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# Multicoalitional solutions

Abstract : The paper proposes a new concept of solution for TU games, called multicoalitional solution, which makes sense in the context of production games, that is, where $v(S)$ is the production or income per unit of time. By contrast to classical solutions where elements of the solution are payoff vectors, multicoalitional solutions give in addition an allocation time to each coalition, which permits to realize the payoff vector. We give two instances of such solutions, called the d-multicoalitional core and the c-multicoalitional core, and both arise as the strong Nash equilibria of two strategic games, where in the first utility per active unit of time is maximized, while in the second it is the utility per total unit of time. We show that the d-core (or aspiration core) of Benett, and the c-core of Guesnerie and Oddou are strongly related to the d-multicoalitional and c-multicoalitional cores, respectively, and that the latter ones can be seen as an implementation of the former ones in a noncooperative framework.
Keywords :
Type de document :
Article dans une revue
Journal of Mathematical Economics, Elsevier, 2016, 64, pp. 1-10
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Littérature citée [25 références]

https://halshs.archives-ouvertes.fr/halshs-01293785
Contributeur : Nelly Wirth <>
Soumis le : mardi 28 février 2017 - 16:41:28
Dernière modification le : mercredi 31 octobre 2018 - 12:24:13
Document(s) archivé(s) le : lundi 29 mai 2017 - 16:22:25

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multicoalitions.pdf
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• HAL Id : halshs-01293785, version 1

### Citation

Stéphane Gonzalez, Michel Grabisch. Multicoalitional solutions. Journal of Mathematical Economics, Elsevier, 2016, 64, pp. 1-10. 〈halshs-01293785〉

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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 13th 2018

started some bare minimum omn RR-field tadpole cancellation. Currently I am using this just to complement discussion at intersecting D-brane models

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 28th 2019

added graphics from Ibanez-Uranga 12.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 30th 2019

added a bunch of references that discuss tadpole cancellation in terms of K-theory charges.

In particular section 4 of

makes explicit the tadpole cancellation condition as equations satisfied by the representation characters corresponding to fractional D-branes.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJan 31st 2019
• (edited Jan 31st 2019)

I started to spell out and discuss some actual mathematical details of tadpole cancellation for the special case of fractional D-branes at orientifold singularities (here), where it turns out to be a neat little thing in elementary representation theory.

It took me a good while to isolate the required general statement from the literature, now referenced to in Marchesano 03.

Beware that in doing so, I am currenly slightly extrapolating: The tadpole cancellation condition as I now state it in the entry is stated in Marchesano 03 (4.9) only for A-type singularities. The formula trivially generalizes to all kinds of singularities, but one should check that in this evident generalization it really still encodes RR-tadpole cancellation. I haven’t yet found any references considering this.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 31st 2019
• (edited Feb 1st 2019)

But in either case, the key point is that for a linear representation $V$, read as a fractional D-brane charge, the character value

$M \coloneqq tr_V(e) =dim(V)$

at the neutral element plays a different role than those at the non-trivial elements

$Q_{g} \coloneqq tr_V(g) \;\;\;\; g \neq e \,.$

The former plays the role of mass, and only the latter should be referred to as RR-charges.

With this finally sorted out, the idea of approaching this via stability conditions becomes viable again…

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeJan 31st 2019

Fixed a few typos. Sounds like

Detailed review of this is in Marchesano 03, Section 4, based on

is missing an end.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJan 31st 2019

Thanks!

Right, I was still going through the references that this section 4 is appearently based on, apparently it’s specifically these two

• G. Aldazabal, D. Badagnani, Luis Ibáñez, Angel Uranga, Tadpole versus anomaly cancellation in $D=4,6$ compact IIB orientifolds, JHEP 9906:031, 1999 (arXiv:hep-th/9904071)

• Gabriele Honecker, Intersecting brane world models from D8-branes on $(T^2 \times T^4\mathbb{Z}_3)/\Omega\mathcal{R}_1$ type IIA orientifolds, JHEP 0201 (2002) 025 (arXiv:hep-th/0201037)

If that is the case, then (4.9) and (4.15) in

is a real achievement in global clarification

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJan 31st 2019

added also the example of tadpole cancellation at a $2 D_4 = Q_8$-singularity, here

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJan 31st 2019

added also the example of tadpole cancellation at a $2 D_8$-singularity, here

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJan 31st 2019
• (edited Jan 31st 2019)

added also the example of tadpole cancellation at a $2 T$-singularity, here

The mass of the generating tadole-free fractional D-brane is… 24.

Interestingly, this still has $\mathbb{Z}$-worth of solutions to the homogeneous tadpole condition, even though (in contrast to the dihedral cases) this system now is overconstrained.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeFeb 2nd 2019
• (edited Feb 2nd 2019)

added the remark here that the homogeneous (no O-planes) tadpole cancellation for fractional D-branes has non-trivial solutions precisely if the “character morphism followed by forgetting the dimension” is still injective away from the regular rep.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeFeb 2nd 2019

Okay, so the operation of forming minus the sum of the truncated character values recovers the dimension of the rep modulo the order of the group (since it gives the actual dimension on the nontrivial irreps and 1 minus the order of the group on the trivial irrep). This means that the virtual reps in the kernel of the truncated character map must have dimension being a multiple of the order of the group. But by injectivity of the full character map, these must be the multiples of the regular rep. So these are generally the only solutions to that homogeneous tadpole cancellation condition.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeFeb 3rd 2019

added statement and proof (here) that the multiples of the regular rep exhaust the space of solutions of the homogeneous tadpole cancellation condition for fractional D-branes

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeFeb 3rd 2019

maded a bunch of little edits here and elsewhere:

• included here graphics for tadpole cancellation from worldsheet perspective

• gave O-plane a stub section on O-plane charge and cross-linked with pointer to the literature

• gave Riemannian orbifold a graphics illustrating $\mathbb{R}^2 \sslash \mathbb{Z}_2$

• added more textbook references

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeMar 16th 2019

added references on C-field tadpole cancellation (here)

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeMay 23rd 2019

added pointer to today’s

• Philip Betzler, Erik Plauschinn, Type IIB flux vacua and tadpole cancellation (arXiv:1905.08823)
• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeAug 30th 2019
• (edited Aug 30th 2019)

I have fixed the statement of the tadpole cancellation condition to match the case of K3-style $\mathbb{T}^4/G^{DE}$-examples actually listed afterwards, which, for orientifold groups $D1:\; \mathbb{Z}_2$ and $D3:\; \mathbb{Z}_4$ is equation (18) in Buchel-Shiu-Tye 99.

The result in this case is, rephrased in maths lingo: The correct element in the $G^{DE}$-equivariant K-theory of the point must be exactly that multiple of the regular representation whose total dimension is 32:

$n \cdot k[G] \in K_G(\ast) \simeq R_G \phantom{AA} \text{such that}\; n \cdot \left\vert G\right\vert = 32 \,.$

I still haven’t found literature that would generalize this beyond the case $D3:\; \mathbb{Z}_4$ to binary dihedral groups. But if the statement remains the same for all finite subgroups of $SU(2)$ in the D- and E-series, then, interestingly, there are only the following possible cases (since in all other cases no multiple of the dimension of the regular rep is 32):

Dynkin label orientifold group $G$ tadpole cancelling charge in $K_G(\ast)$
D0 $1$ $32 \cdot k[1]$
D1 $\mathbb{Z}_2$ $16 \cdot k[\mathbb{Z}_2/1]$
D3 $\mathbb{Z}_4 = Dic_1$ $8 \cdot k[Dic_1/1]$
D4 $2 D_4 = Dic_2 = Q_8$ $4 \cdot k[Dic_2/1]$
D6 $2 D_8 = Dic_4$ $2 \cdot k[Dic_4/1]$
D20 $2 D_{16} = Dic_8$ $1 \cdot k[Dic_8/1]$
• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeAug 31st 2019
• (edited Aug 31st 2019)

Hm, but then I don’t know at the moment whether all these dicyclic groups in #17 are point groups of 4d crystallographic groups.

It’s easy to see to be the case for $\mathbb{Z}_4$ (and of course for $\mathbb{Z}_2$).

Are $Dic_2$ and/or $Dic_4$ the point groups of any 4d crystallographic groups?

[edit: ah, for $Dic_2 = Q_8$ it should be obviously true, too]

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeSep 5th 2019

have been adding references and pointers on explicit examples of worldsheet CFT-computations for tadpole cancellation condition in toroidal orientifolds.

Am beginning to compile this in a table, here

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeSep 6th 2019

Is this what you’d call a bottom-up approach, bottom-up and top-down model building? You have a theory with anomalies and you look to cancel them by adjustment, “tadpole cancellation via orientifolding”. It sounds like things are added in to achieve a goal, rather than because they have to be there.

Is it possible to go top-down and say there can’t be tadpoles because… (Something like the graph complexes arising from the cohomology of configuration spaces don’t admit tadpoles.)?

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeSep 6th 2019

It’s another anomaly cancellation issue:

1) From the point of view of perturbative string theory, one tries to construct a consistent 2d SCFT with certain properties by starting somewhere (with some superstring sigma model), and then incrementally imposing further consistency conditions.

2) From the point of view of a putative non-perturbative formulation, one may ask whether these consistency conditions are all just aspects of the condition that the relevant charges be in a certain generalized cohomology theory.

Here with RR-field tadpolce cancellation in toroidal orientifolds, we are looking for a cohomology theory which says that admissible brane charges on a toroidal orienti 5-fold must be of the form: 1) “multiple of regular rep of orienifold group minus number of fixed points times the trivial rep” with 2) total dimension vanishing.

The question is then: Which cohomology theory has cocycles whose component data is just of this form? The answer is going to be: unstable equivariant Cohomotopy, by the equivariant Hopf degree theorem.

• CommentRowNumber22.
• CommentAuthorDavid_Corfield
• CommentTimeSep 6th 2019

Ok, thanks!

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeSep 6th 2019

By the way, I am still looking for more references for more entries for the table at RR-field tadpole cancellation on toroidal orientifolds – table.

What I’d really like is references computing the tadpole cancellation conditions for the D-brane configurations which correspond to M5-branes in M-theory on S1/G_HW times H/G_ADE, which means: type I’ orientifolds with D4-branes at ADE-singularities and with D8-branes at O8-planes

In Kataoka-Shimojo 01, in a discussion of pure D4-brane orientifolds, the authors actively shy away from diving into the computation involving also D8-branes:

The $\mathbb{Z}_N$ action with even $N$ contains an order 2 element $[ ...]$ Then there will be D8-branes in the type IIA D4-brane theory. Since the concept of intersecting D-branesinvolves use of the same dimensional D-branes, we restrict ourselves to the case that the order $N$ of $\mathbb{Z}_N$ is odd. (p. 4)

And indeed, most (all?) of the ensuing activity on pure D4 or pure D8 orientifolds (Honecker et al.) focuses on the odd-order case $\mathbb{Z}_3$.

Does anyone discuss tadpole cancellation for D4+D8 branes on $\mathbb{T}^{\mathbf{4}_{\mathbb{H}}} \sslash \mathbb{Z}_{2k}$ ??

(It is clear what the result should be, but it would still be good to have a reference for an explicit computation.)

I tend to doubt now that this computation has been published, but if anyone has anything, please drop me a note.

format: plain


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# Optimization of an objective function with noise

I want to estimate the parameters that minimize a multivariate function with noise based on realized observations.

As a simple example, say I have observed the univariate sample below:

The samples were generated from a quadratic function plus noise, but in general the function will be unknown and I don't want to make any assumptions about it other than it is "smooth". In this case, the function at $x \approx 5.8$ happens to be the minimum, but I want to make a statistical argument that the function is minimized at $x=5$.

How should I approach this? Should I fit some kind of nonparametric model to the sample data, and then optimize the fitted, smoothed values? Or are there algorithms that tackle the optimization of noisy function directly?

-
A starting point may be to simply regress $y$ on $x$ and $x^2$ including a constant. You could then apply robust regression techniques to this approach if is necessary. –  John Oct 3 '12 at 18:38
The data in the question in the question was only an example. I am looking for a general method to minimize a function without making model assumptions about the relationship between the dependent and the independent variables. –  user2303 Oct 3 '12 at 18:46
You might be interested in the approach described by Jones et al. (1998). Although it focuses on functions that are expensive to evaluate, its approach is the right one in general because it aims to find the optimal point rather than to achieve some kind of good average fit overall. –  whuber Oct 4 '12 at 17:27

If you truly have a smooth function with noise there are many ways to fit it. Parametric models linear and nonlinear are possibilities, but based on how you ask the question it sounds like you are looking for some sort of a nonparametric regression. I think splines and loess are options. In the end you are searching for a minimum though. For parametric models local and global minima can be estimated and confidence intervals constructed for it. In the nonparametric situation Bill Huber described a procedure in his answer to a similar question on CV involving time series data. I think the idea can apply to your problem as well.

But the way you pose the problem is strange and contradictory. If you have a model for multivariate data with noise without having a way to perfectly filter out the noise you cannot know exactly where the minimum is. No matter what fitting technique you use you can only approximate the point where the minimum occurs. If your estimate is 5.8 the confidence interval could in a certain statistical sense rule out 5 given sufficient data but it could never prove that the minimum is at 5 and your data really seems to be telling you otherwise. Maybe there is an assumption you are making that you haven't told us about. Now that the picture is clear and you have edited your question,

I see that since the data is simulated you know the truth. The true model is a quadratic function that has a U shape and hence has a unique global minimum at 5. Even a statistical technique that assumes a quadratic model with that much noise will give you an estimate different from 5. Nonparametric models that make less restrictive assumptions will probably do worse.

For real data as I said you can't perfectly separate the signal from the noise and you will never get an 'exact right" answer. There may be models that can estimate the minimum and be consistent in a probabilistic sense but not perfect for any finite sample size.

-
Thanks for your answer. I do want to use a nonparametric model and not make any model assumptions other than its smoothness. So if I understand correctly, you are suggesting just fitting with something like LOESS and minimizing the fitted values? –  user2303 Oct 3 '12 at 18:56
That is one way. Huber's suggestion was something like that but I don't remember all the details. For the time series problem we were actually looking for a minimum usingg seasonal data and I recommended harmonic regression and pointed out some advantages. At the time I hadn't read Bill's answer thoroughly and didn't contrast the advantages and disadvantages of each properly. His approach was a little different because of the time series context and amounted to being efficient because it went through the series successively keeping track of the current global minimum. –  Michael Chernick Oct 3 '12 at 19:06


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# Posts tagged as “easy”

Given three integer arrays nums1nums2, and nums3, return distinct array containing all the values that are present in at least two out of the three arrays. You may return the values in any order.

Example 1:

Input: nums1 = [1,1,3,2], nums2 = [2,3], nums3 = [3]
Output: [3,2]
Explanation: The values that are present in at least two arrays are:
- 3, in all three arrays.
- 2, in nums1 and nums2.

Example 2:

Input: nums1 = [3,1], nums2 = [2,3], nums3 = [1,2]
Output: [2,3,1]
Explanation: The values that are present in at least two arrays are:
- 2, in nums2 and nums3.
- 3, in nums1 and nums2.
- 1, in nums1 and nums3.

Example 3:

Input: nums1 = [1,2,2], nums2 = [4,3,3], nums3 = [5]
Output: []
Explanation: No value is present in at least two arrays.

Constraints:

• 1 <= nums1.length, nums2.length, nums3.length <= 100
• 1 <= nums1[i], nums2[j], nums3[k] <= 100

s[x] := bitmask of x in all array[i]

s[x] = 101 => x in array0 and array2

Time complexity: O(n1 + n2 + n3)
Space complexity: O(n1 + n2 + n3)

## C++

There are n seats and n students in a room. You are given an array seats of length n, where seats[i] is the position of the ith seat. You are also given the array students of length n, where students[j] is the position of the jth student.

You may perform the following move any number of times:

• Increase or decrease the position of the ith student by 1 (i.e., moving the ith student from position x to x + 1 or x - 1)

Return the minimum number of moves required to move each student to a seat such that no two students are in the same seat.

Note that there may be multiple seats or students in the same position at the beginning.

Example 1:

Input: seats = [3,1,5], students = [2,7,4]
Output: 4
Explanation: The students are moved as follows:
- The first student is moved from from position 2 to position 1 using 1 move.
- The second student is moved from from position 7 to position 5 using 2 moves.
- The third student is moved from from position 4 to position 3 using 1 move.
In total, 1 + 2 + 1 = 4 moves were used.

Example 2:

Input: seats = [4,1,5,9], students = [1,3,2,6]
Output: 7
Explanation: The students are moved as follows:
- The first student is not moved.
- The second student is moved from from position 3 to position 4 using 1 move.
- The third student is moved from from position 2 to position 5 using 3 moves.
- The fourth student is moved from from position 6 to position 9 using 3 moves.
In total, 0 + 1 + 3 + 3 = 7 moves were used.

Example 3:

Input: seats = [2,2,6,6], students = [1,3,2,6]
Output: 4
Explanation: The students are moved as follows:
- The first student is moved from from position 1 to position 2 using 1 move.
- The second student is moved from from position 3 to position 6 using 3 moves.
- The third student is not moved.
- The fourth student is not moved.
In total, 1 + 3 + 0 + 0 = 4 moves were used.

Constraints:

• n == seats.length == students.length
• 1 <= n <= 100
• 1 <= seats[i], students[j] <= 100

## Solution: Greedy

Sort both arrays, move students[i] to seats[i].

Time complexity: O(nlogn)
Space complexity: O(1)

## C++

A sentence is a list of tokens separated by a single space with no leading or trailing spaces. Every token is either a positive number consisting of digits 0-9 with no leading zeros, or a word consisting of lowercase English letters.

• For example, "a puppy has 2 eyes 4 legs" is a sentence with seven tokens: "2" and "4" are numbers and the other tokens such as "puppy" are words.

Given a string s representing a sentence, you need to check if all the numbers in s are strictly increasing from left to right (i.e., other than the last number, each number is strictly smaller than the number on its right in s).

Return true if so, or false otherwise.

Example 1:

Input: s = "1 box has 3 blue 4 red 6 green and 12 yellow marbles"
Output: true
Explanation: The numbers in s are: 1, 3, 4, 6, 12.
They are strictly increasing from left to right: 1 < 3 < 4 < 6 < 12.

Example 2:

Input: s = "hello world 5 x 5"
Output: false
Explanation: The numbers in s are: 5, 5. They are not strictly increasing.

Example 3:

Input: s = "sunset is at 7 51 pm overnight lows will be in the low 50 and 60 s"
Output: false
Explanation: The numbers in s are: 7, 51, 50, 60. They are not strictly increasing.

Example 4:

Input: s = "4 5 11 26"
Output: true
Explanation: The numbers in s are: 4, 5, 11, 26.
They are strictly increasing from left to right: 4 < 5 < 11 < 26.

Constraints:

• 3 <= s.length <= 200
• s consists of lowercase English letters, spaces, and digits from 0 to 9, inclusive.
• The number of tokens in s is between 2 and 100, inclusive.
• The tokens in s are separated by a single space.
• There are at least two numbers in s.
• Each number in s is a positive number less than 100, with no leading zeros.
• s contains no leading or trailing spaces.

Solution: String

Time complexity: O(n)
Space complexity: O(1)

## C++

You are given a string s consisting of n characters which are either 'X' or 'O'.

move is defined as selecting three consecutive characters of s and converting them to 'O'. Note that if a move is applied to the character 'O', it will stay the same.

Return the minimum number of moves required so that all the characters of s are converted to 'O'.

Example 1:

Input: s = "XXX"
Output: 1
Explanation: XXX -> OOO
We select all the 3 characters and convert them in one move.

Example 2:

Input: s = "XXOX"
Output: 2
Explanation: XXOX -> OOOX -> OOOO
We select the first 3 characters in the first move, and convert them to 'O'.
Then we select the last 3 characters and convert them so that the final string contains all 'O's.

Example 3:

Input: s = "OOOO"
Output: 0
Explanation: There are no 'X's in s to convert.

Constraints:

• 3 <= s.length <= 1000
• s[i] is either 'X' or 'O'.

## Solution: Straight Forward

if s[i] == ‘X’, change s[i], s[i + 1] and s[i + 2] to ‘O’.

Time complexity: O(n)
Space complexity: O(1)

## C++

You are given a string num, representing a large integer. Return the largest-valued odd integer (as a string) that is a non-empty substring of num, or an empty string "" if no odd integer exists.

substring is a contiguous sequence of characters within a string.

Example 1:

Input: num = "52"
Output: "5"
Explanation: The only non-empty substrings are "5", "2", and "52". "5" is the only odd number.

Example 2:

Input: num = "4206"
Output: ""
Explanation: There are no odd numbers in "4206".

Example 3:

Input: num = "35427"
Output: "35427"
Explanation: "35427" is already an odd number.

Constraints:

• 1 <= num.length <= 105
• num only consists of digits and does not contain any leading zeros.

## Solution: Find right most odd digit

We just need to find the right most digit that is odd, answer will be num[0:r].


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+0

# A sealed container containing 4.1 mol of gas is squeezed, changing its volume from 2.5×10−2 m3 to 1.9×10−2 m3. During this process, the temp

0
1056
3

A sealed container containing 4.1 mol of gas is squeezed, changing its volume from 2.5×10−2 m3 to 1.9×10−2 m3. During this process, the temperature decreases by 8.0 K while the pressure increases by 800 Pa.What was the original pressure of the gas in the container?

May 31, 2015

#2
+1036
+10

.

$$\text {}\\ \noindent \text {Set up simultaneous equations using known data set and relations:} \\\\ \noindent \text {P*V = nRT }\\ \text {P = pressure in Pa}\\ \text {V = volume in } m^3\\ \text {T = in Kelvin} \\ \text {n = moles} \\ \text {R = 8.314J mol}^{-1}\\\\ \noindent P2 =P1 + 800 \\ T2 = T1 - 8 \\\\ \noindent P1*(2.5E-2) = 4.1 * 8.314 * T1 \\ \noindent (P1 + 800)*(1.9E-2) = 4.1 * 8.314 *(T1-8) \\\\ \text {Using the site calculator to solve:} \\\\$$

$$\underset{\,\,\,\,{{\rightarrow {\mathtt{p1, t1}}}}}{{solve}}{\left(\begin{array}{l}{\mathtt{P1}}{\mathtt{\,\times\,}}{\mathtt{0.025}}={\mathtt{4.1}}{\mathtt{\,\times\,}}{\mathtt{8.314}}{\mathtt{\,\times\,}}{\mathtt{T1}}\\ \left({\mathtt{P1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{800}}\right){\mathtt{\,\times\,}}{\mathtt{0.019}}={\mathtt{4.1}}{\mathtt{\,\times\,}}{\mathtt{8.314}}{\mathtt{\,\times\,}}\left({\mathtt{T1}}{\mathtt{\,-\,}}{\mathtt{8}}\right)\end{array}\right)} \Rightarrow \left\{ \begin{array}{l}{\mathtt{p1}} = {\frac{{\mathtt{239\,916}}}{{\mathtt{5}}}}\\ {\mathtt{t1}} = {\frac{{\mathtt{5\,997\,900}}}{{\mathtt{170\,437}}}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{p1}} = {\mathtt{47\,983.2}}\\ {\mathtt{t1}} = {\mathtt{35.191\: \!302\: \!358\: \!056\: \!055\: \!9}}\\ \end{array} \right\}$$

$$\text {}\\ \noindent \text {The original pressure of the gas } 47983 { Pa} \\ \text {The original temperature of the gas } 35.91 { K}\\$$

.

Jun 1, 2015

#1
+27374
+5

See Nauseated's solution below.

.

Jun 1, 2015
#2
+1036
+10

.

$$\text {}\\ \noindent \text {Set up simultaneous equations using known data set and relations:} \\\\ \noindent \text {P*V = nRT }\\ \text {P = pressure in Pa}\\ \text {V = volume in } m^3\\ \text {T = in Kelvin} \\ \text {n = moles} \\ \text {R = 8.314J mol}^{-1}\\\\ \noindent P2 =P1 + 800 \\ T2 = T1 - 8 \\\\ \noindent P1*(2.5E-2) = 4.1 * 8.314 * T1 \\ \noindent (P1 + 800)*(1.9E-2) = 4.1 * 8.314 *(T1-8) \\\\ \text {Using the site calculator to solve:} \\\\$$

$$\underset{\,\,\,\,{{\rightarrow {\mathtt{p1, t1}}}}}{{solve}}{\left(\begin{array}{l}{\mathtt{P1}}{\mathtt{\,\times\,}}{\mathtt{0.025}}={\mathtt{4.1}}{\mathtt{\,\times\,}}{\mathtt{8.314}}{\mathtt{\,\times\,}}{\mathtt{T1}}\\ \left({\mathtt{P1}}{\mathtt{\,\small\textbf+\,}}{\mathtt{800}}\right){\mathtt{\,\times\,}}{\mathtt{0.019}}={\mathtt{4.1}}{\mathtt{\,\times\,}}{\mathtt{8.314}}{\mathtt{\,\times\,}}\left({\mathtt{T1}}{\mathtt{\,-\,}}{\mathtt{8}}\right)\end{array}\right)} \Rightarrow \left\{ \begin{array}{l}{\mathtt{p1}} = {\frac{{\mathtt{239\,916}}}{{\mathtt{5}}}}\\ {\mathtt{t1}} = {\frac{{\mathtt{5\,997\,900}}}{{\mathtt{170\,437}}}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{p1}} = {\mathtt{47\,983.2}}\\ {\mathtt{t1}} = {\mathtt{35.191\: \!302\: \!358\: \!056\: \!055\: \!9}}\\ \end{array} \right\}$$

$$\text {}\\ \noindent \text {The original pressure of the gas } 47983 { Pa} \\ \text {The original temperature of the gas } 35.91 { K}\\$$

.

Nauseated Jun 1, 2015
#3
+95356
0

Thanks Nauseated,

Nice use of the site calc to solve the equations simultaneously too.

I always forget how to do things like that. :)

Jun 1, 2015


text:
## Hats-11B and Hats-12B: Two Transiting Hot Jupiters Orbiting Subsolar Metallicity Stars Selected for the K2 Campaign 7

### Description

We report the discovery of two transiting extrasolar planets from the HATSouth survey. HATS-11, a V = 14.1 G0-star shows a periodic $12.9$ mmag dip in its light curve every 3.6192 days and a radial velocity variation consistent with a Keplerian orbit. HATS-11 has a mass of $1.000\pm 0.060$ ${M}_{\odot }$, a radius of $1.444\pm 0.057$ ${R}_{\odot }$ and an effective temperature of $6060\pm 150$ K, while its companion is a $0.85\pm 0.12$ ${M}_{{\rm{J}}}$, $1.510\pm 0.078$ ${R}_{{\rm{J}}}$ planet...[Show more]

Collections ANU Research Publications 2016 Journal article http://hdl.handle.net/1885/152012 Astronomical Journal 10.3847/0004-6256/152/4/88 Open Access


text:
Star Wars: Imperial Bootcamp!

StrifeX

Well-Known Member
Imperial Boot camp I

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?Imperial Boot Camp II

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Imperial Boot Camp III Part 1

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?Imperial Boot Camp III Part 2

In this people just made these action figures into a little movie....

CoZmicShReddeR

Staff member
C7 Commander
Haha, great job on those!! Stop motion animation is funny stuff!!

I didn't know that people can block websites from embeding the video's... :roll:

OP

StrifeX

Well-Known Member
It's surprising how some people have have so much time like this, I thought it was really funny and amazing at the same time to see this.

XxHeroxX

Member
That was a nice star wars toy show NICE!

New Member
haha nice. very funny, very awesome vids! :lol:

Winters

New Member
To bad that the first two are taken off..., but the third one was real awsome

warmarine

New Member
Isn't it weird that george lucas is the sith lord in the 3rd clip? :?

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text:
# Background¶

Understanding the tutorials that follow will require some knowledge of glacier physics. In particular, we’ll expect that you have a basic understanding of:

• mass, momentum, and heat transport

• the Stokes equations and simplified models

• Glen’s flow law

We use a slightly unconventional unit system borrowed from the package Elmer/Ice. The fundamental units are megapascals (MPa), meters (m) and years (yr). In this unit system, most of the quantities we’re interested in, including physical constants, are between 1/10 and 1000.

## Symbols¶

In the tutorials and in the icepack API we’ve tried to use variable names that correspond as closely to the symbols used for each field in the literature and standard textbooks. Our goal is to make the code look as much like the mathematics as possible. The variable names and corresponding fields are:

• u - velocity

• h - thickness

• s - surface elevation

• b - bed elevation

• A - rheology coefficient

• n - exponent in Glen’s flow law

• C - friction coefficient

• m - exponent in Weertman’s sliding law

• E - internal energy density

Many references in the glaciological literature use a lower-case $$h$$ for surface elevation and a capital $$H$$ for thickness. We’ve chosen a slightly different convention, so this is something to be aware of.

The standard textbook on glaciology is The Physics of Glaciers by Cuffey and Paterson [1]. This book is great for breadth and covers many more topics than modeling. Dynamics of Ice Sheets and Glaciers by Greve and Blatter [2] focuses more on continuum mechanics and modeling and has some nice introductions to various numerical methods. While a thorough knowledge of the finite element method shouldn’t be necessary to use icepack, some familiarity is definitely helpful. Understanding and Implementing the Finite Element Method by Gockenbach is a good introduction [3]. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics by Braess goes into much greater depth [4].

1

Kurt M Cuffey and William Stanley Bryce Paterson. The physics of glaciers. Academic Press, 2010.

2

Ralf Greve and Heinz Blatter. Dynamics of ice sheets and glaciers. Springer, 2009.

3

Mark S Gockenbach. Understanding and implementing the finite element method. SIAM, 2006.

4

Dietrich Braess. Finite elements: Theory, fast solvers, and applications in solid mechanics. Cambridge University Press, 2007.


text:
# How to deal with collisions in Bitcoin addresses?

When creating a Bitcoin account, you need to issue a couple of private/public ECDSA keys. Then, you derive your account address by taking a 160-bit hash (through SHA-256 and RIPEMD) of the public key and use a custom Base 58 algorithm to convert it to alphanumeric values (see the picture below, I took it from this page).

My question is that collisions might happen (not necessarily malicious ones, but incidental ones). I would like to know whether there is a specific mechanism in the Bitcoin protocol to sort out these collisions and be ensured that the payment is going to the right place.

In fact, I tried to look through the protocol details but could not find anything dealing with this problem. The only reference I found is on the Bitcoin-wiki and it states that:

## Collisions (lack thereof)

Since Bitcoin addresses are basically random numbers, it is possible, although extremely unlikely, for two people to independently generate the same address. This is called a collision. If this happens, then both the original owner of the address and the colliding owner could spend money sent to that address. It would not be possible for the colliding person to spend the original owner's entire wallet (or vice versa). If you were to intentionally try to make a collision, it would currently take 2^107 times longer to generate a colliding Bitcoin address than to generate a block. As long as the signing and hashing algorithms remain cryptographically strong, it will likely always be more profitable to collect generations and transaction fees than to try to create collisions.

It is more likely that the Earth will be destroyed in the next 5 seconds, than that a collision would occur sometime in the next millennium.

I found this explanation extremely unsatisfactory. If any of you have a better explanation than that, or know about a prevention mechanism that has been planted into the Bitcoin protocol, I would be delighted to read about it.

• Why is "it's very unlikely to happen" unsatisfactory to you? That statement is at the heart of all crypto. – CodesInChaos Mar 19 '16 at 10:21
• Well, I know that occurrences of such collisions will happen with a very low probability. And, this is fine for me. But, at least, I would have expected that, during the creation of a BTC address, one would check that the current address does not appear in the blockchain... But, still, this is not enough as the creation should occurs in a decentralized manner, so two addresses in collision might be created a the same time and checking the blockchain at the same time as well (I agree that we are here speaking of an extremely low probability). – perror Mar 19 '16 at 10:29
• You'd expect a collision after $2^{80}$ public keys. This is only feasible (maybe) for state-sponsored attackers with massive supercomputers. The chance of this happening in the wild is extremely low. – SEJPM Mar 19 '16 at 12:09
• To add to @SEJPM, that is a collision of the public-key hash after $2^{80}$ public keys. It is still likely that you will not have a colliding private key. – mikeazo Mar 19 '16 at 16:56
• Once again, I do not want to consider a malicious attacker, but a "shit happen" case. The fact that you have $1/2^{80}$ chance to get it does not means that this will never happen... – perror Mar 19 '16 at 19:05

Cryptography (and real security in general) offer quantitative analysis of the security provided - Meaning, real security products will describe how long they will resist a certain class of attack.

With cryptography, we select our parameters such that the time required to perform the best attack would exceed the amount considered to be practical, realistic, or feasible.

With a larger input space then output space, collisions must exist - therefore the only solution is to:

• Accept that collisions must exist
• Make collisions so rare as to not be a problem

This is exactly what the bitcoin protocol has more or less done. Granted, $2^{80}$ could arguably be a little bit bigger for more comfort; Bitcoin could have used bigger hash functions, but they may have been limited by availability at the time. If you were going to re-design it, I'm sure you would use larger sizes now. But that's always easy to say in retrospect...

• It takes $2^{80}$ hash invocations to find a collisions, but $2^{-80}$ is not a meaningful probability in this here. – CodesInChaos Jul 4 '17 at 17:18

In this application we don't care about attackers generating collisions with themselves. What we care about is.

2. An attacker deliberately trying to generate collisions with the addresses of existing unspent outputs.

We can't reduce the risks of these to zero but we can reduce them to negligible levels.

Lets consider case 1 first. I don't know a good way to find the total number of addresses that have ever been used but we can get an upper bound by taking the size of the blockchain and dividing it by the size of an address. The blockchain is now about 150GB, divide that by 20 gives us an upper bound of about 7.5 billion addresses. 7.5 billion is approximately $2^{33}$. In reality the blockchain contains a lot more than just addresess and so I expect the real number is going to be lower than this since the blockchain carries a bunch of stuff other than addreses.

The probability of an accidental collision can now be approximated by the equation.

$$p \approx \frac{n^2}{2m}$$

If we assume there are $2^{33}$ addresses in use and we assume that the hash function is uniform then the chance of an inadvertent collision is.

$$p \approx \frac{2^{66}}{2^{161}} = \frac{1}{2^{95}}$$

Now onto the malicious side. The malicious actor is attempting to find an address collision with an "unspent" output. Finding a collision with an output that has already been spent doesn't help him.

According to https://blockchain.info/charts/utxo-count there are about 67 million unspent outputs. 67 million is approximately $2^{26}$ so the probability of the attackers hash matching an existing unspent output is about $\frac{1}{2^{133}}$.

Now of course the attacker can try many times. Lets assume that generating an address takes the same effort as attempting to hash a block (in reality it takes more). Lets assume that the attacker has as much hashing power as the whole bitcoin network put together and that they run their attack for a century. The total bitcoin network hashrate is about $2^{64}$ hashes per second and there are about $2^{32}$ seconds in a century.

$p \approx \frac{1}{2^{133}} \times 2^{64} \times 2^{32} = \frac{1}{2^{37}}$

Bottom line, it's far far easier to mine bitcoin legitimately than to steal it through hash collisions.

Actually you people are missing one simple factor in your calculations that is often omitted. If you were the crypto experts you claim to be you would not so easily brush off his question as stupid. Arrogance and assumption is the calling card of weak security policy.

What are you missing? Some crypto wallets are being generated by seeds from human input, which are MORE likely to produce a collision compared to randomly generated seeds. If you would like to argue the math and common sense of this fact, you will lose.

You should understand the difference between truly random and pseudo-random and how that applies to crypto. A seed is essentially a password in the application of crypto currency wallets. Which is better? A truly random password or a pseudo-random password? You and I know that answer to this question and it's not debatable.

While I'll agree that collision is highly unlikely given the scope of possible key sequences, it IS NOT as unlikely as some have proposed simply because there is a human factor involved.

OP, your question is valuable and so is your concern. The answer to your question is there are no checks in place for this collision problem. Devs have taken the lazy approach and relied on pure probability, even though history is full of examples where such assumptions failed. Instead of actually coming up with a good solution they have kicked the can down the road and left it to chance. A good chance btw but STILL chance.

• This is Crypto.SE where we handle cryptographic things. The fail due to "human factor" you describe is an implementation problem (programming) which merely shows that cryptography, just like everything else, will fail when humans decide to ignore the exact and clearly defined (in this case cryptographic) guidelines and specifications provided. What you are saying compares to blaming breathing from being insufficient to survive because some human might decide to stop doing it. While humanity has definitely evolved during past millennia, there's no failsafe for idiocracy (here: inapt coders). – e-sushi Jan 6 '18 at 2:32


text:
## Ling, Chen

Compute Distance To:
 Author ID: ling.chen Published as: Ling, Chen; Ling, C. External Links: MGP
 Documents Indexed: 72 Publications since 1998 Co-Authors: 49 Co-Authors with 62 Joint Publications 2,034 Co-Co-Authors
all top 5

### Co-Authors

 5 single-authored 31 Qi, Liqun 17 He, Hongjin 8 Zhou, Guanglu 6 Zhang, Xinzhen 5 Caputo, Michael R. 4 Ling, Liyun 4 Wu, Soon-Yi 3 Hu, Yuda 3 Tong, Xiaojiao 3 Yin, Hongxia 2 Bai, Xueli 2 Caccetta, Louis 2 Ni, Qin 2 Peng, Liang 2 Xu, Hong-Kun 2 Yan, Weijie 2 Zheng, Ling’ai 1 Bomze, Immanuel M. 1 Chen, Xiaojun 1 Datta, Sujay 1 Fukushima, Masao 1 Hou, Jiao-Jiao 1 Hsueh, Shao-Chieh 1 Hu, ShengLong 1 Huang, Zheng-Hai 1 Jin, Ping 1 Kang, Hongchao 1 Kravaris, Costas 1 Li, Deyuan 1 Li, Zhening 1 Liu, Jinjie 1 López-Cerdá, Marco Antonio 1 Miran, Seyed M. 1 Nie, Jiawang 1 Ouyang, Chen 1 Pan, Chenjian 1 Shapiro, Alexander 1 Shen, Huifei 1 Teo, Kok Lay 1 Tong, Wanbin 1 Wang, Guifeng 1 Wang, Lingling 1 Wang, Yiju 1 Wu, Ed Xuekui 1 Xu, Yanwei 1 Yang, Lei 1 Yang, Qingzhi 1 Ye, Yinyu 1 Yu, Gaohang
all top 5

### Serials

 7 Pacific Journal of Optimization 6 Computational Optimization and Applications 4 Journal of Optimization Theory and Applications 4 Journal of Global Optimization 3 Journal of Computational and Applied Mathematics 2 Applied Mathematics and Computation 2 Numerical Functional Analysis and Optimization 2 Journal of Systems Science and Mathematical Sciences 2 Journal of Scientific Computing 2 Mathematical Programming. Series A. Series B 2 SIAM Journal on Optimization 2 Applied Mathematics. Series A (Chinese Edition) 2 Journal of Zhejiang Normal University. Natural Sciences 2 Dynamic Games and Applications 2 Numerical Algebra, Control and Optimization 2 Operations Research Transactions 1 Computer Methods in Applied Mechanics and Engineering 1 Journal of Mathematical Analysis and Applications 1 IEEE Transactions on Automatic Control 1 Optimal Control Applications & Methods 1 Acta Mathematicae Applicatae Sinica 1 Operations Research Letters 1 Acta Mathematicae Applicatae Sinica. English Series 1 Optimization 1 Journal of Symbolic Computation 1 Asia-Pacific Journal of Operational Research 1 Journal of Economic Dynamics & Control 1 SIAM Journal on Matrix Analysis and Applications 1 Mathematica Applicata 1 Numerical Algorithms 1 Numerical Linear Algebra with Applications 1 Journal of Convex Analysis 1 European Journal of Mechanics. A. Solids 1 ASTIN Bulletin 1 Communications in Mathematical Sciences 1 North American Actuarial Journal 1 Frontiers of Mathematics in China 1 Optimization Letters 1 Science China. Mathematics 1 Asian Journal of Control 1 Journal of the Operations Research Society of China 1 Japanese Journal of Statistics and Data Science
all top 5

### Fields

 51 Operations research, mathematical programming (90-XX) 26 Numerical analysis (65-XX) 18 Linear and multilinear algebra; matrix theory (15-XX) 10 Calculus of variations and optimal control; optimization (49-XX) 9 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Functional analysis (46-XX) 2 Statistics (62-XX) 2 Mechanics of deformable solids (74-XX) 2 Information and communication theory, circuits (94-XX) 1 Real functions (26-XX) 1 Approximations and expansions (41-XX) 1 Operator theory (47-XX) 1 Convex and discrete geometry (52-XX) 1 Differential geometry (53-XX) 1 General topology (54-XX) 1 Computer science (68-XX) 1 Optics, electromagnetic theory (78-XX) 1 Systems theory; control (93-XX)

### Citations contained in zbMATH Open

47 Publications have been cited 505 times in 371 Documents Cited by Year
On determinants and eigenvalue theory of tensors. Zbl 1259.15038
Hu, Shenglong; Huang, Zheng-Hai; Ling, Chen; Qi, Liqun
2013
Biquadratic optimization over unit spheres and semidefinite programming relaxations. Zbl 1221.90074
Ling, Chen; Nie, Jiawang; Qi, Liqun; Ye, Yinyu
2009
The best rank-1 approximation of a symmetric tensor and related spherical optimization problems. Zbl 1269.15026
Zhang, Xinzhen; Ling, Chen; Qi, Liqun
2012
On the cone eigenvalue complementarity problem for higher-order tensors. Zbl 1339.15008
Ling, Chen; He, Hongjin; Qi, Liqun
2016
Differentiability and semismoothness properties of integral functions and their applications. Zbl 1079.90143
Qi, Liqun; Shapiro, Alexander; Ling, Chen
2005
The $$SC^1$$ 1property of an expected residual function arising from stochastic complementarity problems. Zbl 1155.90461
Ling, Chen; Qi, Liqun; Zhou, Guanglu; Caccetta, Louis
2008
A globally and quadratically convergent algorithm for solving multilinear systems with $$\mathcal {M}$$-tensors. Zbl 1397.65047
He, Hongjin; Ling, Chen; Qi, Liqun; Zhou, Guanglu
2018
On error bounds of polynomial complementarity problems with structured tensors. Zbl 1427.90275
Ling, Liyun; He, Hongjin; Ling, Chen
2018
A smoothing projected Newton-type algorithm for semi-infinite programming. Zbl 1153.90556
Qi, Liqun; Ling, Chen; Tong, Xiaojiao; Zhou, Guanglu
2009
Computation of integrals with oscillatory singular factors of algebraic and logarithmic type. Zbl 1315.65025
Kang, Hongchao; Ling, Chen
2015
A truncated projected Newton-type algorithm for large-scale semi-infinite programming. Zbl 1131.90064
Ni, Qin; Ling, Chen; Qi, Liqun; Teo, Kok Lay
2006
A new smoothing Newton-type algorithm for semi-infinite programming. Zbl 1220.90144
Ling, Chen; Ni, Qin; Qi, Liqun; Wu, Soon-Yi
2010
Higher-degree eigenvalue complementarity problems for tensors. Zbl 1338.15027
Ling, Chen; He, Hongjin; Qi, Liqun
2016
Connectedness of cone superefficient point sets in locally convex topological vector spaces. Zbl 0972.90067
Hu, Y. D.; Ling, C.
2000
Global convergence of a robust smoothing SQP method for semi-infinite programming. Zbl 1140.90044
Ling, C.; Qi, L. Q.; Zhou, G. L.; Wu, S. Y.
2006
Improvement on the 10-node tetrahedral element for three-dimensional problems. Zbl 0982.74067
Lo, S. H.; Ling, Chen
2000
Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints. Zbl 1237.90180
Zhang, Xinzhen; Ling, Chen; Qi, Liqun
2011
A relaxed projection method for split variational inequalities. Zbl 1323.65079
He, Hongjin; Ling, Chen; Xu, Hong-Kun
2015
A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints. Zbl 1160.65029
Tong, Xiaojiao; Ling, Chen; Qi, Liqun
2008
An implementable splitting algorithm for the $$\ell_1$$-norm regularized split feasibility problem. Zbl 1339.65030
He, Hongjin; Ling, Chen; Xu, Hong-Kun
2016
Some properties on Pareto-eigenvalues of higher-order tensors. Zbl 1349.15030
Xu, Feng; Ling, Chen
2015
Semidefinite relaxation approximation for multivariate bi-quadratic optimization with quadratic constraints. Zbl 1274.65170
Ling, Chen; Zhang, Xinzhen; Qi, Liqun
2012
The generalized optimality conditions of multiobjective programming problem in topological vector space. Zbl 1045.90056
Hu, Yuda; Ling, Chen
2004
Generalized tensor equations with leading structured tensors. Zbl 1428.15026
Yan, Weijie; Ling, Chen; Ling, Liyun; He, Hongjin
2019
A smoothing Levenberg-Marquardt algorithm for semi-infinite programming. Zbl 1318.49062
Jin, Ping; Ling, Chen; Shen, Huifei
2015
Further study on tensor absolute value equations. Zbl 1469.15017
Ling, Chen; Yan, Weijie; He, Hongjin; Qi, Liqun
2020
$$l^{k,s}$$-singular values and spectral radius of rectangular tensors. Zbl 1272.15008
Ling, Chen; Qi, Liqun
2013
Standard bi-quadratic optimization problems and unconstrained polynomial reformulations. Zbl 1268.90085
Bomze, Immanuel M.; Ling, Chen; Qi, Liqun; Zhang, Xinzhen
2012
Convergence rate of Newton’s method for $$L_2$$ spectral estimation. Zbl 1111.90113
Yin, Hongxia; Ling, Chen; Qi, Liqun
2006
Properties of the solution set of generalized polynomial complementarity problems. Zbl 07313887
Ling, Liyun; Ling, Chen; He, Hongjin
2020
The measure of diffusion skewness and kurtosis in magnetic resonance imaging. Zbl 1188.90291
Zhang, Xinzhen; Ling, Chen; Qi, Liqun; Wu, Ed Xuekui
2010
Improved approximation results on standard quartic polynomial optimization. Zbl 1410.90167
Ling, Chen; He, Hongjin; Qi, Liqun
2017
A nonnegativity preserving algorithm for multilinear systems with nonsingular $$\mathcal{M}$$-tensors. Zbl 1473.65053
Bai, Xueli; He, Hongjin; Ling, Chen; Zhou, Guanglu
2021
K-T type optimality conditions of multiobjective programming with set-valued maps. Zbl 0965.90046
Ling, Chen
2000
Some advances in tensor analysis and polynomial optimization. Zbl 1313.15045
Li, Zhening; Ling, Chen; Wang, Yiju; Yang, Qingzhi
2014
A qualitative characterization of symmetric open-loop Nash equilibria in discounted infinite horizon differential games. Zbl 1221.91012
Ling, C.; Caputo, M. R.
2011
Multirate sampled-data observer design based on a continuous-time design. Zbl 1482.93364
Ling, Chen; Kravaris, Costas
2019
The intrinsic comparative dynamics of locally differentiable feedback Nash equilibria of autonomous and exponentially discounted infinite horizon differential games. Zbl 1402.91073
Caputo, Michael R.; Ling, Chen
2013
Bias-corrected inference for a modified Lee-Carter mortality model. Zbl 1410.91277
Liu, Qing; Ling, Chen; Li, Deyuan; Peng, Liang
2019
A class of second-order cone eigenvalue complementarity problems for higher-order tensors. Zbl 1372.65174
Hou, Jiao-Jiao; Ling, Chen; He, Hong-Jin
2017
An improved discrete Kirchhoff triangular element for bending, vibration and buckling analyses. Zbl 0995.74075
Soh, Ai-Kah; Ling, Chen
2000
Some results on $$l^k$$-eigenvalues of tensor and related spectral radius. Zbl 1246.15013
Ling, Chen; Qi, Liqun
2011
Smooth and semismooth Newton methods for constrained approximation and estimation. Zbl 1297.90160
Yin, Hongxia; Ling, Chen; Qi, Liqun
2012
Statistical inference for Lee-Carter mortality model and corresponding forecasts. Zbl 1426.91227
Liu, Qing; Ling, Chen; Peng, Liang
2019
A new Levenberg-Marquardt type algorithm for solving nonsmooth constrained equations. Zbl 1364.65121
Ling, Chen; Wang, Guifeng; He, Hongjin
2014
Generalized tensor complementarity problems over a polyhedral cone. Zbl 07309370
Ling, Liyun; Ling, Chen; He, Hongjin
2020
On the properties of tensor complementarity problems. Zbl 1466.15030
Yu, W.; Ling, C.; He, H.
2018
A nonnegativity preserving algorithm for multilinear systems with nonsingular $$\mathcal{M}$$-tensors. Zbl 1473.65053
Bai, Xueli; He, Hongjin; Ling, Chen; Zhou, Guanglu
2021
Further study on tensor absolute value equations. Zbl 1469.15017
Ling, Chen; Yan, Weijie; He, Hongjin; Qi, Liqun
2020
Properties of the solution set of generalized polynomial complementarity problems. Zbl 07313887
Ling, Liyun; Ling, Chen; He, Hongjin
2020
Generalized tensor complementarity problems over a polyhedral cone. Zbl 07309370
Ling, Liyun; Ling, Chen; He, Hongjin
2020
Generalized tensor equations with leading structured tensors. Zbl 1428.15026
Yan, Weijie; Ling, Chen; Ling, Liyun; He, Hongjin
2019
Multirate sampled-data observer design based on a continuous-time design. Zbl 1482.93364
Ling, Chen; Kravaris, Costas
2019
Bias-corrected inference for a modified Lee-Carter mortality model. Zbl 1410.91277
Liu, Qing; Ling, Chen; Li, Deyuan; Peng, Liang
2019
Statistical inference for Lee-Carter mortality model and corresponding forecasts. Zbl 1426.91227
Liu, Qing; Ling, Chen; Peng, Liang
2019
A globally and quadratically convergent algorithm for solving multilinear systems with $$\mathcal {M}$$-tensors. Zbl 1397.65047
He, Hongjin; Ling, Chen; Qi, Liqun; Zhou, Guanglu
2018
On error bounds of polynomial complementarity problems with structured tensors. Zbl 1427.90275
Ling, Liyun; He, Hongjin; Ling, Chen
2018
On the properties of tensor complementarity problems. Zbl 1466.15030
Yu, W.; Ling, C.; He, H.
2018
Improved approximation results on standard quartic polynomial optimization. Zbl 1410.90167
Ling, Chen; He, Hongjin; Qi, Liqun
2017
A class of second-order cone eigenvalue complementarity problems for higher-order tensors. Zbl 1372.65174
Hou, Jiao-Jiao; Ling, Chen; He, Hong-Jin
2017
On the cone eigenvalue complementarity problem for higher-order tensors. Zbl 1339.15008
Ling, Chen; He, Hongjin; Qi, Liqun
2016
Higher-degree eigenvalue complementarity problems for tensors. Zbl 1338.15027
Ling, Chen; He, Hongjin; Qi, Liqun
2016
An implementable splitting algorithm for the $$\ell_1$$-norm regularized split feasibility problem. Zbl 1339.65030
He, Hongjin; Ling, Chen; Xu, Hong-Kun
2016
Computation of integrals with oscillatory singular factors of algebraic and logarithmic type. Zbl 1315.65025
Kang, Hongchao; Ling, Chen
2015
A relaxed projection method for split variational inequalities. Zbl 1323.65079
He, Hongjin; Ling, Chen; Xu, Hong-Kun
2015
Some properties on Pareto-eigenvalues of higher-order tensors. Zbl 1349.15030
Xu, Feng; Ling, Chen
2015
A smoothing Levenberg-Marquardt algorithm for semi-infinite programming. Zbl 1318.49062
Jin, Ping; Ling, Chen; Shen, Huifei
2015
Some advances in tensor analysis and polynomial optimization. Zbl 1313.15045
Li, Zhening; Ling, Chen; Wang, Yiju; Yang, Qingzhi
2014
A new Levenberg-Marquardt type algorithm for solving nonsmooth constrained equations. Zbl 1364.65121
Ling, Chen; Wang, Guifeng; He, Hongjin
2014
On determinants and eigenvalue theory of tensors. Zbl 1259.15038
Hu, Shenglong; Huang, Zheng-Hai; Ling, Chen; Qi, Liqun
2013
$$l^{k,s}$$-singular values and spectral radius of rectangular tensors. Zbl 1272.15008
Ling, Chen; Qi, Liqun
2013
The intrinsic comparative dynamics of locally differentiable feedback Nash equilibria of autonomous and exponentially discounted infinite horizon differential games. Zbl 1402.91073
Caputo, Michael R.; Ling, Chen
2013
The best rank-1 approximation of a symmetric tensor and related spherical optimization problems. Zbl 1269.15026
Zhang, Xinzhen; Ling, Chen; Qi, Liqun
2012
Semidefinite relaxation approximation for multivariate bi-quadratic optimization with quadratic constraints. Zbl 1274.65170
Ling, Chen; Zhang, Xinzhen; Qi, Liqun
2012
Standard bi-quadratic optimization problems and unconstrained polynomial reformulations. Zbl 1268.90085
Bomze, Immanuel M.; Ling, Chen; Qi, Liqun; Zhang, Xinzhen
2012
Smooth and semismooth Newton methods for constrained approximation and estimation. Zbl 1297.90160
Yin, Hongxia; Ling, Chen; Qi, Liqun
2012
Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints. Zbl 1237.90180
Zhang, Xinzhen; Ling, Chen; Qi, Liqun
2011
A qualitative characterization of symmetric open-loop Nash equilibria in discounted infinite horizon differential games. Zbl 1221.91012
Ling, C.; Caputo, M. R.
2011
Some results on $$l^k$$-eigenvalues of tensor and related spectral radius. Zbl 1246.15013
Ling, Chen; Qi, Liqun
2011
A new smoothing Newton-type algorithm for semi-infinite programming. Zbl 1220.90144
Ling, Chen; Ni, Qin; Qi, Liqun; Wu, Soon-Yi
2010
The measure of diffusion skewness and kurtosis in magnetic resonance imaging. Zbl 1188.90291
Zhang, Xinzhen; Ling, Chen; Qi, Liqun; Wu, Ed Xuekui
2010
Biquadratic optimization over unit spheres and semidefinite programming relaxations. Zbl 1221.90074
Ling, Chen; Nie, Jiawang; Qi, Liqun; Ye, Yinyu
2009
A smoothing projected Newton-type algorithm for semi-infinite programming. Zbl 1153.90556
Qi, Liqun; Ling, Chen; Tong, Xiaojiao; Zhou, Guanglu
2009
The $$SC^1$$ 1property of an expected residual function arising from stochastic complementarity problems. Zbl 1155.90461
Ling, Chen; Qi, Liqun; Zhou, Guanglu; Caccetta, Louis
2008
A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints. Zbl 1160.65029
Tong, Xiaojiao; Ling, Chen; Qi, Liqun
2008
A truncated projected Newton-type algorithm for large-scale semi-infinite programming. Zbl 1131.90064
Ni, Qin; Ling, Chen; Qi, Liqun; Teo, Kok Lay
2006
Global convergence of a robust smoothing SQP method for semi-infinite programming. Zbl 1140.90044
Ling, C.; Qi, L. Q.; Zhou, G. L.; Wu, S. Y.
2006
Convergence rate of Newton’s method for $$L_2$$ spectral estimation. Zbl 1111.90113
Yin, Hongxia; Ling, Chen; Qi, Liqun
2006
Differentiability and semismoothness properties of integral functions and their applications. Zbl 1079.90143
Qi, Liqun; Shapiro, Alexander; Ling, Chen
2005
The generalized optimality conditions of multiobjective programming problem in topological vector space. Zbl 1045.90056
Hu, Yuda; Ling, Chen
2004
Connectedness of cone superefficient point sets in locally convex topological vector spaces. Zbl 0972.90067
Hu, Y. D.; Ling, C.
2000
Improvement on the 10-node tetrahedral element for three-dimensional problems. Zbl 0982.74067
Lo, S. H.; Ling, Chen
2000
K-T type optimality conditions of multiobjective programming with set-valued maps. Zbl 0965.90046
Ling, Chen
2000
An improved discrete Kirchhoff triangular element for bending, vibration and buckling analyses. Zbl 0995.74075
Soh, Ai-Kah; Ling, Chen
2000
all top 5

### Cited by 483 Authors

 49 Qi, Liqun 25 Ling, Chen 23 Huang, Zheng-Hai 21 Hu, ShengLong 16 Wei, Yimin 12 Zhou, Guanglu 11 He, Hongjin 11 Wang, Yiju 11 Zhang, Xinzhen 9 Bu, Changjiang 8 Chen, Haibin 8 Kang, Hongchao 8 Li, Chaoqian 8 Li, Yaotang 7 Che, Maolin 7 Du, Shouqiang 7 Li, Zhening 7 Nie, Jiawang 7 Wu, Soon-Yi 7 Zhang, Shuzhong 6 Luo, Meiju 6 Shao, Jiayu 6 Song, Yisheng 6 Tong, Xiaojiao 6 Wang, Changyu 6 Zhang, Liping 5 Bai, Xueli 5 Ferreira, Orizon Pereira 5 He, Jun 5 Jiang, Bo 5 Sun, Hailin 5 Zhou, Jiang 4 Caccetta, Louis 4 Fan, Jinyan 4 Huang, Nan-Jing 4 Liu, Qian 4 Ma, Changfeng 4 Ma, Huiqiang 4 Németh, Sándor Zoltán 4 Ni, Guyan 4 Sun, Lizhu 4 Wang, Xuezhong 4 Xu, Huifu 4 Xu, Zhenhua 4 Yang, Qingzhi 4 Yang, Yuning 4 Yin, Hongxia 4 Zhao, Ruijuan 4 Zheng, Mengmeng 3 Ahmadi, Amir Ali 3 Chen, Zhen 3 Ding, Weiyang 3 Hajarian, Masoud 3 He, Simai 3 Li, Guoyin 3 Li, Suhua 3 Lin, Guihua 3 Liu, Qilong 3 Liu, Yanmin 3 Wen, Zaiwen 3 Xu, Changqing 3 Xu, Guangjun 3 Yue, Junjie 3 Zheng, Bing 2 Bai, Minru 2 Bao, Yanhong 2 Bozorgmanesh, Hassan 2 Caputo, Michael R. 2 Carmon, Yair 2 Chang, Kung-Ching 2 Che, Haitao 2 Chen, Chiyu 2 Chen, Yuanyuan 2 Cui, Lubin 2 Duchi, John C. 2 Fan, Yizheng 2 Fan, Yu-Dong 2 Gautier, Antoine 2 Goberna, Miguel Angel 2 Goldfarb, Donald 2 Gong, Xunhua 2 Gu, Weizhe 2 Guan, Hongbo 2 Hall, Georgina 2 Haskell, William Benjamin 2 He, Guo 2 Hein, Matthias 2 Hieu, Vu Trung 2 Hinder, Oliver 2 Hsu, Daniel 2 Hu, Rong 2 Huang, Yakui 2 Lei, Xiaoqiang 2 Li, Donghui 2 Li, Xia 2 Liang, Maolin 2 Ling, Liyun 2 Liu, Hongwei 2 Liu, Lixia 2 Liu, Sanyang ...and 383 more Authors
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### Cited in 96 Serials

 33 Journal of Optimization Theory and Applications 25 Linear Algebra and its Applications 23 Computational Optimization and Applications 19 Journal of Computational and Applied Mathematics 15 Frontiers of Mathematics in China 13 Applied Mathematics and Computation 13 Journal of Global Optimization 13 Mathematical Programming. Series A. Series B 12 Linear and Multilinear Algebra 11 Journal of Industrial and Management Optimization 9 SIAM Journal on Matrix Analysis and Applications 9 SIAM Journal on Optimization 9 Computational and Applied Mathematics 8 Numerical Algorithms 7 Numerical Linear Algebra with Applications 7 Optimization Letters 6 Journal of Mathematical Analysis and Applications 6 Asia-Pacific Journal of Operational Research 6 Optimization Methods & Software 6 Journal of Inequalities and Applications 5 Optimization 5 Applied Mathematics Letters 5 Journal of the Operations Research Society of China 4 Applied Numerical Mathematics 4 Journal of Scientific Computing 4 Mathematical Problems in Engineering 4 Journal of Applied Mathematics 4 Science China. Mathematics 3 Mathematics of Computation 3 Calcolo 3 Annals of Operations Research 3 Mathematical Methods of Operations Research 2 Computers & Mathematics with Applications 2 International Journal for Numerical Methods in Engineering 2 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 2 Numerical Functional Analysis and Optimization 2 Insurance Mathematics & Economics 2 Foundations of Computational Mathematics 2 Journal of Systems Science and Complexity 2 Journal of Nonlinear Science and Applications 2 SIAM Journal on Applied Algebra and Geometry 1 Discrete Applied Mathematics 1 Indian Journal of Pure & Applied Mathematics 1 International Journal of Systems Science 1 Israel Journal of Mathematics 1 Journal of the Franklin Institute 1 Acta Mathematica Vietnamica 1 The Annals of Probability 1 Mathematics of Operations Research 1 Numerische Mathematik 1 Theoretical Computer Science 1 Transactions of the American Mathematical Society 1 Optimal Control Applications & Methods 1 Systems & Control Letters 1 Operations Research Letters 1 Circuits, Systems, and Signal Processing 1 Acta Applicandae Mathematicae 1 Social Choice and Welfare 1 Acta Mathematicae Applicatae Sinica. English Series 1 Journal of Symbolic Computation 1 Journal of Complexity 1 Computational Mechanics 1 Japan Journal of Industrial and Applied Mathematics 1 Applications of Mathematics 1 Annals of Physics 1 Applied Mathematical Modelling 1 European Journal of Operational Research 1 International Journal of Computer Mathematics 1 Journal of Algebraic Combinatorics 1 Applied Mathematics. Series B (English Edition) 1 ETNA. Electronic Transactions on Numerical Analysis 1 The Electronic Journal of Combinatorics 1 Top 1 Complexity 1 Nonlinear Dynamics 1 Vietnam Journal of Mathematics 1 Positivity 1 Taiwanese Journal of Mathematics 1 Journal of Combinatorial Optimization 1 Mathematical Inequalities & Applications 1 European Journal of Mechanics. A. Solids 1 Engineering Computations 1 Nonlinear Analysis. Real World Applications 1 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 1 Journal of Applied Mathematics and Computing 1 4OR 1 International Journal of Computational Methods 1 Acta Mechanica Sinica 1 Algorithms 1 Journal of Applied Mathematics & Informatics 1 Dynamic Games and Applications 1 Numerical Algebra, Control and Optimization 1 Journal of Applied Analysis and Computation 1 Minimax Theory and its Applications 1 Wavelets and Linear Algebra 1 AIMS Mathematics
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### Cited in 34 Fields

 191 Operations research, mathematical programming (90-XX) 164 Linear and multilinear algebra; matrix theory (15-XX) 150 Numerical analysis (65-XX) 38 Calculus of variations and optimal control; optimization (49-XX) 28 Combinatorics (05-XX) 14 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 13 Computer science (68-XX) 13 Mechanics of deformable solids (74-XX) 10 Approximations and expansions (41-XX) 10 Operator theory (47-XX) 8 Algebraic geometry (14-XX) 7 Convex and discrete geometry (52-XX) 7 Differential geometry (53-XX) 7 Systems theory; control (93-XX) 6 Probability theory and stochastic processes (60-XX) 6 Quantum theory (81-XX) 5 Real functions (26-XX) 5 Information and communication theory, circuits (94-XX) 3 Special functions (33-XX) 3 Partial differential equations (35-XX) 3 General topology (54-XX) 3 Statistics (62-XX) 2 General and overarching topics; collections (00-XX) 2 Field theory and polynomials (12-XX) 2 Commutative algebra (13-XX) 2 Statistical mechanics, structure of matter (82-XX) 1 Functions of a complex variable (30-XX) 1 Ordinary differential equations (34-XX) 1 Integral equations (45-XX) 1 Functional analysis (46-XX) 1 Manifolds and cell complexes (57-XX) 1 Global analysis, analysis on manifolds (58-XX) 1 Optics, electromagnetic theory (78-XX) 1 Biology and other natural sciences (92-XX)


text:
# Secret Order

#### Farmtalk

##### Active member
Here is another challenge that I thought was fun

A man wanted to get into his work building, but he had forgotten his code. However, he did remember five clues. These are what those clues were:

The fifth number plus the third number equals fourteen.

The fourth number is one more than the second number.

The first number is one less than twice the second number.

The second number plus the third number equals ten.

The sum of all five numbers is 30.

What were the five numbers and in what order?

#### MarkFL

Staff member
Let $A,B,C,D,E$ be the five numbers, in that order. Thus, the clues may be written as:

(1) $$\displaystyle C+E=14$$

(2) $$\displaystyle D=B+1$$

(3) $$\displaystyle A=2B-1$$

(4) $$\displaystyle B+C=10$$

(5) $$\displaystyle A+B+C+D+E=30$$

Substituting (1)-(4) into (5), we find:

$$\displaystyle (2B-1)+B+(B+1)+(14)=30$$

$$\displaystyle 4B=16\,\therefore\,B=4\,\therefore\,D=5\, \therefore\,A=7\,\therefore\,C=6\,\therefore\,E=8$$

Hence the code is: $74658$

#### Farmtalk

##### Active member
Correct! I thought this was a rather tough puzzle! It took me forever to figure it out!


text:
# Posterior Predictive Distribution of a Parameter

Suppose I have data, $$y_1, \ldots, y_n \sim \mathcal{N}(\mu, \sigma^2)$$, where $$\mu$$ is an unknown parameter and $$\sigma^2$$ is known. We put a prior on the parameter $$\mu \sim \mathcal{N}(\mu_0, \tau^2_0)$$. Under these assumptions one can compute the posterior

\begin{aligned} p(\mu_n &| y_1, \ldots, y_n; \mu_0, \sigma^2) \sim \mathcal{N}(\alpha_n, \tau_n^2) \\ \alpha_n &= \frac{\frac{\mu_0}{\tau^2_0} + \frac{n\bar{y}}{\sigma^2}}{ \left(\frac{1}{\tau^2_0} + \frac{n}{\sigma^2}\right)},\quad \tau_n^2 = \left(\frac{1}{\tau^2_0} + \frac{n}{\sigma^2}\right)^{-1} \end{aligned}

and the posterior predictive of a new data sample $$y_{n+1}$$

$$p(y_{n+1} | y_1, \ldots, y_n; \mu_0, \sigma^2) \sim \mathcal{N}\left(\alpha_n, \sigma^2 +\tau_n^2\right)$$

I am interested in computing the posterior predictive distribution of the parameter (propagating uncertainty from the posterior predictive of the data to the posterior predictive of the parameter)

I started by writing

\begin{aligned} p(\mu_{n+1} &| y_1, \ldots, y_n, y_{n+1}; \mu_0, \sigma^2) \sim\mathcal{N}(\alpha_{n+1}, \tau_{n+1}^2) \\ \alpha_{n+1} &= \frac{\frac{\alpha_n}{\tau^2_n} + \frac{y_{n+1}}{\sigma^2}}{ \left(\frac{1}{\tau^2_n} + \frac{1}{\sigma^2}\right)},\quad \tau_{n+1}^2 = \left(\frac{1}{\tau^2_n} + \frac{1}{\sigma^2}\right)^{-1} \end{aligned}

\begin{aligned} p(\mu_{n+1}| y_1, \ldots, y_n; \mu_0, \sigma^2) = \int p(\mu_{n+1} &| y_1, \ldots, y_n, y_{n+1}; \mu_0, \sigma^2) p(y_{n+1} | y_1, \ldots, y_n; \mu_0, \sigma^2) dy_{n+1} \\ \end{aligned}

However, there is a $$y_{n+1}$$ in the mean which interacts with $$\mu_{n+1}$$ and is not separable. So I cannot compute an analytical posterior.

I can draw samples $$\tilde{y}^i_{n+1}$$ from the posterior predictive distribution of the data and construct an approximate average $$\frac{1}{S} \sum_{i=1}^S p(\mu_{n+1} | y_1, \ldots, y_n, y^i_{n+1}; \mu_0, \sigma^2)$$ but I want to know if someone has seen an example of this type with an analytical posterior.

How does one propagate uncertainty from the posterior predictive of the data to the posterior predictive of the parameter. Is there any other way to approximate except sampling. I would appreciate any search terms, examples or references

There is no parameter $$\mu_n$$ or $$\mu_{n+1}$$ in your model, only $$\mu$$ which distribution gets updated as you get more data observed. The posterior predictive is defined for future observations not for a parameter. The integrated posterior of $$\mu$$ given $$(y_1,\ldots,y_{n+1})$$ is simply the posterior of $$\mu$$ given $$(y_1,\ldots,y_{n})$$ as shown by inverting the order of conditioning in $$\mu$$ and $$y_{n+1}$$: \begin{align} p(\mu &| y_1, \ldots, y_n, y_{n+1}; \mu_0, \sigma^2) p(y_{n+1} | y_1, \ldots, y_n; \mu_0, \sigma^2)\\ &= p(\mu | y_1, \ldots, y_n; \mu_0, \sigma^2) p(y_{n+1} | y_1, \ldots, y_n,\mu; \mu_0, \sigma^2)\\ &= p(\mu | y_1, \ldots, y_n; \mu_0, \sigma^2) p(y_{n+1} | \mu; \mu_0, \sigma^2) \end{align}


text:
American Institute of Mathematical Sciences

March  2015, 20(2): 599-612. doi: 10.3934/dcdsb.2015.20.599

Exponential decay for linear damped porous thermoelastic systems with second sound

 1 King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, Dhahran 31261 2 Laboratoire de thèorie des operateurs et EDP: fondements et applications, Faculté des Sciences et de technologie, Universit, El Oued 39000, Algeria

Received  March 2014 Revised  July 2014 Published  January 2015

In this paper, we investigate two problems in porous thermoelasticity where the heat conduction is given by Cattaneo's law and prove exponential decay results in the presence of both macro- and micro-dissipations.
Citation: Salim A. Messaoudi, Abdelfeteh Fareh. Exponential decay for linear damped porous thermoelastic systems with second sound. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 599-612. doi: 10.3934/dcdsb.2015.20.599
References:
 [1] P. S. Casas and R. Quintanilla, Exponential decay in one-dimensional porous thermoelasticity, Mech. Res. Comm., 32 (2005), 652-658. doi: 10.1016/j.mechrescom.2005.02.015. [2] H. D. Fernàndez-Sare and R. Racke, On the stability of damped Timoshenko system Cattaneo versus Fourier law$^*$, Arch. Rat. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2. [3] A. Guesmia, S. A. Messaoudi and A. Wehbe, Uniform decay in mildly damped Timoshenko system with non-equal wave speed propagation, Dynamic Systems and Applications, 21 (2012), 133-146. [4] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. appl., 167 (1992), 429-442. doi: 10.1016/0022-247X(92)90217-2. [5] Z. J. Han and G. Q. Xu, Exponential decay result in non-uniform porous-thermo-elasticity model of Lord-Shulman type, Disc. Cont. Dyn. Sys. B, 17 (2012), 57-77. doi: 10.3934/dcdsb.2012.17.57. [6] H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Sol., 15 (1967), 299-309. doi: 10.1016/0022-5096(67)90024-5. [7] A. Magaña and R. Quintanilla, On the time decay of solutions in one-dimensional theories of porous materials, Internat. J. Solids Struct., 43 (2006), 3414-3427. doi: 10.1016/j.ijsolstr.2005.06.077. [8] S. A. Messaoudi and A. Fareh, General decay for a porous thermoelastic system with memory: The case of equal speeds, Nonlinear analysis: TMA, 74 (2011), 6895-6906. doi: 10.1016/j.na.2011.07.012. [9] S. A. Messaoudi and A. Fareh, General decay for a porous thermoelastic system with memory: The case of nonequal speeds, Acta Mathimatica Scientia, 33 (2013), 23-40. doi: 10.1016/S0252-9602(12)60192-1. [10] S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko system with second sound - Global existence and exponential stability, Math. Meth. Appl. Sci., 32 (2009), 505-534. doi: 10.1002/mma.1049. [11] J. E. Muñoz Rivera, Energy decay rate in linear thermoelasticity, Funkcial Ekvac., 35 (1992), 19-30. [12] J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems - global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5. [13] J. E. Muñoz Rivera and R. Quintanilla, On the time polynomial decay in elastic solids with voids, J. Math. Anal. Appl., 338 (2008), 1296-1309. doi: 10.1016/j.jmaa.2007.06.005. [14] R. Quintanilla, Slow decay in one-dimensional porous dissipation elasticity, Applied Math. Letters, 16 (2003), 487-491. doi: 10.1016/S0893-9659(03)00025-9. [15] R. Racke, Thermoelasticity with second sound, exponential stability in linear and non linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298. [16] R. Racke, Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound, Quart. Appl. Math., 61 (2003), 315-328. [17] M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Diff. Eqns., 253 (2012), 2715-2733. doi: 10.1016/j.jde.2012.07.012. [18] A. Soufyane, Energy decay for porous-thermo-elasticity systems of memory type, Appl. Anal., 87 (2008), 451-464. doi: 10.1080/00036810802035634. [19] M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quarterly of Applied Mathematics, 50 (1992), 727-742.

show all references

References:
 [1] P. S. Casas and R. Quintanilla, Exponential decay in one-dimensional porous thermoelasticity, Mech. Res. Comm., 32 (2005), 652-658. doi: 10.1016/j.mechrescom.2005.02.015. [2] H. D. Fernàndez-Sare and R. Racke, On the stability of damped Timoshenko system Cattaneo versus Fourier law$^*$, Arch. Rat. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2. [3] A. Guesmia, S. A. Messaoudi and A. Wehbe, Uniform decay in mildly damped Timoshenko system with non-equal wave speed propagation, Dynamic Systems and Applications, 21 (2012), 133-146. [4] S. W. Hansen, Exponential energy decay in a linear thermoelastic rod, J. Math. Anal. appl., 167 (1992), 429-442. doi: 10.1016/0022-247X(92)90217-2. [5] Z. J. Han and G. Q. Xu, Exponential decay result in non-uniform porous-thermo-elasticity model of Lord-Shulman type, Disc. Cont. Dyn. Sys. B, 17 (2012), 57-77. doi: 10.3934/dcdsb.2012.17.57. [6] H. W. Lord and Y. Shulman, A generalized dynamical theory of thermoelasticity, J. Mech. Phys. Sol., 15 (1967), 299-309. doi: 10.1016/0022-5096(67)90024-5. [7] A. Magaña and R. Quintanilla, On the time decay of solutions in one-dimensional theories of porous materials, Internat. J. Solids Struct., 43 (2006), 3414-3427. doi: 10.1016/j.ijsolstr.2005.06.077. [8] S. A. Messaoudi and A. Fareh, General decay for a porous thermoelastic system with memory: The case of equal speeds, Nonlinear analysis: TMA, 74 (2011), 6895-6906. doi: 10.1016/j.na.2011.07.012. [9] S. A. Messaoudi and A. Fareh, General decay for a porous thermoelastic system with memory: The case of nonequal speeds, Acta Mathimatica Scientia, 33 (2013), 23-40. doi: 10.1016/S0252-9602(12)60192-1. [10] S. A. Messaoudi, M. Pokojovy and B. Said-Houari, Nonlinear damped Timoshenko system with second sound - Global existence and exponential stability, Math. Meth. Appl. Sci., 32 (2009), 505-534. doi: 10.1002/mma.1049. [11] J. E. Muñoz Rivera, Energy decay rate in linear thermoelasticity, Funkcial Ekvac., 35 (1992), 19-30. [12] J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems - global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5. [13] J. E. Muñoz Rivera and R. Quintanilla, On the time polynomial decay in elastic solids with voids, J. Math. Anal. Appl., 338 (2008), 1296-1309. doi: 10.1016/j.jmaa.2007.06.005. [14] R. Quintanilla, Slow decay in one-dimensional porous dissipation elasticity, Applied Math. Letters, 16 (2003), 487-491. doi: 10.1016/S0893-9659(03)00025-9. [15] R. Racke, Thermoelasticity with second sound, exponential stability in linear and non linear 1-d, Math. Meth. Appl. Sci., 25 (2002), 409-441. doi: 10.1002/mma.298. [16] R. Racke, Asymptotic behavior of solutions in linear 2- or 3-D thermoelasticity with second sound, Quart. Appl. Math., 61 (2003), 315-328. [17] M. L. Santos, D. S. Almeida Júnior and J. E. Muñoz Rivera, The stability number of the Timoshenko system with second sound, J. Diff. Eqns., 253 (2012), 2715-2733. doi: 10.1016/j.jde.2012.07.012. [18] A. Soufyane, Energy decay for porous-thermo-elasticity systems of memory type, Appl. Anal., 87 (2008), 451-464. doi: 10.1080/00036810802035634. [19] M. A. Tarabek, On the existence of smooth solutions in one-dimensional nonlinear thermoelasticity with second sound, Quarterly of Applied Mathematics, 50 (1992), 727-742.
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text:
# Numerical solution of PDE:s, Part 7: 2D Schrödinger equation

Haven’t been posting for a while, but here’s something new… Earlier I showed how to solve the 1D Schrödinger equation numerically in different situations. Now I’m going to show how to calculate the evolution of a 2D wavepacket in a potential energy field that has been constructed to mimic the classical “two-slit experiment” which shows how the mechanics of low-mass particles like electrons can exhibit interference similar to the mechanics of classical waves (sound, light, water surface, and so on).

A 2D Schrödinger equation for a single particle in a time-independent background potential V(x,y) is

Where the particle mass has been set to 1 and the Planck’s constant to $2\pi$.

To solve this numerically, we need the Crank-Nicolson method, as was the case when solving the 1D problem. More specifically, the linear system to be solved is

with

where the wavefunction now has two position indices and one time index, and the potential energy has only two position indices.

To form a model of the two-slit experiment, we choose a domain 0 < x < 6; 0 < y < 6 and make a potential energy function defined by

IF (x < 2.2 OR x > 3.8 OR (x > 2.7 AND x < 3.3)) THEN IF (3.7 < y < 4) THEN V(x,y) = 30

IF (x < 0.5 OR x > 5.5 OR y < 0.5 OR y > 5.5) THEN V(x,y) = 30

Otherwise V(x,y) = 0.

which corresponds to having hard walls surrounding the domain and a barrier with two holes around the line y = 3.85

For an initial condition, we choose a Gaussian wavepacket that has a nonzero expectation value of the momentum in y-direction:

An R-Code that solves this problem for a time interval 0 < t < 1 is

library(graphics) #load the graphics library needed for plotting

lx <- 6.0 #length of the computational domain (both x and y)
lt <- 1.0 #length of the simulation time interval
nx <- 47 #number of discrete lattice points in x and y directions
nt <- 60 #number of timesteps
dx <- lx/nx #length of one discrete lattice cell, same in x and y directions
dt <- lt/nt #length of timestep

V = matrix(nrow=nx,ncol=nx) #potential energies at discrete xy points

for(j in c(1:nx)) { #construct the potential field with a double-slit barrier
for(k in c(1:nx)) {
V[j,k] = 0+0i
if((j*dx<2.2)||(j*dx>3.8)||((j*dx>2.7) && (j*dx<3.3))) {
if((k*dx>3.7) && (k*dx<4.0)) {
V[j,k] = 30+0i #No significant density is going to go through these barriers
}
}
if((j*dx>5.5) || (j*dx<0.5) || (k*dx>5.5) || (k*dx<0.5)) {
V[j,k] = 30+0i
}
}
}

kappa1 = (1i)*dt/(2*dx*dx) #an element needed for the matrices
kappa2 <- c(1:(nx*nx)) #another element

for(j in c(1:nx)) {
for(k in c(1:nx)) {
kappa2[(j-1)*nx+k] <- kappa1*2*dx*dx*V[j,k]
}
}

psi = c(1:(nx*nx)) #array for the wave function values

for(j in c(1:nx)) {
for(k in c(1:nx)) {
psi[(j-1)*nx+k] = exp(-2*(k*dx-1.7)*(k*dx-1.7)-2*(j*dx-3)*(j*dx-3)+(2i)*k*dx) #Gaussian initial wavefunction
if((j*dx > 5.5)||(j*dx < 0.5)||(k*dx > 5.5)||(k*dx < 0.5)) {
psi[(j-1)*nx+k] = as.complex(0)
}
}
}

xaxis <- c(1:nx)*dx #the x values corresponding to the discrete lattice points
yaxis <- c(1:nx)*dx #the y values

A = matrix(nrow=nx*nx,ncol=nx*nx) #matrix for forward time evolution
B = matrix(nrow=nx*nx,ncol=nx*nx) #matrix for backward time evolution
P = matrix(nrow=nx,ncol=nx) #matrix for the solution after time stepping
IP = matrix(nrow=4*nx,ncol=4*nx) #matrix for the higher resolution image obtained by interpolation

for(j in c(1:(nx*nx))) { #Set the values for time evolution matrix elements
for(k in c(1:(nx*nx))) {
A[j,k]=0
B[j,k]=0
if(j==k) {
A[j,k] = 1 + 4*kappa1 + kappa2[j]
B[j,k] = 1 – 4*kappa1 – kappa2[j]
}
if((k==j+1) || (k==j-1)) {
A[j,k] = -kappa1
B[j,k] = kappa1
}
if((k==j+nx)||(k==j-nx)) {
A[j,k] = -kappa1
B[j,k] = kappa1
}
}
}

for(k in c(1:nt)) { #main time stepping loop

sol <- solve(A,B%*%psi) #solve the system of equations

for(l in c(1:(nx*nx))) {
psi[l] <- sol[l]
}

for(l in c(1:nx)) {
for(m in c(1:nx)) {
P[l,m] = abs(psi[(l-1)*nx + m])*abs(psi[(l-1)*nx + m]) #square of the absolute value of wave function
if(abs(V[l,m]) > 5) P[l,m] = 2
}
}
for(l in c(1:(nx-1))) {
for(m in c(1:(nx-1))) { #make a bitmap with 4 times more pixels, using linear interpolation
IP[4*l-3,4*m-3] = P[l,m]
IP[4*l-2,4*m-3] = P[l,m]+0.25*(P[l+1,m]-P[l,m])
IP[4*l-1,4*m-3] = P[l,m]+0.5*(P[l+1,m]-P[l,m])
IP[4*l,4*m-3] = P[l,m]+0.75*(P[l+1,m]-P[l,m])
}
}

for(l in c(1:(4*nx))) {
for(m in c(1:(nx-1))) {
IP[l,4*m-2] = IP[l,4*m-3]+0.25*(IP[l,4*m+1]-IP[l,4*m-3])
IP[l,4*m-1] = IP[l,4*m-3]+0.5*(IP[l,4*m+1]-IP[l,4*m-3])
IP[l,4*m] = IP[l,4*m-3]+0.75*(IP[l,4*m+1]-IP[l,4*m-3])
}
}

jpeg(file = paste(“plot_abs_”,k,”.jpg”,sep=””)) #save the image
image(IP, zlim = c(0,0.15))
dev.off()

}

The code produces a sequence of image files, where the probability density is plotted with colors, as an output. Some representative images from this sequence (converted to grayscale) is shown below:

A video of the time evolution is shown below:

The treshold for maximum white color has been chosen to be quite low, to make the small amount of probability density that crosses the barrier visible.

The discrete grid of points has been made quite coarse here to keep the computation time reasonable, and the resolution has been increased artificially by using linear interpolation between the discrete points.

So, now we’ve seen how to solve the motion of 2D wavepackets moving around obstacles. In the next numerical methods post, I’ll go through the numerical solution of a nonlinear PDE.

## 9 thoughts on “Numerical solution of PDE:s, Part 7: 2D Schrödinger equation”

1. Luis says:

The source code for the double slit for schrodinger 2d barrier is not ok.

Like

2. Where is the problem, in the Crank-Nicolson finite differencing or somewhere else? I wrote this three years ago, so I can’t remember whether I checked the conservation of norm of the wave function and so on.

Like

3. Ok, now I noticed it after looking again, there’s something crazy happened when copying that from the source code file. I’ll fix that.

Like

4. Luis says:

Thanks, but how i should simulate the repulsion or atraction from a EM field in the potential matrix.
I mean, if I have the same charge the repulsive forze is like the potential barrier that stops, but is the charge is one postive and the oher negative, it have to atract the wave function, but putting the potential energy negative it will do the same. ¿ or not?

Like

5. Teemu says:

If it’s a static electric field, then it can be modelled with a time-independent potential energy V(x,y) but this ignores radiation emission by the charged particle at high acceleration. If you have a positive and negative charged particle interacting in 2 dimensions, it will be a 4-dimensional problem due to both particles having an x and y coordinate. A Crank-Nicolson time stepping of a 4D wave function usually takes too much time to be done on a home computer.

Like

6. Luis says:

Thanks, but if I replace your barrier with an electron (or proton)I and fill the potential matrix in your code with the values at certain distance in fm from a EM field ( also I will add strong forze too ):like this:
EMPotFromDist(double distMeters){return (2.30668E-28/distMeters)*JulestoMev;}
It will be ok? But I think this is time dependant,, because if I calc the time independant, for each step I have to move the particle in space getting the speed from the E kinetic (and it changes with the EM pot),.. Or not?
And my question before was, what else i have to change in your code to make the barrier atractive, (the potential,,the inital gaussian wavepacket)?.

Like

1. Teemu says:

You can’t represent an interaction with another particle as a static potential energy V(x,y), and not even a time-dependent V(x,y,t) unless the other particle is much heavier and follows some known trajectory.

Making the barrier attractive requires nothing more than changing the sign of function V(x,y).

Like

1. Luis says:

Thanks, but I changed the sign of V(x,y) and it still repulsive. .in
kappa2[(j-1)*nx+k] = kappa1*2*dx*dx*-self.V [j,k]
I have another example of Hydrogen atom, and it is time independant using n,l,m,but i would like to see how the orbit change in time with a gaussian wave packet like yours. With yours I put 2 repulsive particles with a potential spherical probability using the EM forze by distance, placed on the mass center of the probability, but to simulate atraction i need to know how to model this potential or wave equation
Also I translate to phyton embended in c++and use sparse matrix with spsolve(0.02 secs) instead of solve (8 secs) because is much faster.

Like

7. Luis says:

ok, thanks in this link there is source code for 2 particle scatering guassian animation. They calc in 2d when its a 1d correlationm, and then extract the 1d solution for both

Like


text:
antocuni at codespeak.net antocuni at codespeak.net
Sat Dec 20 19:37:47 CET 2008

Author: antocuni
Date: Sat Dec 20 19:37:46 2008
New Revision: 60657

Modified:
Log:
fix another XXX

==============================================================================
+++ pypy/extradoc/talk/ecoop2009/benchmarks.tex	Sat Dec 20 19:37:46 2008
@@ -1,9 +1,6 @@
\section{Benchmarks}
\label{sec:benchmarks}

-\anto{We should say somewhere that flexswitches are slow but benchmarks are so
-  good because they are not involved in the inner loops}
-
In section \ref{sec:tlc-properties}, we saw that TLC provides most of the
features that usaully make dynamically typed language so slow, such as
\emph{stack-based interpreter}, \emph{boxed arithmetic} and \emph{dynamic lookup} of
@@ -39,21 +36,6 @@
multiplication, one subtraction, and one comparison to check if we have
finished the job.

-\commentout{
-\cfbolz{I think we can kill this for space reasons}
-\anto{If we decide to keep them, we must remember to explain the python-like
-  syntax, as it is no longer in tlc.tex}
-
-\begin{lstlisting}
-def main(n):
-    result = 1
-    while n > 1:
-        result = result * n
-        n = n - 1
-    return n
-\end{lstlisting}
-}
-
When doing plain interpretation, we need to create and destroy three temporary
objects at each iteration.  By contrast, the code generated by the JIT does
much better.  At the first iteration, the classes of the two operands of the
@@ -72,21 +54,6 @@
Similarly, we wrote a program to calculate the $n_{th}$ Fibonacci number, for
which we can do the same reasoning as above.

-\commentout{
-\cfbolz{I think we can kill this for space reasons}
-\begin{lstlisting}
-def main(n):
-    a = 0
-    b = 1
-    while n > 1:
-        sum = a + b
-        a = b
-        b = sum
-        n = n - 1
-    return b
-\end{lstlisting}
-}
-
\begin{table}[ht]
\begin{center}

@@ -143,6 +110,15 @@
CLI backend emits slightly non-optimal code and that the underyling .NET JIT
compiler is highly optimized to handle bytecode generated by C\# compilers.

+As we saw in Section~\ref{sec:flexswitches-cli}, the implementation of
+flexswitches on top of CLI is hard and inefficient.  However, our benchmarks
+show that this inefficiency does not affect the overall performances of the
+generated code.  This is because in most programs the vast majority of the
+time is spent in the inner loop: the graphs are built in such a way that all
+the blocks that are part of the inner loop reside in the same method, so that
+all links inside are internal (and fast).
+
+
\subsection{Object-oriented features}

To measure how the JIT handles object-oriented features, we wrote a very

==============================================================================
+++ pypy/extradoc/talk/ecoop2009/clibackend.tex	Sat Dec 20 19:37:46 2008
@@ -66,6 +66,7 @@

\subsection{Implementing flexswitches in CLI}
+\label{sec:flexswitches-cli}

Implementing flexswitches for backends generating machine code is
not too complex: basically, a new jump has to be inserted in the