficulty with the general Burnside problem is that the requirements of being finitely generated and periodic give very little information about the possible structure of a group. Therefore, we pose more requirements on G. Consider a periodic group G with the additional property that there exists a least integer n such that for all g in G, gn = 1. A group with this property is said to be periodic with bounded exponent n, or just a group with exponent n. Burnside problem for groups with bounded exponent asks:
Burnside problem I. If G is a finitely generated group with exponent n, is G necessarily finite?
It turns out that this problem can be restated as a question about the finiteness of groups in a particular family. The free Burnside group of rank m and exponent n, denoted B(m, n), is a group with m distinguished generators x1, ..., xm in which the identity xn = 1 holds for all elements x, and which is the "largest" group satisfying these requirements. More precisely, the characteristic property of B(m, n) is that, given any group G with m generators g1, ..., gm and of exponent n, there is a unique homomorphism from B(m, n) to G that maps the ith generator xi of B(m, n) into the ith generator gi of G. In the language of group presentations, free Burnside group B(m, n) has m generators x1, ..., xm and the relations xn = 1 for each word x in x1, ..., xm, and any group G with m generators of exponent n is obtained from it by imposing additional relations. The existence of the free Burnside group and its uniqueness up to an isomorphism are established by standard techniques of group theory. Thus if G is any finitely generated group of exponent n, then G is a homomorphic image of B(m, n), where m is the number of generators of G. Burnside problem can now be restated as follows:
Burnside problem II. For which positive integers m, n is the free Burnside group B(m, n) finite?
The full solution to Burnside problem in this form is not known. Burnside considered some easy cases in his original paper:
B(1, n) is the cyclic group of order n.
B(m, 2) is the direct product of m copies of the cyclic group of order 2 and hence finite.[note 1]
The following additional results are known (Burnside, Sanov, M. Hall):
B(m, 3), B(m, 4), and B(m, 6) are finite for all m.
The particular case of B(2, 5) remains open: as of 2020[update] it was not known whether this group is finite.
The breakthrough in solving the Burnside problem was achieved by Pyotr Novikov and Sergei Adian in 1968. Using a complicated combinatorial argument, they demonstrated that for every odd number n with n > 4381, there exist infinite, finitely generated groups of exponent n. Adian later improved the bound on the odd exponent to 665.[2] The latest improvement to the bound on odd exponent is 101 obtained by Adian himself in 2015. The case of even exponent turned out to be considerably more difficult. It was only in 1994 that Sergei Vasilievich Ivanov was able to prove an analogue of Novikov–Adian theorem: for any m > 1 and an even n ≥ 248, n divisible by 29, the group B(m, n) is infinite; together with the Novikov–Adian theorem, this implies infiniteness for all m > 1 and n ≥ 248. This was improved in 1996 by I. G. Lysënok to m > 1 and n ≥ 8000. Novikov–Adian, Ivanov and Lysënok established considerably more precise results on the structure of the free Burnside groups. In the case of the odd exponent, all finite subgroups of the free Burnside groups were shown to be cyclic groups. In the even exponent case, each finite subgroup is contained in a product of two dihedral groups, and there exist non-cyclic finite subgroups. Moreover, the word and conjugacy problems were shown to be effectively solvable in B(m, n) both for the cases of odd and even exponents n.
A famous class of counterexamples to the Burnside problem is formed by finitely generated non-cyclic infinite groups in which every nontrivial proper subgroup is a finite cyclic group, the so-called Tarski Monsters. First examples of such groups were constructed by A. Yu. Ol'shanskii in 1979 using geometric methods, thus affirmatively solving O. Yu. Schmidt's problem. In 1982 Ol'shanskii was able to strengthen his results to establish existence, for any sufficiently large prime number p (one can take p > 1075) of a finitely generated infinite group in which every nontrivial proper subgroup is a cyclic group of order p. In a paper published in 1996, Ivanov and Ol'shanskii solved an analogue of the Burnside problem in an arbitrary hyperbolic group for sufficiently large exponents.
Restricted Burnside problem[edit]
Formulated in the 1930s, it asks another, related, question:
Restricted Burnside problem. If it is known that a group G with m generators and exponent n is finite, can one conclude that the order of G is bounded by some constant depending only on m and n? Equivalently, are there only finitely many finite groups with m generators of exponent n, up to isomorphism?
This variant of the Burnside problem can also be stated in terms of certain universal groups with m generators and exponent n. By basic results of group theory, the intersection of two subgroups of finite index in any group is itself a subgroup of finite index. Let M be the intersection of all subgroups of the free Burnside group B(m, n) which have finite index, then M is a normal subgroup of B(m, n) (otherwise, there exists a subgroup g−1Mg with finite index containing elements not in M). One can therefore define a group B0(m, n) to be the factor group B(m, n)/M. Every finite group of exponent n with m generators is a homomorphic image of B0(m, n). The restricted Burnside problem then asks whether B0(m, n) is a finite group.
In the case of the prime exponent p, this problem was extensively studied by A. I. Kostrikin during the 1950s, prior to the negative solution of the general Burnside problem. His solution, establishing the finiteness of B0(m, p), used a relation with deep questions about identities in Lie algebras in finite characteristic. The case of arbitrary exponent has been completely settled in the affirmative by Efim Zelmanov, who was awarded the Fields Medal in 1994 for his work.
^ The key step is to observe that the identities a2 = b2 = (ab)2 = 1 together imply that ab = ba, so that a free Burnside group of exponent two is necessarily abelian.
^ Curtis, Charles; Reiner, Irving (1962). Representation Theory of Finite Groups and Associated Algebras. John Wiley & Sons. pp. 256–262.
^ John Britton proposed a nearly 300 page alternative proof to the Burnside problem in 1973; however, Adian ultimately pointed out a flaw in that proof.
S. I. Adian (1979) The Burnside problem and identities in groups. Translated from the Russian by John Lennox and James Wiegold. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 95. Springer-Verlag, Berlin-New York. ISBN 3-540-08728-1.
S. I. Adian (2015). "New estimates of odd exponents of infinite Burnside groups". Trudy Matematicheskogo Instituta Imeni V. A. Steklova (in Russian). 289: 41–82. doi:10.1134/S0371968515020041. Translation in Adian, S. I. (2015). "New estimates of odd exponents of infinite Burnside groups". Proceedings of the Steklov Institute of Mathematics. 289 (1): 33–71. doi:10.1134/S0081543815040045.
S. V. Ivanov (1994). "The Free Burnside Groups of Sufficiently Large Exponents". International Journal of Algebra and Computation. 04: 1–308. doi:10.1142/S0218196794000026.
S. V. Ivanov; A. Yu. Ol'Shanskii (1996). "Hyperbolic groups and their quotients of bounded exponents". Transactions of the American Mathematical Society. 348 (6): 2091–2138. doi:10.1090/S0002-9947-96-01510-3.
A. I. Kostrikin (1990) Around Burnside. Translated from the Russian and with a preface by James Wiegold. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 20. Springer-Verlag, Berlin. ISBN 3-540-50602-0.
I. G. Lysënok (1996). "Infinite Burnside groups of even exponent" (in Russian). 60 (3): 3–224. doi:10.4213/im77. {{cite journal}}: Cite journal requires |journal= (help) Translation in Lysënok, I. G. (1996). "Infinite Burnside groups of even exponent". Izvestiya: Mathematics. 60 (3): 453–654. Bibcode:1996IzMat..60..453L. doi:10.1070/IM1996v060n03ABEH000077.
A. Yu. Ol'shanskii (1989) Geometry of defining relations in groups. Translated from the 1989 Russian original by Yu. A. Bakhturin (1991) Mathematics and its Applications (Soviet Series), 70. Dordrecht: Kluwer Academic Publishers Group. ISBN 0-7923-1394-1.
E. Zelmanov (1990). "Solution of the restricted Burnside problem for groups of odd exponent". Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya (in Russian). 54 (1): 42–59, 221. Translation in Zel'manov, E I (1991). "Solution of the Restricted Burnside Problem for Groups of Odd Exponent". Mathematics of the USSR-Izvestiya. 36 (1): 41–60. Bibcode:1991IzMat..36...41Z. doi:10.1070/IM1991v036n01ABEH001946. S2CID 39623037.
E. Zelmanov (1991). "Solution of the restricted Burnside problem for 2-groups". Matematicheskii Sbornik (in Russian). 182 (4): 568–592. Translation in Zel'manov, E I (1992). "A Solution of the Restricted Burnside Problem for 2-groups". Mathematics of the USSR-Sbornik. 72 (2): 543–565. Bibcode:1992SbMat..72..543Z. doi:10.1070/SM1992v072n02ABEH001272.
History of the Burnside problem at MacTutor History of Mathematics archive
Retrieved from "https://en.wikipedia.org/w/index.php?title=Burnside_problem&oldid=1086246635"
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Time and Space Complexity of Interpolation Search
Time Complexity Algorithms Search Algorithms
In this post, we discuss interpolation search algorithm, its best, average and worst case time complexity and compare it with its counterpart search algorithms. We derive the average case Time Complexity of O(loglogN) as well.
Basics of Interpolation Search
Difference between an log(n) and log(log(n)) time compexity
Prerequisite: Interpolation Search, Binary Search
Interpolation Search is a search algorithm used for searching for a key in a dataset with uniform distribution of its values. An improved binary search for sorted and equally distributed data.
Uniform distribution means that the probability of a randomly chosen key being in a particular range is equal to it being in any other range of the same length. Therefore we expect to find target element at approximately the slot determined by the probing formula which we shall discuss below.
Comparison with binary search
Binary search chooses the middle element of the search space discarding a half sized chunk of the list depending on the comparison made at each iteration.
Interpolation search goes through different locations according to the search key, meaning it only requires the order of the elements, search space is reduced to the part before or after the estimated position at each iteration.
At each search step, the algorithm will calculate the remaining search space where the target element might be based on the low and high values of the search space and target value.
A comparison is made between the target and the value found at this position, if it is not equal then the search space is reduced to the part before or after the estimated position
The probing formula used is described below;
position(mid) = low + ((target – arr[low]) * (high – low) / (arr[high] – arr[low]))
This formula returns a higher value pos(mid) when target is closer to arr[high] and smaller value of pos(mid) when target is closer to arr[low].
Given a dataset of 15 elements [10, 12, 13, 16, 18, 19, 20, 21, 22, 23, 24, 33, 35, 42, 47] and a target of 18. The algorithm will find the target in two iterations.
mid = 0 + 8 *
\frac{13}{37}
arr[3] = 16 < 18, therefore mid + 1 = 4
After this first step the search space is reduced, we know the target lies between index 3+1 to last index.
\frac{9}{29}
arr[4] = 18 == target, terminate and return mid.
Calculate the value of pos(mid) using the probing formula and start search from there.
If the pos value is equal to target, return index of value and terminate.
If it does not match, probe position to find new mid using probing formula.
If value is greater than arr[pos] search the higher sub-array, right sub-array.
If value is greater than arr[pos] search the lower sub-array, left sub-array.
Repeat until the target is found, terminate when the sub-array reduces to zero.
class InterPolationSearch{
int probePosition(vector<int> arr, int target, int low, int high){
return (low + ((target - arr[low]) * (high - low) / (arr[high] - arr[low])));
int search(vector<int> arr, int target){
//search space
while(arr[high] != arr[low] && target >= arr[low] && target <= arr[high]){
mid = probePosition(arr, target, low, high);
if(target == arr[mid])
else if(target < arr[mid])
InterPolationSearch ips;
vector<int> arr = {10, 12, 13, 16, 18, 19, 20, 21,
cout << ips.search(arr, target) << endl;
//index of 18 in list
The time complexity of this algorithm is directly related to the number of times we execute the search loop because each time we execute the body of the while loop we trigger a probe(comparison) on an element in the list.
In the best case we assume that we find the target in just 1 probe making a constant time complexity O(1).
Lemma: Assuming keys are drawn independently from a uniform distribution then the expected number of probes C is bound by a constant.
Proof: We assume that the current search space is from o - n-1 and keys
{x}_{i}
are independently drawn from a uniform distribution.
The probability of a key to be less than or equal to y is
p=\frac{y-arr\left[0\right]}{arr\left[n-1\right]-arr\left[0\right]}
Generally the probability of exactly i keys being less than or equal to y is
\left(\left(\begin{array}{c}n\\ i\end{array}\right)\right){p}^{i}\left(1-p{\right)}^{n-i}
And because the distribution is binomial we expect
\mu =np
{\sigma }^{2}=np\left(1-p\right)
To determine C we do the following;
C=\sum _{i=1}^{\infty }i·pr\left[exactly i probes are used\right]=\sum _{i=1}^{\infty }·pr\left[at least i probes are used\right]
Observation let f(x)=x(1-x) therefore
f\left(x\right)\le \frac{1}{4}
x\in \mathbb{R}
Proof: f(x) is quadratic with the maximum at x = 1/2 and value f(1/2)=1/4, that is:
\frac{d}{dp}p\left(1-p\right)=\frac{d}{dp}p-{p}^{2}=1-2p=0↔p=\frac{1}{2} and \frac{{d}^{2}}{d{p}^{2}}p\left(1-p\right)=-2<0
By the above observation we have
{\sigma }^{2}=p\left(1-p\right)n\le \frac{1}{4}n
c\le 2+\sum _{i=3}^{\infty }\frac{{\sigma }^{2}}{\left(\left(i-2\right)ceil\left(\sqrt{n}\right){\right)}^{2}}
2+\frac{1}{4}\frac{{\pi }^{2}}{6}\le 2.42
With that we can start to prove the log(log(n)) time complexity for the average case.
Let T(n) be the average number of probes needed to find a key in an array of size n.
Let C be the expected number of probes needed to reduce the search space of size x to
\sqrt{x}
According to the lemma C will be bound by a constant
C\text{'}
Therefore we have the following equation,
T\left(n\right)\le C\text{'}+T\left(\sqrt{n}\right)
T\left(\sqrt{n}\right)
n={{z}^{2}}^{k}
for some k,
z\in \mathbb{N}
and T(z) is small.
We have the following equation;
T\left(n\right)=T\left({{z}^{2}}^{k}\right)\le C\text{'}+T\left({{z}^{2}}^{k-1}\right)\le C\text{'}+C\text{'}+T\left({{z}^{2}}^{k-2}\right)\le C\text{'}+C\text{'}+...+C\text{'}+T\left(z\right)=kC\text{'}+T\left(z\right)
n={{z}^{2}}^{k}
z\ge 2
if n > 1,
{\mathrm{log}}_{2}\left(z\right)\ge 1
k={\mathrm{log}}_{z}\left({\mathrm{log}}_{2}\left(n\right)\right)=\frac{{\mathrm{log}}_{2}\left({\mathrm{log}}_{2}\left(n\right)\right)}{{\mathrm{log}}_{2}\left(z\right)}\le {\mathrm{log}}_{2}\left({\mathrm{log}}_{2}\left(n\right)\right)
T\left(n\right)\le C\text{'}×log\left(log\left(n\right)\right)+T\left(z\right)
And after ingoring T(z) and using the constant from the lemma described we have
T(n) <= 2.42 * log(log(n)) .
Therefore for the average case we have O(log(log(n)) time complexity.
log(n) cuts input size by some constant factor say 2, after every iteration therefore the algorithm will terminate after log(n) iterations when the problem size hah been shrinked down to 1 or 0.
log(log(n)) cuts the input by a square root at each step.
Given an input size of 256.
log(n) will have 8 iterations that is
{2}^{8}
. The problem will be halved at each step.
log(log(n)) will have 3 iterations because at each step the problem is square rooted, 256, 16, 4, 2
In summary, the number of iterations by interpolation search algorithm will be less than half the number of iterations of the binary search if the list has a uniform distribution of elements and is sorted.
Note: If n is 1 billion, log(log(n)) is about 5, log(n) is about 30.
In the worst case, it makes n comparisons. This happens when the numerical values of the targets increase exponentially.
Given the array [1, 2, 3, 4, 5, 6, 7, 8, 9, 100] and target is 10, mid would be repeatedly set to low and target would be compared with every other element in the list. The algorithm degrades to a linear search time complexity of O(n) .
We can improve this complexity to O(log(n)) time if we run interpolation search parallelly with binary search, (binary interpolation search), this is discussed in the paper in the link at the end of this post.
Space complexity is constant O(1) as we only need to store indices for the search in the list.
Can you think of applications of this algorithm? Hint: An ordered dataset with uniform distribution.
In this post, we solve an algebraic geometrical problem using programming whereby we find the number of integral points between two given two points.
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Galois_theory Knowpia
Lattice diagram of Q adjoin the positive square roots of 2 and 3, its subfields, and Galois groups.
Application to classical problemsEdit
Which regular polygons are constructible?[1]
Why is it not possible to trisect every angle using a compass and a straightedge?[1]
Why is doubling the cube not possible with the same method?
Galois' writingsEdit
Permutation group approachEdit
Quadratic equationEdit
{\displaystyle x^{2}-4x+1=0.}
By using the quadratic formula, we find that the two roots are
{\displaystyle {\begin{aligned}A&=2+{\sqrt {3}},\\B&=2-{\sqrt {3}}.\end{aligned}}}
Examples of algebraic equations satisfied by A and B include
{\displaystyle A+B=4,}
{\displaystyle AB=1.}
If the polynomial has rational roots, for example x2 − 4x + 4 = (x − 2)2, or x2 − 3x + 2 = (x − 2)(x − 1), then the Galois group is trivial; that is, it contains only the identity permutation. In this example, if A = 2 and B = 1 then A − B = 1 is no longer true when A and B are swapped.
If it has two irrational roots, for example x2 − 2, then the Galois group contains two permutations, just as in the above example.
Quartic equationEdit
{\displaystyle x^{4}-10x^{2}+1,}
{\displaystyle \left(x^{2}-5\right)^{2}-24.}
{\displaystyle {\begin{aligned}A&={\sqrt {2}}+{\sqrt {3}},\\B&={\sqrt {2}}-{\sqrt {3}},\\C&=-{\sqrt {2}}+{\sqrt {3}},\\D&=-{\sqrt {2}}-{\sqrt {3}}.\end{aligned}}}
Among these equations, we have:
{\displaystyle {\begin{aligned}AB&=-1\\AC&=1\\A+D&=0\end{aligned}}}
{\displaystyle {\begin{aligned}\varphi (B)&={\frac {-1}{\varphi (A)}},\\\varphi (C)&={\frac {1}{\varphi (A)}},\\\varphi (D)&=-\varphi (A).\end{aligned}}}
(A, B, C, D) → (D, C, B, A)
Modern approach by field theoryEdit
It permits a far simpler statement of the fundamental theorem of Galois theory.
The use of base fields other than Q is crucial in many areas of mathematics. For example, in algebraic number theory, one often does Galois theory using number fields, finite fields or local fields as the base field.
It allows one to more easily study infinite extensions. Again this is important in algebraic number theory, where for example one often discusses the absolute Galois group of Q, defined to be the Galois group of K/Q where K is an algebraic closure of Q.
It allows for consideration of inseparable extensions. This issue does not arise in the classical framework, since it was always implicitly assumed that arithmetic took place in characteristic zero, but nonzero characteristic arises frequently in number theory and in algebraic geometry.
It removes the rather artificial reliance on chasing roots of polynomials. That is, different polynomials may yield the same extension fields, and the modern approach recognizes the connection between these polynomials.
Solvable groups and solution by radicalsEdit
A non-solvable quintic exampleEdit
For the polynomial f(x) = x5 − x − 1, the lone real root x = 1.1673... is algebraic, but not expressible in terms of radicals. The other four roots are complex numbers.
Inverse Galois problemEdit
As long as one does not also specify the ground field, the problem is not very difficult, and all finite groups do occur as Galois groups. For showing this, one may proceed as follows. Choose a field K and a finite group G. Cayley's theorem says that G is (up to isomorphism) a subgroup of the symmetric group S on the elements of G. Choose indeterminates {xα}, one for each element α of G, and adjoin them to K to get the field F = K({xα}). Contained within F is the field L of symmetric rational functions in the {xα}. The Galois group of F/L is S, by a basic result of Emil Artin. G acts on F by restriction of action of S. If the fixed field of this action is M, then, by the fundamental theorem of Galois theory, the Galois group of F/M is G.
Inseparable extensionsEdit
In the form mentioned above, including in particular the fundamental theorem of Galois theory, the theory only considers Galois extensions, which are in particular separable. General field extensions can be split into a separable, followed by a purely inseparable field extension. For a purely inseparable extension F / K, there is a Galois theory where the Galois group is replaced by the vector space of derivations,
{\displaystyle Der_{K}(F,F)}
, i.e., K-linear endomorphisms of F satisfying the Leibniz rule. In this correspondence, an intermediate field E is assigned
{\displaystyle Der_{E}(F,F)\subset Der_{K}(F,F)}
. Conversely, a subspace
{\displaystyle V\subset Der_{K}(F,F)}
satisfying appropriate further conditions is mapped to
{\displaystyle \{x\in F,f(x)=0\ \forall f\in V\}}
. Under the assumption
{\displaystyle F^{p}\subset K}
, Jacobson (1944) showed that this establishes a one-to-one correspondence. The condition imposed by Jacobson has been removed by Brantner & Waldron (2020), by giving a correspondence using notions of derived algebraic geometry.
Galois group for more examples
Differential Galois theory for a Galois theory of differential equations
Grothendieck's Galois theory for a vast generalization of Galois theory
^ a b Stewart, Ian (1989). Galois Theory. Chapman and Hall. ISBN 0-412-34550-1.
^ Funkhouser 1930
^ Cardano 1545
^ Tignol, Jean-Pierre (2001). Galois' Theory of Algebraic Equations. World Scientific. pp. 232–3, 302. ISBN 978-981-02-4541-2.
^ Stewart, 3rd ed., p. xxiii
^ Clark, Allan (1984) [1971]. Elements of Abstract Algebra. Courier. p. 131. ISBN 978-0-486-14035-3.
^ Wussing, Hans (2007). The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory. Courier. p. 118. ISBN 978-0-486-45868-7.
^ Scharlau, Winfried; Dedekind, Ilse; Dedekind, Richard (1981). Richard Dedekind 1831–1981; eine Würdigung zu seinem 150. Geburtstag (PDF). Braunschweig: Vieweg. ISBN 9783528084981.
^ Galois, Évariste; Neumann, Peter M. (2011). The Mathematical Writings of Évariste Galois. European Mathematical Society. p. 10. ISBN 978-3-03719-104-0.
^ van der Waerden, Modern Algebra (1949 English edn.), Vol. 1, Section 61, p.191
^ Prasolov, V.V. (2004). "5 Galois Theory Theorem 5.4.5(a)". Polynomials. Algorithms and Computation in Mathematics. Vol. 11. Springer. pp. 181–218. doi:10.1007/978-3-642-03980-5_5. ISBN 978-3-642-03979-9.
^ Lang, Serge (1994). Algebraic Number Theory. Graduate Texts in Mathematics. Vol. 110. Springer. p. 121. ISBN 9780387942254.
Artin, Emil (1998) [1944]. Galois Theory. Dover. ISBN 0-486-62342-4.
Bewersdorff, Jörg (2006). Galois Theory for Beginners: A Historical Perspective. The Student Mathematical Library. Vol. 35. American Mathematical Society. doi:10.1090/stml/035. ISBN 0-8218-3817-2.
Brantner, Lukas; Waldron, Joe (2020), Purely Inseparable Galois theory I: The Fundamental Theorem, arXiv:2010.15707
Cardano, Gerolamo (1545). Artis Magnæ (PDF) (in Latin).
Edwards, Harold M. (1984). Galois Theory. Springer-Verlag. ISBN 0-387-90980-X. (Galois' original paper, with extensive background and commentary.)
Funkhouser, H. Gray (1930). "A short account of the history of symmetric functions of roots of equations". American Mathematical Monthly. 37 (7): 357–365. doi:10.2307/2299273. JSTOR 2299273.
"Galois theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Jacobson, Nathan (1944), "Galois theory of purely inseparable fields of exponent one", Amer. J. Math., 66: 645–648, doi:10.2307/2371772, JSTOR 2371772
Jacobson, Nathan (1985). Basic Algebra I (2nd ed.). W. H. Freeman. ISBN 0-7167-1480-9. (Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)
Janelidze, G.; Borceux, Francis (2001). Galois Theories. Cambridge University Press. ISBN 978-0-521-80309-0. (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.)
Lang, Serge (1994). Algebraic Number Theory. Berlin, New York: Springer-Verlag. ISBN 978-0-387-94225-4.
Postnikov, M. M. (2004). Foundations of Galois Theory. Dover Publications. ISBN 0-486-43518-0.
Rotman, Joseph (1998). Galois Theory (2nd ed.). Springer. ISBN 0-387-98541-7.
Völklein, Helmut (1996). Groups as Galois groups: an introduction. Cambridge University Press. ISBN 978-0-521-56280-5.
van der Waerden, Bartel Leendert (1931). Moderne Algebra (in German). Berlin: Springer. . English translation (of 2nd revised edition): Modern Algebra. New York: Frederick Ungar. 1949. (Later republished in English by Springer under the title "Algebra".)
The dictionary definition of Galois theory at Wiktionary
Media related to Galois theory at Wikimedia Commons
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Numerical Simulation of Convective Heat Transfer Modes in a Rectangular Area With a Heat Source and Conducting Walls | J. Heat Transfer | ASME Digital Collection
Numerical Simulation of Convective Heat Transfer Modes in a Rectangular Area With a Heat Source and Conducting Walls
G. V. Kuznetsov,
G. V. Kuznetsov
Faculty of Thermal Power Engineering,
, 30 Lenin Avenue, 634050 Tomsk, Russia
M. A. Sheremet
Kuznetsov, G. V., and Sheremet, M. A. (May 20, 2010). "Numerical Simulation of Convective Heat Transfer Modes in a Rectangular Area With a Heat Source and Conducting Walls." ASME. J. Heat Transfer. August 2010; 132(8): 081401. https://doi.org/10.1115/1.4001303
Laminar conjugate heat transfer in a rectangular area having finite thickness heat-conducting walls at local heating has been analyzed numerically. The heat source located on the left wall is kept at constant temperature during the whole process. Conjugate heat transfer is complicated by the forced flow. The governing unsteady, two-dimensional flow and energy equations for the gas cavity and unsteady heat conduction equation for solid walls, written in dimensionless form, have been solved using implicit finite-difference method. The solution has been obtained in terms of the stream function and the vorticity vector. The effects of the Grashof number Gr, the Reynolds number Re, and the dimensionless time on the flow structure and heat transfer characteristics have been investigated in detail. Results have been obtained for the following parameters:
103≤Gr≤107
100≤Re≤1000
Pr=0.7
. Typical distributions of thermohydrodynamic parameters describing features of investigated process have been received. Interference of convective flows (forced, natural, and mixed modes) in the presence of conducting solid walls has been analyzed. The increase in Gr is determined to lead to both the intensification of the convective flow caused by the presence of the heat source and the blocking of the forced flow nearby the upper wall. The nonmonotomic variations in the average Nusselt number with Gr for solid-fluid interfaces have been obtained. The increase in Re is shown to lead to cooling of the gas cavity caused by the forced flow. Evolution of analyzed process at time variation has been displayed. The diagram of the heat convection modes depending on the Grashof and Reynolds numbers has been obtained. The analysis of heat convection modes in a typical subsystem of the electronic equipment is oriented not only toward applied development in microelectronics, but also it can be considered as test database at creation of numerical codes of convective heat transfer simulation in complicated energy systems. Comparison of the obtained results can be made by means of both streamlines and temperature fields at different values of the Grashof number and Reynolds number, and the average Nusselt numbers at solid-fluid interfaces.
confined flow, cooling, finite difference methods, flow simulation, forced convection, heat conduction, initial value problems, laminar flow, natural convection, vortices, convection modes, conjugate heat transfer, laminar flow, heat source
Cavities, Convection, Flow (Dynamics), Fluids, Heat, Heat transfer, Temperature, Reynolds number, Heat conduction
Heat Modes of Electronic Devices
Conjugate Heat Transfer in Fractal-Shaped Microchannel Network Heat Sink for Integrated Microelectronic Cooling Application
Conjugate Heat Transfer in Inclined Open Shallow Cavities
G. S. V. L.
Laminar Conjugate Mixed Convection in a Vertical Channel With Heat Generating Components
Muftuoglu
Conjugate Heat Transfer in Open Cavities With a Discrete Heater at Its Optimized Position
Conjugate Mixed Convection With Surface Radiation From a Horizontal Channel With Protruding Heat Sources
Numerical Comparison of Conjugate and Non-Conjugate Natural Convection for Internally Heated Semi-Circular Pools
Conjugate Natural Convection in a Square Enclosure Containing Volumetric Sources
Natural Convection Heat and Mass Transfer
Two-Dimensional Problem of Natural Convection in a Rectangular Domain With Local Heating and Heat-Conducting Boundaries of Finite Thickness
Theory of Thermal Conductivity
Paskonov
Numerical Simulation of Heat and Mass Transfer Processes
Natural Convection of Air in a Square Cavity: A Bench Numerical Solution
Simulation of High Rayleigh Number Natural Convection in a Square Cavity Using the Lattice Boltzmann Method
Conjugate Natural Convection in a Square Enclosure Effect of Conduction on One of the Vertical Walls
Convective Heat Transfer in a Rectangular Channel Filled With Sintered Bronze Beads and Periodically Spaced Heated Blocks
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EUDML | Mappings of finite distortion: Global homeomorphism theorem. EuDML | Mappings of finite distortion: Global homeomorphism theorem.
Mappings of finite distortion: Global homeomorphism theorem.
Holopainen, Ilkka; Pankka, Pekka
Holopainen, Ilkka, and Pankka, Pekka. "Mappings of finite distortion: Global homeomorphism theorem.." Annales Academiae Scientiarum Fennicae. Mathematica 29.1 (2004): 59-80. <http://eudml.org/doc/124282>.
@article{Holopainen2004,
author = {Holopainen, Ilkka, Pankka, Pekka},
keywords = {Zorich theorem; Global homeomorphisms between Riemannian manifolds},
title = {Mappings of finite distortion: Global homeomorphism theorem.},
AU - Holopainen, Ilkka
AU - Pankka, Pekka
TI - Mappings of finite distortion: Global homeomorphism theorem.
KW - Zorich theorem; Global homeomorphisms between Riemannian manifolds
Zorich theorem, Global homeomorphisms between Riemannian manifolds
{𝐑}^{n}
Articles by Holopainen
Articles by Pankka
|
EnWik > Oxygen
Standard atomic weight Ar, std(O)
[15.99903, 15.99977] conventional: 15.999[1]
Phaseat STP
1.4 Lavoisier's contribution
2.4 Isotopes and stellar origin
3 Biological production and role of O2
3.3 Build-up in the atmosphere
3.4 Extraterrestrial free oxygen
6.2 Life support and recreational use
Naturally occurring oxygen is composed of three stable isotopes, 16, 17, and 18O, with 16O being the most abundant (99.762% natural abundance).[55]
Most 16O is synthesized at the end of the helium fusion process in massive stars but some is made in the neon burning process.[56] 17O is primarily made by the burning of hydrogen into helium during the CNO cycle, making it a common isotope in the hydrogen burning zones of stars.[56] Most 18O is produced when 14 (made abundant from CNO burning) captures a 4He nucleus, making 18O common in the helium-rich zones of evolved, massive stars.[56]
(c) IIVQ, CC-BY-SA-3.0
{\displaystyle {\ce {Fe}}_{1-x}{\ce {O}}}
2 but at slightly more than atmospheric pressure, instead of the1⁄3 normal pressure that would be used in a mission.[k][143]
^ Atkins, P.; Jones, L.; Laverman, L. (2016).Chemical Principles, 7th edition. Freeman.ISBN 978-1-4641-8395-9
^ a b Jack Barrett, 2002, "Atomic Structure and Periodicity", (Basic concepts in chemistry, Vol. 9 of Tutorial chemistry texts), Cambridge, UK: Royal Society of Chemistry, p. 153,ISBN 0854046577. See Google Books. Archived May 30, 2020, at the Wayback Machine accessed January 31, 2015.
Stilles Mineralwasser.jpg
Mineral water being poured from a bottle into a glass. Original description “still” in German indicates this particular water is without gas/carbonation or has less than 1 gram CO2 per Liter.
Compressed gas cylinders.mapp and oxygen.triddle.jpg
Oxygen and MAPP gas compresed gas cylinders with regulators. The high pressure side of the oxygen regulator reads 1,000 PSI (69 bar). The tremendous pressure inside the oxygen cylinder (over 2,200 PSI (152 bar) in a full tank) requires extrusion as a manufacturing process and these tanks may never be welded. The lower pressure MAPP cylinder was welded together and the seam is visible where the yellow and white paint meet.
Oxygen molecule orbitals diagram.JPG
Author/Creator: Anthony.Sebastian, Licence: CC BY-SA 3.0
Molecular orbital energy diagram for O2
WOA09 sea-surf O2 AYool.png
Author/Creator: Plumbago, Licence: CC BY-SA 3.0
Annual mean sea surface dissolved oxygen from the World Ocean Atlas 2009. Dissolved oxygen here is in mol O2 m-3. It is plotted here using a Mollweide projection (using MATLAB and the M_Map package).
Clabecq JPG01.jpg
Clabecq (Belgium), continuous casting of the old blast furnace.
Oxygen spectrum visible.png
Oxygen spectrum; 400 nm - 700 nm
Home oxygen concentrator.jpg
Author/Creator: The original uploader was GiollaUidir at English Wikipedia., Licence: CC BY-SA 2.0 uk
A picture I took on 2nd of July, 2007 showing a home en:oxygen concentrator. As stated in the license attribution for the picture must be given to me, Patrick McAleer for use outside of Wikipedia.
STS057-89-067 - Wisoff on the Arm (Retouched).jpg
Author/Creator: Askeuhd, Licence: CC BY-SA 4.0
STS057-89-67 - Against the blackness of space, Mission Specialist Peter J.K. Wisoff, wearing an extravehicular mobility unit, stands on a Portable Foot Restraint (PFR), Manipulator Foot Restraint (MFR) attached to the End Effector of the Remote Manipulator System (RMS), colloquially known as the "robot arm". Wisoff is being maneuvered above the payload bay of Endeavour as part of Detailed Test Objective (DTO) extravehicular activity procedures. DTO results will assist in refining several procedures being developed to service the Hubble Space Telescope on mission STS-61 in December 1993. The Earth's surface and the Endeavour payload bay are reflected in Wisoff's helmet visor.
Apollo 1 fire.jpg
Officially designated Apollo/Saturn 204, but more commonly known as Apollo 1, this close-up view of the interior of the Command Module shows the effects of the intense heat of the flash fire which killed the prime crew during a routine training exercise. While strapped into their seats inside the Command Module atop the giant Saturn rocket, a wire bundle that was worn due to misplacement and a sharp-edged access door, sparked and ignited flammable material. This quickly grew into a large fire in the pure oxygen environment. The speed and intensity of the fire quickly exhausted the oxygen supply inside the crew cabin. Unable to deploy the hatch due to its cumbersome design and lack of breathable oxygen, the crew lost consciousness and perished. They were: astronauts Virgil I. "Gus" Grissom, (the second American to fly into space) Edward H. White II, (the first American to "walk" in space) and Roger B. Chaffee, (a "rookie" on his first space mission).
Goddard and Rocket.jpg
Dr. Robert H. Goddard and a liquid oxygen-gasoline rocket at Auburn, Massachusetts.
Oxygen, Licence: CC-BY-SA-3.0
This is a spoken word version of the Wikipedia article: Oxygen (Intro) Oxygen
Liquid oxygen in a magnet 2.jpg
Author/Creator: Bob Burk, work supported by the National Science Foundation under grant numbers: 1246120, 1525057, and 1413739, Licence: CC BY 3.0
A chemical demonstration of the paramagnetism of molecular oxygen, as shown by the attraction of liquid oxygen to magnets.
Evolved star fusion shells.svg
Author/Creator: User:Rursus, Licence: CC BY 2.5
This diagram shows a simplified (not to scale) cross-section of a massive, evolved star (with a mass greater than eight times the Sun.) Where the pressure and temperature permit, concentric shells of Hydrogen (H), Helium (He), Carbon (C), Neon/Magnesium (Ne), Oxygen (O) and Silicon (Si) plasma are burning inside the star. The resulting fusion by-products rain down upon the next lower layer, building up the shell below. As a result of Silicon fusion, an inert core of Iron (Fe) plasma is steadily building up at the center. Once this core reaches the Chandrasekhar mass, the iron can no longer sustain its own mass and it undergoes a collapse. This can result in a supernova explosion.
Oxygenation-atm.svg
Author/Creator: Heinrich D. Holland, Licence: CC-BY-SA-3.0
Estimated evolution of atmospheric
{\displaystyle \ P_{O_{2}}}
. The upper red and lower green lines represent the range of the estimates. The stages are: stage 1 (3.85–2.45Gyr ago (Ga)), stage 2 (2.45–1.85Ga), stage 3 (1.85–0.85Ga), Stage 4 (0.85–0.54Ga )and stage 5 (0.54Ga–present)
A setup for preparation of Oxygen.jpg
This is an experiment setup with a test tube, a burner, a glass pipe, a glass jar, and a glass container to generate and collect oxygen by heating the potassium chlorate mixed with a small portion of manganese oxide.
PriestleyFuseli.jpg
" Engraving by Charles Turner after a painting by Henry Fuseli. P/P933.28a M (not scanned) is a photomechanical reproduction of this image, originally published in the Journal of Chemical Education, Vol. 16, No. 12, December, 1939, and includes this explanatory paragraph: This, the largest oil portrait of Dr. Priestley, (three feet four inches by four feet two inches) was painted for Joseph Johnson, bookseller and publisher of Priestley's works, in the year 1783, while Dr. Priestley was Johnson's guest. An engraving was made of the Fuseli portrait by C. Turner, and published -- one hundred copies only -- in October, 1836, by Richard Taylor, Red Lion Court, Fleet Street, London. A copy of this rare engraving was recently presented to the Edgar Fahs Smith Memorial Collection by Mrs. Ida M. Priestley of England."
Rust screw.jpg
Author/Creator: User:Paulnasca, Licence: CC BY 2.0
A rusty screw
Phanerozoic Climate Change.png
Hofmann voltameter fr.svg
Werking Toestel van Hoffman Eigengemaakt werk, onder de GFDL geplaatst.
U.S. Air Force Staff Sgt. Dustin Volpi tests liquid oxygen for purity at Eielson Air Force Base, Alaska. Quality liquid oxygen is used to provide oxygen to pilots and aircrews at altitudes above 10,000 feet.
Spectrum = gas discharge tube filled with oxygen O2. Used with 1,8kV, 18mA, 35kHz. ≈8" length.
Acetone-3D-vdW.png
Space-filling model of the acetone molecule, C3H6O.
Bond lengths and angles from CRC Handbook, 88th edition.
Oxygen molecule.png
Author/Creator: Ulflund, Licence: CC0
Space-filling model of the oxygen molecule (O2) created with Jmol (http://chemapps.stolaf.edu/jmol/jmol.php?model=O2).
Symptoms of oxygen toxicity.png
Main symptoms of oxygen toxicity. (See Wikipedia:Oxygen#Toxicity). To discuss image, please see Template talk:Human body diagrams
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MUSIC Super-Resolution DOA Estimation - MATLAB & Simulink - MathWorks Australia
MUltiple SIgnal Classification (MUSIC) is a high-resolution direction-finding algorithm based on the eigenvalue decomposition of the sensor covariance matrix observed at an array. MUSIC belongs to the family of subspace-based direction-finding algorithms.
The signal model relates the received sensor data to the signals emitted by the source. Assume that there are D uncorrelated or partially correlated signal sources, sd(t). The sensor data, xm(t), consists of the signals, as received at the array, together with noise, nm(t). A sensor data snapshot is the sensor data vector received at the M elements of an array at a single time t.
\begin{array}{l}x\left(t\right)=As\left(t\right)+n\left(t\right)\\ s\left(t\right)=\left[{s}_{1}\left(t\right),{s}_{2}\left(t\right),\dots ,{s}_{D}\left(t\right)\right){\right]}^{\prime }\\ A=\left[a\left({\theta }_{1}\right)|a\left({\theta }_{2}\right)|\dots |a\left({\theta }_{D}\right)\right]\end{array}
x(t) is an M-by-1 vector of received snapshot of sensor data which consist of signals and additive noise.
A is an M-by-D matrix containing the arrival vectors. An arrival vector consists of the relative phase shifts at the array elements of the plane wave from one source. Each column of A represents the arrival vector from one of the sources and depends on the direction of arrival, θd. θd is the direction of arrival angle for the dth source and can represents either the broadside angle for linear arrays or the azimuth and elevation angle for planar or 3D arrays.
s(t) is a D-by-1 vector of source signal values from D sources.
n(t) is an M-by-1 vector of sensor noise values.
An important quantity in any subspace method is the sensor covariance matrix,Rx, derived from the received signal data. When the signals are uncorrelated with the noise, the sensor covariance matrix has two components, the signal covariance matrix and the noise covariance matrix.
{R}_{x}=E\left\{x{x}^{H}\right\}=A{R}_{s}{A}^{H}+{\sigma }_{n}^{2}I
where Rs is the source covariance matrix. The diagonal elements of the source covariance matrix represent source power and the off-diagonal elements represent source correlations.
{R}_{s}=E\left\{s{s}^{H}\right\}
For uncorrelated sources or even partially correlated sources, Rs is a positive-definite Hermitian matrix and has full rank, D, equal to the number of sources.
The signal covariance matrix, ARsAH, is an M-by-M matrix, also with rank D < M.
An assumption of the MUSIC algorithm is that the noise powers are equal at all sensors and uncorrelated between sensors. With this assumption, the noise covariance matrix becomes an M-by-M diagonal matrix with equal values along the diagonal.
Because the true sensor covariance matrix is not known, MUSIC estimates the sensor covariance matrix, Rx, from the sample sensor covariance matrix. The sample sensor covariance matrix is an average of multiple snapshots of the sensor data
{R}_{x}=\frac{1}{T}\sum _{k=1}^{T}x\left(t\right)x{\left(t\right)}^{H},
where T is the number of snapshots.
Because ARsAH has rank D, it has D positive real eigenvalues and M – D zero eigenvalues. The eigenvectors corresponding to the positive eigenvalues span the signal subspace, Us= [v1,...,vD]. The eigenvectors corresponding to the zero eigenvalues are orthogonal to the signal space and span the null subspace, Un= [uD+1,...,uN]. The arrival vectors also belong to the signal subspace, but they are eigenvectors. Eigenvectors of the null subspace are orthogonal to the eigenvectors of the signal subspace. Null-subspace eigenvectors, ui, satisfy this equation:
A{R}_{s}{A}^{H}{u}_{i}=0⇒{u}^{H}A{R}_{s}{A}^{H}{u}_{i}=0⇒{\left({A}^{H}{u}_{i}\right)}^{H}{R}_{s}\left({A}^{H}{u}_{i}\right)=0⇒{A}^{H}{u}_{i}=0
Therefore the arrival vectors are orthogonal to the null subspace.
When noise is added, the eigenvectors of the sensor covariance matrix with noise present are the same as the noise-free sensor covariance matrix. The eigenvalues increase by the noise power. Let vi be one of the original noise-free signal space eigenvectors. Then
{R}_{x}{v}_{i}=A{R}_{s}{A}^{H}{v}_{i}+{\sigma }_{0}^{2}I{v}_{i}=\left({\lambda }_{i}+{\sigma }_{0}^{2}\right){v}_{i}
shows that the signal space eigenvalues increase by σ02.
The null subspace eigenvectors are also eigenvectors of Rx. Let ui be one of the null eigenvectors. Then
{R}_{x}{u}_{i}=A{R}_{s}{A}^{H}{u}_{i}+{\sigma }_{0}^{2}I{u}_{i}={\sigma }_{0}^{2}{u}_{i}
with eigenvalues of σ02 instead of zero. The null subspace becomes the noise subspace.
MUSIC works by searching for all arrival vectors that are orthogonal to the noise subspace. To do the search, MUSIC constructs an arrival-angle-dependent power expression, called the MUSIC pseudospectrum:
{P}_{MUSIC}\left(\stackrel{\to }{\varphi }\right)=\frac{1}{{a}^{H}\left(\stackrel{\to }{\varphi }\right){U}_{n}{U}_{n}^{H}a\left(\stackrel{\to }{\varphi }\right)}
When an arrival vector is orthogonal to the noise subspace, the peaks of the pseudospectrum are infinite. In practice, because there is noise, and because the true covariance matrix is estimated by the sampled covariance matrix, the arrival vectors are never exactly orthogonal to the noise subspace. Then, the angles at which PMUSIC has finite peaks are the desired directions of arrival. Because the pseudospectrum can have more peaks than there are sources, the algorithm requires that you specify the number of sources, D, as a parameter. Then the algorithm picks the D largest peaks. For a uniform linear array (ULA), the search space is a one-dimensional grid of broadside angles. For planar and 3D arrays, the search space is a two-dimensional grid of azimuth and elevation angles.
For a ULA, the denominator in the pseudospectrum is a polynomial in
{e}^{ikd\mathrm{cos}\phi }
, but can also be considered a polynomial in the complex plane. In this cases, you can use root-finding methods to solve for the roots, zi. These roots do not necessarily lie on the unit circle. However, Root-MUSIC assumes that the D roots closest to the unit circle correspond to the true source directions. Then you can compute the source directions from the phase of the complex roots.
When some of the D source signals are correlated, Rs is rank deficient, meaning that it has fewer than D nonzero eigenvalues. Therefore, the number of zero eigenvalues of ARsAH exceeds the number, M – D, of zero eigenvalues for the uncorrelated source case. MUSIC performance degrades when signals are correlated, as occurs in a multipath propagation environment. A way to compensate for correlation is to use spatial smoothing.
Spatial smoothing takes advantage of the translation properties of a uniform array. Consider two correlated signals arriving at an L-element ULA. The source covariance matrix, Rs is a singular 2-by-2 matrix. The arrival vector matrix is an L-by-2 matrix
\begin{array}{l}{A}_{1}=\left[\begin{array}{c}1\\ {e}^{ikd\mathrm{cos}{\phi }_{1}}\\ ⋮\\ {e}^{i\left(L-1\right)kd\mathrm{cos}{\phi }_{1}}\end{array}\begin{array}{c}1\\ {e}^{ikd\mathrm{cos}{\phi }_{2}}\\ ⋮\\ {e}^{i\left(L-1\right)kd\mathrm{cos}{\phi }_{2}}\end{array}\right]=\left[a\left({\phi }_{1}\right)|a\left({\phi }_{2}\right)\right]\\ \end{array}
for signals arriving from the broadside angles φ1 and φ2. The quantity k is the signal wave number. a(φ) represents an arrival vector at the angle φ.
You can create a second array by translating the first array along its axis by one element distance, d. The arrival matrix for the second array is
{A}_{2}=\left[\begin{array}{c}{e}^{ikd\mathrm{cos}{\phi }_{1}}\\ {e}^{i2kd\mathrm{cos}{\phi }_{1}}\\ ⋮\\ {e}^{iLkd\mathrm{cos}{\phi }_{1}}\end{array}\begin{array}{c}{e}^{ikd\mathrm{cos}{\phi }_{2}}\\ {e}^{i2kd\mathrm{cos}{\phi }_{2}}\\ ⋮\\ {e}^{iLkd\mathrm{cos}{\phi }_{2}}\end{array}\right]=\left[{e}^{ikd\mathrm{cos}{\phi }_{1}}a\left({\phi }_{1}\right)|{e}^{ikd\mathrm{cos}{\phi }_{2}}a\left({\phi }_{2}\right)\right]
where the arrival vectors are equal to the original arrival vectors but multiplied by a direction-dependent phase shift. When you translate the original array J –1 more times, you get J copies of the array. If you form a single array from all these copies, then the length of the single array is M = L + (J – 1).
In practice, you start with an M-element array and form J overlapping subarrays. The number of elements in each subarray is L = M – J + 1. The following figure shows the relationship between the overall length of the array, M, the number of subarrays, J, and the length of each subarray, L.
For the pth subarray, the source signal arrival matrix is
\begin{array}{c}{A}_{p}=\left[{e}^{ik\left(p-1\right)d\mathrm{cos}{\phi }_{1}}a\left({\phi }_{1}\right)|{e}^{ik\left(p-1\right)d\mathrm{cos}{\phi }_{2}}a\left({\phi }_{2}\right)\right]\\ =\left[a\left({\phi }_{1}\right)|a\left({\phi }_{2}\right)\right]\left[\begin{array}{cc}{e}^{ik\left(p-1\right)d\mathrm{cos}{\phi }_{1}}& 0\\ 0& {e}^{ik\left(p-1\right)d\mathrm{cos}{\phi }_{2}}\end{array}\right]={A}_{1}{P}^{p-1}\\ P=\left[\begin{array}{cc}{e}^{ikd\mathrm{cos}{\phi }_{1}}& 0\\ 0& {e}^{ikd\mathrm{cos}{\phi }_{2}}\end{array}\right].\end{array}
The original arrival vector matrix is postmultiplied by a diagonal phase matrix.
The last step is averaging the signal covariance matrices over all J subarrays to form the averaged signal covariance matrix, Ravgs. The average signal covariance matrix depends on the smoothed source covariance matrix, Rsmooth.
\begin{array}{l}{R}_{s}^{avg}={A}_{1}\left(\frac{1}{J}\sum _{p=1}^{J}{P}^{p-1}{R}_{s}{\left({P}^{p-1}\right)}^{H}\right){A}_{1}^{H}={A}_{1}{R}^{smooth}{A}_{1}^{H}\\ {R}^{smooth}=\frac{1}{J}\sum _{p=1}^{J}{P}^{p-1}{R}_{s}{\left({P}^{p-1}\right)}^{H}.\end{array}
You can show that the diagonal elements of the smoothed source covariance matrix are the same as the diagonal elements of the original source covariance matrix.
{R}_{ii}^{smooth}=\frac{1}{J}\sum _{p=1}^{J}{\left({P}^{p-1}\right)}_{im}{\left({R}_{s}\right)}_{mn}{\left({P}^{p-1}\right)}_{ni}{}^{H}=\frac{1}{J}\sum _{p=1}^{J}{R}_{s}={\left({R}_{s}\right)}_{ii}
However, the off-diagonal elements are reduced. The reduction factor is the beam pattern of a J-element array.
{R}_{ij}^{smooth}=\frac{1}{J}\sum _{p=1}^{J}{e}^{ikd\left(p-1\right)\left(\mathrm{cos}{\phi }_{1}-\mathrm{cos}{\phi }_{2}\right)}{\left({R}_{s}\right)}_{ij}=\frac{1}{J}\frac{\mathrm{sin}\left(kdJ\left(\mathrm{cos}{\phi }_{1}-\mathrm{cos}{\phi }_{2}\right)\right)}{\mathrm{sin}\left(kd\left(\mathrm{cos}{\phi }_{1}-\mathrm{cos}{\phi }_{2}\right)\right)}{\left({R}_{s}\right)}_{ij}
In summary, you can reduce the degrading effect of source correlation by forming subarrays and using the smoothed covariance matrix as input to the MUSIC algorithm. Because of the beam pattern, larger angular separation of sources leads to reduced correlation.
Spatial smoothing for linear arrays is easily extended to 2D and 3D uniform arrays.
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Ordered pair - Citizendium
In mathematics, an ordered pair is a pair of elements in which order is significant: that is, the pair (x,y) is to be distinguished from (y,x). The ordered pairs (a,b) and (c,d) are equal if and only if a=c and b=d.
It would be possible to take the concept of ordered pair as an elementary concept in set theory, but it is more usual to define them in terms of sets. Kuratowksi proposed the definition
{\displaystyle (a,b)=\{\{a\},\{a,b\}\}.\,}
The set of all ordered pairs (x,y) with x in X and y in Y is the Cartesian product of X and Y. A complex number may be expressed as an ordered pair of real numbers, the real and imaginary parts respectively.
Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 9-10. ISBN 0-387-90441-7.
Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold, 22-25.
Ian Stewart; David Tall (1977). The Foundations of Mathematics. Oxford University Press, 62-65. ISBN 0-19-853165-6.
Retrieved from "https://citizendium.org/wiki/index.php?title=Ordered_pair&oldid=550221"
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Sentences - Maple Help
Home : Support : Online Help : Programming : Names and Strings : StringTools Package : English Text : Sentences
approximate segmentation of a string of English text into sentences
Sentences( s )
The Sentences(s) command attempts to split a string, presumed to be composed of English language text, into its constituent sentences. It does this by recognizing sentence boundaries. The beginning and the end of the input string are regarded as sentence boundaries in all cases. Internal sentence boundaries are recognized by the presence of a sentence terminator, which is one of the following:
A small number of built-in patterns are used to recognize some exceptions.
Note that you can also use the RegSplit command with the fixed string "\n\n" as the splitting pattern to segment English text into paragraphs.
8
\mathrm{with}\left(\mathrm{StringTools}\right):
\mathrm{Sentences}\left("This is a\nsentence. Can we have another? Yes, here\text{'}s one more."\right)
\textcolor[rgb]{0,0,1}{"This is a sentence."}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Can we have another?"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"Yes, here\text{'}s one more."}
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Option price and sensitivities by Heston model using finite differences - MATLAB optSensByHestonFD - MathWorks France
\mathrm{max}\left(St-K,0\right)
\mathrm{max}\left(K-St,0\right)
\begin{array}{l}d{S}_{t}=\left(r-q\right){S}_{t}dt+\sqrt{{v}_{t}}{S}_{t}d{W}_{t}\\ d{v}_{t}=\kappa \left(\theta -{v}_{t}\right)dt+{\sigma }_{v}\sqrt{{v}_{t}}d{W}_{t}^{v}\\ \text{E}\left[d{W}_{t}d{W}_{t}^{v}\right]=pdt\end{array}
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Epicyclic gearing - 3D CAD Models & 2D Drawings
Epicyclic gearing (30850 views - Mechanical Engineering)
An epicyclic gear train (also known as planetary gear) consists of two gears mounted so that the centre of one gear revolves around the centre of the other. A carrier connects the centres of the two gears and rotates to carry one gear, called the planet gear, around the other, called the sun gear. The planet and sun gears mesh so that their pitch circles roll without slip. A point on the pitch circle of the planet gear traces an epicycloid curve. In this simplified case, the sun gear is fixed and the planetary gear(s) roll around the sun gear. An epicyclic gear train can be assembled so the planet gear rolls on the inside of the pitch circle of a fixed, outer gear ring, or ring gear, sometimes called an annular gear. In this case, the curve traced by a point on the pitch circle of the planet is a hypocycloid. The combination of epicycle gear trains with a planet engaging both a sun gear and a ring gear is called a planetary gear train. In this case, the ring gear is usually fixed and the sun gear is driven. Epicyclic gears get their name from their earliest application, which was the modelling of the movements of the planets in the heavens. Believing the planets, as everything in the heavens, to be perfect, they could only travel in perfect circles, but their motions as viewed from Earth could not be reconciled with circular motion. At around 500 BC, the Greeks invented the idea of epicycles, of circles travelling on the circular orbits. With this theory Claudius Ptolemy in the Almagest in 148 AD was able to predict planetary orbital paths. The Antikythera Mechanism, circa 80 BC, had gearing which was able to approximate the moon's elliptical path through the heavens, and even to correct for the nine-year precession of that path. (The Greeks would have seen it not as elliptical, but rather as epicyclic motion.)
PARTcloud - gear
Simplified planetary gear animation.
An epicyclic gear train (also known as planetary gear) consists of two gears mounted so that the centre of one gear revolves around the centre of the other. A carrier connects the centres of the two gears and rotates to carry one gear, called the planet gear, around the other, called the sun gear. The planet and sun gears mesh so that their pitch circles roll without slip. A point on the pitch circle of the planet gear traces an epicycloid curve. In this simplified case, the sun gear is fixed and the planetary gear(s) roll around the sun gear.
Epicyclic gears get their name from their earliest application, which was the modelling of the movements of the planets in the heavens. Believing the planets, as everything in the heavens, to be perfect, they could only travel in perfect circles, but their motions as viewed from Earth could not be reconciled with circular motion. At around 500 BC, the Greeks invented the idea of epicycles, of circles travelling on the circular orbits. With this theory Claudius Ptolemy in the Almagest in 148 AD was able to predict planetary orbital paths. The Antikythera Mechanism, circa 80 BC, had gearing which was able to approximate the moon's elliptical path through the heavens, and even to correct for the nine-year precession of that path.[3] (The Greeks would have seen it not as elliptical, but rather as epicyclic motion.)
3 Gear ratio of standard epicyclic gearing
4 Torque ratios of standard epicyclic gearing
5 Fixed carrier train ratio
5.1 Spur gear differential
6 Gear ratio of reversed epicyclic gearing
7 Compound planetary gears
Epicyclic gearing or planetary gearing is a gear system consisting of one or more outer gears, or planet gears, revolving about a central, or sun gear. Typically, the planet gears are mounted on a movable arm or carrier, which itself may rotate relative to the sun gear. Epicyclic gearing systems also incorporate the use of an outer ring gear or annulus, which meshes with the planet gears. Planetary gears (or epicyclic gears) are typically classified as simple or compound planetary gears. Simple planetary gears have one sun, one ring, one carrier, and one planet set. Compound planetary gears involve one or more of the following three types of structures: meshed-planet (there are at least two more planets in mesh with each other in each planet train), stepped-planet (there exists a shaft connection between two planets in each planet train), and multi-stage structures (the system contains two or more planet sets). Compared to simple planetary gears, compound planetary gears have the advantages of larger reduction ratio, higher torque-to-weight ratio, and more flexible configurations.
Epicyclic gearing was used in the Antikythera Mechanism, circa 80 BCE, to adjust the displayed position of the moon for the ellipticity of its orbit, and even for the apsidal precession of its orbit. Two facing gears were rotated around slightly different centers, and one drove the other not with meshed teeth but with a pin inserted into a slot on the second. As the slot drove the second gear, the radius of driving would change, thus invoking a speeding up and slowing down of the driven gear in each revolution.
Richard of Wallingford, an English abbot of St Albans monastery is credited for reinventing epicyclic gearing for an astronomical clock in the 14th century.[4]
In 1588, Italian military engineer Agostino Ramelli invented the bookwheel, a vertically-revolving bookstand containing epicyclic gearing with two levels of planetary gears to maintain proper orientation of the books.[4][5]
{\displaystyle {\begin{aligned}{\text{N}}_{\text{s}}\omega _{\text{s}}+{\text{N}}_{\text{p}}\omega _{\text{p}}-({\text{N}}_{\text{s}}+{\text{N}}_{\text{p}})\omega _{\text{c}}&=0\\{\text{N}}_{\text{r}}\omega _{\text{r}}-{\text{N}}_{\text{p}}\omega _{\text{p}}-({\text{N}}_{\text{r}}-{\text{N}}_{\text{p}})\omega _{\text{c}}&=0\end{aligned}}}
{\displaystyle {\omega _{\text{r}}},{\omega _{\text{s}}},{\omega _{\text{p}}},{\omega _{\text{c}}}}
is the angular velocity of the Ring, Sun gear, Planet gears and planet Carrier respectively, and
{\displaystyle {{\text{N}}_{\text{r}}},{{\text{N}}_{\text{s}}},{{\text{N}}_{\text{p}}}}
is the Number of teeth of the Ring, the Sun gear and each Planet gear respectively.
{\displaystyle {\text{N}}_{\text{s}}\omega _{\text{s}}+{\text{N}}_{\text{r}}\omega _{\text{r}}=({\text{N}}_{\text{s}}+{\text{N}}_{\text{r}})\omega _{\text{c}}}
{\displaystyle -{\frac {{\text{N}}_{\text{r}}}{{\text{N}}_{\text{s}}}}={\frac {\omega _{\text{s}}-\omega _{\text{c}}}{\omega _{\text{r}}-\omega _{\text{c}}}}}
{\displaystyle \omega _{\text{r}}\neq \omega _{\text{c}}}
Alternatively, if the number of teeth on each gear meets the relationship
{\displaystyle N_{\text{r}}=N_{\text{s}}+2N_{\text{p}}}
, this equation can be re-written as the following:
{\displaystyle n\omega _{\text{s}}+(2+n)\omega _{\text{r}}-2(1+n)\omega _{\text{c}}=0}
{\displaystyle n=N_{\text{s}}/N_{\text{p}}}
These relationships can be used to analyze any epicyclic system, including those, such as hybrid vehicle transmissions, where two of the components are used as inputs with the third providing output relative to the two inputs.[6]
One turn of the sun gear results in
{\displaystyle -N_{\text{s}}/N_{\text{p}}}
turns of the planets
One turn of a planet gear results in
{\displaystyle N_{\text{p}}/N_{\text{r}}}
turns of the ring gear
So, with the planetary carrier locked, one turn of the sun gear results in
{\displaystyle -N_{\text{s}}/N_{\text{r}}}
turns of the ring gear.
The ring gear may also be held fixed, with input provided to the planetary gear carrier; output rotation is then produced from the sun gear. This configuration will produce an increase in gear ratio, equal to 1+Nr/Ns.[citation needed]
{\displaystyle \tau _{r}=\tau _{s}{\frac {N_{r}}{N_{s}}}}
{\displaystyle \tau _{r}=-\tau _{c}{\frac {N_{r}}{N_{r}+N_{s}}}}
{\displaystyle \tau _{c}=-\tau _{r}{\frac {N_{r}+N_{s}}{N_{r}}}}
{\displaystyle \tau _{c}=-\tau _{s}{\frac {N_{r}+N_{s}}{N_{s}}}}
{\displaystyle \tau _{s}=\tau _{r}{\frac {N_{s}}{N_{r}}}}
{\displaystyle \tau _{s}=-\tau _{c}{\frac {N_{s}}{N_{r}+N_{s}}}}
{\displaystyle \tau _{r}}
— Torque of ring (annulus),
{\displaystyle \tau _{s}}
— Torque of sun,
{\displaystyle \tau _{c}}
— Torque of carrier. For all three, these are the torques applied to the mechanism (input torques). Output torques have the reverse sign of input torques.
{\displaystyle R={\frac {\omega _{s}}{\omega _{r}}}=-{\frac {N_{r}}{N_{s}}}.}
{\displaystyle R={\frac {\omega _{s}-\omega _{c}}{\omega _{r}-\omega _{c}}}.}
{\displaystyle {\frac {\omega _{s}}{\omega _{r}}}=R,\quad {\mbox{so}}\quad {\frac {\omega _{s}}{\omega _{r}}}=-{\frac {N_{r}}{N_{s}}}.}
{\displaystyle {\frac {\omega _{s}-\omega _{c}}{-\omega _{c}}}=R,\quad {\mbox{or}}\quad {\frac {\omega _{s}}{\omega _{c}}}=1-R,\quad {\mbox{so}}\quad {\frac {\omega _{s}}{\omega _{c}}}=1+{\frac {N_{r}}{N_{s}}}.}
{\displaystyle {\frac {-\omega _{c}}{\omega _{r}-\omega _{c}}}=R,\quad {\mbox{or}}\quad {\frac {\omega _{r}}{\omega _{c}}}=1-{\frac {1}{R}},\quad {\mbox{so}}\quad {\frac {\omega _{r}}{\omega _{c}}}=1+{\frac {N_{s}}{N_{r}}}.}
{\displaystyle {\frac {\omega _{s}-\omega _{c}}{\omega _{r}-\omega _{c}}}=-1,}
{\displaystyle \omega _{c}={\frac {1}{2}}(\omega _{s}+\omega _{r}).}
{\displaystyle (R-1)\omega _{\text{c}}=R\omega _{\text{r}}-\omega _{\text{s}}}
{\displaystyle R=N_{\text{r}}/N_{\text{s}}}
{\displaystyle \omega _{\text{r}}=\omega _{\text{s}}(1/R)}
when the carrier is locked,
{\displaystyle \omega _{\text{r}}=\omega _{\text{c}}(R-1)/R}
when the sun is locked,
{\displaystyle \omega _{\text{s}}=-\omega _{\text{c}}(R-1)}
when the ring gear is locked.
Some designs use "stepped-planet" which have two differently-sized gears on either end of a common shaft. The large end engages the sun, while the small end engages the ring gear. This may be necessary to achieve smaller step changes in gear ratio when the overall package size is limited. Compound planets have "timing marks" (or "relative gear mesh phase" in technical term). The assembly conditions of compound planetary gears are more restrictive than simple planetary gears,[7] and they must be assembled in the correct initial orientation relative to each other, or their teeth will not simultaneously engage the sun and ring gear at opposite ends of the planet, leading to very rough running and short life. Compound planetary gears can easily achieve larger transmission ratio with equal or smaller volume. For example, compound planets with teeth in a 2:1 ratio with a 50T ring gear would give the same effect as a 100T ring gear, but with half the actual diameter.
During World War II, a special variation of epicyclic gearing was developed for portable radar gear, where a very high reduction ratio in a small package was needed. This had two outer ring gears, each half the thickness of the other gears. One of these two ring gears was held fixed and had one tooth fewer than did the other. Therefore, several turns of the "sun" gear made the "planet" gears complete a single revolution, which in turn made the rotating ring gear rotate by a single tooth like a Cycloidal drive.[citation needed]
Planetary gear trains provide high power density in comparison to standard parallel axis gear trains. They provide a reduction volume, multiple kinematic combinations, purely torsional reactions, and coaxial shafting. Disadvantages include high bearing loads, constant lubrication requirements, inaccessibility, and design complexity.[8][9]
One popular use of 3D printed planetary gear systems is as toys for children.[citation needed] Since herringbone gears are easy to 3D print, it has become very popular to 3D print a moving herringbone planetary gear system for teaching children how gears work. An advantage of herringbone gears is that they don't fall out of the ring and don't need a mounting plate.
Split ring, compound planet, epicyclic gears of a car rear-view mirror positioner. This has a ratio from input sun gear to output black ring gear of −5/352.
Reduction gears on Pratt & Whitney Canada PT6 gas turbine engine.
One of three sets of three gears inside the planet carrier of a Ford FMX. Ravigneaux transmission
GearGear bearingGear trainGearboxGearboxesRim (wheel)Mechanical engineeringHerringbone gearRotationRotational speedEgan Bernal
This article uses material from the Wikipedia article "Epicyclic gearing", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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\mathrm{with}\left(\mathrm{EssayTools}\right):
\mathrm{Reduce}\left("The car was super fast. It was rocket screaming fast. Nothing else could touch it."\right)
[\textcolor[rgb]{0,0,1}{"car be super fast"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"car be rocket screaming fast"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"nothing can touch touch"}]
\mathrm{Reduce}\left("The tortoise and hare was a great story because it showed how an underdog can succeed with dedication and perseverance."\right)
[\textcolor[rgb]{0,0,1}{"tortoise hare be great story"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"tortoise show how underdog can succeed dedication perseverance"}]
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Haskell/Next steps - Wikibooks, open books for an open world
Haskell/Next steps
Next steps (Solutions)
2 Introducing pattern matching
3 Tuple and list patterns
4 let bindings
This chapter introduces pattern matching and two new pieces of syntax: if expressions and let bindings.
if / then / elseEdit
Haskell syntax supports garden-variety conditional expressions of the form if... then... else... . For instance, consider a function that returns (-1) if its argument is less than 0; 0 if its argument is 0; and 1 if its argument is greater than 0. The predefined signum function does that job already; but for the sake of illustration, let's define a version of our own:
Example: The signum function.
You can experiment with this:
*Main> mySignum (5 - 10)
The parentheses around "-1" in the last example are required; if missing, Haskell will think that you are trying to subtract 1 from mySignum (which would give a type error).
In an if/then/else construct, first the condition (in this case x < 0) is evaluated. If it results True, the whole construct evaluates to the then expression; otherwise (if the condition is False), the construct evaluates to the else expression. All of that is pretty intuitive. If you have programmed in an imperative language before, however, it might seem surprising to know that Haskell always requires both a then and an else clause. The construct has to result in a value in both cases and, specifically, a value of the same type in both cases.
Function definitions using if / then / else like the one above can be rewritten using Guards.
Example: From if to guards
mySignum x
Similarly, the absolute value function defined in Truth values can be rendered with an if/then/else:
Example: From guards to if
absolute x =
Why use if/then/else versus guards? As you will see with later examples and in your own programming, either way of handling conditionals may be more readable or convenient depending on the circumstances. In many cases, both options work equally well.
Introducing pattern matchingEdit
Consider a program which tracks statistics from a racing competition in which racers receive points based on their classification in each race, the scoring rules being:
10 points for the winner;
6 for second-placed;
4 for third-placed;
3 for fourth-placed;
2 for fifth-placed;
1 for sixth-placed;
no points for other racers.
We can write a simple function which takes a classification (represented by an integer number: 1 for first place, etc.[1]) and returns how many points were earned. One possible solution uses if/then/else:
Example: Computing points with if/then/else
Yuck! Admittedly, it wouldn't look this hideous had we used guards instead of if/then/else, but it still would be tedious to write (and read!) all those equality tests. We can do better, though:
Example: Computing points with a piece-wise definition
Much better. However, even though defining pts in this style (which we will arbitrarily call piece-wise definition from now on) shows to a reader of the code what the function does in a clear way, the syntax looks odd given what we have seen of Haskell so far. Why are there seven equations for pts? What are those numbers doing in their left-hand sides? What about variable arguments?
This feature of Haskell is called pattern matching. When we call pts, the argument is matched against the numbers on the left side of each of the equations, which in turn are the patterns. The matching is done in the order we wrote the equations. First, the argument is matched against the 1 in the first equation. If the argument is indeed 1, we have a match and the first equation is used; so pts 1 evaluates to 10 as expected. Otherwise, the other equations are tried in order following the same procedure. The final one, though, is rather different: the _ is a special pattern, often called a "wildcard", that might be read as "whatever": it matches with anything; and therefore if the argument doesn't match any of the previous patterns pts will return zero.
As for the lack of x or any other variable standing for the argument, we simply don't need that to write the definitions. All possible return values are constants. Besides, variables are used to express relationships on the right side of the definition, so the x is unnecessary in our pts function.
However, we could use a variable to make pts even more concise. The points given to a racer decrease regularly from third place to sixth place, at a rate of one point per position. After noticing that, we can eliminate three of the seven equations as follows:
Example: Mixing styles
So, we can mix both styles of definitions. In fact, when we write pts x in the left side of an equation we are using pattern matching too! As a pattern, the x (or any other variable name) matches anything just like _; the only difference being that it also gives us a name to use on the right side (which, in this case, is necessary to write 7 - x).
We cheated a little when moving from the second version of pts to the third one: they do not do exactly the same thing. Can you spot what the difference is?
Beyond integers, pattern matching works with values of various other types. One handy example is booleans. For instance, the (||) logical-or operator we met in Truth values could be defined as:
Example: (||)
Example: (||), done another way
When matching two or more arguments at once, the equation will only be used if all of them match.
Now, let's discuss a few things that might go wrong when using pattern matching:
If we put a pattern which matches anything (such as the final patterns in each of the pts example) before the more specific ones the latter will be ignored. GHC(i) will typically warn us that "Pattern match(es) are overlapped" in such cases.
If no patterns match, an error will be triggered. Generally, it is a good idea to ensure the patterns cover all cases, in the same way that the otherwise guard is not mandatory but highly recommended.
Finally, while you can play around with various ways of (re)defining (&&),[2] here is one version that will not work:
x && x = x -- oops!
The program won't test whether the arguments are equal just because we happened to use the same name for both. As far as the matching goes, we could just as well have written _ && _ in the first case. And even worse: because we gave the same name to both arguments, GHC(i) will refuse the function due to "Conflicting definitions for `x'".
Tuple and list patternsEdit
While the examples above show that pattern matching helps in writing more elegant code, that does not explain why it is so important. So, let's consider the problem of writing a definition for fst, the function which extracts the first element of a pair. At this point, that appears to be an impossible task, as the only way of accessing the first value of the pair is by using fst itself... The following function, however, does the same thing as fst (confirm it in GHCi):
Example: A definition for fst
It's magic! Instead of using a regular variable in the left side of the equation, we specified the argument with the pattern of the 2-tuple - that is, (,) - filled with a variable and the _ pattern. Then the variable was automatically associated with the first component of the tuple, and we used it to write the right side of the equation. The definition of snd is, of course, analogous.
Furthermore, the trick demonstrated above can be done with lists as well. Here are the actual definitions of head and tail:
Example: head, tail and patterns
The only essential change in relation to the previous example was replacing (,) with the pattern of the cons operator (:). These functions also have an equation using the pattern of the empty list, []; however, since empty lists have no head or tail there is nothing to do other than use error to print a prettier error message.
In summary, the power of pattern matching comes from its use in accessing the parts of a complex value. Pattern matching on lists, in particular, will be extensively deployed in Recursion and the chapters that follow it. Later on, we will explore what is happening behind this seemingly magical feature.
To conclude this chapter, a brief word about let bindings (an alternative to where clauses for making local declarations). For instance, take the problem of finding the roots of a polynomial of the form
{\displaystyle ax^{2}+bx+c}
(in other words, the solution to a second degree equation — think back to your middle school math courses). Its solutions are given by:
{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}
We could write the following function to compute the two values of
{\displaystyle x}
((-b + sqrt(b * b - 4 * a * c)) / (2 * a),
(-b - sqrt(b * b - 4 * a * c)) / (2 * a))
Writing the sqrt(b * b - 4 * a * c) term in both cases is annoying, though; we can use a local binding instead, using either where or, as will be demonstrated below, a let declaration:
let sdisc = sqrt (b * b - 4 * a * c)
in ((-b + sdisc) / (2 * a),
(-b - sdisc) / (2 * a))
We put the let keyword before the declaration, and then use in to signal we are returning to the "main" body of the function. It is possible to put multiple declarations inside a single let...in block — just make sure they are indented the same amount or there will be syntax errors:
twice_a = 2 * a
in ((-b + sdisc) / twice_a,
(-b - sdisc) / twice_a)
Because indentation matters syntactically in Haskell, you need to be careful about whether you are using tabs or spaces. By far the best solution is to configure your text editor to insert two or four spaces in place of tabs. If you insist on keeping tabs as distinct, at least ensure that your tabs always have the same length.
The Indentation chapter has a full account of indentation rules.
↑ Here we will not be much worried about what happens if a nonsensical value (say, (-4)) is passed to the function. In general, however, it is a good idea to give some thought to such "strange" cases, in order to avoid nasty surprises down the road.
↑ If you are going to experiment with it in GHCi, call your version something else to avoid a name clash; say, (&!&).
Retrieved from "https://en.wikibooks.org/w/index.php?title=Haskell/Next_steps&oldid=3676020"
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Quality - Maple Help
Home : Support : Online Help : Programming : ImageTools Package : Quality
compute the quality measure of a reconstructed image
Quality( img_r, img_s, meas, opts )
img_r
Image; reconstructed image
Image; source image
(optional) name; quality measure
(optional) equation(s) of the form option = value; specify options for the Quality command
peak = realcons
Specifies the peak value to use for the peak signal-to-noise ratio (psnr) calculation. It is ignored for the other cases. The default is 1.
The Quality command computes the quality measure of a reconstructed image with respect to a source image.
The img_r and img_s parameters are the reconstructed and the source images, respectively. They must be grayscale images and have the same width, height, and order (C_order or Fortran_order).
The optional meas parameter is a name specifying the quality measure. It can take one of the following values, the default is mse:
mse: mean-squared error.
\mathrm{mse}=\frac{\mathrm{sum}\left({\left({r}_{i,j}-{s}_{i,j}\right)}^{2}\right)}{wh}
r,s
are the reconstructed and source images,
i,j
range over all pixels, and
w,h
are the width and height.
rmse: root-mean-squared error (rms).
\mathrm{rmse}=\sqrt{\mathrm{mse}}
snr: signal-to-noise ratio.
\mathrm{snr}=10{\mathrm{log}}_{10}\left(\frac{\mathrm{sum}\left({s}_{i,j}^{2}\right)}{wh\mathrm{mse}}\right)
psnr: peak signal-to-noise ratio.
\mathrm{psnr}=10{\mathrm{log}}_{10}\left(\frac{{\mathrm{peak}}^{2}}{\mathrm{mse}}\right)
\mathrm{with}\left(\mathrm{ImageTools}\right):
\mathrm{img_s}≔\mathrm{Create}\left(100,200,\left(r,c\right)↦\mathrm{evalf}\left(0.5\cdot \mathrm{sin}\left(\frac{r}{50}\right)+0.5\cdot \mathrm{sin}\left(\frac{c}{30}\right)\right)\right):
\mathrm{img_r}≔0.99\mathrm{img_s}+0.01:
\mathrm{Quality}\left(\mathrm{img_r},\mathrm{img_s},\mathrm{psnr}\right)
\textcolor[rgb]{0,0,1}{42.61176831}
\mathrm{Quality}\left(\mathrm{img_r},\mathrm{img_s},\mathrm{snr}\right)
\textcolor[rgb]{0,0,1}{36.96510324}
\mathrm{Quality}\left(\mathrm{img_r},\mathrm{img_s},\mathrm{rmse}\right)
\textcolor[rgb]{0,0,1}{0.007403065375}
\mathrm{Quality}\left(\mathrm{img_r},\mathrm{img_s},\mathrm{mse}\right)
\textcolor[rgb]{0,0,1}{0.0000548053769525598726}
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Simulating with Continuous Integration Algorithms - MATLAB & Simulink - MathWorks España
State-Space Equations
Choosing an Integration Algorithm
Simulating Switches and Power Electronic Devices
Assuming a circuit containing nx states, ns switches, and ny voltage or current outputs, the software determines:
nx state derivatives to be computed from the A and B matrices of
\stackrel{˙}{x}=A·x+B·u
ns switch variables (either voltages across open switches or currents through closed switches)
ny output variables to be computed from the C and D matrices of
y=C·x+D·u
A total of nx + ns + ny equations is obtained.
Unknown variables are state derivatives dx/dt, outputs y, and switch variables (switch voltages or switch currents). Known variables are state variables x and inputs u (voltage sources or current sources).
As the switch status (open or closed) is undetermined, circuit equations are expressed using both switch voltages (vD1, vD2) and switch currents (iD1, iD2).
These equations express Kirchhoff current laws (KCL) at circuit nodes and Kirchhoff voltage laws (KVL) for the independent loops. These equations are completed by the output equations.
Computation of the state-space model is incorporated in an S-function and performed each time a switch status is changing.
To get a list of the circuit equations in the Diagnostic Viewer, select the Display circuit differential equations check box in the Solver tab of the Powergui block parameters dialog box.
Simulink® software provides a variety of solvers. Most of the variable-step solvers work well with linear circuits. However circuits containing nonlinear models, especially circuits with circuit breakers and power electronics, require stiff solvers.
Best accuracy and fastest simulation speed is usually achieved with ode23tb.
Normally, you can choose auto for the absolute tolerance and the maximum step size. In some instances you might have to limit the maximum step size and the absolute tolerance. Selecting too small a tolerance can slow down the simulation considerably. The choice of the absolute tolerance depends on the maximum expected magnitudes of the state variables (inductor currents, capacitor voltages, and control variables).
For example, if you work with high-power circuit where expected voltage and currents are thousands of volts and amperes, an absolute tolerance of 0.1 or even 1.0 is sufficient for the electric states. However, if your electrical circuit is associated with a control system using normalized control signals (varying around 1), the absolute tolerance is imposed by the control states. In this case, choosing an absolute tolerance of 1e-3 (1% of control signal) would be appropriate. If you are working with a very low power circuit with expected currents of milliamperes, set the absolute tolerance to 1e-6.
Usually, keeping the Solver reset method parameter of the ode23tb solver to its default value (Fast) produces the best simulation performance. However, for some highly nonlinear circuits it might be necessary to set this parameter to Robust. When you build a new model, we recommend that you try both the Robust and the Fast reset methods. If you do not notice a difference in simulation results, then keep the Fast method, which provides fastest simulation speed.
In the Preferences tab of the powergui block, you can select Disable snubbers in switching devices, which disables snubbers of all switches in your model. Otherwise, you may individually disable snubbers of selected switches by specifying Rs=inf in their block menus. You can also simulate perfectly ideal switches by disabling the resistances (Ron) and the forward voltages (Vf).
Eliminating the snubbers reduces the circuit stiffness and lets you use a non-stiff solver, for example, ode45 instead of ode23tb, to achieve correct results and good simulation speed.
If you specify resistive snubber values that are too large, the circuit model might become badly conditioned and cause the simulation to stop. In such a case, reduce snubber resistances so that the resulting leakage current remains acceptable (for example 0.01% to 0.1% of switch nominal current).
In some circuits, using switches with a forward voltage Vf greater than zero and Ron=0 might cause simulation to stop and display an error message due to a State-Source dependency. To avoid this problem, specify a small Ron value.
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Propagate signal from point to point using two-ray channel model - MATLAB - MathWorks España
Compare Two-Ray with Free Space Propagation
Two-Ray Propagation of LFM Waveform
Two-Ray Propagation of LFM Waveform with Atmospheric Losses
System object: twoRayChannel
Propagate signal from point to point using two-ray channel model
prop_sig = step(channel,sig,origin_pos,dest_pos,origin_vel,dest_vel)
prop_sig = step(channel,sig,origin_pos,dest_pos,origin_vel,dest_vel) returns the resulting signal, prop_sig, when a narrowband signal, sig, propagates through a two-ray channel from the origin_pos position to the dest_pos position. Either the origin_pos or dest_pos arguments can have multiple points but you cannot specify both as having multiple points. The velocity of the signal origin is specified in origin_vel and the velocity of the signal destination is specified in dest_vel. The dimensions of origin_vel and dest_vel must agree with the dimensions of origin_pos and dest_pos, respectively.
Electromagnetic fields propagated through a two-ray channel can be polarized or nonpolarized. For, nonpolarized fields, such as an acoustic field, the propagating signal field, sig, is a vector or matrix. When the fields are polarized, sig is an array of structures. Every structure element represents an electric field vector in Cartesian form.
In the two-ray environment, there are two signal paths connecting every signal origin and destination pair. For N signal origins (or N signal destinations), there are 2N number of paths. The signals for each origin-destination pair do not have to be related. The signals along the two paths for any single source-destination pair can also differ due to phase or amplitude differences.
You can keep the two signals at the destination separate or combined — controlled by the CombinedRaysOutput property. Combined means that the signals at the source propagate separately along the two paths but are coherently summed at the destination into a single quantity. To use the separate option, set CombinedRaysOutput to false. To use the combined option, set CombinedRaysOutput to true. This option is convenient when the difference between the sensor or array gains in the directions of the two paths is not significant and need not be taken into account.
channel — Two-ray channel
Two-ray channel, specified as a System object.
Example: twoRayChannel
M-by-N complex-valued matrix | M-by-2N complex-valued matrix | 1-by-N struct array containing complex-valued fields | 1-by-2N struct array containing complex-valued fields
Narrowband nonpolarized scalar signal, specified as an
M-by-N complex-valued matrix. Each column contains a common signal propagated along both the line-of-sight path and the reflected path. You can use this form when both path signals are the same.
M-by-2N complex-valued matrix. Each adjacent pair of columns represents a different channel. Within each pair, the first column represents the signal propagated along the line-of-sight path and the second column represents the signal propagated along the reflected path.
Narrowband polarized signal, specified as a
1-by-N struct array containing complex-valued fields. Each struct contains a common polarized signal propagated along both the line-of-sight path and the reflected path. Each structure element contains an M-by-1 column vector of electromagnetic field components (sig.X,sig.Y,sig.Z). You can use this form when both path signals are the same.
1-by-2N struct array containing complex-valued fields. Each adjacent pair of array columns represents a different channel. Within each pair, the first column represents the signal along the line-of-sight path and the second column represents the signal along the reflected path. Each structure element contains an M-by-1 column vector of electromagnetic field components (sig.X,sig.Y,sig.Z).
For nonpolarized fields, the quantity M is the number of samples of the signal and N is the number of two-ray channels. Each channel corresponds to a source-destination pair.
For polarized fields, the struct element contains three M-by-1 complex-valued column vectors, sig.X, sig.Y, and sig.Z. These vectors represent the x, y, and z Cartesian components of the polarized signal.
origin_pos — Origin of the signal or signals
Origin of the signal or signals, specified as a 3-by-1 real-valued column vector or 3-by-N real-valued matrix. The quantity N is the number of two-ray channels. If origin_pos is a column vector, it takes the form [x;y;z]. If origin_pos is a matrix, each column specifies a different signal origin and has the form [x;y;z]. Position units are meters.
origin_pos and dest_pos cannot both be specified as matrices — at least one must be a 3-by-1 column vector.
dest_pos — Destination position of the signal or signals
Destination position of the signal or signals, specified as a 3-by-1 real-valued column vector or 3-by-N real-valued matrix. The quantity N is the number of two-ray channels propagating from or to N signal origins. If dest_pos is a 3-by-1 column vector, it takes the form [x;y;z]. If dest_pos is a matrix, each column specifies a different signal destination and takes the form [x;y;z] Position units are in meters.
You cannot specify origin_pos and dest_pos as matrices. At least one must be a 3-by-1 column vector.
Velocity of signal destinations, specified as a 3-by-1 real-valued column vector or 3–by-N real-valued matrix. The dimensions of dest_vel must match the dimensions of dest_pos. If dest_vel is a column vector, it takes the form [Vx;Vy;Vz]. If dest_vel is a 3-by-N matrix, each column specifies a different destination velocity and has the form [Vx;Vy;Vz] Velocity units are in meters per second.
prop_sig — Propagated signal
Narrowband nonpolarized scalar signal, returned as an:
Narrowband polarized scalar signal, returned as:
1-by-N struct array containing complex-valued fields. To return this format, set the CombinedRaysOutput property to true. Each column of the array contains the coherently combined signals from the line-of-sight path and the reflected path. Each structure element contains the electromagnetic field vector (prop_sig.X,prop_sig.Y,prop_sig.Z).
1-by-2N struct array containing complex-valued fields. To return this format, set the CombinedRaysOutput property to false. Alternate columns contains the signals from the line-of-sight path and the reflected path. Each structure element contains the electromagnetic field vector (prop_sig.X,prop_sig.Y,prop_sig.Z).
The output prop_sig contains signal samples arriving at the signal destination within the current input time frame. Whenever it takes longer than the current time frame for the signal to propagate from the origin to the destination, the output may not contain all contributions from the input of the current time frame. The remaining output will appear in the next call to step.
Propagate a signal in a two-ray channel environment from a radar at (0,0,10) meters to a target at (300,200,30) meters. Assume that the radar and target are stationary and that the transmitting antenna has a cosine pattern. Compare the combined signals from the two paths with the single signal resulting from free space propagation. Set the CombinedRaysOutput to true to produce a combined propagated signal.
Create a Rectangular Waveform
Set the sample rate to 2 MHz.
waveform = phased.RectangularWaveform('SampleRate',fs);
Create the Transmitting Antenna and Radiator
Set up a phased.Radiator System object™ to transmit from a cosine antenna
Specify Transmitter and Target Coordinates
posTx = [0;0;10];
posTgt = [300;200;30];
Compute the transmitting direction toward the target for the free-space model. Then, radiate the signal.
[~,angFS] = rangeangle(posTgt,posTx);
wavTx = radiator(wavfrm,angFS);
Propagate the signal to the target.
fschannel = phased.FreeSpace('SampleRate',waveform.SampleRate);
yfs = fschannel(wavTx,posTx,posTgt,velTx,velTgt);
release(radiator);
Compute the two transmit angles toward the target for line-of-sight (LOS) path and reflected paths. Compute the transmitting directions toward the target for the two rays. Then, radiate the signals.
[~,angTwoRay] = rangeangle(posTgt,posTx,'two-ray');
wavTwoRay = radiator(wavfrm,angTwoRay);
Propagate the signals to the target.
channel = twoRayChannel('SampleRate',waveform.SampleRate,...
y2ray = channel(wavTwoRay,posTx,posTgt,velTx,velTgt);
Plot the combined signal against the free-space signal
plot(abs([y2ray yfs]))
legend('Two-ray','Free space')
10\mu s
Propagate a linear FM signal in a two-ray channel. The signal propagates from a transmitter located at (1000,10,10) meters in the global coordinate system to a receiver at (10000,200,30) meters. Assume that the transmitter and the receiver are stationary and that they both have cosine antenna patterns. Plot the received signal.
Set up the radar scenario. First, create the required System objects.
waveform = phased.LinearFMWaveform('SampleRate',1000000,...
'CombinedRaysOutput',false,'GroundReflectionCoefficient',0.95);
posTx = [1000;10;10];
Specify the transmitting and receiving radar antenna orientations with respect to the global coordinates. The transmitting antenna points along the +x direction and the receiving antenna points near but not directly in the -x direction.
Compute the transmission angles which are the angles that the two rays traveling toward the receiver leave the transmitter. The phased.Radiator System object™ uses these angles to apply separate antenna gains to the two signals. Because the antenna gains depend on path direction, you must transmit and receive the two rays separately.
Create and radiate signals from transmitter along the transmission directions.
Propagate signals to receiver via two-ray channel.
Collect signals at the receiver. Compute the angle at which the two rays traveling from the transmitter arrive at the receiver. The phased.Collector System object™ uses these angles to apply separate antenna gains to the two signals.
Plot the received signals.
plot((0:(n-1))*dt*1000000,real(yR))
Propagate a 100 Mhz linear FM signal into a two-ray channel. Assume there is signal loss caused by atmospheric gases and rain. The signal propagates from a transmitter located at (0,0,0) meters in the global coordinate system to a receiver at (10000,200,30) meters. Assume that the transmitter and the receiver are stationary and that they both have cosine antenna patterns. Plot the received signal. Set the dry air pressure to 102.5 Pa and the rain rate to 5 mm/hr.
waveform = phased.LinearFMWaveform('SampleRate',1e6,...
Set up the scene geometry giving. the transmitter and receiver positions and velocities. The transmitter and receiver are stationary.
Specify the transmitting and receiving radar antenna orientations with respect to the global coordinates. The transmitting antenna points along the +x-direction and the receiving antenna points close to the –x-direction.
[2] Skolnik, M. Introduction to Radar Systems, 3rd Ed. New York: McGraw-Hill
[3] Saakian, A.Radio Wave Propagation Fundamentals. Norwood, MA: Artech House, 2011.
[4] Balanis, C.Advanced Engineering Electromagnetics. New York: Wiley & Sons, 1989.
[5] Rappaport, T.Wireless Communications: Principles and Practice, 2nd Ed New York: Prentice Hall, 2002.
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EUDML | Some inequalities satisfied by the integrals or derivatives of real or analytic functions EuDML | Some inequalities satisfied by the integrals or derivatives of real or analytic functions
Some inequalities satisfied by the integrals or derivatives of real or analytic functions
G.H., Landau, E. und Li Hardy; J.E., J.E. Littlewood
Hardy, G.H., Landau, E. und Li, and Littlewood, J.E., J.E.. "Some inequalities satisfied by the integrals or derivatives of real or analytic functions." Mathematische Zeitschrift 39 (1935): 677-695. <http://eudml.org/doc/168579>.
author = {Hardy, G.H., Landau, E. und Li, Littlewood, J.E., J.E.},
title = {Some inequalities satisfied by the integrals or derivatives of real or analytic functions},
AU - Hardy, G.H., Landau, E. und Li
AU - Littlewood, J.E., J.E.
TI - Some inequalities satisfied by the integrals or derivatives of real or analytic functions
Imed Feki, Ameni Massoudi, Houda Nfata, A generalization to the Hardy-Sobolev spaces
{H}^{k,p}
{L}^{p}
{L}^{1}
logarithmic type estimate
Slim Chaabane, Imed Feki, Pointwise inequalities of logarithmic type in Hardy-Hölder spaces
Michel Artola, Sur un théorème d'interpolation
Michel Artola, On Derivatives of Complex Order in Some Weighted Banach Spaces and Interpolation
Articles by G.H.
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Set Theory. Christian Christiansen.
2. Equivalence of Sets. The Power of a Set
3. Ordered Sets and Ordinal Numbers
4. Systems of Sets
Definitions of .
Further definitions on , arbitrary, disjoint, pairwise disjoint. Shows how union and intersection are associative, commutative, and distributive. Defines difference, symmetric difference, complement. Shows the duality principle, leading to dual theorems.
Definitions of (real) function on , domain, range, mapping of into , image, preimage, image of , preimage of , map into, map onto, one-to-one, one-to-one correspondence, inverse of .
Note . Also note that Theorems 1-3 hold for an arbitrary number of sets.
Definitions of decomposition or partition into classes, a is related to b by the (binary) relation R, equivalence relation, reflexitivity, symmetry, transitivity.
can be partitioned into classes by a relation is an equivalence relation on .
Show and
since and since . Hence,
Take to be the empty set and to be any non-empty set.
Let and . Find and
Prove that and
Step through using definitions, complements, idempotence, duality principle, distributivity, associativity, commutativity.
Evident once some thought has been applied.
Let be the set of all positive integers divisible by . What are and ?
What are and ?
Let be the set of points lying on the curve with . What is ?
(fractional part of ). Prove every closed interval of length one has the same image under . What is this image? Is one-to-one? What is the preimage of the interval ? Partition the real line into classes of points with the same image.
Given set , let be the set of all ordered pairs with . Let . Interpret .
Find a binary relation which is:
Reflexive and symmetric, but not transitive.
Reflexive, but neither symmetric nor transitive.
Symmetric, but neither reflexive nor transitive.
Transitive, but neither reflexive nor symmetric.
Blood relative. Remembers name of (provided everybody knows their own name). Not equal to. Strictly greater than. (Credit to CoveredInChocolate for the first two answers.)
2 Equivalence of Sets. The Power of a Set
Definitions of finite, infinite, countable, uncountable. Demonstration of how the rationals are countable.
Every subset of a countable subset is countable.
The union of a finite or countable number of countable sets is itself countable.
Line the elements of the sets parallel to each other, and count all of the elements ‘diagonally’.
Every infinite set has a countable subset.
Definition of equivalence.
Demonstration of how the points on a sphere can be mapped one-to-one with the points in the complex plane with stereographic projection.
Every infinite set is equivalent to one of its proper subsets.
By Theorem 3, every infinite set contains a countable subset . Partition this subset into two countable subsets and . A one-to-one correspondence can be set between and . Hence and is a proper subset of .
The set of real numbers in the closed unit interval is uncountable.
Cantor’s diagonal. Make sure to disallow setting a digit as
0
Definition of power, aleph null, continuum.
Given any set , let be the set whose elements are all possible subsets of . Then the power of is greater than the power of the original set .
Attempt to setup to be the set of elements of which do not belong to their ‘associated subsets’. Then we reach an argument similar to Russel’s paradox.
Theorem 7 (Cantor-Bernstein)
Given any two sets and , suppose contains a subset equivalent to , while contains a subset equivalent to . Then and are equivalent.
The proof is beautifully shown in the book. A sketch would be a disservice.
Suppose the set is countable, and very quickly reach a contradiction.
Let be any infinite set and any countable set. Prove that .
Use Theorem 3.
Prove that the following sets are countable:
The set of all numbers with two distinct decimal expansions.
Since all of the numbers can either be shown as or , only consider the representation. Evident that this is a subset of the rationals, hence countable.
The set of all rational points in the plane.
Simply extend the technique used for demonstrating that the rationals are countable.
The set of all rational intervals.
This can be transposed to be a subset of the the set of all rational points in the plane, i.e. the previous question.
The set of all polynomials with rational coefficients.
This can be transposed to a subset of the set of all rational points in dimensions. Extend the argument for problem 3.2. from 2 dimensions to dimensions.
A number is called algebraic if it is a root of a polynomial equation with rational coefficients. Prove that the set of all algebraic numbers is countable.
The set of all polynomials with rational coefficients is countable, and each polynomial has a maximum of solutions. Hence countable by Theorem 2.
Prove that the existence of uncountably many transcendental numbers, i.e., numbers which are not algebraic.
From Theorem 5, we know that there are uncountably many numbers between . And from Theorem 2, we know that no union of countable sets will ever become uncountable. Hence there must exist numbers within which are not algebraic. We call these transcendental.
Prove that the set of all real functions (more generally, functions taking values in a set containing at least two elements) defined on a set is of power greater than the power of . In particular, prove that the power of the set of all real functions (continuous and discontinuous) defined in the interval is greater than .
Just look at the identity functions. The set of all identities has a one-to-one correspondence with the power set.
Give an indirect proof of the equivalence of the closed interval , the open interval and the half-open interval or .
Need to find suitable functions to use with Theorem 7.
Prove that the union of a finite or countable number of sets each of power is itself of power .
Since is uncountable, and by problem 7, so is , map each set to , , … respectively. Evidently the union is a subset within the real line, and is itself of power .
Prove that each of the following sets has the power of the continuum:
The set of all infinite sequences of positive integers.
Place in front of the integers to get the set .
The set of all ordered -tuples of real numbers.
Use a similar argument to problem 8.
The set of all infinite sequences of real numbers.
Need to check how to deal with the uncountable sequences of real numbers.
Develop a contradiction inherent in the notion of the “set of all sets which are not members of themselves”.
Russell's paradox.
3 Ordered Sets and Ordinal Numbers
Definitions of partial ordering, partially ordered (reflexive, transitive, antisymmetric), maximal, minimal.
A bijective function is order-preserving if . An order-preserving function is an isomorphism if . Two partially ordered sets are isomorphic if there exists an isomorphism between them.
Noncomparable if neither nor . A set is ordered if it contains no noncomparable elements. Two isomorphic sets have the same (order) type. Order of (in the normal manner) is . The power corresponds to the order type .
Definition 1 An ordered set is well-ordered if every nonempty subset of has a smallest (or “first”) element, i.e., an element such that for every .
Definition 2 The order type of a well-ordered set is an ordinal number or simply an ordinal. If the set is infinite, the ordinal is transfinite.
The ordered sum of a finite number of well-ordered sets is itself a well-ordered set.
Let be an arbitrary subset of the ordered sum , and let be the first set containing elements of . Then is a subset of the well-ordered set , and therefore has a smallest element . Clearly is also the smallest element of .
Need to check whether this holds for the sum of countably many well-ordered sets, and why?
The ordered sum of a finite number of ordinal numbers is itself an ordinal number.
The ordered product of two well-ordered sets and is well-ordered.
Let be an arbitrary subset of , with such that , . The set of all second elements in is a subset of , and therefore has a least element since is well-ordered. Then look at all . Again, the set of all first elements is a subset of , hence has a smallest element since is well-ordered. is clearly the smallest element of .
Every element of a well-ordered set determines an (initial) section , the set of all such that , and a remainder , the set of all such that . Given two well-ordered sets and of order type and respectively, then:
if and are isomorphic.
if is isomorphic to some section of .
Let be an isomorphism of a well-ordered set onto some subset . Then for all .
If there exists such that , then there is a least such element since is well-ordered. Let , then , and since is an isomorphism. But then is not the smallest element such that . Contradiction.
Two given ordinal numbers and satisfy one and only one of the relations , , .
Let be the set of all ordinals . Any two numbers and in are comparable and the corresponding ordering of makes it a well-ordered set of type . If a set is of type , then by definition, the ordinals less than are the types of well-ordered sets isomorphic to sections of . Hence, the ordinals are in one-to-one correspondence with the elements of .
Let , be two ordinals, then , are well-ordered sets of type , respectively. Let . Then is well-ordered, of type , say. We now show . If , then obviously . Otherwise, if , then is a section of and hence .
[Let , . Then , are comparable, i.e., either or . But is impossible, since then . Therefore and hence is a section of , which implies . Moreover, is the first element of .]
Thus . Similarly . The case , is impossible, since then , . But then on the one hand and on the other hand.
It follows that there are only three possibilities:
i.e., and are comparable.
Let and be well-ordered sets. Then either is equivalent to or one of the sets is of greater power than the other, i.e., the powers of and are comparable.
A definite power corresponds to each ordinal. As ordinals are comparable, so are the corresponding powers.
Given any set , there is a “choice function” such that is an element of for every nonempty subset .
Definition 3 Let be a partially ordered set, and let be any subset of such that and are comparable for every , . Then is a chain (in ). A chain is maximal if there is no other chain in containing as a proper subset.
Definiton 4 An element of a partially ordered set is an upper bound of a subset if for every .
The below two assertions are equivalent to the axiom of choice.
Hausdorff’s maximal principle
Every chain in a partially ordered set is contained in a maximal chain in .
If every chain in a partially ordered set has an upper bound, then contains a maximal element.
Theorem 5 (Mathematical induction)
Given a proposition formulated fo every positive integer , suppose that
The validity of for all implies the validity of .
Then is true for all
Suppose fails to be true for all Let be the smallest integer for which is false ( exists due to the well-ordering of the positive integers). Clearly , so is a positive integer. Therefore is valid for all , but not for . Contradiction.
Theorem 5’ (Transfinite induction)
Given a well-ordered set , let be a proposition formulated for every element . Suppose that
is true for the smallest element of .
Then is true for all .
Suppose fails to be true for all . Then is false for all in some nonempty subset . By well-ordering, has a smallest element . Therefore is valid for all but not for . Contradiction.
Exhibit both a partial ordering and a simple ordering of the set of all complex numbers.
Partial ordering: only order reals. Simple: order reals, then complex.
What is the minimal element of the set of all subsets of a given set , partially ordered by set inclusion? What is the maximal element?
A partially ordered set is a directed set if, given any two element , , there is an element such that , . Are the partially ordered sets in Examples 1-4, Section 3.1 all directed sets?
No, yes, yes, yes.
The greatest lower bound of two elements and of a partially ordered set is such that , and there is no element such that , . Define least upper bound similarly. A lattice is a partially ordered set with both a greatest lower bound and a least upper bound. Prove that the set of all subsets of a given set , partially ordered by set inclusion, is a lattice. What is the set-theoretic meaning of the greatest lower bound and least upper bound of two elements of this set?
All order-mapping preserving mappings are bijective. Hence an inverse exists. Need to show how the inverse also preserves order. Hence isomorphism.
Prove that ordered sums and products of ordered sets are associative.
Just step through to show that they are the same.
Construct well-ordered sets with ordinals , … Show that the sets are all countable.
, … Line each one up on a row, and zigzag up and down.
, … Similar proof to countability of the rationals.
Show that .
Let send to if and to if . Evidently bijective, and order-preserving, hence isomorphic, and hence they have equivalent ordinals too.
Prove that the set of all ordinals less than a given ordinal is well-ordered.
Each ordinal less than corresponds to a section of . There is a least section of , the empty set, as all other sections of are greater than or equal to it. Hence the corresponding ordinal to the empty set is the least element of and we also know that ordinals are comparable. Therefore is well-ordered.
Prove that any nonempty set of ordinals is well-ordered.
subset. Nonempty and ordinals are comparable so exists such that for all . Therefore well-ordered.
Prove that the set of all ordinals corresponding to a countable set is itself countable.
Line them up. Then use Cantor diagonal argument.
Let be the power of the set in the preceding problem. Prove that there is no power such that .
Impossible as there is no set which is strictly bigger than countable and yet strictly smaller than uncountable.
3.2 Open ends
Why do Theorems 1 and 2 hold for just finitely many? Do they also hold for countably many? If not, why not?
What well-ordered set has ordinal ?
I found this link https://en.wikipedia.org/wiki/Ordinal_analysis#Theories_with_proof-theoretic_ordinal_%CF%89%CF%89 but it still leaves me none the wiser. This link https://math.stackexchange.com/questions/1412153/let-omega-be-the-ordinal-of-the-well-ordered-set-mathbbn-what-does-mean is a tiny bit clearer.
For Problem 5, how is it shown that the inverse preserves order?
Is the proof for Problem 13 correct?
4 Systems of Sets
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Memcache Internals | Ri Xu Online
Ri Xu October 7, 2017 Coding No Comments
Memcached, system of the distributed memory caching, is often used to increase the performance and availability of the hosted application through decreasing database load. It creates a common cache for all the application nodes and represents your application short-term memory.
Most of Memcache functionality (add, get, set, flush etc) are
O (1)
. This means they are constant time functions. It does not matter how many items there are inside the cache, the functions will take just as long as they would with just 1 item inside the cache. Memcache uses the LRU algorithm to eliminate data for the slab.
Internally, all objects have a "counter". This counter holds a timestamp. Every time a new object is created, that counter will be set to the current time. When an object gets FETCHED, it will reset that counter to the current time as well. As soon as Memcache needs to "evict" an object to make room for newer objects, it will find the lowest counter. That is the object that isn't fetched or is fetched the longest time ago (and probably isn't needed that much, otherwise the counter would be closed to the current timestamp).
In effect, this creates a simple system that uses the cache very efficient. If it isn't used, it's kicked out of the system.
Memcached system uses the slab instead of the item-by-item memory allocation. As a result, it improves the usage of the memory and protects it from the fragmentation in case the data expires from the cache.
Each slab consists of several 1 MB size pages and each page, in its turn, consists of the equal amount of blocks or chunks. After every data storing the Memcached defines the data size and looks for a suitable allocation in all slabs. If such allocation exists, the data is written to it. If there is no suitable allocation, the Memcached creates a new slab and divides it into the blocks of the necessary size. In the case you update the already stored item and its new value exceeds the size of the block allocation, it was stored in before, Memcached moves it to another, suitable slab.
| Page |
+-------+-------+-------+-------+-------+ < Slab Class #1
| Chunk | Chunk | Chunk | Chunk | Chunk |
As a result, every instance has multiple pages distributed and allocated within the Memcached memory. This method of allocation prevents the memory fragmentation. But on the other hand it can cause the waste of memory if you have not enough amount of items with equal allocation size, i.e. there are only a few filled chunks on every page. Thus one more important point is the distribution of the stored items.
Reference slabs.c
When Memcache starts, it partitions its allocated memory into smaller parts called pages. Each page is 1MB large (coincidentally, the maximum size that an object can have you can store in Memcache). Each of those pages can be assigned to a slab-class or can be unassigned (being a free page). A slab-class decides how large the objects can be that are stored inside that particular page. Each page that is designated to a particular slab-class will be divided into smaller parts called chunks. The chunks in each slab have the same size so there cannot be 2 different sized chunks on the same page. For instance, there could be a page with 64byte chunks (slab class 1), a page with 128byte chunks (slab class 2) and so on, until we get the largest slab with only 1 chunk (the 1MB chunk). There can be multiple pages for each slab-class, but as soon as a page is assigned a slab-class (and thus, split up into chunks), it cannot be changed to another slab-class.
The smallest chunk-size starts at 80 bytes and increases with a factor of 1.25 (rounded up until the next power of 2). So the second smallest chunk size would be 100 etc. You can actually find it out by issuing the "-vv" flag when starting memcache. You can also set the factor (-f) and the initial chunk-size (-s), but unless you really know what you are doing, don't change the initial values.
static void *do_slabs_alloc(const size_t size, unsigned int id, uint64_t *total_bytes,
MEMCACHED_SLABS_ALLOCATE_FAILED(size, 0);
assert(p->sl_curr == 0 || ((item *)p->slots)->slabs_clsid == 0);
if (total_bytes != NULL) {
*total_bytes = p->requested;
assert(size <= p->size);
/* fail unless we have space at the end of a recently allocated page,
we have something on our freelist, or we could allocate a new page */
if (p->sl_curr == 0 && flags != SLABS_ALLOC_NO_NEWPAGE) {
do_slabs_newslab(id);
if (p->sl_curr != 0) {
/* return off our freelist */
it = (item *)p->slots;
p->slots = it->next;
if (it->next) it->next->prev = 0;
/* Kill flag and initialize refcount here for lock safety in slab
* mover's freeness detection. */
it->it_flags &= ~ITEM_SLABBED;
it->refcount = 1;
p->sl_curr--;
ret = (void *)it;
p->requested += size;
MEMCACHED_SLABS_ALLOCATE(size, id, p->size, ret);
MEMCACHED_SLABS_ALLOCATE_FAILED(size, id);
Remainder Hash
What memcache normally does is a simple, yet very effective load balance trick: for each key that gets stored or fetched, it will create a hash (you might see it as md5(key), but in fact, it's a more specialized - quicker - hash method). Now, the hashes we create are pretty much evenly distributed, so we can use a modulus function to find out which server to store the object to:
node_id = hash_key % len(nodes)
The trouble with this system: as soon as node_id (the number of servers) change, almost 100% of all keys will change server as well. Maybe some keys will get the same server id, but that would be a coincidence. In effect, when you change your memcache server count (either up or down, doesn't matter), you get a big stampede on your backend system since all keys are all invalidated at once.
Consistent hashing use a counter that acts like a clock. Hash values
[0, 2^{32}-1]
are distributed over the circle so that objects are instead assigned to the cache server that is closest in the clockwise direction.
All nodes get a relatively equal number of keys, be able to add and remove nodes such as the fewest number of keys are moved around.
cas_badval 0 The cas command is some kind of Memcached's way to avoid locking. cas calls with bad identifier are counted in this stats key.
cas_hits 0 Number of successful cas commands.
cas_misses 0 cas calls fail if the value has been changed since it was requested from the cache. We're currently not using cas at all, so all three cas values are zero.
cmd_flush 0 The flush_all command clears the whole cache and shouldn't be used during normal operation.
cmd_get 1626823 Number of get commands received since server startup not counting if they were successful or not.
cmd_set 2279784 Number of set commands serviced since startup.
curr_connections 34 Number of open connections to this Memcached server, should be the same value on all servers during normal operation. This is something like the count of MySQL's SHOW PROCESSLIST result rows.
decr_hits 0 The decr command decreases a stored (integer) value by 1. A hit is a decr call to an existing key.
delete_hits 138707 Stored keys may be deleted using the delete command, this system doesn't delete cached data itself, but it's using the Memcached to avoid recaching-races and the race keys are deleted once the race is over and fresh content has been cached.
delete_misses 107095 Number of delete commands for keys not existing within the cache. These 107k failed deletes are deletions of non existent race keys (see above).
get_hits 391283 Number of successful get commands (cache hits) since startup, divide them by the cmd_get value to get the cache hitrate: This server was able to serve 24% of it's get requests from the cache, the live servers of this installation usually have more than 98% hits.
get_misses 1235540 Number of failed get requests because nothing was cached for this key or the cached value was too old.
incr_hits 0 Number of successful incr commands processed. incr is a replace adding 1 to the stored value and failing if no value is stored. This specific installation (currently) doesn't use incr/decr commands, so all their values are zero.
incr_misses 0 Number of failed incr commands (see incr_hits).
uptime 1145873 Numer of seconds the Memcached server has been running since last restart. 1145873 / (60 * 60 * 24) = ~13 days since this server has been restarted
Coding, Memcached
← Nicholas Daniel Solo Recital in China Central Conservatory of Music
The 10th Annual International Chamber Music Festival of the China Central Conservatory of Music →
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Electric power conversion - 2D Symbols - 3D Models
Electric power conversion (5509 views - Electrical Engineering)
In electrical engineering, power engineering, and the electric power industry, power conversion is converting electric energy from one form to another such as converting between AC and DC; or changing the voltage or frequency; or some combination of these. A power converter is an electrical or electro-mechanical device for converting electrical energy. This could be as simple as a transformer to change the voltage of AC power, but also includes far more complex systems. The term can also refer to a class of electrical machinery that is used to convert one frequency of alternating current into another frequency. Power conversion systems often incorporate redundancy and voltage regulation. One way of classifying power conversion systems is according to whether the input and output are alternating current (AC) or direct current (DC).
For generic power conversion (e.g., heat to electric energy), see Energy transformation.
In electrical engineering, power engineering, and the electric power industry, power conversion is converting electric energy from one form to another such as converting between AC and DC; or changing the voltage or frequency; or some combination of these. A power converter is an electrical or electro-mechanical device for converting electrical energy. This could be as simple as a transformer to change the voltage of AC power, but also includes far more complex systems. The term can also refer to a class of electrical machinery that is used to convert one frequency of alternating current into another frequency.
Power conversion systems often incorporate redundancy and voltage regulation.
One way of classifying power conversion systems is according to whether the input and output are alternating current (AC) or direct current (DC).
1 DC power conversion
2 AC power conversion
4 Why use transformers in power converters
Main article: DC-to-DC converter
The following devices can convert DC to DC:[further explanation needed]
The following devices can convert DC to AC:[further explanation needed]
The following devices can convert AC to DC:[further explanation needed]
Mains power supply unit (PSU)
Main article: AC-to-AC converter
The following devices can convert AC to AC:[further explanation needed]
Transformer or autotransformer
Main article: Three-phase electric power
There are also devices and methods to convert between power systems designed for single and three-phase operation.
The standard power voltage and frequency varies from country to country and sometimes within a country. In North America and northern South America it is usually 120 volt, 60 hertz (Hz), but in Europe, Asia, Africa and many other parts of the world, it is usually 230 volt, 50 Hz.[1] Aircraft often use 400 Hz power internally, so 50 Hz or 60 Hz to 400 Hz frequency conversion is needed for use in the ground power unit used to power the airplane while it is on the ground. Conversely, internal 400 Hz internal power may be converted to 50 Hz or 60 Hz for convenience power outlets available to passengers during flight.
Certain specialized circuits can also be considered power converters, such as the flyback transformer subsystem powering a CRT, generating high voltage at approximately 15 kHz.
Consumer electronics usually include an AC adapter (a type of power supply) to convert mains-voltage AC current to low-voltage DC suitable for consumption by microchips. Consumer voltage converters (also known as "travel converters") are used when travelling between countries that use ~120 V versus ~240 V AC mains power. (There are also consumer "adapters" which merely form an electrical connection between two differently shaped AC power plugs and sockets, but these change neither voltage nor frequency.)
Why use transformers in power converters
Transformers are used in power converters to incorporate:
Voltage step-down or step up
The secondary circuit is floating, when you touch the secondary circuit, you merely drag its potential to your body potential or the earth potential. There will be no current flowing through your body. That's why you can use your cellphone safely when it is being charged, even if your cellphone has a metal shell and it is connected to the secondary circuit.
Operating at high frequency and supplying low power, power converters have much smaller transformers as compared with those of fundamental frequency, high power applications. Usually, in power systems, transformers transmit power simultaneously, no charge! The current in the primary winding of a transformer plays two roles:
It sets up the mutual flux in accordance with Ampere's law.
It balances the demagnetizing effect of the load current in the secondary winding.
Flyback converter's transformer works differently, like an inductor. In each cycle, flyback converter's transformer first gets charged then releases its energy to the load. Accordingly, flyback converter's transformer air gap has two functions. It not only determines inductance, but also stores energy. For flyback converter, the transformer gap can have the function of energy transmission through cycles of charging and discharging.
{\displaystyle W_{e}={\frac {1}{2}}BH={\frac {1}{2}}{\frac {B^{2}}{\mu }}}
The core's relative permeability
{\displaystyle \mu _{r}}
can be > 1,000, even > 10,000. While the air gap features much lower permeability, accordingly has higher energy density.
AC power plugs and socketsActuatorAlternatorAlternator (automotive)Constant-current diodeCurrent sensorDC-to-DC converterElectric currentElectric power distributionElectric power transmissionElectrical energyElectrical engineeringElectromagnetic fieldElectromechanicsPower electronicsRectifierAlternating currentDirect currentElectrical conductorSemiconductorInsulator (electricity)Electric potential energyMotor-generatorWireless power transferArmature (electrical)Major applianceHome applianceElectric switchboardDistribution boardElectrical roomGrowler (electrical device)
This article uses material from the Wikipedia article "Electric power conversion", which is released under the Creative Commons Attribution-Share-Alike License 3.0. There is a list of all authors in Wikipedia
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torch.kaiser_window — PyTorch 1.11.0 documentation
torch.kaiser_window
torch.kaiser_window¶
torch.kaiser_window(window_length, periodic=True, beta=12.0, *, dtype=None, layout=torch.strided, device=None, requires_grad=False) → Tensor¶
Let I_0 be the zeroth order modified Bessel function of the first kind (see torch.i0()) and N = L - 1 if periodic is False and L if periodic is True, where L is the window_length. This function computes:
out_i = I_0 \left( \beta \sqrt{1 - \left( {\frac{i - N/2}{N/2}} \right) ^2 } \right) / I_0( \beta )
Calling torch.kaiser_window(L, B, periodic=True) is equivalent to calling torch.kaiser_window(L + 1, B, periodic=False)[:-1]). The periodic argument is intended as a helpful shorthand to produce a periodic window as input to functions like torch.stft().
If window_length is one, then the returned window is a single element tensor containing a one.
window_length (int) – length of the window.
periodic (bool, optional) – If True, returns a periodic window suitable for use in spectral analysis. If False, returns a symmetric window suitable for use in filter design.
beta (float, optional) – shape parameter for the window.
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Introductory Real Analysis. Christian Christiansen.
Introductory Real Analysis is a translation, and revision, by Richard A. Silverman of the second edition of Элементы теории функций и функционального анализа (Elements of the Theory of Functions and Functional Analysis), written by Андре́й Никола́евич Колмого́ров (Andrey Nikolaevich Kolmogorov) and Серге́й Васи́льевич Фоми́н (Sergei Vasilyevich Fomin). The original Russian ran to a total of four editions. Based off of a cursory glance, Silverman’s translation is more fluent (and up to date) than the overly literal translation by Leo F. Boron of the first edition called Elements of the Theory of the Theory of Functions and Functional Analysis. However, Silverman appears to have introduced several mathematical errors.
The prose and diagrammes are terse, clear, and enlightening. The problems within bite. What more could a budding mathematician hope for?
As (seemingly) no official errata page is available (though a collection uploaded by Charlie Hoang is available here), I will collate errors Jeffrey Kwan and I come across below:
Page 11: has a rational number height of , not
0
Page 17: On the other hand… should discuss if , then , instead of repeating the same scenario twice.
Page 23: Definition 1. such that not .
Page 27: It follows from the well-ordering theorem and Theorem 4, not Theorem 5.
Page 28, 29: There are two Theorem 4s. The second Theorem 4 (mathematical induction) should be Theorem 5, and accordingly the transfinite induction theorem should be renamed Theorem 5’.
Page 29: Suppose fails to be true for, not tor.
Page 30: Problem 4. By a lattice is meant a partially ordered set which has both a greatest …
Sketched notes
My notes are simply for the purposes of jogging my memory. CoveredInChocolate has more substantial notes here. Nalin Pithwa also has notes here, although Jeffrey has spotted some errors in a couple of Pithwa’s solutions.
My solutions for problems may simply be sketched out too, especially in cases where the LaTeX is particularly cumbersome. Any corrections, or clarification of open ends, would be greatly welcomed.
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Bioeconomics - Wikibooks, open books for an open world
Bioeconomics is the theory of economic exploitation of living resources, dealing with two dynamic systems: population dynamics and the dynamics of economic systems. Bioeconomics therefore leans on two traditional university disciplines, biology and economics.
1 Short reference to production theory
2 Catch production in the short run
2.1 Production of fishing effort
2.2 The concept of efficient production
2.3 Producing fish harvest by two input factors
2.4 Market failures while using common resources
3.2 Biomass growth as a function of time
4 Catch production in the long run
5 Catch Revenue and Cost
6 The Concept of Resource Rent
7 Open access equilibrium
8 Maximum sustainable economic yield
9 Investing in stock by not fishing
10 Social optimum solution
Short reference to production theoryEdit
Production theory is a central element in microeconomics and describes simply the conversion of inputs (v) into outputs (Q):
{\displaystyle Q=Q({v_{1}},{v_{2}}....{v_{n}})\,\!}
There are several ways of specifying this function. One is as an additive production function:
{\displaystyle Q={p_{0}}+{p_{1}}{v_{1}}+{p_{2}}{v_{2}}+...+{p_{n}}{v_{n}}\,\!}
where p0, p1, .... pn are parameters that are determined empirically.
Another is as a Cobb-Douglas production function (multiplicative):
{\displaystyle Q={p_{0}}\cdot {v_{1}}^{p_{1}}\cdot {v_{2}}^{p_{2}}\cdot ...\cdot {v_{n}}^{p_{n}}}
Other forms include the constant elasticity of substitution production function (CES) which is a more general formulation including the Cobb-Douglas function, and the quadratic production function which is a specific type of additive function. The best form of the equation to use and the values of the parameters vary from company to company and industry to industry.
In a short run production function at least one of the (inputs) is fixed. In the long run all factor inputs are variable, in principle at the discretion of management.
In classical theory production may involve three types of input: Labour (L), Capital (K) and Natural resources (R). Some classical works splits the latter into two: Natural resources and Energy resources (E). More often outputs from other production processes are used as inputs others. But in principle it should be possible to separate all inputs down to the three or four basic types of input.
Catch production in the short runEdit
The simplest model of catch production involves only two input factor: A natural resource (x) and a fishing activity (fishing effort, F).
Production of fishing effortEdit
Let us start to discuss the production of a certain quantity of fishing effort. Further let us assume that human labour input (which could be regarded as a natural resource, but is more practically described as labour) and Capital (boats, fishing gears, etc.) is the two basic types of input in the production of fishing effort. Also assume one of the two factors could be perfectly substituted by the other in a certain quantity.
{\displaystyle F=F(L,K)\,\!}
While limiting the input factor to two, the substitution rates can easily be viewed in a contour plot, often referred to as isocurves of production (Fig. 3.1).
Isocurves of fishing effort production.
The curves show at which rate L (Labour) can be substituted by K (Capital) and the other way around, each curve representing a constant output of F. The blue point is representing a certain production of fishing effort involving a high degree of labour inputs, while much of the labour in the red point is substituted by capital, e.g. small boats and hand line (blue point) vs. a few trawlers (red point). The dotted arrow indicates increase in effort production.
The concept of efficient productionEdit
Equation 3.1 is simply giving a technical description on how F is produced by the input factor L and K and does not give us any suggestions on which mix of input factors are to be preferred. In order to prioritize between different alternatives of producing a specific quantity of F, it is convenient to look at the cost of the two input factors. Let the unit cost of labour (L) be w and the unit cost of capital (K) be r. The total cost of production (C) then is:
{\displaystyle C=wL+rK\,\!}
Lagrange's method can be used in order to maximise the production of fishing effort at a cost constraint, which is the dual problem of minimising the cost at a given production. The Lagrange equation will be:
{\displaystyle {\mathcal {L}}=F(L,K)-\lambda ({C_{0}}-wL-rK)}
where C0 is the given cost and
{\displaystyle \lambda }
The first order condition when maximising the Lagrange equation, is that the partial derivatives of the equation with respect of L and K equals 0, from which follows:
{\displaystyle {\frac {\frac {\partial F}{\partial L}}{\frac {\partial F}{\partial K}}}={\frac {w}{r}}}
This expression is referred to as the marginal rate of technical substitution (MRTS). In the most cost-efficient production MRTS should according to Eq. (3.4) equal the price ratio of the two input factors. Cost efficient solutions of different levels of production are shown in Figure 3.2.
The thick line connects the infinite number of points consistent with equation (3.4), when varying the cost constraint C0
Producing fish harvest by two input factorsEdit
By regarding production of fishing effort as an independent production process, production of (h, fish harvest) can be expressed by the two input factors x and F:
{\displaystyle h=h(x,F)\,\!}
It is reasonable to assume that x and F is substitutable in the same way as L and K in Figure 3.1. In order to fish a certain quantity, say one kilo fish, when the stock biomass is low (low x-value), one has to input a larger fishing effort than in the case of a higher stock density (large x-value).
Eq. (3.5) therefore is of the same type as Eq. (3.1) and in principle we have the same type of continuous substitution as indicated in Figure 3.1.
Market failures while using common resourcesEdit
There are however two core issues which turns this production into a drastically different task:
Identifying efficient production levels involves however prices on the input factors from a perfect market. In this case we certainly can calculate a perfect market price on fishing effort (F) based on the paragraphs above. But what is the price of the other input factor, x?
If the resource is regarded as a common property, the price will be zero! In that case there is no limits on how much F (which has a positive price), we want to substitute with the free input factor x, until the resource is fully produced and converted into fish harvest.
The availability (and accessibility) of x is in the short run given. In the long run it will also be given, now as a function of catch production in the past. The two input factors in other words are interrelated to each other! By that traditional methods in production theory break down.
The first problem makes it impossible to continue along the normal methods of identifying cost efficient solutions. Since the scarcity of the natural resource is not reflected in a price, we lack the value information on this factor. The other problem is also corrupting our model, as it attacks the basic assumption of independency in availabilities of the two input factors. In the long run in fact we have:
{\displaystyle x=x(F)\,\!}
The straight forward implication of this is obvious from Eq. (3.5):
{\displaystyle h(x,F)=h(x(F),F)=h(F)\,\!}
showing that in the long run (keeping fishing effort constant over a sufficient period of time) catch will be determined by the fishing effort alone.
The crucial relationship to investigate further therefore is the x-F relationship. How the stock biomass x be defined as a function of fishing effort F? At this point we have to turn to biology and population dynamics.
Population dynamics is the study of marginal and long term changes in numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influence those changes. In idealized population growth models one differs between compensatory growth and decompensatory growth, the first one is regarded as normal growth.
The logistic growth model is a widely-used compensatory growth model.
Logistic growthEdit
Let us assume annual biomass growth of a fish population to follow logistic growth (first proposed as a demographic model by Verhulst, 1838. (Applied as a biomass growth model by Pearl, 1934.) The population dynamics is described by a differential equation where biomass (x) is a function of time (t) and the time derivative of population biomass is:
{\displaystyle {\dot {x}}(t)=r\cdot x(t)\left(1-{\frac {x(t)}{K}}\right)}
Note that the two parameters (constants), r often referred to as the intrinsic growth rate and K the population biomass at natural equilibrium, are not the same as the parameter r and the variable K above (in pnt. 3).
The parabolic function (square function) in Eq. 4.1, shown in Figure 4.1, describe and increasing biomass growth as the population biomass increases, up to a certain population size (which is easy to identify as K/2), where the biomass growth starts declining to reach zero at biomass level K. K therefore represents a natural equilibrium biomass of an unexploited stock.
Equation 4.1, where the time derivative of x is measured along the y-axis
Biomass growth as a function of timeEdit
The differential equation (4.1) has a unique solution ( = INTEGRAL):
{\displaystyle x(t)={\frac {e^{r\,t}\,K\,{x_{0}}}{K+\left(e^{r\,t}-1\right)\,{x_{0}}}}}
x0 being the biomass at t=0.
Catch production in the long runEdit
Let us start with catch production in the short run as discussed above. Eq. (3.5) defines catch as an output from of production process where stock biomass, x, and fishing effort, F, are input factors. x can be substituted by F. This assumption of substitution is taken care of in the so-called Schaefer production equation:
{\displaystyle h(x,F)=q\cdot x\cdot F}
q is a constant (parameter) often referred to as the catchability coefficient. By referring to Eq. (2.3) we see that the Schaeffer production equation is of the Cobb-Douglas type, with powers set equal to 1. Later we will investigate the consequences of the choice of power values.
The biomass growth equation (4.1) now has to be adjusted to include catch. The annual biomass growth will be the natural growth (right hand side of 4.1) minus the harvested biomass (e.g. 5.1):
{\displaystyle {\dot {x}}=r\cdot x\left(1-{\frac {x}{K}}\right)-q\cdot x\cdot F}
As Eq. (4.1) identify K as the natural biomass equilibrium when
{\displaystyle t\rightarrow {\infty }}
, Eq. (5.2) also identify an equilibrium biomass when keeping F constant over an infinite number of years. The equilibrium is defined by
{\displaystyle {\dot {x}}=0}
from which follows (in the case of Eq. 5.2):
{\displaystyle r\cdot x\left(1-{\frac {x}{K}}\right)=q\cdot x\cdot F}
Skipping the trivial solution x=0, the stock biomass - fishing effort relationship is given directly from Eq. (5.4):
{\displaystyle x(F)=K\left(1-{\frac {q}{r}}F\right)}
The long term catch equation is finally found by inserting Eq. (5.5) into the short term catch equation defined by Eq. (5.1):
{\displaystyle h(F)=q\cdot F\cdot K\left(1-{\frac {q}{r}}F\right)}
Catch Revenue and CostEdit
{\displaystyle TR(F)=p\cdot h(F)}
{\displaystyle TC(F)=c\cdot F}
The Concept of Resource RentEdit
Open access equilibriumEdit
Maximum sustainable economic yieldEdit
Investing in stock by not fishingEdit
Social optimum solutionEdit
Retrieved from "https://en.wikibooks.org/w/index.php?title=Bioeconomics&oldid=3440415"
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Exact lower-tail large deviations of the KPZ equation
2022 Exact lower-tail large deviations of the KPZ equation
Li-Cheng Tsai1
1Department of Mathematics, Rutgers University, New Brunswick, New Jersey, USA
Consider the Hopf–Cole solution
h\left(t,x\right)
of the Kandar–Parisi–Zhang (KPZ) equation with the narrow wedge initial condition. Regarding
t\to \mathrm{\infty }
as a scaling parameter, we provide the first rigorous proof of the large deviation principle (LDP) for the lower tail of
h\left(2t,0\right)+\frac{t}{12}
, with speed
{t}^{2}
and an explicit rate function
{\mathrm{\Phi }}_{-}\left(z\right)
. This result confirms existing physics predictions made by Corwin (2011); Sasorov, Meerson, and Prolhac (2017); and Krajenbrink, Le Doussal, and Prolhac (2018). Our analysis utilizes a formula from Borodin and Gorin (2016) to convert the LDP for the KPZ equation to calculating an exponential moment of the Airy point process (PP). To estimate this exponential moment, we invoke the stochastic Airy operator (SAO) and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.
Li-Cheng Tsai. "Exact lower-tail large deviations of the KPZ equation." Duke Math. J. Advance Publication 1 - 44, 2022. https://doi.org/10.1215/00127094-2022-0008
Keywords: Airy point process , Kardar–Parisi–Zhang equation , large deviations , Random operators , Stochastic Airy operator
Li-Cheng Tsai "Exact lower-tail large deviations of the KPZ equation," Duke Mathematical Journal, Duke Math. J. Advance Publication, 1-44, (2022)
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Golden Ratio / Silver Ratio calculator - Hirota Yano
Golden Ratio / Silver Ratio calculator
Round off to the fourth decimal place.
Golden ratio (1:1.618)
Silver ratio (1:1.414)
1:{\dfrac {1+{\sqrt {5}}}{2}}
The approximate value is 1: 1.618, about 5: 8 or 8:13.
It seems Anyway beautiful ratio. It has been used for buildings and works of art since ancient times. Approximate values often appear in nature.
Artificial objects: Parthenon, Mona Lisa, Thirty-six Views of Mt. Fuji, Apple’s logo, etc.
Natural world: Nautilus shell spiral, sunflower seed spiral, typhoon and nebula vortex, etc.
The shape of this tool is a rectangle with a golden ratio of sides.
φ={\dfrac {1+{\sqrt {5}}}{2}}
φ (phi) is the positive solution of
x^{2}-x-1=0
and is called the golden number.
There are the following two silver ratios.
{\textstyle 1:1+{\sqrt {2}}}
- This is one of the precious metal ratios.
{\textstyle 1:{\sqrt {2}}}
- This is used for paper dimensions and so on.
This tool deals with the latter
{\textstyle 1:{\sqrt {2}}}
. For some reason, it is said that the Japanese prefer the silver ratio to the golden ratio, and it has been used in buildings for a long time.
e.g. Horyu-ji and Five-storied Pagoda, Choju-Giga, Tokyo Sky Tree, Doraemon, etc.
The shape of the tool is a rectangle of silver ratio. All rectangles have a silver ratio. This is the standard for paper such as A size and B size.
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Human capital and national power - Nitin, you can't be serious!
June 18, 2021 ☼ economics ☼ geopolitics
Human capital and national power
The biggest change to India's national power in the next decade will come from changes in human capital
I am part of a group of people working on a blueprint for India’s foreign policy for the coming decade. In a discussion this morning, I argued that the biggest change to India’s national power in this timeframe will come from changes in human capital. Even before the Covid-19 pandemic, India — as Puja Mehra argues — had inflicted a Lost Decade upon itself. While the economy had generate the 20 million new jobs per year, it was doing a lot less even if you accept the most optimistic figure of 5 million jobs somewhere between 2017-2019. The difference between requirement and achievement being in the order of magnitude, the politically-loaded debate on whether it was 1 million or 5 million does not matter much.
See Ajit Dasgupta’s A History of Indian Economic Thought, Chapter 7
The pandemic has exacerbated the problem and accelerated the timeline of its consequences. We do not have reliable estimates of its real impact, but we do know that it has caused widespread damage and debilitating setbacks to people’s life chances. As Mahadeo Govind Ranade wrote about poverty over a century ago
“We need only walk through our streets, and study the most superficial aspects of our economic situation and the fact forces itself upon us…”. It is unclear how people can rebound from the blow they have received in 2019-20.
Michael Beckley suggests a simple but very meaningful measure for net national power:
<span class="small-caps">GDP</span> <span class="small-caps">GDP</span>
. Thus national power increases when the economy is larger and when the people are richer. If population grows faster than the economy, then national power might even shrink. Given the massive lead China already enjoys over us, it is clear that India’s power will decline in relative terms with respect to its adversarial neighbour — unless we rapidly raise per capita incomes.
For that we need massive and the right kind of investments in human capital. The need to overhaul healthcare, education and social security have been obvious for a long time. Yet the first two hardly register in electoral politics — voters do not demand them and politicians do not seriously promise them. Social security does figure in political discourse, but as a form of redistribution or gratuity for misfortune.
Employment is certainly politically salient, but the general metaphor is still one of “government giving jobs”, not the economy creating it. That government job model is not sustainable: it’s bad for governance and it’s bad for jobs. The growth-led employment model that captured the national imagination in the 1990s and 2000s, and is still prevalent the South, needs to be brought back into political currency.
These are links to my Business Standard and Mint columns that elaborate on this argument.
I have long argued that India’s foreign policy is 8% economic growth. That will only come if India gets serious about its human capital. If not, we are staring at a relative decline in national power at a time when we cannot afford it.
The banal geopolitical fallout of the laboratory leak hypothesis Next
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Ask Answer - Mensuration - Expert Answered Questions for School Students
Choose any five wildlife sanctuaries out of top 25 sanctuaries of India and find the ratio of the areas of wildlife sanctuaries to the area of the respective states.
6. Choose any five wildlife sanctuaries out of 25 sanctuaries of India and find the ratio of the areas of wildlife sanctuaries to the area of the respective states.
Please answer. Need it for an exam
51. In given figure, if AX = 5 cm, XD= 7 cm, CX= 10 cm, find BX
(A) 3 cm (B) 3.4 cm
(C) 4 cm (D)4.5 cm
Diya Mirani
Can you give me all the formulas of trapezium,rhombus,cylinder, circle, rectangle, square, triangle, total surface area and lateral surface area
The diameter of a roller 120cm long is 48cm. If it takes 350 complete revolutions to level a playground, determine the \mathrm{cos}t of levelling it at the rate of Rs. 15 per {m}^{2}.
28. Samir bought the following articles from a departmental store.
What is the amount Samir has to pay for the departmental store?
30. A train is moving at a uniform speed of 75 km/h.
i) How far will the train travel in 20 minutes.
ii) Find the time required to cover the distance of 250 km.
29. An electric pole 14m high, casts a shadow of 10m. Find the height of tree that casts of
i) 15m
ii) 25m
a. Vol. of cube i\right) 2\left(lb+bh+hl\right) \phantom{\rule{0ex}{0ex}}b. SA of cyclinder ii\right) {\mathrm{\pi r}}^{2}\mathrm{h}\phantom{\rule{0ex}{0ex}}\mathrm{c}. \mathrm{SA} \mathrm{of} \mathrm{cuboid} \mathrm{iii}\right) \mathrm{l}3\phantom{\rule{0ex}{0ex}}\mathrm{d}. \mathrm{Vol}. \mathrm{of} \mathrm{cuboid} \mathrm{iv}\right) \mathrm{lbh}\phantom{\rule{0ex}{0ex}}\mathrm{e}. \mathrm{SA} \mathrm{of} \mathrm{cube} \mathrm{v}\right) 2\mathrm{\pi rC}\phantom{\rule{0ex}{0ex}}\mathrm{f}. \mathrm{Vol}. \mathrm{of} \mathrm{cylinder}.
Find the area of a rhombus whose side is 5cm and whose altitude is 4.8cm. if one of its diagonals is 8cm long, find the length of other diagonal.
a cuboidal tank is 8m long, 6m wide and 2.5 m deep. How many litres of water it can hold?
1cm^3 is how much mm^3 ?
Alex Lohani
Kindly answer the question mention in the image
Q.14. A field is in the shape as shown in the figure. Find the length of the wire required to fence it.
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EUDML | On an inequality of Kolmogorov type for a second-order difference expression. EuDML | On an inequality of Kolmogorov type for a second-order difference expression.
On an inequality of Kolmogorov type for a second-order difference expression.
Delil, A.; Evans, W.D.
Delil, A., and Evans, W.D.. "On an inequality of Kolmogorov type for a second-order difference expression.." Journal of Inequalities and Applications [electronic only] 3.2 (1999): 183-214. <http://eudml.org/doc/120375>.
@article{Delil1999,
author = {Delil, A., Evans, W.D.},
keywords = {difference operators; Kolmogorov type inequalities; Hellinger-Nevanlinna -function; Cauchy transform of the orthogonality measure; Hellinger-Nevanlinna -function},
title = {On an inequality of Kolmogorov type for a second-order difference expression.},
AU - Delil, A.
TI - On an inequality of Kolmogorov type for a second-order difference expression.
KW - difference operators; Kolmogorov type inequalities; Hellinger-Nevanlinna -function; Cauchy transform of the orthogonality measure; Hellinger-Nevanlinna -function
difference operators, Kolmogorov type inequalities, Hellinger-Nevanlinna
m
-function, Cauchy transform of the orthogonality measure, Hellinger-Nevanlinna
m
Articles by Delil
|
EUDML | Singular -harmonic functions and related quasilinear equations on manifolds. EuDML | Singular -harmonic functions and related quasilinear equations on manifolds.
p
-harmonic functions and related quasilinear equations on manifolds.
Véron, Laurent. "Singular -harmonic functions and related quasilinear equations on manifolds.." Electronic Journal of Differential Equations (EJDE) [electronic only] 2002 (2002): 133-155. <http://eudml.org/doc/127602>.
@article{Véron2002,
author = {Véron, Laurent},
keywords = {-harmonic; singularity; degenerate equations; -harmonic},
title = {Singular -harmonic functions and related quasilinear equations on manifolds.},
AU - Véron, Laurent
TI - Singular -harmonic functions and related quasilinear equations on manifolds.
KW - -harmonic; singularity; degenerate equations; -harmonic
p
-harmonic, singularity, degenerate equations,
p
-harmonic
Elliptic equations on manifolds, general theory
Articles by Véron
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A ball of mass m₁ = 50g is moving with velocity 5m/s and another ball of mass 100g is moving - Science - Gravitation - 16911727 | Meritnation.com
A ball of mass m₁ = 50g is moving with velocity 5m/s and another ball of mass 100g is moving in opposite direction with 8 m/s velocity. If they stick together and moves with common velocity find it ? *
a) 5/11 m/s
c) 11/5 m/sec
Solution\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Applying conservation of momentum formuls we have\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Initial momentum of the system = Final momentum of the system\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{50}{1000}kg×5+\frac{100}{1000}kg×\left(-8\right)=\left(\frac{50}{1000}+\frac{100}{1000}\right)×v\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}0.05×5+0.1×-8=0.15×v\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}0.15v=0.25-0.8\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}v=\frac{-0.55}{0.15}=-3.66m/s \left(common velocity\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Negative sign shows that velocity will be in the direction of initial direction of 100g ball.
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Motion Planning in Urban Environments Using Dynamic Occupancy Grid Map - MATLAB & Simulink - MathWorks España
In this example, you represent the surrounding environment as a dynamic occupancy grid map. For an example using the discrete set of objects, refer to the Highway Trajectory Planning Using Frenet Reference Path example. A dynamic occupancy grid map is a grid-based estimate of the local environment around the ego vehicle. In addition to estimating the probability of occupancy, the dynamic occupancy grid also estimates the kinematic attributes of each cell, such as velocity, turn-rate, and acceleration. Further, the estimates from the dynamic grid can be predicted for a short-time in the future to assess the occupancy of the local environment in the near future. In this example, you obtain the grid-based estimate of the environment by fusing point clouds from six lidars mounted on the ego vehicle.
The scenario used in this example represents an urban intersection scene and contains a variety of objects, including pedestrians, bicyclists, cars, and trucks. The ego vehicle is equipped with six homogenous lidar sensors, each with a field of view of 90 degrees, providing 360-degree coverage around the ego vehicle. For more details on the scenario and sensor models, refer to the Grid-Based Tracking in Urban Environments Using Multiple Lidars (Sensor Fusion and Tracking Toolbox) example. The definition of scenario and sensors is wrapped in the helper function helperGridBasedPlanningScenario.
Now, define a grid-based tracker using the trackerGridRFS (Sensor Fusion and Tracking Toolbox) System object™. The tracker outputs both object-level and grid-level estimate of the environment. The grid-level estimate describes the occupancy and state of the local environment and can be obtained as the fourth output from the tracker. For more details on how to set up a grid-based tracker, refer to the Grid-Based Tracking in Urban Environments Using Multiple Lidars (Sensor Fusion and Tracking Toolbox) example.
Define the global reference path using the referencePathFrenet object by providing the waypoints in the Cartesian coordinate frame of the driving scenario. The reference path used in this example defines a path that turns right at the intersection.
At each step of the simulation, the planning algorithm generates a list of sample trajectories that the ego vehicle can choose. The local trajectories are sampled by connecting the current state of the ego vehicle to desired terminal states. Use the trajectoryGeneratorFrenet object to connect current and terminal states for generating local trajectories. Define the object by providing the reference path and the desired resolution in time for the trajectory. The object connects initial and final states in Frenet coordinates using fifth-order polynomials.
The strategy for sampling terminal states in Frenet coordinates often depends on the road network and the desired behavior of the ego vehicle during different phases of the global path. For more detailed examples of using different ego behavior, such as cruise-control and car-following, refer to the "Planning Adaptive Routes Through Traffic" section of the Highway Trajectory Planning Using Frenet Reference Path example. In this example, you sample the terminal states using two different strategies, depending on the location of vehicle on the reference path, shown as blue and green regions in the following figure.
\Delta \mathit{T}
{\mathit{x}}_{\mathrm{Ego}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}\left(\Delta \mathit{T}\right)=\left[\mathrm{NaN}\text{\hspace{0.17em}}\stackrel{˙}{\mathit{s}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\mathit{d}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}0\right];\text{\hspace{0.17em}}
\left\{\Delta \mathit{T}\in \left\{\mathrm{linspace}\left(2,4,6\right)\right\},\text{\hspace{0.17em}}\stackrel{˙}{\mathit{s}}\in \left\{\mathrm{linspace}\left(0,{\stackrel{˙}{\mathit{s}}}_{\mathrm{max},}10\right)\right\},\text{\hspace{0.17em}}\mathit{d}\in \left\{0\text{\hspace{0.17em}}{\mathit{w}}_{\mathrm{lane}\text{\hspace{0.17em}}}\right\}\right\}
{\stackrel{˙}{\mathit{s}}}_{\mathrm{max}}
{\mathit{d}}_{\mathrm{des}}
{\mathit{x}}_{\mathrm{Ego}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}\left(\Delta \mathit{T}\right)=\left[{\mathit{s}}_{\mathrm{stop}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}0\text{\hspace{0.17em}}0\text{\hspace{0.17em}}0\text{\hspace{0.17em}}0\right]
\Delta \mathit{T}
{\mathit{s}}_{\mathrm{stop}}
Further, you set up a collision-validator to assess if the ego vehicle can maneuver on a kinematically feasible trajectory without colliding with any other obstacles in the environment. To define the validator, use the helper class HelperDynamicMapValidator. This class uses the predictMapToTime (Sensor Fusion and Tracking Toolbox) function of the trackerGridRFS object to get short-term predictions of the occupancy of the surrounding environment. Since the uncertainty in the estimate increases with time, configure the validator with a maximum time horizon of 2 seconds.
\mathit{C}={\mathit{J}}_{\mathit{s}}+{\mathit{J}}_{\mathit{d}}+1000{\mathit{P}}_{\mathit{c}}+100{\left({\stackrel{˙}{\mathit{s}}}_{\left(\Delta \mathit{T}\right)}-{\stackrel{˙}{\mathit{s}}}_{\mathrm{Limit}}\right)}^{2}
{\mathit{J}}_{\mathit{s}}
{\mathit{J}}_{\mathit{d}}
{\mathit{P}}_{\mathit{c}}
\Delta \mathit{T}
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(N, C_{\text{in}}, L)
(N, C_{\text{out}}, L_{\text{out}})
\text{out}(N_i, C_{\text{out}_j}) = \text{bias}(C_{\text{out}_j}) + \sum_{k = 0}^{C_{in} - 1} \text{weight}(C_{\text{out}_j}, k) \star \text{input}(N_i, k)
\star
is the valid cross-correlation operator,
N
C
L
is a length of signal sequence.
\frac{\text{out\_channels}}{\text{in\_channels}}
(N, C_{in}, L_{in})
(C_\text{in}=C_\text{in}, C_\text{out}=C_\text{in} \times \text{K}, ..., \text{groups}=C_\text{in})
(N, C_{in}, L_{in})
(C_{in}, L_{in})
(N, C_{out}, L_{out})
(C_{out}, L_{out})
L_{out} = \left\lfloor\frac{L_{in} + 2 \times \text{padding} - \text{dilation} \times (\text{kernel\_size} - 1) - 1}{\text{stride}} + 1\right\rfloor
(\text{out\_channels}, \frac{\text{in\_channels}}{\text{groups}}, \text{kernel\_size})
\mathcal{U}(-\sqrt{k}, \sqrt{k})
k = \frac{groups}{C_\text{in} * \text{kernel\_size}}
\mathcal{U}(-\sqrt{k}, \sqrt{k})
k = \frac{groups}{C_\text{in} * \text{kernel\_size}}
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SGD — PyTorch 1.11.0 documentation
class torch.optim.SGD(params, lr=<required parameter>, momentum=0, dampening=0, weight_decay=0, nesterov=False, *, maximize=False)[source]¶
\begin{aligned} &\rule{110mm}{0.4pt} \\ &\textbf{input} : \gamma \text{ (lr)}, \: \theta_0 \text{ (params)}, \: f(\theta) \text{ (objective)}, \: \lambda \text{ (weight decay)}, \\ &\hspace{13mm} \:\mu \text{ (momentum)}, \:\tau \text{ (dampening)}, \:\textit{ nesterov,}\:\textit{ maximize} \\[-1.ex] &\rule{110mm}{0.4pt} \\ &\textbf{for} \: t=1 \: \textbf{to} \: \ldots \: \textbf{do} \\ &\hspace{5mm}g_t \leftarrow \nabla_{\theta} f_t (\theta_{t-1}) \\ &\hspace{5mm}\textbf{if} \: \lambda \neq 0 \\ &\hspace{10mm} g_t \leftarrow g_t + \lambda \theta_{t-1} \\ &\hspace{5mm}\textbf{if} \: \mu \neq 0 \\ &\hspace{10mm}\textbf{if} \: t > 1 \\ &\hspace{15mm} \textbf{b}_t \leftarrow \mu \textbf{b}_{t-1} + (1-\tau) g_t \\ &\hspace{10mm}\textbf{else} \\ &\hspace{15mm} \textbf{b}_t \leftarrow g_t \\ &\hspace{10mm}\textbf{if} \: \textit{nesterov} \\ &\hspace{15mm} g_t \leftarrow g_{t-1} + \mu \textbf{b}_t \\ &\hspace{10mm}\textbf{else} \\[-1.ex] &\hspace{15mm} g_t \leftarrow \textbf{b}_t \\ &\hspace{5mm}\textbf{if} \: \textit{maximize} \\ &\hspace{10mm}\theta_t \leftarrow \theta_{t-1} + \gamma g_t \\[-1.ex] &\hspace{5mm}\textbf{else} \\[-1.ex] &\hspace{10mm}\theta_t \leftarrow \theta_{t-1} - \gamma g_t \\[-1.ex] &\rule{110mm}{0.4pt} \\[-1.ex] &\bf{return} \: \theta_t \\[-1.ex] &\rule{110mm}{0.4pt} \\[-1.ex] \end{aligned}
Nesterov momentum is based on the formula from On the importance of initialization and momentum in deep learning.
dampening (float, optional) – dampening for momentum (default: 0)
nesterov (bool, optional) – enables Nesterov momentum (default: False)
The implementation of SGD with Momentum/Nesterov subtly differs from Sutskever et. al. and implementations in some other frameworks.
\begin{aligned} v_{t+1} & = \mu * v_{t} + g_{t+1}, \\ p_{t+1} & = p_{t} - \text{lr} * v_{t+1}, \end{aligned}
p
g
v
\mu
denote the parameters, gradient, velocity, and momentum respectively.
This is in contrast to Sutskever et. al. and other frameworks which employ an update of the form
\begin{aligned} v_{t+1} & = \mu * v_{t} + \text{lr} * g_{t+1}, \\ p_{t+1} & = p_{t} - v_{t+1}. \end{aligned}
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Mercury_battery Knowpia
A mercury battery (also called mercuric oxide battery, mercury cell, button cell, or Ruben-Mallory[1]) is a non-rechargeable electrochemical battery, a primary cell. Mercury batteries use a reaction between mercuric oxide and zinc electrodes in an alkaline electrolyte. The voltage during discharge remains practically constant at 1.35 volts, and the capacity is much greater than that of a similarly sized zinc-carbon battery. Mercury batteries were used in the shape of button cells for watches, hearing aids, cameras and calculators, and in larger forms for other applications.
Mercury battery "РЦ-53М"(RTs-53M), Russian manufactured in 1989
For a time during and after World War II, batteries made with mercury became a popular power source for portable electronic devices. Due to the content of toxic mercury and environmental concerns about its disposal, the sale of mercury batteries is now banned in many countries. Both ANSI and IEC have withdrawn their standards for mercury batteries.
Cross section through a button-type mercury battery.
The mercury oxide-zinc battery system was known since the 19th century,[2] but did not become widely used until 1942, when Samuel Ruben developed a balanced mercury cell which was useful for military applications such as metal detectors, munitions, and walkie-talkies.[3][1] The battery system had the advantages of long shelf life (to 10 years) and steady voltage output. After the Second World War the battery system was widely applied for small electronic devices such as cardiac pacemakers and hearing aids. Mercury oxide batteries were made in a range of sizes from miniature button cells used for hearing aids and electric wrist watches, cylindrical types used for portable electronic apparatus, rectangular batteries used for transistor radios,[4] and large multicell packs used for industrial applications such as radio remote control for overhead crane systems. In the United States, mercury oxide batteries were manufactured by companies including P. R. Mallory and Co Inc, (now Duracell), Union Carbide Corporation (whose former battery division is now called Energizer Holdings), RCA Corporation, and Burgess Battery Company.
Mercury batteries use either pure mercury(II) oxide (HgO)—also called mercuric oxide—or a mixture of HgO with manganese dioxide (MnO2) as the cathode. Mercuric oxide is a non-conductor, so some graphite is mixed with it; the graphite also helps prevent collection of mercury into large droplets. The half-reaction at the cathode is:[3]
{\displaystyle {\ce {HgO + H2O + 2e- -> Hg + 2OH-}}}
with a standard potential of +0.0977 V.
The anode is made of zinc (Zn) and separated from the cathode with a layer of paper or other porous material soaked with electrolyte; this is known as a salt bridge. Two half-reactions occur at the anode. The first consists of an electrochemical reaction step:[3]
{\displaystyle {\ce {Zn + 4 OH- -> Zn(OH)4^2- + 2e-}}}
followed by the chemical reaction step: Oxidation occurs at anode:[3]
{\displaystyle {\ce {Zn + 2OH- -> ZnO + H2O + 2e-}}}
{\displaystyle {\ce {Zn(OH)4^2- -> ZnO + 2OH- + H2O}}}
yielding an overall anode half-reaction of:[3]
{\displaystyle {\ce {Zn + 2OH- -> ZnO + H2O + 2e-}}}
The overall reaction for the battery is:
{\displaystyle {\ce {Zn + HgO -> ZnO + Hg}}}
In other words, during discharge, zinc is oxidized (loses electrons) to become zinc oxide (ZnO) while the mercuric oxide gets reduced (gains electrons) to form elemental mercury. A little extra mercuric oxide is put into the cell to prevent evolution of hydrogen gas at the end of life.[3]
Sodium hydroxide or potassium hydroxide are used as an electrolyte. Sodium hydroxide cells have nearly constant voltage at low discharge currents, making them ideal for hearing aids, calculators, and electronic watches. Potassium hydroxide cells, in turn, provided constant voltage at higher currents, making them suitable for applications requiring current surges, e.g. photographic cameras with flash, and watches with a backlight. Potassium hydroxide cells also have better performance at lower temperatures. Mercury cells have very long shelf life, up to 10 years.[3]
Mercuric oxide and cadmiumEdit
A different form of mercury battery uses mercuric oxide and cadmium. This has a much lower terminal voltage around 0.9 volts and so has lower energy density, but it has an extended temperature range, in special designs up to 180 C. Because cadmium has low solubility in the alkaline electrolyte, these batteries have long storage life.[3] A 12 volt battery of this type was formerly used for residential smoke detectors. It was designed as a series stack of cells, where one cell had a reduced capacity resulting in a very distinct two-step voltage discharge characteristic. When reaching the end of its life, this smaller cell would discharge first causing the battery terminal voltage to drop sharply by 0.9 volts. This provided a very predictable and repeatable way to warn users the battery needed replacement while the larger capacity cells kept the unit functioning normally.[5]
Electrical characteristicsEdit
Mercury batteries using a mercury(II) oxide cathode have a very flat discharge curve, holding constant 1.35 V (open circuit) voltage until about the last 5% of their lifetime, when their voltage drops rapidly. The voltage remains within 1% for several years at light load, and over a wide temperature range, making mercury batteries useful as a voltage reference in electronic instruments and in photographic light meters.[6]
Mercury batteries with cathodes made of a mix of mercuric oxide and manganese dioxide have output voltage of 1.4 V and a more sloped discharge curve.[3]
Product banEdit
The 1991 European commission directive 91/157, when adopted by member states, prohibited the marketing of certain types of batteries containing more than 25 milligrams of mercury, or, in the case of alkaline batteries, more than 0.025% by weight of mercury. In 1998 the ban was extended to cells containing more than 0.005% by weight of mercury.[7]
In 1992, the state of New Jersey prohibited sales of mercury batteries. In 1996, the United States Congress passed the Mercury-Containing and Rechargeable Battery Management Act that prohibited further sale of mercury-containing batteries unless manufacturers provided a reclamation facility, effectively banning their sale.[8][9]
The ban on sale of mercury oxide batteries caused numerous problems for photographers, whose equipment frequently relied on their advantageous discharge curves and long lifetime. Alternatives used are zinc-air batteries, with similar discharge curve, high capacity, but much shorter lifetime (a few months), and poor performance in dry climates; alkaline batteries with voltage widely varying through their lifetime; and silver-oxide batteries with higher voltage (1.55 V) and very flat discharge curve, which makes them possibly the best, though expensive, replacement after recalibrating the meter to the new voltage.
Special adapters with voltage dropping Schottky or germanium diodes allow silver oxide batteries to be used in equipment designed for mercury batteries. Since the voltage drop is a non-linear function of the current flow, diodes do not produce a very accurate solution for applications where the current flow varies significantly. Currents drawn by old CdS light meters are typically in the 10 μA to 200 μA range (e.g. Minolta SR-T equipment series). Various kinds of active voltage regulation circuits using SMD transistors[10] or integrated circuits[11] have been devised, however, they are often difficult to integrate into the cramped battery compartment space. Replacements must operate with minimal voltage drop on the already very low voltage produced by a single battery cell, and the lack of a power switch on many traditional light meters and cameras[11] makes an ultra-low power (ULP) or extreme-low power (XLP) design necessary. Many old devices also have their chassis connected to the battery's positive rather than its negative terminal - if this cannot be changed, the necessary negative voltage regulator design further reduces the choice of suitable electronic parts.[11]
Use in zinc batteriesEdit
Formerly, the zinc anodes of dry cells were amalgamated with mercury, to prevent side-reactions of the zinc with the electrolyte that would reduce the service life of the battery. The mercury took no part in the chemical reaction for the battery. Manufacturers have changed to a purer grade of zinc, so amalgamation is no longer required and mercury is eliminated from the dry cell.
^ a b Salkind, Alvin J.; Ruben, Samuel (1986). "Mercury Batteries for Pacemakers and Other Implantable Devices". Batteries for Implantable Biomedical Devices. Springer US. pp. 261–274. doi:10.1007/978-1-4684-9045-9_9. ISBN 978-1-4684-9047-3.
^ Clarke, Charles Leigh (1884-06-06). Galvanic battery. US Patent 298175. [1]
^ a b c d e f g h i Linden, David (2002). "Chapter 11". In Reddy, Thomas B. (ed.). Handbook Of Batteries (3 ed.). New York: McGraw-Hill. ISBN 0-07-135978-8.
^ "Engineering data - Energizer No. E146X" (PDF). Energizer. Archived (PDF) from the original on 2018-11-18. Retrieved 2019-05-11.
^ Crompton, Thomas Roy. Battery reference book. pp. 5–23.
^ Wilson, Anton (2004). "Anton Wilson's Cinema Workshop". American Cinematographer. p. 137. ISBN 0-93557826-9.
^ Hunter, Rod; Muylle, Koen J., eds. (1999). European Community Deskbook. An ELI deskbook - ELR - The environmental law reporter. Environmental Law Institute. p. 75. ISBN 0-911937-82-X.
^ Kreith, Frank; Tchobanoglous, George (2002). Handbook of solid waste management. McGraw-Hill Professional. pp. 6–34. ISBN 0-07-135623-1.
^ "IMERC Fact Sheet: Mercury Use in Batteries". Northeast Waste Management Officials' Association. January 2010. Retrieved 2013-06-20.
^ a b c Paul, Matthias R. (2005-12-12). "Minolta SR-T Batterieadapter" [Using a Bandgap voltage reference as Mercury battery replacement]. Minolta-Forum (in German). Archived from the original on 2016-10-11. Retrieved 2011-02-26.
Wikimedia Commons has media related to Mercury batteries.
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Capture/Probe - Knowledge Base
Capture enables high accuracy position latching based on an external electrical input. Typical sources of external electrical inputs are home switches and encoder index pulses.
Capture will return the position and sample counter (time in servo samples) for each capture position latch. There is usually one capture object per motor.
Position-Based: where the motor number for the input sources must be the same as the feedback motor number.
Time-Based Capture: where the motor number and feedback motor number can be different. This makes it possible to use inputs from one node to capture positions on another node.
Position-Based Capture vs. Time-Based Capture
There are two Capture types --- Position-based and Time-based. The capture methodologies for these types differ and can affect performance. The table below summarizes Position-based and Time-based methodologies and their best use.
Many drives have a proprietary serial encoder that decodes the encoder position and sends the position information to the FPGA once per sample. In these cases, time-based capture is more accurate than position-based capture.
Position-Based Capture
For Position-based Capture, the motor number for the input sources and the feedback motor number must be the same.
Position-based Capture with Quadrature Feedback
Low 10s of nanoseconds capture latency
Most accurate type of position capture
Position-based Capture with Drive Feedback
This type of setup leads to very large position capture errors
RSI strongly discourages using position capture with drive feedback
Position-based capture is very accurate for systems that have continuous position information (i.e., Digital quadrature encoders, in most cases). Any encoder that is samples less frequently than the 10s of nanoseconds accuracy that is inherent to time-based capture will suffer poor accuracy with position-based capture. Any encoder using drive mode will be sampled at the drive update rate (usually 16kHz, or 62.5 microseconds).
Sin/cos encoders are used in Drive Feedback mode.
Position-based capture with encoders in drive feedback mode results in WORSE capture accuracy than drive feedback mode with time-based capture.
Time-Based Capture
For Time-based Capture, the motor number and feedback motor number can be different. This makes it possible to use inputs from one node to capture positions on another node.
Position is linearly interpolated.
Calculated position is truncated to an integer.
It only works correctly if the speed of an axis is less than 344 million counts per second.- Calculation is very accurate even in high acceleration systems.
Time-based capture can be used across SynqNet nodes. (Trigger on one node, feedback position on a different node.)
Time-based capture is ideal for encoders in drive mode.
Time-based capture is accurate to ~40 nanoseconds in time. Latched positions are interpolated between adjacent position samples using this highly accurate time stamp. The position latch is subject to how well the encoder approximates a constant velocity trajectory between two adjacent samples.
Time-based Capture Accuracy
Because the time-based capture uses linear interpolation between two successive controller sample periods, there can be capture inaccuracies during very high acceleration or deceleration trajectories. The maximum capture position error can be calculated from:
Max Capture Position Error =
1/8 * Accel * (1 / Sample Rate)²
Suppose the axis is accelerating at 10,000,000 counts/sec² and the controller sample rate = 2kHz (default).
1 / 8 * 10,000,000 * (1 / 2000)²
= 0.31 counts
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In this study, we derive a new scale parameter φ for the CG method, for solving large scale unconstrained optimization algorithms. The new scale parameter φ satisfies the sufficient descent condition, global convergence analysis proved under Strong Wolfe line search conditions. Our numerical results show that the proposed method is effective and robust against some known algorithms.
Unconstrained Optimization, Hybrid, Conjugate Gradient
Al-Namat, F. and Al-Naemi, G. (2020) Global Convergence Property with Inexact Line Search for a New Hybrid Conjugate Gradient Method. Open Access Library Journal, 7, 1-14. doi: 10.4236/oalib.1106048.
{\mathrm{min}}_{x\in {R}^{n}}f\left(x\right)
x\in {R}^{n}
n\ge 1
f:{R}^{n}\to R
{x}_{k}
{x}_{0}\in {R}^{n}
{x}_{k+1}={x}_{k}+{\alpha }_{k}{d}_{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=0,1,2,\cdots
{\alpha }_{k}
f\left({x}_{k}+{\alpha }_{k}{d}_{k}\right)\le f\left({x}_{k}\right)+\sigma {\alpha }_{k}{d}_{k}
|g\left({x}_{k}+{\alpha }_{k}{d}_{k}\right)|\le \delta |{g}_{k}^{\text{T}}{d}_{k}|
0<\sigma <\delta <\text{1}
{\alpha }_{k}
{x}_{k}
{d}_{k}
{d}_{k}=\left\{\begin{array}{l}-{g}_{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1\\ -{g}_{k}+{\beta }_{k}{d}_{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k>1\end{array}
{g}_{k}
{\beta }_{k}
{x}_{k}
{\beta }_{k}
{\beta }_{k}
{\beta }_{k}
{\beta }_{k}
{B}_{k}^{MMWU}=\frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}
{\beta }_{k}^{RMAR}=\frac{{‖{g}_{k+1}‖}^{2}-\frac{‖{g}_{k+1}‖}{‖{d}_{k}‖}{g}_{k+1}^{\text{T}}{d}_{k}}{{‖{d}_{k}‖}^{2}}
{\beta }_{k}
{\beta }_{k}^{FG}=\left(1-{\phi }_{k}\right){\beta }_{k}^{MMWU}+{\phi }_{k}{\beta }_{k}^{RMAR}
{\phi }_{k}=0
{\beta }_{k}^{FG}={\beta }_{k}^{MMWU}
{\phi }_{k}=1
{\beta }_{k}^{FG}={\beta }_{k}^{RMAR}
{\phi }_{k}
{d}_{k}
{\phi }_{k}
{\phi }_{k}
{\beta }_{k}^{MMWU}
{\beta }_{k}^{RMAR}
{d}_{k+1}
{d}_{k+1}=-{g}_{k+1}+{\beta }_{k}^{HFG}{d}_{k}
{\beta }_{k}^{HFG}
{x}_{1},{x}_{2},{x}_{3},\cdots
{\alpha }_{k}
{\phi }_{k}
0\le {\phi }_{k}\le 1
{\phi }_{k}=0
{\beta }_{k}^{HFG}={\beta }_{k}^{MMWU}
{\phi }_{k}=1
{\beta }_{k}^{HFG}={\beta }_{k}^{RMAR}
0<{\phi }_{k}<1
{\beta }_{k}^{HFG}
{\beta }_{k}^{MMWU}
{\beta }_{k}^{RMAR}
{d}_{k+1}=\left\{\begin{array}{l}-{g}_{k+1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}k=1\\ -{g}_{k+1}+\left(1-{\phi }_{k}\right)\frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}{d}_{k}+{\phi }_{k}\frac{{‖{g}_{k+1}‖}^{2}-\frac{‖{g}_{k+1}‖}{‖{d}_{k}‖}{g}_{k+1}^{\text{T}}{d}_{k}}{{‖{d}_{k}‖}^{2}}{d}_{k},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }k>1\end{array}
{\phi }_{k}
{d}_{k+1}
{d}_{k+1}^{N}=-{\nabla }^{2}f{\left({x}_{k+1}\right)}^{-1}{g}_{k+1}
\begin{array}{l}-{\nabla }^{2}f{\left({x}_{k+1}\right)}^{-1}{g}_{k+1}\\ =-{g}_{k+1}+\left(1-{\phi }_{k}\right)\frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}{d}_{k}+{\phi }_{k}\frac{{‖{g}_{k+1}‖}^{2}-\frac{‖{g}_{k+1}‖}{‖{d}_{k}‖}{g}_{k+1}^{\text{T}}{d}_{k}}{{‖{d}_{k}‖}^{2}}{d}_{k}\end{array}
{s}_{k}^{\text{T}}{\nabla }^{2}f\left({x}_{k+1}\right)
\begin{array}{c}-{s}_{k}^{\text{T}}{g}_{k+1}=-{s}_{K}^{\text{T}}{\nabla }^{2}f\left({x}_{k+1}\right){g}_{k+1}+\left(1-{\phi }_{k}\right)\frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}{s}_{K}^{\text{T}}{\nabla }^{2}f\left({x}_{k+1}\right){d}_{k}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\phi }_{k}\frac{{‖{g}_{k+1}‖}^{2}-\frac{‖{g}_{k+1}‖}{‖{d}_{k}‖}{g}_{k+1}^{\text{T}}{d}_{k}}{{‖{d}_{k}‖}^{2}}{s}_{k}^{\text{T}}{\nabla }^{2}f\left({x}_{k+1}\right){d}_{k}\end{array}
\left({s}_{k},{y}_{k}\right)
{\nabla }^{2}f\left({x}_{k+1}\right){s}_{k}={y}_{k}.
{s}_{k}^{\text{T}}{\nabla }^{2}f\left({x}_{k+1}\right)={y}_{k}^{\text{T}}.
{\phi }_{k}^{FG}={\phi }_{k}
-{s}_{k}^{\text{T}}{g}_{k+1}=-{y}_{K}^{\text{T}}{g}_{k+1}+\frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}{y}_{k}^{\text{T}}{d}_{k}+{\phi }_{k}^{FG}\left(\frac{‖{g}_{k+1}‖\left({g}_{k+1}^{\text{T}}{d}_{k}\right)}{{‖{d}_{k}‖}^{2}}\right)\left({y}_{k}^{\text{T}}{d}_{k}\right)
{\phi }_{k}^{FG}=\frac{\left({s}_{k}^{\text{T}}{g}_{k+1}-{y}_{k}^{\text{T}}{g}_{k+1}\right)\cdot {‖{d}_{k}‖}^{3}+{‖{g}_{k+1}‖}^{2}\cdot ‖{d}_{k}‖\left({y}_{k}^{\text{T}}{d}_{k}\right)}{‖{g}_{k+1}‖\cdot \left({g}_{k+1}^{\text{T}}{d}_{k}\right)\cdot \left({y}_{k}^{\text{T}}{d}_{k}\right)}
{x}_{0}\in {R}^{n}
\in \text{\hspace{0.17em}}>0
k=0
f\left({x}_{0}\right)
{g}_{0}=-\nabla f\left({x}_{0}\right)
{d}_{0}=-{g}_{0}
‖{g}_{k}‖\le \text{\hspace{0.17em}}\in
{\alpha }_{k}
{x}_{k+1}={x}_{k}+{\alpha }_{k}{d}_{k}
{g}_{k+1}=g\left({x}_{k+1}\right)
{s}_{k}={x}_{k+1}-{x}_{k}
{y}_{k}={g}_{k+1}-{g}_{k}
{\phi }_{k}\ge 1
{\phi }_{k}=1
{\phi }_{k}\le 0
{\phi }_{k}=0
{\phi }_{k}
{\beta }_{k}^{FG}
d=-{g}_{k+1}+{\beta }_{k}^{FG}{d}_{k}
|{g}_{k}^{\text{T}}{g}_{k}|\ge 0.2{‖{g}_{k+1}‖}^{2}
{d}_{k}=-{g}_{k+1}
{d}_{k+1}=d
k=k+1
{d}_{k}
S=\left\{x\in {R}^{n},f\left({x}_{n}\right)\right\}
L>0
‖\nabla f\left(x\right)-\nabla f\left(y\right)‖\le L‖x-y‖,\text{\hspace{0.17em}}\forall x,y\in N
\gamma ,\stackrel{¯}{\gamma },\omega
\stackrel{¯}{\omega }
\stackrel{¯}{\gamma }\le \Vert {g}_{k+1}\Vert \le \gamma \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{¯}{\omega }\le \Vert {g}_{k}\Vert \le \omega ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall x\in S
\left\{{g}_{k}\right\}
\left\{{d}_{k}\right\}
{d}_{k}
{g}_{k+1}^{\text{T}}{d}_{k+1}\le -\mu {‖{g}_{k+1}‖}^{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall \mu \ge 0
\mu =1-\left({E}_{4}-{E}_{3}\right)
0<\left({E}_{4}-{E}_{3}\right)<1
{d}_{k}
k=0
{d}_{0}=-{g}_{0}
{g}_{0}^{\text{T}}{d}_{0}=-{‖{g}_{0}‖}^{2}
{d}_{k+1}=-{g}_{k+1}+{\beta }_{k}^{FG}{d}_{k},
{d}_{k+1}=-{g}_{k+1}+\left[\left(1-{\phi }_{k}\right){\beta }_{k}^{MMWU}+{\phi }_{k}{\beta }_{k}^{RMAR}\right]{d}_{k}
{d}_{k+1}=-\left({\phi }_{k}{g}_{k+1}+\left(1-{\phi }_{k}\right){g}_{k+1}\right)+\left(\left(1-{\phi }_{k}\right){\beta }_{k}^{MMWU}+{\phi }_{k}{\beta }_{k}^{RMAR}\right){d}_{k}
{d}_{k+1}={\phi }_{k}\left(-{g}_{k+1}+{\beta }_{k}^{RMAR}{d}_{k}\right)+\left(1-{\phi }_{k}\right)\left(-{g}_{k}+{\beta }_{k}^{MMWU}{d}_{k}\right)
{d}_{k+1}={\phi }_{k}{d}_{k+1}^{RMAR}+\left(1-{\phi }_{k}\right){d}_{k+1}^{MMWU}
{g}_{k+1}^{\text{T}}
{g}_{k+1}^{\text{T}}{d}_{k+1}={\phi }_{k}{g}_{k+1}^{\text{T}}{d}_{k+1}^{RMAR}+\left(1-{\phi }_{k}\right){g}_{k+1}^{\text{T}}{d}_{k}^{MMWU}
{\phi }_{k}=0
{d}_{k+1}={d}_{k+1}^{MMWU}
{g}_{k+1}^{\text{T}}{d}_{k+1}^{MMWU}=-{‖{g}_{k+1}‖}^{2}+\frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}{g}_{k+1}^{\text{T}}{d}_{k}
{g}_{k+1}^{\text{T}}{d}_{k}\le {y}_{k}^{\text{T}}{d}_{k}
{y}_{k}^{\text{T}}{d}_{k}\le {\alpha }_{k}L{‖{d}_{k}‖}^{2}
\begin{array}{c}{g}_{k+1}^{\text{T}}{d}_{k+1}^{MMWU}\le -{‖{g}_{k+1}‖}^{2}+\frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}{\alpha }_{k}L{‖{d}_{k}‖}^{2}\\ =-\left(1-{\alpha }_{k}L\right){‖{g}_{k+1}‖}^{2}\\ =-{E}_{1}{‖{g}_{k+1}‖}^{2}\end{array}
{E}_{1}=\left(1-{\alpha }_{k}L\right)>0
0<{\alpha }_{k}L<1
{d}_{k+1}^{MMWU}
{\phi }_{k}=1
{d}_{k}={d}_{k}^{RMAR}
{d}_{k+1}^{RMAR}=-{g}_{k+1}+{\beta }_{k}^{RMAR}{d}_{k}
{g}_{k+1}^{\text{T}}
{g}_{k+1}^{\text{T}}{d}_{k+1}^{RMAR}=-{‖{g}_{k+1}‖}^{2}+\frac{{‖{g}_{k+1}‖}^{2}-\frac{‖{g}_{k+1}‖}{‖{d}_{k}‖}{g}_{k+1}^{\text{T}}{d}_{k}}{{‖{d}_{k}‖}^{2}}{g}_{k+1}^{\text{T}}{d}_{k}
0\le \frac{{‖{g}_{k+1}‖}^{2}-\frac{‖{g}_{k+1}‖}{‖{d}_{k}‖}{g}_{k+1}^{\text{T}}{d}_{k}}{{‖{d}_{k}‖}^{2}}\le 2\frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}
\begin{array}{c}{g}_{k+1}^{\text{T}}{d}_{k+1}\le -{‖{g}_{k+1}‖}^{2}+2{\alpha }_{k}L{‖{g}_{k+1}‖}^{2}\\ =-\left(1-2{\alpha }_{k}L\right)\cdot {‖{g}_{k+1}‖}^{2}\\ =-{E}_{2}\cdot {‖{g}_{k+1}‖}^{2}\end{array}
{E}_{2}=\left(1-2{\alpha }_{k}L\right)>0
0<2{\alpha }_{k}L<1
0<L<\frac{1}{2}
{d}_{k+1}^{RMAR}
0<{\phi }_{k}<1
\begin{array}{l}\left(1-{\phi }_{k}\right){\beta }_{K}^{MMWU}{g}_{k+1}^{\text{T}}{d}_{k}\\ =\frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}{g}_{k}^{\text{T}}{d}_{k}-\left[\frac{{s}_{k}^{\text{T}}{g}_{k+1}{‖{d}_{k}‖}^{3}-{y}_{k}^{\text{T}}{g}_{k+1}{‖{d}_{k}‖}^{3}+{‖{g}_{k+1}‖}^{2}‖{d}_{k}‖{y}_{k}^{\text{T}}{d}_{k}}{‖{g}_{k+1}‖\left({g}_{k+1}^{\text{T}}{d}_{k}\right){y}_{k}^{\text{T}}{d}_{k}}\right]\ast \frac{{‖{g}_{k+1}‖}^{2}}{{‖{d}_{k}‖}^{2}}{g}_{k+1}^{\text{T}}{d}_{k}\end{array}
{g}_{k+1}^{\text{T}}{d}_{k}<{y}_{k}^{\text{T}}{d}_{k}
-\left(1-\sigma \right)‖{g}_{k}‖\le {y}_{k}^{\text{T}}{d}_{k}\le {\alpha }_{k}L{‖{d}_{k}‖}^{2}
\begin{array}{l}\left(1-{\phi }_{k}\right){\beta }_{k}^{MMWU}{g}_{k+1}^{\text{T}}{d}_{k}\\ \le \left[\frac{{\alpha }_{k}L{\Vert {d}_{k}\Vert }^{2}}{{\Vert {d}_{k}\Vert }^{2}}-\frac{\Vert {s}_{k}\Vert \Vert {g}_{k+1}\Vert \Vert {d}_{k}\Vert -{\alpha }_{k}L{\Vert {d}_{k}\Vert }^{2}+{\Vert {g}_{k+1}\Vert }^{2}{\alpha }_{k}L\Vert {d}_{k}\Vert }{\Vert {g}_{k+1}\Vert \cdot \left(-\left(1-\sigma \right)\Vert {g}_{k}\Vert \right)}\right]{\Vert {g}_{k+1}\Vert }^{2}\\ \le \left[{\alpha }_{k}L+\frac{L}{\left(1-\sigma \right)}\frac{{\Vert {s}_{k}\Vert }^{2}\Vert {d}_{k}\Vert -{\alpha }_{k}L{\Vert {d}_{k}\Vert }^{3}+{\Vert {g}_{k+1}\Vert }^{2}\Vert {d}_{k}\Vert }{\Vert {g}_{k+1}\Vert {\Vert {g}_{k}\Vert }^{2}}\right]{\Vert {g}_{k+1}\Vert }^{2}\\ \le \left[{\alpha }_{k}L+\frac{A\gamma B-{\alpha }_{k}L{B}^{2}+{Y}^{2}{\alpha }_{k}LB}{\left(1-\sigma \right)\stackrel{¯}{Y}{\stackrel{¯}{W}}^{2}}\right]{\Vert {g}_{k+1}\Vert }^{2}\end{array}
{E}_{1}={\alpha }_{k}L+\frac{LAB-{\alpha }_{k}L{B}^{3}+{\gamma }^{2}{\alpha }_{k}LB}{\left(1-\sigma \right)\stackrel{¯}{\gamma }\text{\hspace{0.05em}}{\stackrel{¯}{\omega }}^{2}}
\therefore \text{\hspace{0.17em}}\left(1-{\phi }_{k}\right){\beta }^{MMWU}{g}_{k+1}^{\text{T}}{d}_{k}\le {E}_{3}{‖{g}_{k+1}‖}^{2}
\begin{array}{l}{\phi }_{k}{\beta }_{k}^{RMAR}{g}_{k+1}^{\text{T}}{d}_{k}\\ =\left[\frac{{s}_{k}^{\text{T}}{g}_{k+1}{‖{d}_{k}‖}^{3}-{y}_{k}^{\text{T}}{g}_{k+1}{‖{d}_{k}‖}^{3}+{‖{g}_{k+1}‖}^{2}‖{d}_{k}‖{y}_{k}^{\text{T}}‖{d}_{k}‖}{‖{g}_{k+1}‖\left({g}_{k+1}^{\text{T}}{d}_{k}\right)\left({y}_{k}^{\text{T}}{d}_{k}\right)}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\cdot \left[\frac{{‖{g}_{k+1}‖}^{2}-\frac{‖{g}_{k+1}‖}{‖{d}_{k}‖}{g}_{k+1}^{\text{T}}{d}_{k}}{{‖{d}_{k}‖}^{2}}\right]{g}_{k+1}^{\text{T}}{d}_{k}\end{array}
{s}_{k}^{\text{T}}{g}_{k+1}\le {y}_{k}^{\text{T}}{s}_{k}\le L{‖{s}_{k}‖}^{2}
{s}_{k}={\alpha }_{k}{d}_{k}
{\phi }_{k}{\beta }_{k}^{RMAR}{g}_{k+1}^{\text{T}}{d}_{k}=2\left[\frac{L{‖{s}_{k}‖}^{2}‖{d}_{k}‖-‖{y}_{k}‖‖{g}_{k+1}‖{‖{d}_{k}‖}^{3}+{\alpha }_{k}L{‖{g}_{k+1}‖}^{2}{‖{d}_{k}‖}^{2}}{{‖{g}_{k+1}‖}^{2}\left(-\left(1-\sigma \right)\right){‖{g}_{k}‖}^{2}{‖{d}_{k}‖}^{2}}\right]\cdot {‖{g}_{k+1}‖}^{2}
‖{y}_{k}‖\le ‖{g}_{k+1}‖+‖{g}_{k}‖
\begin{array}{l}{\phi }_{k}{\beta }_{k}^{RMAR}{g}_{k+1}^{\text{T}}{d}_{k}\\ \le \frac{-2}{1-\sigma }\left[\frac{L{\Vert {s}_{k}\Vert }^{2}\Vert {d}_{k}\Vert -0.8{\Vert {g}_{k+1}\Vert }^{2}\Vert {d}_{k}\Vert +{\alpha }_{k}L{\Vert {g}_{k+1}\Vert }^{2}\Vert {d}_{k}\Vert }{{\Vert {g}_{k+1}\Vert }^{2}{\Vert {g}_{k}\Vert }^{2}}\right]\cdot {\Vert {g}_{k+1}\Vert }^{2}\\ \le \frac{-2B}{1-\sigma }\left[\frac{LA-0.8{\gamma }^{2}+{\alpha }_{k}L{\omega }^{2}}{\stackrel{¯}{\gamma }\text{\hspace{0.05em}}{\stackrel{¯}{\omega }}^{2}}\right]\cdot {\Vert {g}_{k+1}\Vert }^{2}\end{array}
{E}_{4}=\frac{2B}{1-\sigma }\left[\frac{LA-0.8{\gamma }^{2}+{\alpha }_{k}L{\omega }^{2}}{\stackrel{¯}{\gamma }\text{\hspace{0.05em}}{\stackrel{¯}{\omega }}^{2}}\right]
\therefore \text{\hspace{0.17em}}{\phi }_{k}{\beta }_{k}^{RMAR}{g}_{k+1}^{\text{T}}{d}_{k}\le -{E}_{4}{‖{g}_{k+1}‖}^{2}
\begin{array}{c}{g}_{k+1}^{\text{T}}{d}_{k+1}\le -{‖{g}_{k+1}‖}^{2}+{E}_{3}{‖{g}_{k+1}‖}^{2}-{E}_{4}{‖{g}_{k+1}‖}^{2}\\ =-\left[1-\left({E}_{4}-{E}_{3}\right)\right]{‖{g}_{k+1}‖}^{2}\\ =-E{‖{g}_{k+1}‖}^{2}\end{array}
E=1-\left({E}_{4}-{E}_{3}\right)
0<{E}_{4}-{E}_{3}<1
{d}_{k+1}
{d}_{k}
{\sum }_{k\ge 1}\frac{1}{{‖{d}_{k}‖}^{2}}=\infty
{\mathrm{lim}}_{k\to \infty }\mathrm{inf}‖{g}_{k}‖=0
0\le {\phi }_{k}\le 1
{\alpha }_{k}
{d}_{k+1}
{\mathrm{lim}}_{k\to \infty }\mathrm{inf}‖{g}_{k}‖=0
|
6.3 Engineering Tools to Maximize Solar Utility | EME 810: Solar Resource Assessment and Economics
6.3 Engineering Tools to Maximize Solar Utility
J.R. Brownson, Solar Energy Conversion Systems (SECS), Chapter 6: "A Comment on Optimal Tilt" (small section at the end of the chapter) and see Figure 6.16.
M. Lave and J. Kleissl. (2011) Optimum fixed orientations and benefits of tracking for capturing solar radiation in the continental United States. Renewable Energy, 36:1145–1152.
C. B. Christensen and G. M. Barker (2001) Effects of tilt and azimuth on annual incident solar radiation for United States locations. In Proceedings of Solar Forum 2001: Solar Energy: The Power to Choose, April 21-25 2001
T. Huld, M. Šúri, T. Cebecauer, E. D. Dunlop (2008) Comparison of electricity yield from fixed and sun-tracking PV systems in Europe. European Commission, Joint Research CentreInstitute for Energy, Renewable Energies Unit, via E. Fermi 2749, TP 450, I-21027 Ispra (VA), Italy (poster, PDF)
Greentech Media Article: Solar Balance-of-System: To Track or Not to Track, Part I (Nov. 2012)
Engineering Approaches to Increase Solar Utility
Locale is the space or an address in time and place within which the client occupies and demands energy resources. Recall that our clients are on the demand side of solar goods and services, and as such they seek maximal utility when making decisions.
for the client or group of stakeholders
We have already learned that the solar resource can be affected by the locale of the site. The solar resource is determined by the locale, as the climate regime affects the seasonal and daily irradiation patterns and frequencies of intermittence. The character or quality of the solar resource will in turn constrain the design team's options for technological solutions that compete with conventional fuel-based technologies.
According to our review of SECS Chapter 6: given that goal for solar project design, we have three main engineering approaches that we can leverage to affect the solar utility for a client in a given locale:
Reduce the cosine projection effect on an aperture/receiver. These are the extreme angles of incidence (also called low glancing angles);
Reduce the angle of incidence (
\theta
) on an aperture/receiver; and
Reduce losses from shading on an aperture/receiver.
These are the three main engineering parameters linked to the locale that will constrain your design options (you can look back to the Angular Solar Symbols guide to refresh your memory). They all affect system performance, without necessarily directly influencing the cost of the system (in the beginning). Let's review how they affect system performance.
Reduce the cosine projection effect
Figure 6.1 The top image shows the effect of inverse square law on the Sun-Earth view factor
{F}_{Sun-Earth}
). The bottom image shows the cosine projection effect as it affects the Sun-Earth view factor (the inverse cosine of the zenith angle reduces the intensity of the Sun's irradiance).
How does the tilt and azimuth each affect the design in SECS, and how does regionality affect the design decisions in solar energy?
We have seen in our reading of Lave and Kleissl that an annual optimum for tilt and azimuth can be selected, while Christensen and Barker demonstrate that annual optimum is not really "peaky," and fixed-tilt systems can be oriented across a broad range of directions in a given locale without dropping solar gains by more than 10-20%. If we were to adjust the tilt for a seasonal optimum, we would select a lower tilt for the summer season and a higher tilt for the winter season. Effectively, we are working to correct for the cosine projection effect of our particular latitude and climate regime (one climate regime per season, recall the "fingerprints").
On broad scales, sites near the equator will have different design constraints than sites near the Arctic Circle, due to the cosine projection effect driving our solar resource across latitudes and the seasons. In this context, the project locale serves as an effective system constraint. The amount of sunlight available on a daily basis and on a seasonal basis differs with locale. Using and implementing the same system design for a client in State College, Pennsylvania (
\varphi =+{40.86}^{°}
\lambda =-{77.83}^{°}
) and another in Lagos, Nigeria (
\varphi =+{6.45}^{°}
\lambda =+{3.39}^{°}
), for example, will yield totally different results and lead to unsatisfied clients.
You see two images of a cartoon Sun, drawn from Ch 4 of the SECS text. The top image shows the effect of inverse square law on the Sun-Earth view factor (
{F}_{Sun-Earth}
). The distance of 150 million km reduces the intensity of the Sun from
6.33×{10}^{7}W/{m}^{2}
W/{m}^{2}
{G}_{s}c
). This effect is fairly uniform year-round. The bottom image shows the cosine projection effect as it affects the Sun-Earth view factor. Here, the inverse cosine of the zenith angle (
{\theta }_{z}
) reduces the intensity of the Sun's irradiance. Hence, the farther away your client is located from the Equator, the more the designer will need to make collector orientation adjustments to compensate for the losses from the cosine projection effect.
Note also that the tilt of the Earth's axis will drive one to consider summer or winter optimized orientations (away from the Equator).
Reduce the angle of incidence
How does tracking affect the design decisions in solar energy?
Well, a fixed axis SECS is often oriented toward the equator at a tilt (
\beta
) somewhat less than the local latitude (do not fall for the latitude = tilt rule of thumb), per our readings from Christensen and Barker, and Lave and Kleissl. When we track the Sun, then more beam is collected (the angle of incidence tends to be consistently lower than for a fixed tilt). By looking at the poster from Huld et al. (2008), we see that a single-axis tracking system, with an axis inclined at an optimum angle towards South, should offer 12-50% improvement over a fixed axis tilted at the optimum, where a 2-axis tracker will offer a very similar solar gain of 13-55%.
So, a tracking system will minimize the angle of incidence (
\theta
), but there will be a cost in terms of land requirements. Why? Because of shading. There will also be a cost in terms of the balance of systems (e.g., the non-SECS trackers). This is why we could read "Solar Balance-of-Systems: To Track or Not to Track, Part I" for more information.
But the reality of solar development (whether on a rooftop or on a field) is that the systems are often "area constrained." We can make certain tradeoffs in systems choices to deliver a better unit cost to the client, but we may not get all the land that we desire to accomplish an optimal tracking system. As such, we must work with the stakeholders to find the highest solar utility solution given the available area.
Reduce losses from shading
Finally, a large group of our SECSs rely on access to the shortwave light from the Sun. If we shade a collector, then we reduce or remove that working energy that we wish to convert to heat or electricity. We performed the shading analysis in Lesson 2 using orthographic and spherical projections specifically to be able to avoid shading of our array over the course of an entire year.
Of course, if we were to design a system to avoid the Sun's rays, that would be different. We have seen examples of solar design for Parasoleil frameworks (shading systems) in the beginning of the textbook (e.g., southern awnings).
1. What are the three goals of solar design and engineering?
To maximize the solar utility
for the client or stakeholders
(may those words be imprinted in your brain forever after!)
2. What are three engineering mechanisms that I can employ in a systems design to maximize solar utility?
Minimize the cosine projection effect (by compensating with tilt $\beta$)
Minimize the angle of incidence $\theta$ over key time intervals (tracking, seasonal tilt adjustments)
Minimize or remove shading (shadows) from neighboring objects (do a shading analysis)
3. What are the tradeoffs that I encounter in systems design when I design a tracking system?
ANSWER: The footprint of the array must be larger in order to avoid shading, and the costs for installation and maintenance will increase with the additional equipment, called the Balance of Systems. In return, I can derive higher annual performance from my system. If my system is not substantially "area constrained," then I might even achieve a lower unit cost.
‹ 6.2 Solar Utility, Locale, and Client up 6.4 Discussion Activity 1 - Communicating Information to Your Client ›
6.2 Solar Utility, Locale, and Client
6.4 Discussion Activity 1 - Communicating Information to Your Client
6.5 The Power Grid System
6.6 Power Grid Pricing and Capacity
6.7 Energy Portfolio Standards and Government Incentives
6.8 Electric Incentives in SAM
6.9 Learning Activity: Pre-Design Charrette Plans
6.10 Discussion Activity 2 - Brainstorming Your Solar Proposal
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Diffraction of singularities for the wave equation on manifolds with corners - Tome (2013) no. 351
Melrose, Richard ; Vasy, András ; Wunsch, Jared
author = {Melrose, Richard and Vasy, Andr\'as and Wunsch, Jared},
title = {Diffraction of singularities for the wave equation on manifolds with corners},
AU - Melrose, Richard
AU - Vasy, András
AU - Wunsch, Jared
TI - Diffraction of singularities for the wave equation on manifolds with corners
Melrose, Richard; Vasy, András; Wunsch, Jared. Diffraction of singularities for the wave equation on manifolds with corners. Astérisque, no. 351 (2013), 141 p. http://numdam.org/item/AST_2013__351__R1_0/
[1] J. Chazarain - "Reflection of
{C}^{\infty }
singularities for a class of operators with multiple characteristics", in Proceedings of the Oji Seminar on Algebraic Analysis and the RIMS Symposium on Algebraic Analysis (Kyoto Univ., Kyoto, 1976), vol. 12, 1976/77, p. 39-52. | Zbl 0365.35050
[2] J. Cheeger & M. Taylor - "On the diffraction of waves by conical singularities. I", Comm. Pure Appl. Math. 35 (1982), p. 275-331. | Article | Zbl 0526.58049
[3] J. Cheeger & M. Taylor, "On the diffraction of waves by conical singularities. II", Comm. Pure Appl. Math. 35 (1982), p. 487-529. | Article | Zbl 0536.58032
[4] J. J. Duistermaat & L. Hörmander - "Fourier integral operators. II", Acta Math. 128 (1972), p. 183-269. | Article | Zbl 0232.47055
[5] F. G. Friedlander - Sound pulses, Cambridge Univ. Press, 1958. | Zbl 0079.41001
[6] P. Gérard & G. Lebeau - "Diffusion d'une onde par un coin", J. Amer. Math. Soc. 6 (1993), p. 341-424. | Article | Zbl 0779.35063
[7] L. Hörmander - The analysis of linear partial differential operators, Springer, 1985. | Zbl 0601.35001
[8] J. B. Keller - "One hundred years of diffraction theory", IEEE Trans. Antennas and Propagation 33 (1985), p. 123-126. | Article | Zbl 0947.78501
[9] P. D. Lax - "Asymptotic solutions of oscillatory initial value problems", Duke Math. J. 24 (1957) p. 627-646. | Article | Zbl 0083.31801
[10] G. Lebeau - "Propagation des ondes dans les variétés à coins", in Séminaire sur les Équations aux Dérivées Partielles, 1995-1996, Sémin. Équ. Dériv. Partielles, École Polytech., 1996, p. Exp. No. XVI, 20. | EuDML 112127 | Zbl 0963.35110
[11] G. Lebeau, "Propagation des ondes dans les variétés à coins", Ann. Sci. École Norm. Sup. 30 (1997), p. 429-497. | Article | EuDML 82439 | Numdam | Zbl 0891.35072
[12] D. Ludwig - "Exact and asymptotic solutions of the Cauchy problem", Comm. Pure Appl Math. 13 (1960), p. 473-508. | Article | Zbl 0098.29601
[13] R. Mazzeo - "Elliptic theory of differential edge operators. I", Comm. Partial Differential Equations 16 (1991), p. 1615-1664. | Article | Zbl 0745.58045
[14] R. Mazzeo & R. B. Melrose - "Pseudodifferential operators on manifolds with fibred boundaries", Asian J. Math. 2 (1998), p. 833-866. | Article | Zbl 1125.58304
[15] R. Melrose, A. Vasy & J. Wunsch - "Propagation of singularities for the wave equation on edge manifolds", Duke Math. J. 144 (2008), p. 109-193. | Article | Zbl 1147.58029
[16] R. Melrose & J. Wunsch - "Propagation of singularities for the wave equation on conic manifolds", Invent Math. 156 (2004), p. 235-299. | Article | Zbl 1088.58011
[17] R. B. Melrose - "Transformation of boundary problems", Acta Math. 147 (1981), p. 149-236. | Article | Zbl 0492.58023
[18] R. B. Melrose, The Atiyah-Patodi-Singer index theorem, Research Notes in Math., vol. 4, A K Peters Ltd., 1993. | Zbl 0796.58050
[19] R. B. Melrose, "Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces", in Spectral and scattering theory (Sanda, 1992), Lecture Notes in Pure and Appl. Math., vol. 161, Dekker, 1994, p. 85-130. | Zbl 0837.35107
[20] R. B. Melrose & P. Piazza - "Analytic
K
-theory on manifolds with corners", Adv. Math. 92 (1992), p. 1-26. | Article | Zbl 0761.55002
[21] R. B. Melrose & J. Sjöstrand - "Singularities of boundary value problems. I", Comm. Pure Appl. Math. 31 (1978), p. 593-617. | Article | Zbl 0368.35020
[22] R. B. Melrose & J. Sjöstrand, "Singularities of boundary value problems. II", Comm. Pure Appl. Math. 35 (1982), p. 129-168. | Article | Zbl 0546.35083
[23] J. Sjöstrand - "Propagation of analytic singularities for second order Dirichlet problems", Comm. Partial Differential Equations 5 (1980), p. 41-93. | Article | Zbl 0458.35026
[24] J. Sjöstrand, "Propagation of analytic singularities for second order Dirichlet problems. II", Comm. Partial Differential Equations 5 (1980), p. 187-207. | Article | Zbl 0534.35030
[25] J. Sjöstrand, "Propagation of analytic singularities for second order Dirichlet problems. III", Comm. Partial Differential Equations 6 (1981), p. 499-567. | Article | Zbl 0524.35032
[26] A. Sommerfeld - "Mathematische Theorie der Diffraction", Math. Ann. 47 (1896), p. 317-374. | Article | EuDML 157794 | JFM 27.0706.03
[27] M. E. Taylor - "Grazing rays and reflection of singularities of solutions to wave equations", Comm. Pure Appl. Math. 29 (1976), p. 1-38. | Article | Zbl 0318.35009
[28] A. Vasy - "Propagation of singularities in many-body scattering", Ann. Sci. École Norm. Sup. 34 (2001), p. 313-402. | Article | EuDML 82545 | Numdam | Zbl 0987.81114
[29] A. Vasy, "Geometric optics and the wave equation on manifolds with corners", in Recent advances in differential equations and mathematical physics, Contemp. Math., vol. 412, Amer. Math. Soc., 2006, p. 315-333. | Article | Zbl 1107.58012
[30] A. Vasy, "Propagation of singularities for the wave equation on manifolds with corners", Ann. of Math. 168 (2008), p. 749-812. | Article | Zbl 1171.58007
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EUDML | An investigation of bounds for the regulator of quadratic fields. EuDML | An investigation of bounds for the regulator of quadratic fields.
An investigation of bounds for the regulator of quadratic fields.
Jacobson, Michael J.jun.; Lukes, Richard F.; Williams, Hugh C.
Jacobson, Michael J.jun., Lukes, Richard F., and Williams, Hugh C.. "An investigation of bounds for the regulator of quadratic fields.." Experimental Mathematics 4.3 (1995): 211-225. <http://eudml.org/doc/222883>.
author = {Jacobson, Michael J.jun., Lukes, Richard F., Williams, Hugh C.},
keywords = {fundamental unit; Dirichlet -function; status report; regulator; quadratic field; class number; Dirichlet -function},
title = {An investigation of bounds for the regulator of quadratic fields.},
AU - Jacobson, Michael J.jun.
AU - Lukes, Richard F.
TI - An investigation of bounds for the regulator of quadratic fields.
KW - fundamental unit; Dirichlet -function; status report; regulator; quadratic field; class number; Dirichlet -function
fundamental unit, Dirichlet
L
-function, status report, regulator, quadratic field, class number, Dirichlet
L
Articles by Jacobson
Articles by Lukes
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Statistical characterization of gas-patch distributions in partially saturated rocksCharacterization of patchy saturated rocks | Geophysics | GeoScienceWorld
Julianna Toms-Stewart;
Julianna Toms-Stewart
Formerly Curtin University of Technology, Perth, Australia; presently ExxonMobil, Upstream Research Company, Houston, Texas, U.S.A. E-mail: julianna.toms@exxonmobil.com.
Tobias M. Müller;
Formerly University of Karlsruhe, Karlsruhe, Germany; presently CSIRO Petroleum, Perth, Australia. E-mail: tobias.mueller@csiro.au.
Curtin University of Technology & CSIRO Petroleum, Perth, Australia. E-mail: b.gurevich@curtin.edu.au.
CSIRO Petroleum, Melbourne, Australia. E-mail: lincoln.paterson@csiro.au.
Julianna Toms-Stewart, Tobias M. Müller, Boris Gurevich, Lincoln Paterson; Statistical characterization of gas-patch distributions in partially saturated rocks. Geophysics 2009;; 74 (2): WA51–WA64. doi: https://doi.org/10.1190/1.3073007
Reservoir rocks are often saturated by two or more fluid phases forming complex patterns on all length scales. The objective of this work is to quantify the geometry of fluid phase distribution in partially saturated porous rocks using statistical methods and to model the associated acoustic signatures. Based on X-ray tomographic images at submillimeter resolution obtained during a gas-injection experiment, the spatial distribution of the gas phase in initially water-saturated limestone samples are constructed. Maps of the continuous variation of the percentage of gas saturation are computed and associated binary maps obtained through a global thresholding technique. The autocorrelation function is derived via the two-point probability function computed from the binary gas-distribution maps using Monte Carlo simulations. The autocorrelation function can be approximated well by a single Debye correlation function or a superposition of two such functions. The characteristic length scales and show sensitivity (and hence significance) with respect to the percentage of gas saturation. An almost linear decrease of the Debye correlation length occurs with increasing gas saturation. It is concluded that correlation function and correlation length provide useful statistical information to quantify fluid-saturation patterns and changes in these patterns at the mesoscale. These spatial statistical measures are linked to a model that predicts compressional wave attenuation and dispersion from local, wave-induced fluid flow in randomly heterogeneous poroelastic solids. In particular, for a limestone sample, with flow permeability of 5 darcies and an average gas saturation of
∼5%
, significant P-wave attenuation is predicted at ultrasonic frequencies.
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Temperature and Load Ratio Effects on Crack-Growth Behavior of Austenitic Superalloys | J. Eng. Mater. Technol. | ASME Digital Collection
Ajit K. Roy,
, 4505 Maryland Parkway, P.O. Box 454027, Las Vegas, NV 89154-4027
e-mail: ajit.roy@srnl.doe.gov
Joydeep Pal,
Muhammad H. Hasan
Roy, A. K., Pal, J., and Hasan, M. H. (November 2, 2009). "Temperature and Load Ratio Effects on Crack-Growth Behavior of Austenitic Superalloys." ASME. J. Eng. Mater. Technol. January 2010; 132(1): 011001. https://doi.org/10.1115/1.3184027
The role of temperature and load ratio
(R)
on the crack propagation rate
(da/dN)
of Alloys 276 and 617 under cyclic loading was investigated. The results indicate that the rate of cracking was gradually enhanced with increasing temperature when the
R
value was kept constant. However, the temperature effect was more pronounced at
100–150°C
. Both alloys exhibited maximum
da/dN
values at a load range of 4.5 kN that corresponds to an
R
value of 0.1. The number of cycles to failure for Alloy 276 was relatively higher compared with that of Alloy 617, indicating its slower crack-growth rate. Fractographic evaluation of the broken specimen surface revealed combined fatigue and ductile failures.
cracks, ductility, fatigue, fractography, nickel alloys, superalloys, crack-growth rate, cyclic loading, temperature, load ratio, superalloys
Alloys, Fracture (Materials), Stress, Superalloys, Temperature, Fatigue
Hydrogen Production as a Major Nuclear Energy Application
Nuclear News, American Nuclear Society
Development of Hydrogen Production Technology by Thermochemical Water Splitting IS Process
Koripelli
High-Temperature Tensile Properties of Nickel-Base Alloys for Hydrogen Generation
Proceedings of the International SAMPE Symposium and Exhibition
Preliminary Consideration of Alloys 617 and 230 for Generation IV Nuclear Reactor Applications
Proceedings of ASME Pressure Vessels and Piping Division Conference, PVP
Dynamic Strain Ageing of an Austenitic Superalloy—Temperature and Strain Rate Effects
Tensile Deformation of a Nickel-Base Superalloy for Application in Hydrogen Generation
Tensile Deformation of a Nickel-Base Alloy at Elevated Temperatures
Corrosion Performance and Fabricability of the New Generation of Highly Corrosion-Resistant Nickel–Chromium–Molybdenum Alloys
, Hastelloy C-276 Alloy Product Brochure, Corrosion-Resistant Alloys, Haynes International, Kokomo, IN.
Tensile and Impact Properties of Candidate Alloys for High-Temperature Gas-Cooled Reactor Applications
Performance of Inconel 617 in Actual and Simulated Gas Turbine Environments
El-Shabasy
Effects of Load Ratio, R, and Test Temperature on Fatigue Crack Growth of Fully Pearlitic Eutectoid Steel
Osinkolu
Fatigue Crack Growth of UDIMET 720 Li Superalloy at Elevated Temperature
Standard Test Methods for Tensile Testing of Metallic Materials
, ASTM Designation E 8-2004, Vol.
, ASTM Designation E 647-2000, Vol.
Qunjia
Stress Corrosion Crack Growth in 316 Stainless Steel in Supercritical Water
Third International Symposium on Supercritical Water-Cooled Reactors-Design and Technology
D-C Electric-Potential Method Applied to Thermal/Mechanical Fatigue Crack Growth
Calibration of Electric Potential Method for Studying Slow Crack Growth
Mater. Res. Stand.
Effects of Temperature and Hold Time on Creep-Fatigue Crack-Growth Behavior of Haynes 230 Alloy
Alloy Design and Development of Inconel 718 Type Alloy
Effect of Cobalt on the Flow Stress of Ni–Co–Al Alloys
Borukhin
Some Structural Effects of Plastic Deformation on Tungsten Heavy Metal Alloys
The Fatigue Behavior of Nickel, Monel, and Selected Superalloys, Tested in Liquid Mercury and Air; A Comparison
TMF Crack Initiation Lifing of Austenitic Carbide Precipitating Alloys
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April, 2002 Convergence rates to viscous shock profile for general scalar viscous conservation laws with large initial disturbance
Kenji NISHIHARA, Huijiang ZHAO
This paper is concerned with the convergence rates to viscous shock profile for general scalar viscous conservation laws. Compared with former results in this direction, the main novelty in this paper lies in the fact that the initial disturbance can be chosen arbitrarily large. This answers positively an open problem proposed by A. Matsumura in [12] and K. Nishihara in [16]. Our analysis is based on the
{L}^{1}\text{-}
stability results obtained by H. Freistiihler and D. Serre in [1].
Kenji NISHIHARA. Huijiang ZHAO. "Convergence rates to viscous shock profile for general scalar viscous conservation laws with large initial disturbance." J. Math. Soc. Japan 54 (2) 447 - 466, April, 2002. https://doi.org/10.2969/jmsj/05420447
Keywords: convergence rate , large initial disturbance , viscous shock profile
Kenji NISHIHARA, Huijiang ZHAO "Convergence rates to viscous shock profile for general scalar viscous conservation laws with large initial disturbance," Journal of the Mathematical Society of Japan, J. Math. Soc. Japan 54(2), 447-466, (April, 2002)
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Huntington–Hill method - Wikipedia
Huntington–Hill method
Method of assigning legislative seats
The Huntington–Hill method is a way of allocating seats proportionally for representative assemblies such as the United States House of Representatives. The method assigns seats by finding a modified divisor D such that each constituency's priority quotient (its population divided by D), using the geometric mean of the lower and upper quota for the divisor, yields the correct number of seats that minimizes the percentage differences in the size of subconstituencies.[1] When envisioned as a proportional electoral system, it is effectively a highest averages method of party-list proportional representation in which the divisors are given by
{\textstyle D={\sqrt {n(n+1)}}}
, n being the number of seats a state or party is currently allocated in the apportionment process (the lower quota) and n + 1 is the number of seats the state or party would have if it is assigned to the party list (the upper quota). Although no legislature uses this method of apportionment to assign seats to parties after an election, it was considered for House of Lords elections under the ill-fated House of Lords Reform Bill.[2]
This method is how the United States House of Representatives assigns the number of representative seats to each state — the purpose for which it was devised — and the Census Bureau calls it the method of equal proportions.[3] It is credited to Edward Vermilye Huntington and Joseph Adna Hill.[4]
In a legislative election under the Huntington–Hill method, after the votes have been tallied, the qualification value would be calculated. This step is necessary because in an election, unlike in a legislative apportionment, not all parties are always guaranteed at least one seat. If the legislature concerned has no exclusion threshold, the qualification value could be a predefined quota, such as the Hare, Droop, or Imperiali quota.
In legislatures which use an exclusion threshold, the qualification value would be equipotent to the threshold, that is:
{\displaystyle {\text{exclusion threshold [percentage]}}\left({\frac {\text{total votes}}{100}}\right).}
total votes is the total valid poll; that is, the number of valid (unspoilt) votes cast in an election.
total seats is the total number of seats to be filled in the election.
Every party polling votes equal to or greater than the qualification value would be given an initial number of seats, again varying if whether or not there is a threshold:
In legislatures which do not use an exclusion threshold, the initial number would be 1, but in legislatures which do, the initial number of seats would be:
{\displaystyle {\text{exclusion threshold [percentage]}}\left({\frac {\text{total seats}}{100}}\right).}
with all fractional remainders being rounded up.
In legislatures elected under a mixed-member proportional system, the initial number of seats would be further modified by adding the number of single-member district seats won by the party before any allocation.
Determining the qualification value is not necessary when distributing seats in a legislature among states pursuant to census results, where all states are guaranteed a fixed number of seats, either one (as in the US) or a greater number, which may be uniform (as in Brazil) or vary between states (as in Canada).
It can also be skipped if the Huntington-Hill system is used in the nationwide stage of a national remnant system, because the only qualified parties are those which obtained seats at the subnational stage.
After all qualified parties or states received their initial seats, successive quotients are calculated, as in other Highest Averages methods, for each qualified party or state, and seats would be repeatedly allocated to the party or state having the highest quotient until there are no more seats to allocate. The formula of quotients calculated under the Huntington–Hill method is
{\displaystyle A_{n}={\frac {V}{\sqrt {s(s+1)}}}}
V is the population of the state or the total number of votes that party received, and
s is the number of seats that the state or party has been allocated so far.
Even though the Huntington–Hill system was designed to distribute seats in a legislature among states pursuant to census results, it can also be used, when putting parties in the place of states and votes in place of population, for the mathematically equivalent task of distributing seats among parties pursuant to an election results in a party-list proportional representation system. A party-list PR system requires large multi-member districts to function effectively.
In this example, 230,000 voters decide the disposition of 8 seats among 4 parties. Unlike the D'Hondt and Sainte-Laguë systems, which allow the allocation of seats by calculating successive quotients right away, the Huntington–Hill system requires each party or state have at least one seat to avoid a division by zero error. In the U.S. House of Representatives, this is ensured by guaranteeing each state at least one seat; in a single-stage PR election under the Huntington–Hill system, however, the first stage would be to calculate which parties are eligible for an initial seat.
This could be done by excluding any parties which polled less than a predefined quota, and giving every party which polled at least the quota one seat.
Is the party eligible or disqualified?
Party A 100,000 Eligible Eligible Eligible
Party B 80,000 Eligible Eligible Eligible
Party C 30,000 Eligible Eligible Eligible
Party D 20,000 Disqualified Disqualified Disqualified
In this case, the qualified parties stay the same regardless of quota.
Each eligible party is assigned one seat. With all the initial seats assigned, the remaining five seats are distributed by a priority number calculated as follows. Each eligible party's (Parties A, B, and C) total votes is divided by 1.41 (the square root of the product of 1, the number of seats currently assigned, and 2, the number of seats that would next be assigned), then by 2.45, 3.46, 4.47, 5.48, 6.48, 7.48, and 8.49. The 5 highest entries, marked with asterisks, range from 70,711 down to 28,868. For each, the corresponding party gets another seat.
For comparison, the "Proportionate seats" column shows the exact fractional numbers of seats due, calculated in proportion to the number of votes received (For example, 100,000/230,000 × 8 = 3.48). If the "Total Seats" column is less than the "Proportionate seats" column (Parties C[a] and D in this example) the party is under-represented. Conversely, if the "Total Seats" column is greater than the "Proportionate seats" column (Parties A and B in this example) the party is over-represented.[b]
won (*)
seats[c]
Party A 70,711* 40,825* 28,868* 22,361 18,257 15,430 13,363 11,785 1 3 4 3.5
Party B 56,569* 32,660* 23,094 17,889 14,606 12,344 10,690 9,428 1 2 3 2.8
Party C 21,213 12,247 8,660 6,708 5,477 4,629 4,009 3,536 1 0 1 1.0
Party D Disqualified 0 0.7
If the number of seats was equal in size to the number of votes cast, this method would guarantee that the apportionments would equal the vote shares of each party.
In this example, the results of the apportionment is identical to one under the D'Hondt system. However, as the District magnitude increases, differences emerge: all 120 members of the Knesset, Israel's unicameral legislature, are elected under the D'Hondt method.[d] Had the Huntington–Hill method, rather than the D'Hondt method, been used to apportion seats following the elections to the 20th Knesset, held in 2015, the 120 seats in the 20th Knesset would have been apportioned as follows:
D'Hondt[d]
Last Priority[e]
Next Priority[f]
Likud 985,408 33408 32313 30 30 0
Zionist Union 786,313 33468 32101 24 24 0
Joint List 446,583 35755 33103 13 13 0
Yesh Atid 371,602 35431 32344 11 11 0
Kulanu 315,360 37166 33242 9 10 –1
The Jewish Home 283,910 33459 29927 9 8 +1
Shas 241,613 37282 32287 7 7 0
Yisrael Beiteinu 214,906 39236 33161 6 6 0
United Torah Judaism 210,143 38367 32426 6 6 0
Meretz 165,529 37013 30221 5 5 0
Compared with the actual apportionment, Kulanu would have lost one seat, while The Jewish Home would have gained one seat.
^ Party C's proportion is actually 1.04
^ While this example favors the largest parties (Parties A and B), if a different number of seats were apportioned, other parties would be favored. In short, the largest party is not always favored.
For example, if there were 12 seats instead of 8, then Party C would be the only over-represented party (since Party D would have qualified) with two full seats while proportionately deserving only 1.6 seats.
^ This proportionality is based on the total votes. If instead it was based on the qualified votes (i.e., reducing the total 230,000 votes by the disqualified 20,000 votes for Party D), the proportionate seats would be: Party A - 3.8 seats, Party B - 3.0 seats, and Party C - 1.1 seats.
^ a b The method used for the 20th Knesset was actually a modified D'Hondt, called the Bader-Ofer method. This modification allows for spare vote agreements between parties.[5]
^ This is each party's last priority number which resulted in a seat being gained by the party. Likud gained the last seat (the 120th seat allocated). Each priority number in this column is greater than any priority number in the Next Priority column.
^ This is each party's next priority number which would result in a seat being gained by the party. Kulanu would have gained the next seat (if there were 121 seats in the Knesset). Each priority number in this column is less than any priority number in the Last Priority column.
^ "Congressional Apportionment". NationalAtlas.gov. Archived from the original on 2009-02-28. Retrieved 2009-02-14.
^ Draft House of Lords Reform Bill: report session 2010-12, Vol. 2. Google Books. 23 April 2012. ISBN 9780108475801. Retrieved 6 November 2017.
^ "Computing Apportionment". United States Census Bureau. Retrieved 2021-04-26. {{cite web}}: CS1 maint: url-status (link)
^ "The History of Apportionment in America". American Mathematical Society. Retrieved 2009-02-15.
^ "With Bader-Ofer method, not every ballot counts". The Jerusalem Post. Retrieved 2021-05-04.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Huntington–Hill_method&oldid=1059880194"
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For the saxophonist, see Carl Magnus Neumann. For people of a similar name, see Karl Neumann (disambiguation).
De problemate quodam mechanico, quod ad primam classem integralium ultraellipticorum revocatur[1]
Friedrich Richelot and Otto Hesse
Carl Gottfried Neumann (also Karl; 7 May 1832 – 27 March 1925) was a German mathematician.
3 Works by Carl Neumann
Neumann was born in Königsberg, Prussia, as the son of the mineralogist, physicist and mathematician Franz Ernst Neumann (1798–1895), who was professor of mineralogy and physics at Königsberg University. Carl Neumann studied in Königsberg and Halle and was a professor at the universities of Halle, Basel, Tübingen, and Leipzig.
While in Königsberg, he studied physics with his father, and later as a working mathematician, dealt almost exclusively with problems arising from physics. Stimulated by Bernhard Riemann's work on electrodynamics, Neumann developed a theory founded on the finite propagation of electrodynamic actions, which interested Wilhelm Eduard Weber and Rudolf Clausius into striking up a correspondence with him. Weber described Neumann's professorship at Leipzig as for "higher mechanics, which essentially encompasses mathematical physics," and his lectures did so.[2] Maxwell makes reference to the electrodynamic theory developed by Weber and Neumann in the Introduction to A Dynamical Theory of the Electromagnetic Field (1864).
Neumann worked on the Dirichlet principle, and can be considered one of the initiators of the theory of integral equations. The Neumann series, which is analogous to the geometric series
{\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+\cdots }
but for infinite matrices or for bounded operators, is named after him.
Together with Alfred Clebsch, Neumann founded the mathematical research journal Mathematische Annalen. He died in Leipzig.
The Neumann boundary condition for certain types of ordinary and partial differential equations is named after him (Cheng and Cheng, 2005).
Liouville–Neumann series
Works by Carl Neumann[edit]
Carl Gottfried Neumann, 1912
Hydrodynamische Untersuchungen, 1883
Lösung des allgemeinen Problemes über den stationären Temperaturzustand einer homogenen Kugel ohne Hülfe von Reihen-Entwicklungen (in German). Halle: Schmidt. 1861.
Allgemeine Lösung des Problemes über den stationären Temperaturzustand eines homogenen Körpers, welcher von irgend zwei nichtconcentrischen Kugelflächen begrenzt wird (in German). Halle: Schmidt. 1862.
Magnetische Drehung der Polarisationsebene des Lichtes (in German). Halle: Buchhandlung des Waisenhauses. 1863.
Theorie der Elektricitäts- und Wärme-Vertheilung in einem Ringe (in German). Halle: Buchhandlung des Waisenhauses. 1864.
Das Dirichlet'sche Princip in seiner Anwendung auf die Riemann'schen Flächen (B. G. Teubner, Leipzig, 1865)
Vorlesungen über Riemann's Theorie der Abel'schen Integrale (B. G. Teubner, 1865)
Die Haupt- und Brenn-Puncte eines Linsen-Systemes (in German). Leipzig: Benedictus Gotthelf Teubner. 1866.
Theorie der Bessel'schen functionen: ein analogon zur theorie der Kugelfunctionen (B. G. Teubner, 1867)
Untersuchungen über das Logarithmische und Newton'sche potential (B. G. Teubner, 1877)
Hydrodynamische Untersuchungen (in German). Leipzig: Benedictus Gotthelf Teubner. 1883.
Die elektrischen Kräfte (Teubner, 1873–1898)
^ Carl Neumann at the Mathematics Genealogy Project
^ Christa Jungnickel, Russell McCormmach, Intellectual Mastery of Nature: Theoretical Physics from Ohm to Einstein (1990) Vol. 1. p. 181.
O'Connor, John J.; Robertson, Edmund F., "Carl Neumann", MacTutor History of Mathematics archive, University of St Andrews
Carl Neumann at the Mathematics Genealogy Project
Retrieved from "https://en.wikipedia.org/w/index.php?title=Carl_Neumann&oldid=1077809766"
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Post 1: A Bayesian 2PL IRT model | R-bloggers
Post 1: A Bayesian 2PL IRT model
In this post, we define the Two-Parameter Logistic (2PL) IRT model, derive the complete conditionals that will form the basis of the sampler, and discuss our choice of prior specification.
2PL IRT Model specification
newcommand{E}[1]{mathbb{E}left[#1right]} newcommand{Egiven}[2]{mathbb{E}left[ left. #1 right| #2right]} newcommand{Var}[1]{text{Var}left[#1right]} newcommand{Vargiven}[2]{text{Var}left[ left. #1 right| #2right]}
Let the probability of person
p
i
pi_{pi}
. We map
pi_{pi}
to a linear scale with a logit link function begin{equation} ln{frac{pi_{pi}}{1+pi_{pi}}} = a_i ( theta_p – b_i) label{eq:pipi} end{equation} where the three parameters
a_i
b_i
theta_p
are interpreted as item discrimination, item difficulty, and person ability parameters respectively. The observed response of person
p
is modeled as a flip of a biased coin begin{aligned} U_{pi} & stackrel{indep}{sim} text{Bernoulli}(pi_{pi}) quad . end{aligned}
This model becomes a Bayesian model when we place prior distributions on the item and person parameters. In the chapter and this supplement, we use the following prior specification begin{aligned} % U_{pi} & stackrel{indep}{sim} text{Bernoulli}(pi_{pi}) \ % ln frac{pi_{pi}}{1+pi_{pi}} & = a_i(theta_p – b_i) \ theta_p & stackrel{iid}{sim} text{Normal}(0,sigma_theta^2) \ sigma_theta^2 & sim text{Inverse-Gamma}(alpha_theta,beta_theta) \ a_i & stackrel{iid}{sim} text{Log-Normal}(mu_a,sigma^2_a) \ b_i & stackrel{iid}{sim} text{Normal}(0,sigma_b^2) end{aligned} where
alpha_theta
beta_theta
mu_a
sigma_a^2
sigma_b^2
are constants that we choose below in the prior specification section.
Derivation of the complete conditionals
The idea of a Gibbs sampler is that we can sample from the joint-posterior by iteratively sampling from conditional posteriors. In this section, we derive the conditional posteriors, which are commonly referred to as complete conditionals because they are conditional on all other parameters and data in the model.
We begin with the likelihood function. Since each
U_{pi}
is an iid Bernoulli, the likelihood function is:
where the bolded quantities
boldsymbol{U}
boldsymboltheta
boldsymbol a
boldsymbol b
are vectors or matrices, and
pi_{pi}
is defined in Equation ref{eq:pipi}. Note that the likelihood is independent of
sigma^2_theta
By applying Bayes rule and using the conditional independence relationships in the model, we find the joint posterior is proportional to: begin{equation} f(boldsymboltheta, sigma^2_theta, boldsymbol a, boldsymbol b | boldsymbol{U} ) propto f(boldsymbol{U} | boldsymboltheta, boldsymbol a, boldsymbol b ) cdot f(boldsymboltheta | sigma^2_theta ) cdot f(sigma^2_theta ) cdot f(boldsymbol a) cdot f(boldsymbol b) label{eq:post} end{equation}
The complete conditionals conditionals are then essentially read off of Equation ref{eq:post} by simply dropping any expression which does not contain the parameter of interest. To illustrate this technique, we derive the complete conditional for
boldsymboltheta
formally: begin{aligned} f(boldsymboltheta | sigma^2_theta, boldsymbol a, boldsymbol b, boldsymbol{U} ) & propto frac{ f(boldsymboltheta, sigma^2_theta, boldsymbol a, boldsymbol b | boldsymbol{U} ) } {f(sigma^2_theta, boldsymbol a, boldsymbol b) } \ & propto frac{ f(boldsymbol{U} | boldsymboltheta, boldsymbol a, boldsymbol b ) cdot f(boldsymboltheta | sigma^2_theta ) cdot f(sigma^2_theta ) cdot f(boldsymbol a) cdot f(boldsymbol b) }{ f(sigma^2_theta ) cdot f(boldsymbol a) cdot f(boldsymbol b) } \ & propto f(boldsymbol{U} | boldsymboltheta, boldsymbol a, boldsymbol b ) cdot f(boldsymboltheta | sigma^2_theta ) end{aligned} which has the same result as the heuristic.
Repeating this process we find the other complete conditionals: begin{aligned} f(theta_p|text{rest}) & propto prod_{i=1}^I pi_{pi}^{u_{pi}} (1 – pi_{pi})^{1-u_{pi}} f_text{Normal}(theta_p| 0,sigma_theta^2) ~, \ f(sigma^2_theta|text{rest}) & propto left[ prod_{p=1}^P f_text{Normal}(theta_p| 0,sigma_theta^2) right] f_text{Inverse-Gamma}left(sigma^2_theta left| alpha_theta, beta_theta right. right) \ & propto f_text{Inverse-Gamma}left(sigma^2_theta left| alpha_theta + frac{P}{2}, beta_theta + frac{1}{2} sum_{p=1}^P theta^2_p right. right) \ f(a_i|text{rest}) & propto prod_{p=1}^P pi_{pi}^{u_{pi}} (1 – pi_{pi})^{1-u_{pi}} f_text{Log-Normal}(a_i| mu_a,sigma_a^2) ~, \ f(b_i|text{rest}) & propto prod_{p=1}^P pi_{pi}^{u_{pi}} (1 – pi_{pi})^{1-u_{pi}} f_text{Normal}(b_i| 0,sigma_b^2) , end{aligned} where
text{rest}
pi_{pi}
is defined in Equation ref{eq:pipi}, and
f_star(dagger|dots)
dagger
star
Details of prior specification
We model the ability of person
p
as being an iid draw from a normal distribution with a zero mean and an unknown variance begin{equation*} theta_i stackrel{iid}{sim} text{Normal}left(0, sigma^2_thetaright) quad . end{equation*}
Although this normal distribution fulfills the mathematical role of a prior for
boldsymboltheta
, its interpretation in IRT is as the population distribution of person ability. Therefore, we are interested in estimating the value of
sigma^2_theta
One approach to estimating
sigma^2_theta
is to build a new layer of the model hierarchy so that our sampler will also give us output for
sigma^2_theta
which we can then use to draw inferences. To do this, we give
sigma^2_theta
a prior. A computationally convenient prior is the conjugate prior for the variance of a normal distribution, the inverse-gamma distribution: begin{equation*} sigma^2_theta sim text{Inverse-gamma}left(alpha_theta, beta_theta right) quad end{equation*} where the
alpha_theta
beta_theta
parameters are the inverse-gamma distribution’s shape and scale parameters respectively. Since the prior is conjugate, the posterior is also inverse-gamma: begin{equation*} sigma^2_theta left| boldsymboltheta right. sim text{Inverse-gamma}left( alpha_theta + frac{P}{2}, beta_theta + frac{1}{2} sum_{p=1}^P theta^2_p right) end{equation*} where
P
is the total number of persons.
To set values for
alpha_theta
beta_theta
, we consider the effect of those values on the posterior of
sigma^2_theta
. Since the posterior is a well known distribution, we can consult the Wikipedia page to find the formula for its mean. The posterior mean is: begin{aligned} Egiven{sigma^2_theta}{boldsymboltheta} %& = frac{beta}{alpha-1} \ & = frac{beta_theta + frac{1}{2} sum_{p=1}^P theta^2_p}{ alpha_theta + frac{P}{2} – 1} end{aligned}
alpha_theta
beta_theta
P
, then the posterior mean will be driven almost entirely by the data (via the
theta_p
estimates): begin{aligned} sum_{p=1}^P theta^2_p & approx sigma^2_theta P & & text{and} & Egiven{sigma^2_theta}{boldsymboltheta} & approx frac{ frac{sigma^2_theta P}{2} }{ frac{P}{2}} = sigma^2_theta end{aligned}
so that the posterior mean will be approximately unbiased.
alpha_theta
beta_theta
P
, then the posterior mean will be approximately begin{aligned} Egiven{sigma^2_theta}{boldsymboltheta} & approx frac{ beta_theta }{ alpha_theta – 1} = E{sigma^2_theta} end{aligned} which is the prior mean.
alpha_theta
beta_theta
are of a similar size as
P
, then the posterior mean will be “shrunk” towards the prior mean. This can be useful for stabilizing estimation or incorporating prior information about the value of
sigma^2_theta
from past studies. For example, if a past study had
n
subjects and estimated
s
sigma^2_theta
, then setting begin{aligned} alpha_theta & = frac{n}{2} & & text{and} & beta_theta & = s cdot frac{n}{2} end{aligned} will incorporate the prior information on
sigma^2_theta
into our study in a principled way.
For our purposes, we choose
alpha_theta = beta_theta = 1
. This leads to a “flat” prior for
sigma^2_theta
We model the difficulty of item
i
as being an iid draw from a normal distribution with a zero mean and an unknown variance begin{equation*} b_i stackrel{iid}{sim} Nleft(0, sigma^2_bright) quad . end{equation*} Since items are often regarded as fixed, we are not usually interested in the variance of the item parameters. Therefore we choose a large value of
sigma^2_b
so that the prior is flat over typical values.
sigma^2_b = 100
, which is flat because it is nearly two orders of magnitude larger than a typical item difficulty parameter.
We model the discrimination of item
i
as being an iid draw from a log-normal distribution with an unknown mean and unknown variance begin{equation*} a_i stackrel{iid}{sim} text{Log-Normal}(mu_a,sigma^2_a) quad. end{equation*} Due to the multiplicative of nature of
a_i
in Equation ref{eq:pipi}, we chose a log-normal distribution for its prior. This is because the log-normal distribution is the limiting distribution for random variables which are multiplied, in the same way that the normal distribution is the limiting distribution for random variables which are added together.
The analogue of a mean zero normal distribution is a mode one log-normal. The mode of a log normal is given by begin{equation*} text{mode}left[ a_i right] = e^{mu_a-sigma^2_a} quad, end{equation*} which is equal to 1 when
mu_a = sigma^2_a
. The variance of a mode one log normal is given by begin{aligned} Var{a_i} & = left( e^{sigma^2_a}-1 right) e^{2mu_a + sigma^2_a} \ & = left( e^{mu_a}-1 right) e^{3mu_a} quad end{aligned} which is zero when
mu_a = sigma^2_a = 0
and can be arbitrarily large as
mu_a = sigma^2_a
Due to the multiplicative and highly skewed nature of the log-normal distribution, it is somewhat difficult to gain an intuition for which values of
mu_a = sigma^2_a
lead to an informative prior, and which values lead to a flat prior. Our approach is to approximately match the spread of the prior on the item difficulty parameter
b_i
, by matching the upper quantile of the log-normal to a truncated version of the prior for
b_i
We can find the appropriate values of
mu_a = sigma^2_a
numerically in R as follows:
require(truncnorm)
## truncnorm library. To do so, un-comment the install.packages
## install.packages('truncnorm')
## We use the quantile function to determine the critical
## value where 95% of the mass of a normal distribution
## truncated at zero (a=0), with a mean of 0 (mean=0) and
## standard deviation of 10 (sd=10) lies.
qtruncnorm(p=.95, mean=0, sd=10, a=0)
## We used a guess and check method to find the
## corresponding quantile for a mode 1 log-normal distribution
xx = 1.185 ;qlnorm(.95, xx, sqrt(xx))
As a visual check, below we compare the CDF of a log-normal with
mu_a = sigma^2_a = 1.185
, with the CDF of a truncated version of the prior for
b_i
(a truncated normal):
Note from the zoomed in CDF that the log-normal puts much less mass on values near zero. This is expected because for a given constant
c
, we want to have values less than
frac{1}{c}
to be rare in the same way that values greater than
c
are rare. Further note that the 95% quantile of both distributions are equal.
The code to produce the graph is given below. Note that it requires you to have run the previous R code from this page.
## Set a 2 by 1 grid for the graphs
## Adjust the margins of the plot
## Plot the zoomed in CDF of the truncated normal
curve( ptruncnorm(q=x, a=0, mean=0, sd=sqrt(100)), from=0, to=1,
main="Zoomed in", lty='dotted', lwd=2,
xlab='quantile', ylab='probability')
## Add the CDF of the log-normal to the graph
curve( plnorm(q=x, mean=1.185, sd=sqrt(1.185)), from=0, to=1,
add=TRUE, lwd=2, col='blue')
## Draw an arrow approximately where the two CDFs cross
arrows( x0=0.325, y0=.038, x1=0.45, y1=.038,
length=1/8)
## Draw the legend
legend( "topleft", c('truncated b_i prior', 'log-normal'),
col=c('black','blue'), lwd=2, lty=c('dotted', 'solid'))
## Draw the zoomed out truncated normal CDF in the second panel
## N.B. It draws in the second panel because add=TRUE is missing.
curve( ptruncnorm(q=x, a=0, mean=0, sd=sqrt(100)), from=0, to=60,
main="Zoomed out", lwd=2, lty='dotted',
## Add the the CDF of the log-normal to the graph
curve( plnorm(q=x, mean=1.185, sd=sqrt(1.185)), from=0, to=60,
arrows( x0=10, y0=.95, x1=16.6, y1=.95,
## Reset to a 1 by 1 grid for any future graphs
For our purposes, we choose choose
mu_a = sigma^2_a = 1.185
for the prior of the item discrimination parameter because those values approximately match it to the prior for the item difficulty parameter.
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When writing an expression for part (a) of problem 2-42, Ricardo wrote
2x-3-(x+1)
, while Francine wrote
-3+2x-(x+1)
. Francine states that their expressions are equivalent. Is Francine’s conclusion true or false? Use algebraic properties to justify your conclusion.
Try taking Ricardo's expression and rearranging it. Can you make it into Francine's expression using only ''legal'' moves? If so, then Francine's conclusion is true.
Due to the Associative Property of Addition, Francine is correct.
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Discrete variable inductor - Simulink - MathWorks Nordic
Minimal inductance absolute value (H)
Discrete variable inductor
The Variable Inductor block represents a linear time-varying inductor. It implements a discrete variable inductor as a current source. The impedance is specified by the Simulink® input signal. The inductance value can be negative.
When you use a Variable Inductor block in your model, set the powergui block Simulation type to Discrete and select the Automatically handle Discrete solver and Advanced tab solver settings of blocks parameter in the Preferences tab. The robust discrete solver is used to discretize the electrical model. Simulink signals an error if the robust discrete solver is not used.
The block uses the following equations for the relationship between the voltage, v, across the device and the current through the inductor, i. The block input specifies the value of the inductance. The flux linkage of the inductor is equal to the specified inductance multiplied by the inductor current:
v=\frac{d\left(phi\right)}{dt},
phi=L\ast i.
L — Inductance
Input port associated with the inductance. The inductance can be negative and must be finite.
Specialized electrical conserving port associated with the inductor positive voltage.
Specialized electrical conserving port associated with the inductor negative voltage.
Discrete Solver — Solver type
Backward Euler (default) | Trapezoidal
Robust integration method used by the block. The discrete solver method is automatically set to Trapezoidal when, in the powergui block, in the Preferences settings, you select Automatically handle discrete solver and Advanced tab solver settings of blocks.
The Trapezoidal robust solver is slightly more accurate than the Backward Euler robust solver, especially when the model is simulated at larger sample times. The Trapezoidal robust solver may produce slight damped numerical oscillations on machine voltage in no-load conditions, while the Backward Euler robust solver prevents oscillations and maintains good accuracy.
Minimal inductance absolute value (H) — Minimal inductance
Lower limit on the absolute value of the signal at port L. This limit prevents the signal from reaching a value that has no physical meaning. The value of this parameter must be greater than 0.
Nonlinear Inductor | Nonlinear Resistor | Variable Capacitor | Variable Resistor | Variable-Ratio Transformer
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Selection Sort - Parth Kabra
Selection sort in pseudocode and Python
Computer Science·June 11, 2021
Find the smallest element in the unsorted portion of the array
Swap the smallest element with the first element in the unsorted portion of the array
The sorted portion of the array is on the left side while the unsorted portion is on the right. In the diagrams below, the blue boxes represent the sorted elements of the array, while the white boxes represent the unsorted elements. Elements that are going to be swapped are shown in green boxes.
On every iteration, an element from the unsorted portion is added to the sorted portion. For example, if we start with this array:
At this stage, the unsorted array is the entire array. The smallest element found is 1, so swap that with the first element of the unsorted array.
So after the first iteration, only the first element is sorted:
Now loop through the unsorted array (the items after the first element, which is now sorted). In the unsorted array, swap the smallest item with the first item
After the second iteration, the first two elements are sorted, as another element from the unsorted portion is added to the sorted portion:
This continues n - 1 times, where n is the length of the list.
Note that it runs n - 1 times, since the last element is already sorted.
function selection_sort(list):
n = list.length
for i from 0 to n-1:
# Initialize smallest element to first one
# Traverse through the unsorted list to find the
# smallest element
for j from i to n:
if collection[j] < collection[smallest_index]:
smallest_index = j
# Swap smallest element with first element of
# unsorted list
swap(collection[smallest_index], collection[i])
While the pseudocode will produce expected results, there is a simple optimization we can make.
smallest_index is already given an initial value of i, so there is no need to start at i in the inner for loop. Even if i happens to be the smallest value, smallest_index is already set to i.
We only need to swap collection[smallest_index] and collection[i] if i is not equal to smallest_index.
if smallest_index != i:
collection[smallest_index], collection[i] = (
collection[i],
collection[smallest_index],
To debug, I am going to run the following and print the list at every iteration:
selection_sort([3, 5, 2, 1, 9, 6])
The worst-case scenario is
O(n^2)
since there is a nested loop.
The first iteration has n swaps, the second has n - 1, the third has n - 2, and so on, so
n + (n - 1) + (n - 2) + (n - 3) + ... + 1 = \cfrac{n^2}{2} + \cfrac{n}{2}
n^2
is the highest magnitude, the time complexity is
O(n^2)
The best case scenario is also
\Omega{(n^2)}
since selection sort will not exit if the list is already sorted.
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Compression with considerable sidelobe suppression effect in weather radar | EURASIP Journal on Wireless Communications and Networking | Full Text
Compression with considerable sidelobe suppression effect in weather radar
Haijiang Wang1,2,
Zhao Shi1,2 &
Jianxin He1,2
Pulse compression is a classical topic. Because of its function in resolution enhancement, pulse compression technology has been applied in many kinds of radar such as pulse Doppler weather radar. In pulse compression, sidelobe suppression plays a key role for reducing clutter. In this article, a combination of two sidelobe suppression techniques for pulse compression is proposed. Simulation results show that the combination of the two techniques has better sidelobe suppression effectiveness.
Radar, serving as a kind of method for detecting, has wide-range application areas such as military, aviation, geosciences, and so on. Pulse Doppler weather radar is an important device to detect clouds, wind fields, and precipitation. So, the resolution is very important for the forecasting of weather phenomena.
Pulse compression is useful for weather radar to increase the average transmitted power by transmitting a longer pulse but without reducing the range resolution of the radar. In general, peak power of solid-state transmitter or millimeter-wave klystron is not sufficient. Pulse compression is required to achieve the desired system sensitivity. Different coding schemes include linear frequency modulation (LFM), nonlinear FM. Frequency-modulated signals are characterized by their time bandwidth (TB) product which represents the ability to multiple signal-to-noise ratio (SNR).
The foundation of pulse compression is matched filtering [1]. In time domain, matched filtering is equivalent to the autocorrelation of the input signal. In frequency domain, the transfer function of the matched filter is the complex conjugate of the input signal’s spectrum. An implementation schedule of the matched filter in frequency domain is shown in Figure 1[1].
Implementation of matched filter by conjugate filters pair.
For a radar pulse signal, after compression, the narrower pulse mainlobe is always accompanied by higher sidelobes. So, all pulse compression radars suffer from range side lobes which cause energy from strong reflections to leak into adjacent range cells. High suppression of side lobes is not required in some other non-meteorological radar, but is important for meteorological radar, because weather phenomena can have significant reflectivity gradients, and ground clutter echo can be between 35 and 55 dB much larger than medium rain. Furthermore, the side lobes of strong signal will falsely be recognized as an existence of small target. Therefore, range side lobes must be suppressed by a large amount to prevent contamination in adjacent range cells.
In this article, compression experiments are conducted on an LFM sample signal extracted from a period of echo of a weather radar. In these experiments, some sidelobe suppression algorithms are used. The algorithms include multiplying window in frequency domain, phase distortion, and frequency modification.
In Section 2, the data are analyzed and compression basing on matched filtering and windowed matched filtering are carried out. In Section 3, spectrum modification technique and phase predistortion technique are utilized, respectively, and the effects are analyzed. Then, the two techniques are combined in Section 4. Experiments are conducted on the real weather echo and the effects are compared. Section 5 is the conclusion.
2. LFM pulse and matched filtering compression
A period of weather echo of a pulse Doppler radar is shown in Figure 2. The sample rate of the echo is 4.8 MHz. The pulse width is 33.3 μs and the spectrum width after LFM is 2 MHz.
A period of weather echo of a pulse Doppler radar.
A section which is a whole LFM pulse signal is intercepted from the echo above. This LFM pulse, shown in Figure 3, can be taken as the sample signal to do pulse compression. Its amplitude spectrum is shown in Figure 4.
The LFM pulse intercept from the echo.
The spectrum of the LFM pulse.
To analyze the frequency modulation quality of the pulse in Figure 3, we extract the phase information and unwrap it. The phases before and after unwrapping are shown in Figures 5 and 6, respectively.
The phase before unwrapping.
The phase after unwrapping.
Quadratic curve fitting is done on the phase, the fitting result is illustrated in Figure 7.
Fitting of the phase.
We found that the phase and the fitting curve superpose each other. This indicates that the linearity of the frequency is good and the pulse is a suitable LFM sample.
Matched filtering [1] is carried out on the LFM pulse directly and the compression result is shown in Figure 8.
Compression result by matched filtering.
The amplitude of the first sidelobe is –17.4827 dB and the compression ratio is about 83.3.
A sidelobe of –17.4827 dB is not satisfying for most target detecting, because there will be serious range ambition if the sidelobe is too high.
During matching filtering, multiplying a window in the frequency domain can suppress the sidelobe to a certain level. But in general, the lower sidelobe is often accompanied by a wider mainlobe after windowed matched filtering.
The general formation of a weighting window can be expressed as follows [2]:
H\left(f\right)=K+\left(1-K\right){cos}^{n}\left(\frac{\mathit{\pi f}}{B}\right)
When K = 0.08 and n = 2, the weighting function is a Hamming window; when K = 0.333 and n = 2, the function is a 3:1 taper weighting, when K = 0 and n = 2, 3, 4, respectively, the functions are cosine square, cosine cube, and cosine quartic weighting.
Weighting the LFM pulse in frequency domain with a Hamming window, the window and the compressed pulse after matched filtering are shown in Figures 9 and 10.
The Hamming window.
Compression result by Hamming windowed matched filtering.
It can be seen from Figure 10 that the sidelobe is suppressed to the level of –37 dB. From theoretical analysis we can educe that the mainlobe broaden to 1.47 times of the one before windowing.
For Doppler weather radar, to enhance the accuracy of weather target detecting, a lower sidelobe is needed. So, further improvements of pulse compression effects are demanded.
3. Sidelobe suppression by phase predistortion and spectrum modification
For an LFM signal with TB product, the cubic phase predistortion technique can be used to suppress the sidelobe [3, 4]. The signal with little TB product has large ripples in its spectrum band, so widowing is not satisfying for sidelobe suppression. In this situation, the sidelobe suppression can be achieved by suppressing the ripples in band through phase predistortion. This method is easy to implement with surface acoustic wave (SAW) technique.
Suppose the complex expression of the LMF signal to carry out phase predistortion on is
\begin{array}{l}s\left(t\right)=exp\left[j2\pi \left({f}_{0}t+\frac{B}{2T}·{t}^{2}+\phi \left(t\right)\right)\right]\phantom{\rule{0.2em}{0ex}},\\ -T/2-\mathit{\Delta T}\le t\le T/2+\mathit{\Delta T}\end{array}
where f0 is the central frequency and B is the bandwidth. And the duration of the pulse is T. Then the additional phase item can be
\phi \left(t\right)=\left\{\begin{array}{ll}\frac{\mathit{\Delta B}}{3\Delta {T}^{2}}·{\left(-t-T/2\right)}^{3},\hfill & -T/2-\mathit{\Delta T}\le t<-T/2\hfill \\ \frac{\mathit{\Delta B}}{3\Delta {T}^{2}}·{\left(t-T/2\right)}^{3},\hfill & T/2<t\le T/2+\mathit{\Delta T}\hfill \\ 0,\hfill & \mathit{elsewhere}\hfill \end{array}\right\
where ΔT = 1/B and ΔB = 0.75B[3].
The spectrum of the pulse after phase predistortion is shown in Figure 11 and after multiplying a Hamming window, the predistorted spectrum is illustrated as shown in Figure 12.
Spectrum after phase predistortion.
Windowed spectrum after phase predistortion.
After matched filtering, the effect of pulse compression is shown in Figure 13.
Compression result by windowed matched filtering after phase predistortion.
Figure 13 demonstrated that the sidelobes neighboring to the mainlobe are suppressed well (3–4 dB) through matched filtering after phase-predistortion. But in the positions away from the mainlobe, there are some sidelobe hunches and this may bring range ambiguity of two targets far away from each other.
In ideal situation, the output of the matched filter for an LFM signal has rectangular spectrum. After weighting, the rectangular spectrum becomes a certain window. But when the TB product of the LFM signal is small, its ripples in band are large, so the weighting has little effects for ripple suppression in band. In this case, spectrum modification technique can be resorted to make the processed spectrum approaches ideal window function mostly [5, 6] and to enhance the main-to-side lobe ratio.
Spectrum modification can be implemented by modifying the transfer function of the matched filter. Suppose the spectrum of the LFM signal is U(f) and the transfer function is H(f), to make the output of the matched filtering to be a rectangular, it is needed that:
U\left(f\right)H\left(f\right)=I\left(f\right){I}^{*}\left(f\right),I\left(f\right)=\mathit{rect}\left(f/B\right)
The modified transfer function of the matched filter is
H\left(f\right)={U}^{*}\left(f\right)\left[I\left(f\right){I}^{*}\left(f\right)/{\left|U\left(f\right)\right|}^{2}\right]
The pulse compression result after using spectrum modification is shown in Figure 14.
Compression result by windowed matched filtering after spectrum modification.
From Figure 14, it can be seen that spectrum modification technique has good suppression effect for the sidelobes neighboring the mainlobe. What is the most important, this technique bring a great advantage that suppress the sidelobes far away from the mainlobe to the level under –62 dB.
It is stressed that both phase predistortion and spectrum modification did not spread the mainlobe apparently.
4. Combination of phase predistortion and spectrum modification
From the simulation results in Section 3, we can find that both phase predistortion and spectrum modification have the ability of sidelobe suppression. But phase predistortion can bring hunches far away from the mainlobe and spectrum modification has good effect of suppression to the sidelobes in the distance. So, we predict that the combination of these two techniques will bring better pulse compression performance. The two techniques are used sequentially as shown in Figure 15.
The flow chart of the combined pulse compression technique.
Figure 16 is the result of the simulation utilizing two techniques sequentially.
Compression result with sidelobe suppression by combination of phase predistortion and spectrum modification.
From Figure 16, it can be seen that
The first sidelobe is suppressed to the level under –45 dB, and the sidelobes attenuate quickly when getting far away from the mainlobe.
The envelope of the sidelobes is monotonically decreasing.
The mainlobe does not spread apparently comparing with the result only using weighting function.
In order to compare the effectiveness for pulse compression and sidelobe suppression of the several methods collectively, the data are listed in Table 1.
Table 1 Effectiveness of pulse compression and sidelobe suppression
The whole sky weather echoes before and after pulse compression are shown in Figure 17.
The whole sky weather radar echoes. (a) Echo with no pulse compression, (b) Echo after compression with phase predistortion, (c) Echo after compression with spectrum modification, (d) Echo after compression with phase predistortion and spectrum modification.
It can be seen from Figure 17 that
SNR statistical comparison between phase predistortion method and combined method of phase predistortion and spectrum modification.
The resolution of weather radar echo is very poor without pulse compression.
Phase predistortion and spectrum modification can bring comparative resolution enhancement through sidelobe suppression. But spectrum modification technique brings some false target echo in the upper left corner of the reflectivity section.
The combination of phase predistortion and spectrum modification has the best resolution.
Furthermore, it is evaluated to statistics the weather echoes SNR of three methods. The mean SNR value of compression with phase predistortion is 0.044 dB less than compression with phase predistortion and spectrum modification, the variance value is 0.4528 dB. The mean SNR value of compression with spectrum modification is 0.017 dB less than compression with phase predistortion and spectrum modification, the variance value is 0.2664 dB.
Figure 18 shows the SNR statistical comparison between phase predistortion method and combined method of phase predistortion and spectrum modification.
Figure 19 shows the SNR statistical comparison between spectrum modification and combined method of phase predistortion and spectrum modification.
SNR statistical comparison between spectrum modification and combined method of phase predistortion and spectrum modification.
Sidelobe suppression is an important part in pulse compression. From the simulations and application on weather radar, it can be seen that the combination of phase predistortion and spectrum modification technique has good sidelobe suppression performance. In some applications involving hardware implementation and real-time processing, the SAW device can be used to carry out impulse compression [7, 8]. Resolution enhancing is important for both weather radar and other kinds of radar, such as the imaging radar [9, 10], so it is still a hotspot for research. The methods development by this article can be used in other kinds of radar with pulse compression systems.
Lin M, Ke Y: Theory of Radar Signal. Beijing: National Defense Industry Press; 1984:87-203.
Cook CE, Bernfeld M, Paolillo J, Palmier CA: Matched filtering, pulse compression and waveform design. Microwave. J. 1965, 8: 73-81.
Cook CE, Paolillo J: A pulse compression predistortion function for efficient sidelobe reduction in a high-power radar. Proc. IEEE 1964, 52: 377-389.
Kowatsch M: Suppression of sidelobes in rectangular linear FM pulse compression radar. Proc. IEEE 1982, 70(3):308-309.
Lv Y, Xiang J, Chen F: A method for reducing range-sidelobes of linear frequency-modulated pulse compression signals. J. UEST China 1993, 22(4):344-349.
Yang B, Wu J: A range sidelobe reduction technique based on modifications to signal spectrum. Syst. Eng. Electron. 2000, 22(9):90-93.
Xianmin Z, Jinlin X, Hongtai W, Qing X: SAW pulse compression systems with lower sidelobes. In Microwave Conference Proceedings, APMC'97, vol. 2. Hongkong; 1997:833-835.
Arthur JW: Modern SAW-based pulse compression systems for radar applications. I. SAW matched filters. Electron. Commun. Eng. 1995, 7(6):236-246. 10.1049/ecej:19950604
Li J, Pi Y, Yang X: Micro-doppler signature feature analysis in Terahertz band. J. Infrared Millim. TE 2010, 31(3):319-328.
Xu J, Pi Y, Cao Z: Bayesian compressive sensing in synthetic aperture radar imaging. IET Radar Sonar Navigat. 2012, 6(1):2-8.
College of Electronic Engineering, Chengdu University of Information Technology, Chengdu, China
Haijiang Wang, Zhao Shi & Jianxin He
CMA, Key Laboratory of Atmospheric Sounding, Chengdu, China
Wang, H., Shi, Z. & He, J. Compression with considerable sidelobe suppression effect in weather radar. J Wireless Com Network 2013, 97 (2013). https://doi.org/10.1186/1687-1499-2013-97
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Contradiction — Wikipedia Republished // WIKI 2
Logical incompatibility between two or more propositions
For other uses, see Contradiction (disambiguation).
This diagram shows the contradictory relationships between categorical propositions in the square of opposition of Aristotelian logic.
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."[1]
In modern formal logic and type theory, the term is mainly used instead for a single proposition, often denoted by the falsum symbol
{\displaystyle \bot }
; a proposition is a contradiction if false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition).[2][3] This can be generalized to a collection of propositions, which is then said to "contain" a contradiction.
PROOF by CONTRADICTION - DISCRETE MATHEMATICS
Proof by Contradiction | Method & First Example
Fundamentals of Marx: Contradictions
2 In formal logic
2.1 Proof by contradiction
2.2 Symbolic representation
2.3 The notion of contradiction in an axiomatic system and a proof of its consistency
3.1 Pragmatic contradictions
3.2 Dialectical materialism
4 Outside formal logic
Note: The symbol
{\displaystyle \bot }
(falsum) represents an arbitrary contradiction, with the dual tee symbol
{\displaystyle \top }
used to denote an arbitrary tautology. Contradiction is sometimes symbolized by "Opq", and tautology by "Vpq". The turnstile symbol,
{\displaystyle \vdash }
is often read as "yields" or "proves".
In classical logic, particularly in propositional and first-order logic, a proposition
{\displaystyle \varphi }
is a contradiction if and only if
{\displaystyle \varphi \vdash \bot }
. Since for contradictory
{\displaystyle \varphi }
{\displaystyle \vdash \varphi \rightarrow \psi }
{\displaystyle \psi }
{\displaystyle \bot \vdash \psi }
), one may prove any proposition from a set of axioms which contains contradictions. This is called the "principle of explosion", or "ex falso quodlibet" ("from falsity, anything follows").[5]
Main article: Proof by contradiction
For a set of consistent premises
{\displaystyle \Sigma }
and a proposition
{\displaystyle \varphi }
, it is true in classical logic that
{\displaystyle \Sigma \vdash \varphi }
{\displaystyle \Sigma }
{\displaystyle \varphi }
{\displaystyle \Sigma \cup \{\neg \varphi \}\vdash \bot }
{\displaystyle \Sigma }
{\displaystyle \neg \varphi }
leads to a contradiction). Therefore, a proof that
{\displaystyle \Sigma \cup \{\neg \varphi \}\vdash \bot }
also proves that
{\displaystyle \varphi }
is true under the premises
{\displaystyle \Sigma }
. The use of this fact forms the basis of a proof technique called proof by contradiction, which mathematicians use extensively to establish the validity of a wide range of theorems. This applies only in a logic where the law of excluded middle
{\displaystyle A\vee \neg A}
is accepted as an axiom.
Using minimal logic, a logic with similar axioms to classical logic but without ex falso quodlibet and proof by contradiction, we can investigate the axiomatic strength and properties of various rules that treat contradiction by considering theorems of classical logic that are not theorems of minimal logic.[6] Each of these extensions leads to an intermediate logic:
Double-negation elimination (DNE) is the strongest principle, axiomatised
{\displaystyle \neg \neg A\implies A}
, and when it is added to minimal logic yields classical logic.
Ex falso quodlibet (EFQ), axiomatised
{\displaystyle \bot \implies A}
, licenses many consequences of negations, but typically does not help inferring propositions that do not involve absurdity from consistent propositions that do. When added to minimal logic, EFQ yields intuitionistic logic. EFQ is equivalent to ex contradictione quodlibet, axiomatised
{\displaystyle A\land \neg A\implies B}
, over minimal logic.
Peirce's rule (PR) is an axiom
{\displaystyle ((A\implies B)\implies A)\implies A}
that captures proof by contradiction without explicitly referring to absurdity. Minimal logic + PR + EFQ yields classical logic.
The Gödel-Dummett (GD) axiom
{\displaystyle A\implies B\vee B\implies A}
, whose most simple reading is that there is a linear order on truth values. Minimal logic + GD yields Gödel-Dummett logic. Peirce's rule entails but is not entailed by GD over minimal logic.
Law of the excluded middle (LEM), axiomatised
{\displaystyle A\vee \neg A}
, is the most often cited formulation of the principle of bivalence, but in the absence of EFQ it does not yield full classical logic. Minimal logic + LEM + EFQ yields classical logic. PR entails but is not entailed by LEM in minimal logic. If the formula B in Peirce's rule is restricted to absurdity, giving the axiom schema
{\displaystyle (\neg A\implies A)\implies A}
, the scheme is equivalent to LEM over minimal logic.
Weak law of the excluded middle (WLEM) is axiomatised
{\displaystyle \neg A\vee \neg \neg A}
and yields a system where disjunction behaves more like in classical logic than intuitionistic logic, i.e. the disjunction and existence properties don't hold, but where use of non-intuitionistic reasoning is marked by occurrences of double-negation in the conclusion. LEM entails but is not entailed by WLEM in minimal logic. WLEM is equivalent to the instance of De Morgan's law that distributes negation over conjunction:
{\displaystyle \neg (A\land B)\iff A\vee B}
In mathematics, the symbol used to represent a contradiction within a proof varies.[7] Some symbols that may be used to represent a contradiction include ↯, Opq,
{\displaystyle \Rightarrow \Leftarrow }
, ⊥,
{\displaystyle \leftrightarrow \ \!\!\!\!\!\!\!}
/ , and ※; in any symbolism, a contradiction may be substituted for the truth value "false", as symbolized, for instance, by "0" (as is common in boolean algebra). It is not uncommon to see Q.E.D., or some of its variants, immediately after a contradiction symbol. In fact, this often occurs in a proof by contradiction to indicate that the original assumption was proved false—and hence that its negation must be true.
In general, a consistency proof requires the following two things:
An axiomatic system
A demonstration that it is not the case that both the formula p and its negation ~p can be derived in the system.
When Emil Post, in his 1921 "Introduction to a General Theory of Elementary Propositions", extended his proof of the consistency of the propositional calculus (i.e. the logic) beyond that of Principia Mathematica (PM), he observed that with respect to a generalized set of postulates (i.e. axioms), he would no longer be able to automatically invoke the notion of "contradiction"—such a notion might not be contained in the postulates:
Post's solution to the problem is described in the demonstration "An Example of a Successful Absolute Proof of Consistency", offered by Ernest Nagel and James R. Newman in their 1958 Gödel's Proof. They too observed a problem with respect to the notion of "contradiction" with its usual "truth values" of "truth" and "falsity". They observed that:
The property of being a tautology has been defined in notions of truth and falsity. Yet these notions obviously involve a reference to something outside the formula calculus. Therefore, the procedure mentioned in the text in effect offers an interpretation of the calculus, by supplying a model for the system. This being so, the authors have not done what they promised, namely, "to define a property of formulas in terms of purely structural features of the formulas themselves". [Indeed] ... proofs of consistency which are based on models, and which argue from the truth of axioms to their consistency, merely shift the problem.[9]
Given some "primitive formulas" such as PM's primitives S1 V S2 [inclusive OR] and ~S (negation), one is forced to define the axioms in terms of these primitive notions. In a thorough manner, Post demonstrates in PM, and defines (as do Nagel and Newman, see below) that the property of tautologous – as yet to be defined – is "inherited": if one begins with a set of tautologous axioms (postulates) and a deduction system that contains substitution and modus ponens, then a consistent system will yield only tautologous formulas.
On the topic of the definition of tautologous, Nagel and Newman create two mutually exclusive and exhaustive classes K1 and K2, into which fall (the outcome of) the axioms when their variables (e.g. S1 and S2 are assigned from these classes). This also applies to the primitive formulas. For example: "A formula having the form S1 V S2 is placed into class K2, if both S1 and S2 are in K2; otherwise it is placed in K1", and "A formula having the form ~S is placed in K2, if S is in K1; otherwise it is placed in K1".[10]
In other words, the notion of "contradiction" can be dispensed when constructing a proof of consistency; what replaces it is the notion of "mutually exclusive and exhaustive" classes. An axiomatic system need not include the notion of "contradiction".[citation needed]
Adherents of the epistemological theory of coherentism typically claim that as a necessary condition of the justification of a belief, that belief must form a part of a logically non-contradictory system of beliefs. Some dialetheists, including Graham Priest, have argued that coherence may not require consistency.[12]
In dialectical materialism: Contradiction—as derived from Hegelianism—usually refers to an opposition inherently existing within one realm, one unified force or object. This contradiction, as opposed to metaphysical thinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not cancelling each other out, but actually defining each other's existence. According to Marxist theory, such a contradiction can be found, for example, in the fact that:
Hegelian and Marxist theory stipulates that the dialectic nature of history will lead to the sublation, or synthesis, of its contradictions. Marx therefore postulated that history would logically make capitalism evolve into a socialist society where the means of production would equally serve the exploited and suffering class of society, thus resolving the prior contradiction between (a) and (b).[14]
Mao Zedong's philosophical essay On Contradiction (1937) furthered Marx and Lenin's thesis and suggested that all existence is the result of contradiction.[15]
Outside formal logic
Colloquial usage can label actions or statements as contradicting each other when due (or perceived as due) to presuppositions which are contradictory in the logical sense.
Proof by contradiction is used in mathematics to construct proofs.
The scientific method uses contradiction to falsify bad theory.
Argument Clinic – Monty Python sketch, a Monty Python sketch in which one of the two disputants repeatedly uses only contradictions in his argument
Auto-antonym – Word that has two opposing meanings
Contrary (logic)
Dialetheism — View that there exists statements which are both true and false
Double standard – Inconsistent application of principles
Graham's hierarchy of disagreement
Irony – Rhetorical device and literary technique
Law of noncontradiction
On Contradiction – 1937 Maoist essay by Mao Zedong
Tautology – In logic, a statement which is always true
TRIZ – Problem-solving tools
^ Horn, Laurence R. (2018), "Contradiction", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-12-10
^ "Contradiction (logic)". TheFreeDictionary.com. Retrieved 2020-08-14.
^ "Tautologies, contradictions, and contingencies". www.skillfulreasoning.com. Retrieved 2020-08-14.
^ Dialog Euthydemus from The Dialogs of Plato translated by Benjamin Jowett appearing in: BK 7 Plato: Robert Maynard Hutchins, editor in chief, 1952, Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago.
^ "Ex falso quodlibet - Oxford Reference". www.oxfordreference.com. Retrieved 2019-12-10.
^ Diener and Maarten McKubre-Jordens, 2020. Classifying Material Implications over Minimal Logic. Archive for Mathematical Logic 59 (7-8):905-924.
^ Pakin, Scott (January 19, 2017). "The Comprehensive LATEX Symbol List" (PDF). ctan.mirror.rafal.ca. Retrieved 2019-12-10.
^ Post 1921 "Introduction to a General Theory of Elementary Propositions" in van Heijenoort 1967:272.
^ boldface italics added, Nagel and Newman:109-110.
^ Nagel and Newman:110-111
^ Nagel and Newman:111
^ In Contradiction: A Study of the Transconsistent By Graham Priest
^ Stoljar, Daniel (2006). Ignorance and Imagination. Oxford University Press - U.S. p. 87. ISBN 0-19-530658-9.
^ Sørensen -, MK (2006). "CAPITAL AND LABOUR: CAN THE CONFLICT BE SOLVED?". Retrieved 28 May 2017. {{cite journal}}: Cite journal requires |journal= (help)
^ "ON CONTRADICTION". www.marxists.org.
Józef Maria Bocheński 1960 Précis of Mathematical Logic, translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.
Wikiquote has quotations related to Contradiction.
"Contradiction (inconsistency)", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
"Contradiction, law of", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Horn, Laurence R. "Contradiction". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
Logical truth ⊤
Common logical symbols
|
How To Understand yVault ROI | Yearn.finance
Trying to understand how ROI is calculated? Skip directly to "Why we should use ROI instead of APY to estimate yVaults returns"
If you are a beginner in DeFi or new to Yearn keep on reading!
ROI definition:#
Return on investment (ROI) is a ratio between net profit (over a period) and cost of investment (resulting from an investment of some resources at a point in time). A high ROI means the investment's gains compare favorably to its cost. As a performance measure, ROI is used to evaluate the efficiency of an investment or to compare the efficiencies of several different investments.[1] In economic terms, it is one way of relating profits to capital invested. Source: Wikipedia
ROI is a key performance indicator (KPI) available in all Yearn Vaults (yVaults) located here as "Estimated Yearly Yield": https://yearn.finance/vaults.
ROI is useful when comparing and assessing vault performance.
ROI presented in Yearn is a yearly ROI. You deposit X and 1 year later you receive X + (X * ROI).
The ROI presented is a current estimation based on data since the yVault's inception. If performance remains constant, after 1 year you will receive the displayed ROI. Rates are unstable currently, and fluctuate based on market/strategy.
yVaults have different yield farming strategies, which determine how assets are moved between liquidity pools. Strategies are created by the Controller who manages the yVault.
New strategies are also voted by the community through governance proposals. A new strategy creates a new challenge in terms of ROI calculation.
Individuals interested in participating in a yVault should monitor the ROI presented in the vault dashboard after a strategy change. The rate presented reflects the most recent ROI.
An individual participating before a strategy change might be interested in comparing ROI before and ROI after. Historic ROI, e.g. since yVault creation, can also help users understand performance and inform future decisions.
ROI calculation#
Even though yVaults have a compounding effect inherently, this compound interest is not fixed like in a CeFi savings account. Hence, the concept of APY and APR is not the most accurate way to estimate yVault gains. They are used by the community and we also added them to our newer interfaces, but interpreting them should always be taken with a grain of salt.
Why we should use ROI to estimate yVaults returns?#
This shows the estimation of an asset that has interest / compounded interest applied to it:
This is the actual, measured performance of an asset in the yUSD vault:
A bank's interest rate is constant: either linear or compounding. The interest rate is multiplied by the asset value each period.
yVaults work differently:
After depositing into a yVault, the user receives 'wrapped' tokens. These tokens start with a 'y' prefix and the depositor receives fewer tokens than were deposited (why is explained below).
Starting from its inception, the vault's input and output are governed by the following equation:
F = I * P
F
is the amount of the tokens in the vault,
I
is the amount of 'wrapped' tokens held by users and
P
is the 'price' of those wrapped tokens.
At the start,
P = 1
I = F
, amount of tokens inside the vault is equal to amount of wrapped tokens.
Almost all strategies are identical: invest deposited tokens, accumulate additional tokens which can be withdrawn in the future.
By keeping track of the number of wrapped tokens, and the amount of tokens inside the vault (which increase the longer they are held in the vault) the price can be calculated as follows:
(vault tokens) / (wrapped tokens) = P
Since the amount of wrapped tokens is constant but the amount of vault tokens increases, the price will increase.
The balance and thus price are therefore constantly increasing.
Therefore, the only thing we can do is:
Try to use data from two different points in time
Assume growth is linear (because simply there is no other information we have)
Construct a line using both points which we then extrapolate
Calculate price at future data to display ROI.
However, as shown below this is dependent on where we take those points.
ROI extrapolation applied to the yUSD Vault chart#
Below are examples to show different results possible by applying linear extrapolation to two points of the price chart for the yUSD yVault.
First of all, why and how is this done?
Vault input/output formula:
F = I * P
Now, we want to calculate the increment of
F
which would be our return.
I
is constant, and only
P
changes, if we know
P
at a future date, we can calulate our return.
In order to get the formula for a line having two points we have to do some math:
Formula for a line:
y = m*x + c
y
x
is the block height and
m
c
are what we are looking for.
First, let's get
c
x = 0
(so at inception of the yVault), we know that the price is 1, hence:
y = m*0+c
(x = 0)
y = c
y = 1
(price is 1 when x = 0)
c = 1
Now m (here we have to apply derivatives):
Using the product formula and the constants formula:
y'(x) = m
Approximating the derivative of a linear function can be done by:
y'(x) = (y2-y1)/(x2-x1) = m
y2
being the price at block
x2
y1
the price at block
x1
Finally we have the complete formula and can estimate the price at any date in the future.
y(x)
is not linear, we get different lines for different points (shown below).
As a result, our estimation for price in future and the return varies greatly depending on which points we choose.
Here we take two points of the performance chart for the yUSD vault (numbered colored points) and apply the above. Notice that the different lines are relatively good indications for the short term, but when we try to use them to predict long term they're totally inaccurate!
Jane has 100 yCRV tokens and decides to invest them in the yUSD yVault.
At that time the price
P
is 1.045, the total number of vault tokens is 10,450 and of wrapped tokens 10,000.
10450 / 10000 = 1.045
Now her yCRV tokens get adjusted according to the formula above which is why she sees her now wrapped tokens (yUSD) reduce in quantity and the tokens in the vault (yCRV) are equal to the amount she invested.
So, how many wrapped tokens will she receive?
I = F/P = 100/1.045 = 95.7 yUSD
This action, did not change the price because she supplied to the number of wrapped tokens and vault tokens each exactly according to the ratio of the current price. This is quite important since it means that deposits and withdrawals will not have any effect on the price of the wrapped token.
Fast forward a few days, the strategy uses Jane's funds to yield farm and uses the profits to purchase more tokens held inside the vault, increasing the balance to 10,500. For simplicity's sake, assume there were no other deposits. The wrapped tokens are still 10,000 and thus:
P = 10500/10000 = 1.05
When Jane now looks at her balance of wrapped tokens, she will see that they have incremented to approx.:
F = I * P = 95.7 * 1.05 = 100.5
At this point, she could withdraw and receive her initial yCRV deposit and an additional amount of yCRV tokens, giving her a return of 0.5% on her initial investment (ignoring the 0.5% withdrawal fee).
The short-term ROI data is a suitable estimation for the short-term (i.e. if we compare the % from the last two days, it's likely that the following two days are going go be similar).
Short-term ROI data is absolutely not accurate when extrapolated in the long-term.
Long-term data (say today and inception of vault) is a good overall estimation of the vaults performance and should be used when comparing different investment opportunities.
In other words, if your goal is to approximate returns in the short-term, you should use datasets that are recent (daily/weekly).
If you would like to make a crude estimation on how returns may look like in a year or longer, the longest possible historic timeframe should be taken.
Other references#
The community has been actively creating tools and guides on this topic.
https://github.com/Zer0dot/yearn_roi/blob/master/yearn_vaults_ROI_calc.ipynb provides a mathemathical explanation on how ROI is calculated with some caveats. (This repository is no longer being maintained).
How Yearn shows APY in the v3 interface
Last updated on 5/23/2022 by pandadefi
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How to Understand yveCRV »
Why we should use ROI to estimate yVaults returns?
ROI extrapolation applied to the yUSD Vault chart
|
Unsteady Pressure Measurements With a Fast Response Cooled Probe in High Temperature Gas Turbine Environments | J. Eng. Gas Turbines Power | ASME Digital Collection
Mehmet Mersinligil,
, 72, Chaussée de Waterloo, B-1640 Rhode-Saint-Genèse, Belgium
e-mail: mersinli@vki.ac.be
Jean-François Brouckaert,
e-mail: brouckaert@vki.ac.be
Julien Desset
e-mail: desset@vki.ac.be
Mersinligil, M., Brouckaert, J., and Desset, J. (April 7, 2011). "Unsteady Pressure Measurements With a Fast Response Cooled Probe in High Temperature Gas Turbine Environments." ASME. J. Eng. Gas Turbines Power. August 2011; 133(8): 081603. https://doi.org/10.1115/1.4002276
This paper presents the first experimental engine and test rig results obtained from a fast response cooled total pressure probe. The first objective of the probe design was to favor continuous immersion of the probe into the engine to obtain a time series of pressure with a high bandwidth and, therefore, statistically representative average fluctuations at the blade passing frequency. The probe is water cooled by a high pressure cooling system and uses a conventional piezoresistive pressure sensor, which yields, therefore, both time-averaged and time-resolved pressures. The initial design target was to gain the capability of performing measurements at the temperature conditions typically found at high pressure turbine exit
(800–1100°C)
with a bandwidth of at least 40 kHz and in the long term at combustor exit (2000 K or higher). The probe was first traversed at the turbine exit of a Rolls-Royce Viper turbojet engine at exhaust temperatures around
750°C
and absolute pressure of 2.1 bars. The probe was able to resolve the high blade passing frequency (≈23 kHz) and several harmonics of up to 100 kHz. Besides the average total pressure distributions rom the radial traverses, phase-locked averages and random unsteadiness are presented. The probe was also used in a virtual three-hole mode yielding unsteady yaw angle, static pressure, and Mach number. The same probe was used for measurements in a Rolls-Royce intermediate pressure burner rig. Traverses were performed inside the flame tube of a kerosene burner at temperatures above
1600°C
. The probe successfully measured the total pressure distribution in the flame tube and typical frequencies of combustion instabilities were identified during rumble conditions. The cooling performance of the probe is compared with estimations at the design stage and found to be in good agreement. The frequency response of the probe is compared with cold shock-tube results and a significant increase in the natural frequency of the line-cavity system formed by the conduction cooled screen in front of the miniature pressure sensor were observed.
combustion, cooling, gas turbines, jet engines, Mach number, pressure measurement, pressure sensors, probes, time series
Pressure, Probes, Temperature, Calibration, Combustion, Gas turbines, Sensors, Cooling
, The Lab Gap Matrix, http://www.evi-gti.comhttp://www.evi-gti.com
, Sensor Specifications, http://www.piwg.orghttp://www.piwg.org
Requirements for Instrumentation Technology for Gas Turbine Propulsion Systems
Advanced Measurement Techniques for Aero Engines and Stationary Gas Turbines
VKI LS
Requirements for Advanced High Temperature Instrumentation and Measurements in Gas Turbine Engines
Advanced High Temperature Instrumentation for Gas Turbine Applications
Sensor Needs for Control and Health Management of Intelligent Aircraft Engines
Gyamarthy
Pressure Instrumentation for Gas Turbine Engines—A Review of Measurement Technology
Measurement Methods in Rotating Components of Turbomachines
Unsteady Aerodynamic Measurement Techniques for Turbomachinery Research
Mersinligil
Development of a High Temperature Fast Response Cooled Total Pressure Probe
Probe Measurements in High-Temperature Gases and Dense Plasmas
AGARDograph on Measurement Techniques in Heat Transfer
Modularized Instrument System for Turbojet Engine Test Facilities
Symposium on Instrumentation for Airbreathing Propulsion, Progress in Astronautics and Aeronautics
High Temperature-High Response Fluctuating Pressure Measurements
Proceedings of the 23rd International Instrumentation Symposium
, Las Vegas, NV, May 1–5,
Seventh International Symposium on Air Breathing Engines
, Beijing, People’s Republic of China, Sep. 2–6, pp.
,” Report No. NLR-TR-F.238.
An Air Cooled Jacket Designed to Protect Unsteady Pressure Transducers at Elevated Temperatures in Gas Turbine Engines
Probe Blockage Effects in Free Jets and Closed Tunnels
|
affect - Simple English Wiktionary
affect is one of the 1000 most common headwords.
Affect is on the Academic Word List.
enPR: əfĕkt', IPA (key): /ə'fɛkt/
SAMPA: /@"fEkt/
Affect is on the Academic Vocabulary List.
{\displaystyle x}
{\displaystyle y}
{\displaystyle x}
causes a change in
{\displaystyle y}
Synonyms: move and change
The movement of the soccer ball was affected by the wind.
Sigmund Freud's ideas were affected by the writings of Frederick Nietzsche.
How do you think the new government will affect the economy?
(transitive) If you affect a way of speaking, walking, dressing, etc., you pretend that it is your normal way, often to impress people.
"No, ma'm," Robert said, affecting a southern accent.
(countable & uncountable) Your affect is the emotion or feeling you have in response to an idea, object, etc.
Actors need to bring out the affect of the scene.
(uncountable) Affect is emotion or emotional response.
When teaching, we need to consider both ideas and affect.
Retrieved from "https://simple.wiktionary.org/w/index.php?title=affect&oldid=489414"
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On Kalman Smoothing for Wireless Sensor Networks Systems with Multiplicative Noises
2012 On Kalman Smoothing for Wireless Sensor Networks Systems with Multiplicative Noises
Xiao Lu, Haixia Wang, Xi Wang
The paper deals with Kalman (or
{H}_{\mathrm{2}}
) smoothing problem for wireless sensor networks (WSNs) with multiplicative noises. Packet loss occurs in the observation equations, and multiplicative noises occur both in the system state equation and the observation equations. The Kalman smoothers which include Kalman fixed-interval smoother, Kalman fixedlag smoother, and Kalman fixed-point smoother are given by solving Riccati equations and Lyapunov equations based on the projection theorem and innovation analysis. An example is also presented to ensure the efficiency of the approach. Furthermore, the proposed three Kalman smoothers are compared.
Xiao Lu. Haixia Wang. Xi Wang. "On Kalman Smoothing for Wireless Sensor Networks Systems with Multiplicative Noises." J. Appl. Math. 2012 (SI08) 1 - 19, 2012. https://doi.org/10.1155/2012/717504
Xiao Lu, Haixia Wang, Xi Wang "On Kalman Smoothing for Wireless Sensor Networks Systems with Multiplicative Noises," Journal of Applied Mathematics, J. Appl. Math. 2012(SI08), 1-19, (2012)
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Comparison theorem for $\lambda$-operations in higher algebraic $K$-theory
Comparison theorem for
\lambda
-operations in higher algebraic
K
Nenashev, A.
K
author = {Nenashev, A.},
title = {Comparison theorem for $\lambda$-operations in higher algebraic $K$-theory},
AU - Nenashev, A.
TI - Comparison theorem for $\lambda$-operations in higher algebraic $K$-theory
Nenashev, A. Comparison theorem for $\lambda$-operations in higher algebraic $K$-theory, dans $K$-theory - Strasbourg, 1992, Astérisque, no. 226 (1994), 35 p. http://archive.numdam.org/item/AST_1994__226__335_0/
[ABW] K. Akin, D. Buchsbaum, and J. Weyman, Schur functors and Schur complexes, Adv. in Math. 44 (1982), n° 3, 207-278. | Article | Zbl 0497.15020
Q
-construction, Illinois J. Math. 31 (1987), n° 4, 574-597. | Zbl 0628.55011
[G1] D. Grayson, Exact sequences in algebraic
K
-theory, Illinois J. Math. 31 (1987), n° 4, 598-617. | Zbl 0629.18010
[G2] D. Grayson, Exterior power operations on higher
K
K
-Theory 3 (1989), n° 3, 247-260. | Article | Zbl 0701.18007
[G3] D. Grayson, Adams operations on higher
K
-theory, preprint, Univ. of Illinois at Urbana-Champaign (1991), 10 p. | Zbl 0776.19001
[Hi] H. Hiller,
\lambda
K
-theory, J. Pure Appl. Algebra 20 (1981), 241-266. | Article | Zbl 0471.18007
[Kr] Ch. Kratzer,
\lambda
K
-théorie algébrique, Comment. Math. Helv. 55 (1980), n° 2, 233-254. | Article | EuDML 139823 | Zbl 0444.18008
[N] A. Nenashev, Simplicial definition of
\lambda
-operations in higher
K
-theory, Adv. in Soviet Math. 4 (1991), 9-20. | Zbl 0735.19004
[W1] F. Waldhausen, Algebraic
K
-theory of generalized free products, Annals of Math. 108 (1978), 135-256. | Article | Zbl 0397.18012
K
-theory of spaces, Lecture Notes in Math. 1126, Springer (1983), 318-419. | Zbl 0579.18006
[Kö] B. Köck, Shuffle products in higher
K
-theory, to appear in J. Pure Appl. Algebra 92 (1994). | Article | Zbl 0797.19003
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DBMS and SQL (OLD) - Test Papers 02
Class-12 Computer Science(SQL)
General Instruction: -
Question No. 1 to 6 carry Two marks,
Question No. 7 to 10 carry Six marks.
What is the difference between column constraint and table constraint? Name some database integrity constrains.
Give examples of some DDL commands and some DML commands.
What is the difference between Unique and Primary Key constraint?
Compare DISTINCT and ALL keywords when used with SELECT command.
How does following constraint work? (iii) Default (iv) Check
Consider the following tables FACULTY and COURSES. Write SQL commands for the statements (i) to (iv) and give outputs for SQL queries (v) to (vi)
i. To display details of those Faculties whose salary is greater than 12000.
ii. To display the details of courses whose fees is in the range of 15000 to 50000 (both values included).
iii. To increase the fees of all courses by 500 of “System Design” Course.
iv. To display details of those courses which are taught by ‘Sulekha’ in descending order of courses.
v. Select COUNT(DISTINCT F_ID) from COURSES;
vi. Select MIN(Salary) from FACULTY,COURSES where COURSES.F_ID = FACULTY.F_ID;
Write the SQL commands and write outputs for SQL commands given below on basis of table MOV
Table: MOV
i. Find the total value of the movie cassettes available in the library.
ii. Display a list of all movies with Price over 20 and sorted by Price.
iii. Display all the movies sorted by Qty in decreasing order.
iv. Display a report listing a movie number, current value and replacement value for each movie in the above table. Calculate the replacement value for all movies as QTY * Price * 1.15
v. Count the number of movies where Rating is not “G”.
vi. Increase the price of Comedy type by 10.
Consider the following tables DRESS and MATERIAL. Write SQL commands for the statements (i) to (iv) and give outputs for SQL queries (v) to (viii).
Table: DRESS
ONFORMAL SHIRT
i. To display DCODE and DESCRIPTION of each dress in ascending order of DCODE.
ii. To display the details of all the dresses which have LAUNCHDATE in between 05-DEC-07 AND 20-JUN-08 (inclusive of both the dates).
iii. To display the average PRICE of all the dresses which are made up of material with MCODE as M003.
iv. To display material we highest and lowest price of dresses from DRESS table. (Display MCODE of each dress along with highest and lowest price)
v. SELECT SUM(PRICE) FROM DRESS WHERE MCODE = ‘M001’;
vi. SELECT DESCRIPTION, TYPE FROM DRESS, MATERIAL WHERE DRESS.MCODE=MATERIAL.MCODE AND DRESS.PRICE >= 1250;
vii. SELECT MAX(MCODE) FROM MATERIAL;
viii. SELECT COUNT(DISTINCT PRICE) FROM DRESS;
The difference between column constraint and table constraint is that column constraint applies only to individual columns, whereas table constraints apply to groups of one or more columns.
Following are the few of database integrity constrains:Unique constraint
- Primary Key constraint
- Default constraint
- Check constraint
Unique: This constraint ensures that no two rows have the same value in the specified columns. For eg , CREATE TABLE employee (ecode integer NOT NULL UNIQUE, ename char(20),Sex char(2) );
Primary Key: Primary key does not allow NULL value and Duplicate data in the column which is declared as Primary Key.
For eg , CREATE TABLE employee (ecode integer NOT NULL PRIMARY KEY, ename char(20),Sexchar(2) );
DISTINCT keyword is used to restrict the duplicate rows from the results of a SELECT statement. ALL keyword retains the duplicate rows, by default ALL keyword is use by SELECT statement.
Places conditions on individual rows.
Places conditions on groups.
Cannot include aggregate function.
Can include aggregate function.
For eg. SELECT * FROM student WHERE Rno >=10;
For eg. SELECT AVG(marks) FROM student GROUP BY grade HAVING grade = ‘B1’;
Default: When a user does not enter a value for the column, automatically the defined default value is inserted in field. A column can have only one default value.
For eg , CREATE TABLE employee (ecode integer NOT NULL PRIMARY KEY, ename char(20), Sexchar(2), Grade char(2) DEFAULT = ‘E1’ );
Check: This constraint limits values that can inserted into a column of table.
For eg , CREATE TABLE employee (ecode integer NOT NULL PRIMARY KEY, ename char(20),Sex char(2) , Grade char(2) DEFAULT = ‘E1’, Gross decimal CHECK (gross > 2000 );
SQL stands for Structured Query Language. It is a unified, non-procedural language used for creating, accessing, handling and managing data in relational databases.
i. SELECT * FROM FACULTY WHERE SALARY > 12000
ii. SELECT * FROM COURSES WHERE FEES BETWEEN 15000 AND 50000
iii. UPDATE COURSES SET FEES = FEES + 500 WHERE CNAME = “System Design”
iv. SELECT * FROM FACULTY FAC,COURSES COUR WHERE FAC.F_ID = COUR.F_ID AND FAC.FNAME = 'Sulekha' ORDER BY CNAME DESC
v. COUNT(DISTINCT F_ID)
vi. MIN(SALARY)
i. SELECT COUNT(TITLE) FROM MOV;
ii. SELECT * FROM MOV WHERE PRICE>20 ORDER BY PRICE;
iii. SELECT * FROM MOV ORDER BY QTY DESC;
iv. SELECT NO,PRICE AS 'CURRENT VALUE',(QTY*PRICE*1.15) AS 'REPLACEMENT VALUE' FROM MOV;
v. SELECT COUNT(TITLE) FROM MOV WHERE RATING<>'G';
vi. UPDATE MOV SET PRICE=PRICE+10 WHERE TYPE=’Comedy’;
i) SELECT DCODE,DESCRIPTION FROM DRESS ORDER BY DCODE;
ii) SELECT * FROM DRESS WHERE LAUNCHDATE BETWEEN '05-DEC-07' AND '20-JUN-08';
iii) SELECT AVG(PRICE) FROM DRESS WHERE MCODE='M003';
iv) SELECT B.MCODE,TYPE,MAX(PRICE) AS "HIGHEST",MIN(PRICE) AS "LOWEST" FROM DRESS A, MATERIAL B WHERE A.MCODE=B.MCODE GROUP BY TYPE;
\frac{SUM\left(PRICE\right)}{2700}
\frac{MAX\left(MCODE\right)}{M004}
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Smallest denormalized quantized number for quantizer object - MATLAB denormalmin - MathWorks Australia
Smallest denormalized quantized number for quantizer object
x = denormalmin(q)
x = denormalmin(q) is the smallest positive denormalized quantized number where q is a quantizer object. Anything smaller than x underflows to zero with respect to the quantizer object q. Denormalized numbers apply only to floating-point format. When q represents a fixed-point number, denormalmin returns eps(q).
When q is a floating-point quantizer object,
x={2}^{{E}_{min}-f}
where Emin is equal to exponentmin(q).
When q is a fixed-point quantizer object,
x=\mathrm{eps}\left(q\right)={2}^{-f}
denormalmax | eps | quantizer
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Popular Science Monthly/Volume 19/October 1881/About Comets - Wikisource, the free online library
Popular Science Monthly/Volume 19/October 1881/About Comets
Increase and Movement of the Colored Population II
About Comets by Aaron Nichols Skinner
The Connection of the Biological Sciences with Medicine
627465Popular Science Monthly Volume 19 October 1881 — About Comets1881Aaron Nichols Skinner
By AARON NICHOLS SKINNER,
UNITED STATES NAVAL OBSERVATORY, WASHINGTON, D. C.
THE study of astronomy reaches back to the very beginnings of history, and through all the ages the ablest intellects have been directed to the wellnigh impossible task of unraveling the celestial motions. The terrestrial observer not being located at the center of the motions of the solar system, the complexity arising from this compounding of the motion of the observer with the motion of the planet observed rendered the problem very difficult.
Copernicus furnished the key, by showing that the sun and not the earth is the center of the solar system. Tycho Brahe soon followed, and furnished an extensive series of accurate observations that afforded Kepler the material upon which he based his studies that developed those immortal laws defining the forms of the orbits of the planets, the character of their motions, and the relation between the dimensions of their orbits and their periods of revolution.
It remained for Newton to discover the existence of the law of universal gravitation, of which Kepler's laws are an immediate sequence.
Thus the secrets of the motions of the planets were explained. But comets, those erratic visitants of our system, whose advent in olden time filled the mind with universal awe, were still an unfathomed mystery. Suddenly they would blaze out in the sky, and as suddenly pass out of sight, and no astronomer could tell whence they came or whither they went, or the laws which governed their motions.
Newton first showed that comets also were obedient to the attraction of gravitation. He demonstrated this fact by means of the comet of 1680. The orbit of this comet he found not to differ perceptibly from a parabola.
After Newton, Edmund Halley, from a careful study of the comets of 1531, 1607, and 1682, ventured the assertion that these were only different appearances of one and the same body, whose period of revolution was about seventy-five years. Halley, consequently, predicted a reappearance of this comet in 1759. This comet was shown to move in a very elongated ellipse. In accordance with prediction, reappearances of this comet occurred in 1759 and 1835.
Since the time of Newton all the comets which have come to view have been submitted to a careful study. To determine the orbit of any newly discovered member of our system, it is necessary that its direction in space from the earth at three dates, as nearly equidistant as may be, should be determined by observation.
The data for the problem are, then, as follows: the positions of the earth with reference to the sun at three different dates, and the positions of the heavenly body with reference to the earth at the same dates. The unknown elements which describe the character of the orbit and its position in space are as follows:
I. The mean longitude of the body at any convenient epoch.
II. The semi-major axis of the orbit.
III. The eccentricity of the orbit.
IV. The longitude of the perihelion.
V. The longitude of the ascending node.
VI. The inclination between the orbit-plane and the plane of the earth's orbit.
Of the above, I indicates the position of the body in the orbit at some definite time; II gives the greatest semi-diameter of the ellipse; III gives the ratio of the distance of the focus from the center divided by the semi-major axis; IV, with VI, gives the position in space of the greatest diameter of the ellipse; Y gives the position of the line of intersection between the plane of the unknown orbit and the plane of the earth's orbit.
From II may be determined immediately the period of revolution by means of Kepler's law as follows: if a and a′ are respectively the semi-major axis of the unknown orbit and the earth's orbit, and t and t′ the respective periods of revolution, then we have from Kepler's law—
{\displaystyle \scriptstyle {\frac {t^{2}}{a^{3}}}={\frac {t'^{2}}{a'^{3}}}}
{\displaystyle \scriptstyle t=\left({\frac {a^{3}t'^{2}}{a'^{3}}}\right)^{\frac {1}{2}}}
If the eccentricity of the orbit is very large, the portion of the ellipse in the vicinity of the perihelion approximates to a parabola, which it becomes when the eccentricity equals unity.
As a matter of history, the great majority of comet orbits hitherto studied are either parabolas or are portions of excessively elongated ellipses, so as to be indistinguishable from parabolas, at least in the part of the orbit traversed during visibility. This portion of the orbit is always adjacent to the perihelion.
From the foregoing fact, and moreover because the computation of a parabolic orbit is much simpler, there being one less unknown quantity, preliminary comet orbits are always parabolic. Subsequent investigations show whether the comet deviates perceptibly from the parabola computed.
On October 10, 1880, Lewis Swift, of Rochester, New York, discovered a comet which has proved to be of peculiar interest. From its first discovery it has presented no brilliancy of appearance, for, during its period of visibility, a telescope of considerable power was necessary to observe it. Since this comet when in close proximity to the earth was very faint indeed, its dimensions must be quite moderate.
As soon after its apparition as the necessary observations of position were obtained, its parabolic elements were computed by several astronomers. After carefully comparing these elements with those of previous comets, Mr. S. C. Chandler, of Boston, remarked the striking similarity between them and those of Comet III of 1869. He immediately suspected them to be one and the same body, revolving in an elongated ellipse, having a period of eleven years, or a sub-multiple of eleven years.
Mr. Chandler hereupon made some extended investigations, to determine which period was the more probable. He showed that the observed positions could be satisfied more closely with a period of five and one half years.
It seemed very desirable that elliptic elements should be determined for this comet without making any previous assumptions in reference to any of the elements; this was undertaken independently by two astronomers of the United States Naval Observatory, each from different data. Professor Frisby made use of observations of October 25th, November 7th, and November 20th. Mr. Upton selected the following dates: October 25th, November 23d, and December 22d.
The results of these two computations agree very closely: the resulting period is only a few days less than six years. The inclination of the plane of the orbit to the plane of the ecliptic is about five and one half degrees.
To show more strikingly the remarkable situation of the comet's orbit with reference to the earth's orbit, the attention of the reader is directed to the accompanying diagram (Fig. 1), which, for the sake of
simplicity, shows the two orbits as if in one plane, when in reality the angle of inclination between them is about five and one half degrees. The line marked "line of nodes" is the line of their mutual intersection, the part of the comet's orbit in the vicinity of the perihelion being north of the plane of the ecliptic.
The relative situations of the earth and comet are shown by their positions in orbit at the date of discovery of the comet, October 10; the date of the perihelion passage, November 8, 1880, and January 1, 1881.
The nearest approach of the comet to the earth was about November 18, 1880, when it was distant from the earth 0·13 of the earth's distance from the sun. The period, as determined by Professor Frisby and Mr. Upton, is probably somewhat too large, owing to the uncertainty arising from the shortness of the arc of observation. The length of the period of revolution affords a reason for the fact that the comet escaped observation at its last return; since then it must have been in the direction of the sun.
It will be seen, from the drawing, that at aphelion the comet passes beyond the orbit of the planet Jupiter.
About the 22d of June last, a comet flashed into view which was unexpected as it was brilliant. It was seen with the unassisted eye by a multitude of persons in widely separated localities. Among the earliest
of those who discovered its presence in the northern sky was Mr. G. W. Simmons, of Boston, Massachusetts, who chanced to be in camp at Morales, Mexico. This gentleman first saw it on the morning of June 20th. It had, however, been discovered nearly one month earlier by Mr. Tebbutt, of New South Wales, Australia, on May 22d. During the interval between these two dates it had moved northward through an arc of more than 60°, which rapid motion accounts for its sudden apparition in our northern sky.
The relative situation of the orbits of the comet and the earth will be best understood by the perspective view of a model of the two orbits constructed to scale (Fig, 2). This model was executed, from elements computed by Messrs. Chandler and Wendell, of Harvard College Observatory, by Ensign S. J. Brown, U.S.N., who kindly placed it at the service of the writer.
In this cut, the horizontal plane represents the position of the earth's orbit, and the plane cutting this at a large angle represents the plane of the comet's orbit. The comet moved from below, which is the southern side, up through the plane of the earth's orbit to the northern side. The dates indicate the positions of the earth and comet at different times in their respective orbits. It passed its perihelion point just before passing through the plane of the earth's orbit.
The orbit of the comet is inclined to the plane of the earth's orbit at an angle of 63°. Its perihelion distance is 0·77 of the earth's distance from the sun. It arrived at its perihelion June 16th, and was nearest the earth June 19th, when its distance from the earth was 0·28 of the earth's distance from the sun.
The nucleus attained fully the brightness of a first-magnitude star, and the length of the tail was variously estimated at from 20° to 30°. This comet is still faintly visible to the naked eye (August 22d).
At first it was suspected that this comet was identical with that of 1807, but later investigation disproved this supposition.
Retrieved from "https://en.wikisource.org/w/index.php?title=Popular_Science_Monthly/Volume_19/October_1881/About_Comets&oldid=8850775"
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(Redirected from Binary notation)
The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implemention.[1]
The Indian scholar Pingala (c. 2nd century BC) developed a binary system for describing prosody.[9][10] He used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.[11][12] They were known as laghu (light) and guru (heavy) syllables.
{\displaystyle \lor }
{\displaystyle 1}
{\displaystyle \lor }
{\displaystyle 1=1}
{\displaystyle 1+1=10}
{\displaystyle \land }
{\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}}
{\displaystyle {\begin{matrix}({\frac {1}{2}})^{2}={\frac {1}{4}}\end{matrix}}}
{\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}}
{\displaystyle {\begin{matrix}({\frac {1}{3}})\end{matrix}}}
{\displaystyle {\begin{matrix}{\frac {1}{3}}\end{matrix}}}
{\displaystyle {\begin{matrix}{\frac {1}{3}}\times 2={\frac {2}{3}}<1\end{matrix}}}
{\displaystyle {\begin{matrix}{\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1\end{matrix}}}
{\displaystyle {\begin{matrix}{\frac {1}{3}}\times 2={\frac {2}{3}}<1\end{matrix}}}
{\displaystyle {\begin{matrix}{\frac {2}{3}}\times 2=1{\frac {1}{3}}\geq 1\end{matrix}}}
{\displaystyle {\begin{aligned}x&=&1100&.1{\overline {01110}}\ldots \\x\times 2^{6}&=&1100101110&.{\overline {01110}}\ldots \\x\times 2&=&11001&.{\overline {01110}}\ldots \\x\times (2^{6}-2)&=&1100010101\\x&=&1100010101/111110\\x&=&(789/62)_{10}\end{aligned}}}
{\displaystyle x}
{\displaystyle x}
{\displaystyle x}
{\displaystyle x}
{\displaystyle {\frac {p}{2^{a}}}}
{\displaystyle {\frac {1_{10}}{3_{10}}}={\frac {1_{2}}{11_{2}}}=0.01010101{\overline {01}}\ldots \,_{2}}
{\displaystyle {\frac {12_{10}}{17_{10}}}={\frac {1100_{2}}{10001_{2}}}=0.1011010010110100{\overline {10110100}}\ldots \,_{2}}
{\displaystyle {\sqrt {2}}}
^ "3.3. Binary and Its Advantages — CS160 Reader". computerscience.chemeketa.edu. Retrieved 22 May 2022.
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Periodic Classification of Elements - Test Papers
CBSE Test Paper 01
How does the valency vary in going down a group? (1)
first increases, then decreases
How many elements are placed in lanthanide and actinide series? (1)
Answer the questions on the basis of the following table, name the family of H, A, C. (1)
What is the number of valence electrons in the last element of 3rd period? (1)
Element X forms a chloride with the formula
\mathrm{X}\mathrm{C}{\mathrm{l}}_{2}
which is a solid with a high melting point. X would most likely to be in the same group of the periodic table as (1)
Which of the following are chemically similar? (1)
7A, 9B, 15C and 18D
Give the name and electronic configuration of second alkali metal? (1)
Helium is an unreactive gas and neon is a gas of extremely low reactivity. What, if anything, do their atoms have in common? (1)
How many periods and groups are present in the long form of periodic table? (1)
In the modern Periodic Table calcium (atomic number 20) is surrounded by elements with atomic number 12, 19, 21 and 38. Which of these have physical and chemical properties resembling calcium? (3)
Lithium, Sodium, Potassium are all metals that react with water to liberate hydrogen gas. Is there any similarity in the atoms of these elements? (3)
Name two elements you would expect to show same kind of chemical reactivity as magnesium. What is the basis for your choice? (3)
Atomic number of an element is 16. Write its electronic configuration. Find the number of valence electrons and its valency. (3)
Why is Long Form of Periodic Table regarded better than Mendeleev's Periodic table? (5)
Given below are few elements of the modern periodic table. Atomic number of the element is given in the parenthesis (5)
A (4) , B(9), C(14), D(19), E(20)
Select the elements that has one electron in the outermost shell. Also write the electronic configuration of this element.
Which two elements amongst these, belong to the same group? Give reason for your answer.
Which two elements amongst these belong to the same period? Which one of the two has bigger atomic radius?
Explanation: Since the number of vallence electrons in a group is the same, all the elements in a group have the same valency.
Explanation: The elements with atomic numbers 57 to 71 are called lanthanide series and the elements with atomic numbers 89 to 103 are called actinide series.
Explanation: The elements of group 1 are called alkali metals as they have one electron in their outermost shell and readily lose that electron to form positive ions.
Explanation: The element in the last of the 3rd period is a noble gas i.e. its octet is complete. So, it has 8 valence electrons in its outermost shell.
Explanation: This is because Mg has a valency of +2 and would easily give its two electrons to complete its outermost electronic configuration.
So, it combines with chlorine to form XCl2
Electronic configuration of A7: (2,5)
Electronic configuration of C15: (2,8,5)
A and C have similar chemical properties because both have same electrons (5 electron) in their outer most shell.
Sodium (Na) (2, 8, 1).It has 1 electron in valence shell with valency is also 1.
Both have the valency zero and are in the same group so they have same chemical property of not reacting with any element.
The vertical columns in the periodic table are known as groups while the horizontal rows are known as periods. The long form of periodic table has seven periods and eighteen groups.
Elements in a group have similar properties. Elements with atomic numbers 12 and 38 lie in the same group as calcium. Therefore, they will have properties resembling calcium.
Lithium, sodium and potassium all react with water to form alkalis, i.e., lithium hydroxide, sodium hydroxide, potassium hydroxide, etc. with the liberation of hydrogen gas.
All these metals have one electron in their respective outermost shells.
Magnesium (Mg) belongs to group 2 known as alkaline earth family. The two other elements belonging to thesame group are calcium (Ca) and strontium (Sr). The basis of choice is the electronic distribution in the valence shell of these elements. All of them have two electrons each.
Mg (Z=12) 2 8 2
Ca(Z=20) 2 8 8 2
Sr(Z=38) 2 8 18 8 2
Since, atomic number of the given element is 16. Hence, the element is sulphur(S). It is kept in group VI A or group 16 in modern periodic table.
Electronic configuration =
\stackrel{K}{2,}\phantom{\rule{thickmathspace}{0ex}}\stackrel{L}{8,}\phantom{\rule{thickmathspace}{0ex}}\stackrel{M}{6}
Number of valence electrons in sulphur are 6, present in M-shell. Its valency is 2 as it requires 2 electrons to complete its octet or achieve the nearest noble gas configuration.
Long form of Periodic table is regarded better than the Mendeleev's periodic table due to the following reasons:
It is based upon atomic number which is considered better than the atomic mass because the properties of the elements are related to the atomic number.
It explains why the elements placed in a group show similar properties but Mendeleev's Periodic Table gives no explanation for the same.
All groups in the Periodic table are independent groups and there are no sub-groups as in Mendeleev's Periodic Table.
Many defects in the Mendeleev's Periodic Table have been removed.
There is no confusion regarding the position of isotopes because all the isotopes of an element have the same atomic number.
The periodic table is more systematic than the Mendeleev's table and is easy to remember.
In Mendeleev's periodic table transition elements included with other elements. In Modern periodic table transition elements placed in separate block.
D (19) has one electron is its outermost shell. Electronic configuration of the element is given below. The element is potassium written as K, it belong to group I A in periodic table.
D\left(18\right)\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\frac{K}{2,}\phantom{\rule{thickmathspace}{0ex}}\frac{L}{8,}\phantom{\rule{thickmathspace}{0ex}}\frac{M}{8,}\phantom{\rule{thickmathspace}{0ex}}\frac{N}{1}
A(4) and E(20) belongs to same group as they have same number of valence electrons. A is beryllium symbol Be and E is calcium Symbol Ca. Both belong to group II A in the periodic table.
A\phantom{\rule{thickmathspace}{0ex}}\left(4\right)\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\frac{K}{2,}\phantom{\rule{thickmathspace}{0ex}}\frac{L}{2,}
E\phantom{\rule{thickmathspace}{0ex}}\left(20\right)\phantom{\rule{thickmathspace}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\frac{K}{2,}\phantom{\rule{thickmathspace}{0ex}}\frac{L}{8,}\phantom{\rule{thickmathspace}{0ex}}\frac{M}{8,}\phantom{\rule{thickmathspace}{0ex}}\frac{N}{2,}
A(4) and B(9) belong to same period and A(4) has bigger radius than B(9). Also the elements D(19) and E(20) belong to same period and D(19) has bigger radius than E(20).
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compressor - zxc.wiki
This article is about fluid power machines - see Compression and Compression (Terminology) for other meanings .
Large compressor in a coking plant
Electric piston compressor with boiler for compressed air production
A compressor ( compressor ) is a machine ( fluid energy machine ) that an enclosed gas mechanical work feeding; Compressors for compressing of gases used. They increase the pressure and density of the gas. Machines in which a low compression is a side effect when transporting gases are called ventilators or blowers and are generally not counted among the compressors. Machines that increase the pressure of liquids are called pumps . Compressors that generate a negative pressure (vacuum) and work against the air pressure are called vacuum pumps .
2 Delivery quantity and operating pressure
4.1 Piston Compressor
4.3 Turbo compressor
4.4 Other types of compressors
If the volume of a gas is reduced by compressing it, one speaks of compression or compression. Corresponding devices are called compressors or compressors. During compression processes, an existing intake volume V 1 is compressed to a smaller volume V 2 with the operating pressure p 1 . There is an increased pressure p 2 in the smaller volume V 2 , and the gas is heated during the compression process.
Since the volume decreases during compression, it is crucial to indicate the respective pressure state for the volume flow. The usual specifications are the suction volume flow (based on suction pressure p 1 ), outlet volume flow (based on the final pressure p 2 ) and the standard volume flow (based on the standard state p = 101.3 kPa, T = 293.15 K = 20 ° C).
For compaction operations, shall remaining constant temperature the Boyle's law .
When applying this law, it must be ensured that p 1 and p 2 are absolute pressures. However, all pressure specifications for pneumatic systems refer to the overpressure Pe compared to atmospheric pressure. Otherwise pressure information will be specially marked.
Pressure specifications in pneumatics relate to overpressure
Pressure gauges in pneumatics are set to overpressure
Delivery quantity and operating pressure
The achievable pressure and the delivery quantity are used to identify a compressor. The delivery quantity is the volume of gas released per period; For small systems it is given in liters / min, otherwise in m 3 / min. Often common but misleading is the specification of the (theoretical) suction power as the product of speed and displacement . It says nothing about the actual delivery rate, since the volumetric degree of filling is neglected.
Delivery quantity - volume of the dispensed fluid per unit of time.
Operating pressure - achievable overpressure.
Pressure ratio = final pressure / suction pressure
{\ displaystyle \ Pi = p_ {2} / p_ {1}}
Delivery rate - Describes the ratio of the conveyed to the theoretically possible (due to the geometry) volume flow.
In order to be able to compare compressors of different types and operating points better, the standard volume flow is often considered. This is the volume flow of the compressor converted to standard conditions (temperature, pressure, humidity).
Mobile construction site compressor, around 1910, with a combustion engine drive
Roots blower for racing use
Main article : reciprocating compressor
A distinction is made between reciprocating compressors and rotary lobe compressors.
In reciprocating compressors, the gas in a cylinder is sucked into the working chamber by a piston moving back and forth, where it is compressed and then expelled again. These compressors work cyclically, have low volume flows and high pressure ratios . The suction and discharge valves are automatically operating plate valves.
There are different types of rotary piston or rotary compressors ( Roots blowers , vane compressors , screw compressors, scroll compressors ). What they have in common is that the working space is formed between the housing and one or more displacers (rotary pistons) that rotate or move on a circular path. The gas is sucked in and expelled through slits that the piston opens and closes as it moves.
Main article : screw pump
The two rotors of a screw compressor
Model of a screw compressor
The screw compressor belongs to the group of rotating, twin-shaft positive displacement compressors with internal compression. It has a simple structure, small dimensions, a small mass, uniform, pulsation-free delivery and smooth running because it lacks oscillating masses and control elements. It reaches up to 30 bar overpressure.
The idea to build a screw compressor came up in 1878, but the geometry of the surfaces could not be created due to technical difficulties. About half a century later, in 1930, the technical prerequisites for manufacturing the complicated screw geometry were in place. In 1955, a Swedish engineer named Alfred Lysholm managed to manufacture and successfully use the world's first screw compressor. Initially, however, the compressor could not prevail over the conventional piston compressor. The internal losses at the screws were too great to speak of an effective compressor and, above all, of an alternative to the piston compressor. Another 40 years had to pass before the decisive point in improving the efficiency was finally found. Oil injection into the compressor stage reduces the loss rate considerably and at the same time serves as cooling for the compressor block. In addition, roller bearings could then be used instead of the plain bearings that had been common up until then . All in all, this knowledge led to very simply built, yet robust compressors. Screw compressors have now proven their worth - about half of all compressors currently in use are screw compressors.
Two rotors arranged in parallel, mechanically (usually by a pair of gears) forcibly coupled rotors with interlocking, helical teeth in a housing are the heart of this system. At the rolling line between the two shafts (the point where the two helical shafts touch) the passage for the medium to be conveyed is mechanically closed (by the toothing). The medium is located in the tooth channels and is held in it by the housing wall. It is conveyed in the axial direction. There are openings for the inlet (suction side) and outlet (pressure side) in the housing on the two end faces of the axles. The length of the rotors, the pitch of the teeth and the inlet and outlet openings must be adjusted so that there is no direct passage from the pressure to the suction side, i.e. no backflow can occur. Apart from the losses, the volume flow of the medium is proportional to the speed .
The medium (for example air) flows into the tooth gears on the suction side until the gear closes at the rolling point when turning further on the suction side. It now forms a helical air hose around the rotor . With further rotation, the toothing opens on the pressure side, and the air is pumped out of the machine by further rotation.
In order to achieve largely impulse-free compression with a high degree of efficiency, the air must already be compressed in the compressor so that the pressure at the end of the compression is as equal as possible to the pressure on the pressure side. In addition, the gas is not immediately released to the pressure side. A wall is opposed to the opening of the tooth passage. As the shaft continues to rotate, the volume of the air hose is reduced because it is practically pressed against the wall, it is compressed. Now, depending on the pressure required, this compressed air hose can be released sooner or later. The degree of compression is therefore determined by the size and arrangement of the outlet opening.
Another possibility for internal compression of the air is to change the pitch of the toothing, which in this case decreases towards the pressure side. When the air hose migrates to the pressure side, its volume is reduced with a decreasing slope.
Axial compressor. The standing blades are the stators .
With turbo compressors , energy is added to the flowing fluid by a rotating rotor according to the laws of fluid mechanics. This design works continuously and is characterized by a low pressure increase per stage and high volume throughput. Radial and axial compressors are the two main types of turbo compressor. With axial compressors, the gas to be compressed flows through the compressor in a direction parallel to the axis. With a radial compressor, the gas flows axially into the impeller of the compressor stage and is then deflected outwards (radially). With multistage centrifugal compressors, a flow deflection is therefore necessary after each stage.
These compressors are used, for example, in exhaust gas turbochargers (mostly as radial compressors) or in turbine jet engines (mostly as axial compressors). Here, however, the pressure does not increase due to the narrowing channel cross section, but rather due to the fact that the space between the blades of such a compressor takes the form of a diffuser . Here the pressure and temperature increase while the speed decreases. In the rotating part of a compressor stage (impeller, rotor), the kinetic energy required for further pressure build-up is returned to the air.
A transonic compressor is understood to be a turbo compressor of axial or radial design, in which the flow velocity in the relative system (the observer “sits on the rotating rotor blade”) at least locally exceeds the speed of sound. The front stages of modern compressors in aircraft engines and gas turbines are usually designed to be transonic, as the temperatures here are still low and a higher Mach number is obtained with the same flow speed . The Mach number is the ratio of the flow velocity to the speed of sound; the latter is a function of the temperature and the chemical composition of the gas.
The advantage of transonic compressors is the high power densities, which is particularly important for aircraft engine compressors, since the system is very compact. These compressors are characterized by complex systems of compression surges, which make the design and stable operation of the compressor much more difficult. Another disadvantage are the high losses associated with the compression shocks, which can only be countered by complex three-dimensional blading and sophisticated profiles. The first stages of the low-pressure compressor in engines are usually designed to be transonic because of the large diameter of the rotors and the equality of speed of all units on the shaft.
Lamellar compressor
Swash plate compressor ( English wobble plate )
Labyrinth piston compressor
Swash plate compressor ( Swash-Plate )
Scroll compressor (" VW G-Lader ")
Junkers free piston compressor , see also free piston machine
Volume flow and final pressure
Turbo compressors deliver large volume flows at low compression end pressures, displacement compressors deliver high compression end pressures at low throughputs.
Compressor types can also be divided into oil-lubricated and oil-free compressors.
Open or hermetic
In refrigeration technology, there is an additional distinction:
Fully hermetic compressors - the motor and compressor are located in an encapsulated, welded housing and are in direct contact with the refrigerant ; the housing cannot be opened for repairs
Semi-hermetic compressors - motor and compressor are in the same housing, the housing can be completely dismantled and repaired using the screwed cover
Open compressor - the compressor is driven by belts, gears or gears, the housing can be completely dismantled and repaired using the screwed cover
The drive by electric motors enables a variable speed in a wide range and thus the controllability of the delivery quantity. Maintenance costs are low, but operating costs depend on the electricity price.
When driven by an internal combustion engine , controllability is severely limited, since internal combustion engines can only be operated efficiently and for a long time in a certain speed range. In the meantime, compressors driven by gas engines are also available as combined heat and power systems. The low gas price in relation to the electricity price means that the operating costs are low. If a commercial enterprise with a continuous need for compressed air can use the waste heat generated by the combustion engine and compressor to heat the building, the higher investment costs pay off due to the savings in heating costs.
Respiratory protection compressor
Engine charging , as an aid to "forced filling" of the engine with additional air or gas mixtures to increase performance
Commons : Compressor - collection of images, videos and audio files
Wiktionary: Verdichter - explanations of meanings, word origins, synonyms, translations
Wiktionary: Compressor - explanations of meanings, word origins, synonyms, translations
↑ " Efficient systems ( Memento of the original from August 4, 2016 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. " Marani Aug 4, 2016. @1@ 2Template: Webachiv / IABot / marani.de
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↑ Reports on compressor systems with waste heat utilization in energy and management , in the Industrieanzeiger and on the Hannover Messe website . accessed in September 2016.
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Equational description of pseudovarieties of homomorphisms
The notion of pseudovarieties of homomorphisms onto finite monoids was recently introduced by Straubing as an algebraic characterization for certain classes of regular languages. In this paper we provide a mechanism of equational description of these pseudovarieties based on an appropriate generalization of the notion of implicit operations. We show that the resulting metric monoids of implicit operations coincide with the standard ones, the only difference being the actual interpretation of pseudoidentities. As an example, an equational characterization of the pseudovariety corresponding to the class of regular languages in
A{C}^{0}
Mots clés : pseudovariety, pseudoidentity, implicit operation, variety of regular languages, syntactic homomorphism
author = {Kunc, Michal},
title = {Equational description of pseudovarieties of homomorphisms},
AU - Kunc, Michal
TI - Equational description of pseudovarieties of homomorphisms
Kunc, Michal. Equational description of pseudovarieties of homomorphisms. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 37 (2003) no. 3, pp. 243-254. doi : 10.1051/ita:2003018. http://www.numdam.org/articles/10.1051/ita:2003018/
[1] J. Almeida, Finite Semigroups and Universal Algebra. World Scientific, Singapore (1995). | MR 1331143 | Zbl 0844.20039
[2] D. Mix Barrington, K. Compton, H. Straubing and D. Thérien, Regular languages in
N{C}^{1}
. J. Comput. System Sci. 44 (1992) 478-499. | Zbl 0757.68057
[3] S. Eilenberg, Automata, Languages and Machines. vol. B, Academic Press, New York (1976). | MR 530383 | Zbl 0359.94067
[4] J.E. Pin, A variety theorem without complementation. Russian Math. (Iz. VUZ) 39 (1995) 74-83.
[5] J. Reiterman, The Birkhoff theorem for finite algebras. Algebra Universalis 14 (1982) 1-10. | Zbl 0484.08007
[6] M.P. Schützenberger, On finite monoids having only trivial subgroups. Inform. and Control 8 (1965) 190-194. | Zbl 0131.02001
[7] H. Straubing, On the logical description of regular languages. in Proc. 5th Latin American Sympos. on Theoretical Informatics (LATIN 2002), edited by S. Rajsbaum, Lecture Notes in Comput. Sci., vol. 2286, Springer, Berlin (2002) 528-538. | Zbl 1059.03034
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Snub_trihexagonal_tiling Knowpia
{\displaystyle s{\begin{Bmatrix}6\\3\end{Bmatrix}}}
Symmetry p6, [6,3]+, (632)
Bowers acronym Snathat
Dual Floret pentagonal tiling
In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol of sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.
Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling (hextille).
There are 3 regular and 8 semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry.
There is only one uniform coloring of a snub trihexagonal tiling. (Naming the colors by indices (3.3.3.3.6): 11213.)
The snub trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. Every circle is in contact with 5 other circles in the packing (kissing number).[1] The lattice domain (red rhombus) repeats 6 distinct circles. The hexagonal gaps can be filled by exactly one circle, leading to the densest packing from the triangular tiling.
There is one related 2-uniform tiling, which mixes the vertex configurations of the snub trihexagonal tiling, 3.3.3.3.6 and the triangular tiling, 3.3.3.3.3.3.
6-fold pentille tilingEdit
Dual semiregular tiling
face-transitive, chiral
In geometry, the 6-fold pentille or floret pentagonal tiling is a dual semiregular tiling of the Euclidean plane.[2] It is one of 15 known isohedral pentagon tilings. Its six pentagonal tiles radiate out from a central point, like petals on a flower.[3] Each of its pentagonal faces has four 120° and one 60° angle.
It is the dual of the uniform tiling, snub trihexagonal tiling,[4] and has rotational symmetry of orders 6-3-2 symmetry.
The floret pentagonal tiling has geometric variations with unequal edge lengths and rotational symmetry, which is given as monohedral pentagonal tiling type 5. In one limit, an edge-length goes to zero and it becomes a deltoidal trihexagonal tiling.
a=b, d=e
A=60°, D=120°
a=b, d=e, c=0
A=60°, 90°, 90°, D=120°
a=b=2c=2d=2e
A=60°, B=C=D=E=120°
a=b=d=e
A=60°, D=120°, E=150°
2a=2b=c=2d=2e
0°, A=60°, D=120°
Related k-uniform and dual k-uniform tilingsEdit
There are many k-uniform tilings whose duals mix the 6-fold florets with other tiles, for example (F for V34.6, C for V32.4.3.4, B for V33.42, H for V36):
Snub Trihexagonal
F, p6 (t=3, e=3) FH, p6 (t=5, e=7) FH, p6m (t=3, e=3) FCB, p6m (t=5, e=6) FH2, p6m (t=3, e=4) FH2, p6m (t=5, e=5)
FH2, p6 (t=7, e=9) F2H, cmm (t=4, e=6) F2H2, p6 (t=6, e=9) F3H, p2 (t=7, e=12) FH3, p6 (t=7, e=10) FH3, p6m (t=7, e=8)
FractalizationEdit
Replacing every V36 hexagon by a rhombitrihexagon furnishes a 6-uniform tiling, two vertices of 4.6.12 and two vertices of 3.4.6.4.
Replacing every V36 hexagon by a truncated hexagon furnishes a 8-uniform tiling, five vertices of 32.12, two vertices of 3.4.3.12, and one vertex of 3.4.6.4.
Replacing every V36 hexagon by a truncated trihexagon furnishes a 15-uniform tiling, twelve vertices of 4.6.12, two vertices of 3.42.6, and one vertex of 3.4.6.4.
In each fractal tiling, every vertex in a floret pentagonal domain is in a different orbit since there is no chiral symmetry (the domains have 3:2 side lengths of
{\displaystyle 1+{\frac {1}{\sqrt {3}}}:2+{\frac {2}{\sqrt {3}}}}
in the rhombitrihexagonal;
{\displaystyle 1+{\frac {2}{\sqrt {3}}}:2+{\frac {4}{\sqrt {3}}}}
in the truncated hexagonal; and
{\displaystyle 1+{\sqrt {3}}:2+2{\sqrt {3}}}
in the truncated trihexagonal).
Fractalizing the Snub Trihexagonal Tiling using the Rhombitrihexagonal, Truncated Hexagonal and Truncated Trihexagonal Tilings
Truncated Hexagonal
Dual uniform hexagonal/triangular tilings
V63 V3.122 V(3.6)2 V36 V3.4.6.4 V.4.6.12 V34.6
Wikimedia Commons has media related to Uniform tiling 3-3-3-3-6 (snub trihexagonal tiling).
^ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern E
^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 "Archived copy". Archived from the original on 2010-09-19. Retrieved 2012-01-20. {{cite web}}: CS1 maint: archived copy as title (link) (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
^ Five space-filling polyhedra by Guy Inchbald
^ Weisstein, Eric W. "Dual tessellation". MathWorld.
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. p. 39
Keith Critchlow, Order in Space: A design source book, 1970, p. 69-61, Pattern R, Dual p. 77-76, pattern 5
Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–56, dual rosette tiling p. 96, p. 114
Klitzing, Richard. "2D Euclidean tilings s3s6s - snathat - O11".
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2.13 Sky Dome and Projections | EME 810: Solar Resource Assessment and Economics
Sun Charts: Projections of Solar Events and Shadowing from the Sky Dome
The emphasis of this lesson is the Sun Chart tool (or Sun Path). These flat diagrams are found in many solar design tools, but may look completely foreign to the new student in solar energy. How do we interpret the arcs and points plotted on a sun chart? Why do we have two different types of plots (one looks like a rectangle, and one looks like a circle)? Why do some plots go from 0-360°, while others go from -180° to +180°?
Video: Angles Sunchart 1 (4:45)
Angles Sunchart 1
Click Here for Transcript of Angles Sunchart 1 video
OK, we're going to do a little preliminary review of where we've been and why these angles are important to the measurement or the plotting of sun charts. So, we know that we have a set of several earth, sun angles. So, earth, sun angles. Excuse my wonderful writing. And we're going to break those down into the latitude phi, the longitude.
So, we've got latitude and longitude. And we're also going to need the declination, the declination which is going to be a function of the day number n, and the hour angle. And the hour angle, again, is omega. And that's going to be just converting time into an angular value, which we do for the location at hand.
So, along with those earth, sun angles, we have our sun observer angles. And the sun observer angles is going to be broken down into just three simple angles which is going to be the solar altitude, altitude angle alpha with the subscript s relating it to the behavior of the sun. Gamma s is going to be the azimuth angle of the sun. And the third angle is going to be theta z which is the zenith angle for the sun.
Now, I'm going to use the information from latitude, longitude, declination, and hour angle to ultimately calculate the sun observer angles over here. The important angles for our future shading plots and for our sun path diagrams are going to boil down to plotting altitude and azimuth.
So, we're going to use these earth, sun angles to calculate sun observer angles right here. And if we plot those angles over the course of the day, over the number of hours in a day, we're going to ultimately plot the altitude and azimuth angles according to their hour angle. And the complement, of course, of the altitude angle is the zenith angle. I forgot to mention that.
But we're going to be able to make these plots that are going to look like this. They're going to be a square plot where on the x-axis, we're going to have a plot of azimuths. And on the y-axis, we're going to have a plot of altitude angles. So, the altitude angle will go from zero to 90 degrees.
And depending on the system that we're using, negative values might be to the west and positive values-- or negative to the east and positive to the west. We could also plot here the complement. From zero down to 90 would be theta z, just so we know.
And again, if I plotted the solar altitude and solar azimuth angles, what I'm going to get is a plot of the arc over the hour angles. And this lower plot will be for winter, when the sun is low in the sky. Low in the sky means a low altitude angle. When the summer comes, the sun is high in the sky. And so, we have a higher angle in the summer.
And basically, this boundary from here to here is going to be the winter solstice and the summer solstice. These plots are basically plotting a series of altitude and azimuth angles for our sun charts. We will next be plotting a series of angles for shading that we will use.
So, that's our basic overview. Let's go on to the next stage.
What are Sun Charts?
If we want to visually convert our observations of the sky-dome onto a two-dimensional medium, we can either use an orthographic projection or a spherical projection on a polar chart. These projections are useful for calculating established times of solar availability or shadowing for a given point of solar collection.
The Sun Path describes the arc of the sun across the sky in relation to an earth-bound observer at a given latitude and time.
Figure 2.17: We display the path of the Sun across two days for the Northern Hemisphere. One day in summer and one in winter, where the trail of the beam has been projected onto the sky dome using angular coordinates of solar azimuth (
{\gamma }_{s}
) and solar altitude (
{\alpha }_{s}
All light incident upon Earth's surface must pass through the atmosphere and be attenuated (lost from absorption or back scattering). In order to simplify the many points of origin of light, we divide the sky and the Earth's surface into components, or spatial blocks of an imaginary hemispherical projection on the sky. The Sky Dome refers to the sum of the components for the entire sky from horizon to zenith, and in all azimuthal directions. In our following sections, a collecting surface is assumed to be horizontal first, as a pyranometer measuring device is mounted horizontally and facing the sky to measure the Global Irradiance/Irradiation in the shortwave band for the sky dome. Most of our solar collectors will be tilted up from horizontal in some way (PV, solar hot water, windows, walls, even your eyes). Those surfaces oriented otherwise are termed a Plane of Array measurement (POA), requiring specific tilt and azimuth information in the description. For those solar collecting surfaces that are not horizontal, the reflectance of the ground is an additional source of light, through the albedo effect. The beam, sky diffuse, and ground diffuse light sources incident upon the tilted collector are estimated using models of light source components.
The sky dome can be projected onto flat surfaces for analysis of shading and sky component behavior.
Figure 2.18: The sky dome as projected to the right in orthographic form, and as projected upward in polar form.
So, if we go back and we think about the Sky Dome-- and I'm going to do this little diagram of-- this is the Sky Dome that we've talked about in the textbook and in the class. And this is going to be the ground. And so we have this Sky Dome.
And you can kind of see that this dome that we have could be-- is a hemisphere, first of all. And we could project that dome onto multiple different surfaces.
So, the first projection that we will do is going to be orthographic. And essentially, an orthographic projection is going to be what happens if I were to try to take a flat piece of paper. maybe like a cylinder of paper, and I were to try to wrap that cylinder right around this Sky Dome.
So, I'm going to have-- everybody see the cylinder that's wrapping around? So, I'm going to want to project points onto that flat surface. And ultimately, that flat surface is going to be printed out.
And we're going to have our basis for our sun charts in terms of the azimuth angle. So, the rotation along here, along the plane or rotation along the horizontal is the azimuth angle. And the angle from the ground up, the vertical rotation, is going to be the altitude angle.
Of course, the complement of that, if we were talking about the sun, would be the zenith angle. Which is why we could represent this is 0 to 90 and 90 down to 0 with the zenith angle.
We have two conventions for plotting the azimuth angle. The one is to make east negative. So, we can go negative 180 degrees. And west positive 180 degrees. Where this 0 is pointed at the equator.
The alternate is to begin in the north with 0 degrees and to work your way clockwise to-- all the way around to 360 degrees, in which case south, not the equator, is going to be positive 180 degrees.
Now, this is the convention that a lot of the solar world has used for some time. However, this is the standard convention that has been established through meteorology. And so, we tend to use both flexibly. But just know that, in general, the 360 degree is an accepted standard.
So, if we then go to the next page and we think about that same Sky Dome. And got our ground. And instead of trying to project it off to the side, we are effectively lying on the ground. And we're going to project upwards.
And this way, the azimuth angles are rotating around in a circle just like the azimuth angles will be doing here on the ground.
The altitude angles, however, are going to be represented as arcs across where the higher in the sky you are, the closer to the center of the circle.
And how do we see that? Well, we look at it like this. And we're going to start to see a center point. And I'm going to put south here. We know that south is in the northern hemisphere where the sun is at its highest point.
So, we're going to see arcs in the sky that look like this going from-- in our case-- east to west. This is going to be summertime. This is going to be wintertime. When the sun is low in the sky versus high in the sky.
Here is going to be an alpha 90 degrees. And the ring around the bottom is going to be an alpha of zero degrees. It's a little different plot. The lines are going to be flipped from what you were used to in an orthographic projection, if you can do both of these at the Oregon site for sun plots.
Orthographic Projection: takes the sky dome and projects altitude and azimuth values outward onto a surrounding vertical cylinder. The cylinder is then opened flat. Figure 2.19, below, shows the sun rising in the East (to the left) and setting in the West (to the right). Proper observation shows that the largest arc in the chart at the top, June 21, is the Northern Solstice, while the smallest, December 21, is the Southern Solstice.
Figure 2.19: Orthographic Projection.
Polar Projection: takes the sky dome and projects altitude and azimuth values down onto a circular plane. However, in the polar projection, the arc for December 21 is at the top while the arc for June 21 is at the bottom. This happens because we are effectively lying on the ground with our heads facing south, and holding that large piece of paper straight up to the sky.
Figure 2.20: Polar Projection
How do We Make and Read Sun Charts?
Go to the University of Oregon Solar Radiation Monitoring Laboratory website. The scientists at the site have provided an excellent tool for plotting sun paths onto orthographic projections or polar/spherical projections. The default page is for creating an orthographic projection of your site of interest. The alternate page for polar projections will use the same data you can input, but will output the alternate form. Note that both use the meteorological standard for azimuth angles, where North is set at 0°, increasing clockwise to 360°.
Specify the location by
\varphi
\lambda
. (Latitude is important for our calculations of sun-observer angles).
Specify the time zone (software does the correction from UTC to local time zone).
Choose data to be plotted (Choose to plot hours in local solar time: default). These plots are symmetrical for half of the year when you plot them in arcs of solar time. Hence, when you plot in solar time, "Plot dates 30 or 31 days apart, between solstices, December through June" will look the same as "Plot dates 30 or 31 days apart, between solstices, June through December." Side note: if you do plot in "local standard time," you will observe half of an analemma each hour, where only the Equation of Time ($E_t$) has not been corrected for in the time correction.
Set chart format parameters: These are your choice to personalize the output file. Be creative, and try to present clear data visualization.
Choose file format for chart. (I prefer the PDF for working.)
Enter the code to make sure you are not a web bot roaming about.
Download the image, print it out, and use it for shading analysis!
When designing a solar energy conversion system for any application, we must pay special attention to the occurrence of shadows throughout the year. We discuss a method to assess the shading using 2-D projections.
The next page gives you an opportunity to print and analyze your own sun chart.
‹ 2.12 Siting using Sun Paths and Spherical Coordinates up 2.14 Try This! Print a Sun Chart ›
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P=\left(a={x}_{0},{x}_{1},...,{x}_{N}=b\right)
\left(a,b\right)
, the left Riemann sum is defined as:
\sum _{i=1}^{N}f\left({x}_{i-1}\right)\left({x}_{i}-{x}_{i-1}\right)
\left({x}_{i-1},{x}_{i}\right)
of the partition is the left-hand point
{x}_{i-1}
10
\mathrm{with}\left(\mathrm{Student}[\mathrm{Calculus1}]\right):
\mathrm{RiemannSum}\left(\mathrm{sin}\left(x\right),x=0...5.0,\mathrm{method}=\mathrm{left}\right)
\textcolor[rgb]{0,0,1}{0.9410826232}
\mathrm{RiemannSum}\left(x\left(x-2\right)\left(x-3\right),x=0..5,\mathrm{method}=\mathrm{left},\mathrm{output}=\mathrm{plot}\right)
\mathrm{RiemannSum}\left(\mathrm{tan}\left(x\right)-2x,x=-1..1,\mathrm{method}=\mathrm{left},\mathrm{output}=\mathrm{plot},\mathrm{partition}=20,\mathrm{boxoptions}=[\mathrm{filled}=[\mathrm{color}="Burgundy"]]\right)
\mathrm{exact}≔\mathrm{int}\left(\mathrm{ln}\left(x\right),x=1..100\right)
\textcolor[rgb]{0,0,1}{\mathrm{exact}}\textcolor[rgb]{0,0,1}{≔}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{99}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{200}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{2}\right)\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{200}\textcolor[rgb]{0,0,1}{}\textcolor[rgb]{0,0,1}{\mathrm{ln}}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{5}\right)
\mathrm{evalf}\left(\mathrm{exact}\right)
\textcolor[rgb]{0,0,1}{361.5170185}
\mathrm{RiemannSum}\left(\mathrm{ln}\left(x\right),1..100,\mathrm{method}=\mathrm{left},\mathrm{outline}=\mathrm{true},\mathrm{output}=\mathrm{animation}\right)
|
How to Calculate Escape Velocity: 10 Steps - wikiHow
1 Understanding Escape Velocity
The escape velocity is the velocity necessary for an object to overcome the gravitational pull of the planet that object is on. For example, a rocket going into space needs to reach the escape velocity in order to make it off Earth and get into space.
Understanding Escape Velocity Download Article
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Define escape velocity. Escape velocity is the velocity of an object required to overcome the gravitational pull of the planet that object is on to escape into space. A larger planet has more mass and requires a much greater escape velocity than a smaller planet with less mass.[1] X Research source
Begin with conservation of energy. Conservation of energy states that the total energy of an isolated system remains unchanged. In the derivation below, we will work with an Earth-rocket system and assume that this system is isolated.
In conservation of energy, we equate the initial and final potential and kinetic energies
{\displaystyle K_{1}+U_{1}=K_{2}+U_{2},}
{\displaystyle K}
is kinetic energy and
{\displaystyle U}
is potential energy.
Define kinetic and potential energy.
Kinetic energy is energy of motion, and is equal to
{\displaystyle {\frac {1}{2}}mv^{2},}
{\displaystyle m}
is the mass of the rocket and
{\displaystyle v}
Potential energy is energy that results from where an object is relative to the bodies in the system. In physics, we typically define the potential energy to be 0 at an infinite distance from Earth. Since the gravitational force is attractive, the potential energy of the rocket will always be negative (and smaller the closer it is to Earth). Potential energy in the Earth-rocket system is thus written as
{\displaystyle -{\frac {GMm}{r}},}
{\displaystyle G}
{\displaystyle M}
is the mass of Earth, and
{\displaystyle r}
is the distance between the two masses' centers.
Substitute these expressions into conservation of energy. When the rocket achieves the minimum velocity required to escape Earth, it will eventually stop at an infinite distance from Earth, so
{\displaystyle K_{2}=0.}
Then, the rocket will not feel Earth's gravitational pull and will never fall back to Earth, so
{\displaystyle U_{2}=0}
{\displaystyle {\frac {1}{2}}mv^{2}-{\frac {GMm}{r}}=0}
{\displaystyle {\begin{aligned}{\frac {1}{2}}mv^{2}&={\frac {GMm}{r}}\\v^{2}&={\frac {2GM}{r}}\\v&={\sqrt {\frac {2GM}{r}}}\end{aligned}}}
{\displaystyle v}
in the above equation is the escape velocity of the rocket - the minimum velocity required to escape the gravitational pull of Earth.
Note that the escape velocity is independent of the mass of the rocket
{\displaystyle m.}
The mass is reflected in both the potential energy provided by Earth's gravity as well as the kinetic energy provided by the movement of the rocket.
Calculating Escape Velocity Download Article
State the equation for escape velocity.
{\displaystyle v={\sqrt {\frac {2GM}{r}}}}
The equation assumes the planet you are on is spherical and has constant density. In the real world, the escape velocity depends on where you are at on the surface because a planet bulges at the equator due to its rotation and has slightly varying density due to its composition.
Understand the variables of the equation.
{\displaystyle G=6.67\times 10^{-11}{\rm {\ N\ m^{2}\ kg^{-2}}}}
is Newton's gravitational constant. The value of this constant reflects the fact that gravity is an incredibly weak force. It was determined experimentally by Henry Cavendish in 1798,[2] X Research source but has proven to be notoriously difficult to measure precisely.
{\displaystyle G}
can be written using only base units as
{\displaystyle 6.67\times 10^{-11}{\rm {\ m^{3}\ kg^{-1}\ s^{-2}}},}
{\displaystyle 1{\rm {\ N}}=1{\rm {\ kg\ m\ s^{-2}}}.}
{\displaystyle M}
{\displaystyle r}
are dependent upon the planet you wish to escape from.
You must convert to SI units. That is, mass is in kilograms (kg) and distance is in meters (m). If you find values that are in different units, such as miles, convert them to SI.
Determine the mass and radius of the planet you are on. For Earth, assuming that you are at sea level,
{\displaystyle r=6.38\times 10^{6}{\rm {\ m}}}
{\displaystyle M=5.98\times 10^{24}{\rm {\ kg}}.}
Search online for a table of masses and radii for other planets or moons.
Substitute values into the equation. Now that you have the necessary information, you can start solving the equation.
{\displaystyle v={\sqrt {\frac {2(6.67\times 10^{-11}{\rm {\ m^{3}\ kg^{-1}\ s^{-2}}})(5.98\times 10^{24}{\rm {\ kg}})}{(6.38\times 10^{6}{\rm {\ m}})}}}}
Evaluate. Remember to evaluate your units at the same time and cancel them out as needed to obtain a dimensionally consistent solution.
{\displaystyle {\begin{aligned}v&={\sqrt {{\frac {2(6.67)(5.98)}{(6.38)}}\times 10^{7}{\rm {\ m^{2}\ s^{-2}}}}}\\&\approx 11200{\rm {\ m\ s^{-1}}}\\&=11.2{\rm {\ km\ s^{-1}}}\end{aligned}}}
In the last step, we converted the answer from SI units to
{\displaystyle {\rm {\ km\ s^{-1}}}}
by multiplying by the conversion factor
{\displaystyle {\frac {\text{1 km}}{\text{1000 m}}}.}
If I start "driving" away from earth at 100 mph and never speed up, won't I eventually get as far away as the rocket and therefore escape?
Yes. What is different, and what was not made clear perhaps, is that escape velocity is the velocity required at the start of movement for un-powered flight, as for example if you were to throw a stone. If you have enough fuel, then you can escape as slowly as you like. When I walk upstairs I am escaping the earths gravity a tiny bit, but I can't go on for long.
I posit that a projectile requires constant propulsion to escape Earth's gravity. A friend maintains that a big enough "bang" at the Earth's surface can achieve the same result. Who is correct?
Both of you are. Conservation of energy does not depend on the propulsion that got you to space. Remember that escape velocity refers to the velocity of an object at sea level. If an explosion sends an object flying away at that speed, it will escape Earth. In your case, constant propulsion generates a constant force which steadily increases velocity, and is another (the practical) way to achieve escape velocity.
Does escape velocity vary on Earth?
In reality, yes, because our planet isn't a perfect sphere, which means the radius varies (lowest at poles & highest at equator). So the escape velocity varies from place to place. But for the sake of school physics problems and such, it's generally assumed that the escape velocity is the same all over the planet.
Because Newton's gravitational constant is so difficult to measure precisely, the standard gravitational parameter
{\displaystyle \mu =GM}
is often known to much greater precision. You can then use this to calculate escape velocity instead.
The standard gravitational parameter of Earth
{\displaystyle \mu =3.986\times 10^{14}{\rm {\ m^{3}\ s^{-2}}}.}
↑ http://www.beaconlearningcenter.com/documents/1483_01.pdf
↑ http://www.physicsclassroom.com/Class/circles/u6l3d.cfm
To calculate escape velocity, multiply 2 times G times M, then divide that by r, and take the square root of the result. In this equation, G is Newton’s gravitational constant, M is the mass of the planet you’re escaping from in kilograms, and r is the radius of the planet in meters. Substitute those values into the equation and solve for v, or escape velocity. For the definition of escape velocity, keep reading!
Español:calcular la velocidad de escape
Italiano:Calcolare la Velocità di Fuga
Français:calculer la vitesse de libération
Bahasa Indonesia:Menghitung Kecepatan Lepas
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"This was very helpful! Even a weak student in physics can understand the concept of escape velocity through these easy steps. Must go through it!"..." more
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"The escape velocity calculation was what I needed to explain to my grandson. Thanks!"
|
Reward your volunteers - OFN User Guide
Would you like to say a little thank you to the people who help run your community food enterprise? One option is to allow staff and volunteers to have a small discount off any shopping they do. Below is a step-by-step guide on how to implement this. The process draws on the highly flexible Customer Management tools available using Tags and Tag Rules.
Ask your volunteers to let you know the email address linked to their OFN account.
Login to your business OFN account and visit Customers page.
Use the ‘Quick Search’ box to identify if the person has shopped with you before.
If their email address doesn’t appear then click + New Customer and add their address.
Add the tag ‘volunteer’ to the customer’s entry.
Visit Enterprises -> Settings and then select ‘Payment Methods’ from the left hand menu.
Click + New Payment Method.
Name: Volunteer 5% Discount Description: Thank you for helping us run our local food hub. Display: Both Checkout and Back Office Active: yes Tags: Add the tag ‘volunteer’ into this space. Provider: choose the most appropriate method for your business. Fee Calculator: Flat Percent
= (100 + Enterprise Fee)*Desired Discount/100
= -(100 + 20) *5/100 = -6
Default: Payment Methods tagged ‘volunteer’ are not visible. For customers tagged ‘volunteer’ payment methods tagged ‘volunteer’ are visible.
Bingo! Only your volunteers will be offered a 5% discount when they shop with you.
Checkout view for volunteer
Checkout view for all other customers
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Algorithms hash map Data Structures
Properties of DHTs
Practical Applications of Distributed Hash Tables
Prerequisite: Hash Map / Hash Table
A hash table is a data structure that stores data in an associative manner. Associative meaning each data value in the table is mapped onto a unique index value.
Access is fast knowing the index of the data.
We can also define a hash table as an implementation of a set or a map using hashing.
Unordered_set in C++.
Set in python.
Unordered_map in C++.
HashMap in Java.
Dict in python.
Hashing is a technique used to convert large sized data into small manageable keys by use of hash functions.
A hash function is a mathematical function that is used to map keys to integers.
An element is converted into a small integer by using a hash function.
The small integer will be used as the key for the original element value in th hash table.
Note: The speed of a hash table depends on the choice of hash function.
Properties of a good hash function.
Fast to compute.
Offers good key distribution.
Fewer collisions.
There are no ways to assure some of the above conditions since the probability distribution from which keys are drawn from is unknown.
Collisions are a common problem encountered in hashing, that is, when two keys are hashed onto the same index in a hash table
This problem is resolved using chaining, open addressing or use of a universal family.
This is where each index in the hash table will have a linked list of elements, elements that compute to same hash value will be chained together in the same index.
Note: A good hash function is still important because we could encounter a case where we have very long chains on one index and no chain in others. We need to keep chains at least balanced because to reach an element in a chain we have to traverse the linked list.
Also called closed hashing is whereby a hash table is maintained as an array of elements(not buckets), each index is initialized to null.
To insert we check if the desired position is empty if so we insert an element otherwise we find another place int the array to insert it.
In this approach, collisions are solved by probing, that is, searching for an alternative position in the array. Probing sequences are used here. You can learn more about probing sequences in the links at the end of this post.
For any deterministic hash function there is a bad input for which there will be a lot of collisions which will lead to O(n) time for access.
In universal hashing, we choose a random hashing function from a family of hash functions that is independent of the keys that will be stored. This will yield good performance no matter the inputs.
We define a universal family, a set of hash functions whereby any two keys in the universe have a small probability of collision.
We select random hash functions from the set of deterministic hash functions and use it throughout the algorithm.
Note: By cleverly randomizing choice of hash function at run time, we guarantee a good average case running time.
You can read more on this in the links at the end of this post.
This is a distributed system that provides lookups similar to hash tables on a larger scale.
Distributed hash tables store big data on many computers and provide fast access as if it was on a single computer.
It does this by use of nodes which are distributed across the network.
To find a node that "owns" an object we hash the object using a hash function which will give a value that represents the node in a cluster.
In effect the hash table buckets are now independent nodes in a network.
Fault tolerance - when a node fails, its data is inherited.
Reliability - nodes fail and new ones are added.
Performance - fast retrieval of information on a larger scale.
Scalability - functions even with multiple nodes being added to it.
Decentralized - nodes form a system but each act independently.
We need to store trillions of objects or even more considering data that gets generated these days.
We also need fast search/lookup once we store this huge amount of data.
We consider a hash table data structure for this case, which provides constant time O(1) lookups, however in this case we are talking about big data whose storage requirements are much larger, n =
{10}^{12}
, the memory requirements become too much and the computational complexity will deteroriate.
Lets say we get 1000 computers.
We need to store a dictionary of key-value pairs (k1, v1), (k2, v2) across these cluster of computers in a way that is easy to manipulate the data without having to think about the details.
Create a hash table on each of them for storing these objects.
We use a hash function to determine which computer "owns" a particular object, h(O) mod 1000, this will give a number from 0 - 999 which will represent a particular computer in the cluster which owns the object.
When a search/modify request is sent from the client, the hash function is computed and sent to the computer that owns the resource.
Computers break occasionally and even if we store several copies of the data of one computer, we need to relocate data from the broken to a new computer.
As the data grows we need more computers and when we add more to the existing cluster, the previous hash function h(O) mod 1000 no longer works when we need to redistribute keys.
We introduce the concept of consistent hashing.
Consistent hashing is one way to determine which computer stores the requested object.
We choose a hash function h with a cardinality m and place points clockwise from 0 - m-1 forming a circle.
Each object and computer in the network will be mapped onto this circle using a universal hash function, that is, h(O) and h(computerId).
Each object mapped onto the circle will be stored by the computer within the same arc as the object.
Each computer will store all objects falling on the same arc in the circle.
When a computer in the cluster fails, its neighbors will inherit its data.
When a new computer is added to the cluster, its hash value is computed and it is placed in the circle, based on its position in the circle, it will inherit data from its neighbors.
When a computer breaks, we need a well defined strategy for relocating the data.
A node needs to know where to send its data.
Overlay network, each node will store addresses of its immediate neighbors and its neighbors of neighbors in distance powers of 2, that is, 1 -> 2 -> 4 -> 8 -> 16
For each key, each node will either store it or know some closer node to this key. When a request for a key is sent to a node in the network, the node will either have the key or it will redirect the request to another node in the network which is closer to this key, these iterations will continue until the request reaches a node that stores the key.
It takes log(n) number of steps to reach the node that actually stores the key.
This is known as an overlay network, a network of nodes that know some other neighbor nodes, we use it to route data to and from nodes within a network of nodes.
Distributed storage, dropbox, yandex disk.
How is a distributed hash table implemented in a distributed storage system such as dropbox to achieve instant uploads and storage optimizations?
Hash Tables - Chapter III section 11 in [CLRS] Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. Introduction to Algorithms.
Jump Consistent Hash: A Fast, Minimal Memory, Consistent Hash Algorithm
In this article, we discuss the jump consistent hashing algorithm which offers improvements compared to previous approaches to consistent hashing for distributed data storage applications.
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EUDML | Measure of nonhyperconvexity and fixed-point theorems. EuDML | Measure of nonhyperconvexity and fixed-point theorems.
Measure of nonhyperconvexity and fixed-point theorems.
Bugajewski, Dariusz; Espínola, Rafael
Bugajewski, Dariusz, and Espínola, Rafael. "Measure of nonhyperconvexity and fixed-point theorems.." Abstract and Applied Analysis 2003.2 (2003): 111-119. <http://eudml.org/doc/51171>.
@article{Bugajewski2003,
author = {Bugajewski, Dariusz, Espínola, Rafael},
keywords = {fixed point; hyperconvex; -measure; -contractive; -measure; -contractive},
title = {Measure of nonhyperconvexity and fixed-point theorems.},
AU - Bugajewski, Dariusz
AU - Espínola, Rafael
TI - Measure of nonhyperconvexity and fixed-point theorems.
KW - fixed point; hyperconvex; -measure; -contractive; -measure; -contractive
fixed point, hyperconvex,
\mu
-measure,
\mu
-contractive,
\mu
\mu
-contractive
Articles by Bugajewski
Articles by Espínola
|
Efficient Portfolio That Maximizes Sharpe Ratio - MATLAB & Simulink - MathWorks Australia
The Sharpe ratio is defined as the ratio
\frac{\mu \left(x\right)-{r}_{0}}{\sqrt{\sum \left(x\right)}}
x\in {R}^{n}
and r0 is the risk-free rate (μ and Σ proxies for portfolio return and risk). For more information, see Portfolio Optimization Theory.
Portfolios that maximize the Sharpe ratio are portfolios on the efficient frontier that satisfy several theoretical conditions in finance. For example, such portfolios are called tangency portfolios since the tangent line from the risk-free rate to the efficient frontier taps the efficient frontier at portfolios that maximize the Sharpe ratio.
To obtain efficient portfolios that maximizes the Sharpe ratio, the estimateMaxSharpeRatio function accepts a Portfolio object and obtains efficient portfolios that maximize the Sharpe Ratio.
Suppose that you have a universe with four risky assets and a riskless asset and you want to obtain a portfolio that maximizes the Sharpe ratio, where, in this example, r0 is the return for the riskless asset.
p = Portfolio('RiskFreeRate', r0);
pwgt = estimateMaxSharpeRatio(p);
If you start with an initial portfolio, estimateMaxSharpeRatio also returns purchases and sales to get from your initial portfolio to the portfolio that maximizes the Sharpe ratio. For example, given an initial portfolio in pwgt0, you can obtain purchases and sales from the previous example:
[pwgt, pbuy, psell] = estimateMaxSharpeRatio(p);
If you do not specify an initial portfolio, the purchase and sale weights assume that you initial portfolio is 0.
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Implement three-phase dynamic load with active power and reactive power as function of voltage or controlled from external input - Simulink - MathWorks Nordic
Implement three-phase dynamic load with active power and reactive power as function of voltage or controlled from external input
The Three-Phase Dynamic Load block implements a three-phase, three-wire dynamic load whose active power P and reactive power Q vary as function of positive-sequence voltage. Negative- and zero-sequence currents are not simulated. The three load currents are therefore balanced, even under unbalanced load voltage conditions.
The load impedance is kept constant if the terminal voltage V of the load is lower than a specified value Vmin. When the terminal voltage is greater than the Vmin value, the active power P and reactive power Q of the load vary as follows:
\begin{array}{c}P\left(s\right)={P}_{0}{\left(\frac{V}{{V}_{0}}\right)}^{{n}_{p}}\frac{1+{T}_{p1}s}{1+{T}_{p2}s}\\ Q\left(s\right)={Q}_{0}{\left(\frac{V}{{V}_{0}}\right)}^{{n}_{q}}\frac{1+{T}_{q1}s}{1+{T}_{q2}s},\end{array}
V0 is the initial positive sequence voltage.
P0 and Qo are the initial active and reactive powers at the initial voltage Vo.
V is the positive-sequence voltage.
np and nq are exponents (usually between 1 and 3) controlling the nature of the load.
Tp1 and Tp2 are time constants controlling the dynamics of the active power P.
Tq1 and Tq2 are time constants controlling the dynamics of the reactive power Q.
For a constant current load, for example, you set np to 1 and nq to 1, and for constant impedance load you set np to 2 and nq to 2.
Nominal L-L voltage and frequency
Specifies the nominal phase-to-phase voltage, in volts RMS, and nominal frequency, in hertz, of the load. Default is [ 500e3 60 ].
Active and reactive power at initial voltage
Specifies the initial active power, Po, in watts, and initial reactive power, Qo, in vars, at the initial voltage, Vo. Default is [50e6 25e6].
When you use the Machine Initialization tool of the powergui block to initialize the dynamic load and start simulation in steady state, these parameters are automatically updated according to the P and Q points specified for the load.
When you use the Load Flow tool of the powergui block to initialize the dynamic load, these parameters represent the P and Q reference powers used by the load flow.
Initial positive-sequence voltage Vo
Specifies the magnitude and phase of the initial positive-sequence voltage of the load. Default is [0.994 -11.8].
When you use the Load Flow tool or the Machine Initialization tool of the powergui block to initialize the dynamic load and start simulation in steady state, this parameter is automatically updated using the values computed by the load flow.
External control of PQ
If selected, the active power and reactive power of the load are defined by an external Simulink® vector of two signals. By default, the check box is cleared.
[np nq]
Specifies the np and nq parameters that define the nature of the load. Default is [1.3 2].
Time constants [Tp1 Tp2 Tq1 Tq2]
Specifies the time constants controlling the dynamics of the active power and the reactive power. Default is [0 0 0 0].
Minimum voltage Vmin
Specifies the minimum voltage at which the load dynamics commences. The load impedance is constant below this value. Default is 0.7.
Filtering time constant
Specifies the time constant, in seconds, of the current filter of the Three-Phase Dynamic Load block. Default is 1e-4.
If External control of PQ is selected, a Simulink input, labeled PQ, appears. This input is used to control the active and reactive powers of the load from a vector of two signals [P, Q].
The m output is a vector containing the following three signals:
positive-sequence voltage (pu)
active power P (W)
reactive power Q (vars)
The power_dynamicload model uses a Three-Phase Dynamic Load block connected on a 500 kV, 60 Hz power network.
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\mathrm{with}\left(\mathrm{EssayTools}\right):
\mathrm{Reduce}\left("The car was super fast. It was rocket screaming fast. Nothing else could touch it."\right)
[\textcolor[rgb]{0,0,1}{"car be super fast"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"car be rocket screaming fast"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"nothing can touch touch"}]
\mathrm{Reduce}\left("The tortoise and hare was a great story because it showed how an underdog can succeed with dedication and perseverance."\right)
[\textcolor[rgb]{0,0,1}{"tortoise hare be great story"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"tortoise show how underdog can succeed dedication perseverance"}]
|
The optional filter parameter, passed as the index to the Map or Map2 command, restricts the application of
\mathrm{with}\left(\mathrm{LinearAlgebra}\right):
A≔\mathrm{Matrix}\left([[1,2,3],[0,1,4]],\mathrm{shape}=\mathrm{triangular}[\mathrm{upper},\mathrm{unit}]\right)
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\end{array}]
M≔\mathrm{Map}\left(x↦x+1,A\right)
\textcolor[rgb]{0,0,1}{M}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{5}\end{array}]
\mathrm{evalb}\left(\mathrm{addressof}\left(A\right)=\mathrm{addressof}\left(M\right)\right)
\textcolor[rgb]{0,0,1}{\mathrm{true}}
B≔〈〈1,2,3〉|〈4,5,6〉〉
\textcolor[rgb]{0,0,1}{B}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map2}[\left(i,j\right)↦\mathrm{evalb}\left(i=1\right)]\left(\left(x,a\right)↦a\cdot x,3,B\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{12}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{6}\end{array}]
\mathrm{Map}\left(x↦x+1,g\left(3,A\right)\right)
\textcolor[rgb]{0,0,1}{g}\textcolor[rgb]{0,0,1}{}\left(\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{,}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\\ \textcolor[rgb]{0,0,1}{0}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{5}\end{array}]\right)
C≔\mathrm{Matrix}\left([[1,2],[3]],\mathrm{scan}=\mathrm{triangular}[\mathrm{upper}],\mathrm{shape}=\mathrm{symmetric}\right)
\textcolor[rgb]{0,0,1}{C}\textcolor[rgb]{0,0,1}{≔}[\begin{array}{cc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{Map}\left(x↦x+1,C\right)
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{4}\end{array}]
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{9}\\ \textcolor[rgb]{0,0,1}{9}& \textcolor[rgb]{0,0,1}{4}\end{array}]
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Standard Deviation Calculator ✔️ ConvertBinary.com
Calculate the Standard Deviation of a population or sample!
Use this statistics calculator to find the Standard Deviation for a set of values representing a population or a sample.
By default the data set is considered a Population. Check the box “Sample” if it is a sample.
What is the Standard Deviation? Standard Deviation definition
Standard Deviation is a measure of dispersion or variation within a data set.
It measures by how much the values in the data set are likely to differ from the mean.
The higher the Standard Deviation, the furthest the data points tend to be to the mean (they will be more dispersed).
Conversely, a low Standard Deviation, indicates that the the data points tend to be close to the mean (they are less dispersed and more concentrated at or around the mean).
Population Standard Deviation is tipically denoted as σ. and it is used when is possible to measure an entire population.
How to Calculate the Standard Deviation for an entire Population
Follow these steps to calculate the Standard Deviation for a population:
Find the arithmetic mean (the average) of the numbers
For each number in the data set:
Subtract the mean.
Add up all the squared differences between each number and the mean.
Divide the sum of squared differences by the data set size (the amount of numbers).
Calculate the square root of the variance: this is the Standard Deviation.
Let’s find the standard deviation for the population 5, 11, 17, 23:
Find the mean: 56 / 4 = 14
Subtract the mean from each number, and we get -9, -3, 3, 9
Square each result, and we get 81, 9, 9, 81
Add up the squared differences: 81 + 9 + 9 + 81 = 180
Divide by the data set size: 180 / 4 = 45 (this is the variance)
Calculate the square root of the variance: 6.708203932
\sigma =\sqrt{\frac{\sum _{i=1}^{n}\left({x}_{i}-\mu {\right)}^{2}}{n}}
Sample Standard Deviation is tipically denoted as s. It is used when it is not possible to measure the entire population, so a random sample is taken into consideration.
Follow these steps to calculate the Standard Deviation for a sample:
Divide the sum of squared differences by the data set size (the amount of numbers) minus 1.
Let’s find the standard deviation for the sample 5, 11, 17, 23:
Divide by the data set size: 180 / ( 4 – 1) = 60 (this is the variance)
s=\sqrt{\frac{\sum _{i=1}^{n}\left({x}_{i}-\overline{x}{\right)}^{2}}{n-1}}
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Discrete variable resistor - Simulink - MathWorks Nordic
Minimal resistance absolute value (Ohms)
Discrete variable resistor
The Variable Resistor block represents a linear time-varying resistor. It implements a discrete variable resistor as a current source. The resistance is specified by the Simulink® input signal. The resistance value can be negative.
The block uses the following equations for the relationship between the voltage, v, across the device and the current through the inductor, i, when the resistance at port R is R:
v=R\ast i.
When you use a Variable Resistor block in your model, set the powergui block Simulation type to Discrete and select the Automatically handle Discrete solver and Advanced tab solver settings of blocks parameter in the Preferences tab. The robust discrete solver is used to discretize the electrical model. Simulink signals an error if the robust discrete solver is not used.
Input port associated with the resistor. The resistance can be negative.
Minimal resistance absolute value (Ohms) — Minimal resistance
Lower limit on the absolute value of the signal at port R. This limit prevents the signal from reaching a value that has no physical meaning. The value of this parameter must be greater than 0.
Nonlinear Inductor | Nonlinear Resistor | Variable Capacitor | Variable Inductor | Variable-Ratio Transformer
|
Hopf structure on the Van Est spectral sequence in $K$-Theory
Hopf structure on the Van Est spectral sequence in
K
K
title = {Hopf structure on the {Van} {Est} spectral sequence in $K${-Theory}},
AU - Tillmann, Ulrike
TI - Hopf structure on the Van Est spectral sequence in $K$-Theory
Tillmann, Ulrike. Hopf structure on the Van Est spectral sequence in $K$-Theory, dans $K$-theory - Strasbourg, 1992, Astérisque, no. 226 (1994), 14 p. http://archive.numdam.org/item/AST_1994__226__421_0/
[Be] E. Beggs, The de Rham complex on infinite dimensional manifolds, Quart. J. Math. Oxford (2) 38 (1987), 131-154.
[BW] A. Borel, N. Wallach, "Continuous Cohomology", Discrete Subgroups, and Representations of Reductive Groups, Princeton UP, Study 94 (1980).
[B] W. Browder, On differential Hopf algebras, Trans. AMS 107 (1963), 153-176.
[BS1] E. H. Brown, R. H. Szczarba, Continuous cohomology and real homotopy type, Trans. AMS 311 (1989), 57-106.
[BS2] E. H. Brown, R. H. Szczarba, Continuous cohomology and real homotopy type II, Astérisque 191 (1990).
[BS3] E. H. Brown, R. H. Szczarba, Split complexes, continuous cohomology, and Lie algebras, to be published.
[DHZ] J. Dupont, R. Hain, S. Zucker, Regulators and characteristic classes of flat bundles, Aarhus Preprint Series (1992).
[K] M. Karoubi, Homologie cyclique et
K
-théorie, Astérisque 149 (1987).
[L] J. L. Loday, "Cyclic Homology", Springer Verlag (1992).
[LQ] J. L. Loday, D. Quillen, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helvetici 59 (1984), 565-591.
[Mi] W. Michaelis, The primitives of the continuous linear dual of a Hopf algebra as the dual Lie coalgebra, in "Lie Algebras and Related Topics", Contemp. Math. 110 (1990), 125-176.
[M] J. Milnor, On the homology of Lie groups made discrete, Comment. Math. Helv. 58 (1983), 72-85.
[MM] J. Milnor, J. C. Moore, On the structure of Hopf algebras, Ann. of Math. 81 (1965), 211-264.
[Ti] U. Tillmann, Relation of the Van Est spectral sequence to
K
-theory and cyclic homology, Ill. Jour. Math. 37 (1993), 589-608.
[T] B. L. Tsygan, Homology of matrix algebras over rings and the Hochschild homology (in Russian), Uspekhi Mat. Nauk. 38 (1983), 217-218.
|
torch.nn.functional.softmax — PyTorch 1.11.0 documentation
torch.nn.functional.softmax¶
torch.nn.functional.softmax(input, dim=None, _stacklevel=3, dtype=None)[source]¶
\text{Softmax}(x_{i}) = \frac{\exp(x_i)}{\sum_j \exp(x_j)}
This function doesn’t work directly with NLLLoss, which expects the Log to be computed between the Softmax and itself. Use log_softmax instead (it’s faster and has better numerical properties).
|
Probability Simulator / Calculator | Loot box, Lottery - Hirota Yano
Probability Simulator / Calculator | Loot box, Lottery
The calculation result of the probability is rounded off to the third decimal place.
(0.01% ~ 100%)
(1 ~ 1000)
Required hits:
Probability of hitting 1 or more times when performed 1% loot box 100 times
Simulate "Continue to trial until 1% chance hits 1 times" for 100 people.
1th person 0 times
10th person 0 times
100th person 0 times
Min: 0 times
Max: 0 times
Avg: 0 times
Within 100 times: 0 people
Probability of hitting 1 or more times when performed p% loot box n times
Probability of hitting 1 or more ＝ 100% － Probability of not hitting even once
P=1-(1-p)^n
e.g. Probability of hitting 1 or more times when performed 1% loot box 1 times
1-0.99=0.01=1\%
-> It hits about 1 in 100 people.
e.g. Probability of hitting 1 or more times when performed 1% loot box 100 times
1-0.99^{100}=1-0.36603...=0.63397...\approx63.4\%
-> It hits about 2 in 3 people.
Probability of hitting m or more times when performed p% loot box n times
If you want to hit more than 2 times, the calculation will be difficult.
Probability of hitting more than 2 times ＝ 100% - Probability of not hitting even once - Probability of hitting only once
Probability of hitting more than 3 times ＝ 100% - Probability of not hitting even once - Probability of hitting only once - Probability of hitting only twice
“Probability of hitting only x times” can be calculated by the following formula.
P=nCx \times p^x \times (1-p)^{(n-x)}
nCx is number of combinations to choose x from n different.
nCx={\dfrac {n!}{x!(n−x)!}}
Probability of not hitting even once:
P_0={}_{100}C_0 \times 0.01^0 \times (1-0.01)^{(100-0)}\\ =1 \times 1 \times 0.99^{100}\\ \approx 0.36603
Probability of hitting only once:
P_1={}_{100}C_1 \times 0.01^1 \times (1-0.01)^{(100-1)}\\ =100 \times 0.01 \times 0.99^{99}\\ \approx 0.36973
Probability of hitting 2 or more times:
P=1-P_0-P_1\\ =1-0.36603-0.36973\\ =0.26424\\ \approx 26.42\%
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EUDML | Higher Order Commutators in Interpolation Theory. EuDML | Higher Order Commutators in Interpolation Theory.
Higher Order Commutators in Interpolation Theory.
M.J. Carro; J. Cerda; J. Soria
Carro, M.J., Cerda, J., and Soria, J.. "Higher Order Commutators in Interpolation Theory.." Mathematica Scandinavica 77.2 (1995): 301-319. <http://eudml.org/doc/167369>.
@article{Carro1995,
author = {Carro, M.J., Cerda, J., Soria, J.},
keywords = {complex method; higher-order commutators; interpolation theory; real method; cancellation properties; boundedness; singular operators between weighted spaces},
title = {Higher Order Commutators in Interpolation Theory.},
AU - Cerda, J.
TI - Higher Order Commutators in Interpolation Theory.
KW - complex method; higher-order commutators; interpolation theory; real method; cancellation properties; boundedness; singular operators between weighted spaces
complex method, higher-order commutators, interpolation theory, real method, cancellation properties, boundedness, singular operators between weighted
{L}^{p}
Articles by M.J. Carro
Articles by J. Cerda
Articles by J. Soria
|
In the Standard and Natural Units environments, you can take the logarithm of a quantity that has a unit of the form
\frac{x}{x\left(\mathrm{base}\right)}
, for any valid unit
x
(see Units,annotations for an explanation of what the symbol in parentheses means), and obtain a result with units in the dimension of logarithmic gain. Similarly, you can raise a unit-free quantity to a quantity with units in the dimension of logarithmic gain, and obtain a result with the dimension
\frac{\mathrm{length}\left(\mathrm{base}\right)}{\mathrm{length}}
. (This corresponds to the case above for
x
being a unit of dimension
\frac{1}{\mathrm{length}}
.) As a special case, you can apply the exponential function to a quantity with units in the dimension of logarithmic gain to the same effect. See the Units[Standard][ln] and Units[Standard][exp] help pages for more details.
A bel is defined as the logarithm base 10 of a power over a base power. Because power is proportional to the square of the voltage, the conversion factor from nepers to bels is
\frac{2}{\mathrm{ln}\left(10\right)}
\mathrm{convert}\left(13.2,'\mathrm{units}','\mathrm{Np}','\mathrm{dB}'\right)
\textcolor[rgb]{0,0,1}{114.6537432}
\mathrm{convert}\left('\mathrm{amperes}','\mathrm{dimensions}','\mathrm{energy}'\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{\mathrm{length}}}
\mathrm{convert}\left('\mathrm{watts}','\mathrm{dimensions}','\mathrm{energy}'\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{\mathrm{length}}}^{\textcolor[rgb]{0,0,1}{2}}}
\mathrm{convert}\left('\mathrm{pascals}','\mathrm{dimensions}','\mathrm{energy}'\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{{\textcolor[rgb]{0,0,1}{\mathrm{length}}}^{\textcolor[rgb]{0,0,1}{4}}}
\mathrm{with}\left(\mathrm{Units}[\mathrm{Standard}]\right):
\mathrm{unit}
\mathrm{ln}\left(3.2\mathrm{Unit}\left(\frac{'W'}{'W\left(\mathrm{base}\right)'}\right)\right)
\textcolor[rgb]{0,0,1}{0.5815754050}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{Np}}⟧
\mathrm{convert}\left(,'\mathrm{units}','\mathrm{dB}'\right)
\textcolor[rgb]{0,0,1}{5.051499784}\textcolor[rgb]{0,0,1}{}⟦\textcolor[rgb]{0,0,1}{\mathrm{dB}}⟧
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Fixed Point Theorems for Geraghty Type Contractive Mappings and Coupled Fixed Point Results in 0-Complete Ordered Partial Metric Spaces
Esra Yolacan, "Fixed Point Theorems for Geraghty Type Contractive Mappings and Coupled Fixed Point Results in 0-Complete Ordered Partial Metric Spaces", International Journal of Analysis, vol. 2016, Article ID 8947020, 5 pages, 2016. https://doi.org/10.1155/2016/8947020
Esra Yolacan 1
1Republic of Turkey Ministry of National Education, 60000 Tokat, Turkey
We establish new fixed point theorems in 0-complete ordered partial metric spaces. Also, we give remark on coupled generalized Banach contraction. Some examples illustrate the usability of our results. The theorems presented in this paper are generalizations and improvements of the several well known results in the literature.
Henceforward, the letters , , and will indicate the set of real numbers, the set of nonnegative real numbers, and the set of positive integer numbers, respectively.
Definition 1 (see [1]). A partial metric on a nonempty set is a function such that, for all , () , () , () , and () . The pair is called a partial metric space.
If is a partial metric on , then the function given by is a metric on . Each partial metric on introduces a topology on which has as a base the family of open balls for all and
Let be a partial metric space, and let be any sequence in and . Then (i) a sequence is convergent to with respect to , if as , (ii) a sequence is a Cauchy sequence in if exists and is finite; (iii) is called complete if for every Cauchy sequence in there exists such that as
Romaguera [2] introduced the notion of 0-Cauchy sequence, 0-complete partial metric spaces and proved some characterizations of partial metric spaces in terms of completeness and 0-completeness. After that many authors extended the results of [2] and studied fixed point theorems in 0-complete partial metric space (see [2–10]).
Definition 2 (see [2]). Let be a partial metric space. A sequence in is called a 0-Cauchy sequence if as . The space is said to be 0-complete if every 0-Cauchy sequence in converges with respect to to a point such that
Remark 3 (see [11, 12]). (1) Let be a partial metric space. If as , then as for all .
(2) If is a continuous at , then for each sequence in , we have as as (see [5]).
Let be the class of functions with implying Amini-Harandi and Emami [13] presented the following results.
Theorem 4 (see [13]). Let be an ordered set endowed with a metric and let be a given mapping. Suppose that the following conditions hold:(i) is complete.(ii) (1) is continuous or(2)if a nondecreasing sequence in converges to some point , then for all .(iii) is nondecreasing.(iv)There exists such that .(v)There exists such that for all with ,Then has a fixed point. Moreover, if for all there exists a such that and , we obtain uniqueness of the fixed point.
In this paper, we establish new fixed point theorems in 0-complete ordered partial metric spaces (briefly 0-COPMS). Also, we give remark on coupled generalized Banach contraction. Some examples illustrate the usability of our results. The theorems presented in this paper are generalizations and improvements of the several well known results in the literature.
Theorem 5. Let be a 0-COPMS. Let be a nondecreasing mapping such thatfor all with and . Also suppose that there exists such that . One assumes (1) is continuous or(2)if a nondecreasing sequence in converges to some point , then for all .Then has a fixed point .
Proof. By assumption there exists such that . Define as . Then we have . In a similar manner, we get as . In that case, . Continuing this procedure we have in such thatIf for some , then the proof is completed. Suppose farther that for each . Consider, as is nondecreasing, we obtain thatFrom (2), (3), and (4), for all , we get thatThen is a monotone decreasing. Hence as . Assume . Then by (2) we haveEquation (6) yields as . By virtue of , this implies thatNow we claim that is a 0-Cauchy sequence. Conversely, suppose thatBy and (2), we have, for ,Owing to (7) and (8), we get thatfrom which we have which implies . Since , we obtain . It is a contradiction. Thus is a 0-Cauchy sequence. As is 0-complete, it follows that there exists such that in and . Furthermore,We will show that . Consider two cases.
Case 1. If is continuous, thenhence .
Case 2. If (2) holds, then,In view of as , then we have
The following is example which illustrate Theorem 5 and that the generalizations are proper.
Example 6. Let , and let be defined by for all . Then is a 0-COPMS. Yet it is not complete partial metric space. We endow with the partial orderLet for all . Then it is clear that . Define asAssume that . Then we have two cases.
Case 1. If , thenTherefore, we haveHence, for , .
Case 2. If , thenHence, we getwhereThus, for , .
Moreover, by Cases 1 and 2, it is clear that both assumptions (1) and (2) of Theorem 5 are satisfied, and for , we have . Hence, all assumptions of Theorem 5 are satisfied, and has a fixed point
On the contrary, consider Example 6 in the standard metric . If and , thenand soThus, is not satisfied.
3. Remark on Coupled Generalized Banach Contraction
The following result generalizes and extends Theorem 2.1. in [14]. When making the proof of the theorem, Radenović’s technique [15] is used.
Theorem 7. Let be a 0-COPMS and let be a mapping. Suppose that, for all and , the following conditionholds. Then has a fixed point.
Proof. Consider the metrics defined byIf is complete, then is complete (resp. 0-complete), too. Now, define the operator byLet be a metric on defined by for all
From (23), for all , with and , we getThis implies that for all , with and ,that is,which is in fact condition (2), for all , with . Hence, all conditions of Theorem 5 are satisfied. In this case, applying Theorem 5, we have that has a fixed point. From the definition of , we have and ; that is to say, is a coupled fixed point of
Remark 8. If , in the inequality (23), then we obtain results of Bhaskar and Lakshmikantham [16] in 0-COPMS.
The following example illustrates the case when Theorem 7 is applicable, while Theorem 2.1. in [14] is not.
Example 9. Let , and let be defined by for all . Then is a 0-COPMS. Yet it is not complete partial metric space. We consider the following order relation on : Let for all . Then it is clear that. Define as for all
We have the following cases.
Case 1. For or or and , we have . Thus, (23) holds.
Case 2. For and , we haveThus, (23) holds.
Therefore, all the conditions of Theorem 7 are satisfied and is a coupled fixed point of .
S. G. Matthews, “Partial metric topology,” Annals of the New York Academy of Sciences, vol. 728, pp. 183–197, 1994. View at: Publisher Site | Google Scholar
S. Romaguera, “A Kirk type characterization of completeness for partial metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 493298, 6 pages, 2010. View at: Publisher Site | Google Scholar
N. Hussain, S. Al-Mezel, and P. Salimi, “Fixed points for
\psi
-graphic contractions with application to integral equations,” Abstract and Applied Analysis, vol. 2013, Article ID 575869, 11 pages, 2013. View at: Publisher Site | Google Scholar
A. G. B. Ahmad, Z. M. Fadail, V. Ć. Rajić, and S. Radenović, “Nonlinear Contractions in 0-complete partial metric spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 451239, 12 pages, 2012. View at: Publisher Site | Google Scholar
H. K. Nashine, Z. Kadelburg, S. Radenović, and J. K. Kim, “Fixed point theorems under Hardy-Rogers contractive conditions on 0-complete ordered partial metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 180, 2012. View at: Publisher Site | Google Scholar
S. Shukla and S. Radenović, “Some common fixed point theorems for F-contraction type mappings in 0-complete partial metric spaces,” Journal of Mathematics, vol. 2013, Article ID 878730, 7 pages, 2013. View at: Publisher Site | Google Scholar
S. Shukla, S. Radenović, and C. Vetro, “Set-valued Hardy-Rogers type contraction in 0-complete partial metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2014, Article ID 652925, 9 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
S. Shukla, “Set-valued Prešić-Ćirić type contraction in 0-complete partial metric spaces,” Matematicki Vesnik, vol. 66, no. 2, pp. 178–189, 2014. View at: Google Scholar | MathSciNet
D. Paesano and C. Vetro, “Multi-valued
F
-contractions in 0-complete partial metric spaces with application to Volterra type integral equation,” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales, Serie A: Matematicas, vol. 108, no. 2, pp. 1005–1020, 2014. View at: Publisher Site | Google Scholar
M. A. Akturk and E. Yolacan, “Generalized (ψ,
\phi
)-weak contractions in 0-complete partial metric spaces,” Journal of Mathematical Sciences and Applications, vol. 4, no. 1, pp. 14–19, 2016. View at: Publisher Site | Google Scholar
A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 5, pp. 2238–2242, 2010. View at: Publisher Site | Google Scholar | MathSciNet
B. S. Choudhury and A. Kundu, “On coupled generalised Banach and Kannan type contractions,” Journal of Nonlinear Science and its Applications, vol. 5, no. 4, pp. 259–270, 2012. View at: Google Scholar | MathSciNet
S. Radenović, “Remarks on some coupled. Fixed point results in Partial metric spaces,” Nonlinear Functional Analysis and Applications, vol. 18, no. 1, pp. 39–50, 2013. View at: Google Scholar
T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis, vol. 65, no. 7, pp. 1379–1393, 2006. View at: Publisher Site | Google Scholar | MathSciNet
Copyright © 2016 Esra Yolacan. 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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Message - Maple Help
Home : Support : Online Help : Programming : Maplets : Examples : Message
display a Maplet application with a user message
Message(msg, opts)
The Message(msg) calling sequence displays a Maplet application with a message for the user.
The Message sample Maplet worksheet describes how to write a Maplet application that behaves similarly to this routine by using the Maplets[Elements] package.
Specifies the Maplet application title. By default, the title is Message.
The message type. By default, the type is plain. This option determines the icon that is located left of the text.
\mathrm{with}\left(\mathrm{Maplets}[\mathrm{Examples}]\right):
\mathrm{Message}\left("Hello world!"\right)
Message Sample Maplet
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This option defines the symbol of the physical constant. It cannot match the name or symbol of a physical constant in the ScientificConstants package (unless the 'check=false' option is given, see below). If the
'\mathrm{symbol}'
is modified, the descriptor must be the full name of the physical constant.
If the pre-existing value of the
'\mathrm{uncertainty}'
is not undefined, this equation is required.
The uncertainty_obj option must be of type constant (or be so after Constant() objects evaluate) or a list of the form
[\mathrm{uncer},\mathrm{uncertainty_opt}]
[\mathrm{uncer},'\mathrm{relative}']
[\mathrm{uncer},'\mathrm{uld}']
, uncer is the uncertainty in "units in the least digit" in the physical constant's value. The quantity
\mathrm{uncer}\mathrm{SFloatExponent}\left(\mathrm{value_obj}\right)
\mathrm{with}\left(\mathrm{ScientificConstants}\right):
\mathrm{GetConstant}\left(c\right)
\textcolor[rgb]{0,0,1}{\mathrm{speed_of_light_in_vacuum}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{symbol}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{c}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{299792458}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{uncertainty}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{units}}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{s}}
\mathrm{ModifyConstant}\left(\mathrm{speed_of_light_in_vacuum},\mathrm{symbol}=c[0]\right)
\mathrm{GetConstant}\left(c[0]\right)
\textcolor[rgb]{0,0,1}{\mathrm{speed_of_light_in_vacuum}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{symbol}}\textcolor[rgb]{0,0,1}{=}{\textcolor[rgb]{0,0,1}{c}}_{\textcolor[rgb]{0,0,1}{0}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{299792458}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{uncertainty}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{units}}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{m}}{\textcolor[rgb]{0,0,1}{s}}
\mathrm{AddConstant}\left(\mathrm{Hooke1},\mathrm{symbol}=\mathrm{k1},\mathrm{value}=2.3,\mathrm{units}=\frac{N}{m}\right)
\mathrm{GetConstant}\left(\mathrm{k1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Hooke1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{symbol}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{k1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2.3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{uncertainty}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{undefined}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{units}}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{N}}{\textcolor[rgb]{0,0,1}{m}}
\mathrm{ModifyConstant}\left(\mathrm{k1},\mathrm{value}=2.3,\mathrm{units}=\frac{N}{m},\mathrm{uncertainty}=[0.1,\mathrm{relative}]\right)
\mathrm{GetConstant}\left(\mathrm{k1}\right)
\textcolor[rgb]{0,0,1}{\mathrm{Hooke1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{symbol}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{k1}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{value}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{2.3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{uncertainty}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{0.23}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{\mathrm{units}}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{N}}{\textcolor[rgb]{0,0,1}{m}}
\mathrm{AddConstant}\left(\mathrm{mass_ratio},\mathrm{symbol}=\mathrm{`m\left[e\right]/m\left[p\right]`},\mathrm{derive}=\frac{m[e]}{m[p]}\right)
\mathrm{evalf}\left(\mathrm{Constant}\left(\mathrm{`m\left[e\right]/m\left[p\right]`}\right)\right)
\textcolor[rgb]{0,0,1}{0.0005446170214}
\mathrm{ModifyConstant}\left(\mathrm{mass_ratio},\mathrm{symbol}=\mathrm{`m\left[p\right]/m\left[e\right]`},\mathrm{derive}=\frac{m[p]}{m[e]}\right)
\mathrm{evalf}\left(\mathrm{Constant}\left(\mathrm{`m\left[p\right]/m\left[e\right]`}\right)\right)
\textcolor[rgb]{0,0,1}{1836.152674}
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What Does Marginal Social Cost Mean?
\begin{aligned} &\text{Marginal Social Cost} = \text{MPC} + \text{MEC} \\ &\textbf{where:} \\ &\text{MPC} = \text{marginal private cost} \\ &\text{MEC} = \text{marginal external cost (positive or negative)} \\ \end{aligned}
Marginal Social Cost=MPC+MECwhere:MPC=marginal private costMEC=marginal external cost (positive or negative)
Marginal social cost reflects the impact that an economy feels from the production of one more unit of a good or service.
Consider, for example, the pollution of a town’s river by a nearby coal plant. If the plant’s marginal social costs are higher than the plant’s marginal private costs, the marginal external cost is positive and results in a negative externality, meaning it produces a negative effect on the environment. The cost of the energy that is produced by the plant involves more than the rate that the company charges because the surrounding environment — the town — must bear the cost of the polluted river. This negative aspect must be factored in if a company strives to maintain the integrity of social responsibility or its responsibility to benefit the environment around it and society in general.
When determining the marginal social cost, both fixed and variable costs must be accounted for. Fixed costs are those that don’t fluctuate — such as salaries, or startup costs. Variable costs, on the other hand, change. For example, a variable cost could be a cost that changes based on production volume.
The Issue With Quantification
Marginal social cost is an economic principle that packs a major global punch, though, it is incredibly difficult to quantify in tangible dollars. Costs incurred by acts of production — such as operational costs and money used for startup capital — are fairly simple to calculate in tangible dollars. The issue comes when the far-reaching effects of production must also be factored in. Such costs are difficult, if not impossible, to pin down with an exact dollar amount, and in many instances, no price tag can be affixed to the effect.
The importance with marginal social cost, then, is that the principle can be used to aid economists and legislators to develop an operating and production structure that invites corporations to cut down on the costs of their actions.
Marginal social cost is related to marginalism, a concept that works to determine the amount of extra use derived from the production of one additional unit. The effects of the extra units on supply and demand are also studied. Marginal social cost can also be compared to the marginal benefit, the principle that determines the amount that consumers will give up to gain one extra unit.
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Trix (technical analysis) - Wikipedia
Trix (or TRIX) is a technical analysis oscillator developed in the 1980s by Jack Hutson, editor of Technical Analysis of Stocks and Commodities magazine. It shows the slope (i.e. derivative) of a triple-smoothed exponential moving average.[1][2] The name Trix is from "triple exponential."
The easiest way to calculate the triple EMA based on successive values is just to apply the EMA three times, creating single-, then double-, then triple-smoothed series. The triple EMA can also be expressed directly in terms of the prices as below, with
{\displaystyle p_{0}}
today's close,
{\displaystyle p_{1}}
yesterday's, etc., and with
{\displaystyle f=1-{2 \over N+1}={N-1 \over N+1}}
(as for a plain EMA):
{\displaystyle TripleEMA_{0}=(1-f)^{3}(p_{0}+3fp_{1}+6f^{2}p_{2}+10f^{3}p_{3}+\dots )}
The coefficients are the triangle numbers, n(n+1)/2. As f is less than 1, the powers
{\displaystyle f^{n}}
decrease faster than the coefficients increase. At a certain point the magnitude of all remaining terms becomes negligible.
^ "TRIX Uptrend & Downtrend | Stock Buy & Sell Signal |Technical Analysis". Web.archive.org. Archived from the original on 2016-03-05. Retrieved 2018-03-22.
^ "TRIX". Web.archive.org. Archived from the original on 2006-01-08. Retrieved 2018-03-22. {{cite web}}: CS1 maint: bot: original URL status unknown (link)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Trix_(technical_analysis)&oldid=1003875537"
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coset - Wiktionary
co- + set; apparently first used 1910 by American mathematician George Abram Miller.
coset (plural cosets)
(algebra, group theory) The set that results from applying a group's binary operation with a given fixed element of the group on each element of a given subgroup.
1970 [Addison Wesley], Frederick W. Byron, Robert W. Fuller, Mathematics of Classical and Quantum Physics, Volumes 1-2, Dover, 1992, page 597,
Theorem 10.5. The collection consisting of an invariant subgroup H and all its distinct cosets is itself a group, called the factor group of G, usually denoted by G/H. (Remember that the left and right cosets of an invariant subgroup are identical.) Multiplication of two cosets aH and bH is defined as the set of all distinct products z = xy, with x ∈ aH and y ∈ bH; the identity element of the factor group is the subgroup H itself.
1982 [Stanley Thornes], Linda Bostock, Suzanne Chandler, C. Rourke, Further Pure Mathematics, Nelson Thornes, 2002 Reprint, page 614,
In general, the coset in row x consists of all the elements xh as h runs through the various elements of H.
2009, Lindsay N. Childs, A Concrete Introduction to Higher Algebra, Springer, 3rd Edition, page 231,
{\displaystyle G=\mathbb {Z} }
(the operation is
{\displaystyle +}
{\displaystyle H=2\mathbb {Z} }
. Then the coset
{\displaystyle 1+2\mathbb {Z} }
is the set of integers of the form
{\displaystyle 1+2k}
{\displaystyle k}
runs through all elements of
{\displaystyle \mathbb {Z} }
Mathematically, given a group
{\displaystyle G}
with binary operation
{\displaystyle \circ }
, element
{\displaystyle g\in G}
and subgroup
{\displaystyle H\subseteq G}
{\displaystyle \left\{g\circ h:h\in H\right\}}
, which also defines the left coset if
{\displaystyle G}
is not assumed to be abelian.
The concept is relevant to the (mathematical) definitions of normal subgroup and quotient group.
result of applying group operation with fixed element from the parent group on each element of a subgroup
Estonian: kõrvalklass
Finnish: sivuluokka
Italian: classe laterale f
Portuguese: classe lateral f
Swedish: sidoklass c
Lagrange's theorem (group theory) on Wikipedia.Wikipedia
Normal subgroup on Wikipedia.Wikipedia
Quotient group on Wikipedia.Wikipedia
Coset on Wolfram MathWorld
Coset in a group on Encyclopedia of Mathematics
Coste, Cotes, OTECs, ScotE, cotes, scote
Retrieved from "https://en.wiktionary.org/w/index.php?title=coset&oldid=62026645"
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Carbon and its Compounds - Revision Notes
Carbon is a versatile element. Found in .02% in form of minerals an 03% in form of
C{O}_{{2}^{.}}
--comparatively weaker intermolecular forces, unlike ionic compounds.
2.TETRA VALENCY : Having a valency of 4, carbon atom is capable of bonding with atoms of oxygen, hydrogen, nitrogen, sulphur, chlorine and other elements. Since it requires four electrons , carbon is said to be tetravalent.
ALKANE :
{C}_{n}{H}_{2n+2}
ALKENE :
{C}_{n}{H}_{{2}^{n}}
ALKYNE :
{C}_{n}{H}_{2n-2}
Apart from branched structures, carbon compounds are present in cyclic form. Example: Electron Dot structure :Lewis structures (\electron dot structures) are diagrams that show the bonding between atoms of a molecule and the lone pairs of electrons that may exist in the molecule.
For instance, the ALCOHOLS :
C{H}_{2}
OH,
{C}_{2}{H}_{5}OH
{C}_{3}{H}_{7}OH
{C}_{4}{H}_{9}OH
C{H}_{4}+{O}_{2}\stackrel{\left\{\phantom{\rule{1pt}{0ex}}}{\to }C{O}_{2}+{H}_{2}O+Heat\phantom{\rule{thickmathspace}{0ex}}and\phantom{\rule{thickmathspace}{0ex}}light
C{H}_{3}-C{H}_{2}OH\underset{Acidified\phantom{\rule{thickmathspace}{0ex}}{K}_{2}C{r}_{2}{O}_{7}+heat}{\overset{AlkalineKMn{O}_{4}+heat}{\to }}\phantom{\rule{0ex}{0ex}}C{H}_{3}COOH
C{H}_{4}+C{l}_{2}\stackrel{}{\to }C{H}_{3}Cl+HCl\left(sunlight\phantom{\rule{thickmathspace}{0ex}}required\right)
{C}_{2}{H}_{5}OH
reacts with Sodium to form Sodium Ethoxide and Hydrogen
{C}_{2}{H}_{5}OH
is heated with concentrated Sulphuric Acid at 443K. it is dehydrated to Ethene
SODIUM ETHANOATE, CARBON
ETHANOL (IN PRESENCE OF CONC. SULPHURIC ACID)
C{H}_{3}-C{H}_{2}OH
SODIUM ETHANOATE AND WATER
ethonol of ester - ethnoic acid - ethyl ethanoate
\underset{ethanol\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}of\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}ester}{{C}_{2}{H}_{5}OH}+\underset{ethanoic\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}acid}{C{H}_{3}COOH}\underset{{H}_{2}S{O}_{4}}{\overset{CONC.}{\to }}\phantom{\rule{0ex}{0ex}}\underset{ethyl\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}ethanoate}{C{H}_{3}COO{C}_{2}{H}_{5}}
C{H}_{3}COO\phantom{\rule{thickmathspace}{0ex}}C{H}_{2}C{H}_{3}+NaOH\stackrel{}{\to }\phantom{\rule{0ex}{0ex}}C{H}_{3}COONa+\phantom{\rule{thickmathspace}{0ex}}C{H}_{3}-C{H}_{2}OH
C{H}_{3}COO\phantom{\rule{thickmathspace}{0ex}}C{H}_{2}C{H}_{3}\underset{Heat}{\overset{Dil.{H}_{2}S{O}_{4}}{\to }}\phantom{\rule{0ex}{0ex}}C{H}_{3}COOH+C{H}_{3}-C{H}_{2}0H
Carbon is a versatile element that forms the basis for all living organisms and man of the things we use.
This large variety of compounds is formed by carbon because of its tetra-valence and the property of catenation that it exhibits.
Covalent bonds are formed by the sharing of electrons between two atoms so that both can achieve a completely filled outermost shell.
Carbon forms covalent bonds with itself and other elements such as hydrogen oxygen, sulphur, nitrogen and chlorine.
Carbon also forms compounds containing double and triple bonds between carbo-atoms. These carbon chains may be in the form of straight chains, branched chain or rings.
The ability of carbon to form chains gives rise to a homologous series of compound in which the same functional group is attached to carbon chains of different lengths.
The functional groups such as alcohols, aldehydes, ketones and carboxylic acid bestow characteristic properties to the carbon compounds that contain them.
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Discrete variable capacitor - Simulink - MathWorks Nordic
Minimal capacitance absolute value (F)
Discrete variable capacitor
The Variable Capacitor block represents a linear time-varying capacitor. It implements a discrete variable capacitor as a current source. The capacitance is specified by the Simulink® input signal. The capacitance value can be negative.
When you use a Variable Capacitor block in your model, set the powergui block Simulation type to Discrete and select the Automatically handle Discrete solver and Advanced tab solver settings of blocks parameter in the Preferences tab. The robust discrete solver is used to discretize the electrical model. Simulink signals an error if the robust discrete solver is not used.
The block uses the following equations for the relationship between the voltage, v, across the device and the current through the capacitor, i, when the capacitance at port C is C:
i=C\frac{dv}{dt}.
C — Capacitance
Input port associated with the capacitance. The capacitance can be negative and must be finite.
Specialized electrical conserving port associated with the capacitor positive voltage.
Specialized electrical conserving port associated with the capacitor negative voltage.
Minimal capacitance absolute value (F) — Minimal capacitance
Lower limit on the absolute value of the signal at port C. This limit prevents the signal from reaching a value that has no physical meaning. The value of this parameter must be greater than 0.
Nonlinear Inductor | Nonlinear Resistor | Variable Inductor | Variable Resistor | Variable-Ratio Transformer
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Serve - Maple Help
Home : Support : Online Help : Connectivity : Web Features : Network Communication : Sockets Package : Serve
establish a Maple server
Serve(port, server)
positive integer; port number on which the service is to be made available
Maple procedure (or other expression); use to service a single request
To establish your own Maple services, you can use the Serve command which runs a server on a specified port. It requires two arguments: a port number port on which to listen for requests, and a Maple procedure server which handles a single incoming request. The call Serve( port, server ) establishes a service on the specified port (assuming that you have permissions to do so), and listens for requests on that port. When a request arrives, the callback server is called with the socket ID of the incoming request as its only argument. The server procedure should service the request and return without error.
Any exception raised in the server callback causes the service to be interrupted and the exception propagates to the Maple process that invoked the Serve call.
Note: Due to licensing considerations, Serve does not establish a true server; you must manually put the Maple server process into the background, and only one incoming request at a time is handled. No new threads or processes are created to service individual requests, even on the UNIX platform.
The procedure Serve is called for effect, and never returns.
The procedures Sockets[GetPeerHost] and Sockets[GetPeerPort] may be useful in authenticating incoming client requests.
If you use a system tool such as ``netstat'' to examine the sockets that are open, you may notice connections allocated in a ``TIME_WAIT'' state after the server has finished servicing a request. This is a normal effect of the TCP protocol, which keeps sockets allocated in a TIME_WAIT state for some period of time (twice the ``maximum segment lifetime'', or four minutes, by default) after the connection is closed. When a socket is in the TIME_WAIT state, the operating system is ensuring that any network packets that arrive with the allocated socket's address are bound to it and not to a new connection that might otherwise be bound to the same address as the old connection. For further details, see RFC 761 (``Transmission Control Protocol'') Most modern TCP implementations use the so-called ``SYN->RCVD'' transition that allows the operating system to open a new connection directly from the TIME_WAIT state providing that certain conditions are met. See RFC 1122 (``Requirements for Internet Hosts -- Communication Layers''), Section 4.2.2.13 for details.
This example illustrates a simple greeting server.
server := proc( sid )
The following call causes the Maple session in which it is invoked to start servicing requests. The call never returns.
\mathrm{Sockets}:-\mathrm{Serve}\left(2525,\mathrm{server}\right)
Sockets[GetPeerHost]
Sockets[GetPeerPort]
Sockets[references]
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Uniform_space Knowpia
Entourage definitionEdit
This definition adapts the presentation of a topological space in terms of neighborhood systems. A nonempty collection
{\displaystyle \Phi }
of subsets
{\displaystyle U\subseteq X\times X}
is a uniform structure (or a uniformity) if it satisfies the following axioms:
{\displaystyle U\in \Phi }
{\displaystyle \Delta \subseteq U}
{\displaystyle \Delta =\{(x,x):x\in X\}}
is the diagonal on
{\displaystyle X\times X}
{\displaystyle U\in \Phi }
{\displaystyle U\subseteq V\subseteq X\times X}
{\displaystyle V\in \Phi }
{\displaystyle U\in \Phi }
{\displaystyle V\in \Phi }
{\displaystyle U\cap V\in \Phi }
{\displaystyle U\in \Phi }
, then there is
{\displaystyle V\in \Phi }
{\displaystyle V\circ V\subseteq U}
{\displaystyle V\circ V}
denotes the composite of
{\displaystyle V}
with itself. (The composite of two subsets
{\displaystyle V}
{\displaystyle U}
{\displaystyle X\times X}
{\displaystyle V\circ U=\{(x,z):\exists y\in X:(x,y)\in U\wedge (y,z)\in V\}}
{\displaystyle U\in \Phi }
{\displaystyle U^{-1}\in \Phi }
{\displaystyle U^{-1}=\{(y,x):(x,y)\in U\}}
is the inverse of U.
One usually writes U[x] = {y : (x,y) ∈ U} = pr2(U ∩ ({ x } × X )), where U ∩ ({ x } × X ) is the vertical cross section of U and pr2 is the projection onto the second coordinate. On a graph, a typical entourage is drawn as a blob surrounding the "y = x" diagonal; all the different U[x]'s form the vertical cross-sections. If (x, y) ∈ U, one says that x and y are U-close. Similarly, if all pairs of points in a subset A of X are U-close (i.e., if A × A is contained in U), A is called U-small. An entourage U is symmetric if (x, y) ∈ U precisely when (y, x) ∈ U. The first axiom states that each point is U-close to itself for each entourage U. The third axiom guarantees that being "both U-close and V-close" is also a closeness relation in the uniformity. The fourth axiom states that for each entourage U there is an entourage V that is "not more than half as large". Finally, the last axiom states that the property "closeness" with respect to a uniform structure is symmetric in x and y.
{\displaystyle U_{a}=\{(x,y)\in X\times X:d(x,y)\leq a\}\quad {\text{where}}\quad a>0}
Pseudometrics definitionEdit
Uniform cover definitionEdit
{X} is a uniform cover (i.e. {X} ∈ Θ).
If P <* Q and P is a uniform cover, then Q is also a uniform cover.
Given a uniform space in the entourage sense, define a cover P to be uniform if there is some entourage U such that for each x ∈ X, there is an A ∈ P such that U[x] ⊆ A. These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of ⋃{A × A : A ∈ P}, as P ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other. [1]
Topology of uniform spacesEdit
Every uniform space X becomes a topological space by defining a subset O of X to be open if and only if for every x in O there exists an entourage V such that V[x] is a subset of O. In this topology, the neighbourhood filter of a point x is {V[x] : V ∈ Φ}. This can be proved with a recursive use of the existence of a "half-size" entourage. Compared to a general topological space the existence of the uniform structure makes possible the comparison of sizes of neighbourhoods: V[x] and V[y] are considered to be of the "same size".
Uniformizable spacesEdit
X is a Kolmogorov space
X is a Hausdorff space
X is a Tychonoff space
for any compatible uniform structure, the intersection of all entourages is the diagonal {(x, x) : x in X}.
Uniform continuityEdit
A Cauchy filter (respectively, a Cauchy prefilter)
{\displaystyle F}
on a uniform space
{\displaystyle X}
is a filter (respectively, a prefilter)
{\displaystyle F}
such that for every entourage
{\displaystyle U,}
{\displaystyle A\in F}
{\displaystyle A\times A\subseteq U.}
In other words, a filter is Cauchy if it contains "arbitrarily small" sets. It follows from the definitions that each filter that converges (with respect to the topology defined by the uniform structure) is a Cauchy filter. A minimal Cauchy filter is a Cauchy filter that does not contain any smaller (that is, coarser) Cauchy filter (other than itself). It can be shown that every Cauchy filter contains a unique minimal Cauchy filter. The neighbourhood filter of each point (the filter consisting of all neighbourhoods of the point) is a minimal Cauchy filter.
Complete uniform spaces enjoy the following important property: if
{\displaystyle f:A\to Y}
is a uniformly continuous function from a dense subset
{\displaystyle A}
of a uniform space
{\displaystyle X}
into a complete uniform space
{\displaystyle Y,}
{\displaystyle f}
can be extended (uniquely) into a uniformly continuous function on all of
{\displaystyle X.}
Hausdorff completion of a uniform spaceEdit
As with metric spaces, every uniform space
{\displaystyle X}
has a Hausdorff completion: that is, there exists a complete Hausdorff uniform space
{\displaystyle Y}
and a uniformly continuous map
{\displaystyle i:X\to Y}
for any uniformly continuous mapping
{\displaystyle f}
{\displaystyle X}
into a complete Hausdorff uniform space
{\displaystyle Z,}
there is a unique uniformly continuous map
{\displaystyle g:Y\to Z}
{\displaystyle f=gi.}
The Hausdorff completion
{\displaystyle Y}
is unique up to isomorphism. As a set,
{\displaystyle Y}
can be taken to consist of the minimal Cauchy filters on
{\displaystyle X.}
As the neighbourhood filter
{\displaystyle \mathbf {B} (x)}
of each point
{\displaystyle x}
{\displaystyle X}
is a minimal Cauchy filter, the map
{\displaystyle i}
can be defined by mapping
{\displaystyle x}
{\displaystyle \mathbf {B} (x).}
{\displaystyle i}
thus defined is in general not injective; in fact, the graph of the equivalence relation
{\displaystyle i(x)=i(x')}
is the intersection of all entourages of
{\displaystyle X,}
{\displaystyle i}s injective precisely when
{\displaystyle X}
is Hausdorff.
The uniform structure on
{\displaystyle Y}
is defined as follows: for each symmetric entourage
{\displaystyle V}
(that is, such that
{\displaystyle (x,y)inV}
{\displaystyle (y,x)\in V}
{\displaystyle C(V)}
be the set of all pairs
{\displaystyle (F,G)}
of minimal Cauchy filters which have in common at least one
{\displaystyle V}
-small set. The sets
{\displaystyle C(V)}
can be shown to form a fundamental system of entourages;
{\displaystyle Y}
is equipped with the uniform structure thus defined.
{\displaystyle i(X)}
is then a dense subset of
{\displaystyle Y.}
{\displaystyle X}
is Hausdorff, then
{\displaystyle i}s an isomorphism onto
{\displaystyle i(X),}
{\displaystyle X}
can be identified with a dense subset of its completion. Moreover,
{\displaystyle i(X)}
is always Hausdorff; it is called the Hausdorff uniform space associated with
{\displaystyle X.}
{\displaystyle R}
denotes the equivalence relation
{\displaystyle i(x)=i(x'),}
then the quotient space
{\displaystyle X/R}
{\displaystyle i(X).}
Every metric space (M, d) can be considered as a uniform space. Indeed, since a metric is a fortiori a pseudometric, the pseudometric definition furnishes M with a uniform structure. A fundamental system of entourages of this uniformity is provided by the sets
{\displaystyle \qquad U_{a}\triangleq d^{-1}([0,a])=\{(m,n)\in M\times M:d(m,n)\leq a\}.}
This uniform structure on M generates the usual metric space topology on M. However, different metric spaces can have the same uniform structure (trivial example is provided by a constant multiple of a metric). This uniform structure produces also equivalent definitions of uniform continuity and completeness for metric spaces.
Using metrics, a simple example of distinct uniform structures with coinciding topologies can be constructed. For instance, let d1(x,y) = | x − y | be the usual metric on R and let d2(x,y) = | ex − ey |. Then both metrics induce the usual topology on R, yet the uniform structures are distinct, since { (x,y) : | x − y | < 1 } is an entourage in the uniform structure for d1 but not for d2. Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet non-uniformly continuous function.
Every topological group G (in particular, every topological vector space) becomes a uniform space if we define a subset V of G × G to be an entourage if and only if it contains the set { (x, y) : x⋅y−1 in U } for some neighborhood U of the identity element of G. This uniform structure on G is called the right uniformity on G, because for every a in G, the right multiplication x → x⋅a is uniformly continuous with respect to this uniform structure. One may also define a left uniformity on G; the two need not coincide, but they both generate the given topology on G.
For every topological group G and its subgroup H the set of left cosets G/H is a uniform space with respect to the uniformity Φ defined as follows. The sets
{\displaystyle {\tilde {U}}=\{(s,t)\in G/H\times G/H:\ \ t\in U\cdot s\}}
, where U runs over neighborhoods of the identity in G, form a fundamental system of entourages for the uniformity Φ. The corresponding induced topology on G/H is equal to the quotient topology defined by the natural map G → G/H.
The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage.
Complete metric space – Metric geometry
Completely uniformizable space
Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
Proximity space – Structure describing a notion of "nearness" between subsets
Topology of uniform convergence
Uniform isomorphism – Uniformly continuous homeomorphism
Uniform property – Object of study in the category of uniform topological spaces
Uniformly connected space – Type of uniform space
Nicolas Bourbaki, General Topology ( Topologie Générale ), ISBN 0-387-19374-X (Ch. 1–4), ISBN 0-387-19372-3 (Ch. 5–10): Chapter II is a comprehensive reference of uniform structures, Chapter IX § 1 covers pseudometrics, and Chapter III § 3 covers uniform structures on topological groups
Ryszard Engelking, General Topology. Revised and completed edition , Berlin 1989.
John R. Isbell, Uniform Spaces ISBN 0-8218-1512-1
I. M. James, Introduction to Uniform Spaces ISBN 0-521-38620-9
I. M. James, Topological and Uniform Spaces ISBN 0-387-96466-5
John Tukey, Convergence and Uniformity in Topology ; ISBN 0-691-09568-X
André Weil, Sur les espaces à structure uniforme et sur la topologie générale , Act. Sci. Ind. 551, Paris, 1937
^ "IsarMathLib.org". Retrieved 2021-10-02.
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torch.linalg.inv — PyTorch 1.11.0 documentation
torch.linalg.inv
torch.linalg.inv¶
torch.linalg.inv(A, *, out=None) → Tensor¶
Computes the inverse of a square matrix if it exists. Throws a RuntimeError if the matrix is not invertible.
\mathbb{K}
\mathbb{R}
\mathbb{C}
, for a matrix
A \in \mathbb{K}^{n \times n}
, its inverse matrix
A^{-1} \in \mathbb{K}^{n \times n}
(if it exists) is defined as
A^{-1}A = AA^{-1} = \mathrm{I}_n
\mathrm{I}_n
is the n -dimensional identity matrix.
The inverse matrix exists if and only if
A
is invertible. In this case, the inverse is unique.
When inputs are on a CUDA device, this function synchronizes that device with the CPU.
Consider using torch.linalg.solve() if possible for multiplying a matrix on the left by the inverse, as:
torch.linalg.solve(A, B) == A.inv() @ B
It is always prefered to use solve() when possible, as it is faster and more numerically stable than computing the inverse explicitly.
torch.linalg.pinv() computes the pseudoinverse (Moore-Penrose inverse) of matrices of any shape.
torch.linalg.solve() computes A .inv() @ B with a numerically stable algorithm.
A (Tensor) – tensor of shape (*, n, n) where * is zero or more batch dimensions consisting of invertible matrices.
RuntimeError – if the matrix A or any matrix in the batch of matrices A is not invertible.
>>> Ainv = torch.linalg.inv(A)
>>> torch.dist(A @ Ainv, torch.eye(4))
>>> A = torch.randn(2, 3, 4, 4) # Batch of matrices
>>> torch.dist(A @ Ainv, torch.eye(4)))
>>> A = torch.randn(4, 4, dtype=torch.complex128) # Complex matrix
tensor(7.5107e-16, dtype=torch.float64)
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Algorithms divide and conquer
What is median of medians heuristic?
Why you may need "Median of Medians"?
Time and Space Complexity of Median of Medians Algorithm
Proofs: Why is this pivot good?
Let us get started with Median of Medians Algorithm.
It is an approximate median selection algorithm that helps in creating asymptotically optimal selection algorithms by producing good pivots that improve worst case time complexities for sorting and selection algorithms.
Some applications of the median of medians heuristic include the following;
Quickselect selects the kth smallest element of an initially unsorted array, it worst case running time is quadratic, when median of medians heuristic is implemented it finds an approximate median which is used as pivot and the worst case time complexity becomes linear.
Quicksort relies on a good pivot element for its performance, the best known approach for finding a pivot is using a randomized pivot element, the running time on average is linear but it becomes quadratic in the worst case.
Using median of medians proves useful in making its worst case O(nlogn).
Unfortunately, implementing this heuristic in Quicksort will actually make it perform a-lot less efficient when compared to the normal randomized pivot selection for most cases.
Introsort on the other hand is a hybrid sorting algorithm that uses both quick sort and the median of medians heuristic to give a fast average performance and an optimal worst case performance, It uses randomized quick sort at the start of the algorithm then based on the pivots thus far selected, it chooses to use the median of medians heuristic to find a good pivot making it asymptotically optimal with O(nlogn) time in the worst case. Introsort is used as a sorting algorithm in c++ stl.
Similarly, introselect uses quickselect and median of medians to select a good pivot at each iteration until a kth element is found.
It is a divide and conquer algorithm in that, it returns a pivot that in the worst case will divide a list of unsorted elements into sub-problems of size
\frac{3n}{10}
\frac{7n}{10}
assuming we choose a sublist size of 5.
It guarantees a good pivot that in the worst case will give a pivot in the range between 30th and 70th percentile of the list of size n.
For a pivot to be considered good it is essential for it to be around the middle, 30-70% guarantees the pivot will be around the middle 40% of the list.
The algorithm is as follows,
Divide the list into sublists if size n, assume 5.
Initialize an empty array M to store medians we obtain from smaller sublists.
Loop through the whole list in sizes of 5, assuming our list is divisible by 5.
\frac{n}{5}
sublists, use select brute-force subroutine to select a median m, which is in the 3rd rank out of 5 elements.
Append medians obtained from the sublists to the array M.
Use quickSelect subroutine to find the true median from array M, The median obtained is the viable pivot.
Terminate the algorithm once the base case is hit, that is, when the sublist becomes small enough. Use Select brute-force subroutine to find the median.
Note: We used chunks of size 5 because selecting a median from a list whose size is an odd number is easier. Even numbers require additional computation. We could also select 7 or any other odd number as we shall see in the proofs below.
Here is the procedure pictorially;
Pseudocode of Median of Medians Algorithm
medianOfMedians(arr[1...n])
if(n < 5 return Select(arr[1...n], n/2))
Let M be an empty list
For i from 0 to n/5 - 1:
Let m = Select(arr[5i + 1...5i+5], 3)
Add m to M
Return QuickSelect(M[1...n/5], n/10)
End medianOfMedians
It is recursive, it calls QuickSelect which in turn will call MedianOfMedians.
This algorithm runs in O(n) linear time complexity, we traverse the list once to find medians in sublists and another time to find the true median to be used as a pivot.
The space complexity is O(logn) , memory used will be proportional to the size of the lists.
Lemma: The median of medians will return a pivot element that is greater than and less than at least 30% of all elements in the whole list.
Array M consists of
\frac{n}{5}
medians of sub lists of size 5, these elements in list M is greater than and less than at-least two elements in the original list.
QuickSelect will return a true median that represents the whole list which is greater than and less than
\frac{\frac{n}{5}}{2}
elements of list M and since each one of the M elements is greater than and less than at least two other elements in their previous sublists, therefore the true median is greater than and less than at least
\frac{3n}{10}
, 30 percentile of elements of the whole list.
Lemma: With that, we guarantee that quickselect finds a good pivot in linear time O(n) which in turn guarantees quickSort's worst case to be O(nlogn).
Recurrence relation;
T\left(n\right) = \left\{\begin{array}{l}c if\left(n \le 1\right)\\ T\left(\frac{n}{5}\right) + T\left(\frac{7n}{10}\right) + dn if\left(n > 1\right)\end{array}\right\\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}
T\left(n\right) \le k . n for n \ge n\text{'}
\mathrm{T}\left(\mathrm{n}\right) = \mathrm{T}\left(\frac{\mathrm{n}}{5}\right) + \mathrm{T}\left(\frac{7\mathrm{n}}{10}\right) + \mathrm{dn}
T\left(1\right) = c \le k . 1 therefore k \ge c
T\left(j\right) \le b . k for\left(k \le n\right)
T\left(n\right) = T\left(\frac{n}{5}\right) + T\left(\frac{7n}{10}\right) + dn
T\left(n\right) \le k . \frac{n}{5} + k . \frac{7n}{10} + dn = \frac{9}{10}kn + dn
By induction;
T\left(n\right) \le k . n for n \ge 1
The above proof worked because
\frac{n}{5} + \frac{7n}{10} < 1
,we split the original list in chunks of 5 assuming the original list is divisible by 5. We could also use another odd number provided the above equation results in a number below 1, then our theorem will perform its operations in O(n) linear time.
Therefore we get a big theta(n) time complexity for QuickSelect which proves using this heuristic for QuickSelect ad QuickSort improves worst case to O(n) and O(nlogn) for the respective algorithms.
In this post at OpenGenus, we explained introsort uses median of medians heuristic to improve the worst case running time for quicksort, Is there another way to improve quicksort worst case run time using another hybrid approach?
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Raúl Montes-de-Oca, Enrique Lemus-Rodríguez, Francisco Sergio Salem-Silva, "Nonuniqueness versus Uniqueness of Optimal Policies in Convex Discounted Markov Decision Processes", Journal of Applied Mathematics, vol. 2013, Article ID 271279, 5 pages, 2013. https://doi.org/10.1155/2013/271279
Raúl Montes-de-Oca ,1 Enrique Lemus-Rodríguez,2 and Francisco Sergio Salem-Silva3
1Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Avenida San Rafael Atlixco 186, Col. Vicentina, 09340 México, DF, Mexico
2Universidad Anáhuac México-Norte, Avenida Universidad Anáhuac 46, Lomas Anáhuac, 52786 Huixquilucan, MEX, Mexico
3Facultad de Matemáticas, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán s/n, Zona Universitaria, 91000 Xalapa, VER, Mexico
Academic Editor: Debasish Roy
From the classical point of view (for instance, in Hadamard’s concept of well-posedness [1]) in a mathematical modeling problem, it is crucial that both the existence and the uniqueness are secured. But, in optimization, neither of these is guaranteed, and even if extra conditions ensure the existence of optimizers, their uniqueness will not automatically follow. For instance, in linear programming, we even have the extreme case that when there are two different optimal vectors all of their convex linear combinations become optimal automatically. But a slight perturbation of the cost functional will “destroy" most of the optimizers. In this sense, nonuniqueness in linear programming is highly unstable. This question is of interest with respect to the standard discounted Markov decision model, as in [2], which presents conditions that guarantee the uniqueness of the optimal policies.
In this paper, we study a family of perturbations of the cost of an MDP and establish that, under convexity and adequate bounds, the value functions of both the original and the cost-perturbed Markov decision processes (MDPs) are uniformly close. This result will eventually help us determine whether both the uniqueness and the nonuniqueness are stable with respect to this kind of perturbation.
The structure of this paper is simple. Firstly, the preliminaries and assumptions of the model are outlined. Secondly, the main theorem is stated and proved, followed by the main example. A brief section with the concluding remarks closes the paper.
2. Preliminaries: Discounted MDPs and Convexity Assumptions
Let be a Markov control model (see [3] for details and terminology) which consists of the state space , the control (or action) set , the transition law , and the cost-per-stage . It is assumed that both and are subsets of (this is supposed for simplicity, but it is also possible to present the theory of this paper considering that and are subsets of Euclidean spaces of the dimension greater than one). For each , there is a nonempty measurable set whose elements are the feasible actions when the state of the system is . Define . Finally, the cost-per-stage is a nonnegative and measurable function on .
Let be the set of all (possibly randomized, history-dependent) admissible policies. By standard convention, a stationary policy is identified with a measurable function such that for all . The set of stationary polices is denoted by . For every and an initial state , let be the total expected discounted cost when using the policy , given the initial state . The number is called the discount factor ( is assumed to be fixed). Here and denote the state and the control sequences, respectively, and is the expectation operator. A policy is said to be optimal if for all , where , . is called the optimal value function. The following assumption will also be taken into consideration.
Assumption 1. (a) is lower semicontinuous and inf-compact on (i.e., for every and the set is compact).
(b) The transition law is strongly continuous, that is, , is continuous and bounded on , for every measurable bounded function on .
(c) There exists a policy such that , for each .
Remark 2. The following consequences of Assumption 1 are well known (see Theorem and Lemma in [3]).(a) The optimal value function is the solution of the optimality equation (OE), that is, for all , There is also such that and is optimal.(b) For every , , with defined as , and . Moreover, for each , there is such that for each ,
Let be a fixed Markov control model. Take as the MDP with the Markov control model . The optimal value function, the optimal policy which comes from (3), and the minimizers in (5) will be denoted for by , , and , , respectively. Also let , , be the value iteration functions for . Let + , .
It will be also supposed that the MDPs taken into account satisfy one of the following Assumptions 3 or 4.
Assumption 3. (a) and are convex.
(b) for all , , , , and . Besides, it is assumed that if and , , then , and are convex for each .
(c) is induced by a difference equation , with , where is a measurable function and is a sequence of independent and identically distributed (i.i.d.) random variables with values in , and with a common density . In addition, we suppose that is a convex function on , for each , and if and , , then for each and .
(d) is convex on , and if and , , then , for each .
Assumption 4. (a) The same as Assumption 3(a).
(b) for all , , , , and . Besides, is assumed to be convex for each .
(c) is given by the relation , , where are i.i.d. random variables taking values in with the density , and are real numbers.
(d) is convex on .
Remark 5. Assumptions 3 and 4 are essentially the same as assumptions C1 and C2 in pages 419–420 of reference [2], with the difference that we are now able to assume that the function is convex and not necessarily strictly convex. (in fact, in [2], Conditions C1 and C2 take into account the more general situation in which both and are subsets of Euclidean spaces of the dimension greater than one). Also note that it is possible to obtain that each of Assumptions 3 and 4 implies that, for each , is convex but not necessarily strictly convex (hence, does not necessarily have a unique optimal policy). The proof of this fact is a direct consequence of the convexity of the cost function and of the proof of Lemma 6.2 in [2].
3. Main Result and an Example
For , consider the following MDP denoted by with the Markov control model , where , , where is the cost function for . Observe that both MDPs and coincide in the components of the Markov control model except for the cost function; moreover, is the same set in both models. Additionally we suppose that.
Assumption 6. There is a policy such that , for each .
Remark 7. Suppose that, for , Assumption 1 holds. Then, it is direct to verify that if satisfies Assumption 6, then it also satisfies Assumption 1.
For , let , , and , , denote the optimal value function, the optimal policy which comes from (3), and the minimizers in (5), respectively. Moreover, let , , be the corresponding value iteration functions for .
Remark 8. Suppose that, for , one of Assumptions 3 or 4 holds. Then, notice that as is a convex function, it is trivial to prove that is strictly convex. Then, under Assumption 6, it follows that satisfies C1 or C2 in [2] and that is strictly convex, where , , so is unique.
Let , so that and , , and take, for each , and .
Remark 9. It is easy to verify, using Assumption 1, that for each , and are nonempty and compact. Moreover, since and from Remark 2, , ; , for each and . It is also trivial to prove that, for each , ; hence , , , , for each and .
Condition 10. There exists a measurable function , which may depend on , such that , and for each and .
Remark 11. With respect to the existence of the function mentioned in Condition 10 that satisfies that for each and , it is important to note that this kind of requirement has been previously used in the unbounded MDPs literature (see, for instance, the Remarks presented on page 578 of [4]).
Theorem 12. Suppose that Assumptions 1 and 6 hold, and that, for , one of Assumptions 3 or 4 holds. Let be a positive number. Then,(a)if is compact, , , where is the diameter of a compact set such that and ;(b)under Condition 10, , .
Proof. The proof of case (a) follows from the proof of case (b) given that , (observe that in this case, if , then ).
(b) Assume that satisfies Assumption 3 (the proof for the case in which satisfies Assumption 4 is similar).
Firstly, for each ,
Secondly, assume that for some positive integer and for each ,
Consequently, using Condition 10, for each ,
On the other hand, from (11) and the fact that , , for each ,
In conclusion, combining (10), (13), and (14), it is obtained that, for each , (11) holds for all . Now, letting in (11), we get , .
Corollary 13. Suppose that Assumptions 1 and 6 hold. Suppose that for one of Assumptions 3 or 4 holds (hence, does not necessarily have a unique optimal policy). Let be a positive number. If is compact or Condition 10 holds, then there exists an MDP with a unique optimal policy , such that inequalities in Theorem 12 (a) or (b) hold, respectively.
Example 14. Let , , for all . The dynamic of the system is given by . Here, are i.i.d. random variables with values in and with a common continuous bounded density denoted by . The cost function is given by , (observe that is convex but not strictly convex).
Lemma 15. Example 14 satisfies Assumptions 1, 3, and 6, and Condition 10.
Proof. Assumption 1 (a) trivially holds. The proof of the strong continuity of is as follows: if is a measurable and bounded function, then, using the change of variable theorem, a simple computation shows that . As is a bounded function and is a bounded continuous function, it follows directly, using the convergence dominated theorem, that is a continuous function on . Hence, is a continuous function on .
By direct computations we get, for the stationary policy , , both and are less or equal to for all (observe that, in this case, and , ); consequently, Assumptions 1 and 6 hold.
On the other hand, Assumptions 3(a), (b), and (d) are immediate. Let . Clearly, is nondecreasing in the first variable.
Now, take and , , and . Then, considering that , , and are less or equal than one, hence, is convex, that is, Assumption 3(c) holds.
Now, for each ,
Hence, taking , , using (20) and, again, that , it is possible to obtain that for each and , and that .
The specific form of the perturbation used in this paper is taken from [5, Exercise 28, page 81], where it is established that a convex function perturbed by a suitable quadratic positive function becomes strictly convex and coercive. In fact, this kind of perturbation is very much related to the one Tanaka et al. propose in their paper [6], and further research in this direction is being conducted.
Both state and action spaces are considered to be subsets of , just for simplicity of exposition. All the results hold in . In this case, if , then it is possible to take , where , (see [5, Exercise 28, page 81]), and all the results on this article remain valid.
Theorem 12, on the closeness of the value functions of the original and the perturbed MDPs, requires conditions that are all very common in the MDPs technical literature. The importance of the result lies in the fact that it is a crucial step to the study of the problem of stability under the cost perturbation of the uniqueness or nonuniqueness of optimal policies.
Finally, we should mention that this research was motivated by our interest in understanding the relationship between nonuniqueness and robustness in several statistical procedures based on optimization.
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D. Cruz-Suárez, R. Montes-de-Oca, and F. Salem-Silva, “Conditions for the uniqueness of optimal policies of discounted Markov decision processes,” Mathematical Methods of Operations Research, vol. 60, no. 3, pp. 415–436, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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Copyright © 2013 Raúl Montes-de-Oca 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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Examine the Gaussian Mixture Assumption - MATLAB & Simulink - MathWorks India
Bartlett Test of Equal Covariance Matrices for Linear Discriminant Analysis
Bartlett Test of Equal Covariance Matrices
Compare Q-Q Plots for Linear and Quadratic Discriminants
Mardia Kurtosis Test of Multivariate Normality
Mardia Kurtosis Test for Linear and Quadratic Discriminants
Discriminant analysis assumes that the data comes from a Gaussian mixture model (see Creating Discriminant Analysis Model). If the data appears to come from a Gaussian mixture model, you can expect discriminant analysis to be a good classifier. Furthermore, the default linear discriminant analysis assumes that all class covariance matrices are equal. This section shows methods to check these assumptions:
The Bartlett test (see Box [1]) checks equality of the covariance matrices of the various classes. If the covariance matrices are equal, the test indicates that linear discriminant analysis is appropriate. If not, consider using quadratic discriminant analysis, setting the DiscrimType name-value pair argument to 'quadratic' in fitcdiscr.
The Bartlett test assumes normal (Gaussian) samples, where neither the means nor covariance matrices are known. To determine whether the covariances are equal, compute the following quantities:
Sample covariance matrices per class Σi, 1 ≤ i ≤ k, where k is the number of classes.
Pooled-in covariance matrix Σ.
Test statistic V:
V=\left(n-k\right)\mathrm{log}\left(|\Sigma |\right)-\sum _{i=1}^{k}\left({n}_{i}-1\right)\mathrm{log}\left(|{\Sigma }_{i}|\right)
where n is the total number of observations, ni is the number of observations in class i, and |Σ| means the determinant of the matrix Σ.
Asymptotically, as the number of observations in each class ni becomes large, V is distributed approximately χ2 with kd(d + 1)/2 degrees of freedom, where d is the number of predictors (number of dimensions in the data).
The Bartlett test is to check whether V exceeds a given percentile of the χ2 distribution with kd(d + 1)/2 degrees of freedom. If it does, then reject the hypothesis that the covariances are equal.
Check whether the Fisher iris data is well modeled by a single Gaussian covariance, or whether it is better modeled as a Gaussian mixture by performing a Bartlett test of equal covariance matrices.
prednames = {'SepalLength','SepalWidth','PetalLength','PetalWidth'};
When all the class covariance matrices are equal, a linear discriminant analysis is appropriate.
Train a linear discriminant analysis model (the default type) using the Fisher iris data.
L = fitcdiscr(meas,species,'PredictorNames',prednames);
When the class covariance matrices are not equal, a quadratic discriminant analysis is appropriate.
Train a quadratic discriminant analysis model using the Fisher iris data and compute statistics
Q = fitcdiscr(meas,species,'PredictorNames',prednames,'DiscrimType','quadratic');
Store as variables the number of observations N, dimension of the data set D, number of classes K, and number of observations in each class Nclass.
[N,D] = size(meas)
K = numel(unique(species))
Nclass = grpstats(meas(:,1),species,'numel')'
Nclass = 1×3
Compute the test statistic V.
SigmaL = L.Sigma;
SigmaQ = Q.Sigma;
V = (N-K)*log(det(SigmaL));
V = V - (Nclass(k)-1)*log(det(SigmaQ(:,:,k)));
nu = K*D*(D+1)/2;
pval1 = chi2cdf(V,nu,'upper')
pval1 = 2.6091e-17
Because pval1 is smaller than 0.05, the Bartlett test rejects the hypothesis of equal covariance matrices. The result indicates to use quadratic discriminant analysis, as opposed to linear discriminant analysis.
A Q-Q plot graphically shows whether an empirical distribution is close to a theoretical distribution. If the two are equal, the Q-Q plot lies on a 45° line. If not, the Q-Q plot strays from the 45° line.
Analyze the Q-Q plots to check whether the Fisher iris data is better modeled by a single Gaussian covariance or as a Gaussian mixture.
Train a linear discriminant analysis model.
Train a quadratic discriminant analysis model using the Fisher iris data.
Compute the number of observations, dimension of the data set, and expected quantiles.
[N,D] = size(meas);
expQuant = chi2inv(((1:N)-0.5)/N,D);
Compute the observed quantiles for the linear discriminant model.
obsL = mahal(L,L.X,'ClassLabels',L.Y);
[obsL,sortedL] = sort(obsL);
Graph the Q-Q plot for the linear discriminant.
gscatter(expQuant,obsL,L.Y(sortedL),'bgr',[],[],'off');
legend('virginica','versicolor','setosa','Location','NW');
xlabel('Expected quantile');
ylabel('Observed quantile for LDA');
line([0 20],[0 20],'color','k');
The expected and observed quantiles agree somewhat. The deviation of the plot from the 45° line upward indicates that the data has heavier tails than a normal distribution. The plot shows three possible outliers at the top: two observations from class 'setosa' and one observation from class 'virginica'.
Compute the observed quantiles for the quadratic discriminant model.
obsQ = mahal(Q,Q.X,'ClassLabels',Q.Y);
[obsQ,sortedQ] = sort(obsQ);
Graph the Q-Q plot for the quadratic discriminant.
gscatter(expQuant,obsQ,Q.Y(sortedQ),'bgr',[],[],'off');
ylabel('Observed quantile for QDA');
The Q-Q plot for the quadratic discriminant shows a better agreement between the observed and expected quantiles. The plot shows only one possible outlier from class 'setosa'. The Fisher iris data is better modeled as a Gaussian mixture with covariance matrices that are not required to be equal across classes.
The Mardia kurtosis test (see Mardia [2]) is an alternative to examining a Q-Q plot. It gives a numeric approach to deciding if data matches a Gaussian mixture model.
In the Mardia kurtosis test you compute M, the mean of the fourth power of the Mahalanobis distance of the data from the class means. If the data is normally distributed with a constant covariance matrix (and is thus suitable for linear discriminant analysis), M is asymptotically distributed as normal with mean d(d + 2) and variance 8d(d + 2)/n, where
d is the number of predictors (number of dimensions in the data).
The Mardia test is two sided: check whether M is close enough to d(d + 2) with respect to a normal distribution of variance 8d(d + 2)/n.
Perform a Mardia kurtosis tests to check whether the Fisher iris data is approximately normally distributed for both linear and quadratic discriminant analyses.
Compute the mean and variance of the asymptotic distribution.
meanKurt = D*(D+2)
meanKurt = 24
varKurt = 8*D*(D+2)/N
varKurt = 1.2800
Compute the p-value for the Mardia kurtosis test on the linear discriminant model.
mahL = mahal(L,L.X,'ClassLabels',L.Y);
meanL = mean(mahL.^2);
[~,pvalL] = ztest(meanL,meanKurt,sqrt(varKurt))
pvalL = 0.0208
Because pvalL is smaller than 0.05, the Mardia kurtosis test rejects the hypothesis of the data being normally distributed with a constant covariance matrix.
Compute the p-value for the Mardia kurtosis test on the quadratic discriminant model.
mahQ = mahal(Q,Q.X,'ClassLabels',Q.Y);
meanQ = mean(mahQ.^2);
[~,pvalQ] = ztest(meanQ,meanKurt,sqrt(varKurt))
pvalQ = 0.7230
Because pvalQ is greater than 0.05, the data is consistent with the multivariate normal distribution.
The results indicate to use quadratic discriminant analysis, as opposed to linear discriminant analysis.
[1] Box, G. E. P. “A General Distribution Theory for a Class of Likelihood Criteria.” Biometrika 36, no. 3–4 (1949): 317–46. https://doi.org/10.1093/biomet/36.3-4.317.
[2] Mardia, K. V. “Measures of Multivariate Skewness and Kurtosis with Applications.” Biometrika 57, no. 3 (1970): 519–30. https://doi.org/10.1093/biomet/57.3.519.
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Convergence of partial sum processes to stable processes with application for aggregation of branching processes
June 2022 Convergence of partial sum processes to stable processes with application for aggregation of branching processes
Mátyás Barczy, Fanni K. Nedényi, Gyula Pap
Mátyás Barczy,1 Fanni K. Nedényi,1 Gyula Pap2
1MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H–6720 Szeged, Hungary
2Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H–6720 Szeged, Hungary
We provide a generalization of Theorem 1 in Bartkiewicz et al. (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index in
\left(0,1\right)\cup \left(1,4/3\right)
in a so-called iterated case, namely when first taking the limit as the time scale and then the number of copies tend to infinity.
We would like to thank Thomas Mikosch for his suggestion to use the anti-clustering type condition (2.6) presented in Lemma 2.5, which will appear in his forthcoming book (2022+) written jointly with Olivier Wintenberger. We would like to thank the referee for the comments that helped us improve the paper. Mátyás Barczy is supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. Fanni K. Nedényi is supported by the UNKP-18-3 New National Excellence Program of the Ministry of Human Capacities. Gyula Pap was supported by grant NKFIH-1279-2/2020 of the Ministry for Innovation and Technology, Hungary.
Mátyás Barczy. Fanni K. Nedényi. Gyula Pap. "Convergence of partial sum processes to stable processes with application for aggregation of branching processes." Braz. J. Probab. Stat. 36 (2) 315 - 348, June 2022. https://doi.org/10.1214/21-BJPS528
Received: 1 April 2021; Accepted: 1 November 2021; Published: June 2022
Keywords: Galton–Watson branching processes with immigration , iterated aggregation , multivariate regular variation , Stable process , strong stationarity
Mátyás Barczy, Fanni K. Nedényi, Gyula Pap "Convergence of partial sum processes to stable processes with application for aggregation of branching processes," Brazilian Journal of Probability and Statistics, Braz. J. Probab. Stat. 36(2), 315-348, (June 2022)
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In this post, we discuss various approaches used to adapt a sequential merge sort algorithm onto a parallel computing platform. We have presented 4 different approaches of Parallel Merge Sort.
Merge sort is a divide and conquer algorithm. Given an input array, it divides it into halves and each half is then applied the same division method until individual elements are obtained. A pairs of adjacent elements are then merged to form sorted sublists until one fully sorted list is obtained.
Sequential merge sort algorithm: MergeSort(arr[], l, r)
Parallel Merge Sort algorithm
Approach 1: Quick Merge sort
Approach 2: Odd-Even merge sort
Approach 3: Bitonic merge sort
Approach 4: Parallel merge sort with load balancing
Prerequisite: Merge Sort Algorithm
Let us get started with Parallel Merge Sort.
if r > l,
Find the mid-point and divide the array into two halves, mid = l + (r-l) / 2.
Mergesort first half, mergeSort(arr, l, mid)
Mergesort second half, mergeSor(arr, mid+1, r)
Merge the two sorted halves, merge(arr, l, mid, r)
The time complexity is O(nlogn) in all cases.
The space complexity is O(n) linear time.
The key to designing parallel algorithms is to find steps in the algorithm that can be carried out concurrently. That is, examine the sequential algorithm and look for operations that could be carried out simultaneously on a given number of processors.
Note: The number of processors will influence the performance of the algorithm.
Ideation. Simple parallel merge sort.
In this approach we can assume we have an unlimited number of processors. Looking at the merge sort algorithm tree in the sequential algorithm we can try to assign simultaneous operations to separate processors which will work concurrently to do the dividing and merging at each level.
Given the list [9, 3, 17, 11, 6, 2, 1, 10] and 8 processors.
Here is how the algorithm would sort it parallelly,pms
Merging would start at the bottom of the tree going up.
Considering the sequential algorithm merge sort tree, tree work would start from the 4th level after the list has been divided into individual sublists of one element, each of the n processors would be divided among the 4 sublists of 2 pairs and merge them together, once done, going down the tree, 2 processors can work on merging the 2 4-element sorted sublists and finally one process can merge the 2 halves to create a final sorted list of 8 items.
Each processor will be executing in parallel at each level of the tree.
A more realistic case is when the number of processors is fewer than the number of elements to be sorted, therefore we need a strategy for dividing work among the given n processors.
Assume an large input of size n, each of the p processes sorts n/p of the original list using a sequential quick sort algorithm. The two sorted halves are then merged by the parent process and this repeated up the tree until there are two halves each of which are merged with one process at the top of the tree to obtain a fully sorted list.
Given a list of 4000 items and 8 processors, we opt to divide the list into halves until we have 8 partitions of size 500.
We sort the sublist of 500 items using a fast sequential algorithm, quick sort.
Recursively merge the sorted sublist items moving up the tree until we achieve a fully sorted list.
The sequential quick sort algorithm sorts in O(nlogn) time and merging is O(logn) steps, total time complexity is
O\left(log\left(n{\right)}^{2}\right)
This is a parallel approach to sorting based on the recursive application of the odd-even merge algorithm.
The odd-even merge algorithm merges sorted sub-lists in a bottom-up manner starting with sublists of size 2 and merging them into larger lists until the final sorted list is obtained.
The odd-even-merge operates on two alternating phases,
Even phase - Even-numbered processes exchange values with their right neighbor.
Odd phase - Odd-numbered processes exchange numbers with their right neighbor.
Given an array [9, 3, 17, 11, 6, 2, 1, 10], the algorithm will divide it into two sorted sublists A[3, 9, 11, 17] and B[1, 2, 6, 10]. The even and odd phases will arrange the list in the following order.
E(A) = [3, 11], O(A) = [9, 17]
E(B) = [1, 6], O(B) = [2, 10]
Recursively merge(E(A), E(B)) = C = oddEvenMerge([3, 11], [1, 6]) = [1, 3, 6, 11]
Recursively merge(O(A), O(B)) = D = oddEvenMerge([9, 17], [2, 10]) = [2, 9, 10, 17]
Interleave C with D, E = [1, 3,2, 6,9, 11,10, 17]
Rearrange unordered neighbors c, d, as (min(c,d), max(c,d)) E = [1, 2, 3, 6, 9, 10, 11, 17] to achieve a fully sorted list.
Divide and merge even and odd element E(A), E(B), O(A), O(B).
Recursively merge(E(A), E(B)) <- C.
Recursively merge(O(A), O(B)) <- D.
Interleave C with D.
Rearrange unordered neighbors C, D as (min(C, D), max(C, D)) <- E(sorted array).
Merging two sorted sublists of size n requires
Two recursive calls to oddEvenMerge each merging n/2 size sublists
oddEvenMerge([3, 11], [1, 6])
oddEvenMerge([9, 17], [2, 10])
n-1 comparison-and-swap operations.
compare and swap (3,2, 6,9, 11,10)
Parallelizing Odd-Even merge sort.
For parallelizing the OddEvenMergSort() algorithm, we can run oddEvenMerge() and compare-and-swap operations concurrently.
2 parallel steps for compare and swap operations for the recursive calls to oddEvenMerge() for lists of size n/2.
logn - 1 parallel steps for remaining compare and swap operations.
There are log(n) sorting steps, oddEvenMerge time complexity is O(logn), total time complexity is
O\left(log\left(n{\right)}^{2}\right)
This is another parallel approach based on the bitonic sequence.
A bitonic sequence is a sequence of numbers which is first increasing up to a certain point and then it starts decreasing.
ie, a1 < a2 < ... < a1 - 1 < ai > a1 + 1 > ai + 2 > ... an.
Following this, a bitonic point is a point in the bitonic sequence before which elements are strictly increasing after which they are strictly decreasing.
A bitonic point can only arise in arrays where there are both increasing and decreasing values.
A list having theses properties can be exploited by a parallel sorting algorithm.
Given the bitonic sequence below.
We can perform a compare and swap operation with elements from both sides to obtain tow bitonic sequences whereby all values in one sequence are smaller than the other values.
Obtaining the result we can further apply compare-and-swap operation, with the following in mind,
All values in left sequence of each bitonic sequence are less than all right values
Finally, we obtain a fully sorted list.
Transform the list into a bitonic sequence, as shown above.
Recursively compare-and-swap then merge the sublists lists to obtain a large fully sorted list.
Convert adjacent pairs of values into increasing and decreasing sequences to form a bitonic sequence.
Sort each bitonic sequence into increasing and decreasing segments.
Merge adjacent sequences into a larger bitonic sequence.
Sort the final bitonic sequence to achive a sorted list.
Note: Bitonic parallel sorting is usually done when a list of size
{2}^{n}
is given as the input otherwise it fails.
We will consider an unsorted list of size
{2}^{n}
in this analysis.
There are n steps, with each step working concurrently.
In total it will take,
\frac{n\left(n+1\right)}{2} = log\left(n\right) . \frac{\mathrm{log}\left(n\right) + 1}{2}
The total time complexity is
O\left(log\left(n{\right)}^{2}\right)
As we have seen in the previous approach, there is a need for load balancing, as we go up or down the tree, some processors will be idle while others are assigned work load at each level.
This approach aims to distribute work onto all processors such that they all partake in merging throughout the execution of the algorithm.
This would achieve more parallelism and achieve a better time computational complexity thereby shortening the sorting time.
The additional complexity with this approach is how to merge two lists that are distributed among multiple processors while minimizing elements movement.
A group handles one sorted list, it will store elements by distributing them evenly to all processors in the group. A histogram is computed which is used for determining the minimum number of elements to be swapped during merging.
In the first step when the algorithm starts, all groups will contain one processor, then in subsequent steps groups are paired together.
Each group will exchange its boundaries(min and max elements) to determine the order of processors according to the minimum elements.
Each group will exchange histograms and a new histogram that covers both pairs is computed.
Each processor divides the histogram bins of the shared histogram so as the lower indexed processor keeps the smaller half and the larger index processor keeps the large half.
Elements are exchanged on the basis of the histogram intervals between partner processors.
As the algorithm proceeds the group size doubles due to merging
To reduce costs during swapping elements, index swapping is used, this is when a processor has to send most of its elements to partner processor and receive the same equal amount, the algorithm will instead swaps the logical ids of the two to minimize element exchange.
Group size doubles as we go up the tree merging.
The larger the number of histogram bins, the better the load balancing, the better the concurrency.
Each process p sorts a list of n/p elements locally to obtain a local histogram.
Iterate log(p) time doing the following;
Exchange boundary values between partner groups and determine logical ids of processors for merged lists.
Exchange histograms with pair group and compute a new histogram and divide accordingly (lesser, larger) into equal parts.
Each processor sends keys to designated processors.
Each processor locally merges its elements with the received elements to obtain a new sorted list
Broadcast logical ids of processors to the next iterations.
You can read more on this approach here (PDF) explained by Minsoo Jeon and Dongseung Kim from Korea University.
With this article at OpenGenus, you must have the complete idea of Parallel Merge Sort.
We all must the idea of the Constructor in Java. But the difference is, the Static Initialization Block is a set of instructions that are run only once when the class is loaded into memory.
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We study several regularizing methods, stationary phase or averaging lemmas for instance. Depending on the regularity assumptions that are made, we show that they can either be derived one from the other or that they lead to different results. Those are applied to Scalar Conservation Laws to precise and better explain the regularity of their solutions.
author = {Jabin, Pierre-Emmanuel},
title = {Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws},
TI - Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws
Jabin, Pierre-Emmanuel. Some regularizing methods for transport equations and the regularity of solutions to scalar conservation laws. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2008-2009), Exposé no. 16, 15 p. http://archive.numdam.org/item/SEDP_2008-2009____A16_0/
{L}^{p}
précisée des moyennes dans les équations de transport. Bull. Soc. Math. France, 122 (1994), 29–76. | Numdam | MR 1259108 | Zbl 0798.35025
[2] F. Bouchut, Hypoelliptic regularity in kinetic equations. J. Math. Pures Appl. (9), 81 (2002), 1135–1159. | MR 1949176 | Zbl 1045.35093
[3] F. Bouchut, F. Golse and M. Pulvirenti, Kinetic equations and asymptotic theories. Series in Appl. Math., no. 4, Elsevier (2000). | Zbl 0979.82048
[4] K.S. Cheng, A regularity Theorem for a Nonconvex Scalar Conservation Law. J. Differential Equations 61 (1986), no. 1, 79–127. | MR 818862 | Zbl 0545.34005
[5] C. Cheverry, Regularizing effects for multidimensional scalar conservation laws. Ann. Inst. H. Poincaré, Analyse Non Linéaire, 17(4) (2000), 413–472. | Numdam | MR 1782740 | Zbl 0966.35074
[6] G. Crippa, F. Otto, and M. Westdickenberg, Regularizing effect of nonlinearity in multidimensional scalar conservation laws. To appear Transport Equations and Multi-D Hyperbolic Conservation Laws, Lecture Notes of the Unione Matematica Italiana. | MR 2409677 | Zbl 1155.35400
[7] C.M. Dafermos, Hyperbolic conservation laws in continuum physics. Springer Verlag, GM 325 (1999). | MR 2169977 | Zbl 0940.35002
[8] C. De Lellis, F. Otto, and M. Westdickenberg, Structure of entropy solutions for multi-dimensional scalar conservation laws. Arch. Ration. Mech. Anal., 170(2) (2003), 137–184. | MR 2017887 | Zbl 1036.35127
[9] R. DiPerna, P.L. Lions and Y. Meyer,
{L}^{p}
regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire, 8 (1991), 271–287. | Numdam | MR 1127927 | Zbl 0763.35014
[10] P. Gérard, F. Golse, Averaging regularity results for PDEs under transversality assumptions. Comm. Pure Appl. Math. 45(1) (1992), 1–26. | MR 1135922 | Zbl 0832.35020
[11] F. Golse, P.L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation. J. Funct. Anal., 26 (1988), 110-125. | MR 923047 | Zbl 0652.47031
[12] F. Golse, B. Perthame, R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport. C.R. Acad. Sci. Paris Série I, 301 (1985), 341–344. | MR 808622 | Zbl 0591.45007
[13] D. Hoff, The sharp form of Oleĭnik’s entropy condition in several space dimensions. Trans. Amer. Math. Soc. 276 (1983), no. 2, 707–714. | MR 688972 | Zbl 0528.35062
[14] P.E. Jabin and B. Perthame, Regularity in kinetic formulations via averaging lemmas. ESAIM Control Optim. Calc. Var. 8 (2002), 761–774. | Numdam | MR 1932972 | Zbl 1065.35185
[15] P.E. Jabin and L. Vega, A real space method for averaging lemmas, J. de Math. Pures Appl., 83 (2004), 1309–1351. | MR 2096303 | Zbl 1082.35043
[16] S.N. Kruzkov, First order quasilinear equations in several independent variables. Math USSR Sb. 10 (1970), 217–243. | Zbl 0215.16203
[17] P.L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related questions. J. Amer. Math. Soc., 7 (1994), 169–191. | MR 1201239 | Zbl 0820.35094
[18] O.A. Oleĭnik, On Cauchy’s problem for nonlinear equations in a class of discontinuous functions. Doklady Akad. Nauk SSSR (N.S.), 95 (1954), 451–454. | Zbl 0058.32101
[19] B. Perthame, Kinetic Formulations of conservation laws, Oxford series in mathematics and its applications, Oxford University Press (2002). | MR 2064166 | Zbl 1030.35002
[20] D. Serre, Systèmes hyperboliques de lois de conservation. Diderot, Paris (1996). | Zbl 0930.35002
[21] E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. | MR 1232192 | Zbl 0821.42001
[22] E. Tadmor, The large-time behavior of the scalar, genuinely nonlinear Lax Friedrichs scheme. Math. Comp. 43 (1984), no. 168, 353–368. | MR 758188 | Zbl 0598.65067
[23] E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal. 28 (1991), no. 4, 891–906 | MR 1111445 | Zbl 0732.65084
[24] A.I. Vol’pert, Spaces
BV
and quasilinear equations.(Russian) Mat. Sb. (N.S.) 73 (115) (1967) 255–302. [English transl.: Math. USSSR-Sb. 2 (1967), 225–267.] | Zbl 0168.07402
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Implement Euler angle representation of six-degrees-of-freedom equations of motion of custom variable mass - Simulink - MathWorks América Latina
Implement Euler angle representation of six-degrees-of-freedom equations of motion of custom variable mass
The Custom Variable Mass 6DOF (Euler Angles) block implements the Euler angle representation of six-degrees-of-freedom equations of motion of custom variable mass. It considers the rotation of a body-fixed coordinate frame (Xb, Yb, Zb) about a flat Earth reference frame (Xe, Ye, Ze). For more information on Euler angles, see Algorithms.
The origin of the body-fixed coordinate frame is the center of gravity of the body. The body is assumed to be rigid, which eliminates the need to consider the forces acting between individual elements of mass. The flat Earth reference frame is considered inertial, an excellent approximation that allows the forces due to the Earth's motion relative to the “fixed stars” to be neglected.
The translational motion of the body-fixed coordinate frame is given below, where the applied forces [Fx Fy Fz]T are in the body-fixed frame. Vreb is the relative velocity in the body axes at which the mass flow (
\stackrel{˙}{m}
) is ejected or added to the body-fixed axes.
\begin{array}{l}{\overline{F}}_{b}=\left[\begin{array}{c}{F}_{x}\\ {F}_{y}\\ {F}_{z}\end{array}\right]=m\left({\stackrel{˙}{\overline{V}}}_{b}+\overline{\omega }×{\overline{V}}_{b}\right)+\stackrel{˙}{m}\overline{V}r{e}_{b}\\ {A}_{be}=\frac{{\overline{F}}_{b}-\stackrel{˙}{m}{\overline{V}}_{r{e}_{b}}}{m}\\ {A}_{bb}=\left[\begin{array}{c}{\stackrel{˙}{u}}_{b}\\ {\stackrel{˙}{v}}_{b}\\ {\stackrel{˙}{w}}_{b}\end{array}\right]=\frac{{\overline{F}}_{b}-\stackrel{˙}{m}{\overline{V}}_{r{e}_{b}}}{m}-\overline{\omega }×{\overline{V}}_{b}\\ {\overline{V}}_{b}=\left[\begin{array}{c}{u}_{b}\\ {v}_{b}\\ {w}_{b}\end{array}\right],\overline{\omega }=\left[\begin{array}{c}p\\ q\\ r\end{array}\right]\end{array}
\begin{array}{l}{\overline{M}}_{B}=\left[\begin{array}{c}L\\ M\\ N\end{array}\right]=I\stackrel{˙}{\overline{\omega }}+\overline{\omega }×\left(I\overline{\omega }\right)+\stackrel{˙}{I}\overline{\omega }\\ \\ I=\left[\begin{array}{ccc}{I}_{xx}& -{I}_{xy}& -{I}_{xz}\\ -{I}_{yx}& {I}_{yy}& -{I}_{yz}\\ -{I}_{zx}& -{I}_{zy}& {I}_{zz}\end{array}\right]\\ \\ \stackrel{˙}{I}=\left[\begin{array}{ccc}{\stackrel{˙}{I}}_{xx}& -{\stackrel{˙}{I}}_{xy}& -{\stackrel{˙}{I}}_{xz}\\ -{\stackrel{˙}{I}}_{yx}& {\stackrel{˙}{I}}_{yy}& -{\stackrel{˙}{I}}_{yz}\\ -{\stackrel{˙}{I}}_{zx}& -{\stackrel{˙}{I}}_{zy}& {\stackrel{˙}{I}}_{zz}\end{array}\right]\end{array}
The relationship between the body-fixed angular velocity vector, [p q r]T, and the rate of change of the Euler angles,
{\left[\stackrel{˙}{\varphi }\stackrel{˙}{\theta }\stackrel{˙}{\psi }\right]}^{\text{T}}
, can be determined by resolving the Euler rates into the body-fixed coordinate frame.
\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ 0\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{c}0\\ \stackrel{˙}{\theta }\\ 0\end{array}\right]+\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right]\left[\begin{array}{ccc}\mathrm{cos}\theta & 0& -\mathrm{sin}\theta \\ 0& 1& 0\\ \mathrm{sin}\theta & 0& \mathrm{cos}\theta \end{array}\right]\left[\begin{array}{c}0\\ 0\\ \stackrel{˙}{\psi }\end{array}\right]={J}^{-1}\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]
\left[\begin{array}{c}\stackrel{˙}{\varphi }\\ \stackrel{˙}{\theta }\\ \stackrel{˙}{\psi }\end{array}\right]=J\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{ccc}1& \left(\mathrm{sin}\varphi \mathrm{tan}\theta \right)& \left(\mathrm{cos}\varphi \mathrm{tan}\theta \right)\\ 0& \mathrm{cos}\varphi & -\mathrm{sin}\varphi \\ 0& \frac{\mathrm{sin}\varphi }{\mathrm{cos}\theta }& \frac{\mathrm{cos}\varphi }{\mathrm{cos}\theta }\end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]
For more information on aerospace coordinate systems, see About Aerospace Coordinate Systems.
6DOF (Euler Angles) | 6DOF (Quaternion) | 6DOF ECEF (Quaternion) | 6DOF Wind (Quaternion) | 6DOF Wind (Wind Angles) | Custom Variable Mass 6DOF (Quaternion) | Custom Variable Mass 6DOF ECEF (Quaternion) | Custom Variable Mass 6DOF Wind (Quaternion) | Custom Variable Mass 6DOF Wind (Wind Angles) | Simple Variable Mass 6DOF (Euler Angles) | Simple Variable Mass 6DOF (Quaternion) | Simple Variable Mass 6DOF ECEF (Quaternion) | Simple Variable Mass 6DOF Wind (Quaternion) | Simple Variable Mass 6DOF Wind (Wind Angles)
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Decline of memory retention in time
Find sources: "Forgetting curve" – news · newspapers · books · scholar · JSTOR (December 2018) (Learn how and when to remove this template message)
Typical Representation of the Forgetting Curve.
The forgetting curve hypothesizes the decline of memory retention in time. This curve shows how information is lost over time when there is no attempt to retain it.[1] A related concept is the strength of memory that refers to the durability that memory traces in the brain. The stronger the memory, the longer period of time that a person is able to recall it. A typical graph of the forgetting curve purports to show that humans tend to halve their memory of newly learned knowledge in a matter of days or weeks unless they consciously review the learned material.
The forgetting curve supports one of the seven kinds of memory failures: transience, which is the process of forgetting that occurs with the passage of time.[2]
2 Increasing rate of learning
From 1880 to 1885, Hermann Ebbinghaus ran a limited, incomplete study on himself and published his hypothesis in 1885 as Über das Gedächtnis (later translated into English as Memory: A Contribution to Experimental Psychology).[3] Ebbinghaus studied the memorisation of nonsense syllables, such as "WID" and "ZOF" (CVCs or Consonant–Vowel–Consonant) by repeatedly testing himself after various time periods and recording the results. He plotted these results on a graph creating what is now known as the "forgetting curve".[3] Ebbinghaus investigated the rate of forgetting, but not the effect of spaced repetition on the increase in retrievability of memories.[4]
Ebbinghaus's publication also included an equation to approximate his forgetting curve:[5]
{\displaystyle b={\frac {100k}{(\log(t))^{c}+k}}}
{\displaystyle b}
represents 'Savings' expressed as a percentage, and
{\displaystyle t}
represents time in minutes, counting from one minute before end of learning. The constants c and k are 1.25 and 1.84 respectively. Savings is defined as the relative amount of time saved on the second learning trial as a result of having had the first. A savings of 100% would indicate that all items were still known from the first trial. A 75% savings would mean that relearning missed items required 25% as long as the original learning session (to learn all items). 'Savings' is thus, analogous to retention rate.
In 2015, an attempt to replicate the forgetting curve with one study subject has shown the experimental results similar to Ebbinghaus' original data.[6]
Ebbinghaus' experiment contributed a lot to experimental psychology. He was the first to carry out a series of well-designed experiments on the subject of forgetting, and he was one of the first to choose artificial stimuli in the research of experimental psychology. Since his introduction of nonsense syllables, a large number of experiments in experimental psychology has been based on highly controlled artificial stimuli.[6]
Increasing rate of learning[edit]
Hermann Ebbinghaus hypothesized that the speed of forgetting depends on a number of factors such as the difficulty of the learned material (e.g. how meaningful it is), its representation and other physiological factors such as stress and sleep. He further hypothesized that the basal forgetting rate differs little between individuals. He concluded that the difference in performance can be explained by mnemonic representation skills.
He went on to hypothesize that basic training in mnemonic techniques can help overcome those differences in part. He asserted that the best methods for increasing the strength of memory are:
His premise was that each repetition in learning increases the optimum interval before the next repetition is needed (for near-perfect retention, initial repetitions may need to be made within days, but later they can be made after years). He discovered that information is easier to recall when it's built upon things you already know, and the forgetting curve was flattened by every repetition. It appeared that by applying frequent training in learning, the information was solidified by repeated recalling.
Later research also suggested that, other than the two factors Ebbinghaus proposed, higher original learning would also produce slower forgetting. The more information was originally learned, the slower the forgetting rate would be.[7]
Spending time each day to remember information will greatly decrease the effects of the forgetting curve. Some learning consultants claim reviewing material in the first 24 hours after learning information is the optimum time to actively recall the content and reset the forgetting curve.[8] Evidence suggests waiting 10–20% of the time towards when the information will be needed is the optimum time for a single review.[9]
However, some memories remain free from the detrimental effects of interference and do not necessarily follow the typical forgetting curve as various noise and outside factors influence what information would be remembered.[10] There is debate among supporters of the hypothesis about the shape of the curve for events and facts that are more significant to the subject.[11] Some supporters, for example, suggest that memories of shocking events such as the Kennedy Assassination or 9/11 are vividly imprinted in memory (flashbulb memory).[12] Others have compared contemporaneous written recollections with recollections recorded years later, and found considerable variations as the subject's memory incorporates after-acquired information.[13] There is considerable research in this area as it relates to eyewitness identification testimony, and eyewitness accounts are found demonstrably unreliable.[13]
Many equations have since been proposed to approximate forgetting, perhaps the simplest being an exponential curve described by the equation[14]
{\displaystyle R=e^{-{\frac {t}{S}}},}
{\displaystyle R}
is retrievability (a measure of how easy it is to retrieve a piece of information from memory),
{\displaystyle S}
is stability of memory (determines how fast
{\displaystyle R}
falls over time in the absence of training, testing or other recall), and
{\displaystyle t}
Simple equations such as this one were not found to provide a good fit to the available data.[15]
Atrophy – Partial or complete wasting away of a part of the body
Learning curve – Relationship between proficiency and experience
^ Curve of Forgetting | Counselling Services
^ Schacter, D. L. (2009). Psychology. New York: Worth Publishers. p. 243. ISBN 978-1-4292-3719-2.
^ a b Ebbinghaus, Hermann (1913). Memory: A Contribution to Experimental Psychology. Translated by Ruger, Henry; Bussenius, Clara. New York city, Teachers college, Columbia university.
^ Wozniak, Piotr. "Did Ebbinghaus invent spaced repetition?". www.supermemo.com. Retrieved 2020-07-11.
^ Ebbinghaus (1913), p. 77
^ a b Murre, Jaap M. J.; Dros, Joeri (2015). "Replication and Analysis of Ebbinghaus' Forgetting Curve". PLOS ONE. 10 (7): e0120644. Bibcode:2015PLoSO..1020644M. doi:10.1371/journal.pone.0120644. PMC 4492928. PMID 26148023.
^ Loftus, Geoffrey R. (1985). "Evaluating forgetting curves" (PDF). Journal of Experimental Psychology: Learning, Memory, and Cognition. 11 (2): 397–406. CiteSeerX 10.1.1.603.9808. doi:10.1037/0278-7393.11.2.397.
^ Pashler, Harold; Rohrer, Doug; Cepeda, Nicholas J.; Carpenter, Shana K. (2007-04-01). "Enhancing learning and retarding forgetting: Choices and consequences". Psychonomic Bulletin & Review. 14 (2): 187–193. doi:10.3758/BF03194050. ISSN 1069-9384. PMID 17694899.
^ Averell, Lee; Heathcote, Andrew (2011). "The form of the forgetting curve and the fate of memories". Journal of Mathematical Psychology. 55: 25–35. doi:10.1016/j.jmp.2010.08.009. hdl:1959.13/931260.
^ Forgetting Curve | Training Industry
^ Paradis, C. M.; Florer, F.; Solomon, L. Z.; Thompson, T. (August 1, 2004). "Flashbulb Memories of Personal Events of 9/11 and the Day after for a Sample of New York City Residents". Psychological Reports. 95 (1): 309. doi:10.2466/pr0.95.1.304-310. PMID 15460385. S2CID 46013520.
^ a b "Why Science Tells Us Not to Rely on Eyewitness Accounts". doi:10.1038/scientificamericanmind0110-68. {{cite journal}}: Cite journal requires |journal= (help)
^ Woźniak, Piotr A.; Gorzelańczyk, Edward J.; Murakowski, Janusz A. (1995). "Two components of long-term memory" (PDF). Acta Neurobiologiae Experimentalis. 55 (4): 301–305. PMID 8713361.
^ Rubin, David C.; Hinton, Sean; Wenzel, Amy (1999). "The precise time course of retention". Journal of Experimental Psychology: Learning, Memory, and Cognition. 25 (5): 1161–1176. doi:10.1037/0278-7393.25.5.1161. hdl:10161/10146.
"Memory: A Contribution to Experimental Psychology -- Ebbinghaus (1885/1913)". Retrieved 2007-08-23.
Schacter, Daniel L (2001). The seven sins of memory: how the mind forgets and remembers. Boston: Houghton Mifflin. ISBN 978-0-618-21919-3.
Baddeley, Alan D. (1999). Essentials of human memory. Hove: Psychology. ISBN 978-0-86377-544-4.
Bremer, Rod. The Manual – A guide to the Ultimate Study Method (USM) (Amazon Digital Services).
Loftus, Geoffrey R. Journal of Experimental Psychology: Learning, Memory, and Cognition11. 2 (Apr 1985): 397–406.
http://www.trainingindustry.com/wiki/entries/forgetting-curve.aspx
Averell, Lee; Heathcote, Andrew (February 2011). "The form of the forgetting curve and the fate of memories". Journal of Mathematical Psychology. 55 (1): 25–35. doi:10.1016/j.jmp.2010.08.009. hdl:1959.13/931260.
https://www.growthengineering.co.uk/what-is-the-forgetting-curve/
Retrieved from "https://en.wikipedia.org/w/index.php?title=Forgetting_curve&oldid=1083400189"
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