text:
# times, .*

Element-wise quaternion multiplication

## Syntax

``quatC = A.*B``

## Description

example

````quatC = A.*B` returns the element-by-element quaternion multiplication of quaternion arrays.You can use quaternion multiplication to compose rotation operators: To compose a sequence of frame rotations, multiply the quaternions in the same order as the desired sequence of rotations. For example, to apply a p quaternion followed by a q quaternion, multiply in the order pq. The rotation operator becomes ${\left(pq\right)}^{\ast }v\left(pq\right)$, where v represents the object to rotate in quaternion form. * represents conjugation.To compose a sequence of point rotations, multiply the quaternions in the reverse order of the desired sequence of rotations. For example, to apply a p quaternion followed by a q quaternion, multiply in the reverse order, qp. The rotation operator becomes $\left(qp\right)v{\left(qp\right)}^{\ast }$. ```

## Examples

collapse all

Create two vectors, `A` and `B`, and multiply them element by element.

```A = quaternion([1:4;5:8]); B = A; C = A.*B```
```C = 2x1 quaternion array -28 + 4i + 6j + 8k -124 + 60i + 70j + 80k ```

Create two 3-by-3 arrays, `A` and `B`, and multiply them element by element.

```A = reshape(quaternion(randn(9,4)),3,3); B = reshape(quaternion(randn(9,4)),3,3); C = A.*B```
```C = 3x3 quaternion array 0.60169 + 2.4332i - 2.5844j + 0.51646k -0.49513 + 1.1722i + 4.4401j - 1.217k 2.3126 + 0.16856i + 1.0474j - 1.0921k -4.2329 + 2.4547i + 3.7768j + 0.77484k -0.65232 - 0.43112i - 1.4645j - 0.90073k -1.8897 - 0.99593i + 3.8331j + 0.12013k -4.4159 + 2.1926i + 1.9037j - 4.0303k -2.0232 + 0.4205i - 0.17288j + 3.8529k -2.9137 - 5.5239i - 1.3676j + 3.0654k ```

Note that quaternion multiplication is not commutative:

`isequal(C,B.*A)`
```ans = logical 0 ```

Create a row vector `a` and a column vector `b`, then multiply them. The 1-by-3 row vector and 4-by-1 column vector combine to produce a 4-by-3 matrix with all combinations of elements multiplied.

`a = [zeros('quaternion'),ones('quaternion'),quaternion(randn(1,4))]`
```a = 1x3 quaternion array 0 + 0i + 0j + 0k 1 + 0i + 0j + 0k 0.53767 + 1.8339i - 2.2588j + 0.86217k ```
`b = quaternion(randn(4,4))`
```b = 4x1 quaternion array 0.31877 + 3.5784i + 0.7254j - 0.12414k -1.3077 + 2.7694i - 0.063055j + 1.4897k -0.43359 - 1.3499i + 0.71474j + 1.409k 0.34262 + 3.0349i - 0.20497j + 1.4172k ```
`a.*b`
```ans = 4x3 quaternion array 0 + 0i + 0j + 0k 0.31877 + 3.5784i + 0.7254j - 0.12414k -4.6454 + 2.1636i + 2.9828j + 9.6214k 0 + 0i + 0j + 0k -1.3077 + 2.7694i - 0.063055j + 1.4897k -7.2087 - 4.2197i + 2.5758j + 5.8136k 0 + 0i + 0j + 0k -0.43359 - 1.3499i + 0.71474j + 1.409k 2.6421 - 5.32i - 2.3841j - 1.3547k 0 + 0i + 0j + 0k 0.34262 + 3.0349i - 0.20497j + 1.4172k -7.0663 - 0.76439i - 0.86648j + 7.5369k ```

## Input Arguments

collapse all

Array to multiply, specified as a quaternion, an array of quaternions, a real scalar, or an array of real numbers.

`A` and `B` must have compatible sizes. In the simplest cases, they can be the same size or one can be a scalar. Two inputs have compatible sizes if, for every dimension, the dimension sizes of the inputs are the same or one of them is 1.

Data Types: `quaternion` | `single` | `double`

Array to multiply, specified as a quaternion, an array of quaternions, a real scalar, or an array of real numbers.

`A` and `B` must have compatible sizes. In the simplest cases, they can be the same size or one can be a scalar. Two inputs have compatible sizes if, for every dimension, the dimension sizes of the inputs are the same or one of them is 1.

Data Types: `quaternion` | `single` | `double`

## Output Arguments

collapse all

Quaternion product, returned as a scalar, vector, matrix, or multidimensional array.

Data Types: `quaternion`

## Algorithms

collapse all

### Quaternion Multiplication by a Real Scalar

Given a quaternion,

`$q={a}_{\text{q}}+{b}_{\text{q}}\text{i}+{c}_{\text{q}}\text{j}+{d}_{\text{q}}\text{k,}$`

the product of q and a real scalar β is

`$\beta q=\beta {a}_{\text{q}}+\beta {b}_{\text{q}}\text{i}+\beta {c}_{\text{q}}\text{j}+\beta {d}_{\text{q}}\text{k}$`

### Quaternion Multiplication by a Quaternion Scalar

The definition of the basis elements for quaternions,

`${\text{i}}^{2}={\text{j}}^{2}={\text{k}}^{2}=\text{ijk}=-1\text{\hspace{0.17em}},$`

can be expanded to populate a table summarizing quaternion basis element multiplication:

 1 i j k 1 1 i j k i i −1 k −j j j −k −1 i k k j −i −1

When reading the table, the rows are read first, for example: ij = k and ji = −k.

Given two quaternions, $q={a}_{\text{q}}+{b}_{\text{q}}\text{i}+{c}_{\text{q}}\text{j}+{d}_{\text{q}}\text{k,}$ and $p={a}_{\text{p}}+{b}_{\text{p}}\text{i}+{c}_{\text{p}}\text{j}+{d}_{\text{p}}\text{k}$, the multiplication can be expanded as:

`$\begin{array}{l}z=pq=\left({a}_{\text{p}}+{b}_{\text{p}}\text{i}+{c}_{\text{p}}\text{j}+{d}_{\text{p}}\text{k}\right)\left({a}_{\text{q}}+{b}_{\text{q}}\text{i}+{c}_{\text{q}}\text{j}+{d}_{\text{q}}\text{k}\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={a}_{\text{p}}{a}_{\text{q}}+{a}_{\text{p}}{b}_{\text{q}}\text{i}+{a}_{\text{p}}{c}_{\text{q}}\text{j}+{a}_{\text{p}}{d}_{\text{q}}\text{k}\\ \text{ }\text{ }+{b}_{\text{p}}{a}_{\text{q}}\text{i}+{b}_{\text{p}}{b}_{\text{q}}{\text{i}}^{2}+{b}_{\text{p}}{c}_{\text{q}}\text{ij}+{b}_{\text{p}}{d}_{\text{q}}\text{ik}\\ \text{ }\text{ }+{c}_{\text{p}}{a}_{\text{q}}\text{j}+{c}_{\text{p}}{b}_{\text{q}}\text{ji}+{c}_{\text{p}}{c}_{\text{q}}{\text{j}}^{2}+{c}_{\text{p}}{d}_{\text{q}}\text{jk}\\ \text{ }\text{ }+{d}_{\text{p}}{a}_{\text{q}}k+{d}_{\text{p}}{b}_{\text{q}}\text{ki}+{d}_{\text{p}}{c}_{\text{q}}\text{kj}+{d}_{\text{p}}{d}_{\text{q}}{\text{k}}^{2}\end{array}$`

You can simplify the equation using the quaternion multiplication table.

`$\begin{array}{l}z=pq\text{\hspace{0.17em}}={a}_{\text{p}}{a}_{\text{q}}+{a}_{\text{p}}{b}_{\text{q}}\text{i}+{a}_{\text{p}}{c}_{\text{q}}\text{j}+{a}_{\text{p}}{d}_{\text{q}}\text{k}\\ \text{ }\text{ }+{b}_{\text{p}}{a}_{\text{q}}\text{i}-{b}_{\text{p}}{b}_{\text{q}}+{b}_{\text{p}}{c}_{\text{q}}\text{k}-{b}_{\text{p}}{d}_{\text{q}}\text{j}\\ \text{ }\text{ }+{c}_{\text{p}}{a}_{\text{q}}\text{j}-{c}_{\text{p}}{b}_{\text{q}}\text{k}-{c}_{\text{p}}{c}_{\text{q}}+{c}_{\text{p}}{d}_{\text{q}}\text{i}\\ \text{ }\text{ }+{d}_{\text{p}}{a}_{\text{q}}k+{d}_{\text{p}}{b}_{\text{q}}\text{j}-{d}_{\text{p}}{c}_{\text{q}}\text{i}-{d}_{\text{p}}{d}_{\text{q}}\end{array}$`

## References

[1] Kuipers, Jack B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton, NJ: Princeton University Press, 2007.

## Version History

Introduced in R2018b


text:
+0

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RainbowPanda  Sep 10, 2018
#1
+2012
+2

it makes me sad that you can't figure out the first one.

any of these will do

$$c_1 a^{m_1}b^{n_1} \cdot c_2 a^{m_2}b^{n_2} \\ \text{where } \dots \\ c_1 \cdot c_2 = -84 \\ m_1 + m_2 = 6 \\ n_1 + n_2 = 10 \\ \text{use your rainbow panda powers to choose two monomial that satisfy these 3 requirements!}$$

for the second one we know (should know) that the volume of a cube is $$s^3 \text{ where s is the length of a side.}$$

our side length is doubled giving us $$s=4x^5$$

Thus the new volume is $$\left(4x^5\right)^3 = 4^3 \cdot x^{5 \cdot 3} = 64 x^8$$

Rom  Sep 10, 2018
#2
+93311
+2

Thanks Rom,

RainbowPanda, maybe it will help you understand what Rom has  said if I tell you this:

A monomial is an eqpression with only one term.    Like   5x   or   -3y^2

Product means multiply.

So you are just looking for 2 things that multiply together to make  $$-84a^6b^{10}$$     there are lots and lots of correct answers.

Rom has introduced all those c's and m's.  They are just numbers.  You have to use numbers that work. Rom is not suggesting that you leave them as c's and m's

Melody  Sep 10, 2018

format: plain


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If (243)^x(463)^y = n, where x and y are positive integers, : GMAT Data Sufficiency (DS)
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# If (243)^x(463)^y = n, where x and y are positive integers,

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If (243)^x(463)^y = n, where x and y are positive integers, [#permalink]

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07 Apr 2010, 22:37
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If (243)^x(463)^y = n, where x and y are positive integers, what is the units digit of n?

(1) x + y = 7
(2) x = 4

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-243-x-463-y-n-where-x-and-y-are-positive-integers-102054.html
[Reveal] Spoiler: OA
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07 Apr 2010, 22:43
I'd go with A

The units digit of 3^n follows the following patterns: 3,9,7,1,3,9,7,1,.....

Thus if we know what both x and y are, we can solve it (statement 1).
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07 Apr 2010, 23:23
nickk wrote:
I'd go with A

The units digit of 3^n follows the following patterns: 3,9,7,1,3,9,7,1,.....

Thus if we know what both x and y are, we can solve it (statement 1).

So how did you find the values of x & y from Stmt 1??
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07 Apr 2010, 23:59
Hussain15 wrote:
nickk wrote:
I'd go with A

The units digit of 3^n follows the following patterns: 3,9,7,1,3,9,7,1,.....

Thus if we know what both x and y are, we can solve it (statement 1).

So how did you find the values of x & y from Stmt 1??

Well we don't know the values of X and Y individually, but all we need to know is how many times a number with 3 in the units digit is multiplied by itself. Since X and Y are both exponents of such numbers, knowing x+y is sufficient.

Of course I might be wrong so the OA would be appreciated.
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08 Apr 2010, 01:02
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In this case, since the base number is different i.e 243 & 463, it makes sense to use the various combinations of x & y:
1. 1&6 or 6&1
2. 2&5 or 5&2
3. 3&4 or 4&3

The units digit of 3^n follows the following patterns: 3,9,7,1,3,9,7,1,.....

Substituting n in the above combinations and multiplying the ending unit digits of each of these numbers, we get the same unit digit i.e., 7.

Choice (B), keeps the n open for y, so the unit digit of the resultant number may vary.

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08 Apr 2010, 01:52
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Expert's post
rohitgoel15 wrote:
If (243)^x(463)^y = n, where x and y are positive integers, what is the units digit of n?

(1) x + y = 7
(2) x = 4

Units digit of $$243^x$$ equals to units digit of $$3^x$$ and units digit of $$463^y$$ equals to units digit of $$3^y$$ (general rule). Hence units digit of $$243^x*463^y$$ equals to units digit of $$3^x*3^y=3^{x+y}$$. So knowing the value of $$x+y$$ is sufficient to determine units digit of $$n$$.

(1) $$x+y=7$$. Sufficient. (As cyclicity of $$3$$ is $$4$$, units digit of $$3^7$$ would be the same as of units digit of $$3^3$$ which is $$7$$)

(2) $$x=4$$. No info about $$y$$. Not sufficient.

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08 Apr 2010, 01:54
Bunuel can you please take a look at probability-question-84062.html
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09 Apr 2010, 22:22
great question ....
I made the mistake of choosing C.
Re: Units digit   [#permalink] 09 Apr 2010, 22:22
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NIPS 2018
Sun Dec 2nd through Sat the 8th, 2018 at Palais des Congrès de Montréal
Paper ID: 5504 A Stein variational Newton method

### Reviewer 1

Summary: SVGD iteratively moves a set of particles toward the target by choosing a perturbative direction to maximumly decrease the KL divergence with the target distribution in RKHS. The paper proposes to add second-order information into SVGD updates, preliminary empirical results show that their method converges faster in few cases. The paper is well written, and the proofs seem correct. An important reason in using second-order information is the hope to achieve a faster convergence rate. My major concern is a lack of theoretical analysis of convergence rate in this paper: 1) An appealing property of SVGD is that the optimal decreasing rate equals to Stein discrepancy D_F(q||p), where F is a function set that includes all possible velocity fields. D(q||p) = 0 iff q = p. Taking F as RKHS, a closed form solution of the perturbation direction is derived in [Liu et al., 2016, Chwialkowski et al., 2016]. However, the authors didn’t draw any connections between their updates and Stein discrepancy. It’s not clear at all why SVN would yield a faster convergence rate than the original SVGD. And if so, in what sense that SVN is faster? Would the updates of SVN equal to a sum of Stein discrepancy and some other positive terms? - Liu et al., 2016, A kernelized Stein discrepancy for goodness-of-fit tests - Chwialkowski et al., 2016. A Kernel Test of Goodness of Fit 2) It would also help l if the authors could discuss the convergence properties of their approach in terms of the number of particles. e.g. Liu et al., 2017, Stein Variational Gradient Descent as Gradient Flow. 3) Calculating the hessian-matrix is computationally expensive. For the purpose of efficiency, the authors used few wild approximations in their update, e.g. (line 127-138) the cross term in H_l is dropped, and the second-order term is replaced by its Gauss-Newton approximation, etc. These simplifications could reduce the computational cost, but it’s not clear how they would affect the convergence analysis. 4) In the current examples, comparisons were only done with low dimensions and toy examples. Probably the results would be more convincing if the authors could test with high dimensions, e.g. the Bayesian neural network example.

### Reviewer 2

The authors propose an version of Stein variational gradient descent (SVGD) that uses approximate second-order information, both to improve the optimization (by analogy with other quasi-Newton optimization methods) and to create a geometry-aware kernel (by analogy with pre-conditioning). The use of second order information improves convergence considerably over standard SVGD, with the greatest benefit provided by both the geometry-aware kernels and second-order optimization steps. Stein methods for approximate inference are an exciting topic these days, the improvements are both practically feasible and significant, and the paper is very well-written. I will argue that the paper should be accepted and really only have minor comments and suggestions. - A more appropriate title is “A Stein variational Quasi-Newton method”, since the authors are not actually able to take a full Newton step, even on small problems, due to the fact that the transforms, and hence push-forward density, are not available in closed form (line 128). - In general, I might replace the phrase “inexact Newton” with “quasi-Newton” for continuity with other optimiziation literature (e.g. line 130). - “There exist infinitely many such maps” (line 62) is repeated from the introduction. - At line 80, you can define equation (6) without Frechet differentiability; it is equation (7) that requires it (specifically, that \nabla J_{p_l} does not depend on V). So for clarity perhaps it would be better to move the assumption of Frechet differentiability to line 81. - I think it might be better to spell out the definition of N(x) at line 131. I gathered that it is the outer product of the gradient of log p(x) at x, but it would be better to say that explicitly since this is an important part of the algorithm. Calling it equivalent to the Fisher information matrix is confusing -- the Fisher information matrix is usually as the covariance of the scores, whereas I think the authors don’t intend any expectation to be calculated. Furthermore, a correctly specified Fisher information matrix could be calculated either from the scores or from the Hessian, and the authors want to emphasize that it is the former, not the latter. - At line 150, the authors define a matrix, not a metric, so it would be better to call it that. This matrix is later used to /form/ a metric. - At line 236, is it supposed to be F(x), not F(u)? In line 238 F is applied to the Brownian motion, not the solution to the SDE. - Line 254, for comparison I’d be interested to know how long HMC took.

### Reviewer 3

Thank you for an interesting read. This paper proposed a newton-like method in function space for approximate Bayesian inference. Specifically, the authors define functional newton direction, and propose Monte Carlo approximation to it. Furthermore the authors also proposes an improvement of the kernel by using the approximately computed second-order information. Experimental results show significant improvement over naive SVGD. I think this paper can be a good contribution to the approximate inference community, however the presentation can be improved. Specifically, the "inexact newton" paragraph needs substantial explanation: 1. Why it is safe to drop the \nabla_x k \nabla_x log p term? 2. I don't understand the justification of approximation (18) to (16). I would also like to see at least a numerical analysis on the error of this inexact Newton approximation. You can construct synthetic densities that you can compute the exact hessian of log \pi(x). Also the experiments, although with significantly better results, seems a bit toy. The original SVGD paper tested the algorithm on Bayesian neural network regression, and the dimensionality there is at least ~300 I believe. ================================================ I have read the feedback. I don't feel the feedback has addressed my concerns on the "inexact Newton" part. While further theoretical analysis might be hard at this point, I would like to see at least a toy example that you can compute exact Newton, and compare it to your approximation method.

### Reviewer 4

The authors proposed a new variational inference method -- a Stein variational Newton (SVN) method. It is based on Stein variational gradient decent (SVGD) method developed in [11], and brings two main contributions. The first contribution is to apply a Newton iteration that exploits the second-order information of the probability distribution to push forward the samples. To avoid the challenge of solving a large coupled system arising from the Newton iteration, the authors present two approximations -- one dropping a cross term involving the current density function, the other using a mass lumping" method to decouple the system. The second contribution is the design of the kernel (of the RKHS used in the method) by incorporating the averaged Hessian of the negative logarithm of the target density function. Compared to a Gaussian kernel with isotropic structure, this rescaled kernel carries the curvature information of the target distribution which prevents samples from collapsing in high dimensions. These contributions largely accelerate the convergence of the samples to the target distribution, thus make SVN method computationally more efficient and tractable. The efficiency and tractability are demonstrated by several numerical examples with parameter dimension ranging from low (2) to high (100). Overall, this new development brings important contribution to the family of variational inference methods; the numerical demonstration is very convincing; the paper is easy to read and well-written. I am confident of the clarity and correctness of the algorithmic development. I also run the code and obtain statistically the same results and conclusions. Therefore, I recommend to accept it after some minor revisions. 1. page 5, Algorithm 2: Changing the step size of the Newton iterations looks important for the fast convergence. A few more details on how the author changed the step size is helpful. 2. page 5, line 174, it is not clear under what technical conditions the g(d)-weighted norm is the square of the discretized L2 norm. 3. Can the author explain why the isotropic kernel takes less time compared to the Hessian kernel, as can be observed from the different number of iterations in Figure 1? 4. supplementary material, page 4, after line 48, should the y" be x"?


text:
DMDA#

“da” - A DM object that is used to manage data for a structured grid in 1, 2, or 3 dimensions. In the global representation of the vector each process stores a non-overlapping rectangular (or slab in 3d) portion of the grid points. In the local representation these rectangular regions (slabs) are extended in all directions by a stencil width. The vectors can be thought of as either cell centered or vertex centered on the mesh. But some variables cannot be cell centered and others vertex centered; see the documentation for DMSTAG, a similar DM implementation which supports these staggered grids.

See Also#

DMType, DMCOMPOSITE, DMSTAG, DMDACreate(), DMCreate(), DMSetType()

intermediate

Location#

src/dm/impls/da/dacreate.c

Edit on GitLab


text:
# Natural Numbers under Addition form Commutative Monoid

## Theorem

The algebraic structure $\left({\N, +}\right)$ consisting of the set of natural numbers $\N$ under addition $+$ is a commutative monoid whose identity is zero.

## Proof

Consider the natural numbers $\N$ defined as the naturally ordered semigroup.

From the definition of the naturally ordered semigroup, it follows that $\left ({\N, +}\right)$ is a commutative semigroup.

From the definition of zero, $\left({\N, +}\right)$ has $0 \in \N$ as the identity, hence is a monoid.

$\blacksquare$


text:
Can you reduce this fraction?

Algebra Level 2

For how many positive integers $n$ is the following fraction reducible?

$\large\frac{8n+3}{4n+2}$.

×


text:
#### 题目列表

For the following question, consider each of the choices separately and select all that apply.

The passage mentions which of the following as an obstacle to using animal bones to reconstruct past climate changes?
The passage mentions "red deer" as an instance of an animal that
The passage mentions that "Earths atmosphere contains oxygen" primarily in order to
According to the passage, those who challenge "Meltzoffs interpretation" do so on the basis that it
The passage implies which of the following about the experimental results reported by Meltzoff and Moore?
The primary purpose of the passage is to
The author of the passage states that which of the following casts doubt on the validity of the "widespread concern"?
The author of the passage mentions The Conscious Lovers in order to
Which of the following, if true, most seriously weakens the argument?
For the following question, consider each of the choices separately and select all that apply. Which of the following, if true, would strengthen the argument that climate change was involved in the disappearance of certain pika populations?
Which of the following best describes the function of the highlighted sentence in the context of the passage as a whole?
The passage implies that Morgans suggestion was
For the following question, consider each of the choices separately and select all that apply. The passage suggests which of the following about the relationship between the theory of plate tectonics and Morgans suggestion?

25000 +道题目

5本备考书籍


text:
# zbMATH — the first resource for mathematics

Weighted likelihood mixture modeling and model-based clustering. (English) Zbl 1436.62255
Summary: A weighted likelihood approach for robust fitting of a mixture of multivariate Gaussian components is developed in this work. Two approaches have been proposed that are driven by a suitable modification of the standard EM and CEM algorithms, respectively. In both techniques, the M-step is enhanced by the computation of weights aimed at downweighting outliers. The weights are based on Pearson residuals stemming from robust Mahalanobis-type distances. Formal rules for robust clustering and outlier detection can be also defined based on the fitted mixture model. The behavior of the proposed methodologies has been investigated by numerical studies and real data examples in terms of both fitting and classification accuracy and outlier detection.

##### MSC:
 62H30 Classification and discrimination; cluster analysis (statistical aspects) 62F35 Robustness and adaptive procedures (parametric inference)
##### Software:
mclust; mtclust; otrimle; R; TCLUST
Full Text:
##### References:
 [1] Agostinelli, C., Robust model selection in regression via weighted likelihood methodology, Stat. Probab. Lett., 56, 3, 289-300 (2002) · Zbl 0998.62034 [2] Agostinelli, C., Notes on pearson residuals and weighted likelihood estimating equations, Stat. Probab. Lett., 76, 17, 1930-1934 (2006) · Zbl 1099.62022 [3] Agostinelli, C.; Greco, L., A weighted strategy to handle likelihood uncertainty in Bayesian inference, Comput. Stat., 28, 1, 319-339 (2013) · Zbl 1305.65018 [4] Agostinelli, C.; Greco, L., Discussion on “The power of monitoring: how to make the most of a contaminated sample”, Stat. Methods Appl. 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Assoc., 111, 516, 1648-1659 (2016) [17] Coretto, P.; Hennig, C., Consistency, breakdown robustness, and algorithms for robust improper maximum likelihood clustering, J. Mach. Learn. Res., 18, 1, 5199-5237 (2017) [18] Day, N., Estimating the components of a mixture of normal distributions, Biometrika, 56, 3, 463-474 (1969) · Zbl 0183.48106 [19] Dempster, A.; Laird, Nm; Rubin, Db, Maximum likelihood from incomplete data via the EM algorithm, J. R. Stat. Soc. Ser. B Methodol., 39, 1-38 (1977) · Zbl 0364.62022 [20] Dotto, F.; Farcomeni, A., Robust inference for parsimonious model-based clustering, J. Stat. Comput. Simul., 89, 3, 414-442 (2019) [21] Dotto, F.; Farcomeni, A.; Garcia-Escudero, La; Mayo-Iscar, A., A reweighting approach to robust clustering, Stat. Comput., 28, 2, 477-493 (2016) · Zbl 1384.62193 [22] Elashoff, M.; Ryan, L., An em algorithm for estimating equations, J. Comput. Graph. 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Lett., 96, 68-74 (2015) · Zbl 1314.62089 [35] Kuchibhotla, A., Basu, A.: A minimum distance weighted likelihood method of estimation. Technical report, Interdisciplinary Statistical Research Unit (ISRU), Indian Statistical Institute, Kolkata, India (2018). https://faculty.wharton.upenn.edu/wp-content/uploads/2018/02/attemptv4p1.pdf. Accessed 17 Jan 2018 [36] Lee, S.; Mclachlan, G., Finite mixtures of multivariate skew t-distributions: some recent and new results, Stat. Comput., 24, 2, 181-202 (2014) · Zbl 1325.62107 [37] Lin, T., Robust mixture modeling using multivariate skew t distributions, Stat. Comput., 20, 3, 343-356 (2010) [38] Markatou, M., Mixture models, robustness, and the weighted likelihood methodology, Biometrics, 56, 2, 483-486 (2000) · Zbl 1060.62511 [39] Markatou, M.; Basu, A.; Lindsay, Bg, Weighted likelihood equations with bootstrap root search, J. Am. Stat. Assoc., 93, 442, 740-750 (1998) · Zbl 0918.62046 [40] Maronna, R.; Jacovkis, P., Multivariate clustering procedures with variable metrics, Biometrics, 30, 3, 499-505 (1974) · Zbl 0285.62036 [41] Mclachlan, G.; Peel, D., Finite Mixture Models (2004), New York: Wiley, New York [42] Mclachlan, Gj; Peel, D.; Bean, R., Modelling high-dimensional data by mixtures of factor analyzers, Comput. Stat. Data Anal., 41, 3-4, 379-388 (2003) · Zbl 1256.62036 [43] Neykov, N.; Filzmoser, P.; Dimova, R.; Neytchev, P., Robust fitting of mixtures using the trimmed likelihood estimator, Comput. Stat. Data Anal., 52, 1, 299-308 (2007) · Zbl 1328.62033 [44] R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2019). https://www.R-project.org/ [45] Rousseeuw, P.; Van Zomeren, B., Unmasking multivariate outliers and leverage points, J. Am. Stat. Assoc., 85, 411, 633-639 (1990) [46] Symon, M., Clustering criterion and multi-variate normal mixture, Biometrics, 77, 35-43 (1977)
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# American Institute of Mathematical Sciences

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## An asymptotic convergence result for a system of partial differential equations with hysteresis

 1 Università degli Studi di Trento, Dipartimento di Matematica, Via Sommarive 14, I–38050 Povo (Trento) 2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D–10117 Berlin

Received  October 2006 Revised  March 2007 Published  September 2007

A partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero.
Citation: Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131
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2017 Impact Factor: 0.884


text:
# Q.6    The English alphabet has 5 vowels and 21 consonants. How many words with            two different vowels and 2 different consonants can be formed from the            alphabet?

S seema garhwal

Two different vowels and 2 different consonants are to be selected from the English alphabets.

Since there are 5 different vowels so the number of ways of selecting two different vowels = $^5C_2$

$=\frac{5!}{2!3!}=10$

Since there are 21 different consonants so the number of ways of selecting two different consonants = $^2^1C_2$

$=\frac{21!}{2!19!}=210$

Therefore, the number of combinations of 2 vowels and 2 consonants $=10\times 210=2100$

Each of these 2100 combinations has 4 letters and these 4 letters arrange among themselves in $4!$ ways.

Hence, the required number of words $=210\times 4!=50400$

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text:
# Stadium Area and Perimeter Calculation

In geometric stadium shape is a circle of radius (r) that has been cut in half through the center and the 2 ends are then separated by a rectangle of height (r) and width (a). therefore :
Area (A) = π r2 + 2ra
Perimeter (P) = 2( π r + a )
 Stadium Calculator Radius value (r) =m Length side (a) =m Area (A) =m2 Perimeter (P) =m


text:
# Bounded function in $[0,1]$ that does not attain its supremum and infimum.

I'm asked to give an example of a function that is bounded in $A=[0,1]$ but it doesn't attain its infimum or supremum, i.e. there is no $y\in A$ such that $f(y)=\sup\{f(x)\mid x\in A\}$ and similarly with infimum.

Clearly, since I have compact and connected domain it is necessary that such function must be discontinuous.

I have got an idea, to define $f$ as it follows: \begin{align} f(x)=\begin{cases} \frac{1}{2^{k}}-1 &\text{if }x=\frac{m}{2^{k}},\, 0<m<2^{k},\,m\neq2\ell\\ \frac{1}{3^{k}}+1 &\text{if }x=\frac{m}{3^{k}},\, 0<m<3^{k},\,m\neq3\ell\\ 0 &\text{anywhere else} \end{cases} \end{align}

Again, is clear that this is not continuous since this rationals are dense in $[0,1]$. The infimum and supremum are $-1,1$ respectively.

But my happiness ended when i see that maybe the supremum and infimum are attained since every rational of that form is in $[0,1]$. If this is true what other example of such a function can be constructed. Thanks in advance.

-
I do think your example works, but it is more complicated than necessary. – Alex Becker May 28 '12 at 17:15
The thing to notice is that open intervals are good for making functions that don't achieve their extrema. – Austin Mohr May 28 '12 at 17:33
Thanks @AlexBecker, for let me see ground safely. I'll remember that. – elessartelkontar May 28 '12 at 17:45

A simple example: \begin{align} f(x)=\begin{cases} x &\text{ if } x\neq 0,1\\ 1/2 &\text{otherwise} \end{cases} \end{align} which has infimum $0$ and supremum $1$ but attains neither.
How about $f(x) = x-\frac{1}{2}$ for $x \in (0,1)$ and $f(0)=f(1)=0$. The supremum is $\frac{1}{2}$ and the infimum is $-\frac{1}{2}$ but by construction, $f$ does not achieve these values.


text:
# Thermo. Relative Error

1. May 3, 2010

### Je m'appelle

1. The problem statement, all variables and given/known data
In a few words, show that

$$\frac{dT_i}{T_i} = 3,73\frac{dr_s}{r_s}$$

Where,

$$r_s = \lim_{P_i\rightarrow 0}\frac{P_s}{P_i}$$

There's a picture below for more details.

2. Relevant equations

$$T_i = \frac{100}{r_s - 1}$$

3. The attempt at a solution

So I tried this,

$$T_i = \frac{100}{r_s - 1}$$

Applying 'ln' to both sides,

$$ln(T_i) = ln(\frac{100}{r_s - 1})$$

And then deriving both sides by $$\frac{d}{dr_s}$$, as $$T_i$$ is a function of $$r_s$$ (right?) I get to

$$\frac{1}{T_i}\frac{dT_i}{dr_s} = \frac{r_s - 1}{100}\frac{-100}{(r_s - 1)^2}\frac{dr_s}{dr_s}$$

$$\frac{1}{T_i}\frac{dT_i}{dr_s} = \frac{-1}{r_s - 1}$$

$$\frac{1}{T_i}\frac{dT_i}{dr_s} = \frac{1}{1 - r_s}$$

$$\frac{dT_i}{T_i} = \frac{dr_s}{1 - r_s}$$

Where

$$r_s = \lim_{P_i\rightarrow 0} (\frac{P_s}{P_i})$$

But this looks completely nonsense when comparing to what I was asked to prove in the first place.

I should have gotten to something like

$$\frac{dT_i}{T_i} = 3,73\frac{dr_s}{r_s}$$

But instead I got to

$$\frac{dT_i}{T_i} = \frac{dr_s}{1 - r_s}$$

Thoughts?

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text:
# Whats the difference between ez isomerism and cis and trans isomers? (they both seem to basically have the same rules so i don't get the difference)

They are the same, $\text{E- "-=" trans-}$ and $\text{Z- "-=" cis-}$. Except E-Z is used for more complex isomers.
Each 'corner' has something different, there is nothing to compare each other too. However, EZ notation uses the total atomic mass of each corner to decide upon E- or Z-. In this case, on the left hand side $C {H}_{3} C {H}_{2}$ has a greater mass than $C {H}_{3}$, on the right hand side, $C {H}_{2} O H$ has a greater mass than $C O H$. As the heaviest 'corner' on each side is on opposite sides, it is an E- isomer.


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# Quaternion.LookRotation() not rotating 360 degrees

In my scene I have two cubes one has a tag "Respawn" and another one has the following script attached:

using UnityEngine;
using System.Collections;

public class NewBehaviourScript : MonoBehaviour {

void Update () {
transform.rotation = Quaternion.LookRotation (
GameObject.FindGameObjectWithTag("Respawn").transform.position
);
}
}

When I run it and move around my second cube with "Respawn" tag the first cube doesn't always looks at it, at certain point it just starts rotating the opposite side as if it has bounds to it's rotation. What am I doing wrong?

• Not a unity expert, so I'm not posting as a for-sure answer... but does LookRotation need to be relative? like Quaternion.LookRotation( otherObject.transform.position - this.transform.position) ? Sep 5, 2015 at 17:46
• Thanx that helped Sep 7, 2015 at 14:40
• @VadimTatarnikov Also dont use GameObject.FindGameObjectWithTag in an update method. You can just find it on Start() or every time it gets removed/instantiated, its not necessary to find it every frame
– Nick
Mar 25, 2020 at 17:50

transform.LookAt(GameObject.FindGameObjectWithTag("Respawn").transform.position);

This will automatically rotate the object to look at the desired position, which in this case is the position of the Respawn object.

I would also really recommend storing the Respawn object as a variable instead of searching for it every frame.

EDIT: Ignore the stuff above. If you want to be able to use Quaternion.LookRotation so you can, for example, use Lerp or Slerp, you just have to make the position relative as david van brink pointed out, so it will look like this:

Vector3 diff = (GameObject.FindGameObjectWithTag("Respawn").transform.position) - transform.position;
transform.rotation = Quaternion.LookRotation (diff);

• Yeah I tried it but the rotation is instant which isn't what I want Sep 5, 2015 at 19:03
• That's why I want to use quaternion.lookrotation() cause I can also use slerp after. Sep 5, 2015 at 19:37


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#### Poisson Distribution: Definition, Properties and applications with real life example

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The Poisson distribution is the discrete probability distribution of the number of events occurring in a given time period, given the average number of times the event occurs over that time period.

A discrete random variable X is said to have Poisson distribution if its probability function is defined as,

$f\left&space;(&space;x;\lambda&space;\right&space;)=\frac{e^{-\lambda&space;}\lambda&space;^{x}}{x!};&space;where&space;;x=0,1,2...,\infty$

where λ is the pararmeter of the distribution and it is the mean number of success. Also е=2.71828.

Mean of the distribution is E[x]= λ and Variance is Var[X]= λ. The mean and variance of the Distribution is equal.

 Poisson curve

### Properties of Poisson Distribution

Following properties are exist,
• Poisson distribution has only one parameter named “λ”.
• Mean of poisson distribution is λ.
• It is only a distribution which variance is also λ.
• Moment generating function is .
• The distribution is positively skewed and leptokurtic.
• It tends to normal distribution if λ⟶∞.

Under following condition , we can derive the distribution from binomial distribution,

• The probability of success or failure in bernoulli trial is very small that means which tends to zero. p⟶0 and q⟶0.
• The number of trial n is very large that is n⟶∞.
• λ=np is finite constant that is average number of success is finite.
we can also derive the it from poisson process.

### Applications

There are some real life example where the distribution may be successively applied-
• Number of death from a disease such as cancer or heart attack.
• Number of suicide reports in a perticular day.
• Number of printing mistake at each page of a book.
• Number of car passing through a certain road in a period of time t.
• Number of road accident in a certain street at a time t etc.

### Releated

#### Probability Distributions in Statistics

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Spread the love          In statistics, probability distributions are very important measure which related to the random variable and statistical data. Every data patern follows a probability distribution. Continuous data follow the continuous distribution and discrete data follow the discrete distributions. According to wikipedia, “In probability theory and statistics, a probability distribution is the mathematical function that […]

#### Gamma Distribution definition, formula and applications

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Spread the love          In probabilistic statistics, the gamma distribution is a two-parameter family of continuous probability distributions which is widely used in different sectors. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of it. Gamma Distribution A continuous random variable X is said to have a generalised gamma distribution with parameter α and […]